text
stringlengths 0
2.11M
| id
stringlengths 33
34
| metadata
dict |
|---|---|---|
L -.5ex arcsarcs...0pt-10pt10pt18027010pt9018010ptarcs'arcs'...=0.5ptby .50pt-441802704901804arcs”arcs”...=0.5ptby .50pt-841802704901804@below @above#1^†#1Departament of Mathematics, IST, Universidade de Lisboa, Portugal{ccal,smarcel}@math.tecnico.ulisboa.pt Lo.L.I.T.A. and DIMAp, UFRN, [email protected] keyword Keywords:Merging fragments of classical logicThis research was done under the scope of R&D Unit 50008, financed by the applicable financial framework (FCT/MEC through national funds and when applicable co-funded by FEDER/PT2020), and is part of the MoSH initiative of SQIG at Instituto de Telecomunicações. Sérgio Marcelino acknowledges the FCT postdoc grant SFRH/BPD/76513/2011.João Marcos acknowledges partial support by CNPq and by the Humboldt Foundation.Carlos Caleiro1 Sérgio Marcelino1 João Marcos2December 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================== We investigate the possibility of extending the non-functionally complete logic of a collection of Boolean connectives by the addition of further Boolean connectives that make the resulting set of connectives functionally complete. More precisely, we will be interested in checking whether an axiomatization for Classical Propositional Logic may be produced by merging Hilbert-style calculi for two disjointincomplete fragments of it.We will prove that the answer to that problem is a negative one, unless one of the components includes only top-like connectives.§ INTRODUCTION Hilbert-style calculi are arguably the most widespread way of defining logics, and simultaneously the least studied one, from the metalogical viewpoint. This is mostly due to the fact that proofs in Hilbert-style calculi are hard to obtain and systematize, in contrast with other proof formalisms such as sequent calculi and their well developed proof-theory, and semantic approaches involving algebraic or relational structures.Still, Hilbert-style calculi are most directly associated with the fundamental notion of logic as a consequence operation and are thus worth studying.Furthermore, merging together Hilbert-style calculi for given logics in order to build a combined logic precisely captures the mechanism for combining logics known as fibring, yielding the least logic on the joint language that extends the logics given as input <cit.>. Fibring fares well with respect to two basic guiding principles one may consider, conservativity and interaction. In contrast, despite their better behaved compositional character,alternative approaches based for instance on sequent calculi are prone to emerging interactions and breaches in conservativity (see, for instance, the collapsing problem <cit.>).In this paper, as an application of recent results about fibred logics, we investigate the modular construction of Hilbert-style calculi for classical logic. Take, for instance, implication and negation. Together, they form a functionally complete set of connectives. However, all suitable axiomatizations of classical logic we have seen include at least one axiom/rule where implication and negation interact. Rautenberg's general method for axiomatizing fragments of classical logic <cit.>, which explores the structure of Post's lattice <cit.>, further confirms the intuition about the essential role of interaction axioms/rules, that one may have drawn from any experience with axiomatizations of classical logic. Additionally, such expectation is consistent with a careful analysis of the characterization of the complexity of different fragments of classical logic and their associated satisfiability problems <cit.>, namely in the light of recent results on the decidability and complexity of fibred logics <cit.>. The question we wish to give a definitive answer to, here, is precisely this: is it possible to recover classical logic byfibring two disjoint fragments of it?We will show thatthe recovery is successful iff one of the logics represents a fragment of classical logic consisting only of top-like connectives (i.e., connectives that only produce theorems, for whichever arguments received as input), while the other results in a functionally complete set of connectives with the addition of ⊤. The paper is organized as follows. In <Ref>, we overview basic notions of logic, including Hilbert calculi and logical matrices, and introduce helpful notation. In <Ref> we carefully review the mechanism for fibring logics, as well as some general results about disjoint fibring that shall be necessary next. Our main results, analyzing the merging of disjointfragments of classical logic, are obtained in <Ref>. We conclude, in <Ref>, with a brief discussion of further work. To the best of our knowledge, <Ref> (<Ref>) and all the characterization results in <Ref> are new. § PRELIMINARIES§.§ Logics in abstractIn what follows, a signature Σ is an indexed set {Σ^(k)}_k∈ℕ, where each Σ^(k) is a collection of k-place connectives.Given a signature Σ and a (disjoint) set P of sentential variables, we denote by L_Σ(P) the absolutely free Σ-algebra generated by P, also known as the language generated by P over Σ.The objects in L_Σ(P) are called formulas, and a formula is called compound in case it belongs to L_Σ(P)∖ P, that is, in case it contains some connective. We will sometimes use (C) to refer to the main connective in a compound formula C, and say that a formula C is Σ-headed if (C)∈Σ. Furthermore, we will use (C) to refer to the set of subformulas of C, and use (C) to refer to the set of sentential variables occurring in C; the definitions ofandare extended to sets of formulas in the obvious way. Given a formula C such that (C)⊆{p_1,…,p_k}, it is sometimes convenient to take it as inducing a k-ary term function φ=λ p_1… p_k.C such that φ(p_1,…,p_k)=C, over which we will employ essentially the same terminology used to talk about connectives and formulas therewith constructed —in particular, a k-ary term function isinduced by a formula generated by k distinct sentential variables over a k-place connective.In such cases we will also say that the corresponding term functions are allowed by the underlying language and expressed by the corresponding logic. We will often employ the appellations nullary for 0-ary and singulary for 1-ary term functions (or for the connectives that induce them). Given signatures Σ⊆Σ^' and sets P⊆ P^' of sentential variables, a substitution is a structure-preserving mapping over the corresponding sets of formulas, namely a function σ:P⟶ L_Σ^'(P^') whichextends uniquely to a homomorphism σ^⋆:L_Σ(P)⟶ L_Σ^'(P^') by setting σ^⋆((C_1,…,C_k)):=(σ^⋆(C_1),…,σ^⋆(C_k)) for every ∈Σ^(k). We shall refer to σ^⋆(C) more simply as C^σ. The latter notation is extended in the natural way to sets of formulas: given Π⊆ L_Σ(P), Π^σ denotes {C^σ:C∈Π}. A logic over the language L_Σ(P) is here a structure L_Σ(P), equipped with a so-called consequence relation ⊆L_Σ(P)× L_Σ(P) respecting (R) Γ∪{C} C;(M) if Γ C then Γ∪Δ C; (T) if Γ D for every D∈Δ and Γ∪Δ C, then Γ C; and (SI) if Γ C then Γ^σ C^σ for any substitution σ:P⟶ L_Σ(P). Any assertion in the form Π E will be called a consecution, and may be read as `E follows from Π (according to )'; whenever Π,E∈ one may say thatsanctions Π E.Henceforth, union operations and braces will be omitted from consecutions, and the reader will be trusted to appropriately supply them in order to make the expressions well-typed.Given two logics =L_Σ(P), and ^'=L_Σ^'(P^'),^',we say that ^' extends in case P⊆ P^', Σ⊆Σ^' and ⊆ ^'.In case Γ C iff Γ^' C, for every Γ∪{C}⊆ L_Σ(P), we say that the extension is conservative.So, in a conservative extension no new consecutions are added in the `reduced language' L_Σ(P) by the `bigger' logic ^' to those sanctioned by the `smaller' logic . Fixed =L_Σ(P),, and given Σ⊆Σ^' and P⊆ P^',letcollect all the substitutions σ:P⟶ L_Σ^'(P^').We say that a formula B of L_Σ^'(P^') is a substitution instance of a formula A of L_Σ(P) if there is a substitution σ∈ such that A^σ=B.A natural conservative extension induced by is given by the logic ^'=L_Σ^'(P^'),^' equipped by the smallest substitution-invariant consequence relation preserving the consecutions of inside the extended language, that is, such that Γ^' C iff there is some Δ∪{D}⊆ L_Σ(P) and some σ∈ such that Δ D, where Δ^σ=Γ and D^σ=C.In what follows, when we simply enrich the signature and the set of sentential variables, we shall not distinguish between a given logic and its natural conservative extension.Two formulas C and D of a logic =L_Σ(P), are said to be logically equivalent according toif C D and D C; two sets of formulas Γ and Δ are said to be logically equivalent according to if each formula from each one of these sets may correctly be said to follow from the other set of formulas (notation: ΓΔ).We call the set of formulas Γ⊆ L_Σ(P) trivial (according to ) if ΓL_Σ(P).We will say that the logic is consistent if its consequence relation does not sanction all possible consecutions over a given language, that is, if there is some set of formulas Π∪{E} such that ΠE, in other words, if contains some non-trivial set of formulas Π; we call a logic inconsistent if it fails to be consistent. We say that a set of formulas Π in =L_Σ(P), is -explosive in case Π^σ E for every substitution σ:P⟶ L_Σ(P) and every formula E.Obviously, an inconsistent logicis one in which the empty set of formulas is -explosive.Fixed a denumerable set of sentential variables P and a non-empty signature Σ, let =⋃Σ.To simplify notation, whenever the context eliminates any risk of ambiguity, we will sometimes refer to L_Σ(P) more simply as L_.For instance, given the 2-place connective , in writing L_ we refer to the language generated by P using solely the connective , and similarly for the 2-place connective and the language L_.Taking the union of the corresponding signatures, in writing L_ we refer to the mixed language whose formulas may be built using exclusively the connectives and . For an illustration involving some familiar connectives, a logic =L_Σ(P), will be said to be -classical if, for every set of formulas Γ∪{A,B,C} in its language (see, for instance, <cit.>): 0.95[=⊤∈Σ^(0)] Γ,⊤ C implies Γ C [=∈Σ^(0)] Γ implies Γ C[=∈Σ^(1)] (i) A, A C;and(ii) Γ,A C and Γ, A C imply Γ C [=∈Σ^(2)] Γ,A B C iff Γ,A,B C [=∈Σ^(2)] Γ,A B C iff Γ,A C and Γ,B C [=∈Σ^(2)] (i) A,A B B;(ii) Γ,A B C implies Γ,B C; and (iii) Γ,A C and Γ,A B C implies Γ COther classical connectives may also be given appropriate abstract characterizations, `upon demand'. If the logic _=L_, is -classical for every ∈, we call it the logic of classical and denoteit by _.Let φ be some k-ary term function expressed by the logic =L_Σ(P),. If φ(p_1,…,p_k) p_j for some 1≤ j≤ k, we say that φ is projective over its j-th component.Such term function is called a projection-conjunction if it is logically equivalent to its set of projective components, i.e., if there is some J⊆{1,2,…,k} such that (i) φ(p_1,…,p_k) p_j for every j∈ J and (ii) {p_j:j∈ J}φ(p_1,…,p_k). In case φ(p_1,…,p_k) p_k+1, we say that φ is bottom-like. We will call φ top-like if φ(p_1,…,p_k); do note that the latter is a particular case of projection-conjunction (take J=∅). Classical conjunction is another particular case of projection-conjunction (take n=2 and J={1,2}); its singulary version (take n=1 and J={1}) corresponds to the so-called affirmation connective.A term function that is neither top-like nor bottom-like will here be called significant;if in addition it is not a projection-conjunction, we will call it very significant; in each case, connectives shall inherit the corresponding terminology from the term functions that they induce. Note that being not very significant means being either bottom-like or a projection-conjunction.§.§ Hilbert-style proof systemsOne of the standard ways of presenting a logic is through the so-called `axiomatic approach'.We call Hilbert calculus over the language L_Σ(P) any structure =L_Σ(P),, presented by a set of inference rules ⊆L_Σ(P)× L_Σ(P).An inference rule =Δ,D∈ is said to have premises Δ and conclusion D, and is often represented in tree-format by writing Δ/D^_, or D_^1 …D_n/D^_ when Δ={D_1,…,D_n}, or /D^_ in case Δ=∅.The latter type of rule, with an empty set of premises, is called axiom.Fix in what follows a Hilbert calculus presentation =L_Σ(P),, and consider signatures Σ⊆Σ^' and sets P⊆ P^' of sentential variables, with the corresponding collectionof substitutions from L_Σ(P) into L_Σ^'(P^').Given formulas Γ∪{C}⊆ L_Σ^'(P^'), a rule application allowing to infer C from Γ according to corresponds to a pair ,σ such that Δ/D^_ is in and σ∈, while Δ^σ=Γ and D^σ=C. Such rule applications are often annotated with the names of the corresponding rules being applied. In case Δ=∅ we may also refer to the corresponding rule application as an instance of an axiom. As usual, an -derivation of C from Γ is a tree with the following features:(i) all nodes are labelled with substitution instances of formulas of L_Σ(P);(ii) the root is labelled with C;(iii) the existing leaves are all labelled with formulas from Γ;(iv) all non-leaf nodes are labelled with instances of axioms, or with premises from Γ, or with formulas inferred by rule applications from the formulas labelling the roots of certain subtrees of , using the inference rules of . It is not hard to see that induces a logic _=L_Σ^'(P^'),_ by setting Γ_ C iff there is some -derivation of C from Γ; indeed, we may safely leave to the reader the task of verifying that postulates (R), (M), (T) and (SI) are all respected by _.We shall say that a logic =L_Σ(P), is characterized by a Hilbert calculus =L_Σ(P), iff = _. We revisit the well-known connectives of classical logic whose inferential behaviors weredescribed in <Ref>. What follows are the rules of appropriate Hilbert calculi for the logics _=L_,__, where p,q,r∈ P: 0.95 [=⊤] /⊤^__𝗍1 [=] /p^__𝖻1 [=]p/ p^__𝗇1 p/p^__𝗇2p p/q^__𝗇3 [=]p q / p^__𝖼1 p q / q^__𝖼2pq/p q^__𝖼3 [=]p/p q^__𝖽1p p/p^__𝖽2p q/q p^__𝖽3p (qr)/(pq) r^__𝖽4 [=]/p (q p)^__𝗂1/(p(q r)) ((pq) (pr))^__𝗂2/((pq)p)p^__𝗂3 p pq /q^__𝗂4 Of course, other classical connectives can also be axiomatized. For instance, the bi-implication defined by the term function λ pq.(p q)(q p) may be presented by:0.95 [=]/(p(q r)) ((p q) r)^__𝖾1/((p r)(q p))(r q)^__𝖾2 p pq /q^__𝖾3 §.§ Matrix semanticsAnother standard way of presenting a logic is through `model-theoretic semantics'. A matrix semantics over the language L_Σ(P) is a collection of logical matrices over L_Σ(P), where by a logical matrix over L_Σ(P) we mean a structure =,, in which the set is said to contain truth-values, each truth-value in ⊆ is called designated, and for each ∈Σ^(k) there is in a k-ary interpretation mapping over . A valuation over a logical matrix is any mapping :L_Σ(P)⟶ such that ((C_1,…,C_k))=((C_1),…,(C_k)) for every ∈Σ^(k). We denote by _ the set of all valuations over , and say that the valuation oversatisfies a formula C∈ L_Σ(P) if (C)∈.Note that a valuationmight be thought more simply as a mapping :P⟶, given thatthere is a unique extension of as a homomorphism from L_Σ(P) into the similar algebra having as carrier and having each symbol ∈Σ^(k) interpreted as the k-ary operator :^k⟶. Analogously, each k-ary term function λ p_1… p_k.φ over L_Σ(P) is interpreted by a logical matrixin the natural way as a k-ary operator φ:^k⟶. We shall call _^Σ the collection of all term functions compositionally derived over Σ and interpreted through ; in the literature on Universal Algebra, _^Σ is known as the clone of operations definable by term functions allowed by the signature Σ, under the interpretation provided by . Given a valuation :L_Σ(P)⟶, where the truth-values ⊆ are taken as designated, and given formulas Γ∪{C}⊆ L_Σ(P), we say that C follows from Γ according to (notation: Γ_ C) iff it is not the case that simultaneously satisfies all formulas in Γ while failing to satisfy C. We extend the definition to a set of valuations by setting Γ_C iff Γ_C for every ∈, that is, _ =⋂_∈(_). On its turn, a matrix semantics defines a consequence relation _ by setting Γ_C iff Γ__C for every ∈, that is, _ = ⋂_∈(__). If we set _:=⋃_∈(_), it should be clear that _ = __. We shall say that a logic =L_Σ(P), is characterized by a matrix semantics iff = _. To make precise what we mean herefrom by a `fragment' of a given logic,given a subsignature Σ^'⊆Σ, a sublogic ^' of is a logic ^'=L_Σ^'(P),^' characterized by a matrix semantics ^' such that the interpretation of the connective is the same at both and ^', for every ∈Σ^' and every ∈^'.It is not hard to see thatwill in this case consist in a conservative extension of ^'.There are well-known results in the literature to the effect that any logic whose consequence relation satisfies (𝐑), (𝐌), (𝐓) and (𝐒𝐈) may be characterized by amatrix semantics <cit.>. We now revisit yet again the connectives of classical logic that received our attention at <Ref>. Let ={0,1} and ={1}.Given alogical matrix ,,, we will call it -Boolean if: 0.95 [=⊤] ⊤=1 [=] =0 [=] (i) (1)=0; and (ii) (0)=1 [=] (i) (1,1)=1; and (ii) (x,y)=0 otherwise [=] (i) (0,0)=0; and (ii) (x,y)=1 otherwise 0.95[=] (i) (1,0)=0; and (ii) (x,y)=1 otherwise0.95[=] (i) (x,y)=1 if x=y; and (ii) (x,y)=0 otherwise It is not difficult to show that, if is a collection of -Boolean logical matrices,the logic _=L_,_ is -classical. Conversely, every -classical logic may be characterized by a single -Boolean logical matrix. We take the chance to introduce a few other connectives that will be useful later on.These connectives may be primitive in some sublogics of classical logic, but can also be defined by term functions involving the previously mentioned connectives, as follows: 0.95 := λ p q.(p q)+ := λ p q.(p q)(q p) := λ p q r.(p q)( p r)T^n_0 :=λ p_1… p_n.⊤, for n≥ 0 T^n_n :=λ p_1… p_n.p_1… p_n, for n>0 T^n_k :=λ p_1… p_n.(p_1 T^n-1_k-1(p_2,…,p_n)) T^n-1_k(p_2,…,p_n), for n>k>0 Note that a logical matrix containing such connectives is -Boolean if: 0.95[=] (i) (1,0)=1; and (ii) (x,y)=0 otherwise[=+] (i) +(x,y)=0 if x=y; and (ii) +(x,y)=1 otherwise [=] (i) (1,y,z)=y; and (ii) (0,y,z)=z [=T_k^n] (i) T_k^n(x_1,…,x_n)=0 if ({i:x_i=1})<k; and (ii) T_k^n(x_1,…,x_n)=1 otherwiseIn what follows we shall use the expression two-valued logic to refer to any logic characterized by the logical matrix {_,_,}, where _={0,1} and _={1}, and use the expression Boolean connectives to refer to the corresponding 2-valued interpretation of the symbols in Σ (see <Ref>).From this perspective, whenever we deal with a two-valued logic whose language is expressive enough, modulo its interpretation through amatrix semantics, to allow for all operators of a Boolean algebra 𝖡𝖠 over _ to be compositionally derived, we will say that we are dealing with classical logic.Alternatively, whenever the underlying signature turns out to be of lesser importance, one might say that classical logic is the two-valued logic that corresponds to the clone _𝖡𝖠containing all operations over _.Due to such level of expressiveness, classical logic is said thus to be functionally complete (over _).On those grounds, it follows that all two-valued logics may be said to be sublogics of classical logic. The paper <cit.> shows how to provide a Hilbert calculus presentation for any proper two-valued sublogic of classical logic.Emil Post's characterization of functional completeness for classical logic <cit.> is very informative. First of all, it tells us that there are exactly five maximal functionally incomplete clones (i.e, co-atoms in Post's lattice), namely:0.95 ℙ_0 = _𝖡𝖠^ℙ_1 = _𝖡𝖠^𝔸 = _𝖡𝖠^ 𝕄 = _𝖡𝖠^⊤𝔻 = _𝖡𝖠^T^3_2The Boolean top-like connectives form the clone 𝕌ℙ_1=_𝖡𝖠^⊤.As it will be useful later on, we mention that an analysis of Post's lattice also reveals that there are also a number of clones which are maximal with respect to ⊤, i.e., functionally incomplete clones that become functionally complete by the mere addition of the nullary connective ⊤ (or actually any other connective from 𝕌ℙ_1).In terms of the Post's lattice, the clones whose join with 𝕌ℙ_1 result in _𝖡𝖠 are:0.95 𝔻 𝕋^∞_0 = _𝖡𝖠^𝕋^n_0 = _𝖡𝖠^T^n+1_n (for n∈)It is worth noting that 𝕋^1_0=ℙ_0.If a logic turns out to be characterized by a single logical matrix with a finite set of truth-values, a `tabular' decision procedure is associable to its consequence relation based on the fact that the valuations over a finite number of sentential variables may be divided into a finite number of equivalence classes, and one may then simply do an exhaustive check for satisfaction whenever a finite number of formulas is involved in a given consecution. More generally, we will say that a logic is locally tabular if the relation of logical equivalencepartitions the language L_Σ({p_1,…,p_k}), freely generated by the signature Σ over a finite set of sentential variables, into a finite number of equivalence classes. It is clear that all two-valued sublogics of classical logic are locally tabular. On the same line, it should be equally clear that any logic that fails to be locally tabular cannot be characterized by alogical matrix with a finite set of truth-values. § COMBINING LOGICSGiven two logics _a=L_Σ_a(P),_a and _b=L_Σ_b(P),_b, their fibring is defined as the smallest logic _a_b=L_a b(P),_a b, where L_a b(P)=L_Σ_a∪Σ_b(P), and where _a ⊆ _a b and _b ⊆ _a b, that is, it consists in the smallest logic over the joint signature that extends both logics given as input.Typically, one could expect the combined logic _a_b to conservatively extend both _a and _b.That is not always possible, though (consider for instance the combination of a consistent logic with an inconsistent logic). A full characterization of the combinations of logics through disjoint fibring that yield conservative extensions of both input logics may be found at <cit.>. The fibring of two logics is called disjoint (or unconstrained) if their signatures are disjoint. A neat characterization of fibring is given by way of Hilbert calculi: Given _a = __a and _b = __b, where _a and _b are sets of inference rules, we may set _a b:=_a∪_b and then note that _a_b=L_a b(P),__a b.Insofar as a logic may be said to codify inferential practices used in reasoning, the (conservative) combination of two logics should not only allow one to faithfully recover the original forms of reasoning sanctioned by each ingredient logic over the respective underlying language, but should also allow the same forms of reasoning —and no more— to obtain over the mixed language. Hence, it is natural to think that each of the ingredient logics cannot see past the connectives belonging to the other ingredient logic —the latter connectives look like `monoliths' whose internal structure is inaccessible from the outside.To put things more formally, given signatures Σ⊆Σ^' and given a formula C∈ L_Σ^'(P), we call Σ-monoliths the largest subformulas of C whose heads belong to Σ^'∖Σ.Accordingly, the set _Σ(C)⊆(C) of all Σ-monoliths of C is defined by setting:This definition may be extended to sets of formulas in the usual way, by setting _Σ(Γ):=⋃_C∈Γ_Σ(C).Note, in particular, that _Σ(Γ)=∅ if Γ⊆ L_Σ(P).From the viewpoint of the signature Σ, monoliths may be seen as `skeletal' (sentential) variables that represent formulas of L_Σ^'(P) whose inner structure cannot be taken advantage of.In what follows, let X^Σ^':={x_D:D∈ L_Σ^'(P)} be a set of fresh symbols for sentential variables.Given C∈ L_Σ^'(P), in order to represent the Σ-skeleton of C wedefine the function _Σ:L_Σ^'(P)⟶ L_Σ^'(P∪ X^Σ^') by setting:Clearly, a skeletal variable x_D is only really useful in case (D)∈Σ^'∖Σ. Recall from <Ref> the inference rules characterizing the logic _ of classical conjunction and the logic _ of classical disjunction.As in <Ref>, we let _ refer to a logic that is at once -classical and -classical, and contains no other primitive connectives besides and . Consider now the fibred logic _:=__.It should be clear that _ ⊆ _.It is easy to see now that p(p q)p (a logical realization of an absorption law of lattice theory).Indeed, a one-step derivation _1 of p from p(p q) in _ is obtained simply by an application of rule 𝖼1 to p(p q), and a two-step derivation _2 of p(p q) from p in _ is obtained by the application of rule 𝖽1 to p to obtain p q, followed by an application of 𝖼3 to p and p q to obtain p(p q). Note that _Σ_(p(p q))={p q} and _Σ_(p(p q))={p(p q)}, and note also that _Σ_(p(p q))=p x_p q and _Σ_(p(p q))=x_p (p q).This means that from the viewpoint of _ the step of _2 in which the foreign rule 𝖽1 is used is seen as a `mysterious' passage from p to a new sentential variable x_p q taken ex nihilo as an extra hypothesis in the derivation, and from the viewpoint of _ the step of _2 in which the foreign rule 𝖼3 is used is seen as the spontaneous introduction of an extra hypothesis x_p(p q). At our next example we will however show that the dual absorption law, represented by p(p q)p, does not hold, even though the corresponding equivalence holds good over all Boolean algebras.This will prove that _ ⊈ _, and thus _⊈__. In a natural conservative extension, where the syntax of a logic is extended with new connectives but no further inference power is added, it is clear that formulas headed by the newly added connectives are treated as monoliths. Hence, the following result from <cit.> applies: Given=L_Σ(P),, Σ⊆Σ' and Δ∪{C,D}⊆ L_Σ'(P) we have Δ D_Σ(Δ)_Σ(D).We will present next a fundamental result from <cit.> that fully describes disjoint mixed reasoning in _a_b, viz. by identifying the consecutions sanctioned by such combined logic with the help of appropriate consecutions sanctioned by its ingredient logics _a and _b.Given that consecutions in _a b are justified by alternations of consecutions sanctioned by _a and consecutions sanctioned by _b, given a set of mixed formulas Δ⊆ L_a b, we define the saturation _a b(Δ) of Δ as ⋃_n∈ℕ_a b^n(Δ), where _a b^0(Δ):=Δ and _a b^n+1(Δ):={D∈(Δ):_a b^n(Δ)_a D_a b^n(Δ)_b D}. In addition, given a set of mixed formulas Δ∪{D}⊆ L_a b, we abbreviate by ^i_a b(Δ,D) the set of Σ_i-monoliths {C∈_Σ_i(D):Δ_a bC}, for each i∈{a,b}.Such ancillary notation helps us stating:Let _a and _b be two logics, each one characterizable by a single logical matrix. If _a and _b have disjoint signatures, the consecutions in the fibred logic _a b are such that Γ_a bC iff the following condition holds good:(𝐙^a)_a b(Γ),^a_a b(Γ,C)_a C or_a b(Γ) is _b-explosive. Note that the roles of a and b may be exchanged in the above theorem, given that the fibring operation is obviously commutative, so we might talk accordingly of a corresponding condition (𝐙^b), in case it turns out to be more convenient.The original formulation of this result in <cit.> was based on a slightly more sophisticated notion of saturation, which reduces to the above one in particular when the logics involved in the combination are characterizable by means of a truth-functional semantics (i.e., a matrix semantics involving a single logical matrix), as it is indeed the case for all sublogics of classical logic.Set a= and b=, E=p(p q), and let Γ={E} and C=p. Note that (i) (Γ)={p,q,p q,p(p q)}. Moreover, it is clear that (ii) _Σ_a(p)=_Σ_b(p)=∅, given that p∈ P, thus ^a_a b(Γ,C)=^b_a b(Γ,C)=∅.We know by the base case of the definition of that (iii) _a b^0(Γ)=Γ.Let us now show that _a b^1(Γ)=Γ, from which it follows that _a b(Γ)=Γ.We shall be freely making use of item (𝖺) of <Ref>.Note first, by (R), that we obviously have Γ_c E, for c∈{a,b}, and note also that (iv) _Σ_a(E)=x_E, (v) _Σ_b(E)=p x_p q, (vi) _Σ_a(p q)=p q, (vii) _Σ_b(p q)=x_p q and (viii) _Σ_c(r)=r when r∈{p,q}, for c∈{a,b}.To see that _a b^0(Γ)_c D for every D∈(Γ)∖{E} in case c is a it suffices to invoke (i), (iii), (iv), (vi) and (viii), and set a valuation v such that (x_E):=1 and (p)=(q):=0; in case c is b it suffices to invoke (i), (iii), (v), (vii) and (viii), and one may even reuse the previous valuation v, just adding the extra requirement that (x_p q):=0.It thus follows from the recursive case of the definition of that _a b^1(Γ)=Γ.It is easy to see, with the help of (iv) and (v), that _a b(Γ)={E} is neither _a-explosive nor _b-explosive.Therefore, according to condition (𝐙^c) in <Ref>, to check whether Γ_a bC one may in this case simply check whether Γ_a C or Γ_b C.From the preceding argument about _a b^0(Γ) we already know that the answer is negative in both cases.We conclude that p(p q)_a bp, thus indeed the fragment of classical logic with conjunction and disjunction as sole primitive connectives must be a non-conservative extension of the fibring of the logic of classical conjunction with the logic of classical disjunction, as we had announced at the end of <Ref>. The following is the first useful new result of this paper, establishing that conservativity is preserved by disjoint fibring, here proved for the (slightly simpler) case where each logic is characterized by a single logical matrix. Let _a and _b be logics with disjoint signatures, each characterizable by means of a single logical matrix. If _a and _bconservatively extend logics _1 and _2, respectively, then _a_b also conservatively extends _1_2.Let Σ_a, Σ_b,Σ_1 and Σ_2,be the signatures of, respectively, _a, _b, _1 and _2. Fix Γ∪{C}⊆ L_Σ_1∪Σ_2(P). From <Ref> we may conclude that: (a) Γ_ab C if and only if either _ab(Γ),^a_ab(Γ,C)_a C, or _ab(Γ) is _b-explosive;(b) Γ_12 C if and only if either _12(Γ),^1_12(Γ,C)_1 C, or _12(Γ) is _2-explosive. Now, from the fact that _ab^n(Γ)∪_12^n(Γ)⊆ L_Σ_1∪Σ_2(P), for all n∈, together with the assumptions that _a conservatively extends _1 and _bconservatively extends _2we conclude that _ab(Γ)=_12(Γ). The assumption about conservative extension also guarantees that (c) _ab(Γ) is _b-explosive if and only if _12(Γ) is _2-explosive. We prove, by induction on the structure of C, that (d) Γ_ab C if and only ifΓ_12 C. If C is a sentential variable then^a_ab(Γ,C)⊆_Σ_a(C)=∅ and, also,^1_12(Γ,C)⊆_Σ_1(C)=∅.We note that (d) then follows from (a), (b) and (c). For the induction step, let C be compound.From the inductive hypothesis we conclude that ^a_ab(Γ,C)=^1_12(Γ,C). Hence, again from (a), (b) and (c), we note that (d) follows.§ MERGING FRAGMENTS This section studies the expressivity of logics obtained by fibring disjoint fragments of classical logic. We start by analyzing the cases in which combining disjointsublogics of classical logic still yields asublogicof classical logic. Let_1 be a Boolean connective and _2 be top-like.We then have that __1__2=__1_2. By assumption, _2 is top-like,hence: (⋆) for any given set of formulas Δ, we have Δ__2ψ iff ψ∈Δ or (ψ)=_2. Let us prove that Γ__1_2φ iff Γ__1_2φ. By <Ref>, we know that Γ__1_2φ iff__1_2(Γ),^_1__1_2(Γ,φ)__1φ or __1_2(Γ) is __2-explosive. By (⋆) it follows that if __1_2(Γ) is __2-explosive then __1_2(Γ)must contain all the sentential variables and {_1}-headed formulas. Furthermore,__2((Γ))⊆__1_2(Γ) and ^_1__1_2(Γ,φ)=__2(φ). Therefore, Γ__1_2φ iff__1_2(Γ),^_1__1_2(Γ,φ)__1φ.Moreover, __1_2(Γ)={ψ∈(Γ):Γ,__1(Γ)__1ψ}.We may then finally conclude that Γ__1_2φ iff Γ,__1(Γ∪{φ})__1φiff Γ__1_2φ._ _⊤=_ ⊤ yields full classical logic, as the set {,⊤} is functionally complete.Let _1 and _2 be Boolean connectives neither of which are very significant. Then,__1__2=__1_2. There are three possible combinations, either (a) both connectives are conjunction-projections, or (b) both are bottom-like, or (c) one connective is bottom-like and the other is a conjunction-projection.[Case (a)] Let J_1 and J_2 be the sets of indices corresponding respectively to the projective components of _1 and of _2. For each ψ∈ L__1_2(P) let us defineP_ψ⊆ P recursively, in the following way: P_ψ:={ψ} if ψ∈ P andP__i(ψ_1,…,ψ_k):=⋃_a∈ J_iP_ψ_a for i∈{1,2}. We claim that ψ is equivalent to P_ψ both according to __1__2 and according to __1_2. Let us prove this by induction on the structure of ψ.For the base case, let ψ be a sentential variable, and note that ψ is equivalent to itself.If ψ is a nullary connective _i, for some i∈{1,2} (and therefore _i is top-like), then _i is equivalent to P__i (namely, the empty set).For the inductive step, consider ψ=_i(ψ_1,…,ψ_k_i) where k_i is the arity of _i. Using the fact that _i is a projection-conjunction we have that _i is equivalent to {ψ_a:a∈ J_i}.By induction hypothesis, each ψ_a is equivalent to P_ψ_a, hence ψ is equivalent to ⋃_a∈ J_iP_ψ_a. Finally, for a set of sentential variables B∪{b} we clearly have that B__1_2b iff B__1_2b iff b∈ B.So, the logics are equal.[Case (b)] This is similar to the previous case. Let ψ∈ L__1_2(P). We now define A_ψ recursively in the following way: A_ψ:={ψ} if ψ∈ P or (ψ)=_2, andA__1(ψ_1,…,ψ_k):=⋃_a∈ J_1A_ψ_a.Again, it is not hard to check that in both __1__2 and __1_2 we have that ψ is equivalent to A_ψ. Moreover, given B∪{b}⊆ P∪{ψ: (ψ)=_2} we clearly have that B__1_2b iff B__1_2b iff b∈ B or there is ψ∈ B such that (ψ)=_2. [Case (c)] It should be clear that according to both __1__2 and __1_2 we may conclude that φ follows from Γ iff either φ∈Γ or there is ψ∈Γ such that ψ∉ P.For any set of Boolean connectives⊆_𝖡𝖠^, we have that __= _∪{}.We first show that __=_. As _ is axiomatized by just the single rule /p, iteasily follows that (a) Γ_∙ C iff Γ_ or Γ_ C. By <cit.>, we note that (b) for every Γ∪{B,C}⊆ L_(P) we have that Γ,B_ C iff Γ_ C or Γ_ BC. Note in addition that (c) _ B ((BA)A).Now, if Γ_∙ A then by (a) we have that Γ_ A and Γ_. Further, using (b) and (c), it follows also that Γ,A_ A and Γ,A_. Now, a straightforward use of the Lindenbaum-Asser lemma shows that there exists a _-theory T extending Γ∪{A} which is maximal relative to A. Obviously ∉ T, and __=_ then follows from the completeness of the axiomatization of _. From this, given ⊆_𝖡𝖠^, we conclude with the help of <Ref> that __=_.For every connective expressed by the logic of classical bi-implication, e.g. ∈{,λ pqr.p+q+r},we have that __=_.We now analyze the cases in which combining disjoint sublogics of classical logic results in a logic strictly weaker than the logic of the corresponding classical mixed language. A detailed analysis of Post's lattice tells us that every clone _𝖡𝖠^Σ that contains the Boolean function of a very significant connective (i.e., _𝖡𝖠^Σ⊈_𝖡𝖠^⊤) must contain the Boolean function associated to at least one of the following connectives: , , , , +, if, T^n+1_n (for n∈), T^n+1_2 (for n∈), λ pqr.p(q r), λ pqr.p(q+r), λ pqr.p(q r), λ pqr.p(q r), λ pqr.p+q+r. Letbe a family of Boolean connectives, and assume that _ expresses at least one among the connectives in <Ref>, distinct from and λ p q r. p+q+r.Then __⊊_∪{}. Letbe one of the aboveBoolean connectives. We show that there are Γ∪{C}⊆ L_(P) and σ:P⟶ P∪{⊤} such that Γ^σ_ C^σ yet Γ^σ_ C^σ, thus concluding that __⊊_. Hence, by applying <Ref>, we obtain that__⊊_∪{} forin the conditions of the statement.We will explain two cases in detail, and for the remaining cases we just present the relevant formulas Γ^σ and C^σ, as the rest of the reasoning is analogous.[Case =] Set Γ:=∅ and C^σ:=.We have that _. However, since _(x_) and _(Γ)=∅ is not _-explosive, we conclude that _() by <Ref>.[Case =] Set Γ^σ:={ q} and C^σ:=q.We have that q_ q. However, since x_ q _ q and_({φ(x_,q)})={φ(x_,q)} is not _-explosive, we conclude that q_q by <Ref>.[Case =+] Set Γ^σ:={ + q} and C^σ:=q. [Case =] Set Γ^σ:=∅ and C^σ:= q. [Case =] let Γ^σ:={p} and C^σ:=p. [Case =λ p q r. p (q + r)] Set Γ^σ:={ (q +)} and C^σ:=q. [Case =λ p q r. p (qr)] Set Γ^σ:={p} and C^σ:=p ( r). [Case =λ p q r. p (qr)] Set Γ^σ:={p ( r)} and C^σ:=r. [Case =λ p q r. p (q r)] Set Γ^σ:={ (qr)} and C^σ:=q.[Case =] Set Γ^σ:={(,q,r)} and C^σ:=r.[Case =T_k^k+1] Set Γ^σ:={T_k^k+1(p,…,p,q,)} and C^σ:=q.[Case =T_2^k+1] Set Γ^σ:={T_2^k+1(p,p,,…,)} and C^σ:=p. Let ∉_𝖡𝖠^ be some very significant Boolean connective. Then, __⊊_.Note, by <Ref> and the fact that both and λ p q r. p+q+r belong to _𝖡𝖠^, thatfulfills the conditions of application of <Ref>. For every connectiveamong, , , +, if, T^n+1_n (for n∈), T^n+1_2 (for n∈), λ pqr.p(q r), λ pqr.p(q+r), λ pqr.p(q r), and λ pqr.p(q r),we have that __⊊_. On a two-valued logic:(i) sentential variables are always significant, every nullary connective is either top-like or bottom-like;(ii) top-like term functions are always assigned the value 1 and bottom-like term functions are always assigned the value 0;(iii) significant singulary term functions all behave semantically either as Boolean affirmation or as Boolean negation.The logic of a significantBoolean k-place connective expresses some 1-ary significant compound term function.Let φ denote the singulary term function induced by the formula (p) obtained by substituting a fixed sentential variable p at all argument positions of (p_1,…,p_k). If φ is significant, we are done.Otherwise, there are two cases to consider. For the first case, suppose that φ is top-like. Thus, given that is significant and the logic is two-valued, we know from <Ref>(ii), in particular, that there must be some valuation v such that ((p_1,…,p_k))=0.Set I:={i:(p_i)=1}, and define the substitution σ byσ(p_j):=φ(p) if j∈ I, and σ(p_j):=p otherwise.Let ψ denote the new singulary term function induced by ((p_1,…,p_k))^σ.On the one hand, choosing a valuation ^' such that ^'(p)=0 we may immediately conclude that ^'(ψ(p))=((p_1,…,p_k))=0. On the other hand, choosing ^'' such that ^''(p)=1 we see that ^''(σ(p_j))=1 for every 1≤ j≤ k.We conclude ^''(ψ(p))=^''((p))=^''(φ(p)), thus ^''(ψ(p))=1, for φ was supposed in the present case to be top-like. It follows that ψ(p) is indeed equivalent here to the sentential variable p.For the remaining case, where we suppose that φ is bottom-like, it suffices to set I:={i:(p_i)=0} and then reason analogously.In both the latter cases our task is seen to have been accomplished in view of <Ref>(i).Let =L_Σ(P), be a two-valued logic whose language allows a very significant k-ary term function φ, let I be the set of indices that identify the projective components of φ,and let σ be some substitution such that σ(p_i)=p_i, for i∈ I, and σ(p_i)=p_k+i, for i∉ I.Then, φ(p_1,…,p_k) (φ(p_1,…,p_k))^σ. By the assumption that φ is very significant, we know that this term function is not a projection-conjunction. Thus, given that I⊆{1,…,k} is the exact set of indices such that φ(p_1,…,p_k) p_i, for every i∈ I, we conclude that {p_i:i∈ I}φ(p_1,…,p_k).There must be, then, some valuation v over {0,1} such that (p_i)=1, for every i∈ I, while (φ(p_1,…,p_k))=0. From the assumption about significance we also learn that φ is not bottom-like, thus, in view of two-valuedness and the <Ref>(ii), we know that there must be some valuation ^' such that ^'(φ(p_1,…,p_k))=1.Using the assumption that φ(p_1,…,p_k) p_i for every i∈ I one may conclude that ^'(p_i)=(p_i)=1 for every i∈ I.Our final step to obtain a counter-model to witness φ(p_1,…,p_k) (φ(p_1,…,p_k))^σis to glue together the two latter valuations by considering a valuation ^'' such that ^''(p_j)=^'(p_j) for 1≤ j≤ k (satisfying thus the premise)and such that^''(p_j)=(p_j) for j>k (allowing for the conclusion to be falsified).The fibring __1__2 of the logic of a very significant classical connective _1 andthe logic of a non-top-like Boolean connective _2 distinct from fails to be locally tabular, and therefore __1__2⊊__1 _2.We want to build over Σ_1∪Σ_2, on a finite number of sentential variables, an infinite family {_m}_m∈ℕ of syntactically distinct formulas that are pairwise inequivalent according to __1__2.In case _2 is significant we know from <Ref> that we can count on a singulary significant term function ψ_0 allowed by L__2({p})∖ P. Set, in this case, ψ_n+1:=ψ_0∘ψ_n.Given the assumption that__2 is a two-valued logic, in view of <Ref>(iii) it should be clear that no such ψ_n+1 can be top-like.To the same effect, in case _2 is bottom-like, just consider any enumeration {ψ_m}_m∈ℕ of the singulary term functions allowed by L__2({p})∖ P. In both cases we see then how to build a family of syntactically distinct {_2}-headed singulary term functions, and these will be used below to build a certain convenient family of ({_1}-headed) formulas in the mixed language.In what follows we abbreviate _1(p_1,p_2,…,p_k_1) to C.We may assume, without loss of generality, that there is some j<k_1 such that C__1p_i for every i≤ j and C__1p_i otherwise.Let σ_n, for each n>0, denote a substitution such that σ_n(p_i)=p_i, for i≤ j, and σ_n(p_i)=ψ_n× i(p) otherwise. We claim that C^σ_a__1_2C^σ_b, for every a≠ b. To check the claim, first note that, for each a>0, we have __1_2({C^σ_a}) ={C^σ_a}∪{p_i:i≤ j}. From the fact that C is a significant term function, it follows that __1_2({C^σ_a}) is neither __1-explosive nor __2-explosive. For arbitrary b>0, since _Σ_2(ψ_b(p))= ∅, we have ^2__1_2({C^σ_a},ψ_b(p))=∅. Therefore, using <Ref>we may conclude that C^σ_a__1_2ψ_b(p) and, given that _Σ_1(C^σ_b)⊆{ψ_k(p):k∈}, it also follows that ^1__1_2({C^σ_a},C^σ_b)=∅. Note, in addition, for each n>0, that _Σ_1(C)=C^σ_n^', where σ_n^'(p_i):=p_i for i∈ I, and σ_n^'(p_i):=x_ψ_n× i for i∉ I.Therefore, given that _1 is very significant, using <Ref> and <Ref> we conclude at last, for every a≠ b,that C^σ_b does not follow from C^σ_a according to __1__2.The latter combined logic, thus, fails to be locally tabular.As a consequence, given thatall two-valued logics arelocally tabularwe see that __1__2 cannot coincide with __1_2. If _1 and _2 are among the Boolean connectives mentioned in <Ref> thenwe have that __1__2⊊__1_2.The following theorem makes use of the previous results to capture the exact circumstances in which the logic that merges the axiomatizations of two classical connectives coincides with the logic of these Boolean connectives. Consider the logic __1 of the classical connective _1 and the logic __2 of the distinct classical connective _2.Then, __1__2=__1_2iffeither: (𝐚)at least one among _1 and _2 is top-like, or (𝐛)neither _1 nor _2 are very significant, or (𝐜) _1∈_𝖡𝖠^ and _2= (or _1= and _2∈_𝖡𝖠^). The direction from right to left follows from <Ref>. The other direction follows from <Ref> and <Ref>. We can finally obtain the envisaged characterization result:Let _1 and _2 be non-functionally complete disjoint sets ofconnectives such that =_1∪_2 is functionally complete. The disjoint fibring of the classical logics of _1 and _2 is classicaliff _𝖡𝖠^_i∈{𝔻,𝕋_0^∞}∪{𝕋_0^k:k∈}_𝖡𝖠^_j= 𝕌ℙ_1, for some i∈{1,2} and j=3-i. Note that if _𝖡𝖠^_i∈{𝔻,𝕋_0^∞}∪{𝕋_0^k:k∈}and _𝖡𝖠^_j= 𝕌ℙ_1, fori≠ j∈{1,2}, then we have thatis functionally complete. For the right to left implication, it suffices to invoke Proposition <ref> and item (𝐚) of Theorem <ref>.As for the converse implication, let us assume that __1__2= _. Using Proposition <ref>, we know that for every pair of connectives _1∈_1 and _2∈_2 one of the items (𝐚), (𝐛) or (𝐜) of Theorem <ref> must hold.If (𝐚) holds in all cases, then, without loss of generality, _𝖡𝖠^_j=𝕌ℙ_1. This, given the functional completeness of , implies that _𝖡𝖠^_i∈{𝔻,𝕋_0^∞}∪{𝕋_0^k:k∈}.Otherwise, we would have _𝖡𝖠^_i and _𝖡𝖠^_j both distinct from 𝕌ℙ_1, and items (𝐛) or (𝐜) of Theorem <ref> would have to hold in all the remaining cases. If (𝐛) holds in all the remaining cases then we would conclude that _i∪_j contains only connectives that are not very significant, and that would contradict the functional completeness of . Thence, without loss of generality, we could say that _𝖡𝖠^_i contains very significant connectives, and item (𝐜) of Theorem <ref> would have to hold in those cases. But this would mean that _𝖡𝖠^_i⊆_𝖡𝖠^⊤=_𝖡𝖠^ and _𝖡𝖠^_j⊆_𝖡𝖠^⊤. Note, however, that neithernorcan coexist in _𝖡𝖠^_i with , or the underlying logic would express some very significant connective not expressible using only . We are therefore led to conclude that _𝖡𝖠^_i⊆_𝖡𝖠^ and _𝖡𝖠^_j⊆_𝖡𝖠^⊤.But this is impossible, as we would then have _𝖡𝖠^⊆𝔸, contradicting the functional completeness of . § CLOSING REMARKS In the present paper, we have investigated and fully characterized the situations when merging two disjoint fragments of classical logic still results in a fragment of classical logic. As a by-product, we showed that recovering full classical logic in such a manner can only be donewhen one of the logics is a fragment of classical logic consisting exclusively of top-like connectives, while the other forms a functionally complete set of connectives with the addition of ⊤. Our results take full advantage of the characterization of Post's lattice, and may be seen as an application of recent developments concerning fibred logics. Though our conclusions cannot be seen as a total surprise, we are not aware of any other result of this kind. Some unexpected situations do pop up, like the fact that __=_, or the fact that _ _⊤ and _ +_⊤ both yield full classical logic. The latter two combinations are particularly enlightening, given that according to <cit.> the complexity of disjoint fibring is only polynomially worse than the complexity of the component logics, and we know from <cit.> that the decision problems for_ or_ + are both 𝐜𝐨-𝐍𝐏-𝐜𝐨𝐦𝐩𝐥𝐞𝐭𝐞,as in full classical logic. As a matter of fact, some of the results we obtained may alternatively be established as consequences of the complexity result in <cit.> together with the conjecture that 𝐏≠𝐍𝐏. In fact, for disjoint sets of Boolean connectives _1 and _2 such that _1∪_2 is functionally complete, if the decision problems for __1 and for __2 are both in 𝐏 then clearly __1__2≠__1∪_2. However, the techniques we use here do not depend on 𝐏≠𝐍𝐏 and allow us to solve also the casesin which the complexity of the components is already in 𝐜𝐨-𝐍𝐏, for which the complexity result in <cit.> offers no hints.Similar studies could certainly be pursued concerning logics other than classical. However, even for the classical case there are some thought-provoking unsettled questions. Concretely, we would like to devise semantical counterparts for all the combinations that do not yield fragments of classical logic, namely those covered by <Ref>. So far, we can be sure that such semantic counterparts cannot be provided by a single finite logical matrix. Additionally, we would like to link the cases yielding fragments of classical logic (as covered by the conditions listed in Theorem <ref>) to properties of the multiple-conclusion consequence relations <cit.> pertaining to such connectives.plain | http://arxiv.org/abs/1706.08689v1 | {
"authors": [
"Carlos Caleiro",
"Sérgio Marcelino",
"João Marcos"
],
"categories": [
"cs.LO",
"math.LO",
"03B05 (Primary), 03B20, 03C05 (Secondary)",
"F.4.1; I.2.3"
],
"primary_category": "cs.LO",
"published": "20170627065308",
"title": "Merging fragments of classical logic"
} |
apalikethmTheorem[section] exampleExample[section] lemma[thm]Lemma defn[thm]Definition cor[thm]Corollary prop[thm]Proposition assumption[thm]Assumptionrem[thm]Remark equationsectionplain[C]plain | http://arxiv.org/abs/1706.08914v2 | {
"authors": [
"Holger Dette",
"Dominik Tomecki"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170627154513",
"title": "Determinants of Random Block Hankel Matrices"
} |
Intrinsic scatter of caustic masses and hydrostatic biasAndreon et al.^1INAF–Osservatorio Astronomico di Brera, via Brera 28, 20121, Milano, Italy [email protected]^2Dep. of Physics and Astronomy, University of the Western Cape, Cape Town 7535, South Africa All estimates of cluster mass have some intrinsic scatter and perhaps some bias with true mass even in the absence of measurement errors for example caused by cluster triaxiality and large scale structure.Knowledge of the bias and scatter values is fundamental for both cluster cosmology and astrophysics. In this paper we show thatthe intrinsic scatter of a mass proxy can be constrained by measurements ofthe gas fraction because masses with higher values of intrinsic scatter with true massproduce more scattered gas fractions. Moreover, the relative bias of two mass estimates can be constrained by comparing the mean gas fraction at the same (nominal) cluster mass. Our observational studyaddresses the scatter between caustic (i.e., dynamically estimated) and true masses, and the relative bias of caustic and hydrostatic masses. For these purposes, we used the X-ray Unbiased Cluster Sample,a cluster sample selected independently from the intracluster medium content with reliable masses:34 galaxy clusters in the nearby (0.050<z<0.135) Universe, mostly with 14<log M_500/M_⊙≲ 14.5, and with caustic masses. We found a 35% scatter between caustic and true masses.Furthermore, we found that the relative biasbetween caustic and hydrostatic masses is small, 0.06±0.05 dex, improving upon past measurements. The small scatter found confirms our previous measurements of a highly variable amount offeedback from cluster to cluster, which is the cause ofthe observed large variety of core-excised X-ray luminosities and gas masses. Intrinsic scatter of caustic masses and hydrostatic bias: An observational study.S. Andreon1 G. Trinchieri1 A. Moretti1 J. Wang2 Accepted ... Received ... ================================================================================== § INTRODUCTION Clusters of galaxies are the largest collapsed objects in the hierarchy of cosmic structures (e.g., Sarazin 1988).They arise from the smooth sea of hot particles and light under the action of gravity modulated by the action of dark energy and dark matter(e.g., Dressler et al., 1996, Weinberg et al.2013). Our understanding of the gravitational processes that shape the cosmic web, which allow us to use galaxy clusters as cosmological probes (e.g., Vikhlinin et al. 2009), and of the interplay between dark matter and baryonic components(e.g., Young et al. 2011 and references therein), relies on scaling relations between halo mass and observable quantities tracing one of their constituting and observable parts, such as galaxies,intracluster medium, or dark matter.All of the methods to weight galaxy clusters using these observables are subject to biases due to scatter between the mass and observable or its dependency on other physical cluster properties.Even the direct observation of the total matter, via weak lensing, is subject to scatter with mass due to cluster triaxiality,large scale structure, and intrinsic alignments (Meneghetti et al. 2010; Becker & Kravtsov 2011).The caustic technique derives masses from measurements of the line-of-sight escape velocity. Caustic masses are unaffected by the dynamical state of the cluster and by large scale structure (Diaferio 1999, Serra et al. 2011), however they are affected by elongation along the line of sight.Previous analyses found that caustic masses have low scatter with true mass, but the results are based on simulations, or are indirect or noisy. In fact, numerical simulations (Serra et al. 2011, Gifford & Miller 2013)showed a 20% scatter,Geller et al. (2013) showed that weak lensing and caustic masses agree within 30%, and Maughan et al. (2016) found a 23±11% scatter with hydrostatic masses. Andreon & Congdon (2014) found a small (≪ 0.1 dex)intrinsic scatter between richnessand weak lensing mass, while Andreon (2012) found a small (≪ 0.1 dex) scatter betweenrichness and caustic masses. Although indirect, the points above suggest thata large scatter between caustic and true mass is unlikely, given the small scatter with weak lensing masses,hydrostatic masses, and richness.The first aim of this paper is a data-driven determination of the intrinsic scatter of caustic masses. We achieve this objective by an innovative approach which can be applied to other types of masses as well. We exploit the factthat a large scatter in halo mass induces a large scatter on gas fraction (cosmic conspiracy notwithstanding); the latter is proportional to one over halo mass. Therefore, a value of scatter of the gas fraction can be converted into an upper limit of the intrinsic scatter of the caustic masses as detailed in Sect. 3.If caustic masses were low scatter proxies of true masses, then they could be useful to calibrate noisier mass proxies. They could also be used to measure mass-related cluster properties free of the large scatter of weak lensing masses due to triaxiality, large scale structure, and intrinsic alignment.A second aim is to measure the relative bias of caustic and hydrostatic masses.As mentioned, different methods to estimate galaxy cluster masses may also return systematically underestimated, or overestimated masses. Hydrostatic masses, i.e., masses derived under the assumption of hydrostatic equilibrium, have been often used both to measure intracluster mediumproperties (e.g., Vikhlinin et al. 2006; Arnaud et al. 2007) and to calibrate other mass proxies such as the integrated pressure (e.g., Arnaud et al. 2010) and integrated pseudo-pressure (Arnaud et al. 2007, Vikhlinin et al. 2009). Hydrostatic masses are known to be slightlybiased estimates of true mass because of deviations from the hydrostatic equilibrium or the presence of nonthermal pressure support such as turbulence, bulk flows, or cosmic rays (e.g., Rasia et al. 2006, Nagai et al. 2007, Nelson et al. 2014). The amount of the hydrostatic bias is uncertain, but usually estimated at10 to 20 %. However, a much larger bias has been invoked to reconcile cosmological parameters derived from the cosmic microwave background and cluster counts (Planck Collaboration2014), although the calibration of the bias by Planck team has been amply discussed (von der Linden 2014, Andreon 2014, Smith et al. 2016). Caustic masses can provide an alternative calibration of the bias of hydrostatic masses. They are almost unused for this purpose (we are only aware of Maughan et al. 2016) and we exploit a new method for using this type of masses: if there is a relative bias between caustic and hydrostatic masses then there should be an offset in gas mass derived using the two masses. Throughout this paper, we assume Ω_M=0.3, Ω_Λ=0.7,and H_0=70 km s^-1 Mpc^-1.Results of stochastic computations are given in the form x± y, where x and y arethe posterior mean and standard deviation. The latter also corresponds to 68% intervals because we only summarize posteriors close to Gaussian in this way. Logarithms are in base 10. § SAMPLE SELECTION, CLUSTER MASSES, AND X-RAY DATA Sample selection, halo mass derivation, and X-ray data are presented anddiscussed in Andreon et al. (2016, Paper I), and gas masses are derived in Andreon et al. (2017, Paper II), to which we refer for details. We summarize here the work done. We used a sample of 34 clusters in the very nearby universe(0.050<z<0.135, XUCS, for X-ray Unbiased Cluster Sample)extracted from the C4 catalog (Miller et al. 2005) in regions of low Galactic absorption.There is no X-ray selection in our sample, meaning that 1) the probability of inclusion of the cluster in the sample is independent of its X-ray luminosity (or count rate), and 2) no cluster is kept or removed on the basis of its X-ray properties, except for two clusters for which we cannot derive gas masses. The impact of this selection is discussed in Sec. 3.3. We collected the few X-ray observations presentin the XMM-Newton or Chandra archives and we observed the remaining clusters with Swift(individual exposure timesbetween 9 and 31 ks), as detailed in Table 1 of Paper I. Swift observations havethe advantage of a low X-ray background (Moretti et al. 2009),making it extremely useful for sampling a cluster population that includes low surface brightness clusters (Andreon & Moretti 2011).Caustic masses within r_200[The radius r_Δ is the radius within which the enclosed average mass density is Δ times the critical density at the cluster redshift.], M_200, have been derivedfollowing Diaferio & Geller (1997), Diaferio (1999), and Serra et al. (2011), then converted into r_500 and M_500 assuming a Navarro, Frenk & White (1997) profile with concentration c=5. Adopting c=3 would change mass estimates by a negligible amount; see Paper I.The median number of members within the caustics is 116 and the interquartile range is 45. The median mass of the cluster sample, log M_500 / M_⊙, is 14.2 and the interquartile range is 0.4 dex. The average mass error is 0.14 dex.Gas masses are derived by projecting a flexible radial profile, fitting its projection to the unbinned X-ray data, and propagating all modeled sources of uncertainties (e.g., spectral normalization, variation in exposure time including those originated by vignetting or excised regions)with their non-Gaussian behavior (when relevant) into the gas mass estimate using Bayesian methods. The spectral normalization, measured in the annulus 0.15 < r/r_500<0.5, is used to convert brightnesses in gas densities. The average gas mass error is 0.10 dex, as detailed and extensively tested, in Paper II. § ANALYSIS AND RESULTSIn Paper II, we fitted the relation betweengas fraction and mass, allowing an intrinsic scatter in f_gas|M,while freezing the scatter between caustic and true masses to 0.08 dex. In this paper, we let it free to vary.In detail, we allow caustic masses to have an additional scatter against true halo masses σ_intr. caus. to be addedto our errors on log M^obs, which already include a 20% intrinsic scatter between caustic masses and true masses (already accounted for in σ_log M,i),log M^obs_i∼ N(log M_i, σ^2_log M,i+σ_addit. scat.^2).We usea linear model with intrinsic scatter σ_intr in (log) gas fractions,log M_i,gas∼ N( a (log M_i-14)+ log M_gas,14 , σ_intr^2).We fit the data in the gas mass versus halo mass plane, where errors are less correlated (see Andreon 2010), i.e., log M^obs_i,gas∼ N(log M_i,gas, σ^2_log M_gas,i) . Since we cannot properly determine the slope of the relation in the limited range covered by XUCS clusters, we adopt as prior the posterior derived inAndreon (2010) for the sample inVikhlinin et al. (2006) and Sun et al. (2009), 0.15±0.03 as follows: a ∼ N(1+0.15,0.03^2) .As shown in Sec. 3.3, the assumption of the slope value does not change the results because the clusters studied have similar masses.For the remaining parameters (namely:additional caustic scatter, intrinsic scatter, andlog of gas fraction at log M/M_⊙=14, log f_gas,14=log M_gas,14-14),we assume a uniform and wide range of values that includes the true value as follows: log M_i∼ U(13.3,15.5) σ_intr∼ U(0.01,1) σ_addit. scat.∼ U(0,3) log M_gas,14∼ U(10,14).The parameter space is sampled by Gibbs sampling using JAGS (see Andreon 2011). For the three clusters for which multiple estimates for gas fraction are available,derived from different telescopes, the fit only uses those with smaller errors. As detailed in Paper II,knowledge of the selection function in the observable and itscovariance with the studied quantity is in general essential to propagate selection effects from the quantity used to select the sampleto the quantity of interest (here gas mass). The use of an X-ray unbiased cluster sample, such that used in our paper, does not require the application of any correction forthe selection function to fit (often difficult to apply). Using core excised gas masses, we foundlog f_gas =(0.15±0.03)(log M_500-14) -1.11±0.04,with the slope posterior largely determined by the slope prior. The results of the fit are plotted in Fig. 1, includingthe mean relation (solid line), its 68% uncertainty (shading), and the mean relation ± 1 σ_intr. §.§ Constraints on the scatter of caustic masses from the gas fraction scatter The top left panel of Figure 2 shows the joint posterior probabilities distributions of intrinsic and additional scatters. The scatter of the data sets a joint constraint on theintrinsic scatter and on the additional caustic scatter (approximatively on their sum). In particular, if σ_intr≳ 0.15, the most probable additional scatter is <0.1, i.e., caustic masses have very littleadditional scatter with true mass above what has already been considered in the mass error (0.08 dex). If σ_intr≈0 then the most probable value of additional scatter is ∼0.15 dex, i.e., scatter with true mass is underestimated by 40% at most.For whatever (positive) intrinsic scatter, the additional caustic scattercannot exceed 0.3 dex (bottom left panel)because higher values require a data scatter that is larger than the one observed.Finally, marginalizing (averaging) over all possible values, the additional caustic scatter is less than 0.19 dexwith 95% probability (see bottom left panel), which brings the posterior mean of the total scatter to 0.13 dex and the 95% upper limit of the total scatter of caustic masses to 0.21 dex.This observation-based 35% scatterconfirms and improves upon other indirect or noisy evidence (see introduction) andmakes caustic masses the prime choice to measuremass, in particularfree of the biases of other mass estimates (such as hydrostatic masses). The idea of exploiting the observed scatter in the gas fraction to derive an upper limit to the scatter of a mass estimate can be applied to other mass estimates, such as hydrostatic or weak lensing masses. However, although the idea is promising, the application to other samples needs to carefully account for complications which are absent in our X-ray Unbiased Cluster Sample but present in other samples such as: a) in samples selected using their content in gas mass (or a quantity showing covariance with it, such as X-ray luminosity; see Paper II); b) in samples including onlya cluster subpopulation; c) in samples selected with unknown or ill-definedselection function, or d) in samples in whichthe reference radius used to determine the gas fraction depends on the X-ray data (as in hydrostatic mass estimates). §.§ The small relative bias of hydrostatic and caustic masses In this section we set a limit to the relative bias of hydrostaticand caustic masses by comparing the average gas fraction of clusters with log M/M_⊙=14, as derived using caustic and hydrostatic masses. If a bias exists, then it should appear as a difference in the average gas fraction of clusters with the same nominal halo mass. The compared fractions are based on XUCS for clusters with caustic masses and on Vikhlinin et al. (2006) and Sun et al. (2009) for clusters with hydrostatic masses.As mentioned in the introduction, a large bias has been invoked to reconcile cosmological parameters derived by the Planck team from different probes. Fig. 3 illustrates the variety of the hydrostatic biases 1-b found; thelefmost solid line is the bias needed to reconcile cosmological parameters(Planck collaboration XXIV, 2016) and the other curves show independently derived mass biases (von der Linden et al. 2014; Hoekstra et al. 2015; Smith et al. 2016), which are often called mass priors in the literature. As shown in the Figure, there is some tension between the various determinations. However, the selection function is either not available or not accounted for in the computation of the bias (except for the black curve), whichmay affect the derived bias (see Andreon 2016). The core-excised gas fraction vs. halo mass mean relation of XUCS has been derived in Sect. 3.1, while the non-core-excised gas fractions were used for clusters in Vikhlinin et al. (2006)and Sun et al. (2009) and the relation was derived in Andreon (2010).The bottom right and top right panels of Fig. 2 show the posterior distribution and joint probability contours for both samples. The gas fractions of the two samples atlog M/M_⊙=14 onlydiffer by 0.06±0.05 dex, where XUCS gas fraction is lower, after accounting for the negligible (0.01 dex) difference between core-excised and non-core excised gas masses for XUCS clusters (derived in Paper II). A similar conclusion may be qualitatively derived from Fig. 4 of Paper II, where we plotted the fit on the Vikhlinin et al. (2006)and Sun et al. (2009) sample on XUCS individual values.The close agreement of the two gas fractions may have two origins. First,there is a negligible relative bias (0.06±0.05 dex)between caustic and hydrostatic mass scales (see Fig. 3). Our result is roughly consistent with that of Maughan et al. (2016) (-0.08±0.04). Second, the agreement is the result of a fine tuning between an hidrostatic mass bias and sample selection.The comparison sample, drawn from clusters in Vikhlinin et al. (2006) and Sun et al. (2009), is formed by clusters selected to be relaxedbut has otherwise an unknown representativeness because the sample has an unknown selection function.Nevertheless, these clusters are those used to calibrate the observable-massrelations used by Vikhlinin et al. (2009) to constrain cosmological parameters, which are found to be in agreement with those based on other probes, suggesting thatthese clusters do not provide biased scaling relations. To have the small offset between the derived gas fraction as result of selection effects compensating real differences, one of the two following conditions are requested. First, hydrostatic masses would be underestimated and gas-rich (and photon-rich) would be clusters preferentially discarded in Vikhlinin et al. (2006) and Sun et al. (2009).However, there is no reason why gas-rich clusters should bepreferentially discarded. Second, hydrostatic masses would be overestimated, but there is noevidence in the literature for that. These make fine tuning an unlikely possibility.We therefore conclude that a relative bias between caustic and hydrostatic masses is small,if it exists at all, at least in the redshift and mass ranges explored by our data, i.e., for clusters in the nearby (0.050<z<0.135) Universe, mostly with 14<log M_500/M_⊙≲ 14.5.If the result can be extrapolated to slightly more massive clusters at intermediate distances (i.e., to the Planck cosmological sample), then the source of the tension between cluster counts and CMB cosmological parameters should be looked for somewhere else, for example invoking a possible non-self-similar evolution of cluster scaling relations (Andreon 2014) or a common bias for caustic and hydrostatic masses. §.§ Sensitivity analysis Our starting sample is formed by 34 clusters that form an X-ray unbiased sample. However, two of these clusters are discarded in the course of the analysis because of their X-ray properties. We verified whether this alters the original properties of the sample. We checked that the intercept of eq. 1 changes byless than 0.01 dex if we reintroduce the two clustersusing a gas mass predicted from L_X (Paper II),which implies that our conclusion on the relative bias of hydrostatic and caustic masses (sec 3.2) is unaffected. Two objects are an unsufficient number to alter the scatter of the whole sample (34 objects) population and therefore our constraint on the scatter of caustic masses from the gas fraction scatter is robust. Finally, our analysis of the gas fraction versus mass assumed as slope prior the posterior of Andreon (2010) based on Vikhlinin et al. (2006)and Sun et al. (2009) clusters because of the limited range in mass covered by XUCS clusters. If we instead took a uniform prior on the angle in order to allowdifferent slopes, we foundidentical scatters and intercept and also joint posterior distributions of the key parameters (gas fraction, intrinsic scatter, and additional scatter of caustic masses) close to those depicted in Fig. 2. This shows the robustness of ourconclusions on assumptions about the slope.§.§ Revisiting our previous papers In Paper I we kept the intrinsic scatter between true and caustic massed fixed at the value of 0.08 dex and wefound a 0.5 dex scatter in L_X|M_c, which is a surprising large value not seen before. Smaller scatters were probably the result of the selection of the samples throughthe ICM content. Paper I assumed, however, that caustic masses have a small scatter with true mass. In the current paper we show that our assumption was correct and therefore our interpretation was correct. Quantitatively, to induce a 0.5 dex scatter in X-ray luminosity at a given mass,caustic masses should have a σ(M_c|M_t)∼0.6 dex scatter, that is ruled out by a large margin by the scatter in the gas fraction (sec. 3.2).Therefore, the big variance in core-excised X-ray luminosities found in Paper Iis real and not induced by an unaccounted scatter between true andcaustic masses.In Paper II, we fitted gas fraction as we do in this paper, but freezing the scatter between true and caustic masses at the value of 0.08 dex. The intercept we derive here allowing caustic masses to have a free amount of intrinsic scatter is in full agreement with that derived in Paper II. The intrinsic scatter in gas fraction derived in the present paper depends on the prior on the intrinsic scatter adopted for caustic masses because, as mentioned above, the data only offer a constraint on the combination of the two. The value of intrinsic scatter found here, 0.12±0.06 dex, is marginally lower than found in Paper II, 0.17±0.04 because the difference has been attributed to the scatter between real and caustic masses. Although lower (by a statistical insignificant difference),the current scatter confirms our conclusion in Paper II ofa highly variable, cluster-to-cluster, amount of gas within r_500. This indicates that the large scatter in gas fraction seen in Paper II is not due to an unaccounted for scatter incaustic masses. Instead, it is due to a highly variable, cluster-to-cluster, amount of gas within r_500. These differences are much larger than found in the literature for subsamples of the whole cluster population. To summarize, the stringent upper limit to the scatter between caustic and truemasses confirms that results in Paper I and II are not due to our assumption of a small scatter between true and caustic masses. § CONCLUSIONS In this paper we introduced an innovative approch to derive an upper limit to the scatter between mass and proxy and to estimate the relative bias of two mass proxies. We applied it tocaustic masses, and we note that it can be applied to other typesof masses. We used a sample of 34 clusters with caustic masses that we observed in X-ray and whose selection is, at a given cluster mass, independent of the intracluster medium content (see Paper I).In Paper II we derived gas masses by projecting a flexible radial profile and fitting its projection to the unbinned X-ray data. We also fit gas mass versus halo mass whilekeeping the intrinsic scatter between true and caustic massed fixed at the value of 0.08 dex. In this paper we modify the fitting model to include an additional source of scatter, i.e. the scatter between true and caustic masses. We then exploit the fact that the observed scatter in the gas fraction is inflated by halo mass errors, and therefore an estimate of the former sets an upper limit on the scatter of caustic mass estimates. We found a 35% scatter betweencaustic and true masses. Then, we set a limit to the relative bias between hydrostaticand caustic masses by comparing the average gas fraction of clusters with log M/M_⊙=14 derived using caustic and hydrostatic masses.We found a small, if any, difference, 0.07±0.05 dex, with caustic massesbeing larger and with the caveat that, as other works in literature, the sample (part of it, in our case) has an unknown representativeness. This amount of bias is, by a large margin, too small to reconcile cosmological parameters derived by the Planck team from different probes.The small scatter measured between caustic and true masses confirms our results in previous papers, which assumed the scatter to be small.We therefore confirm that the overall gas fraction isdifferent from cluster to cluster, indicating a variable amount offeedback from cluster to cluster. By ruling out a large scatter between true and caustic masseswe also confirm that the large variety of core-excised X-ray luminositiesobserved in Paper I is due to scatter into the amount of gas,rather than to the gas mass distribution within r_500.[Andreon(2010)]2010MNRAS.407..263AAndreon, S. 2010, , 407, 263 [Andreon(2010)]2011SpringerAndreon, S. 2011, in "Astrostatistical Challenges for the New Astronomy", ed. J. Hilbe,Springer Series on Astrostatistics, Springer[Andreon(2012)]2012A A...548A..83AAndreon, S. 2012, , 548, A83 [Andreon(2014)]2014A A...570L..10AAndreon, S. 2014, , 570, L10 [Andreon(2016)]2016A A...587A.158AAndreon, S. 2016, , 587, A158 [Andreon & Congdon(2014)]2014A A...568A..23AAndreon, S., & Congdon, P. 2014, , 568, A23 [Andreon & Moretti(2011)]2011A A...536A..37AAndreon, S., & Moretti, A. 2011, , 536, A37 [Andreon et al.(2016)]2016A A...585A.147AAndreon, S., Serra, A. L., Moretti, A., & Trinchieri, G. 2016, , 585, A147(Paper I)[Andreon et al.(2017)]2017papIIAndreon, S., Wang, J., Trinchieri, G., Moretti, A., Serra, A. L.2017, , in press(arXiv:1706.08356) (Paper II)[Arnaud et al.(2007)]2007A A...474L..37AArnaud, M., Pointecouteau, E., & Pratt, G. W. 2007, , 474, L37 [Arnaud et al.(2010)]2010A A...517A..92AArnaud, M., Pratt, G. W., Piffaretti, R., et al. 2010, , 517, A92 [Becker & Kravtsov(2011)]2011ApJ...740...25B Becker, M. R., & Kravtsov, A. V. 2011, , 740, 25 [Diaferio(1999)]1999MNRAS.309..610DDiaferio, A. 1999, , 309, 610 [Diaferio & Geller(1997)]1997ApJ...481..633DDiaferio, A., & Geller, M. J. 1997, , 481, 633 [Dressler & “HST and Beyond” Committee(1996)]1996hstb.rept.....DDressler, A., & “HST and Beyond” Committee 1996, Exploration and the Search for Origins: A Vision for Ultraviolet- Optical-Infrared Space Astronomy, A. Dressler (ed.), Association of Universities for Research in Astronomy, 1996, Washington, D.C. [Geller et al.(2013)]2013ApJ...764...58GGeller, M. J., Diaferio, A., Rines, K. J., & Serra, A. L. 2013, , 764, 58 [Gifford & Miller(2013)]2013ApJ...768L..32GGifford, D., & Miller, C. J. 2013, , 768, L32 [Hoekstra et al.(2015)]2015MNRAS.449..685HHoekstra, H., Herbonnet, R., Muzzin, A., et al. 2015, , 449, 685 [Kravtsov et al.(2006)]2006ApJ...650..128KKravtsov, A. V., Vikhlinin, A., & Nagai, D. 2006, , 650, 128 [Maughan et al.(2016)]2016MNRAS.461.4182MMaughan, B. J., Giles, P. A., Rines, K. J., et al. 2016, , 461, 4182 [Meneghetti et al.(2010)]2010A A...514A..93MMeneghetti, M., Rasia, E., Merten, J., et al. 2010, , 514, AA93 [Miller et al.(2005)]2005AJ....130..968MMiller, C. J., Nichol, R. C., Reichart, D., et al. 2005, , 130, 968 [Moretti et al.(2009)]2009A A...493..501MMoretti, A., Pagani, C., Cusumano, G., et al. 2009, , 493, 501 [Navarro et al.(1997)]1997ApJ...490..493NNavarro, J. F., Frenk, C. S., & White, S. D. M. 1997, , 490, 493 [Nagai et al.(2007)]2007ApJ...668....1NNagai, D., Kravtsov, A. V., & Vikhlinin, A. 2007, , 668, 1 [Nelson et al.(2014)]2014ApJ...782..107NNelson, K., Lau, E. T., Nagai, D., Rudd, D. H., & Yu, L. 2014, , 782, 107 [Planck Collaboration et al.(2014)]2014A A...571A..20PPlanck Collaboration XX,2014, , 571, A20 [Planck Collaboration et al.(2016)]2016A A...594A..24PPlanck Collaboration XXIV,2016, , 594, A24 [Rasia et al.(2006)]2006MNRAS.369.2013RRasia, E., Ettori, S., Moscardini, L., et al. 2006, , 369, 2013 [Sarazin(1988)]1988xrec.book.....SSarazin, C. L. 1988, X-ray emission from clusters of galaxies,Cambridge University Press, Cambridge[Serra et al.(2011)]2011MNRAS.412..800SSerra, A. L., Diaferio, A., Murante, G., & Borgani, S. 2011, , 412, 800 [Smith et al.(2016)]2016MNRAS.456L..74SSmith, G. P., Mazzotta, P., Okabe, N., et al. 2016, , 456, L74 [Sun et al.(2009)]2009ApJ...693.1142SSun, M., Voit, G. M., Donahue, M., et al. 2009, , 693, 1142[Vikhlinin et al.(2006)]2006ApJ...640..691VVikhlinin, A., Kravtsov, A., Forman, W., et al. 2006, , 640, 691 [Vikhlinin et al.(2009)]2009ApJ...692.1033VVikhlinin, A., Burenin, R. A., Ebeling, H., et al. 2009, , 692, 1033 [von der Linden et al.(2014)]2014MNRAS.443.1973Vvon der Linden, A., Mantz, A., Allen, S. W., et al. 2014, , 443, 1973 [Young et al.(2011)]2011MNRAS.413..691YYoung, O. E., Thomas, P. A., Short, C. J., & Pearce, F. 2011, , 413, 691 [Weinberg et al.(2013)]2013Weinberg Weinberg, D., et al.2013, PhR 530, 87 | http://arxiv.org/abs/1706.08353v2 | {
"authors": [
"S. Andreon",
"G. Trinchieri",
"A. Moretti",
"J. Wang"
],
"categories": [
"astro-ph.CO",
"astro-ph.IM"
],
"primary_category": "astro-ph.CO",
"published": "20170626130959",
"title": "Intrinsic scatter of caustic masses and hydrostatic bias: An observational study"
} |
Optimal choice: new machine learning problem and its solution Marina SapirmetaPatternhttp://sapir.usReceived 26th June 2017 / Accepted 25th January 2018 =============================================================§ ABSTRACT The task oflearning to pick a single preferred example out a finite set of examples, an “optimal choice problem”, is a supervised machine learning problemwith complex, structured input.Problems of optimal choice emergeoften in various practical applications.We formalize the problem, show that it does not satisfy the assumptions of statistical learning theory, yet it can be solved efficiently in some cases. Wepropose two approaches to solve the problem. Both of them reach good solutions on real life data from a signal processing application.§ INTRODUCTION Machine learning is described as`automated detection of meaningful patterns in data' <cit.>. No limitations on types of data or patterns are set explicitly.Yet, supervised machine learning includes, mostly, three types of problems: regression, classification and ranking which all havethe relational form:* Given is a finite set of pairs {⟨ x_i, y_i ⟩}, where observations x_i ∈ℝ^n, and y_i are some labels,* The goal isto find a function whichapproximates dependence of labelsy on vectors x.The problems differ by types of labels and measures of success but not by their general structure. The concepts “learning models”, `PAC learnable” in statistical learning theory are developed for the relational data <cit.>.Recently, the concept of “structural output learning” is being developed. There are several specific applied problems being considered (image and string segmentation, string labeling and alignments), but very little general formalizations. Yet, one may observe that in these problems, the inputs x are, usually, reduced to a vectors of certain length, but the labels may have a complex structure and be represented by sequences of different, arbitrary lengths <cit.>.There is unstated assumption that any machine learning problem can be presented in relational form above.However, in some real life learning problems,inputs can not be represented by vectors of fixed lengths.Consider, for example thetask of predicting an outcome of a horse race.Given is the history of horse races,including the information about the participating horses and thewinner of each race. Each competing horse is characterized by certain features. The goal is, knowing the features of the horses running in the next race, predict the winner.This is a supervised machine learning problem, since it requires to automatically learn prediction of the winning horse. Yet, the problem is not equivalentto any relational form problem. The main issue here is thatvictory depends not only onqualities of the winner, but on qualities of other competing horses in the same race. A horse may be sure winner in one race and sure loser in another race, where all the horses are “better” in some important ways. As races are different, so are the winners. The data reflect relationship not between horses and the victories, but between horses and races on one hand,and the victories on another. Since a race may have arbitrary number of running horses, the description of the race can not be presented as a single vector of fixed length.Here, we will study the general problem which requires to learn how to pick the “best” example out of finite sets of examples. We call it an Optimal Choice (OC ) problem, of which horse races is one example. We will start by givingthe formal definition. Then we bring examples of important and popularapplications where such problems appear. We show that the main assumptions of statistical learning theory are violated for OC. Next, we explore two general approaches to solve the problem and test them on a real life data concerning finding the true cycle out of several candidate intervals in a practical signal processing application. §.§ Optimal Choice Problem, Formal Statement Let us introduce the general concepts and the definition of the problem. A choice is vector of values of n features, x⊂^n. A lotis a finite set of choices. Denote D set of possible choices, andΩsetof all possible lots.Each lot has not morethanone choice identified as prime. Conveniently denote X^' the prime in the lot X.If the lot X does not have a prime, it will be denoted as X^' = ∅.Then,training set can be presented as a finite set of pairsZ = {⟨ X_1, I(X_1^', X_1)⟩, ..., ⟨ X_m, I(X_m ^', X_m) ⟩},where all X_i ∈Ω andI(x, X) is identity function on X:* I(x, X) ∈{0, 1}* I(x, X) = 1 ⇔ x = X^'* if X^' = ∅, I(x, X) ≡ 0. For example, in horse races, choices are the horses, a lot is a race, and the prime is the winner of the race.The goal is to build a labeling function f(x, X) →{0, 1} such that, for every X ∈Ω, x ∈ X, f(x, X) = I(x, X).If the condition f(x, X) = I(x, X) holds for a labeling function f and lot X, we say that a lot X is a successof a labeling function f.Success rate of a labeling function is the probability of its successes on Ω. Let us consider a numeric function g(x, X) on pairs X ∈Ω, x ∈ X. The function g correspond to a single labeling functionf(x, X) = 1 ⇔ max_x ∈ X g(x, X) = xif g(x, X) has a single maximum on X; otherwise, f(x, X) ≡∅ on X.The function g(x, X) which identifies the primes by its maxima in the lots will be called scoring function. The success rate of the corresponding labeling function will be associated with the scoring function. §.§ Practical examples of the optimal choice problem OC problem occurs in many practical applications, even though it was not formalized yet, to the best of author's knowledge.* CompetitionsNot only horse races, but most of competitions, including sports, beauty contests, elections,tender awards and so on, give rice to the OC type problems.Every competition with a single winner has the same form of data, essentially. Even when the same participants compete in each round of the competition, they change with time and their features change as well. * Finding cycle in continuous noisy signalThe OCtype problem was discovered in real life data analysis as part of the large signal processing research. The company Predictive Fleet Technology, for which the author did consulting,analyzes signal from piezoelectric sensors, installed in vehicles. The engine's `signature' (recorded signal) continuously reflects changes in pressure in exhaust pipe and crankcase, which occur when engine works. The cylinders in an engine fire consequently, so itshould be possible to identify intervals of work of each cylinder within the signature. The goal is to evaluate the regularity of the engine and identify possible issues.The most important part of the signature interpretation is to find the cycle: interval of time, when every cylinder works once. The problem is difficult when the signature is irregular and curves of consecutive cycles do not look the same, and when the signature is very regular and all the cylinders look identical. Some preliminary work allowed us to find the intervals of potential cycles (choices) for each signature, and each choice is characterized by four “quality criteria” (features).There is a training set, where an expert identified true cycle for each signature. The features are obtained by aggregating several signal characteristics: two features evaluate irregularity of each of the curves (from exhaust pipe and crankcase) would have if the given interval is selected as the cycle. And two other features characterize some measures of complexity of the interval itself. All the features correlate negatively with the likelihood of the choice being the prime. Three features are continuous, and the fourth feature is binary. They are scaled from 0 to 1. The goal is to develop the rule which identifies the true cycle (the prime) among the chosen intervals for each signature. The main property of this problem is that there is no “second best”: only one cycle is correct, the rest are equally wrong. This problem lands naturally into the general OC problem. All the solutions proposed here are applied on the dataset of this problem.The data contain 2453 choices in 114 lots, on average 21 choice per lot, no lot contains less than 2 choices, and every lot has a prime. The data are available upon request. * Image and signal segmentationThe problem of finding a cycle is an example from a large class of problems of image and signal segmentation. Suppose, a preprocessing algorithm can identify variants of segmentation for a particular image. If there are some criteria which can characterize each segmentation, then a lot will consist of the the variants of the segmentation for the given image with the features given by criteria values. If an expert marks a correct segmentation for each image, we are in the situation of OC problem again.* Multiple classes classification with binary classifiersIn many cases,the classification with k > 2 classes is done with a binary classifier. The task is split on classifications for each one class versus all others. But what if some instances do not fit nicely in any of the classes, or found similar to more than one class? What if one uses several classification methods which point to different classes? One still needs to find the “prime” class. In these types of problems, each lot will have k choices, and each choice will be characterized by some criteria of fit between the class and the instance. One needs to find an optimal rule which will aggregate these single class criteria into a rule for all the classes together. This is an OC problem. * Recommender system Suppose, arecommendation system presents a customer with sets of choices each time, and lets him to choose one option he likes the best. The choicesmay be movies, books, real estate, fashions and so on.The goal isto learn from the customer's past choices and recommend him new choices in the order of his preferences.Here, we are in situation of the OC problem again. The training set contains past lots, where each choice is characterized by several features, and the customer's choice (prime) is known. The system needs to develop a personalized scoring function on choices to present them to the customer in order of his preferences. * Rating system Let us consider a ratingproblem. There are two types of such problems, depending on the feedback. The feedback in the training set may be binary, or it may represent ranks. For example, the trainer marks each link as “relevant” to a queryor not. Another option is to have the trainer to assign a rank (or rating) to each object in the training set. In both cases, the goal is to learn to rank the new objects.Both approaches are hard on the trainer. If actual ratings have many values (various degrees of relevancy), it may be difficult to assign just two values correctly. Assigning multiple rating in the training set may be even more difficult.For a normal size training set, it would be too much work for a trainer to check all the comparisons his ratings imply. Besides, some of the objects he rates may not be comparable. It means, the trainer can not guaranty that all the relationships implied by his ranking are true. It leads to inevitable errors, contradictions in labels in training data. The solution may be to ask the trainer to selectgroups of comparable objects (lots), and identify the best choice (prime)in each group.This will lead one to OC problem. Finding robust solutions for any of these practical applications may be very valuable. Yet, it will require some new approaches. §.§ OC and statistical learning theory The problem has some similarity with two popular types of machine learning problems:classification and ranking.As in binary classification, the goal in OC is to learn a rule, which can be applied on the new data to classify each choice in the lot as its prime or not.To see similarity with the ranking problem, let us notice that selecting the prime in each lot establishes partial order on the set of choices of this lot: X^'≻ q for a q ∈ X, q ≠ X^'. So, the OC problem can be considered as a problem of learning the partial order on the samples of the order.Despite the similarities, the problem can not be presented in relational form, and the main explicit and implicit assumptions of statistical learning theory do not holdfor OC. * Labels are not the function of the featuresStatistical learningimplicitly assumes that the values of the features determinethe probability of a label. Essentially, each classification method approximates a random function on D →ℝ given on the training set.In OC, the labels are not a function of the features alone, since they depend both on the choice and its lot. A choice may be the prime in one lot and not in another. For example, the prime horse in a small stable is expected to be a poor competitor in some famous derby. It means that close (or even identical) choices in different lots shall be assigned different labels by the learning algorithm to have satisfactory success rate.* The fit can not be evaluated point-wiseIn classical machine learning,thefit between the true labelsand assigned labels is estimated as a measure of success, accuracy. For example, in classification, the probability of the correct labels is evaluated. In ranking, some measure of the correlation (agreement in order) between the known ranks and predicted scores is estimated.The next examples show that counting correct labels on choices or measuring correlation between the assigned and true labels in the training set can not be used to evaluate the learning success inOC.Suppose, for example, there are 10 choices in every lot. From classification point of view, if the decision rule assigns zero to every choice, the rule is 90% correct. From optimal choicepoint of view,the success rate is 0, because it did not identify any of theprimes.If a scoring function scores a prime in every lot as the “second-best”, many correlation measures used in supervised ranking will be rather high.In this case, on each lot, 8 out of every 9 not-prime choices are below the only prime choice, soAUC = 8/9 <cit.>.Yet, the scoring function fails to findthe prime everywhere, and, accordingly,the success rate of this scoring function is 0.* IndependenceIn machine learning, both feature vectors and feedback are supposed to be taken independently from the same distribution. This is, obviously, not the case here.As for labels, in each lot, only one label is 1, the rest are 0.The distribution of the choice features is expected to depend on thelot.For example,more prestigious races will include better overall participants and have necessarily different from other races distribution of features. However, we can expect that the lots appear independently, according with some probability distribution Pr(X) on Ω. Also, we can assume that there is probability distribution of a choice to be the prime in a given lot: Q(x, X) = Pr(x = X^'). The lots and their primes in the training set are generated in accordance with these two distributions.§ SOLVING THE PROBLEMThe statistical learning assumptions make classic machine learning problems tractable, allow some efficient solutions. We showed that the assumptions are not satisfied in OC type of problems. So, one may wonder, if OC has a decent solution.In fact, identifying these issues helps us to find the ways the problem can be solved. We explore two paths to the solution here.First, we consider the problem in an extended set of features, where the added features characterize lots. If the features are selected successfully, it can make the labels dependent on the choices only. Then, the point-wise fitting machine learning methods can be applied, provided that, in the end, the fit is still evaluated with the success rate.As another possible solution, we explore general optimization methods which can optimize the OC success rate directly in original feature space.We will show here on the example of the real life and difficult problem of finding cycle in continuous noisy signal that a good solution can be found efficiently both ways.§.§ Expandingfeature space to apply popular machine learning methods As we mentioned above, the main issue with OC problem is that the labels depend both on the choice and its lot. To make thelabels less dependent of lots, we add new features about lot as a whole. Denote D the extended feature space. In D, each choice still has its own specific features as well as new features, characterizing its lot,and common for every choice in its lot.The goal of extending the feature space is to have similar inD choicesacross all lots to have identical labels with high likelihood.Selection of the lot features, usually, requires some domain knowledge. However, there may be some empiric considerations which simplify the selection. Let us consider a simple case, whenthere are features F which correlate with the likelihood of a choice to be prime and mutually correlate (if the features F are developed to predict primes, it is the case, usually).Denote m_X vector of maximal values of the features F in lot X, and suppose values ofm_X areused as new features to characterize the lot. Then, a lot with higher values ofm_X will, likely, have a prime with higher values of the features F. It is likely as well that lots with similar values of m_X will have similar primes. Or, at the very least, their primes will be less dissimilar, than primes of the lots with very different features m_X.In finding the cycle problem,two features, cc.d and CrankRegul,have very different distributions from one engine to another because the regularity of engines varies widely. The features have negative correlation with the likelihood of the choice to be prime. Sowe added two new features: min.cc.d, and min.CrankRegul, which equal minimal in the lot values of the features cc.d and CrankRegul respectively. For good engines, the features correlate strongly. They both achieve minima on the intervals multiple of the true cycle.For bad engines, the features do not correlate. It means, for a bad engine, there may be potential cycles with low value of one feature and high value of another. Then, the bad engine has low values of both additional features, as dogood engines. In this case, additional features do not help distinguishing the engines and predicting the prime. Fortunately, this does not happen often. The bad engines have, usually, higher values of these features than good engines.With all the applied methods, the output was interpreted as a scoring function: primes were identified by maximal values of the function in each lot. We used the R implementation of the most popular regressionand classification methods: function SVM with linear kernel from e1071 R package, the function neuralnet from the R package with the same name, function boosting from the R package adabag, function glm from R base to build logistic regression. In all the functions, we used predicted continuous output as a scoring function. Testing was done with “leave one lot out” procedure: each lot, consequently,was removed from training and used for test. Percentages of the test lots, where the prime was correctly foundby each method, are in the table <ref>. Selection of parameters of the neural net is a challenge. We used 1 hidden layer with 3and 4 neurons. Adding more layers or neurons does not help in our experiments. ADAboost strongly depends on the selected method's parameters as well. In our experiments, ADAboost builds 30 decision trees with the depth not more than 5. Changing these parameters did not improve the solution.§.§ Optimization of the success rate criterion We were looking for a linear function of the original features which maximizes the success rate. The dataset at hand is relatively small, so we could usethe most general (slow) optimization methods, which do not use derivatives. §.§.§ Using standard optimization methods We applied the standardNelder - Meadderivative-free optimization method (implemented in the function optim of the R package stats) to find the a linear function of the features optimizing the success rate criterion. Depending on the starting point, the success rate of the found functions on the whole dataset was from 0.82 to 0.86.Other optimization methods implemented in the same R function produced worse results. It is interesting to notice that the the method works amazingly fast, much faster than neural network or ADAboost on the same data. §.§.§ Brute Force Optimization The data for the cycle problem are rather small, so we applied the exhaustive search to find true optimal hypothesis in a narrow class of hypotheses defined by the next rules: * The hypotheses are linear functions of the features f(x) = a_1 · x_1 + … + a_4 · x_4.* All coefficients a_1, …, a_4 have integer valuesfrom 0 to n,where n is a parameter of the algorithm. Out of all the possible rules with the same threshold n and identical (1% tolerance) performance, the algorithms picks the rule with minimal sum of coefficients.For n = 15,the optimal linear function has the maximal value of coefficients equal 5. The rule has success rate 88%. The algorithm with the parameter n = 5 was applied in “leave one lot out” experiment with the same success rate, 88%.§ CONCLUSION Here we formulated the new machine learning problem, Optimal Choice, and proposed two paths to solve it. The problem emerges in various practical applications quite often, but it has undesirable properties from statistical learning theory point of view.Formalization of the problem and close look at its statistical properties helped to find ways to solve it. The solutions were applied to a real life problem of signal processing with rather satisfactory results.The proposed solutions have some shortcomings. The first approach is based on extending feature space, which would require some domain knowledge in each case. The direct optimization of the success rate criterion worked very well on relatively small data, but it may not be scalable.The goal of this article will be achieved, if the OC problem attracts some attention and further research. | http://arxiv.org/abs/1706.08439v2 | {
"authors": [
"Marina Sapir"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170626153233",
"title": "Optimal choice: new machine learning problem and its solution"
} |
Efficient Manifold Approximation with SphereletsDidong Li^1,2, Minerva Mukhopadhyay^3 and David B Dunson^4 Department of Computer Science^1, Princeton University Department of Biostatistics^2, University of California, Los Angeles Department of Mathematics and Statistics^2, Indian Institute of Technology Kanpur Department of Statistical Science^4, Duke University =============================================================================================================================================================================================================================================================================================================================================In statistical dimensionality reduction, it is common to rely on the assumption that high dimensional data tend to concentrate near a lower dimensional manifold.There is a rich literature on approximating the unknown manifold, and on exploiting such approximations in clustering, data compression, and prediction.Most of the literature relies on linear or locally linear approximations. In this article, we propose a simple and general alternative, which instead uses spheres, an approach we refer to as spherelets. We develop spherical principal components analysis (SPCA), and provide theory on the convergence rate for global and local SPCA, while showing that spherelets can provide lower covering numbers and MSEs for many manifolds.Results relative to state-of-the-art competitors show gains in ability to accurately approximate manifolds with fewer components. Unlike most competitors, which simply output lower-dimensional features, our approach projects data onto the estimated manifold to produce fitted values that can be used for model assessment and cross validation. The methods are illustrated with applications to multiple data sets. Key Words: Curvature, Dimensionality reduction, Manifold learning, Spherical principal component analysis. § INTRODUCTIONDimensionality reduction is a key step in statistical analyses of high-dimensional data. If one is willing to assume that data are concentrated close to a lower-dimensional hyperplane, then Principal Components Analysis (PCA) and its variants are natural.The focus of this article is on relaxing the assumption of a lower-dimensional linear structure to allow the true latent structure to be curved; this motivates appropriate generalizations of PCA. This problem relates to manifold learning, which is based on the assumption that higher-dimensional data X_i ∈^D are often concentrated near a d-dimensional Riemannian manifold M with d≪ D.The unknown manifold M is approximately flat in small local neighborhoods but has non-zero curvature.As a simple example to provide motivation, in Figure 1 we plot economics data collected by the U.S. Bureau of Economic Analysis, and retrieved from the Federal Reserve Bank of St. Louis for economic analysis (available from the ggplot2 R package and <https://fred.stlouisfed.org> <cit.>) containing 576 multivariate observations of (1) personal consumption expenditures in billions of dollars (consume), (2) total population in thousands (pop size), (3) personal saving rate (saving), (4) median duration of unemployment in weeks (dur unemploy), and (5) number of unemployed in thousands (num unemploy).The pairwise plots are suggestive that the data may concentrate near a one-dimensional curve. Much of the focus of this paper is on devising a simple method that can parsimoniously fit the data from Figure 1.This rules out most of the existing manifold learning methods, ranging from Isomap <cit.> to Diffusion Maps <cit.>. Such approaches focus on exploiting the manifold structure to replace the original data with lower dimensional features, but do not provide fitted values in the original data space.For the economics data assuming a manifold dimension of d=1, such methods would replace the original 5 features with a single feature.This may provide a useful one-dimensional summary, but much of the interpretability is lost.An alternative general strategy that allows one to obtain fitted values of data lying close to a nonlinear manifold is to rely on local approximations.In particular, if we break up the original data domain into local regions, then within each region we could define a separate local approximation to the manifold. When applying such approaches, by far the most common strategy is to rely on locally linear approximations, fitting a separate hyperplane within each region.Such an approach is intuitively reasonable from a geometric perspective, as it is well known that Riemannian manifolds can be approximated via local tangent planes.Some examples of this strategy include local PCA <cit.> and geometric multiresolution analysis <cit.>.As illustration, consider applying such an approach to the economics data of Figure <ref>. With some risk of misinterpretation in examining pairwise dependence plots, it appears that the one-dimensional manifold the data are concentrated near is very wiggly.This implies that, in using local linear approximations, we will need to rely on a large number of well chosen neighborhoods to obtain adequate performance.However, as the number of neighborhoods increases, the amount of data within each neighborhood decreases, and statistical uncertainty in estimating the local linear parameters increases.The resulting estimator will tend to be noisy, with a tendency towards over-fitting.The second column of Figure <ref> shows the results of applying a state-of-the-art local linear approximation to the economics data, corresponding to an MSE of 2.5× 10^5.The figure shows that the fitted curves are overly jagged.For one-dimensional manifolds there are various possibilities we could consider to improve the performance. If we are willing to assume that there is a single connected manifold, as appears to be the case for the economics data, then we could modify the linear fitting algorithm to include a constraint that the line segments in adjacent neighborhoods are connected.Such a strategy is difficult to extend to higher dimensions, and would perform poorly if the data were instead from multiple disconnected manifolds.Alternatively, one could apply a method designed for fitting a curve through data in ^D, such as principal curves <cit.>. However, we applied principal curves to the economics data and found the curve is dramatically over-smoothed, providing very poor fit with MSE 4.0×10^8; hence, we do not include the plots in Figure <ref> (see Figure <ref> in Appendix).Principal curves cannot be applied beyond the d=1 case, and even for d=1 results are often non-stable and counter-intuitive.Hence, we are motivated to return to the local approximation strategy, but to improve upon local PCA and alternative local linear approaches. Much of the literature on nonlinear extensions of PCA is not relevant to our problem, because it involves replacing the original dataset with higher dimensional modifications based on applying kernels, polynomials, etc.It would be conceptually possible to fit a quadratic or higher order surface within each neighborhood, but then we are substituting the problem of too many neighborhoods for one of too many parameters per neighborhood.The resulting model fitting will be highly complex, and it is not clear that the bias-variance tradeoff will warrant such complexity.Instead, we propose a simple and efficient alternative to PCA, which uses spheres instead of hyperplanes to locally approximate manifolds. Our approachrelies on a spherical generalization of PCA, deemed SPCA. SPCA has a simple analytic form, and can be used as a generalization of PCA to incorporate curvature.We refer to any algorithm that uses spheres for local approximation as spherelets. Spheres provide an excellent basis for locally approximating non-linear manifolds M havingpositive, zero, or negative Gaussian curvature that may vary across regions. A major advantage is the ability to accurately approximate M with dramatically fewer pieces, without significantly increasing the number of parameters per piece. To give a flavor of the possible gains, we applied this approach to the economics data of Figure <ref>.Based on cross validation, the optimal number of neighborhoods for local PCA and spherelets (local SPCA) were both 16.We color code the different pieces and show the fit in Figure <ref> by twopairwise plots to better visualize the learned curve. The fit is visually much better, limiting over-fitting and artifacts and capturing the smooth relationships more accurately.This is also apparent from the out-of-sample MSE, which is 2.5× 10^5 for local PCA and 1.4× 10^5 for spherelets. We provide strong theory support for spherelets. First, as a local result, for data generated from a sphere with measurement error, we show the rate of convergence of SPCA to the true sphere in Hausdorff distance as n increases. As a global result, we provide the convergence rate for spherelets manifold estimates.We additionally show a mathematical result bounding covering numbers for local spherical versus linear approximations, showing that dramatic improvements are possible for manifolds having sufficiently large curvature but not too many subregions having large changes in normal curvature. The sign of the Gaussian curvature has no impact onapproximation performance. In fact, spherelets work well for a broad class of Riemannian manifolds, regardless of whether the Gaussian curvature is positive or negative, or even varying in sign with location. We assume d and D are fixed throughout this paper, and all asymptotic theory is focused on increasing sample size n. Our simulation results show significant reductions in the number of components needed to provide an approximation within a given error tolerance.SPCA has lower MSE than PCA and better preserves clusters. Comparing to local PCA, spherelet manifold estimates have much lower mean square error.Substantial practical gains are shown in multiple examples. The paper is organized as follows. In section 2, we propose spherical principal component analysis (SPCA) and provide convergence rate theory. In section 3, we focus on manifold approximation, prove the convergence rate of spherelets and provide acovering number theorem as additional support.In section 4, we consider simulated and real data applications for both manifold estimation and data visualization. In section 5, we discuss some open problems and future work. Proofs justifying the SPCA estimator are in the Appendix, while other proofs are in the Supplementary Materials. § SPHERICALPRINCIPAL COMPONENT ANALYSIS§.§ SPCA Algorithm Let S_V(c,r) {x: x-c =r, x-c∈ V, (V)=d+1} be the d-dimensional sphere with center c and radius r lying in the d+1 dimensional affine subspace c+V⊂^D. Throughout this paper, we do not distinguish between V as a linear subspace and its matrix representation, that is, a matrix with orthonormal columns. Our goal in this section is to estimate(V,c,r) to obtain the best approximating sphere through data X_1,…,X_n consisting of n>d samples in ^D. We first consider the projection of an arbitrary point x to the sphere S_V(c,r).For any x∈^D, its projection to S_V(c,r), or the closest point y ∈ S_V(c,r), is y∈ S_V(c,r)d^2(x,y)= c+r/VV^⊤(x-c)VV^⊤(x-c) VV^⊤(x-c)≠ 0 S_V(c,r) VV^⊤(x-c)= 0, where c+VV^⊤(x-c) is the projection of x onto the affine subspace c+V.When x=c, the projection is not unique, but the d(x,Proj(x)) is unique, which is the main focus of manifold approximation. As a result, we still treat the projection as unique without any confusion in the remaining sections. The projection of x to the sphere in Lemma <ref> can be conducted by first projecting x to a d+1 dimensional affine subspace and then further projecting to the sphere (see the proof of Lemma <ref>).To find the optimal affine subspace, we use d+1-dimensional PCA, obtaining V = (v_1,⋯,v_d+1), v_i=evec_i{(X-1X̅^⊤)^⊤(X-1X̅^⊤)},where evec_i(S) is the ith eigenvector of S in decreasing order.Letting Y_i =X̅+VV^⊤(X_i-X̅), we then find the optimal sphere through points { Y_i }_i=1^n.A sphere can be expressed as the set of zeros of a quadratic function (y-c)^⊤(y-c)-r^2. When this quadratic function has positive value, y-c>r, so y is outside the sphere, and y is inside the sphere if the function has negative value. Hence, we define the loss function ℒ(c,r)∑_i=1^n((Y_i-c)^⊤(Y_i-c)-r^2)^2. The minimizer of (<ref>) is given by c=1/2H^-1ξ, r=1/n∑_i=1^nY_i-c where H=∑_i=1^n(Y_i-Y̅)(Y_i-Y̅)^⊤ and ξ=∑_i=1^n (Y_i^⊤Y_i-1/n∑_j=1^nY_j^⊤ Y_j)(Y_i-Y̅).We refer to the resulting estimates (V,c,r) as (empirical) Spherical PCA (SPCA). If we replace the sample mean by expectation, thecorresponding estimator is called population SPCA. Alternatively, we could have minimized ∑_i=1^n d^2(X_i,S_V(c,r)), corresponding to the sum of squared residuals, also known as geometric loss.However, the resulting optimization problem is non convex, lacks an analytic solution, and iterative algorithms may be slow to converge, while only producing local minima. Instead, we consider the algebraic loss function in Equation (<ref>), which is more robust with respect to noise <cit.> and admits a closed-form solution. If X_i ∈ S_V(c,r) for all i, SPCA will find the same minimizer as the loss function in the above Remark, corresponding to exactly (V,c,r). The number of unknown parameters of d-dimensional PCA is O(Dd), while the number of unknown parameters of SPCA is O(Dd)+D+1=O(Dd). In addition, the computational cost for the first step of SPCA is the same as PCA. Theadditional cost of SPCA comes from calculating c and r, which are both linear in D and n and hence dominated by PCA complexity. As a result, the two algorithms have the same order of computational cost. The key motivation for SPCA is to maintain simplicity, both conceptually and computationally, while improving performance by relaxing the linearity assumption. §.§ Asymptotics of SPCA In this section, we show that if the data are concentrated around a sphere, then SPCA can recover this sphere with high probability. Assume Y∼ρ where (ρ)=S(V_0,c_0,r_0). Let ϵ∼ N(0,σ^2_D) be Gaussian noise and X=Y+ϵ. Let X_1,⋯,X_n be i.i.d observations and let the empirical solution of SPCA be V_n, c_n, r_n. We denote the population covariance matrix and sample covariance matrix by Σ and Σ̂_n, respectively. It is clear that all eigenvalues of Σ are positive. Furthermore, we rely on the following assumption: * The first d+1 eigenvalues of Σ are all distinct, denoted by λ_1>λ_2>⋯>λ_d+1> σ^2>0 . Under assumption (A), the following hold V_n-V_0≤o_p(σlog n/n^1/2), c_n-c_0≤O(σ^2)+o_p(σlog n/n^1/2), |r_n-r_0|≤O(σ^2)+o_p(σlog n/n^1/2). The above theorem provides an upper bound on the error rate in estimating each sphere parameter. Given observations with measurement error, SPCA can recover the true parameters of the sphere at the parametric rate with respect to the sample size up to a log factor and an asymptotic error depending on the measurement error variance. The following corollary controls the Hausdorff distance (denoted by d_H) between the true sphere and the estimated sphere by SPCA. Under the same assumption as Theorem <ref>, d_H(S(V_0,c_0,r_0),S(V_n,c_n,r_n))≤ Cσ^2+o_p(σlog n/n^1/2).For previous theoretical results on manifold estimation under Hausdorff loss, refer to<cit.>.It is typical in the literature on asymptotic theory for manifold approximation to assume that either the data are noiseless <cit.>, the noise is perpendicular to the manifold and bounded <cit.>, or the level of measurement error decreases with the sample size <cit.>; this would allow us to remove the asymptotic bias in the above bounds.§ MANIFOLD APPROXIMATIONMost manifolds cannot be adequately approximated by a single PCA or SPCA.Hence, in this section, we consider using local SPCA to approximate the manifold locally by spherelets. §.§ Local SPCAAssume Y∼ρ where (ρ)=M. Let ϵ∼ N(0,σ^2_D) be Gaussian noise and X=Y+ϵ with i.i.d observations X_1,⋯,X_n. A single sphere will typically not be sufficient to approximate the entire manifold M, but instead we partition ^D into local neighborhoods and implement SPCA separately in each neighborhood.This follows similar practice to popular implementations of local PCA, but we apply SPCA locally instead of PCA.We divide ^D into non-overlapping subsets C_1,…,C_K.For the kth subset, we let X_[k] = { X_i: X_i ∈ C_k }, (V_k,c_k,r_k) denote the results of applying SPCA to data X_[k], Proj_k denote the projection map from x ∈ C_k to y ∈ S_V_k(c_k,r_k) obtained by Lemma <ref>, and M_k = S_V_k(c_k,r_k)∩ C_k.Then, we approximate M by M = ⋃_k=1^K M_k.In general, M will not be continuous or a manifold but instead is made up of a collection of pieces of spheres chosen to approximate the manifold M.There are many ways in which one can choose the subsets { C_k }_k=1^K, but in general the number of subsets K will be chosen to be increasing with the sample size with a constraint so that the number of data points in each subset cannot be too small, as then M_k cannot be reliably estimated.Below we provide theory on mean square error properties of the estimator M under some conditions on how the subsets are chosen but without focusing on a particular algorithm for choosing the subsets. There are a variety of algorithms for multiscale partitioning of the sample space, ranging from cover trees <cit.> to METIS <cit.>, to iterated PCA <cit.>. As the scale becomes finer, the number of partition sets increases exponentially and the size of each set decreases exponentially. Assume U⊂ M is an arbitrary submanifold of M, ρ_U=ρ|_U is the probability measure of data X_i conditionally on X_i∈ U and diam(U)=sup_x,y∈ U d(x,y)=α_U. For example, if we bisect the unit cube in ^D j times, then the diameter of each piece will be α_U∝ 2^-j which decays to zero with j. The approximation error depends on α_U: as α_U→0, each local neighborhood is smaller so linear or sphericalapproximations perform better. Addition to assumption (A), assume (B): There exists δ>0 such that for any submanifold U⊂ M, r_U≥δ, where r_U is the radius of the sphere obtained by population SPCA (defined after Theorem <ref>) on U with respect to the measure ρ_U. (C): The partition {C_1,⋯,C_K} is regular in the sense that diam(C_k)^d=O(n_k/n), where n_k is the number of samples in C_k. Then the manifold approximation uniform error rate is sup_x∈ Md(x,M)≤Cσ^2+o_p(σlog n/n^2/d+4).As discussed after Corollary <ref>, under stronger assumptions on the noise, the bias term σ^2 can be removed so that the rate is n^-2/d+4 up to a log factor while the optimal rate for manifold approximation is n^-2/d+2 <cit.>.Assumption (B) is a very weak regularity condition on the manifold M, which rules out extreme kinks in M leading to unbounded curvature and hence arbitrarily small radius r_U of the best fitting sphere in a local region U ⊂ M containing the kink. Assumption (C) is also reasonable. For example, suppose the density function of ρ, denoted by f_ρ, is strictly positive. By compactness of M, f_ρ is bounded above and below by a positive number, and hence ρ(C_k)∼Vol(C_k)∼diam(C_k)^d. Since ρ(C_k)∼ n_k/n, (C) follows.In addition, throughout this paper, we assume D and d are fixed and all asymptotic theories are for n→∞. §.§ Covering NumbersTheorem <ref> does not imply that applying SPCA in local neighborhoods leads to a better rate than applying PCA. This is not surprising since we are not restricting the curvature. When the curvature is zero, we expect SPCA and PCA to have similar performance. However, when curvature is not approximately zero, SPCA is expected to have notably improved performance except for very small local regions. This is consistent with the empirical results in the following section. In this section, we provide geometric evidence in favor of spherelets over local PCA through covering numbers. This covering number theory is mathematical and does not involve data or distributional assumptions.We define the covering number as the minimum number of local bases needed to approximate the manifold within ϵ error.Our main theorem shows the covering number of spherelets is smaller than that of hyperplanes. We assume M to be compact, otherwise the covering number is not well-defined. Let M denote a d-dimensional compact C^3 Riemannian manifold embedded in ^D, and ℬ a collection of d-dimensional subsets of ^D.Then the ϵ>0covering number N_ℬ(ϵ,M) is defined as N_ℬ(ϵ,M)inf_K∈_+{K: ∃{C_k, Proj_k,B_k}_k=1^K s.t. x-Proj(x)≤ϵ, ∀ x∈ M}, where {C_k}_k=1^K is a partition of ^D, B_k∈ℬ, Proj_k: C_k→ B_k, x↦y∈ B_kx-y^2 is the corresponding local projection and Proj(x)∑_k=1^K 1_{x∈ C_k}Proj_k (x) is the global projection.The above covering number is the minimal number of bases in dictionary ℬ needed to approximateM with ϵ error. We focus on two choices of ℬ: all d-dimensional hyperplanes in ^D, denoted by ℋ, and all d-dimensional spheres in ^D, denoted by 𝒮.Including spheres with infinite radius in 𝒮, we haveℋ⊂𝒮 implying the following Proposition.For any compact C^3 Riemannian manifold M, and ϵ>0, N_𝒮(ϵ,M)≤ N_ℋ(ϵ,M).The proposition implies that an oracle focused on approximating M with error ≤ϵ using either spheres or hyperplanes will never need to use more spheres than hyperplanes.The following Theorem provides an improved comparison. Assume M is a compact C^3 d-dimensional Riemannian manifold. Then there exists constants C=C(M) and δ=δ(M)>0 such that ∀ϵ≤δ N_ℋ(ϵ,M)≤ Cϵ^-d/2, N_𝒮(ϵ,M)≤ CV_ϵϵ^-d/3+C(V-V_ϵ)ϵ^-d/2, where V is the (Riemannian) volume of M while V_ϵ∈[0,V] is the volume of a submanifold of M that is locally well approximated by spheres; see Appendix <ref> for more details.When d=1, M=γ is a curve and we have the following Corollary.For any ϵ>0 and compact C^3 curve γ, N_𝒮(ϵ,γ)≤ Cϵ^-1/3. Thecurse of dimensionality comes in through the term ϵ^-d/2, but we can decrease its impact from ϵ^-d/2 to ϵ^-d/3 using spheres instead of planes. The upper bounds of covering numbersN_ℋ(ϵ,M) and N_𝒮(ϵ,M) are both tight in terms of the rate in ε. The covering number depends on the geometry of the manifold, particularly the curvature of geodesics on the manifold and not the sectional curvature or Ricci curvature.Although spheres have positive Gaussian/sectional curvature, they can be used as a dictionary to estimate manifolds with negative Gaussian/sectional curvature. For example when the manifold is a 2-dimensional surface, V_ϵ is determined by the difference of the two principal curvatures, not the Gaussian curvature or mean curvature.When d=1, the absolute curvature determines the radius of the osculating circle and the sign of the curvature determines which side of the tangent line the center of the circle is on. This provides intuition for why spherelets works well for both positive and negative curvature spaces. The bounds in our main theorem are tight, implying that spherelets often require many fewer pieces than locally linear dictionaries to approximate M to any fixed accuracy level ϵ; particularly large gains occur when a non-negligible subset of M is covered by the closure of points having not too large change in curvature of geodesics along different directions.As each piece involves O(D) unknown parameters, these gains in covering numbers should lead to real practical gains in statistical performance; indeed this is what we have observed in applications. § APPLICATIONS, ALGORITHMS AND EXAMPLESThis section contains a variety of simulation studies and real data applications of SPCA and spherelets. To measure performance in analyzing data, we focus on the mean squared error (MSE),1/n∑_i=1^n X_i-X_i, where X_i is the fitted value of X_i by a dimensionality reduction method. Most dimension reduction methods do not provide X_i but instead replace X_i with a lower-dimensional summary; such methods are not directly comparable to global or local SPCA. §.§ Simulation study of global SPCA We first verify the convergence rates in Theorem <ref> on simulated data. For a specific ambient and intrinsic dimension pair (D,d),we sample Y_1,⋯,Y_n from a von Mises-Fisher distribution on S^d with sample size n. Then we sample Gaussian noise ϵ_i∼ N(0,σ^2 _D), and let X_i = c_0+r_0V_0Y_i+ϵ_i, where c_0∈^D, r_0>0 and V_0∈^D×(d+1) with orthonormal columns. The X_i are close to the sphere centered at c_0 with radius r_0 in linear subspace V_0. The true parameters are generated randomly. First we fix σ^2 = {0.1,0.01} (the results are similar for larger σ^2 so we present the plots for two different σ^2 to make the figure clearer). Let V, c and r be the estimated parameters by SPCA. Figure <ref> shows the error rate with respect to sample size n for two choices of (D,d). Thex-axis is log sample size while the y-axis is log(V_0-V), log(c_0-c) and log(|r_0-r|). The rates for all three parameters are n^-1/2. In Theorem <ref>, when σ^2 is fixed, the upper bound of the error rate for V is n^-1/2 up to a log n factor, which is asymptotically negligible compare to √(n). While the rates for c and r are affected by the bias term O(σ^2), the rates are very close to n^-1/2 when C_vlog n/n^1/2≫ C_bσ, where C_b and C_v are the constants in Theorem <ref> for the bias and variance, respectively. There is a subtle difference between the rate of c and r because of the constant C_b that controls the bias. For the center c, C_b∼ 1/λ_d+1^2 where λ_d+1 is the (d+1)-th eigenvalue of the covariance matrix of Y, which is supported on the d-dimensional sphere without noise. As a result, λ_d+1is often much larger than σ^2; otherwise Y will be concentrated around the equator of S^d or S^d-1.λ_d+1/σ can be viewed as the “signal-to-noise ratio” (SNR). The constant C_b in the bias term for c is proportional to 1/SNR^2, which is usually very small; hence the rate for c is observed as n^-1/2 in our experiments. For radius r, the constant C_b∼√(d+1), which is independent of the SNR and often larger than the constant for c. As a result, we may observe attenuation of the n^-1/2 rate for r when n is large enough so that C_vlog n/n^1/2≲ C_bσ. For more details of the constant C_b, see the proof of Theorem <ref> in the Supplementary Materials. Next we fix n∈{100,1000} and vary σ^2. Figure <ref> shows that the error rates in estimating the sphere parameters as σ^2 varies are all (σ^2)^1/2. In Theorem <ref>, when n is fixed, the upper bound of the error rate with respect to σ is O(σ^2+σ)=O(σ) since σ dominates σ^2 when σ→ 0.§.§ Application to data visualization A common focus of dimensionality reduction algorithms is on data visualization.To illustrate the use of SPCA for data visualization we consider an application to a banknote dataset.The data consist of 1372 400 × 400 pixel images of genuine and fake banknotes.Based on these images, 4 features are extracted using a wavelet analysis and it is of interest to investigate differences between authentic and fake banknotes.For data visualization, we choose d=2 for all algorithms, and compare SPCA with PCA, t-distributed stochastic neighbor embedding (tSNE, <cit.>), uniform manifold approximation and projection (UMAP, <cit.>), locally linear embedding (LLE, <cit.>) and multidimensional scaling (MDS, <cit.>). In Figure <ref> we plot the 2-dimensional embedding of tSNE, UMAP, LLE, MDS, PCA, and SPCA. For PCA we show the first two principal components.For SPCA, we first project the data to the 2-dimensional sphere and then obtain the polar angle and azimuthal angle.The results are shown in Figure <ref>, which shows a clear separation between the genuine and fake banknotes for SPCA.Interesting, (locally) linear methods including PCA, LLE and MDS fail to separate the two groups, at least based on only two components.The popular tSNE and UMAP methods do well at separating genuine and forged banknotes, but in a highly complex way that shows many sub-clusters in the data.These approaches are much more complex that SPCA, including computationally, and there has been concern in the literature that tSNE and UMAP may show artifactual structure in the data <cit.>. §.§ Manifold approximation via spherelets In manifold approximation we attempt to find an estimator of the unknown manifold M, say M. When the underlying manifold is complex, a single sphere is not enough, motivating local SPCA or spherelets. As local SPCA provides an estimator of a submanifold U⊂ M in a neighborhood, we split ^D into subsetsC_1,…,C_k and apply local SPCA to estimate the manifold in each subset. Let M_k=C_k∩ M be the sub-manifold of M restricted to C_k. LetM_k denote the estimate of M_k based on applying SPCA to the data within C_k, andset M=⋃_k=1^K M_k. The map which projects a data point x to the estimated manifold M is denoted by Proj: ℝ^D→M. Algorithm <ref> describes the calculation of Proj and M given a partition of ^D.We apply spherelets to multiple examples.The first two (Euler spiral, cylinder) are toy examples usingknowledge of the manifold to choose the partition. The subsequent examples (Euler spiral, economics, user knowledge) use a multiscale scheme to choose C_1,…,C_k, and compare with local PCA.Euler Spiral. The Euler spiral, γ(s)=[ ∫_0^s cos(t^2)dt, ∫_0^s sin(t^2)dt], s∈ [0,2], is a common example in the manifold learning literature, having curvature linear with respect to the arc length s. We generate s_i uniformly from [0,2] and then add Gaussian noise to γ(s_i).We uniformly partition s∈[0,2] to obtainC_1,…,C_k. The first panel in Figure <ref> shows the convergence rate is n^-0.57, which is better than the n^-0.4 rate in Theorem <ref>.The remaining panels show the projected data with different sample size and different number of partitions, along with the true spiral. When the number of partitions is 3, the approximation performance is excellent.Cylinder. The cylinder {(x=cosθ,y=sinθ,z):θ∈[0,2π],z∈[0,1]} is a surface with zero Gaussian curvature. The principal curvatures are 0 and 1. We independently sample θ_i and z_i uniformly from [0,π] and [0,1], respectively. C_1,…,C_k are obtained by uniformly partitioning z∈[0,1]. The first panel in Figure <ref> shows the convergence rate is n^-0.35, slightly better than the n^-1/3 rate in Theorem <ref>. The last two panels show how spherelets approximate the cylinder, where the number of partitions is 2 and 3, respectively. Estimating the partitionNext, we study the relation between number of partitions and MSE and compare with PCA. There are many existing partitioning algorithms for subdividing the sample space into local neighborhoods. Popular algorithms, such as cover trees, iterated PCA and METIS, have a multi-scale structure, repeatedly partitioning ^D until a stopping condition is achieved.For simplicity, for local SPCA and PCA, we consider a multi-scale partitioning scheme which iteratively splits the sample space based on the first principal component score until a predefined bound ϵ on MSE is met for each of the partition sets or the sample size n_k no longer exceeds a minimal value n_0.If MSE_k>ϵ and n_k>n_0, we calculate PC_1=(X_[k]-μ_k) v_1,k, where μ_k=X̅_[k] and v_1,k is the first eigenvector of the covariance matrix of X_[k]. Next we split C_k into two sub-partitions C_k,1 and C_k,2 based on the sign of PC_1, i.e, i^th sample of X_[k] is assigned to C_k,1 if PC_1,i>0 and to C_k,2 otherwise. We estimate the intrinsic dimension d as corresponding to the elbow point in the dimension v.s. MSE plot for PCA or SPCA, following common practice in PCA. Estimating the intrinsic dimension d is known to be a difficult problem for algorithms lackingfitted values X_i for X_i; see <cit.>. Euler Spiral. We generate 2500 training and 2500 test samples from the Euler spiral with Gaussian noise with variance 0.01. Figure <ref>(a) and (b) show the projected test dataset with different partitions described in different colors. It is clear that there are fewer pieces of circles than lines and the estimated M is smoother in the second panel, reflecting better approximation by SPCA. Figure <ref>(c) shows the comparative performance of local PCA and SPCA with respect to log of number of partitions vs log of MSE. Clearly SPCA has much better performance than PCA, as it requires only 14 partitions to achieve an MSE of about 10^-7, while PCA requires 120 partitions to achieve a similar error. Economics. As introduced in Section <ref>, the Economics dataset has D=5 attributes and 576 samples. We choose 460 samples randomly as the training set and the remaining 116 samples as the test set for cross validation. We choose d=1 as the first principal component explains 99.7% of the variance. Figure <ref>(d) shows that SPCA has much better performance than PCA, as it requires only 8 partitions to achieve an MSE of 0.81, while PCA requires 26 partitions to achieve a similar error. User knowledge. These data were collected to assess students’ knowledge of Electrical DC Machines <cit.>. There were n=258 students who were studied to obtain D=5 attributes including study time for goal object materials, repetition number for goal object materials, study time for objects related to the goal object, exam performance for related objects and exam performance for the goal object. We choose 206 samples randomly as the training set and the remaining 52 samples as the test set for cross validation. We chose d=4 because we did not observe any sudden drop (also known as elbow point) in the MSE plot in Figure <ref> left panel. Figure<ref> right panel compares performance of PCA and SPCA. Overfitting can occur as the number of partitions becomes too large, forcing the sample size per partition to be too low.The optimal number of partitions for SPCA is 8, which is significantly lower than the optimal number for PCA.In addition, the MSE for spherelets with 8 partitions is 0.25, which is significantly lower than the minimal value obtained by local PCA.The above examples show that spherelets has smaller MSE than local PCA given the same number of partitions. Equivalently, given a fixed error ϵ, the number of spheres needed to approximate the manifold is smaller than that of hyperplanes. This coincides with the statement of Theorem <ref>. When training sample size is small to moderate, there will be limited data available per piece and local PCA will have high error when the number of pieces is sufficiently large to obtain an accurate approximation to the manifold.§ DISCUSSION There are several natural next directions building on the spherelets approach. The current version of spherelets is not constrained to be connected, so that the estimate M of the manifold M will in general be disconnected.We view this as an advantage in many applications, because it avoids restricting consideration to manifolds that have only one connected component and instead accommodates true disconnectedness.Nevertheless, in certain applications it is very useful to obtain a connected estimate; for example, when we have prior knowledge that the true manifold is connected and want to use this knowledge to improve statistical efficiency and produce a more visually appealing and realistic estimate. A typical case is in imaging when D is 2 or 3 and d is 1 or 2 and we are trying to estimate a known object from noisy data.Possibilities for incorporating connectedness constraints include (a) producing an initial M using spherelets and then closing the gaps through linear interpolation; and (b) incorporating a continuity constraint directly into the objective function, to obtain essentially a type of higher dimensional analogue of splines. An additional direction is improving the flexibility of the basis by further broadening the dictionary beyond simply pieces of spheres.Although one of the main advantages of spherelets is that we maintain much of the simplicity and computational tractability of locally linear bases, it is nonetheless intriguing to include additional flexibility in an attempt to obtain more concise representations of the data with fewer pieces.Possibilities we are starting to consider include the use of quadratic forms to obtain a higher order local approximation to the manifold and extending spheres to ellipses.In considering such extensions, there are major statistical and computational hurdles; the statistical challenge is maintaining parsimony, while the computational one is to obtain a simple and scalable algorithm.As a good compromise to clear both of these hurdles, one possibility is to start with spherelets and then perturb initial sphere estimates (e.g., to produce an ellipse) to better fit the data.Another important direction is to study the optimal partitioning for spherelets. Existing partitioning algorithms are locally linear, mainly designed for local PCA. A spherical partitioning algorithm needs to be developed to improve spherelets. With such partitioning, it might be possible to verify the covering number upper bound numerically.Finally, there is substantial interest in scaling up to very large D cases; the current algorithm will face problems in this regard similar to issues faced in applying usual PCA to high-dimensional data.To scale up spherelets, one can potentially leverage on scalable extensions of PCA, such as sparse PCA (<cit.>, <cit.>).The availability of a very simple closed form solution to spherical PCA makes such extensions conceptually straightforward, but it remains to implement such approaches in practice and carefully consider appropriate asymptotic theory. In terms of theory, it is interesting to consider optimal rates of simultaneously estimating M and the density of the data on (or close) to M, including in cases in which D is large and potentially increasing with sample size.§ ACKNOWLEDGEMENTDL and DD were supported by United States Office of Naval Research, N00014-14-1-0245 and N00014-16-1-2147; United States National Institutes of Health, 5R01ES027498-02. § APPENDIX §.§ Economics data fitAs the Economics data have clear evidence in favor of d=1, we also consider applying principal curves in addition to local PCA and local SPCA (spherelets).We present the data and fitted values for representative pairs of variables in Figure <ref>. We find that principal curves over-smooths the data and does not have competitive performance relative to the local PCA-based methods.In addition, spherelets clearly have the best performance for all pairs of features, with more gain when the data exhibit more curvature. §.§ SPCA solutionIn this section we prove the SPCA solution shown inSection <ref>. Let Φ_V,c be the orthogonal projection to the affine subspace c+V; that is, Φ_V,c(x)=c+VV^⊤ (x-c). Then observe that x-Φ_V,c(x)⊥Φ_V,c(x)-y, ∀ y∈ S_V(c,r), so x-y^2=x-Φ_V,c(x)+Φ_V,c(x)-y^2=x-Φ_V,c(x)^2+Φ_V,c(x)-y^2. That is, the optimization problem y∈ S_V(c,r)x-y^2 is equivalent to y∈ S_V(c,r)Φ_V,c(x)-y^2. Since the second problem only involves the affine subspace c+V, we can translate it to the following problem: y∈ S(c,r)⊂^d+1x-y^2, where x is any point in ^d+1 and S(c,r)={y∈^d+1: y-c=r}. So we only need to prove Ψ_V,c(x)y∈ S(c,r)x-y^2=c+r/x-c(x-c). On one hand, x-Ψ_V,c(x)^2 =x-c-r/x-c(x-c)^2=(1-r/x-c)(x-c)^2=(1-r/x-c)^2x-c^2=(x-c-r)^2. On the other hand, for any y∈ S(c,r), x-y^2 =x-c+c-y^2=x-c^2+c-y^2-2(x-c)^⊤(y-c)=x-c^2+r^2-2(x-c)^⊤ (y-c)≥x-c^2+r^2-2x-cy-c=x-c^2+r^2-2rx-c=(x-c-r)^2=x-Ψ_V,c(x)^2. By simple calculation, we can show that r^2r ℒ(c,r)=1/n∑_i=1^nY_i-c^2. Hence, if we adopt the same notation: ℒ(c)∑_i=1^n(Y_i-c^2-r^2)^2=∑_i=1^n{Y_i-c^2-1/n∑_j=1^nY_j-c^2)}^2, it suffices to minimize ℒ(c) to obtain c. ℒ(c)=∑_i=1^n{ Y_i^⊤ Y_i-2c^⊤ Y_i-1/n∑_j=1^n(Y_j^⊤ Y_j-2c^⊤ Y_j)}^2. Letting l_i=Y_i^⊤ Y_i, l̅=1/n∑_i=1^n l_i and Y̅=1/n∑_i=1^n Y_i, then ℒ(f) =∑_i=1^n{ Y_i^⊤ Y_i-2c^⊤ Y_i-1/n∑_j=1^n(Y_j^⊤ Y_j-2c^⊤ Y_j)}^2=∑_i=1^n(l_i-2c^⊤ Y_i-l̅+2c^⊤Y̅)^2=∑_i=1^n( (l_i-l̅)-2c^⊤ (Y_i-Y̅))^2=∑_i=1^n {4c^⊤ (Y_i-Y̅)(Y_i-Y̅)^⊤ c-4(l_i-l̅)c^⊤ (Y_i-Y̅)+(l_i-l̅)^2}=4c^⊤∑_i=1^n(Y_i-Y̅)(Y_i-Y̅)^⊤ c-4c^⊤∑_i=1^n(l_i-l̅)(Y_i-Y̅)+∑_i=1^n(l_i-l̅)^2 is a quadratic function. So c=1/2H^-1ξ where H=∑_i=1^n(Y_i-Y̅)(Y_i-Y̅)^⊤,ξ=∑_i=1^n (Y_i^⊤Y_i-1/n∑_j=1^nY_j^⊤ Y_j)(Y_i-Y̅). Then we prove Corollary <ref>, that is, when the data are sampled from some sphere S_V_0(c_0,r_0), then SPCA and the sum of squared residuals have the same minimizer:(V_0,c_0,r_0)=(V,c,r) =V,c,r ∑_i=1^n d^2(Y_i,S_V(c,r))V,c,r ℒ(c). When X_i∈ S_V_0(c_0,r_0), both geometric and algebraic loss functions are zero at (V_0,c_0,r_0), which is the minimizer of both algorithms. §.§ A deeper look at Theorem 8 We define some key geometric features of the manifold M involved in the covering number proof. Let M be a d-dimensional C^3 Riemannian manifold. The volume of M is denoted by V. Let κ:T^1M→ denote the curvature of the geodesic on M starting from point p with initial direction v: κ(p,v):=d^2 exp_p(tv)/dt^2|_t=0, where T^1M⋃_p∈ M{v∈ T_pM|v=1} is the unit sphere bundle over M and exp_p(·) is the exponential map from the tangent plane at p to M. Then κ_maxsup_(p,v)∈ T^1M|κ(p,v)|<∞ is the maximum curvature. Similarly, Tsup_(p,v)∈ T^1Md^3 exp_p(tv)/dt^3|_t=0 is the maximum of the absolute rate of change of the curvature. Given any ϵ>0 and letting T_pM denote the tangent plane to M at p ∈ M, F_ϵ{p∈ M: sup_v∈ T^1_pMκ(p,v)-inf_v∈ T^1_pMκ(p,v) ≤(2ϵ/κ_max)^1/2} is called the set of ϵ-spherical points on M, where T^1_pM is the unit ball in tangent space T_pM. Let B(p,ϵ) be the geodesic (open) ball centered at p with radius ϵ, then M_ϵ:=⋃_p∈ F_ϵ B(p,1/2(6ϵ/3+T)^1/3) is called the spherical submanifold of M, and the volume is V_ϵVol(M_ϵ). M is called an ϵ sphere if V_ϵ=V. V_ϵ is non-decreasing with respect to ϵ, and it is possible that V_ϵ=V but M_ϵ≠ M. In this case, M∖ M_ϵ has zero Riemannian measure so it does not impact the manifold approximation given observations from the manifold. As a result, we will not consider zero measure sets in the following sections. A space form, a complete, simply connected Riemmanian manifold of constant sectional curvature, is an ϵ sphere for any manifold dimension d and ϵ>0. A one dimensional manifold (a curve) is an ϵ sphere for any ϵ>0. Now we present a nontrivial surface, called the Enneper's surface (a minimal surface) and calculate the spherical points and spherical submanifold explicitly. Let M={(u-1/3u^3+uv^2,-v-u^2v+1/3v^3,u^2-v^2)∈^3|u^2+v^2≤ R^2} be the compact truncation of the Enneper surface, which is an interesting surface in differential geometry that has varying curvature. In fact this surface is a minimal surface, that is, the mean curvature is zero and the two principal curvatures are mutually additive inverse to each other. By definition, M is a compact smooth surface. We calculate the spherical points, spherical submanifold as well as its volume. For the Enneper surface, when ϵ <4/(R^2+1)^2, there are no spherical point so F_ϵ=M_ϵ=∅ and V_ϵ=0. When ϵ≥4/(R^2+1)^2, F_ϵ={(u-1/3u^3+uv^2,-v-u^2v+1/3v^3,u^2-v^2):2/√(ϵ)-1≤ u^2+v^2≤ R^2}, and T= 1/1024√(7)R≥1/√(7) 4R/(R^2+1)^4R<1/√(7). When R≥1/√(7), M_ϵ={(u-1/3u^3+uv^2,-v-u^2v+1/3v^3,u^2-v^2)|u^2+v^2≥α^2}, where α=max{0,√(2/√(ϵ)-1)-(6ϵ/3+1/1024√(7))^1/3}, then V_ϵ=π(R^2+1/2R^4-α^2-1/2α^4), V_ϵ/V=2R^2+R^4-2α^2-α^4/2R^2+R^4. An extreme case is ϵ≥4/(R^2+1)^2 and √(2/√(ϵ)-1)-(6ϵ/3+1/1024√(7))^1/3≤0. In this case, although F_ϵ≠ M, the union of small geodesic balls centered on the spherical points is M, that is, M_ϵ=M, so α=0 and V_ϵ/V=1, which means M is an ϵ sphere. The first fundamental form of Enneper's surface is E=(1+u^2+v^2)^2, F = 0, G = (1+u^2+v^2)^2, the two principal curvatures are k_1 = 2/(1+u^2+v^2)^2,k_2 = -2/(1+u^2+v^2)^2,k_2. So the Gaussian curvature is -4/(1+u^2+v^2)^4 and the mean curvature is 0 so K=4. Then all other quantities including F_ϵ and M_ϵ can be directly calculated, see <cit.> for more details. chicago§ SUPPLEMENTARY MATERIALS§ SPCA ASYMPTOTICSIn this section, we prove the SPCA asymptotic results from Section 2.2. Let V^*,c^*,r^* be the population SPCA solution, then the error term splits into bias and variance: c_0-c_n≤c_0-c^*_Bias+c^*-c_n_Variance. It suffices to bound the bias and variance separately. We handle the bias term in Lemma <ref> and variance in Lemma <ref>, and then Theorem 4 follows. Let V^*,c^*,r^* be the population SPCA solution, then V^*=V_0, c^*-c_0≤σ^2/2λ_d+1(λ_d+1+σ^2)ξ_Y+σ^2/2(λ_d+1+σ^2) V_0V_0^⊤ Y, |r^*^2-r_0^2|≤ 2 Y-c_0c_0-c^*+c_0-c^*^2+(d+1)σ^2, where ξ_Y = [(Y^⊤ Y-(Y^⊤ Y))(Y- Y)] and λ_d+1 is the d+1th eigenvalue of the covariance matrix of Y. Let Σ_X=(X) and Σ_Y=(Y). First observe thatΣ_Y = [v_1,⋯,v_D]{λ_1,⋯,λ_d+1,0,⋯,0}[v_1,⋯,v_D]^⊤with eigenvectors v_1,⋯,v_D since Y is supported on V, a d+1-dimensional subspace. It isclear that V=span{v_1,⋯,v_d+1}. Recall that Σ_X = Σ_Y+σ^2_D, then Σ_X =[v_1,⋯,v_D] {λ_1+σ^2,⋯,λ_d+1+σ^2,σ^2,⋯,σ^2}[v_1,⋯,v_D]^⊤ and the eigenvectors of X are also v_1,⋯,v_D. Recall that V^* consists of the first d+1 eigenvectors of Σ_Y, so V^*=V_0. Let X= X+ V^*V^*^⊤(X- X) and Y= Y+ V_0V_0^⊤(Y- Y) be the projection of X and Y to the linear space V^*=V_0, then X=Y+ϵ where ϵ=V_0V_0^⊤ϵ. The projected noise ϵ∼ N(0,σ^2 V_0V_0^⊤). The population solution of SPCA is then given by c^* = -1/2[(X-X)(X-X)^⊤]^-1 [(X^⊤X-(X^⊤X))(X-X)]. Recall that c^*=c(X-c^2) and c_0=c(Y-c^2) since (Y-c_0^2)=0 a.s. Letting σ^2→ 0, all moments of X converge to the corresponding moments of Y, so (X-c^2)(Y-c_0^2). By convexity of the objective function, the minimizer is unique so c^*→ c_0 as σ^2→ 0 and Equation (<ref>) becomes c_0 = -1/2[(Y-Y)(Y-Y)^⊤]^-1 [(Y^⊤Y-(Y^⊤Y))(Y-Y)]. First observe that Y=Y, X= X =Y = Y, (Y)=Σ_Y=Σ_Y,(X)=Σ_X=V^*V^*^⊤Σ_X V^*V^*^⊤=Σ_Y+σ^2V_0V_0^⊤. Then observe that ξ_X= [(X^⊤X-(X^⊤X))(X-X)] = [(Y+ϵ)^⊤ (Y+ϵ)-((Y+ϵ)^⊤ (Y+ϵ)))(Y+ϵ- (Y+ϵ))] = [(Y^⊤Y-Y^⊤Y+ϵ^⊤ϵ-ϵ^⊤ϵ+2ϵ^⊤Y)(Y-Y+ϵ)] =[(Y^⊤Y-(Y^⊤Y))(Y-Y)]+[(Y^⊤Y-Y^⊤Y)ϵ] + [(ϵ^⊤ϵ-ϵ^⊤ϵ+2ϵ^⊤Y)(Y-Y)]+ [(ϵ^⊤ϵ-ϵ^⊤ϵ+2ϵ^⊤Y)ϵ] = ξ_Y +2[Y^⊤ϵϵ]= ξ_Y+2σ^2V_0V_0^⊤ Y. Let U=[v_1,⋯,v_D], then Σ_Y = U{λ_1,⋯,λ_d+1,0⋯,0}U^⊤. Now we can compare c^* and c_0: c^*-c_0 = -1/2Σ_X^-1ξ_X+1/2Σ_Y^-1ξ_Y = 1/2(Σ_Y+σ^2V_0V_0^⊤)^-1(ξ_Y+σ^2V_0V_0^⊤ Y)-Σ_Y^-1ξ_Y = 1/2U{1/λ_1+σ^2,⋯,1/λ_d+1+σ^2,0,⋯,0}U^⊤ξ_Y-Σ_Y^-1ξ_Y+(Σ_Y+σ^2V_0V_0^⊤)^-1σ^2V_0V_0^⊤ Y≤1/2U{-σ^2/λ_1(λ_1+σ^2),⋯,-σ^2/λ_d+1(λ_d+1+σ^2),0,⋯,0}U^⊤ξ_Y+1/2U{σ^2/λ_1+σ^2,⋯,σ^2/λ_d+1+σ^2,0,⋯,0}U^⊤ V_0V_0^⊤ Y≤σ^2/2λ_d+1(λ_d+1+σ^2)ξ_Y+σ^2/2(λ_d+1+σ^2) V_0V_0^⊤ Y. Then we consider the radius: |r^*^2-r_0^2| = |[X-c^*^2]-[Y-c_0^2]| = |[Y+ϵ-c^*^2]-[Y-c_0^2]| = |Y-c_0+c_0-c^*+ϵ^2-Y-c_0^2| = |[2(Y-c_0)^⊤(c_0-c^*+ϵ)+c_0-c^*^2+ϵ^2+2(c_0-c^*)^⊤ϵ]| = |2( Y-c_0)^⊤ (c_0-c^*)+c_0-c^*^2+(d+1)σ^2|≤ 2 Y-c_0c_0-c^*+c_0-c^*^2+(d+1)σ^2, Let v_i be eigenvectors of Σ and v_i be the corresponding eigenvectors of Σ, then under assumption (A), _n -=o_p( σ n^-1/2log n), |r_n-r |=o_p( σ n^-1/2log n), v_i-v_i=o_p(n^-1/2σlog n), ∀ i. Under assumption (A) <cit.> showed asymptotic normality of the first d+1 eigenvalues and corresponding eigenvectors, when the underlying distribution is normal. <cit.> extended that result to general populations with finite fourth moment. Although the explicit forms of the joint distribution of the eigenvalues or eigenvectors are not available<cit.>, the asymptotic distributions ofeigenvalues with multiplicities one, and the corresponding eigenvectors, are explicitly obtained. Under assumption (A), let λ_j be the j-th largest eigenvalue of the sample covariance matrix, for j=1,…,d+1, then √(n)( λ_j-λ_j) asymptotically follows a normal distribution with mean 0 and variance κ_jjjj+2λ_j^2, where κ_jjjj depends on the fourth order moment of W=Γ^T(X-E(X)), where Γ is the full population loading matrix. Further, if v_j is the sample eigenvector corresponding to the j-th largest eigenvalue, and v_j is the j-th population eigenvector,√(n)(v_j-v_j) asymptotically follows a normal distribution with mean 0 and variance Ξ_j, where Ξ_j is a positive definite matrix with finite components depending on the fourth cumulant of W. Finally, if √(n)(x-μ) is asymptotically normal with mean 0 and covariance matrix Σ with finite components, then it is easy to see that √(n) x-μ/(log n)=o_p(1). The remainder of the proof is split into three sections. * We first show that √(n)(σlog n)^-1( - )0. The proof is split into three sub-parts. * Showing √(n)(σlog n)^-1H^+ - H^+ 0, with the spectral norm defined as A=sup{Ax_2: x=1 }. Let U=(V W), Λ=(Λ_0, Λ_1). Then H=VV^TΣ VV^T=VΛ_0 V^T. Similarly, H=Σ_d+1^T, where Σ_d+1 is the vector of first (d+1) largest eigenvalues of Σ. Further, from the properties of Moore-Penrose inverse, we have H^+=VΛ^-1_0V^T, and similarly H^+=Λ^-1^T. Let ·_F denote the Frobenius norm. Note that by triangle inequality Λ^-1^T - VΛ^-1_0V^T≤(-V) Λ^-1^T+ V (Λ^-1-Λ^-1_0) ^T+VΛ_0^-1(-V)^T. The first term in right hand side (RHS) of (<ref>), (-V) Λ^-1^T =Λ^-1 (-V)^T=Λ^-1 (-V)^T≤(-V)Λ^-1_F^2 ≤λ_d+1^-1(-V) _F^2 where λ_d+1 is the (d+1)^th largest eigenvalue of Λ, as ^T=I_d+1. Further, √(n) (σlog n)^-1-V _F= √( n (σlog n)^-2∑_i=1^d+1_i-_i^2 ) 0. As λ_d+1 is bounded away from zero, by (A), λ_d+1^-1=O_p(1). Thus, the first part of (<ref>) converges to zero in probability after multiplying by √(n)(σlog n)^-1. Consider the second term in RHS of (<ref>). We haveV(Λ^-1-Λ^-1)^T≤V(Λ^-1-Λ^-1)^T_F=Λ^-1-Λ^-1_F ,as V^TV=^T=I_d+1. By the delta method, we have √(n){(λ_d+1)^-1-(λ_d+1)^-1}=O_p(1). Thus, the second part of RHS of (<ref>) converges to zero in probability after multiplying by √(n)(σlog n)^-1. Finally, following a similar argument as before, it can be shown that the third part of RHS of (<ref>) converges to zero in probability after multiplying by √(n)(σlog n)^-1. * We next show that √(n)(σlog n )^-1ξ - ξ 0. As E(_i)=0, we have ξ=E( V^T ^2VV^T ) - E (V^T ^2) E(VV^T ). First we will show the following three convergences in order: √(n)(σlog n )^-1| 1/n∑_i=1^n^T _i ^2- E( V^T ^2 ) |0 √(n)(σlog n )^-1| 1/n∑_i=1^n^T _i - E(V V^T ) |0√(n)( σlog n )^-1|1/n∑_i=1^n^T _i ^2 ^T _i- E( V^T ^2VV^T ) |0 The convergence in (<ref>) is proved in two steps: √(n)(σlog n )^-1|1/n∑_i=1^n^T _i ^2 -1/n∑_i=1^nV^T_i ^2|0 √(n)( σlog n )^-1| 1/n∑_i=1^nV^T _i ^2-E( V^T ^2 )|0 To see (<ref>), observe that the left hand side (LHS) of (<ref>) is proportional to | 1/n∑_i=1^n_i^2 {(_i/_i)^T ( ^T -VV^T ) _i/_i }| ≤| λ_max(^T-VV^T )|1/n∑_i=1^n_i^2 . Note that _i^2s, i=1,2,…,n, are independent random variables having mean E( ^T)=E(Σ)=(Λ_D) and finite variance by assumption (A). Therefore, by the Strong Law of Large Numbers (SLLN) (see, e.g., <cit.>) the following holds: 1/n∑_i=1^n_i^2 (Λ_D)⟹1/n∑_i=1^n_i^2=O_p(1). Further, recall that for a symmetric A, λ_max(A) ≤σ_max(A)≤A_F, where σ_max(A) is the largest singular value of A. Thus λ_max(^T-VV^T )≤^T-VV^T _F≤^T-V^T _F +V^T-VV^T _F. Now, √(n)( σlog n )^-1(^T-V ) ^T_F = √(n)(σlog n )^-1^T-V _F 0, as shown in part i. To see (<ref>) note that _i^T VV^T _i=w_i, for i=1,2,…, n, where w_is are independent random variables having finite moments by (A). Therefore, by the Lideberg-Feller CLT (see <cit.>) the following holds: √(n)(σlog n )^-1| 1/n∑_i=1^nV^T _i ^2 - E( V^T ^2 ) |0. Thus (<ref>) is proved. 10pt Next consider the convergence in (<ref>). As before we split the proof in two parts: √(n)(σlog n )^-11/n∑_i=1^n(^T -VV^T ) _i ^20,√(n)(σlog n )^-11/n∑_i=1^n VV^T _i - E ( VV^T ) ^20. To see (<ref>) observe as before that the LHS of (<ref>) is less than or equal to 1/n∑_i=1^n(^T -VV^T ) _i ^2 ≤^T -VV^T_F^2 ( 1/n∑_i=1^n_i^2 ), as A X ^2≤λ_max (A^T A) X^2 =σ_max^2 (A) X^2 ≤A_F^2x^2 for any symmetric matrix A. As before, we write ^T -VV^T_F^2 ≤^T -V^T_F^2+V^T -VV^T_F^2. Thus following similar arguments, as provided earlier, one can show that (<ref>) holds. Next consider (<ref>). Let u_i=VV^T _i. As u_is are independent and third order moments of u_is are finite, by Lindeberg Feller CLT, (<ref>) is satisfied. Finally, we show (<ref>). Again, we split the proof in two main steps: √(n) (σlog n)^-1 n^-1∑_i=1^n( ^T _i ^2 ^T _i - V^T _i ^2 VV^T _i )0√(n) (σlog n)^-1 n^-1∑_i=1^nV^T _i ^2 VV^T _i - E( V^T ^2 VV^T )0 To show (<ref>), we split the problem into two parts √(n) (σlog n)^-1 n^-1∑_i=1^n( ^T _i ^2 ^T _i - ^T _i ^2 VV^T _i )0√(n) (σlog n)^-1 n^-1∑_i=1^n( ^T _i ^2 VV^T_i - V^T _i ^2 VV^T _i )0 The proof of (<ref>) is similar to that of (<ref>), except here we have to show that n^-1∑_i=1^n^T_i ^2 _i is bounded in probability. To show the boundedness, it is enough to show that n^-1∑_i=1^n_i^3 converges (as _i^T^T _i≤λ_max( ^T)=λ_max(^T) =1). Again, the random variables _i^3, i=1,2,…,n, are independent and have finite third order moments by assumption (A). Therefore n^-1∑_i=1^n_i^3EX ^3, by SLLN, and hence is bounded in probability. Finally, observe that the LHS of (<ref>) is less than or equal to n^-1/2 (σlog n)^-1∑_i=1^n{^T _i ^2 - V^T _i ^2} VV^T _i= n^-1/2 (σlog n)^-1∑_i=1^n| _i^T(^T -VV^T ) _i |V^T _i≤n^-1/2 (σlog n)^-1|λ_max(^T -VV^T )|∑_i=1^n_i^3≤n^-1/2 (σlog n)^-1σ_max(^T -VV^T )∑_i=1^n_i^3≤n^-1/2 (σlog n)^-1^T -VV^T _F∑_i=1^n_i^3. As before, one can show that (<ref>) converges in probability to zero. 10pt * Remaining steps towards showing √(n) (σlog n)^-1(- )0. Observe that, - =1/2{( H^+ - H^+) ξ + H^+(ξ -ξ) }. Therefore, 2√(n) (σlog n)^-1-≤√(n) (σlog n)^-1H^+ - H^+ξ +H^+√(n) (σlog n)^-1ξ -ξ Thus, to show that √(n) (σlog n)^-1(- )0, it is enough to show that ξ is bounded in probability, and H^+ is bounded. From (ii) we observe that each component of ξ converges in probability to ξ. Consider a number N=N_ϵ such that P(ξ -ξ>N/2)≤ϵ, and ξ<N/2, then P( ξ >N ) ≤ P( ξ -ξ+ξ >N ) ≤P(ξ -ξ > N/2)≤ϵ. Finally, we have already seen that H^+=VΛ^-1V^T. Thus, H^+≤H^+_F=√((Λ^-2))≤λ_d+1^-1√(d+1) = √(d+1)/σ^2, as λ_d+1 is bounded away from zero, the proof follows. 15pt * The next step is to show √(n) (σlog n)^-1(-r)0. Recall that r=E - c, and =n^-1∑_i=1^n_i-. We will prove this in the following two steps: √(n) (σlog n)^-1| 1/n∑_i=1^n( _i- -_i-) |0√(n) (σlog n)^-1| 1/n∑_i=1^n_i- - E-|0. We first prove (<ref>). Observe that, by Jensen's inequality _i- -_i-≤- . Thus, by the previous part (<ref>) holds. Equation (<ref>) can be shown by an application of Lideberg-Feller CLT, as the components are independent and identically distributed with finite second order moments. * Showing √(n) (σlog n)^-1v_i-v_i 0, for i=1,2,…, d+1. This directly follows from the results of <cit.>. Before proving Corollary 5, we show the following two Lemmas first. Given two hyper-spheres S(c_1,r_1) and S(c_2,r_2), their Hausdorff distance is d_H(S(c_1,r_1),S(c_2,r_2))=c_1-c_2+|r_1-r_2|. First assume c_1=c_2=c. Then by definition, for any x∈ S(c,r_1), the distance between x and S(c,r_2) isd(x,S(c,r_2))=|x-c-r_2|=|r_1-r_2|. Then we assume c_1≠ c_2. Observe that for any x∈ S(c,r_1), the distance between x and S(c_2,r_2) is d(x,S(c_2,r_2))=|x-c_2-r_2|. Then observe thatx∈ S(c_1,r_1)d(x,S(c_2,r_2))=c_1+r_1c_1-c_2/c_1-c_2. As a result, sup_x∈ S(c_1,r_1)inf_y∈ S(c_2,r_2)x-y=sup_x∈ S(c_1,r_1)|x-c_2-r_2| = |c_1+r_1c_1-c_2/c_1-c_2-c_2-r_2| = |c_1-c_2+r_1-r_2|. By switching the indices, we have the Hausdorff distance: d_H(S(c_1,r_1),S(c_2,r_2))=max{|c_1-c_2+r_1-r_2|,|c_2-c_1+r_2-r_1|}=c_1-c_2+|r_1-r_2|. Given two spheres S(V_1,c_1,r_1) and S(V_2,c_2,r_2), their Hausdorff distance is d_H(S(V_1,c_1,r_1),S(V_2,c_2,r_2))≤(I-V_1V_1^⊤+V_1V_1^ ⊤)c_1-c_2+|r_1-r_2|+r_1 V_1-V_2. By triangular inequality, d_H(S(V_1,c_1,r_1),S(V_2,c_2,r_2))≤ d_H(S(V_1,c_1,r_1),S(V_1,c_2,r_2))+d_H(S(V_1,c_2,r_2),S(V_2,c_2,r_2)) We start with the first term by assuming V_1=V_2=V and observe that d^2(x,S(V,c_2,r_2))=x-c_2-VV^⊤(x-c_2)^2+(c_2+VV^⊤(x-c_2)-c_2-r_2)^2. For any x∈ S(V,c_1,r_1), x=c_1+VV^⊤(x-c_1) so the first term in Equation (<ref>) becomes x-c_2-VV^⊤(x-c_2)^2=x-c_2-VV^⊤ x+VV^⊤ c_2^2= x-c_2-x+c_1-VV^⊤ c_1+VV^⊤ c_2^2 = c_1-c_2-VV^⊤(c_1-c_2)^2, which does not depend on x. Let c_1 = c_2+VV^⊤(c_1-c_2) be the projection of c_1 to c_2+V. Then the above term can be simplied as c_1-c_1^2. Similarly, let x=c_2+VV^⊤(x-c_2) be the projection of x to c_2+V, then the second term in Equation (<ref>) can be written as (x-c_2-r_2)^2. Recall that x∈ S(V,c_1,r_1), then x∈ S(V,c_1,r_1) so by Lemma <ref>, x∈ S(V,c_1,r_1)(x-c_2-r_2)^2=c_1+r_1c_1-c_2/c_1-c_2 if c_1≠ c_2, otherwise thecan be any point on S(V,c_1,r_1). The corresponding maximizer x∈ S(V,c_1,r_1) is determined by c_2+VV^⊤(x-c_2)=x=c_1+r_1c_1-c_2/c_1-c_2. Recall that VV^⊤ x = x-c_1+VV^⊤ c_1, so we have x= c_1-VV^⊤ c_1+VV^⊤ x= c_1-VV^⊤ c_1-c_2+VV^⊤ c_2+c_1+r_1c_1-c_2/c_1-c_2 = c_1-c_1+c_1+r_1c_1-c_2/c_1-c_2= c_1+r_1c_1-c_2/c_1-c_2 = x∈ S(V,c_1,r_1)(x-c_2-r_2)^2 = x∈ S(V,c_1,r_1)d(x,S(V,c_2,r_2)). As a result, sup_x∈ S(V,c_1,r_1)inf_y∈ S(V,c_2,r_2) = √(c_1-c_1^2+(c_1+r_1c_1-c_2/c_1-c_2-c_2-r_2)^2) =√(c_1-c_2-VV^⊤(c_1-c_2)^2+((1+r_1/c_1-c_2)c_1-c_2-r_2)^2) = √((I-VV^⊤)(c_1-c_2)^2+(VV^⊤(c_1-c_2)+r_1-r_2)^2)≤(I-VV^⊤)(c_1-c_2)+|VV^⊤(c_1-c_2)+r_1-r_2|≤I-VV^⊤c_1-c_2+VV^ ⊤c_1-c_2+|r_1-r_2|, where ·=·_2 is the induced 2-norm for matrices. Note that this norm can also be replaced by the Frobenius norm since A_2≤A_F. By symmetry, we conclude that d_H(S(V_1,c_1,r_1),S(V_1,c_2,r_2))≤I-V_1V_1^⊤c_1-c_2+V_1V_1^ ⊤c_1-c_2+|r_1-r_2|. Next, we assume c_1=c_2=c and r_1=r_2=r. The Hausdorff distance is invariant under translation so we can assume c=0 without loss of generality.Recalling that for any x∈ S(V_1,0,r) with V_2V_2^⊤ x≠ 0, then d^2(x,S(V_2,0,r)) =x-r/V_2V_2^⊤ xV_2V_2^⊤ x^2= r^2+r^2-2r x·V_2V_2^⊤ x/V_2V_2^⊤ x = 2r^2-2r^2x/x·V_2V_2^⊤ x/V_2V_2^⊤ x = r^2(x/x-V_2V_2^⊤ x/V_2V_2^⊤ x^2). As a result, the Hausdorff distance is given by d_H(S(V_1,c_1,r_1),S(V_2,c_1,r_1)) =r_1min{min_x∈ V_1x/x-V_2V_2^⊤ x/V_2V_2^⊤ x,min_y∈ V_2y/y-V_1V_1^⊤ x/V_1V_1^⊤ y}≤ r_1 V_1-V_2. To conclude, the Hausdorff distance between two d-dimensional spheres is d_H(S(V_1,c_1,r_1),S(V_2,c_2,r_2))≤(I-V_1V_1^⊤+V_1V_1^ ⊤)c_1-c_2+|r_1-r_2|+r_1 LV_1-V_2. The corollary is a direct consequence of Lemma <ref> and Theorem 4. § LOCAL SPCAIn this section, we prove results related to local SPCA from Section 3.1. We first consider a single piece M_k=M with radius α and sample size n. Let S_V_y(c_y,r_y) be the population solution of SPCA based on y∼ρ with (ρ)=M and diam(M)=α, then there exists C>0 such that sup_x∈ M d(x,S_V_y(c_y,r_y))≤ Cα^2.Before proving Lemma <ref>, we consider the following lemmas. Recall that a hyperplane can be viewed as a sphere with infinite radius, which is rigorously stated in the following lemma. Let V be a d-dimensional subspace of ^d+1 and α>0 is a fixed positive real number. Then for any ϵ, there exists a sphere S(c,r) such that sup_x∈ V, x≤αd(x,S(c,r))<ϵ. This is a direct corollary of Taylor expansion. In fact we can let c=r n where n is the unit normal vector of V and r=α^2/ϵ. Let (μ^*,V^*) be the best approximating hyperplane of M (the population solution of PCA), then there exists C>0 such that sup_x∈ M d^2(x,μ^*+V^*)≤ Cα^4. This is another direct consequence of Taylor expansion, see also Proposition <ref>. Now we prove Lemma <ref>. Let (c_y,r_y) be the solution of SPCA, then (x-c_y-r_y)^2=(x-c_y^2-r_y^2)^2/(x-c_y+r_y)^2≤(x-c_y^2-r_y^2)^2/r_y^2≤(x-c_y^2-r_y^2)^2/δ^2, where δ is defined in Assumption (B) in Theorem 6. As a result, it suffices to find the upper bound of (x-c_y^2-r_y^2)^2 which is the loss function of SPCA. Lemma<ref> implies that there exists an affine subspace μ+V and C>0 such that d^2(x,μ+V)≤ Cα^4 for any x∈ U. Then set ϵ=Cα^2 in Lemma <ref> so there exists c,r such that d(y,c+r/y-c(y-c))≤ϵ=Cα^2 for any y=μ+VV^⊤ x where x∈ U. In fact, r=α^2/ϵ=1/C. For convenience, when x is the original point in U, let y be the linear projection of x onto the affine subspace μ+V and z be the spherical projection to sphere S(c,r). By the triangular inequality, we have x-z≤x-y+y-z≤ Cα^2+Cα^2=2Cα^2=Cα^2 by Lemma <ref> and <ref>, where we are abusing C for all constants without confusion. Then we evaluate the loss function at such (c,r): (x-c^2-r^2)^2 =(x^⊤ x-2c^⊤ x+c^⊤ c-r^2)^2=((z+x-z)^⊤ (z+x-z)-2c^⊤ (z+x-z)+c^⊤ c-r^2)^2 =(0+x-z^2+2(z-c)^⊤ (x-z))^2≤ (x-z^2+2|(z-c)^⊤ (x-z)|)^2≤ (C^2α^4+2z-cx-z)^2≤(C^2α^4+2rCα^2)^2∼ 4r^2C^2α^4=Cα^4 when α is sufficiently small. To conclude, sup_x∈ M d^2(x,S_V_y(c_y,r_y))≤sup_x∈ M(x-c_y^2-r_y^2)^2/δ^2≤sup_x∈ M(x-c^2-r^2)^2/δ^2≤ Cα^4. Then we consider the following Lemma regarding Hausdorff distance: For any x∈^D and two compact sets A,B⊂^D, |d(x,A)- d(x,B)|≤ d_H(A,B). Assume x∈ A, then d(x,A)=0, so |d(x,A)- d(x,B)|=d(x,B)≤sup_x∈ Ad(x,B)≤ d_H(A,B). A similar proof holds if x∈ B. Then we assume x∉ A∪ B. Let b_0∈ B such that d(x,B)=d(x,b_0) and a_0∈ A such that d(b_0,A)=d(b_0,a_0), then d(x,A)- d(x,B)≤ d(x,a_0)-d(x,b_0)≤ d(a_0,b_0)=d(b_0,A)≤ d_H(A,B). Similarly we can show d(x,B)- d(x,A)≤ d_H(A,B). Now we are ready to prove Theorem 6. For the single piece case, let M = S_V_n(c_n,r_n), where V_n, c_n and r_n are the solution of empirical SPCA on data x_1,⋯,x_n. Similarly, let V_z, c_z, r_z be the solution of population SPCA based on z∼ρ and V_x, c_x, r_x be the solution of population SPCA based on x∼ρ*N(0,σ^2 I_D). Then we have sup_y∈ Md(x,S_V_n(c_n,r_n)) Lemma <ref>≤sup_y∈ Md(y,S_V_z(c_z,r_z))+d_H(S_V_z(c_z,r_z),S_V_n(c_n,r_n))Lemma <ref>≤Cα^2+d_H(S_V_z(c_z,r_z),S_V_n(c_n,r_n))tri. ineq.≤ Cα^2+d_H(S_V_z(c_z,r_z),S_V_x(c_x,r_x))+d_H(S_V_x(c_x,r_x),S_V_n(c_n,r_n))Lemma <ref>≤ Cα^2+Cσ^2+d_H(S_V_x(c_x,r_x),S_V_n(c_n,r_n))Lemma <ref>≤ Cα^2+Cσ^2+C σlog n/n^1/2≤ C(α^2+σ^2+σlog n/n^1/2). Then consider the general case, when C_1,⋯,C_K are partitions and M_k∩ C_k is the submanifold of M with radius α_k containing n_k samples. Let S_V_n^k(c_n,r_n) be the empirical solution of SPCA for the k-th partition. Then by the above argument, sup_y∈ M_kd(y,S_V_n^k(c_n,r_n))≤ C(α_k^2+σ^2+σlog n_k/n_k^1/2). Then by Assumption (C) and the bias-variance trade off, we know that log n_k/n_k^1/2=α_k^2 = (n_k/n)^2/d⟹ n_k =O( n^4/d+4), α_k =O( n^-1/d+4). As a result, sup_y∈ M_kd(y,S_V_n^k(c_n,r_n))≤ C(σ^2+σlog n/n^2/(d+4)). So we conclude that sup_y∈ Md(y,M) =sup_k=1,⋯,Ksup_y∈ M_kd(y,M)≤sup_k=1,⋯,Ksup_y∈ M_kd(y,S_V_n^k(c_n,r_n)) ≤sup_k=1,⋯,K C(σ^2+σlog n/n^2/(d+4)) = C(σ^2+σlog n/n^2/(d+4)). § COVERING NUMBERIn this section we prove the covering number theorem in Section 3.2. We split the proof into three cases: curves with d=1, D=2, surfaces with d=2,D=3, and the general case with any d and D. The proof for curves is the simplest, while the proof for surfaces is different from the curve case and motivates the proof of the general case. §.§ Curves (d=1)Throughout this section, γ:[0,V]→^2 is a C^3 curve with input s, the arc length parameter. Let s_0∈[0,V] be fixed. Let κ(s) be the curvature at point γ(s). Let L(s) be the tangent line of γ at point γ(s_0), then L is the unique line such that d(γ(s),L)≤K/2|s-s_0|^2, so lim_s→ s_0L(s)-γ(s)/|s-s_0|=0, where κ_max=sup_s |κ(s)|. This is a standard result so we will not prove it in this paper. Let ϵ>0, then there exists N≤ V(2ϵ/κ_max)^-1/2 tangent lines L_1,⋯,L_N at points γ(s_1),⋯,γ(s_n) such that ∀ p∈γ, ∃ i s.t. d(p,L_i)≤ϵ. Let s_1=0, then by Proposition <ref>, γ(s)-L_1(s)< κ_max/2 |s|^2, so if s< (2ϵ/κ_max)^1/2, γ(s)-L_1(s)<ϵ. Then starting from γ((2ϵ/κ_max)^1/2), we can approximate the next segment by another tangent line. We can repeat this process N times to approximate each segment by a tangent line, so N≤ V/(2ϵ/κ_max)^1/2=V(2ϵ/κ_max)^-1/2. Let C(s) be the osculating circle of γ at point γ(s_0). Then if κ(s_0)≠0, C is the unique circle such that d(γ(s),C)≤T+2κ_max/6|s-s_0|^3, so lim_s→ s_0C(s)-γ(s)/|s-s_0|^2=0, where T=sup_s|γ^(3)(s)|. Proposition <ref> holds only when the osculating circle is non degenerate. If the curvature of γ at γ(s_0) is κ(s_0)=0, the osculating circle C degenerates to tangent line L. In this case, Proposition <ref> applies. As osculating circle is a local approximation of the curve, |s-s_0| is assumed to be small in the following proof. The osculating circle C has radius r=1/|κ(s_0)| and center γ(s_0)+1/κ(s_0)𝐧, where 𝐧=γ”(s_0)/γ”(s_0) is the unit normal vector. Let {-𝐧,t} be the Frenet frame, where t=γ'(s_0). Without loss of generality, assume s_0=0 and γ(s_0)=0. Under the Frenet frame, we can rewrite the osculating circle as C(s)=[ -1/κ;0 ]+1/κ[ cos(κ s); sin(κ s) ]=[ r(-1+cos(κ s));rsin(κ s) ]. The Taylor expansion for γ can be written as γ(s) =γ(0)+γ'(0)s+1/2γ”(0)s^2+R_2(s)=0+s[ 0; 1 ]+ s^2/2[ -κ;0 ]+R_2(s) =[ -κ s^2/2;s ]+R_2(s), where |R_2(s)|≤T/6 |s|^3, solim_s→ s_0R_2(s)/s^2=0. As a result, C(s)-γ(s) =[ 1/κ(-1+cos(κ s))+κ s^2/2;1/κsin(κ s)-s ]-R_2(s)=[ 1/κ(-1+1-κ^2s^2/2+o(s^3))+κ s^2/2; 1/κ(κ s-1/3κ^3s^3+o(s^3))-s ]-R_2(s) =[o(s^3); -1/3κ^2s^3+o(s^3) ] -R_2(s). As a result, C(s)-γ(s)≤T+2|κ|/6|s|^3≤T+2κ_max/6|s|^3 . Now we prove the uniqueness. Observe the first entry of C(s)-γ(s): 1/κ'(-1+1-κ^2s^2/2+o(s^3))+κ s^2/2=o(s^3)⟺κ^2s^2/κ'=κ s^2⟺κ'=κ, which means C(s) is the osculating circle. Let ϵ>0, then there exists N≤ V(6ϵ/T+2κ_max)^-1/3 osculating circles C_1,⋯,C_N at points γ(s_1),⋯,γ(s_n) such that ∀ p∈γ, ∃ i s.t. d(p,C_i)≤ϵ. Let s_1=0, then by Proposition <ref>, γ(s)-C_1(s)< T+2κ_max/6 |s|^3, so if s< (6ϵ/T+2κ_max)^1/3, γ(s)-C_1(s)<ϵ. Then starting from (6ϵ/T+2κ_max)^1/3, repeat this process to find N osculating circles, so N≤ V/(6ϵ/T)^1/3=V(6ϵ/T+2κ_max)^-1/3.Note that k(s_0)≠ 0 implies κ_max>0, so T+2κ_max>0 and 1/T+2κ_max is well-defined. §.§ Surfaces (d=2)Throughout this section, M: U→^3 is a regular C^3 surface parametrized by x=x(u,v),y=y(u,v),z=z(u,v) where U is a compact subset of ^2.Without loss of generality, assume X_0=(x(0,0),y(0,0),z(0,0))=(0,0,0)∈ M is fixed. Letting H(u,v) be the tangent plane of M at point X_0, H is the unique plane such that H(u,v)-M(u,v)^2≤κ_max/2(u,v)^2, so lim_(u,v)→ (0,0)H(u,v)-M(u,v)/(u,v)=0. This is a higher dimensional analogue of Proposition <ref>, but a similar analogue of Proposition <ref> does not exist, which is a direct result from the following lemma. There exists a sphere S(u,v) such that lim_(u,v)→ (0,0)S(u,v)-M(u,v)/(u,v)^2=0 if and only if X_0 is an umbilical point. Recall that X_0 is an umbilical point ⇔ L/E=M/F=N/G=α, where 1=[ E F; F G ] is the first fundamental form and 2=[ L M; M N ] is the second fundamental form. Without loss of generality, assume (u,v) are locally orthonormal parameters, which means M_u,M_v=0 and M_u=M_v=1 hence E=G=1, F=0. If X_0 is an umbilical point, L=N=α, M=0. Observe the Taylor expansion dX=X_0+M_udu+M_vdv+L du^2+Mdudv+N dv^2+o(√(u^2+v^2)^2). Plugging in L=N=α and M=0, it is clear that there exists a sphere S(u,v) such that (<ref>) holds. If there exists a sphere S(u,v) such that (<ref>) holds, then the quadratic terms in the Taylor expansion must satisfy L=N and M=0, so X_0 is umbilical.Although we can't find a sphere so that the error is third order in general, we can still find a sphere with this property in some direction, as shown in the following proposition. Let k_1≥ k_2 be the principal curvatures of M at X_0, e_1, e_2 be the corresponding principal directions and n be the normal vector at X_0. Then for any k∈[k_2,k_1], let S_k be the sphere centered at c=X_0-1/k n with radius 1/|k|, there exists a curve γ on M and a constant T such that d(γ(s),S_k)≤T/6|s|^3. Since k∈[k_2,k_1], there exists a direction, represented by unit vector ξ at (0,0)∈ U so that k is the normal curvature in direction ξ. To be more specific, let γ_ξ be the curve on M in direction ξ; that is, γ_ξ(s)=M(sξ),s∈[L_1,L_2], where L_1=inf{s|sξ∈ U }, L_2=inf{s|sξ∈ U }. By the above construction, the curvature of γ_ξ at X_0 is just k, so from Proposition <ref>, we know thatd(γ_ξ(s),C_ξ)≤T/6|s|^3, where T=sup_s|γ_ξ^(3)(s)| and C_ξ is a circle centered at c=X_0-1/k n with radius 1/k. Since C_ξ is just a great circle of S_k, we have the desired inequality: d(γ_ξ(s),S_k)≤ d(γ_ξ(s),C_ξ) ≤T/6 |s|^3. §.§ General CasesThroughout this section, M is a d-dimensional C^3 compact manifold embedded in ^D. Let p∈ M be a fixed point and we can assume p=0 without loss of generality. Then we have the following proposition that is similar to Proposition <ref> and Proposition <ref>. Let T_pM be the tangent space of M at p, then d(x,T_pM)≤κ_maxx^2.Before proving Theorem 8, we need a lemma regarding covering numbers and packing numbers of metric spaces; for more properties of these two numbers, see <cit.>. Let (X,d) be a metric space and δ>0, then 𝒩⊂ X is called a δ-net if ∀ x∈ X, ∃ y∈𝒩: d(x,y)<δ. The covering number of X denoted by 𝒩(X,d,δ) is defined to be the smallest cardinality of an δ-net of (X,d). 𝒩⊂ X is said to be δ-separated ifd(x,y) > δ for all distinct points x,y ∈𝒩. The packing number of X denoted by 𝒫(X,d,ϵ) is defined to be the largest cardinality of an δ-separated subset of X. Let (M,g) be a compact d-dimensional Riemannian manifold embedded in ^D and d_g be the geodesic distance on M, then there exists constant C=C(M) and δ>0 such that ∀ r<δ, 𝒩(M,d_g,r)≤ CVr^-d, where V=Vol_g(M) is the Riemannian volume of M. First we claim that 𝒩(M,d_g,r)≤𝒫(M,d_g,r). Let 𝒩={x_1,⋯, x_N} be an r-separated subset of M whose cardinality is N=𝒫(M,d_g,r), then for any y∈ M, if d_g(y,x_i)≥ r for all i=1,⋯,N, then we can add y to 𝒩 to get another r-separated subset with cardinality N+1 which contradict the assumption. As a result, there exists x_i_0∈𝒩 such that d_g(y,x_i_0)< r, which means 𝒩 is a r-net of M so the claim is true. Then we show that there exists C=C(M)>0 and δ=δ(M)>0 such that ∀ r≤δ, 𝒫(M,d_g,r)≤ CVr^-d. For any x∈ M, there exists δ_x>0 such that exp_x is homeomorphic on B(0,δ_x)⊂ T_xM. By compactness of M, there exists δ>0 such that δ_x≥δ for any x∈ M. Denote the Riemannian volume form dV_g and the Lebesgue measure on T_xM by dV, then Vol_g(B_d_g(x,r))=∫_B_d_g(x,r)dV_M(y)=∫_B(0,r)|J_x(v)|dV(v), where J_x(v) is the Jacobian of exp_x at v. By the compactness of {v∈ T_xM| x∈ M, v≤δ}, there exists constant C such that |J_x(v)|≥ C for any x∈ M and v∈ T_xM with v≤δ. As a result, Vol_g(B_d_g(x,r))≥ CC_dr^d=C(M)r^d, where C_d is the volume of the d-dimensional unit ball. Again, let 𝒩={x_1,⋯, x_N} be a r-separated subset of M whose cardinality is N=𝒫(M,d_g,r), then {B_d_g(x_i,r/2)}_i=1^N are disjoint geodesic balls so V=Vol_g(M)≥∑_i=1^N Vol_g(B_d_g(x_i,r/2))≥ NC(r/2)^d, hence N=𝒫(M,d_g,r)≤Vol_g/(Cr^d)=CVr^-d, as desired. Now we can prove Theorem 8. Firstly, we prove the first inequality in Equation (4).First we focus on one local neighborhood of p∈ M. Proposition <ref> shows that there exists a hyperplane H such that when x-p<(2ϵ/κ_max)^1/2=r, d(x,H)≤ϵ. Since x-p≤ d_g(x,p), for any x∈ B(p,r), d(x,H)≤ϵ. By Lemma <ref>, there exists C=C(M) and δ>0 such that ∀ r≤δ, 𝒩(M,d_g,r)≤ CVr^-d=CVϵ^-d/2. As a result, when ϵ≤δ^2κ_max/2, the manifold M can be covered by at most CVr^-d geodesic balls and if we approximate each geodesic ball by a hyperplane, the approximation error is no more than ϵ, which means N_ℋ(ϵ,M)≤ CVϵ^-d/2, as desired. Then we prove the second inequality in Equantion (4) by considering two submanifolds M_ϵ and M-M_ϵ, which are both compact. 1. M_ϵ^𝖼. Firstly we consider the worst part: the complement of M_ϵ, which is compact as a closed subset of M. Let ι>0 and let M_ϵ^𝖼(ι)={x∈ M: d(x,M_ϵ^𝖼)<ι}. It is clear that M_ϵ^𝖼(ι) is a open submanifold of M, hence is C^3. In addition, M_ϵ^𝖼(ι)⊃ M_ϵ^𝖼, so any cover of M_ϵ^𝖼(ι) also covers M_ϵ^𝖼.We cover this subset of M by geodesic balls with radius (2ϵ/κ_max)^1/2. The first part of the proof shows that this covering exists with approximation error no more than ϵ, and the number of balls is less than or equal to CVol(M_ϵ^𝖼(ι))ϵ^-d/2, where the constant C is the same as the constant in the hyperplane case according to the proof of Lemma <ref>. Then let ι→0, by the continuity of Riemannian volume, Vol(M_ϵ^𝖼(ι))→ V-V_ϵ, so the number of balls to cover M_ϵ^𝖼 is no more than C(V-V_ϵ)ϵ^-d/2. 2. M_ϵ. We cover this part by bigger geodesic balls. For any point p∈ F_ϵ, we have sup_v∈ T^1_pMκ(p,v)-inf_v∈ T^1_pMκ(p,v)≤(2ϵ/κ_max)^1/2. Let k^*∈[inf_v∈ T^1_pMκ(p,v),sup_v∈ T^1_pMκ(p,v)] be the curvature of a sphere to approximate U B(p,(6ϵ/3+T)^1/3). Then for any q∈ U, if q∈ B(p,(2ϵ/κ_max)^1/2), when case 1 shows that the error is less than or equal to ϵ, so we only need to consider q∈ U-B(p,(2ϵ/κ_max)^1/2), that is, (2ϵ/κ_max)^1/2≤ d(p,q)≤ (6ϵ/3+T)^1/3. Let γ_q(s)=exp_p(slog q) be the geodesic connecting p and q, assume γ_q at p is k_q. Since both k_q and k^* are in [k_d(p),k_1(p)], we have the following relation: |k_q-k^*|≤sup_v∈ T^1_pMκ(p,v)-inf_v∈ T^1_pMκ(p,v)≤(2ϵ/κ_max)^1/2. Recall in the proof of Proposition <ref>,only the first three terms in the Taylor expansion of γ_q matter; that is, γ_q(0), γ_q^'(s) and γ_q^”(s), so we only need to consider the first two coordinates of γ_q(s) and C(s) while other coordinates are all o(s^3). Similar to the proof of Proposition <ref>, the first two coordinates of C(s)-γ_q(s) are [ 1/k_q(-1+cos(k_q s))+k^* s^2/2;1/k_qsin(k_q s)-s ]-R_2(s)=[ 1/k_q(-1+(1-k_q^2s^2/2+o(s^3))+k^* s^2/2;1/k_q(k_q s-o(s^3))-s ]-R_2(s) =[ s^2/2(k_q-k^*)+o(s^3);o(s^3) ] -R_2(s), where |R_2(s)|≤T/6|s|^3. We claim that C(s_q)-q≤ϵ where s_q=d(p,q). Since |k_q-k^*|≤(2ϵ/κ_max)^1/2≤|s|≤(6ϵ/3+T)^1/3, C(s_q)-q≤s^2/2|k_q-k^*|+T/6|s|^3 ≤(1/2+T/6)|s|^3≤3+T/66ϵ/3+T=ϵ. As a result, we can cover M_ϵ by geodesic balls with radius r=(6ϵ/3+T)^1/3 so that the error is less than or equal to ϵ. On M_ϵ the centers of geodesic balls are not arbitrary, but restricted to be in F_ϵ. Define the smallest cardinality of an r-net in F_ϵ of (M_ϵ,d_g) by 𝒩(M_ϵ,F_ϵ,d_g,r), then we have 𝒩(M_ϵ,d_g,r)≤𝒩(M_ϵ,F_ϵ,d_g,r) because any r-net in F_ϵ is automatically an r-net in M_ϵ. In this situation, a similar claim in the proof of Lemma <ref> does not hold; however, we claim that 𝒩(M_ϵ,F_ϵ,d_g,r)≤𝒩(M_ϵ,d_g,r/2)≤ CV_ϵϵ^-d/3 for r=(6ϵ/3+T)^1/3 and the theorem follows. To prove the claim, let 𝒩={x_1,⋯, x_N}⊂M_ϵ be an r-net of M_ϵ whose cardinality is N=𝒩(M_ϵ,d_g,r/2). Recall the definition of M_ϵ=∪_x∈ F_ϵB(x,r/2) so there exists {y_1,⋯,y_N}⊂ F_ϵ such that d_g(x_i,y_i)≤ r/2. Then for any y∈M_ϵ, there exists x_i_0 such that d_g(y,x_i_0)≤ r/2. By triangle inequality, we have d_g(y,y_i_0)≤ d_g(y,x_i_0)+d_g(x_i,y_i_0)<r. This implies {y_i}_i=1^N is an r-net of M_ϵ where y_i∈ F_ϵ for i=1,⋯,N, which means 𝒩(M_ϵ,F_ϵ,d_g,r)≤𝒩(M_ϵ,d_g,r/2). As a result, the number of balls needed to cover M_ϵ is less than or equal to CV_ϵϵ^-d/3. Based on the above two cases, the total number of balls N_S(ϵ,M)≤ CV_ϵϵ^-d/3+C(V-V_ϵ)ϵ^-d/2. 1. Corollary <ref> is a special case of Theorem 8. 2. When d increases, the performance will be worse and worse, which is another representation of the curse of dimensionality.Fortunately, d is the intrinsic dimension of M, which is assumed to be small in most cases. It suffices to provide two manifolds with covering numbers achieving the upper bounds in Theorem 8. * Let γ(t)=(t,t^2), t∈(0,1) so γ^”(t)=(0,2) is constant. The covering number N_ℋ(ϵ,γ) follows. * Let γ(t)=(t,t^3), t∈ (0,1) so γ^(3)(t)=(0,6) is constant. The covering number N_𝒮(ϵ,γ) follows. | http://arxiv.org/abs/1706.08263v4 | {
"authors": [
"Didong Li",
"Minerva Mukhopadhyay",
"David B. Dunson"
],
"categories": [
"stat.ML"
],
"primary_category": "stat.ML",
"published": "20170626074555",
"title": "Efficient Manifold and Subspace Approximations with Spherelets"
} |
Critical side channel effects in random bit generation with multiple semiconductor lasers in a polarization-based quantum key distribution system Heasin Ko,1† Byung-Seok Choi,1 Joong-Seon Choe,1 Kap-Joong Kim,1 Jong-Hoi Kim,1 and Chun Ju Youn1,2* June 25, 2017 ================================================================================================================================================= The stationary states of nonlinear Schrödinger equation on a ring with a defect is numerically analyzed.Unconventional connection conditions are imposed on the point defect, and it is shown that the system displays energy level crossings and level shifts and associated quantum holonomies in the space of system parameters, just as in the corresponding linear system.In the space of nonlinearity parameter, on the other hand, the degeneracy occurs on a line, excluding the possibility of any anholonomies.In contrast to the linear case, existence of exotic phenomena such as disappearance of energy level and foam-like structure are confirmed.§ INTRODUCTIONGross and Pitaevskii found a striking fact that many boson system with the short-range interaction in its dilute limit is effectively described under Hartree-Fock approximation by a single wave function that satisfies an wave equation similar to the Schrödinger equation with a critical difference of nonlinearity <cit.>.It is natural to ask a question how different the properties of nonlinear Schrödinger waves are from its linear counterparts.Such question should be most easily asked in its simplest setting in one-dimension.In light of experimental progress in recent years on Bose-Einstein condensate in one dimensional devices, such study has obtained unexpected urgency <cit.>.The nonlinear Schrödinger equation appears not only in model equations describing Bose-Einstein condensate, but also in various other physical systems such as vortex motion <cit.>, precession of spin <cit.>, pulse wave in optical fiber <cit.>. There is a great significance in examining the physical and mathematical properties of nonlinear Schrödinger equation in detail.It is known that the inverse scattering method, or the method of Hirota lead to an exact n-soliton solution of the nonlinear Schrödinger equation<cit.>, in which there are infinite numbers of conserved quantities.Moreover, Cazenave and others have found critical results on the stability of the stationary wave of the nonlinear Schrödinger equation on a one-dimensional infinite line <cit.>.Although the solution of nonlinear Schrödinger equation in infinite line has been known for quite some time, its study on graphs made up of one-dimensional finite lines and vertices has not started until recently.Some interesting properties are now being uncovered <cit.>.Most studies concentrates on different graph topologies.The implication of generalized connection condition, which is known to bring quite a richness to linear Schrödinger waves on graphs <cit.>, has been mostly overlooked until now, with a few exception <cit.>.In this article, we place the nonlinear wave described by the Schrödinger equation with cubic nonlinearity on a ring with a single defect that is described by the connection condition specified by two parameters, Fulop-Tsutsui scaling factor and the delta potential strength.The Fulop-Tsutsui scaling factor connects the wave function at the vertex with the self-adjoint condition and has an interesting property of being scale invariant <cit.>. Our choice of connection condition should be considered as a starting trial for the exploration of complete self-adjoint connection condition, which is described by four parameters.It is found that this nonlinear system displays the quantum holonomies, both Berry phase <cit.> and exotic types <cit.>, just like its linear counterpart.It is also found that the nonlinear system has its unique feature of possessing foam-like energy surface in parameter space.§ NONLINEAR SCHRÖDINGER EQUATION ON A RING WITH A DEFECTWe consider the cubic nonlinear Schrödinger equation i Ψ̇(x,t) = - Ψ”(x,t) + g Ψ^*(x,t)Ψ(x,t)^2 , on a ring with circumference 2π, namely, a finite line x ∈ [0, L) with both ends of the line, x=0 and x=L identified.The dot and prime signify temporal and spatial derivatives, respectively.At this identified end point, we impose the connection condition Ψ'(0,t) - t Ψ'(L,t) = v Ψ(0,t) t Ψ(0,t) - Ψ(L,t) = 0 , in which v and t are real numbers <cit.>. This amounts to considering a ring with a point defect located at x=0 (or identically, x=L). This is a subset of the connection condition that guarantees the conservation of the wave function norm,μ = ∫_0^L dx Ψ^*(x,t) Ψ(x,t), which sometime is also termed the mass, in the literature. With an Ansatz Ψ(x,t) = ψ(x) e^-i E t where E is a real number and ψ a real function, we have an eigenvalue equation -ψ”(x) + g ψ^3(x) = E ψ(x) The connection condition (<ref>) is expressible, in terms of time independent wave function ψ(x) as ψ'(0) - t ψ'(L) = v ψ(0) t ψ(0) - ψ(L) = 0 and the mass is also expressed as μ = ∫_0^L dx ψ^*(x) ψ(x) With the rescaling ψ=√(μ)ψ̃, we have -ψ̃”(x) + g̃ψ̃^3(x) = E ψ̃(x) with ∫_0^L dxψ̃^*(x) ψ̃(x) =1 and rescaled coupling constant g̃ = g μ.Therefore we loose no generality by setting the mass to be one,μ=1.We shall adopt this convention in the rest of this paper.The eigenvalue problem (<ref>) with the mass constraint μ = 1and the connection condition (<ref>) at the defect yield eigenvalues E_n and ψ_n(x) (n=1, 2, ...) which are the functions of the connection parameter t, v, and the coupling g.How the energy levels E_n(t, v, g) and the eigenfunctionsψ_n(x; t, v, g) behave as in the parameter space (t, v, g) is the matter of our interest.Identifying the location and the mode of level degeneracies in the parameter space is the key to our study.§ SOLUTIONS IN TERMS OF ELLIPTIC INTEGRALS The eigenvalue equation (<ref>) is fully integrable and its solution can be expressed in terms of elliptic functions sn , cn and dn.For g>0, E>0the solution takes the form ψ (x)=k_-sn[√(g/2)k_+(x-x_0),k_-^2/k_+^2]e^i η_0 (0≦ c ) , orψ (x)=k_+/sn[√(g/2)k_+(x-x_0),k_-^2/k_+^2]e^i η_0 (0≦ c ≦E^2/4g) ,and ψ (x)=k_+/cn[√(g(k_+^2-k_-^2)/2)(x-x_0),-k_-^2/k_+^2-k_-^2]e^i η_0 (c ≦ 0). Here, k_± are defined by k_± = √(μ/g)√(1 ±√(1-4gc/μ^2)) , and two real numbers c and x_0 are the constants of integral. For g>0, E<0, it is given by ψ (x)=k_-sn[√(g/2)k_+(x-x_0),k_-^2/k_+^2]e^i η_0 (E^2/4g≦ c ) , ψ (x)=k_-/sn[√(g/2)k_-(x-x_0),k_+^2/k_-^2]e^i η_0(0 ≦ c ≦E^2/4g) ,and byψ (x)=k_+/cn[√(g(k_+^2-k_-^2)/2)(x-x_0),-k_-^2/k_+^2-k_-^2]e^i η_0 (c ≦ 0) .For g<0, E>0, it takes the formψ (x)=k_+sn[√(g/2)k_-(x-x_0),k_+^2/k_-^2]e^i η_0(0 ≦ c) ,and no solution exist for c > 0.Finally, for g<0, E<0, it is given byψ(x) = k_+cn[√(-g(k_+^2-k_-^2)/2)(x-x_0),k_+^2/k_+^2-k_-^2]e^i η_0 (0 ≦ c)and byψ(x) = k_+dn[√(-g/2)k_+(x-x_0),k_+^2-k_-^2/k_+^2]e^i η_0 (E^2/4g≦ c ≦ 0) ,and no solution exist for c < E^2/4g.Finally, for g<0, E=0, we haveψ(x) =√(2/g)1/x-x_0e^i η_0 .By imposing the connection condition (<ref>), and the normalization condition μ = 1 (<ref>) as self consistent constraints, we can numerically obtain allowable sets of x_0, c, and E simultaneously, for a given numbers of t, v, and g.The linear limit g=0 is solvable in terms of trigonometric functions ψ(x) = sink (x-x_0) with the integral constantx_0 = -1/karctansink L/t-cosk L , and with k=√(E) which is given as the solution of t cosk L = 1 ±√( 1 - cosk L(cosk L + v/ksink L) ) . From (<ref>), we immediately obtain the function t=t(k, v), from which E=E(t,v) is obtainable.§ ENERGY NUMERICS§.§ Energy levels as functions of nonlinearity parameterWith the method detailed in the previous section, we evaluate the energy eigenvalues numerically with Newton's method. Figure 1 shows the energy eigenvalues as functions of nonlinearity parameter g with a defect parameters v=0 and t=1, namely, the free connection condition. All energy eigenvalues are seen to increase monotonically with g.The level intersection and branching, characteristic to nonlinear system, are observed.There are two types of energy lines in the figure; first is the ones obtained fromthe equations (<ref>), (<ref>), (<ref>) and (<ref>), and the other is ones obtained from the equations (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>). The straight line passing through the origin in Figure 1 belong to the first type, while the energy level branching from the fisrt type in the negative side of gbelong to the second type.Since the energy level grows with nonlinearity, the second harmonic of the wave function and the third harmonic satisfy the connection conditions (<ref>) and the mass constraint μ=1, so for the arbitrary nonlinearity and energy eigenvalue pair (g,E) in Figure 1, the pair (n^2g, n^2E) also satisfies the conditions (<ref>) and μ=1. §.§ Energy levels of linear system as functions of connection condition parameters We now calculate the energy eigenvalue E as a function of connection parameters t or v, with fixed nonlinearity parameter g. We start by looking at the linear case g=0, as the reference to be compared to the nonlinear cases. The results are shown in Figure 2, in which the vertical axis represents the scaled energy eigenvalue sgn(E) √(E) and the horizontal axis represents the scaled connection parameter 2arctan(t).The δ-strength parameter v is fixed to be v=-1, 0 and 1 for the left, middle, and right graphs, respectively. With the connection condition v=0,the level crossing occurs at t = 1 and 1.With other values for v, the level repulsion occurs near t=1 and -1.§.§ Energy levels of nonlinear system as functions of connection condition parameters The energy levels as functions of connection parameters t and v are calculated for non-zero nonlinearity parameter.The result for g=5 is shown in Figure 3, while one for g=-5 is shown in Figure 4.Scaling conventions are the same as in the linear case, Figure 2. The energy eigenvalues were plotted on the t-E plane for δ-strength v=-1 (left figure), 0 (middle figure), 1 (right figure) . For g=5, Figure 3, the energy levels get deformed and are generally pushed upward, compared to the linear case, Figure 2.All solutions are in the region E>0 and thus obtained from (<ref>), (<ref>) and (<ref>).The existence of the level crossing at {t, v} = { 1, 0} and { -1, 0} remains intact, along with the avoided crossing around those points in parameter space {t, v}.A very notable difference in g=5 case is the sudden disappearanceof energy levels. Eigenvalues that disappear are those calculated from (<ref>), and beyond the value of t that corresponds to the disappearing point,the condition c<E^2/4g is violated and no solution exists.For g=-5, Figure 4, the energy levels get deformed and are generally pushed downward, compared to the linear case, Figure 2.For this case also, the existence of the level crossing at {t, v} = { 1, 0} and { -1, 0} remains intact, along with the avoided crossing around those points in parameter space {t, v}. Unlike the case of positive g, we have, for g=-5, no disappearance of the energy level is observed.Instead, we observe a characteristic closed-ring shaped energy level in t-E plane.The ring-shaped levels correspond to the branched-out lines in Figure 1, that are obtained from the solutions(<ref>), while the normal energy levels are obtained from the solutions(<ref>) and (<ref>).In the three-dimensional plane t-v-E, these ring-shaped levels form a foam like structure, which has contact to the normal energy surface at the single point that lie on the line v=0.Finally, it is also noteworthy that the lowest energy level appears almost symmetrically with respect to the reflection at t=0 axis.§ QUANTUM HOLONOMY§.§ Berry phase Consider a closed path in the parameter space { v, t} that goes around the point where energy is degenerate. Berry has proven that, for the system giverned by Schrödinger equation, an extra nontrivial phase might appear for the wave functions of the system, when the parameter is adiabatically changed along the path and brought back to the original value after the circulation around the degeneracy point <cit.>. It is interesting to check whether analogous phenomenon is observed in the system governed by nonlinear Schrödinger equation.We can be assured that the answer is positive, from the Figure 5 andFigure 6, in which adiabatic change of wave functions are drawn around the degeneracy points { t, v }={ 1, 0} and { t, v }={ -1, 0}, respectively. It is to be noted that no Berry phase is observed for the path going around the degeneracy point connecting the ring-shapedlevels and normal energy level.§.§ Exotic quantum holonomy It is known that an adiabatic motion along a parametric cycle can bring not only the Berry phase, but also the exchange of different energy eigenvalues.This phenomenon is known as the exotic quantum holonomy <cit.>. In figure 7, numerically calculated eigenvalues of nonlinear Schrödinger equation are drawn as functions of δ-strength parameter v with fixed value for the scale-invariant parameter t=1, for fifferent nonlinearity parameters.The conditions v=∞ and v=- ∞ are essentially equivalent, signifying the disconnected Dirichlet boundary conditions at the defect.Therefore we can identify the left and right edges of each graphs in Figure 7, and regard the motion from the left edge, v=-∞ to right edge v=∞ as a cycle.We can then recognize the existence of exotic quantum holonomy in each graphs.In all three systems with different nonlinearity parameters g, the ground state and fist, third and fifth excited states gets the energy shift after the completion of adiabatic cycle v=-∞ → v=∞. Thus, we have shown that exotic holonomy exists in nonlinear Schrödinger system as well as in normal linear Schrödinger system. § SUMMARY In this article, we have numerically analyzed the energy eigenvalues of a system described by cubic nonlinear Schrödinger equation with coupling g on a ring with a defect, which is described by two parameters v, the δ-strength, and t, the scale invariant Fulop-Tsutsui coupling. We have found that, for all values of nonlinear coupling parameter g, the degeneracy occurs at the points {t, v}={1, 0} and {t, v}={-1, 0}.Around thedegeneracy points, we have confirmed the existence of Berry phase e^i π.We have also found that there is a exotic quantum holonomy expressed as the level flow as we increase v and pass though v=∞ and come back to finite v from negative infinity v=-∞. We have also identified, in our system, the disappearing energy levels and the foam like energy surface in parametric space, the phenomena that are characteristic to nonlinear systems.AcknowledgementsThis research was supported by the Japan Ministry of Education, Culture, Sports, Science and Technology under the Grant number 15K05216.We thank Dr. Riccardo Adami, Dr. Atushi Tanaka, and Dr. Ondrej Turek for stimulating discussions.99 GR61 E. P. Gross, Structure of a quantized vortex in boson systems,Nuovo Cim. 20 (1961) 454–457. PI61 L. P. Pitaevskii, Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13 (1961) 451–454.EXP M. Eckart, R. Walser, W.P. Schleich, Exploring the growth of correlations in a quasi one-dimensinal trapped Bose gas, New J. Phys. 10 (2008) 045024 (28pp). VOF H. Hasimoto, A solliton on a vortex filament, J. Fluid Mech. 51 (1972) 477–485.SPI R. Myrzakulov, G. K. Mamyrbekova, G. N. Nugmanova, K. R. Yesmakhanova, M. Lakshmanan, Integrable Motion of Curves in Self-Consistent Potentials : Relation to Spin Systems and Soliton Equations, J. Phys. Lett. A, 378 (2014) 2118–2123.OPF D. Felice, A Study of a Nonlinear Schrödinger Equation for Optical Fibers, http://arxiv.org/abs/1612.00358 (2016)ISM V. F. Zakharov, A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69.STA H. Berestycki and T. Cazenave, Instsbilite des etats stationnaires dans les equations de Schrödinger et de Klein-Gordon non lineares, C. R. Acad. Sci. Paris: I Math, 293 (1981) 489–492.STA2 T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982) 549-561.DLT Reika Fukuizumi, Masahito Ohta, Tohru Ozawa, Nonlinear Schrödinger equation with a point defect, Comm. Math. Phys. 318 (2013) 247-289TU03 B. P. Venkatesh, J. Larson, and D. H. J. ODell, Band-structure loops and multistability in cavity QED, Phys. Rev.A83 (2011) 063606 (22pp).TU08 R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Fast solitons on star graphs,Rev. Math. Phys. 23 (2011) 409–451. AK96 R. Adami, E. Serra, P. Tilli, NLS ground states on graphs, Calc. Var. PDEs 54 (2015) 743–761.TH03 R. Adami, F. Golse, A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 (2007) 1193–1220.TN08 D. Pelinovsky, G. Schneider, Bifurcations of standing localized waves on periodic graphs, http://arxiv.org/abs/1603.05463 (2016) ST94 S. Gnutzmann and D. Waltner, Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory Phys. Rev. E93 (2016) 032204 (19pp).TC12 O. Turek and T. Cheon,Threshold resonance and controlled filtering in quantum star graphs, Europhys. Lett. 98 (2012) 50005 (5pp).TC13 O. Turek and T. Cheon,Potential-controlled filtering in quantum star graphs, Ann. of Phys. (NY)330 (2013) 104–141.DLTP R. Adami, D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ' interaction, Ann. H. Poincaré 25 (2008) 837–845.SP18 K. Nakamura, D. Babajanov, D. Matrasulov, M. Kobayashi, P. Muruganandam, Dynamics of trapped interacting vortices in Bose-Einstein condensates: Role of breathing degree of freedom, J. Phys. A: Math. Theor. 49 (2016) 315102 (20pp).FT T. Fulop, I. Tsutsui, A Free Particle on a Circle with Point Interaction, Phys. Lett. A 264 (2000) 366-374.BE84 M. V. Berry.Quantal Phase Factors Accompanying Adiabatic Changes,Proc. Roy. Soc. A 392 (1984) 45?57.CT09 T. Cheon and A. Tanaka, New anatomy of quantum holonomy,Europhys. Lett. 85 (2009) 20001 (5pp). | http://arxiv.org/abs/1706.08695v1 | {
"authors": [
"Takaaki Nakamura",
"Taksu Cheon"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170627071933",
"title": "Spectral properties of nonlinear Schrödinger equation on a ring"
} |
empty empty©IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Pre-print of article that will appear at the 2017 IEEE International Conference on Intelligent Robots and Systems.Please cite this paper as:A. Dewan, G. Oliveira and W. Burgard, "Deep Semantic Classification for 3D LiDAR Data" in Intelligent Robots and Systems (IROS), 2017 IEEE/RSJ International Conference on. IEEE, 2017. bibtex:@inproceedings{ dewan17iros, author = { Ayush Dewan, Gabriel L. Oliveira and Wolfram Burgard}, booktitle = { Intelligent Robots and Systems (IROS), 2017 IEEE/RSJ International Conference on}, organization = {IEEE}, title = { {Deep Semantic Classification for 3D LiDAR Data} }, year = {2017} }Double crystallographic groups and their representations on the Bilbao Crystallographic Server Mois I. Aroyo December 30, 2023 ============================================================================================== Robots are expected to operate autonomously in dynamic environments. Understanding the underlying dynamic characteristics of objects is a key enabler for achieving this goal. In this paper, we propose a method for pointwise semantic classification of 3D LiDAR data into three classes: non-movable, movable and dynamic.We concentrate on understanding these specific semantics because they characterize important information required for an autonomous system. Non-movable points in the scene belong to unchanging segments of the environment, whereas the remaining classes corresponds to the changing parts of the scene. The difference between the movable and dynamic class is their motion state. The dynamic points can be perceived as moving, whereas movable objects can move, but are perceived as static. To learn the distinction between movable and non-movable points in the environment, we introduce an approach based on deep neural network and for detecting the dynamic points, we estimate pointwise motion. We propose a Bayes filter framework for combining the learned semantic cues with the motion cues to infer the required semantic classification. In extensive experiments, we compare our approach with other methods on a standard benchmark dataset and report competitive results in comparison to the existing state-of-the-art. Furthermore, we show an improvement in the classification of points by combining the semantic cues retrieved from the neural network with the motion cues. § INTRODUCTIONOne of the vital goals in mobile robotics is to develop a system that is aware of the dynamics of the environment. If the environment changes over time, the system should be capable of handling these changes. In this paper, we present an approach for pointwise semantic classification of a 3D LiDAR scan into three classes: non-movable, movable and dynamic. Segments in the environment having non-zero motion are considered dynamic, a region which is expected to remain unchanged for long periods of time is considered non-movable, whereas the frequently changing segments of the environment is considered movable. Each of these classes entail important information. Classifying the points as dynamic facilitates robust path planning and obstacle avoidance, whereas the information about the non-movable and movable points can allow uninterrupted navigation for long periods of time. To achieve the desired objective, we use a Convolutional Neural Network (CNN) <cit.> for understanding the distinction between movable and non-movable points. For our approach, we employ a particular type of CNNs called up-convolutional networks <cit.>. They are fully convolutional architectures capable of producing dense predictions for a high-resolution input. The input to our network is a set of three channel 2D images generated by unwrapping 360 3D LiDAR data onto a spherical 2D plane and the output is the objectness score, where a high score corresponds to the movable class. Similarly, we estimate the dynamicity score for a point by first calculating pointwise 6D motion using our previous method <cit.> and then comparing the estimated motion with the odometry to calculate the score. We combine the two scores in a Bayes filter framework for improving the classification especially for dynamic points. Furthermore, our filter incorporates previous measurements, which makes the classification more robust. In fig:summary we show the classification results of our method. Black points represent non-movable points, whereas movable and dynamic points are shown in green and blue color respectively. Other methods <cit.> for similar semantic classification have been proposed for RGB images, however, a method solely relying on range data does not exist according to our best knowledge. For LiDAR data, separate methods exists for both object detection <cit.> and for distinguishing between static and dynamic objects in the scene <cit.>. The two main differences between our method and the other object detection methods is that the output of our method is a pointwise objectness score, whereas other methods concentrate on calculating object proposals and predict a bounding box for the object. Since our objective is pointwise classification, the need for estimating a bounding box is alleviated as a pointwise score currently suffices. The second difference is that we utilize the complete 360 field of view (FOV) of LiDAR for training our network in contrast to other methods which only use the points that overlap with the FOV of the front camera. The main contribution of our work is a method for semantic classification of a LiDAR scan for learning the distinction between non-movable, movable and dynamic parts of the scene. As mentioned above, these three classes encapsulate information which is critical for robust autonomous robotic system. A method for learning the same classes in LiDAR scans has not been proposed before, even though different methods exists for learning other semantic level information. Unlike other existing methods, we use the complete range of the LiDAR data. For training the neural network we use the KITTI object benchmark <cit.> and compare our results on this benchmark with the other methods. We also test our approach on the dataset by <cit.>.§ RELATED WORKSWe first discuss methods which have been proposed for similar classification objectives. Then, we discuss other methods proposed especially for classification in LiDAR scans and briefly discuss the RGB image based methods.For semantic motion segmentation in images, <cit.> proposed a dense CRF based method, where they combine semantic, geometric and motion constraints for joint pixel-wise semantic and motion labeling. Similar to them <cit.> proposed a neural network based method. Their method is closest to our approach as they also combine motion cues with the object information. For retrieving the object level semantics, they use a deep neural network <cit.>. Both of these methods show results on the KITTI sceneflow benchmark for which ground truth is only provided for the images, thus making a direct comparison difficult. However, we compare the performance of our neural network with the network used by <cit.>.For LiDAR data, a method with similar classification objectives does not exist, however different methods for semantic segmentation <cit.>, object detection <cit.> and moving object segmentation <cit.> have been proposed. Targeting semantic segmentation in 3D LiDAR data, Wang et al. proposed a method <cit.> for segmenting movable objects. More recentlyproposed a method <cit.> for object segmentation, primarily concentrating on objects with missing points and <cit.> discusses a method for hierarchical semantic segmentation of a LiDAR scan. These methods report results on different datasets and since we use the KITTI object benchmark for our approach we restrict our comparison to other recent methods that use the same benchmark. For object detection, <cit.> extends their previous work <cit.> and propose a CNN based method for detecting objects in 3D LiDAR data.proposed a Fully Convolutional Network based method <cit.> for detecting objects, where they use two channel (depth + height) 2D images for training the network and estimate 3D object proposals.The most recent approach for detecting objects in LiDAR scans is proposed by <cit.>. Their method leverages over both multiple view point information (front camera view + bird eye view) and multiple modalities (LiDAR + RGB). They use a region based proposal network for fusing different sources of information and also estimate 3D object proposals. For RGB images, approaches by <cit.> are the two recent methods for object detection. In All of these methods, the neural network is trained for estimating bounding boxes for object detection, whereas, our network is trained for estimating pointwise objectness score; the information necessary for pointwise classification. In the results section, we discuss these differences in detail and present comparative results. Methods proposed for dynamic object detection include our previous work <cit.> and other methods <cit.>. Our previous method and <cit.> are model free methods for detection and tracking in 3D and 2D LiDAR scans respectively. For detecting dynamic points in a scene, <cit.> proposed a method that relies on a visibility assumption, i.e., the scene behind the object is observed, if an object moves. To leverage over this information, they compare an incoming scan with a global map and detect dynamic points. For tracking and mapping of moving objects a method was proposed by <cit.>. The main difference between these methods and our approach is that we perform pointwise classification and these methods reason at object level. § APPROACHIn this paper, we propose a method for pointwise semantic classification of a 3D LiDAR scan. The points are classified into three classes: non-movable, movable and dynamic. In fig:overview we illustrate a detailed overview of our approach. The input to our approach consists of two consecutive 3D LiDAR scans. The first scan is converted into a three-channel 2D image, where the first channel holds the range values and the second and third channel holds the intensity and height values respectively. The image isprocessed by an up-convolutional network called Fast-Net <cit.>. The output of the network is the pointwise objectness score. Since, in our approach points on an object are considered movable, the term object is used synonymously for movable class. For calculating the dynamicity score, our approach requires two consecutive scans. As a first step, we estimate pointwise motion using our RigidFlow <cit.> approach. The estimated motion is then compared with the odometry to calculate the dynamicity score. These scores are provided to the Bayes filter framework for estimating the pointwise semantic classification.§.§ Object Classification Up-convolutional networks are becoming the foremost choice of architectures forsemantic segmentation tasks based on their recentsuccess <cit.>. These methods are capableof processing images of arbitrary size, are computationally efficient andprovide the capability of end-to-end training. Up-convolutional networks havetwo main parts: contractive and expansive. The contractive part is aclassification architecture, for example AlexNet <cit.> orVGG <cit.>. They are capable of producing a dense prediction for ahigh-resolution input. However, for a low-resolution output of the contractivepart, the segmentation mask is not capable of providing the descriptivenessnecessary for majority of semantic segmentation tasks.The expansive partsubdues this limitation by producing an input size output through themulti-stage refinement process. Each refinement stage consists of an upsamplingand a convolution operation of a low-resolution input, followed by the fusion ofthe up-convolved filters with the previous pooling layers. The motivation ofthis operation is to increase the finer details of the segmentation mask at eachrefinement stage by including the local information from pooling. In our approach we use the architecture called Fast-Net <cit.> (see fig:overview). It is an up-convolutional network designed for providing near real-time performance. More technical details and a detailed explanation of thearchitecture is described in <cit.>. §.§ Training InputFor training our network we use the KITTI object benchmark. The network is trained for classifying points on cars as movable. The input to our network are three channel 2D images and the corresponding ground truth labels. The 2D images are generated by projecting the 3D data onto a 2D point map. The resolution of the image is 64×870. Each channel in an image represents a different modality. First channel holds the range values, second channel holds the intensity values, corresponding to the surface reflectance and the third channel holds the height values for providing geometric information. The KITTI benchmark provides ground truth bounding boxes for the objects in front of the camera, even though the LiDAR scanner has 360 FOV. To utilize the complete LiDAR information we use our tracking approach <cit.> for labeling the objects that are behind the camera by propagating the bounding boxes from front of the camera.§.§.§ TrainingOur approach is modeled as a binary segmentation problem and the goal is to predict the objectness score required for distinguishing between movable and non-movable points. We define a set of training images T = (𝑋_n, 𝑌_n), n=1,…, N, where 𝑋_𝑛 = { x_k,k=1,…,|𝑋_𝑛|} is a set of pixels in an example input image and 𝑌_n = { y_k,k=1,…, |𝑌_𝑛|} is the corresponding ground truth, where y_k={0,1}.The activation function of our model is defined as f(x_k,θ), where θ is our network model parameters. The network learns the features by minimizing the cross-entropy(softmax) loss in (<ref>) and the final weights θ^* are estimated by minimizing the loss over all the pixels as shown in (<ref>). L(p,q)=- ∑_c∈{0,1} p_clogq_cθ^* = θargmin∑_k=1^N×|𝑋_𝑛|L ( f ( x_k,θ ),y_k) We perform a multi-stage training, by using one single refinement at a time. Suchtechnique is used based on the complexity of single stage training and on thegradient propagation problems of training deeper architectures. The processconsists of initializing the contractive side with the VGG weights.After that the multi-stage training begins and each refinement is trained until wereach the final stage that uses the first pooling layer.We use Stochastic Gradient Descent with momentum as the optimizer, a minibatch of size one and a fixed learningrate of 1e^-6. Based on the mini batch size we set the momentum to 0.99, allowing us to use previous gradients as much as possible. Since the labels in our problemare unbalanced because the majority of the points belong to the non-movableclass, we incorporate class balancing as explained by <cit.>. The output of the network is a pixel wise score a^k_c for each class c. The required objectness score ξ^k∈[0,1] for a point k is the posterior class probability for the movable class.ξ^k = exp(a^k_1)/exp(a^k_1) + exp(a^k_0)§.§ RigidFlowIn our previous work <cit.>, we proposed a method for estimating pointwise motion in LiDAR scans. The input to our method are two consecutive scans and the output is the complete 6D motion for every point in the scan. The two main advantages of this method is that it allows estimation of different arbitrary motions in the scene, which is of critical importance when there are multiple dynamic objects in the scene and secondly it works for both rigid and non-rigid bodies.We represent the problem using a factor graph G= (Φ,𝒯,ℰ) with two node types: factor nodes ϕ∈Φ and state variables nodes τ_k ∈𝒯. Here, ℰ is the set of edges connecting Φ and state variable nodes 𝒯. The factor graph describes the factorization of the functionϕ(𝒯) = ∏_i∈ I_dϕ_d(τ_i) ∏_l∈ N_pϕ_p(τ_i,τ_j), where 𝒯 is the following rigid motion field:𝒯 = {τ_k |τ_k∈𝑆𝐸(3), k=1,…,K } {ϕ_d, ϕ_p}∈ϕ are two types of factor nodes describing the energy potentials for the data termand regularization termrespectively. The term I_d is the set indices corresponding to keypoints in the first frame and N_p={⟨ 1,2 ⟩, ⟨ 2,3 ⟩,…, ⟨ i,j⟩} is the set containing indices of neighboring vertices. The data term, defined only for keypoints is used for estimating motion, whereas the regularization term asserts that the problem is well posed and spreads the estimated motion to the neighboring points. The output of our method is a dense rigid motion field 𝒯^∗, the solution of the following energy minimization problem:𝒯^∗ = *arg min_𝒯 E(𝒯),where the energy function is:E(𝒯)=- lnϕ(𝒯)A more detailed explanation of the method is presented by <cit.>. §.§ Bayes Filter for Semantic Classification The rigid flow approach estimates pointwise motion, however it does not provide the semantic level information. To this end, we propose a Bayes filter method for combining the learned semantic cues from the neural network with the motion cues for classifying a point as non-movable,movable and dynamic. The input to our filter is the estimated 6D motion, odometry and the objectness score from the neural network. The dynamicity score is calculated within the framework by comparing the motion with the odometry.The objectness score from the neural network is sufficient for classifying points as movable and non-movable, however, we still include this information in filter framework for the following two reasons:* Adding object level information improves the results for dynamic classification because a point belonging to a non-movable object has infinitesimal chance of being dynamic, in comparison to a movable object. * The current neural network architecture does not account for the sequential nature of the data. Therefore, having a filter over the classification from the network, allows filtering of wrong classification results by using the information from the previous frames. The same holds for classification of dynamic points as well. For every point P^k_t∈ℝ^3 in the scan, we define a state variable x_t = {}. The objective is to estimate the belief of the current state for a point P^k_t. Bel(x^k_t) = p(x_t^k | x^k_1:t-1, τ^k_1:t,ξ^k_1:t,o_t^k)The current belief depends on the previous states x_1:t-1^k, motion measurements τ^k_1:t, object measurements ξ^k_1:t and a Bernoulli distributed random variable o^k_t. This variable models the object information, where o^k_t = 1 means that a point belongs to an object and therefore it is movable. For the next set of equations we skip the superscript k that represents the index of a point.Bel (x_t) = p(x_t | x_1:t-1, τ_1:t, o_t,ξ_1:t) = η p(τ_t,o_t| x_t,ξ_1:t)∫ p(x_t| x_t-1)Bel(x_t-1)dx_t-1 = η p(τ_t| x_t)p(o_t| x_t,ξ_1:t) ∫ p(x_t|x_t-1)Bel(x_t-1)dx_t-1In (<ref>) we show the simplification of the (<ref>) using the Bayes rule and the Markov assumption. The likelihood for the motion measurement is defined in (<ref>).p(τ_t| x_t) = 𝒩(τ_t;τ̂_̂t̂,Σ)It compares the expected measurement τ̂_̂t̂ with the observed motion. In our case the expected motion is the odometry measurement. The output of the likelihood function is the required dynamicity score. In (<ref>) we assume the independence between the estimated motion and the object information. To calculate the object likelihood we first update the value of the random variable o_t by combining the current objectness score ξ_t with the previous measurements in a log-odds formulation ((<ref>)).l(o_t|ξ_1:t) = l(ξ_t) + l(o_t|ξ_1:t-1) - l(o_0) The first term on the right side incorporates the current measurement, the second term is the recursive term which depends on the previous measurements and the last term is the initial prior. In our experiments, we set o_0 = 0.2 because we assume that the scene predominately contains non-moving objects. p(o_t| x_t,ξ_1:t) =p( o_t|ξ_1:t) if x_t = p(o_t |ξ_1:t) if x_t = s· p(o_t|ξ_1:t) if x_t =The object likelihood model is shown in (<ref>).As the neural network is trained to predict the non-movable and movable class, the first two cases in (<ref>) are straightforward. For the case of dynamic object, we scale the prediction of movable class by a factor s∈ [0,1] since all the dynamic objects are movable, however, not all movable object are dynamic. This scaling factor approximates the ratio of number of dynamic objects in the scene to the number of movable objects. This ratio is environment dependent for instance on a highway, value of s will be close to 1, since most of movable objects will be dynamic. For our experiments, through empirical evaluation, we chose the value of s=0.6. § RESULTSTo evaluate our approach we use the dataset from the KITTI object benchmark and the dataset provided by <cit.>. The first dataset provides object annotations but does not provide the labels for moving objects and for the second dataset we have the annotations for moving objects <cit.>. Therefore to analyze the classification of movable and non-movable points we use the KITTI object benchmark and use the second dataset for examining the classification of dynamic points. For all the experiments, Precision and Recall are calculated by varying the confidence measure of the prediction. For object classification the confidence measure is the objectness score and for dynamic classification the confidence measure is the output of the Bayes filter approach. The reported F1-score <cit.> is always the maximum F1-score for the estimated Precision Recall curves and the reported precision and recall corresponds to the maximum F1-score.§.§ Object ClassificationOur classification method is trained to classify points on cars as movable. The KITTI object benchmark provides 7481 annotated scans. Out of these scans we chose 1985 scans and created a dataset of 3789 scans by tracking the labeled objects. The implementation of Fast-Net is based on a deep learning toolbox Caffe <cit.>. The network was trained and tested on a system containing an NVIDIA Titan X GPU. For testing, we use the same validation set as mentioned by <cit.>. We provide quantitative analysis of our method for both pointwise prediction and object-wise prediction. For object-wise prediction we compare with these methods <cit.>. Output for all of these methods is bounding boxes for the detected objects. A direct comparison with these methods is difficult since output of our method is pointwise prediction, however, we still make an attempt by creating bounding boxes out of our pointwise prediction as a post-processing step. We project the predictions from 2D image space to a 3D point cloud and then estimate 3D bounding boxes by clustering points belonging the to same surface as one object. The clustering process is described in our previous method <cit.>.For object-wise precision, we follow the KITTI benchmark guidelines and report average precision for easy, moderate and hard cases. The level of difficulty depends on the height of the ground truth bounding box, occlusion level and the truncation level. We compare the average precision for 3D bounding boxes AP_3D and the computational time with the other methods in tab:object_bbox. The first two methods are based on RGB image, third method is solely LiDAR based, and the last method combines multiple view points of LiDAR data with RGB data. Our method outperforms the first three methods and an instance of the last method (front view) in terms of AP_3D. The computational time for our method includes the pointwise prediction on a GPU and object-wise prediction on CPU. The time reported for all the methods in tab:object_bbox is the processing time on GPU. The CPU processing time for object-wise prediction of our method is 0.30s. Even though performance of our method is not comparable with the two cases where LiDAR front view (FV) data is combined with bird eye view (BV) and RGB data, the computational time for our method is nearly 10× faster.In tab:pointvsobject we report the pointwise and object-wise recall for the complete test data and for the three difficulty levels. The object level recall correspond to the AP_3D results in tab:object_bbox. The reported pointwise recall is the actual evaluation of our method. The decrease in recall from pointwise prediction to object-wise is predominantly for moderate and hard case because objects belonging to these difficulty levels are often far and occluded therefore discarded during object clustering. The removal of small clusters is necessary because minimal over segmentation in image space potentially results in multiple bounding boxes in 3D space as neighboring pixels in 2D projected image can have large difference in depth, this is especially true for pixels on the boundary of an object. The decrease in performance from pointwise to object-wise prediction should not be seem as a drawback of ourapproach since our main focus is to estimate precise and robust pointwise prediction required for the semantic classification. We show the Precision Recall curves for pointwise object classification in fig:pr (right). Our method outperforms Seg-Net and we report an increase in F1-score by 12% (see tab:segnet). This network architecture was used by <cit.> in their approach. To highlight the significance of class balancing, we trained a neural network without class balancing. Inclusion of this information increases the recall predominantly at high confidence values (see fig:pr).§.§ Semantic Classification For the evaluation of semantic classification we use a publicly available dataset <cit.>. In our previous work <cit.> we annotated the dataset for evaluating moving object detection. The dataset consists of two sequences: Scenario-A and Scenario-B, of 380 and 500 frames of 3D LiDAR scans respectively.We report the results for the dynamic classification for three different experiments. For first experiment we use the approach discussed in sec:Bayes filter. In the second experiment, we skip the step of updating the object information (see (<ref>)) and only use the current objectness score within the filter framework. For the final experiment, object information is not included in the filter framework and the classification of dynamic points rely solely on motion cues. We show the Precision Recall curves for classification of dynamic points for all the three experiments for Scenario-A in fig:pr (right). The PR curves illustrates that the object information affects the sensitivity (recall) of the dynamic classification, for instance when the classification is based only on motion cues (red curve), recall is better among all the three cases. With the increase in object information sensitivity decreases, thereby causing a decrease in recall. In tab:bayes_filter we report the F1-score for all the three experiments on both the datasets. For both the scenarios, F1-score increases after adding the object information which shows thesignificance of leveraging the object cues in our framework. In fig:bayes_filter, we show a visual illustration for this case. For the Scenario-A, the highest score is for the second experiment. However, we would like to emphasize that the affect of including the predictions from the neural network in the filter is not only restricted to classification of dynamic points. In fig:bayes_filter_object, we show the impact of our proposed filter framework on the classification of movable points. § CONCLUSIONIn this paper, we present an approach for pointwise semantic classification of a 3D LiDAR scan. Our approach uses an up-convolutional neural network for understanding the difference between movable and non-movable points and estimates pointwise motion for inferring the dynamics of the scene. In our proposed Bayes filter framework, we combine the information retrieved from the neural network with the motion cues to estimate the required pointwise semantic classification. We analyze our approach on a standard benchmark and report competitive results in terms for both, average precision and the computational time. Furthermore, through our Bayes filter framework we show the benefits of combining learned semantic information with the motion cues for robust and precise classification. For both the datasets we achieve a better F1-score. We also show that introducing the object cues in the filter improves the classification of movable points.plainnat | http://arxiv.org/abs/1706.08355v1 | {
"authors": [
"Ayush Dewan",
"Gabriel L. Oliveira",
"Wolfram Burgard"
],
"categories": [
"cs.RO",
"cs.CV"
],
"primary_category": "cs.RO",
"published": "20170626131657",
"title": "Deep Semantic Classification for 3D LiDAR Data"
} |
^1 Epidemiology and Biostatistics Section,Rehabilitation Medicine Department, The National Institutes of Health, Clinical Center, Bethesda, Maryland 20892, U.S.A.Joshua C. Chang [email protected] Consider the problem of modeling memory effects in discrete-state random walks using higher-order Markov chains. This paper explores cross validation and information criteria as proxies for a model's predictive accuracy. Our objective is to select, from data, the number of prior states of recent history upon which a trajectory is statistically dependent. Through simulations, I evaluate these criteria in the case where data are drawn from systems with fixed orders of history, noting trends in the relative performance of the criteria. As a real-world illustrative example of these methods, this manuscript evaluates the problem of detecting statistical dependencies in shot outcomes in free throw shooting. Over three NBA seasons analyzed, several players exhibited statistical dependencies in free throw hitting probability of various types – hot handedness, cold handedness, and error correction. For the 2013–2014 through 2015–2016 NBA seasons, I detected statistical dependencies in 23% of all player-seasons. Focusing on a single player, in two of these three seasons, LeBron James shot a better percentage after an immediate miss than otherwise. In those seasons, conditioning on the previous outcome makes for a more-predictive model than treating free throw makes as independent. When extended to data from the 2016–2017 NBA season specifically for LeBron James, a model depending on the previous shot (single-step Markovian) is approximately as predictive as a model with independent outcomes. An error-correcting variable length model of two parameters, where James shoots a higher percentage after a missed free throw than otherwise, is more predictive than either model. Predictive Bayesian selection of multistep Markov chains, applied to the detection of the hot hand and other statistical dependencies in free throws Joshua C. Chang^1 Received 26th June 2017 / Accepted 25th January 2018 ====================================================================================================================================================§ INTRODUCTIONMultistep Markov chains (also known as N-step or higher-order Markov chains) are flexible models that are useful for quantifying any discrete-state discrete-time phenomenon. They have appeared in limitless contexts such as analysis of text <cit.>, human digital trails <cit.>, DNA sequences, protein folding <cit.>,eye movements <cit.>, and queueing theory <cit.>. In these models, transition probabilities between states depend on the recent history of states visited. In order to learn these models from data, a choice for the number of states of history to retain must be made. In this manuscript we evaluate contemporary Bayesian methods for making this choice, from the perspective of predictive accuracy.Bayesian model selection is an active field with many recent theoretical and computational advancements. In the broad setting of Markov Chain Monte Carlo (MCMC) inference,a significant advance has been approximation of leave one out (LOO) cross validation through use of Pareto smoothed importance sampling <cit.>, which has made LOO computationally feasible for a large class of problems. As explored in <cit.>, many methods similar to LOO exist. This manuscript adapts these methods to the context of selection for multistep Markovian models, providing closed-form expressions for computing information criterion that summarize the evaluation made by each method.As a concrete illustration of these methods, I examinethe problem of the detection of statistical dependencies broadly using free throw data from the National Basketball Association (NBA). A particular type of statistical dependency is known as the hot hand phenomenon. This phenomenon implies that recent success is indicative of success in the immediate future. While controversial in analytical circles, belief in the hot hand phenomenon is certainly widespread in both the general public and in athletes <cit.>. Empirically, the phenomenon has proven to be elusive <cit.>. In the 1980s, examinations of the phenomenon in basketball based on analysis of shooting streaks yielded negative results <cit.>, failing to reject null hypotheses of statistically independent shot outcomes.Based on these early analyses, some studies have dismissed the widespread belief in the hot hand by relating it to the Gambler's fallacy <cit.>. The gambler's fallacy refers to the seemingly mistaken belief that “random” events such as roulette spins exhibit autocorrelation <cit.>. In the context of the hot hand, an autocorrelation would involve increased probability of making a shot when one is in a “hot” state. Follow-up studies have examined the effects of belief in the hot hand under the supposition that it is a fallacy <cit.>.Ignoring the fact that statistical dependencies in even intended games of chance can exist <cit.>, one might reasonably suspect that various latent factors can affect the accuracy of an individual, where the outcome is the result of physical processes. These latent factors, modeled for instance by hidden Markov models <cit.>, would manifest as statistical dependencies in outcomes. Additionally, there were weaknesses in the prior research efforts that failed to find the hot hand effect. Recent analyses, using multivariate methods that can account for factors such as shot difficulty <cit.>, have supported the phenomenon, finding the original studies to be underpowered <cit.>, or to suffer from methodological issues regarding the weighting of expectation values <cit.>. A statistical testing approach that did not share this methodological issue <cit.> found evidence of the hot hand in aggregategame data but also raised the question of whether the observed patterns were a result of the hold/cold hand or of other individual-level states that imply statistical dependencies. In this manuscript I focus on detecting individual-level effects.As an illustration of model selection for multistep Markov chains, this manuscript re-examines the hot hand phenomenon from a different analytical philosophy. Presently, a broad class of analyses of the phenomenon <cit.> have been rooted in null hypothesis statistical testing.Rather than follow this approach, which requires a subjective choice of a cut off p-value to assess “significance,” this manuscript frames this problem as a model selection task. We are interested in whether models that encompass statistical dependencies like hot handedness are better at predicting free throw outcomes for individual players than models without such effects. § QUANTITATIVE METHODS§.§ Probabilistic modeling Multistep Markov chains (also known as n-step Markov Chains) are factorized probability models for discrete-state trajectories, where the probability of a particular trajectory is the product of conditional transition probabilities between possible states. The conditions pertain to the prior locations that a trajectory has visited, or its recent history. In our model of free throw shooting there are two states (make and miss), however, let us consider the more general problem of a model with any number M states. Assume that a trajectory ξ consists of steps ξ_l, where each step takesa value x_l taken from the set {1,2,…,M}. We are interested in representations for the trajectory probability of theform (ξ) =∏( ξ_l = x_l| previous h states)=∏_l=1^L (ξ_l = x_l | ξ_l-1 = x_l-1, …, ξ_l-h = x_l-h ) = ∏_l^L p_x_l-h,x_l-h+1,⋯,x_l, where h, a non-negative counting number, represents the number of states worth of memory needed to predict the next state, with appropriate boundary conditions for the beginning of the trajectory. In the context of the hot hand effect, models with h≥1 encompass statistical dependencies between shot outcomes. A hot hand would correspond to higher make probabilities after recent makes and cold hands correspond to lower probabilities after misses.Mathematically, the stochastic process underlying discrete-time Markov chains (implicitly h=1) is represented by a transition matrix, where each entry is a conditional probability of a transition from a state (row) to a new state (column). Multi-step Markov chains are no different in this respect. Each row corresponds to a given history of states and the corresponding matrix entries provide conditional probabilities of transitioning at the next step to a new state (column).In the case of absolutely no memory (h=0),the path probability is simply the product of the probabilities of being in each of the separate states in a path, p_x_1p_x_2,… p_x_L, and there are essentially M-1 free model parameters, where M is the number of states. The memoryless property of Markov chains refers to h=1.It should be noted that h=0 is a special sub-case of h=1, where the associated transition matrix has identical rows. If h=1, the model is single-step Markovian (memoryless) in that only the current state is relevant in determining the next state. These models involve M(M-1) free parameters. Knowledge of prior states beyond the current state is considered “memory.”Generally, if h states of history are required, then the model is h-step Markovian, and M^h(M-1) parameters are needed (see Fig <ref>). Hence, the size of the parameter space grows exponentially with memory. Our objective is to determine, based on observational evidence, an appropriate value for h.Note that multistep-Markovian models are nested. Lower-order models (smaller-h) can be represented by higher order models (larger-h) but not visa-versa. Variable-length models <cit.> also fit into this paradigm, as pictured in Fig. <ref>. A model might have an effective order of 1<h<2 for instance if many of its parameter vectors _x are identical.For a fixed degree of memory h, we may look at possible history vectors = [x_1,x_2,…,x_h] of length h taken from the set 𝐗_h = {1,2,…,M }^h. For each , denote the vector 𝐩_ = [p_,1,p_,2,…p_,M], where p_,m is the probability that a trajectory goes next to state m given thatrepresents its most recent history. For convenience, we denote the collection of all _ as 𝐩 (see below for an example of the notation). Generally one has availableJ∈ℤ^+ trajectories. Assuming independence between trajectories, one may write the joint probability, or likelihood, of observing these trajectories as({ξ^(j)}_j=1^J |) = ∏_j=1^J (ξ^(j)|) = ∏_j=1^J ∏_∈𝐗_h∏_m=1^M p_,m^N^(j)_,m = ∏_∈𝐗_h∏_m=1^M p_,m^N_,m , where N^(j)_,m is the number of times that the transition → m occurs in trajectory ξ^(j), and N_,m = ∑_j N^(j)_,m is the total number of times the transition is seen.For convenience,denote N_ =∑_mN_,m,_= [N_,1,N_,2,…,N_,M], and the collection {{^(j)_}_}_j as . The sufficient statistics of the likelihood are the counts, so we will refer to the likelihood as (|). The maximum likelihood estimator for each parameter vector _ is found by maximizing the probability in Eq. <ref>, and can be written easily as p̂^MLE_ = 𝐍_ / N_. Example:The outcomes of free throws for a player in a particular game can berepresented as string or trajectory of states (miss or make). For example, using “+” to denote makes and “-” to denote misses, a trajectory of “++-+” corresponds to a game where a player makes the first two free throws, misses the third, and makes the fourth. To clarify our notation, consider a model for free throw shooting informed using J=2 observed trajectories (games) given: {ξ^(1) = +-++-++, ξ^(2) =+–+-+++++-}. Suppose that we set h=1 in this model. This choice implies that we need the counts N_++^(1)=2, N^(1)_+-=2, N^(1)_–=0,N^(1)_-+=2, N_++^(2)=4, N^(2)_+-=3, N^(2)_–=1, N^(2)_-+=2 coinciding to the number of makes following makes, misses following makes, misses following misses and makes following misses respectively for each trajectory (game). In addition, since the first state in each trajectory is stochastic as well, we add two special states representing the outcome of the initial free throw, N_· - = 0, N_· + = 2. Aggregating the counts across the trajectories, in the vector notation above, we have 𝐍_+ = [N_+-, N_++]= [5,6], 𝐍_-=[1,4], 𝐍_· = [0,2], where the indices are all of length one since our choice of h=1 means that we only consider history vectors of length one. Using maximum likelihood, we arrive at the parameter estimates 𝐩̂_-^MLE = [p̂_–^MLE , p̂_-+^MLE ] = [1/5, 4/5], 𝐩̂_+^MLE = [5/11, 6/11], 𝐩̂_·^MLE = [0, 1].It is notable that our estimate of the probability of missing the first free throw in a game is null – this is probably an unrealistic inference. Fundamentally, the maximum likelihood estimator precludes the existence of unobserved transitions – a property that is problematic ifthe sample size J is small, as already seen in this example. This problemamplifies when increasing h. It is desirable to regularize the problem by allowing a nonzero probability that transitions that have not yet been observed will occur. This manuscript's approach to rectifying these issues is Bayesian.§.§ Bayesian modeling A natural Bayesian formulation of the problem of determining the transition probabilities is to use the Dirichlet conjugate prior on each parameter vector 𝐩_∼Dirichlet(α), hyper-parameterized by α, a vector of size M. The Dirichlet probability distribution is a distribution over finite-dimensional probability distributions. It has the probability density function π(𝐩_) = 1/B()∏_m=1^M p_,m^α_m-1, where B:ℝ^M→ℝ refers to the multivariate beta function <cit.>,B(𝐱) = ∏_m=1^M Γ(x_m)/Γ(∑_m=1^M x_m ),and Γ refers to the gamma function. This manuscript assumes that =1, corresponding to a uniform prior. This prior, paired with the likelihood of Eq. <ref>, yields the posterior distribution on the probabilities, 𝐩_|𝐍_∼Dirichlet( + 𝐍_ ).In effect, one is assigning a mean probability of α_m/(∑_mα_m+N_) to any unobserved transition, where || can be made small if it is expected that the transition matrix should be sparse. Other values ofare possible, for instance =1/2 corresponds to the Jeffreys' prior. Note that other Bayesian treatments of Markov chain inference have used priors within the Dirichlet family <cit.>. In the large-sample limit, as long as the components ofare bounded, the posterior distribution is not sensitive to the choice ofas the posterior density of Eq. <ref>becomes tightly concentrated about the maximum likelihood estimates. This fact is evident by observing that in Eq. <ref>, α_m + N_≈ N_ as N_→∞. §.§ Model selection criteria The parameter h controls the trade-off between complexity and fitting error. From a statistical viewpoint, complexity results in less-precise determination of model parameters, leading to larger prediction errors (overfitting). Conversely, a simple model may not capture the true probability space where paths reside, and fail to catch patterns in the real process (underfit).There are various existing generalized methods for evaluating how well models predict. Each of these methods summarizes a model using a single quantity. To facilitate comparison between the methods themselves, this manuscript scales the output of all methods to the deviance scale as used in the AIC. The deviance is a measure of information loss when going from a full model to an alternative model <cit.>.The Akaike Information Criterion (AIC), <cit.>, defined through the formula AIC= -2 ∑_log(_|_MLE) + 2k, where k is the number of parameters in the model, is an estimate of deviance between an unknown true model and a given model. For the selection of h, it may be computed exactly in closed form AIC= -2∑_∑_m=1^M N_,mlog( N_,m/N_)+ 2M^h(M-1), where in this context we define 0×log(0)≡0.Rooted in information theory, the AIC is an asymptotic approximation of the deviance <cit.>. The model with the smallest AIC is chosen. A limitation of the AIC is inaccuracy for small datasets. A correction to the AIC known as the AICc exists <cit.>, however, its exact form is problem specific <cit.>.The Bayesian Information Criterion (BIC) is closely related to the AIC but differs in the form of complexity penalty, taking sample size into account. The BIC, obeying the general formula BIC= -2 ∑_log(_|_MLE) + log(N)k, is also available in closed formBIC=-2∑_∑_m=1^M N_,mlog( N_,m/N_)+ log(∑_ N_)M^h(M-1).Despite its name, the formulation of the BIC is not Bayesian, using neither the prior nor posterior distributions. Under some conditions, however, the BIC can be seen as an asymptotic approximation of a Bayes factor <cit.>.Many of the Bayesian evaluation criteria feature the multivariate beta function (Eq. <ref>), as found in the normalization constant of the Dirichlet distribution (Eq. <ref>). To understand the large-sample properties of these methods, asymptoticexpansion of the multivariate beta function can help. In the case where |𝐱|→∞, assuming that all components of 𝐱 become unbounded,by Stirling's approximation the log multivariate beta function has the behavior log B(𝐱) = ∑_m=1^M [ x_mlog(x_m) - x_m - 1/2logx_m/2π + 1/12x_m + 𝒪(x_m^-3) ] - [log(∑_m x_m)∑_m x_m - ∑_m x_m-1/2log∑_m x_m/2π + 1/12∑_m x_m + 𝒪((∑_m x_m)^-3)]= ∑_mx_mlog(x_m/∑_x_m) -1/2( ∑_m logx_m/2π -log∑_m x_m/2π)+1/12(∑_m 1/x_m - 1/∑_m x_m)+ 𝒪((∑_m x_m)^-3) .Bayes factors are ratios of the probability of the dataset given two models averaged over their corresponding prior parameter distributions <cit.>.In the case of Markov chains, the likelihood completely factorizes intoa product of transition probabilities and each model's corresponding term in a Bayes factor is the exponential of its logmarginal likelihood (LML) LML =∑_log( B(_+)/B( )). If the expectation is computed instead against a posterior distribution |, one arrives at the expected log predictive density (LPD)LPD =∑_log( B(2_+)/B(_ + )). Related to the LPD is the expected log pointwise predictive density (LPPD), where the expectation in the LPD is broken down “point-wise.” For our application, we will consider trajectories to be points and write the LPPD asLPPD = ∑_j ∑_log(B(_ +_^(j) +)/B(_ +)).The LPPD features in alternatives to Bayes factors and the AIC <cit.>.The Widely Applicable Information Criterion <cit.> (WAIC) is a Bayesian information criterion with two variants, each featuring the LPPD but differing in how they compute model complexity. The WAIC is defined asWAIC = -2LPPD+2k_WAIC,where the effective model sizes are computed exactly ask_WAIC1 = 2 LPPD- 2∑_∑_m=1^M N_,m[ ψ(N_,m+ α_m ) - ψ(N_+ ∑_mα_m ) ],andk_WAIC2 =∑_j ∑_[∑_m=1^M [N^(j)_,m]^2ψ^'(α_m+N_,m) -[N^(j)_]^2ψ^'(N_ + ∑_mα_m)].In each of these expressions, Ψ refers to the digamma function, the derivative of the Gamma function <cit.>.The WAIC, unlike the AIC, is applicable to singular statistical models and is asymptotically equivalent to Bayesian leave-one-out cross-validation <cit.>.The two effective model size estimates for the WAIC are posterior expectations of equivalent estimates used in the Deviance Information Criterion (DIC).The DIC,DIC = -2∑_log p(_|_ = 𝔼__|_(_ )) + 2k_DIC,also resembles the WAIC. It consists of two variants in the computation of model complexity,k_DIC1=2{∑_∑_m=1^M N_,mlog( N_,m +α_m/N_+ ∑_mα_m)- ∑_∑_m=1^M N_,m[ ψ(α_m+N_,m) - ψ(N_ + ∑_mα_m) ] } , and k_DIC2 = 2var_|[ log( |) ] , which may be computed k_DIC2 =2∑_(∑_m N_,m^2 ψ^'(α_m+N_,m)- N_^2ψ^'(∑_mα_m+N_) ).The two estimates of complexity can be derived asymptotically from the LPD <cit.>, both reducing exactly to the number of predictors in the case of linear regression models using uniform priors.Finally Bayesian variants of cross-validation have recently been proposed as alternatives to information criterion <cit.>.In our problem, k-fold CV, where data is divided into k partitions, can be evaluated in closed form without repeated model fitting. Using -2×LPPD as a metric, this manuscript also evaluates two variants of k-fold CV: leave-one-out cross validation (LOO)LOO = -2∑_j ∑_log(B(_+)/B(_ -_^(j) +)),and two-fold (leave-half-out, LHO) cross validation,LHO= -2∑_j=1^J/2∑_log(B(^+_ +_^(j) +)/B(^+_+)) -2 ∑_j=J/2^J∑_log(B( ^-_ +_^(j) +)/B( ^-_ +)),where ^±_ constitute the transition counts of the last J/2 trajectories or the first J/2 trajectories respectively, so that ^-_ + ^+_ = _, and B refers to multivariate beta function.§ EVALUATION OF SELECTION CRITERIASimulations provided a means for testing how the criteria mentioned perform in finite-sample settings typical of most learning tasks. For the large-sample characteristics of cross-validation, I refer the reader to <cit.>.As a test system, consider a system of M=8 states, with designated start and absorbing states.For each given value of h_true, I generated for each ∈𝐗_h, a single set of fixed true transition probabilities {_ : ∈𝐗_h } drawn from Dirichlet(1) distributions. For each of these random networks of a fixed h, I randomly sampled sets of J trajectories, 10^4 times – each trajectory terminating when hitting a designated absorbing state.Note that the number of steps in a given trajectory is itself stochastic and determined by the statistics of the first-passage time to an absorbing state given the true transition probabilities. Then for each sample of J trajectories, I computed all the criteria mentioned in the previous section for various values of h.Fig. <ref> provides the frequency that each of six models (h=0,1,…,5,6) was chosen based on the selection criteria compared. Each row corresponds to a given true degree of memory h_true∈{0,1,2,3,4,5} and sample sizes increase along columns when viewed from left to right. Generally, as the number of samples increases, all selection criteria except for the LML (Bayes factors) and LPPD improve in their ability to select the true model. The LML consistently selects a more-complex (higher-h) model. The AIC does well if h_true is small, but requires more data than many of the competing methods in order to resolve larger degrees of memory.The BIC behaves like a more-conservative version of the AIC, requiring more data to select the more-complex but true generating model than the other methods.LOO, the two variants of the WAIC,and DIC_1 perform roughly on par. Since one uses each criterion by choosing the model of the lowest value, it is desirable that ΔCriterion(h)=Criterion(h)-Criterion(h_true)>0, for h≠ h_true. Fig. <ref> explores the distributions of these quantities in the case where h_true=2.As sample size J increases, there is clearer separation of these quantities from zero. By J=64, no models where h=1 are selected using any of the criteria. The WAIC_2 and LOO criteria perform about the same whereas the WAIC_1 criteria and the DIC_1 criteria lag behind in separating themselves from zero.In the Supplemental Materials, theaforementioned experiment is repeated for M=4 state systems, finding consistent results. This consistency is evident when comparing Supplemental Fig. <ref> to Fig. <ref>, where the same trends are present in both sets of results.Informed by these tests, this manuscript recommends the use of leave-one-out cross validation (LOO). LOO performed slightly better than WAIC_2 in the included tests, while being somewhat simpler to compute. Eq. <ref> decomposes completely into a sum of logarithms of gamma functions, and is hence easy to implement in standard scientific software packages.§ THE HOT HAND PHENOMENON I used the methodology in this manuscript to evaluate the hot hand effect in the controlled context of free throws. The free throw data was manually scraped from game-logs publicly available on the ESPN website. In Fig. <ref>, LOO is evaluated for h∈{0,1,2,3,4}, for all player-seasons between 2013–2014 and 2015–2016. LOO favored a model where h>0 for approximately 23% of all player-seasons tested. Restricted to player-seasons where at least 82 free throws are attempted, ΔLOO = LOO(h) - LOO(0) is presented in Fig. <ref>. The finding that h>0 does not by itself present information about the nature of statistical dependencies. To examine their nature, one must examine the inferred probabilities. In Fig. <ref>, conditional hitting probabilities are presented, for the same player-seasons represented in Fig. <ref>, where h=1 is favored over h=0. It is evident that many of the players exhibit what one might call a “hot hand” by shooting a better percentage after a previous make than otherwise. Likewise, many players exhibit cold hands, shooting worse after misses, or by shooting a worse free throw percentage on the first attempt in a game. Notably, some players, such as LeBron James, seem to shoot better after a miss than otherwise which would represent a form of error correction. LeBron James is a volume free throw shooter who appears in Fig. <ref> so I singled him out for further analysis. Looking at Fig. <ref>, James appears in the first two seasons but is not present in the third season (2015–2016). Examining his shooting splits from Fig. <ref> two trends stand out. First, he shoots a lower percentage on the first free of the game than otherwise. Second, he appears to consistently hit a higher percentage after a miss than otherwise. In the 2015–2016 season, those patterns do not survive and LOO does not favor h=1 over h=0. During the 2016-2017 season, in 91 games, LeBron James attempted at least a single free throw, hitting 471 of 693 overall (Fig. <ref>).Conditioning the hit probabilities by the outcome of the preceding free throw in the same game, James shot a slightly better percentage after missing a free throw than otherwise. However, the h=0 model is favored slightly over h=1(Model selection pane of Fig <ref>).As in Fig. <ref> for the M=8 test system, we can evaluate the performance of the model selection criteria using simulations. Assuming that the h=1 model is true, sets of 91 strings of free throw outcomes were simulated. The length of each string was chosen by drawing from a Poisson distribution where the expectation matched the mean number of free throws attempted by James per game (≈ 7.6). The overall hitting percentage in these simulations was matched to 68%, as found in the original game data, and the transition probabilities were matched overall to those in Fig. <ref>. Despite the fact that h=1 was the underlying true model, it was chosen slightly under half the time (Fig. <ref>).Another variant of the simulated power analysis is given in Fig. <ref>, where within each game James' free throw outcomes are resampled from the actual game data, in effect scrambling the order of makes and misses. Shuffling of shot outcomes destroys the correlations between consecutive shots. It is seen in Fig. <ref> that the information criteria match up both qualitatively and quantitatively in their model choice with the h=0 simulated game data presented in Fig. <ref>. Examining the model parameters in the case of h=1, one sees that the hitting probabilities are similar in all cases except after a miss (Fig. <ref>). This observation suggestsamodel with jagged variable-length memory: independence of outcome except after a miss. Having one fewer parameter than the full h=1 model, this variable-length model is favorable to both the h=0 and h=1 models (Fig. <ref>). Hence, at least for this season and the two present in Fig. <ref>, the most predictive model ofJames' free throw shooting tells a story of error correction rather than a story of hot hand. § DISCUSSION AND SUMMARY This manuscript addressed general methods of degree selection for multistep Markov chain models. While the example provided was related to the evaluation of the hot hand phenomenon, the class of models where the included methodology can be used is broad. Notably, multistep Markov Chains have applications in queueing models which feature heavily in operations research. The simulations yielded insight on the performance of the criteria in the typical small-sample setting. This manuscript provided simulations for M=2,4,8 state systems, finding consistent results throughout. Importantly, both the AIC and LML (Bayes factors) are biased in opposite situations, in opposite directions. For small datasets, the AIC tends to sparsity, which runs counter to the typical situation in linear regression problems where the AIC can favor complexity with too few data, a situation ameliorated by the more-stringent AICc <cit.>. The BIC is the most-conservative of the methods tested, however, requires more data to accept the veracity of any given higher-order model. Bayes factors with flat modelpriors (via the AIC-scaled log marginal likelihood) as investigated here, on the other hand, consistently select a higher value of h given more data. Due to the nested nature of these models (depicted in Fig. <ref>), such behavior may not be undesirable. One may still learn an effective lower-order model within a higher-order model, finding that the higher-order model makes effectively the same predictions.Notably, alternative Bayes factors methods for selecting the degree of memory also include model-level priors that behave like the penalty term in the AIC <cit.>. Since the upper bound of the LML is the logarithm of the likelihood found from the MLE procedure, this selection method is more stringent in the low sample-size regime than the pure AIC and hence will suffer from the same bias towards selecting models with less memory.Additionally, it is known that Bayes factors are sensitive to the choice of prior <cit.>, since they involve an expectation relative to the prior distribution. Examining Eq. <ref>, the denominator of the term within the logarithm of Eq. <ref> is invariant to observations.In contrast, Bayesian alternatives where expectations are taken with respect to the posterior should not be as sensitive to the choice of prior. When looking at LOO of Eq. <ref>, the prior comes into the formulation only to increment the overall count of a given pattern. Asymptotically, _ quickly overwhelms . Hence, LOO is not as sensitive to the exact choice of . §.§ Limitations and extensions This manuscript addressed only a limited aspect of the overall model selection task – the evaluation of competing models on the basis of predictive accuracy. This manuscript does not tackle the parallel task of model searching, outside of the context of fixed-order multistep models. For fixed order models, search is easy. One fits models by order sequentially. We have seen, however, that at times variable-length histories are appropriate. Notably, in LeBron James' 2016–2017 free throws, a variable length model is favored over a larger encompassing model, which is itself disfavored relative to a smaller fixed-length model. In that example, with the small number of states, one could easily detect the variable-length model directly. However, when the number of states increases, the number of variable-length models also increases exponentially. While out of the intended scope of this manuscript, I note that projective search methods <cit.> may have promise for adaptation to the search for variable-length Markov chains. In these methods, one searches for submodels nested within a larger encompassing model.As a baseline for such a procedure, one may choose to begin with a model of slightly higher order than that selected by LOO. As pertains to the hot hand and related phenomena, fundamentally, these phenomena manifest as observable correlations in shot outcomes. However, the “generating distributions” for free throw outcomes are likely not in the class of multistep Markov models. From a modeling standpoint, a hidden Markov model with “hot” and “cold” states, as implemented by <cit.>, may be more mechanistically-valid. Hidden Markov models map to multistep Markov models of perhaps infinite order at arbitrarily high precision. Hence, this manuscript focused on the detection of any statistical dependency in free throws. However, one could make an argument, as I have done, that the various patterns of statistical dependencies detected: cold first shot, error correction, etc, have real-world physical interpretations. Such an argument may not hold in generality for other processes modeled using multistep Markov chains. Hence, I would like to stress that the focus of the manuscript is on finding predictive models, rather than on mechanistic certainty.§.§ The hot hand phenomenon From the modeling perspective, for a given player, there can be large fluctuations in free throw shooting percentage between seasons. It is common, particularly early in careers, for players to drastically improve their accuracy after an offseason of training. However, changes occur often in the other direction as well. Hence, one either needs to model non-stationarity in the percentages or restrict the time interval of the applicability of any given model. In this manuscript I have chosen to restrict modeling to the interval of a single season, ignoring non-stationarity within the season, a trade-off common to other analyses <cit.>. The downside of such restrictions is that they limit the volume of data that may be used in detecting effects. LeBron James, a volume free throw shooter who does not miss many games and plays well into the playoffs, is perhaps a best-case scenario for detection of statistical dependencies in free throws using such models.Even for James, the detection of these effects can be difficult. Judging from simulations (Fig <ref>), it appears that the dataset is underpowered for the selection of a pure model where h=1. On the other hand, in simulations where h_true=0, h=0 is correctly chosen approximately 83% of the time. The h=0 and h=1 models are both approximately as predictive. In fact, from the model averaging perspective, they would be weighted approximately the same as weights are exponential in the gap between the selection criteria <cit.>.For James' 2016–2017 attempts, the variable length model (Fig <ref>) is more predictive than either of the pure h=0 and h=1 models. While his 2013–2014 and 2014–2015 seasons do favor h=1 over h=0, the effective models have the same error-correcting behavior as the variable-length model of 2016–2017. Hence, based on finding the model with the best prediction, one would predict that LeBron James is often more likely to make a free throw after a miss than otherwise.I have no competing interests. All data has been deposited on DryadThe author declares no competing interests.This section does not apply. The sole author is responsible for the entirety of the work.This work is supported by the Intramural Research Program of the National Institutes of Health Clinical Center and the US Social Security Administration.This manuscript is dedicated to the memory of Dr. Robert M. Miura. He is survived by his loving family, his numerous mathematical contributions, and the many generations of researchers that he has mentored and advised throughout his decades of generous service. I also thank members of the Biostatistics and Rehabilitation Section in the Rehabilitation Medicine Department at NIH, John P. Collins in particular, and also Carson Chow at NIDDK for helpful discussions. § DERIVATIONS The marginal likelihood, also known as Bayes factor, is the expectation of the likelihood under the prior distribution. The logarithm of this quantity (LML) isLML = log𝔼_[ (𝐍|) ]= log𝔼_( ∏_∏_m=1^M p_,m^N_,m)= ∑_log( B(_+)/B( )). The log predictive density (LPD) given a model defined by an inferred posterior distribution 𝐩|𝐍 may be computed LPD = log𝔼_|[ (𝐍|) ]=log𝔼_|( ∏_∏_m=1^M p_,m^N_,m)= ∑_log( B(2_+)/B(_ + )). The log pointwise predictive density (LPPD), requires partition of data into disjoint “points.” Treating trajectories as points yieldsLPPD = ∑_j∑_log𝔼__|_[ (𝐍^(j)_|_) ]= ∑_j∑_𝐱log𝔼__|_( ∏_m=1^M p_,m^N^(j)_,m)=∑_j ∑_log(B(_ +_^(j) +)/B(_ +)). The LOO, as defined in this manuscript, is similar to the LPPD. For the LOO, each of the pointwise posterior distributions is computed after leaving out the corresponding trajectory. Hence, LOO =-2∑_j∑_log𝔼__|_∖_^(j)[ (𝐍^(j)_|_) ]=-2 ∑_j∑_𝐱log𝔼__|_∖_^(j)( ∏_m=1^M p_,m^N^(j)_,m)=-2∑_j ∑_log(B(_ -_^(j) +_^(j)+)/B(_ -_^(j) +)). For the WAIC, the two variants of complexity parameters are k_WAIC1 = 2LPPD-2∑_j ∑_𝔼__|[log_^_^(j)] = 2 LPPD -∑_j∑_∑_m=1^M N_,m^(j)𝔼__|_(log p_,m)= 2 LPPD - 2∑_j ∑_∑_m=1^MN_,m^(j)[ ψ(N_,m+ α_m ) - ψ(N_+ ∑_mα_m) ] = 2 LPPD - 2∑_∑_m=1^M N_,m[ ψ(N_,m+ α_m ) - ψ(N_+ ∑_mα_m) ],andk_WAIC2 =∑_j ∑___[ log(𝐍^(j)_|_) ]=∑_j∑___{log( ∏_m=1^M p_,m^N^(j)_,m) } =∑_j∑___[∑_m=1^M N^(j)_,mlog p_,m]= ∑_j∑_∑_m=1^M ∑_n=1^M N^(j)_,m N^(j)_,n( log p_,m, log p_,n)= ∑_j∑_∑_m=1^M ∑_n=1^M N^(j)_,m N^(j)_,n[ψ^'( α_n+N_,n)δ_nm - ψ^'( ∑_mα_m+N_) ]=∑_j ∑_[∑_m=1^M [N^(j)_,m]^2ψ^'(α_m+N_,m) -[N^(j)_]^2ψ^'(∑_mα_m+N_)]. The commonly used Deviance Information Criterion (DIC)DIC = -2∑_log p(_|_ = 𝔼__|__) + 2k_DICalso resembles the WAIC,consisting of two variants in the computation of model complexity,k_DIC1= -2{∑_∑_m=1^M N_,mlog( N_,m +α_m/N_+ ∑_mα_m) -∑_j ∑_𝔼__|log_^_^(j)} =2{∑_∑_m=1^M N_,mlog( N_,m +α_m/N_+ ∑_mα_m)-∑_j ∑_∑_m=1^M _,m^(j)[ ψ(α_m+N_,m) - ψ(∑_mα_m+N_) ] } =2{∑_∑_m=1^M N_,mlog( N_,m +α_m/N_+ ∑_mα_m)- ∑_∑_m=1^M N_,m[ ψ(α_m +N_,m) - ψ(∑_mα_m+N_) ] } , and k_DIC2 = 2var_|[ log( |) ] , which may be computed k_DIC2 =2 __[ ∑_∑_m N_,mlog p_,m]=2∑___( ∑_m N_,mlog p_,m)=2∑_∑_m∑_n N_x,mN_x,n(log p_,m,log p_,n)=2∑_∑_m∑_n N_x,mN_x,n[ ψ^'(α_m+N_,m)δ_mn - ψ^'(∑_mα_m+N_)]=2∑_(∑_m N_,m^2 ψ^'(α_m+N_,m)- (N_)^2ψ^'(∑_mα_m+N_) ),where δ_mn refers to the Kronecker delta function.§ SUPPLEMENTAL RESULTS In Fig. <ref>, it is apparent that models where h=0 and h=1 have comparable predictive power. As a form of permutation test, we consider resamplings without replacement of James' 2016–2017 free throws where within each game the order of his shot outcomes are scrambled. The results here are similar to those found in Fig. <ref>, where the statistical power of the information criteria are evaluated using simulated data.Fig. <ref> presents a version of the same tests performed in the main manuscript (where an eight state system is used) on a four-state system. In comparing Fig. <ref> to Fig. <ref>, one finds consistent results. Note that on average the trajectory lengths are shorter in the four state system relative to the eight state system due to the fact that there are fewer possible states relative to the number of absorbing states. | http://arxiv.org/abs/1706.08881v3 | {
"authors": [
"Joshua C. Chang"
],
"categories": [
"stat.ME",
"physics.pop-ph"
],
"primary_category": "stat.ME",
"published": "20170626143536",
"title": "Predictive Bayesian selection of multistep Markov chains, applied to the detection of the hot hand and other statistical dependencies in free throws"
} |
http://arxiv.org/abs/1706.08318v2 | {
"authors": [
"Daniel Alsina"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170626110043",
"title": "PhD thesis: Multipartite entanglement and quantum algorithms"
} |
|
[pages=1-20]IJSS-D-17-00698R1 | http://arxiv.org/abs/1706.08754v2 | {
"authors": [
"Marco Lepidi",
"Andrea Bacigalupo"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170627094307",
"title": "Multi-parametric sensitivity analysis of the band structure for tetrachiral acoustic metamaterials"
} |
[email protected] COSMO – Centro Brasileiro de Pesquisas Físicas, Xavier Sigaud, 150, Urca, Rio de Janeiro, Brasil [email protected] COSMO – Centro Brasileiro de Pesquisas Físicas, Xavier Sigaud, 150, Urca, Rio de Janeiro, Brasil [email protected] COSMO – Centro Brasileiro de Pesquisas Físicas, Xavier Sigaud, 150, Urca, Rio de Janeiro, Brasil Center for Cosmology, Particle Physics and Phenomenology, Institute of Mathematics and Physics, Louvain University, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, BelgiumWe investigate cosmological scenarios containing one canonical scalar field with an exponential potential in the context of bouncing models, where the bounce happens due to quantum cosmological effects. The only possible bouncing solutions in this scenario (discarding an infinitely fine tuned exception) must have one and only one dark energy phase, either occurring in the contracting era or in the expanding era. Hence, these bounce solutions are necessarily asymmetric. Naturally, the more convenient solution is the one where the dark energy phase happens in the expanding era, in order to be a possible explanation for the current accelerated expansion indicated by cosmological observations. In this case, one has the picture of a Universe undergoing a classical dust contraction from very large scales, the initial repeller of the model, moving to a classical stiff matter contraction near the singularity, which is avoided due to the quantum bounce. The Universe is then launched to a dark energy era, after passing through radiation and dust dominated phases, finally returning to the dust expanding phase, the final attractor of the model. We calculate the spectral indexes and amplitudes of scalar and tensor perturbations numerically, considering the whole history of the model, including the bounce phase itself, without making any approximation or using any matching condition on the perturbations. As the background model is necessarily dust dominated in the far past, the usual adiabatic vacuum initial conditions can be easily imposed in this era. Hence, this is a cosmological model where the presence of dark energy behavior in the Universe does not turn problematic the usual vacuum initial conditions prescription for cosmological perturbation in bouncing models. Scalar and tensor perturbations end up being almost scale invariant, as expected. The background parameters can be adjusted, without fine tunings, to yield the observed amplitude for scalar perturbations, and also for the ratio between tensor and scalar amplitudes, r = T/S ≲ 0.1. The amplification of scalar perturbations over tensor perturbations takes place only around the bounce, due to quantum effects, and it would not occur if General Relativity has remained valid throughout this phase. Hence, this is a bouncing model where a single field induces not only an expanding background dark energy phase, but also produces all observed features of cosmological perturbations of quantum mechanical origin at linear order.Consistent Scalar and Tensor Perturbation Power Spectra in Single Fluid Matter Bounce with Dark Energy Era Sandro Dias Pinto Vitenti December 30, 2023 ==========================================================================================================§ INTRODUCTIONBouncing models have been proposed as cosmological scenarios without an initial singularity. Instead, the Universe had at least a preceding contracting phase from very large length scales, shrinking the space until the scale factor reaches a minimum value in which some new physics takes place, mainly related to gravity modifications at very small length scales, halting the contraction and launching the Universe into the expanding phase we are living in.In the standard cosmological model, inflation is responsible for exponentially increase the particle horizon after the Big Bang in order to explain why regions, which are not in causal contact at the last scattering surface, present a highly correlated temperature distribution, as observed today in the Cosmic Microwave Background radiation (CMB). Without inflation, these regions would be causally disconnected in a purely Big Bang model. This puzzle is the so called horizon problem, and it does not exist in bouncing models. Since the Universe had a very large period of contraction in the past, there is no limit to the particle horizon (if the fluids dominating the contracting phase satisfy the strong energy condition). Another puzzle of a purely Big Bang scenario is the flatness problem, i.e., considering a Friedmann metric in a expanding phase, the spatial curvature dilutes slower than any other matter content (assuming again the strong energy condition) of the model. Hence, unless the spatial curvature is strongly fine tuned to zero initially, it would quickly dominates the expansion afterwards. When inflation is added to the Big Bang scenario, it turns out that the Universe is driven dynamically to an almost flat hypersurface, avoiding the initial curvature fine tuning problem. In the bounce scenario this issue is not posed, since flat space-like hypersurfaces are dynamical attractors during contraction <cit.>.Despite the fact that inflation has not yet any consensual fundamental physics behind it, a simple slow-roll prescription for the inflation scalar field is enough to solve the above mentioned puzzles, and to amplify quantum vacuum fluctuations after the Big Bang, thus giving rise to an almost scale invariant adiabatic power spectrum, in good agreement with CMB observations <cit.>. It is a challenge for bounce cosmologies to reproduce a competitive fit for the observations, and many models have been scrutinized over the years with this aim.In what concerns the primordial phase of bounce cosmologies, it has been shown that perturbations originated from quantum vacuum fluctuations during a matter dominated contraction phase become almost scale invariant <cit.> in the expanding phase. Whether the linear perturbation theory remains valid for a specific gauge choice through the bounce is a subtle question addressed by many authors <cit.>, and finally clarified in Refs.<cit.>, confirming the validity of linear perturbation theory up to the expanding phase. The background scenarios used to developed those investigations are matter contractions driven either by a canonical scalar field with an exponential potential, a K-essence scalar field representing a hydrodynamical fluid, or a relativistic perfect fluid using Schutz formalism <cit.>. This scenario with a contracting phase dominated by a dust-like fluid is called matter bounce scenario. This class of models is an interesting new approach to the Big Bang/Inflation scenario <cit.>, and they have been extensively studied over the past 15 years. A weak point for any model including a contracting phase, where all the matter content satisfy the dominant energy condition, is the presence of Belinsky-Khalatnikov-Lifshitz (BKL) instabilities <cit.>, the fast growth of anisotropies during the contraction. Some proposals inspired in the Ekpyrotic Model <cit.> address this problem by means of an ad-hoc ekpyrotic type potential <cit.>. It is not the aim of this work to address this kind of issue, since it is possible to overcome it in more complex scenarios without completely spoiling out a suitable primordial power spectra. Note that any cosmological model, either inflationary, bouncing, or any other, has a much more serious problem to deal with: the large degree of initial homogeneity necessary to turn all these models compatible with observations. This is largely more serious than the BKL problem. Note also that once one assumes an initial homogeneous and isotropic Universe, one can show for the models we are considering that the shear perturbation will never overcome the background degrees of freedom, even growing as fast as a^-6 in the contracting phase <cit.>, and hence the BKL problem is not present once such assumption is made. For a discussion on that, see also Ref. <cit.>.In this paper we will carefully study the physical properties of primordial quantum perturbations in a matter bounce realized by a canonical scalar field with an exponential potential. Our starting point are the results obtained in Refs. <cit.>. In our scheme, we will argue that, approaching the singularity during contraction, quantum effects as calculated in Ref. <cit.> become relevant, and a bounce arises naturally in the context of the canonical quantization of gravity, connecting the classical contracting and expanding phases described in Ref. <cit.>. The known cosmological solutions obtained through the de Broglie-Bohm (dBB) formulation of quantum mechanics can be applied to this system since the classical singularity takes place when the kinetic term dominates the scalar field dynamics, and the potential becomes negligible, exactly as in the model investigated in Ref. <cit.>. Hence, one has classical contracting and expanding phases connected by a quantum bounce. Our background model avoids the need of a ghost scalar field and is sustained by the fact that, in the regime where the curvature scale is 10^2 Planck length or larger, the canonical quantization we implement is expected to be an effective limit of more fundamental theories of quantum gravity. Finally, since the perturbations evolve through the background quantum phase, we use the right action for the perturbations when the background is not assumed to be classical <cit.>.Our dynamical system analysis shows that this scenario carries an interesting feature: as we will see, if a bounce takes place in between classical contracting and expanding phases, the scalar field presents an effective transient dark energy-type Equation of State (EoS), either in the past of the contracting phase or in the future of the expanding phase. This addresses an aspect of bouncing cosmologies that has gained increased attention <cit.>: the role of the dark energy (DE) in the contracting phase of bouncing models. In the expansion history probed by current observations, the effects due to the existence of a DE component can only be felt when the scale factor is of about 3-4 e-folds from the last scattering surface <cit.>. Whether the DE is a cosmological constant or a quintessence field, it should be present during the contracting phase too. Thus, it may affect the contracting phase evolution of perturbations sensible to scales around the same Hubble radius as today. The presence of dark energy in the contracting phase of bouncing models may turn problematic the imposition of vacuum initial conditions for cosmological perturbations in the far past of such models. For instance, if dark energy is a simple cosmological constant, all modes will eventually become larger than the curvature scale in the far past, and an adiabatic vacuum prescription becomes quite contrived, see Ref. <cit.> for a discussion on this point. However, in the case of a scalar field with exponential potential, which contains a transient dark energy phase, the Universe will always be dust dominated in the far past, and adiabatic vacuum initial conditions can be easily imposed in this era, as usual. Hence, this is a situation where the presence of dark energy does not turn problematic the usual initial conditions prescription for cosmological perturbations in bouncing models.The bounce solutions obtained here are necessarily asymmetric, i.e., the transient DE is effective either in the contracting phase or in the expanding phase, but not in both. Here we will study the more realistic solution where the dark energy phase happens in the expanding era, connecting the bounce model with the current accelerated expansion phase. Hence one has the picture of a Universe which realizes a dust contraction from very large scales, the initial repeller of the model, moves to a stiff matter contraction near the singularity and realizes a quantum bounce that ejects the Universe in a stiff matter expanding phase. The latter moves to a dark energy era, finally returning to the dust expanding phase, the final attractor of the model. The other possibility, DE in the contracting phase, is more academic, and we let it for a future work. The background solutions are constructed numerically, matching the classical and quantum eras in the phase where both have the similar dynamics.Note that it can already be found in the literature exact solutions of the full Wheeler-DeWitt equation for a canonical scalar field with exponential potential, which is not neglected in the quantum phase <cit.>. These solutions were obtained without matchings. However, as they have exactly the same physical features as the solutions described above (classical behavior up to stiff matter domination, where quantum effects begin to be important and the potential is negligible, and one and only one DE energy phase all along), we preferred to adopt the above matching procedure, in which the numerical calculations are simpler to handle.After the background construction, scalar and tensor perturbations are calculated numerically, and the results are understood analytically. Depending on the parameters of the background, they turn out to be almost scale invariant, with the right observed amplitude for scalar perturbations, and also for the ratio between tensor and scalar amplitudes, r = T/S ≲ 0.1. The amplification of scalar perturbations over tensor perturbations takes place only around the bounce, and we explicitly show that it happens due to the quantum effects on the background model producing the bounce. There are many papers pointing out the difficulties of producing this amplification of scalar perturbations over tensor perturbations in the framework of General Relativity (GR), see Refs. <cit.>. Indeed, the amplification we will present would not occur if GR has remained valid all along the bounce. Hence, our result shows that when GR is violated around the bounce, the influence of this phase on the evolution of cosmological perturbations can be nontrivial, and must be evaluated with care. These effectsprovide a counter example to the usual case where the perturbations are unaffected by the details of the bounce and their amplitude are determined by the bounce depth (the ratio between the value of the scale factor during the potential crossing and its value at the bounce).The paper will be divided as follows: in Sec. <ref>, based on Ref. <cit.>, we summarize the classical minisuperspace model and its full space of solutions. In Sec. <ref>, we present the quantum background near the singularity, as presented in Ref. <cit.>. The matching of the classical and quantum solutions is explained in Sec. <ref>, where we obtain the full background model with reasonable observational properties. Section <ref> describes the equations of motion for the quantum perturbations with suitable vacuum initial conditions, and we perform the numerical calculations in Sec. <ref>, exhibiting our final results for both scalar and tensor perturbations. We conclude in Sec. <ref> with discussions and perspectives for future work.In what follows we will consider ħ = c = 1 and the reduced Planck mass ≡ 1 / κ≡ 1 / √(8 π). The metric signature is (+, -, -, -).§ BACKGROUNDWe consider a canonical scalar field ϕ whose Lagrangian density is given by= √(-g)[∇^νϕ∇_νϕ - V(ϕ)].The potential V(ϕ) is chosen to be the exponential, i.e.,V(ϕ) = V_0 ^-λκϕ,where the constant V_0 has units mass^4 and λ is dimensionless.The exponential potential has vastly assisted cosmologists to address puzzles of the standard model because of its rich dynamics. In Refs. <cit.> we have some heterogeneous collection of what was published with exponential potential only in 2016. For the background dynamics, we will use results from <cit.>.In a flat, homogeneous and isotropic Universe, the Friedmann-Lamâitre-Robertson-Walker metric iss^ 2 =N^2(τ)τ^2 - a(τ)^2 ( x^2 +y^2 +z^2 ),where N(τ) is the lapse function and a(τ) is the scale factor. The evolution of the scale factor in cosmic time (N(τ) = 1,τ = t) is given by the Friedmann equation coupled to the Klein-Gordon equation, respectively,ȧ = a H,Ḣ = - κ^2/2ϕ̇^2 ,ϕ̈ = - 3 H ϕ̇ -V /ϕ,where the dot operator represents the derivative with respect to the cosmic time t. The Hubble function, H, must satisfy the Friedmann constraintH^2 = κ^ 2/3[ ϕ̇^ 2/2 + V(ϕ) ]. The background dynamics can be made simpler through a choice of dimensionless variables that allows us to rewrite the coupled second order equations (<ref>) and (<ref>) as a planar system <cit.>, i.e.,x = κϕ̇/√(6)H,y = κ√(V)/√(3)H.In those new variables, the Friedmann constraint, Eq. (<ref>), and the effective EoS read,x^2 + y^2 = 1, w = 2x^2-1.Applying the above definitions to the system of Eqs. (<ref>) and (<ref>) leads to the planar system:x/α = - 3 x (1-x^2)+ λ√(3/2)y^2, y/α = x y (3x-λ√(3/2)),where α≡ln (a). This system is supplemented by the equationsα̇ = H, Ḣ = -3H^2x^2.The critical points are listed in Tab. <ref>.Near the critical points where w = 1, the effective energy density of the scalar field evolves close to a^-6, i.e., it behaves approximately like a stiff-matter fluid. For those where the effective EoS is w = 1/3(λ^2 - 3), it evolves as a^-λ^2. The qualitative behavior of the system can be studied with the tools described in <cit.>, for a detailed analysis see <cit.>.In the contracting phase H < 0, we can see by the definition of y, Eq. (<ref>), that y < 0. Note that y is completely determined by the value of x through the constraint (<ref>) and the sign of H. This phase is, therefore, tied to the lower quadrants of the phase space, while the upper quadrants depict the expanding phase, see Fig. <ref>. Using the Friedmann constraint Eq. (<ref>) yieldsx/α = -3 (x-λ/√(6)) (1-x)(1+x).For λ < √(6), the first two critical points, x_±c = ± 1 and y_±c = 0, are unstable (repellers) during expansion and stable (attractors) in the contraction phase, i.e.,. x/α|_x=1-ϵ < 0, . x/α|_x=-1+ϵ > 0,for 0 < ϵ≪1. For λ > 0, the critical point x_λ = λ/√(6) has the following behavior,. x/α|_x=λ/√(6)+ϵ < 0, . x/α|_x=λ/√(6)-ϵ > 0.It is, therefore, an attractor during the expanding phase and an a repeller in a contracting phase.For the purpose of this work, we will choose λ = √(3). As a consequence, the scalar field behaves as a matter-fluid, w = 0, around the critical point x_λ. Then, choosing the initial conditions for x, asx_λ = λ/√(6)±ϵ,leads to a matter-fluid contracting phase. Since x_λ is the only attractor in the expanding phase, the model will always end in matter-fluid dominated epoch.In Fig. <ref> we have the phase space for the planar system. The critical points in which the scalar field behaves as a stiff-matter fluid are marked as S_± and the ones it behaves as a matter-like fluid are M_±. The contraction history goes as follows: the Universe starts close to the critical point M_- and, depending on the choice of sign in Eq. (<ref>), arrives at the stable point S_+ or S_- for, respectively + and -. The evolution then ends up in a singularity (if no quantum effects are included). The final EoS parameter is w = 1. Classically, there is no possible bounce solution when the system arrives in the critical points S_±.In the trajectories M_- → S_-and S_-→ M_+, the Universe passes through a transient DE epoch, since-1 < x < λ/√(6)⇒ -1 ≤ w < 1/3(λ^2 - 3).Both possible expanding trajectories start in a stiff-matter epoch, S_±, andends up in the matter epoch, M_+.It will be useful to analyze what happens with H and ϕ̇ when the system is close to the critical points S_±. Around these points, we get from Eq. (<ref>) thatH ∝^-3α,as it is expected for a stiff-like fluid, which diverges when approaching the singularity. Consequently, using Eq. (<ref>) we deduce that in the neighborhood of these points the asymptotic behavior islim_x→± 1{[ H → -∞,; ϕ →±√(6)/κα→∓∞,;ϕ̇ →±√(6)/κH→∓∞, ].in the contracting phase, while in the expanding phase one getslim_x→± 1{[ H →∞,; ϕ →±√(6)/κα→∓∞,;ϕ̇ →±√(6)/κH→±∞. ]. Figure <ref> summarizes what we presented above qualitatively. Nevertheless, this figure represents only the critical points and flow resulting from the classical equations of motion. The quantum dynamics takes place in the portion of phase space where the Ricci scalar is close to the Planck length, near the singularities. As in Fig. <ref> we set κ=√(6), the S_± critical points representing the classical singularities are depicted in this figure by the lines H = ±ϕ̇ when |H|→∞ and |ϕ̇|→∞. Hence, quantum effects can modify Fig. <ref> only around these regions, with a quantum bounce connecting the regions around S_± in the lower quadrants to the regions around S_± in the upper quadrants. However, trajectories connecting the neighborhood of S_+ in the lower quadrant to the neighborhood of S_+ in the upper quadrant, or similarly connecting the neighborhoods of S_- in both quadrants, necessarily cross the classical M_± line, H = √(2)ϕ̇. As we will see, in the de Broglie-Bohm quantum theory, which we will use in this paper, velocity fields yielding the quantum Bohmian trajectories are single valued functions on phase space, as they arise from well defined functions on this space (the phase's gradient from the corresponding Wheeler-DeWitt wave equation's solution). Consequently, such trajectories cannot cross each other. Hence, the only way to connect the almost singular contracting and expanding behaviors without crossing the line M_± is through a connection from the regions around S_± to the regions around S_∓, in this reversed order, necessarily. Indeed, as we will see below [see Eq. (<ref>)], around the critical points S_± the quantum bouncing trajectories have a well defined sign for ϕ̇, which is determined completely from the model parameters. As H necessarily changes sign through a bounce, then these bouncing trajectories can only connect the neighborhoods of S_± to the ones of S_∓. Concluding, there is no solution in the complete phase space where the Universe contracts in the direction of S_- (S_+) and expands from S_- (S_+). A bounce is only possible in the phase space of H and ϕ̇ if it connects the contracting phase ending in S_+ (S_-) to the expansion starting from S_- (S_+). For the model in consideration, we have then two possible histories of the Universe, depicted in Figs. <ref> and <ref>. In the case of Fig. <ref>, a contracting phase starts close to M_- (matter-fluid era), passes through a DE phase and ends in S_- [as described in Eq. (<ref>)], where the scalar field behaves as an stiff-matter fluid. At this point, new physics takes place performing a bounce and the Universe starts expanding from S_+ (scalar field as stiff-matter) and ends in matter-like expansion in M_+. There is no DE epoch in the expanding phase in this scenario.The case of Fig. <ref> goes in the opposite direction. Contraction happens from a matter epoch, M_-, to a stiff-matter one, S_+. The new physics avoids the singularity and brings the Universe to an expansion that starts in S_- followed by a DE epoch, ending finally in M_+, a matter-like epoch.The two above-mentioned scenarios have the interesting feature of a transient DE-like phase. For the sake of future reference along the article, let us call the case with DE during contraction, Fig. <ref>, case , and the one with DE during expansion, Fig. <ref>, case .Case shows the less compelling situation, where the background performs a matter contraction followed by a DE epoch before the bounce, but the expanding phase has no DE epoch. Previous works considering the presence of DE during contraction used a ghost field to perform the bounce. In such scenarios the phase space is very different from the one presented here, and it can have a DE epoch in both contracting and expanding phases <cit.>. Nevertheless, a complete and rigorous calculation of the primordial power spectra in such scenarios have not yet been performed.Case is the one we will explore in this work. As we will see, the same scalar field with exponential potential that realizes the matter bounce scenario yielding an adiabatic scale invariant power spectrum also produces a DE epoch in the expanding phase. Furthermore, there is no need of an extra ghost potential nor any auxiliary scalar fields in order to yield the physical conditions that produce the bounce.As we mentioned earlier, there is no classical bounce in the previous described backgrounds. Close to the attractor of the contracting phase, S_+ in case and S_- in case , H ∝ - a ^-3, and when a →0 we reach a singularity. This happens because the kinetic term dominates the Lagrangian of the scalar field and diverges, but in these cases it has already been proved that quantum bounce solutions may arise. In the next section we present the results from <cit.>, and show how they can by applied to our case to avoid the singularity.§ THE QUANTUM BOUNCEQuantum cosmology is the field of research in which quantum theory is applied to the Universe, and should have the standard cosmological model as its classical limit. This interesting and challenging topic, not only whitens fundamental problems of cosmology, as the singularity problem, but also allows fundamental quantum mechanics to be tested at the cosmological level <cit.>.The quantum description of gravity, besides facing many difficulties with the non-renormalizable aspect of GR <cit.>, also suffer from fundamental conceptual issues in what concerns the application of a quantum theory to the description of the whole Universe. In order to construct a quantum theory for the Universe, the traditional Copenhagen interpretation of quantum mechanics has to be replaced. Its main limitation is the postulate of the collapse of the wave function <cit.>, where an outside classical system is necessary to perform the collapse. This does not make sense if the whole Universe is quantized.The quantum theory which will replace the traditional Copenhagen point of view must of course be able to reproduce the results of quantum experiments already performed, but it must also dispense this exterior classical world, or collapse recipes, in order to be applicable to quantum cosmology. There are many proposals of quantum theories that satisfy these criteria, and were already applied to Cosmology: the consistent histories approach <cit.>, collapse models for the wave-function <cit.>, the many worlds interpretation <cit.>, and the dBB quantum theory <cit.>, which is the one we will adopt here.The canonical quantization of gravity obtained through the ADM formalism <cit.>, which should be an effective limit of a more fundamental theory, can be interpreted using the dBB formulation of quantum mechanics. The dynamics of the wave function of the Universe is given by the Wheeler-DeWitt equation from where, in this formulation, one can obtain Bohmian trajectories with objective reality describing the evolution of the whole system. In this approach, there is no need to postulate any collapse of the wave function of the Universe <cit.>.Both models described in this paper depicted in Figs. <ref> and <ref> present the same feature: the end of the classical contraction and the beginning of classical expansion happen when the kinetic energy overcomes the scalar field potential V(ϕ). A system consisting of a flat, homogeneous and isotropic space-time in the presence of a scalar-field with a dominant kinetic term has already been quantized. The Bohmian trajectories resulting from the Gaussian superposition of plane wave functions led to bounce solutions. Details of this construction can be found in <cit.>. We will summarize their results in what follows.In the case where the dynamics is dominated by the kinetic term, the Hamiltonian for a scalar field in the metric (<ref>) readsH = Nℋ =Nκ^2/12 V^3 α(- Π_α^2 + Π_ϕ^2),where V is the volume of the conformal hypersurface, and from here on we will be using the dimensionless scalar fieldϕ→κϕ/√(6).The associated momenta to the canonical variables α and ϕ are, respectively,Π_α = - 6V/Nκ^2^3 αα̇, Π_ϕ = 6V/Nκ^2^3 αϕ̇. Finally, we choose the conformal hypersurface volume as V = 4π^3 / 3, where ≡√() = 1/(√(8π)) is the Planck length. With this choice, the scale factor value has an absolute meaning, i.e., when a = 1 the Universe has approximately the Planck volume.Performing the Dirac quantization procedure, we can write the Wheeler-DeWitt equation asℋ̂Ψ(α,ϕ) = 0 ⇒[- ∂^2/∂α^2 + ∂ ^2/∂ϕ] Ψ(α,ϕ) = 0.The general solution isΨ (α,ϕ)= F(ϕ + α) + G(ϕ - α) ≡∫ k {f(k) exp[ik(ϕ + α)] +g(k) exp[ik(ϕ - α)]},where f and g are arbitrary functions of k.Writing the wave-function in polar form, Ψ = R exp (i S), where R is the amplitude and S is the phase in units of ħ, and substituting into Eq. (<ref>) leads to the Hamilton-Jacobi like equation for the phase S,(∂ S/∂α)^2 - (∂ S/∂ϕ)^2 - 1/R(∂^2 R/∂α^2 - ∂ ^2 R/∂ϕ) = 0.When the last term of Eq. (<ref>), the so called quantum potential, is negligible with respect to the others, we get the usual classical Hamilton-Jacobi equation for the minisuperspace model at hand. Assuming the ontology of the trajectories α(t) and ϕ(t), this classical limit suggests the imposition of the so called dBB guidance relations in order to determine the trajectories, in correspondence to the usual classical Hamilton-Jacobi theory, and they read, in cosmic time N=1,Π_α = ∂ S/∂α = - ^3 αα̇,Π_ϕ = ∂ S/∂ϕ =^3 αϕ̇ .When the quantum potential is not negligible in Eq. (<ref>), these Bohmian quantum trajectories may be different from the classical ones, and may present a bounce.In Ref. <cit.> it was chosen the simple appealing prescription of a Gaussian superposition of plane waves in Eq. (<ref>) given byf(k) = g(k) = exp[- (k - d)^2/σ^2].Calculating the phase S of the aforementioned solution, and substituting it into the guidance relations (<ref>) and (<ref>), we find the planar system that describes the Bohmian trajectoriesα̇ = ϕσ ^2 sin (2dα) + 2dsinh (σ ^2 αϕ)/2^3α[ cos (2dα) + cosh (σ ^2 αϕ)],ϕ̇ = -ασ ^2 sin (2dα) +2dcos(2dα) + 2dcosh (σ ^2 αϕ)/2^3α[ cos (2dα) + cosh (σ ^2 αϕ)].When solving the equations above we have a time definition in units of Planck time (essentially putting = 1 in the above equations). However, since the scales of interest for the perturbations are those around the Hubble radius today, R_H ≡ 1/H_0 (here we adopt the current value H_0 = 67.8 ± 0.9 km s^-1Mpc^-1 <cit.>), we convert back using the factor R_H / when matching with the classical solution.In Fig. <ref> we have the phase space for Eqs. (<ref>) and (<ref>). We can notice the presence of bounce and cyclic solutions. It is easy to calculate the nodes and the centers. They happen all along the line ϕ =0: the nodes for d α = (2n +1)π / 2 and the centers forσ^2 α / 2 d = (d α).The classical limits of Eqs. (<ref>) and (<ref>) are obtained for large α, when the hyperbolic function dominates. From the definition of x in that limit, it is straightforward to obtain the relationsx≈(σ^2 αϕ),H/H_0 ≈R_H/d ^-3 α/x,ϕ̇ ≈ d^-3α.These equations imply that ϕ and x have the same sign, and ϕ̇ the same sign as d in the classical limit. This means that in case , since its contraction ends in x → -1, Eq. (<ref>) is satisfied only if d > 0. Similarly, case requires d < 0. This result is consistent with our discussion in Sec. <ref>. In case , the quantum dynamics starts with x ≈ -1 (ϕ≪ -1), ending in x ≈ 1 (ϕ≫ 1). The opposite happens in case , i.e., our bouncing dynamics always connects the classical critical points S_- (S_+) with S_+ (S_-). In practice, the sign of d determines which case is being evolved.§ MATCHING OF BACKGROUNDIn the previous section we presented the quantum corrections to the system when the kinetic term of the scalar field dominates, yielding a bounce. In order to construct a complete background, we should be able to match the solutions from the classical evolution, described in Sec. <ref>with the quantum solution from the system of Eqs. (<ref>) and (<ref>).In a complete formulation of this problem, the designation of “classical” and “quantum” solutions should not be taken strictly. In thecomplete dBB formulation, we have the Bohmian trajectories that accounts for both regimes. In our hybrid background, we make this distinction only to emphasize the fact that we do not have a complete Bohmian trajectory and to make explicit which equations of motion are being used. The full quantum description of this system can be found in Ref. <cit.>, where full Bohmian bounce solutions are exhibited. However, to perform the calculation of cosmological perturbations around some full Bohmian trajectory is quite cumbersome, hence we prefer to adopt this simpler method of matching classical to quantum solutions, with no loss of relevant information.The nomenclature in what follows may be tricky, and in order to avoid confusion we will adopt the expressions quantum/classical solutions, regimes or branches to distinguish the dynamics described by Eqs. (<ref>) and (<ref>) (quantum) from the one determined by Eqs. (<ref>) and (<ref>) (classical). To make reference to the period at which the quantum potential isrelevant, we will adopt quantum phase in opposition to classical phase, in which the quantum potential is irrelevant.The complete background solution has three branches. The first one is the classical contraction that starts with x ≈ 1 / √(2) and ends in x →± 1. The second branch is the quantum background that starts in x ≈± 1 and bounces to x →∓ 1. The third branch, the classical expansion, starts with x ≈∓ 1 and ends with x → 1/√(2). The lower signs stand for case while the upper signs for the case . The matching is performed guaranteeing continuity of the solutions up to the first derivative at the time when they move from the quantum to the classical regimes. This happens when the quantum solutions reach their classical limit given in Eqs. (<ref>) and (<ref>).However, the classical limit of the quantum regime happens when x = ± 1, which is a critical point of the classical equations, Eqs. (<ref>) and (<ref>). To start the classical phase exactly in a critical point means that the Universe would be stuck in the stiff-matter epoch. Nonetheless, the initial classical epoch is unstable, hence we always start it with a small shift around the critical points. We will parametrize x in the proximity of the stiff critical points by x = ± (1 - ϵ_±), 0 < ϵ_±≪ 1. In the matching point, ϵ_± should be small enough to justify the classical limit of the quantum regime.If we had a complete Bohmian trajectory, it would only be necessary to set initial conditions in the far past. For instance, we would give an initial x_λ close to the unstable point M_-. The proximity to the critical point and from which side of the critical point it begins dictates the duration of the matter contraction and selects between cases and (which must be compatible with the choice of sign for d). We would also give an initial scale factor, a_ini, and a Hubble constant, H_ini, (ϕ_ini and ϕ̇_ini are constrained by the value of x_ini and V_0 through the Friedmann equation). With that all set, the Bohmian trajectories would handle the whole evolution until the expanding phase. Naturally, parameters as the minimum scale factor at the bounce, the energy scale of the DE epoch, and the duration of the quantum bounce would be obtainable from the model parameter V_0, the system wave-function and the initial conditions.Because of our matching procedure, things are not so simple. We have not only the choice of initial conditions and the quantum bounce parameters, d and σ (extracted from the wave-function), but also two new variables, namely, the contraction and expansion matching parameters, denoted by ϵ_c and ϵ_e, respectively. In what follows, we will rewrite these two variables in terms of new parameters which also controls the number of e-folds between the bounce and a given Hubble scale.In order to connect the classical solution parameters at the matching point with its quantum evolution, we integrate Eqs. (<ref>) and (<ref>) analytically. However, it is not possible to write explicit functions for x(α) and H(α). The best we can do is to obtain implicit functions which, apart from two integration constants, read3α = - √(2)tanh^-1(x) - ln[ (1/√(2) - x)^2/1-x^2] + cte,lnH= √(2)tanh^-1(x) + ln(1/√(2) - x /1-x^2) + cte.We begin by recasting these solutions in a more convenient form(a/a̅_0)^6 (H/H_0)^2= C_1/( 1/√(2)-x)^2,(a/a̅_0)^3= C_2 (1-x)^γ_+(1+x)^γ_-/( 1/√(2)-x)^2,whereγ_±≡ 1 ±1/√(2),and C_1 and C_2 are constants. We introduced H_0, the Hubble parameter today, and a̅_0 for mere convenience. Note that these constants can be absorbed in C_1 and C_2, and they do not represent any additional freedom of the system. We can calculate the number of e-folds between the critical points S_± and M_±. Expanding Eq. (<ref>) around x = ±(1-ϵ_±) and x =(1/√(2)±ϵ_λ) at leading order, yields, respectively(a_±/a̅_0)^3≈C_2 2^γ_∓ϵ_±^γ_±/γ_∓^2,(a_λ/a̅_0)^3≈C_2 γ_-^γ_+γ_+^γ_-/ϵ_λ^2,where we note that, at leading order, the second expression does not depend from which side we approach M_±. From their ratio we get(a_±/a_λ)^3 ≈2^γ_±/γ_-^γ_+γ_+^γ_-γ_∓^2ϵ_±^γ_±ϵ_λ^2.Imposing that we must be close enough to the critical points, i.e., ϵ_i < 10^-4 (for i = ±, λ), Eq. (<ref>) leads toa_+/a_λ≲^-10, a_-/a_λ≲^-6.This means that the trajectories M_- → S_- or S_- → M_+, depicted in the lower quadrants of Fig. <ref> and the upper quadrants of Fig. <ref>, respectively, must have a minimum of 6 e-folds. The remaining trajectories (M_- → S_+ and S_+ → M_+) must have a minimum of 10 e-folds.Analogously, we can also calculate the number of e-folds until the DE phase (x = 0), yielding(/a̅_0)^3 ≈ 2C_2, (a_-/)^3 ≈2^γ_+ϵ_-^γ_-/2γ_+^2, a_-/≲^-1. The background will be constructed numerically from the quantum bounce to the classical contracting and expanding phases. As the solutions are necessarily asymmetric, the matching between the quantum and classical regimes can be arranged in two possible ways, depending on whether we want to write ϵ_c and ϵ_e in terms of the number of e-folds between the bounce and a point in the matter-fluid domination or the DE phase. Let us now describe this construction in the following sub-sections. §.§ Initial conditions at the bounce Our numerical calculation consists in solving the background, starting from the bounce and evolving to the expanding and contracting phases. In order to accomplish this, we start the calculation around the bounce with slightly positive (negative) time for the expansion (contraction) phase. A convenient time variable is τ defined byα =+ τ^2/2,which leads to τ/t = H / τ and α = ττ. We have already included in this choice of time variable the initial condition for α, i.e., we always set the bounce at τ = 0. We can see from Eq. (<ref>) that the initial value α(t_0) = 0 induces a trivial solution α(t) = 0. As trajectories in the (α , ϕ) plane cannot cross, this implies that α cannot chance sign along the possible trajectories. Hence, the choice of time above select the positive branch of the phase space (α , ϕ), Fig. <ref>. For a single bounce, αattains its smallest value at the bounce, which provides the last justification for our parametrization (<ref>).We need now the initial condition for the field ϕ. Examining Eq. (<ref>), we realize that the bounce can only take place when ϕ = 0. Indeed, the denominator is always positive, and it diverges to the classical limit, where no bounce is possible. Hence the necessary condition for the bounce α̇=0 can occur if and only if the numerator is zero. If there were a root of the numerator of Eq. (<ref>) different from the trivial one ϕ = 0, it would satisfysinh(A)/A = -sin(B)/B,where we have defined A = σ^2 αϕ and B = 2dα, both different from zero, by assumption. However, this equation cannot be solved for any real A and B. Therefore, for the quantum system, we will always use the only possible initial conditions α(0) = and ϕ(0) = 0.Expanding Eqs. (<ref>) and Eq. (<ref>) about the bounce, we get the leading order approximationsτ/ t_Q = ϕ/τD_1,ϕ/ t_Q = D_2,where we rewrote the equation in terms of τ and the convenient dimensionless time variable ^3α t_Q =t. The two constants D_1 and D_2 areD_1= σ^2[sin(2d)+2d]/2[2cos(2d) + 1], D_2= -σ^2 sin(2d) + 2dcos(2d)+2d/2[2cos(2d) + 1].These equations can be easily integrated to giveτ = t_Q√(D_1D_2), ϕ = t_Q D_2,where we have also chosen the sign of τ coinciding with the sign of t_Q. These solutions hold close to the bounce, hence we chose a very small value of t_Q, for instance t_Q^ini∝±𝒪(10^-50), to start the numerical evolution of Eqs. (<ref>) and (<ref>), using the new time t_Q. In order to do that, one must know D_1 and D_2, hence α_b, d, σ must be given. The above choice of initial t_Q gives well defined numerical results for the whole range of parameters studied in this work. The time variable t_Q is then used to solve the quantum dynamics until the matching with the classical phases, in the contracting branch and in the expanding branch. For d<0, the positive time direction in the integration moves the solution to the DE branch, while for d>0, the positive time direction moves the solution to the branch without DE. From there on, we have two possibilities to parametrize the matching, depending on whether there is a DE behavior in the classical dynamics or not. §.§ Matching with matter domination scale Evolving the quantum era as described above, we arrive at a nearly classical evolution with stiff matter behavior at some a_± = ln(α_±) with x(a_±) = ± [1 - ϵ_±(a_±)]. At this point, we match the quantum evolution with the classical one. In order to control this matching and its cosmological meaning, we will write a_± and its corresponding ϵ_± (a_±) in terms of the number of e-folds between the bounce and the matter-fluid domination, where x = 1 / √(2) + ϵ_λ, with 0 < ϵ_λ≪ 1. We will suppose that the free constant a̅_0 belongs to the infinity open set of real numbers satisfying x(a̅_0) = 1 / √(2) + ϵ_λ(a̅_0), with 0 < ϵ_λ(a̅_0) ≪ 1.The first step is to impose continuity of the Hubble function at the matching point. Expanding Eq. (<ref>) about x_± gives(H_±/H_0)^2 ≈C_1/γ_∓^2(a̅_0/a_±)^6.Equating this expression to Eq. (<ref>) yieldsC_1 = R_H^2/^2d^2 γ_∓^2/a̅_0^6.This equation relates the free constant C_1 of the classical system to a̅_0. As the scale factor at the bounce a_b was already chosen, giving the physical parameter≡a̅_0/,is equivalent to fixing C_1. The parameteryields the number of e-folds from the bounce to the momentof the matter-fluid domination determined by a̅_0, which, as commented above, is still rather arbitrary: the only constraint on it is to satisfy x(a̅_0) = 1/ √(2) + ϵ_λ(a̅_0), with 0 < ϵ_λ(a̅_0) ≪ 1. For the second constant C_2, we obtain from Eq. (<ref>) thatC_2 = γ_∓^2/2^γ_∓ϵ_±^γ_±(a_±/a̅_0)^3.This equation relates the matching point a_± and its corresponding ϵ_± (a_±) to C_2. Note, however, that the end of the quantum evolution does not designate any specific value of a_± as long as the quantum evolution stays very close to the stiff matter classical evolution. Hence, all points where 0 < ϵ_±≪ 1 are acceptable. Here lies the ambiguity of the matching.We could arbitrarily choose a_± in order to fix C_2. However, we will do the reverse: we will connect C_2 with sensible cosmological parameters associated with physical features of the classical branch, and after a judicious choice of them determining C_2, we use Eq. (<ref>) to finally find the matching point a_±. This can be done as follows: close to the matter-fluid epoch, the zero order term of the Hubble function reads(H/H_0)^2 ≈C_1/C_2γ_-^γ_+γ_+^γ_-( a̅_0/a)^3.The above result motivates the definition of the arbitrary constantΩ_d = C_1/C_2 γ_-^γ_+γ_+^γ_- = R_H^2/^2d^2 γ_∓^2/C_2 γ_-^γ_+γ_+^γ_-a̅_0^6.This constant is very useful, as long as it gives a precise meaning to . Indeed, from Eq. (<ref>) and Eq. (<ref>), we getH^2(a = a̅_0) ≈ H_0^2Ω_d.The parametercan now be understood as yielding the number of e-folds between the bounce and the moment where the Hubble radius is R_H / √(Ω_d). Hence, once Ω_d is given,acquires a very precise meaning.In terms of the physical variables Ω_d and , the constant C_2 reads,C_2 = R_H^2/^2d^2 γ_∓^2/Ω_d γ_-^γ_+γ_+^γ_-^6^6.This expression is completely determined by our choices of quantum initial condition a_b and the constants Ω_d and . Plugging it into Eq. (<ref>), we obtain our matching timea_±/ϵ_±^γ_±/3≈1/(R_H^2d^22^γ_∓/^2γ_-^γ_+γ_+^γ_-Ω_d)^1/3.Then, given a value forand Ω_d, with the cosmological meanings described above, we must evolve the quantum branch until some values of α_± = ln a_± and ϵ_± where Eq. (<ref>) is satisfied. From this point on, we continue using the classical equations (<ref>) with these matching values for α_± and x_± as initial conditions.Summarizing, from the classical dynamics we introduced four (redundant) constants to control the initial conditions (a̅_0,H_0,C_1,C_2). We chose H_0 to match today's value of the Hubble parameter in order to have the problem scaled to the perturbation scales of cosmological interest. To fix the other arbitrary parameters, we studied the classical solution around the matter-fluid critical point, introducing the matter parameter Ω_d, which only serves to give a meaning to . Note that the matching (<ref>) depends only on the product ^3Ω_d, evincing that one of these parameters is arbitrary. The first actual initial condition we impose by matching the value of the Hubble function in the end of a quantum branch with its value in the beginning of a classical branch. The second initial condition is, however, not completely determined by the quantum phase, as we can choose any small value of ϵ_± as we want. For this reason, we choose a value for(keeping Ω_d fixed), which has a clear cosmological meaning, to completely determine a_± and ϵ_± through Eq. (<ref>), and the subsequent system evolution .Finally, it is useful to study the allowed values for . Using for the moment Ω_d = 1, i.e., choosingto represent the number of e-folds between the bounce and the instant where the Hubble radius matches today's value, we get≈ (a_b a_±)^-1(R_H^2d^22^γ_∓ϵ_±^γ_±/^2γ_-^γ_+γ_+^γ_-)^1/3.The value ofis inversely proportional to a_ba_±, thus, to maximizewe need to minimize a_ba_±. Using the fact that α_b > 0, we get a_b = 1 as the minimum value of a_b. Assuming that the quantum phase is fast enough so that it has approximately zero e-folds of duration, we get a_± = 1 as the minimum value of a_±. Then, ignoring the order one factors we get≲ 10^40 d^2/3ϵ_±^γ_±/3,which shows, that given the value of d, we have a maximum for the number of e-folds. This fact is very relevant for the perturbations, since their amplitudes are determined partially by . Note also that this kind of constraint is present in any matter bounce, and it can make difficult to generate enough amplitudes for the perturbations without approaching too much the Planck scale. Moreover,is proportional to ϵ_±^γ_±, thus, a long quantum branch (ϵ_±^γ_± very small) results in a shallow bounce. §.§ Matching with the dark energy scale For trajectories containing the DE epoch, we have an alternative way to give meaning for a reference scale. We can choose a̅_0 to represent the exact point where w = -1, i.e., x = 0. At this point we have[H(a=a̅_0)/H_0]^2 = 2C_1 ≡Ω_Λ,2C_2 = 1,where we have introduced the parameter Ω_Λ (in the same way and with similar characteristics to Ω_d). There is an important distinction to make in comparison with the other case. Here, we are matching with a fixed point in time, whereas, in the last matter-fluid we can match to any time where 0 < ϵ_λ≪ 1. For this reason, the value of Ω_Λ completely determines the matching and in this caseis obtained from it. For Ω_Λ = 1, DE domination takes place around our present Hubble time.Since we have fully determined C_1 and C_2 all we need to do is substitute them on Eqs. (<ref>) and (<ref>) from which we obtain the matching conditiona_-/ϵ_-^γ_-/3≈(R_H | d| 2^γ_+/√(2Ω_Λ)γ_+)^1/3,where we specialized in the _- branch since it is the only one containing a DE phase. The number of e-folds in this case is given by the logarithm of≈1/[2 R_H^2 γ_+^2 d^2 /^2Ω_Λ]^1/6.It is worth noting that the number of e-folds between the bounce and the | H| = H_0 scale, assuming Ω_Λ = 1, is different for the different matching procedures, showing the asymmetry of the model. Note also that the main factor in determiningin the matter-fluid matching is (R_H/)^2/3 while for the DE matching we have (R_H/)^1/3, i.e., the DE matching produces a smaller number of e-folds in the branch where it is applied. It is also clear from the equation above that larger (smaller) Ω_Λ results in fewer (more) e-folds between the DE phase and the bounce.One last comment: when the DE phase happens during expansion (the case ) it is natural to use this procedure to match, using Ω_Λ≈ 1, which guarantees us that we could model the current accelerated expansion using this field. However, if the DE phase happens during the contraction (case ), we do not have any reason to choose a priori a given scale to match. §.§ Summarizing the background reconstruction As explained in this section, the numerical integration used to construct the background model is initiated at the bounce itself, using the quantum guidance equations. For that, one should give the values of α_b, d and σ. The system is evolved until reaching the classical limit, where it is matched with the classical evolution. This matching is controlled by the cosmological parameters Ω_Λ and . In the branch with a DE phase, the quantum evolution is halted when Eq. (<ref>) is satisfied, while in the branch without a DE phase the quantum dynamics is stopped when condition (<ref>) is reached. This gives the values of α_± and x_± to be used as initial conditions to the subsequent integration of the classical equations (<ref>).Hence, the complete collection of parameters needed to fix the background model is (α_b, d, σ, Ω_Λ, ), all of them with clear cosmological significance. For d>0 we have case , and the DE phase is in the contracting phase, while for d<0 we are in case , and the DE phase is in the expanding phase.As a last remark, the classical branch could, in principle, be calculated using Eq. (<ref>), and all other quantities are obtained by simple integrals of x. However, the variable x is not well behaved numerically, for instance, to represent a point very close to S_+, say x = 1 - 10^20, we would need at least a 20 decimal digits floating point number, well beyond the default double precision (about 16 decimal digits), present in today's computers. To avoid this numerical pitfall, we make the following change of variablesx_±(q_±) = ±(1-γ_∓ q_±/1+q_±),which maps the x range (±1, 1/√(2)) into the numerically well defined interval (0, ∞).[The default double precision float point number can represent numbers from ≈ 10^-300 to ≈ 10^+300.] For this reason we solve numerically Eq. (<ref>) written in terms of q_± for the paths ending/beginning on S_±.§ NUMERICAL SOLUTIONS FOR THE BACKGROUNDIn this section we will explore the parameter space of the theory, and see its influence in the complete background behavior, which will be used in the perturbative analysis. When one parameter is varied in the figures below, the bold values in Tab. <ref> designate the other parameters which get fixed in the corresponding figures. For example, thedifferent curves appearing in Fig. <ref> for the case corresponding to the different values of d present in Tab. <ref> were calculated with the parameters σ = 0.5, α_b = 10^-40, 𝒳_b= 10^30 and Ω_Λ = 1. Also, since we are interested in perturbations at scales near the Hubble radius today, we from here on use Ω_d = 1. In what concerns the study of the cosmological perturbations, the background dynamics can be fully understood by the plot of H with α, Figs. <ref> to <ref>. We show clearly the bounce asymmetry by choosing the horizontal axis as being sign(τ)(α-). In that way, the negative interval represents the contracting phase while the positive interval represents the expanding phase. For a perfect fluid with p = w ρ, the evolution of H isln| H|∝3/2(w + 1) α,thus, in the intervals with a well defined w (stiff and matter-fluid phases) H behaves as a power-law. In the matter epoch, the effective EoS of the scalar field is w = 0, and in the stiff matter one it is w=1, implying in different slopes in Figs. <ref> to <ref>. The duration of the epochs are connected with the size of the Universe at which we see the transition from one slope to another. The closer to the bounce that transition happens, longer is the matter epoch. This is very important since we are interested in controlling the matter contraction to confirm its influence in the relevant mode scales.The DE epoch happens when x = 0 corresponding to a short plateau between the matter and stiff-matter phases, for example, in Fig. <ref> around α = 20.[This could be deduced directly from Eq. (<ref>) since R_H/≈ 10^60 and all other constants are of order one.] In Fig. <ref> we included a zoom-in for this interval, showing the transient DE phase.We plotted Figs. <ref> to <ref> for case , however, equivalent figures for case can be obtained by simple mirror symmetrysign(τ)(α-) → - sign(τ)(α-)As we have mentioned before, it makes no sense to change Ω_Λ away from ≈ 1 in case , since it is an observational constraint and we expect to use this transient DE phase to model the current accelerated expansion. On the other hand, in case , this is exactly the parameter we are interested in order to study perturbations in a DE epoch. Therefore, Fig. <ref> is a plot from case .The bounce happens in α =, and the two peaks are the highest values of H reached by the system, further on referred as H_max.The peaks happen when Ḣ = 0 and we can use them to define the duration of the bounce, δ_b. They can be better noticed in the zooms depicted in Figs. <ref> and <ref>. The closer the peaks are in the plots, the smaller is δ_b (faster bounce).In what concerns the quantum solutions, the variation of the parameters d, σ and α_b directly changes the time and energy scales of the bounce. When increasing d, the frequency in Eqs. (<ref>) and (<ref>) is higher and it is possible that the background oscillates close to the bounce, Fig. <ref>.Another important influence of d is in the duration of the matter phase in contraction. Using Eq. (<ref>), we can mark the onset of the matter-fluid phase by choosing a fixed ϵ_λ (for example 10^-4). Then, it is clear from Eq. (<ref>) that a_λ (marking the beginning of the matter-fluid phase) is proportional to | d|^2/3. Hence, larger d yields longer stiff-matter phases. This effect can be seen in Fig. <ref>. In the expanding era, which contains a DE phase, the same Eq. (<ref>) can be used, but now C_2 = 1/2, and the connection to d is through a̅_0 given by Eq. (<ref>). Consequently, a_λ∝| d|^1/3. In the expanding era of Fig. <ref> we can see that the DE plateau shifts by a smaller δα than the matter-fluid era shift in the contraction phase.The parameter σ is relevant only in the quantum phase. Figure <ref> shows that larger σ's implies in higher energy and in shorter time scales in the bounce. This can be easily understood looking again to Eqs. (<ref>) and (<ref>). The hyperbolic functions have the argument σ^2αϕ [see for instance Eq. (<ref>)] and they saturate when the argument is of the order of 11 [x ≈(11) ≈ 1 + 𝒪(10^-10)]. Hence, a larger value for σ leads to a faster saturation of the hyperbolic functions and, consequently, to an earlier stiff-fluid epoch. Furthermore, in the small α and ϕ approximation, the value of |H_max| is also proportional to σ, meaning that larger values of σ generate more energetic bounces.A similar argument leading to higher values of | H_max| can be used when analyzing the influence of , Fig. (<ref>). More profound bounces imply in a longer stiff-matter epoch, as we can see in the transitions of the slopes in Fig. (<ref>). During that period | H| has more time to increase leading to high-energy scale bounces.Finally, in Fig. (<ref>) we observe that , controlling the matching point between the classical and quantum branches, influences the duration of the dust domination era. This turns out to be crucial for the perturbations, not only because the power-spectrum slope depends on which fluid dominates when the mode leaves the Hubble radius, but also because the spectrum amplitude depends on how long the mode evolves in each fluid-like epoch.§ PRIMORDIAL PERTURBATIONThe perturbed Einstein's equationscan be recast in a very simple and objective manner combining the scalar perturbation in the metric and in the matter component by means of the gauge invariant curvature perturbation ζ. The action that gives the dynamic for the perturbations comes from the first-order perturbed Einstein-Hilbert action and reads <cit.>S = ∫τx^3 z^2/2(ζ^'2 + N^2ζΔζ/a^2), z^2 ≡3a^3x^2/κ^2N,where Δ is the spacial Laplacian, ζ is the gauge invariant dimensionless curvature perturbation, and the time operator is defined by '≡/τ.[Note that, our definition of z is different from the one in Ref. <cit.>, z^2 = 1/(Nκ^2z̅^2), where z̅ denotes the z appearing in Ref. <cit.>.] The equation of motion for ζ is obtained by the variational principle, and after the field decomposition it readsζ^''_k + 2z^'/zζ_k^' + N^2k^2/R_H^2a^2ζ_k = 0.In the above equations, we used the dimensionless time τ [Eq. (<ref>)],N =t/τ = τ/H.Here k is measured in units of R_H^-1,[The eigenvalues of the Laplacian Δ are given by -R_H^-2k^2.] z and ζ_k have dimensions of lenght^-3/2 and lenght^3/2 respectively.We are in the domain of linear perturbation theory, in which the scalar, vector and tensor perturbations decouple. The tensor perturbation h_ij, whose amplitude of any of its two polarizations will be refereed by h, presents a similar action, i.e.,S = ∫τx^3 z_h^2/2(h^'2 + N^2 h Δ h/a^2), z_h^2 ≡a^3/4κ^2N.Its equation of motion can be easily deduced from the action above,h_k^'' + 2z_h^'/z_hh_k^' + N^2k^2/R_H^2a^2h_k = 0,where h_k also has dimension of length^3/2. The above formulation can be found in the literature, for instance in <cit.>. Here we introduced only the necessary information in order to define the observational probes that we will calculate. The quantities constrained by the observations are the power spectra,Δ_ζ_k ≡k^3 |ζ_k|^2/R_H^3 2 π^2 = ^2/R_H^24k^3 |_k|^2/3π, Δ_h_k ≡k^3| h_k|^2/R_H^3 2π^2 = ^2/R_H^216k^3|_k|^2/π,where we introduced the dimensionless mode functionsζ_k ≡√(κ^2R_H/3)_k, h_k ≡√(4κ^2 R_H)_k.The spectral indexes for the scalar curvature perturbation ζ and tensor perturbation h labeled by the subscripts s and T are, respectively,n_s, T - 1 ≡.log (Δ_ζ_k, h_k)/log k|_k = k_*and thetensor-to-scalar ratior ≡ 2 .Δ_h_k/Δ_ζ|_k = k_*,where the factor 2 comes to account for the two polarizations of the tensor perturbation. We use the same pivotal scale as in <cit.>, k_* = 0.05R_H Mpc^-1. The latest Planck release estimates for long wave-lengths Δ_ζ_k≈ 10^-10, n_s≈ 0.96 and r < 0.1 <cit.>.Following the results from <cit.>, the spectral index for the modes entering the horizon during the domination of a fluid with EoS w isn_s = 1 + 12w/1 + 3 w.In case , the spectrum is scale invariant when entering the horizon while w = 0, matter epoch. Depending on the duration of the matter contraction, some very small scale modes may enter during the transition to the stiff-matter phase, w = 1, leading to a blue spectrum. For case , besides the above two possibilities, there is also the influence of the transient DE epoch in large scale modes. Hence, although case is more academic, it should be quite interesting to evaluate the exact spectral index in this case in order to estimate the impact of the presence a transient DE in the contracting phase of a matter bounce in the standard results. §.§ Initial vacuum perturbations It is usually proposed in current cosmological models that the inhomogeneities in the Universe have their origin in primordial vacuum quantum fluctuations. In the inflationary scenario, the exponential growth of the scale factor are responsible for amplifying those quantum fluctuations. After a 60 e-fold expansion they have enough amplitude to fit the CMB observations.Bouncing models assume the same mechanism for the origin of inhomogeneities, but replaced in the far past of the contracting phase. Some scenarios may find difficulty in providing the Minkowsky vacuum as initial conditions. This is the case when the cosmological constant is considered <cit.>.In the present case, the scalar field behaves like a matter fluid and the usual quantization of the adiabatic vacuum fluctuations in a Minkowsky space-time coincides with the WKB solution with positive energy for the Mukhanov-Sasaki variable v ≡ zζ. The equation of motion (<ref>) can be writtenv_k^'' + w_k^2 v_k = 0,where,w_k^2(η, k) ≡N^2k^2/a^2R_H^2 - z^''/z.Note that the expressions above reduce to the usual conformal time equations for N = a.A solution of the above equation can be expressed in terms of the WKB approximation (see for example <cit.>), which has a certain limit of validity. Let us define,Q_WKB = 3/4(w_k^'/w_k)^2- 1/2w_k^ ''/w_k.In the regime in which|Q_WKB/w_k^2|≪ 1,the solution isṽ_k^WKB≈1/√(2 w_k)^± i ∫τ w_k.The matter contraction satisfies(<ref>) and Eq. (<ref>) not only gives the initial conditions but also a good approximation for the solution of Eq. (<ref>) while condition (<ref>) is satisfied.For N^2k^2/(a^2R_H^2) ≫| z^''/z| the initial vacuum conditions are reduced tov_ini = 1/√(2k)√(aR_H/N),.v/τ|_ini = i √(2 k)√(N/aR_H),where we have set the initial phase factor equal to zero. Again, we can recover the usual vacuum conditions choosing the conformal time lapse function N = a. The tensor modes h can be expressed in terms of the variableμ = z_h h,which satisfies similar equations as v, but with z_h. The same treatment given to the quantization of v can be performed for μ and the initial conditions are equivalent for the tensor modes.In the adiabatic limit, the perturbations are in a high oscillatory regime and the numerical calculations become contrived. A very common approach to numerically solve the perturbations is to consider the WKB solution until N^2k^2/(a^2R_H^2) > | z^''/z| and to switch to the numerical calculation just before the saturation of the inequality. The problem with this approach is that, by construction, the WKB approximation is worse and worse as we approximate the saturation point. Thus, we must balance two problems, first if we start the numerical evolution very close to the saturation we would need a high order WKB approximation to get a reasonable initial condition. The high order WKB approximation needs high order time derivatives of the background functions, which in turn is badly defined numerically or involves complicated background functions also numerically error prone. If we decide to start away from the saturation point we need to deal with the highly oscillatory period, in which the processing time is longer and the numerical errors accumulate with the oscillations.[One should also note that, the usual approximation, k^2 = V, used to calculate the power-spectrum, underestimates its amplitude <cit.>.]To circumvent the above mentioned problems, we will use the action angle variables to rewrite the perturbations dynamics and numerically solve the new sets of equations. These variables are better suited to the high oscillatory regime and allows us to use initial conditions away from the saturation point. §.§ Action angle variables The Action Angle (AA) variables can be used to solve high oscillatory differential equations <cit.>. General linear oscillatory systems have a quadratic Hamiltonian in the formℋ = Π__k^2 /2 m +m ν^2/2_k^2,where m is the associated “mass” of the system and ν the frequency. The generalized variable and associated momenta are _k and Π__k, respectively.From action (<ref>) and the definition of the dimensionless variables (<ref>), we can easily deduce thatm = κ^2R_Hz^2/3 = a^3x^2R_H/N,ν = N k/a R_H.The Hamilton equations areζ^'_k = Π_ζ_k/m, Π^' = - m ν^2 ζ_k.AA variables are based on the adiabatic invariant of oscillatory systems <cit.>. A real solution for the Hamilton equations above can be rewritten in terms of the variables (I, θ), implicit defined by^a_k= √(2 I/m ν)sin( θ),Π^a__k = √(2 I m ν)cos( θ). Deriving the above expressions and using the Hamilton equations, we find the equation of motion in terms of the new variables θ and I are, respectively, I^' = - I (m ν)^'/m νcos( 2 θ), θ^' = ν + 1/2(m ν)^'/m νsin(2 θ). The second real solution ζ^b introduces another pair of AA variables,(J, ψ), again implicit defined by ^b_k= √(2 J/m ν)sin( ψ),Π__k^b = √(2 J m ν)cos( ψ). They follow the same equations of motion J^' = - J (m ν)^'/m νcos( 2 ψ) , ψ^' = ν + 1/2(m ν)^'/m νsin(2 ψ). The final complex solution is defined by_k = _k^a +i_k^b/2i,with the real and imaginary parts satisfying the normalization condition imposed by the initial quantum vacuum perturbations, i.e.,√(I J)sin( ψ - θ) = 1We will define the variables ϵ, θ̅ and Δθ in order to re-write the above equations so the constraint will be automatically satisfied all along the evolution. Definingsinh( ϵ)= ( Δθ),Δθ = ψ - θ,θ̅ = ψ + θ/2,we can easily demonstrate the relations√(I J) = cosh( ϵ),sin( Δθ)= 1/cosh( ϵ),cos( Δθ)= tanh( ϵ).Differentiating Eqs. (<ref>) and (<ref>) and using Eqs. (<ref>) and(<ref>) to rewrite ψ and θ in terms of ϵ and θ̅ we findθ̅^' = ν + (m ν)^'/m νtanh( ϵ) sin(θ̅)cos(θ̅), ϵ^' = -(m ν)^'/mνcos(2θ̅).The system is not yet fully described, since we have only the dynamics for a composition of I and J through ϵ. Introducing γ by the relation^γ = √(I/J),we easily obtainγ^' = - 2(m ν)^'/m νsin(θ̅)cos(θ̅)/cosh( ϵ),and I and J can be recovered usingI=^γcosh( ϵ),J=^- γcosh( ϵ) .Finally the complete set of equations that replaces Eqs. (<ref>), (<ref>), (<ref>), (<ref>) already accounting the constraint (<ref>) isθ̅^' = ν + (m ν)^'/m νtanh(ϵ) sin( θ̅)cos( θ̅) ,ϵ^' = -(m ν)^'/m νcos(2 θ̅),γ^' = - 2(m ν)^'/m νsin(θ̅)cos(θ̅)/cosh( ϵ).Using these new variables, the two real linearly independent solutions can be recast as_k^a = ^γ/2/√(m ν)[ ^ϵ/2sin(θ̅) - ^-ϵ/2cos(θ̅)],_k^b = ^-γ/2/√(m ν)[ ^ϵ/2sin(θ̅) + ^-ϵ/2cos(θ̅)]. Now we have to rewrite the adiabatic vacuum initial conditions, for that purpose we use the adiabatic limit (mν)^'/mν→ 0. Expanding in the leading order in (mν)^'/mν the set of equations providesϵ ≈ϵ_0,γ ≈γ_0,θ̅ ≈θ̅_0 + kη.Using the above approximations to calculate the complex _k, we have, as an initial condition, that the following choice recovers the leading order WKB approximation, Eqs. (<ref>) and (<ref>):ϵ_0 = γ_0 = 0,which, naturally, satisfies the Eq. (<ref>). The real solutions using this choice are_k^a = √(2/m ν)sin(θ̅-π/4),_k^b = √(2/m ν)cos(θ̅-π/4), then, consequently, the complex solution is _k = ^-i(θ̅-π/4)/√(2m ν).Because it is just a phase, we can choose θ̅_0 = π/4. The same treatment follows through in the case of the dimensionless tensor perturbation, with the only difference being the “mass” definitionm_h = 4κ^2R_Hz_h^2 = a^3R_H/N,ν_h = N k/a R_H. A general purpose integrator for systems described by an action in the form (<ref>) was implemented as part of the(numerical cosmology library) <cit.>. The abstract implementation of the harmonic oscillator action angle is provided by the object <cit.>. The adiabatic and tensor mode objects, respectivelyand , are built on top of , connecting it to the exponential potential background model . We should stress that these codes can be used to any background model, all one needs to do is to implement the interface which calculates the mass m and frequency ν of the model.§ NUMERICAL SOLUTIONS FOR THE PERTURBATIONSUsing the AA variables, we can calculate the power spectra for both scalar and tensor perturbations. In this section, we will present these solutions, and discuss how the basic parameters of the model influence the primordial perturbations. We will focus in case , which is a complete background that addresses the problem of DE in bounce models by means of a single scalar field, in this analysis we keep fixed both Ω_Λ = 1 and Ω_d = 1. Here we analyze in detail four parameter sets defined in Tab. <ref>.Before moving to the solutions themselves, it is worth to take a look at the differences between the scalar mode and the tensor mode evolution. The super Hubble solutions for Eqs. (<ref>) are, at leading order, _k= A^1_k[1 + 𝒪(ν^2)] + A^2_k[ ∫τ/m + 𝒪(ν^2)],_k= B^1_k[1 + 𝒪(ν_h^2)] + B^2_k[ ∫τ/m_h + 𝒪(ν_h^2)], where changing the integration limits in the integrals above result in a redefinition of A^1_k and B^1_k. Using the expression for the masses in Eqs. (<ref>) and (<ref>), we recast the integral as ∫τ/m = 1/R_H∫τN/x^2a^3 = 1/R_H∫t/x^2a^3,∫τ/m_h = 1/R_H∫τN/a^3 = 1/R_H∫t/a^3. In the classical contracting branch of case , x varies between (1/√(2),1), while the scale factor goes through a large contraction. In other words, the value of this integral will be dominated by the values of a near the bounce phase, where a attains its smallest value (see <cit.> for a detailed analysis of this integral). Nonetheless, when the quantum phase begins, the value of x is no longer restricted to (1/√(2),1). For example, in Fig. <ref>, we show three time evolutions for 1/x^2 using four different sets of parameters.During the matter phase 1 / x^2 ≈ 2, and during the stiff matter domination, 1/x^2 ≈ 1. Therefore, in the classical phase, the presence of 1/x^2 in the integral (<ref>) increases its value by a maximum factor of two. On the other hand, throughout the quantum phase, different parameter sets result in quite different behaviors, as can be seen in Fig. <ref>. The set1 curve shows that the presence of 1/x^2 in the aforementioned integral will result in a sharp increase in the spectrum amplitude around |α-|≈ 10^-1. This effect takes place slightly closer to the bounce in set2. Furthermore, in set3 we show a solution where the peaks are negligible, and hence there is no further increase of the perturbation amplitudes around the bounce. The phase space plot in Fig. <ref> elucidates what is happening. The set1 and set2 configurations are such that the system passes closes to the cyclic solutions (see Fig. <ref>), and the curve is near vertical, α changes abruptly with ϕ, making x close to zero during this interval. The scalar field shortly behaves as a DE fluid, implying a momentarily large deceleration (acceleration), which enhance the scalar perturbations. We also added the set3 and set4 to show solutions which pass far from these cyclic solutions. In this case, the evolution of α with respect to ϕ is smoother, resulting in bigger values of x.On the other hand, the tensor mode amplitudes do not depend on x, see Eq. (<ref>). In Fig. <ref>, we present the evolution of both integrands of Eqs. (<ref>) and (<ref>). Note that the first peak to the left in both figures is just the usual increase in amplitude related to the contraction, while during the bounce itself we have two different behaviors depicted. The tensor modes, which depend only on the lapse function N and the scale factor, are sensitive to the peaks of N at the bounce, whereas for scalar modes, the presence of the 1/x^2 term in the integrand overcomes the N dependence around the bounce. Here we would like to emphasize that the exact dynamics controlling the bounce is extremely important to determine the amplitude of the spectrum. Integrating only the classical phase, the first left peak in both integrands would provide similar amplitudes for both scalar and tensor modes, implying in a larger than one tensor-to-scalar ratio (see, for example, Ref. <cit.>). However, one cannot stop at the classical phase, since the bounce evolution will left a definite imprint on the amplitudes: an increase in the tensor mode amplitude at the bounce, and two amplifications of the scalar modes at two symmetric points around the bounce. Hence, any new physics around the bounce producing this kind of effect can be physically relevant, and its consequences must be evaluated with care.The contracting phase with a matter era puts this model in the category of the matter bounce scenarios. Previous works on the field obtained the bounce by means of a second scalar field with a ghost-type Lagrangian. Choosing wisely the parameters of the ghost scalar field, the bounce takes place only very close to the singularity, and the perturbations are studied, sometimes, without taking it into account. The results obtained in the literature about matter bounces can be summarized as follows: the spectrum is scale invariant; the tensor-to-scalar ratio is usually larger then measured in CMB if the scalar field is canonical and the bounce is symmetric; attempts to solve this problem assuming the validity of GR all along results in the increase of non-Gaussianities, which seems to suggest a no-go theorem for bounce cosmologies <cit.>. Our results point to a new direction: bounces which are out of the scope of GR can lead to new ways to decrease r, and they should be investigated with care. In our model, the decrease of the tensor-to-scalar ratio relies on the actual bounce dynamics in a non-trivial way, due to quantum effects. The increase in amplitude of the scalar and tensor modes take place at different times and are controlled by distinct parameters. The consequences of that for non-Gaussianities is something yet to be investigated.Concerning the amplitude growth in the classical regime, a known result is that they grow more substantially during a matter epoch then during the stiff matter one. This can be seen by looking at the super Hubble approximations in Eqs. (<ref>) and (<ref>). During the matter domination, N/a^3 ≈τ / a^3/2, while for stiff matter N/a^3 ≈τ, where we are using that N = τ/H, and H ∝ a^-3/2 in the matter phase and H ∝ a^-3 at the stiff phase. Since this part of the amplitude growth is determined by the matter epoch duration, it is closely connected with the parameters d and , which also control the bounce depth.We emphasize that the parameter choices are implicit determinations of the background model initial conditions (including the wave function parameters), as already mentioned. The results for the power spectra at the pivotal mode k_* are:set1: .Δ_ζ_k|_k=k_* = 1.4 × 10^-10,r = 1.9 × 10^-7, set2: .Δ_ζ_k|_k=k_* = 4.6 × 10^-11,r = 1.3 × 10^-5, set3: .Δ_ζ_k|_k=k_* = 1.2 × 10^-14,r = 56, set4: .Δ_ζ_k|_k=k_* = 1.7 × 10^-14,r = 59.The time evolution for this same pivotal mode is shown in Fig. <ref>. Observe that, for set1 and set2, the extra enhancement of the scalar amplitude due to the quantum effects takes it to a value close to the observed one .Δ_ζ_k|_k=k_*≈ 10^-10. On the contrary, the power spectra obtained from set3 and set4 have an amplitude smaller than the required by observations, even though their bounces are deeper (≈ 10^36 for set1 and set2 and ≈ 10^37 for set3 and set4). In principle, one could choose the parameters in order to make the bounce deeper, hoping to get the right amplitude. Nevertheless, we must take care to not go beyond the scale of validity of these models. One should verify whether the energy scale of the bounce is not dangerously close to the Planck energy scale, where our simple approach would not be appropriate. The curvature scale at the bounce is given by the Ricci scalar,R = 12 H^2 + 6Ḣ,L_R = 1/√(R),and Ricci scale L_R should not be smaller than the Planck length. Figure <ref> displays the Ricci scale evolution for all parameter sets. This figure shows that the absolute value of d controls the minimum scale L_R attained around the bounce. This means that we could not increase the amplitudes of set3 and set4 by increasing | d| without violating the validity of our approach.In what concerns the spectral index, our modes of interest cross the potential during the matter domination phase. As such, their spectrum are very close to scale invariant. Again, this can be changed using a slight negative value for the matter phase EoS.§ CONCLUSIONWe have studied the evolution of cosmological perturbations of quantum mechanical origin in a nonsingular cosmological model containing a single scalar field with exponential potential. The bounce is driven by quantum corrections of gravity in high-energy scales, but still smaller than the Planck energy scale, in which the canonical quantization of gravity may be applied.In Secs. <ref>, <ref> and <ref> we presented two possibilities for the homogeneous and isotropic background dynamics: case , which contains a DE epoch only in the contracting phase, and case , in which the DE epoch happens only in the expanding phase. In both cases, the scalar field behaves like a dust fluid in the asymptotic past and future. The free parameters at our disposal were mapped to quantities with physical significance, as the duration of the matter contraction, the energy scales of the DE epoch and of the bounce phase, and their selection was equivalent to choose initial conditions for the numerical integration, hence allowing a better physical control of the whole scenario.We restricted our attention mainly to case , which is a matter bounce model with a DE epoch in the expanding phase, offering a complete background solution in which the DE epoch arises naturally in the expanding phase by means of the same scalar field that drives the bounce and the matter contraction. This is a bouncing model with DE where its presence does not cause any trouble in defining adiabatic vacuum initial conditions in the far past of the contracting phase (as described in Ref. <cit.>), because it is dominated by dust. We numerically calculated the evolution of cosmological perturbations of quantum mechanical origin in such backgrounds. Scalar and tensor perturbations result to be almost scale invariant, and the parameters of the background can be adjusted to yield the good amplitudes for scalar and tensor perturbations, producing r < 0.1. We have also seen that it was in the quantum bounce that the scalar perturbations were amplified with respect to tensor perturbations. Hence, the same quantum effects which produces the background bounce do also induce the property r < 0.1. Our result shows that when GR is violated around the bounce, the influence of this phase on the evolution of cosmological perturbations can be nontrivial, and must be evaluated with care. Since we have only one scalar field, the perturbations were solved numerically for the whole background history, without approximations and matching conditions. Using the AA variables, we were able to construct a robust code where the details of the bounce effects on the perturbations could be appreciated for different background features. We have also found that longer the dust contraction, bigger are the amplitudes, a fact that was not noticed in other investigations.We have thus obtained a bouncing model with a single canonical scalar field which produces the observed features of cosmological perturbations at linear order. Usually, canonical scalar fields in the framework of GR produces r ≥ 1 for symmetric bounces <cit.>. We should emphasize that, in our model, we get r < 1 not because it is asymmetric, but because of violations of GR around the bounce due to quantum effects. The next step should be to evaluate non-Gaussianities in this model. Note that, as long as GR is not satisfied around the bounce, arguments based on the full validity of GR even at the bounce suggesting that non-Gaussianities should be huge in single scalar field bouncing models do not apply <cit.>. The evaluation of the non-Gaussianities must be made with care through the quantum bounce, as long as GR is not valid there. A consistent framework must be developed in order to perform this calculation correctly. This is one of our future investigations.Note that this is a very simple model, with a single scalar field, but it is astonishing that it can produce alone the right amount of cosmological perturbations, and also yield a future DE phase. Hence, it is quite reasonable to pursue this route and try to complete the model in order to obtain more accurate scenarios for the real Universe. One possibility is to perform the same calculations when other fluids are present, as the classical extension of this model presented in Ref. <cit.>. This is a much more involved calculation, where entropy perturbations must be considered.In a future work, we will also study more deeply case , which is a very interesting theoretical laboratory to investigate more precisely the influence of DE in a contracting phase. In that scenario, the cosmological perturbations evolve in a situation where usual adiabatic vacuum initial conditions can be posed normally, since DE behavior is transient and the background is matter dominated in the far past.Finally, let us emphasize that the present model can accommodate non negligible primordial gravitational waves which might be detected in the near future. Hence, these models are observationally testable.APB and NPN would like to thank CNPq of Brazil for financial support. SDPV acknowledges the financial support from a PCI postdoctoral fellowship from Centro Brasileiro de Pesquisas Físicas of Brazil, and from BELSPO non-EU postdoctoral fellowship.apsrev | http://arxiv.org/abs/1706.08830v1 | {
"authors": [
"Anna Paula Bacalhau",
"Nelson Pinto-Neto",
"Sandro Dias Pinto Vitenti"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170627131325",
"title": "Consistent Scalar and Tensor Perturbation Power Spectra in Single Fluid Matter Bounce with Dark Energy Era"
} |
A Cryptographic Approach for Steganography Jacques M. Bahi, Christophe Guyeux, and Pierre-Cyrille Heam*FEMTO-ST Institute, UMR 6174 CNRSComputer Science Laboratory DISCUniversity of Franche-Comté, France{jacques.bahi, christophe.guyeux, pierre-cyrille.heam}@femto-st.fr *Authors are cited in alphabetic orderDecember 30, 2023 ============================================================================================================================================================================================================================================================================================We consider an optimal control on networks in the spirit of the works of Achdou et al. (2013) and Imbert et al. (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. (2014) and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis (2016). 34H05, 35F21, 49L25, 49J15, 49L20, 93C30 § INTRODUCTION A network (or a graph) is a set of items, referred to as vertices or nodes, which are connected by edges (see Figure <ref> for example). Recently, several research projects have been devoted to dynamical systems and differential equations on networks,in general or more particularlyin connection with problems of data transmission or traffic management (see for example Garavello and Piccoli <cit.> and Engel et al <cit.>). An optimal control problem is an optimization problem where an agent tries to minimize a cost which depends on the solution of a controlled ordinary differential equation (ODE). The ODE is controlled in the sense that it depends on a function called the control. The goal is to find the best control in order to minimize the given cost. In many situations, the optimal valueof the problem as a function of the initial state (and possibly of the initial time when the horizon of the problem is finite) is a viscosity solution of a Hamilton-Jacobi-Bellman partial differential equation(HJB equation). Under appropriate conditions, the HJB equation has a unique viscosity solution characterizing by this way the value function. Moreover, the optimal control may be recovered from the solution of the HJB equation, at least if the latter is smooth enough. The first articlesabout optimal control problems in which the set of admissible statesis a network (therefore the state variable is a continuous one) appeared in 2012:in <cit.>, Achdou et al. derived theHJB equation associated to an infinite horizonoptimal control on a network and proposed a suitable notion of viscosity solution. Obviously, the main difficulties arise at the vertices where the network does not have a regular differential structure. As a result, the new admissible test-functions whose restriction to each edge is C^1 are applied.Independently and at the same time, Imbert et al. <cit.> proposed an equivalent notion of viscosity solution for studying a Hamilton-Jacobi approach to junction problems and traffic flows. Both <cit.> and <cit.> contain first resultson comparison principles which were improved later. It is also worth mentioning the work by Schieborn and Camilli <cit.>, in which the authors focus on eikonal equations on networks and on a less general notion of viscosity solution. In the particular case of eikonal equations,Camilli and Marchi established in <cit.> the equivalence betweenthe definitions given in <cit.>. Since 2012, several proofs of comparison principles for HJB equations on networks, giving uniqueness of the solution, have beenproposed. * In <cit.>, Achdou et al. give a proof of a comparison principle fora stationary HJB equation arising from an optimal control with infinite horizon, (therefore the Hamiltonian is convex) by mixing arguments from the theory of optimal control and PDE techniques. Such a proof was much inspired by works of Barles et al. <cit.>, onregionaloptimal control problems in ^d, (with discontinuous dynamics and costs).* A different and more general proof, using only arguments from the theory of PDEs was obtained by Imbert and Monneau in <cit.>. The proof works for quasi-convex Hamiltonians, and for stationary and time-dependentHJB equations. It relies on the construction of suitable vertex test functions. * A very simple and elegant proof, working for non convex Hamiltonians, has been very recently givenbyLions and Souganidis <cit.>.The goal of thispaper is to consider an optimal control problem on a network in which there are entry (or exit)costs at each edge of the network and to study the related HJB equations. The effect of the entry/exit costs is to make the value function of the problemdiscontinuous.Discontinuous solutions of Hamilton-Jacobi equation have been studied by various authors, see for example Barles <cit.>, Frankowska and Mazzola <cit.>, and in particular Graber et al. <cit.> for different HJB equations on networks with discontinuous solutions.To simplify the problem, we will first study the case of junction, i.e., a networkof the form 𝒢= ∪_i=1^N Γ_i withN edges Γ_i (Γ_i is theclosed half line ^+ e_i) andonly one vertex O, where {O}= ∩_i=1^N Γ_i.Later, we willgeneralize our analysis tonetworks with an arbitrary number of vertices.In the case of thejunction described above, our assumptions about the dynamics and the running costsare similar to those made in <cit.>, except that additional costs c_i for enteringthe edge Γ_i at O ord_i for exiting Γ_i at O are addedin the cost functional. Accordingly, the value function is continuous on 𝒢, but is in general discontinuous at the vertex O. Hence, instead of considering the value function 𝗏, we split it into thecollection (v_i)_1≤ i≤ N, where v_i iscontinuous function defined on the edgeΓ_i. More precisely,v_i(x)=𝗏(x)if x∈Γ_i\{ O} , lim_δ→ 0 ^+𝗏(δ e_i)if x=O.Our approach is therefore reminiscent of optimal switching problems (impulsional control):in the present case the switches can only occur at the vertex O. Note that our assumptions will ensure that𝗏|_Γ_i∖{O} is Lipschitz continuous near O and that lim_δ→ 0 ^+𝗏(δ e_i) does exist.In the case of entry costs for example, our first main result will be to findthe relation between𝗏(O), v_i(O) and v_j(O)+c_j for i,j=1,N.This will show that the functions (v_i)_1≤ i≤ N are (suitably defined) viscosity solutionsof the following system[ λ u_i(x)+H_i(x,d u_id x_i(x))=0; λ u_i(O)+max{ -λmin_j i{ u_j(O)+c_j} ,H_i^+(O,d u_idx_i(O)),H_O^T} =0 . ]Here H_i is the Hamiltonian corresponding to edge Γ_i. At vertex O,the definition of the Hamiltonian has to be particular, in order to consider all the possibilities when x is close to O. More specifically, if x is close to O and belongs to Γ_i then:* The term min_j i{ u_j(O)+c_j} accounts for situations in which the trajectory enters Γ_i_0 where u_i_0(O)+c_i_0=min_j i{ u_j(O)+c_j}.* The term H_i^+(O,d u_idx_i(O)) accounts for situations in which the trajectory does not leave Γ_i.* The term H_O^T accounts forsituations in whichthe trajectory stays at O. The most important part of the paper will be devoted to two different proofs of a comparison principle leading to the well-poseness of (<ref>): the first one uses arguments from optimal control theory coming from Barles et al. <cit.> andAchdou et al. <cit.>; the second one is inspired by Lions and Souganidis <cit.> and uses arguments from thetheory of PDEs.The paper is organized as follows: Section <ref> deals with the optimal control problems with entry and exit costs: we give a simple example in which the value function is discontinuous at the vertex O, and also prove results on the structure of the value function near O.In Section <ref>, the new system of (<ref>) is definedand a suitable notion ofviscosity solutions is proposed. In Section <ref>,we prove our value functions are viscosity solutions of the above mentioned system.In Section <ref>, some properties of viscosity sub and super-solution are given andused to obtain the comparison principle. Finally, optimal control problems withentry costs which may be zero and related HJB equationsare consideredin Section <ref>. § OPTIMAL CONTROL PROBLEM ON JUNCTION WITH ENTRY/EXIT COSTS§.§ The geometry We consider the model case of the junction in ℝ^d with N semi-infinite straight edges, N>1. The edges are denoted by (Γ_i)_i=1,N where Γ_i is the closed half-line ℝ^+e_i. The vectors e_i are two by two distinct unit vectors in ℝ^d.The half-lines Γ_i are glued at the vertex O to form the junction 𝒢 𝒢=⋃_i=1^NΓ_i.The geodetic distance d(x,y) between two points x,y of 𝒢 isd(x,y)=|x-y|x,yΓ_i, |x|+|y|x,yΓ_iΓ_j. §.§ The optimal control problem We consider infinite horizon optimal control problems which have different dynamic and running costs for each and every edge. For i=1,N, * the set of control on Γ_i is denoted by A_i* the system is driven by a dynamics f_i* there is a running cost ℓ_i.Our main assumptions, referred to as [H] hereafter,are as follows: * [H0] (Control sets)Let A be a metric space (one can take A=ℝ^d. For i=1,N, A_i isa nonempty compact subset of A and the sets A_i are disjoint.* [H1] (Dynamics) For i=1,N, the function f_i:Γ_i× A_i→ℝ is continuous and bounded by M. Moreover, there exists L>0 such that |f_i(x,a)-f_i(y,a)|≤ L|x-y| x,y∈Γ_i, a∈ A_i.Hereafter, we will use the notation F_i(x) for the set { f_i(x,a)e_i:a∈ A_i}.* [H2] (Running costs)For i=1,N, the function ℓ_i:Γ_i× A_i→ℝ is a continuous function bounded by M>0. There exists a modulus of continuity ω such that|ℓ_i(x,a)-ℓ_i(y,a)|≤ω(|x-y|) x,y∈Γ_i,a∈ A_i. * [H3] (Convexity of dynamic and costs) For x∈Γ_i, the following set_i(x)={(f_i(x,a)e_i,ℓ_i(x,a)):a∈ A_i} is non-empty, closed and convex.* [H4] (Strong controllability) There exists a real number δ>0 such that[-δ e_i,δ e_i]⊂ F_i(O)={ f_i(O,a)e_i:a∈ A_i} . The assumption that thesets A_i are disjoint is not restrictive. Indeed, if A_i are not disjoint, then we define Ã_i=A_i×{ i} and f̃_i(x,ã)=f_i(x,a),ℓ̃_i(x,ã)=ℓ_i(x,a)with ã=(a,i) with a∈ A_i. The assumption [H3] is made to avoid the use of relaxed control. With assumption [H4], one gets that the Hamiltonian which will appear later is coercive for x close to the O. Moreover, [H4] is an important assumption to prove Lemma <ref> and Lemma <ref>. Letℳ={(x,a):x∈𝒢, a∈ A_ix∈Γ_i\{ O} ,a∈∪_i=1^NA_ix=O} .Then ℳ is closed. We also define the function on ℳ by(x,a)∈ℳ, f(x,a)= f_i(x,a)e_i x∈Γ_i\{ O}a∈ A_i,f_i(O,a)e_i x=Oa∈ A_i.The function f is continuous on ℳ since the sets A_i are disjoint. The set F̃(x) which contains all the “possible speeds” at x is defined byF̃(x)= F_i(x)x∈Γ_i\(O), ⋃_i=1^NF_i(O).For x∈𝒢, the set of admissible trajectories starting from x isY_x={ y_x∈ Lip(ℝ^+;𝒢):ẏ_x(t)∈F̃(y_x(t)) t>0 y_x(0) =x } . According to <cit.>, a solution y_x can be associated with several control laws. We introduce the set of admissible controlled trajectories starting from x𝒯_x={(y_x,α)∈ L_loc^∞(ℝ^+;ℳ):y_x∈ Lip(ℝ^+;𝒢)y_x(t)= x+∫_0^tf(y_x(s),α(s))ds} .Notice that, if (y_x,α)∈𝒯_x then y_x∈ Y_x. Hereafter, we will denote y_x by y_x,α if (y_x,α)∈𝒯_x. For any y_x,α, we can define the closed set T_O={ t∈ℝ^+:y_x,α(t)=O} and the open set T_iin ℝ^+=[0,+∞)by T_i={ t∈ℝ^+:y_x,α(t)∈Γ_i\{ O}}. The set T_i is a countable union of disjoint open intervals T_i=⋃_k∈ K_i⊂ℕT_ik=[0,η_i0)∪⋃_k∈ K_i⊂ℕ^⋆(t_ik,η_ik)x∈Γ_i\{ O} , ⋃_k∈ K_i⊂ℕ^⋆(t_ik,η_ik)x∉Γ_i\{ O} ,where K_i=1,n if the trajectory y_x,α enters Γ_i n timesand K_i=ℕ if the trajectory y_x,α enters Γ_i infinite times.From the above definition, one can see that t_ik is an entry time in Γ_i\{ O} and η_ik is an exit time from Γ_i\{ O} . Hencey_x,α(t_ik)=y_x,α(η_ik)=O. Let C={ c_1,c_2,…,c_N} be a set of entry costsand D={ d_1,d_2,…,d_N} be a set of exit costs.We underline that, except inSection <ref>, entry and exist costs are positive.In the sequel, we define two different cost functionals (the first one corresponds to the case when there is a cost for entering the edges and the second onecorresponds to the case when there is a cost for exiting the edges):The costs associated to trajectory (y_x,α,α)∈𝒯_x are defined byJ(x;(y_x,α,α))=∫_0^+∞ℓ(y_x,α(t),α(t))e^-λ tdt+∑_i=1^N∑_k∈ K_ic_ie^-λ t_ik,andJ(x;(y_x,α,α))=∫_0^+∞ℓ(y_x,α(t),α(t))e^-λ tdt+∑_i=1^N∑_k∈ K_id_ie^-λη_ik,where the running cost ℓ:ℳ→ℝ isℓ(x,a)=ℓ_i(x,a)a∈ A_i, ℓ_i(O,a)x=0a∈ A_i.Hereafter, to simplify the notation, we will useJ(x,α) and J(x,α) instead of J(x;(y_x,α,α)) and J(x;(y_x,α,α)), respectively.The value functions of the infinite horizon optimal control problemaredefined by:𝗏(x)=inf_(y_x,α,α)∈𝒯_xJ(x;(y_x,α,α)),and𝗏(x)=inf_(y_x,α,α)∈𝒯_xJ(x;(y_x,α,α)).By the definition of the value function, we are mainly interested in a control law α such that J(x,α)<+∞. In such acase, if |K_i|=+∞, then we can order { t_ik,η_ik:k∈ℕ} such thatt_i1<η_i1<t_i2<η_i2<…<t_ik<η_ik<…,andlim_k→∞t_ik=lim_k→∞η_ik=+∞.Indeed, assuming iflim_k→∞t_ik=t<+∞, thenJ(x,α)≥-Mλ+∑_k=1^+∞e^-λ t_ikc_i=-Mλ+c_i∑_k=1^+∞e^-λ t_ik=+∞,in contradiction with J(x,α)<+∞. This means that the state cannot switch edges infinitely many times in finite time, otherwise the cost functional is obviously infinite. The following example shows that the value function with entry costs is possibly discontinuous (The same holds for the value function with exit costs).Consider the network 𝒢=Γ_1∪Γ_2 where Γ_1=ℝ^+ e_1=(-∞,0] and Γ_2=ℝ^+e_2=[0,+∞). The control sets are A_i=[-1,1]×{ i} with i∈{1,2}. Set(f(x,a),ℓ(x,a))=(f_i(x,(a_i,i))e_i,ℓ_i(x,(a_i,i)))if x∈Γ_i\{ O} and a=(a_i,i)∈ A_i, (f_i(O,(a_i,i))e_i,ℓ_i(O,(a_i,i)))if x=O and a=(a_i,i)∈ A_i,where f_i(x,(a_i,i))=a_i and ℓ_1≡1,ℓ_2(x,(a_2,2))=1-a_2. For x∈Γ_2\{ O}, then𝗏(x)=v_2(x)=0 with optimal strategy consists in choosing α(t)≡(1,2). For x∈Γ_1, we can check that𝗏(x)=min{1λ,1-e^-λ|x|λ+c_2e^-λ|x|}. More precisely, for all x∈Γ_1, we have𝗏(x)=1λ if c_2≥1λ, with the optimal control, 1-e^-λ|x|λ+c_2e^-λ|x| if c_2<1λ, with the optimal controlSummarizing, we have the two following cases * If c_2≥1λ, then 𝗏(x)= 0x∈Γ_2\{ O} , 1λ x∈Γ_1.The graph of the value function with entry costs c_2≥1λ=1 is plotted in Figure <ref>.* If c_2<1λ, then 𝗏(x)= 0x∈Γ_2\{ O} , 1-e^-λ|x|λ+c_2e^-λ|x| x∈Γ_1.The graph of the value function with entry costs c_2=12<1=1λ is plotted in Figure <ref>. Under assumptions [H1] and [H4], there exist twopositive numbers r_0 andC such that for all x_1,x_2∈ B(O,r_0)∩𝒢, there exists (y_x_1,α_x_1,x_2,α_x_1,x_2)∈𝒯_x_1 and τ_x_1,x_2≤ Cd(x_1,x_2) such that y_x_1(τ_x_1,x_2)=x_2. This proof is classical. It is sufficient to consider the case when x_1 and x_2 belong to same edge Γ_i, since in the other cases, we will use O as a connecting point between x_1 and x_2. According to Assumption [H4], there exists a∈ A_i such that f_i(O,a)=δ. Additionally, by the Lipschitz continuity of f_i,|f_i(O,a)-f_i(x,a)|≤ L|x|,hence, if we choose r_0:=δ2L>0, then f_i(x,a)≥δ2 for all x∈ B(O,r_0)∩Γ_i. Let x_1,x_2 be in B(O,r_0)∩Γ_i with |x_1|<|x_2|: there exist a control law α and τ_x_1,x_2>0 such that α(t)=a if 0≤ t≤τ_x_1,x_2 and y_x_1,α(τ_x_1,x_2)=x_2. Moreover, since the velocity f_i(y_x_1,α(t),α(t)) is always greater than δ2 when t≤τ_x_1,x_2, then τ_x_1,x_2≤2δd(x_1,x_2). If |x_1|>|x_2|,the proof is achieved by replacing a∈ A_i by a∈ A_i such that f_i(O,a)=-δ and applying the same argument as above.§.§ Some properties of value function at the vertex Under assumption [H], 𝗏|_Γ_i\{ O} and 𝗏|_Γ_i\{ O} are continuous for any i=1,N. Moreover, there exists ε>0 such that 𝗏|_Γ_i\{ O} and 𝗏|_Γ_i\{ O} are Lipschitz continuous in (Γ_i\{ O})∩ B(O,ε). Hence, it is possible to extend 𝗏|_Γ_i\{ O} and 𝗏|_Γ_i\{ O} at O into Lipschitz continuous functions in Γ_i ∩ B(O,ε). Hereafter, v_i and v_i denote these extensions.The proof of continuity inside the edgeis classical by using [H4], see <cit.> for more details. The proof of Lipschitz continuity is a consequence of Lemma <ref>. Indeed, for x,y belong to Γ_i∩ B(0,ε), by Lemma <ref> and the definition of value function, we have𝗏(x)-𝗏(z) = v_i(x)-v_i(z)≤∫_0^τ_x,zℓ_i(y_x,α_x,z(t),α_x,z(t))e^-λ tdt+v_i(z)(e^-λτ_x,z-1).Since ℓ_i is bounded by M (by [H2]), v_i is bounded in Γ_i∩ B(O,ε) and e^-λτ_x,z-1 is bounded by τ_x,y, there exists a constant C such thatv_i(x)-v_i(z)≤Cτ_x,z≤CC|x-z|.The last inequality follows from the Lemma <ref>. The inequality v_i(z)-v_i(x)≤CC|x-z| is obtained in a similar way. The proof is done.Let us define the tangential Hamiltonian H_O^T at vertex O byH_O^T=max_i=1,Nmax_a_i∈ A_i^O{ -ℓ_j(O,a_j)}= -min_i=1,Nmin_a_i∈ A_i^O{ℓ_j(O,a_j)},where A_i^O={ a_i∈ A_i:f_i(O,a_i)=0} . The relationship between the values 𝗏(O), v_i(O) and H_O^T will be givenin the next theorem. Hereafter, the proofs of the results will be supplied only for the value function with entry costs𝗏, the proofs concerning the value function with exit costs 𝗏 are totally similar. Under assumption [H],the value functions 𝗏 and 𝗏 satisfy𝗏(O)=min{min_i=1,N{ v_i(O)+c_i} ,-H_O^Tλ} ,and𝗏(O)=min{min_i=1,N{v_i(O)} ,-H_O^Tλ} .Theorem <ref> gives us the characterization of the value function at vertex O. The proof of Theorem <ref>, makes use ofLemma <ref> and Lemma <ref> below. Under assumption [H], thenmax_i=1,N{ v_i(O)}≤𝗏(O)≤min_i=1,N{ v_i(O)+c_i} ,andmax_i=1,N{v_i(O)-d_i}≤𝗏(O)≤min_i=1,N{v_i(O)} .We divide the proof into two parts. * Prove that max_i=1,N{ v_i(O)}≤𝗏(O).First, we fix i∈{ 1,…,N} and any control law α such that (y_O,α̅,α̅)∈𝒯_O. Let x∈Γ_i\{ O} such that |x| is small. From Lemma <ref>, there exists a control law α_x,O connecting x and O and we considerα(s)=α_x,O(s)s≤τ_x,O, α̅(s-τ_x,O)s>τ_x,O.It means that the trajectory goes from x to O with the control law α_x,O and then proceeds with the control law α̅. Therefore𝗏(x) =v_i(x)≤ J(x,α)=∫_0^τ_x,Oℓ_i(y_x,α(s))e^-λ sds+e^-λτ_x,OJ(O,α̅).Since α is chosen arbitrarily and ℓ_i is bounded by M, we getv_i(x)≤ Mτ_x,O+e^-λτ_x,O𝗏(O).Let x tend to O then τ_x,O tend to 0 from Lemma <ref>. Therefore, v_i(O)≤𝗏(O). Since the above inequality holds for i=1,N, we obtain thatmax_i=1,N{ v_i(O)}≤𝗏(O). * Prove that 𝗏(O)≤min_i=1,N{ v_i(O)+c_i}. For i=1,N; we claim that 𝗏(O)≤ v_i(O)+c_i. Consider x∈Γ_i\{ O} with |x|small enough and any control law α̅_x such that (y_x,α̅_x,α̅_x)∈𝒯_x. From Lemma <ref>, there exists a control law α_O,x connecting O and x and we consider α(s)=α_O,x(s)s≤τ_O,x, α̅_x(s-τ_O,x)s>τ_O,x.It means that the trajectory goes from O to x using thecontrol law α_O,x then proceeds with the control law α̅_x. Therefore𝗏(O)≤ J(O,α)=c_i+∫_0^τ_O,xℓ_i(y_O,α(s))e^-λ sds+e^-λτ_O,xJ(x,α̅_x).Since α_x is chosen arbitrarily and ℓ_i is bounded by M, we get𝗏(O)≤c_i+Mτ_O,x+e^-λτ_O,xv_i(x)Let x tend to O then τ_O,x tends to 0 from Lemma <ref>, then 𝗏(O)≤ c_i+v_i(O). Since the above inequality holds for i=1,N, we obtain that𝗏(O)≤min_i=1,N{ v_i(O)+c_i} .The value functions 𝗏 and 𝗏 satisfy𝗏(O),𝗏(O) ≤-H_O^Tλwhere H_O^T is defined in (<ref>).From (<ref>), there exists j∈{ 1,…,N} and a_j∈ A_j^O such thatH_O^T=-min_i=1,Nmin_a_i∈ A_i^O{ℓ_i(O,a_i)} =-ℓ_j(O,a_j)Let the control law α be defined by α(s)≡ a_j for all s, then𝗏(O)≤ J(O,α)=∫_0^+∞ℓ_j(O,a_j)e^-λ sds=ℓ_j(O,a_j)λ=-H_O^Tλ. We are ready to prove Theorem <ref>.According to Lemma <ref> and Lemma <ref>,𝗏(O)≤min{min_i=1,N{ v_i(O)+c_i} ,-H_O^Tλ} .Assuming that𝗏(O)<min_i=1,N{ v_i(O)+c_i} ,it is sufficient to prove that𝗏(O)=-H_O^Tλ. By (<ref>), there exists a sequence {ε_n} _n∈ℕ such that ε_n→0 and 𝗏(O)+ε_n<min_i=1,N{ v_i(O)+c_i} n∈ℕ.On the other hand, there exists an ε_n-optimal control α_n, 𝗏(O)+ε_n>J(O,α_n). Let us define the first time that thetrajectory y_O,α_n leaves Ot_n:=inf_i=1,NT_i^n,where T_i^n is the set of times t for which y_O,α_n(t) belongs to Γ_i\{ O}. Notice that t_n is possibly +∞, in which case y_O,α_n(s)=O for all s∈[0,+∞).Extracting a subsequence if necessary, we may assume that t_n tends to t∈[0,+∞] when ε_n tends to 0. If there exists a subsequence of { t_n} _n∈ℕ (which is still noted{ t_n} _n∈ℕ) such that t_n=+∞ for all n∈ℕ,then for a.e. s∈[0,+∞)f(y_O,α_n(s),α_n(s)) =f(O,α_n(s))=0, ℓ(y_O,α_n(s),α_n(s)) =ℓ(O,α_n(s)).In this case, α_n(s)∈∪_i=1^NA_i^O for a.e. s∈[0,+∞). Therefore, for a.e. s∈[0,+∞)ℓ(y_O,α_n(s),α_n(s))=ℓ(O,α_n(s))≥-H_O^T,and𝗏(O)+ε_n>J(O,α_n)=∫_0^+∞ℓ(O,α_n(s))e^-λ sds≥∫_0^+∞(-H_O^T)e^-λ sds=-H_O^Tλ.By letting n tend to ∞, we get 𝗏(O)≥-H_O^Tλ. On the other hand, since 𝗏(O)≤-H_O^Tλ by Lemma <ref>, this implies that 𝗏(O)=-H_O^Tλ.Let us now assume that 0≤ t_n<+∞ for all n large enough. Then, for a fixed n and for any positive δ≤δ_n where δ_n small enough, y_O,α_n(s) still belongs to some Γ_i(n)\{ O} for all s∈(t_n,t_n+δ]. We have𝗏(O)+ε_n> J(O,α_n) =∫_0^t_nℓ(y_O,α_n(s),α_n(s))e^-λ sds+c_i(n)e^-λ t_n+∫_t_n^t_n+δℓ_i(n)(y_O,α_n(s),α_n(s))e^-λ sds+e^-λ(t_n+δ)J(y_O,α_n(t_n+δ),α_n(·+t_n+δ))≥ ∫_0^t_nℓ(y_O,α_n(s),α_n(s))e^-λ sds+c_i(n)e^-λ t_n+∫_t_n^t_n+δℓ_i(n)(y_O,α_n(s),α_n(s))e^-λ sds+e^-λ(t_n+δ)v(y_O,α_n(t_n+δ)) =∫_0^t_nℓ(y_O,α_n(s),α_n(s))e^-λ sds+c_i(n)e^-λ t_n+∫_t_n^t_n+δℓ_i(n)(y_O,α_n(s),α_n(s))e^-λ sds+e^-λ(t_n+δ)v_i(n)(y_O,α_n(t_n+δ)).By letting δ tend to 0, 𝗏(O)+ε_n≥∫_0^t_nℓ(y_O,α_n(s),α_n(s))e^-λ sds+c_i(n)e^-λ t_n+e^-λ t_nv_i(n)(O).Note that y_O,α_n(s)=O for all s∈[0,t_n], i.e., f(O,α_n(s))=0 a.e. s∈[0,t_n). Hence𝗏(O)+ε_n ≥ ∫_0^t_nℓ(O,α_n(s))e^-λ sds+c_i(n)e^-λ t_n+e^-λ t_nv_i(n)(O)≥ ∫_0^t_n(-H_O^T)e^-λ sds+c_i(n)e^-λ t_n+e^-λ t_nv_i(n)(O) =1-e^-λ t_nλ(-H_O^T)+c_i(n)e^-λ t_n+e^-λ t_nv_i(n)(O).Choose a subsequence {ε_n_k} _k∈ℕ of {ε_n} _n∈ℕ such thatfor some i_0∈{ 1,…,N}, c_i(n_k)=c_i_0 for all k. By letting k tend to ∞, recall that lim_k→∞t_n_k=t, we have three possible cases * If t=+∞, then 𝗏(O)≥-H_O^Tλ. By Lemma <ref>, we obtain 𝗏(O)=-H_O^Tλ.* If t=0, then 𝗏(O)≥ c_i_0+v_i_0(O). By (<ref>), we obtain acontradiction.* If t∈(0,+∞), then 𝗏(O)≥1-e^-λtλ(-H_O^T)+[c_i_0+v_i_0(O)]e^-λt. By (<ref>), c_i_0+v_i_0(O)>𝗏(O), so𝗏(O)>1-e^-λtλ(-H_O^T)+𝗏(O)e^-λt.This yields 𝗏(O)>-H_O^Tλ, and finally obtain a contradiction by Lemma <ref>.§ THE HAMILTON-JACOBI SYSTEMS. VISCOSITY SOLUTIONS§.§ Test-functions A function φ:Γ_1×…×Γ_N→ℝ^N is an admissible test-function if there exists(φ_i)_i=1,N, φ_i∈ C^1(Γ_i), such thatφ(x_1,…,x_N)=(φ_1(x_1),…,φ_N(x_N)). The set of admissible test-function is denoted by ℛ(𝒢). §.§ Definition of viscosity solution We define the Hamiltonian H_i:Γ_i×ℝ→ℝ byH_i(x,p)=max_a∈ A_i{ -pf_i(x,a)-ℓ_i(x,a)}and the Hamiltonian H_i^+ (O,·) : ℝ→ℝ byH_i^+(O,p)=max_a∈ A_i^+{ -pf_i(O,a)-ℓ_i(O,a)} ,where A_i^+={ a_i∈ A_i:f_i(O,a_i)≥0}. Recall thatthe tangential Hamiltonian at O, H_O^T, has been defined in (<ref>).We now introduce the Hamilton-Jacobi system for the case with entry costs[λ u_i(x)+H_i(x,d u_id x_i(x))=0 ; λ u_i(O)+max{ -λmin_j i{ u_j(O)+c_j} ,H_i^+(O,d u_id x_i(O)),H_O^T} =0]for all i=1,N and the Hamilton-Jacobi system with exit costs[λu_i(x)+H_i(x,du_id x_i(x))=0 ; λu_i(O)+max{ -λmin_j i{u_j(O)+d_i} ,H_i^+(O,du_idx_i(O)),H_O^T-λ d_i} =0]for all i=1,N and their viscosity solutions.∙ A function u:=(u_1,…,u_N) where u_i∈ USC(Γ_i;ℝ) for all i=1,N, is called a viscosity sub-solution of (<ref>) if for any (φ_1,…,φ_N)∈ℛ(𝒢), any i=1,N and any x_i∈Γ_i such that u_i-φ_i has a local maximum point on Γ_i at x_i, then[λ u_i(x_i)+H_i(x,dφ_id x_i(x_i))≤0; λ u_i(O)+max{ -λmin_j i{ u_j(O)+c_j} ,H_i^+(O,dφ_idx_i(O)),H_O^T}≤0 ] ∙ A function u:=(u_1,…,u_N) where u_i∈ LSC(Γ_i;ℝ) for alli=1,N, is called a viscosity super-solution of (<ref>) if for any (φ_1,…,φ_N)∈ℛ(𝒢), any i=1,N and any x_i∈Γ_i such that u_i-φ_i has a local minimum point on Γ_i at x_i, then[λ u_i(x_i)+H_i(x_i,dφ_id x_i(x_i))≥0; λ u_i(O)+max{ -λmin_j i{ u_j(O)+c_j} ,H_i^+(O,dφ_idx_i(O)),H_O^T}≥0 ] ∙ A functions u:=(u_1,…,u_N) where u_i∈ C(Γ_i;ℝ) for all i=1,N, is called a viscosity solution of (<ref>) if it is both a viscosity sub-solution and a viscosity super-solution of (<ref>).∙ A function u:=(u_1,…,u_N) where u_i∈ USC(Γ_i;ℝ) for all i=1,N, is called a viscosity sub-solution of (<ref>) if for any (ψ_1,…,ψ_N)∈ℛ(𝒢), any i=1,N and any y_i∈Γ_i such that u_i-ψ_i has a local maximum point on Γ_i at y_i, then[ λu_i(y_i)+H_i(y_i,dψ_id x_i(y_i))≤0; λu_i(O)+max{ -λmin_j i{u_j(O)} -λ d_i,H_i^+(O,dψ_idx_i(O)),H_O^T-λ d_i}≤0 ] ∙ A function u:=(u_1,…,u_N) where u_i∈ LSC(Γ_i;ℝ) for all i=1,N, is called a viscosity super-solution of (<ref>) if for any (ψ_1,…,ψ_N)∈ℛ(𝒢), any i=1,N and any y_i∈Γ_i such that u_i-ψ_i has a local minimum point on Γ_i at y_i, then[ λu_i(y_i)+H_i(y_i,dψ_id x_i(y_i))≥0; λu_i(O)+max{ -λmin_j i{u_j(O)} -λ d_i,H_i^+(O,dψ_idx_i(O)),H_O^T-λ d_i}≥0 ] ∙ A functions u:=(u_1,…,u_N) where u_i∈ C(Γ_i;ℝ) for all i=1,N, is called a viscosity solution of (<ref>) if it is both a viscosity sub-solution and a viscosity super-solution of (<ref>). This notion of viscosity solution is consitent with the one of <cit.>. It can be seen in Section <ref> when all the switching costs are zero, our definition and the one of <cit.> coincide. § CONNECTIONS BETWEEN THE VALUE FUNCTIONS AND THE HAMILTON-JACOBI SYSTEMS.Let 𝗏 be the value function of the optimal control problem with entry costs and 𝗏 be a value functionof the optimal control problem with exit costs. Recall that v_i,v_i:Γ_i→ℝ are defined in Lemma <ref> byv_i(x)=𝗏(x)x∈Γ_i\{ O} ,v_i(O)=lim_Γ_i\{ O}∋ x→ O𝗏(x), andv_i(x)=𝗏(x)x∈Γ_i\{ O} , v_i(O)=lim_Γ_i\{ O}∋ x→ O𝗏(x). We wish to prove that v:=(v_1,v_2,…,v_N) and v:=(v_1,…,v_N) are respectivelyviscosity solutions of (<ref>) and (<ref>). In fact, since 𝒢\{ O} is a finite union of open intervals in which the classical theory can be applied,we obtain that v_i and v_i are viscosity solutions of λ u(x)+H_i(x,Du(x))=0Γ_i\{ O} .Therefore, we can restrict ourselves to prove the following theorem.For i=1,N, the function v_i satisfiesλ v_i(O)+max{ -λmin_j i{ v_j(O)+c_j} ,H_i^+(O,d v_id x_i(O)),H_O^T} =0in the viscosity sense. The function v_i satisfiesλv_i(O)+max{ -λmin_j i{v_j(O)+d_i} ,H_i^+(O,dv_id x_i(O)),H_O^T-λ d_i} =0in the viscosity sense.The proof of Theorem <ref> follows from Lemmas <ref> and <ref> below. We focus on v_i since the proof for v_i is similar.For i=1,N, the function v_i is a viscosity sub-solution of (<ref>) at O. From Theorem <ref>,λ v_i(O)+max{ -λmin_j i{ v_j(O)+c_j} ,H_O^T}≤ 0.It is thus sufficient to prove that λ v_i(O)+H_i^+(O,d v_id x_i(O))≤ 0in the viscosity sense. Let a_i∈ A_i be such that f_i(O,a_i)>0. Setting α(t)≡ a_i then (y_x,α,α)∈𝒯_x for all x∈Γ_i. Moreover, for all x∈Γ_i\{ O}, y_x,α(t)∈Γ_i\{ O} (the trajectory cannot approach O since the speed pushes it away from O for y_x,α∈Γ_i∩ B(O,r)). Note that it is not sufficient to choose a_i∈ A_i such that f(O,a_i)=0 since it can lead to f(x,a_i)<0 for all x∈Γ_i\{ O}. Next, for τ>0 fixed and any x∈Γ_i, if we choose α_x(t)=α(t)=a_i0≤ t≤τ, â(t-τ) t≥τ, then y_x.α_x(t)∈Γ_i\{ O} for all t∈[0,τ]. It yields v_i(x)≤J(x,α_x)=∫_0^τℓ_i(y_x,α(s),a_i)e^-λ sds+e^-λτJ(y_x,α(τ),α). Since this holds for any α (α_x is arbitrary for t>τ), we deduce thatv_i(x)≤∫_0^τℓ_i(y_x,α_x(s),a_i)e^-λ sds+e^-λτv_i(y_x,α_x(τ)).Since f_i(·,a) is Lipschitz continuous by [H1], we also have for all t∈[0,τ],|y_x,α_x(t)-y_O,α_O(t)| =|x+∫_0^tf_i(y_x,α(s),a_i)e_ids-∫_0^tf_i(y_O,α(s),a_i)e_ids|≤ |x|+L∫_0^t|y_x,α(s)-y_O,α(s)|ds,where α_0 satisfies (<ref>) with x=O. According to Grönwall's inequality, |y_x,α_x(t)-y_O,α_O(t)|≤|x|e^Lt,for t∈[0,τ], yielding that y_x,α_x(t) tends to y_O,α_O(t) when x tends to O. Hence, from (<ref>), by letting x→ O, we obtainv_i(O)≤∫_0^τℓ_i(y_O,α_O(s),a_i)e^-λ sds+e^-λτv_i(y_O,α_O(τ)).Let φ be a function in C^1(Γ_i) such that 0=v_i(O)-φ(O)=max_Γ_i(v_i-φ). This yieldsφ(O)-φ(y_O,α_O(τ))τ≤1τ∫_0^τℓ_i(y_O,α_O(s),a_i)e^-λ sds+(e^-λτ-1)v_i(y_O,α_O(τ))τ.By letting τ tend to 0, we obtain that- f_i(O,a_i)dφd x_i(O)≤ℓ_i(O,a_i)-λ v_i(O).Hence,λ v_i(O)+sup_a∈ A_i:f_i(O,a)>0{ -f_i(O,a)d v_id x_i(O)-ℓ_i(O,a)}≤0in the viscosity sense. Finally, from Corollary <ref> in Appendix, we havesup_a∈ A_i:f_i(O,a)>0{ -f_i(O,a)dφ_id x_i(O)-ℓ_i(O,a)} =max_a∈ A_i:f_i(O,a)≥0{ -f_i(O,a)dφ_id x_i(O)-ℓ_i(O,a)} .The proof is complete.Ifv_i(O)<min{min_j i{ v_j(O)+c_j} ,-H_O^Tλ},then there exist τ̅>0,r>0 and ε_0>0 such that for any x∈(Γ_i\{ O})∩ B(O,r), any ε<ε_0 and any ε-optimal control law α_ε,x for x, y_x,α_ε,x(s)∈Γ_i\{ O} ,s ∈[0,τ̅].Roughly speaking, this lemma takes care of the case λ v_i+H_i^+(x,dv_idx_i(O))≤0, i.e., the situation when the trajectory does not leave Γ_i, see introduction. Suppose by contradiction that there exist sequences {ε_n},{τ_n}⊂ℝ^+ and { x_n}⊂Γ_i\{ O} such that ε_n↘0, x_n→ O,τ_n↘0and a control law α_n such that α_n is ε_n-optimal control law and y_x_n,α_n(τ_n)=O. This implies thatv_i(x_n)+ε_n>J(x_n,α_n)=∫_0^τ_nℓ(y_x_n,α_n(s),α_n(s))e^-λ sds+e^-λτ_nJ(O,α_n(·+τ_n)).Since ℓ is bounded by M by [H1], then v_i(x_n)+ε_n≥-τ_nM+e^-λτ_n𝗏(O). By letting n tend to ∞, we obtainv_i(O)≥𝗏(O).From (<ref>), it follows thatmin{min_j i{ v_j(O)+c_j} ,-H_O^Tλ} >𝗏(O).However, 𝗏(O)=min{min_j{ v_j(O)+c_j} ,-H_O^Tλ} by Theorem <ref>. Therefore, 𝗏(O)=v_i(O)+c_i>v_i(O), which is a contradiction with (<ref>).The function v_i is a viscosity super-solution of (<ref>) at O. We adapt the proof ofOudet <cit.> and start by assuming thatv_i(O)<min{min_j i{ v_j(O)+c_j} ,-H_O^Tλ}.We need to prove thatλ v_i(O)+H_i^+(O,d v_id x_i(O))≥0in the viscosity sense. Let φ∈ C^1(Γ_i) be such that0=v_i(O)-φ(O)≤ v_i(x)-φ(x) x∈Γ_i,and { x_ε}⊂Γ_i\{ O} be any sequence such that x_ε tends to O when ε tends to 0. From the dynamic programming principle and Lemma <ref>, there exists τ̅ such that for any ε>0, there exists (y_ε,α_ε):=(y_x_ε,α_ε,α_ε)∈𝒯_x_ε such that y_ε(τ)∈Γ_i\{ O} for any τ∈[0,τ̅] and v_i(x_ε)+ε≥∫_0^τℓ_i(y_ε(s),α_ε(s))e^-λ sds+e^-λτv_i(y_ε(τ)).Then, according to (<ref>)v_i(x_ε)-v_i(O)+ε ≥ ∫_0^τℓ_i(y_ε(s),α_ε(s))e^-λ sds+e^-λτ[φ(y_ε(τ))-φ(O)]-v_i(O)(1-e^-λτ).Next,∫_0^τℓ_i(y_ε(s),α_ε(s))e^-λ sds=∫_0^τℓ_i(y_ε(s),α_ε(s))ds+o(τ), [φ(y_ε(τ))-φ(O)]e^-λτ=φ(y_ε(τ))-φ(O)+τ o_ε(1)+o(τ),and v_i(x_ε)-v_i(O) =o_ε(1),v_i(O)(1-e^-λτ) =o(τ)+τλ v_i(O),where the notation o_ε(1) is used for a quantity which is independent on τ and tends to 0 as ε tends to 0. For k∈ℕ^⋆ the notation o(τ^k) is used for a quantity that is independent on ε and such that o(τ^k)τ^k→0 as τ→0. Finally, 𝒪(τ^k) stands for a quantity independent on εsuch that 𝒪(τ^k)τ^k remains bounded as τ→0.From (<ref>), we obtain thatτλ v_i(O)≥∫_0^τℓ_i(y_ε(s),α_ε(s))ds+φ(y_ε(τ))-φ(O)+τ o_ε(1)+o(τ)+o_ε(1).Since y_ε(τ)∈Γ_i for all ε, one hasφ(y_ε(τ))-φ(x_ε)=∫_0^τdφd x_i(y_ε(s))ẏ_ε(s)ds=∫_0^τdφd x_i(y_ε(s))f_i(y_ε(s),α_ε(s))ds.Hence, from (<ref>)[ τλ v_i(O)-∫_0^τ[ℓ_i(y_ε(s),α_ε(s))+dφd x_i(y_ε(s))f_i(y_ε(s),α_ε(s))]ds ≥ τ o_ε(1)+o(τ)+o_ε(1). ]Moreover, φ(x_ε)-φ(O)=o_ε(1) and that dφd x_i(y_ε(s))=dφd x_i(O)+o_ε(1)+𝒪(s). Thus [ λ v_i(O)-1τ∫_0^τ[ℓ_i(y_ε(s),α_ε(s))+dφd x_i(O)f_i(y_ε(s),α_ε(s))]ds ≥ o_ε(1)+o(τ)τ+o_ε(1)τ. ] Let ε_n→0 as n→∞ and τ_m→0 as m→∞such that (a_mn,b_mn):=(1τ_m∫_0^τ_mf_i(y_ε_n(s),α_ε_n(s))e_ids,1τ_m∫_0^τ_mℓ_i(y_ε_n(s),α_ε_n(s))ds)⟶(a,b)∈ℝe_i×ℝas n,m→∞. By [H1] and [H2]f_i(y_ε_n(s),α_ε_n(s))e_i=f_i(O,α_ε_n(s))+L|y_ε_n(s)|=f_i(O,α_ε_n(s))e_i+o_n(1)+o_m(1), ℓ_i(y_ε_n(s),α_ε_n(s))e_i=ℓ_i(O,α_ε_n(s))+ω(|y_ε_n(s)|)=ℓ_i(O,α_ε_n(s))e_i+o_n(1)+o_m(1).It follows that(a_mn,b_mn) =(1τ_m∫_0^τ_mf_i(O,α_ε_n(s))e_ids,1τ_m∫_0^τ_mℓ_i(O,α_ε_n(s))ds)+o_n(1)+o_m(1)∈FL_i(O)+o_n(1)+o_m(1),since FL_i(O) is closed and convex. Sending n,m→∞, we obtain (a,b)∈FL_i(O) so there exists a∈ A_i such thatlim_m,n→∞(1τ_m∫_0^τ_mf_i(y_ε_n(s),α_ε_n(s))e_ids,1τ_m∫_0^τ_mℓ_i(y_ε_n(s),α_ε_n(s))ds)=(f_i(O,a)e_i,ℓ_i(O,a)). On the other hand, from Lemma <ref>, y_ε_n(s)∈Γ_i\{ O} for all s∈[0,τ_m]. This yieldsy_ε_n(τ_m)=[∫_0^τ_nf_i(y_ε_n(s),α_ε_n(s))ds]e_i+x_ε_n.Since |y_ε_n(τ_m)|>0, then1τ_m∫_0^τ_mf_i(y_ε_n(s),α_ε_n(s))ds≥-|x_ε_n|τ_m.Let ε_n tend to 0, then let τ_m tend to 0, one gets f_i(O,a)≥0, so a∈ A_i^+. Hence, from (<ref>) and (<ref>), replacing ε by ε_n and τ by τ_m, let ε_n tend to 0, then let τ_m tend to 0, we finally obtainλ v_i(O)+max_a∈ A_i^+{ -f_i(O,a)dφd x_i(O)-ℓ_i(O,a)}≥λ v_i(O)+[-f_i(O,a)dφd x_i(O)-ℓ_i(O,a)]≥0.§ COMPARISON PRINCIPLE AND UNIQUENESS Inspired by <cit.>, we begin by proving some properties of sub and super viscosity solutions of (<ref>). Thefollowing three lemmas are reminiscent of Lemma 3.4, Theorem 3.1 and Lemma 3.5 in <cit.>. Let w=(w_1,…,w_N) be a viscosity super-solution of (<ref>). Let x∈Γ_i\{ O} and assume thatw_i(O) <min{min_j i{ w_j(O)+c_j} ,-H_O^Tλ} .Then for all t>0,w_i(x)≥ inf_α_i(·),θ_i(∫_0^t∧θ_iℓ_i(y_x^i(s),α_i(s))e^-λ sds+w_i(y_x^i(t∧θ_i))e^-λ(t∧θ_i)),where α_i∈ L^∞(0,∞;A_i), y_x^i is the solution of y_x^i(t)=x+[∫_0^tf_i(y_x^i(s),α_i(s))ds]e_i and θ_i satisfies y_x^i(θ_i)=0 and θ_i lies in [τ_i,τ_i], where τ_i is the exit time of y_x^i from Γ_i\{ O} and τ_i is the exit time of y_x^i from Γ_i. According to (<ref>), the function w_i is a viscosity super-solution of the following problem in Γ_iλ w_i(x)+H_i(x,d w_id x_i(x)) =0x∈Γ_i\{ O} , λ w_i(O)+H_i^+(O,d w_idx_i(O)) =0x=O. Hence, we can apply the resultin <cit.>. We refer to <cit.> for a detailed proof. The main point of that proof uses the results of Blanc <cit.> on minimal super-solutions of exit time control problems.Under assumption [H], let w=(w_1,…,w_N) be a viscosity super-solution of (<ref>) that satisfies (<ref>); then there exists a sequence {η_k} _k∈ℕ of strictly positive real numbers such that lim_k→∞η_k=η>0 and a sequence x_k∈Γ_i\{ O} such that lim_k→∞x_k=O,lim_k→∞w_i(x_k)=w_i(O) and for each k, there exists a control law α_i^k such that the corresponding trajectory y_x_k(s)∈Γ_i for all s∈[0,η_k] andw_i(x_k)≥∫_0^η_kℓ_i(y_x_k(s),α_i^k(s))e^-λ sds+w_i(y_x_k(η_k))e^-λη_k.According to (<ref>) w_i(O)<-H_O^Tλ. Hence, this proof is complete by applying the proof ofin <cit.>. Under assumption [H], let u=(u_1,…,u_N) be a viscosity sub-solution of (<ref>). Then u_i is Lipschitz continuous in B(O,r)∩Γ_i.Therefore, there exists a test function φ_i∈ C^1(Γ_i) which touches u_i from above at O. Since u is a viscosity sub-solution of (<ref>), u_i is a viscosity sub-solution of (<ref>).Recal that H_i(x,·) is coercive for any x∈Γ_i∩ B(O,r), we can apply the proofin <cit.>, which is based on arguments due to Ishii and contained in <cit.>.Under assumption [H], let u=(u_1,…,u_N) be a viscosity sub-solution of (<ref>). Consider i=1,N ,x∈Γ_i\{ O} and α_i∈ L^∞(0,∞;A_i). Let T>0 be such that y_x(t)=x+[∫_0^tf_i(y_x(s),α_i(s))ds]e_i belongs to Γ_i for any t∈[0,T], thenu_i(x)≤∫_0^Tℓ_i(y_x(s),α_i(s))e^-λ sds+u_i(y_x(T))e^-λ T.Since u is a viscosity sub-solution of (<ref>), u_i is a viscosity sub-solution of (<ref>).and satisfies u_i(O)≤-H_O^Tλ. Hence, we can apply the proof in <cit.>. Under assumption [H], Lemmas <ref>, <ref>, <ref> and <ref> hold for vicosity sub- and super-solution û and ŵ repestively, of the exit cost control problem if (<ref>) replaced byw_i(O)<min{min_j i{w_j(O)} +d_i,-H_O^Tλ+d_i} . Under assumption [H], let u be a bounded viscosity sub-solution of (<ref>) and w be a bounded viscosity super-solution of (<ref>); then u≤ w in 𝒢, componentwise. This theorem also holds for viscosity sub- and super-solution u and w, respectively, of the exit cost control problem (<ref>).We give two proofs of Theorem <ref>. The first one is inspired by <cit.> and uses thepreviously stated lemmas. The second one uses theelegant arguments proposed in <cit.>. We focus on u and w, the arguments used for the comparison of u and w are totally similar. Suppose by contradiction that there exists x∈Γ_i such that u_i(x)-w_i(x)>0. By classical comparison arguments for the boundary value problem, see <cit.>, sup_∂Γ_i{ u_i-v_i} ^+≥sup_Γ_i{ u_i-v_i} ^+, so we haveu_i(O)-w_i(O)=max_x∈Γ_i{ u_i(x)-w_i(x)} >0.By definition of viscosity sub-solutionλ u_i(O)+H_O^T≤0.This implies λ w_i(O)+H_O^T<0. We now consider the two following cases.* Case 1: If w_i(O)<min_j i{ w_j(O)+c_j}, from Lemma <ref> (using the same notations),w_i(x_k)≥∫_0^η_kℓ_i(y_x_k(s),α_i^k(s))e^-λ sds+w_i(y_x_k(η_k))e^-λη_k.Moreover, according to Lemma <ref>, we also haveu_i(x_k)≤∫_0^η_kℓ_i(y_x_k(s),α_i^k(s))e^-λ sds+u_i(y_x_k(η_k))e^-λη_k.This yieldsu_i(x_k)-w_i(x_k)≤[u_i(y_x_k(η_k))-w_i(y_x_k(η_k))]e^-λη_k≤[u_i(O)-w_i(O)]e^-λη_k.By letting k tend to ∞, one getsu_i(O)-w_i(O)≤[u_i(O)-w_i(O)]e^-λη.This implies that u_i(O)-w_i(O)≤0 and leads to a contradiction.* Case 2: If w_i(O)≥min_j i{ w_j(O)+c_j}, then there exists j_0 i such thatw_j_0(O)+c_j_0=min_j=1,N{ w_j(O)+c_j} =min_j i{ w_j(O)+c_j}≤ w_i(O),because c_i>0. Since c_j_0 is positivew_j_0(O)<min_j j_0{ w_j(O)+c_j}. Next, by Lemma <ref>, there exists a test function φ_iin C^1(J_i) that touches u_i from above at O, it yields λ u_i(O)-λmin_j i{ u_j(O)+c_j}≤λ u_i(O)+max{ -λmin_j i{ u_j(O)+c_j} ,H_i^+(O,dφ_idx_i(O)),H_0^T}≤0.Thereforew_j_0(O)+c_j_0≤ w_i(O)<u_i(O)≤min_j i{ u_j(O)+c_j}≤ u_j_0(O)+c_j_0.Thus w_j_0(O)<u_j_0(O).Replacing index i by j_0 in (<ref>), we get λ w_j_0(O)+H_O^T<0.By (<ref>) and (<ref>), (<ref>) holds true. Repeating the proof of Case 1 with j_0, we reach a contradiction with (<ref>). It ends the proof.The comparison principle can also be obtained alternatively, usingthe arguments which were very recently proposed by Lions and Souganidisin <cit.>. This new proof is self-combined and the arguments do not rely at all on optimal control theory, but are deeply connected to the ideas used bySoner <cit.> and Capuzzo-Dolcetta and Lions <cit.> for proving comparison principles for state-constrained Hamilton-Jacobi equationsWe start as in first proof. We argue by contradiction without loss of generality, assuming that there exists i such thatu_i(O)-w_i(O)=max_Γ_i{ u_i(x)-w_i(x)} >0.Therefore w_i(O)<-H_O^Tλ. We now consider the two following cases. * Case 1:If w_i(O)<min_j i{ w_j(O)+c_j}, then w_i is a viscosity super-solution of (<ref>). Recall that by Lemma <ref>, there exists a positive number L such that for i=1,N, u_i is Lipschitz continuous with Lipschitz constant L in Γ_i∩ B(0,r). We consider the functionΨ_i,ε:Γ_i×Γ_i ⟶ℝ (x,y)⟶ u_i(x)-w_i(y)-12ε[-|x|+|y|+δ(ε)]^2-γ(|x|+|y|),where δ(ε)=(L+1)ε and γ∈(0,12). It is clear thatΨ_i,ε attains its maximum M_ε,γ at (x_ε,γ,y_ε,γ)∈Γ_i×Γ_i. By classical techniques,we check that x_ε,γ,y_ε,γ→ O and that (x_ε,γ-y_ε,γ)^2ε→0 as ε→0. Indeed, one hasu_i(x_ε,γ)-w_i(y_ε,γ)-[-|x_ε,γ|+|y_ε,γ|+δ(ε)]^22ε-γ(|x_ε,γ|+|y_ε,γ|)≥ max_Γ_i{ u_i(x)-w_i(x)-2γ|x|} -δ^2(ε)2ε≥u_i(O)-w_i(O)-(L+1)^22ε.Since u_i(O)-v_i(O)>0, the term in (<ref>) is positive when ε is small enough. We also deduce from the above inequality and from the boundedness of u_i and w_i that, maybe after the extraction of a subsequence,x_ε,γ,y_ε,γ→ x_γ as ε→0, for somex_γ∈Γ_i.From (<ref>),u_i(x_ε,γ)-w_i(y_ε,γ)-(|x_ε,γ|-|y_ε,γ|)^22ε-(-|x_ε,γ|+|y_ε,γ|)δ(ε)ε≥max_Γ_i{ u_i(x)-w_i(x)-2γ|x|} .Taking the lim sup on both sides of this inequality when ε→0,u_i(x_γ)-w_i(x_γ)-2γ|x_γ|≥max_Γ_i{ u_i(x)-w_i(x)-2γ|x|} +lim sup_ε→0(|x_ε,γ|-|y_ε,γ|)^22ε≥ u_i(O)-w_i(O)+lim sup_ε→0(|x_ε,γ|-|y_ε,γ|)^22ε≥ u_i(O)-w_i(O)+lim inf_ε→0(|x_ε,γ|-|y_ε,γ|)^22ε≥ u_i(O)-w_i(O).Recalling that u_i(O)-w_i(O)=max_Γ_i(u_i-w_i), we obtain from the inequalities above thatx_γ=O and thatlim_ε→0(|x_ε,γ|-|y_ε,γ|)^22ε=0.We claim that if ε>0, then x_ε,γ O.Indeed, assume by contradiction that x_ε,γ=O: * if y_ε,γ>0, thenM_ε,γ=u_i(O)-w_i(y_ε,γ)-12ε[|y_ε,γ|+δ(ε)]^2-γ|y_ε,γ|≥ u_i(y_ε,γ)-w_i(y_ε,γ)-δ^2(ε)2ε-2γ|y_ε,γ|.Since u_i is Lipschitz continuous in B(O,r)∩Γ_i,we see that for ε small enoughL|y_ε.γ|≥ u_i(O)-u_i(y_ε,γ)≥|y_ε,γ|^22ε+|y_ε,γ|δ(ε)ε-γ|y_ε,γ|≥|y_ε,γ|δ(ε)ε-γ|y_ε,γ|.Therefore, if y_ε,γ≠O, then L≥ L+1-γ which gives a contradiction since γ∈(0,12).* Otherwise, if y_ε,γ=O, then M_ε,γ=u_i(O)-w_i(O)-δ^2(ε)2ε≥ u_i(εe_i)-w_i(O)-12ε[-ε+δ(ε)]^2-γε.Since u_i is Lipschitz continuous in B(O,r)∩Γ_i, we see that for ε small enough, Lε≥ u_i(O)-u_i(ε e_i)≥|y_ε.γ|^22ε+|y_ε.γ|δ(ε)ε-2γ|y_ε.γ|≥|y_ε.γ|δ(ε)ε-2γ|y_ε.γ|.This implies that L≥-12+L+1-γ, which gives a contradiction since γ∈(0,12). Therefore the claim is proved. It follows that we can apply the viscosity inequality for u_i at x_ε,γ. Moreover, notice that the viscosity super-solution inequality (<ref>) holds also for y_ε,γ=0 since H_i(O,p)≤ H^+_i(O,p) for any p. Thereforeu_i(x_ε,γ)+H_i(x_ε,γ,-x_ε,γ+y_ε,γ+δ(ε)ε+γ)≤0,w_i(y_ε,γ)+H_i(y_ε,γ,-x_ε,γ+y_ε,γ+δ(ε)ε-γ)≥0. Subtracting the two inequalities,u_i(x_ε,γ)-w_i(y_ε,γ)≤ H_i(y_ε,γ,-x_ε,γ+y_ε,γ+δ(ε)ε+γ)-H_i(x_ε,γ,-x_ε,γ+y_ε,γ+δ(ε)ε-γ). Using [H1] and [H2], it is easy to see that there exists M_i>0 such that for any x,y∈Γ_i,p,q∈ℝ|H_i(x,p)-H_i(y,q)|≤|H_i(x,p)-H_i(y,p)|+|H_i(y,p)-H_i(y,q)|≤M_i|x-y|(1+|p|)+M_i|p-q|.It yieldsu_i(x_ε,γ)-w_i(y_ε,γ)≤M_i[|x_ε,γ-y_ε,γ|(1+|-x_ε,γ+y_ε,γ+δ(ε)ε-γ|)+2|γ|]≤M_i[|x_ε,γ-y_ε,γ|(γ+1+δ(ε)ε)+|x_ε,γ-y_ε,γ|^2ε+2|γ|].Applying (<ref>),let ε tend to 0 and γ tend to 0, we obtain that u_i(O)-w_i(O)≤ 0, the desired contradiction.* Case 2:w_i(O)≥min_j i{ w_j(O)+c_j} =w_j_0(O)+c_j_0. Using the same arguments as in Case 2 of the first proof, we getw_j_0<min{min_j j_0{ w_j(O)+c_j} ,-H_O^Tλ}and w_j_0(O)<u_j_0(O). Repeating Case 1, replacing the index i by j_0, implies that w_j_0(O)≥ u_j_0(O), the desired contradiction. If 𝗏 is the value function (with entry costs) and(v_1,…,v_N) is defined byv_i(x)=𝗏(x)x∈Γ_i\{ O} , lim_δ→0^+𝗏(δ e_i)x=O,then(v_1,…,v_N) is the unique bounded viscosity solution of (<ref>). Similarly, if 𝗏 is the value function (with exit costs) and (v_1,…,v_N) is defined by v_i(x)=𝗏(x)x∈Γ_i\{ O} , lim_δ→0^+𝗏(δ e_i)x=O,then (v_1,…,v_N) is the unique bounded viscosity solution of (<ref>).From Corollary <ref>, we see thatin order to characterize the original value function with entry costs, we need to solve first the Hamilton-Jacobi system (<ref>) and find the unique viscosity solution (v_1,…,v_N). The original value function𝗏 with entry costssatisfies𝗏(x)= v_i(x)if x∈Γ_i\{ O} , min{min_i=1,N{ v_i(O)+c_i} ,-H_O^Tλ} ,if x=O.The characterization of 𝗏(O) follows from Theorem <ref>. The characterization of the original value function with exit costs 𝗏 is similar.§ A MORE GENERAL OPTIMAL CONTROL PROBLEMIn what follows, we generalize the control problem studied in the previous sectionsby allowing some of the entry (or exit) costs to be zero. The situation can be viewed as intermediary between the one studied in <cit.> when all the entry (or exit) costs werezero, and that studied above when all the entry or exit costs were positive. Accordingly, every result presented below will mainly be obtained by combining the arguments proposed above with those used in <cit.>. Hence, we will present the results and omit the proofs.To be more specific, weconsider the optimal control problemswith non-negative entry costC={c_1,…c_m,c_m+1,…c_N} where c_i=0 if i≤ m and c_i>0 if i>m, keeping all the assumptions and definitions of Section <ref> unchanged. The value function associated to C will be denoted by 𝖵. Similarly to Lemma <ref>, 𝖵|_Γ_i\{ O}is continuous and Lipschitz continuous near O: therefore, it is possible to extend 𝖵|_Γ_i\{ O} at O. This extension will be noted𝒱_i. Moreover, one can check that 𝒱_i(O)=𝒱_j(O) for all i,j≤ m, which means that 𝖵|_∪_i=1^mΓ_i is a continuous function which will be noted𝒱_c hereafter.Combining the argumentsin <cit.> and in Section <ref> leads us to the following theorem.The value function 𝖵 satisfiesmax_i=m+1,N{𝒱_i(O)}≤𝖵(O)=𝒱_c(O)≤min{min_i=m+1,N{𝒱_i(O)+c_i} ,-H_O^Tλ} .In the case when c_i=0 for i=1,N,𝖵 is continuous on 𝒢 and it is exactly the value function of the problem studied in <cit.>.We now define a set of admissible test-functionand the Hamilton-Jacobi equation that will characterize 𝖵.A function φ:(∪_i=1^mΓ_i)×Γ_m+1×…×Γ_N→ℝ^N-m+1 of the formφ(x_c,x_m+1,…,x_N)=(φ_c(x_c),φ_m+1(x_m+1),…,φ_N(x_N)) is an admissible test-function if* φ_c is continuous and for i≤ m, φ_c|_Γ_i belongs to C^1(Γ_i),* for i>m, φ_i belongs to C^1(Γ_i), * the space of admissible test-function is noted R(𝒢). A function U=(U_c,U_m+1,…,U_N) where U_c∈ USC(∪_j=1^mΓ_j;ℝ),U_i∈ USC(Γ_i;ℝ) is called a viscosity sub-solution of the Hamilton-Jacobi system iffor any (φ_c,φ_m+1,…,φ_N)∈ R(𝒢): * ifU_c-φ_c has a local maximumat x_c∈∪_j=1^mΓ_j and if * x_c∈Γ_j\{ O} for some j≤ m, then[ λ U_c(x_c)+H_j(x,dφ_cd x_j(x_c))≤0, ] * x_c=O, thenλ U_c(O)+max{ -λmin_j>m{ U_j(O)+c_j} ,max_j≤ m{ H_j^+(O,dφ_cd x_j^+(O))} ,H_O^T}≤0;* if U_i-φ_i has a local maximum pointat x_i∈Γ_i for i>m, and if * , then [ λ U_i(x_i)+H_i(x,dφ_id x_i(x_i))≤0, ] * x_i=O, thenλ U_i(O)+max{ -λmin_j>m,j i{ U_j(O)+c_j} ,-λ U_c(O),H_i^+(O,dφ_idx_i(O)),H_O^T}≤0.A function U=(U_c,U_m+1,…,U_N) where U_c∈ LSC(∪_j=1^mΓ_j;ℝ),U_i∈ LSC(Γ_i;ℝ) is called a viscosity super-solution of the Hamilton-Jacobi system if U_c(O)≥ U_i(O), fori=m+1,N,and for any (φ_c,φ_m+1,…,φ_N)∈ R(𝒢): * ifU_c-φ_c has a local maximumat x_c∈∪_j=1^mΓ_j and if * x_c∈Γ_j\{ O} for some j≤ m, then[ λ U_c(x_c)+H_j(x,dφ_cd x_j(x_c))≥0, ] * x_c=O, thenλ U_c(O)+max{ -λmin_j>m{ U_j(O)+c_j} ,max_j≤ m{ H_j^+(O,dφ_cd x_j^+(O))} ,H_O^T}≥0;* if U_i-φ_i has a local minimum pointat x_i∈Γ_i for i>m, andif * , then[ λ U_i(x_i)+H_i(x,dφ_id x_i(x_i))≥0, ] * x_i=O for i>m thenλ U_i(O)+max{ -λmin_j>m,j i{ U_j(O)+c_j} ,-λ U_c(O),H_i^+(O,dφ_idx_i(O)),H_O^T}≥0.A function U=(U_c,U_1,…,U_m) where U_c∈ C(∪_j≤ mΓ_j;ℝ) and U_i∈ C(Γ_i;ℝ) for all i>m is called a viscosity solution of the Hamilton-Jacobi system if it is both a viscosity sub-solution and a viscosity super-solution of the Hamilton-Jacobi system.The term -λ H_C(O) in the above definition accounts for the situation in which the trajectory enters ∪_j=1^mΓj. The term max_j≤ m{ H_j^+(O,dφ_cd x_j^+(O))} accounts for the situation in which the trajectory enters Γ_i_0 where H_i_0^+(O,dφ_cd x_j^+(O))=max_j≤ m{ H_j^+(O,dφ_cd x_j^+(O))}.In the case when c_i=0 for i=1,N, i,e., m=N, the term -λmin_j>mU_j ( O )+c_j vanishes. This implies thatmax{ -λmin_j>m{ U_j(O)+c_j} ,max_j≤ m{ H_j^+(O,∂φ_c∂ e_j^+(O))} ,H_O^T} =max_j=1,N{ H_j^+(O,∂φ_c∂ e_j^+(O))}= H_O(∂φ_c∂ e_1^+(O),…,∂φ_c∂ e_N^+(O)).where H_O ( p_1,…,p_N ) is defined in <cit.>. This means that, in the case when all the entry costs c_j vanish, we recover the notion of viscosity solution proposedin <cit.>. We now study the relationship between the value function 𝖵 and the Hamilton-Jacobi system. Let 𝖵 be the value function corresponding to the entry costs C, then(𝒱_c,𝒱_m+1,…,𝒱_N) is a viscosity solution of the Hamilton-Jacobi system.Let usstate the comparison principle for theHamilton-Jacobi system.Let U=(U_c,U_m+1,…,U_N) and W=(W_c,W_m+1,…,W_N) be a bounded viscosity sub-solution and a viscosity super-solution, respectively,of the Hamilton-Jacobi system. The following holds:U≤ W in 𝒢, i.e., U_c≤ W_c on ∪_j=1^mΓ_j, and U_i≤ W_i in Γ_i for all i>m. Suppose by contradiction that there exists i∈{ 1,…,N} and x∈Γ_i such thatU_c(x)-W_c(x)>0if i≤ m,U_i(x)-W_i(x)>0if i>m,thenU_c(O)-W_c(O)=max_∪_j=1^mΓ_j{ U_c-W_c} >0if i≤ m,U_i(O)-W_i(O)=max_Γ_i{ U_i-W_i} >0if i>m,since the case where the positive maximum is achieved outside the junction leads to a contradition by classical comparison results.* Case 1: U_c(O)-W_c(O)=max_∪_i=1^mΓ_i(U_c-W_c)>0 * Sub-case 1-a:W_c(O)<min_j>m{ W_j(O)+c_j}. Since W_c(O)<U_c(O)≤-H_O^Tλ, the function W_c is a viscosity super-solution ofλ W_c(x)+H_i(x,d W_cd x_i(x)) = 0i≤ m,x∈Γ_i\{ O} , λ W_c(O)+H_c(d W_cd x_1^+(O),…,d W_cd x_m^+(O)) = 0x=O.where H_c(p_1,…,p_m)=max_i≤ mH_i^+(O,p_i). Applying Lemma <ref> in the Appendix, we obtain thatU_c(O)≤ W_c(O) in contradiction with the assumption.* Sub-case 1-b: W_c(O)≥min_j>m{ W_j(O)+c_j} =W_i_0(O)+c_i_0. Since c_i_0>0, we first see that W_i_0(O)<min{min_j>m{ W_j(O)+c_j} ,W_c(O),-H_O^Tλ}. Hence, W_i_0 is a viscosity super-solution of (<ref>) replacing i by i_0. Moreover, sinceU_i_0(O)+c_i_0≥min_j>m(U_j(O)+c_j)≥ U_c(O)> W_c(O)>W_i_0(O)+c_i_0,then U_i_0(O)>W_i_0(O). Applying the same argument as Case 1 in the second proof of Theorem <ref> replacing i by i_0,we obtain that U_i_0(O)≤ W_i_0(O), which is contradictory. * Case 2: U_i(O)-W_i(O)=max_Γ_i(U_i-W_i)>0 for some i>m. Using the definition of viscosity sub-solutions and Case 1,we see that W_i(O)<U_i(O)≤ U_c(O)≤ W_c(O).* Sub-case 2-a: W_i(O)<min_j>m{ W_j(O)+c_j}. Since U_i(O)<-H_O^Tλ, we first see that W_i(O)<min{min_j>m{ W_j(O)+c_j} ,W_c(O),-H_O^Tλ}. Hence, W_i is a viscosity super-solution of (<ref>). Applying the same argument as in Case 1 in the second proof of Theorem <ref>, we see that U_i(O)≤ W_i(O), which iscontradictory.* Sub-case 2-b: W_i(O)≥min_j>m{ W_j(O)+c_j} =W_i_0(O)+c_i_0. Since c_i_0>0, we can check that W_i_0(O)<min{min_j>m{ W_j(O)+c_j} ,W_c(O),-H_O^Tλ}. Hence, W_i_0 is a viscosity super-solution of (<ref>) replacing i by i_0. Moreover, sinceU_i_0(O)+c_i_0≥min_j>m(U_j(O)+c_j)≥ U_c(O) > W_i(O)>W_i_0(O)+c_i_0,then U_i_0(O)>W_i_0(O). Applying the same argument as Case 1 in the second proof of Theorem <ref> replacing i by i_0, we obtain that U_i_0(O)≤ W_i_0(O) which is contradictory.§ APPENDIX For any a∈ A_i^+, there exists a sequence { a_n} such that a_n∈ A_i andf_i(O,a_n)≥ δn>0, |f_i(O,a_n)-f_i(O,a)|≤ 2Mn, |ℓ_i(O,a_n)-ℓ_i(O,a)|≤ 2Mn.From assumption [H4], there exists a_δ∈ A_i such that f_i(O,a_δ)=δ. Since _i(O) is convex (by assumption [H3]), for any n∈ℕ,a∈ A^+_i1n(f_i(O,a_δ)e_i,ℓ_i(O,a_δ))+(1-1n)(f_i(O,a),ℓ_i(O,a)e_i)∈_i(O).Then, there exists a sequence { a_n} such that a_n∈ A_i and1n(f_i(O,a_δ),ℓ_i(O,a_δ))+(1-1n)(f_i(O,a),ℓ_i(O,a))=(f_i(O,a_n),ℓ_i(O,a_n))∈_i(O).Notice that f_i(O,a)≥0 since a∈ A_i^+, this yieldsf_i(O,a_n)≥f_i(O,a_δ)n=δn>0.From (<ref>), we also have|f_i(O,a_n)-f_i(O,a)|=1n|f_i(O,a_δ)-f_i(O,a)|≤2Mn,and|ℓ_i(O,a_n)-ℓ_i(O,a)|=1n|ℓ_i(O,a_δ)-ℓ_i(O,a)|≤2Mn. We can state the following corollary of Lemma <ref>:Fori=1,N and p_i∈ℝ,max_a∈ A_if_i(O,a)≥0{ -f_i(O,a)p_i-ℓ_i(O,a)} =sup_a∈ A_if_i(O,a)>0{ -f_i(O,a)p_i-ℓ_i(O,a)} .If U_c and W_c are respectively viscosity sub and super-solution ofλ U_c(x)+H_i(x,d U_cd x_i(x))≤0 if x∈Γ_i\{ O} , λ U_c(O)+H_c(d U_cd x_1(O),…,d U_cd x_m(O))≤0 if x=O,and λ W_c(x)+H_i(x,d W_cd x_i(x))≥0 if x∈Γ_i\{ O} , λ W_c(O)+H_c(d W_cd x_1(O),…,d W_cd x_m(O))≥0 if x=O,then U_c(x)≤ W_c(x) for all x∈⋃_i=1^mΓ_i. Assume that there exists x∈Γ_i where 1≤ i≤ m and U_c(x)-W_c(x)>0. By classical comparison principle for the boundary problem on Γ_i, one gets U_c(O)-W_c(O)=max_Γ_i{ U_c(x)-W_c(x)} >0.Applying again classical comparison principle for the boundary problem for each edge Γ_jU_c(O)-W_c(O)=max_⋃_i=1^mΓ_i{ U_c(x)-W_c(x)} >0.For j=1,N, we consider the functionΨ_j,ε,γ:Γ_j×Γ_j ⟶ℝ (x,y)⟶ U_c(x)-W_c(y)-12ε[-|x|+|y|+δ(ε)]^2-γ(|x|+|y|),where δ(ε)=(L+1)ε, γ∈(0,12).The function Ψ_j,ε attains its maximum at (x_j,ε,γ,y_j,ε,γ)∈Γ_j×Γ_j. Applying the same argument as in the second proof of Theorem <ref>, we have x_j,ε,γ,y_j,ε,γ→ O and (x_j,ε,γ-y_j,ε,γ)^2ε→0 as ε→0. Moreover, for any j=1,m, x_j,ε,γ O.We claim that y_j,ε,γ must be O for ε small enough . Indeed, if there exists a sequence ε_n such that y_j,ε_n,γ∈Γ_j\{O }, then applying viscosity inequalities, we haveU_c(x_j,ε_n,γ)+H_j(x_j,ε_n,γ,-x_j,ε_n,γ+y_j,ε_n,γ+δ(ε_n)ε_n+γ)≤0,W_c(y_j,ε_n,γ)+H_j(y_j,ε_n,γ,-x_j,ε_n,γ+y_j,ε_n,γ+δ(ε_n)ε_n-γ)≥0.Subtracting the two inequalities and using (<ref>) with H_j, we obtainU_c(x_j,ε_n,γ)-W_c(y_j,ε_n,γ)≤M_j|x_j,ε_n,γ-y_j,ε_n,γ|(1+|-x_j,ε_n,γ+y_j,ε_n,γ+δ(ε_n)ε_n-γ|)+M_j2γ.Recall that we already have (x_j,ε_n,γ-y_j,ε_n,γ)^2ε_n→0 as n→∞. Let n tend to ∞ and γ tend to 0 then we obtain U_c(O)-W_c(O)≤0. It leads us to a contradiction. So this claim is proved.Define the functionΨ:⋃_j=1^mΓ_j→ℝ byΨ|_Γ_i(y)=12ε∑_j i{[-|x_i,ε,γ|+δ(ε)]^2-γ|x_i,ε,γ|} +12ε[-|x_i,ε,γ|+|y|+δ(ε)]^2+γ(-|x_i,ε,γ|+|y|).We can see that Ψ is continuous on ⋃_j=1^mΓ_j and belongs to C^1(Γ_j) for j=1,m. Moreover, for j=1,m and for ε small enough, y_j,ε,γ=Othen the function Ψ + W_c has a minimum point at O. It yieldsλ W_c(O)+H_c(-x_1,ε,γ+δ(ε)ε,…,-x_m,ε,γ+δ(ε)ε)≥0.By definition of H_c, there exists j_0∈{ 1,…,m} such thatλ W_c(O)+H_j_0^+(O,-x_j_0,ε,γ+δ(ε)ε)≥0.This impliesλ W_c(O)+H_j_0(O,-x_j_0,ε,γ+δ(ε)ε)≥0On the other hand, since x_j_0,ε,γ∈Γ_j_0\{ O}, we have λ U_c(x_j_0,ε,γ)+H_j_0(x_j_0,ε,γ,-x_j_0,ε,γ+δ(ε)ε)≤0.Subtracting the two inequalities and using properties of Hamiltonian H_j_0, let ε tend to 0 then γ tend to 0, we obtain that U_c(O)-W_c(O)≤0, which is contradictory.§ ACKNOWLEDGEMENTI would like to express my thanks to my advisors Y. Achdou, O. Ley and N. Tchou for suggesting me this work and for their help. I also thank the hospitality of Centre Henri Lebesgue and INSA de Rennes during the preparation of this work. Moreover, this work was partially supported by the ANR (Agence Nationale de la Recherche) through HJnet project ANR-12-BS01-0008-01 and MFG project ANR-16-CE40-0015-01. 10ACCT2011 Y. Achdou, F. Camilli, A. Cutri, and N. Tchou. Hamilton-Jacobi equations on networks. IFAC proceedings volumes, 44(1):2577–2582, 2011.ACCT2013 Y. Achdou, F. Camilli, A. Cutrì, and N. Tchou. Hamilton-Jacobi equations constrained on networks. NoDEA Nonlinear Differential Equations Appl., 20(3):413–445, 2013.AOT2015 Y. Achdou, S. Oudet, and N. Tchou. Hamilton-Jacobi equations for optimal control on junctions and networks. ESAIM Control Optim. Calc. Var., 21(3):876–899, 2015.Barles1993 G. Barles. Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit. Nonlinear Anal., 20(9):1123–1134, 1993.Barles2013 G. Barles. An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications. In Hamilton-Jacobi equations: approximations, numerical analysis and applications, volume 2074 of Lecture Notes in Math., pages 49–109. Springer, Heidelberg, 2013.BBC2013 G. Barles, A. Briani, and E. Chasseigne. A Bellman approach for two-domains optimal control problems in ℝ^N. ESAIM Control Optim. Calc. Var., 19(3):710–739, 2013.BBC2014 G. Barles, A. Briani, and E. Chasseigne. A Bellman approach for regional optimal control problems in ℝ^N. SIAM J. Control Optim., 52(3):1712–1744, 2014.Blanc1997 A.-P. Blanc. Deterministic exit time control problems with discontinuous exit costs. SIAM J. Control Optim., 35(2):399–434, 1997.Blanc2001 A.-P. Blanc. Comparison principle for the Cauchy problem for Hamilton-Jacobi equations with discontinuous data. Nonlinear Anal., 45(8, Ser. A: Theory Methods):1015–1037, 2001.CM2013 F. Camilli and C. Marchi. A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks. J. Math. Anal. Appl., 407(1):112–118, 2013.DL1990 I. Capuzzo-Dolcetta and P.-L. Lions. Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc., 318(2):643–683, 1990.EKNS2008 K.-J. Engel, M. Kramar Fijavvz, R. Nagel, and E. Sikolya. Vertex control of flows in networks. Netw. Heterog. Media, 3(4):709–722, 2008.FM2013 H. Frankowska and M. Mazzola. Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints. Calc. Var. Partial Differential Equations, 46(3-4):725–747, 2013.GP2006 M. Garavello and B. Piccoli. Traffic flow on networks, volume 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. Conservation laws models.GHZ2017 P. J. Graber, C. Hermosilla, and H. Zidani. Discontinuous solutions of Hamilton-Jacobi equations on networks. J. Differential Equations, 263(12):8418–8466, 2017.IM2017 C. Imbert and R. Monneau. Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks. Ann. Sci. Éc. Norm. Supér. (4), 50(2):357–448, 2017.IMZ2013 C. Imbert, R. Monneau, and H. Zidani. A Hamilton-Jacobi approach to junction problems and application to traffic flows. ESAIM Control Optim. Calc. Var., 19(1):129–166, 2013.Ishii2013 H. Ishii. A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton-Jacobi equations. In Hamilton-Jacobi equations: approximations, numerical analysis and applications, volume 2074 of Lecture Notes in Math., pages 111–249. Springer, Heidelberg, 2013.LS2016 P.-L. Lions and P. Souganidis. Viscosity solutions for junctions: well posedness and stability. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27(4):535–545, 2016.LS2017 P.-L. Lions and P. Souganidis. Well posedness for multi-dimensional junction problems with kirchoff-type conditions. preprint arXiv: 1704.04001, 2017.Oudet2014 S. Oudet. Hamilton-Jacobi equations for optimal control on multidimensional junctions. preprint arXiv: 1412.2679, 2014.SC2013 D. Schieborn and F. Camilli. Viscosity solutions of Eikonal equations on topological networks. Calc. Var. Partial Differential Equations, 46(3-4):671–686, 2013.Soner1986a H. M. Soner. Optimal control with state-space constraint. I. SIAM J. Control Optim., 24(3):552–561, 1986.Soner1986b H. M. Soner. Optimal control with state-space constraint. II. SIAM J. Control Optim., 24(6):1110–1122, 1986. | http://arxiv.org/abs/1706.08748v2 | {
"authors": [
"Manh-Khang Dao"
],
"categories": [
"math.OC",
"math.AP"
],
"primary_category": "math.OC",
"published": "20170627092601",
"title": "Hamilton-Jacobi equations for optimal control on networks with entry or exit costs"
} |
A Cryptographic Approach for Steganography Jacques M. Bahi, Christophe Guyeux, and Pierre-Cyrille Heam*FEMTO-ST Institute, UMR 6174 CNRSComputer Science Laboratory DISCUniversity of Franche-Comté, France{jacques.bahi, christophe.guyeux, pierre-cyrille.heam}@femto-st.fr *Authors are cited in alphabetic orderDecember 30, 2023 ============================================================================================================================================================================================================================================================================================In this research work, security concepts are formalized in steganography,and the common paradigms based on information theory are replaced by another onesinspired from cryptography, more practicable are closer than what is usually done in other cryptographic domains. These preliminaries lead to a first proof of a cryptographically secure information hiding scheme.Information hiding; Steganography; Security; Cryptographic proofs. § INTRODUCTION The usual manner for preserving privacy when communicating over public channelsis by using cryptographic tools.Users cipher the data and send them over possibly insecure networks.Even if a third party intercepts these data, he or she will not understand themwithout having the secret key for deciphering. In that well investigated scenario, anyone knows that a private message istransmitted through the public channel, but only authorized individuals (i.e., owners of the secret key) can understand it. A second approach investigated over two decades <cit.>, and usually referred as information hiding or steganography <cit.>, aims at inserting a secret message into aninnocent cover, in such a way that observers cannot detect the existence of this hidden channel (for instance, images sent through the Internet).The goal in this field is to appear as innocent as possible: observers should not think that something goes wrong with this public channel. It must not cross their mind that sometimes the public channel is used to transmit hidden messages. In that context, an attack is succeeded whenthe sleazy character of the channel is detected. Tools used in that field are mainly based on artificial intelligence. Theyare called steganalyzers, and their main objective is to detect whether agiven communication channel is possibly steganographied, or if it only contains “natural” images. In case of detection, the unique countermeasureproposed by the literature is to stop the sleazy communication by closing the channel. To sum up, the steganography community currently only focuses on the ability to detect hidden channels, without investigating the consequences of this detection <cit.>.However, observers have not necessarily the ability or the desire to stop the communication. For instance, who can switch off the Internet? Furthermore, by stopping the faked channel, attackers miss the opportunity to obtain more information about the secret message and the intended receiver. Finally, if attackers observe the communication, man can reasonably think that they already knew in advance that this channel is sleazy (if not, why they observe it?). The use of a steganalyzer on a channel only appears in the bestsituation as a reinforcement of their doubts or fears. In most operational contexts, only sleazy channels are observed, and the questions arefinally to determine <cit.>: (1) when the hidden messages have been transmitted in this channel (among all the possibly faked images, how to determinethe ones that really contain hidden information?), (2) what was the content of this message,and perhaps (3) who was the receiver among the observers. These questions make sense only within a stegranographic context, that is, when the channel is not ciphered. However, these important questionings have never been regarded by the information hiding community.In this paper, authors provide a cryptographic theoretical framework to study this scenario related to steganography. Concrete illustrative examples of this frameworkof study are given thereafter. A first toy example is the hypothetical case of a dissident blogger in a totalitarian state, who posts regularly and publiclyinformation in his or her blog, while being severely watched by the authorities. This blogger wants to transmit one day a secret message or a signal toan observer into confidence, without sounding the alarm in the authorities side. Another example is an individual who is invigilated, because he is correctly suspectedto be a spy. This agent cannot be arrested on a simple presumption, or on the claim that the images he sent in his emails look sleazy. Despite this surveillance, this spy wants to transmit one day a message to his sponsor. The observers want to determine if an hidden message isreally transmitted or not, to have a proof of such a transmission, together with the content of the message, the date of transmission, and thetargeted receiver if possible.Obviously, these situations are related to both cryptography and steganography, however there is currently a lack of tool allowing their study. The key idea of this research work is to propose algorithms such that observers cannot switch from doubts (sleazy channels)to certainties or proofs.The remainder of this article is organized as follows.In Section <ref>, generalities from steganography are discussed. The key concepts and main results arepresented in Section <ref>.Finally, Section <ref> concludesthis research work and details further investigations. § NOTIONS AND TERMINOLOGIES IN INFORMATION HIDING In the following some common notions in the field of information hiding are recalled.We refer to <cit.> for a complete survey of this subject. §.§ Information Hiding SecurityRobustness and security are two major concerns in information hiding. These two concerns have been defined in <cit.> as follows.“Robust watermarking is a mechanism to create a communication channel that is multiplexed into original content [...]. It is required that, firstly, the perceptual degradation of the marked content [...] is minimal and, secondly, that the capacity of the watermark channel degrades as a smooth function of the degradation of the marked content. [...]. Watermarking security refers to the inability by unauthorized users to have access to the raw watermarking channel [...] to remove, detect and estimate, write or modify the raw watermarking bits.” In the framework of watermarking and steganography, security has seen several important developments since the last decade <cit.>.The first fundamental work in security was made by Cachin in the context of steganography <cit.>.Cachin interprets the attempts of an attacker to distinguish between an innocent image and a stego-content as a hypothesis testing problem.In this document, the basic properties of a stegosystem are defined using the notions of entropy, mutual information, and relative entropy.Mittelholzer, inspired by the work of Cachin, proposed the first theoretical framework for analyzing the security of a watermarking scheme <cit.>. These efforts to bring a theoretical framework for security in steganography and watermarking have been followed up by Kalker, who tries to clarify the concepts (robustness vs. security), and the classifications of watermarking attacks <cit.>.This work has been deepened by Furon et al., who have translated Kerckhoffs' principle (Alice and Bob shall only rely on some previously shared secret for privacy), from cryptography to data hiding <cit.>.They used Diffie and Hellman methodology, and Shannon's cryptographic framework <cit.>, to classify the watermarking attacks into categories, according to the type of information Eve has access to <cit.>, namely: Watermarked Only Attack (WOA), Known Message Attack (KMA), Known Original Attack (KOA), and Constant-Message Attack (CMA). Levels of security have been recently defined in these setups. The highest level of security in WOA is called stego-security <cit.>, recalled below.In the prisoner problem of Simmons <cit.>, Alice and Bob are in jail, and they want to, possibly, devise an escape plan by exchanging hidden messages in innocent-looking cover contents.These messages are to be conveyed to one another by a common warden, Eve, who over-drops all contents and can choose to interrupt the communication if they appear to be stego-contents.The stego-security, defined in this framework, is the highest security level in WOA setup <cit.>. To recall it, we need the following notations:* K is the set of embedding keys, * p(X) is the probabilistic model of N_0 initial host contents, * p(Y|K_1) is the probabilistic model of N_0 watermarked contents. Furthermore, it is supposed in this context that each host content has been watermarked with the same secret key K_1 and the same embedding function e. It is now possible to define the notion of stego-security.The embedding function e is stego-secure if and only if:∀ K_1 ∈K, p(Y|K_1)=p(X).This definition is almost always considered as not really tractable in practice, reasons explaining this mistrust are outlined in the following section.This is the reason why the information hiding community majorly focuses on the construction of steganalyzers, supposed to be able to determine whether a given communication channel appears to transmit steganographied messages or not.§.§ Drawbacks of the Stego-Security NotionTheoretically speaking, the stego-security notion matches well with the idea of a perfectsecrecy in the WOA category of attacks. However, its concrete verification raises several technical problems difficult to get around. These difficulties impact drastically the effective security of the scheme.For instance, in a stego-secure scheme, the distribution of the set of watermarkedimages must be the same than the one of the original contents, no matter the chosen keys. But how to determine practically the distribution of the original contents?Furthermore, claiming that Alice can constitutes her own subset of well-chosen images havingthe same “good” distribution is quite unreasonable in several contexts ofsteganography: Alice has not always the choice of the supports. Moreover,it introduces a kind of bias, as the warden can find such similarities surprising.Suppose however that Alice is in the best situation for her, that is, she has the possibility toconstitute herself the set of original contents. How can she proceed practicallyto be certain that all media into the set follow a same distribution p(X)? According to the authors opinion, Alice has two possible choices: * Either she constitutes the set by testing, for each new content, whether this media has a same distribution than the ones that have been already selected.* Or she forges directly new images by using existing ones. For instance, she can replace all the least significant bits of the original contents by using a good pseudorandom number generator. In the first situation, Alice will realize a χ^2 test, or other statistical tests of this kind,to determine if the considered image (its least significant bits, or its low frequency coefficients, etc.) has a same distribution than images already selected. In that situation, Alicedoes not have the liberty to choose the distribution, and it seems impossible to find a scheme being able to preserve any kind of distribution, for all secret keys and all hidden messages. Furthermore, such statistical hypothesis testing are not ideal ones, as they only regard if a result is unlikely to have occurred by chance alone according to a pre-determined threshold probability (the significance level). Errors of the first (false positive) and second kind (false negative) occur necessarily, with a certain probability. In other words, with such an approach, Alice cannot design a perfect set of cover contents having all the same probability p(X).This process leads to a set of media that follows a distribution Alice does not have access to.The second situation seems more realistic, it will thus befurther investigated in the next section. § TOWARD A CRYPTOGRAPHICALLY SECURE HIDING In this section a theoretical framework for information hiding security is proposed, which is more closely resembling that of usual approaches in cryptography. It allows to define thenotion of steganalyzers, it is compatible with the new original scenarios of information hiding that have been dressedin the previous sections, and it does not have the drawbacksof the stego-security definition. §.§ Introduction Almost all branches in cryptology have a complexity approach for security. For instance, in a cryptographic context, a pseudorandom number generator (PRNG) is a deterministic algorithm G transforming strings of length ℓ into strings of length M, with M> ℓ.The notion of secure PRNG can be defined as follows <cit.>.Let 𝒟: B^M ⟶B be a probabilistic algorithm that runs in time T.Let ε > 0.𝒟 is called a (T,ε)-distinguishing attack on pseudorandom generator G if | Pr[𝒟(G(k)) = 1 | k ∈_R {0,1}^ℓ ] - Pr[𝒟(s) = 1 | s ∈_R B^M ]| ⩾ε, where the probability is taken over the internal coin flips of 𝒟, and the notation “∈_R” indicates the process of selecting an element at random and uniformly over the corresponding set. Let us recall that the running time of a probabilistic algorithm is defined to be the maximum of the expected number of steps needed to produce an output, maximized over all inputs; the expected number is averaged over all coin flips made by the algorithm <cit.>. We are now able to recall the notion of cryptographically secure PRNG. A pseudorandom generator is (T,ε)-secure if there exists no (T,ε)-distinguishing attack on this pseudorandom generator.Intuitively, it means that no polynomial-time algorithm can make adistinction, with a non-negligible probability, between a trulyrandom generator and G.Inspired by these kind of definitions, we propose what follows. §.§ Definition of a stegosystem Let 𝒮, ℳ, and 𝒦=B^ℓthree sets of words on B called respectively the sets ofsupports, of messages, and of keys (of size ℓ).A stegosystem on (𝒮, ℳ, 𝒦)is a tuple (ℐ,ℰ, inv) such that: * ℐ is a function from 𝒮×ℳ×𝒦 to 𝒮,(s,m,k) ⟼ℐ(s,m,k)=s', * ℰ is a function from 𝒮×𝒦 to ℳ,(s,k) ⟼ℰ(s,k) = m'.* inv is a function from 𝒦 to 𝒦, s.t. ∀ k ∈𝒦, ∀ (s,m)∈𝒮×ℳ, ℰ(ℐ(s,m,k),inv(k))=m.* ℐ(s,m,k) and ℰ(c,k^') can be computed inpolynomial time.ℐ is called the insertion or embedding function, ℰ the extraction function, s the host content, m the hidden message, k the embedding key, k'=inv(k) the extraction key, and s' is the stego-content. If ∀ k ∈𝒦, k=inv(k), the stegosystem is symmetric (private-key), otherwise it is asymmetric (public-key). §.§ Heading Notions Let S=(ℐ,ℰ, inv)a stegosystem on (𝒜, ℳ, 𝒦), with𝒜⊂B^M. A (T,ε)-distinguishing attack on the stegosystem S is a probabilisticalgorithm 𝒟:𝒜⟶{0,1} in running time T, such that there exists m ∈ℳ, | Pr[𝒟(ℐ(s,m,k))=1 | k ∈_R 𝒦, s ∈_R 𝒜]. . -Pr[𝒟(x)=1 | x ∈_R 𝒜]|⩾ε, where the probability is also taken over the internal coin flips of 𝒟,and the notation ∈_R indicates the process of selecting an element atrandom and uniformly over the corresponding set. A stegosystem is (T,ε)-undistinguishable if there exists no(T,ε)-distinguishing attack on this stegosystem. Intuitively, it means that there is no polynomial-time probabilistic algorithm being able to distinguish the host contents from the stego-contents§.§ A Cryptographically Secure Information Hiding SchemeLet𝒮 = {s_1^1, s_2^1,…,s_2^N^1, s_1^2,s_2^2,…,s_2^N^2,…,s_1^r, s_2^r,…,s_2^N^r}a subset of B^M = 𝒜. Consider G:B^L ⟶B^N a (T,ε)-secure pseudorandom number generator, and ℐ(s_j^i,m,k)=s^i_m⊕ G(k). Assuming that r is a constant, and that from i,j one can compute the image s^i_j in time T_1, the steosystem is (T-T_1-N-1,ε)-secure.Intuitively, 𝒮 is built from r images containing N bits of low information. The image s^i_j corresponds to the i-th image where the N bits are set to j. Assume there exists a (T^',ε) distinguisher 𝒟^'for the stego-system. Therefore, there exists m_0 such that[ | Pr(𝒟^'(ℐ(s,m_0,k))=1 | k ∈_R B^ℓ, s ∈_R 𝒮).;. -Pr(𝒟^'(x)=1 | x ∈_R 𝒮)|⩾ε ]Choosing randomly and uniformly s∈𝒮 is equivalent to choose uniformly and randomly i∈{1,…,r} and j∈{1,…,2^N}. Therfore (<ref>) is equivalent to[ | Pr(𝒟^'(s^i_m_0⊕ G(k))=1 | k∈_R B^ℓ, i ∈_R {1,…,r}).; .-Pr(𝒟^'(x)=1 | x ∈_R 𝒮)|⩾ε ] Let 𝒟 be the distinguisher for G defined for y∈{0,1}^N into {0,1} by: * Pick randomly and uniformly i∈{1,…,r}.* Compute s=s^i_m_0⊕ y.* Return 𝒟^'(s).The complexity of this probabilistic algorithm is 1 for the first step since r is a constant, T_1+N for the second step, and T^' for the last one. Thus it works in thimeT^'+T_1+1+N.Now we claim that 𝒟 is a (T^'+T_1+1+N,ε)-distinguisher for G.Indeed, Pr(𝒟(y)=1 | y ∈_R {0,2^N})=Pr(𝒟^'(s^i_y)=1 | y ∈_R {0,2^N},i∈_R{1,…,r})= Pr(𝒟^'(x)=1 | x ∈_R 𝒮).Moreover,Pr(𝒟(G(k))=1 | k ∈_R {0,1}^ℓ)=Pr(𝒟^'(s^i_m_0⊕ G(k))=1 | k ∈_R {0,1}^ℓ, i∈_R{1,…,r}).Therefore, using (<ref>), one has[ | Pr[𝒟(G(k)) = 1 | k ∈_R {0,1}^ℓ ].; .- Pr[𝒟(s) = 1 | s ∈_R B^M ]| ⩾ε, ]proving that 𝒟 is a (T^'+T_1+1+N,ε)-distinguisher for G, which concludes the proof. § CONCLUSIONIn this research work, a new rigorous approach forsecure steganography, based on the complexity theory, has been proposed. This work has beeninspired by the definitions of security that can usually be found in other branches of cryptology.We have proposed a new understanding for the notion of secure hiding and presented a first secure information hiding scheme.The intention was to prove the existence of such a schemeand to give a rigorous cryptographical framework for steganography.In future work, we will investigate the situation where detection is impossible. In that case, we will considerboth weak indistinguability (using a statistical or a complexity approach, with the cryptographically securedefinition of PRNGs) and strong indistinguability (using the well known CC1 and CC2 sets). Additionally, we will reconsider and improve the definitions of security in the information hiding literature that are based on the signal theory. Among other thing, we willtake into account a Shannon entropy that is not reduced to simple 1-bit blocs.Finally, we will show that tests using generators allow to attack information hiding schemes that are secure for the statistical approach, as LSB are not uniform in that situation. plain | http://arxiv.org/abs/1706.08752v1 | {
"authors": [
"Jacques M. Bahi",
"Christophe Guyeux",
"Pierre-Cyrille Heam"
],
"categories": [
"cs.CR"
],
"primary_category": "cs.CR",
"published": "20170627094008",
"title": "A Cryptographic Approach for Steganography"
} |
Astronomy Letters, 2017, Vol. 43, No 8, pp. 559–566.0.5cmSearching for Stars Closely Encountering with the Solar SystemBased on Data from the Gaia DR1 and RAVE5 CataloguesV.V. Bobylev[e-mail: [email protected]] andA.T. BajkovaCentral (Pulkovo) Astronomical Observatory, Russian Academy of Sciences, Pulkovskoe sh. 65, St. Petersburg, 196140 RussiaAbstract—We have searched for the stars that either encountered in the past or will encounter in the future with the Solar system closer than 2 pc. For this purpose, we took more than 216 000 stars with the measured proper motions and trigonometric parallaxes from the Gaia DR1 catalogue and their radial velocities from the RAVE5 catalogue. We have found several stars for which encounters closer than 1 pc are possible. The star GJ 710, for which the minimum distance is d_m=0.063±0.044 pc at time t_m=1385±52 thousand years, is the record-holder among them. Two more stars, TYC 8088-631-1 and TYC 6528-980-1, whose encounter parameters, however, are estimated with large errors, are of interest. DOI: 10.1134/S1063773717080011 §.§ INTRODUCTIONAccording to the hypothesis of Oort (1950), the Solar system is surrounded by a comet cloud. Although there are little reliable data on this cloud, it is highly likely that it has a spherical shape and a radius 1×10^5 AU (0.49 pc). The total number of comets is supposed to be 10^11. The flybys of Galactic field stars near the Oort cloud can trigger the formation of comet showers moving into the region of the major planets (Hills 1981). In the long run, the possibility that the Moon and the Earth are bombarded with such comets is not ruled out (Wickramasinghe and Napier 2008).The long-term evolution of the Oort cloud was considered on the basis of numerical simulations, for example, in Emelyanenko et al. (2007), Leto et al. (2008), Rickman et al. (2008), and Dybczyński and Królikowska (2011). In particular, the Jupiter–Saturn system was shown to be a tangible barrier leading to a redistribution of the density of comets in the cloud. Apart from the flybys of stars, the Oort cloud is perturbed by giant molecular clouds and the gravitational tide produced by the Galactic attraction (Dybczyński 2002, 2005; Martinez-Barbosa et al. 2017).Matthews (1994), Mülläri and Orlov (1996), Garcia-Sánchez et al. (1999, 2001), Bobylev (2010a, 2010b), Anderson and Francis (2012), Dybczyński and Berski (2015), Bailer-Jones (2015), Feng and Bailer-Jones (2015), and Mamajek et al. (2015) searched for the close encounters of stars with the solar orbit using various observational data. As a result, ∼200 Hipparcos (1997) stars that either encountered or would encounter with the Solar system closer than 5 pc in the time interval from -10 to +10 Myr were revealed. Several candidates have a high probability of their penetration into the Oort cloud region.For example, the star HIP 85605 (a dwarf of spectral type ∼K4) may encounter with the Solar system within a distance d_m∼0.1 pc at tm ∼330 thousand years (Bailer-Jones 2015); for the star HIP 63721 (F3V) these parameters are d_m∼0.2 pc and t_m∼150 thousand years (Bailer-Jones 2015; Dybczyński and Berski 2015). At the same time, all authors point out that the parallaxes of the stars HIP 85605 and HIP 63721 are very unreliable.The low-mass binary system WISE J072003.20-084651.2 (M9.5 + T5) with a total mass of ∼0.15 M_⊙ is of interest. Mamajek et al. (2015) estimated d_m=0.25^+0.11_-0.07 pc and t_m=-70^+0.15_-0.10 thousand years for it.The star GL 710 (K7V), for which the encounter parameters found from the Hipparcos data are d_m=0.31±0.17 pc and t_m=1447±60 thousand years (Garcia-Sánchez et al. 2001; Bobylev 2010a), is well known. Completely new estimates of these parameters have recently been obtained by Berski and Dybczyński (2016) using the parallaxes and proper motions measured in the Gaia experiment (Prusti et al. 2016): d_m=0.065±0.030 pc and t_m=1350±50 thousand years. Thus, the star GJ 710 remains the record-holder in terms of encounters among the candidate stars with more or less reliable measurements.New possibilities in searching for stars closely encountering with the Solar system are associated with the appearance of the first version of the Gaia catalogue. This catalogue was produced from a combination of the data in the first year of Gaia observations with the positions and proper motions of Tycho-2 stars (Hog et al. 2000). It is designated as TGAS (Tycho–Gaia Astrometric Solution, Michalik et al. 2015; Brown et al. 2016; Lindegren et al. 2016) and contains the parallaxes and proper motions of ∼2 million bright stars. The TGAS version has no stellar radial velocities; therefore, specialized RAVE type catalogues of radial velocities should be invoked to calculate the total space velocities of stars.The goal of this paper is the search for candidate stars closely encountering with the Sun based on the present-day data on stars. For this purpose, we use the Gaia DR1 and RAVE5 catalogues (Kunder et al. 2017). We construct the orbits of stars using an improved model Galactic gravitational potential (Bajkova and Bobylev 2016).§.§ DATAThe random errors of the parameters included in the Gaia DR1 catalogue are either comparable to or smaller than those given in the Hipparcos and Tycho-2 catalogues. The mean parallax errors are ∼0.3 mas (milliarcseconds). For most stars of the TGAS version the mean proper motion error is ∼1 mas yr^-1 (milliarcseconds per year), but for quite a few(∼94 000) stars common to the Hipparcos catalogue this error is smaller by an order of magnitude, 0.06 mas yr^-1.The RAVE (RAdial Velocity Experiment) project (Steinmetz et al. 2006) is devoted to determining the radial velocities of many faint stars. The observations in the southern hemisphere at the 1.2-m Schmidt telescope of the Anglo-Australian Observatory started in 2003. Five data releases of this catalogue (DR1–DR5) have been published since then. The mean radial velocity error is ∼3 km s^-1. The RAVE DR5 version (Kunder et al. 2017) contains data on 457 588 stars; there is an overlap with the TGAS catalogue for about half of these stars.In this paper we use the data set from Hunt et al. (2016), where the common stars from the TGAS and RAVE DR5 catalogues were studied. There are trigonometric parallaxes and proper motions from the TGAS catalogue and radial velocities from the RAVE DR5 catalogue for 216 201 stars in this list. The stars with a relative distance error of more than 10% were excluded when the sample was produced. Hunt et al. (2016) used both photometric distance estimates from the RAVE catalogue and trigonometric parallaxes from the TGAS catalogue. In this paper we calculate all distances to the stars using their trigonometric parallaxes.§.§ METHODS §.§.§ Model Galactic Gravitational PotentialThe expressions for the potentials are considered in a cylindrical coordinate system (R,ψ,z) with the coordinate origin at the Galactic center. In a rectangular coordinate system (x,y,z) with the coordinate origin at the Galactic center the distance to a star (spherical radius) will be r^2=x^2+y^2+z^2=R^2+z^2.The equations of motion for a test particle in an axisymmetric gravitational potential Φ can be derived (see Appendix A to Irrgang et al. (2013)) from the Lagrangianof the system:[ (R,z,Ṙ,ψ̇,ż)=0.5(Ṙ^2+(Rψ̇)^2+ż^2)-Φ(R,z). ]Introducing the canonical momenta[ p_R=∂/∂Ṙ=Ṙ,p_ψ=∂/∂ϕ̇=R^2ψ̇, p_z=∂/∂ż=ż, ]we will obtain the Lagrange equations as a system of six first-order differential equations: [Ṙ=p_R,; ψ̇=p_ψ/R^2,;ż=p_z,; ṗ_̇Ṙ=-∂Φ(R,z)/∂ R +p_ψ^2/R^3,;ṗ_̇ψ̇=0,;ṗ_̇ż=-∂Φ(R,z)/∂ z. ]The fourth-order RungeKutta algorithm was used to integrate Eqs. (3).In this paper we use a three-component model Galactic gravitational potential: Φ=Φ_b+Φ_d+Φ_h,where the subscripts denote the bulge, disk, and halo, respectively.In accordance with the convention adopted in Allen and Santillán (1991), we express the gravitational potential in units of 100 km^2 s^-2, the distances in kpc, and the masses in units of the Galactic mass M_gal=2.325× 10^7 M_⊙, corresponding to the gravitational constant G=1.The bulge, Φ_b(r), and disk, Φ_d(r(R,z)), potentials are represented by the expressions from Miyamoto and Nagai (1975): Φ_b(r)=-M_b/(r^2+b_b^2)^1/2,Φ_d(R,z)=-M_d/{R^2+[a_d+(z^2+b_d^2)^1/2]^2}^1/2,where M_b and M_d are the masses of the components, b_b, a_d, and b_d are the scale lengths of the components in kpc.The expression for the halo potential was derived by Irrgang et al. (2013) based on the expression for the halo mass from Allen and Martos (1986):m_h(<r) = {[ M_h(r/a_h)^γ/1+(r/a_h)^γ-1,if r≤Λ; M_h(Λ/a_h)^γ/1+(Λ/a_h)^γ-1=const, ifr>Λ ]},It slightly differs from that given in Allen and Santillán (1991) and isΦ_h(r) = {[ M_h/a_h( 1/(γ-1)ln(1+(r/a_h)^γ-1/1+(Λ/a_h)^γ-1)- (Λ/a_h)^γ-1/1+(Λ/a_h)^γ-1),ifr≤Λ; -M_h/r(Λ/a_h)^γ/1+(Λ/a_h)^γ-1,if r>Λ, ].where M_h is the mass, a_h is the scale length, the Galactocentric distance is Λ=200 kpc, and the dimensionless coefficient γ=2.0.The model parameters M_b, M_d, M_h, b_b, a_d, b_d, and a_h were taken from Bajkova and Bobylev (2016), where they were refined based on a large set of present-day observational data. Their values are given in Table 1.The circular rotation velocity of the Galaxy at the adopted Galactocentric distance of the Sun R_0=8.3 kpc is 244 km s^-1 (Bajkova and Bobylev 2016). The peculiar velocity components of the Sun relative to the local standard of rest were taken to be (U_⊙,V_⊙,W_⊙)_LSR=(10,11,7) km s^-1 based on the results from Bobylev and Bajkova (2014a) in agreement with the results from Schönrich et al. (2010). We take into account the Suns height above the Galactic plane Z_⊙=16 pc (Bobylev and Bajkova 2016). We neglect the star–Sun gravitational interaction. In the case where the spiral density wave is taken into account (Lin and Shu 1964; Lin et al. 1969), the following term (Fernandez et al. 2008) is added to the right-hand side of Eq. (2):Φ_sp (R,θ,t)= Acos[m(Ω_p t-θ)+χ(R)].Here, A= (R_0Ω_0)^2 f_r0tan i/m,χ(R)=- m/tan iln(R/R_0)+χ_⊙,where A is the amplitude of the spiral potential, f_r0 is the ratio between the radial component of the force due to the spiral arms and that due to the general Galactic field, Ω_p is the angular velocity of the spiral pattern, m is the number of spiral arms, i is the pitch angle of the arms (i<0 for a wound pattern), χ is the radial phase of the spiral wave (the arm center then corresponds to χ=0^∘), and χ_⊙ is the radial phase of the Sun in the spiral wave.The following spiral wave parameters were taken as a first approximation: [ m=4,i=-13^∘,f_r0=0.05,χ_⊙=-140^∘,Ω_p=20 km s^-1 kpc^-1 ]We used this set of parameters in Bobylev and Bajkova (2014b), where a broad overview of the parameter selection problem is given. If necessary, some of them, in particular, χ_⊙ and Ω_p, can be varied. §.§.§ Monte Carlo SimulationsIn accordance with the method of Monte Carlo simulations, for each object we calculate a set of orbits by taking into account the random errors in the input data. For each star we calculate the parameter of the encounter between the stellar and solar orbits d(t)=√(Δ x^2(t)+Δ y^2(t)+Δ z^2(t)). The closest encounter is characterized by two parameters, d_m and t_m. The errors of the stellar parameters are assumed to be distributed as a normal law with a dispersion σ. The errors are added to the stellar equatorial coordinates, proper motion components, parallaxes, and radial velocities.§.§ RESULTS AND DISCUSSIONFor each of the 216 201 stars we constructed its orbit relative to the Sun in the time interval from -15 to +15 Myr. From the entire list we selected the stars with an encounter parameter d_m<1 pc. We divided them into two samples, depending on the quality of the measured initial velocities and positions.Sample 1 includes the stars for which the relative errors of the measured initial velocities and positions do not exceed 10% and the measurement error of the radial velocity <15 km s^-1. The parameters of the stars from sample 1 are given in Table 2, where columns 1–9 give, respectively, the ordinal star number, the Tycho identification number (the Hipparcos number is also provided, if available), the stellar equatorial coordinates α and δ, the proper motion components μ_αcosδ, and μ_δ with their measurement errors, the trigonometric parallax with its measurement error, the radial velocity with its measurement error, the signal-to-noise (S/N) ratio of the spectrum when determining the radial velocity copied from column 12 of the RAVE5 catalogue, and the encounter parameters d_m and t_m we found. Many stars with huge radial velocities, which is most likely due to the erroneous measurements in the RAVE catalogue, enter into the sample. We collected them into separate Table 3. As can be seen, the S/N ratios for the stars from Table 2 exceed those for the stars from Table 3 by an order of magnitude.Several radial velocity determinations are given for some of the stars in the RAVE5 catalogue. These include, for example, TYC 5116-143-1, TYC 7567-304-1, TYC 5302-849-1, TYC 9163-286-1 or TYC 7978-659-1. The radial velocities usually differ by an order of magnitude. The encounter parameters calculated for several known radial velocity measurements are given in Bailer-Jones (2015) in such cases for each star. In contrast, we give only one value at which the closest encounter is obtained.Many of the stars from sample 2 have huge (more than 600 km s^-1) space velocities. By this parameter they can be attributed to hypervelocity stars that are capable of escaping from the Galactic attractive field. The escape velocity at the Galactocentric distance of the Sun slightly depends on the model gravitational potential and is ∼550 km s^-1 (see, e.g., Bajkova and Bobylev 2016). The following fact forces us to doubt that such high velocities are realistic. According to the well-known Kleiber theorem (Agekyan et al. 1962), the mean tangential, V_t, and radial, V_r, velocities are related by the relation |V_t|=0.5π|V_r|. Although this relation is valid in the statistical sense, it does not hold at all in our case, because for all stars from sample 2 |V_t|=4.74 r |μ|<100 km s^-1, where μ=√(μ^2_αcosδ+μ^2_δ).Note that the flags c_1–c_20 describing the morphology of the spectra are specified in the RAVE catalogues. According to these characteristics, all of the detected stars with radial velocities |V_r|>300 km s^-1 have very low signal-to-noise ratios, and the spectra for all these stars are either with problems in their continuum (c_1,2,3=“c”) or peculiar (c_1,2,3=“p”). This leads us to conclude that the radial velocities of such stars have been measured very poorly.Note the star TYC 4888-146-1 (absent in our tables), for which four radial velocities found from four good (c_1,2,3=“n”, the spectrum of a normal star) spectra taken at different epochs are given in the RAVE5 catalogue. All four values are close to V_r=-15 km s^-1, and one value (V_r=1897 km s^-1) was found from a spectrum with problems in its continuum (c_1,2,3=“c”). All of this reinforces our attitude to the stars with huge radial velocities from the RAVE5 catalogue as problem ones.The measurement error of the radial velocity σ_V_r is unknown for several stars. In such cases, we adopted σ_V_r=±30 km s^-1 for them when estimating the errors of the encounter parameters d_m and t_m by the Monte Carlo method.Figure 1 gives a histogram of the distribution of model minimum distances d_m obtained for the star GJ 710 by the Monte Carlo method for 1000 trajectories, 100 model trajectories of GJ 710 relative to the Sun.Figure 2 gives the trajectories of the encounter of the nine stars from Table 3 with the solar orbit. All these trajectories resemble thin vertical lines attributable to large flyby velocities. Therefore, the impact of these stars on Oort cloud comets can only be very brief with minor consequences.Bailer-Jones (2015) analyzed a large sample of stars with the radial velocities from the RAVE4 catalogue for close encounters. For example, for the star TYC 5116-143-1 (HIP 91012) (Table 3) he found d_m=0.48 pc and t_m=302 thousand years when using the radial velocity V_r=-364 km s^-1 (as we did). To integrate the stellar orbits, Bailer-Jones (2015) used a model Galactic gravitational potential different from ours. In spite of this, we can conclude that we have results very close to those obtained by other authors when using the same observational data. As can be seen from Table 3, interesting encounter parameters, with small random errors, were obtained for this star. However, the RAVE5 catalogue provides another radial velocity for it, V_r=-36.5±18.6 km s^-1, obtained from a different but, just as in the former case, poor spectrum. In addition, according to the measurements by Nordström et al. (2004), V_r=-16.8±0.4 km s^-1. We found that with such a radial velocity the encounter of TYC 5116-143-1 (HIP 91012) would not be close (d_m>5 pc).The star GJ 710 is known quite well. For example, T_ eff= 4109 K and log(g)=4.91 cm s^-2 for it (Franchini et al. 2014), i.e., this is an orange dwarf with a mass of ∼0.6M_⊙. Since there is no radial velocity for it in the RAVE catalogues, we took its previously known value from the catalogue by Gontcharov (2006). Using the new trigonometric parallax and proper motions from the Gaia DR1 catalogue, we found the encounter parameters d_m=0.063±0.044 pc and t_m=1385±52 thousand years, which are in excellent agreement with the estimates obtained by Berski and Dybczyński (2016) using the same data and a similar technique for calculating the Galactic stellar orbits, d_m=0.065±0.030 pc and t_m=1350±50 thousand years.The following parameters are given in the RAVE5 catalogue for the other stars from Table 2:T_ eff=5940 K, log(g)=4.08 cm s^-2 for TYC 8088-631-1 andT_ eff=4750 K, log(g)=4.000 cm s^-2 for TYC 6528-980-1. They show that these stars are dwarfs.Previously, Dybczyński (2006) and Jimeénez-Torres et al. (2011) performed numerical simulations of the evolution of comet orbits using the penetration of a star like GJ 710 with a mass of 0.6M_⊙ and d_m=0.3 pc into the Oort cloud as an example and found a small stream of comets toward the major planets from the impact of a model star that is difficult to separate from the stream of comets caused by a Galactic tide. However, using a closer encounter parameter, d_m=0.065 pc, Berski and Dybczyński (2016) showed that a noticeable stream with a density of about ten comets per year with a duration of 2–4 Myr could emerge.§.§ CONCLUSIONSWe searched for the stars that encountered or would encounter with the Solar system closer than 1 pc. For this purpose, we took more than 216 000 stars with the measured proper motions and trigonometric parallaxes from the Gaia DR1 catalogue and their radial velocities from the RAVE5 catalogue. The orbits were integrated over the time interval from -15 to +15 Myr using a model Galactic gravitational potential that includes an axisymmetric part (bulge, disk, and halo)with the addition of a nonaxisymmetric component that allows for the influence of the Galactic spiral density wave.We found the stars for which encounters with the Solar system closer than 1 pc are possible. For the bulk of this list such an analysis has been made for the first time. We divided all of the stars found into two samples.Sample 1 contains the stars with small errors of the input data and low radial velocities. The star GJ 710, for which the minimum distance is d_m=0.063±0.044 pc at time t_m=1385±52 thousand years, is the record-holder in this sample. This confirms the estimates that have recently been obtained for GJ 710 from similar data by Berski and Dybczyński (2016). The first sample includes two more stars, TYC 8088-631-1 and TYC 6528-980-1, with d_m<1 pc, which, however, are estimated with large errors. For example, d_m=0.37±1.18 pc and t_m=-2792±66 thousand years were found for TYC 8088-631-1.The remaining stars enter into sample 2. They are characterized by unrealistically large space velocities and their random errors. This is because poor quality spectra were used for these stars in the RAVE catalogues. Therefore, the results obtained from the stars of sample 2 are much less trustworthy than the previous ones. §.§.§ ACKNOWLEDGMENTSWe are grateful to the referee for the useful remarks that contributed to an improvement of the paper. We are thankful to J. Hunt for the provided data. This work was supported by the Basic Research Program P–7 of the Presidium of the Russian Academy of Sciences, the “Transitional and Explosive Processes in Astrophysics” Subprogram. REFERENCES 1. T.A. Agekyan, B.A. Vorontsov-Velyaminov, V.G. Gorbatskii, A.N. Deich, V.A. Krat, O.A. Melnikov, and V.V. Sobolev, Course of Astrophysics and Stellar Astronomy (Fizmatgiz, Moscow, 1962) [in Russian]. 2. C. Allen and M.A. Martos, Rev. Mex. Astron. Astrofis. 13, 137 (1986). 3. C. Allen and A. Santillán, Rev. Mex. Astron. Astrofis. 22, 255 (1991).4. E. Anderson and C. Francis, Astron. Lett. 38, 331 (2012).5. C.A.L. Bailer-Jones, Astron. Astrophys. 575, 35 (2015).6. A.T. Bajkova and V.V. Bobylev, Astron. Lett. 42, 567 (2016).7. F. Berski and P.A. Dybczyński, Astron. Astrophys. 595, L10 (2016).8. V.V. Bobylev, Astron. Lett. 36, 220 (2010a).9. V.V. Bobylev, Astron. Lett. 36, 816 (2010b). 10. V. Bobylev and A. Bajkova, Mon. Not. R. Astron. Soc. 441, 142 (2014a).11. V.V. Bobylev and A.T. Bajkova, Astron. Lett. 40, 352 (2014b).12. V.V. Bobylev and A.T. Bajkova, Astron. Lett. 42, 1 (2016).13. A.G.A. Brown, A. Vallenari, T. Prusti, J. de Bruijne, F. Mignard, R. Drimmel, et al. (GAIA Collab.), Astron. Astrophys. 595, 2 (2016).14. P.A. Dybczyński, Astron. Astrophys. 396, 283 (2002).15. P.A. Dybczyński, Astron. Astrophys. 441, 783 (2005).16. P.A. Dybczyński, Astron. Astrophys. 449, 1233 (2006).17. P.A. Dybczyński and M. Królikowska, Mon. Not. R. Astron. Soc. 416, 51 (2011).18. P.A. Dybczyński and F. Berski, Mon. Not. R. Astron. Soc. 449, 2459 (2015).19. V.V. Emelyanenko, D. J. Asher, and M. E. Bailey, Mon. Not. R. Astron. Soc. 381, 779 (2007).20. F. Feng and C.A.L. Bailer-Jones, Mon. Not. R. Astron. Soc. 454, 3267 (2015).21. D. Fernandez, F. Figueras, and J. Torra, Astron. Astrophys. 480, 735 (2008).22. M. Franchini, C. Morossi, P. di Marcantonio, M.L. Malagnini, and M. Chavez, Mon. Not. R. Astron. Soc. 442, 220 (2014).23. J. Garcia-Sánchez, R.A. Preston, D. L. Jones, P.R. Weissman, J.-F. Jean-Francois, D.W. Latham, and R.P. Stefanik, Astron. J. 117, 1042 (1999).24. J. Garcia-Sánchez, P.R. Weissman, R.A. Preston, D.L. Jones, J.-F. Lestrade, D.W. Latham, R.P. Stefanik, and J.M. Paredes, Astron. Astrophys. 379, 634 (2001).25. G. A. Gontcharov, Astron. Lett. 32, 759 (2006).26. J. G. Hills, Astron. J. 86, 1730 (1981).27. The Hipparcos and Tycho Catalogues, ESA SP–1200 (1997).28. E. Hog, C. Fabricius, V.V. Makarov, U. Bastian, P. Schwekendiek, A. Wicenec, S. Urban, T. Corbin, and G. Wycoff, Astron. Astrophys. 355, L27 (2000).29. J.A.S. Hunt, J. Bovy, and R.G. Carlberg, Astrophys. J. 832, L25 (2016).30. A. Irrgang, B. Wilcox, E. Tucker, and L. Schiefelbein, Astron. Astrophys. 549, 137 (2013).31. J.J. Jiménez-Torres, B. Pichardo, G. Lake, and H. Throop, Mon. Not. R. Astron. Soc. 418, 1272 (2011).32. A. Kunder, G. Kordopatis, M. Steinmetz, T. Zwitter, P. McMillan, L. Casagrande, H. Enke, J. Wojno, et al., Astron. J. 153, 75 (2017).33. G. Leto, M. Jakubik, T. Paulech, L. Neslušan, and P.A. Dybczyński, Mon. Not. R. Astron. Soc. 391, 1350 (2008).34. C.C. Lin and F.H. Shu, Astrophys. J. 140, 646 (1964). 35. C.C. Lin, C. Yuan and F.H. Shu, Astrophys. J. 155, 721 (1969).36. L. Lindegren, U. Lammers, U. Bastian, J. Hernandez, S. Klioner, D. Hobbs, A. Bombrun, D. Michalik, et al., Astron. Astrophys. 595, 4 (2016).37. E.E. Mamajek, S.A. Barenfeld, V.D. Ivanov, A.Y. Kniazev, P. Väisänen, Y. Beletsky, and H.M.J. Boffin, Astrophys. J. Lett. 800, L17 (2015).38. C. A. Martinez-Barbosa, L. Jýlková, S. Portegies Zwart, and A.G. A. Brown, Mon. Not. R. Astron. Soc. 464, 2290 (2017). 39. R. A. J. Matthews, R. Astron. Soc. Quart. J. 35, 1 (1994). 40. D. Michalik, L. Lindegren, and D. Hobbs, Astron. Astrophys. 574, 115 (2015). 41. M. Miyamoto and R. Nagai, Publ. Astron. Soc. Jpn. 27, 533 (1975). 42. A. A. Mülläri and V. V. Orlov, Earth, Moon, Planets 72, 19 (1996).43. B. Nordström, M. Mayor, J. Andersen, J. Holmberg, F. Pont, B. R. Jorgensen, E. H. Olsen, S. Udry, and N. Mowlavi, Astron. Astrophys. 418, 989 (2004).44. J.H. Oort, Bull. Astron. Inst. Netherland 11 (408), 91 (1950).45. T. Prusti, J.H. J. de Bruijne,A.G. A. Brown, A. Vallenari, C. Babusiaux, C. A. L. Bailer-Jones, U. Bastian, M. Biermann, et al. (GAIA Collab.), Astron. Astrophys. 595, 1 (2016).46. H. Rickman, M. Fouchard, C. Froeschlé, and G. B. Valsecchi, Sel. Mech. Dyn. Astron. 102, 111 (2008).47. R. Schönrich, J. Binney, and W. Dehnen, Mon. Not. R. Astron. Soc. 403, 1829 (2010).48. M. Steinmetz, T. Zwitter, A. Siebert, F. G. Watson, K. C. Freeman, U. Munari, R. Campbell, M. Williams, et al., Astron. J. 132, 1645 (2006).49. J. T. Wickramasinghe and W. M. Napier, Mon. Not. R. Astron. Soc. 387, 153 (2008). | http://arxiv.org/abs/1706.08867v1 | {
"authors": [
"V. V. Bobylev",
"A. T. Bajkova"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170627141044",
"title": "Searching for Stars Closely Encountering with the Solar System Based on Data from the Gaia DR1 and RAVE5 Catalogues"
} |
1Photonic/Wireless Convergence Components Research Division, Electronics and Telecommunications Research Institute, Daejeon, 34129, South Korea2School of Advanced Device Technology, University of Science & Technology, Daejeon, 34113, South Korea*[email protected],†[email protected] polarization-based BB84 quantum key distribution (QKD) systems utilize multiple lasers to generate one of four polarization quantum states randomly. However, random bit generation with multiple lasers can potentially open critical side channels, which significantly endangers the security of QKD systems. In this paper, we show unnoticed side channels of temporal disparity and intensity fluctuation, which possibly exist in the operation of multiple semiconductor laser diodes. Experimental results show that the side channels can enormously degrade security performance of QKD systems. An important system issue for the improvement of quantum bit error rate (QBER) related with laser driving condition is furtherly addressed with experimental results. (270.5568) Quantum cryptography; (060.5565) Quantum communication; (140.5960) Semiconductor lasers.99lydersen2010hacking Lydersen, L., Wiechers, C., Wittmann, C., Elser, D., Skaar, J., and Makarov, V., "Hacking commercial quantum cryptography systems by tailored bright illumination", Nat. Photon. 4, 686-689 (2010).ko2016informatic Ko, H., Lim, K., Oh, J., and Rhee, J. K. K., "Informatic analysis for hidden pulse attack exploiting spectral characteristics of optics in plug-and-play quantum key distribution system", Quant. Inf. Proc. 15, 4265-4282 (2016).nauerth2009information Nauerth, S., Furst, M., Schmitt-Manderbach, T., Weier, H., and Weinfurter, H., "Information leakage via side channels in freespace BB84 quantum cryptography", New J. Phys. 11, 065001 (2009).rau2015spatial Rau, M., Vogl, T., Corrielli, G., Vest, G., Fuchs, L., Nauerth, S., and Weinfurter, H., "Spatial mode side channels in free-space QKD implementations", IEEE J. Sel. Top. Quantum Electron. 21, 187-191 (2015).nakata2016intensity Nakata, K., Tomita, A., Fujiwara, M., Yoshino, K. I., Tajima, A., Okamoto, A., and Ogawa, K., "Intensity fluctuation of a gain-switched semiconductor laser for quantum key distribution systems", Opt. Express 25, 622-634 (2017).BB84 Bennett. C. H, and Brassard. G., "Quantum cryptography: Public key distribution and coin tossing", in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, pp. 175-179 (1984).nauerth2013air Nauerth, S., Moll, F., Rau, M., Fuchs, C., Horwath, J., Frick, S., and Weinfurter, H., "Air-to-ground quantum communication", Nat. Photon. 7, 382-386 (2013).wang2013direct Wang, J. Y., Yang, B., Liao, S. K., Zhang, L., Shen, Q., Hu, X. F.,Wu, J. C., Yang, S. J., Jiang, H., Tang, Y. L., Zhong, B., Liang, H. Liu, W. Y., Hu, Y. H., Huang, Y. M., Qi, B., Ren, J. G., Pan, G. S., Yin, J. Jia, J. J., Chen, Y. A., Chen, K., Peng, C. Z., and Pan, J. W., "Direct and full-scale experimental verifications towards ground-satellite quantum key distribution", Nat. Photon. 7, 387-393 (2013).peng2007experimental Peng, C. Z., Zhang, J., Yang, D., Gao, W. B., Ma, H. X., Yin, H., Zeng, H. P., Yang, T., Wang, X. B., and Pan, J. W., "Experimental long-distance decoy-state quantum key distribution based on polarization encoding", Phys. Rev. Lett. 98, 010505 (2007).colren2012diode Coldren, L. A., Corzine, S. W., and Mashanovitch, M. L., Diode lasers and photonic integrated circuits (John Wiley and Sons, 2012).Dušek2000Unambiguous Dusek, M., Jahma, M., and Lutkenhaus, N., "Unambiguous state discrimination in quantum cryptography with weak coherent states", Phys. Rev. A 62, 022306 (2000).brassard2000limitations Brassard, G., Lutkenhaus, N., Mor, T., and Sanders, B. C., "Limitations on practical quantum cryptography", Phys. Rev. Lett. 85, 1330 (2000).hwang2003quantum Hwang, W. Y., "Quantum key distribution with high loss: toward global secure communication", Phys. Rev. Lett. 91, 057901 (2003). lo2005decoy Lo, H. K., Ma, X., and Chen, K., "Decoy state quantum key distribution", Phys. Rev. Lett. 94, 230504.(2005).wang2008general Wang, X. B., Peng, C. Z., Zhang, J., Yang, L., and Pan, J. W., "General theory of decoy-state quantum cryptography with source errors", Phys. Rev. A 77, 042311 (2008).hayashi2014security Hayashi, M., and Nakayama, R., "Security analysis of the decoy method with the Bennett-Brassard 1984 protocol for finite key lengths", New J. Phys. 16, 063009 (2014).mizutani2015finite Mizutani, A., Curty, M., Lim, C. C. W., Imoto, N., and Tamaki, K., "Finite-key security analysis of quantum key distribution with imperfect light sources", New J. Phys. 17, 093011 (2015). § INTRODUCTIONTrials of secure information transfer between two distant parties have driven the development of the field of quantum key distribution (QKD) whose security is guaranteed by the nature of quantum physics. Sender and receiver, normally called Alice and Bob, can notice the existence of an unauthenticated eavesdropper, Eve, by investigating quantum bit error rate (QBER) of some fractions of their distilled keys. However, unconditional security of QKD is only guaranteed under the assumption of perfect implementations. In practical systems, many unnoticed degree of freedoms, so called side channels, usually open loopholes for eavesdroppers, which degrades the performance of secret key exchanges <cit.>. Polarization is one of the most prevalent physical observables utilized in the implementations of BB84 <cit.> QKD protocol especially for free-space links <cit.>. In most polarization-based QKD systems, Alice randomly turns on one of four semiconductor laser diodes for each time slot to generate one of four polarization quantum states. Previously, several loopholes possibly occurred in the configuration of laser diodes have been studied. In <cit.>, physical quantities such as spatial, temporal, and spectral characteristics which are possibly different among multiple laser diodes were discussed, which need to be identical to avoid information leakage. Also, side information caused by the disparity of spatial mode due to misalignment among multiple lasers was addressed in <cit.>. Very recently, intensity fluctuation of a single semiconductor laser diode was reported in periodic pulse generation with a single laser for phase-encoding QKD system <cit.>. In this paper, we firstly report how random bit generation with multiple semiconductor laser diodes opens critical side channels in a polarization-based QKD system. In a semiconductor laser diode, the level of driving current and initial carrier density are key parameters which directly define the shape of output pulses. Contrary to periodic pulse generation, aperiodic pulses by random bit generation with multiple laser diodes result in different initial carrier density conditions for each pulse. It causes two side channels, which are temporal disparity and intensity fluctuation among output pulses. We discuss how these phenomena threat the security performance of QKD systems based on simulation and experimental results. Moreover, we furtherly address an important operation issue of multiple laser diodes in terms of QBER performance in QKD systems. While previously discussed loopholes by utilization of multiple laser diodes were focused on discriminating four laser diodes <cit.>, side information covered in this paper is provided by correlations between pulses coming from the same laser, even though Alice turns on one of four lasers with true randomness. This characteristics unavoidably appears in QKD systems utilizing multiple semiconductor lasers with direct modulation. The remaining part of this paper is structured as follows. In sect. 2, configuration of polarization-based free-space QKD system and operational principles of semiconductor laser focusing on carrier density and photon density are described. In sect. 3, mechanisms of output pulse variations by random bit generation with multiple lasers are discussed with simulation and experimental results. Also, diverse aspects of information leakage caused by the aforementioned loophole are described. An operation issue of multiple laser diodes considering system performance is furtherly addressed in sect.4. Finally, we finish this paper in sect 5. with some concluding remarks.§ POLARIZATION-BASED FREE-SPACE BB84 QKD SYSTEM§.§ General configuration Schematic diagram of a practical free-space BB84 QKD system is shown in Fig. <ref>. Alice randomly turns on one of four lasers for each time slot to generate polarized photon pulses which are passively created through optical components such as polarizing beam splitter, beam splitter, and half-wave plate. The polarized photon pulses are transmitted to Bob through a free-space channel after they attenuated to the single photon level. Each polarization state is passively decoded as it passes through the Bob's optical system. In a polarization-based BB84 QKD system, four polarization states are generated from four different laser diodes, whereas only single laser diode with a phase modulator is utilized in phase-encoding QKD systems. This is because polarization states used in BB84 protocol are technically much easier to be configured with passive optics, not with a polarization modulator. In decoy BB84 QKD systems, the number of laser diode can be increased to eight or more according to the system configuration <cit.>. §.§ Operation principle of multiple semiconductor laser diodes An example of L-I curve of a semiconductor laser is shown in Fig. <ref>. When driving current is lower than the threshold level I_th, spontaneous emission is a major process which generates incoherent photons with small output power. If driving current becomes higher than I_th, stimulated emission starts which generates high power coherent photons. When a laser is under stimulated emission, carrier density goes higher than its threshold level, N_th, and injected carriers are changed into output photons with certain efficiency. Note that increasing rate of output power under stimulated emission decreases as current increases due to the thermal roll-over effect <cit.> as shown in Fig. <ref>. Short optical pulses are produced by injection of short current pulses into a semiconductor laser. After the injection of current pulses stops, stimulated photon power rapidly decreases and carrier density becomes lower than N_th. Carrier density keeps decreasing according to the carrier lifetime if no additional driving current pulse is injected. After a period of time much longer than the carrier lifetime, carrier density becomes a steady level, N_DC, which is determined by the level of DC biased current.In a phase-encoding BB84 QKD system, a single laser source has nothing to do with random bit generation because quantum phase information is determined by a phase modulator. Thus, we periodically inject same electrical pulses into a single laser to generate identical photon pulses as shown in Fig. <ref>(left). On the other hand, in a polarization-based BB84 QKD system using four lasers, each laser diode is switched on with 25% probability for each time slot to generate random bit information. Thus, each laser diode can be operated in aperiodic manner as shown in Fig. <ref>(right). Even though we injected the identical electrical pulses to the same laser, output photon pulses are no longer identical as shown in Fig. <ref>(right), which were almost indistinguishable in the periodic case Fig. <ref>(left). Output pulses are different from one another in two perspectives, which are temporal disparity and intensity fluctuation. Temporal disparity indicates that starting points of output pulses are different with respect to current injection in time domain. Even though it is not clearly seen in the Fig. <ref>, one can see the phenomenon in simulated and experimental results in the following sections. Another point, intensity fluctuation among output pulses, is easily seen in the Fig. <ref>. These two behaviors among output pulses open unnoticed side channels in QKD systems, which must be properly compensated to guarantee the security. § BEHAVIOR OF PHOTON PULSES CAUSED BY RANDOM BIT GENERATION§.§ Behavior of output photon pulses Even though Alice drives the same level of current pulses to the same laser diode, temporal behavior of the output pulses can be varied if the initial carrier density is different. Dynamics of carrier and photon density can be simulated with the following rate equations of semiconductor laser diode using common parameters <cit.>.dN/dt = η_iI/qV - (R_sp+R_nr) - v_ggN_p, dN_p/dt = (Γ v_gg - 1/τ_p)N_p + Γ R_sp^',where N is carrier density, N_p is photon density, I is current, q is electrical charge, and τ_p is photon lifetime. R_sp, R_nr, and R_sp^' are recombination terms. η_i, V, Γ, g, and v_g are material parameters of a laser diode. Simulation result of the rate equations is shown in Fig. <ref> with the assumption of the steady initial carrier density, N_DC. Here, Alice generates current pulses of I_AC with bias current I_DC as described in the Fig. <ref>. As the first current pulse is injected, carrier density, the solid line, reaches to N_th after t_1 and stimulated emission process begins, which results a photon pulse after a short time. After the AC current pulse disappears, carrier density rapidly decreases, which stops stimulated emission. In this region, carrier density decreases down to the steady level of DC point, N_DC, after a period of time much longer than the carrier lifetime. If the next electrical pulse injection occurs after the carrier density becomes the steady level, output pulse will be identical with the previous one.However, the situation changes if the following electrical pulse is injected before the carrier density reaches down to the steady level as shown in Fig. <ref>. We injected the second current pulse after 2ns from the first one, which drives carrier density to rise again. Here, initial carrier density is somewhat higher than the steady level, N_DC, as shown in the Fig. <ref>. In this case, it takes a shorter time for carrier density to reach N_th (t_2 < t_1), which makes stimulated emission process occur earlier. It causes output pulse to appear earlier than the previous one with respect to current injection time. Moreover, the power of the output pulse is relatively stronger than the previous one because carrier density rises relatively higher. Aforementioned phenomena may not be a problem if lasers are operated in a periodic manner because initial carrier density can be same for each repetition even if it is somewhat higher than the N_DC level. Also, even in aperiodic operations, temporal disparity and intensity fluctuation could not appear under low-speed systems where minimum time interval between two pulses is much longer than the carrier lifetime. However, under aperiodic operation in high-speed systems, as shown in the Fig. <ref>, initial carrier densities are different according to the time interval between two consecutive pulses, which causes temporal disparity and intensity fluctuation. If the time interval is short, initial carrier density for the following pulse is relatively high whereas it would be relatively low if the time interval becomes longer. Due to this mechanism, random bit generation by multiple laser diodes can cause temporal disparity and intensity fluctuation in high-speed QKD systems, which opens side channels for potential eavesdroppers. §.§ Experimental setup Experimental setup for measuring the side channel effects is shown in Fig. <ref>. A polarization beam splitter (PBS) and a beam splitter (BS) are included in order to make the setup similar with the transmitter of polarization-based BB84 QKD system. We adopted a single longitudinal mode VCSEL with lasing wavelength of 787nm. We generated electrical AC current pulses of 500ps duration with the level of I_AC = 4I_th (FWHM) and three different levels of DC bias current I_DC∈{0,0.6I_th,0.9I_th}. AC and DC signals are combined with a wideband bias-tee. AC pulses are created by a FPGA system and properly amplified to meet the level of pulse current I_AC. The temporal shape of output pulses is measured by a photo-detector of 12GHz bandwidth and a oscilloscope (OSC) of 13GHz bandwidth. Also, half of the output power split at BS is monitored with a free-space type power-meter with the resolution of 100pW power.To investigate how time interval between two consecutive pulses impacts on the second output pulses, initial carrier density of the first pulse must be equally maintained because it affects the dynamics of the second pulses. Thus, we set repetition time block as 1us to make the initial carrier density to be in a steady level N_DC. Note that 1us is much longer than the carrier lifetime of InGaAs/GaAs lasers <cit.> used in the experiments. Under this condition, we can safely ensure that carrier density becomes N_DC for each iteration. Within the time block of 1us, we generated two consecutive current pulses with different time intervals from 2ns to 40ns and monitored the output behavior of the second pulses. §.§ Experimental results Behaviors of output pulses in time domain are shown in Fig. <ref>. Profiles of the second pulses for DC=0, DC=0.6I_th, and DC=0.9I_th are shown in Fig. <ref>(a), <ref>(b), and <ref>(c), respectively. Within the order of carrier lifetime, second pulses are generated in different time position with respect to the current injection time for DC=0 and DC=0.6I_th. Especially for DC=0 case, output pulses are temporally distinguishable when the time interval is smaller than the 20ns, which corresponds to 50MHz or higher speed operations. Moreover, pulse intensity becomes evidently smaller as time interval becomes longer. Temporal disparity and intensity fluctuation seems to be negligible after 20ns interval, which corresponds to 50MHz or lower speed systems.Temporal disparity and intensity fluctuation are diminished as DC bias level increases as shown in Fig. <ref>(b) and Fig. <ref>(c). This is because that N_DC level increases as DC level rises, which curtails time for carrier density to reach down to N_DC. In case of DC=0.6I_th, the phenomena become very weak after 10ns interval, which corresponds to 100MHz or lower speed operation. For DC=0.9I_th, temporal disparity and intensity fluctuation among the second pulses are hardly observed even in the short time interval of 2ns. Note that all photon pulses regardless of time intervals must be temporally overlapped to close the temporal side channels. Intensity of the second pulses was also measuredby a power-meter as shown in Fig. <ref>. We normalized output pulse powers with the one at 10 ns time interval. Optical power caused by spontaneous emission and the first pulse output are subtracted to properly measure the second output pulses exclusively. For a given DC bias current, intensity of the second pulse decreases as time interval increases. For a given time interval, intensity fluctuation evidently decreases as DC bias current goes close to threshold current level. In case of DC = 0, intensity fluctuation becomes within the range of ±2.5% for the time intervals longer than 20ns. For DC=0.6I_th, it becomes within the same range for time intervals longer than 10ns. Intensity fluctuation was almost negligible under DC=0.9I_th.§.§ Security threat Aforementioned phenomena can be extremely critical in terms of security of QKD. The most critical side information is correlation between consecutive pulses possibly revealed by investigating time position of each photon. For example, in case of Fig. <ref>(a), if a photon is detected in the time position around 250ps, Eve can know that the current pulse was injected to the same laser diode 2ns earlier with high probability. Thus, Eve can guess that the same polarization state must be generated at 2ns earlier. Such correlation between consecutive pulses caused by this behavior destroys randomness even if Alice generates quantum states with true randomness. Under this condition, unambiguous state discrimination (USD) <cit.> attack becomes possible even for the single photon states because Eve can collect the same polarization states from other time positions. Note that USD attack is only valid for the multi-photon states. Also, photon number splitting (PNS) <cit.> attack becomes even powerful because Eve can attain additional correlated bits in other time position by investigating a single multi-photon state. Thus, single photon states guaranteed to be secure by decoy methods <cit.> can be no longer safe, which enormously degrades security performance of QKD systems.Intensity fluctuation is also a potential threat as already discussed in <cit.> especially for decoy-BB84 QKD protocol. In <cit.>, general theory of security performance of QKD under intensity fluctuation is described with assumption that Eve can knows the relatively strong and weak pulses. It claims that secure key rate described in <cit.> can be decreased to 70.8 bits/s under intensity error upper bound of 5%, which was originally 136.3 bits/s. Security analysis with intensity fluctuation has been studied <cit.> in diverse situations because it can seriously decrease secure key rates. § LASER OPERATION CONSIDERING QKD PERFORMANCE To guarantee the unconditional security, all photon pulses must be temporally overlapped with same intensity. It seems that condition of high DC bias current improves the quality of temporal overlap and intensity fluctuation as shown in Fig. <ref> and Fig. <ref>. However, background photons caused by spontaneous emission process become non-negligible as DC bias current increases as shown in Fig. <ref>, which can directly increase QBER of QKD systems. We estimated this negative effect of spontaneous emission on the performance of BB84 QKD system as shown in Fig. <ref>. Four polarization sources of a BB84 QKD system were implemented using four VCSELs with the single longitudinal mode at the wavelength of 787nm. Laser diode1 is operated under the condition of bias current DC=0.9I_th, pulse current AC=4I_th with 500ps (FWHM), and clock rate of 100 MHz with 0.6 mean photon number per pulse. Note that decoy-BB84 QKD system normally uses mean photon number per pulse around 0.6 photon for signal states <cit.>. Other three lasers are biased at DC=0.9I_th without injection of AC current pulses. Additional ND filter of 18.5dB is adopted for channel loss. Detection outputs are counted using a Si-APD based single photon detector (PerkinElmer SPCM-AQ4C). We repeated the experiment without AC current pulsesinjection under the condition of bias current DC=0.9I_th, DC=0.6I_th, and DC=0 exclusively to estimate photon counts caused by the DC bias. The experiments were performed in a dark room condition to eliminate noise by other photons. Experimental results are measured for 100 seconds as shown in Fig. <ref>. Count rates of 481kHz (Avg.: 480,924 counts/s, Std. Dev.: 1347) and 22kHz (Avg.: 21,642 counts/s, Std. Dev.: 242) are recorded under the condition of DC=0.9I_th+AC and DC=0.9I_th, respectively. Thus, the count rate of 22kHz out of 481kHz can be roughly interpreted as photon counts caused by spontaneous emission under DC=0.9I_th, which spread in all time domain. We measured photon counts for the case of DC=0.6I_th and DC=0 for comparison which are 14kHz (Avg.: 13,756 counts/s, Std. Dev.: 161) and 180Hz (Std. Dev.: 12), respectively. Count rate at DC=0 indicates dark count rate of the single photon detector. Noise photon count of 22kHz caused by spontaneous emission is not negligible compared to the signal count of 481kHz. It can definitely increase the QBER of the system, which directly decreases secret key rates. If a system utilizes eight or more laser diodes to configure the decoy states <cit.>, the noise count can become double or more. Considering the fact that polarization-based systems are mostly used for free-space channel where other photon noises exist including the sunlight, noise effect caused by the source itself must be minimized to improve the system performance. Thus, high DC bias level may not be a good solution for the performance of QKD systems. Steps for finding desirable DC bias condition considering both information leakage and system performance are as follows. First, one needs to figure out minimum time interval between two consecutive pulses from a single laser diode, which is determined by the system clock rate. Second, one should investigate minimum DC bias current level eliminating temporal disparity and intensity fluctuation among the pulses from the minimum time interval to longer ones. Third, one must check whether QBER caused by noise photon counts due to DC bias current is negligible or not. If negative effects caused by the DC bias current is non-negligible, system operation should be slow down to decreases the DC bias level condition. Optimal DC bias current for a fixed system clock rate can be different according to the structure of the semiconductor laser diode and its material characteristics.§ CONCLUSION In this paper, we discussed critical side channel effects possibly occurred in random bit generation with multiple semiconductor laser diodes, which are temporal disparity and intensity fluctuation among photon pulses. We showed that the phenomena are severe under low DC bias condition, which allows an eavesdropper to obtain correlations between consecutive pulses from the same laser diode. The negative effects become worse for the high-speed operation where time interval between two consecutive pulses is within the order of carrier lifetime of the laser diode. The information leakage caused by the behaviors could be alleviated as DC bias current increases close to the threshold level. However, we furtherly addressed that QBER performance can be degraded under high DC current situation due to background photons caused by spontaneous emission, which almost linearly increases with DC current level. Thus, AC pulse and DC bias current must be elaborately controlled with the system operation speed, considering both information leakage and system performance. § ACKNOWLEDGMENTSThis work was supported by Electronics and Telecommunications Research Institute (ETRI) grant funded by the Korean government. [Development of preliminary technologies for transceiver key components and system control in polarization based free space quantum key distribution] | http://arxiv.org/abs/1706.08705v1 | {
"authors": [
"Heasin Ko",
"Byung-Seok Choi",
"Joong-Seon Choe",
"Kap-Joong Kim",
"Jong-Hoi Kim",
"Chun Ju Youn"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170627080347",
"title": "Critical side channel effects in random bit generation with multiple semiconductor lasers in a polarization-based quantum key distribution system"
} |
School of Mathematics and Computer Sciences, Jianghan University, Wuhan 430056, PR [email protected] study the stability of Riemann solutions to pressureless Euler equations with Coulomb-like friction under the nonlinear approximation of flux functions with one parameter. The approximated system can be seen as the generalized Chaplygin pressure Aw-Rascle model with Coulomb-like friction, which is also equivalent to the nonsymmetric system of Keyfitz-Kranzer type with generalized Chaplygin pressure and Coulomb-like friction. Compared with the original system, The approximated system is strictly hyperbolic, which has one eigenvalue genuinely nonlinear and the other linearly degenerate. Hence, the structure of its Riemann solutions is much different from the ones of the original system. However, it is proven that the Riemann solutions for the approximated system converge to the corresponding ones to the original system as the perturbation parameter tends to zero, which shows that the Riemann solutions to nonhomogeneous pressureless Euler equations is stable under such kind of flux approximation. The result in this paper generalizes the stability of Riemann solutions with respect to flux perturbation from the well-known homogeneous case to the nonhomogeneous case. stability of Riemann solutions; pressureless Euler equations; delta shock wave;Coulomb-like friction; flux approximation.[2010] 35L65 35L6735B30 § INTRODUCTIONNon-strictly hyperbolic systemhave important physical background, which is also difficult and interesting in mathematics and attract many people to study them. It is well known that their Cauchy problem usually does not have a weak L^∞-solution. A typical example is the Cauchy problem for pressureless Euler equations (which is also called as zero pressure flow or transportation equations) <cit.>. Therefore, the measure-value solution should be introduced to this nonclassical situation, such as delta shock wave <cit.> and singular shock <cit.>, which can also provide a reasonable explanation for some physical phenomena. However, the mechanism for the formation of delta shock wave cannot be fully understood, although the necessity of delta shock wave is obvious for Riemann solutions to some non-strictly hyperbolic system. Now there are some related results for homogenous equations <cit.>, but few results have been shown for nonhomogeneous equations. In this paper, we are mainly concerned with zero pressure flow with Coulomb-like friction{[ρ_t+(ρ u)_x=0,; (ρ u)_t+(ρ u^2)_x=βρ, ].where the state variable ρ>0, u denote the density and velocity, respectively, and β is a frictional constant. The motivation of study (<ref>) comes from the violentdiscontinuities in shallow flows with large Froude number <cit.>. Itcan also be derived directly from the so-called pressurelessEuler/Euler-Possion systems <cit.>. Moreover, the system (<ref>) canalso be obtained formally from the model proposed by Brenier et al.<cit.> todescribe the sticky particle dynamics with interactions. Recently,the Riemann problem and shadow wave for (<ref>) have been studied respectively in<cit.>and <cit.>. Remarkably, in <cit.>, it is shown that the Riemann problem for the nonhomogeneous equations (<ref>) has delta shock wave solutions in some situations.Delta shock wave is a kind of nonclassical nonlinear wave on which at least one of the state variables becomes a singular measure. Korchinski <cit.> firstly introduced the concept of the δ-function into the classical weak solution in his unpublished Ph.D. thesis. In 1994, Tan, Zhang and Zheng <cit.> considered some 1-D reduced system and discovered that the form of δ-functions supported on shocks was used as parts in their Riemann solutions for certain initial data. Since then, delta shock wave has been widely investigated, see <cit.> and references cited therein.The formation of delta shock wave has been extensively studied by the vanishing pressure approximation for zero pressure flow <cit.> and Chaplygin gas dynamics <cit.>. Recently, the flux approximation with two parameters <cit.> and three parameters <cit.> has also been carried out for zero pressure flow. In the present paper, we consider the nonlinear approximation of flux functions for zero pressure flow with coulomb-like friction which has not been paid attention before.Specifically, we introduce the nonlinear approximation of flux functions in (<ref>) as follows:{[ρ_t+(ρ u)_x=0,; (ρ( u+P))_t+(ρ u( u+P))_x=βρ, ].where P is given by the state equation for generalized Chaplygin gas <cit.>P=-A/ρ^α, A>0,0<α<1,with α a real constant and the parameter A sufficiently small. System (<ref>) and (<ref>) can be seen as the generalized Chaplygin pressure Aw-Rascle model with Coulomb-like friction. By taking u=w-P, (<ref>) can be writtenas follows:{[ρ_t+(ρ (w-P))_x=0,; (ρ w)_t+(ρ w( w-P))_x=βρ, ].with a pure flux approximation. (<ref>) together with (<ref>) can also be seen as the nonsymmetric system of Keyfitz-Kranzer type with generalized Chaplygin pressure and Coulomb-like friction <cit.>. Recently, for β=0, Cheng has shown that the structure of the Riemann solutions to (<ref>) and (<ref>) were very similar <cit.>. More precisely, we are only concerned with the Riemann problem, i.e. the initial data taken as follows:(ρ,u)(x,0)={[(ρ_-,u_-),x<0,; (ρ_ +,u_+),x>0, ].where ρ_± and u_± are all given constants.In this paper, we will find that the delta shock wave also appears in the Riemann solutions to (<ref>) for some specific initial data. We are interested in how the delta-shock solution of (<ref>) and (<ref>) develops under the influence of the Coulomb-like friction. The advantage of this kind source term is in that (<ref>) can be written in a conservative form such that exact solutions to the Riemann problem (<ref>) and (<ref>) can be constructed explicitly. We shall see that the Riemann solutions to (<ref>) and (<ref>) are not self-similar any more, in which the state variable u varies linearly along with the time t under the influence of the Coulomb-like friction. In other words, the state variable u-β t remains unchanged in the left, intermediate and right states. In some situations, the delta-shock wave appears in the Riemann solutions to (<ref>) and (<ref>). In order to describe the delta-shock wave, the generalized Rankine-Hugoniot conditions are derived and the exact position, propagation speed and strength of the delta shock wave are obtained completely. It is shown that the Coulomb-like friction term make contact discontinuities, shock waves, rarefaction waves and delta shock waves bend into parabolic shapes for the Riemann solutions.Furthermore, it is proven rigorously that the limits of Riemann solutions to (<ref>) and (<ref>) converge to the corresponding ones to (<ref>) and (<ref>) when the perturbation parameter A tends to zero. In other words, the Riemann solutions (<ref>) and (<ref>) is stable with respect to the nonlinear approximations of flux functions in the form of (<ref>). Actually, for the case α=1 in (<ref>), system (<ref>) becomes the Chaplygin pressure Aw-Rascle model with Coulomb-like friction <cit.>. Similar result can be easily got, so we do not focus on it here. Moreover, the results got in this paper can also be generalized to the nonsymmetric system of Keyfitz-Kranzer type (<ref>) with the same generalized Chaplygin pressure and Coulomb-like friction. This paper is organized as follows. In section 2, we describe simply the solutions of the Riemann problem (<ref>) and (<ref>) for completeness. In Section 3, the approximated system (<ref>) is reformulated into a conservative form and some general properties of the conservative form are obtained. Then, the exact solution to the Riemann problem for the conservative form are constructed explicitly, which involves the delta shock wave. Furthermore, the generalized Rankine-Hugoniot conditions are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. In Section 4, the generalized Rankine-Hugoniot conditions and three kinds of Riemann solutions to the approximated system (<ref>) and (<ref>) are given. Furthermore, it is proven rigorously that the delta-shock wave is indeed a week solution to the Riemann problem (<ref>) and (<ref>) in the sense of distributions. In Section 5, the limit of Riemann solutions to the approximated system (<ref>) is taken by letting the perturbation parameter A tends to zero, which is identical with the corresponding ones to the original system. Finally, conclusions and discussions are drawn in Section 6.§ PRELIMINARIESIn this section, we simply describe the results on the Riemann problem (<ref>) and (<ref>), which can be referred to <cit.> in details.Let us first state some known fact about elementary waves of the given system. The system (<ref>) is weakly hyperbolic with the double eigenvalue λ_1=λ_2=u. Let us first look for a solution to (<ref>) when initial data are constants, (ρ(x,0),u(x,0))=(ρ_0,u_0). For smooth solutions, one can substitute ρ_t from the first equation of (<ref>) into the second one and eliminate ρ from it by division (provieded that we are away from a vacuum state). So, we have now the equation u_t+uu_x=β that can be solved by the method of characteristics: u=u_0+β t, x=x_0+u_0t+1/2β t^2. The first equation then becomes ρ_t+(u_0+β t)ρ_x=0 with a solution ρ=ρ_0 on each curve x=x_0+u_0t+1/2β t^2. So, the solution for constant initial data is (ρ,u)=(ρ_0,u_0+β t).For the case u_-<u_+, there is no characteristic passing through the region {(x,t): u_-t+1/2β t^2<x<u_+t+1/2β t^2}, so the vacuum should appear in the region. The solution can be expressed as(ρ,u)(x,t)={[ (ρ_-,u_-+β t),-∞<x<u_-t+1/2β t^2,; vacuum,u_-t+1/2β t^2<x<u_+t+1/2β t^2,;(ρ_+,u_++β t),u_+t+1/2β t^2<x<∞. ].For the case u_-=u_+, it is easy to see that the two states (ρ_±,u_±+β t) can be connected by a contact discontinuity x=u_±t+1/2β t^2. So the solution can be expressed as(ρ,u)(x,t)={[ (ρ_-,u_-+β t), x<u_-t+1/2β t^2,; (ρ_+,u_++β t), x>u_+t+1/2β t^2, ]. For the case u_->u_+,the characteristics originating from the origin overlap in the domain{(x,t): u_+t+1/2β t^2<x<u_-t+1/2β t^2}, which means that there exists singularity. A solution containing a weighted δ-measure supported on a curve will be constructed.In order to define the measure solution as above, like as in <cit.>, the two-dimensional weighted δ-measure w(t)δ_S supported on a smooth curve S={(x(s),t(s)):a≤ s≤ b} should be introduced as follows:⟨ w(s)δ_S,ψ(s,s)⟩=∫_a^bw(s)ψ(x(s),t(s))√(x'(s)^2+t'(s)^2)ds,for any ψ∈ C_0^∞(R× R_+). Let x=x(t) be a discontinuity curve, we consider a piecewise smooth solution of (<ref>) in the form(ρ,u)(x,t)={[ (ρ_-,u_-+β t), x<x(t),; (w(t)δ(x-x(t)),u_δ(t)),x=x(t),;(ρ_+,u_++β t),x>x(t), ].in which u_δ (t) is the assignment of u on this delta shock wave curve and u_δ (t)-β t is assumed to be a constant. The delta shock wave solution of the Riemann problem (<ref>) and (<ref>) must obey the following generalized Ranking-Hugoniot conditions:{[ dx(t)dt=σ(t)=u_δ(t),;dw(t)dt= σ(t)[ρ]-[ρ u],; d(w(t) u_ δ(t))dt= σ(t)[ρ u]-[ρ u^2]+β w(t), ].and the over-compressive entropy conditionλ(ρ_+,u_+)<σ(t)<λ(ρ_-,u_-),namelyu_++β t<u_δ(t)<u_-+β t.In(<ref>), it should be remarkable that[ρ u]=ρ_+(u_++β t)-ρ_-(u_-+β t), [ρ u^2]=ρ_+(u_++β t)^2-ρ_-(u_-+β t)^2. Through solving (<ref>) with x(0)=0, w(0)=0, we obtain{[ u_δ(t)=σ(t)=σ_0+β t,;x(t)=σ_0t+1/2β t^2,; w(t)=-√(ρ_-ρ_+)(u_+-u_-)t, ].withσ_0= √(ρ_-)u_-+√(ρ_+)u_+√(ρ_-)+√(ρ_+).It is easy to prove that the delta shock wave solution (<ref>) with(<ref>) satisfy the system(<ref>) in the distributional sense. That is to say, the following identities{[ ⟨ρ,ψ_t⟩+⟨ρ u,ψ_x⟩=0,; ⟨ρ u,ψ_t⟩+⟨ρ u^2,ψ_x⟩=-⟨βρ,ψ⟩, ].holds for any test function ψ∈ C_0^∞(R× R_+),in which⟨ρ u,ψ⟩=∫_0^∞∫_-∞^∞ρ̂_0 û_0ψ dxdt+⟨ w(t)u_δ(t)δ_S,ψ⟩,withρ̂_0=ρ_-+[ρ]H(x-σ t),û_0=u_--β t+[u]H(x-σ t). From the above discussions, we can concluded that the Riemann problem (<ref>) and (<ref>) can be solved by three kinds of solutions: one contact discontinuity, two contact discontinuities with the vacuum state between them (see Fig.2.1), or the delta shock wave (see Fig.2.2) connecting two states (ρ_±,u_±+β t).1mm0.4pt(169,69.25)(10,0)(82,18)(1,0).07 (15,18)(1,0)67 (9,68)(0,1).07 (9,20.25)(0,1)47.75(40,18)(49.75,47.63)(75.5,57.25) (40,18)(17.13,34.75)(26.5,65.75)(166.25,18.25)(1,0).07 (96,18.5)(1,0)68.78125 (91.25,68)(0,1).07 (91.25,21)(0,1)47(125,18.5)(126.63,54.25)(144.75,68) (125,18.5)(138.75,44.63)(161.25,55.5) (5.5,65.25)t (84,18)x (88,65)t (169,18)x (19,64.25)J_1 (77,59.5)J_2 (39.75,14.75)0 (125,15)0 (136.25,66.75)J_1 (164.5,57)J_2 (10,23)(ρ_-,u_-+β t) (53.25,32.75)(ρ_+,u_++β t) (32,46)Vac. (100,40)(ρ_-,u_-+β t) (134,51.75)Vac. (145,32)(ρ_+,u_++β t) (45,11.5)(0,0)[cc](a)u_-<0<u_+ (130,12.75)(0,0)[cc](b) 0<u_-<u_+ (90,6)(0,0)[cc] Fig.2.1 The Riemann solution to (1.1) and (1.5) when β>0.1mm0.5pt(182,63)(10,0)(85.5,12.5)(1,0).07 (12,12.5)(1,0)73.5 (4.75,61)(0,1).07 (4.75,15)(0,1)46(38.25,12.5)(39.75,40.25)(67.25,55.25)(176,13.5)(1,0).07 (104,13.5)(1,0)72 (94.75,63)(0,1).07 (94.75,14.5)(0,1)48.5(1.75,58.75)t (87.75,11.75)x (66,56)δS (92,61.75)t (177,13)x (110,50)δS (21,35.75)(ρ_-,u_-+β t) (55,35.75)(ρ_+,u_++β t) (106.75,35)(ρ_-,u_-+β t) (150,35.25)(ρ_+,u_++β t) (38,8.5)0 (115,9)0 (45,8.5)(a)β>0 (127.5,8.5)(b)β<0 (90,5)(0,0)[cc] Fig.2.2 The delta shock wave solution to (1.1) and (1.5) when u_+<u_- and σ_0>0.(115,13.5)(182,22.75)(105.5,57.75) § RIEMANN PROBLEM FOR A MODIFIED CONSERVATIVE SYSTEM OF (<REF>)In this section, we are devoted to the study of the Riemann problem for a conservative system of (<ref>) in detail. Let us introduce the new velocity v(x,t)=u(x,t)-β t, then the system (<ref>) can be reformulated into a conservative form as follows: {[ ρ_t+(ρ (v+β t))_x=0,; (ρ( v+P))_t+(ρ (v +P)(v+β t))_x=0. ].In fact, the change of variable was introduced by Faccanoni and Mangeney <cit.> to study the shock and rarefaction waves of the Riemann problem for the shallow water equations with a with Coulomb-like friction. Here, we use this transformation to study the delta shock wave for the system (<ref>).Now we want to deal with the Riemann problem for the conservative system (<ref>) with the sameRiemann initial data (<ref>) as follows:(ρ,v)(x,0)={[(ρ_-,u_-),x<0,; (ρ_ +,u_+),x>0. ].We shall see hereafter that the Riemann solutions to (<ref>) and (<ref>) can be obtained immediately from the Riemann solutions to (<ref>) and (<ref>) by using the transformation of state variables (ρ,u)(x,t)=(ρ,v+β t)(x,t).The system (<ref>) can be rewritten in the quasi-linear form([10; v+P+ρ P'ρ ]) ([ ρ; v ])_t +([ v+β t ρ; (v+P+ρ P')(v+β t) ρ(2v+β t+P) ]) ([ ρ; v ])_x =([ 0; 0 ]).It can be derived directly from (<ref>) that the conservative system (<ref>) has two eigenvaluesλ_1(ρ,v)=v+β t-Aα/ρ^α,λ_2(ρ,v)=v+β t,whose corresponding right eigenvectors can be expressed respectively byr_1=(ρ,-Aα/ρ^α)^T,r_2=(1,0)^T.So (<ref>) is strictly hyperbolic for ρ>0. Moreover, ▽λ_1· r_1≠0 and ▽λ_2· r_2=0. Then it can be concluded that λ_1 is genuinely nonlinear whose associated waves are shock waves denoted by S_1 or rarefaction waves denoted by R_1, see <cit.>. Then the Riemann invariants along the characteristic fields may be chosen asw=v-A/ρ^α,z=v,which should satisfy ▽ w· r_1=0 and ▽ z· r_2=0, respectively.Let us draw our attention on the elementary waves for the system (<ref>) in detail. We first consider the rarefaction wave which is a one-parameter family of states connecting a given state. This kind of continuous solution satisfying the system (<ref>) can be obtained by determining the integral curves of the first characteristic fields. It is worthwhile to notice that the 1-Riemann invariant is conserved in the 1-rarefaction wave.For a given left state (ρ_-,u_-), the 1-rarefaction wave curve R_1(ρ_-,v_-) in the phase plane which is the set of states connected on the right, should satisfyR_1(ρ_-,u_-): {[ dx/dt=λ_1(ρ,v)=v+β t-Aα/ρ^α,; v-A/ρ^α=u_--A/ρ_-^α=w_-,; λ_1(ρ_-,u_-)≤λ_1(ρ,v). ]. By differentiating v with respect to ρ in the second equation in (<ref>), we havedv/dρ=-Aα/ρ^α+1<0, d^2v/dρ^2=Aα(α+1)/ρ^α+2>0.Thus, the 1-rarefaction wave is made up of the half-branch of R_1(ρ_-,u_-) satisfying v≥ u_- and ρ≤ρ_-, which is convex in the (ρ,v) plane.Let us compute the solution (ρ,v) at a point in the interior of the 1-rarefaction wave, then it follows from the first equation in (<ref>), we havev-Aα/ρ^α=x/t-β t.By combining (<ref>) with the second equation in (<ref>), we get(ρ,v)(x,t)=((A(1-α)/x/t-β t-w_-)^1/α,x/t-β t-α w_-/1-α). Let us return our attention on the shock wave which is a piecewise constant discontinuous solution, satisfying the Rankine-Hugoniot conditions and the entropy condition. Herethe Ranking-Hugoniot conditions can be derived in a standard method as in <cit.>, since the parameter t only appearsin the flux functions in the conservative system (<ref>). For a bounded discontinuity at x=x(t), let us denote σ(t)=x'(t), then the Ranking-Hugoniot conditions for the conservative system (<ref>) can be expressed as{[-σ(t)ρ+[ρ (v+β t)]=0,; -σ(t)[ρ (v+P)]+[ρ(v+P) (v+β t)]=0, ].where [ρ]=ρ_r-ρ_l with ρ_l=ρ(x(t)-0,t), ρ_r=ρ(x(t)+0,t), in which [ρ] denote the jump of ρ across the discontinuity, etc. It is clear that the propagation speed of the discontinuity depends on the parameter t, which is obviously different from classical hyperbolic conservation laws.If σ(t)≠0, then it follows from (<ref>) that ρ_rρ_l(v_r-v_l)((v_r-A/ρ_r^α)-(v_l-A/ρ_l^α))=0,from which we have v_r=v_l or v_r-A/ρ_r^α=v_l-A/ρ_l^α.Thus, for a given left state (ρ_-,u_-), with the latex entropy condition in mind, the 1-shock wave curve S_1(ρ_-,u_-) in the (ρ,v) plane which is the set of states connected on the right, should satisfyS_1(ρ_-,u_-): {[ σ_1(t)=ρ v-ρ u_-/ρ-ρ_-+β t,;v-A/ρ^α=u_--A/ρ_-^α=w_-,; ρ>ρ_-, v<u_-, ].which indicates the 1-rarefaction wave and 1-shock wave are different branch of the same curve.Moreover, from (<ref>),for a given left state (ρ_-,u_-), the 2-contact discontinuity curve J(ρ_-,u_-) in the (ρ,v) plane which is the set of states connected on the right, should satisfy J(ρ_-,u_-): σ(t)=v+β t=u_-+β t.0.9mm0.4pt(200,69)(-15,0)(95.75,13.75)(1,0).07 (15.75,13.75)(1,0)80 (5.75,69)(0,1).07 (5.75,14)(0,1)5550(35.25,67.75)(35.25,23.25)(35.25,13.75) (65,68)(0,-1)54(37.5,65.5)(46,23.25)(88.5,21) (12.5,67.5)ρ (99,13.75)v(26,41.25)III (55,46.25)II (76.5,45.5)I (44.75,53.75)S_1 (69,55.25)J (80,23.25)R_1 (31,50.75)S_δ (30,9.5)u_–A/ρ_-^α (63.5,11)u_- (67,30)(ρ_-,u_-) (60.75,5.5)(0,0)[cc] Fig. 3.1the (ρ,v) phase plane for the conservative system (<ref>).Let us now consider the Riemann problem (<ref>) and (<ref>). In the (ρ,v) phase plane, for a given left state (ρ_-,u_-), the set ofstates connected on the right consist of the 1-rarefaction wave R_1(ρ_-,u_-), the 1-shock wave S_1(ρ_-,u_-) and the 2-contact discontinuity curve J(ρ_-,u_-). It is clear to see that R_1(ρ_-,u_-) has the line S_δ:v=u_--A/ρ_-^α and S_1(ρ_-,u_-) has the positive v-axis as their asymptotic lines, respectively.In view of the right state (ρ_+,u_+) in different positions, one wants to construct the unique global Riemann solution of (<ref>) and (<ref>). However, as in <cit.>, if u_+≤ u_--A/ρ_-^α is satisfied, the Riemann solution of (<ref>) and (<ref>) can not be constructed by using only the elementary waves including shocks, rarefaction waves and contact discontinuities. In this nonclassical situation, the concept of delta shock wave should be introduced such as in <cit.> and be discussed later.Draw all the curvesR_1(ρ_-,u_-), S_1(ρ_-,u_-)J(ρ_-,u_-) and S_δ in the the (ρ,v) phaseplane, thus the phase plane is divided into three regionsI, II and III (See Fig.3.1), whereI={(ρ,v)|v≥ u_-}, II={(ρ,v)|u_--A/ρ_-^α<v< u_-}, III={(ρ,v)|v≤ u_–A/ρ_-^α}.According to the right state(ρ_+,u_+) in different regions, the uniqueglobal Riemann solution of (<ref>) and (<ref>) can be constructed connecting two constant states (ρ_-,u_-) and (ρ_+,u_+) If (ρ_+,u_+)∈ I, namely u_+>u_-, then the Riemann solution consists of 1-rarefaction wave R_1 and a 2-contact discontinuity J with an intermediate constant state (ρ_∗,v_∗) determined uniquelyby{[ v_∗-A/ρ_∗^α=u_–A/ρ_-^α=w_-,;u_+=v_∗. ].which immediately leads to (A/ρ_∗^α,v_∗)=(u_+-u_-+A/ρ_-^α,u_+),or(ρ_∗,v_∗)=((A/(u_+-u_-+A/ρ_-^α)^1/α,u_+),Thus, the Riemann solution of (<ref>) and (<ref>) can be express as(ρ,v)(x,t)={[ (ρ_-,u_-+β t),x<x_1^-(t),; R_1, x_1^-(t)<x<x_1^+(t),; (ρ_∗,u_∗+β t), x_1^+(t)<x<x_2(t),; (ρ_+,u_++β t),x>x_2(t),; ].in whichx_1^-(t)=(u_–A/ρ_-^α)t+1/2β t^2, x_1^+(t)=(u_∗-A/ρ_∗^α)t+1/2β t^2, x_2(t)=u_+t+1/2β t^2,and the state (ρ_1,u_1) in R_1 can be calculated by (<ref>).If (ρ_+,u_+)∈ II, namely u_--A/ρ_-^α<u_+< u_-, then the Riemann solution consists of 1-shock wave S_1 and a 2-contact discontinuity J with an intermediate constant state (ρ_∗,v_∗) determined uniquely by (<ref>). Thus, the Riemann solution of (<ref>) and (<ref>) can be express as(ρ,v)(x,t)={[ (ρ_-,u_-+β t),x<x_1(t),; (ρ_∗,u_∗+β t), x_1(t)<x<x_2(t),; (ρ_+,u_++β t),x>x_2(t),; ].in which the position of S_1 is given byx_1(t)=ρ_∗v_∗-ρ_-u_-/ρ_∗-ρ_-t+1/2β t^2, and x_2(t) is given by (<ref>).On the other hand, when (ρ_+,u_+)∈ III, namely u_+≤ u_--A/ρ_-^α, then there exist a nonclassical situation where the Cauchy problem does not own a weak L^∞-solution. In order to solve the Riemann problem (<ref>) and (<ref>) in the framework of nonclassical solution, a solution containing a weighted δ-measure supported on a curve should be defined such as in <cit.>. In what follows, let us provide the definition of delta shock wave solution to the Riemann problem (<ref>) and (<ref>). Let us also refer to <cit.> about the more exact definition of generalized delta shock wave solution for related systems with delta measure initial data.Let (ρ,v) be a pair of distributions in which ρ has the form ofρ(x,t)=ρ̂(x,t)+w(x,t)δ_S,in which ρ̂,v∈ L^∞(R× R_+). Then, (ρ,v) is called as the delta shock wave solution to the Riemann problem (<ref>) and (<ref>) if it satisfies{[ ⟨ρ,ψ_t⟩+⟨ρ (v+β t),ψ_x⟩=0,; ⟨ρ( v+P)),ψ_t⟩+⟨ρ ( v+P)(v+β t)),ψ_x⟩=0, ].for any ψ∈ C_0^∞(R× R^+). Here we take ⟨ρ (v+P)(v+β t)),ψ⟩=∫_0^∞∫_-∞^∞(ρ(v-A/ρ^α)(v+β t))ψ dxdt+⟨ w(t)v_δ(t)(v_δ(t)+β t)δ_S,ψ⟩,as an example to explain the inner product, in which we use the symbol S to express the smooth curve with the Dirac delta function supported on it, v_δ is the value of v and A/ρ^α is equal to zero on this delta shock wave S.With the above definition, if (ρ_+,u_+)∈ III and u_+<u_--A/ρ_-^α, a piecewise smooth solution of the Riemann problem (<ref>) and (<ref>) should be introduced in the form(ρ,v)(x,t)={[ (ρ_-,u_-),x<x(t),; (w(t)δ(x-x(t)),v_δ),x=x(t),; (ρ_+,u_+),x>x(t), ].where x(t), w(t) and σ(t)=x'(t) denote respectively the location, weight and propagation speed of the delta shock, and v_δ indicates the assignment of v on thisdelta shock wave. It is remarkable that the value of v should be given on the delta shock curve x=x(t) such that the product of ρ and v can be defined in the sense of distributions. When u_+=u_--A/ρ_-^α, it can be discussed similarly and we omit it.The delta shock wave solution of the form (<ref>) to the the Riemann problem (<ref>) and (<ref>) should obey the generalized Rankine-Hugoniot conditions{[ dx(t)dt=σ(t)=v_δ+β t,;dw(t)dt=σ(t)[ρ]-[ρ (v+β t)],; d(w(t)v_δ)dt=σ(t)[ρ( v-A/ρ^α)]-[ρ( v-A/ρ^α) (v+β t)], ].with initial data x(0)=0 and w(0)=0. In addition, for the unique solvability of the above Cauchy problem, it is necessary to require that the value of v_δ to be a constant along the trajectory of delta shock wave (see <cit.> for details). The derivation process of the generalized Rankine-Hugoniot conditions is similar to that in <cit.> and we omit it here. In order to ensure the uniqueness of Riemann solutions, an over-compressive entropy condition for the delta shock wave should be proposed byλ_1(ρ_+,u_+)<λ_2(ρ_+,u_+)<σ(t)<λ_1(ρ_-,u_-)<λ_2(ρ_-,u_-),such that we haveu_+<v_δ<u_--A/ρ_-^α,which implies that all the characteristics on both sides of the delta shock are in-coming.It follows from (<ref>) thatdw(t)dt=v_δ(ρ_+-ρ_-)-(ρ_+u_+-ρ_-u_-), v_δdw(t)dt=v_δ((ρ_+u_+-ρ_-u_-)-(A/ρ_+^α-1-A/ρ_-^α-1))- (ρ_+u_+^2-ρ_-u_-^2)+(Au_+/ρ_+^α-1-Au_-/ρ_-^α-1),Thus, we have(ρ_+-ρ_-)v_δ^2-(2(ρ_+u_+-ρ_-u_-)-(A/ρ_+^α-1-A/ρ_-^α-1))v_δ+ (ρ_+u_+^2-ρ_-u_-^2)-(Au_+/ρ_+^α-1-Au_-/ρ_-^α-1)=0, For convenience, let us denotew_0=√(ρ_+ρ_-(u_+-u_-)((u_+-u_-)-(A/ρ_+^α-A/ρ_-^α))+ 1/4(A/ρ_+^α-1-A/ρ_-^α-1)^2)- 1/2(A/ρ_+^α-1-A/ρ_-^α-1)>0,If ρ_+≠ρ_-, with the entropy condition (<ref>) in mind, one can obtain directly from (<ref>) thatv_δ=ρ_+ u_+-ρ_-u_-+w_0/ρ_+-ρ_-,which enables us to getσ(t)=v_δ+β t,x(t)=v_δt+1/2β t^2,w(t)=w_0t. Otherwise, if ρ_+=ρ_-, then we have v_δ=1/2(u_++u_–A/ρ_-^α).In this particular case, we can also getσ(t)=1/2(u_++u_–A/ρ_-^α)+β t,x(t)=1/2(u_++u_–A/ρ_-^α)t+1/2β t^2,w(t)=(ρ_-u_–ρ_+ u_+)t. § RIEMANN PROBLEM FOR THE APPROXIMATED SYSTEM (<REF>)In this section, let us return to the Riemann problem (<ref>) and (<ref>). If (ρ_+,u_+)∈ I, the Riemann solutions to (<ref>) and (<ref>)R_1+J can be represented as(ρ,u)(x,t)={[ (ρ_-,u_-+β t), x<x_1^-(t),; (ρ_1,v_1+β t),x_1^-(t)<x<x_1^+(t),; (ρ_*,v_*+β t),x_1^+(t)<x<x_2(t),; (ρ_+,u_++β t), x>x_2(t), ].where x_1^-(t), x_1^+(t) and x_2(t) are given by (<ref>) and (<ref>) respectively, and the states (ρ_1,v_1) and (ρ_*,v_*) can be calculated as (<ref>) and (<ref>). Let us use Fig.4.1(a) to illustrate this situation in detail, where all the characteristics in the rarefaction wave fans R_1 and contact discontinuity curve J are curved into parabolic shapes.If (ρ_+,u_+)∈ II, the Riemann solutions to (<ref>) and (<ref>) S_1+J can be represented as(ρ,u)(x,t)={[ (ρ_-,u_-+β t), x<x_1(t),; (ρ_*,v_*+β t),x_1(t)<x<x_2(t),; (ρ_+,u_++β t), x>x_2(t), ].where x_1(t) and x_2(t) are given by (<ref>) and (<ref>) respectively and the states (ρ_*,v_*) can be calculated as (<ref>). Let us use Fig.4.1(b) to illustrate this situation in detail, where both the shock wave curve S_1 and the contact discontinuity curve J are curved into parabolic shapes.Analogously, if (ρ_+,u_+)∈ III, namely u_+≤ u_--A/ρ_-^α, then we can also define the weak solutions to the Riemann problem (<ref>) and (<ref>) in the sense of distributions below. Let (ρ,u) be a pair of distributions in which ρ has the form of (<ref>), then it is called as the delta shock wave solution to the Riemann problem (<ref>) and (<ref>) if it satisfies{[ ⟨ρ,ψ_t⟩+⟨ρ u,ψ_x⟩=0,; ⟨ρ( u+P)),ψ_t⟩+⟨ρ u( u+P)),ψ_x⟩=-⟨βρ,ψ⟩, ].for any ψ∈ C_0^∞(R× R^+), in which ⟨ρ u( u+P)),ψ⟩=∫_0^∞∫_-∞^∞(ρu(u-A/ρ^α))ψ dxdt+⟨ w(t)(u_δ(t))^2δ_S,ψ⟩,and u_δ(t) is the assignment of u on this delta shock wave curve.1mm0.4pt(169,69.25)(10,0)(82,18)(1,0).07 (15,18)(1,0)67 (9,68)(0,1).07 (9,20.25)(0,1)47.75(40,18)(49.75,47.63)(75.5,57.25) (40,18)(16.13,34.75)(22.5,65.75) (40,18)(15.13,34.75)(18.5,65.75) (40,18)(17.13,34.75)(26.5,65.75) (40,18)(19.13,34.75)(31.5,65.75) (40,18)(21.13,34.75)(34.5,65.75)(166.25,18.25)(1,0).07 (96,18.5)(1,0)68.78125 (91.25,68)(0,1).07 (91.25,21)(0,1)47(125,18.5)(126.63,54.25)(144.75,68) (125,18.5)(138.75,44.63)(161.25,55.5) (5.5,65.25)t (84,18)x (85.75,68.75)t (169,18)x (23.5,69.25)R_1 (77,59.5)J (39.75,14.75)0 (125,15)0 (136.25,66.75)S_1 (164.5,56)J (10,23)(ρ_-,u_-+β t) (53.25,32.75)(ρ_+,u_++β t) (32,46)(ρ_*,v_*+β t) (100,40)(ρ_-,u_-+β t) (134,51.75)(ρ_*,v_*+β t) (145,32)(ρ_+,u_++β t) (45,11.5)(0,0)[cc](a)u_–A/ρ_-^α<u_-<u_+ (130,12.75)(0,0)[cc](b) u_–A/ρ_-^α<u_+<u_- (90,6)(0,0)[cc] Fig.4.1 The Riemann solution to (1.2) and (1.5) when u_–A/ρ_-^α<u_+ and β>0, (59,2.5)(0,0)[cc] where (ρ_*,v_*) is given by (3.13). With the above definition in mind, if u_+< u_--A/ρ_-^α is satisfied, then we look for a piecewise smooth solution to the Riemann problem (<ref>) and (<ref>) in the form(ρ,u)(x,t)={[ (ρ_-,u_-+β t), x<x(t),; (w(t)δ(x-x(t)),u_δ(t)),x=x(t),; (ρ_+,u_++β t), x>x(t), ].It is worthwhile to notice that u_δ(t)-β t is assumed to be a constant based on the result in Sect.2. With the similar analysis and derivation as before, the delta shock wave solution of the form (<ref>) to the Riemann problem (<ref>) and (<ref>) should also satisfy the following generalized Rankine-Hugoniot conditions{[ dx(t)dt=σ(t)=u_δ(t),; dw(t)dt=σ(t)[ρ]-[ρ u],; d(w(t)u_δ(t))dt=σ(t)[ρ( u-A/ρ^α)]-[ρ u( u-A/ρ^α) ]+β w(t). ].in which the jumps across the discontinuity are[ρ u]=ρ_+(u_++β t)-ρ_-(u_-+β t), [ρ u( u-A/ρ^α)]=ρ_+(u_++β t)(u_++β t-A/ρ_+^α)-ρ_-(u_-+β t)(u_-+β t-A/ρ_-^α). In order to ensure the uniqueness to the Riemann problem (<ref>) and (<ref>), the over-compressive entropy condition for the delta shock waveu_++β t<u_δ(t)<u_--A/ρ_-^α+β t.should also be proposed when u_+< u_--A/ρ_-^α.Like as before, we can also obtain x(t),σ(t) and w(t) from (<ref>) and (<ref>) together. In brief, we have the following theorem to depict the Riemann solution to (<ref>) and (<ref>) when the Riemann initial data (<ref>) satisfy u_+< u_--A/ρ_-^α and ρ_+≠ρ_-. If both u_+< u_--A/ρ_-^α and ρ_+≠ρ_- are satisfied, then the delta shock solution to the Riemann solutions to (<ref>) and (<ref>) can be expressed as{[ dx(t)dt=σ(t)=u_δ(t),; dw(t)dt=σ(t)[ρ]-[ρ u],; d(w(t)u_δ(t))dt=σ(t)[ρ( u-A/ρ^α)]-[ρ u( u-A/ρ^α) ]+β w(t). ].in whichσ(t)=u_δ(t)=v_δ+β t, x(t)=v_δt+1/2β t^2,w(t)=w_0t,in which w_0 and v_δ are given by (<ref>) and (<ref>) respectively.Let us check briefly that the above constructed delta shock wave solution (<ref>) and (<ref>) should satisfy (<ref>) in the sense of distributions. The proof of this theorem is completely analogs to those in <cit.>. Therefore, we only deliver the main steps for the proof of the second equality in (<ref>) for completeness. Actually, one can deduce thatI = ∫_0^∞∫_-∞^∞(ρ( u-A/ρ^α)ψ_t+ρ u( u-A/ρ^α) ψ_x)dxdt= ∫_0^∞∫_-∞^x(t)(ρ_-( u_-+β t-A/ρ_-^α)ψ_t+ρ_-(u_-+β t)(u _-+β t-A/ρ_-^α)ψ_x)dx dt+∫_0^∞∫^∞_x(t)(ρ_+( u_++β t-A/ρ_+^α)ψ_t+ρ_+(u_++β t)(u _++β t-A/ρ_+^α) ψ_x)dx dt+∫_0^∞w_0t(v_δ+β t)(ψ_t(x(t),t)+(v_δ+β t)ψ_x(x(t),t))dt. It can be derived from (<ref>) that the curve of delta shock wave is given byx(t)=v_δt+1/2β t^2. 1mm0.5pt(182,63)(10,0)(85.5,12.5)(1,0).07 (12,12.5)(1,0)73.5 (4.75,61)(0,1).07 (4.75,15)(0,1)46(38.25,12.5)(39.75,40.25)(67.25,55.25)(176,13.5)(1,0).07 (104,13.5)(1,0)72 (94.75,63)(0,1).07 (94.75,14.5)(0,1)48.5(1.75,58.75)t (87.75,11.75)x (66,56)δS (92,61.75)t (177,13)x (110,50)δS (21,35.75)(ρ_-,u_-+β t) (55,35.75)(ρ_+,u_++β t) (106.75,35)(ρ_-,u_-+β t) (150,35.25)(ρ_+,u_++β t) (38,8.5)0 (115,9)0 (45,8.5)(a)β>0 (127.5,8.5)(b)β<0 (90,5)(0,0)[cc] Fig.4.2 The delta shock wave solution to (1.1) and (1.2) when u_+<u_–A/ρ_-^α and v_δ>0,(90,1)(0,0)[cc]where v_δ is given by (3.28) for ρ_-≠ρ_+ and (3.30) for ρ_-= ρ_+.(115,13.5)(182,22.75)(105.5,57.75) For β>0 (see Fig.4.2(a)), there exists an inverse function of x(t) globally in the time t, which may be written in the formt(x)=√(v_δ^2/β^2+2x/β)-v_δ/β.Otherwise, for β<0 (see Fig.4.2(b)), there is a critical point (-v_δ^2/2β,-v_δ/β) on the delta shock wave curve such that x'(t) change its sign when across the critical point. Thus, the inverse function of x(t) is needed to find respectively for t≤-v_δ/β and t>-v_δ/β, which enable us to havet(x)={[ -√(v_δ^2/β^2+2x/β)-v_δ/β, t≤-v_δ/β,; √(v_δ^2/β^2+2x/β)-v_δ/β,t>-v_δ/β. ].Without loss of generality, let us assume that β>0 for simplicity. Actually, the other situation can be dealt with similarly. Under our assumption, it follows from (<ref>) that the position of delta shock wave satisfies x=x(t)>0 for all the time. It follows from (<ref>) thatdψ(x(t),t)/dt = ψ_t(x(t),t)+dx(t)/dtψ_x(x(t),t)= ψ_t(x(t),t)+(v_δ+β t)ψ_x(x(t),t)= ψ_t(x(t),t)+u_δ(t)ψ_x(x(t),t). By exchanging the ordering of integrals and using integration by parts, we have I = ∫_0^∞∫^∞_t(x)ρ_-( u_-+β t-A/ρ_-^α)ψ_tdt dx +∫_0^∞∫^∞_t(x)ρ_-(u_-+β t)(u _-+β t-A/ρ_-^α)ψ_xdt dx+∫_0^∞∫_0^t(x)ρ_+( u_++β t-A/ρ_+^α)ψ_tdt dx+∫_0^∞∫_0^t(x)ρ_+(u_++β t)(u _++β t-A/ρ_+^α) ψ_xdtdx +∫_0^∞w_0t(v_δ+β t)dψ(x(t),t)= ∫_0^∞(ρ_+( u_++β t(x)-A/ρ_+^α)-ρ_-( u_-+β t(x)-A/ρ_-^α))ψ(x,t(x))dx +∫_0^∞(ρ_-(u_-+β t)( u_-+β t-A/ρ_-^α)-ρ_+(u_++β t)( u_++β t-A/ρ_+^α))ψ(x(t),t)dt-∫_0^∞∫^∞_t(x)βρ_-ψ dt dx-∫_0^∞∫_0^t(x)βρ_+ψ dt dx-∫_0^∞w_0(v_δ+2β t)ψ(x(t),t)dt= ∫_0^∞C(t)ψ(x(t),t)dt-β(∫_0^∞∫_-∞^x(t)ρ_-ψ dx dt+∫_0^∞∫^∞_x(t)ρ_+ψ dx dt),in whichC(t) = (ρ_+( u_++β t-A/ρ_+^α)-ρ_-( u_-+β t-A/ρ_-^α))(v_δ+β t) +(ρ_-(u_-+β t)( u_-+β t-A/ρ_-^α)-ρ_+(u_++β t)( u_++β t-A/ρ_+^α)) -w_0(v_δ+2β t).By a tedious calculation, we haveC(t)=-β w_0t=-β w(t).Thus, it can be concluded from (<ref>) and (<ref>) together that the second equality in (<ref>) holds in the sense of distributions. The proof is completed. If both u_+< u_--A/ρ_-^α and ρ_+=ρ_- are satisfied, then the delta shock solution to the Riemann problem (<ref>) and (<ref>) can be expressed in the form (<ref>) whereσ(t)=u_δ(t)=1/2(u_++u_–A/ρ_-^α)+β t,x(t)=1/2(u_++u_–A/ρ_-^α)t+1/2β t^2, w(t)=(ρ_-u_–ρ_+ u_+)t.The process of proof is completely similar and we omit the details.If u_+=u_--A/ρ_-^α, then the delta shock solution to the Riemann problem (<ref>) and (<ref>) can be also expressed as the form in Theorem <ref> and Remark <ref>. The process of proof is easy and we omit the details. § THE FLUX APPROXIMATION LIMITS OF RIEMANN SOLUTIONS TO (<REF>) In this section, we are concerned that the flux approximation limits of Riemann solutions to (<ref>) and (<ref>) converge to the corresponding ones to (<ref>) and (<ref>) or not when the perturbation parameter A tends to zero. According to the relations between u_- and u_+, we will divide our discussion into the following three cases: (1) u_-<u_+; (2) u_-=u_+;(3) u_->u_+.Case 5.1.u_-<u_+In this case, (ρ_+,u_+)∈ I in the (ρ,v) plane, so the Riemann solutions to (<ref>) and (<ref>) R_1+J is given by (<ref>), where x_1^-(t), x_1^+(t) and x_2(t) are given by (<ref>) and (<ref>) respectively and the states (ρ_1,v_1) and (ρ_*,v_*) can be calculated as (<ref>) and (<ref>). From (<ref>) and (<ref>) we havelim_A→0ρ_1=lim_A→0(A(1-α)/x/t-β t-w_-)^1/α=0, lim_A→0ρ_*=lim_A→0(A/u_+-u_-+A/ρ_-^α)^1/α=0, which indicate the occurrence of the vacuum states. Furthermore, the Riemann solutions to (<ref>) and (<ref>) converge to lim_A→0(ρ,u)(x,t)={[(ρ_-,u_-+β t), x<u_-t+1/2β t^2,; vacuum, u_-t+1/2β t^2<x<u_+t+1/2β t^2,;(ρ_+,u_++β t), x>u_+t+1/2β t^2, ].which is exactly the corresponding Riemann solutions to the pressureless Euler equations with the same source term and the same initial data.Case 5.2.u_-=u_+In this case, (ρ_+,u_+) is on the J curve in the (ρ,v) plane, so the Riemann solutions to (<ref>) and (<ref>) is given as follows:(ρ,u)(x,t)={[ (ρ_-,u_-+β t), x<u_-t+1/2β t^2,; (ρ_+,u_++β t), x>u_+t+1/2β t^2, ].which is exactly the corresponding Riemann solutions to the pressureless Euler equations with the same source term and the same initial data . Case 5.3.u_->u_+If u_->u_ +, there exists A_1>A_0>0, such that (ρ_+,u_+)∈ II as A_0<A<A_1, and (ρ_+,u_+)∈ III as A≤ A_0. Proof. If (ρ_+,u_+)∈ II , then 0<u_--A/ρ_-^α<u_+<u_-, which gives ρ_-^α(u_–u_+)<A<ρ_-^α u_-. Thus we take A_0=ρ_-^α(u_–u_+) and A_1=ρ_-^α u_-, then (ρ_+,u_+)∈ II as A_0<A<A_1 and (ρ_+,u_+)∈ III as A≤ A_0. When A_0<A<A_1, (ρ_+,u_+)∈ II in the (ρ,v) plane, so the Riemann solutions to (<ref>) and (<ref>) is given by (<ref>), where x_1(t) and x_2(t) are given by (<ref>) and (<ref>) respectively and the states (ρ_*,v_*) can be calculated as (<ref>). From (<ref>) we havelim_A→ A_0ρ_*=lim_A→ A_0(A/u_+-u_-+A/ρ_-^α)^1/α =lim_A→ A_0(ρ_-A/A-A_0)^1/α=∞. Furthermore, we have the following result.Let dx_1(t)/dt=σ_1(t), dx_2(t)/dt=σ_2(t), then we havelim_A→ A_0v_*+β t=lim_A→ A_0σ_1(t)=lim_A→ A_0σ_2(t)=(u_--A_0/ρ_-^α)t+β t=u_++β t=:σ(t), lim_A→ A_0∫_x_1(t)^x_2(t)ρ_*dx= ρ_-(u_–u_+)t, lim_A→ A_0∫_x_1(t)^x_2(t)ρ_*(v_*+β t)dx=ρ_-(u_–u_+)(u_++β t) t.Proof. (<ref>) is obviously true. We will only prove (<ref>) and (<ref>). lim_A→ A_0∫_x_1(t)^x_2(t)ρ_*dx=lim_A→ A_0ρ_*(x_2(t)-x_1(t))=lim_A→ A_0ρ_*(u_+-ρ_∗v_∗-ρ_-u_-/ρ_∗-ρ_-)t=ρ_-(u_–u_+)t, lim_A→ A_0∫_x_1(t)^x_2(t)ρ_*(v_*+β t)dx=(u_++β t)lim_A→ A_0∫_x_1(t)^x_2(t)ρ_*dx=ρ_-(u_–u_+)(u_++β t) t.The proof is completed. It can be concluded from Lemma <ref> that the curves of the shock wave S_1 and the contact discontinuity J will coincide when A tends to A_0 and the delta shock waves will form.Next we will arrange the values which gives the exact position, propagation speed and strength of the delta shock wave according to Lemma <ref>. From (<ref>) and (<ref>), we letw(t)=ρ_-(u_–u_+) t,, w(t)u_δ(t)=ρ_-(u_–u_+)(u_++β t)t,thenu_δ(t)=(u_++β t),which is equal to σ(t). Furthermore, by letting dx(t)/dt=σ(t), we havex(t)=u_+ t+1/2β t^2. From (<ref>)-(<ref>), we can see that the quantities defined above are exactly consistent with those given by (<ref>)-(<ref>) or (<ref>) in which we takeA=A_0. Thus, it uniquely determines that the limits of the Riemann solutions to the system (<ref>) and (<ref>) when A→ A_0 in the case (ρ_+,u_+)∈ II is just the delta shock solution of (<ref>) and (<ref>) in the case (ρ_+,u_+)∈ S_δ, where S_δ is actually the boundary between the regions II and III. So we get the following results in the case u_+<u_-. If u_+< u_-, for each fixed A with A_0<A<A_1, (ρ_+,u_+)∈ II, assuming that (ρ,u) is a solution containing a shock wave S_1 and a contact discontinuity J of (<ref>) and (<ref>) which is constructed in Section 4, it is obtained that when A→ A_0, (ρ,u) converges to a delta shock wave solution of (<ref>) and (<ref>) when A=A_0. When A≤ A_0, (ρ_+,u_+)∈ III, so the Riemann solutions to (<ref>) and (<ref>) is given by (<ref>) with (<ref>) or (<ref>), which is a delta shock wave solution. It is easy to see that when A→ 0, for ρ_+≠ρ_-,x(t)→σ_0 t+1/2β t^2,w(t)→√(ρ_+ρ_-)(u_– u_+)t, σ(t)=u_δ(t)→σ_0 +β t,whereσ_0=√(ρ_-)u_-+√(ρ_+)u_+/√(ρ_-)+√(ρ_+), for ρ_+= ρ_-,x(t)→1/2(u_++u_-)t+1/2β t^2,w(t)→ρ_+(u_– u_+)t, σ(t)=u_δ(t)→1/2(u_++u_-)+β t,which is exactly the corresponding Riemann solutions to the pressureless Euler equations with the same source term and the same initial data <cit.>. Thus, we have the following result: If u_+< u_-, for each fixed A<A_0, (ρ_+,u_+)∈ III, assuming that (ρ,u) is a delta shock wave solution of (<ref>) and (<ref>) which is constructed in Section 4, it is obtained that when A→ 0, (ρ,u) converges to a delta shock wave solution to the pressureless Euler equations with the same source term and the same initial data <cit.>. Now we summarize the main result in this section as follows.As the perturbed parameter A→0, the Riemann solutions to the approximated nonhomogeneous system (<ref>) tend to the three kinds of Riemann solutions to the Riemann solutions to nonhomogeneous pressureless Euler equations with the same source term and the same initial data, which include a delta shock wave and a vacuum state. That is to say, the Riemann solutions to the transportation equations with Coulomb-like friction is stable under this kind of flux perturbation.§ CONCLUSIONS AND DISCUSSIONSIt can be seen from the above discussions that the limits of solutions to the Riemann problem (<ref>) and (<ref>) converge to the corresponding ones of the Riemann problem (<ref>) and (<ref>) as A→0. The approximated system (<ref>) is strictly hyperbolic. Although the characteristic field for λ_1 is genuinely nonlinear, the characteristic field for λ_2 is still linearly degenerate and (<ref>) still belongs to the Temple class. Thus, this perturbation does not totally change the structure of Riemann solutions to (<ref>).If we also consider the approximation of the flux functions for (<ref>) in the form{[ ρ_t+(ρ u)_x=0,; (ρ( u+1/1-αP))_t+(ρ u( u+P))_x=βρ, ].where P is also given by (<ref>). We can check that (<ref>) has two different eigenvalues λ=u±√(α Bρ^-αu), and the characteristic fields for both the two eigenvalues are genuinely nonlinear. Hence, (<ref>) is strictly hyperbolic and by simple calculation, it can be seen that (<ref>) does not belong to the Temple class anymore. It is clear to see that the Riemann solutions for the approximated system (<ref>) have completely different structures from those for the original system (<ref>). Similar to <cit.>, we can construct the Riemann solutions to the Riemann problem (<ref>) and (<ref>) in all situations and prove them converge to the corresponding ones to the Riemann problem (<ref>) and (<ref>) as A→0.00 Bilic-Tupper-Viollier N.Bilic, G.Tupper and R.Viollier. Unification of dark matter and dark energy: the inhomogeneous Chaplygin gas. Phys. Lett. B, 2002, 535: 17-21.Bouchut F.Bouchut. On zero-pressure gas dynamics//Advances in Kinetic Theory and Computing. Ser Adv Math Appl Sci 22. River Edge, NJ: World Scientific, 1994: 171-190.Brenier-etc. Y.Brenier,W.Gangbo, G.Savage and M.Westdickenberg. The sticky particle dynamics with interactions. J Math. Pure Appl., 2013, 99: 577-617. Chen-Liu1 G.Q.Chen and H.Liu. Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34 (2003), 925-938.Cheng1 H.Cheng. Delta shock waves for a linearly dengenerate hyperbolic system of conservation laws of Keyfitz-Kranzer type. Advances in mathmatical Physics, 2013, Article ID 958120, 10 pages.Cheng2 H.Cheng. On a nonsymmetric Keyfitz-Kranzer system of conservation laws with generalizedand modified Chaplygin gas pressure law. Advances in mathmatical Physics, 2013, 14 pages.Cheng-Yang H.Cheng and H.Yang. Approaching Chaplygin pressure limit to the Aw-Rascle model. J. Math. Anal. Appl., 2014, 416: 839-854. Danilvo-Shelkovich1V.G.Danilvo and V.M.Shelkovich. Dynamics of propagation and interaction of δ-shock waves in conservation law system. J. Differential Equations 221 (2005), 333-381.Danilvo-Shelkovich2V.G.Danilvo and V.M.Shelkovich. Delta-shock waves type solution of hyperbolic systems of conservation laws. Q. Appl. Math. 63 (2005), 401-427.Daw-NedeljkovD.A.E.Daw,M.Nedeljkov. Shadow waves for pressureless gas balance laws. Appl. Math. Lett. 57 (2016), 54-59.Edwards-etc.C.M.Edwards, S.D.Howison, H.Ockendon and J.R.Ockendon. Non-classical shallow water flows. IMA J. Appl. Math. , 2008, 73: 137-157. Faccanoni-Mangeney G.Faccanoni, A.Mangeney. Exact solution for granular flows. Int. J. Numer. Anal. Mech. Geomech., 2012, 37: 1408-1433.Guo L.Guo. The Riemann problem of the transport equations for the generalized Chaplygin gas. J. Xinjiang University (Natuatural Science Edition), 2013, 30(2): 170-176. Guo-Sheng-Zhang L.Guo, W.Sheng and T.Zhang. The two-dimensional Riemann problelm for isentropic Chaplygin gas dynamic system. Comm. Pure Appl. Anal., 2010, 9(2): 431-458. Huang-Wang F.Huang and Z.Wang. Well posedness for pressureless flow. Comm. Math. Phys., 2001, 222: 117-146.Kalisch-Mitrovic1 H.Kalisch and D.Mitrovic. Singular solutions of a fully nonlinear 2×2 system of conservation laws. Proceedings of the Edinburgh Mathematical Society, 2012, 55: 711-729.Kalisch-Mitrovic2 H.Kalisch and D.Mitrovic. Singular solutions for shallow water equations. IMA J. Appl. Math. 2012, 77: 340-350.Keyfitz-Kranzer B.L.Keyfitz and H.C.Kranzer. Spaces of weighted measures for conservation laws with singular shock solutions. J. Differential Equations, 1995, 118: 420-451.Korchinski D.Korchinski. Solutions of a Riemann problem for a system of conservation laws possessing no classical weak solution. Thsis, Adelphi University, 1977.Li-Zhang-Yang J.Li, T.Zhang and S.Yang. The Two-dimensional Riemann Prolem in Gas Dynamics. Pitman Monogr Surv Pure Appl Math 98. Longman Scientific and Technical, 1998.Nedeljkov M.Nedelikov. Shadow waves: entropies and interactions for delta and singular shocks. Arch.Rational Mech. Anal., 2013, 264: 2457-2468. Nguyen-TudorascuT.Nguyen and A.Tudorascu. Pressureless Euler/Euler-Poisson systems wia adhesion dynamics and scalar conservation laws. SIAM J. Math. Anal., 2008, 40: 754-775.Pan-Han L.Pan and X.Han. The Aw-Rascle traffic model with Chaplygin pressure.J. Math. Anal. Appl., 2013, 401: 379-387.Setare M.Setare. Interacting holographic generalized Chaplygin gas model. Phys. Lett. B, 654(2007), 1-6. Shen1 C.Shen. The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IAM J. Appl. Math., 2015:1-24,doi:10.1039/imamat/hxv028.Shen2 C.Shen. The Riemann problem for the Chaplygin gas equations with a source term. Z. Angew. Math. Mech., 2015:1-15,doi:10.1002/zamm.201500015. Shen-Sun C.Shen and M.Sun. Formation of delta-shocks and vacuum states in the vanishing pressue limit of solutions to theAw-Rascle model.J. Differential Equations, 2010, 249: 3024-3051.Smoller Smoller J. Shock Waves and Reaction-Diffusion Equation. New York: Springer-Verlag, 1994.Sheng-Wang-Yin W.Sheng, G.Wang and G.Yin. Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes. Nonlinear analysis: real world Appl., 2015, 22: 115-128.Sheng-Zhang W.Sheng and T.Zhang. The Riemann problem for transportation equations in gas dynamics. Mem. Amer. Math. Soc., 1999, 137(654).Sun M.Sun. The exact Riemann solutions to the generalized Chaplygin gas equations with friction. Commun. Nonlinear Sci. Numer. Simulat., 2016, 36: 342-353.Tan-Zhang-Zheng D.Tan, T.Zhang, Y.Zheng. Delta-shock wave as limits of vanishing viscosity for hyperbolic system of conservation laws. J. Differential Equations, 1994, 112: 1-32.Wang G.Wang. The Riemann problem for one dimensional generalized Chaplygin gas dynamics. J. Math. Anal. Appl., 2013, 403: 403-450. Wang-Ding Z.Wang, and X.Ding. Uniqueness of generalized solution for the Cauchy problem of transportation equations. Acta Math Scientia, 1997, 17(3): 341-352.Wang-Huang-Ding Z.Wang, F.Huang and X.Ding. On the Cauchy problem of transportation equations. Acta Math. Appl. Sinica, 1997, 13(2): 113-122.Yang-Liu H.Yang and J.Liu. Delta-shocks and vacuum states in zero-pressue gas dynamics by the flux approximation. Science China (Mathematics), 2015, 58:2329-2346.Yang-Wang1 H.Yang and J.Wang. Delta-shocks and vacuum states in the vanishinig pressure limit of solutions to the isentopic Euler equations for modified Chaplygin gas. J. Math. Anal. Appl., 2014, 413:800-820. Yang-Wang2 H.Yang and J.Wang. Concentration in vanishinig pressure limit of solutions to the modified Chaplygin gas equations. J. Math.Phys., 2016, 57:doi:10.1063/1.4967299. | http://arxiv.org/abs/1706.08882v1 | {
"authors": [
"Qingling Zhang"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170626083526",
"title": "Stability of Riemann solutions to pressureless Euler equations with Coulomb-like friction by flux approximation"
} |
On Probability of Support Recovery for Orthogonal Matching Pursuit Using Mutual Coherence Ehsan Miandji^†, Student Member, IEEE, Mohammad Emadi^†, Member, IEEE, Jonas Unger, Member, IEEE, and Ehsan Afshari, Senior Member, IEEE Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. E. Miandji and J. Unger are with the Department of Science and Technology, Linköping University, Sweden (e-mail:{ehsan.miandji, jonas.unger}@liu.se). M. Emadi is with Qualcomm Technologies Inc., San Jose, CA USA (e-mail: [email protected]). And E. Afshari is with the Department of Electrical Engineering and Computer Science, University of Michigan, MI USA (e-mail: [email protected]). † Equal contributer December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ In this paper we present a new coherence-based performance guarantee for the Orthogonal Matching Pursuit (OMP) algorithm. A lower bound for the probability of correctly identifying the support of a sparse signal with additive white Gaussian noise is derived. Compared to previous work, the new bound takes into account the signal parameters such as dynamic range, noise variance, and sparsity. Numerical simulations show significant improvements over previous work and a closer match to empirically obtained results of the OMP algorithm. Compressed Sensing (CS), Sparse Recovery, Orthogonal Matching Pursuit (OMP), Mutual Coherence§ INTRODUCTION Let s∈ℝ^N be an unknown variable that we would like to estimate from the measurements y = As+w, where A∈ℝ^M× N is a deterministic matrix and w∈ℝ^M is a noise vector, often assumed to be white Gaussian noise with mean zero and covariance σ^2I, where I is the identity matrix. The matrix A is called a dictionary. We consider the case when A is overcomplete, i.e. N>M, hence uniqueness of the solution of (<ref>) cannot be guaranteed. However, if most elements of s are zero, we can limit the space of possible solutions, or even obtain a unique one, by solving ŝ = xmin x_0s.t. y-Ax_2^2 ≤ϵ,where ϵ is a constant related to w. The location of nonzero entries in s is known as the support set, which we denote by Λ. In some applications, e.g. estimating the direction of arrival in antenna arrays <cit.>, correctly identifying the support is more important than accuracy of values in ŝ. When the correct support is known, the solution of the least squares problem y-A_Λx_Λ_2^2 gives ŝ, where A_Λ is formed using the columns of A indexed by Λ, see <cit.>. Solving (<ref>) is an NP-hard problem and several greedy algorithms have been proposed to compute an approximate solution of (<ref>); a few examples include Matching Pursuit (MP) <cit.>, Orthogonal Matching Pursuit (OMP) <cit.>, Regularized-OMP (ROMP) <cit.>, and Compressive Sampling Matching Pursuit (CoSaMP) <cit.>. In contrast to greedy methods, convex relaxation algorithms <cit.> replace the ℓ_0 pseudo-norm in (<ref>) with an ℓ_1 norm, leading to a convex optimization problem known as the Basis Pursuit (BP) problem <cit.>. While convex relaxation methods require weaker conditions for exact recovery <cit.>, they are computationally more expensive than greedy methods, specially when N≫ M <cit.>. The most important aspect of a sparse recovery algorithm is the uniqueness of the obtained solution. Mutual Coherence (MC) <cit.>, cumulative coherence <cit.>, the spark <cit.>, Exact Recovery Coefficient (ERC) <cit.>, and Restricted Isometry Constant (RIC) <cit.> are metrics proposed to evaluate the suitability of a dictionary for exact recovery. Among these metrics, RIC, spark, and ERC achieve better performance guarantees; however, computing RIC and the spark is in general NP-hard and calculating ERC is a combinatorial problem. In contrast, MC can be efficiently computed and has shown to provide acceptable performance guarantees <cit.>. In this paper, we derive a new lower bound for the probability of correctly identifying the support of a sparse signal using the OMP algorithm. Our main motivation is that previous methods do not directly take into account signal parameters such as dynamic range, sparsity, and the noise characteristics in the computed probability. We will elaborate on this in section <ref>, where we discuss the most recent theoretical analysis for OMP based on MC. The main result of the paper will be presented in section <ref>, followed by numerical evaluation of the new performance guarantee in section <ref>.§ MOTIVATION The mutual coherence of a dictionary A, denoted μ_max(A), is the maximum absolute cross correlation of its columns <cit.>: μ_i,j(A)= ⟨A_i,A_j⟩,μ_max(A)= 1≤ i≠ j ≤ Nmax|μ_i,j(A)|, where we have assumed, as with the rest of the paper, that A_i_2=1, i∈{1,…,N}. Apart from MC and sparsity,s_min = min(|s_i|),ands_max = max(|s_i|), ∀ i∈Λ, which define the dynamic range of the signal, also affect the performance of OMP. The following theorem establishes an important coherence-based performance guarantee for OMP. Let y = As+w, where A∈ℝ^M× N, s_0=τ and w∼𝒩(0,σ^2I). If s_min-(2τ-1)μ_max s_min≥ 2β, where β≜σ√(2(1+α)log N) is defined for some constant α>0, then with probability at least 1 - 1/N^α√(π (1+α)log N), OMP identifies the true support, denoted Λ. The proof involves analyzing the probability event Pr{|⟨A_j,w⟩|≤β}, for some constant β > 0 and for all j=1,…,N (see <cit.> for details). They show that with the lower bound probability of (<ref>), the inequality |⟨A_j,w⟩|≤β holds. It is then shown that if |⟨A_j,w⟩|≤β and (<ref>) hold, then OMP identifies the correct support in each iteration. Moreover, it is assumed that the elements of the sparse vector s are deterministic variables. Hence a strong condition such as (<ref>) is required to determine if the support of s can be recovered. Our analysis removes the condition stated in (<ref>) and introduces a probabilistic bound that depends on N, τ, μ_max, s_max, s_min, and the signal noise. Hence we derive a probability bound that directly takes into account signal parameters and MC. Moreover, unlike <cit.>, we assume that the nonzero elements of s are centered independent random variables with arbitrary distributions. This enables the derivation of a more accurate bound for the probability of exact support recovery.§ OMP CONVERGENCE ANALYSISIn this section we present and prove the main result of the paper. Numerical results will be presented in section <ref>. Let y = As+w, where A∈ℝ^M× N, τ=s_0 and w∼𝒩(0,σ^2I). Moreover, assume that the nonzero elements of s are independent centered random variables with arbitrary distributions. Let λ=Pr{|⟨A_j,w⟩|≤β}, for some constant β≥ 0 and ∀ j∈{1,…,N}. If s_min/2≥β, then OMP identifies the true support with lower bound probabilityλ(1-2N exp(-N(s_min/2-β)^2/2τ^2γ^2+2N γ(s_min/2-β)/3)), where γ=μ_maxs_max. Moreover, λ is lower bounded by 1-N√(2/π)σ/βe^-β^2/2σ^2. Before presenting the proof, let us compare Theorems <ref> and <ref> analytically. It is important to note that (<ref>) is indeed equivalent to (<ref>). The apparent difference is only attributed to the use of α or β from the definition β≜σ√(2(1+α)log N). For instance, using the aforementioned definition of β on (<ref>) leads to (<ref>). As a result, the second term of (<ref>) can be interpreted as a probabilistic representation of the condition imposed by (<ref>) in Theorem <ref>. Moreover, because (<ref>) is equal to (<ref>) and the second term of (<ref>) is in the range [0,1], therefore (<ref>) is always smaller or equal to (<ref>). However, as it will be seen in section <ref>, since the condition of Theorem <ref> in (<ref>) is not satisfied in many scenarios, our results match the empirical results more closely. Evidently, the condition s_min/2≥β in Theorem <ref> is more relaxed compared to (<ref>). Our numerical results in Section <ref> also verify this fact.The following lemma will provide us with the necessary tool for the proof of Theorem <ref>. The proof of the lemma is postponed to the Appendix. Define Γ_j = |⟨A_j,As+w⟩|, for any j∈{1,…,N}, where w∼𝒩(0,σ^2I) and |⟨A_j,w⟩|≤β. Then for some constant ξ≥ 0, and assuming ξ≥β, we have Pr{Γ_j ≥ξ}≤ 2exp(-(ξ-β)^2/2(Nν+c(ξ-β)/3)), where|μ_j,ns_n| ≤ c,E{μ_j,n^2s_n^2}≤ν, ∀ n∈{1,…, N} We can now state the proof of Theorem <ref>.It was shown in <cit.> that OMP identifies the true support Λ if j∈Λmin |⟨A_j,A_Λs_Λ+w⟩| ≥k∉Λmax |⟨A_k,A_Λs_Λ+w⟩|. The term on the left-hand side of (<ref>) can be rewritten asj∈Λmin |⟨A_j,A_Λs_Λ+w⟩| = j∈Λmin|s_j + ⟨A_j,A_Λ∖{j}s_Λ∖{j}+w⟩|≥j∈Λmin|s_j| - j∈Λmax|⟨A_j,A_Λ∖{j}s_Λ∖{j}+w⟩|. From (<ref>) and (<ref>), we can see that the OMP algorithm identifies the true support if k∉Λmax{Γ_k} < j∈Λmin|s_j|/2,j∈Λmax|⟨A_j,A_Λ∖{j}s_Λ∖{j}+w⟩| < j∈Λmin|s_j|/2. Using (<ref>), we can define the probability of error as Pr{error} ≤Pr{j∈Λmax|⟨A_j,A_Λ∖{j}s_Λ∖{j}+w⟩| ≥s_min/2}+Pr{k∉Λmax{Γ_k}≥s_min/2}≤∑_j∈Λ^Pr{|⟨A_j,A_Λ∖{j}s_Λ∖{j}+w⟩| ≥s_min/2}+∑_k∉Λ^Pr{Γ_k ≥s_min/2}. For the first term on the right-hand side of (<ref>), excluding the summation over the indices in Λ, from Lemma <ref> we have j∈ΛPr{|⟨A_j,A_Λ∖{j}s_Λ∖{j} + w⟩| ≥s_min/2} ≤2exp(-ρ^2/2((τ-1)ν+ cρ/3))_P_1, where ρ=s_min/2-β is defined for notational brevity. Note that the dictionary A in (<ref>) is supported on Λ∖{j}, i.e. all the indices in the true support excluding j. Therefore the term (τ-1), instead of N, appears in the denominator of (<ref>). Similarly, for the second term of (<ref>) we have k∉ΛPr{Γ_k ≥s_min/2}≤2exp(-ρ^2/2(τν+cρ/3))_P_2. Substituting (<ref>) and (<ref>) into (<ref>) yields Pr{error}≤τP_1 + (N-τ)P_2 ≤ NP_2, where the last inequality follows since P_2>P_1. Moreover, for the upper bounds c and ν in (<ref>) we have |μ_j,ns_n|≤μ_maxs_max,E{μ_j,n^2s_n^2} ≤1/N∑_n=1^Nμ_max^2 E{s_n^2}≤τ/N s_max^2μ_max^2, Combining (<ref>) and (<ref>) with (<ref>), the following is obtained Pr{error}≤ 2Nexp(-Nρ^2/2τ^2γ^2+2N γρ/3), where we have defined γ=μ_maxs_max for notational brevity. So far we have assumed that |⟨A_j,w⟩| ≤β, ∀ j. Therefore, the probability of success is the joint probability of Pr{|⟨A_j,w⟩| ≤β} and the inverse of (<ref>). For the former, a lower bound was formulated in <cit.> as follows Pr{|⟨A_j,w⟩| ≤β}≥ 1 - √(2/π)σ/βe^-β^2/2σ^2_P_3. Since |⟨A_j,w⟩| ≤β should hold ∀ j∈{1,…, N}, we have j=1,…,NPr{|⟨A_j,w⟩| ≤β}≥ (1-P_3)^N ≥ 1-NP_3. Inverting the probability event in (<ref>) and multiplying by the lower bound in (<ref>) yields (<ref>), which completes our proof. § NUMERICAL RESULTSIn this section we compare numerical results of Theorem <ref> (Ben-Haim et al. <cit.>), and Theorem <ref> (proposed herein) with the empirical results of OMP. Indeed we only consider probability of successful recovery of the support. An upper bound for the MSE of the oracle estimator has been previously established, see e.g. Theorem 5.1 in <cit.> or Lemma 4 in <cit.>. The oracle estimator knows the support of the signal, a priori.All the empirical results are obtained by performing the OMP algorithm 5000 times using a random sparse signal with additive white Gaussian noise in each trial. The probability of success is computed as the ratio of successful trials to the total number of trials; note that a trial is successful if Λ=Λ̂, where Λ̂ is the support of ŝ obtained from OMP by solving (<ref>). Moreover, the number of trials was empirically set such that the probability of success for the OMP algorithm was stable across different parameters. For comparison, we use the dictionary of <cit.> defined as A = [I,H], where I is an identity matrix and H is a Hadamard matrix, hence we have N=2M. The sparse signal in each trial, denoted s in (<ref>), is constructed as follows: The support of the sparse signal, Λ=supp(s), is constructed by uniform random permutation of the set {1,…,N} and taking the first τ indices. The nonzero elements located at Λ are drawn randomly from a uniform distribution on the interval [s_min,s_max], multiplied randomly by +1 or -1. Once the sparse signal is constructed, the input of the OMP algorithm, y, is obtained by evaluating (<ref>). In order to facilitate the comparison of Theorems <ref> and <ref>, we need to fix the value of β. To do this, we empirically calculate β as wmax jmax|⟨A_j,w⟩|, where the maximum over w is computed using 10^4 vectors w∼𝒩(0,σ^2I), as assumed by both theorems. Given β, we can calculate α for Theorem <ref> from the definition β≜σ√(2(1+α)log N). Indeed, a lower value of β leads to better results for both theorems, see (<ref>) and (<ref>). As a result, here we consider the worst-case scenario. When (<ref>) is not satisfied for Theorem <ref>, we set the probability of success to zero. We use the same procedure for the condition of Theorem <ref>; i.e. the probability of success is set to zero when s_min/2<β.Numerical results are summarized in Figure <ref>. We analyze the effect of sparsity on the probability of successful support recovery in plots <ref>, <ref>, and <ref>. Three signal dimensionalities and three noise variances: σ_1^2=10^-6, σ_2^2=2.5× 10^-5, and σ_3^2=10^-4, are considered. For all these cases we set s_min=0.5 and s_max=1. In Fig. <ref> we see that Theorem <ref> achieves a higher probability for σ_3 and small values of τ, while Theorem <ref> leads to more accurate results for larger values of τ. Additionally, for σ_1 and σ_2, Theorem <ref> is much closer to empirical results. Most importantly, the shape of the probability curves for Theorem <ref> matches the empirical curves. In contrast, Theorem <ref> produces a step function due to the fact that condition (<ref>) is not satisfied for a large range of values for τ, even though the success probability in (<ref>) is close to one for different values of σ. The condition of Theorem <ref> is satisfied across all the parameters for figures <ref>-<ref>. We discussed in section <ref> that (<ref>) is always smaller than (<ref>) due to the second term of (<ref>). We expect this term to become more accurate as the signal dimensionality grows since it is exponential in N; moreover, β and μ_max become smaller as N grows. This is confirmed in figures <ref> and <ref>. As we increase N, the gap between theorems <ref> and <ref> increases, confirming that the second term of (<ref>) is becoming more accurate compared to (<ref>). The empirical probability is close to one for all the values of τ plotted in figures <ref> and <ref>. The effect of s_min on the probability of success is demonstrated in figures <ref>, <ref>, and <ref>. For each plot, we consider τ_1=16, τ_2=32, and τ_3=64, while setting σ^2=10^-4. The empirical results show a probability of success close to one across the parameters considered. In Fig. <ref> we see a significant difference between Theorems <ref> and <ref>. The condition of Theorem <ref> is not satisfied for any value of s_min and τ. In contrast, Theorem <ref> shows high probabilities for all three values of τ. The dynamic range (DR) of the signal can be defined as s^2_max/s^2_min. As we increase the signal dimensionality (N), Theorem <ref> reports larger probability for larger values of DR and all three values of τ. On the other hand, the condition of Theorem <ref> fails for τ_2 and τ_3, even when we have M=4096. For τ_1, Theorem <ref> can produce valid results for a slightly higher DR. Lastly, in plots <ref>, <ref>, and <ref>, we analyze the effect of noise variance on the probability of success for τ_1=16, τ_2=32, and τ_3=64. In Fig. <ref>, where M=1024, both theorems fail to produce valid results for τ_3=64. However, Theorem <ref> reports acceptable results for τ_1 and τ_2, while the condition of Theorem <ref> is not satisfied. As the signal dimensionality grows, see Fig. <ref> and <ref>, Theorem <ref> becomes more tolerant of higher noise variances. The results for Theorem <ref> also improves with increasing signal dimensionality, however only for τ_1. This shows the robustness of Theorem <ref> to larger values of sparsity. § CONCLUSIONS We presented a new bound for the probability of correctly identifying the support of a noisy sparse signal using the OMP algorithm. Compared to the analysis of Ben-Haim et al. <cit.>, our analysis replaces a sharp condition with a probabilistic bound. Comparisons to empirical results obtained by OMP show a much improved correlation than previous work. Expanding Γ_j, we can show that Γ_j= |∑_m=1^MA_m,j(∑_n=1^NA_m,ns_n+w_m)|=|∑_n=1^N{∑_m=1^MA_m,jA_m,ns_n + 1/N∑_m=1^MA_m,jw_m}|.=|∑_n=1^N{μ_j,ns_n+1/N⟨A_j,w⟩}|. We are interested in tail bounds for sum of random variables μ_j,ns_n+ N^-1⟨A_j,w⟩, for n=1,…,N. Let us define x_n = μ_j,ns_n. Using the assumption |⟨A_j,w⟩|≤β we have Pr{Γ_j ≥ξ} ≤Pr{|∑_n=1^Nx_n| + |1/N∑_n=1^N⟨A_j,w⟩| ≥ξ}≤Pr{|∑_n=1^Nx_n| ≥ξ - β}. Since {s_n}_n=1^N,and hence {x_n}_n=1^N, are centered independent real random variables, according to Bernstein's inequality <cit.>, ifE{x_n^2}≤ν, and Pr{|x_n| < c}=1, then for a positive constant δ we have Pr{|∑_n=1^Nx_n| ≥δ} ≤ 2 exp(-δ^2/2(∑_n=1^NE{x_n^2}+cδ/3)) ≤ 2 exp(-δ^2/2(Nν+cδ/3)). Setting δ=ξ-β in (<ref>) completes the proof.IEEEtran | http://arxiv.org/abs/1706.09288v2 | {
"authors": [
"Ehsan Miandji",
"Mohammad Emadi",
"Jonas Unger",
"Ehsan Afshari"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170626225535",
"title": "On Probability of Support Recovery for Orthogonal Matching Pursuit Using Mutual Coherence"
} |
CISUC, Department of Informatics Engineering,University of Coimbra, Portugal fga, naml, machado, [email protected] grammar-based NeuroEvolution approaches have several shortcomings. On the one hand, they do not allow the generation of ANN composed of more than one hidden-layer. On the other, there is no way to evolve networks with more than one output neuron. To properly evolve ANN with more than one hidden-layer and multiple output nodes there is the need to know the number of neurons available in previous layers. In this paper we introduce DSGE: a new genotypic representation that overcomes the aforementioned limitations. By enabling the creation of dynamic rules that specify the connection possibilities of each neuron, the methodology enables the evolution of multi-layered ANN with more than one output neuron. Results in different classification problems show that DSGE evolves effective single and multi-layered ANN, with a varying number of output neurons. <ccs2012> <concept> <concept_id>10010147.10010257.10010293.10010294</concept_id> <concept_desc>Computing methodologies Neural networks</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257.10010293.10011809.10011813</concept_id> <concept_desc>Computing methodologies Genetic programming</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257.10010258.10010259.10010263</concept_id> <concept_desc>Computing methodologies Supervised learning by classification</concept_desc> <concept_significance>300</concept_significance> </concept> <concept> <concept_id>10003752.10003809.10003716.10011804.10011813</concept_id> <concept_desc>Theory of computation Genetic programming</concept_desc> <concept_significance>100</concept_significance> </concept> </ccs2012> [500]Computing methodologies Neural networks [500]Computing methodologies Genetic programming [300]Computing methodologies Supervised learning by classification [100]Theory of computation Genetic programming Towards the Evolution of Multi-Layered Neural Networks: A Dynamic Structured Grammatical Evolution Approach Filipe Assunção, Nuno Lourenço, Penousal Machado, Bernardete Ribeiro June 23, 2017 ===========================================================================================================§ INTRODUCTIONML approaches, such as ANN, are often used to learn how to distinguish between multiple classes of a given problem. However, to reach near-optimal classifiers a laborious process of trial-and-error is needed to hand-craft and tune the parameters of ML methodologies. In the specific case of ANN there are at least two manual steps that need to be considered: (i) the definition of the topology of the network, i.e., number of hidden-layers, number of neurons of each hidden-layer, and how should the layers be connected between each other; and (ii) the choice and parameterisation of the learning algorithm that is used to tune the weights and bias of the network connections (e.g., initial weights distribution and learning rate).EANN or NeuroEvolution refers to methodologies that aim at the automatic search and optimisation of the ANN' parameters using EC. With the popularisation of DL and the need for ANN with a larger number of hidden-layers, NeuroEvolution has been vastly used in recent works <cit.>.The goal of the current work is the proposal of a novel GGP methodology for evolving the topologies, weights and bias of ANN. We rely on a GGP methodology because in this way we allow the direct encoding of different topologies for solving distinct problems in a plug-and-play fashion, requiring the user to set a grammar capable of describing the parameters of the ANN to be optimised. One of the shortcomings of previous GGP methodologies applied to NeuroEvolution is the fact that they are limited to the evolution of network topologies with one hidden-layer. The proposed approach, called DSGE solves this constraint.The remainder of the document is organised as follows: In Section <ref> we detail SGE and survey the state of the art on NeuroEvolution; Then, in Sections <ref> and <ref>, DSGE and its adaption to the evolution of multi-layered ANN are presented, respectively; The results and comparison of DSGE with other GGP approaches is conducted in Section <ref>; Finally, in Section <ref>, conclusions are drawn and future work is addressed.§ BACKGROUND AND STATE OF THE ARTIn the following sub-sections we present SGE, which serves as base for the development of DSGE. Then, we survey EC approaches for the evolution of ANN. §.§ Structured Grammatical EvolutionSGE SGE was proposed by Lourenço et al. <cit.> as a new genotypic representation for GE <cit.>. The new representation aims at solving the redundancy and locality issues in GE, and consists of a list of genes, one for each non-terminal symbol. Furthermore, a gene is a list of integers of the size of the maximum number of possible expansions for the non-terminal it encodes; each integer is a value in the interval [0,non_terminal_possibilities-1], where non_terminal_possibilities is the number of possibilities for the expansion of the considered non-terminal symbol. Consequently, there is the need to pre-process the input grammar to find the maximum number of expansions for each non-terminal (further detailed in Section 3.1 of <cit.>).To deal with recursion, a set of intermediate symbols are created, which are no more than the same production rule replicated a pre-defined number of times (recursion level). The direct result of the SGE representation is that, on the one hand, the decoding of individuals no longer depends on the modulus operation and thus its redundancy is removed; on the other hand, locality is enhanced, as codons are associated with specific non-terminals.The genotype to phenotype mapping procedure is similar to GE, with two main differences: (i) the modulus operation is not used; and (ii) integers are not read sequentially from a single list of integers, but are rather read sequentially from lists associated to each non-terminal symbol. An example of the mapping procedure is detailed in Section 3 of <cit.>. §.§ NeuroEvolutionWorks focusing the automatic generation of ANN are often grouped according to the aspects of the ANN they optimise: (i) evolution of the network parameters; (ii) evolution of the topology; and (iii) evolution of both the topology and parameters.The gradient descent nature of learning algorithms, such as backpropagation, makes them likely to get trapped in local optima. One way to overcome this limitation is by using EC to tune the ANN weights and bias values. Two types of representation are commonly used: binary <cit.> or real (e.g., CoSyNE <cit.>, Gravitational Swarm and Particle Swarm Optimization applied to OCR <cit.>, training of deep neural networks <cit.>).When training ANN using EC the topology of the network isprovided and is kept fixed during evolution. Nevertheless, approaches tackling the automatic evolution of the topology of ANN have also been proposed. In this type of approaches for finding the adequate weights some authors use a off-the-shelf learning methodology (e.g., backpropagation) or evolve both the weights and the topology simultaneously.Regarding the used representation for the evolution of the topology of ANN it is possible to divide the methodologies into two main types: those that use direct encodings (e.g., Topology-optimization Evolutionary Neural Network <cit.>, NeuroEvolution of Augmenting Topologies <cit.>) and those that use indirect encodings (e.g., Cellular Encoding <cit.>). In the first two works the genotype is a direct representation of the network, and in the latter a mapping procedure has to be applied to transform the genotype into an interpretable network. Focusing on indirect representations, in the next section we further detail grammar-based approaches.§.§ Grammar-based NeuroEvolutionOver the last years several approaches applying GE to NeuroEvolution have been proposed. However, the works that focus the evolution of the networks topology are limited to generating one-hidden-layered ANN, because of the difficulties in tracking the number of neurons available in previous layers.Tsoulos et al. <cit.> and Ahmadizar et al. <cit.> describe two different approaches based on GE for the evolution of both the topology and parameters of one-hidden-layered ANN. While the first evolves the topology and weights using GE, the latter combines GE with a GA: GE is applied to the evolution of the topology and the GA is used for searching the real values (i.e., weights and bias). The use of a GA to optimise the real values is motivated by the fact that GE is not suited for the evolution and tuning of real values. For that reason, Soltanian et al. <cit.> just use GE to optimise the topology of the network, and rely on the backpropagation algorithm to train the evolved topologies.Although GE is the most common approach for the evolution of ANN by means of grammars it is not the only one. Si et al. in <cit.> present GSNN: an approach that uses Grammatical Swarm for the evolution of the weights and bias of a fixed ANN topology, and thus, EC is used just for the generation of real numbers. In <cit.>, Jung and Reggia detail a method for searching adequate topologies of ANN based on descriptive encoding languages: a formal way of defining the environmental space and how should individuals be formed; the RPROP algorithm is used for training the ANN. SGE has also been used for theevolution of the topology and weights of ANN <cit.>. § DYNAMIC STRUCTURED GRAMMATICAL EVOLUTIONDSGEDSGE is our novel GGP approach. With the proposed methodology the gain is twofold: (i) all the genotype is used, i.e., while in GE and SGE the genotype encodes the largest allowed sequence, in DSGE the genotype grows as needed; and (ii) there is no need to pre-process the grammar in order to compute the maximum tree-sizes of each non-terminal symbol, so that intermediate grammar derivation rules are created. In the next sections we describe in detail the components that make these gains possible. §.§ RepresentationEach candidate solution encodes an ordered sequence of the derivation steps of the used grammar that are needed to generate a specific solution for the problem at hand. The representation is similar to the one used in SGE, with one main difference: instead of computing and generating the maximum number of derivations for each of the grammar's non-terminal symbols, a variable length representation is used, where just the number of needed derivations are encoded. Consequently, there is no need to create intermediate symbols to deal with recursive rules. To limit the genotype size, a maximum depth value is defined for each non-terminal symbol. Allowing different limits for each non-terminal symbol provides an intuitive and flexible way of constraining the search space by limiting, for instance, the maximum number of hidden-layers, neurons or connections.Figure <ref> represents an example grammar for the generation of real numbers in the [0, 3[ interval, along with the representation of a candidate solution encoding the number 1.259. The genotype is encoded as a list of genes, where each gene encodes an ordered sequence of choices for expanding a given non-terminal symbol, as a list of integers. The genotype to phenotype mapping is detailed in Section <ref>. §.§ InitialisationAlgorithm <ref> details the recursive function that is used to generate each candidate solution. The input parameters are: the grammar that describes the domain of the problem; the maximum depth of each non-terminal symbol; the genotype (which is initially empty); the non-terminal symbol that we want to expand (initially the start symbol is used); and the current sub-tree depth (initialised to 0). Then, for the non-terminal symbol given as input, one of the possible derivation rules is selected (lines 2-11) and the non-terminal symbols of the chosen derivation rule are recursively expanded (lines 12-18). However, when selecting the expansion rule there is the need to check whether or not the maximum sub-tree depth has already been reached (lines 3-5). If that happens, only non-recursive derivation rules can be selected for expanding the current non-terminal symbol (lines 6-7). This procedure is repeated until an initial population with the desired size is created.§.§ Mapping FunctionTo map the candidate solutions genotype into the phenotype that will later be interpreted as an ANN we use Algorithm <ref>. The algorithm is similar to the one used to generate the initial population but, instead of randomly selecting the derivation rule to use in the expansion of the non-terminal symbol, we use the choice that is encoded in the individual's genotype (lines 12-22). During evolution the genetic operators may change the genotype in a way that requires a larger number of integers than the ones available. When this happens, the following genotype's repair procedure is applied: new derivation rules are selected at random and added to the genotype of the individual (lines 3-11). In addition to returning the genotype the algorithm also computes which genotype integers are being used, which will later help in the application of the genetic operators. Table <ref> shows an example of the mapping procedure applied to the genotype of Figure <ref>. §.§ Genetic OperatorsTo explore the problem's domain and therefore promote evolution we rely on mutation and crossover.Mutation is restricted to integers that are used in the genotype to phenotype mapping and changes a randomly selected expansion option (encoded as an integer) to another valid one, constraint to the restrictions on the maximum sub-tree depth. To do so, we first select one gene; the probability of selecting the i-th gene (p_i) is proportional to the number of integers of that non-terminal symbol that are used in the genotype to phenotype mapping (read_integers):p_i = read_integers_i/∑_j=1^n read_integers_j,where n is the total number of genes. Additionally, genes where there is just one possibility for expansion (e.g, <start> or <float> of the grammar of Figure <ref>) are not considered for mutation purposes. After selecting the gene to be mutated, we randomly select one of its integers and replace it with another valid possibility.Considering the genotype of Figure <ref>, a possible result from the application of the mutation operator is [[0], [0], [2], [0,0,1], [2,5,9]], that represents the number 2.259. Crossover is used to recombine two parents (selected using tournament selection) to generate two offspring. We use one-point crossover. As such, after selecting the cutting point (at random) the genetic material is exchanged between the two parents. The choice of the cutting point is done at the gene level and not at the integers level, i.e., what is exchanged between parents are genes and not the integers.Given two parents [[0], [0], [2], | [0,0,1], [2,5,9]] and [[0], [0], [1], | [0,0,0,1], [1,0,2,4]], where | denotes the cutting point, the generated offspring would be[[0], [0], [2], [0,0,0,1], [1,0,2,4]] and [[0], [0], [1], [0,0,1], [2,5,9]].§.§ Fitness Evaluation To enable the comparison of the evolved ANN with the results from a previous work <cit.> the performance is measured as the RMSE obtained while solving a classification task. In addition, for better dealing with unbalanced datasets, this metric considers the RMSE per class, and the resulting fitness function is the multiplication of the exponential values of the multiple RMSE per class, as follows:fitness = ∏_c=1^m exp(√(∑_i=1^n_c (o_i - t_i)^2 /n_c) ),where m is the number of classes of the problem, n_c is the number of instances of the problem that belong to class c, o_i is the confidence value predicted by the evolved network, and t_i is the target value. This way, higher errors are more penalised than lower ones, helping the evolved networks to better generalise to unseen data.§ EVOLUTION OF MULTI-LAYERED ANNSFigure <ref> shows the grammar that was used in <cit.>, which is similar to those used in other works focusing the grammar-based evolution of ANN <cit.>. The rationale behind the design of this grammar is the evolution of networks composed of one hidden-layer, where only the hidden-neurons as well as the weights of the connections from the input and to the output neurons are evolved. Three major drawbacks can be pointed to the previous grammatical formulation: (i) it only allows the generation of networks with one hidden-layer; (ii) the output neuron is always connected to all neurons in the hidden-layer; and (iii) there is no way to define multiple output nodes, at least one that reuses the hidden-nodes, instead of creating new ones.These drawbacks are related to limitations of the evolutionary engines that are overcome by the approach presented in this paper. Figure <ref> represents a grammar capable of representing multi-layered ANN, with a variable number of neurons in each layer. However, there is no way of knowing how many neurons are in the previous layer, so that the established connections are valid. In the next sections we detail the adaptions we introduce to allow the evolution of multi-layered networks.§.§ Dynamic Production Rules To know the neurons available in each hidden-layer we create new production rules, in run-time. More precisely, considering the grammar of Figure <ref>, for each i-th <layer> non-terminal symbol we create a <features-i> production rule, encoding the features that layer can use as input. For the first hidden-layer, <features-1> has the available features as expansion possibilities, i.e., the ones that are initially defined in the grammar (x_1, …, x_n, where n represents the number of features). Then, for the next hidden-layers there are two possibilities: (i) let the connections be established to all the neurons in previous layers (including input neurons); or (ii) limit the connections to the neurons in the previous layer. In the current work we have decided for the first option, with the restriction that the neurons in the output layer can only be connected to hidden-nodes. When establishing the connections between neurons, the probability of choosing a neuron in the previous layer is proportional to the number of previous hidden-layers. More specifically:P(neuron_i-1) = ∑_j=1^j=i-2 P(neuron_j),i.e., when establishing the connections of the i-th layer, the probability of linking to a neuron in the previous layer (i-1) is equal to the probability of linking to a neuron in the remaining layers (1, …, i-2). The rationale is to minimise the emergence of deep networks with useless neurons, in the sense that they are not connected (directly or indirectly) to output nodes. | http://arxiv.org/abs/1706.08493v1 | {
"authors": [
"Filipe Assunção",
"Nuno Lourenço",
"Penousal Machado",
"Bernardete Ribeiro"
],
"categories": [
"cs.NE",
"cs.AI"
],
"primary_category": "cs.NE",
"published": "20170626173414",
"title": "Towards the Evolution of Multi-Layered Neural Networks: A Dynamic Structured Grammatical Evolution Approach"
} |
SUNNY-CP and the MiniZinc Challenge[ December 30, 2023 =====================================We study billiard dynamics inside an ellipse for which the axes lengthsare changed periodically in time and an O(δ)-small quartic polynomial deformation is added to the boundary. In this situation the energy of the particle in the billiard is no longer conserved. We show a Fermi accelerationin such system: there exists a billiard trajectory on which the energy tends to infinity. The constructionis based on the analysis of dynamics in the phase space near a homoclinic intersectionof the stable and unstable manifolds of the normally hyperbolic invariant cylinder Λ,parameterised by the energy and time, that corresponds to the motion along the major axis of the ellipse.The proof depends on the reduction of the billiard map near the homoclinic channel to an iterated function systemcomprised by the shifts along two Hamiltonian flows defined on Λ. The two flows approximatethe so-called inner and scattering maps, which are basic tools that arise in the studies of the Arnold diffusion;the scattering maps defined by the projection along the strong stable and strong unstable foliations W^ss,uuof the stable and unstable invariant manifolds W^s,u(Λ) at the homoclinic points. Melnikov type calculations imply that the behaviour of the scattering map in this problem is quite unusual: it is only defined on a smallsubset of Λ that shrinks, in the large energy limit, to a set of parallel lines t=const as δ→ 0.§ INTRODUCTION AND MAIN RESULTS Billiards are Hamiltonian dynamical systems, representing the motion of a point particle inside a domainQ (the billiard table) in a straight line with constant speed and elastically bouncing off the boundary of the domain ∂Q. The study of billiard systems was initiated by Birkhoff <cit.>.Depending on the boundary shape, billiard's dynamical behaviour may range from completely integrable to chaotic.The billiard inside an ellipse is the only known integrable strictly convex billiard <cit.>.The integrability of elliptic billiard is closely connected to the existence of a continuous family of caustics.A caustic is a curve such that if a billiard trajectory segment is tangent to it, all other segments of the trajectoryare also tangent to the same curve. For the trajectories that do not intersect the segment connecting the foci of theellipse, the caustics are confocal ellipses while for the trajectories that intersect this segment the causticsare confocal hyperbolas. The period two trajectory along the major axis is hyperbolic, with stable and unstablemanifolds that coincide. The corresponding motions repeatedly gothrough the foci and converge to the major axis.Billiards with time-dependent boundaries have received much attention in recent years <cit.>.One of the fundamental issues here is determining whether the particle energy may grow without bound as a resultof repeated elastic collisions with the moving boundary. This phenomenon is called Fermi acceleration, afterFermi who first proposed it in his studies of highly energetic cosmic rays <cit.>. The existenceof Fermi acceleration has been investigated theoretically and numerically in various billiard geometries.The simplest one-dimensional case corresponding to a particle bouncing between two periodically moving walls (Fermi-Ulam model) and its variants is already very subtle and the existence of Fermi acceleration has been shown to depend on the class of smoothness of the motion of the wall <cit.>.For domains in two dimensions and higher, it has been observed<cit.> that the acceleration depends on the structure of the phase space of the static ”frozen" billiard. It has beenconjectured by Losktutov, Ryabov and Akinshin (LRA) <cit.> and consequently provedin <cit.> that a sufficient condition for Fermi acceleration is the presence of a Smale horseshoe in the phase space of the frozen billiard.On the other hand, it has been shown <cit.> that the energy of trajectories in the time-dependent circle billiard stays bounded due to the angular momentum conservation. Earlier Koiler et al. <cit.> numerically studied time-dependent perturbations of elliptic billiards and did not observe sustained energy growth. However, more detailed numerical simulations byLenz et al. <cit.> showed slow growth of the particle speedwhen initial conditions belong to the separatrix region. An elliptic billiard with a slow boundary perturbation and a slow angular velocity was also studied by Itin and Neishtadt <cit.> who investigatedthe destruction of adiabatic invariants near a separatrix. In this paper we further push the study of Fermi acceleration in periodically perturbed ellipse.Fermi acceleration question is a part of the general question of energy growth in a priori unstable Hamiltonian systems, that also includes Mather acceleration problem <cit.>.Sincetime-dependent billiards on a plane are given by a nonautonomous Hamiltonian systems with two and a half degrees offreedom, the billiard map is exact symplectic four-dimensional diffeomorphism <cit.>. In particular, invariant KAM-tori, if exist, do not divide the phase space into invariant regionsand Arnold diffusion <cit.> may occur. Arnold diffusion refers to the instabilityof action variables in a Hamiltonian system with n>2 degrees of freedom of the formH = H_0(I) + ϵ H_1(I, φ, ϵ) where (I, φ) are action-angle variables, ϵis small, and H_0 is integrable. Following terminology in <cit.>, a Hamiltonian systemis called a-priori unstable if the integrable part H_0 has a normally hyperbolic invariantmanifold Λ with stable and unstable manifolds W^s,u(Λ) that coincide in a homoclinic loop.Under small perturbations, Λ and W^s,u(Λ) persist but W^s,u(Λ) may intersect transversally along a homoclinic set Γ. In this case the diffusing orbit stays near Λ most of the time,occasionally making a trip near Γ. It was shown by Treschev <cit.> and Delshams, de la LLave and Seara <cit.> that such homoclinic excursions can lead to a systematic drift of the action variable in the a priori unstable case.The technique for the analysis of such excursions, which is also used in this paper, goes back tothe works of Delshams, de la LLave and Seara <cit.> where notions of the inner and scattering maps have been introduced and studied in detail. The inner map isthe restriction of the dynamicsto Λ, and the scattering map relates two points on Λ that are connected asymptoticallyin the past and future if the intersection of W^s,u(Λ) isstrongly transverse <cit.> along Γ. The iteration function system obtainedby successive application of these two maps in an arbitrary order gives the diffusing orbitif they do not have common invariant curves <cit.>. It was shown e.g. in <cit.> that the finite-length diffusing orbits of the iteratedfunction system on Λ correspond to Arnold diffusion in the original diffeomorphism near Λ∪Γ,under the assumption of strong transversality of homoclinic intersections. This result was generalised byGidea, de la Llave and Seara <cit.> to the orbits of semi-infinite length.In this paper we study the time-dependent four-dimensional billiard map B (defined in Section 3)describing the motion of a billiard inside the planar domain with the time-dependent boundaryx^2/a^2(t) + y^2/b^2(t) = 1 +2 δ y^4/b^4(t),where 0 < | δ | ≪ 1 is a constant parameter and 0<b(t)<a(t) are periodic C^r+1-smooth functions(r ≥ 4) of time (so t ∈𝕊^1), and (x,y) ∈ℝ^2. The boundary (<ref>)may be viewed as an ellipse with time-periodically changing semi-axes a(t) and b(t) plusan O(δ)quartic polynomial perturbation superimposed at each fixed value of t. Let ℰ(t) be the kinetic energy of the particle in the billiard bounded by (<ref>). We prove here the following Let 0 < b(t) < a(t) be time-periodic C^r+1-functions (r ≥ 4) such thatthe function a(t)/b(t) has a nondegenerate critical point. Then, there exists a constant C>0,independent of δ, such that for any ℰ_0≥C/|δ|, there existsa billiard trajectory for which the energy ℰ(t) grows from ℰ_0 to infinity. We note that if one replaces the O(δ) quartic polynomial perturbation in the right-hand side of (<ref>) by another O(δ) perturbation that also destroys integrability of the static frozen ellipse for every fixed t(for instance symmetric entire perturbations studied in <cit.>), then Theorem 1.1 should still hold. We however restrict ourselves to a particular form of the perturbation, in order to keep the computations explicit. We do not know whether the measure of the set of orbits for which the energy grows to infinity is positive. However, the same construction we use in the proof can show the existence of “diffusing” orbits which takeevery sufficiently large value of energy infinitely many times, following an arbitrary given itinerary, so the evolution of energy is sensitive to initial conditions and has to be described by some random process.The proof of Theorem 1.1 is based on the study of inner and scattering maps and an application of the theorydeveloped in <cit.>.The scheme of the proof is as follows.Each time the particle hits the boundary of (<ref>), one records the collision time moment t∈𝕊^1, the particle kinetic energy ℰ, the angular variable φ that determines the position of the collision point on the billiard boundary, and the post-collision reflection angle θ. Then, the particle motion is described by the billiard map B in the four-dimensional space of variables (φ, θ, ℰ, t). We assume that the speed w= √(2ℰ) of the particle is large compared to the speed with which the boundary moves. This invokes the presence of two time scales in the problem: the variables ℰ and t vary slowly,while (φ, θ) change fast. To make the presence of different time scales more apparent, we scale variables like it was done in <cit.>: take largespeed w^*, introduce a small parameter ε=1/w^* and the rescaled energyE=ε^2 ℰ. Then, for any bounded interval of E, the map B becomes near-identityin terms of (E,t), i.e., it becomes ε-close to the two-dimensional billiard map B_scorresponding to static boundary (<ref>) at fixed t, with augmented phase space to account for (E,t). Hence, the map B may be expanded in series of ε=1/w^* and δ. We will give full details of this construction in section 3. If the particle moves along the major axis, it will never leave the major axis. This motion corresponds to an invariant manifold Λ in the phase space of the billiard map B, a two-dimensional cylinder parametrised by(ℰ,t). In the static billiard, the motion along the major axis is a saddle periodic orbit for each frozen value of ℰ and t. Therefore, because the map B is close so the static billiard map in the rescaled variables, it follows that the cylinder Λ is normally hyperbolic. In particular, it hasthree-dimensional stable and unstable manifolds W^s,u(Λ) foliated by the strong stable and unstable foliations W^ss,uu(Λ). These geometric objects are inherited by B from the stable and unstable separatrices of the static billiard's motion along the major axis. The restriction of B to Λ is close to identity (when written in the coordinates (E,t) where E is the rescaled energy). It is well-known <cit.> that a near-identity C^l-smooth (or analytic) symplectic map x_1 = x_0 + ν f(x_0) with small ν is approximated by a time-ν shift along the orbits of an autonomous Hamiltonian system up to accuracy O(ν^l+1) (or exponential accuracy in ν for analytic case). Therefore, the mapB | _Λ (which we call the inner map) is approximated to a high level of accuracy bythe time-shift along level curves of a certain Hamiltonian H_in. As we mentioned, the billiard in ellipse is integrable, so the stable and unstable manifolds of the periodic orbit that corresponds to the motion along the major axis coincide. It is well-known that the resulting separatrix surface is a graph of a function θ of φ, where φ∈(0,π). Therefore,for any small β>0, the (perturbed) stable and unstable manifolds W^s,u(Λ) at sufficiently small ε and δ can be expressed asgraphs θ=θ^s,u_ε, δ(φ, E, t) (see section 4.3)over the interval φ∈ (β, π - β). They are O(ε, δ)-close to the unperturbed manifolds θ=θ^s,u_0,0, therefore they can be expanded in seriesof ε, δ. The zeroes of the difference (see section 4.3)θ^s_ε, δ - θ^u_ε, δcorrespond to the primaryhomoclinic intersections (φ_0, θ_0, ℰ_0, t_0) ∈Γ where Γ is the homoclinic set. If the corresponding unstable leaf of the foliation W^uu(Λ) intersects transversely the stable manifold W^s(Λ) at the homoclinic point (φ_0, θ_0, ℰ_0, t_0), the intersection is called strongly transversal (see section 4). When strong transversality condition is satisfied, projecting from the homoclinic point to Λ alongthe corresponding unstable fiber of W^u(Λ) and stable fiber of W^s(Λ) produces a pair of points in Λ which are related by what is called the scattering map S_Γ. Its domain of definitionΛ̅⊂Λ is the projection of the set of strong-transverse primary homoclinic pointsby the strong unstable fibers; the image S_Γ (Λ̅)⊂Λ is the projection ofthe set of primary homoclinic points by the strong stable fibers.The scattering map is exact symplectic <cit.>. For any bounded interval of the rescaled energy E, this map is close to identity, so it is well approximated by the time-ε shift along the level curves of a Hamiltonian H_out. We build the trajectory whose energy grows to infinity by following levelcurves of H_in and H_out, and switching to the level curve which leads to the larger energy gain in the immediate future. This construction is similar to <cit.>, however the application of inner and scattering maps is novel. Formally, we define H_in and H_out everywhere on Λ butthe switch of the orbit of B to the level curve of H_out is only allowed at the domain ofdefinition of the scattering map. We find that this domain has a non-trivial structure in our problem.Namely, we find that the strong transversality is only satisfied in the limit of large energy if δ> 0, andthat at δ=0 the projection of the primary homoclinic set to the cylinder Λ by the strong unstablefibers shrinks to a set of parallel lines t=const as ε=0. To this aim, we put δ=0and investigate the splitting of invariant manifoldsW^s,u(Λ). The first term of the power expansionin ε for the distance between (θ^s_ε, 0 - θ^u_ε,0) between perturbed W^s,u(Λ) is given by the so-called Melnikov function M_1(φ, θ, E, t). Non-degenerate zeros of M_1 correspond to transverse primary intersections ofW^s and W^u, if ε is small enough. We showConsider the time-dependent elliptic billiard map B without the quartic perturbation (i.e. δ=0).The Melnikov function associated to the splitting of invariant manifoldsW^s,u(Λ) has zeroes only for the times t^* such that d/dt(a(t^*)/b(t^*)) = 0, for all values ofenergy and reflection angle. If t^* is a nondegenerate critical point of a(t)/b(t), there exists a corresponding transverse intersection of W^s,u(Λ) along a 2-dimensional homoclinic surface where(θ,t)=(θ^s,u_0,0, t^*)+O(ϵ) are smooth functions of (ℰ,φ). In the cylinder Λ, the image of the two-dimensional homoclinic intersection found in this theorem by theprojection along the strong-unstable fibers is confined in a narrow strip around the critical lines t=t^*. This means that the scattering map is not properly defined if δ=0 (to estimate the domain of definition of the scatteringmap we would need further expansion of the separatrix splitting function in powers of ε, but wesuspect that it is small beyond all orders). We conjecture that the same structure is characteristic of a more general case of an integrable system with slowly varied parameters.In order to have a scattering map defined, we add the δ-dependent term in (<ref>). The nonintegrability of static elliptic billiards subject to polynomial perturbations was studiedin <cit.>.While we use the Melnikov function calculations from these works, we also develop a novel Melnikov function technique for the computation of the scattering map for systems with normally-hyperbolic invariant manifolds(e.g. time-dependent billiards). In particular, we show that the domain Λ̅ of definition of the scattering map S_Γin our situation has an unusual shape at small δ - it contains essential curves only at very large energies.Namely, to the first order in 1/√(ℰ) and δ the domain Λ̅is given by √(ℰ) > |adb/dt - b da/dt|/| δ| ϕ(t),where ϕ(t), defined by (<ref>) is a strictly positive, continuous, periodic function of t.More precisely, we have the following For any constant k>0 there exists C>0 such thatall points (ℰ,t) in the cylinder Λ, which satisfy√(ℰ) > |adb/dt - b da/dt|/| δ|ϕ(t) + k, ℰ≥C(k)/δ,belong to the domain Λ̅ of the definition of the scattering map S_Γ. Vice versa, the points which satisfy√(ℰ) < |adb/dt - b da/dt|/| δ|ϕ(t) - k,do not belong to Λ̅∩{ℰ≥C(k)/δ}.It is seen from these formulas that Λ̅ contains a circle ℰ=const only ifℰ> C_1δ^-2 where C_1 is some constant. In this region of energies techniques of <cit.> can be applied in order to prove the existence of orbits for which the energy tends toinfinity starting from ℰ∼δ^-2. Our Theorem 1.1 gives a stronger result by allowing to start at much lower energiesℰ∼δ^-1. Our paper is organised as follows. In section 2 we review the known facts about the static elliptic billiard. Section 3 introduces the time-dependent, perturbed billiard map, where we show how the rescaling of billiard speed gives rise to a slow-fast billiard map. In Section 4 we study the inner and scattering maps. We compute the splitting of stable and unstable invariant manifolds of Λ and use this result to derive a first order formula for the scattering map, and therefore give proofs of Theorems 1.2 and 1.3. We also derive the Hamiltonians H_in and H_out that give the first order approximations to inner and scattering maps. We provide the estimates on energy growth via asymptotic study of inner and outer Hamiltonian vector fields and provide a proof of Theorem 1.1 in Section 5. The Appendices A, B, C, D and E contain the derivation of the Melnikov function giving the first order distance between perturbed invariant manifolds W^s,u(Λ) and its explicit computation with elliptic functions; they also provide computations for the scattering map.§ STATIC ELLIPTIC BILLIARDThe following facts are well-known, see for example <cit.>. Our exposition follows <cit.>. Let us consider a billiard inside a static ellipse.In Cartesian coordinates, we may define the analytic boundary Q of the ellipse by Q = {(x,y) ∈ℝ^2: x^2/a^2 + y^2/b^2 = 1},where 0<b<a. Here a is the half-length of the major axis and b is half-length of the minor axis. The foci are at (± c,0) where c = √(a^2-b^2). Let us parameterise the ellipse as γ(φ): [0,2π) → Q, whereγ (φ) = {(acos (φ), bsin (φ)): φ∈ [0,2π)}.Let us introduce the angle of reflection θ∈ (0,π) of the particle velocity vector made with the positive tangent to γ(φ) at the collision point.Wedefine the static billiard map B_s:(φ_n, θ_n) ↦ (φ_n+1, θ_n+1) (with subscript s for `static'). Observe that Q is symmetric with regard to the origin. As in the work by Tabanov <cit.>, we may exploit this symmetry for B_s, by identifying the points on the ellipse that are π across, hence defining φπ. The following formulas for B_s are known <cit.>:φ_n+1 = -φ_n + 2 arctan(b(a tan(φ_n)+ b tan (θ_n))/a(b -a tan (φ_n) tan(θ_n))) π, θ_n+1 = -θ_n + arctan(b/atan(φ_n)) - arctan(b/atan(φ_n+1)) π.The map B_s is analytic and preserves the symplectic form |γ'(φ)| dφ∧ dθ, that becomes standard symplectic form ds ∧ d(cos (θ)) in coordinates (s, cos (θ)), where s is the arc length associated to Q. The map B_s has a hyperbolic saddle fixed point z=(0,π/2) with eigenvalues {λ, 1/λ}, whereλ = a+c/a-c > 1.The other fixed point (π/2, π/2) is elliptic.The elliptic billiard is integrable: the first integral I of B_s may be physically interpreted as the conservation of the inner product of angular momenta about the foci. The integral I may be written asI(φ, θ) = b^2cos^2(θ) - c^2sin^2 (θ) sin^2 (φ).Tabanov <cit.> gives the integral as Ĩ = cosh^2μcos^2 (θ) + cos^2 (φ) sin^2 (θ) in the elliptical coordinates x = h coshμcos (φ),y = h sinhμsin (φ), where h^2 = a^2-b^2. Upon changing from elliptical coordinates to the parameterisation γ (φ) above and using a^2-b^2=c^2, we haveĨ=a^2/c^2cos^2 (θ) + cos^2(φ) sin^2 (θ),which gives usc^2Ĩ(φ, θ) = a^2cos^2 (θ) + c^2cos^2 (φ) sin^2 (θ) = b^2cos^2 (θ) - c^2sin^2 (φ) sin^2 (θ) +c^2.Rearranging, we haveI(φ,θ) = c^2Ĩ - c^2 = b^2cos^2 (θ) - c^2sin^2 (φ) sin^2 (θ),which gives us (<ref>). The level set I=-c^2 corresponds to the elliptic fixed point; for -c^2<I<0 the billiard trajectories cross the major axis between the foci and have confocal hyperbolas as caustics, and for 0<I<b^2 trajectories cross the major axis outside the foci and have confocal ellipses as caustics. Zero level set, I(φ,θ)=0, corresponds to the union of homoclinic orbits that comprise two coincident branches W_1,2=W^s_1,2(z)=W^u_1,2(z), thestable and unstablemanifolds of z. From I(φ, θ)=0, the union W(z)= W_1(z) ⋃ W_2(z) is given by the expression <cit.>:sin^2(φ) = b^2/c^2tan^2(θ). Physically, W_1(z) corresponds to the billiard trajectory segments repeatedly passing through the focus at (c,0) while W_2(z) correspond to trajectories passing the focus at (-c,0). These trajectories asymptotically tend to the major axis of the ellipse, which corresponds to the saddle fixed point z of B_s(recall that we take φ by modulo π).One can obtain explicit expressions for B_s^n|_W_1,2(z) for n∈ℤ. We haveB_s^n(φ_0,θ_0)|_W_1(z) = (2 arctan(λ^ntan( φ_0/2) ),arctan(-b/csin (φ_n)) ), B_s^n(φ_0,θ_0)|_W_2(z) = (2 arctan(λ^-ntan( φ_0/2)),arctan(b/csin (φ_n)) ). Let us introduce the variable ξ∈ (0, ∞) such that ξ_n=tan(φ_n/)2, n ∈ℤ, so that (<ref>) givesξ_n = λ^nξ_0, tan (θ_n) = -b(1+ξ^2_n)/2c ξ_n,while (<ref>) givesξ_n=λ^- nξ_0, tan (θ_n) = b(1+ξ^2_n)/2cξ_n.Upon making the change of coordinatesν = γ'(φ) cos (θ), as in <cit.>, the expression (<ref>) becomes ν = ± c sin (φ) and the phase portrait of B_sresembles the one of the pendulum Hamiltonian H = p^2/2 + cos (q) - 1. In spite of the integrability, the existence of the hyperbolic fixed point with a homoclinic orbit implies that global action-angle variables cannot be introduced in an elliptic billiard: it is an example of an apriori unstable system.§ TIME-DEPENDENT PERTURBED ELLIPTIC BILLIARD§.§ Billiard map setup Let us consider a billiard inside a time-dependent convex curveQ(q,t, δ) that is O(δ) quartic polynomial perturbation of the ellipse for each fixed time t:Q(q,t; δ) := { q = (x,y) ∈ℝ^2,t ∈𝕊^1: x^2/a^2(t) + y^2/b^2(t) = 1 + 2 δ y^4/b^4(t)}, where a and b are periodic C^r+1 (r ≥ 4)functionsof time t, such that 0<b(t)<a(t) for all t, and 0 <| δ | ≪ 1. Let us parameterise Q(q,t; δ) asQ(q,t; δ) = {(a(t)cos (φ), b(t)sin (φ) (1+ δsin^2 (φ) ) ) + O(δ^2),φ∈ [0, 2 π), t ∈𝕊^1}.The O(δ^2) terms do not play any role in our work. Polynomial perturbations of billiards in ellipses were considered previously in a number of works <cit.>, however these works considered static perturbations only, not time-dependent ones. Assuming the billiard reflection at the moment of collision with the moving boundary is elastic, we define the time-dependent billiard map B: (φ_n, θ_n, ℰ_n, t_n) ↦ (φ_n+1, θ_n+1, ℰ_n+1, t_n+1). Here φ_n is the collision point on the boundary at the n-th collision, θ_n is the reflection angle of the post-collision particle velocity vector made with the positively oriented tangent to the boundary at the collision point, ℰ_n is the particle post-collision energy, and t_n is the time of the n-th collision. It is known that B is symplectic hence volume-preserving <cit.>. Since the boundary curve (<ref>) is analytic with respect to φ and C^r+1 in t, it is known that B is a C^r diffeomorphism <cit.>.Denote the speed of the particle as w, its corresponding velocity as 𝐰, its energy as ℰ= w^2/2; the speed of the boundary in the direction of outward normal is given by u(q,t)= -1 /∇_q Q(q,t)∂ Q(q,t; δ)/∂ t, and the unit outward normal is 𝐧 = ∇_q Q(q,t; δ)/∇_q Q(q,t; δ). We assume that positive u corresponds to outward motion of the boundary. The following formula <cit.> gives the change in velocity at the boundary collision:𝐰_n+1 = 𝐰_n-2⟨𝐰_n,𝐧_n+1⟩ 𝐧_n+1 +2u(φ_n+1, t_n+1) 𝐧_n+1. By analogy with <cit.>, let us introduce the auxilliary variable θ^*, which denotes the angle of incidence at the (n+1)-th impact with the tangent to the boundary, and let α denote the angle between the tangent to Q(q,t; δ) and the x-axis, given by tan (α) = y'(φ)/x'(φ), with ' = d/dφ (at each fixed t) and x(φ), y(φ) defined by (<ref>). Since (<ref>) is symmetric with respect to the origin, let us define φπ as in section 2, thus identifying points on the boundary that are π across.Define u( φ_n+1, t_n+1) to be the normal speed of the boundary at (n+1)-th impact. In this way,we obtain an implicit form for the billiard mapB: (φ_n, θ_n, ℰ_n, t_ n) ↦ (φ_n+1, θ_n+1, ℰ_n+1, t_n+1), given by the following formulas (more details can found in <cit.>):a(t_n+1) cos (φ_n+1) = a(t_n) cos (φ_n) + √(2ℰ_n)(t_n+1 - t_n) cos( α_n+ θ_n), b(t_n+1)sin (φ_n+1) (1+ δsin^2 (φ_n+1) ) = b(t_n)sin (φ_n) (1+ δsin^2 (φ_n) ) + √(2ℰ_n)(t_n+1-t_n) sin (α_n + θ_n), θ_n + α_n + θ^*_n+1 - α_n+1= 0, √(2ℰ_n+1)cos (θ_n+1) = √(2ℰ_n)cos(θ^*_n+1), √(2ℰ_n+1)sin(θ_n+1) = √(2ℰ_n)sin(θ^*_n+1)-2u(φ_n+1, t_n+1). The first pair of equations of (<ref>) implicitly defines t_n+1 and φ_n+1, while the last three give ℰ_n+1 and θ_n+1 after expressing θ^*_n+1 in terms of φ_n+1, θ_n and φ_n. The last pair of equations in (<ref>) corresponds to (<ref>) written in components normal and tangential to the boundary, and they give the expression for the change of energyℰ_n+1 = ℰ_n - 2 √(2E_n)u(φ_n+1, t_n+1) sin (θ^*_n+1) + 2u^2(φ_n+1, t_n+1). Let us denote by D the Euclidean distance between φ_n and φ_n+1, then we havet_n+1 = t_n+D/√(2ℰ_n), where the expression D = √([a(t_n+1) cos (φ_n+1) - a(t_n)cos (φ_n)]^2 + [b(t_n+1)sin (φ_n+1) (1+ δsin^2 (φ_n+1) )) - b(t_n)sin (φ_n) (1+ δsin^2 (φ_n) )]^2)is obtained from the first two equations of (<ref>). We assume that the initial speed of the particle is much larger than the speed of the boundary, so that the shape of the billiard table does not change significantly from one impact to the next. This implies that the time interval between two consecutive collisions is small and the change in the speed of the particle due to a single collision is small compared to the initial speedof the particle. Motivated by this, let us write the billiard map in a “slow-fast" form. Let us take an initial large value of speed w^* and introduce a small parameter ε = 1/w^* such that 0 < ε≪ 1.Let us introduce the scaled speed v that is related to the original physical variable w through v = w/w*. In terms of ε this givesw =v/ε. This transformation is equivalent to ℰ= E/ε^2, where E is the rescaled energy. The billiard map in the rescaled energy and time variables becomes close to identity, since(<ref>) transforms toE_n+1 = E_n - 2 ε√(2E_n)u(φ_n+1, t_n+1) sin (θ^*_n+1) + 2 ε^2 u^2(φ_n+1, t_n+1),or, in terms of v,v_n+1 = v_n - 2ε u(φ_n+1, t_n+1) sin (θ^*_n+1) + O(ε^2)Using (<ref>) transforms the equation (<ref>) tot_n+1 = t_n+ ε D/√(2E_n).Note that in the limit ε→ 0 the variables (t,E) become constants, i.e., the billiard map coincides with the frozen billiard map in the domain bounded by (<ref>).Now, by virtue of C^r-smoothness of the boundary and smallness of ε and δwe write the time-dependentbilliard map B = B_ε, δ (φ_n, θ_n, E_n, t_n) ↦ (φ_n+1,ε, δ, θ_n+1,ε, δ, E_n+1,ε, δ, t_n+1,ε, δ) in the form B_ε, δ= B_0 + ε B_1 + δ B_2 + O(ε^2 + δ^2). We defineφ_n+1,ε, δ=φ_n+1+ ε f_1(φ_n,θ_n, E_n,t_n) + δ g_1(φ_n,θ_n, t_n) + O(ε^2 + δ^2),θ_n+1,ε, δ=θ_n+1+ ε f_2(φ_n,θ_n, E_n, t_n) + δ g_2(φ_n, θ_n, t_n) + O(ε^2 +δ^2), E_n+1,ε=E_n + ε f_3(φ_n,θ_n,E_n,t_n)+ ε O(ε +|δ|), t_n+1,ε=t_n + ε f_4(φ_n,θ_n,E_n,t_n)+ ε O(ε +|δ|).Here B_1 = (f_1,f_2,f_3,f_4)^⊤ and B_2 = (g_1,g_2,0,0)^⊤ (with ^⊤ denoting the transpose). We use the notation (φ_n+1, θ_n+1) = B_s(φ_n, θ_n), i.e. B_0 is the same as the static two-dimensional billiard map B_s in the ellipse (<ref>), with the increase of the phase space dimension to account for the two "frozen" variables E and t; so, B_0 ( φ, θ, E, t) = (B_s(φ, θ), E,t). Let us call B_0 the unperturbed time-dependent elliptic billiard map. The map B_0 is integrable with the first integral (<ref>) where b = b(t) and c=c(t) are fixed, and two more trivial first integrals I_2=E, I_3=t. The phase space of B is [0, π) × (0, π) ×ℝ^+×𝕊^1, with φ∈ [0,π), θ∈ (0, π), E ∈ℝ^+, and t ∈𝕊^1.Since we consider high billiard energies, we assume that the billiard reflection angle θ∈ (0, π), i.e. the situations described in <cit.> where the billiard trajectory continues in a tangential or `outward' direction to the boundary at the moment of collision do not occur.By substituting the expression for t_n+1, ε, δ and φ_n+1, ε, δ from (<ref>) into (<ref>) and expanding in Taylor series, we find the zero-th order in ε and δ free-flight distanceD_0 = √(a^2[cos(φ_n) - cos(φ_n+1)]^2 + b^2[sin(φ_n) - sin(φ_n+1)]^2). Upon substituting expansion (<ref>) into (<ref>), and examining the coefficients of the order ε terms, we find that B_1 = (f_1,f_2,f_3,f_4)^⊤ is given by the following expressions:f_1= D_0/√(2E_n)(ȧcos(φ_n+1) tan(θ_n+α_n) - ḃsin(φ_n+1)/asin(φ_n+1) tan(θ_n+α_n) + bcos(φ_n+1)), f_2 = -2u cos (θ_n+1)/√(2E_n) + a^2sin^2 (φ_n+1)/a^2sin^2 (φ_n+1) + b^2cos^2 (φ_n+1) (D_0(ȧa^-1b - ḃ)/a √(2E_n)tan (φ_n+1) + b f_1/a sin^2 (φ_n+1)), f_3= -2√(2E_n) u sin (θ_n+1), f_4 = D_0/√(2E_n). The dot abovea and b denotes the derivative with respect to time evaluated at time t_n. We also denote a = a(t_n), b=b(t_n), and u =ȧbcos^2(φ_n+1) + aḃsin^2(φ_n+1)/√( a^2sin^2(φ_n+1) + b^2cos^2(φ_n+1)) is the normal speed of the boundary. Similarly, comparing the coefficients of the first order in δ, we find that the expressions for g_i for i=1,..,4 are independent of Eand ȧ, ḃ.Observe that f_1, f_2, f_3, f_4 are written in a certain “cross-form" as functions of the image of (φ, θ, E, t) under B_0 as well as the initial point (φ, θ, E, t) itself; however since B is a C^r diffeomorphism, one may express the functions f_i in form (<ref>).Observe that the Taylor series expansion of the map B_ε, δ consists of two perturbations that may be considered independently in the first order of the perturbation parameters: the O(ε) perturbation terms that arise due to rescaling of energy (this is B_1), and the O(δ) perturbation terms that correspond to the polynomial perturbation of the boundary (this is B_2, which is independent of t). For ε = 0, the variables E and t are constant and thus B_0,δ becomes abilliard map corresponding to an ellipse with a quartic polynomial perturbation, with the semi-axes lengths fixed at a(t_n), b(t_n). This is a twist map and thus has a certain generating function L(φ_n, φ_n+1) <cit.> . The nonintegrability of such convex billiard curves and the relation between generating function, Melnikov function and Melnikov potential was investigated in detail in <cit.>. Thus, for the computation of the Melnikov function we do not require the knowledge of the explicit form of B_2, as we will be using the generating function formulation and Melnikov potential <cit.> for the polynomial part of the perturbation. §.§ Phase space geometry of the time-dependent billiard If a point in our billiard moves along the major semi-axis, it will continue to move along this semi-axis forever. In other words, this motion is confined to an invariant manifold Λ in the phase space of the billiard map B. It is given by (φπ,θ)=(0,π/2) and is parameterized by the energy ℰ∈ℝ^+ and the timet∈𝕊^1, so Λ is a cylinder. If we rescale the energy and take the limit ε=0, the cylinder Λ corresponds to the hyperbolic saddle fixed point z=(0,π/2) of the static billiard map B_s, so Λ is a normally-hyperbolic invariant manifold of B_0,δ.It has two branches of stable and unstable three-dimensional invariant manifolds W^s,u_i(Λ) with i = 1,2. At δ=0, they coincide and form two symmetric three-dimensional homoclinic manifolds W_i that are inherited from one-dimensional separatrices of z (see (<ref>)). Denoting W = ∪_i=1^2 W_i, we haveW = {(φ, θ, E,t):φ∈ [0,π), θ∈ (0,π), E ∈ℝ^+, t ∈𝕊^1; sin^2(φ) = b^2(t)/c^2(t)tan^2(θ) }. By the theory of Fenichel <cit.>, the normal hyperbolicity of Λ persists as the map B_0,δ is perturbed. In particular, the stable and unstable manifolds of any subset of Λ that corresponds to a bounded set of values of the rescaled energy E persist at all small ε, and depend continuously on ε (and δ). If we return to the non-rescaled energy variable ℰ, this gives us, for all small δ, the normal hyperbolicity, for the time-dependent billiard, of the piece of Λ that corresponds to sufficiently large values of ℰ; with the stable and unstable manifolds W^s,u(Λ) close to those for the frozen billiard map and depending continuously on δ.At ε=0, the stable and unstable manifolds W^s,u_i are foliated by strong stable and strong unstable one-dimensional fibers W^ss, uu_(E,t);i: (E,t)=const. They form smooth invariant foliations transverse to Λ; such foliations are unique and persist at small smooth perturbations of the system <cit.>. Thus, these invariant foliations persist for the time-dependent billiard as well, and depend continuously on δ and, when the rescaling of the energy variable is introduced, on ε.The closeness of B to identity (in the rescaled energy E and time t) implies a large spectral gap <cit.> for the normally-hyperbolic cylinder Λ. Thus, the stable and unstable manifolds W^s,u(Λ) are C^r and their leaves W^ss,uu are C^r-1 in (φ, θ,E,t) and also in the parameters (ε, δ), for r ≥ 4 <cit.>.Note also, that for the unperturbed system (i.e., at ε=0 and δ=0) the fibers W^ss_(E,t);i and W^uu_(E,t);i coincide for each (E,t). § INNER AND OUTER DYNAMICSIn this section we will define and study the inner and scattering (outer)maps associated to Λ. Iterations of these maps will be the main tool for constructing a billiard orbit with growing energy in section 5.§.§ Inner map The inner map is the restriction of B to Λ. Physically, the inner map describes the billiard motion along the major axis of the billiard domain. Let us denote the inner map by Φ and hence Φ = B|_Λ writes as B(0, π/2, E_n, t_n) = (0, π/2, E_n+1, t_n+1) (recall that E is the rescaled energy). Therefore, using (<ref>) and (<ref>), the inner map can be given in an implicit formE_n+1 =E_n - 2 ε√(2E_n)ȧ(t_n+1) + 2 ε^2ȧ^2(t_n+1), t_n+1 = t_n + ε( a(t_n) + a(t_n+1))/√(2E_n),The map Φ which defines a C^r diffeomorphism (E_n, t_n) ↦ (E_n+1, t_n+1) for small ε, preserves the symplectic form (1+ εȧ(t)/√(2E)) dE ∧ dt that becomes standard form dF ∧ dt upon defining F = √(E)(√(E) + ε2ȧ(t)/√(2)). The symplecticity of the inner map and its closeness to identity imply that it may be approximated by the time-ε shift along a level curve of an autonomous Hamiltonian H_in(t, E ; ε)= H_in(t,E) + O(ε) defined on Λ. Let us find H_in.A series expansion in εyields the following approximation of (<ref>):E_n+1=E_n - 2εȧ(t_n) √(2E_n) +O(ε^2) ,t_n+1 = t_n + √(2)ε a(t_n)/√(E_n) + O(ε^2).Since (<ref>) givesE_n+1 - E_n/t_n+1-t_n = -2ȧ(t_n)E_n/a(t_n) + O(ε),we see that Φ is approximated to O(ε^2) by a time-ϵ shift along a trajectory of the solution of the differential equation dE/dt = -2ȧE/a. Its integral √(E)a(t)= gives the zero-th order approximation of H_in. Thus,map (<ref>) up to O(ε^2) is given by the time-ϵ shift along a level curve of the HamiltonianH_in(t,E)= 2√(2E)a(t).The corresponding Hamiltonian vector field is:dt/ds= ∂ H_in/∂ E = √(2)a(t)/√(E), dE/ds = - ∂ H_in/∂ t = -2√(2E)ȧ(t),where s is an auxiliary time variable. Let (E̅, t̅) be the image of (E, t ) under Φ^p where p = [1/ε]. Let us show the followingThe inner map Φ^p: (E, t ) ↦ (E̅, t̅) satisfies the twist condition, i.e., ∂t̅/∂ E≠ 0.Let us denote by ϕ the time-1 map of H_in(t,E). Observe that ϕ is O(ε) close to Φ^p in (E,t) coordinates. Let us verify that ϕ has the twist property.Since H_in is integrable, ϕ is also integrable, i.e., it preserves H_in.From the first equation of (<ref>), we have ds/dt = 1/a(t)√(E/2). Expressing E in terms of H_in from (<ref>) on this curve yields ds/dt = H_in/4a^2(t). Then we have 1 = H_in∫_t^t̅dt/4a^2(t) = √(E/2) a(t) ∫_t^t̅dt/a^2(t)for the map ϕ. Differentiating both sides of (<ref>) with respect to E yields ∂t̅/∂ E < 0. Thus, by definition ϕ is a twist map. Since Φ^p is an O(ε) perturbation of ϕ, it also has the twist property.Since all orbits of H_in are invariant closed curves forming a continuous foliation of Λ, and ϕ is integrable twist map, it follows from the KAM theory that Φ^p: (E, t ) ↦ (E̅, t̅) has closely spaced invariant curves on Λ. Hence, the energy of the billiard motion with the initial conditions on the major axis is always bounded. It follows from standard results <cit.> that the inner map (<ref>) coincides up to O(ε^r+1) with ε-time shift along a level curve of some Hamiltonian H_in(t,E;ε), which is given by (<ref>) at ε = 0, a fact that we will make use of in section 5.§.§ Scattering map: theory The so-called outer dynamics on Λ is defined by the scattering map (also called the outer map), studied in detail in <cit.>. It is obtained by an asymptotic process: to construct it, one starts infinitesimally close to the normally hyperbolic invariant manifold, moves along its unstable manifold up to a homoclinic intersection and, then, back to Λ along its stable manifold.Let us define the scattering map in a general setup.Let T: M ↦ M be a C^r (here we take r ≥ 1)diffeomorphism on a compact manifold M, and let Λ⊂ M be a compact normally-hyperbolic invariant manifold of T. By the normal hyperbolicity, there exits stable W^s(Λ) and unstable W^u(Λ) manifolds of Λ with strong stable and strong unstable foliations W^ss,uu(x) for each x ∈Λ.Let us assume thatstable and unstable manifolds of Λ intersect transversally along a homoclinic manifoldΓ⊂ W^s(Λ) ∩ W^u(Λ): for all z ∈Γ we haveT_zW^s_Λ + T_zW^u_Λ = T_zM,T_zW^s_Λ∩ T_zW^u_Λ = T_zΓ. For a given z ∈Γ, there are unique points x_±∈Λ satisfying z ∈ W^ss_x_+ and z ∈ W^uu_x_-.Following <cit.> we call the homoclinic intersection at the point z strongly transverse if the leaf W^ss_x_+ is transverse to W^s(Λ) and the leaf W^uu_x_- is transverse to W^u(Λ) at z:T_zW^ss_x_+⊕ T_zΓ = T_zW^s_Λ,T_zW^uu_x_-⊕ T_zΓ = T_zW^u_Λ. Conditions (<ref>) are used to locally define the scattering map S_Γ. We say that Γ is a homoclinic channel if it satisfies (<ref>) and (<ref>). Let π^s: Γ↦Λ and π^u: Γ↦Λ be the projections by the strong stable leaves of stable manifolds, and strong unstable leaves of unstable manifolds, respectively. If we have a sufficiently large spectral gap (the expansion in the strong unstable directions is much larger than the possible expansion we have in the directions tangent to Λ), then the strong-stable and strong-unstable foliations are C^r-1-smooth <cit.>. This condition is obviously satisfied in our case, since the restriction of our billiard map to Λ is close to identity, i.e. the expansion in the directions tangent to Λ can be made as weak as we want. Thus, when conditions (<ref>) are fulfilled, the projections π^s,u are local C^r-1 diffeomorphisms. Then, the scattering map S_Γ: Λ↦Λ defined asS_Γ = π^s∘ (π^u)^-1is a C^r-1 diffeomorphism (which is symplectic if T is symplectic): <cit.>. The invariance property of the strong stable and strong unstable foliations (i.e. the property T(W^ss,uu_x) = W^ss,uu_T(x) for all x ∈Λ) implies thatπ^s,u = T^-1∘π^s,u∘ T, π^s,u = T^-n∘π^s,u∘ T^n,n ≥ 1.Hence, the dependence of the scattering map on the choice of a particular homoclinic channel obeys the ruleS_Γ = T^-1∘ S_T( Γ)∘ T.§.§ Scattering map for B and splitting of invariant manifolds Let us take a sufficiently large compact subset Λ̃⊂Λ of the normally hyperbolic invariant cylinder Λ:Λ̃ = {(φ, θ, E, t): (φ, θ) = (0, π/2), E ∈ [E_1, E_2], t ∈𝕊^1}.In this section we will find a subset Λ̅⊂Λ̃ where thescattering map S_Γ is well-defined, i.e. we will find a set Γ consisting of homoclinic points for which the transversality conditions (<ref>), (<ref>) are satisfied - then the projection π^u(Γ) is the set Λ̅. We will derive perturbatively an explicit formula for S_Γ up to ε O(ε + |δ|) and then provide an approximation to it by a time-ε shift along a level curve of a certain Hamiltonian H_out. In the process, we will also prove Theorems 1.2 and 1.3. To study the properties of the scattering map, we need to determine whether the perturbed stable and unstable manifolds W^s,u_i(Λ̃) of Λ̃ intersect transversally, and furthermore, if these intersections are strongly transverse.We will use Melnikov-type method to determine the existence of transverse intersections of W^s,u_i(Λ̃). Since unperturbed W^s,u_1(Λ̃) and W^s,u_2(Λ̃) are symmetric to each other, we will study W^s,u_2(Λ̃) only. From (<ref>), the unperturbed invariant manifolds W^s,u_2(Λ̃) coincide and form a 3-dimensional (unperturbed) homoclinic manifold W_2 given byW_2(Λ̃) = {(φ, θ, E,t):φ∈ [0,π)π/2, θ∈ (0,π), E ∈ [E_1, E_2], t ∈𝕊^1; sin(φ) = b(t)/c(t)tan(θ) }.We note that W^s,u_2(Λ̃) are co-dimension 1 manifolds. Therefore, in order to determine whether they split or not for nonzero ε, δ≠ 0. we only need one measurement in the normal direction to the coincident tangent spaces of unperturbed W^s,u_2(Λ̃).Later in this section we will introduce the distance function d̅(x_0, ε, δ) that measures the splitting of W^s,u_2(Λ̃) for ε, δ≠ 0 in the normal direction to x_0= (φ_0, θ_0, E_0, t_0) ∈ W_2(Λ̃) and will show that if ∂d̅/∂φ_0≠ 0, then the strong transversality condition (<ref>) is satisfied. Let us consider strong stable and unstable foliations W^ss,uu_(E̅, t̅) ∈Λ̃;2 of thestable/unstable manifolds W^s,u_2(Λ̃) (with notation as in section 3.2). Since we study W^s,u_2(Λ̃), we will drop the subscript 2 and write W^ss,uu_(E̅, t̅) ∈Λ̃ from now on. Since B_0 is integrable and is identity in (E,t) variables, the unperturbed one-dimensional strong stable and unstable fibers W^ss,uu_(E̅, t̅) ∈Λ̃with the same basepoint (E̅, t̅) coincide, and may be trivially expressed as graphs over φ variable with E = E̅ = and t = t̅ =, with θ as a function of φ determined from (<ref>). Because the spectral gap is large in our situation, the strong stable and unstable fibers are C^r-1-smooth functions of parameters <cit.>. Therefore, for small ε and δ, the fibersW^ss,uu_(E̅, t̅) ∈Λ̃ and perturbed stable and unstable manifolds W^s,u(Λ̃) may be written asE = E^ss,uu(φ; E̅, t̅, ε, δ) = E̅ + εϕ_1^ss,uu(φ; E̅, t̅, 0 ,0) + ε O(ε + |δ|), t = t^ss,uu(φ; E̅, t̅, ε, δ)= t̅ + εϕ_2^ss,uu(φ; E̅, t̅, 0, 0) + ε O(ε + |δ|), θ = θ^s,u(φ, E, t, ε, δ),where ϕ_1,2^ss,uu are certain C^r-1 functions, and the graphs of the C^r-functions θ^s,u define sufficiently large pieces of the stable and unstable manifolds W^s,u_2(Λ̃) at small ε and δ.Observe that in the formulas (<ref>), (<ref>) the O(δ) terms do not appear, since the billiard map is identity in (E,t) at ε=0, so the strong-stable and strong-unstable fibers are given by E = and t = at ε=0 even if δ≠ 0. We note that if we know φ (and thus θ) as a function of (E,t), then the formulas (<ref>), (<ref>) will provide us with a formula for the scattering map S_Γ.In what follows, we start with the analysis of the splitting of W^s_2(Λ̃) and W^u_2(Λ̃) for the case δ = 0, ε≠ 0. Let us set δ = 0 in (<ref>). It can be shown <cit.> that the distance d(x_0, ε, 0) (measured in the normal direction at the point x_0 = (φ_0, θ_0, E_0, t_0) ∈ W_2(Λ̃)) between the perturbed invariant manifoldsW^s,u_2(Λ̃) is given by d(x_0, ε, 0) = ε M_1(x_0) + O(ε^2), where M_1 is the Melnikov function corresponding to the time-dependent ellipse only (without the quartic polynomial part). The following formula for M_1 is derived in Appendix A: M_1(x_0) := ∑_n = -∞^∞⟨∇ I(B_0(x_n), B_1(x_n) ⟩, where x_n=(φ_n,θ_n,E_n,t_n) ∈ W_2(Λ̃) is the orbit of the point x_0∈ W_2(Λ̃) under the map B_0 (i.e., x_n= B^n_0(x_0)),and I is the first integral given by (<ref>). Since the separatrices of the static billiard are one-dimensional, we have θ_n = θ^0(φ_n), a smooth function of φ given by (<ref>). Also observe that E_n=E_0, t_n=t_0. Hence we evaluate the series (<ref>) with slow variables held constant and equal to their initial value.Let us set d̅(x_0, ε, 0) = d(x_0, ε, 0)/ε for ε≠ 0, and d̅(x_0, 0, 0) = M_1(x_0). Then d̅(x_0, ε, 0) = M_1(x_0) + O(ε) and the zeroes of d̅(x, ε, 0) correspond to intersection of W^s_2(Λ̃) and W^u_2(Λ̃). The implicit function theorem implies that if M_1(x_0)=0 and DM_1(x_0) ≠ 0, thenW^s,u_2(Λ̃) intersect transversally at x_0 along a 2-dimensional homoclinic manifold.We are able to compute M_1analytically (see Appendix B). Note that sincewe express φ and θ through ξ on homoclinic manifolds using parametrisation (<ref>), effectively we have M_1(x) = M_1(φ, θ(φ),E,t) = M_1(ξ,E,t). We define h = logλ where λ from (<ref>) is the largest eigenvalue of the linearization of the static map B_s at the saddle point z. For the map B, the value of λ depends on t and is given by λ = λ(t) = a+c/a-c with a=a(t), b=b(t) and c = c(t). We also define the variable τ in terms of ξ from (<ref>): exp(τ):= ξ := tan (φ/2). The method developed in <cit.> enables us to compute the sum of the series (<ref>) in terms of elliptic functions - the computations are in Appendix B. Using the definitions of the elliptic functions , ,and complete elliptic integrals E, E', K, K' depending on parameter m ∈ [0,1] whose dependence on h is K'/K = π/h as in Appendix B, we find:M_1 =4b/v(-ȧb + ḃa)(2K/h)^2(E'/K' - 1 +^2(2Kτ/h)),where v = √(2E) is the (rescaled) speed of the particle. Since (2K/h)^2(E'/K' - 1 +^2(2Kτ/h)) = ∑_n= -∞^n= ∞^2(τ + nh)>0, as known from <cit.>, the zeroes of M_1 only exist for values of time t^* satisfying (-ȧb + ḃa)=0, i.e. whend/dt(a/b) = 0. For such t^*, the values of v and φ = φ(τ) for which the Melnikov function vanishes can be arbitrary. Hence the zeroes are of the form (φ, θ(φ), t^*, v). By implicit function theorem, if DM_1≠ 0, then the zeros of the Melnikov function correspond to zeros of the splitting function d for all small ε, hence to the homoclinic intersections, and these intersections are transverse. The condition DM_1≠ 0 implies d^2/dt^2(a(t^*)/b(t^*)) ≠ 0 which is satisfied if (and only if) t^* is a nondegenerate critical point of a(t^*)/b(t^*). This gives us the existence, for all small ε, of a transverse intersection Γ̃ of W^u(Λ̃) and W^s(Λ̃) along a smooth two-dimensional surface close to the surface {t=t^*, θ=θ^s,u_0,0(φ, E, t^*)}. Since ε is just an energy scaling parameter, we obtain the transverse homoclinic intersection Γ of W^u(Λ) and W^s(Λ) for all sufficiently large values of the non-rescaled energy ℰ.In the limit ε=0 (i.e., in the limit ℰ→+∞) the projection π^u of this homoclinic surface to the cylinder Λ by the strong-unstable fibers, i.e., the domain of definition of the scattering map S_Γ̃ shrinks to just the vertical line t=t^* (recall that the strong-unstable fibers are close to the lines (E,t)=const at small ϵ). In other words, we cannot define the scattering map at δ = 0 by first order expansion in ε. We, therefore, proceed to the case δ≠ 0.Let us introduce the Melnikov function M_2 corresponding to the quartic polynomial perturbation O(δ) only, with the time and speed variables frozen. The following formula for M_2 is again derived in Appendix A:M_2(x_0) := ∑_n = -∞^∞⟨∇ I(B_0(x_n), B_2(x_n) ⟩,where the series is evaluated as before over the orbit of x_0∈ W_2(Λ̃) under the map B_0 and I is the first integral given by (<ref>). Using the same notation as for M_1 above, we find from Appendix B that (see also <cit.>):M_2 = -4m ab^2/c^2(2K/h)^3(2Kτ/h) (2Kτ/h) (2Kτ/h).It is known <cit.> that for each fixed t, the function M_2 is h-periodic in τ and has two simple, in terms of τ, zeroes in the period [0,h).Before we proceed to prove Theorem 1.3, let us introduce some notation. Rewrite (<ref>) as M_1 = f(t)g(τ,t)/v, and(<ref>) as M_2 = j(τ,t), wheref(t) =-ȧb + ḃa, g(τ,t) = 4b(2K/h)^2(E'/K' - 1 +^2(2Kτ/h)) > 0, j(τ,t) = -4m ab^2/c^2(2K/h)^3(2Kτ/h) (2Kτ/h) (2Kτ/h). Let us also define the function ϕ(t) by:ϕ(t) = min_τ| g(τ,t)/j(τ, t)| /√(2).Note that it is seen from the properties of elliptic functions ,andthat for each t, the function g(τ,t)/j(τ, t) is non-zero, odd and periodic in τ, with one maximum and one minimum point in the period τ∈ [0, h(t)). Let (v,t) be such that ε |f(t)|/| δ| v < max_τ |j(τ, t)/g(τ, t)| -k̃ where k̃>0 is constant and v=√(2E) is the rescaled particle speed. Let Λ̅ be the subset of Λ̃ corresponding to these values of (E,t). Then the stable and unstable manifolds W^s,u_2(Λ̅) have a strong-transverse intersection along a 2-dimensional homoclinic manifoldΓ for all small ε, δ≠ 0 such that |δ|≫ε^2.As shown in Appendix A, the distance d(x_0, ε, δ) measured in the normal direction at the point x_0 = (φ_0, θ_0, E_0, t_0) ∈ W_2(Λ̃) between the perturbed manifolds W^s,u_2(Λ̃) is given by d(x_0, ε, δ) = ε M_1(x_0) + δ M_2(x_0) + O(ε^2 + δ^2). To deal with two small parameters ε and δ, let us considerd(x_0, ε, δ)/δ = ε/δ M_1(x_0)+ M_2(x_0) + O(ε^2/δ + δ) =ε f(t)g(τ,t)/δ v +j(τ,t) + O(ε^2/δ + δ). We have from the assumption of the proposition that ε M_1(x_0)/δ is bounded for all small nonzero δ and that terms of order O(ε^2/δ + δ) are uniformly small. Then, if we drop the O(ε^2/δ + δ) terms in (<ref>) and define d̅ = ε f(t)g(τ,t)/δ v +j(τ,t),it follows from the implicit function theorem that if ∂d̅/∂τ|_d̅=0≠ 0, then the zeroes of d̅ correspond to a transverse intersection of W^s,u_2(Λ̃) along a 2-dimensional homoclinic manifold Γ for all small ε^2/δ and δ. This manifold is a graph of a smooth function τ of (v,t), i.e., it is a graph of a smooth function φ of (E,t). Since the strong-stable and strong-unstable fibers are ε-close to (E,t)=const, we immediately have that the conditions (<ref>),(<ref>) ofthe strong transverse intersection are fulfilled at the points of Γ.Thus, to prove the proposition, we need to investigate for which values of v and t the function d̅(τ) hassimple zeroes. Rearranging (<ref>) yieldsε f(t)/δ v = -j(τ, t)/g(τ, t).Let us fix any t and v. Since g(τ,t) >0 and v>0, solutions of this equation belong to the interval of values of τ for which the sign of j(τ, t) is the same as the sign of f(t). Degenerate zeros correspond to critical points ofthe function_tj(τ,t)/g(τ,t). It can be checked that it has exactly one (positive) maximum and one (negative) minimum in the period τ∈ [0, h(t)); since this function is odd, the maximum and minimum are the same in the absolute value. It follows that for any constant k̃>0, given t and v such thatϵ |f(t)|/| δ | v < max_τ |j(τ, t)/g(τ, t)| -k̃,equation (<ref>) has exactly two solutions τ(t,v), both of them non-degenerate.As we explained above, these solutions correspond to strong-transverse homoclinic intersections, provided δ, ε and ε^2/δ are small enough. Now we can prove Theorem 1.3.We have shown in Proposition 4.3 that the values of (v,t) satisfying (<ref>) correspond to a strong-transverse homoclinic intersection, i.e., they lie in the domain of definition of the scattering map S_Γ. It is also seen from (<ref>) that if, for some k̃>0,ε |f(t)|/| δ | v > max_τ |j(τ, t)/g(τ, t)| + k̃,then the function d does not have zeros for small δ, ε^2/δ and ε, sothe region (<ref>) is not included in the domain of the scattering map. Now, by noting that ε/v=1/√(2ℰ) where ℰ is the non-rescaled energy, we immediately obtain the statement of the theorem from estimates (<ref>),(<ref>). Now we proceed to obtain a first order expression in perturbation parameters ε, δ for S_Γ in terms of E and t. Let (E̅_0, t̅_0) ∈Λ̅ and (Ẽ_0, t̃_0) ∈Λ̅ be two points in the domain of S_Γ given by (<ref>). We will obtain a perturbative expression up to ε O(ε + |δ|) for the scattering map S_Γ: Λ̅↦Λ̅:S_Γ: (E̅_0, t̅_0; ε, δ ) ↦ (Ẽ_0, t̃_0; ε, δ).We will call the first order approximation of S_Γ the truncated scattering map. Let us derive a formula for ϕ^ss,uu_1,2 given by (<ref>), (<ref>). Let us take a point (φ_0, θ_0, E_0, t_0) ∈Γ⋔ W^ss_(E̅_0, t̅_0). Its energy component E_0 is given by (<ref>):E_0 = E^ss(φ_0; E̅_0, t̅_0, ε, δ) = E̅_0 + εϕ_1^ss(φ_0; E̅_0, t̅_0,0, 0) + ε O(ε + |δ|). Let B (W^ss_(E̅_0, t̅_0)) denote the action of map B on the point (φ_0, θ_0, E_0, t_0) on the leaf W^ss_(E̅_0, t̅_0).Let (φ_n, ε, δ, θ_n, ε, δ, E_n, ε, δ, t_n, ε, δ) be the orbit of the point (φ_0, θ_0, E_0, t_0) under B, and let (E̅_n, ε, t̅_n, ε) be the orbit of (E̅_0, t̅_0) ∈Λ̅ under B (the notation is the same as in (<ref>)). Note that on Λ̅ we have (φ_n, ε, θ_n, ε) = (0, π/2)for all n, and there is no dependence on δ of B restricted to Λ.Consider the energy E component of B (W^ss_(E̅_0, t̅_0)).Using the notation (<ref>) givesE_1, ε = E_1, 0 + ε f_3(φ_0, θ_0(φ_0), E_0, t_0) + ε O(ε + |δ|) =E_0 + ε f_3(φ_0, θ_0(φ_0),E_0, t_0) + ε O(ε + |δ|) =E̅_0 +εϕ_1^ss(φ_0(φ_0), E̅_0, t̅_0, 0) + ε f_3(φ_0, θ_0(φ_0), E_0, t_0) + ε O(ε + |δ|),where f_3(φ_0, θ_0(φ_0), E_0, t_0) is evaluated on the unperturbed homoclinic trajectory on the unperturbed homoclinic manifold W_2(Λ̅) given by (<ref>). The invariance property of the stable and unstable foliations gives B (W^ss_(E̅_0, t̅_0)) = W^ss_B(E̅_0, t̅_0) and, by definition of the map B, we have W^ss_B(E̅_0, t̅_0)) = W^ss_(E̅_1, ε, t̅_1, ε). Therefore, we have E_1, ε = E(φ_1, ε; E̅_1, ε, t̅_1, ε, ε, δ) = E̅_1, ε + εϕ_1^ss(φ_1, E̅_1, t̅_1, 0,0) + ε O(ε + |δ|) = E̅_0 + ε f_3(·, E̅_0, t̅_0)|_Λ̅ +εϕ_1^ss(φ_1, E̅_0, t̅_0, 0, 0) + ε O(ε + |δ|),where we used E̅_1 = E̅_0, t̅_1 = t̅_0 and E̅_1, ε = E̅_0 + ε f_3(·, E̅_0, t̅_0)|_Λ̅ in the last line above, with f_3(·,E̅_0, t̅_0)| _Λ denoting the restriction of f_3 to Λ̅. The notation f_3(·,E̅_0, t̅_0)| _Λ̅ signifies that the variables (φ, θ) on Λ̅ are fixed at (0, π/2). Examining the coefficients of O(ε) terms in (<ref>) and (<ref>) gives ϕ_1^ss(φ_0; E̅_0, t̅_0,0, 0) = ϕ_1^ss(φ_1; E̅_0, t̅_0, 0, 0) + f_3(· , E̅_0, t̅_0)|_Λ - f_3(φ_0,θ_0(φ_0), E_0, t_0).Upon iterating (<ref>), we obtain an expression for ϕ^ss_1(φ_0; E̅_0, t̅_0,0, 0):ϕ^ss_1(φ_0; E̅_0, t̅_0,0, 0) = ϕ^ss_1(φ_∞; E̅_∞, t̅_∞,0, 0) + ∑_i=0^∞{ f_3(·, E̅_i, t̅_i)|_Λ- f_3(φ_i, θ_i(φ_i), E_i, t_i)|_W_2(Λ̅)},where φ_∞ = 0, E̅_∞ = E̅_0 and t̅_∞ = t̅_0, and f_3(φ_i, θ_i(φ_i), E_i, t_i) is evaluated over the unperturbed homoclinic trajectory on W_2(Λ̅) given by (<ref>). Therefore the slow variables (E,t) in the summation are held constant with E̅ = E_i = E_0, t̅ = t_i = t_0. Hence, the equation for the energy E component of the strong stable leaf W^ss_(E̅_0, t̅_0) through (E_0, t_0) isE_0 = E̅_0 + εϕ^ss_1(0;E̅_0, t̅_0,0, 0) + ε∑_i=0^∞{ f_3(· , E̅_i, t̅_i)|_Λ- f_3(φ_i, θ_i(φ_i), E_i, t_i)|_W_2(Λ̅)} + ε O(ε + |δ|).Similarly, we express the E component of the strong unstable leaf W^uu_(Ẽ_0, t̃_0) through (φ_0, θ_0, E_0, t_0) ∈Γ⋔ W^uu_(Ẽ_0, t̃_0)with the base point (Ẽ_0, t̃_0) asE_0 = Ẽ_0 + εϕ^uu_1(0;Ẽ_0, t̃_0,0, 0) - ε∑_i=- ∞^-1{ f_3(·, Ẽ_i, t̃_i)|_Λ - f_3(φ_i, θ_i(φ_i),E_i, t_i)|_W_2(Λ̅)} + ε O(ε + |δ|). Using that ϕ^ss_1(0;E̅_0, t̅_0,0, 0) -ϕ^uu_1(0;Ẽ_0, t̃_0,0, 0)=0 since the unperturbed stable and unstable foliations coincide, and subtracting(<ref>) from (<ref>) gives an expression forthe E component of the truncated scattering map up to ε O(ε + |δ|):Ẽ_0 = E̅_0 + ε∑_i=- ∞^∞{ f_3(·, E_i, t_i)|_Λ - f_3(φ_i, θ_i(φ_i), E_i, t_i)|_W_2(Λ̅)}.Analogously, for the t components of the truncated map S_Γ we obtain:t̃_0 = t̅_0 + ε∑_i = - ∞^∞{f_4(·, E_i, t_i)|_Λ - f_4(φ_i, θ_i(φ_i), E_i, t_i)|_W_2(Λ̅)}. We see that S_Γ is close to identity and may be approximated by a time-ε shift of some Hamiltonian flowH_out(t,E) with the accuracy ε O(ε + |δ|). From (<ref>) and (<ref>), we have Hamilton's equations for H_out(t, E) with ' = d/ds denoting differentiation with respect to auxiliary time s: E' = ∑_i=-∞^∞{ f_3(·, E_i, t_i)|_Λ - f_3(φ_i, θ_i(φ_i), E_i, t_i)|_W_2(Λ̅)},t' = ∑_i=-∞^∞{f_4(·, E_i, t_i)|_Λ -f_4(φ_i, θ_i(φ_i), E_i, t_i)|_W_2(Λ̅)}. Let us compute the above infinite sums. Recalling the definitions of f_3 and f_4 given by (<ref>, <ref>) and that the summation is performed over the fast variables (φ, θ) while the slow variables (E,t) are held fixed at their initial value (E_i=E_0, t_i = t_0),it turns out (see Appendix C) that the sum for E' may be computed to give ∑_i=-∞^∞{ f_3(·, E_i, t_i)|_Λ - f_3(φ_i, θ_i(φ_i), E_i, t_i)|_W_2(Λ̅)} = 2√(2E_0)∑_i = -∞^∞{- ȧ(t_i+1) + u_i+1sin (θ_i+1) } = 2 √(2E_0)(bḃ-aȧ/c),where u_i+1 is the normal speed of the boundary at (i+1)-th impact. The expression for t' may be computed using the geometric property of the ellipse <cit.> that the sum of homoclinic lengths converges to -2c(t_0):∑_i=-∞^∞{ f_4(·, E_i, t_i)|_Λ̅ - f_4(φ_i, θ_i(φ_i), E_i, t_i)|_ W_2(Λ̅)} = 1/√(2E_0)∑_i = -∞^∞{2a(t_i) - D_0|_ W_2(Λ̅)} = 2c(t)/√(2E),where D_0|_ W_2(Λ̃) is flight distanceD_0 (see (<ref>)) evaluated onW_2(Λ̃). Therefore the scattering map S_Γ is approximated up to O(ε(ε + |δ|)) by a time-ϵ shift along a trajectory of the solution of the differential equationdE/dt = -2E ċ(t)/c(t).This corresponds tothe Hamiltonian vector fieldt'= ∂ H_out/∂ E= √(2)c(t)/√(E),E'= -∂ H_out/∂ E = -2√(2E)ċ(t).Therefore we obtain the following Hamiltonian H_out(t,E) defined on Λ that approximates S_Γ:H_out(t,E) = 2√(2E)c(t). § ENERGY GROWTH In <cit.> it is proved (Lemma 4.4) that if two points on a normally-hyperbolic invariant manifold of a symplectic diffeomorphism are connected by an orbit of the iterated function system (IFS) formed by the inner and scattering maps, then there exists a trajectory of the original diffeomorphism that connects arbitrarily small neighbourhoods of those two points. Lemmas 3.11 and 3.12 of <cit.> show that the same is true when orbits of an IFS are infinite (in one direction).In this section we prove Theorem 1.1 using these facts. Namely, we consider an iterated function system {Φ, S_Γ}comprised of the inner map Φ defined by (<ref>) and the scattering map S_Γ defined by (<ref>), and show that it has an orbit with the energy ℰ tending to infinity. By the above quoted results, theexistence of the orbit of the map B for which the energy grows to infinity follows too.We start with the analysis of the behaviour of the IFS in the rescaled coordinates.For any initial condition (E,t) ∈Λ̃ with |δ| ≫ε^2, there exist positive integersn_1, n_2, where n_1 + n_2= O(1/ε), such thatthe gain Δ H_in of the Hamiltonian H_in(t,E; ε) (given in Remark 4.2) along the orbit S_Γ^n_2∘Φ^n_1 of the IFS{Φ), S_Γ} is Δ H_in≥ K_1min(δ^2/ε^2, 1 ) where K_1 is a strictly positive constant. Suppose that t^* is a nondegenerate critical point of a/b. Then f(t^*) = 0, which corresponds to ċ/c = ȧ/a, where f(t) is defined in (<ref>). Therefore (E,t^*) ∈Λ̅ where Λ̅ is the domain of S_Γ as in (<ref>) for all E. There exists an interval in t near t^*, denote it [t_1, t_2], such that ċ/c < ȧ/a and S_Γ is defined att ∈ [t_1, t_2].Denote Δ t = t_2 - t_1. Let us obtain the lower bound on Δ t.Taking (<ref>) and rewriting it in scaled variable v using the definition (<ref>), then Taylor expanding about t^*, we obtain that small Δ t can be chosen such thatv< c_2εΔ t/δ for a constant c_2 > 0. In other words, we can always choose the interval [t_1,t_2] such thatΔ t = c_1(δ/ε, 1 )where c_1>0 is constant. Let us take a point (E,t) ∈Λ̃.Iterate (E,t) under the inner map Φ until the image Φ^n_1(E,t) = (E_n_1, t_n_1)enters the domain [t_1, t_2], i.e., t_n_1∈ [t_1, t_2], and thus the point (E_n_1, t_n_1) ∈Λ̅ (if t is originally in the interval [t_1, t_2], then iterate until it gets out of this interval and, then, returns to it again). Note that the change in t during one iteration of Φ is of order ε which is much smaller than Δ t (because δ/ε≫ε by assumption). Therefore, the iterates of Φ cannot "miss" [t_1, t_2] and the number of iterations n_1 is bounded from above as O(ε^-1).There exists a level curve h_0 of H_in(t, E ;ε) passing through (E,t). The orbit of (E,t) under Φ will follow the level curve h_0.Indeed, the number of iterations n_1 of Φ is bounded by a number of order O(1/ε). Denote by ϕ^n_1ε_h_0 the time-n_1ε shift along h_0 with the initial condition (E,t). It follows from standard mean value theorem estimates and the Remark 4.2 that Φ^n_1 coincides withϕ^n_1ε_h_0 up to O(ε^r). Hence we have the following bound for the difference between the level of H_in at (E_n_1, t_n_1)andh_0=H_in(ϕ^n_1ε_h_0(E,t); ε):||H_in(E_n_1,t_n_1; ε) - H_in(ϕ^n_1ε_h_0(E,t); ε)||≤_(E,t) ∈Λ̃ || DH_in(E,t; ε)|| ||(E_n_1, t_n_1) - ϕ_h_0^n_1ε(E,t) || ≤_(E,t) ∈Λ̃ || DH_in(E,t; ε)||C̃_1ε^r < C_2ε^r,where _(E,t) ∈Λ̃ || DH_in(E,t; ε)|| is bounded as Λ̃ is compact; C̃_1, C_2>0 are constants. Since r≥ 4, we have that the error in the difference of H_in(t, E;ε) following Φ is maximumO(ε^4).Whent ∈ [t_1, t_2], following the level curve of H_out(t, E; ε, δ) will give a greater gain of energy than following H_in(t, E;ε). Therefore we iterate S_Γ while t ∈ [t_1, t_2] and its orbit will follow the level curve of H_out(t, E; ε, δ). Following H_in(t, E;ε) will switch to following H_out(t, E;ε, δ) when t = t_1 + O(ε) and then switch back to following another level curve of H_in(t, E;ε) when t = t_2 + O(ε). As Δ t≫ε, it follows that the number n_2 of iterates of S_Γ is n_2∼ O(Δ t/ε). Let us consider the Hamiltonian flow given by H_out(t, E) = 2 √(2E)c(t) as in (<ref>). Since the scattering map S_Γ and the time-ε flow map ϕ_H_out of H_out(t, E) coincide up to O(ε(ε + |δ|)), we have an upper bound for the difference between the values of H_in evaluated at (E_n_2, t_n_2) and at the time-n_2ϵ shift by the flow of H_out with the initial condition (E_n_1, t_n_1) (below C_3 >0 is an irrelevant constant):||H_out(E_n_2,t_n_2) - H_out(ϕ^n_2ε_H_out(E_n_1,t_n_1)) || < C_3Δ t (ε + |δ|)= O(|δ|(ε+|δ|)).Let us compute the change Δ H_in following the level curve of H_out(t,E). Using (<ref>), we haveΔ H_in = ∫_s_1^s_2dH_in(t, E;ε)/ds| _H_out= ds =∫_t_1^t_2dH_in(t, E;ε)/ds| _H_out=ds/dtdt.where t_1,2 = t_1,2(s_1,2) and s is the auxiliary time variable. Since dH_in(t, E;ε)/ds = ∂ H_in(t, E;ε)/∂ EdE/ds + ∂ H_in(t, E;ε)/∂ tdt/ds where we use (<ref>) for dE/ds and dt/ds, yielding dH_in(t, E;ε)/ds = 4(ȧc - ċa) + O(ε ). ThusΔ H_in = ∫_t_1^t_2(4(ȧc - ċa) + O(ε ) ) ds/dtdt=∫_t_1^t_2[H_in(t, E; 0) (ȧ/a - ċ/c) + O(ε)] dt.Note that ȧ/a - ċ/c>0 for all t ∈ [t_1, t_2]. We expand the above integral in Taylor series to obtain the estimate from belowΔ H_in∼ H_in(t,E; 0)(Δ t)^2 + O(ε) Δ t > K_1(δ^2/ε^2, 1 ),where K_1>0 is constant.The error estimates (<ref>), (<ref>) imply that for n_1 iterations of Φ followed by n_2 iterations of S_Γ, the increase of H_in is given byΔ H_in > K_1(δ^2/ε^2, 1 ) + O(ε^4, |δ| (ε + |δ|) ). Since |δ|≫ε^2, the net gain of H_in is strictly positive and is given by (<ref>) indeed. Take any initial condition (ℰ,t) on the cylinder Λ, where ℰ is the non-rescaled kinetic energy of the billiard particle. If ℰ≥C/| δ | for a sufficiently large constant C>0, then one can find the scaling parameter ε such that the scaled energy E=ℰε^2 lies in the middle of the interval [E_1, E_2] corresponding to the compact piece Λ̃ considered in the lemma above, and | δ| ≫ε^2. Suppose that Λ̃ is sufficiently large, E_2 - E_1 is sufficiently large. Since the function a(t) is bounded, the ratio of H_in/E is bounded away from zero and infinity. Therefore, by repeated application of Lemma 5.1 we find that the iterated function system {Φ, S_Γ} has an orbit, starting with our initial conditions (E=(E_1+E_2)/2, t) for which the value of H_in(t, E, ε) increases without bound until the orbit stays in Λ̃. In other words, this orbit will eventually get to the values of E larger than E_2. For the non-rescaled energy ℰ this means the multiplication at least to 2E_2/(E_1+E_2)>1. Thus, we have shown that for every initial condition (ℰ,t) with ℰ≥C/| δ | there exists an orbit of the IFSwith the end point (ℰ̅,t̅) such that ℰ̅>q ℰ where the factor q>1 is independent of the initial point. By taking the end point of the orbit we just constructed as the new initial point, and so on, we continue the process up to infinity and obtain the orbit of the IFS for which the energy ℰ tends to infinity.The shadowing Lemmas 3.11, 3.12 of <cit.> imply the existence of a true orbit of the map Bthat shadows the orbit of the IFS {Φ, S_Γ}, so the energy ℰ tends to infinity along this true orbit too (the shadowing lemmas of <cit.> require compactness of Λ, but it is easy to see that the result remains valid also in the situation where every orbit of the inner map is bounded - so the Poincare recurrence theorem can be used, and this property holds true in our case, as the KAM curves bound every orbit of Φ).§.§ AcknowledgementsThe authors would like to thank Vassili Gelfreich, Anatoly Neishtadt and Rafael Ramirez Ros for useful discussions. This work was supported by the grant 14-41-00044 of the Russian Science Foundation. Carl Dettmann's research is supported by EPSRC grant EP/N002458/1. Vitaly Fain's research is supported by University of Bristol Science Faculty Studentship grant. Dmitry Turaev's research is supported by EPSRC grant EP/P026001/1.§ MELNIKOV FUNCTION DERIVATION In this section we provide a derivation of the Melnikov function M_1 given by (<ref>); the Melnikov function M_2 given by (<ref>) is derived in the same manner. Melnikov theory for n-dimensional diffeomorphisms with hetero-homoclinic connections to normally hyperbolic invariant manifolds has been developed in <cit.>.Let us briefly review this construction and adapt it for our slow-fast setup. Since for the map B the invariant manifolds W^s,u(Λ) are three-dimensional while the phase space is four-dimensional, one only needs a scalar Melnikov function to measure their splitting for small nonzero ε, δ. Let us consider the case ε >0 and δ = 0. Take Λ̃ as in (<ref>). By symmetry we only need to consider the splitting of W^s,u_2(Λ̃). Theorem 3.4 in <cit.> gives the following expression for the Melnikov function M_1: ∑_n= -∞^n= ∞⟨ DB_0^n(x_-n)B_1(x_-n-1), ν(x_0) ⟩,where x_0 = (φ_0, θ_0, E_0, t_0) ∈ W_2(Λ̃) and ν(x_0) is the vector forming a basis of an orthogonal space to the tangent space of the unperturbed three dimensional homoclinic manifold W_2(Λ̃). Since B_0 has a first integral I, we take ∇ I(x_0) = ν(x_0). By the property of first integrals, observe that ∇ I(x_0) = (DB_0(x_0))^T∇ I(x_1) and by induction ∇ I(x_0) =(DB_0^n(x_0))^T∇ I(x_n). Then rewriting (<ref>) yieldsM_1 =∑_n= -∞^n= ∞⟨ DB_0^n(x_-n) B_1(x_-n-1), ∇ I(x_0) ⟩= ∑_n= -∞^n= ∞⟨B_1(x_-n-1), (DB_0^n(x_-n))^T∇ I(x_0) ⟩= ∑_n= -∞^n= ∞⟨B_1(x_n-1), (DB_0^-n(x_n))^T∇ I(x_0) ⟩ = ∑_n= -∞^n= ∞⟨B_1(x_n-1), (DB_0^-n(x_n))^T (DB_0^n(x))^T∇ I(x_n) ⟩=∑_n= -∞^n= ∞⟨B_1(x_n-1), (DB_0^n(x) DB_0^-n(x_n))^T∇ I(x_n) ⟩= ∑_n =-∞^n = ∞⟨ B_1(x_n-1), ∇ I(x_n) ⟩,which gives (<ref>). Note that since (E,t) are close to identity, effectively the summation above is performed only over the fast variables (φ, θ), while (E,t) are held at an initial value, hence they enter the sum as “fixed coefficients". For (<ref>) to converge, we require the restriction of the perturbed components of fast variables (φ, θ) (i.e. f_1, f_2, g_1, g_2) to Λ to vanish, and the form of I given by (<ref>) was chosen to ensure that ∇_E,t I(x) = 0 on Λ (i.e. the gradient of I with respect to slow variables (E,t) ).The Melnikov function (<ref>) is obtained by repeating the same steps above, by setting ε =0, and δ≠ 0. Note that the B_2 terms are independent of E and are evaluated at a given fixed moment of time t, hence effectively (<ref>) corresponds to the Melnikov function for the δ-polynomial perturbation of the elliptic billiard that has been studied by Delshams and Ramirez Ros in <cit.>. Since at O(ε + | δ|) the components of B_1 and B_2 given by formula (<ref>) simply add, the distance between perturbed invariant manifolds W^s,u(Λ̃) for ε, δ≠ 0 is given by (<ref>).§ COMPUTATION OF THE MELNIKOV FUNCTION M_1 FOR TIME-DEPENDENT ELLIPSE The Melnikov functions (<ref>), (<ref>) can be computed analytically in terms of elliptic functions using the theory developed in <cit.> to give the formulas (<ref>) and (<ref>) respectively. In this appendix we will derive the formula (<ref>). First, let us introduce some notation following <cit.> and quote the Proposition 3.1 we use from <cit.>.Given a parameter m ∈ [0,1], we have the following complete elliptic integrals of the first and second kind respectively: K=K(m)=∫^π/2_0(1-m sin(θ))^-1/2 dθ, E=E(m)=∫^π/2_0(1-msin(θ))^1/2 dθ. The incomplete elliptic integral of the second kind is E(u)=E(u| m):=∫^u_0^2(v| m) dv, where the functionis one of the Jacobian elliptic functions. Further, K'=K'(m):=K(1-m), E'=E'(m):=E(1-m) and if one of m, K, K', E, E', K'/K is given, all the rest are determined. We determine the parameter m for a given T, h >0by relationK'/K = T/h.From now on, we do not explicitly write the dependence of K, K', E, E', m on T and h. We introduce a function χ_T(z), χ_T(z) = (2K/h)^2(E'/K' - 1)z + (2K/h)E(2Kz/h + K'i),with the properties: (1): χ is meromorphic on ℂ, (2): χ is Ti-periodic with h-periodic derivative, (3): the poles of χ are in the set hℤ + Tiℤ, all simple with residue 1. The following formula is easily derived using the properties of elliptic functions: χ(iπ/2 - τ) -χ(h+iπ/2 - τ) = -2. It is also easily shown <cit.> that the following relation holds: χ (z +h) - χ(z) = 2π/T. For an isolated singularity z_0∈ℂ of a function q, we denoteby a_j(q,z_0) the coefficient of (z-z_0)^-j in the Laurent series of q around z_0.Then the following result holds. <cit.>, <cit.>Let q be a function satisfying: * q is analytic in ℝ, with only isolated singularities in ℂ* q is Ti-periodic for some T>0,* |q(τ)| ≤ A e^-c|τ| when | τ| →∞, for some constants A,c≥0. Then, Q(τ) = ∑_n = -∞^∞q(τ+hn) is analytic in ℝ, has only isolated singularities in ℂ, and is doubly periodic with periods h ≠ 0, where h ∈ℝ and Ti. Furthermore, Q(τ) may be expressed as Q(τ) = -∑_z ∈_T(q)∑_j≥0a_j+1(q,z)/j!χ_T^j(z-τ),where _T(q) is the set of singularities of q in I_T = {z ∈ℂ: 0<z<T}, and χ^j denotes the j-th derivative of χ. We note that if q is meromorphic, then Q(τ) is elliptic and can be computed analytically. Now we proceed to give the derivation of (<ref>). We will show that the sum (<ref>) is an elliptic function with two periods logλ and π i where λ given by (<ref>), and then apply the above proposition to compute (<ref>).Let us denote by I_φ, θ, E, t the partial derivatives of integral I given by (<ref>) with respect to φ, θ, E and t respectively. We will evaluate (<ref>) over the unperturbed homoclinic W_2(Λ̃) for fixed (E,t), hence we use the parameter ξ = tanφ_n/2 and the equation of W_2(Λ̃) given by (<ref>). Then the formula (<ref>) expresses θ as a function of φ to give W_2(Λ) in terms of ξ, holding (E,t) fixed at some initial value (E_0,t_0) = (E_n, t_n). Using B_1 = (f_1,f_2,f_3,f_4)^T from (<ref>) and expressing f_1,2,3,4 in terms of ξ we write (<ref>) as M_1:= ∑_n = -∞^∞⟨∇ I(B_0(x_n),B_1(x_n) ⟩ = ∑_n = -∞^∞f_1(ξ_n) I_φ(ξ_n+1) + f_2(ξ_n) I_θ(ξ_n) + f_4(ξ_n) I_t(ξ_n+1). Note we have writtenφ and θ in terms of ξ by virtue of (<ref>), and x_n = (φ_n, θ_n, E_n, t_n) as before. We have suppressed the dependence of functions f_1,2,3,4 and integral I on (E,t). Note that the term f_3I_E is identically zero since I is independent of energyE and thus we omit it from the above sum. Since parametrisation (<ref>) yields ξ_n+1 = λ^-1ξ_n, we may express the sum above purely in terms of ξ_n, i.e.M_1 = ∑_n = -∞^∞f_1(ξ_n) I_φ(λ^-1ξ_n) + f_2(ξ_n) I_θ(λ^-1ξ_n) + f_4(ξ_n) I_t(λ^-1ξ_n)= ∑_n = -∞^∞F(ξ_n) , for certain function F. Introduce a change of variables τ defined by ^τ=ξ. Since ξ∈ (0, ∞), then τ∈ (-∞, ∞), and for brevity put h = logλ, as in <cit.>. Then M_1 becomes M_1 = ∑ F(e^τ + nh) = ∑F̃(τ + nh), after swapping n ↦ -n. Here F̃(τ) = f_1(τ)I_φ(τ) +f_2(τ)I_θ(τ) + f_4(τ)I_t(τ).Using the formulae (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) together with (<ref>), we obtain the following expressions for f_1I_φ|_W_2, f_2I_θ|_W_2 and f_4I_t|_W_2 in terms of τ: f_1(τ)I_φ(τ)|_W_2 = -8 ȧc^2λ(λ+1)^2τ(λ^2 - ^2τ)^2/v_0(1+^2τ)(^2τ + λ^3)(^2τ+λ^2)^2 + 16ḃac^2λ^2^2τ(λ-^2τ)(λ^2-^2τ)/bv_0(1+^2τ)(^2τ+λ^3)(^2τ+λ^2)^2, f_2(τ)I_θ(τ )|_W_2=-16a^2cλ^2(ȧba^-1-ḃ)^2τ(λ^2-^2τ)(λ+^2τ)/v_0b(λ^3+^2τ)(λ^2+^2τ)^2(^2τ+1)_1 + 8acλ^2τ(-ȧ(λ+1)(λ^2-^2τ) +2λḃab^-1(λ - ^2τ)) /v_0(^2τ+λ^3)(^2τ+λ^2)(^2τ+1)_2 + 16c^2λ^2^2τ(ȧb(λ^2-^2τ)^2 + 4ḃaλ^2^2τ)/v_0b(^2τ+λ^3)(^2τ+λ^2)^2(^2τ+λ)_3, f_4(τ) I_t(τ)|_W_2 = 16a^2λ^2^2τ(aḃb^-1 - ȧ)(^2τ+λ)/(^2τ+λ^3)(^2τ+λ^2)(^2τ+1).Since in this appendix we are only considering the orbit of map B restricted to W_2, we drop the subscript W_2 for brevity. It is clear that f_1I_φ, f_2I_θ, f_4I_t are analytic on ℝ, exponentially decay at infinityand are π i-periodic on ℂ with isolatedsingularities that are poles, hence they are meromorphic, satisfying Proposition <ref>. Therefore we may apply Proposition <ref> to computeM_1 = ∑F(τ). We therefore take T = π,K'/K = π/h and χ_π(z) = χ(z).In the following computations, we will be using the following formula that can be shown using properties of elliptic functions: χ(z + lh + h/2) - χ(z+lh) =1+ 2K/hm (2Kτ/h) (2Kτ/h) = 1 + Y(τ), where l ∈ℤ, and we defined Y(τ) = 2K/hm (2Kτ/h) (2Kτ/h) for brevity. Also, we have χ^'(π i/2 - τ) =(2K/h)^2( E'/K' - 1 +^2(2Kτ/h)) = X(τ), where for brevity we put X(τ) =(2K/h)^2( E'/K' - 1 +^2(2Kτ/h)) and ' denotes differentiation w.r.t. τ. In (<ref>) and (<ref>) we have used the identities (u) = dn(-u), (u) = -(u + 2K + 2K'i), E(-u) = -E(u), E(u+2K + 2K'i) = E(u) + 2E + 2i(K' - E'), E(K - u) = E - E(u) + m (u) (u) and the Legendre equality EK' + E'K - K'K = π/2 together with the formulas (<ref>), (<ref>). We will compute separately the three components ∑ f_1I_φ, ∑ f_1 I_θ and ∑ f_4I_t of theexpression for M_1 given by (<ref>). Further, wherever it facilitates the computations, will consider individually the ȧ and ḃ components of the sums∑ f_1I_φ, ∑ f_1 I_θ and f_4I_t. Consider∑ f_1I_φ. Take the ȧ component of ∑ f_1I_φ with the coefficient -8c^2λ(λ+1)/v_0 factored out. Denote by a_j(f_1I_φ, z_0; ȧ) the corresponding Laurent series coefficient of the ȧ component of f_1I_φ at z_0 (here-8c^2λ(λ+1)/v_0 is factored out).We have: * Simple pole at z= iπ/2 with a_1(f_1I_φ, iπ/2; ȧ)=1+2λ + λ^4/2(-1+λ)^3(1+λ)^2(1+λ + λ^2),* simple pole at z = 3h/2 + iπ/2witha_1(f_1I_φ, 3h /2 + iπ/2; ȧ)=-1-2λ - λ^2/2(-1+λ)^3(1+λ + λ^2),* double pole at z = h + iπ/2 with a_1(f_1I_φ, h + iπ/2; ȧ) = 2λ/(-1+λ)^3(1+λ)^2 and a_2(f_1I_φ, h + iπ/2; ȧ) = 1/(λ - 1)^2(1+λ).Observe that a_1(f_1I_φ, iπ/2; ȧ) + a_1(f_1I_φ, 3 h/2 + iπ/2; ȧ) = -a_1(f_1I_φ, h+iπ/2; ȧ). Thus we obtain the following formula for ȧ component of∑ f_1I_φ: ∑ f_1 I_φ|_ȧ =8c^2λ(λ+1)/v_0(-2a_1(iπ/2) + a_1(3h/2 + iπ/2)(1 + Y(τ) ) + a_2(h + iπ/2)X(τ) ).Similarly, considering the ḃ component of∑ f_1I_φ (see (<ref>)), with factored out 16ḃac^2λ^2/bv_0, we have:* Simple pole at z = i π/2 with a_1(f_1I_φ,iπ/2; ḃ) = λ^2 +1/2(λ+1)(λ-1)^3(λ^2 + λ+1),* simple pole at z = 3h/2 + iπ/2 witha_1(f_1I_φ, 3h/2 + iπ/2; ḃ) = -(λ+1)(λ^2+1)/2λ(λ-1)^3(λ^2+λ+1),* double pole at h + iπ/2 with a_1(f_1I_φ, h + iπ/2; ḃ) = λ^2+1/2λ(λ+1)(λ-1)^3 and a_2(f_1I_φ, h + iπ/2; ḃ)= 1/2λ(λ-1)^2. Here a_1(f_1I_φ, iπ/2; ḃ) + a_1(f_1I_φ, 3h/2 + iπ/2; ḃ) = -a_1(f_1I_φ, h + iπ/2; ḃ). Thus we have the following formula for ḃ component of ∑ f_1 I_φ: ∑ f_1 I_φ|_ḃ = -16ḃac^2λ^2/bv_0(-2a_1(iπ/2) + a_1(3h/2 + iπ/2)(1+ Y(τ)) + a_2(h + iπ/2)X(τ)).Adding the ȧ and ḃ contributions and simplifying gives the formula ∑ f_1I_φ= ȧ/v(-2b^2(3a^4+2a^2c^2+c^4)/ac(3a^2 + c^2) + 2b^2X(τ) - 4a^3b^2/c(3a^2+c^2)Y(τ) )+ ḃ/bv(2b^2(3a^2-c^2)(a^2+c^2)/c(3a^2+c^2) - 2ab^2X(τ) + 4a^2b^2(a^2+c^2)/c(3a^2+c^2)Y(τ)). Next we will compute ∑ f_2I_θ; the corresponding expression (<ref>) consist of the sum of three parts. Take the part (1) defined by braces, and factor out -16a^2c λ^2(ȧba^-1 - ḃ)/v_0b. Denote by a_j(f_1I_θ, z_0; (1)) the corresponding Laurent series coefficient of the (1) component of f_1I_φ at z_0, (without coefficient -16a^2c λ^2(ȧba^-1 - ḃ)/v_0b). We have* Simple pole at z = iπ/2 with a_1(f_2I_θ, iπ/2; (1)) = λ^2+1/2(λ^2 - 1)^2 (λ^2 + λ + 1),* simple pole at z = 3h/2 + iπ/2 with a_1(f_2I_θ, 3h/2 + i π/2; (1)) = (λ +1)^2/2λ (λ-1)^2(λ^2 + λ + 1),* double pole at z = h + iπ/2 with a_1(f_2I_θ, h + iπ/2; (1)) = -1-4 λ - λ^2/2λ(λ^2 - 1)^2 and a_2(f_2I_θ, h + iπ/2; (1) = 1/2λ(λ^2-1),where a_1(f_2I_θ, iπ/2; (1)) + a_1(f_2I_θ, 3h/2 + iπ/2; (1)) = -a_1(f_2I_θ, h + iπ/2; (1)). Hence the sum for the part (1) is:∑ f_2 I_θ|_(1) = 16a^2cλ^2(ȧb a^-1 - ḃ)/v_0b( -2a_1(f_2I_θ, iπ/2; (1) ) + a_1(f_2I_θ, 3h/2 + iπ/2; (1) ) (1 + Y(τ) ))+16a^2cλ^2(ȧb a^-1 - ḃ)/v_0b (a_2(f_2I_θ, h + iπ/2; (1)) X(τ)).Let us consider the ȧ and ḃ components of the second component (2) of (<ref>) separately. Take contribution in ȧ. Factor out -8aȧcλ(λ+1)/v_0. Denote by a_j(f_1I_θ, z_0; ȧ, (2)) the corresponding Laurent series coefficient of the ȧ coefficient of the (2) component of f_1I_φ at z_0. We have* Simple pole at z = iπ/2 with a_1(f_2I_θ, iπ/2; ȧ, (2)) = λ^2+1/2(λ^2-1)(λ-1)(λ^2 + λ+1),* simple pole at z = 3h/2 + iπ/2 with a_1(f_2I_θ, 3h/2 + iπ/2; ȧ, (2)) = λ+1/2(λ-1)^2 (λ^2+λ+1),* Simple pole at z = h + iπ/2 with a_1(f_2I_θ, h + iπ/2; ȧ, (2)) = -1/(λ-1)^2 (λ+1),where a_1(f_2I_θ, iπ/2; ȧ, (2)) + a_1(f_2I_θ, 3h + iπ/2; ȧ, (2)) = -a_1(f_2I_θ, h + iπ/2; ȧ, (2)). Thus the corresponding sum for part (2) for ȧ contribution is ∑ f_2 I_θ|_ȧ, (2) =8aȧcλ(λ+1)/v_0(-2a_1(f_2I_θ, iπ/2; ȧ) + a_1(f_2I_θ, 3h/2 + iπ/2; ȧ)(1+ Y(τ)) ).For ḃ component of (2) of (<ref>), first factor out 16a^2cλ^2ḃ/bv. Using the notation as above, we find:* Simple pole at z = iπ/2 with a_1(f_2I_θ, iπ/2; ḃ, (2)) = 1/2(λ-1)^2(λ^2 + λ + 1),* simple pole at z = 3h/2 + iπ/2 with a_1(f_2I_θ, 3h/2 + iπ/2; ḃ, (2)) = λ^2 +1/2λ(λ-1)^2(λ^2 + λ+1),* simple pole at z = h +iπ/2 with a_1(f_2I_θ, h + iπ/2; ḃ, (2)) = -1/2λ(λ-1)^2,where a_1(f_2I_θ, iπ/2; ḃ, (2)) + a_1(f_2I_θ, 3h/2 + iπ/2; ḃ, (2)) = -a_1(f_2I_θ, h + iπ/2; ḃ, (2)). Thus the second component (2) sum for ḃ is ∑ f_2 I_θ|_ḃ,(2) =-16a^2cλ^2ḃ/bv(-2a_1(f_2I_θ, iπ/2; ḃ) + a_1(f_2I_θ, 3h/2 + iπ/2; ḃ)(1+ Y(τ)) ).Finally, let us consider the ȧ and ḃ coefficients of part (3) of (<ref>) separately. For ȧ component of (3), we factor out 16ȧc^2λ^2/v_0 and obtain* Simple pole at z = 3h/2 + iπ/2 with a_1(f_2I_θ, 3h/2 + iπ/2; ȧ, (3)) = -a_1(h/2 + iπ/2) = -1 - λ/2λ(λ-1)^3,* Simple pole at z = h + iπ/2 with a_1(f_2I_θ, h + iπ/2; ȧ, (3)) = 0, a_2(h + iπ/2) = 1/λ(λ-1)^2. Hence we get the corresponding sum∑ f_2 I_θ|_ȧ,(3) =-16ȧc^2λ^2/v_0(2a_1(f_2I_θ, 3h/2 + iπ/2; ȧ, (3)) + a_2(f_2I_θ, h + iπ/2; ȧ, (3)) X(τ) ).For third component (3) coefficient in b, we factor out 64aḃc^2λ^4/bv_0 to yield* Simple pole at z = 3h/2 + iπ/2 with a_1(f_2I_θ, 3h/2 + iπ/2; ḃ, (3) ) = -a_1(h/2 + iπ/2)= 1/2λ^2(λ+1)(λ-1)^3,* simple pole at z = h + iπ/2 with a_1(f_2I_θ, h + iπ/2; ḃ, (3)) = 0, a_2(f_2I_θ, h+ iπ/2; ḃ, (3)) = -1/4λ^3(λ-1)^2,which gives the sum∑ f_2 I_θ|_ḃ,(3) =-64aḃc^2λ^4/bv_0(2a_1(f_2I_θ, 3h/2 + iπ/2; ḃ, (3)) + a_2(f_2I_θ, h+ iπ/2; ḃ, (3))X(τ) ).Adding the expressions (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>)and simplifying gives the formula ∑ f_2I_θ = ȧ/v(2b^2(9a^4+c^4)/ac(3a^2+c^2) - 6b^2X(τ) + 12a^3b^2/c(3a^2+c^2)Y(τ)) + ḃ/bv(6ab^2 X(τ) - 2b^2(3a^2-c^2)/c - 4a^2b^2/cY(τ)). Finally we compute ∑_n = -∞^∞ f_4I_t. Let us take out the factor 16a^2λ^2(aḃb^-1 - ȧ). Then we obtain* Simple pole at z = iπ/2 with a_1(f_4I_t, iπ/2)=1/2(-1-λ + λ^3 + λ^4),* simple pole at z = 3h/2 + iπ/2 witha_1(f_4I_t, 3logλ /2 + iπ/2)=-1-λ/2λ(-1+λ^3),* simple pole at z = h + iπ/2 witha_1(f_4I_t, logλ + iπ/2) = 1/2λ(-1+λ^2). Observe that a_1(f_4I_t, logλ +iπ/2) + a_1(f_4I_t, 3 logλ/2 + iπ/2) = -a_1(f_4I_t, iπ/2). Therefore the sum is ∑f_4I_t = -(aḃ/bv - ȧ/v)(16a^2λ^2) (a_1(3h/2 + iπ/2)(3 + Y(τ)) + 2a_1(h + iπ/2)).Upon simplifying, this yields ∑f_4I_t = (aḃ/bv - ȧ/v)(4ab^2(3a^2-c^2)/c(3a^2+c^2) + 8a^3b^2/c(3a^2+c^2)Y(τ)). Finally adding (<ref>), (<ref>) and (<ref>) yields (<ref>).§ COMPUTATION OF ZERO ORDER DISTANCE D_0 In this appendix we show that the free flight distance (<ref>) that is incorporated in the formulas for f_1, f_2, f_4 turns out to be an elliptic function when evaluated on the homoclinic manifold W_2(Λ̃) (see formula (<ref>)). The zero order in ε, δ free flight distance D_0 given by equation (<ref>) evaluated on W_2(Λ̃) expressed in terms of ξ is D_0|_W_2(Λ̃)=2a(λ +ξ^2_n)^2/(λ^2+ξ^2_n)(1+ ξ^2_n) = 2a(1+λξ^2_n+1)^2/(1+ξ^2_n+1)(1+λ^2ξ^2_n+1). Recall that W_2(Λ̃) corresponds to the billiard orbit passing the focus at (-c,0), and therefore each consecutive collision point on the boundary (<ref>) is in alternate halves of the ellipse. In terms of variable φπthe negative signs in the square brackets in (<ref>) change to positive and gives D_0|_W_2(Λ̃)=√(a^2[cos(φ_n) + cos(φ_n+1)]^2 + b^2[sin(φ_n) + sin(φ_n+1)]^2).Taking the parametrisation (<ref>) to express trigonometric functions of φ_n+1 and φ_n in terms of ξ_n and using the identity b^2/a^2(1+λ)^2 ≡ 4λ, with λ given by (<ref>), we obtain: D_0|_W_2(Λ̃) = √( a^2( 4λ/(1+λ)^2( 2ξ_n/1 + ξ^2_n+ 2λξ_n/λ^2 + ξ^2_n) ^2 + ( 1 - ξ^2_n/1 + ξ^2_n + λ^2 - ξ^2_n/λ^2 + ξ^2_n)^2)). Taking a positive root of this expression since it is Euclidean distance and simplifying yields: D_0|_W_2(Λ̃)= √( a^2(4(λ + ξ_n^2)^4/(1+ξ_n^2)^2(λ^2+ ξ_n^2)^2)). Note that in the limit of n →∞ , (<ref>) yields D_0|_W_2(Λ̃) =2a, i.e. the length of the major axis of the ellipse,which corresponds the hyperbolic fixed point z.Analogously we may calculate D_0|_W_1(Λ̃), which is: D_0|_W_1(Λ̃)=2a(1+λξ^2_n)^2/(1+ξ^2_n)(1+λ^2ξ^2_n) =2a(λ +ξ^2_n+1)^2/(λ^2+ξ^2_n+1)(1+ ξ^2_n+1).§ SCATTERING MAP COMPUTATIONS We provide a derivation of truncated scattering map formula (<ref>). From the E component of S_Γ given by (<ref>), we need to compute the sum ∑_n=-∞^∞f_3(·, E_n, t_n)|_Λ - f_3(φ_n, θ_n(φ_n), E_n, t_n)|_W_2(Λ̅) = 2√(2E_0)∑_n = -∞^∞- ȧ(t_n+1) + u_n+1sin (θ_n+1) , where u_n+1 = ȧbcos^2(φ_n+1) + aḃsin^2(φ_n+1)/√( a^2sin^2(φ_n+1) + b^2cos^2(φ_n+1)) is the normal boundary speed at boundary point φ_n+1 evaluated on the homoclinic manifold W_2(Λ̃). Rewriting φ in terms of ξ using (<ref>) gives u_n+1 =ȧb(1-ξ^2_n+1)^2 + 4aḃξ^2_n+1/(1+ξ^2_n+1)√(4c^2ξ^2_n+1 + b^2(1+ξ^2_n+1)^2).Now4c^2ξ^2_n+1 + b^2(1+ξ^2_n+1)^2 ≡b^2(ξ^2_n+1+λ)(λξ^2_n+1+1)/λ,and expressing sin(θ) in terms of ξ using (<ref>), we obtain u_n+1sin (θ_n+1) - ȧ(t_n+1) = λ(ȧb(1-ξ_n+1^2)^2+4aḃξ_n+1^2)/b(ξ^2_n+1 + λ)(λξ^2_n+1+1) - ȧ(t_n+1) = ȧξ^2_n+1(λ+1)^2/(ξ^2_n+1+λ)(λξ^2_n+1+1) + 4aḃλξ^2_n+1/b(ξ^2_n+1+λ)(λξ^2_n+1+1). It is clear that we need to calculate ∑_n = -∞^∞ξ^2_n/(ξ^2_n+λ)(λξ^2_n+1). After a change of variables ξ = ^τ, h = logλ as before, the function ^2τ/(^2τ+λ)(λ^2τ+1) is bounded for τ→∞, is periodic in the complex plane with period iπ, and it has simple poles at τ = ±logλ /2 + iπ / 2 with residues ∓ 1/2(λ^2-1), and by using the result χ(logλ/2 +iπ/2 - τ) -χ(logλ/2+iπ/2 - τ) = 2, we have that ∑_n = -∞^∞ξ^2_n/(ξ^2_n+λ)(λξ^2_n+1) = 1/λ^2-1. Therefore after some manipulations we get u_n+1sin (θ_n+1) - ȧ(t_n+1) = bḃ-aȧ/c, which gives (<ref>).The t component of S_Γ is computed using the geometric property of the ellipse, as given in Section 4.3.§ COMPUTATION OF MELNIKOV FUNCTION M_2 As remarked previously, M_2 corresponds to the Melnikov function for the O(δ) quartic polynomial perturbation of the ellipse, evaluated for the 'fixed' boundary at time t = t_n. Therefore M_2 may be evaluated by standard methods developed for static perturbations of elliptic billiards using the generating function approach <cit.>. The parametrisation (<ref>) of the boundary at fixed time t_n for first order in δ coincides with the parametrisation in <cit.> given by equation (4.6) up to the adjustment of a constant factor. Thus, it is easy to essentially re-write the derivation of the Melnikov function in <cit.> to obtainM_2 = -4m ab^2/c^2(2K/h)^3(2Kτ/h) (2Kτ/h) (2Kτ/h).Notice that (<ref>) coincides with the Melnikov function for the perturbed ellipse in <cit.>, (see section 4.4.2 of <cit.>), upon setting c^2 = 1, c_1=b.10birkhoff1927periodic G. D. Birkhoff, “On the periodic motions of dynamical systems,” Acta Mathematica, vol. 50, no. 1, pp. 359–379, 1927.avila2014integrable A. Avila, J. De Simoi, and V. Kaloshin, “An integrable deformation of an ellipse of small eccentricity is an ellipse,” Annals of Mathematics, vol. 184, p. 1–32, 2016.koiller1995time J. Koiller, R. Markarian, S. O. Kamphorst, and S. P. de Carvalho, “Time-dependent billiards,” Nonlinearity, vol. 8, no. 6, p. 983, 1995.fermi1949mesons E. Fermi and C.-N. Yang, “Are mesons elementary particles?,” Physical Review, vol. 76, no. 12, p. 1739, 1949.pustyl1983ulam L. D. Pustyl'nikov, “On Ulam's problem,” Theoretical and Mathematical Physics, vol. 57, no. 1, pp. 1035–1038, 1983.kruger1995acceleration T. Kruger, L. Pustyl'nikov, and S. Troubetzkoy, “Acceleration of bouncing balls in external fields,” Nonlinearity, vol. 8, no. 3, p. 397, 1995.zharnitsky1998instability V. Zharnitsky, “Instability in Fermi-Ulam ping-pong problem,” Nonlinearity, vol. 11, no. 6, p. 1481, 1998.loskutov2000properties A. Loskutov, A. Ryabov, and L. Akinshin, “Properties of some chaotic billiards with time-dependent boundaries,” Journal of Physics A: Mathematical and General, vol. 33, no. 44, p. 7973, 2000. bolotin1999unbounded S. Bolotin and D. Treschev, “Unbounded growth of energy in nonautonomous Hamiltonian systems" Nonlinearity,vol. 12, no. 2, p. 365, 1999.treschev2004evolution D. Treschev, “Evolution of slow variables in a priori unstable Hamiltonian systems"Nonlinearity, vol. 17, no. 5, p. 1803, 2004.loskutov1999mechanism A. Y. Loskutov, A. Ryabov, and L. Akinshin, “Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries,” Journal of Experimental and Theoretical Physics, vol. 89, no. 5, pp. 966–974, 1999.kamphorst1999bounded S. O. Kamphorst and S. P. de Carvalho, “Bounded gain of energy on the breathing circle billiard,” Nonlinearity, vol. 12, no. 5, p. 1363, 1999.de2006fermi R. E. de Carvalho, F. C. Souza, and E. D. Leonel, “Fermi acceleration on the annular billiard,” Physical Review E, vol. 73, no. 6, p. 066229, 2006.shah2010exponential K. Shah, D. Turaev, and V. Rom-Kedar, “Exponential energy growth in a Fermi accelerator,” Physical Review E, vol. 81, no. 5, p. 056205, 2010.gelfreich2008fermi V. Gelfreich and D. Turaev, “Fermi acceleration in non-autonomous billiards,” Journal of Physics A: Mathematical and Theoretical, vol. 41, no. 21, p. 212003, 2008.gelfreich2008unbounded V. Gelfreich and D. Turaev, “Unbounded energy growth in Hamiltonian systems with a slowly varying parameter,” Communications in Mathematical Physics, vol. 283, no. 3, p. 769–794, 2008.koiller1996static J. Koiller, R. Markarian, S. Oliffson Kamphorst, and S. Pinto de Carvalho, “Static and time-dependent perturbations of the classical elliptical billiard,” Journal of Statistical Physics, vol. 83, no. 1, pp. 127–143, 1996.lenz2008tunable F. Lenz, F. K. Diakonos, and P. Schmelcher, “Tunable Fermi acceleration in the driven elliptical billiard,” Physical Review Letters, vol. 100, no. 1, p. 014103, 2008.lenz2009evolutionary F. Lenz, C. Petri, F. Koch, F. Diakonos, and P. Schmelcher, “Evolutionary phase space in driven elliptical billiards,” New Journal of Physics, vol. 11, no. 8, p. 083035, 2009.lenz2010phase F. Lenz, C. Petri, F. K. Diakonos, and P. Schmelcher, “Phase-space composition of driven elliptical billiards and its impact on Fermi acceleration,” Physical Review E, vol. 82, no. 1, p. 016206, 2010.itin2003resonant A. Itin and A. Neishtadt, “Resonant phenomena in slowly perturbed elliptic billiards,” Regular and Chaotic Dynamics, vol. 8, no. 1, pp. 59–66, 2003.delshams2008geometric A. Delshams, R. De La Llave, and T. M. Seara, “Geometric properties of the scattering map of a normally hyperbolic invariant manifold,” Advances in Mathematics, vol. 217, no. 3, pp. 1096–1153, 2008.le2007drift P. Le Calvez, “Drift orbits for families of twist maps of the annulus,” Ergodic Theory and Dynamical Systems, vol. 27, no. 3, p. 869–879, 2007. moeckel2002generic R. Moeckel, “Generic drift on Cantor sets of annuli”, Celestial Mechanics (Evanston, Il, 1999), Contemporary Mathematics, vol. 292, p. 163–171, 2002.nassiri2012robust M. Nassiri and E.R. Pujals, “Robust transitivity in Hamiltonian dynamics”, Ann. Sci. Norm. Sup., vol. 45, no. 2, p. 191-239, 2012. gelfreich2012fermi V. Gelfreich, V. Rom-Kedar, and D. Turaev, “Fermi acceleration and adiabatic invariants for non-autonomous billiards,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 22, no. 3, p. 033116, 2012.chierchia1994drift L. Chierchia and G. Gallavotti, “Drift and diffusion in phase space,” Ann. Inst. H. Poincaré Phys. Théor, vol. 60, no. 1, p. 144, 1994.arnold1964instability V. I. Arnold, “Instability of dynamical systems with several degrees of freedom(instability of motions of dynamic system with five-dimensional phase space),” Soviet Mathematics, vol. 5, pp. 581–585, 1964.delshams2000geometric A. Delshams, R. de la Llave, and T. M. Seara, “A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of 𝕋^2,” Communications in Mathematical Physics, vol. 209, no. 2, pp. 353–392, 2000.Delshams2006 A. Delshams, R. de la Llave, and T. M. Seara, “A geometric mechanism for diffusionin Hamiltonian systems overcoming the large gap problem: heuristics and rigorousverification on a model” Mem. Am. Math. Soc. 179, 20006.gelfreich2017arnold V. Gelfreich and D. Turaev, “Arnold diffusion in A Priori Chaotic Symplectic Maps,”Communications in Mathematical Physics, vol. 353, no. 2, p. 507–547, 2017. gidea2014general M. Gidea, R. de la Llave, and T. Seara, “A general mechanism of diffusion in Hamiltonian systems: qualitative results,” arXiv preprint arXiv:1405.0866, 2014.benettin1994hamiltonian G. Benettin and A. Giorgilli, “On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms,” Journal of Statistical Physics, vol. 74, no. 5, pp. 1117–1143, 1994.neishtadt1984separation A. I. Neishtadt, “The separation of motions in systems with rapidly rotating phase,” Journal of Applied Mathematics and Mechanics, vol. 48, no. 2, pp. 133–139, 1984.delshams1996poincare A. Delshams and R. Ramírez-Ros, “Poincaré - Melnikov - Arnold method for analytic planar maps,” Nonlinearity, vol. 9, no. 1, p. 1, 1996.lomeli1996perturbations H. E. Lomeli, “Perturbations of elliptic billiards,” Physica D: Nonlinear Phenomena, vol. 99, no. 1, pp. 59–80, 1996.tabanov1994separatrices M. Tabanov, “Separatrices splitting for Birkhoff’s billiard in symmetric convex domain, closed to an ellipse,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 4, no. 4, pp. 595–606, 1994.levallois1993separation P. Levallois and M. Tabanov, “Séparation des séparatrices du billard elliptique pour une perturbation algébrique et symétrique de l'ellipse,” Comptes rendus de l'Académie des sciences. Série 1, Mathématique, vol. 316, no. 6, pp. 589–592, 1993.delshams2001homoclinic A. Delshams, Y. Fedorov, and R. Ramírez-Ros, “Homoclinic billiard orbits inside symmetrically perturbed ellipsoids,” Nonlinearity, vol. 14, no. 5, p. 1141, 2001.fenichel1971persistence N. Fenichel, “Persistence and smoothness of invariant manifolds for flows,” Indiana Univ. Math. J, vol. 21, no. 193-226, p. 1972, 1971.baldoma1998poincare M. Baldoma and E. Fontich, “Poincaré- Melnikov theory for n-dimensional diffeomorphisms,” Applicationes Mathematicae, vol. 25, no. 2, pp. 129–152, 1998.delshams1997melnikov A. Delshams and R. Ramírez-Ros, “Melnikov potential for exact symplectic maps,” Communications in mathematical physics, vol. 190, no. 1, pp. 213–245, 1997.ramirez2005exponentially R. Ramírez-Ros, “Exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables,” Physica D: Nonlinear Phenomena, vol. 210, no. 3, pp. 149–179, 2005. | http://arxiv.org/abs/1706.08623v2 | {
"authors": [
"Carl P. Dettmann",
"Vitaly Fain",
"Dmitry Turaev"
],
"categories": [
"math.DS"
],
"primary_category": "math.DS",
"published": "20170626230757",
"title": "Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse"
} |
Ames Laboratory, U.S. DOE, Ames, IA 50011, USA Department of Chemistry, Iowa State University, Ames, Iowa 50011, USAAmes Laboratory, U.S. DOE, Ames, IA 50011, USAAmes Laboratory, U.S. DOE, Ames, IA 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USAAmes Laboratory, U.S. DOE, Ames, IA 50011, USA Ames Laboratory, U.S. DOE, Ames, IA 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USAInstitute of Theoretical Physics, Goethe University Frankfurt am Main, D-60438 Frankfurt am Main, GermanyAmes Laboratory, U.S. DOE, Ames, IA 50011, USAAmes Laboratory, U.S. DOE, Ames, IA 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USAInstitute of Theoretical Physics, Goethe University Frankfurt am Main, D-60438 Frankfurt am Main, GermanyAmes Laboratory, U.S. DOE, Ames, IA 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USAAmes Laboratory, U.S. DOE, Ames, IA 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA We report ^75As nuclear magnetic resonance (NMR) studies on a new iron-based superconductor CaKFe_4As_4 with T_ c = 35 K. ^75As NMR spectra show two distinct lines corresponding to the As(1) and As(2) sites close to the K and Ca layers, respectively, revealing that K and Ca layers are well ordered without site inversions. We found that nuclear quadrupole frequencies ν_ Q of the As(1) and As(2) sites show an opposite temperature (T) dependence.Nearly T independent behavior of the Knight shifts K are observed in the normal state,and a sudden decrease in K in the superconducting (SC) state suggests spin-singlet Cooper pairs.^75As spin-lattice relaxation rates 1/T_1 show a power law T dependence with different exponents for the two As sites.The isotropic antiferromagnetic spin fluctuations characterized by the wavevector q = (π, 0) or (0, π) in the single-iron Brillouin zone notationare revealed by 1/T_1T and K measurements. Such magnetic fluctuations are necessary to explain the observed temperature dependence of the ^75As quadrupole frequencies, as evidenced by our first-principles calculations. In the SC state, 1/T_1 shows a rapid decrease below T_ c without a Hebel-Slichter peak and decreases exponentiallyat low T, consistent with an s^± nodeless two-gap superconductor.Magnetic fluctuations and superconducting properties of CaKFe_4As_4 studied by ^75As NMR Y. Furukawa, December 30, 2023 ========================================================================================§ I. INTRODUCTIONFollowing the discovery of unconventional superconductivity (SC) in the so-called 1111 and 122 systems exemplified by LaFeAsO <cit.> and BaFe_2As_2 <cit.>, respectively, many pnictides have been investigated in the search for unconventional superconductivity and other novel properties <cit.>. Quite recently, a new Fe-based SCCaKFe_4As_4 (in short, CaK1144) with a transition temperature of T_ c ∼ 33.1 K has been discovered <cit.>.In CaK1144, Ca^2+ and K^1+ occupy different atomic positions in its crystal structure due to the different ionic radii, forming alternating Ca and K planes along the crystallographic caxis separated by FeAs layers <cit.>. This is contrary to the doped 122 system such as Ba_1-xK_xFe_2As_2 where alkaline-earth metal and alkaline ions occupy randomly the same atomic position. The ordering of the K and Ca ions changes the space group from I4/mmmsystem to P4/mmm. Consequently, as shown in the inset ofFig. <ref>(a), there are two different As sites: As(1) and As(2) sites close to the K and Ca layers, respectively.Soon after the discovery of CaK1144, thermodynamic and transport measurements on single crystals revealed that CaK1144 is an ordered stoichiometric superconductor with T_ c ∼ 35 K and a very high upper critical field of 92 T <cit.> with no other phase transition from 1.8 K to room temperature, and shows similar physical properties to the optimally doped (Ba_1-xK_x)Fe_2As_2.Muon spin rotation and relaxation (μSR) measurements show two nodeless gaps in CaK1144 with gap sizes of 2.5 and 8.6 meV <cit.>, consistent with tunnel-diode resonator (TDR) and scanning tunneling microscope (STM) measurements <cit.>.The multiple superconducting gaps in CaK1144 are also revealed by high resolution angle resolved photoemission spectroscopy (ARPES) measurements and density functional theory (DFT) calculations <cit.>.However, studies of magnetic fluctuations in CaK1144 have not been carried out yet.Since the magnetic fluctuations are considered to be one of the keys in driving SC in iron pnictide superconductors <cit.>, it is important to characterize the dynamical properties of the Fe moments. Nuclear magnetic resonance (NMR) is a microscopic probe suitable for investigating static spin susceptibility andlow-energy spin excitations for pnictide superconductors <cit.>.The temperature T dependence of the nuclear spin-lattice relaxation rate (1/T_1)is known to reflect the wave vector q-summed dynamical susceptibility. On the other hand, NMR spectrum measurements, in particular the Knight shift K, give us information on static magnetic susceptibility χ. Thus from the T dependence of 1/T_1T and K, one can obtain valuable insights about magnetic fluctuations in materials.Furthermore, 1/T_1 measurements in the SC state provide important information in understanding the gap structure in SCs. In this paper, we carried out ^75As NMR measurements to investigate the electronic and magnetic properties of single crystalline CaKFe_4As_4 with T_ c = 35 K. We observed two ^75As NMR signals corresponding to the two different As sites in the crystal structure, showing the ordering of Ca and K ions on each layer. The antiferromagnetic (AFM) spin fluctuations are clearly evidenced by ^75As spin-lattice relaxation rate 1/T_1 and Knight shift K measurements. Ab initio density functional theory calculations of the^75As nuclear quadrupole frequency (ν_ Q) confirm the importance of such fluctuations for thedescription of the observed temperature evolution of ν_Q.In the superconducting state, 1/T_1 shows a rapid decrease below T_ c without showing a Hebel-Slichter coherence peak and decreases exponentially at low temperatures, which is explained by a two-nodeless-gap model, consistent with previous reports.§ II. EXPERIMENTAL METHODSThe single crystals ofCaKFe_4As_4for NMR measurementswere grown by high temperature solution growth out of FeAs flux <cit.>and were extensively characterized by thermodynamic and transport measurements <cit.>. NMR measurements were carried out on ^75As(I = 3/2, γ/2π = 7.2919 MHz/T, Q =0.29 Barns)by using a lab-built, phase-coherent, spin-echo pulse spectrometer.The ^75As-NMR spectra were obtained by sweeping the magnetic field at a fixed frequency f = 53 MHz. The magnetic field was applied parallel to either the crystalline c axis or the ab plane. The ^75As 1/T_ 1 was measured with a recovery method using a single π/2 saturation pulse.In the normal state above T_ c, the 1/T_1 at each T was determined by fitting the nuclear magnetization M versus time tusing the exponential function 1-M(t)/M(∞) = 0.1 e^ -t/T_1 +0.9e^ -6t/T_1 for ^75As NMR, where M(t) and M(∞) are the nuclear magnetization at time t after the saturation and the equilibrium nuclear magnetization at t → ∞, respectively. In the SC state, the function could not reproduce the M versust data. Then we fitted the data with the two T_1 component function1-M(t)/M(∞) = M_ S [0.1 e^ -t/T_1S +0.9e^ -6t/T_1S] + M_ L [0.1 e^ -t/T_1L +0.9e^ -6t/T_1L] where M_ S+M_ L = 1.The M_ Sand M_ L correspond to the fraction for the short relaxation time T_ 1S and the long relaxation time T_ 1L, respectively.The M_ L was estimated to be ∼ 0.5 in a temperature range of 4.3 - 25 K and then starts to increase slowly up to 1 at T_ c. The T_ 1S and T_ 1L could be attributed to T_1 for the As(2) and As(1) sites, respectively, in the SC state.§ III. RESULTS AND DISCUSSION §.§ A. ^75As NMR spectrumFigure <ref>(a)shows typical field-swept ^75As-NMR spectra of the CaKFe_4As_4 single crystal atT = 36 K for the two magnetic field directions: H ∥ c axisand H ∥ ab plane. The typical spectrum for a nucleus with spin I=3/2 with Zeeman and quadrupolar interactions can be described by a nuclear spin Hamiltonian H=-γħ(1+K)HI_z+hν_ Q6[3I_z^2-I(I+1)+1/2η(I^2_+ + I^2_-)], where K is the Knight shift, h is Planck's constant, and ħ = h/2π. The nuclearquadrupole frequency for I=3/2 nuclei is given by ν_ Q = eQV_ ZZ/2h√(1+η^2/3), where Q is the nuclear quadrupole moment, V_ ZZ is the electric field gradient (EFG) along z at the As site and η = (V_ YY - V_ XX)/V_ ZZ is the in-plane asymmetry of the EFG tensor. For I=3/2 nuclei, when the Zeeman interaction is much greater than the quadruple interaction, this Hamiltonian produces a spectrum with a sharp central transition line flanked by single satellite peak on each side for each equivalent As site.As shown in the crystal structure [inset of Fig. <ref>(a)], CaKFe_4As_4 has two inequivalent As sites: As(1) and As(2) close to the K and Ca layers, respectively. As seen inFig. <ref>(a), we indeed observed two sets of I =3/2 quadrupole split lines corresponding to the two As sites for each H direction. The observed ^75As NMR spectra are well reproduced by simulated spectra [the red and blue lines in Fig. <ref>(a)] from the above simple Hamiltonian where the Zeeman interaction is greater than the quadrupole interaction. Since the distance between the As(1) and Fe ions is 2.395(2) Å which is slightly greater than 2.387(2) Å for the As(2) sites at 300 K <cit.>, the As(1) site is expected to be less strongly coupled to the Fe ions than the As(2) sites. Thus we assign the lower field central peak with a greater Knight shift K to the As(2) site and higher field central peak with a smaller K to the As(1) sites, respectively [see, Fig.<ref>(b)]. The well separated As(1) and As(2) lines indicate that K and Ca ions are not randomlydistributed but are well ordered making the Ca and K layers without a significantsite inversion.The full-width at half-maximum (FWHM) of each satellite line is estimated to be 1.35 kOe and 1.40 kOe for the As(1) and As(2) sites at 36 K and H ∥ ab plane, respectively.Since the FWHM of the satellite line is proportional to the distribution of ν_ Q and thus V_ZZ, the results indicate no large difference in inhomogeneity for the local As environments, again consistent with the ordering of the Ca and K ions for each plane. It is noted that these values are greater than ∼ 0.75 kOe for the satellite line of ^75As NMR at 200 K and H ∥abplane in CaFe_2As_2 which exhibits a stripe-type antiferromagnetic order below T_ N ∼ 170 K <cit.> and are also greater than ∼ 0.4 kOe at 10 K in the superconductor KFe_2As_2 with T_ c= 3.5 K <cit.>. On the other hand, these values are smaller than the K-doped BaFe_2As_2 case, for example, ∼ 2.5 kOe at 50 K inBa_0.42K_0.58Fe_2As_2 <cit.>. The T dependence of the central transition lines of the As(1) and As(2) sites under H parallel to the c axis is shown inFig. <ref>(b) where no obvious change in the spectra has been observed down to T_ c = 33 K which is slightly reduced from the zero-field transition temperature due to the application of H ∼ 7.23 T.The FWHM of the central line is nearly independent of T with ∼46 and ∼85 Oe for the As(1) and As(2) sites, respectively, in the normal state, indicating no magnetic and structural phase transitions above T_ c. The difference in FWHM originates from the distribution of the hyperfine field at the As sites.The small hump between the As(1) and As(2) lines is not intrinsic, probably due to unknown impurity, since no angle and temperature dependences are observed. Below T_ c, the intrinsic two peaks shift to higher magnetic field and, at the same time, the lines became broad due to the SC transition. Even at 30 K, just 3 K below T_ c, the two lines were smeared out and no clear separation of the two lines is observed as seen in Fig.<ref>(b). Despite a very low signal intensity, we were able to measure the spectrum at 4.2 Kwhich is very broad but still allows to determine the peak position. Figure <ref>(a) shows the T dependence of NMR shifts K_ab (H∥ ab plane)and K_c(H∥ c axis) for the two As sites.For both As sites, K's are nearly independent of T above ∼130 K and decrease very slightly down to T_ c upon cooling. Similar weak T dependence of K is observed in (Ba_1-xK_x)Fe_2As_2 <cit.>. BelowT_ c, K_c for boththe As sites show a sudden drop, indicating spin-singlet Cooper pairing.§.§ B. ^75As quadrupole frequency Figure <ref>(b) shows the T dependence ofν_ Q for the two As sites estimated from the fit of the NMR spectra, together with the ^75As-ν_ Q data in KFe_2As_2 and CaFe_2As_2.Theν_ Qs for the As(1) and the As(2) sites are also obtained by nuclear quadrupole resonance (NQR) measurements under zero magnetic field.For the As(2) site, with decreasing T, ν_ Q increases from 13.8 MHz at 205 K to 14.8 MHz at 36 K, while for the As(1) site an opposite trend is observed and ν_ Q decreases from 13.2 MHz at 205 K to 12.2 MHz at 36 K with decreasing T.It is interesting to note that the T dependences of ν_ Q of As(1) and As(2) are similar to those of ν_ Q of the As site in KFe_2As_2 <cit.> and CaFe_2As_2 <cit.>, respectively, as shown in the figure.These results are also consistent with our site assignment.In order to understand the peculiar temperature dependence ofν_ Q for As(1) and As(2), we have performed a density functional study of the electric-field gradients inunder different assumptions concerning the structural and spin degrees of freedom. By making use of the experimental lattice parameters measured between 6 K and 300 K <cit.>, two types of structures were created due to the present lack of knowledge of the internal positions of As and Fe; one type was obtained by fixing the z_Feand z_As to their values known for the 300 K structure (z_Fe = 0.2319; As:z_Ca = 0.1235 and z_K = 0.3415) and the volume adapted to the experimentaldata at each temperature. This procedure is equivalent to a uniform stretching of the crystalwhen going to higher temperatures. The second type of crystal structures was obtained by fixing the volume given at each temperature and relaxing the internal atomic positions.Magnetic fluctuations are taken into account -in a first approximation- by imposing different types of frozen magneticorder.This approach has been successfully applied to the recently discovered half-collapse transition in CaKFe_4As_4 where a specific type of spin configuration reproduces the structural behavior under external pressure <cit.>. In the present study, we analyze the effect of such (fluctuating) spin configurations on the electric field gradients (EFG) V_ ZZ on As sites.The latterwere calculated fully relativisticallyin the generalized-gradient approximation (GGA) using the projector-augmented wave pseudopotentialmethod of the Vienna Ab initio Simulation Package(VASP) <cit.>.The integration over the irreducible Brillouin zone was realized on the Γ-centered(10× 10× 10) k-mesh. The V_ ZZ values are converted to the quadrupole frequenciesusing the relation ν_ Q = eQV_ ZZ/2h√(1 + η^2/3), discussed already in Sec. III A.For clarity, we first consider here in detail the spin-vortex hedgehog phase <cit.> and compareits behavior to the purely non-magnetic case. This type of order has one of the lowest energies amongthe configurations preserving the C_4 symmetry and has been determined already to play a role forpressure-induced structural collapse transitions in <cit.>, as well as in Co and Ni-doped CaK1144 where a hedgehog long range order has been identified <cit.>. Calculations of ν_ Q in the hedgehog phase for both sets of structures show the same temperature trend as observed in our measurements [compare Fig. <ref>(b) with Fig. <ref>(b)].The first set of structures [z_As fixed structures,solid lines in Fig. <ref>(b)] indicate a significantly non-zero splitting between the two As frequencies at low temperature [Δν_ Q = 2.5 MHz at 6 K and ν_ Q(Ca) > ν_ Q(K)],while the splitting is almost zero in the room-temperature structure [ν_ Q(Ca) ≈ν_ Q(K)].Our low-temperature estimate ofΔν_ Q is in a good agreement with the experimental result Δν_ Q = 2.5 MHz,although both calculated frequencies are ∼2.5 MHz lower than in experiment. The second set of structures (z_As relaxed structures, see Table I)show that relaxationof the internal positions induces slight changes of the z_As for As(1) and As(2) whichtranslates into a shortening of the As-As bonds across the Ca layer and their stretching across the K layer.Accordingly, in the optimized structures, the Fe magnetic moment is slightly diminished compared to the non-relaxed structures which is correlated with the overall smaller Fe-As distances (Table I).As a function of temperature, the relaxed Fe-As distances barely change, which is in contrast to thecase of fixed z_Aswhere these distances decrease upon cooling. Upon structure optimization,ν_ Q for the Ca layer is almost uniformly increased by ∼1.4 – 1.8 MHz at all temperatures[compare dashed and solid lines for As(2) in Fig. <ref>(b)], while ν_ Q for the K layer [As(1)] is much less affected. The effect of z_As optimizationon the NQR properties can be interpreted in terms of the Fe moments which change by only0.04 μ_B with temperature, while they increase by 0.11 μ_B and have, in general, a larger magnitude when the z_As is fixed at all temperatures,as highlighted in Table I. Furthermore, the Fe-As and As-As bonding certainly affects the sizeand ordering of ν_ Qs, but it is not the only determining factor.Illustrative of the role of magnetic degrees of freedom is a comparison with the results for the truly non-magnetic state with zero Fe local moments [Fig. <ref>(c)].The order of the quadrupole frequencies is reversed leading to ν_ Q(Ca) < ν_ Q(K)and no crossing is observed at higher temperatures, which contradicts the experimental observations. Optimizing z_Asfor this non-magnetic state restores the correct orderν_ Q(Ca) > ν_ Q(K) but the splitting Δν_ Q is severelyoverestimated compared to the spin-polarized calculations and experiment. In addition, one of the quadrupole frequencies is significantly underestimated by a factorof two compared to the measured values of ν_ Q.Finally, from plot (d) in Fig. <ref>, we see that the stripe configuration provides similar values of the quadrupole frequencies as the hedgehog order. The stripe order is close in energy to the hedgehog order, which is discussed in detail in Ref. Meier2017. The quadrupole frequencies resulting from the magnetic fluctuations may be approximated by an average of the data shown in Figs. <ref> (b) and (d). Such average would stay in good agreement with the measured ν_ Q.These results show that both temperature-dependent structural features, non-zero Fe momentand the type of magnetic fluctuations shape the temperature evolution of NQR in .In this respect, different temperature behavior of the quadrupole frequencies for the two crystallographically inequivalent As sites might indicate the presence of C_4-type magnetic fluctuations competing with the usual stripe fluctuations, as suggested by the first-principles analysis. Additional calculations (not shown here) suggest that the splitting between the quadrupole frequencies of As(1) and As(2) is sensitive topressure, especially near the half-collapse transition around 4 GPa <cit.>.Finally it is noted that the origin of the opposite T dependence of ν_ Q in CaFe_2As_2 and KFe_2As_2 is still an open question. It is interesting to see that our approach based on the first-principles calculation explains this behavior, however disentangling the details of the various trends requires further investigations that are beyond the scope of the present study. §.§ C. ^75As spin-lattice relaxation rate 1/T_1In order toinvestigate magnetic fluctuations in CaK1144, we measured 1/T_1 at various temperatures. Figure <ref> shows the T dependences of 1/T_1 of both As sites for magnetic field directions, H ∥ c axis and H ∥ ab plane. 1/T_1 of the As(2) sites is nearly T independent above T^* ∼ 130 K for both H directions.Below T^*, 1/T_1 decreases with a power law T dependence of 1/T_1 ∝ T^0.6 as shown by the black line in Fig. <ref>.Similar T dependenceof ^75As 1/T_1 has been reported in AFe_2As_2 (A = K, Rb, Cs) where 1/T_1 increases with a power law 1/T_1 ∝ T^0.75 and levels off above T^* ∼ 165 K, 125 K, and 85 K for A = K, Rb, and Cs, respectively <cit.>.Such a characteristic T dependence of 1/T_1 is often observed in heavy fermion systems where the exponent strongly depends on the type of quantum criticality nearby, such as 1 for URu_2Si_2 <cit.>, 0.25 for CeCoIn_5 <cit.> and 0 for YbRh_2Si_2 <cit.>.As for the As(1) sites, a similar T dependence is also observed, but the exponent is found to be ∼ 0.2 which is smaller than 0.6 for the As(2) sites.Since the As(1) and As(2) are expected to probe the same spin dynamics from the Fe layers, one naively expects a similar T dependence of 1/T_1 for both sites, although the magnitude can be different due to different hyperfine coupling constants. The results suggest that the Fe spins produce different magnetic fluctuations atthe As(1) and As(2) sites in CaK1144 and a possible origin of the different T dependence of 1/T_1 is discussed later.In the SC state below T_ c ∼ 33 K, we attempted to measure 1/T_1 at the As(1) site for H ∥c axis.As described above, however, we could not distinguish the two As sites due to the broadening of the spectra.Therefore, 1/T_1 is measured at the peak position of the spectrum at low T.Because of the overlap of the spectra, we found two T_1 components: short (T_ 1S) and long (T_ 1L) relaxation times, which could be attributed to T_1 for the As(2) and As(1) sites, respectively. Just below T_ c, both 1/T_ 1L and 1/T_ 1Sdecrease rapidly without any coherence peak, consistent with an s^± model.With further decreaseof T,both 1/T_ 1L and 1/T_ 1S showbroad humps around ∼ 15 K and exponential decreases at lower T, which can be clearly seen in the semi-log plot of 1/T_1 vs. 1/T shown in the inset of Fig. <ref>. Similar behavior of 1/T_1 in SC state has been observed in LaFeAsO_1-xF_x <cit.> and K-doped BaFe_2As_2 superconductors <cit.>. These results suggest a multiplefully-gapped SC state, consistent with previous measurements.In fact, the T dependence of 1/T_1 is roughly reproduced by a two full gap model with superconducting gap parameters Δ = 2.0 ± 0.4 meV and 20 ± 9 meV for 1/T_ 1L and Δ = 1.9 ± 0.4 meV and 16 ± 8 meV for 1/T_ 1S as shown by the solid lines. The smaller gap ∼ 2.0meV is not far from 2.5 meV obtained from the μSR measurements <cit.> and in good agreement with 1-4 meV from the TDR and STM measurements <cit.>.On the other hand, the larger gaps of 20 ± 9 and 16 ± 8 meVseem to be greater than 6-9 meV from the other measurements <cit.>. However, it should be noted that our estimates of the larger gap have a large uncertainty due to the poor statistics accuracy and relatively narrow T range for the fit, therefore it should be regarded as tentative estimate. Now we discuss magnetic fluctuations in the normal state. Figures <ref> (a) and (b) show the T dependences of 1/T_1T for H ∥ c and H ∥ abin CaK1144.For comparison, we also plot the T_1 data ofKFe_2As_2 (T_ c = 3.5 K) <cit.>,Ba_0.45K_0.55Fe_2As_2(T_ c = 34 K) <cit.> and Ba_0.68K_0.32Fe_2As_2(T_ c = 38.5 K) <cit.>. 1/T_1T at the As(1) site is comparable to those of Ba_0.45K_0.55Fe_2As_2 and Ba_0.68K_0.32Fe_2As_2, while for the As(2) sites 1/T_1T is enhancedlarger than the KFe_2As_2 case.All 1/T_1T show a monotonic increase down to T_ c with decreasing T except forBa_0.45K_0.55Fe_2As_2 where 1/T_1 decreases at low Tshowing gap-like behavior <cit.>.In general, 1/T_1T is related to the dynamical magnetic susceptibility as 1/T_1T=2γ^2_Nk_ B/N_ A^2∑_q⃗|A(q⃗)|^2χ^''(q⃗, ω_0)/ω_0, where A(q⃗) is the wave-vector q⃗ dependent form factor and χ^''(q⃗, ω_0) is the imaginary part of the dynamic susceptibility at the Larmor frequency ω_0. Although the exponents of the power-law behavior of 1/T_1 for As(1) and As(2) are different, we canattribute the increase of 1/T_1T to the growth of AFM spin fluctuations at q≠0since both K_c and K_ab, reflecting χ^'(q=0, ω_0=0), are nearly independent of T.In order to see the nature of spin correlations, we plotted the ratio of 1/T_1 for the two field directions, r ≡ T_1,c/T_1, ab.According to previous NMR studies performed on Fe pnictides and related materials <cit.>theratio rdepends on AFM spin correlation modes as r ={[ 0.5 + (S_ab/S_c) ^2 q = (π,0) (0, π);0.5 q = (π,π);] . whereS_α is the amplitude of the spin fluctuation spectral density at the NMR frequency along the α direction.The wavevectors are given in the single-iron Brillouin zone notation. As plotted in Fig. <ref>(a), the ratios for both the As(1) and As(2) sites are close to r ∼ 1.5. This indicates that the spin fluctuations in CaKFe_4As_4 are characterized by the AFM spin fluctuations withq = (π, 0) or (0, π) and S_c ∼ S_ab indicatingthe isotropic spin fluctuations. In order to confirm the AFM spin correlations, we have calculated χ(q) <cit.>using the full-potential linearized augmented plane wave method <cit.> with a generalized gradient approximation <cit.>. The calculated χ(q) is shown in the inset of Fig. <ref>(a), which actually exhibits the AFM spin correlations where a peak aroundthe wavevector q = (π, 0) or (0, π)is observed with a small structure. A similar structure in χ(q) was reported in CaFe_2As_2 <cit.>.In most of the Fe pnictide SCs, the AFM fluctuations are described in terms of the stripe-type spin fluctuations.Recently another possible AFM fluctuation withq vectors [(π, 0) and (0, π)] has been proposed in the spin vortex crystalstate where Fe spins are non-collinear and form spin vortices staggered across the iron square lattices <cit.>.Since the stripe- and spin vortex-type states produce the AFM spin correlations with the same q, in general, it would be difficult to distinguish between the two different spin correlations from the T_1 measurements.However, provides us a special occasion to distinguish between them.In the hedgehog vortex ordered state, the internal fields at As(1) is calculated to be finite along the c axis while the internal field at As(2) is zerodue to a cancellation originating from the characteristic spin structure <cit.>.Therefore, one expects that 1/T_1T for As(1) is more enhanced than that for As(2) if the AFM spin fluctuations originate from the spin vortex hedgehog-type spin correlations.On the other hand, in the case of stripe-type AFM fluctuations, the temperature dependence of 1/T_1T for As(1) should scale to that of 1/T_1T for As(2) sites since there is no cancellation of the internal field at both As sites<cit.>. 1/T_1T for As(1) divided by 1/T_1T for As(2) for H ∥ c and H ∥ ab are shown in Fig. <ref>(b) where clear enhancements are observed below ∼ 150 K.Above ∼ 150 K, the ratios of T_1[ As(2)]/T_1[ As(1)]show a nearly temperature independent value of ∼ 0.35 which could be determined by the different hyperfine coupling constants for the As(1) and As(2) sites.The increases below ∼ 150 K indicatethat the As(1) sites pick up the AFMspin fluctuations more strongly than the As(2) sites, consistent with the case of hedgehog-type spin fluctuations. Our ab initio DFT total energy calculationsshow that both stripe order and spin-vortex order are energetically competing in CaKFe_4As_4 and the temperature dependence of ν_ Q, as presented above, is well described by assuming a hedgehog spin vortex order.Furthermore,a hedgehog spin vortex order has been recently observed in Co-doped and Ni-doped CaKFe_4As_4 <cit.>.Therefore, our results strongly indicate the existence of the spin vortex hedgehog-type spin correlations in the paramagnetic state.Further experimental and theoretical studies are important to elucidate more details of the characteristic of the hedgehog-type spin correlations and also its relationship to superconductivity in CaKFe_4As_4 and the carrier-doped CaKFe_4As_4. § IV. SUMMARYIn conclusion, we have carried out ^75As NMR measurements in the single crystalline CaKFe_4As_4.The observation of two distinct ^75As NMR lines indicates that K and Ca ions are not randomlydistributed but are well ordered forming the Ca and K layers without a significant site inversion.We found that ν_ Q of the As(1) and As(2) sites show an opposite T dependence.Based on our ab initio density functional theory calculations, this behavior might originate from the presenceof spin vortex hedgehog-type spin fluctuations which represent a superposition of (π, 0) and (0, π)stripe fluctuations. We find that the calculated quadrupole frequencies are comparably affectedby the Fe-As and As-As bonding, non-zero Fe moments and the underlying magnetic order whichsimulates the spin fluctuations to a first approximation.In the normal state, we also observe evidenceof the enhancement of isotropic AFM spin fluctuations withwavevector q = (π, 0) or (0, π) inthe T dependences of 1/T_1T and K.It is suggested that the different T dependence of 1/T_1 for the As(1) and As(2) sites is explained by the peculiar spin fluctuations due to the spin vortex hedgehog-type spin correlations.In the SC state, we observeda sudden decrease in K, indicating spin-singlet Cooper pairs. 1/T_1 shows a rapid decreasebelow T_ c without showinga Hebel-Slichter peak and decreases exponentially at lowtemperatures. These results indicate an s^± nodeless two-gap SC state. § V. ACKNOWLEDGMENTSThe authors would like to thank Khusboo Rana for his assistance with NQR experiments.We also thank Paul Wiecki, Rafael Fernandes, Peter Orth, Igor Mazin and Andreas Kreyssig for helpful discussions.The research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.WRM was supported by the Gordon and Betty Moore Foundations EPiQS Initiative through Grant GBMF4411. VB and RV are supported by the Deutsche Forschungsgemeinschaft through grant SFB/TRR 49 and the computer time was allotted by the centre for supercomputing (CSC) in Frankfurt. * present address: Department of Chemistry, Princeton University, Princeton, NJ 08544, USA99 Kamihara2008 Y. Kamihara, T. Watanabe, M. Hirano and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008). Rotter2008 M. Rotter, M. Tegel, and D. Johrendt, Phys. Rev. Lett. 101 107006 (2008). Johnston2010 D. C. Johnston, Adv.Phys. 59, 803 (2010). Canfield2010 P. C. Canfield and S. L. Bud'ko, Annu. Rev. Condens. Matter Phys. 1, 27 (2010). Ni2008 N. Ni, S. Nandi, A. Kreyssig, A. I. Goldman, E. D. Mun, S. L.Bud'ko, and P. C. Canfield, Phys. Rev. B 78, 014523 (2008).Chu2009 C.W. Chu, F. Chen, M. Gooch, A.M. Guloy, B. Lorenz, B. Lv, K. Sasmal, Z.J. Tang, J.H. Tapp and Y.Y. Xue,Physica C469, 326 (2009). Stewart2011 G. R. Stewart, Rev. Mod. Phys. 83, 1589 (2011).Iyo2016 A. Iyo, K. Kawashima, T. Kinjo, T. Nishio, S. Ishida, H. Fujihisa, Y. Gotoh, K. Kihou, H. Eisaki, and Y. Yoshida, J. Am. Chem. Soc. 138, 3410 (2016), Meier2016 W. R. Meier, T. Kong, U. S. Kaluarachchi, V. Taufour, N. H. Jo, G. Drachuck, A. E. Böhmer, S. M. Saunders, A. Sapkota and A. Kreyssig, M. A. Tanatar, R. Prozorov, A. I. Goldman, F. F. Balakirev, A. Gurevich, S. L. Bud'ko, and P. C. Canfield, Phys. Rev. B 94,064501 (2016). uSR2017 P. K. Biswas, A. Iyo, Y. Yoshida, H. Eisaki, K. Kawashima and A. D. Hillier, Phys. Rev. B95,140505 (2017). Cho2017 K. Cho, A. Fente, S. Teknowijoyo, M. A. Tanatar, K. R. Joshi, N. M. Nusran, T. Kong, W. R. Meier, U. Kaluarachchi, I. Guillamón, H. Suderow, S. L. Bud'ko, P. C. Canfield and R. Prozorov, Phys. Rev. B 95, 100502 (2017). ARPES2016 D. Mou, T. Kong, W. R. Meier, F. Lochner, L.-L. Wang, Q. Lin, Y. Wu, S. L. Bud'ko, I. Eremin, D. D. Johnson, P. C. Canfield and A. Kaminski, Phys. Rev. Lett. 117, 277001 (2016). Ishida2009 K. Ishida, Y. Nakai, and H. Hosono, J. Phys. Soc. Jpn. 78 062001 (2009).Paglione2010 J. Paglione and R. L. Greene, Nat. Phys.,6,645 (2010). PaulPRL P. Wiecki, B. Roy, D. C. Johnston, S. L. Bud'ko, P. C. Canfield, and Y. Furukawa, Phys. Rev. Lett. 115, 137001 (2015). JeanPRB J. Cui, P. Wiecki, S. Ran, S. L. Bud'ko, P. C. Canfield, and Y. Furukawa, Phys. Rev. B 94, 174512 (2016).Furukawa2014 Y. Furukawa, B. Roy, S. Ran, S. L. Bud'ko, and P. C. Canfield, Phys. Rev. B 89, 121109 (2014).Wiecki_unpublish P. Wiecki, P. C. Canfield, Y. Furukawa, unpublished.Hirano2012 M. Hirano, Y. Yamada, T. Saito, R. Nagashima, T. Konishi, T. Toriyama, Y. Ohta, H. Fukazawa, Y. Kohori, Y. Furukawa, K. Kihou, C-H Lee, A. Iyo and H. Eisaki, J. Phys. Soc. Jpn. 81, 054704(2012). Cui2015 J. Cui, B. Roy, M. A. Tanatar, S. Ran, S. L. Bud'ko, R. Prozorov, P. C. Canfield and Y. Furukawa, Phys. Rev. B92, 184504 (2015). Kaluarachchi2017 U. S. Kaluarachchi, V. Taufour, A. Sapkota, V. Borisov, T. Kong, W. R. Meier, K. Kothapalli, B. G. Ueland, A. Kreyssig, R. Valentí, R. J. McQueeney, A. I. Goldman, S. L. Bud'ko, P. C. Canfield, arXiv:1702.06622.Kresse1993PRBG. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). Kresse1996PRBG. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). Kresse1996G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996). Bloechl1994P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). Kresse1999G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).Meier2017 W. R. Meier, Q.-P. Ding, A. Kreyssig, S. L. Bud'ko, A. Sapkota, K. Kothapalli, V. Borisov, R. Valentí, C. D. Batista, P. P. Orth, R. M. Fernandes, A. I. Goldman, Y. Furukawa, A. E. Böhmer, P. C. Canfield,arXiv:1706.01067. Kondo2016 Y. P. Wu, D. Zhao, A. F. Wang, N. Z. Wang, Z. J. Xiang, X. G. Luo, T. Wu and X. H. Chen, Phys. Rev. Lett.116,147001 (2016). Kohara1986 T. Kohara, Y. Kohori, K. Asayama, Y. Kitaoka, M. B. Maple, and M. S. Torikachvili, Solid State Commun. 59, 603 (1986). Kohori2001 Y. Kohori, Y. Yamato, Y. Iwamoto, T. Kohara, E. D. Bauer, M. B. Maple, and J. L. Sarrao, Phys. Rev. B 64, 134526 (2001). Ishida2002 K. Ishida, K. Okamoto, Y. Kawasaki, Y. Kitaoka, O. Trovarelli, C. Geibel, and F. Steglich, Phys. Rev. Lett. 89, 107202 (2002).Oka2012 T. Oka, Z. Li, S. Kawasaki, G. F. Chen, N. L. Wang, and G.-q. Zheng, Phys. Rev. Lett. 108, 047001 (2012). Li2011 Z. Li, D. L. Sun, C. T. Lin, Y. H. Su, J. P. Hu and G.-q. Zheng, Phys. Rev. B83,140506 (2011). Matano2009 K. Matano, Z. Li, G. L. Sun, D. L. Sun, C. T. Lin, M. Ichioka and G.-q. Zheng, EPL 87, 27012 (2009). KitagawaSrFe2As2 K. Kitagawa, N. Katayama, K. Ohgushi, and M. Takigawa, J. Phys. Soc. Jpn. 78, 063706 (2009). KitagawaAFSF S. Kitagawa, Y. Nakai, T. Iye, K. Ishida, Y. Kamihara, M. Hirano, and H. Hosono, Phys. Rev. B 81, 212502 (2010).PandeySrCo2As2 A.Pandey, D. G. Quirinale, W. Jayasekara, A. Sapkota, M. G. Kim, R. S. Dhaka, Y. Lee, T. W. Heitmann, P. W. Stephens, V. Ogloblichev, A. Kreyssig, R. J. McQueeney, A. I. Goldman, A. Kaminski, B. N. Harmon, Y. Furukawa, and D. C. Johnston, Phys. Rev. B 88, 014526 (2013).YongbinFor obtaining self-consistent charge density, we employed R_ MTk_ max=8.0 and muffin-tin radii R_ MT=2.0 a.u. for all atoms. We selected 364 k-points in the irreducible Brillouin zone (BZ) and calculated the band structure with 67,626 k-points which was resulted by 100*100*100 division of whole BZ.Blaha2001 P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnick, and J. Luitz,WIEN2K, an Augmented Plane Wave + Local Orbitals Program for Calculation Crystal Properties (Karlheinz Schwarz, Technical Universität Wien, Austria), 2001. Perdew1996 J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).Goldman2009A. I. Goldman, A. Kreyssig, K. Prokes̆, D. K. Pratt, D. N. Argyriou, J. W. Lynn, S. Nandi, S. A. J. Kimber, Y. Chen, Y. B. Lee, G. D. Samolyuk, J. B. Leão, S. J. Poulton, S. L. Bud'ko, N. Ni, P. C. Canfield, B. N. Harmon, and R. J. McQueeney, Phys. Rev. B 79, 024513 (2009). Fernandes2016 R. M. Fernandes, S. A. Kivelson, E. Berg, Phys. Rev. B 93, 014511 (2016). | http://arxiv.org/abs/1706.09075v2 | {
"authors": [
"J. Cui",
"Q. -P. Ding",
"W. R. Meier",
"A. E. Böhmer",
"T. Kong",
"V. Borisov",
"Y. Lee",
"S. L. Bud'ko",
"R. Valentí",
"P. C. Canfield",
"Y. Furukawa"
],
"categories": [
"cond-mat.supr-con",
"cond-mat.str-el"
],
"primary_category": "cond-mat.supr-con",
"published": "20170627233636",
"title": "Magnetic fluctuations and superconducting properties of CaKFe4As4 studied by 75As NMR"
} |
Model reduction of Fokker–Planck and quantum Liouville equations]Modelreduction of controlled Fokker–Planck and Liouville–von Neumann equationsP. Benner]Peter Benner Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical SystemsSandtorstr. 1,D-39106 Magdeburg, Germany [email protected]. Breiten]Tobias Breiten Institut für Mathematik, Karl-Franzens-UniversitätHeinrichstr.36/III, A-8010 Graz, Austria [email protected]. Hartmann]Carsten Hartmann Institut für Mathematik, Brandenburgische TechnischeUniversität Konrad-Wachsmann-Allee 1, D-03046 Cottbus, Germany [email protected]. Schmidt]Burkhard Schmidt Institut für Mathematik, Freie Universität Berlin Arnimallee 6,D-14195 Berlin, Germany [email protected] Model reduction methods for bilinear control systems are compared by meansof practical examples of Liouville–von Neumann and Fokker–Planck type.Methods based on balancing generalized system Gramians and on minimizing anℋ_2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of themethods. Structure and stability preservation are investigated, and thecompetitiveness of the approaches is shown for practically relevant,large-scale examples.[ [ December 30, 2023 =====================§ INTRODUCTION Due to the growing ability to accurately manipulate single molecules byspectroscopic techniques, numerical methods for the control of molecularsystemshave recently attracted a lot of attention<cit.>. Key applications involve probing ofmechanical properties of biomolecules by force microscopy and optical tweezers<cit.>, or the control of chemical reaction dynamics bytemporally shaped femtosecond laser pulses in femtochemistry<cit.>.A key feature of these small systems is that they are open systems, in thatthey are subject to noise and dissipation induced by the interaction with theirenvironment, as a consequence of which the dynamics are inherently random andthe description is on the level of probability distributions or measures ratherthan trajectories <cit.>.Depending on whether or not quantum effects play a role, the evolution of thecorresponding probability distributions is governed by parabolic partialdifferential equations of either Liouville–von Neumann or Fokker–Planck type.The fact that the dynamics are controlled implies that the equations arebilinear as the control acts as an advection term that is coupled linearly tothe probability distribution, but the main computational bottleneck clearly isthat the equations, in spatially semi-discretized form, are high-dimensionalwhich explains why model reduction is an issue; for example, in catalysis,optimal shaping of laser pulses requires the iterated integration of thedissipative Liouville–von Neumann (LvN) equation for reduced quantummechanicaldensity matrices, the spatial dimension of which grows quadratically with thenumber of quantum states involved <cit.>; cf. <cit.>.Many nonlinear control systems can be represented as bilinear systems by asuitable change of coordinates (as well as linear parametric systems), and ittherefore does not come as a surprise that model reduction of bilinear controlsystems has recently been a field of intense research; see<cit.> and the references therein. In recent years,variousmodel reduction techniques that were only available for linear systems havebeenextended to the bilinear case, among which are Krylov subspace techniques<cit.>, interpolation-basedapproaches <cit.>, balanced modelreduction <cit.>, empirical POD<cit.>, or ℋ_2-optimal modelreduction <cit.>.The downside of many available methods is their lack of structure preservation,most importantly, regarding asymptotic stability. In our case, positivity is anissue too, as we are dealing with probability distributions.In this paper we compare two different model reduction techniques thatrepresentdifferent philosophies of model order reduction, with the focus being onpractical computations and numerical tests rather than a theoretical analysis.The first approach is based on the interpolation of the Volterra seriesrepresentation of the system's transfer function and gives a localℋ_2-optimal approximation, because the interpolation is chosen sothat the system satisfies the necessary ℋ_2-optimality conditionsupon convergence of the algorithm; see <cit.> for details. Thesecond approach is based on balancing the controllable and observablesubspaces, and exploits the properties of the underlying dynamical system in that it usesthe properties of the controllability and observability Gramians to identifysuitable small parameters that are sent to 0 to yield a reduced-order system;for details, we refer to <cit.>. Both methods require the solution of large-scale matrix Sylvester or Lyapunovequations. While the computational effort of balanced model reduction isessentially determined by the solution of two generalized Lyapunov equationsforcontrollability and observability Gramians, the effort of theℋ_2-optimal interpolation method is mainly due to the solution oftwo generalized Sylvester equations in each step of the bilinear iterativerational Krylov algorithm (B-IRKA). We stress that both generalized Lyapunov orSylvester equations can be solved iteratively at comparable numerical cost (fora given accuracy), but they all require the dynamics of the uncontrolled systemto be asymptotically stable <cit.>. However, as both thedissipative LvN and Fokker-Planck operators have a simple eigenvalue zero,stability has to be enforced before solving Lyapunov or Sylvester equations,andin this paper we systematically compared stabilization techniques for bothapproaches. The outline of the article is as follows: In Section <ref> we brieflydiscuss the basic properties of bilinear systems and set the notation for theremainder of the article. Model reduction by ℋ_2-norm minimizationand balancing are reviewed in Sections <ref> and <ref>,along with some details regarding the numerical implementation for the specificapplications considered in this paper in Section <ref>.Finally,inSection <ref> we study model reduction of the Fokker-Planck equationcomparing balancing and ℋ_2-norm minimization, and in Section<ref> we carry out a similar study for the dissipative Liouville–vonNeumann equation. We discuss our observations in Section <ref>.The article contains an appendix, Appendix <ref>, that recordssome technical lemmas related to the asymptotic stability of bilinear systems.§ BILINEAR CONTROL SYSTEMS We start by setting the notation that will be used throughout this article. Letx(t)∈^n begovernedby the time-inhomogeneous differential equationx/ t = A x + ∑_k=1^m(N_k x+b_k)u_k , x(0)=x_0 ,with coefficients A, N_k∈^n× n, b_k∈^n andu=(u_1,…, u_m)^T being a vector of bounded measurable controlsu_i(t)∈ U⊂. We assume that not all state variables x arerelevant or observable, so we augment(<ref>) by a linear outputequation y =C x ,with C∈^l× n, l≤ n. The systems of equations(<ref>)–(<ref>) is called a bilinear control system withinputs u(t)∈ U^m⊂^m and outputs y(t)∈^l.As is well-known, see e.g. <cit.>, an explicit outputrepresentation for (<ref>) can be obtained by means of successiveapproximations. The resulting so-called Volterra series is given asy(t)= ∑_k=1^∞∫_0^∞⋯∫_0^∞∑_ℓ_1,…,ℓ_k=1^m C e^As_k N_ℓ_1 e^As_k-1 N_ℓ_2⋯ e^As_2N_ℓ_k-1e^As_1 b_ℓ_k× u_ℓ_1(t-s_k)u_ℓ_2(t-s_k-s_k-1) ⋯u_ℓ_k(t- ∑_j=1^k s_j) s_1⋯ s_k.Moreover, based on a multivariate Laplace transform of these integrands, thesystem can alternatively be analyzed in a generalized frequency domain by meansof generalized transfer functions. Since this will not be essential for theresults presented here, we refrain from a more detailed discussion and referto, e.g., <cit.>.§.§ Reduced-order modelsWe seek coefficients Â,N̂_k∈^d× d,b̂_k∈^d and Ĉ∈^l× d with d≪ n such that ξ/ t=Âξ + ∑_k=1^m(N̂_kξ + b̂_k)u_k ,ξ(0) = ξ_0 , ŷ= Ĉξhas an input-output behavior that is similar to (<ref>)–(<ref>).In otherwords, we seek a reduced-order model with the property that for anyadmissible control input u (to be defined below), the error in the outputsignal,δ(t) = ŷ(t)-y(t) ,is small, relative to u (in some norm) and uniformly on bounded timeintervals.As will be outlined below, both model reduction schemes considered in this paperare closely related to thesolutions of the following adjoint pair of generalized Lyapunov equations:AP+ P A^* + ∑_k=1^mN_k P N_k^* + BB^* = 0and A^*Q + Q A + ∑_k=1^mN^*_k Q N_k + C^*C = 0 ,where, in the first equation, we have introduced the shorthandB=(b_1,…,b_m)∈^n× m. The Hermitian and positivesemi-definite matrices P,Q∈^n× n are called thecontrollability and observability Gramians associated with(<ref>)–(<ref>)—assuming well-posedness of the Lyapunovequations and hence existence and uniqueness of P and Q. The relevance ofthe Gramians for model reduction is related to the fact that the nullspace ofthe controllability Gramian contains only states that cannot be reached by anybounded measurable control and that the system will not produce any outputsignal, if the dynamics is initialized in the nullspace of the observabilityGramian <cit.>; as a consequence one can eliminate states thatbelong to (P)∩(Q) without affecting the input-output behavior of(<ref>)–(<ref>); cf. <cit.>.§.§ Standing assumptions The following assumptions will be used throughout to guarantee existence anduniqueness of the solutions to the generalized Lyapunov equations (Assumption1)and existence and uniqueness of the solution of the bilinear system(<ref>) for all t≥ 0 (Assumptions 2 and 3):Assumption 1: There exists constants λ,μ>0, such that exp(A t)≤λexp(-μ t) and λ^2/2μ∑_k=1^m N_k^2 < 1 ,where ·=·_2 is the matrix 2-norm that is induced by theEuclidean norm |·|. Assumption 2: The bilinear system (<ref>)–(<ref>) isbounded-input-bounded-output (BIBO) stable, i.e., there exists M<∞, suchthat for any input u with u_∞ = sup_t∈[0,∞)|u(t)| ≤ Mthe output y(t) is uniformly bounded.Assumption 3: The admissible controls u[0,∞)→U^m⊂^m are continuous, bounded and square integrable, i.e., u∈_b([0,∞),U^m) withu_2 = (∫_0^∞ |u(t)|^2t )^1/2 <∞ .Specifically, we require that the admissible controls are uniformly bounded by M < μ/λ∑_k=1^m N_k ,with λ,μ as in Assumption 1, which by BIBO stability (Assumptions 2)implies that the output y(t) is bounded for all t≥ 0(cf. <cit.>). § ℋ_2 OPTIMAL MODEL REDUCTION OF BILINEARSYSTEMS In this section, we recall some existing results onℋ_2-optimal model order reduction for bilinear systems. Fora more detailed presentation, see <cit.>. For a better understanding of the subsequent concepts, let us briefly focus on the linear case, i.e., N_k=0 in (<ref>). Here, the Volterra seriesrepresentation (<ref>) simplifies to y(t) = ∫_0^∞Ce^AsBu(t-s) s. If the input signal is a Dirac mass at 0, weobtain the impulse response h(t)=Ce^AtB. The ℋ_2-normfor linear systems now is simply defined as the L_2-norm of the impulseresponse, i.e.,h_L^2(0,∞;^m)^2=∫_0^∞tr(B^*e^A^*tC^*Ce^AtB)t. Based on the latter definition and the Volterra series, in <cit.>, the ℋ_2-norm has been generalized for bilinearsystems as follows.Let Σ=(A,N_1,…,N_k,B,C) denote a bilinear system as in(<ref>). We then define its ℋ_2-norm byΣ_ℋ_2^2 = tr(∑_k=1^∞∫_0^∞⋯∫_0^∞∑_ℓ_1,…,ℓ_k=1^mg_k^(ℓ_1,…,ℓ_k) (g_k^(ℓ_1,…,ℓ_k))^*s_1⋯s_k), with g_k^(ℓ_1,…,ℓ_k) (s_1,…,s_k) = Ce^As_kN_ℓ_1e^As_k-1 N_ℓ_2⋯ e^As_1 b_ℓ_k.Obviously, for a bilinear system having a finite ℋ_2-norm, it isrequired that the system is stable in the linear sense, i.e., A has onlyeigenvalues in _-. Moreover, the matrices N_k have to be sufficientlybounded. From <cit.>, let us recall that Assumption 1 ensures thatthe bilinear system under consideration has a finite ℋ_2-norm,which, moreover, can be computed by means of the solution P and Q of thegeneralized Lyapunov equations (<ref>) and (<ref>), respectively. Inparticular, we have thatΣ_ℋ_2^2 = tr(CPC^*) = tr(B^*QB).Given a fixed system dimension lthe goal of ℋ_2-optimal model order reduction now is to construct areduced-order bilinear system Σ̃ such that Σ-Σ̃_ℋ_2=min_(Σ̂) =l Σ̂ stableΣ-Σ̃_ℋ_2.Unfortunately, already in the linear case this is a highly nonconvexminimization problem such that finding a global minimizer is out of reach.Instead, we aim at constructing Σ̃ such that first-ordernecessary conditions for ℋ_2-optimality are fulfilled. In<cit.>, the optimality conditions from<cit.> are extended to the bilinear case. More precisely,it isshown that anℋ_2-optimal reduced-order model is defined by a Petrov-Galerkinprojection of the original model. Given a reduced-order system Σ̂,let us consider the associated error systemA_e = [ A 0; 0  ],N_k,e = [N_k0;0 N̂_k ],B_e = [B; B̂ ],C_e = [ C - Ĉ ],as well as the generalized Lyapunov equations associated with itA_e P_e + P_e A_e^* + ∑_k=1^m N_k,e P_e N_k,e^* + B_e B_e^*= 0, A_e^* Q_e + Q_e A_e + ∑_k=1^m N_k,e^* Q_e N_k,e + C_e^* C_e= 0.Assuming the partitioning P_e = [ P X; X^*P̂ ], Q_e = [ Q Y; Y^*Q̂ ],the first-order necessary optimality conditions now areY^* A X + Q̂^*ÂP̂ = 0,Y^* N_k X + Q̂^*N̂_k P̂ = 0, Y^* B + Q̂^*B̂ = 0,CX -ĈP̂ =0.In <cit.> the authors have proposed a gradient flow technique toconstruct a reduced-order model satisfying (<ref>). Since here we areinterested in computations for large-scale systems for which this technique isnot feasible, we instead use the iterative method from <cit.>. Themain idea is inspired by the iterative rational Krylov algorithm from<cit.> and relieson solving generalized Sylvester equations of the formAX + XÂ^* + ∑_k=1^m N_k X N̂_k^* + B B̂^*= 0, A^*Y + Y + ∑_k=1^m N_k^* Y N̂_k - C^* Ĉ = 0.Based on a given reduced-order model(Â_i,N̂_k,i,B̂_i,Ĉ_i), the subspaces spanned bycolumns of thesolutions X_i,Y_i ∈^n× l are used to generate an updatedreduced-order model. More precisely, given unitary matrices V_i,W_i ∈ℂ^n× l such that span(V_i)=span(X_i) andspan(W_i)=span(Y_i), we setÂ_i+1 = (W_i^*V_i)^-1W_i^*AV_i,N̂_i+1 =(W_i^*V_i)^-1W_i^*N_kV_i,B̂_i+1 = (W_i^*V_i)^-1W_i^*B , Ĉ_i+1=CV_i.This type of fixed-point iteration is repeated until thereduced-order model is numerically converged up to a prescribed tolerance. Formore details on the iteration, we also refer to<cit.>.§ BALANCED MODEL REDUCTION FOR BILINEAR SYSTEMS We shall briefly explain model reduction based on balancing controllability andobservability. To this end we assume that the generalized Gramian matricesP,Q are both Hermitian positive definite which is guaranteed by theassumptionthatthe bilinear system (<ref>)–(<ref>) is completely controllableand observable:Assumption 4: The matrix pair (A,B) is controllable, i.e., rank(B AB A^2B… A^n-1B)=n . Assumption 5: The matrix pair (A,C) is observable, i.e., rank(C^* A^*C^* A^2C^*… A^n-1C^*)=n .§.§ Singularly perturbed bilinear systems We consider a balancing transformation x↦ T^-1x under which theGramians transform according to <cit.>T^-1Q (T^-1)^*=Σ =T^*PT ,where the diagonal matrix Σ=diag(σ_1,σ_2,…,σ_n) with σ_1≥σ_2≥…≥σ_n>0 contains the real-valued Hankel singular values(HSV) of the system. Under the linear map T, the coefficients of (<ref>)–(<ref>)transform according to (A,N_k,B,C)↦ (T^-1AT, T^-1N_kT, T^-1B,CT) ,k=1,…,m .As the Hankel singular values are the square roots of the eigenvalues of theproduct QP, they are independent of the choice of coordinates. It can be shown (e.g. <cit.>) that a balancing transformation thatmakes the two Gramians Q and P equal and diagonal is given by the matrixT=Σ^-1/2 V^T R with inverse T^-1=S^TUΣ^-1/2 wherethe matrices U,V,S,R are defined by the Cholesky decompositionsP=S^TS andQ=R^TR of the twoGramians solving (<ref>) and (<ref>), and theirsingular value decomposition SR^T=UΣ V^T.Now suppose that Σ=(Σ_1,Σ_2) with Σ_1∈^d×d and Σ_2∈^(n-d)×(n-d) corresponding to the splitting ofthe system states into relevant and irrelevant states. Further assume thatΣ_2≪Σ_1 in the sense that the smallest entryof Σ_1 is much larger than the largest entry of Σ_2. The rationale of balanced model reduction is based on a continuity argument: ifthespace of the uncontrollable and unobservable states is spanned by the singularvectors corresponding to Σ_2=0, then, by continuity of the solution of(<ref>)–(<ref>) on the system's coefficients, small singularvalues should indicate hardly controllable and observable states that do notcontribute much to the input-output behavior of the system.Using the notation Σ_2=() with 0<≪ 1 and partitioningthe balanced coefficients according to the splitting into large and small HSV,then yields the following singularly perturbed system of equations (see<cit.>): z^_1/ t= Ã_11z^_1 +1/√()Ã_12z^_2 +∑_k=1^m(Ñ_k,11z^_1 + 1/√()Ñ_k,12z^_2 + b̃_k,1) u_k√() z^_2/ t= Ã_21z^_1 +1/√()Ã_22z^_2 + ∑_k=1^m(Ñ_k,21z^_1 + 1/√()Ñ_k,22z^_2 + b̃_k,2) u_ky^= C̃_1z^_1 +1/√()C̃_2z^_2Here z=T^-1x, with z=(z_1,z_2)∈^d×^n-d, denotes thebalanced state vector where the splitting into z_1, z_2 is inaccordancewith the splitting of the HSV into Σ_1 and Σ_2. The splittingof the balanced coefficients Ã=T^-1AT, Ñ_k=T^-1N_kT, b̃_k=T^-1b_k, C̃=CTinto Ã_11, Ã_12 etc. can be understood accordingly. §.§ An averaging principle for bilinear systems In order to derive reduced-order models of (<ref>)–(<ref>), weconsider the limit → 0 in (<ref>). This amounts to the limit ofvanishing small HSV Σ_2 in the original bilinear system. We suppose that Assumptions 1–5 hold for all >0. As we will show inAppendix <ref>, the results in <cit.> can be modified toshow that thematrices Ã_11 and Ã_22 are Hurwitz, andthat their eigenvaluesare bounded away from the imaginary axis. In this case, BIBO stability of thesystem together with the assumptions on the admissible controls imply thatz^_2→ 0 pointwise for all t> 0 as → 0. However, the rateat which z^_2 tends to zero and hence the limiting bilinear systemsclearly depends on the controls u, especially when u depends on . Wegive only a formal justification of the different candidate equations that canbe obtained in the limit of vanishing small HSV and refer to <cit.>for further details.§.§ Balanced truncationIf z^_2=o(√()), we expect that the first two equations in(<ref>) decouple as → 0, which implies that the limiting bilinearsystem will be of the form z_1/ t= Ã_11z_1 +∑_k=1^m(Ñ_k,11z_1 + b̃_k,1) u_ky = C̃_1z_1 .The assumption that z^_2 goes to zero faster than √() isthe basis of the traditional balanced truncation approach in which theweakly controllable and observable degrees of freedom are eliminated byprojecting the equations to the linear subspace S_1 = {(z_1,z_2)∈^n z_2=0}≃^d.The validity of the approximation for all t≥ 0 requires thatz_2^(0)=0; cf. Remark <ref> below.§.§ Singular perturbation If z^_2=(√()) the z_1, z_2 equations do notdecouple as → 0, and the limiting equation turns out to be differentfrom (<ref>). To reveal it, it is convenient to introduce scaledvariablesby z_2= √()ζ by which (<ref>) becomes z^_1/ t= Ã_11z^_1 +Ã_12ζ^ + ∑_k=1^m(Ñ_k,11z^_1 +Ñ_k,12ζ^ + b̃_k,1) u_kζ^/ t= Ã_21z^_1 +Ã_22ζ^ +∑_k=1^m(Ñ_k,21z^_1+Ñ_k,22ζ^ + b̃_k,2) u_ky^= C̃_1z^_1 + C̃_2ζ^ .Equation (<ref>) is an instance of a slow-fast system with z_1 being theslow variable and ζ=z_2/√() being fast, andfor non-pathological controls u, the averaging principle applies<cit.>. The idea of the averaging principle is to average the fastvariables in the equation for z_1 against their invariant measure, becausewheneveris sufficiently small, the fast variables relax to theirinvariant measure while the slow variables are effectively frozen, andthereforethe slow dynamics move under the average influence of the fast variables. Thisclearly requires that the convergence of the fast dynamics is sufficiently fastand independent of the initial conditions. The auxiliary fast subsystem forfrozen slow variable z_1 readsζ̃/τ = Ã_22(ζ̃+Ã_22^-1Ã_21z_1) + ∑_k=1^m(Ñ_k,22(ζ̃+ Ã_22^-1Ã_21z_1) + B̃_k,2) ũ_k ,with B̃_k,2 = (Ñ_k,21 - Ñ_k,22Ã_22^-1Ã_21z_1) + b̃_k,2 .It is obtained from (<ref>) by rescaling the equations according to τ=t/ andζ̃(τ)=ζ^(τ),ũ(τ)=u(τ) and sending → 0. Since the admissiblecontrols decay on time scales that are of order one in t (i.e. (1/)in τ), it follows that lim_τ→∞ζ̃(τ;z_1) = - Ã_22^-1Ã_21z_1 .In other words, for fixed z_1 the fast dynamics converge to the Dirac massδ_m at m=-Ã_22^-1Ã_21z_1. This can berephrased by saying that for all admissible controls and in the limit →0 the dynamics (<ref>) collapse to the invariant subspace S_2 = {(z_1,z_2)∈^n z_2 = - Ã_22^-1Ã_21z_1}≃^d.Averaging the fast variables in (<ref>) against their invariant measureδ_m, then yields the averaged equation for the slow variables: z_1/ t=  z_1 + ∑_k=1^m(N̂_kz_1 + b̃_1,k) u_ky = Ĉz_1 ,with the coefficients Â= Ã_11 - Ã_12Ã_22^-1Ã_21 N̂_k= Ñ_k,11 -Ñ_k,12Ã_22^-1Ã_21 Ĉ= C̃_1 - C̃_2Ã_22^-1Ã_21 . The situation here is special, in that the controls decay sufficiently fast sothat the invariant measure of the fast variables is independent of u. Forother choices of admissible controls, however, the invariant measure may dependon u, which then gives rise to averaged equations with measure-valued righthand side <cit.>. The followingapproximation result has been proved in <cit.>;cf. <cit.>.Let u=u^,γ in (<ref>) be admissible, satisfying u(t) =u(t/^γ) for some 0<γ<1. Further let y^(t)be theobserved solution of (<ref>) with consistent initial conditions(z_1^(0),ζ^(0))=(η,-Ã_22^-1Ã_21η), and let y̅(t) denote the output of the averagedequation (<ref>) on the bounded time interval [0,T], starting from thesame z_1(0)=η. Then there exists a constant C=C(T), such that sup_0≤ t≤ T|y^(t) - ŷ(t) | ≤ C^γ . We should stress that it is possible to relax the condition on the initialconditions that guarantees that (z_1^(0),ζ^(0))∈ S_2.Inthis case there will be a transient initial layer of thickness(√()), in which there is a rapid adjustment of the initialconditions to the invariant subspace S_2 and during which the averageddynamics deviates from the original dynamics, with an (1) error. A uniformapproximation on [0,T] can then be obtained by a so called matchedasymptotic expansion that matches an initial layer approximation with theaveraged dynamics <cit.>. For single-input systems (m=1), a sufficient condition for BIBO stability of(<ref>) is that A is Hurwitz, in which case there exists a δ>0,such that A+s N is Hurwitz for all s∈[-δ,δ]. The stability of A isinherited by the Schur complement Â, in (<ref>) and consequently + sN̂ inherits stability, with a possibly smaller stabilityregion. (See Appendix <ref> for details.) Hence reducedsingle-input systems are again BIBO stable. § NUMERICAL DETAILSBeforetesting ℋ_2 and balanced model reduction for examples fromstochastic control and quantum dynamics, see Secs. <ref> and<ref>, respectively, we will first focuson the numerical issues related to the scaling of the controls and thepreprocessing of the unstable A matrix.§.§ Structured bilinear systems The subsequent numerical examples share several special properties that result from a physical interpretation and that require a careful numericaltreatment. In this section, we provide some insight in how the model reductionmethods are applied to the particularly structured bilinear systems. In fact,in the FPE as well as in the LvNE context, the initial setup leadsto a purely bilinear system of the formẋ(t)= Ax(t) + ∑_k=1^m N_kx(t)u_k(t),x(0)=x_0, y(t) =Cx(t).In either case, the system exhibits a nontrivial stationary solution x_ecorresponding to a simple eigenvalue 0 of the system matrix A, i.e.,Ax_e=0. For the applications we are interested in the deviation of the statex from the stationary solution. Let us therefore introduce the referencestate x̃=x-x_e that is governed by the bilinear systemẋ̃̇(t) = Ax̃(t) + ∑_k=1^m N_k x̃(t) u_k(t) +[ N_1 x_e,…,N_kx_e ]_Bu(t), x̃(0) = x_0-x_e,y(t)= Cx̃(t)+Cx_e,where the term Cx_e can be interpreted as a constant nonzero feedthrough Dof the system. For the reduced-order model, we thus may simply setD̂=Cx_esuch that we can simply focus on the output operator C. In accordance withstandard model reduction concepts that assume a homogeneous initial condition,here we assume that the initial state of the original system is theequilibrium, i.e., x̃(0)=x_e-x_e=0. While the system now has beentransformed from a purely bilinear into a standard bilinear system, we stillhave to deal with the problem of a system matrix that is not asymptoticallystable. In what follows, we present two different techniques that bypass thisproblem.§.§ Sparsity preserving projection In our examples, the system matrices are mass and positivity preserving.Numerically this is reflected in the fact that the system matrix A as well asthe bilinear coupling matrices have zero row sum. In other words, thevector 1_n:=[ 1,…,1 ]^*∈^nsatisfies 1_n^* A = 1_n^*N_k = 0. The intuitive idea now is splitting the stateinto the direct sum of the asymptotically stable subspace and theeigenspace associated with the eigenvalue 0. Since a straightforwardimplementation in general will destroy the sparsity pattern of the matrices, wesuggest to use a particular decomposition that has been introduced in a similar setup in <cit.>. Define the matrixR = [ I 0; 0 0 ]+ x_e e_n^* - e_n[ 1_n-1^* 0 ],where e_n denotes the n-th unit vector in ^n. An easy calculation nowshows that the inverse R^-1 is given asR^-1 =[ I 0; 0 0 ] + e_n 1^* -[ x̃_e;0 ]1^*,where the vector x̃_e ∈^n-1 consists of the first n-1components of x_e ∈^n. Assume that the matrices A,N_k and B arepartitioned as followsA = [ à A_(1:n-1,n); * ],N_k = [ Ñ_k N_k,(1:n-1,n); * ], B=[ B̃;],with Ã,Ñ_k ∈^n-1× n-1 and B̃∈^n-1× m. Finally, a state space transformation z:=R^-1x̃yields the equivalent bilinear systemż(t)= (R^-1AR) z(t) + ∑_k=1^m (R^-1N_k R) z(t) u_k(t) +(R^-1B) u(t),z(0)= 0, y(t)= (CR) z(t) + Cx_e.Making use of the relations Ax_e=0=A^* 1_n=N_k^*1_n, weconclude that the last row of R^-1AR,R^-1N_kR and R^-1B=R^-1N_k x_eis zero. This implies that the last component of z(t) is constant, and, dueto z(0)=0 vanishes for all times t. As a consequence, we can focus on thefirst n-1 components z̃(t) of z(t) which, after some calculations,can be shown to satisfyz̃(t)=( à - A_(1:n-1,n)1_n-1^* ) z̃(t)+ ∑_k=1^m ( Ñ_k - N_k,(1:n-1,n)1_n-1^* )z̃(t)u_k(t) + B̃u(t), y(t)= C̃z̃(t) + Cx_e, z̃(0) = 0.Typically, the matrices A and N_k result from finite difference or finiteelement discretization, respectively, and thus are sparse. The previousprojection in fact only slightly increases the number of nonzero entries.Moreover, the matrices are given as the sum of the original data and a low rankupdate which can be exploited in a numerical implementation as well.§.§ Discounting the system state An ad-hoc alternative to the decomposition of the state space into stableand unstable directions is the “shifting” of the A-matrix by a translationA↦ A-α I for someα>0. If A has a simple eigenvalue zero,as in our case, there exists an α>0, such that the matrix A-α I isHurwitz. For linear systems the shifting can be interpreted as a discounting of thecontrollability and observability functionals that renders the associatedGramians finite <cit.>. As the controllability and observability Gramians in the bilinear case arelacking a similar interpretation, the shifting has no clear functional analogue(cf. <cit.>). It is still possible to stabilize the system by a joint state-observabletransformation(x,y)↦ (e^-α tx,e^-α ty)=:(x̃,ỹ)under which the system (<ref>)–(<ref>) transforms according tox̃/ t= (A - α I) x̃ +∑_k=1^m(N_k x̃+b_k)u_k ,x̃(0)=x_0 ỹ= Cx̃ .Even though (<ref>) and (<ref>)–(<ref>) areequivalent as state space systems, the shifting clearly affects the Hankelsingular value spectrum and, as a consequence, the reduced system. (As a matterof fact, the Hankel singular values do not even exist in case of theuntransformed system.) Hence the parameter α should be regarded as aregularization parameters that must chosen as small as possible. Later on we compare stabilization of the A matrix by state space decompositionand shifting in terms of the achievable state space reduction (i.e., decay ofHankel singular values) and fidelity of the reduced models.§.§ Scaling the control fieldsAssumption 1 in Sec. <ref> deals with the existence anduniqueness of controllability and observability Gramians which are obtained assolutions to the generalized Lyapunov equations. The criterion given there involves an upper bound for the matrix 2-norm of thecontrol matrices N_k. In the examples of model order reduction shown below, this can be achieved by asuitable scaling u ↦η u, N_k↦ N_k/η, B ↦ B/η withreal η>1 which leaves the equations of motion invariant but, clearly, notthe Gramians. Hence, by increasing η, we drive the system to its linearcounterpart. For the limit η→∞, the system matrices N andB vanish and we obtain a linear system. For this reason, η should not bechosen too large.§.§ Calculation of the ℋ_2 errorTo quantify the error introduced by dimension reduction, we use theℋ_2-norm introduced in Sec. <ref>. We emphasize that theeffort requiredfor computing the ℋ_2-error is negligible when compared to solvingthe generalized Lyapunov equations arising for balanced truncation and singularperturbation, respectively, which is seen as follows. Given a reduced-ordersystem Σ̂, the associated ℋ_2-error is given asΣ -Σ̂_ℋ_2^2 = tr(C_e P_e C_e^*),where P_e solves (<ref>). Using the particular structure ofthe error system, this is obviously the same asΣ -Σ̂_ℋ_2^2 = tr(CPC^*) -2tr(CXĈ^*) + tr(ĈP̂Ĉ^*).However, the term tr(CPC^*) now can be precomputed since P isrequired for the balancing-based methods anyway. What remains is thecomputation of the solutions X and P̂ of the following the generalizedSylvester and Lyapunov equations, respectivelyAX + XÂ^* + ∑_k=1^m N_k X N̂_k^* + BB̂^*=0, ÂP̂ + P̂Â^* + ∑_k=1^m N̂_k P̂N̂_k^* + B̂B̂^*=0.Based on the results from <cit.>, we can compute X=lim_i→∞X_i and P̂=lim_i→∞P̂_i as the limits ofsolutions to standard Sylvester and Lyapunov equations AX_1 + X_1 Â^* + BB̂^*=0, AX_i + X_iÂ^* + ∑_k=1^m N_k X_i-1N̂_k^* + B B̂^*=0, i≥ 2,ÂP̂_1 + P̂_1 Â^* + B̂B̂^*=0,ÂP̂_i + P̂_iÂ^* + ∑_k=1^m N̂_k P̂_i-1N̂_k^* + B̂B̂^*=0, i≥ 2.§.§ Software All of the numerical tests of the dynamical systems presented in the followinghave been carried out using theWavePacket software project whichencompasses all numerical methods for model order reduction as discussed above. Being hosted at the open–source platform Sourceforge.net, this program packageis publicly available, along with many instructions and demonstration examples,seeandRefs. <cit.>. In addition to a mature version, there is also a C++ version currentlyunder development.§ FOKKER–PLANCK EQUATIONWe start off with an example from stochastic control in classical mechanics: asemi-discretized Fokker–Planck equation (FPE) with external forcing. To thisend, we consider the stochastic differential equationX_t = (u_t - ∇ V(X_t)) t + σ W_t , X_0=x ,that governs the motion of a classical particle with position X_t∈^n attime t>0. The motion is influenced by the gradient of a smooth potential V,a deterministic control force u and a random forcing coming from theincrements of the Brownian motion (W_t)_t≥ 0 in ^n. For simplicity weassume that the potential V is C^∞, with V(x)∼ |x|^2kas |x|→∞.Note that X_t=X_t(ω) is a random variable for every t>0, and anequivalent characterization of the diffusion process X_t is in terms of itsprobability distribution∫_A ρ(y,t)y = Prob[X_t ∈ A|X_0=x]where A⊂^n is any measurable (Borel) subset of ^n, and ρ^n×_+→_+ is the associated probability density whosetime evolution is governed by the Fokker–Planck equation∂ρ/∂ t = ∇·(β^-1∇ρ +ρ(∇ V - u)) ,lim_t↘ 0ρ(·,t) = δ_x , with the shorthand β=2/σ^2 for the inverse temperature.The limit in the last equation, that must be understood in the sense of weakconvergence of probability measures (or, equivalently, weak-* convergence),reflects our choice of deterministic initial condition X_0=x; theregularization property of the parabolic FPE guarantees that ρ(·,t) isC^2 for any t>0; moreover the solution stays non-negative. Later on, we will consider the case that the initial conditions are drawn fromaprobability density ρ_0 and thus replace δ_x by ρ_0. Note that by the divergence theorem, / t∫ρ(y,t)y = 0 ,hence the total probability is conserved along the solution of (<ref>). When u=u_0 is constant, the properties of the potential V entail that thesolution to the FPE converges exponentially fast to a stationary solutionρ_∞ as t→∞ (see, e.g., <cit.>). The stationarysolution is then given as the unique normalized solution to the ellipticpartialdifferential equation0= ∇·(β^-1∇ρ + ρ∇ V_u )and has the form μ(x) = 1/Z_ue^-β V_u(x) , Z_u = ∫_^ne^-βV_u(x)x ,where we have introduced the shorthand V_u(x)=V(x)-u_0· x for the tiltedpotential.Later on we will study the convergence towards the stationary distribution thatis exponential with a rate essentially given by the first non-zero eigenvalue-λ_1>0, and compare the fully discretized model with its reduced-orderapproximant.§.§ Metastable model systemWe consider the situation of a diffusive particle in ^2 that is confined bythe following periodically perturbed quadruple-well potential[EricBarth, private communication.] shown inFigure <ref>V = 0.01( (x_1-0.1)^4 - 20x_1^2+ (x_2+0.4)^4 - 20 x_2^2. + .10sin(5x_1)cos(5x_2) + x_1 x_2 + 290.4 )The potential has a deep energy well in the south-east of thex_1-x_2-plane, one slightly shallower well in the south-west and two evenshallower wells in the north-west and north-east.The system is metastable, in that the time scale to reach the deepest potentialenergy well from any of the other three wells is of the order of the Arrheniustimescale e^βΔ V_ min≫ 1 where Δ V_ min denotesthe minimum energy barrier that a particle going from one well to thesouth-eastwell would have to overcome <cit.>. The various local minima ofthe potential energy surface that originate from the periodic perturbation do nothave any significant effect on the transition rates between the main wells. The corresponding stationary density μ is shown in the upper left panel ofFig. <ref>. For moderate temperature (β=4.0) essentiallyonly the two main wells are populated, with considerably more weight on thedeepest minimum (SE). §.§ Finite difference discretization Sine all coefficients in the FPE (<ref>) are sufficiently smooth, we candiscretize it using finite differences. Let Ω = (a, b)× (c, d), andconsider the solution domain D=Ω̅× [0,T]⊂^2×_+.On a bounded domain, probability conservation (<ref>) requires that theoutwards probability flux J_u(ρ) = β^-1∇ρ + ρ(∇ V - u)across the boundary of the spatial domain is zero at any time. Letting νdenote the outward pointing normal to ∂Ω, the FPE (<ref>)on D reads∂ρ/∂ t = ∇·(β^-1∇ρ +ρ(∇ V - u)) ,(x,t) ∈Ω× (0,T]0= ν· J_u(ρ) ,(x,t)∈∂Ω× [0,T] ρ_0 = ρ , (x,t)∈Ω×{0} . For simplicity we will discretize the equation on the uniform mesh Ω_h := {(a+ih_1,c+jh_2)1<i<n_1-1, 1<j<n_2-1}, ∂Ω_h := {(a+ih_1,c+jh_2)0≤ i ≤ n_1, 0≤ j≤n_2}∖Ω_h,where h_1=(b-a)/(n_1+1) and h_2=(d-c)/(n_2+1) are the mesh sizes in x_1and x_2 direction. Letting w_i,j=ρ(x_1,i,x_2,j) with(x_1,i,x_2,j)∈Ω, we approximate the first and second derivativesinthe usual way by centered finite differences, e.g..∂ρ/∂ x_1|_x=(x_1,i,x_2,j) ≈w_i+1,j-w_i-1,j/2h_1 .∂^2ρ/∂ x_1^2|_x=(x_1,i,x_2,j) ≈w_i+1,j-2w_i,j + w_i-1,j/h_1^2 . For sufficiently small mesh size h=(h_1,h_2), the finite differencediscretization is known to preserve positivity, norm and stochastic stability.As a consequence, the stationary distribution of the discretized equation istheunique asymptotically stable fixed point and approximately equal to thestationary solution μ of the original equation, evaluated at the gridpoints; cf. <cit.>. In matrix-vector notation, the discretization of (<ref>) can becompactly written asv̇ = Av + ∑_k=1^2u_k N_k v, v(0)=v_0where v∈^n with n=n_1 n_2 is the column-wise tensorization of(w_i,j)_i,j, i.e. v_i + (j-1)n_1 = w_i,j, A∈^n× n isthe discretization of the Fokker–Planck operator ∇·(β^-1∇ρ + ρ∇ V ) =β^-1Δρ + ∇ V ·∇ρ + (Δ V)ρof the uncontrolled dynamics, and the N_i are the discretization of thepartial derivatives ∂/∂ x_i on the tensorized grid, u_1 andu_2 are the components of u. By construction, -A is an M-matrix with a simple eigenvalue 0 thatcorresponds to the discretized unique stationary distributionπ≈μ|_Ω_h, all other eigenvalues have strictly negative realparts. This is in contrast to the spectral properties of the original operatorthat is symmetric (essentially self-adjoint) when considered on theappropriately weighted Hilbert space, i.e., all its eigenvalues are real. Weobserve, however, that the dominant eigenvalues are real when thediscretizationis sufficiently fine. Tab. <ref> gives the 12 smallest eigenvalues (by theirmagnitude) of the matrix A andfor a discretization of the domainΩ=(-6.0, 6.0)× (-5.5, 6.5) with uniform mesh size h_1=h_2=0.25;the size of the resulting matrix A is 2401× 2401. The L^1-deviation between the eigenvector π to the eigenvalueλ_0=0 and μ evaluated at the grid points is smaller than 0.007.Asthe theory predicts, the matrix has 4 dominant eigenvalues close to 0(including λ_0=0) that are separated from the rest of the spectrum.Figure <ref> shows the 4 dominant eigenvectors of A, thefirst one being the stationary distribution that is essentially supported bythetwo deepest minima, the second one describing the dominant transition processbetween the deepest and the second deepest minimum, the third one representingthe transitions between the second and the third deepest minimum and so on. The absolute values of the corresponding eigenvaluesλ_1, λ_2, λ_3<0 represent (up to an error of order√()) the transition rates between the dominant potential energywells.The fact that the subdominant eigenvalues appear in clusters of 4has to dowith the approximate four-fold symmetry of the potential. By tilting the potential towards one or several of the minima (thus flatteningsome of the other minima) the number of eigenvalues in the dominant clusterchanges according to the number of resulting wells.§.§ Stable input-output system in standard form We first augment (<ref>) by an output equation. To this end weintroducethe observable y=(y_1,…,y_4) ≥ 0 denoting the probability for each ofthe four energy wells. The y_i are given by summation of the density v overall mesh points corresponding to the four quadrants of the x_1-x_2-plane,which, using the tensorized form of the equation, can be written as y = Cx for a matrix C∈^4× n. The discretized FPE is bilinear, but it ishomogeneous, i.e., it does not contain a purely linear term “Bu”, whichimplies that no state is reachable from the origin v(0)=0.[Note thatv(0)=0 is not a probability density, hence not an admissible starting pointfrom a probabilistic point of view.] To transform (<ref>) into thestandard form (<ref>), we follow the procedure described in Sec.<ref>. §.§ Numerical resultsHere and throughout the following we will use the following short-hand notationwhen comparing results for the three approaches to model order reduction: BT stands for balanced truncation, as given by equation (<ref>) inSec. <ref> whereas SP symbolizes the averaging principle derived fromsingular perturbation theory, as given by equations(<ref>)–(<ref>) in Sec. <ref>.Finally, H2 is the ℋ_2-optimal model order reduction ofSec. <ref>.The details of the following comparisons depend sensitively on the value of theparameter η used for scaling of the control field u(t) and matricesN_kand B, which is necessary to guarantee existence and uniqueness ofcontrollability and observability Gramians, see Sec. <ref>. For the particular example of the FPE dynamics for inverse temperature β=4investigated here, we use a value of η=10 consistently for all threeapproaches to model order reduction. Moreover, to stabilize the A matrix we use here the projection method fromSec. <ref>.However, our results are practically unchanged when using the discountingapproach described in Sec. <ref> instead, assuming that theregularization parameter α is within a reasonable range. The behavior of the ℋ_2-error defined in (<ref>) for thediscretized FPE is shown in Fig. <ref>.Similarly for all of the three methods, this error displays a plateau value ofapproximately 10^-5 for a reduced dimensionality of about d ≳ 60.Upon further reduction of the dimensionality we observe a rapid increase overseveral orders of magnitude indicating a decreased quality when reducing overly. In most cases it is found that the ℋ_2-error for the H2 method isslightly lower than for BT, which in turn is slightly lower than for the SPmethod.While the ℋ_2-error characterizes the error of model orderreductionfor the limiting case of an infinitely short pulse (Dirac-like) control field,it may be also of interest to compare full versus reduced order models for morerealistically shaped control fields. As an example we consider here the Fokker–Planck dynamics, again for β=4,induced by a Gaussian-shaped control pulse along the x_2-directionu_2(t) = a exp( -(t-t_0)^2/2σ^2)centered at t_0=150.Here σ=τ/√(8log2)is chosen to yield a full width at half maximum of τ=100which is on the same order of magnitude as the relaxation time to equallyaccount for the aspects of controllability and observability. The time evolution of the four above-mentioned observables (populations of thequadrants of the x_1-x_2 plane) is shown in Fig. <ref>. The amplitude a=0.5 of the pulse has been determined to drive approximatelyone half of the density from the lower minima (south) to the higher minima(north) at t≈ 200.At later times, the populations return exponentially to their original valuesdefined by the canonical density of Eq. (<ref>).Our numerical experiments show that the population dynamics for d=100 isstillpractically indistinguishable from calculations in full dimensionality. Whenfurther reducing the model order down to d=50 and d=30, we observe that thequality of the SP method is superior to the BT or H2 method. However, despite of some minor differences, the overall performance of allthreemodel order reduction schemes is impressive when considering that the originaldimension of the problem is n=2401. We observe that the ℋ_2 erroroccasionally drops below machine precision. These occurrences appear at randomand are not reproducible (depending e.g. on the computer used for the numericalcalculation) and therefore we attribute them to numerical artifacts and excludethe values in the corresponding plots. We emphasize that replacing the reduced-order model by a coarsefinite-difference discretization of the advection-dominated Fokker-Planckequation is not advisable. For example, using a mesh size h_1=h_2=1.25, andthus11 grid points per dimension, corresponding to a system of dimensiond=121, we find that the error in the stationary distribution (i.e. theeigenvector to the eigenvalue λ_0=0) is of order 1 and that none of thedominant eigenvalues is approximated. For even larger mesh size, theeigenvaluesof the matrix A cross the imaginary axis, resulting in an unstable system.Hence the recommended reduction strategy consists in first generating asufficiently fine discretization of the original system and thenreducingthe dimension. § LIOUVILLE–VON NEUMANN EQUATIONAs a second example we choose the dynamics of open q–state quantum systems. Usually those are formulated in terms of a matrix representation of the reduceddensity operator, ρ∈^q× q,the diagonal and off-diagonalentries of which stand for populations and coherences, respectively.The time–evolution of ρ is governed by a quantum master equation which,due to a formal similarity with the Liouville equation in classical mechanics,is termed Liouville–von Neumann (LvNE) equation <cit.>i ∂/∂ tρ(t) = ℒ_Hρ(t) + ℒ_Dρ(t) ,where we have used atomic units (ħ=1). The first Liouvillian on the right hand side represents the closed systemquantum dynamics ℒ_Hρ(t)=-i[H_0 - ∑_kF_k(t)μ_k,ρ(t)]_-where [·,·]_- stands for a commutator and where the field–free systemis expressed in terms of its Hamiltonian matrix H_0. The system can be controlled through the interaction of its dipole momentmatrices μ_k with electric field components F_k(t) which is thelowest–order semiclassical expression for the interaction of a quantum systemwith an electromagnetic field. The second Liouvillian on the right hand side of (<ref>) represents theinteraction of the system with its environment thus accounting fortime-irreversibility, i.e., dissipation and/or dephasing.A commonly used model for these processes is the Lindblad form<cit.> ℒ_Dρ = i ∑_c (C_cρ C_c^†-1/2[C_c^† C_c,ρ]_+) ,where the index c runs over all dissipation channels <cit.> andwhere [·,·]_+ stands for an anti–commutator. The Lindblad operators C_c describe the coupling to the environment inBorn-Markov approximation (weak coupling, no memory), typically chosen to beprojectorsC_c=C_i← j=√(Γ_i← j) |i⟩⟨ j|with rate constants (inverse times) Γ_i← j. In order to cast the evolution equation (<ref>) into the standard formof bilinear input-output systems (<ref>) for deviations fromthe stationary solution, the density matrix ρ has to be mapped onto avector x with n=q^2 components. Choosing the vectorization such that populations go in front of coherencesoffers the advantage that A is blockdiagonal with block sizes q and (n-q)where the latter block is diagonal.Moreover, the upper left submatrix of N is a zero matrix of size q× q. We note that typically both A and N are sparse matrices whereas B and Care not. For more details of the vectorization procedure and the associated constructionof matrices A, N, B, and C from the LvNE, see Appendix A ofRef. <cit.>.With the Lindblad model introduced above, the LvNE (<ref>) istrace-preserving (i.e., the sum of populations remains constant) and completelypositive (i.e., the individual populations remain positive) thus ensuringthe probabilistic interpretation of densities in quantum mechanics. Despite ofthe different discretization schemes used, the model bears many similaritieswith the discretized Fokker–Planck equation considered in Sec. <ref>,including the simple zero eigenvalue of the matrix A.§.§ Double well model systemWe apply our model reduction approaches to dissipative quantum dynamicsdescribed by a (one–dimensional) asymmetric double well potential as presentedin our previous work <cit.>. Our parameters are chosen such that there are six (five) stationary quantumstates which are essentially localized in the left (right) well. We also include the first ten eigenstates above the barrier separating the wellswhich are delocalized while even higher states are not considered forsimplicity.In total, the q=21 considered states lead to a density matrix with dimensionn=441.Thus, model order reduction can be mandatory, e.g., during a refinement offields in optimal control. In the present model simulations, the dependence of rate constants Γ_i← j with j>i describing the decay of populations (and associateddecoherence) are obtained from the model of Ref. <cit.> whichemploys only one adjustable parameter which we choose asΓ≡Γ_0← 2; the rates for upward transitions (i>j)are calculated from those for downward ones using the principle of detailedbalanceΓ_j← i = exp( - E_j-E_i/Θ)Γ_i← j, j>iwhere E are the eigenvalues of the unperturbed Hamiltonian H_0. Hence, thetemperature Θ is the second parameter needed to set up matrix A(assuming Boltzmann constant k_B=1). The external control of the quantum system is modeled within the semi-classicalapproximation of Eq. (<ref>): The electric field F(t) interactslinearly with the dipole moment μ which is assumed to be proportionate tothe system coordinate of the double well system which is used to set upmatricesN and B describing the controllability. To observe the system dynamics, we monitor the sums of the populations of thequantum states localized in the left and right well, and of the delocalizedstates over the barrier. These three quantities are used to construct the matrix C describing theobservability <cit.>. §.§ Numerical resultsAs was already noted for the FPE example, the performance of the model orderreduction schemes depends sensitively on the value of the parameter η usedfor scaling of the control field u(t) and matrices N_k and B, see Sec. <ref>. For all examples from LvNE dynamics discussed here, we use a value of η=3. In addition, the A matrices are stabilized using the projection methodintroduced in Sec. <ref>.Again, all results are practically unchanged when using the discounting approachdescribed in Sec. <ref> instead.We begin our discussion by considering the spectrum of the A-matrix asdisplayed in Fig. <ref>. With increasing dimension reduction, more and more of the eigenvalues withlowest (most negative) real parts are eliminated first.As has been detailed in Appendix A of Ref. <cit.>, those correspondtoquantum states which decay fastest. Hence, the eigenvalues of A with lowest real part are associated with lowestobservability. At the same time, the order reduction tends to eliminate stateswith large imaginary part first.Those correspond to coherences between quantum states with large energy gapsforwhich the Franck-Condon (FC) factors are typically very low. Hence, the eigenvalues of A with largest imaginary part are associated withlowest controllability. A noteworthy exception are the results for d=30 (green dots inFig. <ref>) with real parts near zero. There, the imaginary parts (energy differences) near even multiples of ≈0.1 can be assigned to ladder climbing within each of the wells of the doublewell potential, while odd multiples correspond to transitions between theminima. Because the FC factors for the former ones are larger, they are more likely tobe preserved in dimension reduction due to their higher controllability. This is seen most clearly in the left panel of Fig. <ref>,i.e., for the BT method. In summary, the model order reduction confines the spectrum of A to the lower(most controllable) and to the right (most observable) part of the complexnumber plane.In general, the results of the three different approaches (BT, SP, and H2method) are very similar to each other. To quantify the error introduced by model order reduction of the LvNE system,weconsider the behavior of the ℋ_2-error as defined in(<ref>). Our results for various values of the relaxation rateΓ (but constant temperature, Θ=0.1) are shown in the left half ofFig. <ref>. The higher the value of the relaxation rateΓ, the smaller is the ℋ_2 error and the earlier the errorreaches a plateau at about 10^-10… 10^-9.Hence, dimension reduction is more effective for open quantum systems withlarger rate constants for relaxation (and associated decoherence). Furthermore,it is noted that the BT and the H2 method yield similar ℋ_2 errorsat comparable computational effort so that there is no clear preference foreither one of them.Our results for various values of the temperature Θ (but constantrelaxation, Γ=0.1) are shown in the right half of Fig.<ref>. For low (Θ=0.07) and for medium (Θ=0.1)temperatures, the ℋ_2 error decreases with increasingdimensionality r and again reaches a plateau. However, at higher temperature(Θ=0.2) the error decreases rapidly and reaches machine precision atr≈ 100. Again, in most cases the results for the different methods are close to eachother, with the only exception being the lower temperature (Θ=0.07),where the ℋ_2-error for the BT method is often found below that forthe H2 method. At low temperature the system becomes less controllable, andthissuggests that H2 does not always correctly capture the controllablestates—which BT does by construction.As before in the Fokker–Planck example, we observe that the ℋ_2error occasionally drops below machine precision. As these occurrences appearatrandom and are not reproducible (depending e.g. on the computer used for thenumerical calculation), we attribute them to numerical artifacts and excludethe values in the corresponding plots. Finally, an example for the time evolution of the three above-mentionedobservables (populations) in the asymmetric double well system (relaxation rateΓ=0.1 and temperature Θ=0.1) is investigated for the controlfieldgiven in Eq. (<ref>), here with a=3, t_0=15, and τ=10. The pulse drives the population, which is initially mainly in the left well ofthe potential, to delocalized quantum states over the barrier from wheretransitions to the right well are induced. The subsequent relaxation to the thermal distribution proceeds on a much longertime scale not shown here. In Fig. <ref> we compare the results for fulldimensionality (n=441) with reduced dimensionality d.While the results for d=100 are still essentially exact, the results ford=50 start to deviate notably. For d=30 only the BT method (left panel ofFig <ref>) reproduces the full dimensional onesqualitatively while SP method (center panel) as well as H2 method (right panel)fail completely. § CONCLUSIONS In this paper, model reduction methods for bilinear control systems arecompared, with a special focus on Fokker–Planck and Liouville–von Neumannequation. The methods can be categorized into balancing based (balancedtruncation, singular perturbation) and interpolation based (ℋ_2optimization) reduction methods. While these methods have already beendiscussed in <cit.>,our focus is on a direct and thorough comparison between all of them.Particularly, we draw the following conclusionswith regard tocomputational complexity, accuracy and applicability to realistic bilineardynamics. §.§ Computational complexity The computational effort of BT and SP is essentially determined by thesolution of the two generalized Lyapunov equations (<ref>) and (<ref>).From a theoretical point of view, the complexity for solving these equations explicitly is 𝒪(n^6). On the other hand, an iterative approximation(<cit.>) as described in Sec. <ref> with r iteration stepsonly requires 𝒪(r n^3) operations (due to solving the standardLyapunov equations in each step by a direct solver such as the Bartels-Stewartalgorithm byin ). As an alternative, the generalizedequations can be rewritten as a linear problem which can be solved, e. g., by the bi–conjugate gradient method whereit is advantageous to use the solutions of the corresponding ordinary equationsfor pre-conditioning.The effort of H2 is mainly dueto the solution of two generalized Sylvester equations in each step of thebilinear iterative rational Krylov algorithm (BIRKA). In contrast to BT/SP, adirect solution of these equations requires “only” 𝒪(l^3 n^3)operations (l denoting the dimension of the reduced model). Similarly, thecost for an iterative procedure is less since the standard Sylvester equationscan be handled efficiently for sparse system matrices. Hence, a single step ofBIRKA is computationally less expensive than performing the balancing stepin BT/SP. However, the overall cost for BIRKA obviously depends on the numberof iteration steps that is needed until the fixed point iteration is(numerically) converged, see Sec. <ref>. Based on the numerical examplesstudied here, we can not report significant differences between all threemethods. §.§ Accuracy of reduced models The overall performance of all three methods is very satisfactory. Bothtransient responses as well as spectral properties of the original model arefaithfully reproduced by all reduced models (seeFigs. <ref>–<ref> and<ref>–<ref>. Despite the nature of H2, asignificant difference of the quality (w.r.t. the ℋ_2-norm)of thereduced models cannot be observed. Also, the (moderate) additional effort forSP instead of BT does not seem to lead to more accurate reduced models. §.§ Unstable bilinear dynamics and scaling Both BT/SP and H2 require the dynamics of the unperturbed systemto be stable.The spectrum of the matrix A representing the field-free FPE / LvNE dynamicsis in the left half of the complex number plane, however, with an additionalsingle eigenvalue zero. The effects of two differentstabilization techniques, i.e. a shift of the spectrum of A versus asplittingof stable and unstable parts leads to similarly accurate results (seeSecs. <ref> and <ref>). The latterapproach however has the benefit that the bilinear dynamics are not changed byprojecting onto the asymptotically stable part. For the generalized Lyapunov and/or Sylvester equations to be solvable, thenorms of the matrices B and N_k have to be kept below certain thresholdswhich is achieved by down-scaling these matrices and corresponding up-scalingofthe control fields, cf. Sec. <ref>. This leaves the equations ofmotion invariant (but not the Gramians). Here we observe significantlydifferent results depending on the choice/size of the scaling factor. In somecases, good results are obtained only for large scaling factors. However, weemphasize that large scaling factors drive the Gramians to those appearing forthe linear(ized) system. For this reason, an automatic (large) choice of thesefactors is not recommended but has to be investigated for the problem underconsideration on a case by case basis. From the numerical example, we believethat the scaling is a very important point for obtaining “optimal” reducedmodels. §.§ Further issuesAnother aspect related to the computation of the balancing transformation thatwe mention only for the sake of completeness is that it is oftenadvisable toexploit sparsity and to use low-rank techniques that do not require to computethe full Gramians and their Cholesky factorization, one such example being thelow-rank Cholesky factor ADI method<cit.>. These methodsrequire some fine tuning of the parameters to enforce convergence, but forexample, in case of the Fokker–Planck equation for which the matrices A andN that are extremely sparse and the rank of the matrix -BB^T is muchsmallerthan the size of the matrices A,N, there can be a considerable gain fromusinglow-rank techniques.§ STABILITY OF BALANCED AND REDUCED SYSTEMS We now prove that the balancing transformation(<ref>)–(<ref>) preserves the stability of the submatricesÃ_11 and Ã_22. The idea of the proof essentially follows<cit.>; see also <cit.>.We confine our attention to Ã_22, the stability of which is neededfor the averaging principle to apply, and we stress that the proof readilycarries over to the proof that Ã_11 is stable (Hurwitz). Letà =([ Ã_11 Ã_12; Ã_21 Ã_22 ]),Ñ_k=([ Ñ_k,11 Ñ_k,12; Ñ_k,21 Ñ_k,22 ]), B̃=([ B̃_1; B̃_2 ]),C̃=([ C̃_1 C̃_2 ])denote the coefficients of the balanced bilinear system for =1.Suppose that Assumptions 1–5 from pages ass and ass2 hold,and let the matrix of Hankel singular values Σ be defined as in(<ref>). If the submatrices Σ_1 and Σ_2 have disjointspectra, λ(Σ_1)∩λ(Σ_2)=∅, then λ(Ã_22)⊂_- ,where _- denotes the open left complex half-plane. We first prove that the spectrum of Ã_22 lies in the closed leftcomplex half-plane (including the imaginary axis). To this end note that(<ref>) implies that Ã_22Σ_2 + Σ_2Ã_22^* +∑_k=1^m(Ñ_k,22Σ_2Ñ^*_k,22 +Ñ_k,21Σ_1Ñ^*_k,12) +B̃_2B̃_2^* = 0 .Now let v∈^n-d be an eigenvector of Ã^*_22 to theeigenvalue λ∈, i.e. Ã^*_22v=λ v. Multiplicationof the last equation with v^* and v from the both sides yields 2 (λ) |Σ^1/2_2v|^2+∑_k=1^m(|Σ_2^1/2Ñ^*_k,22v|^2 +|Σ_1^1/2Ñ^*_k,12v|^2) +|B̃_2^*v|^2 = 0.Noting that both Σ_1 and Σ_2 are positive definite, it followsthat (λ)≤ 0, thus the eigenvalues of Ã_22 are in theleft complex half-plane or on the imaginary axis. As a second step we will demonstrate that indeed (λ) < 0. We proceedby contradiction and suppose the contrary. Following <cit.>, thereexists a linear change of variables x↦ V x, x∈^n, such that V= ([;V_22 ]) , V_22Ã_22V_22^-1 =([ Â_22 ;Â_33 ]) ,with Â_22 having eigenvalues in _- while the eigenvalues ofÂ_33 are pure imaginary. Under the change of variables, the balancedcoefficients transform as follows:  =([ Â_11 Â_12 Â_13; Â_21 Â_22 ; Â_31Â_33 ]),N̂_k =([ N̂_k,11 N̂_k,12 N̂_k,13; N̂_k,21 N̂_k,22 N̂_k,23; N̂_k,31 N̂_k,32 N̂_k,33 ]),B̂ =([ B̂_1; B̂_2; B̂_3 ]),Ĉ=([ Ĉ_1 Ĉ_2 Ĉ_3 ]).Here Â_11=Ã_11, N̂_k,11=Ñ_k,11,B̂_1=B̃_1, and Ĉ_1=C̃_1.Accordingly, we have Q̂ =([ Σ_1; Q̂_22 Q̂_23; Q̂_32 Q̂_33 ]),P̂ =([ Σ_1; P̂_22 P̂_23; P̂_32 P̂_33 ]).Now consider the (3,3) block of the generalized Lyapunov equation (<ref>)for the controllability Gramian that readsÂ_33Q̂_33 + Q̂_33Â_33^* + ∑_k=1^m(N̂_k,31 N̂_k,32 N̂_k,33)Q̂(N̂_k,31 N̂_k,32 N̂_k,33)^T +B̂_3B̂_3^* = 0Now let w be an eigenvector of Â_33 to a pure imaginary eigenvalueλ=iσ. Then sandwiching the last equation with w^* and w fromthe left and from the right and iterating the argument from above, it followsthat ∑_k=1^m|Q̂^1/2 (N̂_k,31 N̂_k,32 N̂_k,33)^Tw|^2 + |B̂_3^*w|^2 = 0 ,which, by complete controllability and thus positivity of the matrix Q̂implies that (N̂_k,31 N̂_k,32 N̂_k,33)^Tw=0 for allk=1,…,m. Therefore B̂_3^*w=0, and as we can pick w to be anyof the linearly independent eigenvectors of Â_33 we conclude that B̂_3 = ,N̂_k,31 =,N̂_k,32= ,N̂_k,33=, k=1,…,m .By the same argument, using the adjoint Lyapunov equation (<ref>) for thepositive definite observability Gramian, it follows thatĈ_3 = ,N̂_k,13 =,N̂_k,23= , k=1,…,m .This entails that the (2,3) block of the Lyapunov equation for Q̂ hasthe formÂ_22Q̂_23 = Q̂_23Â_22^* = 0 .Hence Q̂_23= and the analogous argument for the observabilityGramian yields that P̂_23=. Note that the Gramians are hermitian,i.e., Q̂_23=Q̂_32^* and P̂_23=P̂_32^*,whichimplies that the Gramians are block diagonal:Q̂ =([ Σ_1; Q̂_22; Q̂_33 ]),P̂ =([ Σ_1; P̂_22; P̂_33 ]).The Lyapunov equations for the (1,3) blocks thus readsÂ_13Q̂_33 + Σ_1Â_31^* = 0 ,Â_31^*P̂_33 + Σ_1Â_13 = 0Now multiplying the first of the two equations by Σ_1 from the left andsubstituting Σ_1Â_13 by -Â_31^*P̂_33 yieldsÂ_31P̂_33Q̂_33 = Σ_1^2Â_31^*.Interchanging the two Lyapunov equations we can show thatΣ_1^2Â_13^*=Â_13Q̂_33P̂_33. Now recall that the diagonal matrix Σ^2 contains the eigenvalues ofP̂Q̂ or Q̂P̂, and since the Gramians are block diagonal,it follows that Σ_2^2 contains the eigenvalues ofP̂_33Q̂_33 or Q̂_33P̂_33. By the assumption thatΣ_1 and Σ_2 have no eigenvalues in common, we conclude thatÂ_13 =,Â_31 =.This shows that the matrix  the form  =([ Â_11 Â_12 ; Â_21 Â_22 ; Â_33 ]),which together with B̂_3= and Ĉ_3= violates theassumption of complete controllability and observability on ass2.Hence Ã_22 cannot have eigenvalues on the imaginary axis, in otherwords: λ(Ã_22)⊂_-.Consequences of Lemma <ref> are the analogous statements for thematrix Ã_11 and the Schur complement of Ã_22. Under the assumptions of Lemma <ref> it holds that λ(Ã_11) ⊂_- . The proof is a simple adaption of the one of Lemma <ref> and<cit.>. Under the assumptions of Lemma <ref> it holds that λ(Ã_11 - Ã_12Ã_22^-1Ã_21)⊂_- . The assertion follows from Corollary <ref> by noting that thereciprocal system (Â,N̂_k,B̂,Ĉ):=(Ã^-1,Ã^-1 Ñ_k,Ã^-1B̃,-C̃Ã^-1)is balanced if and only if (Ã,Ñ_k,B̃,C̃) isbalanced, with Â_11 = Ã_11 -Ã_12Ã_22^-1Ã_21 . Note that N̂_k = Ñ_k,11 - Ñ_k,12Ã_22^-1Ã_21 is not the (1,1)block ofthe matrix Ã^-1Ñ_k, but rather the (1,1)coefficient of the matrix Ñ_kÃ^-1 unless à andÑ commute. This has been pointed out in <cit.>, and as aconsequence, the singular perturbation approximation(<ref>)–(<ref>) is not the truncation of the reciprocalsystem as is the case for linear systems. Yet this does not affect the aboveargument and hence thestability of the Schur complementÂ=Ã_11 - Ã_12Ã_22^-1Ã_21 .abbrv | http://arxiv.org/abs/1706.09882v1 | {
"authors": [
"Peter Benner",
"Tobias Breiten",
"Carsten Hartmann",
"Burkhard Schmidt"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170626150247",
"title": "Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations"
} |
0 1 0 0 1 1 0 0theoremTheorem[section] *theorem*TheoremsubclaimClaim[theorem] proposition[theorem]Proposition *proposition*Proposition lemma[theorem]Lemma *lemma*Lemma corollary[theorem]Corollary *conjecture*Conjecture fact[theorem]Fact *fact*Fact hypothesis[theorem]Hypothesis *hypothesis*Hypothesis conjecture[theorem]Conjecture definition definition[theorem]Definition construction[theorem]Construction example[theorem]Example question[theorem]Question openquestion[theorem]Open Question algorithm[theorem]Algorithm problem[theorem]Problem protocol[theorem]Protocol assumption[theorem]Assumptionremark claim[theorem]Claim *claim*Claim remark[theorem]Remark *remark*Remark observation[theorem]Observation *observation*Observation 1.1 =1=1=0 eq#1 eqn#1 lem#1Lemma <ref>#1 obs#1Observation <ref>#1 def#1Definition <ref>#1 thm#1Theorem <ref>#1 cor#1Corollary <ref>#1 cha#1Chapter <ref>#1 sec#1Section <ref>#1 app#1Appendix <ref>#1 tab#1Table <ref>#1 fig#1Figure <ref>#1 hyp#1Hypothesis <ref>#1 alg#1Algorithm <ref>#1 rem#1Remark <ref>#1 item#1Item <ref>#1 step#1step <ref>#1 conj#1Conjecture <ref>#1 fact#1Fact <ref>#1 prop#1Proposition <ref>#1 prob#1Problem <ref>#1 claim#1Claim <ref>#1 relax#1Relaxation <ref>#1 red#1Reduction <ref>#1 part#1Part <ref>#1=/= =1 =1 =0mybox 1.0∵ =1ϵ=ε equationsection =Fast and robust tensor decomposition with applications to dictionary learningTselil SchrammUC Berkeley, [email protected]. T. S. is supported by an NSF Graduate Research Fellowship (NSF award no. 1106400). David SteurerCornell University, [email protected]. D. S. is supported by a Microsoft Research Fellowship, a Alfred P. Sloan Fellowship, NSF awards (CCF-1408673,CCF-1412958,CCF-1350196), and the Simons Collaboration for Algorithms and Geometry.December 30, 2023 ====================================================================================================================================================================================================================================================================================================================================================================================================================empty We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy.Our algorithms can decompose a 4-tensor with n-dimensional orthonormal components in the presence of error with constant spectral norm (when viewed as an n^2-by-n^2 matrix). The running time is n^5 which is close to linear in the input size n^4.We also obtain algorithms with similar running time to learn sparsely-used orthogonal dictionaries even when feature representations have constant relative sparsity and non-independent coordinates.The only previous polynomial-time algorithms to solve these problem are based on solving large semidefinite programs. In contrast, our algorithms are easy to implement directly and are based on spectral projections and tensor-mode rearrangements.Or work is inspired by recent of Hopkins, Schramm, Shi, and Steurer (STOC'16) that shows how fast spectral algorithms can achieve the guarantees of SOS for average-case problems. In this work, we introduce general techniques to capture the guarantees of SOS for worst-case problems.=1empty § INTRODUCTIONTensor decomposition is the following basic inverse problem: Given a k-th order tensor T∈ (^d)^⊗ k of the formT=∑_i=1^n a_i^⊗ k + E,we aim to approximately recover one or all of the unknown components a_1,…,a_n∈^d. The goal is to develop algorithms that can solve this problem efficiently under the weakest possible assumptions on the order k, the components a_1,…,a_n, and the error E.Tensor decomposition is studied extensively across many disciplines including machine learning and signal processing. It is a powerful primitive for solving a wide range of other inverse / learning problems, for example: blind source separation / independent component analysis <cit.>, learning phylogenetic trees and hidden Markov models <cit.>, mixtures of Gaussians <cit.>, topic models <cit.>, dictionary learning <cit.>, and noisy-or Bayes nets <cit.>.A classical algorithm based on simultaneous diagonalization <cit.> (often attributed to R. Jennrich) can decompose the input tensor eq:1 when the components are linearly independent, there is no error, and the order of the tensor is at least 3. Current research on algorithms for tensor decomposition aims to improve over the guarantees of this classical algorithm in two important ways:Overcomplete tensors: What conditions allow us to decompose tensors when components are linearly dependent? Robust decomposition: What kind of errors can efficient decomposition algorithms tolerate? Can we tolerate errors E with “magnitude” comparable to the low-rank part ∑_i=1^n a_i^⊗ k? The focus of this work is on robustness. There are two ways in which errors arise in applications of tensor decomposition. The first is due to finite samples. For example, in some applications T is the empirical k-th moment of some distribution and the error E accounts for the difference between the empirical moment and actual moment (“population moment”). Errors of this kind can be made smaller at the expense of requiring a larger number of samples from the distribution. Therefore, robustness of decomposition algorithms helps with reducing sample complexity.Another way in which errors arise is from modeling errors (“systematic errors”). These kinds of errors are more severe because they cannot be reduced by taking larger samples. Two important applications of tensor decomposition with such errors are learning Noisy-or Bayes networks <cit.> and sparsely-used dictionaries <cit.>. For noisy-or networks, the errors arise due to non-linearities in the model. For sparsely-used dictionaries, the errors arise due to unknown correlations in the distribution of sparse feature representations. These examples show that robust tensor decomposition allows us to capture a wider range of models.Robustness guarantees for tensor decomposition algorithms have been studied extensively (e.g., the work on tensor power iteration <cit.>). The polynomial-time algorithm with the best known robustness guarantees for tensor decomposition<cit.> are based on the sum-of-squares (SOS) method, a powerful meta-algorithm for polynomial optimization problems based on semidefinite programming relaxations. Unfortunately, these algorithms are far from practical and have polynomial running times with large exponents. The goal of this work is to develop practical tensor decomposition algorithms with robustness guarantees close to those of SOS-based algorithms.For the sake of exposition, we consider the case that the components a_1,…,a_n∈^d of the input tensor T are orthonormal. (Through standard reductions which we explain later, most of our results also apply to components that are spectrally close to orthonormal or at least linearly independent.) It turns out that the robustness guarantees that SOS achieves for the case that T is a 4-tensor are significantly stronger than its guarantees for 3-tensors. These stronger guarantees are crucial for applications like dictionary learning. (It also turns out that for 3-tensors, an analysis of Jennrich's aforementioned algorithm using matrix concentration inequalities gives robustness guarantees that are similar to those of SOS <cit.>.)In this work, we develop an easy-to-implement, randomized spectral algorithm to decompose 4-tensors with orthonormal components even when the error tensor E has small but constant spectral norm as a d^2-by-d^2 matrix. This robustness guarantee is qualitatively optimal with respect to this norm in the sense that an error tensor E with constant spectral norm (as a d^2-by-d^2 matrix) could change each component by a constant proportion of its norm. To the best of our knowledge, the only previous algorithms with this kind of robustness guarantee are based on SOS.[We remark that the aforementioned analysis <cit.> of Jennrich's algorithm can tolerate errors E if its spectral norm as a non-square d^3-by-d matrix is constant. However, this norm of E can be larger by a √(d) factor than its spectral norm as a square matrix.] Our algorithm runs in time d^2+ω≤ d^4.373 using fast matrix multiplication. Even without fast matrix multiplication, our running time of d^5 is close to linear in the size of the input d^4 and significantly faster than the running time of SOS. As we will discuss later, an extension of this algorithm allows us to solve instances of dictionary learning that previously could provably be solved only by SOS.A related previous work <cit.> also studied the question how to achieve similar guarantees for tensor decomposition as SOS using just spectral algorithms. Our algorithms follow the same general strategy as the algorithms in this prior work: In an algorithm based on the SOS semidefinite program, one solves a convex programming relaxation to obtain a “proxy” for a true, integral solution (such as a component of the tensor). Because the program is a relaxation, one must then process or “round” the relaxation or proxy into a true solution. In our algorithms, as in <cit.>, instead of finding solutions to SOS semidefinite programs, the algorithms find “proxy objects” that behave in similar ways with respect to the rounding procedures used by SOS-based algorithms. Since these rounding procedures tend to be quite simple, there is hope that generating proxy objects that “fool” these procedures is computationally more efficient than solving general semidefinite programs.However, our algorithmic techniques for finding these “proxy objects” differ significantly from those in prior work. The reason is that many of the techniques in <cit.>, e.g., concentration inequalities for matrix-valued polynomials, are tailored to average-case problems and therefore do not apply in our setting because we do not make distributional assumption about the errors E.The basic version of our algorithm is specified by a sequence of convex sets _1,…,_r⊆ (^d)^⊗ 4 of 4-tensors and proceeds as follows:Given a 4-tensor T∈(^d)^⊗ 4, compute iterative projections T 1,…, T r to the convex sets _1,…,_r (with respect to euclidean norm) and apply Jennrich's algorithm on T r.It turns out that the SOS-based algorithm correspond to the case that r=1 and _1=_SOS is the feasible region of a large semidefinite program. For our fast algorithm, _1,…,_r are simpler sets defined in terms of singular values or eigenvalues of matrix reshapings of tensors. Therefore, projections boil down to fast eigenvector computations. We choose the sets _1,…,_r such that they contain _SOS and show that the iterative projection behaves in a similar way as the projection to _SOS. In this sense our algorithm is similar in spirit to iterated projective methods like the Bregman method (e.g., <cit.>). Dictionary learningIn this basic unsupervised learning problem, the goal is to learn an unknown matrix A∈^d× n from i.i.d. samples y 1 = Ax 1, …,y m = Ax m, where x 1,…, x m are i.i.d. samples from a distribution {x} over sparse vectors in ^n. (Here, the algorithm has access only to the vectors y 1,…, y m but not to x 1,…, x m.)Dictionary learning, also known as sparse coding, is studied extensively in neuroscience <cit.>, machine learning <cit.>, and computer vision <cit.>. Most algorithms for this problem used in practice do not come with strong provable guarantees. In recent years, several algorithms with provable guarantees have been developed for this problem <cit.>.For the case that the coordinates of the distribution {x} are independent (and non-Gaussian) there is a well-known reduction[The reduction only requires k-wise independence where k is the order to tensor the reduction produces.] of this problem to tensor decomposition where the components are the columns of A and the error E can be made inverse polynomially small by taking sufficiently many samples. (In the case of independent coordinates, dictionary learning becomes a special case of independent component analysis / blind source separation, where this reduction originated.) Variants of Jennrich's spectral tensor decomposition algorithm (e.g., <cit.>) imply strong provable guarantees for dictionary learning in the case that {x} has independent coordinates even in the “overcomplete” regime when n≫ d (using a polynomial number of samples). More challenging is the case that {x} has non-independent coordinates, especially if those correlations are unknown. We consider a model proposed in <cit.> (similar to a model in <cit.>): We say that the distribution {x} is τ-nice if * x_i^4=1 for all i∈ [n],* x_i^2 x_j^2 ≤τ for all i≠ j∈[n],* x_i x_j x_k x_ℓ=0 unless x_i x_j x_k x_ℓ is a square.The conditions allow for significant correlations in the support set of the vector x. For example, we can obtain a τ-nice distribution {x} by starting from any distribution over subsets S⊆ [n] such that { i∈ S}=p for all i∈ [n] and {j∈ S| i∈ S }≤τ for all i≠ j and choosing x of the form x_i =p^-1/4·σ_i if i∈ S and x_i=0 if i∉S, where σ_1,…,σ_n are independent random signs.An extension of our aforementioned algorithm for orthogonal tensor decomposition with spectral norm error allows us to learn orthonormal dictionaries from τ-nice distributions. To the best of our knowledge, the only previous algorithms to provably solve this problem use sum-of-squares relaxations <cit.>, which have large polynomial running time. Our algorithm recovers a 0.99 fraction of the columns of A up to error τ from (n^3) samples, and runs in time n^3+O(τ)d^4. By a standard reduction, our algorithm also works for non-singular dictionaries and the running time increases by a factor polynomial in the condition number of A.§.§ Results Tensor decomposition For tensor decomposition, we give an algorithm with close to linear running time that recovers the rank-1 components of a tensor with orthonormal components, so long as the spectral norm of the square unfoldings of the error tensor is small.There exists a randomized spectral algorithm with the following guarantees: Given a tensor ∈ (^d)^ 4 of the form = ∑_i=1^n a_i^ 4 + such that a_1,…,a_n ∈^d are orthonormal andhas spectral norm at mostas a d^2-by-d^2 matrix, the algorithm can recover one of the components with ℓ_2-error O() in time (d^2+ω + O()) with high probability, and recover 0.99n of the components with ℓ_2-error O() in time (d^2+ω + O()) with high probability.Furthermore, if ϵ≤ O(loglog n/log^3 n), the algorithm can recover all components up to error O(ϵ) in time (d^2+ω + nd^4) with high probability. Here, ω≤ 2.373 is the matrix multiplication exponent. The orthogonality condition may at first seem restrictive, but for most applications it is possible to take a tensor with linearly independent components and transform it to a tensor with orthogonal components, as we will do for our dictionary learning result below. Furthermore, our algorithm works as-is if the components are sufficiently close to orthonormal: If we have that a_1,…,a_n are only approximately orthonormal in the sense that the a_i are independent and ∑_i a_ia_i^⊤ -_S≤η, where _S is the identity in the subspace spanned by the a_i, then we can recover b_i so that a_i,b_i^2 ≥ 1 - O(√(η)) with the same algorithm and runtime guarantees.These robustness guarantees are comparable to those of the sum-of-squares-based algorithms in <cit.> for the undercomplete case, which are the best known. Meanwhile, the sum-of-squares based algorithms require solving large semidefinite programs, while the running time of our algorithms is close to linear in the size of the input, and our algorithms are composed of simple matrix-vector multiplications.On the other hand, our algorithms fail to work in the overcomplete case, when the rank grows above n, and the components are no longer linearly independent. One interesting open question is whether the techniques used in this paper can be extended to the overcomplete case. Dictionary learningUsing our tensor decomposition algorithm as a primitive, we give an algorithm for dictionary learning when the sample distribution is τ-nice. Suppose that A ∈^d× n is a dictionary with orthonormal columns, and that we are given random independent samples of the form y = Ax for x ∼. Suppose furthermore thatis τ-nice, as defined above, for τ < c^* for some universal constant c^*.Then there is a randomized spectral algorithm that recovers orthonormal vectors b_1,…,b_k ∈^d for k ≥ 0.99 n with b_i,a_i^2 ≥ (1 - O(τ)), and with high probability requires m = (n^3) samples and time (d^2+ω + n^1+O(τ)d^4 + md^4). The total runtime is thus (n^3 d^4)—in the theorem statement, we write it in terms of the number of samples m in order to separate the time spent processing the samples from the learning phase. We note that the sample complexity bound that we have, m = (n^3), may very well be sub-optimal; we suspect that m = (n^2) is closer to the truth, which would yield a better runtime.We are also able to apply standard whitening operations (as in e.g. <cit.>) to extend our algorithm to dictionaries with linearly-independent, but non-orthonormal, columns, at the cost of polynomially many additional samples. If A ∈^d× n is a dictionary with linearly independent columns, then there is a randomized spectral algorithm that recovers the columns of A with guarantees similar to thm:dict-informal given (n^2 · f(μ)) additional samples, where μ=λ_max(AA^⊤)/λ_min(AA^⊤) is the condition number of the covariance matrix, and f is a polynomial function. To our knowledge, our algorithms are the only remotely efficient dictionary learning algorithms with provable guarantees that permit τ-nice distributions in which the coordinates of x may be correlated by constant factors, the only other ones being the sum-of-squares semidefinite programming based algorithms of <cit.>. § PRELIMINARIESThroughout the rest of this paper, we will denote tensors by boldface letters such as , matrices by capital letters M, and vectors by lowercase letters v, when the distinction is helpful. We will use A^ k/u^ k to denote the kth Kronecker power of a matrix/vector with itself. To enhance legibility, for u ∈^d we will at times abuse notation and use u^ 4 to denote the order-4 tensor u uu u, the d^2 × d^2 matrix (u^ 2)(u^ 2)^⊤, and the dimension d^4 vector u^ 4—we hope the meaning will be clear from context.For a tensor ∈ (^d)^⊗ 4 and a partition of the modes {1,2,3,4} into two ordered sets A and B, we let T_A,B denote the reshaping ofas d^ A-by-d^ B matrix, where the modes in A are used to index rows and the modes in B are used to index columns. For example, T_{1,2}, {3,4} is a d^2-by-d^2 matrix such that the entry at row (i,j) and column (k,ℓ) contains the entry T_i,j,k,ℓ of T. We remark that the order used to specify the modes matters—for example, for the rank-1 tensor = abab, we have that T_{1,2}{3,4} = (ab)(a b)^⊤ is a symmetric matrix, while T_{2,1}{3,4} = (ba)(a b)^⊤ is not. We use T_A,B to denote the spectral norm (largest singular value) of the matrix T_A,B.We will also make frequent use of the following lemma, which states that the distance between two points cannot increase when both are projected onto a closed, convex set.Let ⊂^n be a closed convex set, and let Π:^n → be the projection operator ontoin terms of norm ·_2, i.e. Π(x) _c ∈x-c_2. Then for any x,y ∈^n,x-y_2 ≥Π(x) - Π(y)_2. This lemma is well-known (see e.g. <cit.>), but we will prove it for completeness in app:tools. § TECHNIQUESIn this section we give a high-level overview of the algorithms in our paper, and of their analyses. We begin with the tensor decomposition algorithm, after which we'll explain the (non-trivial) extension to the dictionary learning application. At the very end, we will discuss the relationship between our algorithms and sum-of-squares relaxations.Suppose have a tensor ∈ (^d)^ 4, and that =+ where the signalis a low-rank tensor with orthonormal components, = ∑_i∈[n] a_i^ 4, and the noiseis an arbitrary tensor of noise with the restriction that for any reshaping ofinto a square matrix E, E≤ϵ. Our goal is to (approximately) recover the rank-1 components, a_1,…,a_n ∈^d, up to signs. Failure of Jennrich's algorithmTo motivate the algorithm and analysis, it first makes sense to consider the case when the noise component = 0. In this case, we can run Jennrich's algorithm: if we choose a d^2-dimensional random vector g∼(0,), we can compute the contractionM_g ∑_i,j=1^n g_ij T_ij = ∑_i=1^n g,a_i^ 2· a_i a_i^⊤,where T_ij is the i,jth d × d matrix slice of the tensor . Since the a_i are orthogonal, the coefficients g,a_i^ 2 are independent, and so we find ourselves in an ideal situation—M_g is a sum of the orthogonal components we want to recover with independent Gaussian coefficients. A simple eigendecomposition will recover all of the a_i.On the other hand, when we have a nonzero noise tensor , a random contraction along modes {1,2} results in the matrixM_g = ∑_i=1^n g, a_i^⊗ 2 a_i a_i^⊤ + ∑_i,j=1^n g_ij· E_ij,where the E_ij are d× d slices of the tensor . The last term, composed of the error, complicates things. Standard facts about Gaussian matrix series assert that the spectral norm of the error term behaves like E_{1,2,3}{4}, the spectral norm of a d^3 × d reshaping of E, whereas we only have control over square reshapings such as E_{1,2}{3,4}.[In fact, this observation was crucial in the analysis of <cit.>—in that work, semidefinite programming constraints are used to control the spectral norm of the rectangular reshapings.] These can be off by polynomial factors. If the Frobenius norm ofis _F^2 ≈ϵ^2 d^2, which is the magnitude one would expect from a tensor whose square reshapings are full-rank matrices with spectral norm ϵ, then we have that necessarilyE_{1,2,3}{4}^2 ≥_F^2/(E_{1,2,3}{4})≥ϵ^2 d,since there are at most d nonzero singular values of rectangular reshapings of . In this case, unless ϵ≪ 1/√(d), the components a_ia_i^⊤ are completely drowned out by the contribution of the noise, and so the robustness guarantees leave something to be desired. Basic ideaThe above suggests that, as long as we allow the errorto have large Frobenius norm, an approach based on random contraction will not succeed. Our basic idea is to take , whose error has small spectral norm, and transform it into a tensor ' whose error has small Frobenius norm.Because we do not know the decomposition of , we cannot access the errordirectly. However, we do know that for any d^2 × d^2 reshaping T of ,T = S + E,where S = ∑_i=1^n a_i^ 4, and E≤ϵ. The rank of S is n ≤ d, and all eigenvalues of S are 1. Thus, if we perform the operationT^> = (T-)_+,where (·)_+ denotes projection to the cone of positive semidefinite matrices, we expect that the signal term S will survive, while the noise term E will be dampened. More formally, we know that T has n eigenvalues of magnitude 1 ±ϵ, and d^2 - n eigenvalues of magnitude at most ϵ, and therefore (T^>) ≤ n. Also by definition, T - T^>≤ϵ. Therefore, we have thatS + E = T = T^> + E'with E'≤ϵ, and thusT^> - S = E - E'.Since S,T^> are both of rank at most n, and E',E have spectral norm bounded by , we have thatT^>-S^2_F ≤((S) + (T^>)) ·(E + E')^2≤ 2n · 4ϵ^2.So, the Frobenius norm is no longer an impassable obstacle to the random contraction approach—using our upper bound on the Frobenius norm of our new error Ẽ T^> - S, we have that the average squared singular value of Ẽ_{1,2,3}{4} will beσ^2_avg(Ẽ_{1,2,3}{4}) = Ẽ_F^2/d = O(n/dϵ^2).So while Ẽ_{1,2,3}{4} may have large singular values, by Markov's inequality it cannot have too many singular values larger than O(ϵ).Finally, to eliminate these large singular values, we will project ^>ϵ into the set of matrices whose rectangular reshapings have singular values at most 1—because S is a member of this convex set, the projection can only decrease the Frobenius norm. After this, we will apply the random contraction algorithm, as originally suggested.Variance of Gaussian matrix seriesRecall that we wanted to sample a random d^2-dimensional Gaussian vector g, and the perform the contractionM_g ∑_i,j=1^d g_ij T^>_ij = ∑_i=1^d g,a_i^ 2· a_i a_i^⊤ + ∑_ij g_ij·Ẽ_ij.The error term on the right is a matrix Gaussian series. The following lemma describes the behavior of the spectra of matrix Gaussian series: [See e.g. <cit.>] Let g∼(0,), and let A_1,…,A_k be n× m real matrices. Define σ^2 = max{∑_i A_i A_i^⊤,∑_i A_i^⊤ A_i }. Then( ∑_i=1^k g_i A_i ≥ t ) ≤ (n+m)·exp(-t^2/2σ^2)For us, this means that we must have bounds on the spectral norm of both ∑_ijẼ_ijẼ_ij^⊤ and ∑_ijẼ_ij^⊤Ẽ_ij. This means that if we have performed the contraction along modes 1 and 2, so that the index i comes from mode 1 and the index j comes from mode 2, then it is not hard to verify that ∑_ijẼ_ijẼ_ij^⊤ = Ẽ_{1,2,3}{4}^2, and ∑_ijẼ_ij^⊤Ẽ_ij = Ẽ_{1,2,4}{3}^2. So, we must control the maximum singular values of two different rectangular reshapings of Ẽ simultaneously.It turns out that for us it suffices to perform two projections in sequence—we first reshape ^> to the matrix ^>_{123}{4},project it to the set of matrices with singular values at most 1, and then reshape the result along modes {124}{3}, and project to the same set again. As mentioned before, because projection to a convex set containing S cannot increase the distance from S, the Frobenius norm of the new error can only decrease. What is less obvious is that performing the second projection will not destroy the property that the reshaping along modes {123}{4} has spectral norm at most 1. By showing that each projection corresponds to either left- or right- multiplication of ^>_{123}{4} and ^>_{124}{3} by matrices of spectral norm at most 1, we are able to show that the second projection does not create large singular values for the first flattening, and so two projections are indeed enough. Call the resulting tensor (^>)^≤ 1. Now, if we perform a random contraction in the modes 1,2, we will haveM_g = ∑_ij g_ij(^>)^≤ 1_ij = ∑_i a_i^ 2,g a_i a_i^⊤ + Ê_g,where the spectral norm of Ê_g≤√(log n) with good probability. So, ignoring for the moment dependencies between a_i^ 2,g and E_g, max_i|g,a_i^ 2| > 1.1·E_g with probability at least n^-.2, which will give a_i correlation 0.9 with the top eigenvector of M_g with good probability. Improving accuracy of componentsThe algorithm described thus far will recover components b_i that are 0.9-correlated with the a_i, in the sense that b_i,a_i^2 ≥ 0.9. To boost the accuracy of the recovered components, we use a simple method which resembles a single step of tensor power iteration.We'll use the closeness of our original tensorto ∑_i a_i^ 4 in spectral norm. We let T be a d^2 × d^2 flattening of , and compute the vectorv = T(b_ib_i) = 0.9 · a_i a_i + ∑_j≠ ib_i,a_j^2 a_ja_j + E(b_ib_i).Now, when the vector v is reshaped to a d × d matrix V, the term E(b_i b_i) is a matrix of Frobenius norm (and thus spectral norm) at most ϵ. By the orthonormality of the a_j, the sum of the coefficients in the second term is at most 0.1, and so a_i is ϵ-close to the top eigenvector of V.Recovering every component Because the Frobenius norm of the error in(^>)^≤ 1 is O(ϵ√(n)), there may be a small fraction of the components a_i^ 4 that are “canceled out” by the error—for instance, we can imagine that the error term is = - ∑_i=1^ϵ^2 n a_i^ 4. So while only a constant fraction of the components a_i^ 4 can be more than ϵ-correlated with the error, we may still be unable to recover some fixed ϵ^2-fraction of the a_i via random contractions.To recover all components, we must subtract the components that we have found already and run the algorithm iteratively—if we have found m = 0.99n components, then if we could perfectly subtract them from , we would end up with an even lower-rank signal tensor, and thus be able to make progress by truncating all but 0.1n eigenvalues in the first step.The challenge is that we have recovered b_1,…,b_m that are only (1-ϵ)-correlated with the a_i, and so naively subtracting - ∑_i=1^m b_i^ 4 can result in a Frobenius and spectral norm error of magnitude ϵ√(m)—thus the total error is still proportional to √(n) rather than √(0.1n).In order to apply our algorithm recursively, we first orthogonalize the components we have found b_1,…,b_m to obtain new components b̃_1,…, b̃_m. Because the b_i are close to the truly orthonormal a_i, the orthogonalization step cannot push too many of theb̃_̃ĩ more than O(ϵ)-far from the b_i—in fact, letting B be the matrix whose columns are the b_i, and letting A be the matrix whose columns are the corresponding a_i, we use that A - B_F ≤ O(ϵ)√(m), and that the matrix B̃ with columns b̃_i is closer to B than A. We keep only the b̃_i for which b̃_i^ 4,≥ 1 - O(ϵ), and we argue that there must be at least 0.9m such b̃_i. Let K ⊂ [m] be the set of indices for which this occurred.Now, given that the two sets of orthogonal vectors {b̃_i}_i∈ K and {a_i}_i∈ K are all O(ϵ) close, we are able to prove that∑_i∈ Kb̃_i^ 4 - a_i^ 4≤ O(√(ϵ)).So subtracting the b̃_i^ 4 will not introduce a large spectral norm! Since we will only need to perform this recursion O(log n) times, allowing for some leeway in ϵ (by requiring √(ϵ)log n = o(1)), we are able to recover all of the components. Dictionary learningIn the dictionary learning problem, there is an unknown dictionary, A ∈^d× n, and we receive independent samples of the form y = Ax for x ∼ for some distributionover ^n. The goal is, given access only to the samples y, recover A.We can use our tensor decomposition algorithm to learn the dictionary A, as long as the columns of A are linearly independent. For the sake of this overview, assume instead that the columns of A are orthonormal. Then given samples y^(1),…,y^(m) for m = (n), we can compute the 4th moment tensor to accuracy ϵ in the spectral norm,1/m∑_j=1^m (y^(j))^ 4≈_x∼[(Ax)^ 4]If the right-hand side were close to ∑_i a_i^ 4 in spectral norm, we would be done. However, for almost any distributionwhich is supported on x with more than one nonzero coordinate, this is not the case. If we assume that [x_ix_jx_kx_ℓ] = 0 unless x_ix_jx_kx_ℓ is a square, then we can calculate that any square reshaping of this tensor will have the form_x∼[(Ax)^ 4]= ∑_i [x_i^4] · a_i^ 4 + ∑_i≠ j[x_i^2 x_j^2] · ( a_ia_j^⊤ a_ia_j^⊤ + a_ia_i^⊤ a_ja_j^⊤ + a_ia_j^⊤ a_ja_i^⊤)If [x_i^4] = [x_j^4] for all i,j, then the first term on the right is exactly the 4th order tensor that we want. The second term on the right can be further split into three distinct matrices, one for each configuration of the a_i,a_j. The a_ia_i^⊤ a_ja_j^⊤ term and the a_ia_j^⊤ a_ja_i^⊤ terms can be shown to have spectral norm at most max_i≠ j[x_i^2x_j^2], and so as long as we require that max_i≠ j[x_i^2x_j^2]≪ϵ[x_k^4], these terms have spectral norm within the allowance of our tensor decomposition algorithm.The issue is with the a_ia_j^⊤ a_ia_j^⊤ term. This term factors into (a_i a_i)(a_j a_j)^⊤, and because of this the entire sum ∑_i≠ j[x_i^2x_j^2] · a_ia_j^⊤ a_ia_j^⊤ has rank at most n≪ d^2, but Frobenius norm as large as max_i≠ j[x_i^2 x_j^2] · n. If we require that the coordinates of x∼ are independent, then we can see that this is actually close to a spurious rank-1 component, which can be easily removed without altering the signal term too much.[As was done in <cit.>, for example, albeit in a slightly different context.] However, if we wish to let the coordinates of x exhibit correlations, we have very little information about the spectrum of this term.In the sum-of-squares relaxation, this issue is overcome easily: because of the symmetries required of the SDP solution matrix X, X,a_ia_j^⊤ a_ia_j^⊤ = X, a_ia_i^⊤ a_ja_j^⊤, so by linearity this low-rank error term cannot influence the objective function any more than the a_ia_i^⊤ a_ja_j^⊤ term.Inspired by this sum-of-squares analysis, we remove these unwanted directions as follows. Given the scaled moment matrix M = 1/[x_1^4]_x∼[(Ax)^ 4] (where the scaling serves to make the coefficients of the signal 1), we truncate the small eigenvalues:M^> = (M_{1,2}{3,4} - )_+.This removes the spectrum in the direction of the “nice” error terms, corresponding to a_ia_i^⊤ a_ja_j^⊤ and a_ia_j^⊤ a_ja_i^⊤. The fact that the rest of the matrix is low-rank means that we can apply an analysis similar to the analysis in the first step of our tensor decomposition to argue that M^>-∑_i a_i^ 4_F ≤M- ∑_i a_i^ 4_F.Now, we re-shape M^>, so that if initially we had the flattening → M_{1,2}{3,4}, we look at the flattening M^>_{1,3}{2,4}. In this flattening, the term a_ia_j^⊤ a_ia_j^⊤ from M^> is transformed to a_ia_i^⊤ a_ja_j^⊤, and so the problematic error term from the original flattening has spectral norm ϵ in this flattening! Applying the projection(M^>_{1,3}{2,4} - )_+eliminates the problematic term, and brings us again closer to the target matrix ∑_i a_i^ 4. Thus, we end up with a tensor that is close to ∑_i a_i^ 4 in Frobenius norm, and we can apply our tensor decomposition algorithm. Connection to sum-of-squares algorithms We take a moment to draw parallels between our algorithm and tensor decomposition algorithms in the sum-of-squares hierarchy. In noisy orthogonal tensor decomposition, we want to solve the non-convex programX, s.t.{X∈^d^2× d^2, X ≽ 0, X= 1, (X) = n,X ∈{u^ 4 : u ∈^d}}The intended solution of this program is X = (after which we can run Jennrich's algorithm to recover individual components). At first it may not be obvious that the maximizer of the above program is close to , but for any unit vector x ∈^d,x^ 4,= ∑_i=1^n x,a_i^4 + (x x)^⊤ E (xx).The error term is at mostby our bound on E, and the first term is x^⊤ A_4^4, where A is the matrix whose columns are the a_i. Since by the orthonormality of the a_i, x^⊤ A_2 ≤ 1, and the ℓ_4 norm is maximized relative to the ℓ_2 for vectors supported on a single coordinate, the x that maximize this must be ϵ-close to one of the a_i. In conjunction with the X≤ 1 constraint and the (X) = n constraint, we have that X := is the maximizer.The (somewhat simplified) corresponding sum-of-squares relaxation is the semidefinite programmaxX,s.t.{X ∈^d^2× d^2, X ≽ 0, X≤ 1, X^2_F = n,X_ijkℓ = X_π(ijkℓ) ∀π∈_4},[ The program is actually over matrices indexed by all subsets of d of size at most 2, d≤ 2, but for simplicity in this description we ignore this (and the interaction with the SOS variables corresponding to lower-degree monomials or lower-order moments).]The constraints X_F^2 = n and X_ijkℓ = X_π(ijkℓ) together are a relaxation of the constraint that X be a rank-n matrix in the symmetric subspace {u^ 4}. Further, the constraint X≤ 1 is enforced in every rectangular d × d^3 reshaping of X (this consequence of the SOS constraints is crucially used in <cit.>).To solve this semidefinite program, one should projectinto the intersection of all of the convex feasible regions of the constraints. However, projecting to the intersection is an expensive operation in terms of runtime. Instead, we choose a subset of these constraints, and projectinto the set of points satisfying each constraint sequentially, rather than simultaneously. These are not equivalent projection operations, but because we select our operations carefully, we are able to show that ouris close to the SDP optimum in a sense that is sufficient for successfully running Jennrich's algorithm.In the first step of our algorithm, we change the objective function from X, to X, -. In the sum-of-squares SDP, this does not change the objective value dramatically—because the original objective value is at least n, and because the Frobenius norm constraint X_F^2 = n constraint in conjunction with the sum-of-squares constraints implies that X, =n. Therefore this perturbation cannot decrease the objective by more than a multiplicative factor of ϵ. Then, we project a square reshaping of the objective to the PSD cone, (T - )_+—this corresponds to the constraint that X ≽ 0.[For a proof that truncating the negative eigenvalues of a matrix is equivalent to projection to the PSD cone in Frobenius norm, see fact:projcont.] Finally, we project first to the set of matrices that have spectral norm at most 1 for one rectangular reshaping, then repeat for another rectangular reshaping. So after perturbing the objective very slightly, then choosing three of the convex constraints to project to in sequence, we end up with an object that approximates the maximizer of the SDP in a sense that is sufficient for our purposes.Our dictionary learning pre-processing can be interpreted similarly. We first perturb the objective function by ϵ, and project to the PSD cone. Then, in reshaping the tensor again, we choose another point that has the same projection onto any point in the feasible region (by moving along an equivalence class in the symmetry constraint). Finally, we perturb the objective by ϵ again, and again project to the PSD cone. § DECOMPOSING ORTHOGONAL 4-TENSORSRecall our setting: we are given a 4-tensor ∈ (^d)^⊗ 4 of the form =+ whereis a noise tensor and = ∑_i=1^n a_i^⊗ 4 for orthonormal vectors a_1,…,a_n. (We address the more general case of nearly orthonormal vectors in sec:northo.). First, we have a pre-processing step, in which we go from a tensor with low spectral norm error to a tensor with low Frobenius norm error.[We are approximating T = S+E with (S) = n ≪ d^2, and E≤ϵ, so for example if E = then we may have T - S_F^2 = ϵ^2d^2, which is too large for us.]Suppose that for some square reshaping E of(without loss of generality along modes {1,2},{3,4}), E≤ϵ. Say we are given access to =+, and we produce the matrix T^> = (T - ϵ·)_+ as described in alg:preproc. Then T' - S_F ≤ 2ϵ√(2n). This operation requires time (min{nd^4,d^2+ω}). Because E≤ϵ, T = S + E has only n eigenvalues of magnitude more than ϵ. So (T^>)≤ n, and therefore (T^> - S)≤ 2n. Furthermore, S + E = T = T^> + Ẽ for a matrix Ẽ of spectral norm at most ϵ. Therefore T^> - S≤ 2ϵ, and T^> - S_F ≤ 2ϵ√(2n). To compute T^>ϵ, we can compute the top n eigenvectors of T. Since O(ϵ)·λ_n ≥λ_n+1, where λ_n,λ_n+1 are the nth and (n+1)st eigenvalues, we can compute this in time (min{nd^4,d^2+ω}) via subspace power iteration (see for example <cit.>). Now, we can run our main algorithm: Under the appropriate conditions on , with probability (n^-), alg:orthog will output a vector that is 0.9-correlated with a_j^⊗ 2 for some j∈[n]. Suppose we are given a 4-tensor ∈ (^d)^ 4, and = ∑_i a_i^⊗ 4 + E, where a_1,…,a_n ∈^d are orthonormal vectors and E_F ≤η√(n).Then running alg:orthog (n^1+O(η)) times allows us to recover m ≥ 0.99n unit vectors u_1,…,u_m∈^d such that for each i ∈ [m], there exists j ∈ [n] such thatu_i,a_j^2 ≥ 0.99.Recovering one component requires time (d^2+ωn^O(η)), and recovering m components requires time (max{mn^O(η)d^4,d^2+ω}). We can then post-process the vectors to obtain a vector that has correlation 1-ϵ with a_j—the details are given in sec:fullrecovery below.We will prove thm:orthog momentarily, but first, we bring the reader's attention to a nontrivial technical issue left unanswered by thm:orthog. The issue is that we can only guarantee that alg:orthog recovers 0.99 n of the vectors, and the set of recoverable vectors is invariant under the randomness of the algorithm. That is, as a side effect of the error-reducing step 2, thepart ofmay also be adversely affected. For that reason, in Õ(n) runs of alg:orthog, we can only guarantee that recover a constant fraction of the components, and we must iteratively remove the components we recover in order to continue to make progress. This removal must be handled delicately to ensure that the Frobenius norm of the error shrinks at each step, so the conditions of thm:orthog continue to be met (for shrinking values of n). The overall algorithm, which uses alg:orthog as a subroutine, will be given in sec:fullrecovery.We will now prove the correctness of alg:orthog step-by-step, tying details together at the end of this subsection. First, we argue that in step 1, the truncation of the large eigenvalues cannot increase the Frobenius norm of the error. Suppose that we define ^≤ 1 to be the result of projecting =+ to the set of tensors whose rectangular reshapings along modes {1,2,3},{4} have spectral norm at most 1, then projecting the result to the set of tensors whose rectangular reshapings along modes {1,2,3},{4} have spectral norm at most 1. Then ^≤ 1 - _F ≤E_F, and^≤ 1_{1,2,3}{4}≤ 1, and^≤ 1_{1,2,4}{3}≤ 1 This operation requires time (d^2+ω). To establish the first claim, we note that the tensor ^≤ 1 was obtained by two projections of different rectangular reshapings of the matrix S+E to the set of rectangular matrices with singular value at most 1. This set is closed, convex, and contains S, and so the error can only decrease in Frobenius norm (see lem:proxproj in app:tools for a proof),^≤ 1 - _F ≤ - _F = _F. It is not hard to see that each projection step can be accomplished by reshaping the tensor to the appropriate rectangular matrix, then truncating all singular values larger than 1 to 1. Now, we establish the remaining claims. For convenience, define T̂^≤ 1 to be the matrix (S+E)_{123}{4} after restricting singular values of magnitude >1 to 1. Now, we reshape T̂^≤ 1 to a new matrix B T̂^≤ 1_{1,2,4}{3}, which has the d^2 blocks B_1 = (T̂^≤ 1)_1^⊤,…,B_d^2=(T̂^≤ 1)_d^2^⊤. Say that the singular value decomposition of B is B = U Σ V^⊤. Define Σ to be the diagonal matrix with entries equal to those of Σ when the value is < 1 and with ones elsewhere. When we truncate the large singular values of B this is equivalent to multiplying by P = UΣU^⊤. The result is the matrix PB, with blocks PB_1 = P(T̂^≤ 1)_1^⊤,…,PB_d^2 = P(T̂^≤ 1)_d^2^⊤. By definition, the singular values of PB = T^≤ 1_{3}{1,2,4} are at most 1. Also, T^≤ 1_{4}{123} = T̂^≤ 1(P⊗_d^2), and by the submultiplicativity of the norm, T̂^≤ 1(P⊗_d^2)≤T̂^≤ 1·P⊗_d^2≤ 1, and the first reshaping still has spectral norm at most 1.Finally, each reshaping step takes O(d^4) time. Since we are only interested in truncating large singular values, it suffices for us to compute the SVD corresponding to singular values between √(n) and 1. This can be done via subspace power iteration, which here involves the multiplication of a d^3 × d matrix and a d × d matrix (with intermediate orthogonalization steps for the d × d matrix, see <cit.>), which requires time (d^2+ω), where ω is the matrix multiplication constant. Going forward the representation of the matrix will be as the original matrix, with the subtracted SVD corresponding to large singular values. We will need to argue that if the Frobenius norm of the error matrix is small, this is a sufficient condition under which we succeed. For this, we will use the following two lemmas. The first tells us that with probability (n^-O()) over the choice of g, we will have for some i∈[n] thatM_g = c· a_i a_i^⊤ + N,where |c| ≥N and furthermore Na_i and N^⊤a_i are small: Let g ∼(0,_d^2). Suppose that T_{1,2,3}{4},T_{1,2,3}{4}≤ 1, and that T - ∑_i a_i^ 4_F ≤ϵ√(n). Define the matrix M_g to be the flattening of = ∑_i a_i^⊗ 4 + along g in the modes {1} and {2}, and let |c| be the magnitude of a_j's projection onto M_g, i.e.M_g ∑_j=1^d^2e_j,g· T_j = c · a_j a_j^⊤ + NThen for a 1-3δ fraction of j ∈ [n],_g[ |c| ≥ (1+β)N, N^⊤a_j,Na_j≤ (ϵ/δ)(c+√(2)+o(1))] = (n^-(1+β/1-(1+β)ϵ/δ)^2)In particular, if δ = Ω(1), β = O(ϵ), β < 1, then this probability is (n^-(1+O(ϵ))).The proof consists primarily of the application of concentration inequalities, and we provide it below in sec:supporting.The second lemma states that if M_g indeed has the form above, the top singular vectors of M_g must be close to the component a_i.Let M_g be an n × n matrix, and a_1 ∈^n, and suppose thatM_g = c · a_1 a_1^⊤ + Nwith |c| ≥ (1+β)N for β > 0, and N a_1, N^⊤ a_1≤ϵ|c| so that the relationship 2ϵ(1+β)/β <0.01 holds. Then letting u be a top singular vector of M_g, it follows thatu,a_1^2 ≥ 0.99. The proof requires some careful calculations, but is not complicated, and we will prove it below in sec:supporting.Finally, we are ready to stitch these arguments together and prove that alg:orthog works. After reshaping and truncatingin step 1 of the algorithm, by lem:step2 the matrix T^≤ 1 = ∑_i a_i^⊗ 4 + E has the properties that T^≤ 1_{1,2,3}{4}, T^≤ 1_{1,2,4}{3}≤ 1, and also that still E_F ≤η√(n). We can now apply lem:algsuccess with δ = 1/300 and β = 400 η/δ = O(η) to conclude that for at least a 0.99-fraction of the i∈[n], with probability at least (n^-1 - O(η)), we will have M_g = c · a_i a_i^⊤ + N, where N≤ (1+β)c and Na_i,N^⊤ a_i≤ 48η· |c|. Applying lem:topeig, we have that either the left- or right- top unit singular vector u of M_g has correlation at least u, a_i^2 ≥ 0.99, as desired. For runtime, by our arguments in lem:step2 step 1 takes time (d^2+ω). After this, with either representation of our matrix T^≤ 1 (whether we compute the full truncated SVD or have the original matrix minus the subtracted SVD), performing power iteration to find the top eigenvector with the flattening M_g takes time (d^4), and finding a single component takes (n^-O(η)) samples of random contractions. Since we can reuse T^≤ 1 with new random contractions, the total runtime for recovering one component is (d^2+ωn^-O(η)), and by the independence of the runs recovering m components requires (d^2+ω) + mn^-O(η)·(d^4) time.With the core of our algorithm in place, we now take care of the remaining technical issues: recovery precision, working with near-orthonormal vectors, and recovering the full set of component vectors. §.§ Postprocessing for closer vectors Because the precision of recovery will be important in not amplifying the error, we begin with our precision-amplifying postprocessing algorithm. Suppose that v is a unit vector with v,a_i^2 ≥ 0.99, and T = ∑_i a_i^ 4 + E for E≤ϵ and a_1,…,a_n orthonormal. Then if we let A be the reshaping of T(v v) to a d× d matrix, and if we let u_L,u_R be the top left- and right- unit singular vectors of M, then u_L,a_i^2 ≥ 1-3ϵoru_R,a_i^2 ≥ 1-3ϵ. In other words, alg:post succeeds. Further, the time required is (d^4). For convenience and without loss of generality, let i := 1, and let α 1 - a_1,v^2 ≤ 0.01. Because a_1,…, a_n are orthonormal, we can write v = ∑_j a_j,v· a_j + w, where w ⊥ a_j for all j ∈ [n]. By assumption, a_1,v^2 ≥ 1-α, and therefore ∑_j>1v,a_j^2 + w^2 ≤α. Now, (T-E) (vv) = ∑_j a_j^ 2(a_j^ 2)^⊤(∑_ja_j,v a_j ∑_ja_j,v a_j)= ∑_j,k,ℓ a_j^ 2a_j,a_ka_j,a_ℓa_k,va_ℓ,vby the orthonormality of the a_i, = ∑_j a_j^ 2a_j,v^2 Therefore, defining M to be the n × n reshaping of T(v v) and defining N to be the n × n reshaping of E(v v), M = (1-α)a_1a_1^⊤ + ∑_j > 1a_j,v^2 a_j a_j^⊤ + N, where N_F ≤ϵ, since E≤ϵ. Now, we have that M≥1-α - ϵ, and that if we choose η so that 1-η≥ 2α≥ 1/50, M - η· a_1a_1^⊤≤ 1-α- η + ϵ. Thus, M - η a_1 a_1^⊤≤M - η + 2ϵ, and we have by fact:topeig (see app:tools for a proof) that the top unit eigenvector u of M is such that u,a_1^2 ≥η - 2ϵ/η≥ 1 - 2ϵ/η. Choosing η = 49/50, the result follows. §.§ Near-orthonormal components We'll now dispense with the discrepancy between the orthonormal and near-orthonormal cases.If S = ∑_i a_i a_i^⊤ =+ E for E≤ϵ, then ã_1 =S^-1/2a_1,…,ã_n = S^-1/2a_n are orthonormal, ã_̃ĩ,a_i^2 ≥ (1-ϵ)a_i^2, and∑_i ã_i^ 4 - a_i^ 4≤ 4√(ϵ).The fact that the ã_i are orthonormal follows because they are independent and have Gram matrix . Using the fact that the eigenvalues of S are between (1-ϵ)^-1/2 and (1+ϵ)^-1/2, quantity a_i,ã_i^2 = (a_i^⊤ S^-1/2a_i)^2 ≥a_i^2/1+ϵ≥a_i^2(1-ϵ). Finally,∑_i ã_i^ 4 - a_i^ 4 = (S^-1/2)^ 2(∑_i a_i^ 4)(S^-1/2)^ 2-∑_i a_i^ 4and because (S^-1/2)^ 2 - ≤√(ϵ), and ∑_i a_i^ 4≼∑_ij a_ia_i^⊤ a_ja_j^⊤, this difference has spectral norm at most 3ϵ∑_i a_i^ 4≤ 3√(ϵ) (1+ϵ)^2 ≤ 4√(ϵ).§.§ Full RecoveryNow we give the full algorithm, which will remove the components we find in each step from the tensor without amplifying the spectral norm of the error too much.Given = ∑_i=1^n a_i^ 4 + E where the a_i are orthonormal and E_{1,2}{3,4}≤ϵ, then if ϵ < O(η^2/log^2 n), with probability 1-o(1), alg:overall recovers orthonormal vectors b_1,…,b_n so that there exists a permutation π:[n]→ [n] such that for each i ∈ [n],a_i,b_π(i)^2 ≥ 1-3ϵ.Furthermore, this requires runtime (n^1+O(η)d^2+ω). First, we prove that if we have an orthonormal basis that approximates a_1,…,a_k, we can subtract it without introducing a large spectral norm error—this motivates and justifies steps 3(b)–3(e). Let a_1,…,a_k∈^d and b_1,…,b_k∈^d be two sets of orthonormal vectors, such that a_i,b_i^2 ≥ 1-ϵ. Then∑_i a_i^ 4 - b_i^ 4_2 ≤ 4√(ϵ)Define the matrices U,V ∈^d^2 × k so that the ith column of U (or V) is equal to a_i^ 2 (b_i^ 2 respectively). We have that∑_i a_i^ 4 - b_i^ 4 = UU^⊤ - VV^⊤ = (U-V)(U+V)^⊤.So it suffices for us to bound U-V·U+V.By the subadditivity of the norm, U+V≤U + V = 2. Meanwhile, the singular values of U-V are the square roots of the eigenvalues of (U-V)^⊤(U-V), and so we bound(U-V)^⊤ (U-V) = UU^⊤ + VV^⊤ - U^⊤ V - V^⊤ U= 2_k - U^⊤ V - V^⊤ Uwhere the second line follows because U and V have orthonormal columns. Now, by assumption we know thatU^⊤ V = (1-ϵ)·_k + E,where for i≠ j, E_ij = b_i,a_j^2 and E_ii = b_i,a_i^2 - (1-ϵ), and by the orthonormality of the a_j,∑_j |E_ij| = ∑_j b_i,a_j^2 = ϵ.So the 1-norm of the rows of E is at most ϵ. By the orthonormality of the b_j, the same holds for the 1-norm of the columns, ∑_i |E_ij|. It follows that E≤ϵ. Therefore,U^⊤ V + V^⊤ U = 2(1-)_k + Ê,where Ê≤ 2ϵ. Returning to eq:sqbd, we can conclude that (U-V)≤ 2√(ϵ), and we have our result. Now, we will prove that by orthogonalizing, we do not harm too many components b̃_i.Suppose a_1,…,a_k ∈^d are orthonormal vectors, and u_1,…,u_k ∈^d are unit vectors such that u_i - a_i_2^2 ≤ϵ. Let U be the d× k matrix whose ith column is u_i, U = XΣ Y be the singular value decomposition of U, and let ũ_i = XΣ^-1X^⊤ u_i. Then for a 1-δ fraction of i∈[k],u_i,ũ_i≥ 1-ϵ/2δ.For convenience, let A be the d × k matrix whose ith column is a_i, and let Ũ = XΣ^-1X^⊤ U, and ũ_i = XΣ^-1X^⊤ u_i. Let 𝕏 be the space of all real d× k real matrices with orthonormal columns, and notice that Ũ, A ∈𝕏 and that Ũ is closer to U than A. Indeed, for any matrix X with orthonormal columns,X - U_F^2 = k + U_F^2 - 2 U,X ≥ k + U_F^2 - 2XU_* = k + U_F^2 - 2U_* = Ũ - U_F^2,Where we have used that the spectral norm and nuclear norm are dual. Therefore,Ũ-U_F^2 ≤ A - U_F^2 = ∑_i a_i - u_i_2^2 = ϵ· kAnd on average, ϵ≥ũ_i - u_i_2^2, so by Markov's inequality, for at least (1-δ)k of the u_i, u_i,ũ_i≥ 1 - ϵ/2δ. Finally, we are ready to prove that alg:overall works. We claim that in the tth iteration of step 3, with high probability we have at most 0.45^tn components remaining to be found, and that _work^(t) =+ F^(t) where F^(t)≤ 8t √(ϵ). For t = 0, this is easily true.Now assume this holds for t, and we will prove it for t+1. Since by assumption t√(ϵ)log n ≤ 100√(ϵ)log n ≤ O(η), applying lem:step1 and thm:orthog to the running of preprocessing alg:preproc and the main step alg:orthog with _work^(t) and lem:postproc to the running of the postprocessing alg:post with _clean, in step 3(a) with high probability we will find m ≥ 0.9n_t vectors b_1,…,b_m so that b_i,a_i^2 ≥ 1-3ϵ. Furthermore, this takes a total of (m n^O(η) d^2+ω) time.By lem:orthonormalize, in step 3(c) we will remove no more than a half of the b̃_i, while maintaining b̃_i,a_i^2 ≥ 1-3ϵ (where we are abusing notation by re-indexing conveniently), so that n_t+1≥ 0.45 n_t. Finally, by lem:subtraction, we have that∑_i=1^|B̃| a_i^ 4 -b̃_̃ĩ^ 4_2 ≤ 4√(3ϵ),So that in step 3(e),_work^(t+1) = (_work^(t) - ∑_i=1^|B̃| a_i^⊗ 4) + (∑_i=1^|B̃| a_i^⊗ 4 - ∑_b̃_i ∈B̃b̃_i^ 4)= _work^(t-1) + F,for a matrix F with F≤ 8√(ϵ). By induction, this implies that _work^(t+1) =+ F^(t+1) where F^(t+1)≤F +F^(t)≤ (t+1)8√(ϵ). Taking a union bound over the high-probability success of alg:orthog, we have that after t = O(log n) steps we have found all of the components. We have spent a total of (n^1+O(η)d^2+ω) time in step 3(a). Finally, steps 3(b)-3(e) of alg:overall require no more than (d^3) time, and since the entire loop runs (1) times, we have our result. §.§ Supporting Lemmas Now we circle back and prove the omitted supporting lemmas. For convenience, fix j := 1. Let g^(1) be the component of g in the direction a_1^⊗ 2, and let g^(>1) be the component of g orthogonal to a_1^⊗ 2. Notice that g^(1),g^(>1) are independent.By the orthogonality of the a_i, our matrix M_g can be written asM_g = g^(1),a_1^⊗ 2· a_1 a_1^⊤ + ∑_j=1^d^2 (g^(1)_j+g^(>1)_j) · (S - a_i^⊗ 4 + E)_j= g^(1),a_1^⊗ 2· a_1 a_1^⊤ + (∑_j=1^d^2 g^(1)_j · E_j) + (∑_j=1^d^2 g^(>1)_j · T_j)where T_j is the jth matrix slice of T. For convenience, we can refer to the two sums on the right asN =(∑_j=1^d^2 g^(1)_j · E_j) + (∑_j=1^d^2 g^(>1)_j · T_j).First, we get a lower bound on the probability that the coefficient of a_1a_1^⊤ is large. Let _1(α) be the event that |g^(1),a_1^⊗ 2| = g^(1)≥√(2αlog n). By standard tail estimates on univariate Gaussians, we have that[_1(α)] ≥Õ(n^-α). Now, we bound N. Define the event _>1(ρ) to be the even that_>1(ρ) {∑_j=1^d^2 g^(>1)_j · T_j≤√(2(1+ρ)log d)}By lem:gflat, we can conclude that[_>1(ρ)] ≥ 1 - d^-ρ To bound N, it thus remains to understand the term∑_j g_j^(1)E_j = g,a_1^⊗ 2·∑_j a_i^⊗ 2(j) · E_j = g,a_1^⊗ 2· (a_ia_i^⊤⊗_d^2)E,where the quantity (a_ia_i^⊤⊗_d^2)E corresponds to the contraction of E along two modes by the vector a_i^⊗ 2. We make the following observation: If P_1,…,P_n are orthogonal projections from ^n^4→ K for some convex set K, then for a 1-δ fraction of i ∈ [n], P_iE_F ≤ϵ/δ. This follows from the fact that ϵ^2 n ≥E_F^2 ≥∑_i P_i E_F^2, and then by an application of Markov's inequality.Now, note that ∑_j a_ia_i^⊤_d^2 for i ∈ [n] are orthogonal projectors from ^n^4 to ^n^2. Thus it follows that for a 1-δ fraction of i ∈ [n], and without loss of generality assuming that i=1 is among them, (a_1a_1^⊤⊗_d^2)E_F ≤ϵ/δ. Therefore for any unit vectors u,v ∈^d, returning to eq:equality,|u^⊤(∑_j=1^d^2 g_j^(1)· E_j)v| = |g,a_1^⊗ 2·uv^⊤, (a_1a_1^⊤_d^2)E|≤g^(1)·(a_1a_1^⊤⊗_d^2)E_F ·uv^⊤_F ≤ ϵ/δg^(1)Thus, combining the above we have a two-part upper bound on N.Finally, define the event _a_1,E(θ) to be the event that_a_1,E(θ) {(∑_j=1^d^2 g^(>1)_j · T_j)a_1_2, (∑_j=1^d^2 g^(>1)_j · T_j)^⊤ a_1_2 ≤ϵ/δ·√(2(1+θ))}Examining this form, we can split(∑_j=1^d^2 g^(>1)_j · T_j)a_1 =∑_j=1^d^2 g^(>1)_j · (S-a_1^⊗ 4)_ja_1 +∑_j=1^d^2 g^(>1)_j · E_ja_1=∑_j=1^d^2 g^(>1)_j · E_ja_1where the last line follows because the a_i are orthogonal. We note that ∑_j g^(>1)_j E_j a_1 is a Gaussian contraction of the form (a_1 _d^3)E. Again appealing to obs:frobs and to the fact that the a_i _d^3 are orthogonal projections, we conclude that for a 1-δ fraction of i ∈ [n], (a_1 _d^3)E_F ≤ϵ/δ. Without loss of generality we assume that this is true for i=1, from which it follows by lem:gflat that[∑_j=1^d^2 g^(>1)_j · E_ja_1_2 ≤ϵ/δ√(2(1+θ))] ≥ 1 - d^-θ.We can apply the same arguments to (∑_j g_j^(>1)E_j)^⊤ a_1, and we conclude that[_a_1,E(θ)] ≥ 1-2d^-θ. Now, by the union bound _>1(ρ) and _a_1,E both occur with probability at least 1-d^-ρ-2d^-θ. Also, we notice that _>1∪_a_1,E and _1 are independent. Therefore, for ρ,θ≥loglog n/log n,[_1(α),_>1(ρ),_a_1,E(θ)] ≥Õ(n^-α)Conditioning on _>1 and _1,M_g = c · a_1 a_1^⊤ + N,where |c| ≥√(2αlog n), and N is a matrix of norm at most (ϵ/δ) c + √(2(1+ρ)log d), such that N a_1, N^⊤ a_1≤ (ϵ/δ) (c+√(2(1+θ))).We now set α so that |c| ≥βN. This occurs whenα ≥(β1/1-β(ϵ/δ))^2(1+ρ).Choosing β = 1 + β', ρ,θ = loglog n/log n, we have our conclusion for a_1, and by symmetry for all other a_i in the 1-3δ fraction of i ∈ [n] for which the Frobenius norms of the contractions are small. Assume without loss of generality that c ≥ 0. Choose κ = 2ϵ|c|/δ≤ |c|(1-1/1+β). When κ· a_1a_1^⊤ is subtracted from M_g, then given any unit vector v ∈^d with |v,a_1| = α, we can write v = α a_1 + w where w,a_1 = 0 and w = √(1-α^2). Examining the action of M_g - κ a_1a_1^⊤ on v,(M_g - κ·a_1 a_1^⊤)v_2^2 =(c - κ)α a_1 + Nv_2^2 ≤ (c-κ)^2α^2 + (c-κ)α a_1^⊤ N v + (c-κ)α v^⊤ N a_1 + v^⊤ N^⊤ N vapplying the Cauchy-Schwarz inequality and our bounds on N^⊤ a_1,Na_1, ≤ (c-κ)^2α^2 + 2(c-κ)αϵ c + v^⊤ N^⊤ N v Now expanding the v^⊤ N^⊤ N v term along the components of v,v^⊤ N^⊤ N v = (α a_1 + w)^⊤ N^⊤ N (α a_1 + w)≤ (αNa_1 + wN)^2 and since w = √(1-α^2), Na_1≤ϵ c, and N≤ c/(1+β)≤ c(1-β+2β^2), ≤(αϵ c + √(1-α^2)(1-β+2β^2)c)^2and putting these together,(M_g - κ·a_1 a_1^⊤)v_2^2 ≤ (c-κ)^2α^2 + 2(c-κ)αϵ c + (αϵ c + √(1-α^2)/1+βc)^2It is easy to see that when c/(1+β) < c-κ, this quantity is maximized at α = 1, and so by our choice of κ we have that(M_g - κ·a_1 a_1^⊤)v_2^2 ≤ (c-κ)^2 +2ϵ c(c-κ) +ϵ^2 c^2 = (c(1+ϵ) -κ)^2and thus M_g -κ a_1a_1^⊤≤ (1+ϵ)c - κ.Now we will lower bound M_g.M_g ≥ a_1^⊤M_ga_1 = c + a_1^⊤Na_1≥ c -a_1Na_1≥ c(1-ϵ) Where we have applied the Cauchy-Schwarz inequality, and the assumption that Na_1≤ϵ c. It follows that M_g - κ a_1 a_1^⊤≤M_g + 2ϵ c - κ.Finally applying fact:topeig, we can conclude that for either the left- or right-singular unit vector u of M_g,a_i,u^2 ≥κ - 2ϵ c/κ ≥1 - δChoosing δ = 2ϵ(1+β)/β as small as possible, we have our result.§ LEARNING ORTHONORMAL DICTIONARIESHere, we show how to use our tensor decomposition algorithm to learn dictionaries with orthonormal basis vectors. Given access to a dictionary A ∈^d× d with independent columns a_1,…,a_d, in the form of samples y^(1) = Ax^(1),…,y^(m) = Ax^(m) for independent x^(i), recover A. Below, in sec:samplecomplex, we will prove that (n^3) samples suffice to estimate the 4th moment tensor within o(1) spectral norm error. Computing this matrix from (n^3) samples takes (n^3d^4) time. Thus we can equivalently formulate the problem as follows: Given access to A ∈^d× n with independent columns a_1,…,a_n, via a noisy copy of the 4th moment tensor = [(Ax)^ 4] + E, recover the columns a_1,…, a_n. Finally, given access to a sufficiently large number of samples, we can reduce to the case where the columns of A are orthogonal:Suppose that the samples y = Ax are generated from a distribution over x for which [x_i^2] = [x_j^2] for all i,j∈[n], and that the columns of A are independent. Then there exists an efficient reduction from the case when A has independent columns to the case when A has orthogonal columns, with sample complexity growing polynomially with the condition number of Σ = AA^⊤.The proof is straightforward, involving a transformation by the empirical covariance matrix, and we give it below in sec:indtoorth.Now, we reduce the dictionary learning problem to tensor decomposition. In sec:techniques, we explained that the4th moment tensor itself may be far from our target tensor ∑_i a_i^ 4, due to the presence of a low-rank, high-Frobenius norm component. The sum-of-squares algorithm for this problem can overcome this difficulty by exploiting the SDP's symmetry constraints. In our algorithm, we will exploit this symmetry manually to go from [(Ax)^ 4] to a tensor that approximates ∑_i a_i^ 4 well in Frobenius norm.Our claim is that this produces a tensor that is close to ∑_i[x_i^4]· a_i^ 4 in Frobenius norm. If the x are independent and distributed so that [x_ix_jx_kx_ℓ] = 0 unless x_ix_jx_kx_ℓ is a square, and so that for all i,j∈[n], [x_i^2x_j^2]≤α[x_1^4] for α < 1, then given access to = [(Ax)^ 4] + E where E≤α, alg:cleanmoment with ϵ = 3α returns a tensor T̃ such thatT̃ - ∑_i [x_i^4] · a_i^ 4_F ≤ 9α√(n). For convenience denote by T([(Ax)^ 4] + E)_{1,2}{3,4}, and define S = ∑_i[x_i^4] · a_i^ 4. We have thatT - E = [(Ax)^ 4] = ∑_i,j,k,ℓ=1^d [x_ix_jx_kx_ℓ]· (a_i a_j)(a_k a_ℓ)^⊤= ∑_i[x_i^4] · a_i^ 4 + ∑_i≠ j[x_i^2x_j^2] ·((a_i^ 2)(a_j^ 2)^⊤ + a_ia_i^⊤ a_ja_j^⊤ +a_ja_i^⊤ a_ia_j^⊤)The latter term can be split into three distinct matrices—the first is a potentially low-rank matrix, and may have large eigenvectors.[For instance, in the case when [x_i^2x_j^2] = p^2, this term is rank-1 and has spectral norm pn.] The latter two terms have small spectral norm. ∑_i≠ j[x_i^2x_j^2] · a_ia_i^⊤ a_ja_j^⊤≤α[x_1^4], and∑_i≠ j[x_i^2x_j^2]· a_ja_i^⊤ a_ia_j^⊤≤α[x_1^4] We'll prove this claim below. Now, again for convenience define the matrix N to be the remaining term, N = ∑_i≠ j[x_i^2x_j^2] (a_i^ 2)(a_j^ 2)^⊤. From claim:spectrals and by our assumption on E,T = S + N + Ê,for Ê≤ 3α. On the other hand, if we let B be the d × n matrix whose ith column is a_i^ 2, and we let X be the n × n matrix whose i,jth entry is [x_i^2x_j^2], then S + N = BXB^⊤, and so (S+N) ≤ n.It follows that when we perform the eigenvalue truncation in step 1 of alg:cleanmoment,T^< = (T - 3α·)_+,then we have that (T^<) ≤ n as well. Also by definition of truncation, T = T^< + Ẽ, and because to begin with we had T≽ 0, Ẽ≤ 3α. Putting the above together, it follows thatT^< - (S+N)_F = Ẽ - Ê_F ≤ 6α√(2n),where we have used that (T^< - (S+N)) ≤ 2n and Ẽ - E≤ 6α. Now, we recall the reshaping operation on tensors from step 2 of alg:cleanmoment—in going from the reshaping {1,2}{3,4} to {1,3}{2,4}, the rank-1 tensor (ab)(cd)^⊤) is reshaped to (a c)(b d)^⊤. Let σ(·) denote this reshaping operation. Reshaping does not change the Frobenius norm. So by linearity, and since σ fixes S,σ(T^<) - S - σ(N)_F = T^< - (S+N)_F ≤ 6α√(2n).We now remark that σ maps N to one of the bounded-norm matrices from claim:spectrals,σ(N) =∑_i≠ j[x_i^2x_j^2]· a_i a_i^⊤ a_j a_j^⊤≼α·. Furthermore, because the positive semidefinite cone is a closed convex set, and because projection to closed convex sets can only decrease distances (see lem:proxproj), σ(T^<) - S - σ(N) _F = σ(T^<) - S - α· - (σ(N) - α·)_F≥(σ(T^<) - S - α·)_+ - (σ(N) - α·)_+ _F ≥(σ(T^<) - S - α·)_+_F≥(σ(T^<) - α·)_+- S _F where to obtain the last inequality we used that S is positive semidefinite. Therefore, step 3 of the algorithm ensures that T̃ is close to S in Frobenius norm, as desired.Now we prove that the spectral norms of the symmetrizations of the tensor have small spectral norm.The first matrix that we are interested in is PSD, and can dominated by a tensor power of the identity:0≼∑_i≠ j[x_i^2x_j^2]/[x_1^4]· a_i a_i^⊤ a_j a_j^⊤≼α∑_i,j a_i a_i^⊤ a_j a_j^⊤≼α·(∑_i a_i a_i^⊤) (∑_j a_j a_j^⊤) ≼α·For the second matrix, if we let A be the d^2 × n^2 matrix whose i,jth column is a_ia_j, and let M be the n^2 × n^2 matrix whose i,jth diagonal entry is [x_i^2 x_j^2], then∑_i≠ j[x_i^2x_j^2]· a_j a_i^⊤ a_i a_j^⊤ = AMΠ A^⊤,where Π is the permutation matrix that takes the i,jth row to the j,ith row. By assumption, M≤max_i≠ j[x_i^2x_j^2] ≤α[x_1^2], and the columns of A are orthonormal, so A = 1. It follows by the submultiplicativity of the spectral norm that∑_i≠ j[x_i^2x_j^2]· a_j a_i^⊤ a_i a_j^⊤≤α[x_1^4]This gives us the claim. When [x_i^4] = p for all i ∈ [n], applying alg:orthog with T̃ a total of (n) times will allow us to recover m ≥ n/2 vectors b_1,…,b_m with a_i,b_i^2 ≥ 0.99. The following subsections contain the details regarding the refinement of the approximation, and the sample complexity bounds for estimating the 4th moment tensor. §.§ Postprocessing to refine approximation We now analyze the postprocessing algorithm alg:post for the context of dictionary learning, in which our tensor has the form = [(Ax)^ 4]. We claim that, despite not having bounded spectral norm error away from ∑_i[x_i^4] · a_i^ 4, the postprocessing algorithm still succeeds. Suppose that we are given = [(Ax)^ 4], where x is distributed so that [x_ix_jx_kx_ℓ] = 0 unless x_ix_jx_kx_ℓ is a square, [x_i^4] = p for all i∈[n], and [x_i^2x_j^2]≤α p. Suppose furthermore that we have a unit vector u such that b,a_i^2 ≥ 0.99 for some i∈[n]. Then applying alg:post to u andwith error parameter 1/2 returns a vector v such thatv,a_i^2 ≥ 1-16α.Without loss of generality, let i:=1 so that b,a_i^2 = 1-η≥ 0.99 (henceforth, we use i as an ordinary index). We have that[(Ax)^ 4](u u) = p ∑_iu,a_i^2 a_i^ 2 + ∑_i≠ j[x_i^2 x_j^2](u,a_j^2· a_i^ 2 + u,a_iu,a_j (a_ia_j + a_ja_i))Define M_u to be the reshaping of [(Ax)^ 4](u u) to a d^2 × d^2 matrix. We must understand the spectrum of M_u, and for now we turn our attention to the second sum. Splitting the second sum into distinct parts, we have by the orthonormality of the a_i that∑_i≠ j[x_i^2 x_j^2]·u,a_j^2· a_ia_i^⊤ ≼max_i≠ j[x_i^2x_j^2]·∑_i≠ j a_ia_i^⊤ ≼ α p ·,where the last line is by our assumption on [x_i^2x_j^2]. Finally, for any w ∈^d,w^⊤(∑_i≠ j[x_i^2 x_j^2]·u,a_iu,a_j (a_i a_j^⊤ + a_ja_i^⊤))w = ∑_i≠ j[x_i^2x_j^2] · 2u,a_iu,a_jw,a_iw,a_j Applying Cauchy-Schwarz and pulling out the maximum multiplier, ≤ 2max_i≠ j(E[x_i^2x_j^2])·( ∑_i≠ ju,a_i^2w,a_j^2)^1/2(∑_i≠ ju,a_j^2w,a_i^2)^1/2 Now noticing that the two parenthesized terms are actually identical, then adding a positive quantity and factoring, ≤ 2max_i≠ j(E[x_i^2x_j^2])·( (∑_iu,a_i^2)(∑_jw,a_j^2)) = 2max_i≠ j[x_i^2x_j^2] ≤ 2pα.An identical proof, up to signs, gives us a lower bound of 2pα. Therefore,1/p M_u = ∑_iu,a_i^2 a_ia_i^⊤ + E,For a matrix E with E≤ 4α.It remains to argue that the top eigenvector of M_u is a_1. We have thatp^-1M_u ≥ a_1^⊤ M_u a_1 a_1^⊤∑_iu,a_i^2 a_ia_i^⊤ a_1 + a_1^⊤ E a_1≥ 1-η - 4α,where the last line follows from the orthonormality of the a_i and our bound on E. Meanwhile, for any unit vector w ∈^d and any ϵ < 1-2η,w^⊤(p^-1M - ϵ· a_1a_1^⊤)w = (1-η - ϵ)a_1,w^2 + ∑_i>1u,a_i^2a_i,w^2 + w^⊤ E w and since max_i>1a_i,u^2 ≤η, ≤ (1-η-ϵ)a_1,w^2 + η· (1-a_1,w^2) + 4 α≤ 1 - η - ϵ + 4α,where the last line follows because w is a unit vector, and we chose ϵ so that 1-η-ϵ > η. Thereforep^-1M_u - ϵ a_1a_1^⊤≤p^-1 M_u - ϵ +8α. Applying fact:topeig, we conclude that if v is the top eigenvector of M_u, then v,a_1^2 ≥ϵ - 8α/ϵ≥ 1 - 8α/ϵ. Since we can choose ϵ = 1/2 and still have that ϵ < 0.98 < 1-2η, the conclusion follows.§.§ From independent columns to orthonormal columns We now use standard techniques to prove that one can reduce from a dictionary with independent columns to a dictionary with orthogonal columns, given sufficiently many samples.Note that the expected covariance matrix of the samples is equal to a scaled version of the covariance matrix,[(Ax)(Ax)^⊤] = [x_1]^2 ·Σ.Since we have assumed that x_2^2 ≤ n, the Frobenius norm of (Ax)(Ax)^⊤ is bounded by n for every sample, and [((Ax)(Ax)^⊤)^2] ≤ n^2. By applying a matrix Bernstein inequality (see e.g. <cit.>), we have that so long as we have m ≥((n/β)^2) samples, the empirical covariance matrix Σ̂= 1/m∑_i=1^m y^(i)(y^(i))^⊤ will approximate Σ within β in spectral norm.So given sufficiently many samples, we can compute a good spectral approximation of Σ^-1/2, Σ̂^-1/2 withΣ̂^-1/2 - Σ^-1/2≤ϵ.Assuming access to such a Σ̂^-1/2, we can transform A to à = Σ̂^-1/2A. Now, for matrices X,Y,Z of suitable dimensions,YXY - ZXZ = 1/2((Y-Z)X(Y+Z) + (Y+Z)X(Y-Z))Applying this to ÃÃ^⊤ -= Σ̂^-1/2ΣΣ̂^-1/2 - Σ^-1/2ΣΣ^-1/2,ÃÃ^⊤ -≤Σ̂^-1/2 - Σ^-1/2·Σ· (Σ̂^-1/2 + Σ^-1/2)≤ϵ·Σ· (2+ϵ)Σ^-1/2≤ O(ϵλ_max(Σ)/λ_min(Σ)^1/2)Defining ηÃÃ^⊤ -, we have that ÃÃ^⊤ = (1±η), and the columns of à are near-orthonormal. Similarly, we can transform our samplesy^(i) = Ax^(i)→ỹ^(i) = Σ̂^-1/2 Ax^(i).Now, define S_diff = ((Σ̂^-1/2)^ 2 - (Σ^-1/2)^ 2) and S_sum = ((Σ̂^-1/2)^ 2 + (Σ^-1/2)^ 2). We can factor the difference[(Ãx)^ 4 - (Σ^-1/2Ax)^ 4] = 1/2S_diff[(Ax)^ 4]S_sum+1/2S_sum[(Ax)^ 4]S_diff≤S_sum·S_diff·[(Ax)^ 4]≤(Σ̂^-1/2^2+ Σ^-1/2^2)·S_diff·[(Ax)^ 4]≤ (2+)Σ^-1·S_diff·[(Ax)^ 4]Applying the identityA^ 2 - B^ 2 = 1/2(A-B) (A+B) + 1/2 (A-B) (A+B),to S_diff, we can get that[(Ãx)^ 4 - (Σ^-1/2Ax)^ 4] ≤ (2+ϵ)Σ^-1·Σ̂^-1/2 - Σ^-1/2(Σ̂^-1/2+Σ^-1/2)[(Ax)^ 4]≤ 9·Σ^-3/2·ϵ[(Ax)^ 4]Since we can choose the number of samples so as to make this last quantity as small as we would like, as a function of the condition number, and then appealing to fact:orthog, the reduction is complete. §.§ Sample complexity boundsBelow is our bound on the sample complexity of approximating the 4th moment tensor, which we believe may be loose. Given samples of the form y^(i) = Ax^(i) for x^(i)∼, as long as β≥[x_i^8] dominates the expectation of any other order-8 monomial in x, and any monomial with odd multiplicity has expectation 0, and the entries of x are bounded by κ, then with high probability given m ≥(max{β n^3, (κ n)^2}) samples,1/m∑_i=1^m (y^(i))^ 4 - _x∼[(Ax)^ 4] ≤ o(1).Our matrix has the formM = ∑_ijkℓ x_ix_jx_kx_ℓ· (a_ia_j)(a_k a_ℓ)^⊤,And the a_i are orthonormal, so[MM^⊤] = ∑_i,j,i',j' k,ℓ[x_ix_jx_i'x_j'x_k^2x_ℓ^2] · (a_ia_j)(a_i' a_j')^⊤.This is because of the orthonormality of the a_i, which guarantees that terms in the product MM^⊤ in which we have an inner product between two non-identical vectors drop out to 0.If we define A to be the d^2 × n^2 matrix whose columns are the Kronecker products a_ia_j for all i,j∈[n], and if for each pair k,ℓ∈ [n] we define X^(k,ℓ) be the n^2 × n^2 matrix whose (i,j),(i',j')th entry is [x_ix_jx_ix_j'x_k^2x_ℓ^2], we can realize E[MM^⊤] as[MM^⊤] = A (∑_k,ℓ X^(k,ℓ)) A^⊤.Because we assumed that E[x_ix_jx_i'x_j'x_k^2x_ℓ^2] = 0 unless every index appears with even multiplicity, the entry of X^(k,ℓ) will be 0. This happens only on the diagonal, unless i'=j' and i=j, or for X^(k,ℓ) in the intersection of the (k,ℓ)th row and the (ℓ,k)th column and the (ℓ,k)th row and the (k,ℓ)th column. So we split each X^(k,ℓ) into a diagonal part D^(k,ℓ), an intersection part corresponding to the (k,ℓ) and (ℓ,k) intersections C^(k,ℓ), and the rest of the off-diagonal part R^(k,ℓ). and it follows that[MM^⊤]≤ n^2 A^2 max_k,ℓ(D^(k,ℓ) + C^k,ℓ+ R^(k,ℓ)).Because every entry is bounded by [x_i^8] ≤β, and the D^(k,ℓ) are diagonal, the D term contributes β. The C matrices have Frobenius norm 2β, and the R matrices have only n^2 nonzero entries, so R_F ≤β n. Therefore,[MM^⊤]≤ 3β n^3.For each sample x^(i), we have that 1/m(Ax^(i))^ 4_F ≤κ^2n^2/m, and we have by the above reasoning that [1/m^2(Ax^(i))^ 4(Ax^(i))^ 4] ≤ 3[x_i^8] n^3/m^2. So the variance of the empirical 4-tensor is √(n^3/m), and the absolute bound on the norm of any summand is n^2/m. Applying a matrix Bernstein inequality (see e.g. <cit.>), we have that as long as we have m ≫max{κ^2n^2log n,β n^3log n} samples, with high probability we approximate [(Ax)^ 4] within spectral norm o(1). amsalpha§ USEFUL TOOLSLet g be a standard Gaussian vector in ^k, g ∼(0,_k). Let A be a tensor in (^k)⊗ (^ℓ) ⊗ (^m), and call the three modes of A α,β,γ respectively. Let A_i be a ℓ× m slice of A along mode α. Then,[∑_i=1^k g_iA_i≥ t ·max{A_{αβ}{γ}, A_{αγ}{β}}] ≤ (m+ℓ)exp(-t^2/2)We compute the expectation and variance of our matrix,_g[∑_i=1^k g_iA_i] = 0, and_g[∑_i=1^k g_iA_i]= max{∑_i=1^k A_iA_i^⊤, ∑_i=1^k A_i^⊤A_i},The two variance terms correspond to A_{αβ}{γ}^2 and A_{αγ}{β}^2 respectively. We can now apply concentration results for matrix Gaussian series to conclude the proof <cit.>. The following lemma states that distances can only decrease under projections to a convex set, and is well-known (see e.g. <cit.>).Let ⊂^n be a closed convex set, and let Π:^n → be the projection operator ontoin terms of norm ·_2, i.e. Π(x) _c ∈x-c_2. Then for any x,y ∈^n,x-y_2 ≥Π(x) - Π(y)_2.If we let D_x = x - Π(x), D_y = y - Π(y),x-y^2 = D_x - D_y + Π(x) - Π(y)^2= D_x- D_y^2 + Π(x) - Π(y)^2 + 2D_x - D_y, Π(x) - Π(y)Now the conclusion will follow from the fact thatD_x - D_y, Π(x) - Π(y)≥ 0.This is because, by definition of Π,Π(x) = _c ∈x-c_2^2 = _p ∈^n1/2x-p_2^2 + _(p)where _(·) is the convex function defined to be ∞ on elements not inand 0 otherwise. From the strong convexity of the last expression the projection is unique. From the optimality conditions, it follows that Π(x) is the unique point p ∈^n such that x - p ∈∂_(p), where ∂_(p) is the set of subgradients of _ at p.By definition of the subgradient and by the convexity of _, for any p,q ∈^n and for g_p ∈∂_(p),g_q ∈∂_(q),_(p) + g_p,q-p ≤_(x) g_p ,q-p ≤_(q) - _(p) -g_q ,q-p ≤ -_(q) + _(p) g_p - g_q, p-q ≥ 0Now, taking p = Π(x) and q = Π(y), we have that D_x = x - Π(x) ∈∂_(Π(x)), and D_y = y - Π(y) ∈∂_(Π(y)), so from the above,D_x - D_y, Π(x) - Π(y) ≥ 0,as desired.Let v ∈^n, and suppose that M - vv^⊤≤M - ϵv^2. Then if u,w are the top unit left- and right-singular vectors of M,u,v^2 ≥ϵ·v^2 orw,v^2 ≥ϵ·v^2 We have thatM - ϵv^2 ≥ |u^⊤(M-vv^⊤)w| ≥ |u^⊤ M w| - |u^⊤ v v^⊤ w| = M - |u,vw,v|,where the second inequality is the triangle inequality. Rearranging, the conclusion follows.If A is an n × n symmetric matrix with eigendecomposition ∑_i∈[n]λ_i u_i u_i^⊤ for orthonormal u_1,…,u_n ∈^n and eigenvalues λ_1≥⋯≥λ_n, then the projection of A to the PSD cone is equal to ∑_i ∈ [n][λ_i ≥ 0]·λ_i u_i u_i^⊤. Let  = A + B be the projection (in Frobenius norm) of A to the PSD cone. Because u_1,…,u_n are orthonormal, we may choose an orthonormal basis V={v_i}_i=1^n^2 for ^n^2 that includes v_1 = u_1u_1,v_2 = u_2u_2,…, v_n = u_nu_n, as the first n basis vectors. Now, viewing A,B as vectors in ^n^2, we can write A = ∑_i=1^n λ_i v_i for λ_i the eigenvalues of A, and write B = ∑_i=1^n^2β_i v_i for some scalars β_1,…,β_n^2.For any eigenvector u_i of A, we have that u_i^⊤ u_i = v_i, A + B = λ_i + β_i by the orthonormality of the v_i. Therefore, if λ_i < 0 we must have β_i ≥ |λ_i|, since  is PSD. We also have that A - Â_F^2 = B_F^2 = ∑_i=1^n^2β_i^2, so B = ∑_i=1^n 𝕀[λ_i < 0] · |λ_i|· u_i u_i^⊤ minimizes the Frobenius norm of the difference, which (after checking to see that A + B has all non-negative eigenvalues) concludes the proof. | http://arxiv.org/abs/1706.08672v1 | {
"authors": [
"Tselil Schramm",
"David Steurer"
],
"categories": [
"cs.LG",
"cs.DS",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170627051239",
"title": "Fast and robust tensor decomposition with applications to dictionary learning"
} |
Controlled Tactile Exploration and Haptic Object RecognitionThis research has received funding from the European Union's Seventh Framework Programme for research, technological development and demonstration under grant agreement No. 610967 (TACMAN).Massimo Regoli, Nawid Jamali, Giorgio Metta and Lorenzo Natale iCub FacilityIstituto Italiano di Tecnologiavia Morego, 30, 16163 Genova, Italy{massimo.regoli, nawid.jamali, giorgio.metta, lorenzo.natale}@iit.it December 30, 2023 ========================================================================================================================================================================================================================================================= In this paper we propose a novel method for in-hand object recognition. The method is composed of a grasp stabilization controller and two exploratory behaviours to capture the shape and the softness of an object. Grasp stabilization plays an important role in recognizing objects. First, it prevents the object from slipping and facilitates the exploration of the object. Second, reaching a stable and repeatable position adds robustness to the learning algorithm and increases invariance with respect to the way in which the robot grasps the object. The stable poses are estimated using a Gaussian mixture model (GMM).We present experimental results showing that using our method the classifier can successfully distinguish 30 objects. We also compare our method with a benchmark experiment, in which the grasp stabilization is disabled. We show, with statistical significance,that our method outperforms the benchmark method.Tactile sensing, grasping § INTRODUCTION Sense of touch is essential for humans. We use it constantly to interact with our environment. Even without vision, humans are capable of manipulating and recognizing objects. Our mastery of dexterous manipulation is attributed to well developed tactile sensing <cit.>. To give robots similar skills, researchers are studying use of tactile sensors to help robots interact with their environment using the sense of touch. Furthermore, different studies show the importance of tactile feedback when applied to object manipulation <cit.><cit.>. Specifically, in the context of object recognition, tactile sensing provides information that cannot be acquired by vision. Indeed, properties such object texture and softness can be better investigated by actively interacting with the object. In order to detect such properties, different approaches have been proposed. Takamuku et al. <cit.> identify material properties by performing tapping and squeezing actions. Johansson and Balkenius <cit.> use a hardness sensor to measure the compression of materials at a constant pressure, categorizing the objects as hard and soft. Psychologists have shown that humans make specific exploratory movements to get cutaneous information from the objects <cit.>, that include, pressure to determinate compliance, lateral sliding movements to determinate surface texture, and static contact to determine thermal properties. Hoelscher et al. <cit.> use these exploratory movements to identify objects based on their surface material, whereas other researchers have focused on how to exploit them to reduce the uncertainty in identifying object properties of interest <cit.>. All these approaches carry out exploratory movements using a single finger and assume that the object does not move. Conversely, other works recognize an object by grasping the object, putting less restrictions on the hand-object interaction. Schneider et al. <cit.> propose a method in which each object is grasped several times, learning a vocabulary from the tactile observations. The vocabulary is then used to generate a histogram codebook to identify the objects. Chitta et al. <cit.> propose a method that, using features extracted while grasping and compressing the object, can infer if they are empty or full and open or close. Chu et al. <cit.> perform exploratory movements while grasping the object in order to find a relationship between the features extracted and haptic adjectives that humans typically use to describe objects.However, most of these approaches do not deal with the stability problem and assume that the object is laying on, or are fixed to a surface such as a table. When the object has to be held in the robot's hand, stability problems such as preventing it from falling, make the task of extracting features through interactions more challenging. Kaboli et al. <cit.> recognise objects using their surface texture by performing small sliding movements of the fingertip while holding the object in the robot's hand. Gorges et al. <cit.> merge a sequence of grasps into a statistical description of the object that is used to classify the objects. In a recent work Higy et al. <cit.> propose a method in which the robot identifies an object by carrying out different exploratory behaviours such as hand closure, and weighing and rotating the object. In their method the authors fuse sensor data from multiple sensors in a hierarchical classifier to differentiate objects. In these approaches the stability is typically managed by performing a power grasp, that is,wrapping all the fingers around the object. This means that in general, the final hand configuration after the grasp is not controlled. It strictly depends on the way the object is given to the robot. Due to this, the tactile and proprioceptive feedback suffer from high variability. This requires a larger number of grasps to be performed and negatively affects the performance. Moreover, performing power grasps may limit further actions that could help in extracting other object features such as softness/hardness.In this work we propose a novel method for in-hand object recognition that uses a controller proposed by Regoli et al <cit.> to stabilize a grasped object. The controller is used to reach a stable grasp and reposition the object in a repeatable way.We perform two exploratory behaviours: squeezing to capture the softness/hardness of the object; and wrapping all of the fingers around the object to get information about its shape. The stable pose achieved is unique given the distance between the points of contact (related to the size of the object), resulting in high repeatability of features, which improves the classification accuracy of the learned models. Differently from other methods, we do not put any restrictions on the objects.We validated our method on the iCub humanoid robot <cit.> (Fig. <ref>). We show that using our method we can distinguish 30 objects with 99.0% ± 0.6% accuracy. We also present the results of a benchmark experiment in which the grasp stabilization is disabled. We show that the results achieved using our method outperforms the benchmark experiment. In the next section we present our method for in-hand object recognition. In section <ref> we describe the experiments carried out to validate our method, while in section <ref> we present our results. Finally, in section <ref> we conclude the paper and provide future directions.§ METHODOLOGYHere we present the method used to perform the in-hand object recognition task. We use an anthropomorphic hand, but the method can be easily extended to any type of hand that has at least two opposing fingers. We use the tactile sensors on the fingertips of the hand <cit.>, which provide pressure information on 12 taxels for each fingertip. An important assumption in this work is that the object is given to the robot by a collaborative operator, in such a way that the robot can grasp it by closing the fingers. The remaining steps are performed by the robot autonomously, namely: * grasping the object using a precision grasp, that is, using the tip of the thumb and the middle finger,* reaching an optimal stable pose,* squeezing the object to get information about its softness,* wrapping all the fingers around the object to get information about its shape. We start by giving an overview of the grasp stabilizer component. This is followed by a description of the feature space, and then we give a brief overview of the machine learning algorithm used to discriminate the objects.§.§ Grasp stabilizationGrasp stabilization is a crucial component of our method for two reasons. First, it is needed to prevent the object from falling, for example, when executing actions like squeezing. Second, reaching a stable and repeatable pose for a given object improves the classifier accuracy. We use our previously developed method to stabilize the object <cit.>. In the rest of this section we quickly revise this method and explain how we apply it to our problem (details of the controller can be found in <cit.>). In this paper we use two fingers instead of three, namely, the thumb and the middle finger. Figure <ref> shows the controller, which is made of three main components:§.§.§ Low-level controller it is a set of P.I.D. force controllers responsible for maintaining a given force at each fingertip. The control signal is the voltage sent to the motor actuating the proximal joint, while the feedback is the tactile readings at the fingertip. We estimate the force at each fingertip by taking the magnitude of the vector obtained by summing up all the normals at the sensor locations weighted by the sensor response.§.§.§ High-level controller it is built on top of the low-level force controllers. It stabilizes the grasp by coordinating the fingers to a) control the object position, and b) maintain a given grip strength. The object position α_o is defined as in Fig. <ref>, and it is controlled using a P.I.D. controller in which the control signals are the set-points of the forces at each finger, while the feedback is the object position error.The grip strength is the average force applied to the object. It is defined as: g = f_th+f_mid/2,where f_th and f_mid are the forces estimated at the thumb and the middle finger, respectively. The target grip strength is maintained by choosing set-points of the forces that satisfy (<ref>). §.§.§ Stable grasp model it is a Gaussian mixture model, trained by demonstration. The robot was presented with stable grasps using objects of different size and shape. The stability of a grasp was determined by visual inspection. A stable grasp is defined as one that avoids non-zero momenta and unstable contacts between the object and the fingertips. We also preferred grasp configurations that are far from joint limits (details are in <cit.>). Given the distance, d, between the fingers, the model estimates the target object position, α_o^r, and the target set of non-proximal joins, Θ_𝐧𝐩, to improve grasp stability and make it robust to perturbations. The target α_o^r is used as the set-point of the high-level controller, while the Θ_𝐧𝐩 is set directly using a position controller. §.§ The Feature Space Once a stable grasp is achieved, the robot manipulates the object to capture its softness and shape by performing two exploratory behaviours: a) squeezing the object between the thumb and the middle finger, and b) wrapping all the fingers around the object. The softness of the object is captured both by the distribution of the forces in the tactile sensor and the deflection of the fingers when the object is squeezed between the fingers of the robot. The shape of the object is captured by wrapping all of the fingers of the robot around it.As mentioned earlier, the grasp stabilization implies a high degree of repeatability of the achieved pose, independent of the way the object is given to the robot. Thereby, the features produced during the exploratory behaviours exhibit low variance between different grasps of the same object. Which, in turn, increases the accuracy of the classifier.§.§.§ Tactile responsesthe distribution of forces in the tactile sensors is affected by the softness of an object. A hard object will exert forces that are strong and concentrated in a local area. A soft object, in contrast, will conform to the shape of the fingertip and exert forces across all tactile sensors. The tactile sensors also capture information on the local shape of the object at the point of contact. We use the tactile responses from the thumb and the middle finger, , in our feature space, since the objects are held between these two fingers.§.§.§ Finger encodersthe finger encoders are affected by the shape and the harness/softness of the object. When the robot squeezes the object, a hard object will deflect the angles of the finger more than a softer object. Since we use only the thumb and the middle finger during the squeezing action, we use both the initial and the final encoder values for these fingers – and , respectively.To capture the shape of the object, the robot wraps the rest of its fingers around the object. We also include the encoder data, Θ_𝐰𝐫𝐚𝐩,of these fingers in our feature space.§.§ The learning algorithmIn order to train the classifier, we used as features the data acquired during the grasping, squeezing and enclosure phase, as described in the previous section. We simply concatenated the collected values, obtaining the feature vector [] composed of 45 features, 21 related to the encoders and 24 related to the tactile feedback.As learning algorithm we adopted Kernel Regularized Least-Squares using the radial basis function kernel. For the implementation we used GURLS <cit.>, a software library for regression and classification based on the Regularized Least Squares loss function. § EXPERIMENTS To test our method, we used the iCub humanoid robot. Its hands have 9 degrees-of-freedom. The palm and the fingertips of the robot are covered with capacitive tactile sensors. Each fingertip consists of 12 taxels <cit.>. §.§ The objects We used a set of 30 objects shown in Fig. <ref>, of which, 21 were selected from the YCB object and model set <cit.>. Using a standard set helps in comparing the results of different methods. The objects were selected so that they fit in the iCub robot's hand without exceeding its payload. The YCB object set did not have many soft objects fitting our criteria, hence, we supplemented the set with 9 additional object with variable degree of softness. We also paid attention to choose objects with similar shape but different softness, as well as objects with similar material but different shapes.§.§ Data collectionThe dataset to test our method was collected using the following procedure (depicted in Fig. <ref>):* The iCub robot opens all of its fingers.* An object is put between the thumb and the middle finger of the robot. The robot starts the approach phase, which consists of closing the thumb and the middle finger until a contact is detected in both fingers. A finger is considered to be in contact with an object when the force estimated at its fingertip exceeds a given threshold. To capture variations in the position and the orientation of the object, each time the object is given to the robot, it is given in a different position and orientation. *At this point the grasp stabilizer is triggered with a given grip strength. The initial value of the grip strength is chosen as the minimum grip strength needed to hold all the objects in the set. The method described in section <ref> is used to improve the grasp stability. When the grasp has been stabilized, the robot stores the initial values of the encoders of the thumb and the middle finger. * Then the robot increases the grip strength to squeeze the object and waits for 3 seconds before collecting the tactile data for the thumb and the middle finger. At this point the robot also records the encoder values for the thumb and the middle finger. * Finally, the robot closes all of the remaining fingers around the object until all fingersare in contact with the object. At this point, the robot collects the values of the encoders of the fingers.These steps were repeated 20 times for each object. To test our algorithm we use a fourfold cross-validation. That is, we divide the dataset into 4 sets. We hold one of the sets for testing and use the other three to train a classifier. This is repeated for all 4 sets. We compute the accuracy and the standard deviation of our classifier using the results of these 4 learned classifiers. §.§ Benchmark experiment To test our hypothesis that reaching a stable pose improves the classification results we carried out an experiment in which we disable part of the grasp stabilization. As described earlier and depicted in Fig. <ref>, the grasp stabilization consists of three modules: the low-level force controller, the high-level controller and the stable grasp model. We only disable the stable grasp model. The other two components are needed to stop the object from slipping and to control the grip strength.The stable grasp model produces two outputs: a) the target object position, α_o^r,and the target set of non-proximal joints, Θ_𝐧𝐩. In the benchmark experiment we calculate the value of α_o^r and the Θ_𝐧𝐩 when the thumb and the middle finger make contact with the object. That is, the alpha is set to the current position of the object and the theta is set to the current joint configuration. Apart from this difference, the high-level controller and the low-level force controller are still active, controlling grip strength and maintaining a stable grasp. However, without the stable grasp model, the grasp is less robust to perturbations. Henceforth, unless stated otherwise, when we mention that the grasp stabilization is disabled, we mean that we only disable the repositioning based on the GMM. Hence, we collected the data for the benchmark experiment following the same steps as described in section <ref> where the grasp stabilization was disabled. § RESULTSIn this section we present the results of our method and show how each of the selected features in our feature space helps in capturing different properties of the objects, namely, the softness/harness and the shapeof the object. This will be followed by a comparison between our method and the benchmark method in which the grasp stabilization is disabled. When reporting the results for brevity we concatenated some of the features: = [ ], and = [].§.§ Finger encoders To study the effectiveness of the encoder features, we trained a model using different combinations of thesefeatures. Table <ref> reports the results of these experiments. We notice that using only the initial encoder values, the accuracy is already quite high, 80.5% ± 2.0%, while including the final encoder values of the thumb and the middle finger after squeezing it increases to 93.3% ± 0.8%. This is because the fingers will move considerably if the object is soft, thereby, capturing the softness of the object. Figure <ref> shows the confusion matrices for the experiments. We notice that several pairs of objects such as the tennis ball (11) and the tea box (30) or the sponge (26) and the soccer ball (28) are sometimes confused if only the initial encoders values are used as features, while they are discriminated after the squeezing action.Finally we analysed the results of including all encoder data, that is, including the data when the robot wraps its fingers around the object. This improved the classification accuracy to 96.3% ± 0.7%. From the confusion matrices we notice that adding such features resolves a few ambiguities, such as the one between the soccer ball (28) and the water bottle (22) and the one between the yellow cup (24) and the strawberry Jello box (19). Indeed, these pairs of objects have similar distance between the points of contact when grasped, and cause similar deflections of the fingers when squeezed, but have different shapes.§.§ Tactile responses As discussed earlier the tactile sensors are useful in capturing the softness of the objects as well as the local shape of the objects. In Fig. <ref> we can see that using only the tactile feedback we get an accuracy of 95.0% ± 0.8%, which is comparable with the 96.3% ± 0.7% obtained using the encoder values. Although they have similar classification accuracy, studying the confusion matrices reveals that objects confused by them are different. For example, the classifier trained using only the tactile data often confuses the Pringles can (1) and the tomato can (7), since they are hard and share similar local shape. Conversely, due to their slightly different size they are always distinguished by the classifier trained using only encoder data. This means that combining the two feature spaces can further improve the accuracy of the learned classifiers. §.§ Combining the two features Finally, using the complete feature vector we get an accuracy of 99.0% ± 0.6%. We also notice that the standard deviation in our experiments is decreasing as we add more features. From the confusion matrix we can see that several ambiguities characterizing each individual classifier are now solved. A few objects are still confused due to their similar shape and softness, namely the apple (5) and the orange (6), and the apricot (16) and the prune (10). Less intuitively, the classifier once confuses the apricot with the SPAM can (21), and once it confuses the apricot with the brown block (18).To explain the confusion between these objects, we notice that there is a particular way to grasp them such that the joints configuration is very similar. This happens when the middle finger touches the flat side of the apricot, and the little finger misses both objects.§.§ Comparison with the benchmark experiment Figure <ref> shows the results of running the same analysis on the data collected in the benchmark experiment where the grasp stabilization was removed. The results show that the proposed method performs significantly better than the benchmark experiment, achieving 99.0% ± 0.6%, compared to the benchmark experiment which achieved an accuracy of 69.9% ± 1.4%. This is because the stabilization method proposed in this paper increases the repeatability of the exploration, which makes the feature space more stable. Indeed, the initial position of the object in the hand strongly affects the collected tactile and encoders data. This variability is reduced using the grasp stability controller. Note that the accuracy of the benchmark experiment increases as more features are added, showing that the feature space is able to capture the object properties.We run a further analysis to study the effect of increasing the number of trials in the training set. In this case we always trained the classifier with the complete feature vector and considered 5 trials per object for the test set, while we varied the number of trials in the training set between 3 and 15. Figure <ref> shows the results of this analysis. The results show that the proposed method boosts the accuracy of the classification, requiring less samples to be able to distinguish the objects. The trend of the accuracy obtained using the benchmark method suggests that it may improve by increasing the number of samples in the training set. However, this is not preferred because it makes it impractical to collect data on large sets of objects, adversely affecting the scalability of the learned classifier.§.§ Results using objects form the YCB set only Finally, in table <ref> we provide the results of our method using only the object from the YCB object set, in order to let researchers having the same dataset compare their results with ours. § CONCLUSIONS In this work we proposed a method for in-hand object recognition that makes use of a grasp stabilizer and two exploratory behaviours: squeezing and wrapping the fingers around the object. The grasp stabilizer plays two important roles: a) it prevents the object from slipping and facilitates the application of exploratory behaviours, and b) it moves the object to a more stable position in a repeatable way, which makes the learning algorithm more robust to the way in which the robot grasps the object. We demonstrate with a dataset of 30 objects and the iCub humanoid robot that the proposed approach leads to a remarkable recognition accuracy (99.0% ± 0.6%), with a significant improvement of 29% with respect to the benchmark, in which the grasp stabilizer is not used.This work demonstrates that a reliable exploration strategy (e.g. squeezing and re-grasping) is fundamental to acquiring structured sensory data and improve object perception. In future work we will employ an even larger set of objects and explore the use of other control strategies and sensory modalities.IEEEtran | http://arxiv.org/abs/1706.08697v1 | {
"authors": [
"Massimo Regoli",
"Nawid Jamali",
"Giorgio Metta",
"Lorenzo Natale"
],
"categories": [
"cs.RO",
"cs.LG"
],
"primary_category": "cs.RO",
"published": "20170627072533",
"title": "Controlled Tactile Exploration and Haptic Object Recognition"
} |
[email protected] of Earth and Planetary Science, University of California, Berkeley, California 94720, USA (Current address: Lawrence Livermore National Laboratory, Livermore, California 94550, USA)Department of Earth and Planetary Science, University of California, Berkeley, California 94720, [email protected] Department of Earth and Planetary Science, University of California, Berkeley, California 94720, USA Department of Astronomy, University of California, Berkeley, California 94720, USA We use path integral Monte Carlo and density functional molecular dynamics to construct a coherent set of equation of state for a series of hydrocarbon materials with various C:H ratios (2:1, 1:1, 2:3, 1:2, and 1:4) over the range of 0.07-22.4 g cm^-3 and 6.7×10^3-1.29×10^8 K.The shock Hugoniot curve derived for each material displays a single compression maximum corresponding to K-shell ionization.For C:H=1:1, the compression maximum occurs at 4.7-fold of the initial density and we show radiation effects significantly increase the shock compression ratio above 2 Gbar, surpassing relativistic effects.The single-peaked structure of the Hugoniot curves contrasts with previous work on higher-Z plasmas, which exhibit a two-peak structure corresponding to both K- and L-shell ionization.Analysis of the electronic density of states reveals that the change in Hugoniot structure is due to merging of the L-shell eigenstates in carbon, while they remain distinct for higher-Z elements. Finally, we show that the isobaric-isothermal linear mixing rule for carbon and hydrogen EOSs is a reasonable approximation with errors better than 1% for stellar-core conditions.First-principles Equation of State and Shock Compression Predictions of Warm Dense Hydrocarbons Burkhard Militzer December 30, 2023 =============================================================================================== Introduction.Hydrocarbon ablator materials are of primary importance for laser-driven shock experiments, such as those central to the study of inertial confinement fusion (ICF) <cit.> and the measurement of high energy density states relevant to giant planets <cit.> and stellar objects <cit.>. Accurate knowledge of the equation of state (EOS) of the hydrocarbon ablator is essential for optimizing experimental designs to achieve desired density and temperature states in a target. Consequently, a number of planar-driven shock wave experiments have been performed on hydrocarbon materials, including polystyrene (CH) <cit.>, glow-discharge polymer (GDP) <cit.>, and foams <cit.>, to measure the EOS. The highest pressure achieved among these experiments is 40 Mbar <cit.>, which is yet not high enough to probe the effects of K-shell ionization on the shock Hugoniot curve.Since the first X-ray scattering results on CH at above 0.1 Gbar(1 Gbar=100 TPa) <cit.>, ongoing, spherically-converging shock experiments using the Gbar platform at the National Ignition Facility (NIF) <cit.> and the OMEGA laser <cit.> will extend measurements of the shock Hugoniot curve of polystyreneto pressures above 0.35 Gbar and into the K-shell ionization regime <cit.>.These experiments provide an important benchmark for the theoretical community working on models for EOSs of warm dense matter (WDM). EOS tables, such as SESAME <cit.> and QEOS <cit.>, which are largely based on variations of Thomas-Fermi (TF) models, are outdated and yet still often used in hydrodynamic simulations for the design of shock experiments.There have been ongoing efforts to develop efficient first-principles methods for WDM that maintain an accurate treatment of the many-body and shell-ionization effects that TF method neglects <cit.>. Standard Kohn-Sham density functional theory molecular dynamics (DFT-MD) is a suitable method for low and intermediate temperatures, but becomes computationally intractable beyond temperatures of 100 eV, where K-shell ionization becomes important in mid-Z elements.The efficiency limitation of thermal DFT-MD has largely been addressed by the development of orbital-free (OF) <cit.> and average-atom <cit.> approximations. Indeed, a number of calculations employing both DFT-MD and OF-DFT have been used to study the EOS of hydrocarbon materials, including polystyrene <cit.>, polyethylene <cit.>,and GDP <cit.>, in the WDM regime. The highest density and temperature simulations among these calculations have been performed with OF-DFT, up to 100 g cm^-3 and 345 eV <cit.>. At low temperatures, these approximate DFT-based simulations predict the shock Hugoniot curve in good agreement with experiments. However, there are important limitations to their accuracy. OF-DFT replaces the orbital-based kinetic energy functional with a density-based TF functional and, therefore, is also unable to account for shell ionization effects <cit.>. On the other hand, DFT-based average atom methods only compute shell structure for an average ionic state and, subsequently, it is not well-suited for studies of compounds.As an alternative to DFT-based methods, the path integral Monte Carlo (PIMC) method <cit.> offers an approach to explicitly treat all the many-body and ionization effects as long as a suitable nodal structure is employed. Early developmental work established the accuracy of the method for fully-ionized hydrogen <cit.> and helium <cit.> plasmas using free-particle nodes.In recent works, we have further developed free-particle <cit.> and localized <cit.> nodal structures, which has allowed us to compute first-principles EOSs across a wide range of density-temperature regimes for heavier, first- and second-row, elements.In this work, we combine low-temperature DFT-MD data with high-temperature PIMC data to compute coherent EOSs for several hydrocarbon materials across a wide density-temperature range. We aim to provide a highly accurate theoretical benchmark for the shock Hugoniot curves, which can help guide hydrodynamic target designs and interpret ongoing Gbar spherically-converging shock experiments in the WDM regime, particulary where K-shell ionization effects arise. While such state-of-the-art shock experiments maintain exquisite control over many experimental parameters, difficulties can remain in the interpretation of results due to insufficient knowledge of the opacity in the density unfolding process of radiographic measurements <cit.>, preheating effects, or shock uniformity and stability.Comparing both theoretical and experimental benchmarks can offer great insight into narrowing down and eliminating future sources of error. Methods.We consider five different C:H ratios of 2:1, 1:1, 2:3, 1:2, and 1:4, in order to cover the full range of interest in future shock experiments. Depending on the C:H ratio, our simulation cells contain between 30 and 50 nuclei, as well as between 100 and 130 electrons. In order to eliminate the finite-size effects at low temperatures,we use four times larger cells at temperatures up to 2×10^5 K.Above 2×10^4 K and 400 GPa, the Hugoniot curves derived with thesmall- and large-cell results are indistinguishable.Using the CUPID code <cit.>, we perform PIMC simulations within the fixed node approximation <cit.>. Similar to the PIMC simulations of hydrogen <cit.>, helium <cit.>, H-He mixtures <cit.>, carbon <cit.>, nitrogen <cit.>, oxygen <cit.>, neon <cit.>, and water <cit.>, we employ a free-particle nodal structure.We enforce fermion nodes at a small imaginary timeinterval of 1/8192 Hartree^-1 (Ha^-1) while pair density matrices <cit.> are evaluated in largerstep of 1/1024 Ha^-1 <cit.>.DFT-MD simulations use the Vienna Ab initio Simulation Package (VASP) <cit.> and exchange-correlation functionals within the local density approximation (LDA) <cit.>.We use all-electron projector augmented wave (PAW) pseudopotentials <cit.> with a 1.1 and a 0.8 Bohr radius core for carbon and hydrogen, respectively.We use a plane wave basis with 2000 eV cutoff, the Γ-point for sampling the Brillouin zone, a MD timestep of 0.05-0.2 fs, and a NVT ensemble controlled with a Nosé thermostat <cit.>.Typical MD trajectories consist of more than 1000 steps.Longer simulations of up to 2 ps show that the energies and pressures are converged. In order to put the DFT-MD pseudopotential energies on the same all-electron scale as PIMC calculations, we shifted all of our VASP energies by -37.4243 Ha/C and -0.445893 Ha/H. These shifts were determined by performing all-electron calculations for isolated C and H atoms with the OPIUM code <cit.>.Results and discussion. We performed DFT-MD at 6.7×10^3-10^6 K and 2-12 timesthe ambient density of ρ_ambient = 1.05 gcm^-3. PIMC simulations were performed for 10^6-1.29×10^8 Kand a much wider density range of 0.1-20 ρ_ambientsince this method does not rely on plane waves expansions nor pseudopotentials.At 10^6 K, the internal energyobtained from the two methods agreed to within 0.8 Ha/CH while pressures agreed to within 2%. We thus obtain a coherent first-principles EOS table <cit.>for hydrocarbon compounds over a wide range of temperatures and densitiesand show that of polystyrene (CH) in Fig. <ref>.We then use the EOS to determine the Hugoniot curves, by computing the P-V-T conditions that satisfy the Hugoniot equation (E-E_0) + (P+P_0)(V-V_0)/2 = 0, where (E_0, P_0, V_0) and (E, P, V) denote the initial and final internal energies, pressures, and volumes in a shock experiment, respectively.The initial conditions were determined based on thermo-physical and thermo-chemical data at1 bar <cit.>. For instance, the density of polystyrene at ambient is ρ_0=1.05 g/cm^3 <cit.> and using the enthalpy of combustion <cit.> we determined E_0=-38.3224 Ha/CH. The principal Hugoniot curve of polystyrene is plotted in Fig. <ref>.Besides the dependence of shock velocity, the shock compression is controlled by the excitation of internal degrees of freedom, which increases the compression, and interaction effects, which decrease it <cit.>. With increasing temperature and pressure, polystyrene is increasingly compressed untila maximum density of 4.9 gcm^-3 reached at 2.0×10^6 K, which corresponds to the excitation of K shell electrons of carbon ions, as we will explain below.At temperatures above 10^6 K, radiation effects can no longer be neglected. We re-construct the Hugoniot curve by considering the contribution of an ideal black body radiation to the EOS via P_photon=4σ T^4/3c and E_photon=3P_photonV, where σ is the Stefan-Boltzmann constant and c is the speed of light in vacuum. This is only an upper limit as the system is more likely to be a gray body with unknown efficiency. With the radiation contribution,the Hugoniot curve shiftsto significantly higher densities at above 10^7 K and2 Gbar, while the K-shell compression peak remains unchanged (see Fig. <ref>). This shift can primarily be attributed to the photon contribution to the internal energy.A key result of shock experiments is the relation between the compression ratio (ρ/ρ_0) and pressure. In Fig. <ref>, we compare our CH calculations with several experimental results and theoretical predictions. Our results are in very good agreement with experiments up to the highest pressure (4 TPa). We predict a maximum compression ratio of 4.7 at 47 TPa, which is higher than results from OF-DFT simulations (∼4.4) <cit.> imply and SESAME 7593 (∼4.3) <cit.> but similar to predictions froma semi-analytical EOS model LEOS 5400 for GDP <cit.>.Still, all methods predict the compression maximum to occur at very similar pressures. Our DFT-MD results imply there is a small shoulder in the Hugoniot curve at 4.1-fold compression, 10^4 GPa, and 6× 10^5 K, which separates in temperature the excitation regimes of the K and L shell electrons. Such a shoulder is absent from OF-DFT predictions because this method predicts the ionization to occur gradually and underestimates shell effects <cit.>. The shape of SESAME Hugoniot curve is similar to that of OF-DFT, since it is largely based on TF models.The structure of the LEOS Hugoniot curve is similar to the first-principles curve, butshows significant differences at low pressures due to the existence of oxygen in GDP. Figure <ref> also shows that the radiation contributions dominate over relativistic effects even though both lead to a compression ratio of 7 in the high-temperature limit, while the limit for a non-relativistic gas is 4.In Fig. <ref>, we compare our predictions for the shock Hugoniot curves of C_2H, CH, C_2H_3, CH_2, and CH_4 compounds.The initial conditions are determined by referring to representativehydrocarbon materials at ambient or cryogenic conditions <cit.>. For all C-H materials, we find the compression maximum to occur at very similar pressures. At the same time, we see a trend that lets the maximal compression ratio gradually decrease from 4.7 to 4.4 as the hydrogen contents is increased from C_2H to CH_4. Our Hugoniot curves of graphite and diamond do not follow this trend because the initial density of both materials is much higher. This implies that the particles interact more strongly under shock conditions, which reduces the shock compression ratio and shifts compression maximum towards higher pressures <cit.>. We also see this trend when we compare the CH Hugoniot curves for different initial densities and that of graphite and diamond with each other in Fig. <ref>.The compression maximum appear at a similar temperature (∼2×10^6 K) for all C-H compounds, graphite, and diamond (see <cit.>). This corresponds to the thermal ionization ofthe K shell of carbon, as we will discuss later. The magnitude of the shift in the compression maxima, ∼0.1, of the CH materials is small compared to the deviations between predictions from various EOS models in Fig. <ref>.When the EOS of mixtures needs to be derived for astrophysical applications or to design shock wave experiments, one typically invokes the ideal mixing approximation because the EOS of the fully interacting systems, which we have computed here, is often not available. One simply approximates the properties of the mixture as a linear combination of the endmember properties at the same pressure and temperature. For C-H mixtures, all interactions between C and H particles are thus neglected. Furthermore, the ionization fraction of carbon atoms in the C-H mixtures is set equal to the ionization fraction of carbon at the same conditions. The presence of hydrogen does not affect the ionization of carbon atoms and vice versa. For a mixture of heavier species and metallic hydrogen at conditions in gas giant interiors, it has been shown that a linear mixing approximation was not accurate <cit.> while it worked very well for molecular H_2-H_2O mixtures in ice giant envelopes <cit.>.To test the validity of the linear mixing approximation at higher temperatures, relevant for stellar cores, we performed additional PIMC and DFT-MD simulations for pure H and C and combined them with EOS tables from Refs. <cit.>. Our results in Figs. <ref> and <ref> show that the linear mixing approximation works exceptionally well for all C-H compounds. We only see a small underestimation of the compression-ratio maximum of less than 1% (∼0.035). Under these conditions, the shock compression is controlled by the ionization equilibrium of K-shell electrons of the C ions. This appears to be rather insensitive to whether a C ion is surrounded by a C-H mixture or just by other C ions. We can thus anticipate that the maximum uncertainty induced by using an ideal mixing rule is below the 1% level for stellar core conditions.Figure <ref> also shows that carbon and all C-H compounds exhibit only a single compression maximum while nitrogen, oxygen, and neon <cit.> display two that have been attributed to the excitation of K- and L-shell electrons. The fact that carbon materials do not show the lower L-shell compression maximum of ∼3 TPa requires further investigation. In order to better understand the difference in the Hugoniot curve shapes, we compare the electronic density of states (DOS) in Fig. <ref> that we have derived from DFT-MD simulations of oxygen and polystyrene at 4-fold compression and 10^5 K.Both DOSs show an isolated peak at low energy, which corresponds to the electrons in K shells of oxygen and carbon. Their thermal ionization leads to a pronounced compression maximum along the Hugoniot curve.However, while oxygen DOS shows another set of sharp peaks corresponding to the L-shell, the eigenstates of polystyrene are even distributed and partially merged with the continuum.It is the excitation of electrons in these well-defined L-shell states that leads to the second compression maximum for oxygen, nitrogen, and neon.For carbon and hydrocarbons, the L-shell ionization is much more gradual and already starts at much lower temperatures <cit.>than for oxygen. This does not lead to a well-defined compression peak but only to the shoulder in Hugoniot curve that we have discussed in Fig. <ref>.Conclusions.We performed the first entirely first-principles determination of hydrocarbon mixtures in the WDM regime by including all non-ideal effects. Based on PIMC and DFT-MD, we obtained coherent sets of EOS over wide range of density and temperature conditions and derived the shock Hugoniot curves of a series of hydrocarbon materials. For polystyrene, we predict a maximum shock compression ratio of 4.7 while earlier estimates range from 4.3 to 4.7. Our calculated Hugoniot curve agrees very well with experimental measurements and provides guidance for the interpretation of experiments on the Gbar platform at NIF. We observe a single compression maximum for hydrocarbon materials while there are two compression maxima in the Hugoniot curve of nitrogen, oxygen and neon. We have shown that this difference is related to the properties of the L-shell ionization, which is much more gradual for carbon. We found that the linear isobaric-isothermal mixing approximation works very well, resulting in a discrepancy in the density of CH of 1% or less under stellar core conditions. This implies that it is sufficient to derive only accurate EOS tables for the endmembers in order to provide a thermodynamic description of deep stellar interiors. Acknowledgments. This research is supported by DOE grants DE-SC0010517 and DE-SC0016248. FS and BM acknowledge partial support from DOE Grant DE-NA0001859.SZ is partially supported by the PLS-Postdoctoral Grant of the Lawrence Livermore National Laboratory. Computational support was provided by the Blue Waters sustained-petascale computing project (NSF ACI-1640776). We thank L. Benedict for helpful discussions. 92 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Betti and Hurricane(2016)]Betti2016 author author R. Betti and author O. Hurricane, @noopjournal journal Nature Phys. volume 12, pages 435 (year 2016)NoStop [Meezan et al.(2016)Meezan, Edwards, Hurricane, Patel, Callahan, Hsing, Town, Albert, Amendt, Hopkins et al.]Meezan2016 author author N. Meezan, author M. Edwards, author O. Hurricane, author P. Patel, author D. Callahan, author W. Hsing, author R. Town, author F. Albert, author P. Amendt, author L. B. Hopkins,et al., @noopjournal journal Plasma Phys. Contr. Fusion volume 59, pages 014021 (year 2016)NoStop [Goncharov et al.(2016)Goncharov, Regan, Campbell, Sangster, Radha, Myatt, Froula, Betti, Boehly, Delettrez et al.]Goncharov2016 author author V. Goncharov, author S. Regan, author E. Campbell, author T. Sangster, author P. Radha, author J. Myatt, author D. Froula, author R. Betti, author T. Boehly, author J. Delettrez,et al., @noopjournal journal Plasma Phys. Contr. Fusion volume 59, pages 014008 (year 2016)NoStop [Guillot(2005)]Guillot2005 author author T. Guillot, @noopjournal journal Annu. Rev. Earth Planet. Sci. volume 33, pages 493 (year 2005)NoStop [Wallerstein et al.(1997)Wallerstein, Iben, Parker, Boesgaard, Hale, Champagne, Barnes, Käppeler, Smith, Hoffman et al.]Wallerstein1997 author author G. Wallerstein, author I. Iben, author P. Parker, author A. M. Boesgaard, author G. M. Hale, author A. E. Champagne, author C. A. Barnes, author F. Käppeler, author V. V. Smith, author R. D. Hoffman,et al.,@noopjournal journal Rev. Mod. Phys.volume 69, pages 995 (year 1997)NoStop [Hauver and Melani(1964)]Hauver1964 author author G. Hauver and author A. Melani, @nooptitle Shock Compression of Plexiglas and Polystyrene, type Tech. Rep. (institution DTIC Document, year 1964)NoStop [Dudoladov et al.(1969)Dudoladov, Rakitin, Sutulov, andTelegin]Dudoladov1969 author author I. Dudoladov, author V. Rakitin, author Y. N. Sutulov,andauthor G. Telegin, @noopjournal journal J. Appl. Mech. Tech. Phys.volume 10, pages 673 (year 1969)NoStop [Lamberson et al.(1972)Lamberson, Asay, and Guenther]Lamberson1972 author author D. L. Lamberson, author J. R. Asay,and author A. H. Guenther,@noopjournal journal J. Appl. Phys.volume 43, pages 976 (year 1972)NoStop [Marsh(1980)]Marsh1980 author author S. P. Marsh, @nooptitle LASL shock Hugoniot data, Vol. volume 5 (publisher Univ of California Press, year 1980)NoStop [Nellis et al.(1984)Nellis, Ree, Trainor, Mitchell, andBoslough]Nellis1984 author author W. J. Nellis, author F. H. Ree, author R. J. Trainor, author A. C. Mitchell,and author M. B. Boslough, 10.1063/1.447027 journal journal J. Chem. Phys. volume 80, pages 2789 (year 1984)NoStop [Kodama et al.(1991)Kodama, Tanaka, Nakai, Nishihara, Norimatsu, Yamanaka, and Nakai]Kodama1991 author author R. Kodama, author K. Tanaka, author M. Nakai, author K. Nishihara, author T. Norimatsu, author T. Yamanaka,and author S. Nakai, @noopjournal journal Phys. Fluids B: Plasma Phys. volume 3, pages 735 (year 1991)NoStop [Bushman et al.(1996)Bushman, Lomonosov, Fortov, Khishchenko, Zhernokletov, and Sutulov]Bushman1996 author author A. Bushman, author I. Lomonosov, author V. Fortov, author K. Khishchenko, author M. Zhernokletov,and author Y. N. Sutulov, http://www.jetp.ac.ru/cgi-bin/dn/e_082_05_0895.pdf journal journal J. Exp. Theor. Phys. volume 82, pages 895 (year 1996)NoStop [Koenig et al.(1998)Koenig, Benuzzi, Faral, Krishnan, Boudenne, Jalinaud, Rémond, Decoster, Batani, Beretta et al.]Koenig1998 author author M. Koenig, author A. Benuzzi, author B. Faral, author J. Krishnan, author J. Boudenne, author T. Jalinaud, author C. Rémond, author A. Decoster, author D. Batani, author D. Beretta,et al., @noopjournal journal Appl. Phys. Lett. volume 72, pages 1033 (year 1998)NoStop [Cauble et al.(1997)Cauble, Silva, Perry, Bach, Budil, Celliers, Collins, Ng, T. W. Barbee, Hammel, Holmes, Kilkenny, Wallace, Chiu, and Woolsey]Cauble1997 author author R. Cauble, author L. B. D. Silva, author T. S. Perry, author D. R. Bach, author K. S. Budil, author P. Celliers, author G. W. Collins, author A. Ng, author J. T. W. Barbee, author B. A.Hammel, author N. C.Holmes, author J. D.Kilkenny, author R. J.Wallace, author G. Chiu,and author N. C. Woolsey, 10.1063/1.872362 journal journal Phys. Plasmas volume 4,pages 1857 (year 1997)NoStop [Cauble et al.(1998)Cauble, Perry, Bach, Budil, Hammel, Collins, Gold, Dunn, Celliers, Da Silva et al.]Cauble1998 author author R. Cauble, author T. Perry, author D. Bach, author K. Budil, author B. Hammel, author G. Collins, author D. Gold, author J. Dunn, author P. Celliers, author L. Da Silva,et al., @noopjournal journal Phys. Rev. Lett. volume 80, pages 1248 (year 1998)NoStop [Koenig et al.(2003)Koenig, Philippe, Benuzzi-Mounaix, Batani, Tomasini, Henry, andHall]Koenig2003 author author M. Koenig, author F. Philippe, author A. Benuzzi-Mounaix, author D. Batani, author M. Tomasini, author E. Henry,and author T. Hall, @noopjournal journal Phys. Plasmas volume 10, pages 3026 (year 2003)NoStop [Ozaki et al.(2005)Ozaki, Ono, Takamatsu, Tanaka, Nakano, Kataoka, Yoshida, Wakabayashi, Nakai, Nagai et al.]Ozaki2005 author author N. Ozaki, author T. Ono, author K. Takamatsu, author K. Tanaka, author M. Nakano, author T. Kataoka, author M. Yoshida, author K. Wakabayashi, author M. Nakai, author K. Nagai,et al., @noopjournal journal Phys. Plasmas volume 12, pages 124503 (year 2005)NoStop [Hu et al.(2008)Hu, Smalyuk, Goncharov, Knauer, Radha, Igumenshchev, Marozas, Stoeckl, Yaakobi, Shvarts et al.]Hu2008 author author S. Hu, author V. Smalyuk, author V. Goncharov, author J. Knauer, author P. Radha, author I. Igumenshchev, author J. Marozas, author C. Stoeckl, author B. Yaakobi, author D. Shvarts,et al., @noopjournal journal Phys. Rev. Lett. volume 100, pages 185003 (year 2008)NoStop [Ozaki et al.(2009)Ozaki, Sano, Ikoma, Shigemori, Kimura, Miyanishi, Vinci, Ree, Azechi, Endo et al.]Ozaki2009 author author N. Ozaki, author T. Sano, author M. Ikoma, author K. Shigemori, author T. Kimura, author K. Miyanishi, author T. Vinci, author F. Ree, author H. Azechi, author T. Endo,et al., @noopjournal journal Phys. Plasmas volume 16, pages 062702 (year 2009)NoStop [Barrios et al.(2010)Barrios, Hicks, Boehly, Fratanduono, Eggert, Celliers, Collins, and Meyerhofer]Barrios2010 author author M. Barrios, author D. Hicks, author T. Boehly, author D. Fratanduono, author J. Eggert, author P. Celliers, author G. Collins,and author D. Meyerhofer, @noopjournal journal Phys. Plasmas volume 17,pages 056307 (year 2010)NoStop [Shu et al.(2010)Shu, Fu, Huang, Jia, Zhou, Wu, Ye, and Gu]Shu2010 author author H. Shu, author S. Fu, author X. Huang, author G. Jia, author H. Zhou, author J. Wu, author J. Ye,andauthor Y. Gu, @noopjournal journal Chin. Opt. Lett. volume 8, pages 1142 (year 2010)NoStop [Shang et al.(2013)Shang, Wei, Li, Yi, Zhu, Song, Huang, and Yang]Shang2013 author author W. Shang, author H. Wei, author Z. Li, author R. Yi, author T. Zhu, author T. Song, author C. Huang,andauthor J. Yang, @noopjournal journal Phys. Plasmas volume 20, pages 102702 (year 2013)NoStop [Shu et al.(2015)Shu, Huang, Ye, Wu, Jia, Fang, Xie, Zhou, andFu]Shu2015 author author H. Shu, author X. Huang, author J. Ye, author J. Wu, author G. Jia, author Z. Fang, author Z. Xie, author H. Zhou,and author S. Fu, @noopjournal journal Eur. Phys. J. D volume 69,pages 1 (year 2015)NoStop [Barrios et al.(2012)Barrios, Boehly, Hicks, Fratanduono, Eggert, Collins, andMeyerhofer]Barrios2012 author author M. Barrios, author T. Boehly, author D. Hicks, author D. Fratanduono, author J. Eggert, author G. Collins,and author D. Meyerhofer, @noopjournal journal J. Appl. Phys. volume 111,pages 093515 (year 2012)NoStop [Huser et al.(2013)Huser, Ozaki, Sano, Sakawa, Miyanishi, Salin, Asaumi, Kita, Kondo, Nakatsuka et al.]Huser2013 author author G. Huser, author N. Ozaki, author T. Sano, author Y. Sakawa, author K. Miyanishi, author G. Salin, author Y. Asaumi, author M. Kita, author Y. Kondo, author K. Nakatsuka, et al., @noopjournal journal Phys. Plasmas volume 20, pages 122703 (year 2013)NoStop [Huser et al.(2015)Huser, Recoules, Ozaki, Sano, Sakawa, Salin, Albertazzi, Miyanishi, and Kodama]Huser2015 author author G. Huser, author V. Recoules, author N. Ozaki, author T. Sano, author Y. Sakawa, author G. Salin, author B. Albertazzi, author K. Miyanishi,and author R. Kodama, @noopjournal journal Phys. Rev. E volume 92, pages 063108 (year 2015)NoStop [Moore et al.(2016)Moore, Prisbrey, Baker, Celliers, Fry, Dittrich, Wu, Kervin, Schoff, Farrell et al.]Moore2016 author author A. S. Moore, author S. Prisbrey, author K. L. Baker, author P. M. Celliers, author J. Fry, author T. R. Dittrich, author K.-J. J. Wu, author M. L.Kervin, author M. E.Schoff, author M. Farrell,et al., @noopjournal journal High Energ. Dens. Phys. volume 20,pages 23 (year 2016)NoStop [Hamel et al.(2012)Hamel, Benedict, Celliers, Barrios, Boehly, Collins, Döppner, Eggert, Farley, Hicks et al.]Hamel2012 author author S. Hamel, author L. X. Benedict, author P. M. Celliers, author M. Barrios, author T. Boehly, author G. Collins, author T. Döppner, author J. Eggert, author D. Farley, author D. Hicks,et al., @noopjournal journal Phys. Rev. B volume 86, pages 094113 (year 2012)NoStop [Bolkhovitinov and Khvostov(1978)]Bolkhovitinov1978 author author L. Bolkhovitinov and author Y. B. Khvostov, @noopjournal journal Nature volume 274, pages 882 (year 1978)NoStop [Koenig et al.(1999)Koenig, Benuzzi, Philippe, Batani, Hall, Grandjouan, and Nazarov]Koenig1999 author author M. Koenig, author A. Benuzzi, author F. Philippe, author D. Batani, author T. Hall, author N. Grandjouan,and author W. Nazarov, @noopjournal journal Phys. Plasmas volume 6, pages 3296 (year 1999)NoStop [Takamatsu et al.(2003)Takamatsu, Ozaki, Tanaka, Ono, Nagai, Nakai, Watari, Sunahara, Nakano, Kataoka et al.]Takamatsu2003 author author K. Takamatsu, author N. Ozaki, author K. Tanaka, author T. Ono, author K. Nagai, author M. Nakai, author T. Watari, author A. Sunahara, author M. Nakano, author T. Kataoka,et al., @noopjournal journal Phys. Rev. E volume 67, pages 056406 (year 2003)NoStop [Koenig et al.(2005)Koenig, Benuzzi-Mounaix, Batani, Hall, and Nazarov]Koenig2005 author author M. Koenig, author A. Benuzzi-Mounaix, author D. Batani, author T. Hall,andauthor W. Nazarov, @noopjournal journal Phys. Plasmas volume 12, pages 012706 (year 2005)NoStop [Kraus et al.(2016)Kraus, Chapman, Kritcher, Baggott, Bachmann, Collins, Glenzer, Hawreliak, Kalantar, Landen, Ma, Le Pape, Nilsen, Swift, Neumayer, Falcone, Gericke, and Döppner]Kraus2016 author author D. Kraus, author D. A. Chapman, author A. L. Kritcher, author R. A. Baggott, author B. Bachmann, author G. W. Collins, author S. H. Glenzer, author J. A. Hawreliak, author D. H. Kalantar, author O. L. Landen, author T. Ma, author S. Le Pape, author J. Nilsen, author D. C.Swift, author P. Neumayer, author R. W. Falcone, author D. O. Gericke,and author T. Döppner, 10.1103/PhysRevE.94.011202 journal journal Phys. Rev. E volume 94, pages 011202 (year 2016)NoStop [Swift et al.(2012)Swift, Hawreliak, Braun, Kritcher, Glenzer, Collins, Rothman, Chapman, and Rose]Swift2011 author author D. C. Swift, author J. A. Hawreliak, author D. Braun, author A. Kritcher, author S. Glenzer, author G. Collins, author S. D. Rothman, author D. Chapman,and author S. Rose, @noopjournal journal AIP Conf. Proc. volume 1426, pages 477 (year 2012)NoStop [Döppner et al.(2014)Döppner, Kritcher, Kraus, Glenzer, Bachmann, Chapman, Collins, Falcone, Hawreliak, Landen et al.]Doppner2014 author author T. Döppner, author A. Kritcher, author D. Kraus, author S. Glenzer, author B. Bachmann, author D. Chapman, author G. Collins, author R. Falcone, author J. Hawreliak, author O. Landen,et al., @noopjournal journal J. Phys.: Conf. Ser. volume 500, pages 192019 (year 2014)NoStop [Kritcher et al.(2014)Kritcher, Döppner, Swift, Hawreliak, Collins, Nilsen, Bachmann, Dewald, Strozzi, Felker et al.]Kritcher2014 author author A. Kritcher, author T. Döppner, author D. Swift, author J. Hawreliak, author G. Collins, author J. Nilsen, author B. Bachmann, author E. Dewald, author D. Strozzi, author S. Felker,et al., @noopjournal journal High Energ. Dens. Phys. volume 10, pages 27 (year 2014)NoStop [Kritcher et al.(2016)Kritcher, Doeppner, Swift, Hawreliak, Nilsen, Hammer, Bachmann, Collins, Landen, Keane et al.]Kritcher2016 author author A. Kritcher, author T. Doeppner, author D. Swift, author J. Hawreliak, author J. Nilsen, author J. Hammer, author B. Bachmann, author G. Collins, author O. Landen, author C. Keane,et al., @noopjournal journal J. Phys.: Conf. Ser. volume 688, pages 012055 (year 2016)NoStop [Nilsen et al.(2016)Nilsen, Bachmann, Zimmerman, Hatarik, Döppner, Swift, Hawreliak, Collins, Falcone, Glenzer et al.]Nilsen2016 author author J. Nilsen, author B. Bachmann, author G. Zimmerman, author R. Hatarik, author T. Döppner, author D. Swift, author J. Hawreliak, author G. Collins, author R. Falcone, author S. Glenzer,et al., @noopjournal journal High Energ. Dens. Phys. volume 21, pages 20 (year 2016)NoStop [Nora et al.(2015)Nora, Theobald, Betti, Marshall, Michel, Seka, Yaakobi, Lafon, Stoeckl, Delettrez et al.]Nora2015 author author R. Nora, author W. Theobald, author R. Betti, author F. Marshall, author D. Michel, author W. Seka, author B. Yaakobi, author M. Lafon, author C. Stoeckl, author J. Delettrez,et al.,@noopjournal journal Phys. Rev. Lett.volume 114, pages 045001 (year 2015)NoStop [CHG()]CHGbarExpts @noopnote T. Doeppner and A. Kritcher, private communication.Stop [Lyon and Johnson(1992)]sesame editor S. P. Lyon and editor J. D. Johnson, eds., @nooptitle SESAME: The Los Alamos National Laboratory Equation of State Database (publisher Group T-1, Report No. LA-UR-92-3407, year 1992)NoStop [More et al.(1988)More, Warren, Young, and Zimmerman]More1988 author author R. More, author K. Warren, author D. Young,and author G. Zimmerman, @noopjournal journal Phys. Fluids volume 31, pages 3059 (year 1988)NoStop [Graziani et al.(2014)Graziani, Desjarlais, Redmer, andTrickey]FCWDM editor F. Graziani, editor M. P. Desjarlais, editor R. Redmer,and editor S. B. Trickey, eds., @nooptitle Frontiers and Challenges in Warm Dense Matter(publisher Springer International Publishing, year 2014)NoStop [Lambert et al.(2006)Lambert, Clérouin, and Zérah]Lambert2006 author author F. Lambert, author J. Clérouin,and author G. Zérah, @noopjournal journal Phys. Rev. E volume 73, pages 016403 (year 2006)NoStop [Wilson et al.(2006)Wilson, Sonnad, Sterne, and Isaacs]Purgatorio2006 author author B. Wilson, author V. Sonnad, author P. Sterne,and author W. Isaacs, @noopjournal journal J. Quant. Spectrosc. Radiat. Transfer volume 99, pages 658 (year 2006)NoStop [Mattsson et al.(2010)Mattsson, Lane, Cochrane, Desjarlais, Thompson, Pierce, andGrest]Mattsson2010 author author T. R. Mattsson, author J. M. D. Lane, author K. R. Cochrane, author M. P. Desjarlais, author A. P. Thompson, author F. Pierce,and author G. S. Grest, @noopjournal journal Phys. Rev. B volume 81, pages 054103 (year 2010)NoStop [Wang et al.(2011)Wang, He, and Zhang]Wang2011 author author C. Wang, author X.-T. He,andauthor P. Zhang, @noopjournal journal Phys. Plasmas volume 18, pages 082707 (year 2011)NoStop [Lambert and Recoules(2012)]Lambert2012 author author F. Lambert and author V. Recoules, @noopjournal journal Phys. Rev. E volume 86, pages 026405 (year 2012)NoStop [Hu et al.(2014)Hu, Boehly, Collins et al.]Hu2014 author author S. Hu, author T. Boehly, author L. Collins,et al.,@noopjournal journal Phys. Rev. Evolume 89, pages 063104 (year 2014)NoStop [Hu et al.(2015)Hu, Collins, Goncharov, Kress, McCrory, Skupsky et al.]Hu2015 author author S. Hu, author L. Collins, author V. Goncharov, author J. Kress, author R. McCrory, author S. Skupsky,et al., @noopjournal journal Phys. Rev. E volume 92, pages 043104 (year 2015)NoStop [Hu et al.(2016a)Hu, Collins, Goncharov, Kress, McCrory, and Skupsky]Hu2016 author author S. Hu, author L. Collins, author V. Goncharov, author J. Kress, author R. McCrory,and author S. Skupsky, @noopjournal journal Phys. Plasmas volume 23, pages 042704 (year 2016a)NoStop [Knyazev and Levashov(2015)]Knyazev2015 author author D. Knyazev and author P. Levashov, @noopjournal journal Phys. Plasmas volume 22, pages 053303 (year 2015)NoStop [Colin-Lalu et al.(2016)Colin-Lalu, Recoules, Salin, Plisson, Brambrink, Vinci, Bolis, and Huser]Colin2016 author author P. Colin-Lalu, author V. Recoules, author G. Salin, author T. Plisson, author E. Brambrink, author T. Vinci, author R. Bolis,and author G. Huser, @noopjournal journal Phys. Rev. E volume 94, pages 023204 (year 2016)NoStop [Hu et al.(2016b)Hu, Militzer, Collins, Driver, and Kress]PhysRevB.94.094109 author author S. X. Hu, author B. Militzer, author L. A. Collins, author K. P. Driver,and author J. D. Kress, 10.1103/PhysRevB.94.094109 journal journal Phys. Rev. B volume 94, pages 094109 (year 2016b)NoStop [D. M. Ceperley(1995)]Ceperley1995 author author D. M. Ceperley,@noopjournal journal Rev. Mod. Phys.volume 67, pages 279 (year 1995)NoStop [Ceperley(1996)]Ce96 author author D. M. Ceperley, in @noopbooktitle Monte Carlo and Molecular Dynamics of Condensed Matter Systems, Vol. volume 49, editor edited by editor K. Binder and editor G. Ciccotti (publisher Editrice Compositori, Bologna, Italy, year 1996) p.pages 443NoStop [Pierleoni et al.(1994)Pierleoni, Ceperley, Bernu, andMagro]Pierleoni1994 author author C. Pierleoni, author D. Ceperley, author B. Bernu,and author W. Magro, @noopjournal journal Phys. Rev. Lett. volume 73, pages 2145 (year 1994)NoStop [Militzer and Ceperley(2001)]PhysRevE.63.066404 author author B. Militzer and author D. M. Ceperley, 10.1103/PhysRevE.63.066404 journal journal Phys. Rev. E volume 63, pages 066404 (year 2001)NoStop [Militzer(2009)]Militzer2009 author author B. Militzer, @noopjournal journal Phys. Rev. B volume 79, pages 155105 (year 2009)NoStop [Driver and Militzer(2012)]Driver2012 author author K. P. Driver and author B. Militzer, @noopjournal journal Phys. Rev. Lett. volume 108, pages 115502 (year 2012)NoStop [Militzer and Driver(2015)]Militzer2015Silicon author author B. Militzer and author K. P. Driver, @noopjournal journal Phys. Rev. Lett. volume 115, pages 176403 (year 2015)NoStop [Militzer(2000)]militzerphd author author B. Militzer, @nooptype Ph.D. Thesis, school University of Illinois at Urbana-Champaign (year 2000)NoStop [Ceperley(1991)]Ceperley1991 author author D. M. Ceperley, @noopjournal journal J. Stat. Phys. volume 63, pages 1237 (year 1991)NoStop [Magro et al.(1996)Magro, Ceperley, Pierleoni, and Bernu]PhysRevLett.76.1240 author author W. R. Magro, author D. M. Ceperley, author C. Pierleoni,andauthor B. Bernu, 10.1103/PhysRevLett.76.1240 journal journal Phys. Rev. Lett. volume 76, pages 1240 (year 1996)NoStop [Militzer et al.(2001)Militzer, Ceperley, Kress, Johnson, Collins, and Mazevet]PhysRevLett.87.275502 author author B. Militzer, author D. M. Ceperley, author J. D. Kress, author J. D. Johnson, author L. A. Collins,and author S. Mazevet, 10.1103/PhysRevLett.87.275502 journal journal Phys. Rev. Lett. volume 87, pages 275502 (year 2001)NoStop [Hu et al.(2010)Hu, Militzer, Goncharov, and Skupsky]PhysRevLett.104.235003 author author S. X. Hu, author B. Militzer, author V. N. Goncharov,andauthor S. Skupsky, 10.1103/PhysRevLett.104.235003 journal journal Phys. Rev. Lett. volume 104, pages 235003 (year 2010)NoStop [Militzer and Ceperley(2000)]PhysRevLett.85.1890 author author B. Militzer and author D. M. Ceperley, @noopjournal journal Phys. Rev. Lett. volume 85, pages 1890 (year 2000)NoStop [Hu et al.(2011)Hu, Militzer, Goncharov, and Skupsky]PhysRevB.84.224109 author author S. X. Hu, author B. Militzer, author V. N. Goncharov,andauthor S. Skupsky, 10.1103/PhysRevB.84.224109 journal journal Phys. Rev. B volume 84, pages 224109 (year 2011)NoStop [Militzer et al.(1999)Militzer, Magro, and Ceperley]CTPP:CTPP2150390137 author author B. Militzer, author W. Magro, and author D. Ceperley, 10.1002/ctpp.2150390137 journal journal Contrib. Plasm. Phys. volume 39, pages 151 (year 1999)NoStop [Militzer and Graham(2006)]Militzer20062136 author author B. Militzer and author R. L. Graham, http://dx.doi.org/10.1016/j.jpcs.2006.05.015 journal journal J. Phys. Chem. Solids volume 67, pages 2136(year 2006)NoStop [Militzer(2006)]PhysRevLett.97.175501 author author B. Militzer, 10.1103/PhysRevLett.97.175501 journal journal Phys. Rev. Lett. volume 97, pages 175501 (year 2006)NoStop [Militzer(2005)]Militzer2005 author author B. Militzer, 10.1007/s10909-005-5485-8 journal journal J. Low Temp. Phys. volume 139, pages 739 (year 2005)NoStop [Benedict et al.(2014)Benedict, Driver, Hamel, Militzer, Qi, Correa, Saul, and Schwegler]Benedict2014C author author L. X. Benedict, author K. P. Driver, author S. Hamel, author B. Militzer, author T. Qi, author A. A. Correa, author A. Saul,and author E. Schwegler, @noopjournal journal Phys. Rev. B volume 89, pages 224109 (year 2014)NoStop [Driver and Militzer(2016)]Driver2016Nitrogen author author K. P. Driver and author B. Militzer, @noopjournal journal Phys. Rev. B volume 93, pages 064101 (year 2016)NoStop [Driver et al.(2015)Driver, Soubiran, Zhang, and Militzer]Driver2015Oxygen author author K. P. Driver, author F. Soubiran, author S. Zhang,and author B. Militzer, @noopjournal journal J. Chem. Phys. volume 143, pages 164507 (year 2015)NoStop [Driver and Militzer(2015)]Driver2015Neon author author K. P. Driver and author B. Militzer, @noopjournal journal Phys. Rev. B volume 91, pages 045103 (year 2015)NoStop [Pollock(1988)]pollock1988 author author E. L. Pollock, @noopjournal journal Comput. Phys. Commun. volume 52, pages 49 (year 1988)NoStop [Militzer(2016)]pdm author author B. Militzer, @noopjournal journal Comput. Phys. Commun. volume 204, pages 88 (year 2016)NoStop [Kresse and Furthmüller(1996)]kresse96b author author G. Kresse and author J. Furthmüller, @noopjournal journal Phys. Rev. B volume 54, pages 11169 (year 1996)NoStop [Perdew and Zunger(1981)]Perdew81 author author J. P. Perdew and author A. Zunger, @noopjournal journal Phys. Rev. B volume 23, pages 5048 (year 1981)NoStop [Ceperley and Alder(1980)]Ceperley1980 author author D. M. Ceperley and author B. J. Alder, @noopjournal journal Phys. Rev. Lett. volume 45, pages 566 (year 1980)NoStop [Blöchl et al.(1994)Blöchl, Jepsen, and Andersen]Blochl1994 author author P. E. Blöchl, author O. Jepsen, and author O. K. Andersen,@noopjournal journal Phys. Rev. Bvolume 49, pages 16223 (year 1994)NoStop [Nosé(1984)]Nose1984 author author S. Nosé, 10.1063/1.447334 journal journal J. Chem. Phys. volume 81,pages 511 (year 1984)NoStop [opi()]opium @noopnote http://opium.sourceforge.netNoStop [sup()]supp @noopnote See Supplemental Material for the way of determining the initial conditions for the shock Hugoniot curves and details of the EOS data.Stop [NIS()]NIST @noopnote NIST/TRC Web Thermo Tables (WTT) NIST Standard Reference Subscription Database 3 - Professional Edition Version 2-2012-1-Pro, http://wtt-pro.nist.gov/wtt-pro/NoStop [Walters et al.(2000)Walters, Hackett, and Lyon]walters_2000 author author R. N. Walters, author S. M. Hackett,and author R. E. Lyon, @noopjournal journal Fire Mater. volume 24, pages 245 (year 2000)NoStop [Sterne et al.(2016)Sterne, Benedict, Hamel, Correa, Milovich, Marinak, Celliers,and Fratanduono]Sterne2016 author author P. A. Sterne, author L. X. Benedict, author S. Hamel, author A. A. Correa, author J. L. Milovich, author M. M. Marinak, author P. M. Celliers,and author D. E. Fratanduono, http://stacks.iop.org/1742-6596/717/i=1/a=012082 journal journal J. Phys. Conf. Ser. volume 717, pages 012082 (year 2016)NoStop [Young and Corey(1995)]Young1995 author author D. A. Young and author E. M. Corey, @noopjournal journal J. App. Phys. volume 78, pages 3748 (year 1995)NoStop [Soubiran and Militzer(2016)]Soubiran2016 author author F. Soubiran and author B. Militzer, @noopjournal journal Astrophys. J. volume 829, pages 14 (year 2016)NoStop [Soubiran and Militzer(2015)]Soubiran2015 author author F. Soubiran and author B. Militzer, @noopjournal journal Astrophys. J. volume 806, pages 228 (year 2015)NoStop [Potekhin et al.(2005)Potekhin, Massacrier, and Chabrier]Potekhin2005 author author A. Y. Potekhin, author G. Massacrier,and author G. Chabrier, @noopjournal journal Phys. Rev. E volume 72, pages 046402 (year 2005)NoStop | http://arxiv.org/abs/1706.09073v1 | {
"authors": [
"Shuai Zhang",
"Kevin P. Driver",
"François Soubiran",
"Burkhard Militzer"
],
"categories": [
"physics.plasm-ph",
"astro-ph.SR",
"cond-mat.other",
"physics.chem-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170627233241",
"title": "First-principles Equation of State and Shock Compression Predictions of Warm Dense Hydrocarbons"
} |
[email protected] for Microgravity Research, Department of Physics and Florida Space Institute, University of Central Florida, 4111 Libra Drive, Orlando FL-32816, USA, Phone: +1-407-823-6168 In an effort to better understand the early stages of planet formation, we have developed a 1.5U payload that flew on the International Space Station (ISS) in the NanoRacks NanoLab facility between September 2014 and March 2016. This payload, named NanoRocks, ran a particle collision experiment under long-term microgravity conditions. The objectives of the experiment were (a) to observe collisions between mm-sized particles at relative velocities of<1 cm/s, and (b) to study the formation and disruption of particle clusters for different particle types and collision velocities. Four types of particles were used: mm-sized acrylic, glass, and copper beads, and 0.75 mm-sized JSC-1 lunar regolith simulant grains. The particles were placed in sample cells carved out of an aluminum tray. This tray was attached to one side of the payload casing with three springs. Every 60 s, the tray was agitated and the resulting collisions between the particles in the sample cells were recorded by the experiment camera. During the 18 months the payload stayed on ISS, we obtained 158 videos, thus recording a great number of collisions. The average particle velocities in the sample cells after each shaking event were around 1 cm/s. After shaking stopped, the inter-particle collisions damped the particle kinetic energy in less than 20 s, reducing the average particle velocity to below 1 mm/s, and eventually slowing them to below our detection threshold. As the particle velocity decreased, we observed the transition from bouncing to sticking collisions. We recorded the formation of particle clusters at the end of each experiment run. This paper describes the design and performance of the NanoRocks ISS payload.NanoRocks: Design and Performance of an Experiment Studying Planet Formation on the International Space Station Doug Maukonen December 30, 2023 ===============================================================================================================§ INTRODUCTION The current state of knowledge on the early stages of planet formation includes the growth of dust grains around newly formed stars inside of protoplanetary disks (PPDs) <cit.>. These grains are observed to be about a micrometer in size after condensation from the gaseous phase <cit.>. Numerical simulations and laboratory experiments have shown that they then proceed to grow inside the PPD through sticking collisions <cit.>, reaching mm to cm sizes. The progress of grain growth can be observed until this stage in the latest ALMA (Attacama Large Millimeter Array) pictures <cit.>. However, growth through sticking cannot account for the formation of m- to km-sized bodies, the so-called planetesimals. Indeed, at sizes of a mm to a cm, the grain relative velocities in the PPD are expected to increase <cit.>, and collisions between particles cease to result in sticking. As shown in previous laboratory experiments <cit.>, these particle sizes and velocities also lead to bouncing and fragmentation of the collision partners. Numerical simulations have shown that these collisional outcomes can stall the growth of particles inside the PPD, a phenomenon called the “bouncing barrier” <cit.>. Several planetesimal formation processes have been put forward to reconcile this bouncing barrier with the observed presence of planetesimal remnants such as asteroids and comets in our Solar System. In particular, Johansen et al. (2014) <cit.> proposed that gas turbulences and instabilities inside the PPD lead to concentrations of mm- to cm-sized particles into dense clouds, which eventually collapse under their own weight, forming a m- to km-sized planetesimal. The collisional behavior of dust particles at the bouncing barrier and inside multi-particle environments, such as these concentrated clouds, therefore plays an essential role in the processes leading to planetesimal formation. In an effort to better understand the early stages of planet formation, dust collision experiments were performed around the sticking to bouncing transition. Weidling et al. (2011) <cit.> studied bouncing collisions between mm-sized SiO_2 aggregates and Kothe et al. (2013) <cit.> and Brisset et al. (2017) <cit.> observed the collisional behavior of 100-μm dust aggregates. In order to achieve the range of relative velocities between colliding particles relevant to the bouncing transition (1 cm/s and below), all of these experiments were performed under microgravity conditions. The platform used to run these experiments was the drop tower in Bremen, Germany, which allows microgravity experiments of 9 s duration. We designed the NanoRocks experiment to use the NanoRacks platform on the International Space Station (ISS) to perform a long-duration microgravity experiment. In its 1.5U format (10×10×15 cm^3) it recorded the collisional behavior of several types of particles during repeated 5-minute experiment runs over a period of18 months between October 2014 and March 2016. In this paper, we describe the design and performance of the experiment. In Section <ref>, we describe the NanoRocks experimental setup. In Section <ref> we discuss the performance of the experiment onboard the ISS, and our conclusion are in Section <ref>. Scientific results of the experiment will be published in a separate paper. § THE NANOROCKS EXPERIMENT §.§ Scientific BackgroundThe objective of the NanoRocks experiment was to study low-energy collisions of mm-sized particles of different shapes and materials. In particular, the experiment was designed to study the bouncing-to-sticking transition for collisions with decreasing collision velocity. Current state-of-the-art research on the very early stages of planet formation relies on the understanding of dust particle behavior upon collisions inside the PPD. Güttler et al. (2009) <cit.> have developed a dust collision model predicting the outcome of collisions between dust aggregates that depends on their masses and relative velocities. The model predicts that small particles colliding at low relative velocities always stick to each other (“hit-and-stick” behavior <cit.>). However, with increasing particle mass and relative velocities, collision outcomes transition to bouncing. Figure <ref> shows the collision outcomes for dust aggregates of masses from 10^-4 to 1 grams and relative velocities from 10^-3 to 10 cm/s. This region of the parameter field covers the transition between sticking (green) and bouncing (yellow). The NanoRocks parameter space is indicated by the entire length of the red arrow in Figure <ref>. The low relative velocities required for these collisions (from a few cm/s to under a mm/s) can only be obtained under long-term microgravity conditions. Residual accelerations up to 10^-4g, with g being the acceleration of gravity at the surface of the Earth, were acceptable for the intended data collection. For these reasons, the ISS was the ideal platform to fly the NanoRocks experiment. §.§ Interfacing with the NanoRacks ISS Payload Platform The NanoRacks NanoLab provides power (5 V at 400 mA per line) and data routing to ground. The NanoRocks payload was allocated a 1.5U and 1.5 kg mass limit. Both power and data were provided via USB 2.0 connections. Further details on the payload requirements to fly on the NanoRacks NanoLab can be found in the NanoRacks Internal Platforms 1A/2A and NanoLab Modules Interface Control Document <cit.>.All hardware and flight vehicle compatibility tests were provided and performed by NanoRacks. Figure <ref>b shows an exterior view of the NanoRocks experiment flight unit with its USB connections. For this experiment, the use of two connectors provided additional power for the camera. Once on the ISS, data exchange with the payload was possible via NanoRacks. Video data downloaded from the ISS laptop was made available via an ftp repository. Command file upload was also possible via NanoRacks. §.§ Experiment Setup The main component of the experiment is an aluminum tray (∼8×8×2 cm^3) which was divided into eight sample cells each holding different types and combinations of particles (Figure <ref>). Each sample cell was 3 mm deep, while the other dimensions varied (Figure <ref>). The tray was sealed with a transparent polycarbonate top plate to allow for observation of the particles inside the sample cells. The tray was evacuated via an attached vacuum valve to ∼10^-4 bar. The evacuation of the experiment tray was performed at the Center for Microgravity Research (CMR) before shipping the payload to NanoRacks for integration. During the development of the experiment, the particle tray was tested to ensure that it could be safely evacuated, and to determine if it would maintain vacuum during its operation on station.While we were able to achieve high vacuum while the tray was connected to a pump (<10^-6 bar), the tray did not hold this quality of vacuum once disconnected, stabilizing around 10^-4 bar. Our longest test revealed that, after several days, the pressure in the tray was on the order of 10^-2 bar. Most of the increase in pressure came within the first day after it was removed from vacuum, and naturally slowed after that. Therefore, we expect that the tray gradually leaked, increasing the pressure over the course of the flight, but no measurements of the tray pressure were possible during the mission.The experiment tray was mounted on three springs to allow for shaking to induce particle velocities via collisions with the walls. During an experiment run, an electromagnet grabbed and released the bottom of the tray every 60 s, resulting in shaking of the particle samples. An array of 4 LEDs (2 on each side of the camera) provided the required illumination, while a Hack HD camera <cit.> recorded the motion of the particles (Figure <ref>). A light diffuser for the LED array was originally planned in the form of a layer of blurred paper between the LEDs and the experiment tray. Upon completion of hardware assembly this diffuser was not implemented as it could not be fixed reliably to the side walls and a displacement of the diffuser sheet during handling or launch would have significantly impaired scientific data collection. Losing partial data return through LED glare (< 5 % of the observed area) was considered an acceptable mitigation to the risk of losing the entire science data return. The video recording was performed at a resolution of 1080P and 30 fps, allowing for the observation of particle motion from one frame to the next and the determination of collision parameters. The experiment data recorded during an experiment run was stored on the experiment memory card, a 32GB microSDHC class 4 card. NanoRocks was also outfitted with its own electronics to allow for autonomous operation of the experiment during its stay on ISS. §.§ Electronics and Data Collection The NanoRocks electronics were composed of two boards: the NanoRacks Control Module 001 (NCM) provided by Celestial Circuits and a custom electronics board produced at UCF. The NCM provided power switching to the different electronic components (the camera, electro-magnet, and LED array) as well as a data connection to the ISS laptop, which could read the microSD card on the board. For use of the NCM board with its compatible camera, no additional electronics would have been required. However, during the design and testing of the hardware at the CMR, we recognized that the resolution and frame rate provided by the NCM camera were not sufficient for the NanoRocks science goals. Therefore, an alternative camera, the HackHD 1080P, was chosen for the experiment. The HackHD can record frames autonomously to its own micro SD card upon power on, following a script loaded on the card.However, it was not possible to read the data collected by the camera on its SD card via the NCM board, as the SD card was inserted in the camera card reader which had no connection to the NCM card reader. Therefore it was necessary to add a custom board that served as a switch for the micro SD card. The experiment card was physically inserted into the UCF board but connected to both the camera and the NCM board via the appropriate data lines and read alternately by each of them. Before the start of an experiment run, the microSD card was connected to the camera for data recording. Once the experiment run was completed, the UCF board switched the card back to the NCM board, which was then able to transfer the recorded data to the ISS laptop. The experiment sequence (see Section <ref>) was run by the NCM board, reading the experiment parameters from a command text file. Every power cycle of the payload reinitiated the NCM, thus restarting an experiment sequence. It was possible to control the experiment parameters from one run to the next by uploading an updated parameter file to the NCM before a payload power cycle. The parameters included in the command file were the total number of shaking events and the time lapse between two shaking events, thus determining the length of the experiment run.The recording parameters such as frame rate, resolution, and recording mode were set by a text file on the micro SD card. As it was not possible to update that file autonomously through the experiment electronics, it was prepared before payload delivery and thus determined the recording parameters for the entire flight. The resolution was set in order to distinguish particles and features down to 75 μm, which might result from the breakup of the smallest JSC-1 aggregates during collisions. With an experiment tray side of 8 cm, this required a pixel array of at least 1,067 px. Therefore, a resolution of 1080P, i.e. 1,920×1,080 px, met our experiment requirements. In the same way, the frame rate was set to allow for particle tracking from one frame to another. We limited this requirement to the accurate tracking of the mm-sized beads during the shaking events, as tracking individual JSC-1 grains in these phases has proven to be difficult<cit.> and was considered to be outside of the scope of this experiment. In order to track particles reliably throughout the experiment run, each particle should be seen moving less than its diameter from one frame to the other. To track the motion of a bead of 2 mm in diameter from one frame to another, while it is moving at a speed of 5 cm/s (expected maximum particle speed during shaking events), the time between two frames has to be 40 ms or less. Therefore, a frame rate of 30 fps was acceptable. The camera was set to record continuously.The shortest possible time between shaking events was set by the corresponding time unit (seconds, minutes, hours) that we hardcoded on the NCM. We chose the time unit to be minutes as we expected particle settling times of >200 s and thus did not deem it necessary for shaking events to be more frequent than a few minutes. This assumption was calculated from the evolution profile of the average collision particle velocity, v=v_0ϵ^N, v_0 being the average initial velocity of the particles after each shaking event, ϵ the average coefficient of restitution, and N the average number of collisions that happened to each particle. An estimation of the coefficient of restitution was obtained from a simple laboratory experiment, in which a bead was dropped 50 times onto a surface covered with the same type of beads. High-speed cameras recorded the rebound of the dropped bead and the coefficient of restitution of the collision was determined for each test. For the acrylic beads and at the lowest drop heights (lowest collision velocities of <10 cm/s), the average measured coefficient of restitution was 0.9. This was the highest compared to the glass and copper beads. Therefore, the calculated settling times calculated for the acrylic beads represented an upper limit to and we initially set the spacing between shaking events to 3 minutes. For an expected initial velocity of v_0∼5 cm/s (induced by a shaking event) and an estimated coefficient of restitution of ϵ=0.9, we calculated that average particle velocities would decrease to under 1 mm/s after 38 collisions and under 0.1 mm/s for 59 collisions. The time T_N required for N collisions to happen to a particle depends on the average collision frequency f_n after the n^th collision, where f_n is given by f_n=v_n/λ, with v_n=v_0ϵ^n the average particle velocity after the n^th collision and λ, the average particle mean free path. Therefore, T_N=∑_n=0^N1/f_n=∑_n=0^Nλ/v_0ϵ^n=λ/v_0×∑_n=0^N(1/ϵ)^n=λ/v_0×1-(1/ϵ)^N+1/1-1/ϵFor the 15 2 mm in diameter acrylic beads in their 15×15×3 mm^3 experiment cell, we have λ = (nσ)^-1 = 1.5 cm, σ = π r^2 being the cross section of each particle, with r = 1 mm the radius of the acrylic beads. With our estimate of v_0 and ϵ, we calculated the time needed for 38 and 59 collisions to happen to be T_38 = 162 s and T_59 = 1,500 s, respectively.In the continuous recording mode, the camera started recording upon power-on for a chosen overall duration, dividing the recorded data into a chosen number of video data files. Both the overall duration of the recording and the length of the data chunks were indicated in the NCM parameter file. We could change these parameters during the experiment's time on-board ISS by uploading a text file to the NCM. As described in the previous paragraph, during the development phase of the payload we calculated the expected particle settling time after each shaking event to range from 200 to 2,000 s. Therefore, we chose an overall recording duration of 60 minutes. The limitations imposed by ISS power and communication resources did not allow for a continuous (24/7) operation of the NanoRocks payload during its time on-board the space station. Instead, discrete periods of operations were planned every few months. As an unexpected power-off of the camera during a recording would produce a corrupted video file and result in the loss of that entire data chunk, we set the individual video file length to 60 seconds. An unexpected loss of power to the payload would therefore only erase the last minute of recording. Data transfer from the NanoRocks experiment to UCF was by way of planned routine data downlink from ISS; retrieving the NanoRocks payload including its micro SD card was not required for the success of the experiment. § NANOROCKS EXPERIMENT RUNS ONBOARD THE INTERNATIONAL SPACE STATION §.§ Observed Particles We chose particle samples to maximize the scientific return of the experiment. They are listed in Figure <ref>: red spherical acrylic beads, blue aspherical glass beads, aspherical copper beads, and angular JSC-1 lunar simulant grains. The acrylic beads were 2 mm in average diameter, the glass and copper beads were 1 mm in average diameter, and the JSC-1 particles had a size distribution around 0.75 mm in diameter. Additionally, some particles were coated with a layer of fine chalk powder. The size and composition of the particles were chosen to have particle masses around a few milligrams. Theacrylic, glass, and copper beads, and the JSC-1 grains at mm sizes have masses ranging from 1 to 35 mg. The density of the beads were 1.18, 2.6, 8.96, and 2.9 g/cm^3 for acrylic, glass, copper, and JSC-1, respectively. At the relative velocities induced by the shaking mechanism of the experiment (1 cm/s and below), observing this particle size allowed us to monitor the transition between bouncing and sticking collisions (see Figure <ref>). In addition, we varied the shape and surface of the particles. The acrylic beads were spherical, the glass and copper beads were aspherical (ellipsoidal), the JSC-1 grains were angular, and we coated the particles in trays 2 and 4 with a chalk layer.These different particle properties allowed us to determine the influence of (1) particle size (by comparing the behavior between beads and JSC-1 grains in trays 1 and 8, for example), (2) particle shape (by comparing the spherical beads of tray 1 with the non-spherical beads of trays 3 and 5, as well as the JSC-1 grains of tray 8), (3) coefficient of restitution (by comparing between trays 1, 3, and 5), and (4) surface texture (by comparing between coated and non-coated beads in trays 2 and 3, and 4 and 5) on the collision behavior. Due to the limited space available on this platform, the influence of some factors had to be combined. In particular, it will be difficult to distinguish the influence of size and shape between trays 1 and 8 on the particle behavior and additional experiments will be required in the future.The observation of spherical and ellipsoidal particles is of particular interest to be able to compare our experimental results to numerical simulations reproducing the same initial conditions. Being able to calibrate such simulations with experimental data will allow for more accurate simulations of dust grain behavior at PPD size scales. §.§ Experiment SequenceEach experiment sequence consisted of a series of experiment runs during which the samples were observed while they were shaken. At the start of an experiment run, the micro SD card was switched to the camera and the camera powered on. The recording then started automatically and lasted 60 minutes, dividing the data into 60-second video files. During an experiment run, the particles were agitated regularly.After the first data returned from ISS, we recognized that the damping time for the particle velocities after each shaking event was ∼20 s. We then set the shaking interval to its minimum, 60 s, and all further experiment runs were performed with this shaking interval. Once each 60-minute recording was completed, the NCM board cut the power to the camera and LED array until the start of the next experiment run. The microSD card was then switched back to the NCM to grant access from the ISS laptop. We first chose to run the experiment 3 times in a row every other day to complete one experiment run. This would allow us to check on the scientific data after a week and, if necessary, update the NCM parameter file for future experiment runs. A new experiment run was triggered by a power cycle of the payload, so that every power cycle reinitiated the NCM and restarted an experiment sequence. §.§ Experiment Performance The video data we received from the payload demonstrated the nominal operation of all the experiment components, the LED arrays, camera, and shaking magnet. The particle velocities induced by the shaking events were measured by tracking a subset of particles using the manual tracking tool Spotlight <cit.>. Figure <ref> shows an example of the tracking results for 10 particles in tray 1 (acrylic beads, see Figure <ref>). The measured maximum particles velocities peaked at 2 cm/s with an average of ∼1 cm/s, which was somewhat lower than our goal of ∼5 cm/s. We were able to observe particle collisions at relative velocities decreasing from 1 cm/s to <1 mm/s (our motion detection threshold with the camera was around 1 mm/s). In addition, we observed the formation of particle clusters at very low collision velocities (under 1 mm/s). Figure <ref> shows the particle clustering in tray 3 as an example. The only unexpected factor was the time the particle systems required to damp the velocities after each shaking event. We had predicted damping times of at least 200 s based on expected collision parameters, but the NanoRocks data showed that the particle velocities were significantly damped after only 20 s. This was due to a lower initial average particle velocity after each shaking event and average coefficients of restitution different than assumed. As described above, the data collected by NanoRocks was in the form of video files. Figure <ref>a shows an example of a frame captured during an experiment run. In this image we see the distortion introduced by the fisheye lens of the HackHD camera. Figure <ref>b shows another frame after distortion correction using the GIMP software package. Three LEDs can also be seen reflected by the tray cover. The specular light reflection creates areas of the tray where particles cannot be seen thus reducing the area usable for data analysis. However, the surface affected is limited to three distinct and immobile spots, which allows for easy elimination during image processing. A light diffusing screen was considered but not implemented due to concerns that launch vibrations would dislodge it and obstruct the camera. Data analysis from this experiment will be published separately.During its stay on-board the ISS we received a total of 158 video files from the NanoRocks payload through direct download from the ISS laptop. The experiment recorded more data, but not all the files could be downlinked to ground due to limitations in the available downlink time (the ISS downlink is a resource shared with other facilities). Out of the 158 video files received, the two first videos were from the first experiment sequence and each had a duration of 3 minutes. The other 156 videos were each 60 s long due to adjustment of the shaking interval and video length after the first sequence. The NanoRocks payload was retrieved from ISS and returned to the CMR, where we had the opportunity to retrieve the flight SD card from the UCF board to collect the remaining video files. Unfortunately, the SD card was damaged upon payload disassembly, therefore no further data could be retrieved from the flight hardware.The amount of data retrieved from NanoRocks was considerably higher than the amount of data collected from previous microgravity particle collision experiments <cit.>. These former experiments were flown on platforms with limited available microgravity time (a few seconds in the Bremen drop tower, a few minutes on suborbital rockets). In fact, the amount of data produced by NanoRocks obliged our team at UCF to develop new and automated data analysis methods for the video analysis. Previous data analysis could be performed manually (particle tracking in particular) due to the limited amount of data collected, but manual analysis was not a viable option for the amount of data produced by NanoRocks. Automated tracking has enabled us to validate statistical analysis of the evolution of particle velocities <cit.>. § SUMMARY AND CONCLUSION The NanoRocks payload was a 1.5U experiment designed to study low-energy collisions in multi-particle environments that flew on the International Space Station from September 2014 to March 2016. In this paper, we described the science objectives, experiment setup, and performance of the payload in order to illustrate the possibilities that orbital miniaturized payloads offer for planetary science.During its time on-board the ISS, NanoRocks functioned nominally and collected over 158 video files of scientific data. This very successful experiment also allowed us to benefit from several lessons learned. In addition to small hardware caveats like the lack of a light diffuser, we gained insight into the multi-particle collisional environment and thus ideas on how to optimize follow-up experiments. In particular, the upcoming CubeSat Particle Aggregation and Collision Experiment (Q-PACE) developed at the CMR will study multi-particle collisional systems on an orbital platform and will benefit from the hardware experience and data analysis performed with NanoRocks. Finally, it can be noted that the microgravity platform of the ISS was perfectly suited for the NanoRocks experiment and might be used in the future for further multi-particle collision experiments.This work is based in part upon research supported by NASA through the Origins of Solar Systems Program grant NNX09AB85G and by NSF through grant 1413332. The NanoRocks experiment was supported by Space Florida, the ISSRDC, NanoRacks LLC, and the University of Central Florida. | http://arxiv.org/abs/1706.08625v1 | {
"authors": [
"J. Brisset",
"J. Colwell",
"A. Dove",
"D. Maukonen"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170626233633",
"title": "NanoRocks: Design and Performance of an Experiment Studying Planet Formation on the International Space Station"
} |
Cache-enabled Wireless Networks with Opportunistic Interference Alignment Y. He and S. Hu Department of Systems and Computer Eng., Carleton University, Ottawa, ON, CanadaDecember 30, 2023 ===================================================================================================== Both caching and interference alignment (IA) are promising techniques for future wireless networks. Nevertheless, most of existing works on cache-enabled IA wireless networks assume that the channel is invariant, which is unrealistic considering the time-varying nature of practical wireless environments. In this paper, we consider realistic time-varying channels. Specifically, the channel is formulated as a finite-state Markov channel (FSMC). The complexity of the system is very high when we consider realistic FSMC models. Therefore, we propose a novel big data reinforcement learning approach in this paper. Deep reinforcement learning is an advanced reinforcement learning algorithm that usesdeep Q networkto approximate the Q value-action function.Deep reinforcement learning is used in this paper to obtain the optimal IA user selection policy in cache-enabled opportunistic IA wireless networks. Simulation results are presented to show the effectiveness of the proposed scheme.Caching, interference alignment, deep reinforcement learning § INTRODUCTION Recently, information-centric networking (ICN) has attracted great attentions from both academia and industry <cit.>. In ICN, in-network caching can efficiently reduce the duplicate content transmissions in networks. Caching has been recognized as one of the promising techniques for future wireless networks to improve spectral efficiency, shorten latency, and reduce energy consumption <cit.>. Another new technology called interference alignment (IA) has been studied extensively as a revolutionary technique to tackle the interference issue in wireless networks <cit.>. IA exploits the cooperation of transmitters to design the precoding matrices, and thus eliminating the interferences. IA can benefit mobile cellular networks <cit.>. Due to the large number of users in cellular networks, multiuser diversity has been studied in conjunction with IA, called opportunistic IA, which further improves the network performance <cit.>. Jointly considering these two important technologies, caching and IA, can be beneficial in IA-based wireless networks <cit.>. The implementation of IA requires the channel state information (CSI) exchange among transmitters, which usually relies on the backhaul link. The limited capacity of backhaul link has significant impacts on the performance of IA <cit.>. Caching can relieve the traffic loads of backhaul links, thus the saved capacity can be used for CSI exchange in IA.In <cit.>, the authorsinvestigate the benefits of caching and IA in the context of mutiple-input and multiple-output (MIMO) interference channels, and maximize the average transmission rate by optimizing the number of the active transceiver pairs. In <cit.>, it is shown that by properly placing the content in the transmitters' caches, the IA gain can be increased.Although some excellent works have been done on caching and IA, most of these previous works assume that the channel is block-fading channel or invariant channel, where the estimated CSI of the current time instant is simply taken as the predicted CSI for the next time instant. Considering the time-varying nature of wireless environments, this kind of memoryless channel assumption is not realistic<cit.>. In addition, it is difficult to obtain the perfect CSI due to channel estimation errors, communication latency, handover and backhaul link constraints <cit.>.In this paper, we consider realistic time-varying channels, and propose a novel big data deep reinforcement learning approach in cache-enabled opportunistic IA wireless networks. Cache-enabled opportunistic IA is studied under the condition of time-varying channel coefficients. The channel is formulated as a finite-state Markov channel (FSMC) <cit.>. The complexity of the system is very high when we consider realistic FSMC models. Therefore, we propose a novel big data reinforcement learning approach in this paper. Deep reinforcement learning is an advanced reinforcement learning algorithm that usesdeep Q networkto approximate the Q value-action function <cit.>. Google Deepmind adopts this method on some games <cit.>, and gets quite good results. Deep reinforcement learning is used in this paper to obtain the optimal IA user selection policy in cache-enabled opportunistic IA wireless networks. Simulation results are presented to illustrate that the performance of cache-enabled opportunistic IA networks can be significantly improved by using the proposed big data reinforcement learning approach.The rest of this paper is organized as follows. Section II presents the system model. The deep reinforcement learning algorithm is presented in Section III. In Section IV, the system is formulated. Simulation results are discussed in Section V. Finally, Section VI gives the conclusions.§ SYSTEM MODELIn this section, we describe the model of IA, followed by the time-varying channel. Then, cache-equipped transmitters are described. §.§ Interference Alignment We consider a L-user MIMO interference network with limited backhaul capacity and caches equipped at the transmitter side, as illustrated in Fig. <ref>.There is a central scheduler who is responsible for collecting the channel state and cache status from each user, scheduling the users and allocating the limited resources. All theusers are connected to the central scheduler via a backhaul link for CSI share and Internet connection, and the total capacity is limited.IA is a revolutionary interference management technique, which theoretically enables the network's sum rate grow linearly with the cooperative transmitter and receiver pairs. That is to say, each user can obtain the capacity 1/2()+o(()), which has nothing to do with the interferences.Consider a K-user MIMO interference channel. N_t^[k] and N_r^[k] antennas are equipped at the kth transmitter and receiver, respectively. The number of data streams of the kth user is denoted asd^[k]. The received signal at the kth receiver can be written asy^[k](t) =U^[k]†(t)H^[kk](t)V^[k](t)x^[k](t)+∑_j = 1,j≠ k^KU^[k]†(t)H^[kj](t)V^[j](t)x^[j](t)+U^[k]†(t)z^[k](t),where the first term at the right side represents the expected signal, and the other two terms mean the inter-user interference and noise, respectively. H^[kj](t) is the N_r^[k]×N_t^[j] matrix of channel coefficients from the jth transmitter to the kth receiver over the time slot t. Each element of H^[kj](t) is independent and identically distributed (i.i.d) complex Gaussian random variable, with zero mean and unit variance. V^[k](t) and U^[k](t) are the unitary N_t^[k]×d^[k] precoding matrix and N_r^[k]×d^[k] interference suppression matrix of the kth user, respectively. x^[k](t) and z^[k](t) are the transmitted signal vector of d^[k] DoFs and the N^[k]×1 additive white Gaussian noise (AWGN) vector whose elements have zero mean and σ^2 variance at the kth receiver, respectively. The interference can be perfectly eliminatedonly when the following conditions can be satisfiedU^[k]†(t)H^[kj](t)V^[j](t)=0,∀ j≠ k, rank(U^[k]†(t)H^[kk](t)V^[k](t))=d^[k].Under this assumption, the received signal at the kth receiver can be rewritten asy^[k](t)=U^[k]†(t)H^[kk](t)V^[k](t)x^[k](t)+U^[k]†(t)z^[k](t). To meet Condition (3), the global CSI is required at each transmitter. Each transmitter can estimate its local CSI (i.e., the direct link), but theCSI of other links can only be obtained by CSI share with other transmitters via the backhaul link<cit.>. Thus, in IA network, the backhaul link is more than a pipeline for connecting with Internet. The limited capacity should be made optimum use of.The recent advances focus on the benefits of edge caching, which is capable to decrease the data tranfer and leave more capacity for CSI share. The detail is described in the following subsection. In this paper, we assume the total backhaul link capacity of all the users is C_total, and the CSI estimation is perfect with no errors and no time delay.§.§ Time-varying Channel We consider realistic time-varying channels in this paper. Since finite-state Markov channel (FSMC) is an effective model to characterize the fading nature of wireless channels<cit.>, we choose FSMC model in this paper. Specifically, the first-order FSMC is used in this paper. The received SNR is a proper parameter that can be used to reflect the quality of a channel. We model SNRas a random variable, partition and quantize the range of the SNR into H Levels, which is characterized by a set of states Υ={Υ_0,Υ_1, Υ_2,…, Υ_H-1}. We consider T time slots over a period of wireless communication.Let's denote t∈{0,1,2,…,T-1} as the time instant, and the SNR varies from one state to another state when one time slot elapses.Actually, SNR plays a crucial role in determining the IA results.Cadambe and Jafar pointed out that IA performs better at very high SNR, and suffers from low quality at moderate SNR levels.Meanwhile higher and higher SNR is required to approach IA network's theoretical maximum sumrate as the number of IA users increases<cit.>.Thus, there exist competitions among users for accessing to IA network.§.§ Cache-equipped Transmitters In the era of explosive information, the vast amount of content makes it impossible for all of them gain popularity. As a matter of fact, only a small fraction becomes extensively popular. That means certain content may be requested over and over during a short time span, which gives rise to the network congestion and transmission delay. We assume that each transmitter is equipped with a cache unit that has certain amount of storage space. The stored content may follow a certain popularity distribution. For consistency, the cache of each transmitter stores the same content,usually the web content, and thus alleviating the backhaul burden and shorten delay time.In <cit.>, the authors survey on the existing methods for predicting the popularity of different types of web content. Specifically, they show that different types of content follow different popularity distributions. For example, the popularity growth of online videos complies with power-law or exponential distributions, that of the online news can be represented by power-law or log-normal distributions, etc. Based on the content popularity distribution and cache size, cache hit probability P_hit and cache miss probability P_miss can be derived<cit.>. In this paper, the specific popularity distribution is not the focus, and we just concentrate on two states, whether the requested content is within the cache or not. We describe the two states as Λ={0,1}, where 0 means the requested content is not within the cache, and 1 indicates it is within the cache. § DEEP REINFORCEMENT LEARNINGIn this section, we first present reinforcement learning. Then, deep Q-learning is described. §.§ Reinforcement Learning Reinforcement learning is an important branch of machine learning, where an agent makes interactions with an environment trying to control the environment to its optimal states that receive the maximal rewards.The task of reinforcement learning can usually be described as a Markov Decision Process (MDP), however, state space, explicit transition probability and reward function are not necessarily required<cit.>. Therefore, reinforcement learning is promising in handling tough situations that approach real-world complexity<cit.>.Let X = {x_1, x_2, ...,x_n} bethe state space, and A = {a _1, a _2, ...a_m} be the action set. Based on the current state x(t) ∈ X, theagent takes an action a(t)∈ A on the environment and then the system transfers to a new state x(t+1)∈ X according to the transition probability P_x(t) x(t+1)(a). The immediate reward is denoted as r(x(t),a(t)).Taking into the long-term returns, the agent should not only consider the immediate rewards, but also the future rewards. The more into the future, the more discounts the reward may get. Thus, the future rewards are discounted with a discount factor 0<ϵ<1.The aim of the reinforcement learning agent is to find an optimal policy a^*=π^*(x)∈ A for each state x, which maximizes the cumulative reward over a long time. The cumulative discounted reward at state x can be expressed by the state value function:V^π(x) = E[ ∑_t=0^∞ϵ^tr(x(t),a(t))|x(0)=x],whereE denotes the expectation, and it is considered over an infinite time horizon.Due to the Markov property, i.e., the state at the subsequent time instant is only determined by the current state, irrelevant to the former states,the value function can be rewritten asV^π(x) = R(x,π(x))+ϵ∑_x'∈ XP_x x^'(π(x))V^π(x^'),whereR(x,π(x)) is the mean value of the immediate reward r(x,π(x)), and P_x x^'(π(x)) is the transition probability from x to x^', when action π(x) is executed. The optimal policy π^* follows Bellman's criterionV^π^*(x) = max_a^'∈ A[R(x,a)+ϵ∑_x'∈ XP_x x^'(a)V^π^*(x^')]. Given the reward R and transition probability P, the optimal policy can be obtained. §.§ Deep Q-learning When R and P are unknown, Q-learning is one of the most widely-used strategies to determine the best policy π^*. A state-action function, i.e., Q-function is defined asQ^π(x,a) = R(x,a)+ϵ∑_x'∈ XP_x x^'(a)V^π(x^'),which represents the discounted cumulative reward when action a is performed at state x and continues optimal policy from that point on.The maximum Q-function will beQ^π^*(x,a) = R(x,a)+ϵ∑_x'∈ XP_x x^'(a)V^π^*(x^'),then the discounted cumulative state function can be written asV^π^*(x) =max_a∈ A[Q^π^*(x,a)]. Up to now, the objective can change from finding the best policy to finding the proper Q-function. Usually, Q-function is obtained in a recursive manner using the available information (x, a, r, x^', a^'), i.e., the state x, the immediate reward r, the action a at the current time instant t, and the state x^' and action a^' at the next time instant t+1. The Q-function isupdated asQ_t+1(x,a)=Q_t(x,a) +α(r+ϵ [max_a^' Q_t(x^',a^')]-Q_t(x,a)),where α is the learning rate. Utilizing proper learning rate, Q_t(x,a) will definitely converges to Q^*(x,a)<cit.>. As a matter of fact, the Q-function is commonly estimated by a function approximator, sometimes a nonlinear approximator, such as a neural network Q(x,a;θ) ≈Q^*(x,a). This neural network is named Q network. The parameter θ are the weights of the neural network, and the network is trained by adjusting θ at each iteration to reduce the mean-squared error. However,Q-network exhibits some instabilities, and the causes are provided in <cit.>. Deep Q learning, in which deep neural network is used to approximate the Q-function,is proposed recently, and it is proven to be more advantageous<cit.>.Two techniques were used by deep Q-learning to modify the regular Q-learning. The first one is experience replay. At each time instant t, an agent stores its interaction experience tuple e(t) =(x(t) ,a(t) ,r(t) ,x(t+1))into a replay memory D(t)= {e(1) ,...,e(t)}. Then it randomly samples from the experience pool to train the deep neural network's parameters rather than directly using the consecutive samples as in Q-learning.The other modification is that deep Q-learning adjusts the target value to update several time steps, instead of updating every time step. The target value is expressed as y=r+ϵmax_a^'Q(x^',a^',θ_i^-).In the Q-learning, the weights θ_i^- are updated as θ_i^-=θ_i-1, whereas in the deep Q-learningθ_i^-=θ_i-N, i.e.,the weights update every N time steps. Such modification can make the learning process more stable. The deep Q function is trained towards the target value by minimizing the loss function L(θ) at each iteration, the loss function can be written asL(θ)=E[(y-Q(x,a,θ)^2)].We use deep reinforcement learning in optimizing the performance of the cache-enabled IA network, and the formulation process is described in the following section.§ PROBLEM FORMULATIONIn this section, we formulate the cache-enabled IA network optimization problem as a deep Q-learning process, which can determine the optimal policy for IA user grouping.In our system, there are L candidates that want to join in the IA network to communicate wirelessly. We assume that the IA network size is always smaller than the number of candidates, which is in accordance with the fact that a large number of users expect wireless communications anytime and anywhere.As aforementioned, the value of SNR affects theperformance of interference alignment, and the candidates who occupy the better channels are more advantageous for accessing to the IA network.Therefore, we make an action at each time slot to decide which candidates are the optimal users for constructing an IA network based on their current states.Here, a central scheduler is responsible for acquiring each candidate's CSI and cache status, then it assembles the collected information into a system state. Next, the controller sends the system state to the agent, i.e., the deep Q network, and then the deep Q network feeds back the optimal action max_πQ^*(x,a) for the current time instant. After obtaining the action, the central scheduler will send a bit to inform the users to be active or not, and the corresponding precoding vector will be sent to each active transmitter.The system will transfer to a new state after an action is performed, and the rewards can be obtained according to the reward function. Inside the deep Q network, the replay memory stores the agent's experience of each time slot. The Q network parameter θis updated at every time instant with samples from the replay memory. The target Q network parameter θ^- is copied from the Q network every N time instants. The ε-greedy policy is utilized to balance the exploration and exploitation, i.e., to balance the reward maximization based on the knowledge already known with trying new actions to obtain knowledge unknown. In order to obtain the optimal policy, it is necessary to identifythe actions, states and reward functionsin our deep Q learning model, which will be described in the next following subsections.§.§.§ System StateThe current system state x(t) is jointly determined by the states of L candidates. The system state at time slot t is defined as,x(t) = {γ_1(t),c_1(t),γ_2(t),c_2(t),…, γ_L(t),c_L(t)},where each candidate contains two states: the channel state γ_i(t)∈Υ={Υ_0,Υ_1,…,Υ_H-1}, andthe cache state c_i(t)∈Λ={0,1 }, the index i means the ith candidate, and i=1,2,…, L.The number of possible system states is (2× H)^L, and this number can be very large asL increases. Due to the curse of dimensionality, it is difficult for traditional approaches handle our problem. Fortunately, deep Q network is capable of successfully learning directly from high-dimensional inputs<cit.>, thus it is proper to be used in our system.§.§.§ System ActionIn the system, the central scheduler has to decide which candidates to be set active, and the corresponding resources will be allocated to the active users. The current composite action a(t) is denoted bya(t) = {a_1(t),a_2(t),… , a_L(t) },where a_i(t) represents the control of the ith candidate, andeach element a_i(t)∈{0,1}, and a_i(t)=0 means the candidate i is passive (not selected) at time slot t, and a_i(t)=1 means it is active (selected). §.§.§ Reward FunctionReward function indicates the received reward when a certain action is performed under a certain state. The system reward represents the optimization objective, and we take the objective to maximize the IA network's throughput,and the reward function of the nth candidate is defined as Eq. (<ref>) on the top of the next page. Here, C_total is the total capacity of the backhaul link, and C_c is the fixed capacity allocated to each active user to exchange CSI with other active users. For the nth candidate, if the requested content is not in the local cache, it can only acquire the content from the backhaul link, and equal capacity (the total capacity minus the total capacity for CSI exchange) is allocated among the active users. If the requested content is within the cache, the nth candidate can get the maximum rate that an IA user can achieve. Note that, for simplicity we assume the interference can be perfectly eliminated, and each active user's sum rate is approaching half the capacity that the user could achieve without interferers.The immediate system reward is the sum of all the candidates' immediate rewards, i.e., r(t)=∑_l=1^l=Lr_l(t). The central scheduler gets r(t) in state x(t) when action a(t) is performed in time slot t. However, a maximum immediate value does not mean the maximum long-term cumulative rewards. Therefore, we should also think about the future rewards. The more into the future, the more uncertainty there exists. A discounted future reward with a discount factor ϵ is much more reasonable. The goal of using deep Q network into our system model is to find a selection policy that maximizes the discounted cumulative rewards during the communication period T, and the cumulative reward can be expressed as R = max_π E[∑_t=0^t=T-1ϵ^t r(t)] ,where ϵ^t approaches to zero when t is large enough.In practice, a threshold for terminating the process can be set. § SIMULATION RESULTS AND DISCUSSIONSIn this section, computer simulations are carried out to demonstrate the performance of the proposed big data deep reinforcement learning approach to the optimization of cache-enabled opportunistic IA wireless networks. We compare the proposed scheme with two other schemes: 1) The same proposed approach without caching and 2) An existing user selection approach without cache <cit.>, in which invariant channels are assumed. The performance improvements of the proposed scheme are present. In the simulations, we consider a cache-enabled opportunistic IA network, in which L=5 candidates want to access to. Due to the feasibility of IA<cit.>, i.e., N_t+N_r≥d(L+1), we assume that each candidate is equipped with three antennas at both the transmitter node and the receiver node, and DoF is set to be 1. We quantize and partition the received SNR into 10 levels, i.e., [-∞, 5], [5, 10], [10, 15], [15, 20], [20, 25], [25, 30], [30, 35], [35, 40], [40, 45] and [45, +∞].We assume that the channel state transition probability is identical for all the candidates. In one simulation scenario, the transition probability of remaining in the same state is set to be 0.489, and the probability of transition to the adjacent state to be twice that of transition to a nonadjacent state.The cache at each transmitter includes two states: existence and nonexistence of the requested content. The implementation of the big data deep reinforcement learning algorithm is based on the TensorFlow to derive the optimal policy for IA user selection. The discount factor ϵ is set to be 0.5, and the learning rate α is designed to be state-action dependent varying with time. In the ε-greedy exploration, ε is initially set to be 0.1, and finally to be 1. The Q value update frequency N is set to be 4, and the relay memory size is 100K.Fig. <ref> shows the convergence performance of the proposed scheme. From this figure, we canobserve that the sum rate of the proposed scheme is low at the beginning of the learning process. During the learning process, the sum rate increases, and converges after about 3500 episodes. Please note that the learning is done off-line to train the deep neural network parameters. Fig. <ref> shows the network's average sum rate with different state-transition probabilities of staying in the same state.It can be seen that the proposed OIA with cache scheme can achieve the highest sum rate compared to the other two schemes. This is because the channel is time-varying, and the proposed scheme can obtain the optimal IA user selection policy in the realistic time-varying channel environment using the big data deep reinforcement learning algorithm. We can also observe that the performance of the existing selection method is getting closer to the proposed OIA without cache scheme as the transition probability increases, and this method performs the same when the channel remains absolutely static, i.e., the transition probability that the channel will be in the same state is 1. § CONCLUSIONS AND FUTURE WORKIn this paper, we studied cache-enabled opportunistic IA under the condition of time-varying channel coefficients. The system complexity is very high when we model the time-varying channel as a finite-state Markov channel. Thus, we exploited the recent advances, and formulated the system as a big data deep reinforcement learning problem. A central scheduler is responsible for collecting the CSI from each candidate, and then sends the integral system state to the deep Q network to derive the optimal policy for user selection. Simulation results were presented to show that the performance of cache-enabled opportunistic IA networks can be significantly improved by using the proposed big data reinforcement learning approach. Future work is in progressed to consider wireless virtualization in the proposed framework. | http://arxiv.org/abs/1706.09024v1 | {
"authors": [
"Y. He",
"S. Hu"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170627193702",
"title": "Cache-enabled Wireless Networks with Opportunistic Interference Alignment"
} |
[Electronic address: ][email protected] Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 4056, 2600 GA Delft, The Netherlands[Electronic address: ][email protected] Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 4056, 2600 GA Delft, The NetherlandsCenter for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, DenmarkResearch Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAKavli Institute of Nanoscience, Delft University of Technology, P.O. Box 4056, 2600 GA Delft, The NetherlandsThe proximity effect in hybrid superconductor-semiconductor structures, crucial for realizing Majorana edge modes, is complicated to control due to its dependence on many unknown microscopic parameters. In addition, defects can spoil the induced superconductivity locally in the proximitised system which complicates measuring global properties with a local probe. We show how to use the nonlocal conductance between two spatially separated leads to probe three global properties of a proximitised system: the bulk superconducting gap, the induced gap, and the induced coherence length. Unlike local conductance spectroscopy, nonlocal conductance measurements distinguish between non-topological zero-energy modes localized around potential inhomogeneities, and true Majorana edge modes that emerge in the topological phase. In addition, we find that the nonlocal conductance is an odd function of bias at the topological phase transition, acting as a current rectifier in the low-bias limit. More generally, we identify conditions for crossed Andreev reflection to dominate the nonlocal conductance and show how to design a Cooper pair splitter in the open regime.The Andreev rectifier: a nonlocal conductance signature of topological phase transitions A. R. Akhmerov October 24, 2017 ========================================================================================§ INTRODUCTION The proximity effect induces superconductivity in non-superconducting materials, which among other things gives Majoranas. The superconducting proximity effect occurs when a normal material (metal) is placed in contact with a superconductor. The resulting transfer of superconducting properties to the normal material <cit.> makes it possible to explore induced superconductivity in a range of materials that are not intrinsically superconducting, for example in ferromagnetic metals <cit.> and in graphene <cit.>. Another recent application of the proximity effect is the creation of the Majorana quasiparticle <cit.>, which is a candidate for the realization of topological quantum computation <cit.>, and a focus of research efforts in recent years <cit.>.The quality of induced superconductivity depends on uncontrollable and unknown factors, and at the same time it is hard to probe. The proximity effect is due to the Andreev reflection of quasiparticles at the interface with the superconductor <cit.>, which forms correlated electron-hole pairs that induce superconductivity in the normal material. This makes the proximity effect in real systems sensitive to microscopic interface properties, such as coupling strength, charge accumulation and lattice mismatch <cit.>. Spatial inhomogeneities in the proximitised system, such as charge defects, may furthermore spoil the induced correlations locally. In a typical proximity setup, the superconductor proximitises an extended region of a normal material, as shown in Fig. <ref>. A normal lead attachedto one of the ends of the proximitised region probes the response to an applied voltage. When the coupling between the lead and the proximitised region is weak, the lead functions as a tunnel probe of the density of states in the latter. Since induced superconductivity may be inhomogeneous, and Andreev reflection happens locally, such an experiment only probes the region in direct vicinity of the normal lead, and not the overall properties of the proximitised region. For example, if the electrostatic potential is inhomogeneous, it may create accidental low-energy modes that are nearly indistinguishable from Majoranas <cit.>.The nonlocal response reveals properties of induced superconductivity. We show how the nonlocal response between two spatially separated normal leads (see Fig. <ref>) may be used to probe both the bulk superconducting gap Δ and the induced gap Δ_ind, as well as the induced coherence length ξ. At subgap energies, quasiparticles propagate as evanescent waves with the decay length ξ in the proximitised system. This suppresses the nonlocal response with increasing separation L between the two normal leads <cit.>. Therefore, the length dependence of the nonlocal conductance measures when two ends of a proximitised system are effectively decoupled. When L/ξ≳ 1,the nonlocal conductance is only possible in the energy window between the bulk superconducting gap Δ and the induced gap Δ_ind. The sensitivity to an induced gap allows one to use nonlocal conductance to distinguish between a induced gap closing and an Andreev level crossing at zero energy. In contrast, a local measurement may produce a similar result in both cases.With Zeeman field, we create a Cooper pair splitter and an Andreev rectifier Two processes constitute the nonlocal response: direct electron transfer between the normal leads, and the crossed Andreev reflection (CAR) of an electron from one lead into a hole in the second lead <cit.>. Experimental <cit.> and theoretical <cit.> studies of CAR-dominated signals aim at producing a Cooper pair splitter <cit.>, which has potential applications in quantum information processing. We show that applying a Zeeman field in the proximitised system creates wide regions in parameter space where CAR dominates the nonlocal response. Furthermore, we demonstrate how to obtain a CAR dominated signal in the absence of a Zeeman field in the low-doping regime. Finally, we prove that at the topological phase transition and with L/ξ≳ 1, the nonlocal conductance is an approximately odd function of bias. This phenomenon only relies on particle-hole symmetry, and hence manifests both in clean and disordered junctions. Therefore, a proximitised system coupled to normal leads acts as a rectifier of the applied voltage bias universally at the topological phase transition.There are other proposals for probing the bulk topological phase transition instead of the edge states. Our method is new and all-electrical, allows to extract other parameters, and can be implemented in ongoing experiments. Our method is based on probing the bulk topological phase transition in Majorana devices, instead of the Majoranas themselves. Several other works propose different methods to probe the bulk instead of the edge states in one-dimensional topological superconductors. Quantized thermal conductance and electrical shot noise measurements are predicted signatures of a bulk topological phase transition <cit.>, and here we present a different route based on straightforward electrical conduction measurements in already available experimental systems. Further work predicts bulk signatures of a topological phase transition in the difference between the local Andreev conductances at each end of the proximitised region <cit.>, or in the spin projection of bulk bands along the magnetic field direction <cit.>. In addition to probing the bulk topological phase transition, our proposed method allows to probe a number of relevant physical parameters, and can be implemented in ongoing experiments, providing a novel technique to use in the hunt for Majoranas.Descriptions of sections. This paper is organized as follows. In Sec. <ref>, we give an overview of our model and discuss the relevant energy and length scales. In Sec. <ref>, we study how nonlocal conductance measures superconductor characteristics. We investigate the effects of a Zeeman field in homogeneous and inhomogeneous systems in Sec. <ref>. In Sec. <ref> we consider the possible application of the proximitised system as a Cooper pair splitter. We finish with a summary and discussion of our results in Sec. <ref>. § MODEL AND PHYSICAL PICTUREThe geometry consists of a proximitised normal material attached to two spatially separated normal leads. We consider a three terminal device sketched in Fig. <ref>, with a normal central region of lateral length L and width W separating two normal leads of width W_L. The device has a grounded superconducting lead of width L attached to the central region perpendicularly to the other two leads. This geometry models the proximity effect of a lateral superconductor on a slab of normal material, with normal leads probing the transport properties, and is therefore relevant both for heterostructures based on nanowires and quantum wells. The Hamiltonian consists of a kinetic term, chemical potential, spin-orbit interaction, superconducting pairing and Zeeman field. We model the hybrid system using the Bogoliubov-de Gennes Hamiltonian. For a semiconductor electron band with effective mass m^* and Rashba spin-orbit interaction (SOI) with strength α, it reads H =( p_x^2 + p_y^2/2m^* - μ)τ_z+ Δ (y) τ_x + α/ħ( p_yσ_x - p_x σ_y )τ_z + E_Z (y) σ_x, with p_x,y = -i ħ∂_x,y, μ the equilibrium chemical potential and E_Z the Zeeman energy due to an in-plane magnetic field parallel to the interface between the central region and the superconductor. We assume a constant s-wave pairing potential that is nonzero only in the superconductor, Δ (y) = Δθ (y-W) with θ(y) a step function, and choose Δ to be real since only one superconductor is present. We neglect the g-factor in the superconductor since it is much smaller than in the adjacent semiconductor, such that E_Z (y) = E_Zθ(W-y), and our conclusions are not affected by this choice. The Pauli matrices σ_i and τ_i act in spin and particle-hole space, respectively. The Hamiltonian acts on the spinor Ψ = ( ψ_e↑, ψ_e↓, ψ_h↓, -ψ_h↑), which represent the electron (e) or hole (h) components of spin up (↑) or down (↓).The superconductor induces a gap in the proximitised system, the size of which is determined by the bulk gap and Thouless energy. The superconductor induces an energy gap Δ_ind in the heterostructure. If L ≫ W, the larger of two energy scales, namely the bulk gap Δ and the Thouless energy E_Th, determines the magnitude of Δ_ind, with E_Th at low μ given by E_Th = γδ, δ = ħ^2 π^2/2m^* (2W)^2, where γ is the transparency of the interface with the superconductor and δ the level spacing. Our emphasis is on short and intermediate junctions, for which E_Th≫Δ and E_Th≲Δ, respectively, such that Δ_ind≲Δ. A brief review of normal-superconductor junctions in different limits and the relevant length and energy scales is given in Appendix <ref>. We keep μ constant in the entire system, but assume an anisotropic mass<cit.> in the superconductor with a component parallel to the interface m_∥→∞. This approximation results in a transparent interface γ=1 at normal incidence and at E_Z=0, and is motivated by recent advances in the fabrication of proximitised systems with a high-quality superconductor-semiconductor interface <cit.>.Conductance is calculated using a scattering matrix formalism. We compute differential conductance using the scattering formalism. The scattering matrix relating all incident and outgoing modes in the normal leads of Fig. <ref> is S = [ S_11 S_12; S_21 S_22 ],S_ij = [ S_ij^ee S_ij^eh; S_ij^he S_ij^hh ]. Here, the S_ij^αβ is the block of scattering amplitudes of incident particles of type β in lead j to particles of type α in lead i. Since quasiparticles may enter the superconducting lead for |E|> Δ, the scattering matrix (<ref>) is unitary only if |E|< Δ. The zero-temperature differential conductance matrix equals <cit.>G_ij (E)≡∂ I_i/∂ V_j = e^2/h( T_ij^ee- T_ij^he - δ_ij N_i^e),with I_i the current entering terminal i from the scattering region and V_j the voltage applied to the terminal j, andN_j^e the number of electron modes at energy E in terminal j, and finally the energy-dependent transmissions are T_ij^αβ = Tr( [ S_ij^αβ(E)]^† S_ij^αβ). The blocks of the conductance matrix involving the superconducting terminal are fixed by the condition that the sum of each row and column of the conductance matrix has to vanish. The finite temperature conductance is a convolution of zero-temperature conductance with a derivative of the Fermi distribution function f(E) = (1+exp(E/k_BT))^-1: G_ij (eV_j, T) = -∫_-∞^∞dE d f(E-eV_j, T)/dE G_ij(E).We use Kwant to compute the scattering matrix and ξ, using material parameters for an InAs 2DEG with epixatial Al. We discretize the Hamiltonian (<ref>) on a square lattice, and use Kwant <cit.> to numerically obtain the scattering matrix of Eq. (<ref>), see the supplementary material for source code <cit.>. The resulting data is available in Ref. suppl_data. We obtain ξ numerically by performing an eigendecomposition of the translation operator in the x-direction for a translationally invariant system and computing the decay length of the slowest decaying mode at E=0 <cit.>. We use the material parameters[All parameters are provided per figure in a text file as supplementary material.] m^* = 0.023 m_e, α = 28 meVnm, and unless otherwise specified Δ = 0.2 meV, typical for an InAs two-dimensional electron gas with an epitaxial Al layer <cit.>. All transport calculations are done using T = 30 mK unless stated otherwise. § NONLOCAL CONDUCTANCE AS A MEASURE OF SUPERCONDUCTOR PROPERTIESLocal conductance tunnel spectroscopy is not a reliable probe of the bulk properties of induced superconductivity, since it only probes the density of states close to the probe. On the other hand, nonlocal conductance probes the bulk properties. In the tunnelling regime, the local conductance in a normal lead probes the density of states in the proximitised region, which is commonly used to measure the induced gap in experiment. However, such a measurement only probes the region near the tunnel probe, but fails to give information about the density of states in the bulk of the proximitised region. The tunnelling conductance is thus not a reliable probe of the entire proximitised region if the density of states varies spatially over the proximitised region, for example due to an inhomogeneous geometry. As an illustration, Fig. <ref> compares the local conductance G_11 in the tunnelling limit to the nonlocal conductance G_21 in the open regime for a proximitised system that is inhomogeneous and in a magnetic field.Inhomogeneous systems are further treated in Sec. <ref>. The combination of an inhomogeneous system and broken time-reversal symmetry creates low-energy states localized near the junctions with the normal leads, which appear as peaks in the tunnelling conductance. However, away from the junctions with the normal leads, the proximitised system remains close to fully gapped, the induced gap matching the energies at which the nonlocal conductance becomes finite in Fig. <ref>(b). Therefore, the nonlocal conductance is better than the local tunnelling conductance as a probe for the induced gap in the bulk of the proximitised region. In the following, we describe three ways in which the nonlocal conductance probes induced superconductivity. Nonlocal conductance is suppressed at E=0 for L > ξ, hence it measures the ratio between L and ξ. First of all, the nonlocal conductance measures the induced decay length ξ in the bulk of the proximitised region between the two normal leads. To understand this, consider a nonlocal process at a subgap energy | E | < Δ_ind. An electron injected from a normal lead must propagate as an evanescent wave ∝ e^-x/ξ + ikx through the gapped central region to the second normal lead, with ξ the decay length. Accordingly, as shown in Fig. <ref>, increasing L suppresses the nonlocal conductance at E = 0 exponentially <cit.>. Therefore, the suppression of the nonlocal conductance with increasing length L at E=0 is a measure of the induced decay length ξ.For L ≳ξ, nonlocal conductance is zero for E>Δ, hence it measures the bulk superconducting gap Furthermore, the nonlocal conductance measures the bulk gap Δ of the superconductor. Increasing L also suppresses the nonlocal conductance G_21 for |E| > Δ, as the right column of Fig. <ref> shows. For energies above the bulk superconducting gap Δ, the superconductor increasingly absorbs quasiparticles when the length is increased, and suppresses the nonlocal conductance to zero when L ≫ξ. Hence, the energy above which nonlocal conductance is suppressed at large lengths is a measure of Δ.Since, for L ≳ξ, nonlocal conductance is only nonzero for Δ_ind < E < Δ, it also measures the induced gap. In addition, the nonlocal conductance measures the induced superconducting gap Δ_ind. When L ≳ξ, the nonlocal conductance is suppressed at E = 0 but grows in a convex shape with E and peaks around | E | ≈Δ_ind, as shown in the right column of Fig. <ref>. This is due to a divergence in ξ, since the system is no longer gapped. To illustrate the correspondence between the nonlocal conductance and Δ_ind, the left column of Fig. <ref> shows the dispersions of the corresponding proximitised systems that have the normal leads removed and are translationally invariant along the x direction, such that k = p_x/ħ is conserved. Because the system is not gapped for |E | > Δ_ind, G_21 is generally nonzero at these energies. Note that aside from occasional dips to negative G_21, direct electron transfer dominates the nonlocal response (we investigate this in more detail in Sec. <ref>).All of the above still holds in the presence of disorder, such that the nonlocal conductance remains a reliable probe of induced superconductivity. The presence of finite nonlocal conductance in the energy rangeΔ_ind < |E | < Δ depends only on density of states of the proximitised system, and therefore still holds in the presence of disorder. In Fig. <ref>, we show the effects of disorder on the transport signatures of Δ and Δ_ind for short and intermediate junctions when L ≳ξ. We include onsite disorder in the central region, and vary the elastic mean free path l_e from l_e = L to l_e = 0.1L <cit.>. Even in the presence of disorder, all of the aforementioned qualities are still apparent in the nonlocal conductance (a) and (b), namely suppression for | E | < Δ_ind, a finite signal for Δ_ind < | E | < Δ and vanishing conductance for |E | > Δ. Therefore, the nonlocal conductance remains a reliable probe of induced superconductivity even in the presence of disorder.The open-regime local conductance is Andreev enchanced and smooth at subgap energies, and the bulk and induced gaps are visible in the local conductance. However, this is only because there are no extended potential inhomogeneities. Lastly, in the absence of extended potential inhomogeneities, Δ and Δ_ind may also be inferred from the local conductance G_11 in the open regime. As Figs. <ref>(c) and (d) show, G_11≲ 4e^2/h in the ballistic case l_e = L for | E | < Δ_ind, which indicates that Andreev reflection is the dominant local process. This is the expected behavior for a normal-superconductor junction with high interface transparency <cit.>, and is consistent with our results. Reducing the mean free path makes normal reflection more likely and hence lowers G_11, similar to an ideal normal-superconductor junction with a reduced interface transparency. Here, comparing G_11 and G_21 shows that Δ_ind and Δ may also be inferred from the local conductance, because it changes smoothly with bias only outside the interval Δ_ind< | E | < Δ. However, the signatures are clearer in G_21, where it is a transition between finite and vanishing conductance that indicates the gaps. Furthermore, the induced gap observed in the local and nonlocal conductances coincide here only due to the absence of extended potential inhomogeneities. For the case of an inhomogeneous geometry as in Fig. <ref>, only the nonlocal conductance correctly measures Δ_ind in the bulk of the proximitised system. § ANDREEV RECTIFIER AT THE TOPOLOGICAL PHASE TRANSITION §.§ Andreev rectification as a measure of the topological phaseThe proximitised system acts as a rectifier of the applied voltage at the topological phase transition.In order to study nonlocal conductance at the topological phase transition, we apply an in-plane Zeeman field along the x-direction of the proximitised system. Figure <ref> shows the nonlocal conductance G_21 as a function of bias E and Zeeman energy E_Z, for short and intermediate junctions in (a) and (b) with L = 10ξ and L = 3ξ, respectively, such that the two normal leads are well decoupled, and the nonlocal conductance is exponentially suppressed at subgap energies. Increasing the magnetic field closes the induced gap and the system is driven into a topological phase. The line cuts of Fig. <ref>(c), taken at the critical magnetic field E_Z =E_Z^c, show that at the topological phase transition the nonlocal conductance is a linear function of energy, G_21(E) ∝ E around E=0. At the topological phase transition, the current I ∝ V^2 with V the voltage bias, and the system functions as a current rectifier due to crossed Andreev reflection. The linear behaviour of the conductance is due to topology and symmetry, and makes the junction a rectifier at the topological phase transition. This Andreev rectifier manifests due to the topology and symmetry of the proximitised system. The system only has particle-hole symmetry and is therefore in class D <cit.>. Expanding G_21(E, E_Z) = c_0(E_Z) + c_1(E_Z) E + O(E^2) around E = 0, the exponential suppression of G_21 at subgap energies means that the coefficients c_0 and c_1 are exponentially suppressed at magnetic fields before the topological phase transition. In class D systems, if G_21 is exponentially suppressed at subgap energies, it is guaranteed to remain exponentially suppressed across the topological phase transition <cit.>. At the critical magnetic field E_Z = E_Z^c, G_21(E=0, E_Z^c) = c_0(E_Z^c) is therefore also exponentially suppressed. However, the system is gapless at the topological phase transition, such that G_21 is generally finite at any nonzero E, and c_1(E_Z^c) thus not exponentially suppressed. At the topological phase transition, we therefore have G_21∝ E in the limit E → 0, where higher order contributions are negligible. Consequently, rectifying behavior in the nonlocal conductance is an indication of a topological phase transition. This makes the nonlocal conductance not only a probe of the bulk properties of induced superconductivity as discussed in Sec. <ref>, but also makes it selectively sensitive to topological phase transitions. The Andreev rectifier is robust against disorder. The rectifying behavior G_21∝ E at the topological phase transition in Fig. <ref> is grounded in the symmetry classification of the channel. As a result, we expect it to be robust to the presence of onsite disorder, so long as it does not alter the symmetry class. Figure <ref> shows G_21 as a function of E and E_Z for systems with the same widths as in Fig. <ref>. In the left column, parameters are chosen identical to those in Fig. <ref>, with the addition of onsite disorder to give an elastic mean free path l_e = 0.2 L<cit.>, bringing the systems well into the quasiballistic regime. In the right column of Fig. <ref>, we investigate G_21 when the central region is in the diffusive limit, with l_e = 0.2 W. The widths are the same as in the quasiballistic (and clean) case, but μ is increased such that several modes are active. We gate the leads into the single mode regime using quantum point contacts at the junctions with the scattering region. In each case we pick L ≳ξ̃, since in the diffusive limit ξ̃ = √(ξ l_e) governs the range of the coupling between the two normal terminals at subgap energies <cit.>. In both quasiballistic and diffusive cases, G_21 remains an approximately odd function of E around the gap closing, and the proximitised system therefore acts as a rectifier even in the presence of disorder. §.§ Distinguishing the topological phase transition in spatially inhomogeneous devices Recent literature points out that local conductance measurements do not distinguish between trivial and topological zero-energy states in spatially inhomogeneous devices. Several works <cit.> discuss the emergence of zero-energy modes in the trivial phase of a hybrid semiconductor-superconductor device with an extended, spatially inhomogeneous potential. Local conductance measurements do not distinguish between these modes and well-separated Majorana modes at the endpoints of the proximitised region in the topological phase, since both give rise to zero-bias conductance features.To study this issue, we consider the nonlocal conductance in systems with a potential inhomogeneity. To study this problem, we include an extended inhomogeneous potential ϕ(x, y) = V_0 exp[-1/2(x - x_0/d_x)^2] exp[ -1/2(y - y_0/d_y)^2], in the setup shown in Fig. <ref>, with V_0 the potential amplitude, x_0 and y_0 the coordinates of the potential center, and d_x and d_y parameters to control the smoothness in x- and y-direction, respectively. We compare conductance for an amplitude V_0 = -4.5 mV to conductance in a homogeneous system V_0 = 0 V. We calculate the local conductance in the tunneling regime, with tunnel barriers at both wire ends x=0 and x=L, and the nonlocal conductance in the open regime, with the system length fixed to L = 8ξ and the width to W = 100 nm.The spectrum of such system confirms that such potentials can lead to gap closings and zero-energy modes in the trivial regime. To confirm that such a spatially inhomogeneous system can indeed exhibit trivial zero-energy modes, we calculate the low-energy spectrum of our system when decoupled from the leads, forming a closed superconductor-semiconductor system. The phase transition is computed from the absolute value of the determinant of the reflection matrix in the open system at E=0, with|det(r)| = 1 everywhere for L ≫ξ, except at the phase transition, where it drops to zero <cit.>. Fig. <ref>(a) shows the spectrum as a function of E_Z in the homogeneous case (V_0=0), Fig. <ref>(b) for the inhomogeneous case (V_0 = -4.5 mV). While in the first case the closing of the induced superconducting gap coincides with the topological phase transition, in the second case an extended topologically trivial region exists with states around zero energy (yellow region).Local conductance can not distinguish topological from non-topological gap closings. Comparing the local conductance with and without an inhomogeneous potential, we find that zero-energy modes appear regardless of whether they are topological or trivial.Panels (c) and (d) of Fig. <ref> show the local response as a function of bias and Zeeman energy when leads are connected to the central region via tunnel barriers. Since the system is ballistic and long (L ≫ξ), the local conductance agrees well with the spectra presented in panels (a) and (b). Accordingly, the local conductance in panel (d) for V_0 = -4.5 mV shows zero-energy modes in the topologically trivial regime. Therefore, a gap closing and the emergence of zero-energy modes in the local conductance is not a sufficient sign of a topological phase transition. However, due to its rectifying behavior, nonlocal conductance is a more reliable measure of a topological phase transition. On the other hand, nonlocal conductance has a much clearer signature of the topological transition than the local conductance. To demonstrate this, in panels (e) and (f) of Fig. <ref> we show the nonlocal conductance as a function of bias and Zeeman energy. Both for the homogeneous and the inhomogeous case, the appearance of nonlocal conductance around E=0 coincides with the change of the topological invariant. In other words, the appearance of finite nonlocal conductance around E = 0 implies a global closing of the induced gap. Additionally, the nonlocal conductance shows rectifying behavior around E=0 at the gap closing. These two features of the nonlocal conductance are strong evidence of a topological phase transition. Therefore, due to its insensitivity to spatial inhomogeneities in the potential and the additional feature of Andreev rectification, nonlocal conductance is a more reliable measure of a topological phase transition. § COOPER PAIR SPLITTER CAR dominated nonlocal conductance is important, however it does not happen under normal circumstances. A negative nonlocal conductance, dominated by CAR, is of fundamental interest, since the proximitised system then functions as a Cooper pair splitter <cit.>. In Sec. <ref>, we observed that the nonlocal conductance in clean systems at zero magnetic field is generally positive, and a CAR-dominated signal (G_21<0) is rare.The reason for this is shown schematically in Fig. <ref>: an electron entering the proximitised region usually converts into an electron-like quasiparticle.Andreev reflection changes both the quasiparticle charge and velocity, so that the resulting hole-like quasiparticle returns to the source. Therefore under normal circumstances Andreev reflection alone is insufficient to generate a negative nonlocal current. In clean systems, CAR dominates when the energy is aligned with hole-like states in the proximitised region. Despite G_21 stays predominantly positive in clean systems, in Sec. <ref> we found that a magnetic field can make the nonlocal conductance negative in large regions of parameter space. We identify these regions with the presence of only hole-like bands in the proximitized region at the relevant energy, as shown in Fig. <ref>. If only hole-like states are present in the proximitized region, the incoming electron may only convert into a right-moving hole-like quasiparticle, which in turn converts predominantly into a hole when exiting the proximitized region. To confirm this argument, we compare the energy ranges where only hole-like quasiparticles are present with the regions of negative G_21. Our results are shown in Fig. <ref>, and they exhibit a very good agreement. Since the only required property to get a negative nonlocal conductance is a hole-like dispersion relation, this phenomenon does not require SOI, or even Zeeman field. Indeed, our calculations (not shown here) reveal that it is possible to extend the energy ranges over which CAR dominates by filtering the nonlocal conductance by spin, e.g. by using magnetically-polarized contacts <cit.>. In the low doping-regime, we can engineer a hole-like band structure without a Zeeman field, but to see this manifest as negative nonlocal conductance, a larger chemical potential must be chosen in the normal leads. It is possible to systematically obtain a negative nonlocal conductance in the low-doping regime without using a Zeeman field if Δ > Δ_ind. This is shown in Fig. <ref>(c) and (d), where we have also neglected SOI for simplicity. By choosing μ comparable to the band offset of the lowest mode in the proximitised channel, at negative energies we obtain an energy range in which the band structure is only hole-like [Fig. <ref>(c)]. However, the small μ implies that no electron modes are active in the normal leads in this energy range. To observe negative nonlocal conductance here, it is therefore necessary to have a larger chemical potential in the normal leads than in the proximitised region, which ensures the presence of propagating electron modes at the relevant energies. Doing so, we indeed observe a negative nonlocal conductance in the expected energy range of Fig. <ref>(d).Disorder does not spoil CAR dominance, and may even be beneficial for CAR, since it enhances the probability of quasiparticle momentum flipping which is needed for CAR. Disorder provides an alternative mechanism to obtain negative nonlocal conductance. Unlike direct electron transfer, which generally conserves the sign of quasiparticle momentum, CAR often requires a sign change of the quasiparticle momentum. Since disorder breaks momentum conservation, the probabilities of CAR and direct electron transfer become comparable once the system length exceeds the mean free path, and CAR thus more prominent than in a clean system. Indeed, as shown in Fig. <ref>, in disordered systems the nonlocal conductance becomes positive or negative with approximately equal probability.§ SUMMARY AND OUTLOOKProbing induced superconductivity is hard, but we show that nonlocal conductance works for it. The standard experimental tool for probing induced superconductivity in a Majorana device is a tunnelling conductance measurement using an attached normal lead. While this approach detects the density of states, its usefulness is limited because it cannot distinguish the properties in close vicinity of the lead from the properties of the bulk system. We studied how the nonlocal conductance between two spatially separated normal leads attached to the proximitised region overcomes this limitation. We find that the nonlocal conductance is selectively sensitive to the bulk properties of a proximity superconductor, and allows to directly measure the induced and the bulk superconducting gaps as well as the induced coherence length of the proximitised region. While we focused on the quasi 1D-systems suitable for the creation of Majorana states, our conclusions are applicable to general proximity superconductors, including 2D materials like graphene covered by a bulk superconductor.Beyond being able to probe basic properties of proximity superconductivity, nonlocal conductance is sensitive to topological phase transitions, and can result in CAR-dominated response. When the probability of CAR is larger than that of electron transmission, the nonlocal conductance turns negative. While this does not happen normally, we identified conditions that allow CAR to dominate. This may happen due to disorder, which breaks the relation between quasiparticle charge, velocity and momentum and makes the nonlocal conductance zero on average. We identified another, systematic way of obtaining dominant CAR by ensuring that the only available states in the proximitised region are hole-like. A special case of this behaviour is the vicinity of the topological phase transition, where the nonlocal conductance becomes proportional to voltage, resulting in a linear relation between the differential conductance and voltage, or in other words a positive nonlocal current regardless of the sign of the voltage. This behavior is specific to topological phase transitions, and we showed how it can be used to distinguish accidental low energy states from Majorana states, resolving a potential shortcoming of Majorana tunneling experiments identified in Refs. Kells2012,mi2014,Prada2012,Moore2016,liu2017.As a follow-up one could investigate other geometries, Josephson junctions, and better ways to generate CAR by dispersion and geometry. Our setup can be used with trivial adjustments to probe the properties of Josephson junctions, proposed as a promising alternative platform for the creation of Majorana states<cit.>. Further work could investigate interaction effects on the the nonlocal response <cit.>. An alternative promising avenue of follow-up work is to consider a multiterminal generalization of a nonlocal setup in order to combine local and global sensitivity within the same device. In Fig. <ref> we show a possible experimental realization of such a multiterminal device, where the effective length can be adjusted with gates. Finally, our results regarding control of the CAR dominance can be used to design devices with a large electron-hole conversion efficiency. We thank D. Sticlet, M. P. Nowak and M. Wimmer for fruitful discussions. This work was supported by ERC Starting Grant 638760, the Netherlands Organisation for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience program and the US Office of Naval Research. MK gratefully acknowledges support from the Carlsberg Foundation.§ SHORT, INTERMEDIATE AND LONG JUNCTION LIMITS FOR HYBRID STRUCTURES We consider the different limits of a one-dimensional junction that consists of a normal part in contact with a much larger superconductor. In this appendix, we briefly discuss the subgap spectral characteristics of normal-superconductor junctions in different limits, using heuristic arguments to highlight the essential physics. For a more rigorous study, we refer the interested reader to e.g. Refs. beenakker1992, volkov1995, pilgram2000, tkachov2005, reeg2016. Consider a quasi-one dimensional channel of length L →∞ that consists of a junction between a normal part of width W and a superconductor of width W_sc≫ W. The Hamiltonian is the same as in Eq. (<ref>), but with p_x →ħ k and as before Δ≠ 0 only in the superconductor. Furthermore, we consider only E_Z = 0 and neglect SOI (α = 0) and disorder for simplicity.The relative size of the superconducting gap and the Thouless energy of the normal part, or equivalently the relative size of the normal part and the BCS coherence length, determines which limit the hybrid system is in. The hybrid structure generally has an energy gap Δ_ind, the size of which is determined by two competing energy scales, namely the bulk gap Δ and the Thouless energy E_Th≈ħ/ τ, with τ the quasiparticle dwell time in the normal part of the junction. A short junction has Δ≪ E_Th and a long junction Δ≫ E_Th, while Δ≳ E_Th for an intermediate junction. Alternatively, these conditions are expressed in terms of W and the BCS coherence length ξ_0 = ħ v_F / Δ, where v_F is the Fermi velocity. For a quasiparticle incident perpendicularly from the normal part to the interface with the superconductor and assuming perfect interface transparency, we have τ∝ W/v_F and thus E_Th∝ħ v_F/W. The conditions for short, intermediate and long junctions then become W ≪ξ_0, W ≳ξ_0 and W ≫ξ_0, respectively. In the short junction limit, we have Δ_ind≈Δ, while for long and intermediate junctions Δ_ind∝ E_Th.Level spacing in the normal part and the transparency of the interface with the superconductor determine the Thouless energy. We now derive a lower bound for E_Th in terms of the level spacing δ in the normal part of the junction. A quasiparticle exiting the superconductor has the dwell time τ∝ 2W/γ v_⊥(k) in the normal part. Here, v_⊥ (k) = ħ k_⊥(k)/m^* and k_⊥ = √(k_F - k^2) are respectively the velocity and momentum projections perpendicular to the interface with the superconductor at the parallel momentum k, with k_F the Fermi momentum, and 2W is the distance the quasiparticle travels before colliding with the superconductor again. The dwell time scales inversely with the transparency γ of the interface between the normal part and the superconductor. In practice, the transparency is determined by interface properties, such as the presence of a barrier or velocity mismatch, which we parametrize with 0 ≤γ≤ 1 for simplicity. We thus obtain E_Th(k) ∝γħ^2 √(k_F^2 - k^2)/2m^* W. Observe that E_Th decreases with k and tends to vanish as k → k_F since then v_⊥→ 0. However, v_⊥ is bounded from below in a finite geometry by the momentum uncertainty associated with the band offset, which corresponds to the velocity dv_⊥≈ħπ/m^*W in a square-well approximation. Using v_⊥ = dv_⊥ gives the lower bound for the Thouless energy E_Th∝γħ^2 π/2m^*W^2. The preceding discussion implies that in the absence of magnetic fields, the gap in the spectrum of such a junction decreases with momentum to a minimum ∝ 1/m^*W^2 at k = k_F [see left column Fig. <ref>]. Since Δ_ind is the energy of the lowest Andreev bound state in the junction, we define E_Th = γδ, δ = ħ^2 π^2/2m^* (2W)^2 as the Thouless energy of the junction. Observe that we use 2W in the denominator, since that is the distance normal to the interface a quasiparticle travels between successive Andreev reflections <cit.>. The limit in which the hybrid junction is in strongly affects the dispersion and density of states, with subgap states appearing in the intermediate and long junction regimes. The spectral characteristics of a proximitised system strongly depend on which regime the system is in. Figure <ref> shows the dispersion ϵ_n(k) and density of states ρ per unit length for junctions in the short, intermediate and long regimes. The density of states is given by ρ(E)= 1/2πħ∑_n ∫δ[ E - ϵ_n(k) ] dE/|v(E) |= 1/2πħ∑_n | dϵ_n(k)/dk|^-1. Here, n is the subband index including spin and we have used ħ v = dE/dk for the velocity v. In the left column, the solid lines give the dispersion of the hybrid structure, while the dotted lines show the electron and hole dispersions of the normal channel only (with W_sc = 0 or γ = 0). In all cases, μ≫Δ, and ρ has been broadened by convolution with a Lorentzian of full width at half maximum Γ≪Δ. For the short junction, we indeed have Δ_ind≈Δ, which manifests as an essentially hard superconducting gap for | E | < Δ_ind. We have verified that ρ vanishes identically in this regime with Γ→ 0. In the intermediate and long regimes, subgap states exist at energies smaller than Δ, which manifests as a nonzero subgap ρ (soft gap). The difference between the two regimes is the number of these states: in an intermediate junction, they are few, but multiple in the long junction limit, as the conditions Δ≳ E_Th and Δ≫ E_Th indicate. Observe that in both cases, the subgap bands are flat around k = 0 and drop towards a minimum in energy as k increases before rising sharply again <cit.>. Superimposed on this are intraband oscillations that happen on a smaller energy scale. In principle, oscillations thus manifest in ρ on two energy scales: the larger energy scale is the interband spacing around k = 0 (∝ 1/W^2), and the smaller the scale of intraband oscillations. Overall, the former has a larger contribution to ρ due to the small curvature in the dispersion. Oscillations on both scales are clearly visible for the intermediate junction. However, increasing Γ further (dashed curve) washes out the fine structure due to intraband oscillations. As a result, ρ gradually increases towards a maximum, when E aligns with the energy of the subgap state around k = 0. On the other hand, in the long junction there are multiple states at subgap energies, and the most prominent feature in ρ is the peaks associated with the flat parts of those bands. The fine structure due to intraband oscillations is superimposed, but masked by the broadening.apsrev4-1 | http://arxiv.org/abs/1706.08888v4 | {
"authors": [
"T. Ö. Rosdahl",
"A. Vuik",
"M. Kjaergaard",
"A. R. Akhmerov"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170627143514",
"title": "Andreev rectifier: a nonlocal conductance signature of topological phase transitions"
} |
OT1pzcmitcapbtabboxtable[][] arrows,decorations.pathmorphing,backgrounds,positioning,fit,petri,automata,shadows,calendar,mindmap, decorations.markings,calc 𝗒 a]Ilaria Brivio and Michael Trott[a]Niels Bohr International Academy & Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen, DenmarkProjecting measurements of the interactions of the known Standard Model (SM) states into an effective field theory (EFT) framework is an important goal of the LHC physics program. The interpretation of measurements of the properties of the Higgs-like boson in an EFT allows one to consistently study the properties of this state, while the SM is allowed to eventually break down at higher energies. In this review, basic concepts relevant to the construction of such EFTs are reviewed pedagogically. Electroweak precision data is discussed as a historical example of some importance to illustrate critical consistency issues in interpreting experimental data in EFTs. A future precision Higgs phenomenology program can benefit from the projection of raw experimental results into consistent field theories such as the SM, the SM supplemented with higher dimensional operators (the SMEFT) or an Electroweak chiral Lagrangian with a dominantly J^P = 0^+ scalar (the HEFT). We discuss the developing SMEFT and HEFT approaches, that are consistent versions of such EFTs, systematically improvable with higher order corrections, and comment on the pseudo-observable approach. We review the challenges that have been overcome in developing EFT methods for LHC studies, and the challenges that remain.The Standard Model as an Effective Field Theory [===============================================§ INTRODUCTIONOur understanding of the nature of fundamental interactions can advance through a direct discovery of a new particle, or indirectly. Knowledge gathered through indirect methods has historically been the leading indication of a new particle or theoretical framework. It has also been the case that such indirect knowledge is usually ambiguous in that it can be an indication of several possible models. This is essentially due to the decoupling theorem <cit.> which formalizes how the non-analytic structure of correlation functions due to heavy states are projected out when matching onto a low energy effective field theory (EFT). The discovery of a new particle is a clarifying event, as it usually removes such ambiguities.The Large Hadron Collider (LHC) was constructed based on the expectation that the functional mechanism by which SU_L(2) × U_Y(1) → U_em(1) below the unitarity violation scale(s) dictated by the massive W^±, Z vector bosons <cit.> would be revealed by probing the TeV energy range, with strong theoretical prejudice in favour of the Brout-Englert-Higgs mechanism influencing design choices <cit.>. In addition, it was expected that other beyond the Standard Model (SM) states involved in this mechanism could also be discovered in the characteristic energy range that LHC is exploring. The first expectation for LHC has been met to date with the discovery of a dominantly J^P = 0^+ boson <cit.> consistent with the SM Higgs boson <cit.>.The lack of additional new state discoveries at LHC (to date) is perhaps unsurprising considering the large global data set consistent with the SM. In recent years, the direct discovery of new[At least arguably fundamental.] states has become less frequent; the last three such discoveries being the top quark in 1995 <cit.>, the reporting of direct evidence of the tau neutrino in 2000 <cit.>, and the Higgs-like boson discovered in 2012 <cit.>. The possibility that the next direct discovery of a new particle is a prospect for the far experimental future is manifest. This expectation is supported by the lack of statistically significant deviations from SM predictions in the global data set, which can be largely a result of at least a moderate degree of decoupling of physics beyond the SM to higher energy scales (≫ m_Z,W,h). Avoiding unproductive melancholy, this is motivation for increasing our understanding of all manner of SM physics to improve our theoretical predictions of experimental results. Thereby we sharpen the theoretical tools that allow us to indirectly search for physics beyond the SM.This motivation is supported by the fact that all of these latest discoveries of new states {t, ν_τ, h} were preceded by decades of indirect evidence gathered using EFT techniques, requiring precise SM predictions. Further, this argument also supports developing EFT methods to capture the low energy effect of physics beyond the SM, as only focusing efforts on improving SM predictions is insufficient the moment a real deviation is discovered. In addition, the unavoidable theoretical ambiguity associated with indirect knowledge of physics beyond the SM means that a singular theoretical explanation of such a deviation from the SM is unlikely to be epistemologically assured. It is important to be able to systematically understand such an anomalous measurement in a well defined field theory framework, that also dictates correlated deviations in other processes to distinguish between competing explanations. After all, any successful explanation of such a deviation must be consistent with the global data set, not just the observable deviating from the SM.It can be remarkably efficient to approach such tests of consistency by projecting a particular model into an EFT framework in the presence of some degree of decoupling. This is the case so long as the EFT is well developed, so that properly interfacing with the global data set can be done in a one time matching calculation. To this end, it is essential to systematically improve our understanding of the EFTs that can accommodate SM deviations in advance of any such discovery of the SM breaking down in describing the data. It is also critical to encode the current data set into a form that maximizes its future utility when the SM can no longer successfully describe higher energy measurements.This review is focused on these tasks. We discuss the recent developments in using indirect methods to study the Higgs-like scalar, related signals, and the development of two EFT frameworks. Both of these frameworks describe the known SM particles that lead to non-analytic structure in the correlation functions measured in particle physics experiments to date, in some region of phase space.These theories are distinguished by the nature of the low energy (infrared -IR) limit of physics beyond the SM being assumed. When the SM Higgs doublet is present in the EFT construction, the EFT is known as the Standard Model Effective Field Theory (the SMEFT). Conversely, when the SM Higgs doublet is not present, the Higgs Effective Field Theory (the HEFT) is constructed. Due to the lack of any clear experimental indication to choose between these approaches at this time, it is important to minimize theoretical bias when reporting LHC data. For this reason, it can also be advantageous to project raw experimental data in terms of cross section measurements into gauge invariant pseudo-observables in some cases, that are constructed by expanding around the poles of the SM states. These pseudo-observable decompositions can be related to multiple theoretical frameworks, such as these two EFTs. We also discuss this developing paradigm and the relationship between these various approaches.The outline of this review is as follows. In Section <ref> we briefly review historical, and more traditional uses of EFT. In Section <ref> we then discuss the current pressing motivation for a precision Higgs phenomenology program using EFT methods. In Section <ref> we review pedagogically the key points leading to the structure of EFTs. We then turn to discussing the candidate field theories to use to interpret the global data set. First the SM is reviewed in Section <ref> to fix notation. We then discuss the SMEFT in Section <ref> and the HEFT in Section <ref>. Some issues that are being currently debated in the literature are reviewed in Section <ref>. In Section <ref> we discuss and review pedagogically the distinction between S matrix elements, Lagrangian parameters and pseudo-observables with an emphasis on the differences between these concepts that are accentuated in the presence of an EFT such as the SMEFT or HEFT. We apply this understanding to LEPI-II pseudo-observable measurements and interpretations in Section <ref>. In Section <ref> we discuss how many of these concepts and subtleties appear again in the interpretation of the measurements of the Higgs-like scalar at LHC, and review the κ formalism and proposals to go beyond this formalism in the long term LHC program. In Section <ref> we discuss the application of EFTs to top quark measurements at LHC. Finally, in Section <ref> we summarize the state of affairs early in LHC Run II and sketch out some expected future developments. The Appendix presents a series of LO results in a unified notation for SMEFT shifts to ψ̅ψ→ V →ψ̅ψ scattering through V = {Z,W} gauge boson scattering, Higgs production and decay, vector boson scattering and h V production using a {, Ĝ_F, m̂_Z } input parameter set.The intended audience for this review is a mixture of experts and novices and both theorists and experimentalists. The presentation is geared to aid a new Ph.D. student with a solid quantum field theory background to jump into this area of research. We hope that experts in the field will also benefit from some of the discussion on the conceptual and technical aspects of these interesting examples of EFTs incorporating the presence of the Higgs-like boson.[We apologize in advance for overlooking any references in this literature.]§ FEATURES OF THE EFT LANDSCAPEEFT is now a common tool used in many areas of particle physics, and increasingly in related areas of physics. The main reason EFT has become a standard theoretical tool is that it allows one to study large sets of experimental data in a systematically improvable field theory approach. This is the case withouthaving to assume the theory used is valid to arbitrarily high energies. We discuss some of the physics underlying this view in Section <ref>. Considering that the SM will eventually break down at higher energies/shorter distance scales, this makes EFT extensions of the SM key tools to develop in the modern "data rich" era. EFT methods generally come to the fore when large amounts of data are at hand to constrain the many parameters that usually arise in such a construction. This is now occurring for studies of the Higgs boson and the top quark, due to the successful operation of LHC. This is the reason that this review is more focused on studying these particles using EFT methods.These efforts are beginning in earnest using an approach to EFT that has a long history. An early influential example of EFT is given in Fermi's theory of β decay. In retrospect, Fermi theory is an effective operator approach to describe μ^- → e^- + ν̅_e+ ν_μ via an assumed Lagrangian <cit.>ℒ_GF = - G̃_F (ψ̅_i γ^μ P_L ψ_j)(ψ̅_k γ^μ P_L ψ_l).The left handed structure of the interactions was only fixed in due time with experimental input, and Fermi's approach was even more general in its initial formulation. The postulated interaction was introduced with a free coupling fit to data –G̃_F.[We turn our attention back to Fermi theory in Section <ref>.] This EFT description of a decay was advanced and found to be manifestly useful to study experimental results, before any solid experimental evidence of the existence of a W boson, or the SM itself, was at hand. This illustrates a key point of underlying the power of EFT: it is not required to know the underlying UV completion to use EFT methods. For more discussion on this point for the case of Fermi theory, see Refs. <cit.>.The amount of data available in high energy collisions √(s)≫ m_W,Z has historically been limited compared to lower energy collisions or decays, such as β decay. The historical use of EFT has been focused on using the relatively larger data sets gathered on lower energy phenomena, such as in flavour physics and/or studies of bound states of Quantum Chromodynamics (QCD) as a result. Another important example of an early application of EFT was the study of bound states of QCD in the 1960's, and Weinberg's calculation of pion scattering lengths. This occurred before any clear experimental evidence of the existence of quarks or any understanding of QCD was at hand. These calculations were first performed using assumed symmetry principles (partially conserved axial currents) in Ref. <cit.> and otherwise free parameters were again fit to the data. It was soon understood that a non-linear realization of SU(2) × SU(2) chiral symmetry allowed a more elegant and general understanding of the physics at work in Ref. <cit.>. A clear discussion on this point is presented in Ref. <cit.>. The understanding of non-linear realizations of symmetries, commonly used in EFT applications, was simultaneously advanced and generalized in the classic works of Coleman et al. in Refs. <cit.>.Once supplied with the solid theoretical hammer that is EFT, and a clear conceptual foundation of this approach developed coincident with the resurgent interest in field theory in the 1970's, the theoretical community found many nails. An explosion of applications and well defined EFT's has emerged in the past few decades. An incomplete summary of some important applications of EFT include the following.* ChPT. Following the pioneering studies of the 1960's the study of the π, K and ηmesons using chiral perturbation theory (ChPT) methods has been systematically developed for decades. Some of the key papers of this development are Refs. <cit.>. For reviews of this approach see Ref. <cit.>.* LEFT. Fermi's theory has been systematically extended into a complete description of a low energy phenomena where higher dimensional operators are used to describe flavour conserving and violating contact interactions. The operators of this EFT are generated when the W,Z,h,t particles of the SM are integrated out, and can also have beyond the SM matching contributions. This EFT is used extensively in studies of flavour transitions of QCD bound states at low energies for decades. Recently, this approach has been further systematized in Ref. <cit.>, which determined the complete one loop anomalous dimension of the theory and the matching onto the framework of the SMEFT. Some of the results in the Appendix are defined in this EFT. * HQET. A systematic expansion in the ratio Λ_QCD/m_b ≪ 1 underlies the Heavy Quark Effective Theory. This theory describes the interactions of a heavy quark with soft partons and has been applied to describe B meson decays and oscillations. An excellent resource to learn and calculate in HQET is Ref. <cit.>. * SCET. Building on the idea of the large energy effective field theory (LEET) <cit.>, and initially motivated out of the study of summing Sudakov logarithms in B → X_s γ decay that had failed in the LEET formalism, the Soft Collinear Effective Field Theory was developed in Refs. <cit.>. This EFT describes the interactions of particles of relatively different energies Q and the small expansion parameter is Λ_QCD/Q in most applications. A good introduction to SCET is given in Ref. <cit.>.EFT's continue to be developed and added to this incomplete list. In this review, we mostly focus on the SMEFT and HEFT effective field theories that are currently experiencing an intense development related to LHC experimental results. First we set the stage by discussing the strong motivation for precision Higgs studies and lay out the basic ideas underlying EFT.§ THE NEED FOR A PRECISION HIGGS PHENOMENOLOGY PROGRAMFurther developing the theoretical methods discussed in this review is not an idle pursuit. It is reasonable to expect that the properties of the dominantly J^P = 0^+ scalar boson could be perturbed by physics beyond the SM, and a precision Higgs phenomenology program could uncover such perturbations.The reason for this expectation is the idea that the SM Higgs mechanism describing SU_L(2) × U_Y(1) → U_em(1) is likely to be only an effective description. This belief is deeply rooted in the historical origin of the Higgs mechanism itself. The Higgsed phase of the SM (see discussion in Refs. <cit.>) can be understood to be analogous to the ideas that first emerged in the Landau-Ginzburg effective model of superconductivity <cit.>. The Landau-Ginzburg action functional is given by <cit.> LG(s) = ∫_^3 d x^3 [1/2|(d - 2 i e A)s|^2 + γ/2(|s|^2 - a^2 )],where A is the vector potential of Electromagnetism, d is a derivative defined on ^3, e is the electric charge and s is a section of the (squared) unitary complex line bundle of Electromagnetism. This action has an energetically favoured minimum for |s| = a and (d - 2 i e A)s = 0when γ>0, leading to a topologically flat line bundle in the superconducting phase, and the exclusion of the magnetic field from the superconducting material.[Interestingly, the topology of the scalar manifold defined by the Higgs doublet H will form a crucial discriminant between the SMEFT and HEFT theories in the discussion that follows.]The (partial) action of the SM Higgs <cit.> is directly analogousS_H = ∫d^4 x(|D_μ H|^2 -λ(H^† H -1/2 v^2)^2 ),with H is the Higgs doublet and D is the covariant derivative of the SU_L(2) × U_Y(1) theory. This theory has an energetically favoured minimum at ⟨ H^† H⟩ = v^2/2. Expanding around the minimum of the potential (that defines the EW vacuum background field) leads to the massive W^±_μ,Z_μ vector bosons, due to the Higgsing of SU_L(2) × U_Y(1) → U_em(1). Massless SU_L(2) × U_Y(1) vector boson field configurations are then energetically excluded.Landau-Ginzburg theory is not a fundamental theory. It is a functional effective description of superconductivity that can be related to a theory of Cooper pairs, such as BCS <cit.> theory. The connection drawn between Landau-Ginzburg theory and the Higgsed phase in Yang-Mills theory by Anderson <cit.>, leads to an expectation of a shorter distance (or higher energy i.e ultraviolet – UV) completion/origin of the Higgs mechanism.[ A direct analogy would have the Higgs as a composite field, similar to the Cooper pair of BCS theory.] In this manner,the curious appearance of a classical Higgs potential with a chosen “Mexican hat” form and an explicit scale v in the SM Lagrangian is understandable as a general low energy parameterization of underlying physics leading to an effective SU_L(2) × U_Y(1) → U(1)_em. It is possible that this parameterization is not appropriate as an IR limit of a UV sector leading to the observed massive gauge bosons. This is a way to understand the difference between HEFT and SMEFT that will be discussed in more detail below. §.§ Quantum corrections to Higgs properties and potentialThe previous section advanced an argument in support of a more fundamental description of Electroweak symmetry breaking (EWSB) in that the Higgs potential has no quantum or dynamical origin in the SM; it is a parameterization. Functionally, it is a directly assumed classical potential – that is extremely sensitive to UV physics including quantum corrections. The reason for this is that the dimension of the (H^† H) operator is two. Dimensional analysis indicates that this operator can receive dimensionful corrections due to heavy states that extend the SM. Such heavy states generally couple to the field H, or composite operators involving H, to perturb the properties of the Higgs when integrated out,[In some cases low energy effects can be present modifying Higgs properties that do not satisfy this requirement, due to reducing the field theory with the Equations of Motion (EOM) to a minimal basis reshuffling the appearance of IR physics effects.] see Fig. <ref>.The Higgs field transforms as an SU_L(2) doublet and has hypercharge _H = 1/2, so composite operators involving H that allow dimension ≤ 4 couplings to singlet fermion (F), scalar bi-linear (S^† S), and (non-gauged) vector fields V_μ are of the form[H^† D^μ H V_μ + h.c can be directly shown to not lead to a threshold correction, by integrating by parts.]Δℒ =|κ_v|^2 (H^† H) V^†_μV^μ - |κ_s|^2 (H^† H) (S^† S) + κ_FF H̃^†ℓ_L+ h.c.where ℓ is a lepton SU_L(2) doublet with hypercharge _ℓ = -1/2. These interaction terms lead to threshold matching contributions in the Higgs potential[The threshold matching contributions are obtained by calculating in dimensional regularization (DR) in d = 4 - 2ϵ dimensions using MS subtraction, and Taylor expanding the resulting amplitudes in the limit v^2/m_i^2 < 1.]Δ V(H^† H) ≃ H^† H (3 |κ_v|^2 m_v^2N_v/16π^2 + |κ_s|^2 m_s^2 N_s/16π^2 - |κ_F|^2 m_F^2 N_F/16π^2) + ⋯,where m_i is the mass of the corresponding field, and N_i can result from the sum over the degrees of freedom in an internal (flavour) symmetry group of the field i. When avoiding tuning the bare Higgs mass in the classical Lagrangian against quantum corrections,[Tuning parameters at the Lagrangian level to avoid these conclusions can be understood to be best avoided in the following way. As Lagrangian parameters can always be related to measured quantities through S matrix elements, and eliminated in relationships between S matrix elements, parameter tuning can be understood to be equivalent to assuming precise relationships between a series of independent measured quantities, in order to have a further S matrix element take on a value not expected by naive dimensional analysis.] then it follows that m_h ∼ |κ_i| m_i √(N_i)/4π. This is the reason the mass of the SM Higgs is expected to be proximate (up to a loop factor) to beyond the SM mass scales.One can turn the relation between m_h and m_i around. Then corrections to cross sections that are probed through a measurement exploiting a propagating SM state (that goes on-shell) scale as σ_SM + i/σ_SM≃1/16π^2 (N_i^2 |κ_i|^2 κ'/g_SM λ) + ⋯ Here we have introduced another coupling between the new physics state and the SM (κ') and Taylor expanded out the non-analytic structure of the tree level propagating state i. The non-analytic structure of the propagating SM state is essentially unchanged in this limit and cancels out in the ratio. g_SM corresponds to a generic SM coupling. This result argues that ∼% level deviations in Higgs properties can occur in scenarios that avoid parameter tuning. This estimate is subject to the following qualifications:* Differences between coupling constants can lead to further enhancements/suppressions. * This estimate implicitly assumed one local contact operator was introduced (at tree level) interfering with the SM. It has been proven <cit.> that when considering tree level effects, subject to the condition that a flavour symmetry is not explicitly broken and the UV scales have a dynamical origin, multiple operators are always present. * If the state i does not lead to any corrections to σ_SM + i at tree level, then a further ∼ 1/16 π^2 suppression occurs. On the other hand, when considering one loop effects, the multiplicity of operators present is generically very large due to one loop mixing, see Section <ref> for more detail.[In some exceptional cases, ℒ_6 operators do not mix. This can be understood at an operator level using operator weights and related helicity and unitarity arguments <cit.>.]This rough and schematic understanding is nevertheless validated in exact calculations in some popular new physics models, see Refs. <cit.> for more discussion. §.§ Higgs substructureEFTs capture the Taylor expanded effects of particle exchange at tree and loop level, but also encode multi-pole expansions of underlying structure <cit.> that are generic in field configurations set up by charge distributions that are spatially separated. The classic example is the multi-pole expansion of an electrostatic charge distribution <cit.>. This physics is not directly or trivially identified in general with tree or loop level particle exchange diagrams in the presence of non-perturbative bound states (such as a composite Higgs), and offers further hopes for perturbations of Higgs properties that could be experimentally measured.When an EFT is capturing the consistent low energy limit of a strongly interacting light composite Higgs, the multi-pole expansion should be considered <cit.>.[This point has recently been re-emphasized in Ref. <cit.>.] The possibility that the Higgs field is composite underlies a significant fraction of the interest in the EFT methods discussed in this review.Composite Higgs models are still of experimental interest. These ideas emerged in the 80's in Refs. <cit.> with early studies also exploring the possibility of dynamical mass generation in extra dimensional scenarios <cit.>. These ideas re-emerged and were extended in the late 90's in the context of Little Higgs constructions <cit.>, and extra dimensional models aimed at dynamical mass generation <cit.> with the Holographic composite Higgs models consolidating many of these developments in Refs. <cit.>.In analyzing scattering off of non-perturbative bound states, leading to a multi-pole expansion, perturbative methods fail by definition. One can gain some intuition on how this scattering is represented in an EFT by considering the solutions to the time independent Schrödinger equation of a non-local potential of a fixed target, represented as V(r,r'). Such a potential can mimic the extended nature of the composite particle. With appropriate boundary conditions this scattering is described by the Lippmann–Schwinger equation <cit.>; outgoing wavefunctions are related to those incoming by a partial wave transition matrix that satisfies the integral equationT_ℓ(k,k';E) = V_ℓ(k,k') + 2/π∫_0^∞ d |q| q^2 V_ℓ(k',q) T_ℓ(q,k;E)/E - q^2/μ + i ϵ.Here k,k' are the Fourier momentum of the radial coordinate r,r' and μ is the reduced mass. The asymptotic scattering is described by a partial wave scattering matrix: S_ℓ(k) = 1 + 2 i k f_ℓ(k), which can be parameterized by a partial wave phase shift S_ℓ(k) = e^2 i δ_ℓ(k), for spherically symmetric potentials. The multi-pole expansion in this case is the fact that the phase shift parameter characterizing the scattering matrix has a power series expansion in k^2. For the partial wave ℓ = 0, this expansion is given ask δ_0(k) = -1/a_0 + 1/2r_0 k^2 - C_2 r_0^3 k^4 + ⋯ There is an expansion in derivatives acting on a field F of the state associated with asymptotic wavefunction scattering off of the fixed target field 𝒯 as a series of interactions of the form ∼{𝒯 F F, 𝒯 F ∇^2 F, 𝒯 F ∇^4 F}. The effective range expansion in the parameters {a_0,r_0,C_2 r_0^3} are analogous to Wilson coefficients in the relativistic EFT. The bound state substructure generates a series of scales that characterize the multi-pole expansion. This is generic in EFTs describing the scattering off of bound states. See Refs. <cit.> for related discussion on non-relativistic bound states in EFTs.In nucleon-nucleon (NN) EFT <cit.> an analogy to the non-relativistic scattering case is extensive. The time independent Schrödinger equation corresponds to a summation of an infinite set of Feynman diagrams in the EFT, defining a scattering amplitude 𝒜 as shown in Fig. <ref>.The Born series expansion of the related Lippmann–Schwinger equation can be mapped to an infinite sum of ladder diagrams that depends on the particles exchanged in the EFT and the kinematics of the poles dominating the convolution integrals between the nucleon potential and the non-relativistic propagators. The amplitude defined by this Born series is related to the phase shift <cit.> |p|δ(p) = i |p| + 4π/M1/𝒜.Here p is the three momentum of the NN system and M is the mass scale of the nucleon. 𝒜 is a scattering amplitude that corresponds to the infinite sum of ladder diagrams. Again a power series expansion of the phase shift leads to the multi-pole expansion. The parameters characterizing the effective range expansion of nucleon scattering take on values that differ by an order of magnitude <cit.> and depends in a non-trivial manner on the spectrum of states retained in the EFT as the bound state is near threshold.For a composite Higgs, one can consider the Higgs to be analogous to the nucleon, and the convolution integral of the Higgs self interaction potential to be made with Higgs propagators and beyond the SM states involved in EW symmetry breaking, such as vector resonances analogous to the ρ meson in chiral perturbation theory. Considering current LHC data, there is little motivation to assume that such a ρ state is accidentally lighter than the remaining states in the strong sector. Scattering results developed in analogy to NN scattering EFT then lead to a multi-pole expansion in derivative operators involving the Higgs field.In addition, when considering the multi-pole expansion in terms of the SU_L(2) × U_Y(1) gauge fields, scattering off the bound constituents that make up the composite Higgs can occur. This is the case if the constituents are charged under the SM gauge groups, or a larger group that contains the SM as a subgroup as illustrated in Fig.<ref> (right).[We restrict our attention to the EW interactions due to the expectation that new physics underlying Higgs compositeness would be associated with EW symmetry breaking.]A composite Higgs has an associated multi-pole expansion. Unfortunately, using crossing symmetry (i.e. rotating Fig.<ref> (right) 90^∘ counterclockwise) to consider the interaction potential as only describing the composite Higgs state is inconsistent with a non-relativistic EFT approach related to the Schrödinger equation.[As crossing symmetry relations are between Mandelstam variables constructed out of full four vectors.] Furthermore, the summation of subsets of diagrams in “ladder approximations” to the convolution integrals is not valid in general. Noting all of these concerns, the multi-pole expansion can be associated with the suppression scale and Wilson coefficients of the U(3)^5 symmetric operators[See Table <ref> for the operator definitions corresponding to these Wilson coefficients.]λ_Mul^2 ≃{C_H /Λ^2, C_HD/Λ^2, C_HWB/Λ^2,C_HW/Λ^2,C_HB/Λ^2}.When these operators are all constrained so that λ_Mul≪λ_h, where the Compton wavelength of the Higgs λ_h = ħ/m_h c, the Higgs boson is effectively interacting as a point-like particle, when considering these dimension six operators. As we have scaled the operators by Λassociated with particles integrated out of the spectrum, the Wilson coefficients of the operators involved in the multi-pole expansion can be expected to differ from order one values – if the scales characterizing the effective range expansion are distinct from the mass scale of the states integrated out of the theory constructing the EFT.To summarize: considering the possibility of compositeness and the related multi-pole expansion, the UV sensitivity of Higgs properties, and the classical nature of the assumed SM EW symmetry breaking potential, a precision Higgs phenomenology program to probe for indirect hints of physics beyond the SM is well motivated. § BASICS OF EFTA Taylor expansion in dimensionless ratios was used in the previous sections to simplify the results. That such a simplification can occur is consistent with the intuitive understanding that IR physics can be calculated without reference to the details of all UV physics. This is generic in observables calculated in a Quantum Field Theory (QFT)so long as limited theoretical precision is all that is demanded. Manifestly this is true for the SM; which despite being a QFT that is not well defined to arbitrarily high energies,[Due to the presence of Landau poles in the SU_L(2) × U_Y(1) theory.] has still been validated to be an adequate description of LHC data considering current experimental precision. §.§ Separation of scales, renormalization and local/analytic expansionsEFT is a set of ideas that justifies why this systematic separation of the physics of different scales can be true in field theory.[For excellent reviews on EFT see Refs. <cit.>. The pioneering works developing the modern understanding of EFT include Refs. <cit.>.] Renormalization also separates IR and UV physics, but EFT is more than a statement that QFTs are systematically renormalizable. Furthermore, the success of renormalization programs in QFTs can be understood as an EFT consequence in an intuitive way.[The old field theory approach of stressing of the distinction between bare and renormalized parameters is drawn when correlation functions involving the parameters are considered to be measured or predicted to arbitrary precision. In this sense, the EFT understanding of renormalization is consistent with such lore.] When renormalizing a QFT the short distance physics in the theory one calculates in is modified, and the effects of regularizing such physics is absorbed into the low energy parameters of the effective theory. How this modification takes place is illustrative of scale separation in EFTs.Consider calculating an amplitude at one loop using dimensional regularization in d = 4 - 2ϵ dimensions <cit.>.[We use MS subtraction, by introducing n powers of μ̂^(4-d)/2 = (μ √(e^γ/4 π))^(4-d)/2 for each power of the coupling present defining the amplitude, so that the renormalized coupling remains dimensionless. Here γ is the Euler-Mascheroni constant. The arguments in this section are formulated for a one loop example but they generalize to higher loop orders directly.] The amplitude can be expressed by using the master formula for Minkowski space momentum integralsM_I = ∫d^d q/(2π)^d (μ̂^2)^n(d-4)/4 (q^2)^α/(q^2 - Δ^2)^β = i(-1)^α-β/(Δ^2)^β - α - d/2 Γ(α + d/2)Γ(β - α - d/2)/Γ(d/2)Γ(β),with a four momentum q^μ and a factor Δ introduced for the characteristic scales (masses, kinematic invariants) in the amplitude. Following the discussion of Georgi <cit.>, consider integrating over the ϵ momentum space of such an amplitude after Wick rotating to Euclidean momentum space and factorizing the loop momentum as q^2 = q^2_ϵ + q^2_E. Restricting one's attention to divergent terms one finds <cit.> M_I ∝∫d^4 q_E/(2π)^4(q_E^2)^α/(q_E^2 + Δ^2)^β[Γ(β +ϵ)/Γ(β)] [4πμ^2/q_E^2 + Δ^2]^ϵ.The last two factors in square brackets are both finite as ϵ→ 0, but there is an important difference between them. The final term does not significantly change the amplitude so long as all the scales are similar μ^2 ∼Δ^2 ∼ q^2_E. On the other hand, this term leads to a modification (an introduced regularization) of the amplitude that can become significant for small ϵ if μ^2 ≫Δ^2 + q_E^2 or μ^2 ≪Δ^2 + q_E^2, i.e. when highly separated scales are present in the amplitude. In this manner, the universal subtractions present in systematically renormalizing a QFT are understood to correspond to UV physics effects that have been systematically removed out of the lower energy theory by such a regularization for μ^2 ≪Δ^2 + q_E^2. That such a separation of IR and UV physics can occur is the key idea of EFT and this can be understood to be an underlying reason for renormalization to work. The case μ^2 ≫Δ^2 + q_E^2 has a different meaning, it corresponds to an IR divergence, and we discuss this divergence below.Counterterms are of a simple universal analytic form when using DR. This is also the case in other regularization schemes, such as schemes with dimensionful regulators that directly satisfy the decoupling theorem <cit.>. One might doubt if the regularization of divergences due to arbitrary UV physics sectors can be subtracted out of a prediction of a lower energy observable in this simple manner. Formally, this can be understood to follow from a proof supplied by Bogoliubov and Parasiuk[We thank F. Herzog for this reference.] on the analytic nature of counterterms in 1957 <cit.>. Renormalization Group (RG) based arguments also support this understanding, as demonstrated by Polchinski in Ref. <cit.>, as do the diagrammatic arguments of Weinberg's power counting theorem <cit.>. Recently the formal proof of the renormalizability in effective field theories has also been studied with increased mathematical rigor in Ref. <cit.>.A less formal and more intuitive understanding of the universal nature of the subtractions follows from considering the constraints of the global symmetries in the EFT, Lorentz invariance, and the fact that the non-analytic structure of correlation functions[Here we refer to the poles and cuts in the momentum space of the spectral function defined in analogy to the Källén-Lehmann <cit.> two point spectral function.] is only generated when intermediate states propagate on-shell. When subtracting the effects of UV physics acting to regularize divergences in the full theory systematically out of a lower energy amplitude, far below the characteristic mass scales of such UV states, these states are off-shell. The correlation functions can be simplified by Taylor expanding in the ratio of the separated scales, and are well approximated by the first fewanalytic terms in the expansion. Any divergence thereby has an analytic form. The locality of the subtractions is because off-shell exchange of the virtual particles (of mass ∼ M) occurs, but it is local as it is limited to short times and distances by the uncertainty principle <cit.>Δ tΔ E∼Δ t M> 1 →Δ t ∼1/M,Δ |x|Δ |p| ∼Δ |x|M >1 →Δ |x| ∼1/M.Here we are using units where ħ = 1 = c.[Unless otherwise noted we use such “God-given" units in this review.] Renormalization understood in this manner does not draw any fundamental distinction between theories with only interaction terms limited to mass dimension d ≤ 4 and EFTs with a tower of higher dimensional operators. The renormalizability is understood to be possible due to the fact that the only way that high energy physics integrated out of the low energy theory can modify the lower energy construction is through a tower of local analytic operators. In both cases the renormalizability of the theories follows from the separation of scales that allows the Taylor expansion.This reasoning also holds for the non-divergent contributions of UV physics approximated in an EFT by expanding in a ratio of scales. As a result, an EFT is a field theory with a tower of local analytic operators of dimension d divided by d-4 powers of a suppression scale characteristic of the UV physics removed from the EFT construction.[In some exceptional cases EFTs can be constructed with non-local operators. This is usually due to distinguishing field excitations as retained or removed from the EFT by assigning a four momentum of a particle excitation of a field (not the p^2 Lorentz invariant used to distinguish on or off-shell) some scaling rules. See Refs. <cit.> for famous examples.] A well constructed EFT is designed to capture the relevant low energy physics to predict a set of experimental measurements, while exploiting the simplifications that result from such expansions as soon as possible. The essential and key idea is to separate the description of the processes under study into IR (i.e. infrared or long distance) propagating states and their interactions, captured by the local and analytic operator expansion, and the UV dependent short distance Wilson coefficients, i.e. construct ℒ_EFT≃∑_i C_i^UV(μ) O_i^IR(μ).Taking this reasoning to its logical conclusion gives a commonly held set of “prime directives” of effective field theorists:[We acknowledge M. Luke for this nomenclature.] * Isolate and separate a series of characteristic scales in observables. * Construct the Lagrangian of the EFT only out of the degrees of freedom that lead to non-trivial structure in the correlation functions. These are propagating on-shell states (i.e. with p^2 ∼ m^2), ideally with only one scale defining the EFT closely related to the scales identified in the previous step. In short, expand ASAP, at the Lagrangian level. * Calculate in the EFT without unnecessary reference to the UV physics that isis decoupled. Some UV dependence is present, but it is sequestered in the EFT into the short distance Wilson coefficients in the matching procedure. In contrast, in the EFT the operators encode the IR physics describing long distance propagating states. * Use a mass independent renormalization scheme, such as dimensional regularization.The last two points are related to the requirement of matching and some technical consequences that result from a renormalization and subtraction scheme choice that we discuss in more detail in the following sections.The physics that can be captured in the SMEFT in this manner as a consistent IR limit is only limited by the assumptions that Λ > v, and the existence of a Higgs doublet in the EFT construction. Various cases of beyond the SM physics are illustrated in Fig. <ref>.When this strict separation of scales is maintained, i.e. the SMEFT is treated as a general EFT retaining all of the operators that are allowed by the assumed symmetries, powerful model independent conclusions can result. All of the cases in Fig. <ref>, and combinations of such cases in possible UV physics sectors, have to project onto a series of local and analytic operators with various Wilson coefficients in the EFT expansion. This point holds even when the UV physics cannot be calculated with known field theory techniques.If this separation of scales is violated, then the resulting statements and analysis, even if constructed and framed in EFT language, are not EFT conclusions. Such conclusions can be model dependent or simply ill-defined. This issue is very well known in research areas where EFT techniques have been dominant for decades, but is less appreciated when applying EFT techniques to characterize and constrain new physics at LHC, which is a continual source of debates in the literature. The key problem that can be introduced when going beyond EFT is the introduction of an assumption heavy hypothetical UV physics sector, which can result in the lack of a consistent IR limit, rendering the EFT framework used inconsistent and without meaning. To avoid this problem the key requirements that UV assumptions must address are* The IR limit of a UV theory must be well defined, which requires that the UV theory is written down. In particular, the origin of the scales in a UV theory should be specified to have a possibility of a meaningful IR limit. * If a strong interaction is present in a UV completion, and the mass scale characterizing bound states is Λ≫ v, then non-perturbative contributions can exist (see Fig.<ref> right) and should not be assumed to vanish without a precise justification. Assuming that such non-perturbative effects are absent, or negligible, in the EFT projects into a strong, and at times undefined, condition on UV completions that can generate the EFT. Again we emphasize that one of the core strengths of the standard approach to EFT is the ability to characterize and constrain such physics rendering UV assumptions on strongly coupled physics completely avoidable. §.§ The decoupling theoremThe previous section outlines the basic intuition underlying EFT methods that is formalized in the Appelquist-Carazzone decoupling theorem <cit.> (see also Symanzik <cit.>). This theorem played an important role in the emergence of EFT methods in the 1970s. Examining this result in detail shows how renormalization scheme choice is also a technical issue of some importance when calculating in EFTs, as in the SM. The decoupling theorem is developed studying a set of massless gauge fields, denoted A_μ (and referred to as “vector mesons” at times in Ref. <cit.>), that are coupled to a set of massive fermions, denoted Ψ. The Lagrangian is ℒ_dc = - 1/4 F_μν^a F_a^μν + Ψ̅ iDΨ - Ψ̅ m Ψ - δ m Ψ̅Ψ,whereF^a_μν = ∂_μA_ν^a - ∂_νA_μ^a - g f^abc A_b, μ A_c, ν, (D_μΨ)_n= ∂_μΨ_n + i [T_aA_μ^a]_n,and finally [T_a,T_b] = i f_abc T^a defines the Lie algebra of the gauge group, with coupling g.[ We have modified some notational conventions compared to Ref. <cit.> to maintain a common notation throughout the review.] Here δ m explicitly denotes the mass counterterm. The decoupling theorem is stated as <cit.>: For any 1PI Feynman graph with external vector mesons only but containing internal fermions, when all external momenta (i.e. p^2) are small relative to m^2, then apart from coupling constant and field strength renormalization the graph will be suppressed by some power of m relative to a graph with the same number of external vector mesons but no internal fermions.[Here the exact wording ofRef. <cit.> is edited for clarity.] Removing the field whose quantum is a heavy particle from the Lagrangian used to calculate experimental observables, based on the decoupling theorem, is known as “integrating out” a particle from the theory. The decoupling theorem is stated and proven for the specific field theory in Eq. <ref>, but the arguments used to prove it generalize directly to other theories.[ For example, for a detailed discussion and proof on decoupling for scalar field theory with two fields when d=6 see Collins <cit.>.] The generalization of this result to arbitrary field theories can be given in terms of all n-point Green's functions G^n as ∏_i^N Z_i G_full^n(p_1,p_2 ⋯ p_n; μ) = ∏_j^M Z_j G_EFT^n(p_1,p_2 ⋯ p_n; μ) + 1/m^2∏_i^k Z_i G_EFT^' n(p_1,p_2 ⋯ p_n; μ)+ ⋯ where Z_i..N is the set of renormalizations required to render the full theory finite, Z_i..M is the set of renormalizations required to render the leading d ≤ 4 terms in the effective theory finite in an on-shell scheme. Z_i..M..k includes these renormalizations and the additional renormalizations required to also render the local contact operators suppressed by 1/m^2 finite. This theorem is formally establishing that if the intermediate heavy fields do not go on-shell (i.e never satisfy p^2 ≃ m^2) they modify the leading local analytic operator structures through renormalization in the lower energy Lagrangian, or add additional interactions suppressed by powers of 1/m. This is expected considering the schematic arguments of the previous section. The proof of the decoupling theorem is non-trivial. The statement is forany 1PI graph, i.e. can be an arbitrarily high order in perturbation theory. Due to this, the renormalization scheme chosen has an important impact on the arguments required to establish the proof. In Ref. <cit.> the scheme used defines δ m to fix the fermion self energy to vanish at k = m. The remaining counterterms are subtractions defined at off-shell Euclidean momentum subtraction points (p^2 = - μ^2). Wavefunction renormalization conditions fix the propagator to have its tree level form. Using this scheme Ref. <cit.> considered divergent and finite terms in arbitrary 1PI Feynman graphs and established the decoupling theorem exhaustively. Careful attention is paid in the proof to ensure that a well defined IR limit (p^2/m^2 <1) is under consideration, by examining IR safe observables consistent with the KLN theorem <cit.>. Equally important is a careful consideration of sub-divergences (that are sensitive to the regularization scheme used) in establishing the theorem. An implicit dependence on an off-shell subtraction scheme is present in the decoupling theorem.One of the “prime directives” of EFT is a direct consequence of the decoupling theorem: calculate in the EFT without unnecessary reference to the UV physics. This is required as the UV physics is decoupled and simply removed from an EFT in a controlled fashion. Its IR effects are reproduced to a limited precision and encoded in the matching procedure in the Wilson coefficients of the EFT. §.§ Non-decoupling physics The decoupling theorem has some exceptions. This should be surprising considering the generality of the arguments that have been advanced in the previous sections. Calculating in field theories to an approximate precision, in the presence of separated scales, is usefully thought of using the techniques of EFT. Such EFTs are constructed based on the separation of scales that underlies decoupling. However, no theorem can escape the constraints of its exact wording and assumptions, and this is also true for the decoupling theorem. Several examples of “non-decoupling" effects are discussed in the literature. Heavy physics of this form does not imply that an EFT is impossible to construct to capture an IR limit of some UV physics. It just enforces the construction of the EFT to take on a particular form, usually by requiring that a non-linear representation of a symmetry be used.[Again the existence of the SMEFT and the HEFT can be understood to be related to this fact, as non-decoupling effects in the scalar sector are a possibility.]§.§.§ The ρrho parameter Non-decoupling effects can occur when heavy states and the light states are embedded in the same representations of a symmetry group in the full theory. Divergences can be forbidden by the linearly realized symmetry,due to cancellation between the particles of different masses embedded in such a (softly broken) representation of a symmetry group. Whensome of the states are no longer in the spectrum in the EFT, the counterterms are no longer forbiddenby the linearly realized symmetry.Then perturbative corrections can grow with the mass of the state removed from the theory. The practical signal of this physics can be the appearance of numericallylarger perturbative corrections when the heavy state is still retained in the theory,and at times an additional mass dependence outside of logarithms in such corrections. This can be the case as in the loop corrections the masses sometimes act to regulate the divergences when the symmetry is linearly realized. Several historical examples of this form of non-decoupling are present in the literature, in ν e scattering <cit.>,in large 𝒪(α_s) corrections (due to quark doublet mass splittings)to the axial neutral current <cit.> and in the behavior of one loop corrections<cit.>to the ratio of charged and neutral currents in the SM, due to the diagrams shown in Fig. <ref>.We discuss this latter case of the ρ parameter <cit.>, defined as the ratio of charged and neutral currents at low energies, as an example. The ρ parameter has the perturbative expansion, with one loop contributions shown in Fig.<ref> (in MS) which give ρ≃g̅_Z^2m̅_W^2/g̅_2^2m̅_Z^2+ N_cĜ_F/8π^2 √(2)(m_t^2 + m_b^2 - 2 m_t^2 m_b^2/m_t^2 - m_b^2log(m_t^2/m_b^2) ) - 11 Ĝ_FM̂_Z^2 s_θ̅^2/24 √(2)π^2log(m_h^2/m_Z^2). The Higgs mass dependent correction is not exceedingly large and it is not related in mass to another particle in the spectrum by a linear realization of a symmetry. The limit m_h →∞ can be taken, which still leads to a non-linear realization of SU_L(2) × U_Y(1) as the Higgs field and its vacuum expectation value are related when this symmetry is linearly realized. The effective theory construction of Refs <cit.> results when the limit m_h →∞ is taken. The corrections due to the heavy Higgs matched onto this EFT are not suppressed by explicit powers of m_h^2 in their leading contributions. The full results of this form are given in Refs. <cit.>. This is an example of non-decoupling effects that deviate from a naive expectation formed from the decoupling theorem.Even larger corrections come about due to splitting the quark masses in the limit m_t ≫ m_b. Note thatm_t^2 + m_b^2 - 2 m_t^2 m_b^2/m_t^2 - m_b^2log(m_t^2/m_b^2) → 0,in the limit m_t→ m_b. Integrating out the top, while leaving the b quark in the spectrum, leads to a theory without a linearly realized SU_L(2) symmetry. Furthermore, m_t is acting to regulate an integral, which is a reason that it appears as a polynomial outside of the logarithm. As m_t = y_tv/√(2) the limit m_t →∞ must correspond to v →∞, y_t →∞, or both. The former limit is interesting, as the corrections given Eq. <ref> vanish if m_t/m_b → 1 as v →∞. Then SU_L(2) can again be linearly realized. The limit y_t ≫ 1 is a strong coupling limit, leading to a breakdown of perturbation theory. Then the decoupling theorem's implicit assumption of a valid perturbation theory no longer holds. In the case of the ρ parameter, the non-decoupling effects come about due to this strong coupling limit in addition to the differences in the realization of the symmetries of the full theory and the low energy EFT.[In the case of the ρ parameter the corrections shown are also the leading violations of custodial symmetry, as an additional subtlety.]§.§.§ Weak interactionsWhen exact symmetries are present in a subset of interactions in the EFT, such symmetries can first be broken explicitly by the heavy fields integrated out. Then the leading operator mediating a process can be due to the local contact operator correction to the EFT suppressed by m^2 (in the case of weak interactions a suppression by m_W^2), but with norelative suppression compared to any leading order effect, which is absent. This is another way in which non-decoupling effects can come about.The weak interactions are an important example of this form of non-decoupling. The SM is defined in Section <ref>. Flavour violating effects in the SM due to the weak interactions having an intricate pattern that encodes non-decoupling physics of this form. In the limit that the Yukawa interactions of the SM vanish, Y_u,d,e→ 0 a U(3)^5 global flavour symmetry group of the SM is present. We define this group through the relation between the weak (unprimed) basis and the mass (primed) basis asu_L= 𝒰(u,L)u_L^', u_R= 𝒰(u,R)u_R^',ν_L= 𝒰(ν,L) ν_L^', d_L= 𝒰(d,L)d_L^', d_R= 𝒰(d,R)d_R^', e_L= 𝒰(e,L)e_L^', e_R= 𝒰(e,R)e_R^'.Each 𝒰 rotation defines a U(3) flavour group. The U(3)^5 group of the SM is defined asU(3)^5 = 𝒰(u,R) ×𝒰(d,R) ×𝒰(Q,L)×𝒰(ℓ,L) ×𝒰(e,R).The relative 𝒰 rotations between components of the lepton and quark SU_L(2) doublet fields define the PMNS and CKM matrices asV_ CKM = 𝒰(u,L)^† 𝒰(d,L),U_ PMNS = 𝒰(e,L)^† 𝒰(ν,L).Consistent with the discussion in the previous section, the unbroken U(3)^5 flavour symmetry of the SM forbids divergences corresponding to flavour violating interactions.The 𝒰(u,R) ×𝒰(d,R) ×𝒰(e,R) rotations commute with the weak interaction generators. At tree level the neutral current interactions to the left handed doublet fields (ψ_L) couple to the diagonal generators _iψ_L = y_i( [ 1 0; 0 1 ])ψ_L , τ^3 ψ_L = ( [10;0 -1 ]) ψ_L.which also commutes with the 𝒰(Q,L)×𝒰(ℓ,L) rotations between the weak and mass eigenstates. No tree level flavour changing neutral currents follows.This symmetry is broken when the charged currents that interact in the weak eigenbasis of the SM propagate, and quark mass differences are retained in the resulting amplitudes. This distinguishes the mass and weak eigenstates of the SM. Flavour violating effects come about due to the relative rotation of the SM states in ψ_L proportional to V_ CKM,U_ PMNS, that distinguishes the components of the ψ_L doublets, and appear in interactions proportional to τ^1,2 through which the charged currents couple. This leads to flavour changing charged currents at tree level in the SM. Nevertheless, if all weak or mass eigenstates are summed over, and flavour violating spurions are neglected, the flavour symmetry is again restored due to the rotation between the eigenbases being unitary.Consider the diagrams in Fig. <ref> as an illustrative example. The leftmost diagram gives a contribution to the decay K_L →μ^+μ^-.[The Kaon mesons are defined by their quark content as K^0 = ds̅, K̅^̅0̅ = d̅s and K_L = (ds̅ - s d̅)/√(2), K_S = (ds̅ + s d̅)/√(2).] Constructing the EFT useful for the measurement scale μ^2 ≃ m_K^2, both the top quarkand the W boson are not on-shell fields propagating for longer distances (compared to the measurement scale), and hence integrated out. The EFT so constructed is the LEFT, briefly introduced in Section <ref>.The leading operator mediating the decay is given by <cit.>ℒ_K_L →μ^+μ^- = Ĝ_F/√(2)/2π^2V_ts^⋆V_td Y(x_t)(s̅ γ^μ P_L d ) (μ̅ γ^μ P_Lμ) + h.c + ⋯ Here x_t = m_t^2/m_W^2 and retained is the contribution from the top quark in the loop that breaks flavour symmetry. We neglect higher order (and penguin diagram) contributions to the decay. See Refs. <cit.> for more discussion on such corrections.The “non-decoupling" effects are indicated by the presence of polynomial powers of m_t^2 in the numerator, similar to the case of the ρ parameter. Again, integrating out the top quark will lead to a non-linearly realized SU_L(2) symmetry, but the situation is different than in the case of the ρ parameter, where the top mass scale also regulates the integral. Fig. <ref> (left) is a naively convergent integral (in an appropriately chosen gauge). The m_t,c,udependence as a polynomial mass contribution outside of a logarithm follows from the flavour breaking pattern of the SM matched onto the lower scale EFT.The result reflects the Glashow-Iliopoulos-Maiani <cit.> (GIM) mechanism describing the relevant phenomenological suppression by powers of the quark masses in addition to the weak couplings as a result of the U(3)^5symmetry breaking pattern of the SM. The GIM mechanism is also a statement that in the limit of vanishing quark masses, Eq. <ref> (and similar flavour changing amplitudes for other processes) exactly vanish as due to unitarity ∑_iV^⋆_isV_id = 0. One loop contributions to Kaon mixing also respect the GIM mechanism, and include the contributions shown in Fig. <ref> (right two diagrams). Reproducing the SM amplitudes following from the weak interactions at low energies in an EFT is non-trivial. A proper treatment summing all logarithms again uses the LEFT by integrating out a series of SM fields {h,W,Z,t,c} in sequence, in the renormalization group evolution down to the experimental scale of K^0 -K̅^̅0̅ mixing. For a discussion on the reproduction of the SM result in the LEFT see Ref. <cit.>.The sum of these graphs (combined with Goldstone boson diagrams for gauge independence) gives the leading result <cit.>ℒ_K^0 -K̅^̅0̅ = Ĝ_F/√(2)/4π^2 ∑_i V_is^⋆V_id ∑_j V_jsV_jd^⋆ E̅(x_i,x_j)(s̅ γ^μ P_L d ) (d̅ γ^μ P_L s ) + h.c + ⋯ where E̅(x_i,x_j)=- x_i x_j(1/x_i - x_j[1/4 - 3/21/x_i-1 - 3/41/(x_i-1)^2]log x_i, .+ .1/x_j - x_i[1/4 - 3/21/x_j-1 - 3/41/(x_j-1)^2]log x_j - 3/41/(x_i-1)(x_j-1))Again x_i,j = m_i,j^2/m_W^2 and these indices sum over up quark flavours. We neglect here higher order corrections, see Refs. <cit.> for further discussion. The “non-decoupling" flavour breaking structure of the SM interactions is present in that the result is proportional to four powers of quark masses in Eq. <ref>.The many instances of non-decoupling effects in the SM should generate caution when choosing between the SMEFT and HEFT formalisms to capture the low energy limit of physics beyond the SM. The possibility of non-decoupling effects related to the discovered 0^+ boson with mass ∼ 125 GeV is pressing and well motivated. One of the key distinctions between the SMEFT and HEFT constructions is the latter is arguably more appropriate to capture the IR limit of such non-decoupling UV physics coupled to the 0^+ boson.§.§.§ Renormalization scheme dependence and decouplingA renormalization scheme is composed of a method to regulate divergent integrals, and a subtraction scheme choice. When relationships between physically measured S matrix elements are determined in perturbation theory, the regularization and subtraction scheme choice has no physical effect. When discussing renormalized Lagrangian parameters per se and intermediate results for observables in terms of these parameters, scheme dependence is present. Ref. <cit.> used a renormalization scheme which performs subtractions at an off-shell Euclidean momentum point. This approach to renormalization has the benefit of making decoupling manifest, which is not the case in dimensional regularization (DR) when MS is used as a subtraction scheme.Consider the Lagrangian for quantum electrodynamics (QED), and the running of the QED coupling e due to fermions ψ_f. The Lagrangian is given by ℒ^0_QED = -1/4F^μ ν_0F_μ ν^0 + ψ̅_f^0γ_μ (i∂^μ - e_0 Q_f A^μ_0) ψ_f^0 - m_f^0 ψ̅_f^0 ψ_f^0,with F^μ ν_0 =∂^μA^ν_0-∂^νA^μ_0 the QED field strength tensor composed of bare fields, indicated with 0 labels. The one loop diagram shown in Fig. <ref> (left) givesi𝒜 = - e_0^2 Q^2μ^4-d ∫d^dq/(2 π)^d Tr(γ^μ (q+ p + m_f)γ^ν (q+ m_f))/((q+p)^2- m_f^2)(q^2-m_f^2),which is divergent. The bare fields are related to the renormalized fields (denoted with r labels) by introducing the renormalization constants Z_i A^ν_0= √(Z_A) A^ν_r,ψ_f^0= √(Z_ψ_f)ψ_f^r, e^0= Z_e μ^ϵ e^r, m^0_f= Z_m_fm^r_f. μ^ϵ is introduced as dimensional regularization with d= 4 - 2 ϵ is used to regulate the divergent integrals. The counterterm that performs the subtraction for Π_AA(p^2) is indicated inFig. <ref>(right) and gives-i/4Z_A (p^μp^ν - p^2 g^μ ν).Choosing the manner in which the divergence is subtracted fixes Z_A and defines the subtraction scheme. Defining a renormalization condition for Π_AA(p^2) where the diagram is subtracted at the Euclidean momentum point p^2 = - M_f^2 removes the divergence, and gives the sum for Fig. <ref> i𝒜_M_f = - ie_0^2 Q^2μ^2 ϵ/2π^2(p^μp^ν - p^2 g^μ ν) ∫_0^1 dx logm_f^2 - p^2 x (1-x)/m_f^2 + M_f^2 x (1-x).Alternatively, the MS scheme subtracts the ϵ poles and a set of constant terms due to the rescaling μ→μ (e^γ/4 π)^1/2. Using MS the sum for Fig. <ref> is given byi𝒜_MS = - ie_0^2 Q^2/2π^2(p^μp^ν - p^2 g^μ ν) ∫_0^1 dx logm_f^2 - p^2 x (1-x)/μ^2.In either case, the Ward identities of the theory due to unbroken U(1)_em fixZ_e = 1/√(Z_A),to order e^2_0. The bare coupling is independent of the renormalization scheme choice so expanding the derivative of the bare coupling with respect to M_f gives β(e) = e_0^3 Q^2/2π^2 ∫_0^1 dxM_f^2 x^2 (1-x)^2/m_f^2 + M_f^2 x (1-x).The running of β(e) calculated in this manner exhibits manifest decoupling. For m_f < M_f one finds β(e) ≃e_0^3 Q^2/12π^2,while for M_f < m_f one has β(e) ≃e_0^3 Q^2/60π^2 M_f^2/m_f^2.When renormalizing the theory for measurements made at scales ∼ - M^2_f < m^2_f, decoupling of the effects of the heavy fermion is manifest. For MS one finds the result in Eq. <ref> for all μ. In this case, to implement decoupling appropriately one must set β(e) ≃ 0 for μ≲ m_f by hand.[This point holds for β functions, and also for other theoretical quantities such as the effective potential <cit.>.]The effects of heavy particles contributing to an experimental measurement that do not propagate on shell are still encoded in local contact operators in the EFT, no matter what subtraction scheme is chosen. Connecting back to the initial regularization result in Eq. <ref>, large logarithms can be present expanding this equation when using DR and MS, if μ^2 ≪ m_f^2 (or μ^2 ≪Δ^2 in the notation of Section <ref>), indicating the regularization of an amplitude. A poorly behaved perturbative expansion results if decoupling is not imposed by hand in this scheme.Considering the requirement of modifying the beta functions by hand, it could be surprising that using MS and DR is strongly preferred in modern EFT calculations. This renormalization scheme makes the power counting of the EFT manifest and directly preserved in loop calculations. This is an important technical simplification that overwhelms the drawback of having to impose decoupling by hand. See Section <ref> for further discussion on this point. §.§ MatchingAn EFTs dynamics is defined without the need to extensively reference the details of any UV completion. This is fortunate, as the EFT and the UV completion are quite different. They do not have the same high energy behavior and each theory is renormalized separately, with a different set of counterterms.An EFT is a self-consistent field theory capable of predicting S matrix elements for a range of energies where the expansion leading to the EFT is convergent. At times an EFT can be constructed to faithfully reproduce the predictions of a UV completion in a low energy limit. As the correspondence between the EFT and the UV theory is limited, when this is done, the theories must be matched to ensure the predictions agree. This procedure fixes the free parameters (the Wilson coefficients) of the tower of higher dimensional operators that make up ℒ_EFT. When the UV completion is weakly coupled, the matching procedure can be directly carried out in perturbation theory. To fix a set of n free parameters in the EFT, a set of n linearly independent S matrix elements[We define the S matrix precisely below in Section <ref>.] are calculated in both the EFT and in the UV completion. The results are equated in the IR limit that defines the EFT. Denote this limit as p^2 ≪ M^2 with M^2 some heavy mass scale of a state in the UV completion and not in the EFT. Fixing ⟨ p_1 ⋯ p_a|S_1..n|k_1 ⋯ k_b ⟩^ UV_p^2 ≪ M^2≡⟨ p_1 ⋯ p_a|S_1..n|k_1 ⋯ k_b ⟩^ EFT,so defines the matching conditions that fixes the Wilson coefficients in terms of the parameters of the UV completion, and the couplings of the perturbative expansion. This procedure works at tree level, and order by order in perturbation theory where both UV and IR divergences can occur. The divergences can be neglected in practice and the matching condition is defined by the finite parts of Eq. <ref>. This follows from the UV divergences being subtracted by the corresponding counterterms on each side of Eq. <ref>. The IR divergences correspond to the case μ^2 ≫Δ^2 + q_E^2 in Section <ref>. These divergences are also regulated in dimensional regularization but a key defining condition of the EFT construction is that the IR physics of the EFT and the UV completion is the same. Matching fixes the Wilson coefficients to reflect the short distance UV physics integrated out of the EFT. The reason is simple and intuitive, the IR physics that is not modified in transitioning from the UV theory to an EFT description cancels in the matching. This includes the IR divergences themselves. The correction that remains to match onto the Wilson coefficients is then only due to the UV physics integrated out of the EFT. When the matching procedure is carried out in DR and MS, the following technical simplifications also occur:* Scaleless integrals vanish and the IR and UV divergences in such integrals cancel. A simple example of this is given by the wavefunction renormalization factor of a massless fermion in QCD with the ψ two point function having divergences α_s C_F/4πi p[1/ϵ_IR - 1/ϵ_UV].This can simplify matching calculations dramatically, as diagrams that are scaleless in the EFT can be neglected when using DR. * In calculating one loop matchings, expanding intermediate results in small ϵ, or dimensionless ratios, before Feynman parameter integrals are carried out, can be justified so long as the modifications in the results cancel in the matching condition. * Quadratic divergences are represented as ϵ poles. Dimensionful threshold corrections in the matching conditions occur at tree level (in the EFT), and also as one loop running corrections to EFT parameters <cit.>. * Usually the matching conditions are evaluated at the scale of particles integrated out of the theory to define the EFT. This is not required, but is advantageous as it acts to minimize potentially large logs in the perturbatively expanded matching equations, when such matching is combined with Renormalization Group Evolution (RGE) running.§.§.§ Matching examples “Integrating out a field" as nomenclature follows from the path integral approach to defining an effective action as developed by Wilson <cit.>. The effective action S_ eff[ϕ]retaining a light field ϕ, and removing the heavy field Φ is defined such thate^iS_ eff[ϕ] =∫ dΦe^iS[ϕ,Φ]/∫ dΦe^iS[ϕ,0] ,where the integral is over all field values Φ. Hence the heavy field is “integrated out". At times this procedure can be carried out formally in the path integral while using the Equations of Motion (EOM) for the theory. Such manipulations can require the interactions to be of a limited form to be formally justified.Integrating out a heavy field and determining the matching conditions for the Wilson coefficients at tree level can always be done using Feynman diagram techniques directly. Again the EOM are used and this approach can be easier in some cases than directly determining S matrix elements in the full and effective theories as in Eq. <ref>, and solving the resulting system of equations.As a set of examples, consider the case of a heavy SM singlet scalar field (S) and a heavy SM singlet (Weyl) fermion (N). The Lagrangian for the former case for d ≤ 4 interactions is ℒ_SM+S = ℒ_SM + 1/2((∂_μ S)(∂^μ S) - m_S^2 S^2 ) - κ_1/2S^2 H^† H - Λ_1 S H^†H - Λ_2 S^3 - κ_2 S^4.The SM is defined in Section <ref>, κ_1,2 are dimensionless couplings while Λ_1,2 have mass dimension one. For field values ⟨ H^†H ⟩ < ⟨ S^2 ⟩ < m_S^2 and p^2 < m_S^2 one can solve the equation of motion for S and Taylor expand around the classical solution findingS ≃ - Λ_1 H^† H/m_S^2 + ⋯,which substituted back into the initial Lagrangian gives ℒ_SMEFT = ℒ_SM + Λ_1^2/2 m_S^2(H^†H)^2 + ℒ^(6) + ⋯.The leading correction term shown can be absorbed into a finite shift of the SM Higgs self coupling, consistent with the decoupling theorem. The terms in ℒ_6 are given by ℒ^(6) = - Λ_1^2/m_S^4𝒬_H+ (Λ_2Λ_1/m_S^2 -κ_1/2)Λ_1^2/m_S^4 𝒬_Husing the Warsaw basis for ℒ_6.[See Section <ref> for details on basis choice and ℒ_6.] The matching contributions are organized due to the IR operator forms that result, not naive scalings in m_S. The coefficients of terms in ℒ_6 are expected to be overall ∝ 1/m_S^2. The naive expectation of the ordering of the operator forms in powers of m_S is generically upset due to the presence of dimensionful couplings, as purposefully illustrated here. Naturalness considerations imply that Λ_1,2^2 ≲ m_S^2 16π^2. When this bound is saturated, large Wilson coefficients result. Even when the bound is not saturated, the presence of such dimensionful couplings can lead to 𝒪(1) Wilson coefficients(× 1/m_S^2). This is a particular concern when considering matching to strongly interacting UV physics sectors, below the scale present in the confining phase of such a sector. Such dimensionful couplings can then be present in the interactions of the resulting bound states, unless forbidden by a symmetry, and can directly upset any intuition based on perturbative matching in a UV coupling g^⋆ and taking a limit g^⋆→ 4π.[See Section <ref> for more discussion on such an approach.]As another example, consider the case of a SM singlet Weyl fermion integrated out in the UV. This scenario corresponds to the Seesaw model <cit.> for generating massive Neutrino's, and has been studied in an EFT context in Refs.<cit.>. The Lagrangian can be defined as ℒ_SM + ℒ_N_p where2ℒ_N_p=N_p (i∂ - m_p)N_p - ℓ_L^βH̃ω^p,†_βN_p -ℓ_L^c βH̃^*ω^p,T_β N_p - N_p ω^p,*_βH̃^T ℓ_L^c β- N_p ω^p_βH̃^†ℓ_L^β. The couplings ω^p_β = {x_β,y_β,z_β} are complex vectors in flavour space that absorbed the Majorana phases. p={1,2,3} is summed over. Integrating out the N_p at tree level by taking the p^2 < m_p^2 limit of the tree level exchange diagram gives the result ℒ_SMEFT = ℒ_SM + ℒ^(5) + ⋯ where ℒ^(5) = c_β κ/2 (ℓ^c, β_L H̃^⋆) (H̃^† ℓ_L^κ)+ h.c.and c_β κ =(ω^p_β)^Tω^p_κ/m_p. The matching is onto the leading correction to the SM dimension four Lagrangian <cit.>. The notation used here is that the c superscript in Eq. <ref> corresponds to a charge conjugated Dirac four component spinor defined as ψ^c= C ψ^T with C= - i γ_2γ_0 in the chiral basis. ℓ_L^c denotes the doublet lepton field that is chirally projected and subsequently charge conjugated. See Ref. <cit.> for further notational details. Expanding the result around the vacuum expectation value for the Higgs field gives experimentally required Neutrino masses.Calculating higher order corrections to the matching results can also be directly determined using standard Feynman diagram techniques. Conversely, a naive path integral approach to integrating out a field can be practically limited to leading order calculations. Determining higher order perturbative matching corrections is straightforward in the case of SM singlet fields in a UV sector. The required loop corrections are only present on the left or the right hand side of Eq. <ref> due to the different symmetry groups of the SM and the UV sector in this case. Alternatively, when UV field content integrated out is charged under the SM gauge groups, loop corrections in the EFT and in the full theory (on both sides of Eq. <ref>) are required at each order in perturbation theory. The S matrix elements are calculated to higher orders in perturbation theory in the full theory and the EFT. UV divergences are canceled by counterterms dictated by the subtraction and regularization scheme chosen. IR divergences and constant terms cancel in the matching calculations, and the Wilson coefficients are then determined to the desired order by the UV physics removed from the EFT. Higher order terms in the operator expansion (which is usually referred to as the non-perturbative expansion of the EFT in the literature) can also be determined using these techniques, and mixed perturbative and non-perturbative contributions. For a sample of excellent examples of matching see Ref. <cit.>.§.§.§ Covariant Derivative Expansion matchingMatching typically requires the computation of a large number of diagrams in the full theory and the EFT, which is done choosing a convenient gauge, and subsequently recombining the results into gauge invariant effective operators. This can be cumbersome when determining matching calculations to higher orders in the expansions present in the EFT. A recently developed technique, that goes under the name of the Covariant Derivative Expansion (CDE), is aimed at simplifying this computation by resorting to more advanced functional methods. This technique has two main advantages: it does not require the evaluation of Feynman diagrams because the matching is done at the action level and, at the same time, it seeks to preserve manifest gauge invariance at all the stages of the calculation. The CDE method has been introduced in the modern EFT context in Ref. <cit.> reviving an approach previously explored in the 80's <cit.> for other applications. In the following we summarize the main argument of <cit.>. At the action level, the matching of a theory containing both heavy fields Φ and light fields ϕ onto an EFT thatdescribes only the ϕ degrees of freedom again amounts to constructing an effective action S_ eff[ϕ] such thate^iS_ eff[ϕ] =∫ dΦe^iS[ϕ,Φ]/∫ dΦe^iS[ϕ,0] .Using a saddle-point approximation, which is valid for a perturbative expansion up to one loop, and expanding the heavy fields around their background values Φ=Φ_c+η one obtainsS_ eff.[ϕ] ≃ S(Φ_c) + i/2log(-.^̣2 S/Φ̣^2|_Φ_c),where the first term contains the structures obtained integrating out the heavy field in tree level diagrams, while the second encodes the contributions generated at one-loop. Eq. <ref> can be evaluated explicitly assuming a generic (universal) structure for the Lagrangian of the UV model. For example, if the heavy field is a complex scalar, one hasℒ_UV⊇ -Φ^†(D^2+M^2+U(x))Φ + (Φ^† B(x)+)+𝒪(Φ^3)where D_μ = _μ-i A_μ is a covariant derivative and U(x), B(x) are arbitrary model-dependent expressions containing light fields. The tree-level matching contribution S(Φ_c) is derived in a standard fashion replacing Φ→Φ_c in ℒ_UV, where Φ_c is the solution of the EOM for Φ, and expanding the resulting Lagrangian in inverse powers of M. The final result isΔℒ_ eff, tree =-B^†[-D^2-M^2-U]^-1B+𝒪(Φ_c^3)≃ B^† M^-2 B + B^† M^-2[-D^2-U] M^-2 B + …where we have dropped the x dependence. Note that the field Φ is generally a multiplet, so that the mass term M is a matrix, which does not necessarily commute with (D^2+U).The evaluation of the one loop piece is slightly more involved. The most general result can be found in Ref. <cit.> together with a detailed derivation. The final expression obtained expanding up to dimension 6 is Δ ℒ_eff,1-loop =c_s/(4π)^2{+M^4[-1/2(logM^2/μ^2 -3/2) ] +M^2[-(logM^2/μ^2 - 1) U] +M^0[-1/12(logM^2/μ^2 - 1) G_μν'^2 - 1/2logM^2/μ^2U^2 ]+ 1/M^2[ 1/60 (D_μG_μν')^2 - 1/90G_μν'G_ν'G_μ' +1/12(D_μU)^2 - 1/6U^3 - 1/12U G_μν'G_μν' ] +1/M^4[1/24U^4 - 1/12U (D_μU)^2 + 1/120 (D^2U)^2 +1/24 ( U^2 G'_μνG'_μν) + 1/120 [(D_μU),(D_νU)] G'_μν - 1/120 [U[U,G'_μν]] G'_μν] +1/M^6[-1/60U^5 + 1/20U^2(D_μU)^2 + 1/30 (UD_μU)^2 ] + 1/M^8[ 1/120U^6 ]} .Here c_s={1/2,1} for a real and complex scalar Φ respectively and G'_μν=[D_μ,D_ν]. Finally, the trace is over internal indices, i.e. Lorentz, flavour, gauge indices etc. Inserting into Eq. <ref> the expressions of U and G_μν defined in a specific model, one immediately obtains a sum of dimension six operators whose coefficients are automatically matched with the UV model. Eq. <ref> is universal in the sense that it can be applied not only to the case of scalar Φ but also when the heavy field is a fermion or vector boson, as detailed in Ref. <cit.>. Although extremely practical, this expression has two main defects:* It holds only in the case of degenerate heavy states, in which the mass matrix M is diagonal and commutes with the other structures; * The Δℒ_eff,1-loop computed in this way accounts only for loop diagrams in which all the internal lines are heavy. Mixed heavy-light loops are missing because the light field have been treated as background fields and therefore only enter as external lines <cit.>. Point 1 was addressed in Ref. <cit.>, that generalized Eq. <ref> to the case of a non-degenerate multiplet Φ obtaining an expression for the effective action at one loop that was named the UOLEA (Universal One Loop Effective Action). Point 2 represents a deeper problem in the basic CDE technique described above, which requires a modification of the functional treatment. Different solutions have been proposed in Refs. <cit.>. Both Ref. <cit.> and Ref. <cit.> expand the functional analysis of Ref. <cit.> with the inclusion of the fluctuations around the background fields for the light degrees of freedom ϕ and both suggest a method that requires the subtraction of non-local terms from the functional determinant. This step is avoided with the alternative method proposed in Refs. <cit.>, that builds upon Refs. <cit.> and employs the “expansion-by-regions” technique for the evaluation of the loop integrals <cit.>. One defines the multiplet φ=(Φ,ϕ) andΔℒ_eff,1-loop = 1/2φ^†.^̣2 ℒ_UV/φ̣^* φ̣|_φ_c =φ^†[Δ_H X_HL^†; X_HLΔ_L ]φ,where the last term contains a block matrix so that the heavy fields Φ are contracted by Δ_H, the light ones by Δ_L and X_HL is a mixed term. The key idea of Ref. <cit.> is to perform a field transformation that brings the matrix to a block diagonal form diag(Δ̃_H,Δ_L), shifting the effect of X_HL into the heavy-particle contribution while leaving the Δ_L unchanged. This procedure gives Δ̃_H = Δ_H-X_HL^†Δ_L^-1 X_HL which, in the scalar case, can be expressed in the notation of Ref. <cit.> asΔ̃_H = -(D^2+M^2+Ũ), Ũ = U(x)+U_HL(x,p) ,where U_HL comes from the field redefinition and carries a dependence on the loop momentum p. The effective action takes the form S_ eff[ϕ] = i c_s logΔ̃_H and it generates all the loop diagrams with at least one heavy internal propagator. The desired result for the mixed heavy-light one loop contributions to the EFT matching are obtained performing the loop momentum integrals in Δ̃_H only in the “hard” region, i.e. first expanding out all the low-energy scales, that are small in the limit p∼ M, and then integrating over the full d-dimensional p space. This method makes use of dimensional regularization and is known as “expansion-by-regions” <cit.>discussed in more detail in Section <ref>. As a result, the contribution to the dimension six effective Lagrangian in Eq. <ref> is extended by the inclusion of <cit.>:Δℒ_eff,1-loop^HL =-ic_s ∫d^dp/(2π)^d{ 1/p^2-M^2 _ s(Ũ)+1/21/(p^2-M^2)^2 _ s(Ũ^2) +1/31/(p^2-M^2)^3 [_ s(Ũ^3)+_ s(ŨD^2Ũ)+2ip^μ _ s(ŨD_μŨ)] +1/41/(p^2-M^2)^4 [_ s(Ũ^4)+2ip^μ _ s(Ũ^2 D_μŨ)+2ip^μ _ s(Ũ D_μŨ^2) + _ s(Ũ^2 D^2 Ũ)+ _ s(Ũ D^2 Ũ^2)-4 p^μ p^ν _ s(Ũ D_μ D_νŨ)+2ip^μ _ s(Ũ D^2D_μŨ)+2ip^μ _ s(Ũ D_μ D^2 Ũ) +_ s(Ũ (D^2)^2 Ũ)] +1/51/(p^2-M^2)^5 [_ s(Ũ^5)+2ip^μ _ s(Ũ^3D_μŨ)+2ip^μ _ s(Ũ^2D_μŨ^2)+2ip^μ _ s(ŨD_μŨ^3)-4p^μ p^ν _ s(Ũ^2 D_μ D_νŨ)-4p^μ p^ν _ s(Ũ D_μŨ D_νŨ)-4p^μ p^ν _ s(Ũ D_μ D_νŨ^2)-8i p^μ p^ν p^ρ _ s(Ũ D_μ D_ν D_ρŨ}]} +ℒ_ EFT^F+𝒪(M^-3) .Here _ s is a subtracted trace defined as_ sf(Ũ,D_μ)≡(f(Ũ,D_μ)-f(U,D_μ)-Θ_f) ,where f is an arbitrary function of Ũ and covariant derivatives, while Θ_f generically denotes all the terms with covariant derivatives at the rightmost of the original trace. The subtraction of the terms containing only U avoids a double counting, as these contributions are already contained in Eq. <ref>. Finally, ℒ_ EFT^F contains all the terms containing open derivatives, that eventually combine into operators with field-strength tensors. The evaluation of this piece truncated at dimension-six terms is quite complex: a universal expression, although achievable in principle, is not yet available to date.Although not definitive, the results presented in Ref. <cit.> represented a key step in the development of CDE techniques. In particular, they highlighted that the heavy-light structures could be directly inferred from the heavy-only ones and they paved the way for the development of a covariant diagrammatic formalism <cit.>. The latter allows an immediate and efficient evaluation of functional quantities in terms of gauge-invariant operators, while providing a graphical representation that keeps track of the CDE expansion[Analogously, Feynman diagrams allow to compute correlation functions and make manifest the organization of different contributions in a perturbative expansion.]. This powerful formalism has been adopted recently in Ref. <cit.> to compute explicitly all the universal terms (operators and coefficients) in the UOLEA functional in <cit.>, with the addition of the heavy-light ones and for both the degenerate and non-degenerate cases. These results extend the construction in Ref. <cit.> and supersede those of Ref. <cit.>, providing universal expressions for CDE matching that can be applied to a given model without the need of reiterating the functional procedure.As mentioned above, the evaluation of open derivative terms is still missing at this stage, but it is expected to become available in the near future, again thanks to covariant diagrams techniques. The same holds for mixed bosonic-fermionic loops. The calculation of these two categories of effects will complete the “UOLEA program” to supply a completely universal, gauge-invariant result for EFT matching.The power of the CDE approach has recently been illustrated in many examples in the literature. A stand out example is the demonstration of how the known one loop matching MSSM results of Ref. <cit.> can be elegantly determined using the modern CDE approach <cit.>.§.§.§ Method of regionsSince the first version of this review was written, a rather comprehensive tree level matching dictionary for integrating out UV field content (when the underlying theory is perturbative) has been reported in Ref. <cit.>. This set of matching results is for the SMEFT up to ℒ^(6).Building further on this result is theoretically well motivated. The power of EFT is most apparent when going beyond leading order in the EFT expansion parameter in matching, and in determining perturbative corrections to observable processes. For studies to advance beyond leading order in perturbation theory consistentlyrequires that matching calculations be performed beyond leading order. Recent advances to this end include the one loop CDE matching results discussed in the previous section, where the technique of “expansion-by-regions” (or method of regions) was also used. In this section we summarize some of the physics underlyingthis latter approach.The method of regions <cit.> is an elegant way to determine matching coefficients. The utility of the method of regions relies on the point that non-analytic structure of a full theory is projected out in matching. Using this fact actively simplifies a matching calculation. This simplification can be important to enable the determination of a matching coefficient at one loop, or higher orders in the EFT expansion. The method of regions allows one to evaluate Eqn. <ref> without performing the full theory calculation first with all mass scales retained. This can be done by expanding a loop result in the EFT expansionbefore integrating.Consider matching the seesaw model given in Eqn. <ref> at one loop to the SMEFT. A one loop calculation to determine the matching to the Higgs two point function is definedby expanding a result in the ratio of scales v_T^2/m_r^2 <1, where m_r is the mass of the N_r Majorana particle. The mass of the charged lepton field is denoted m_ℓ and we retain both mass scales for illustrative purposes initially. The only diagram contributing is drawn in Fig. <ref> and it givesi Π^full _H H^†(p^2) = - 2 |ω_r^2|∫d^4 ℓ/(2 π)^4∫_0^1 dx ℓ^2 + x (x-1)p^2/[ℓ^2 - Δ]^2,= - i |ω_r^2|/16π^2ϵ (2 m_ℓ^2 + 2 m_r^2 - p^2)- i |ω_r^2|/48 π^2 (3 m_ℓ^2 + 3 m_r^2 - p^2), + i|ω_r^2|/8π^2∫_0^1 dx (2m_r^2 (x-1) - x (2m_ℓ^2 + 3 p^2 (x-1)) log[μ^2/m_ℓ^2 x + (x-1) (p^2 x - m_r^2)] ,where Δ = m_ℓ^2 x + (x-1) (p^2 x - m_r^2).Recall that the UV theory is renormalized directly with its own set of counterterms. This effectively removes the first term in the expression. The second term in the expression leads to a threshold matching correction.This physics is the origin of what is called the Hierarchy problem, and we will return to this physics in the following sections. The difficulties of loop calculations with multiple scales are illustrated in the last term of this expression. At higher loop orders, this complication grows and becomes a serious technical hurdle. On the other hand, expanding in ratios of the scales present in the problem simplifies the result directly. This expansion can be done when matching onto the EFT even before integrating is the key point.First consider the dependence on the scale m_ℓ in this last term. This is an IR scale that is present in the SMEFT and in the full theory. The full theory in this case is ℒ_SM + ℒ_Nand has non-analytic dependence on this scale in predicted S matrix elements. This occurs when the intermediate ℓ state goes on-shell. This non-analytic behavior is common to the EFT and the UV theory and so cancels out in the matching. By definition, the EFT (in this case the SMEFT) reproduces the IR of the full theory and this statement holds at arbitrary orders in perturbation theory, including for the non-analytic behavior of the full theory loop diagrams dependence on the scale m_ℓ. For this reason one can expand in m_ℓ/(p,m_r) before integrating and neglect m_ℓ in determining the matching coefficient.Now, consider expanding in the remaining scales of the problem, p^2 and m_r^2. The propagator of the heavy Majorana field can be expanded as 1/k^2 - m_r^2 = - 1/m_r^2[ 1+ k^2/m_r^2 +⋯]where k^2 is the loop momentum when considering the loop integral in the SMEFT for this matrix element. Using this result one can simplify the expression for the two point function in the SMEFT toi Π^EFT _H H^†(p^2) = |ω_r^2|/2 m_r^2∫d^4 k/(2 π)^4k^2/k^2 + ⋯.Here the loop momentum has been shifted after the propagator expansion in Eqn. <ref> has been performed. All of the terms that are present are scaleless integrals that vanish in dimensional regularization when all IR scales are expanded out. For this reason, the matching result in Eqn. <ref> can be directly determinedby performing the simpler calculation in the full theory, where all IR mass scales such as m_ℓ was expanded out from the start, and the one loop contribution from the SMEFT in the matching is dropped. One only needs to perform the much simpler loop integral that remains, and expand in p^2/m_r^2 the result. This determines the matching coefficient. For the two point function one finds the contributioni Π^full,exp _H H^†(p^2) = -i |ω_r^2| m_r^2/8 π^2(1 + logμ^2/m_r^2)+i |ω_r^2| p^2/32 π^2(1 + 2logμ^2/m_r^2).A finite field redefinition is used to cancel the last term in this expression, and the final matching result to the Higgs two point function is the first term.This procedure illustrates the utility of expanding in the scales of the problem before integrating when performing matching calculations. The wide use of the method of regions is essentially due to it using this physics systematically. For more discussion, see Refs. <cit.>. §.§ Choose any scheme, so long as it is dimensional regularization and MSMSIn matching calculations, divergences can be dropped as the UV divergences in each theory are canceled by UV counterterms, and the IR divergences in the full theory and the EFT coincide by definition. If such divergences are retained in the calculation, they must be regulated. Any physical conclusion is independent of a regulator choice, in the limit the regulator is taken to infinity.[For a recent discussion on this point in more detail see Ref.<cit.>.] Nevertheless, the ease of obtaining physical conclusions, and performing loop calculations depends on the regulator choice. The discussions in the previous sections were made usingDR. Using a dimensionless regulator is now standard in most EFT studies, and can be essential in EFT studies of the Higgs boson. The reason for this is that the SM is classically scaleless in the limit v → 0 (m_h → 0 as a result) and a dimensionful regulator explicitly breaks this (anomalous) symmetry by introducing a dimensionful cut off scale.It took decades for the benefits of DR to be fully appreciated in the EFT community. This was due to some history and some misunderstanding. The history is due to the key initial ideas of the RG emerging into broad use[A precursor of these ideas were presented by Stückelberg and Petermann in a prescient work <cit.>.] following the pioneering condensed matter studies of Kadanoff <cit.>, Wilson <cit.> and Wilson and Fisher <cit.>. The partition function in the early cases studied using the RG was constructed out of a reduced number of degrees of freedom by integrating out a series of high energy modes asZ = ∫^Λ_1𝒟 ϕ^1_i e^- S_1(ϕ^2_i) = ∫^Λ_2𝒟 ϕ^2_i e^- S_2(ϕ^2_i)⋯ = ∫^Λ_n𝒟 ϕ^n_i e^- S_n(ϕ^n_i),where Λ^n > Λ^n-1. At each step in integrating out a momentum shell, the action is matched by fixing its parameters and redefining the fields, to reproduce the low energy physics of the action with the higher energy modes removed. The differential version of this procedure gives the RG Equations which are local and analytic, as can be understood from the general arguments of the previous sections. That the initial applications <cit.> of the RG was formulated with a dimensionful regulator makes perfect sense as the modes integrated out were discretizing physically separated spin configurations, with a corresponding dimensionful Fourier transformed momentum. The misunderstanding that slowed the adoption of DR was the perception that as it integrates over all momenta, it does not faithfully limit the EFT to momenta where it is defined, introducing an inconsistency. This is incorrect, as discussed in Section <ref>.A key step forward in the development of EFTs was the realization that such a dimensionful regularization is not necessary, and best avoided. The main reasons for this are that such regulators make power counting suspect, and calculations technically challenging. For example, the method of regions approach in the previous section relies on using dimensional regularization and the fact that scaleless integrals vanish when using this regulator. §.§.§ ZZ decay and dimensionful regulatorsThe problems of dimensionful regulators are well illustrated by the example of four fermion operators generating the decay Z →ℓ ℓ at one loop. The results dependent on y_t are known <cit.> in the U(3)^5 limit. The Effective Lagrangian generated has the terms ℒ_Z,eff = - 2 2^1/4 √(Ĝ_F) m̂_Zℓ̅_sγ_μ[(g̅^ℓ_L)_ssP_L +(g̅^ℓ_R)_ssP_R ] ℓ_s,where at one loop the four fermion operators give a correction <cit.>Δ (g^ℓ_L)_ss = N_cm̂_t^2/16π^2Λ^2 log[μ^2/m̂_t^2][C_ℓ qss33^(1) - C_ℓ qss33^(3) - C_ℓ uss33],Δ(g^ℓ_R)_ss = N_cm̂_t^2/16π^2Λ^2 log[μ^2/m̂_t^2],[-C_euss33+ C_q e 33ss].Here we are using the notation of Ref. <cit.> where hat superscripts correspond to measured quantities at tree level. These results come about in DR and MS in the following manner. The anomalous dimensions required were determined in Ref. <cit.> using DR and exploiting the Background Field method <cit.>. This method preserves the symmetries of the SMEFT when gauge fixing as fields in the action are split into classical (ϕ) and quantum (Q) componentsS(Q) → S(Q+ ϕ).A gauge fixing term then breaks the gauge invariance of the quantum fields while maintaining the gauge invariance of the classical background fields, this makes gauge independent counterterms easier to determine. The Background Field method can be directly implemented in DR which also preserves the symmetries of the Lagrangian <cit.>. The direct calculation of Fig. <ref> leads to contributions of the formi𝒜 ∝4 N_c (g̅^ℓ_L,SM g̅_L,Q + g̅^ℓ_R,SM g̅_R,Q) ∫d^D l/(2 π)^D(D-2)/D l^2/(l^2 - m̂_t^2)^2,-4 N_c(g̅^ℓ_L,SM g̅_R,Q + g̅^ℓ_R,SM g̅_L,Q)∫d^D l/(2 π)^D m̂_t^2/(l^2 - m̂_t^2)^2.The notation g̅_L/R,Q corresponds to P_L/R in the four fermion operators Q_i inserted in the loop diagrams. In DR and MS the only scale in the loops to make up the dimensions of the 1/Λ^2 suppression is the top quark mass, and the results combine in a non-trivial manner to cancel the pole of Ref. <cit.>, leaving the finite terms in Eq. <ref>. In the case of a dimensionful regulator where D=4 a cut off is introduced to the loop momentum ∼Λ. Generally such regulators violate gauge invariance, as they regulate p^μ and not D^μ. Furthermore, translation invariance of the momenta in the propagators can be broken, which stands in the way of using the Feynman/Schwinger trick to combine propagators in loop calculations. This makes identifying gauge independent counter-terms a challenge and the use of the Background Field method unfeasible. Assuming that the gauge invariant anomalous dimensions were determined using a dimensionful regulator (somehow), the first term in Eq. <ref> gives ∫^Λd^4 l/(2π)^4l^2/(l^2 - m̂_t^2)^2≃Λ^2/16π^2.The Λ^2 dependence acts to cancel the 1/Λ^2 suppression of the operators ing̅_L/R,Q. This results in an 𝒪(1) shift of the amplitude, due to a violation of the power counting.[DR is not without its own challenges, in particular the definition of γ_5 in d dimensions requires a scheme choice. See the recent discussion in Ref. <cit.> on the appearance of this issue in Fig. <ref>.] Dimensionful regulators are a nuisance to be avoided.§.§.§ Avoiding regulator dependenceIn DR, power counting violations due to the regulator choice do not occur in the insertion of higher dimensional operators in loop diagrams. The μ scale introduced in the regulation procedure only appears in the Logs as in Eq. <ref>. Large mass scales that are present in the UV theory can appear outside of Logs, in threshold matching corrections to (H^†H), as shown in Section <ref>. This is the appearance of the Hierarchy problem using DR.The Hierarchy problem is the need to stabilize the scale invariance violating coefficient of (H^†H) against perturbations proportional to scales Λ≫m̂_h. A concrete example of these perturbations was given in Section <ref> where a correction to Π_H H^†∝ |ω_r|^2 m_r^2 was found. Here m_r ≫m̂_h is the heavy mass scale of putative Majorana states leading to Neutrino masses. This result was developed in dimensional regularization. Conversely, using a hard cut off regulator makes it challenging to disentangle the perturbation to(H^†H) due to UV physics from unphysical regulator dependence. Explicit breaking of scale invariance by the regulator is unfortunate, as the mass parameter of the Higgs is the only classical source of the violation of scale invariance in the SM.[See Refs. <cit.> for related discussion on scale invariance and the Hierarchy problem.] This can lead to a different point of view as to what the Hierarchy problem is, and what can solve the Hierarchy problem.A regulator dependent argument can be formulated focused on the effect of the top quark on the operator (H^† H) at one loop. The relevant diagram is the leftmost entry in Fig. <ref> which gives a coefficient to this operator of the formi𝒜 =-|y_t|^2 N_c/2∫^Λd^4 l/(2 π)^4l^2 + 4m̂_t^2/(l^2 - m̂_t^2)^2, =-|y_t|^2 N_cΛ^2/32π^2 + ⋯ Such a regulated quadratic divergence in the case of hard cut off can be compared to DR where the mass scale in the loop is m̂_t^2 and the ϵ pole is subtracted. The appearance of the quadratic divergence can be interpreted as a signal of the Hierarchy problem that is regulator independent, when it is indicating a threshold correction in a manner consistent with a dimensionless regulator such as DR.The standard conclusion that TeV scale states are motivated to appear in multiplets that stabilize (H^† H) is valid and regulator independent. In DR, this is the statement that one loop threshold corrections to (H^† H) scale ∝ (y_a^2 m_a^2 ± y_b^2 m_b^2)/(16π^2) where m_a,b are states in such a multiplet with couplings y_a,b to H. Symmetries can be built into models to suppress such threshold corrections, and stabilize the dimension two operator. For example, unbroken supersymmetry by construction fixes y_a = y_b and m_a =m_b with the difference in sign between terms being present due to the different spin states in supermultiplets. In a regulator independent manner, a precision Higgs phenomenology program is well motivated to search for the low energy signatures of physics beyond the SM that acts to stabilize the Higgs mass. § CANDIDATE FIELD THEORIES: THE SM, SMEFT AND HEFTThe main field theoriesdiscussed in this review used to interpret LHC, LEP and other low energy data, are the SM, the SMEFT or the HEFT. The choice of which field theory to use is distinguished by the assumption on the size of possible new physics effects compared to the achievable experimental resolution (Δ E_r) at current and future facilities, and an assessment of the propagating states in the particle spectrum. By using the SM, one assumes that it will always hold when interpreting the data that Δ E_r ≫C_i v^2/g_SM Λ^2, C_i p^2/g_SM Λ^2.Here each C_i is a Wilson coefficient in the SMEFT or HEFT that corresponds to the “pole expansion” ratio v^2/Λ^2 or the derivative expansion ratio p^2/Λ^2. We define the SM to fix our notation in Section <ref>. Both the HEFT and the SMEFT follow from the expectation that it is possible that Δ E_r ≲C_i v^2/g_SM Λ^2, C_i p^2/g_SM Λ^2,will occur in the near future, or in the longer term. This assumption is reasonable to adopt. These EFTs are further distinguished by the presence of a Higgs doublet (or not) in the construction. In the SMEFT (see Section <ref>) the EFT is constructed with an explicit Higgs doublet, while in the HEFT (see Section <ref>) no such doublet is included. §.§ The Standard ModelWe define the SM Lagrangian <cit.>, with conventions consistent with Refs. <cit.>, as ℒ _ SM =-1/4 G_μν^A G^Aμν-1/4 W_μν^I W^I μν -1/4 B_μν B^μν + ∑_ψ=q,u,d,ℓ,eψi D ψ+ (D_μ H)^†(D^μ H) -λ(H^† H -1/2 v^2)^2- [ H^† jdY_dq_j + H^† juY_uq_j + H^† jeY_e ℓ_j + h.c.],where H is an SU_L(2) scalar doublet.[The alert reader will notice the lack of dual field strength terms of the form Tr[F^μ νF̃_μ ν] for the Yang-Mills fields F = {W,G}, which can be present <cit.>. Here and below the dual fields are defined with the convention F_μν =(1/2) ϵ_μναβ F^αβ with ϵ_0123=+1. The measurements of the electric dipole moment of the neutron indicate that such topological terms <cit.> are strongly suppressed for QCD, and the accidental conservation of B+L in the SM allows the neglect of such terms for electroweak theory in the SM <cit.>.]With this normalization convention, the Higgs boson mass is m_H^2=2λv^2. The vacuum expectation value (vev) acts to break SU_L(2) × U_Y(1) → U_em(1) and is defined as ⟨ H^†H ⟩= v^2/2 in the SM, with v ∼ 246 GeV. The gauge covariant derivative is defined by the states its SU_c(3) × SU_L(2) × U_Y(1) generators act on, and we use the conventional formD_μ = ∂_μ + i g_3 T^A A^A_μ + i g_2t^I W^I_μ + i g_1 _i B_μ.The _i is the U_Y(1) hypercharge generator. The T^A are the SU_c(3) generators, the Gell-Mann matrices, with normalization Tr(T^A T^B) = 2 δ^AB. The t^I=τ^I/2 are the SU_L(2) generators, the Pauli matrices, taken to be τ^1 = ( [ 0 1; 1 0 ]), τ^2 = ( [0 -i;i0 ]), τ^3 = ( [10;0 -1 ]).For example, H has hypercharge _H=1/2, is a SU_c(3) singlet and a SU_L(2) doublet so that D acting on H is given by the matrix equation D_μ H = (∂_μ +i g_2t^I W^I_μ + i g_1 B_μ/2) H. SU_L(2) indices are usually denoted as {i,j,k} and {I,J,K} in the fundamental and adjoint representations respectively. The SU_c(3) indices in the adjoint representation are instead denoted as {A,B,C}, each of which runs from {1..8}. H is defined by H_j = ϵ_jk H^†k where the SU_L(2) invariant tensor ϵ_jk is defined by ϵ_12=1 and ϵ_jk=-ϵ_kj, j,k={1,2}.All fermion fields have a suppressed flavour index in Eq. <ref>. We conventionally denote these indices by {p,r,s,t} that each run over {1,2,3} for the three generations. The fermion mass matrices are M_u,d,e=Y_u,d,ev /√(2). Y_u,d,e and M_u,d,e are complex Yukawa matrices in flavour space. Explicitly reintroducing flavour indices one hasH^† j dY_dq_j = H^† j d_p[Y_d]_prq_rj.The fermion fields q and ℓ are left-handed fields, i.e. they transform as (1/2,0) under the restricted Lorentz group SO^+(3,1).[Formally the spinors are defined as finite dimensional representations of the orthochronous Lorentz transformations. The label (a,b) corresponds to the spin of the representation, where the SU(2) × SU(2) group that is locally isomorphic to SO^+(3,1) has been used to label the 2 component Weyl spinor subgroups of a four component Dirac spinor. As the Lorentz group is connected it is related to the universal cover group SL(2, C) which is sometimes itself referred to as the Lorentz group in other applications.] The u, d and e are right-handed fields and transform as (0,1/2). The chiral projectors have the convention ψ_L/R = P_L/R ψ where P_R/L = (1 ±γ_5 )/2. The matter fields of the SM have the charges and representations [ Field SU_c(3) SU_L(2)U_Y(1) SO^+(3,1); q_i = (u_L^i,d_L^i)^T 3 2 1/6 (1/2,0); u_i = {u_R,c_R,t_R} 3 1 2/3 (0,1/2); d_i = {d_R,s_R,b_R} 3 1-1/3 (0,1/2); ℓ_i = (ν_L^i,e_L^i)^T 1 2-1/2 (1/2,0); e_i = {e_R,μ_R,τ_R} 1 1-1 (0,1/2); H 1 2 1/2 (0,0); ] The fields in Eq. <ref> are in the weak eigenbasis, whereq_1= [ [ u_L; d_L^' ]], q_2= [ [ c_L; s_L^' ]], q_3= [ [ t_L; b_L^' ]],ℓ_1= [ [ ν^e'_L;e_L ]],ℓ_2= [ [ ν^μ '_L; μ_L ]],ℓ_3= [ [ ν^τ '_L; τ_L ]],[ [ d_L^'; s_L^'; b_L^' ]]= V_ CKM [ [ d_L; s_L; b_L ]],[ [ ν^e'_L;ν^μ '_L;ν^τ '_L ]]= U_ PMNS [ [ ν^e_L; ν^μ_L; ν^τ_L ]].To date, the SM is a successful EFT for physics at and below the energies probed by LHC √(s)≲ 3 TeV. The SM does not explain the experimental evidence for dark matter <cit.>, Baryogenesis <cit.> and neutrino masses <cit.>. These experimental facts, and the theoretical arguments of Section <ref>, argue for embedding the SM in a more complete model of fundamental interactions and particles – of some unknown form.§.§.§ The Standard Model EOMThe SM equations of motion (EOM) play a role in the choice of a SMEFT operator basis, and the removal of redundant operators. We summarize the well known SM EOM here, which follow from demanding a stationary action S_SM =∫ℒ_SMd^4 x with respect to a variation due to each SM field. We follow the notation and presentation of Ref. <cit.>. For the Higgs field one hasD^2 H_k -λ v^2 H_k +2 λ (H^† H) H_k + q^jY_u^†u ϵ_jk + dY_dq_k +eY_e l_k=0,while the fermion field EOM are given byiDq_j= Y_u^†u H_j + Y_d^†dH_j, iD d= Y_d q_jH^†j , iDu= Y_uq_j H^†j ,iDl_j= Y_e^†eH_j , iDe= Y_el_j H^†j ,and the gauge field EOM are given as[D^α , G_αβ]^A= g_3j_β^A,[D^α , W_αβ]^I= g_2j_β^I, D^α B_αβ = g_1j_β.Note that [D^α , F_αβ] is the covariant derivative in the adjoint representation in the notation above. Hermitian derivative notation is introduced asH^†iD_β H= i H^† (D_β H) - i (D_β H)^† H, H^†iD_β^I H= i H^†τ^I (D_β H) - i (D_β H)^†τ^I H.Using this notation, the gauge currents arej_β =∑_ψ=u,d,q,e,lψ _i γ_βψ + 1/2 H^†iD_β H, j_β^I= 1/2q τ^I γ_βq + 1/2l τ^I γ_βl + 1/2 H^†iD_β^I H ,j_β^A=∑_ψ'=u,d,qψT^A γ_βψ.We use the notation ψ={u,d,q,e,l} to sum over all SM fermions, and V = {B,W,G} to sum over the SM gauge fields. Note that these EOM have corrections due to ℒ^(5)+ℒ^(6) + ⋯ in the SMEFT, that must be included for a consistent matching to higher orders in the non-perturbative expansion. Such corrections are also ℒ^(n) basis dependent. §.§ The Standard Model Effective Field Theory The SMEFT is a consistent EFT generalization of the SM constructed out of a series of SU_c(3) × SU_L(2) × U_Y(1) invariant higher dimensional operators, built out of SM fields and including an H field as defined in Table <ref>. The idea of the SMEFT is that extensions to the SM are assumed to involve massive particles heavier than the measured vev, which sets the scale (up to coupling suppression) of the SM states. In addition, it is assumed that any non-perturbative matching effects are characterized by a scale parametrically separated from the EW scale and the observed Higgs-like boson is embedded in the SU_L(2) Higgs doublet.The SMEFT follows from these assumptions and is defined as ℒ_SMEFT = ℒ_SM + ℒ^(5) + ℒ^(6) + ℒ^(7) + ..., ℒ^(d)= ∑_i = 1^n_dC_i^(d)/Λ^d-4 Q_i^(d) for d > 4. The operatorsQ_i^(d) are suppressed by d-4 powers of the cutoff scale Λ, and the C_i^(d) are the Wilson coefficients. The number of non-redundant operators in ℒ^(5), ℒ^(6), ℒ^(7) and ℒ^(8) is known <cit.>. Furthermore, the general algorithm to determine operator bases at higher orders developed in Refs. <cit.> makes the SMEFT defined to all orders in the expansion in local operators. Note that when transitioning to the SMEFT, symmetry arguments leading to a neglect of dual field strength terms in ℒ_SM should be reformulated, as such terms multiplied by (H^† H) appear in ℒ_6. The dual field strength terms should not be casually neglected.In the SMEFT SU_L(2) × U_Y(1) → U(1)_em is Higgsed as in the SM. The minimum of the Higgs potential is now determined including the effect of the operator Q_H ≡(H^† H )^3, which modifies the scalar doublet potential to the form <cit.> V(H^† H)= λ(H^† H -1/2 v^2)^2 - C_H ( H^† H )^3,yielding the new minimum⟨ H^† H ⟩ = v^2/2( 1+ 3 C_H v^2/4 λ) ≡1/2 v_T^2,on expanding the exact solution (λ- √(λ^2-3 C_H λ v^2))/(3 C_H) to first order in C_H. This expansion assumes a mass gap to the scale(s) of new physics (referred to schematically as Λ) which leads to the expansion parameter ∼ C_Hv̅_T^2/Λ^2 < 1. The dependence on Λ was suppressed in the previous equations. We absorb the cut off scale into the Wilson coefficients as a notational choice unless otherwise noted. The SMEFT is an enormously powerful consistent field theory to use to characterize the low energy limit of physics beyond the SM. Even if a full model extension of the SM becomes experimentally supported in the future, the SMEFT can still be a useful and appropriate tool to use to interface with large swaths of experimental data below the characteristic scale(s) Λ of a new physics sector.It cannot be emphasized too strongly that the systematic development of this framework is expected to have an important return on investment of the time expended on it. The payoff in terms of improved scientific conclusions being enabled from the ever growing data set of measurements of SM states below the scale Λ is clear. This payoff can be starkly contrasted to the return on time invested when developing the predictions of a particular model, or even a set of models, if the many assumptions of the model are not experimentally validated. Considering the current global data set of particle physics, adopting the IR assumptions that define the SMEFT seems to be a very reasonable compromise between utility and generality of the theoretical framework assumed to accommodate the certain fact that the SM is an incomplete description of reality, while the LHC data set is indicating some degree of decoupling is present to the scales involved with extending the SM. §.§.§ Operator bases in the SMEFTIn this work we use the first[This basis was built using the foundation laid down in Refs. <cit.>.] non-redundant operator basis for ℒ^(6) determined in the literature, as given in Ref. <cit.>. This construction has come to be known as the “Warsaw basis”. To fix notation we include the complete summary of the baryon number conserving operators in this basis in Table <ref>, defined using notational conventions consistent with the previous sections.The development of the Warsaw basis <cit.> underlies the systematic development of the SMEFT in recent years. It is a surprising fact that the full reduction of an operator basis for ℒ^(6) to a non-redundant form took to 2010, a full twenty four years after Ref. <cit.> was published. Preceding the development of the Warsaw basis, innumerable works employed subsets of operators to perform phenomenological studies of the low energy effects of various models. Examples of this form of analysis include Refs. <cit.>. In the literature, the Buchmüller and Wyler result <cit.> is frequently referred to as an operator basis, we note that it is fully specified and well defined, but overcomplete, unlike the Warsaw basis. In addition the SILH subset of operators <cit.> is sometimes referred to as an operator basis in some literature, as is the HISZ subset of operators <cit.>. This is perhaps due to the influential nature of these works. To avoid confusion we stress that neither of these works contains a complete set of ℒ^(6) operators, and certainly not a well defined minimal non-redundant basis. Subtleties involving flavour indices are present in extending the subset of operators in Ref. <cit.> to a full basis, see Ref. <cit.> for a discussion. As a result, the emergence of a full basis including these “SILH operators" required some further efforts to resolve these issues <cit.>. A full basis was defined in Ref. <cit.> for the first time incorporating these flavour index subtleties.§.§.§ Removing redundant operatorsRef. <cit.> reported a complete set of operators that satisfies SU_c(3) × SU_L(2) × U_Y(1) symmetry. This was the first step in constructing an ℒ_6 operator basis, but such a construction is overcomplete. Combinations of operators are present, whose Wilson coefficients vanish when observables are calculated. This is due to the EOM relating the field variables when external states go on-shell. Making small field redefinitions on the SM fields 𝒪(1/Λ^2) one can remove the combination of ℒ_6 operators that will vanish in this manner directly in the Lagrangian, instead of having the cancellation occur when the S matrix element is constructed.[ This procedure can be understood heuristically as aligning the field variables in the SMEFT more directly with classical (asymptotic) poles and the residues of the propagators at these poles, when the external particles go on-shell.] When this is done, the SM fields have a meaning that is contextual in the SMEFT and are fixed in a particular operator basis with a chosen set of ℒ_6 operators. The set of 𝒪(1/Λ^2) bosonic field redefinitions that preserve SU_c(3) × SU_L(2) × U_Y(1) are[Here we restrict ourselves to field redefinitions that have only dynamical field content, neglecting explicit v^2/Λ^2 corrections.] H_j'→H_j + h_1 D^2 H_j/Λ^2 + h_2 e̅ ℓ_j Y_e/Λ^2 + h_3 d̅q_j Y_d/Λ^2 + h_4 (u̅ϵq_j)^⋆Y_u^⋆/Λ^2 + h_5 H^† H H_j/Λ^2, B'_μ →B_μ + b_1 ψ̅γ_μψ/Λ^2 + b_2 H^†iD_μ H/Λ^2 + b_3 D^α B_αμ/Λ^2 + b_4H^† H B_μ/Λ^2,W^I '_μ →W^I_μ + w_1 q̅σ^I γ_μ q/Λ^2 + w_2 ℓ̅σ^I γ_μℓ/Λ^2 + w_3 H^†D^I_μ H/Λ^2 + w_4 [D^α , W_αμ]^I/Λ^2 + w_5 H^† H W^I_μ/Λ^2, G^A '_μ →G^A_μ + g_1 q̅ T^A γ_μ q/Λ^2 + g_2 d̅ T^A γ_μ d/Λ^2 + g_3 u̅ T^A γ_μ u/Λ^2 + g_4 [D^α , G_αμ]^A/Λ^2 + g_5 H^† H G^A_μ/Λ^2,while the corresponding transformations on the right handed fermion fields aree'→e + e_1 ℓ̅ iD H Y_e^†/Λ^2+ e_2 ℓ̅ i D H Y_e^†/Λ^2 + e_3 H^† H e/Λ^2 + e_4 D^2 e/Λ^2,d'→d + d_1 q̅ iD H Y_d^†/Λ^2+ d_2 q̅ i D H Y_d^†/Λ^2 + d_3 H^† H d/Λ^2 + d_4 D^2 d/Λ^2,u'→u + u_1 q̅ iDH̃ Y_u^†/Λ^2 + u_2 q̅ i DH̃ Y_u^†/Λ^2 + u_3 H^† H u/Λ^2 + u_4 D^2 u/Λ^2,and finally the redefinitions of the left handed fermion fields areq'_j→q_j + q_1 u iDH̃_j Y_u^†/Λ^2 + q_2 u i DH̃_j Y_u^†/Λ^2 + q_3 d iD H _j Y_d^†/Λ^2 + q_4 d i D H_j Y_d^†/Λ^2 + q_5 H^† H q_j/Λ^2 + q_4 D^2 q_j/Λ^2,ℓ'_j→ ℓ + l_1 e iD H_j Y_e^†/Λ^2 + l_2 e i D H_j Y_e^†/Λ^2 + l_3 H^† H ℓ_j/Λ^2 + l_4 D^2 ℓ_j/Λ^2.Here {h_a,b_a,w_a,e_a,d_a,u_a,q_a,l_a} are free variables. Performing field redefinitions with only a single 𝒪(1/Λ^2) term on the right hand side of each equation one can choose to cancel an operator out of a full set of operators reported in Ref. <cit.>. This is known as removing redundant operators. An example of this procedure is as follows. The B^μ dependent, flavour symmetric terms in an overcomplete ℒ_SMEFT are ℒ_B' =-1/4 B'_μν B^'μν - g_1_ψ ψ B' ψ + (D^μ H)^† (D_μ H) + 𝒞_B(H^† D^μ H) (D^ν B_μ ν), + 𝒞_BH (D^μ H)^† (D^ν H)B'_μ ν + C_H l tt^(1) Q_H l tt^(1) + C_H e ttQ_H e tt + C_H q tt^(1) Q_H q tt^(1) + C_H u ttQ_H u tt,+C_H d ttQ_H d tt + C_HBQ_HB + C_T(H^† D^μ H)(H^† D^μ H).Performing the small field redefinitionB'_μ →B_μ + b_2 H^†iD_μ H/Λ^2,yields the result ℒ _B -g_1 b_2 Δ B where Δ B= _l Q_H l tt^(1) +_e Q_H e tt + _q Q_H q tt^(1)+_u Q_H u tt + _d Q_H d tt,+ _H (Q_H+4 Q_H D) + 1/g_1B^μν∂_μ (H^† i D_ν H).Choosing b_2 to cancel one of the ℒ_B' operators introduces a shift in the Wilson coefficients of the remaining operators. When the full set of such field redefinitions has been performed, this corresponds to choosing a non-redundant basis. The removal of redundant operators is always done in a gauge independent manner, as this procedure is only justified by the invariance of observables (i.e. S matrix elements) under gauge independent field redefinitions.[See Section <ref> for more details.]Many field redefinitions are possible and Eq. <ref>-<ref> can introduce or remove the same operators. It is essential to have a gauge independent algorithm to employ to systematically remove operator forms to obtain a minimal non-redundant basis. In the Warsaw basis, the algorithm is the systematic removal of derivative operators and an equally careful application of Fierz identities to reduce out redundant four fermion operators (see also Ref. <cit.>). This reflects the approach of an on-shell EFT construction <cit.>, so named because the on-shell EOM are used to reduce out explicit factors of D^2 H and Dψ in the higher order terms. The derivative removing algorithm of the Warsaw basis was used to help develop the results defining higher order corrections in the SMEFT operator expansion reported in Refs. <cit.>. Building on past works enumerating flavour invariants in the SM <cit.> and group invariants in SUSY theories <cit.> it has been found to be beneficial to employ Hilbert series and a conformal algebra to systematize the counting of the number of operators at even higher orders than ℒ_6 in the SMEFT, while systematically removing derivative operators as in the Warsaw basis construction. Finally, the Warsaw basis algorithm was also essential to enabling the one loop renormalization of the operators in ℒ_6, that was developed in Refs. <cit.>, as discussed in Section <ref>.Although the Warsaw basis removes derivative terms systematically, not all of the derivative invariants acting on H can be removed with such 𝒪(1/Λ^2) field redefinitions. For example, H does not appear in Eq. <ref> as it does not satisfy SU_L(2)_L × U_Y(1) invariance as H is not a singlet. The distinction between D^2 H and H is a way to understand the relevance of the scalar manifold topology in determining what derivative terms can be removed.[It is possible to misunderstand Ref. <cit.> on this point as only a singlet scalar field is discussed in detail.] Defining the doublet field H to be decomposed into the real scalar fields ϕ⃗^T = {ϕ_1,ϕ_2,ϕ_3,ϕ_4} asH = ([ ϕ_2 +i ϕ_1; ϕ_4- i ϕ_3 ]),the Lagrangian derivative terms can be expressed as ℒ_derv = 1/2 (∂_μϕ⃗)· (∂^μϕ⃗) + C_H/Λ^2 ϕ⃗^2 ϕ⃗^2 + C_HD/Λ^2 (ϕ⃗·(∂^μϕ⃗))^2 + ⋯ This defines a target space metric for the scalar manifold that acts on ∂^μϕ_i ∂_μϕ_j/2 asR_ij = δ_ij + 2 ϕ_iϕ_j/Λ^2 (C_HD - 4 C_H ) + ⋯ The Riemann tensor R^i_jkl associated with this manifold can be directly determined from R_ij, and it does not vanish <cit.>. Since the target space metric is not flat, there does not exist a field redefinition which everywhere sets R_ij = δ_ij. Not all of the H self interaction derivative terms can be removed with a gauge independent field redefinition as a result.When expanding around the vev in unitary gauge, a distinction between D^2 h andh is absent. As expected, the on-shell effective field theory approach of removing all p^2 invariants for h can be achieved with agauge dependent field redefinition, and h can be canonically normalized for calculations with <cit.> h→ h( 1 + (C_H- 1/4 C_HD) v̅_T^2(1 + h/v̅_T + h^2/3 v̅_T^2) ).This distinction between the ability to remove derivative terms on h and not H is problematic as SU_L(2) × U_Y(1) is Higgsed, and one uses SU_L(2) global symmetry rotations to rotate the theory to a form where only the h field takes on a vev. This is also done when including corrections in the EFT. If this symmetry is broken by assumption, or an ill defined Lagrangian convention choice, a mismatch between the vev and the fluctuations around the vev can be introduced in a gauge dependent manner. This in turn can render a parameterization of new physics effects intrinsically gauge dependent and unsuitable to use in the event that real (gauge independent) deviations from the SM are found using EFT techniques. Such a parameterization can break structures in the EFT intrinsic to its construction and well defined nature, leading to stronger constraints that are misleading on the Wilson coefficient space. See Section <ref> for a discussion on such a structure - that is known as a SMEFT reparameterization invariance, which when not broken by assumption, or an inconsistent parameterization, requires that combination of data sets be used to lift degeneracies between parameters. This issue can change bounds on Wilson coefficients by orders of magnitude <cit.>, so great care should be taken to avoid introducing inconsistencies of this form in SMEFT studies.In addition to preforming the field redefinitions in Eqs. <ref>-<ref> one can directly translate between operator bases once a minimal non-redundant basis is found. This is done by constructing ℒ_6 operator relations directly out of the SM EOM's in Section <ref> and then employing them to transition between bases. As illustrated by Eq. <ref> there is a gauge independent field redefinition that underlies the EOM relations in Eq. <ref>.[We explicitly separate out the integration by parts identity P_T = - Q_H-4 Q_H D.] An example of these relationships is between the flavour singlet operator forms appearing in Ref. <cit.> 𝒫_HW =-i g_2 (D^μ H)^† τ^I (D^ν H)W^I_μ ν,𝒫_HB =-i g_1 (D^μ H)^† (D^ν H)B_μ ν,𝒫_W =-i g_2/2(H^† τ^ID^μ H)(D^ν W^I_μ ν),𝒫_B =-i g_1/2(H^† D^μ H)(D^ν B_μ ν), 𝒫_T =(H^† D^μ H)(H^† D^μ H),and the Warsaw basis operators in Table <ref>, given by <cit.> 𝒫_B = 1/2_Hg_1^2 Q_H+2g_1^2 _H Q_H D+ 1/2g_1^2[_l Q_H l tt^(1) +_e Q_H e tt + _q Q_H q tt^(1)+_u Q_H u tt+ _d Q_H d tt], 𝒫_W = 3/4 g_2^2 Q_H -1/2 g_2^2 m_H^2 (H^† H)^2 +2 g_2^2 λ Q_H +1/4g_2^2[Q_H l tt^(3) + Q_H q tt^(3)]+1/2 g_2^2( [Y_u^†]_rs Q_ uH rs + [Y_d^†]_rs Q_ dH rs+ [Y_e^†]_rs Q_ eH rs+h.c. ), 𝒫_HB = 1/2 g_1^2 _H Q_H +2g_1^2 _H Q_H D - 1/2_H g_1^2 Q_H B - 1/4 g_1g_2Q_H WB, +1/2g_1^2[_l Q_H l tt^(1) +_e Q_H e tt + _q Q_H q tt^(1)+_u Q_H u tt+ _d Q_H d tt] ,𝒫_HW = 3/4 g_2^2 Q_H-1/2 g_2^2 m_H^2 (H^† H)^2 +2 g_2^2 λ Q_H -1/4 g_2^2 Q_H W-1/2_H g_1 g_2Q_H WB + 1/4g_2^2[Q_H l^(3) + Q_H q^(3)]+1/2 g_2^2( [Y_u^†]_rs Q_ uH rs + [Y_d^†]_rs Q_ dH rs+ [Y_e^†]_rs Q_ eH rs+h.c. ).Note that the operators in Eq. <ref> do not carry flavour indices while the operators in Eq. <ref> do carry flavour indices. One needs to define the flavour indices of the operators removed when changing basis in order to avoid ambiguities <cit.>. Respecting flavour symmetry is the reason Yukawa matrices appear in Eqs. <ref>-<ref> instead of arbitrary flavour matrices. §.§.§ Ad-hoc phenomenological LagrangiansThere is a distinction between the concept of an operator basis in the SMEFT and an incomplete ad-hoc phenomenological Lagrangian used to characterize a subset of some SM deviations. Such an ad-hoc formalism can be constructed in a manner akin to a coordinate system basis choice for ^3, in a beyond the SM “deviation space”. A core characteristic of ad-hoc constructions is that the full Lagrangian is not specified, only a few terms, or a sector are defined. An ad-hoc approach can have some uses as discussed in Refs. <cit.>, but referring to such a construction as a SMEFT operator basis has lead to enormous confusion in recent literature.The following problems can also occur when using ad-hoc phenomenological Lagrangians:* Utilizing unitary gauge to perform gauge dependent field redefinitions on only parts of a full SMEFT Lagrangian does not satisfy the equivalence theorem <cit.>.[See Section <ref> for more discussion on the equivalence theorem.] There is no formal expectation of gauge independent results being obtained making such a transformation, even in LO results. An ad-hoc construction developed in unitary gauge is very susceptible to gauge dependence for this reason. See Section <ref> and Refs. <cit.> for more discussion. * Particular UV scenarios can lead to the expectation that only certain operators (times Wilson coefficients) are present due to a vanishing of all other Wilson coefficients at tree level, and can also lead to the expectation that the Wilson coefficients of different operators differ by a loop factor. This cannot formally break the field redefinition relations in Eqs. <ref>-<ref> or the EOM relations Eq. <ref> when the separation of scales present in the EFT construction is adhered to, as these are IR operator relations. At times such UV assumptions are imposed removing operators, before a non-redundant basis is defined as the suppression is associated with the operator, not the UV dependent Wilson coefficient. This is a mistake to avoid. * A clear sign of a phenomenological Lagrangian is the inability of such an approach to accommodate and aid in determining loop corrections. To decide if a construction is truly an operator basis one can examine if the field redefinitions in Eqs. <ref>-<ref> can be used to transform a complete set of SU_c(3) × SU_L(2) × U_Y(1) operators to the particular chosen set or form. If this is not possible, then such a construction is not an operator basis according to the definition above. The “Higgs Basis” construction in Section II.2.1 of Ref. <cit.> has not been shown to satisfy this definition of a basis.§.§.§ One loop running of ℒ_SM +ℒ_6L6 Determining the closure of the full one loop anomalous dimension matrix of ℒ_6 is an important check of the consistency of a basis. The counterterm structure of ℒ_6 can be determined without expanding around the vev of the Higgs boson, as the scales introduced when the Higgs takes on a vev regulate the IR of the theory. Using DR and MS the full renormalization of the ℒ_6 Warsaw basis was reported in Refs. <cit.> using this approach. These results built upon the past results reported in Refs. <cit.>. It was found that the Warsaw basis closes at one loop and the full 2499 × 2499 anomalous dimension matrix is now determined. This is the only basis for which this has been demonstrated to date.[Interest in the renormalization of ℒ_6 continues. These results were distilled into the tool reported in Ref. <cit.>. Some results reported in an alternate scheme appeared in Ref. <cit.> as did some partial results in an alternate basis in Ref. <cit.>. This continuing interest is also due to the curious structure of the anomalous dimension matrix <cit.>. This structure was vigorously misunderstood in the literature until it was explained in Ref. <cit.> as being due to helicity and unitarity in the SMEFT. The explanation in Ref. <cit.> is UV independent, it is a statement on the IR physics captured in the operator forms in the SMEFT, so it is a valid EFT understanding of this structure.]Several aspects of the results in Refs. <cit.> can be understood to follow directly from the procedure to construct an operator basis described in Section <ref>. The renormalization of operators in ℒ_6 mix down, modifying the running of the ℒ_SM parameters <cit.>. This is a scale dependent result that indicates that the ℒ^(6) operator bases are only defined when performing small field redefinitions on the SM fields of 𝒪(1/Λ^2). Similarly, as is well known in past calculations in NRQCD <cit.> the anomalous dimensions of redundant operators exhibit scheme and gauge dependence that only cancel once an operator basis is reduced to its non-redundant form <cit.>. This is another scale dependent sign that the field variables and operators are not independent andrelated by the EOM (as discussed in Section <ref>) when a redundant basis is used. This is a reason to report results in a non-redundant basis.For the same reason, one cannot use only the results of Eq. <ref> to translate the anomalous dimensions determined in the Warsaw basis to an alternate basis constructed out of the operators in Eq. <ref>. The removal of operator forms when defining the Warsaw basis used the full set of EOM results, not just the EOM field redefinitions related to Eq. <ref>. To map the anomalous dimension results of the Warsaw basis to another basis, all of these EOM reductions must be undone, mapping the complete set of divergences to an overcomplete ℒ_SMEFT. Subsequently, a gauge independent algorithm must be defined that maps the full set of overcomplete divergences to a chosen non-redundant basis. If no such algorithm exists, then this procedure cannot be carried out. This is probably the reason that only the Warsaw basis has been completely renormalized to date.Gauge independent field redefinitions have a central role in defining the SMEFT, and this is manifest in the counterterm structure of ℒ_6. An ad-hoc phenomenological Lagrangian as discussed Section <ref>, that is not obtained with such field redefinitions, is a challenge to ever renormalize for this reason.[See also the discussion in Ref. <cit.>.]§.§.§ Functional redundancy/factorizability of the SMEFTA consequence of how operator bases are defined is that specifying an incomplete parameterization in only a few possible interaction terms, or in a sector of interactions (like the Higgs sector) while simultaneously assuming “SM-like" interactions in other sectors, generally introduces inconsistencies into the SMEFT.For example, the parameter b_2 can be chosen in Eq. <ref> so that the operator Q_H l^(1) is absent in a basis. This can only be done at the cost of shifting the Wilson coefficients of the remaining operators in Eq. <ref>. Attempting to study the effects of the operator Q_H in the process h →ψ̅ ψ, that receives such a shift, must be done with care. This operator's Wilson coefficient leads to a shift in the effective Yukawa coupling ℒ = - 𝒴_ψhψ̅ ψ of the form 𝒴_ψ = m_ψ/v_T[1 + C_H v_T^2 ] + ⋯.Studying deviations in Higgs properties due to this operator, while simultaneously assuming a “SM-like”Z boson interaction with ψ̅ ψ to accommodate Electroweak precision data (EWPD) constraints in a global analysis is inconsistent in general. The same shift in the Wilson coefficients of the operators Q_H e,Q_H q^(1), Q_H u,Q_H d related to the shift of Q_H, that is introduced to remove the operator Q_H l^(1) from a basis, could be uncovered in a study of h →ψ̅ ψ or Z →ψ̅ ψ. This shift is required for consistency to be present in anomalous Z couplings, unless a series of other Wilson coefficients are tuned to cancel out the expected correction.If this cancellation is assumed one has stepped outside of a general SMEFT analysis. If a parameterization is constructed that is designed to directly hide such correlations, enormous confusion can be introduced into SMEFT studies. Considering such shifts as independent one must impose the constraint of the chosen Wilson coefficients canceling in all other processes in a global analysis. Even more arduous is to impose this assumption on all other equivalent combinations of Wilson coefficients (due to EOM relations) when using a non-redundant minimal basis. Not imposing this assumption introduces an inconsistency which is referred to as a functional redundancy in Ref. <cit.>. Due to the large number of parameters present in the SMEFT, it is required to combine a series of measurements to constrain the SMEFT Wilson coefficient space. Functional redundancies can block this combination of measurements being performed in a consistent manner and lead to spurious results.Directly following from this subtlety is in what sense the SMEFT is factorizable into a subset of contributions when predicting S matrix elements. Approximations must be made in interfacing with experimental results. The approximations that are safe to impose while maintaining a model independent SMEFT analysis areIR assumptions. For more discussion on this point see Section <ref>Conversely, naively imposing UV assumptions can make the SMEFT inconsistent as a field theory construction, and incapable of capturing the IR limit of a UV physics sector. This can render such a SMEFT study ambiguous, with unclear implications for a UV physics sector, as it only uncovers a distorted approximation of the true low energy constraints that a UV sector will face. SMEFT studies walk a fine line between being powerful model independent conclusions, and statements based on inconsistent field theory without any true UV implication or meaning. This fine line is defined by consistency in the SMEFT analysis being enforced (or violated) to the precision and accuracy demanded by the data.§.§ The Higgs Effective Field Theory §.§.§ Minimal assumptions in the scalar sectorBoth the SM and the SMEFT constructions are field theories that assume the existence of the scalar complex field H defined in Table <ref> in the constructed Lagrangian. The introduction of such a field is a consequence of requiring:* (i) Three Goldstone bosons π^I, the longitudinal components of the EW gauge bosons. * (ii) One singlet scalar h, corresponding to the physical Higgs boson, that ensures the exact unitarity at all energies of scattering amplitudes with external π^I fields. Pedagogical illustrations of this argument can be found in Refs. <cit.>.Condition (i) is an IR assumption that is imperative for a correct description of the EW symmetry breaking. Requirement (ii) can be relaxed and the EFT can remain self-consistent for lower energies scattering events where the EFT is well defined. An EFT does not have to exactly preserve unitarity, it only has to be unitary up to the cut off scale where the Taylor expansion used in its construction breaks down. An EFT is dictated by describing the long distance propagating states that lead to non-analytic structure in the correlation functions of scattering amplitudes. From this perspective, the ideal theoretical tool to describe scenarios without assumption (ii) being made is the Higgs Effective Field Theory (HEFT).In the HEFT the long distance propagating states are again the massive SM fermions and gauge bosons. Instead of a H field, a dominantly J^P = 0^+ singlet scalar state h is included in the EFT with free couplings to the remaining SM states.The HEFT is based on the Callan-Coleman-Wess-Zumino (CCWZ) formalism <cit.> and provides a parameterization of the scalar sector with minimal IR assumptions.This approach has been continually rediscovered over the years as it adheres to the EFT “prime directive” in the scalar sector. Several ad-hoc parameterizations have independently emerged over the years along these lines <cit.> but none of these works developed a complete self-consistent EFT.[Ref. <cit.> appears after Refs. <cit.> but states that it introduces this parameterization to the literature, which has caused some confusion, as Ref. <cit.> cites Ref. <cit.>.] It is an important development that in recent years in Refs. <cit.> this theory has been advanced to a degree that it is now a consistent EFT description.The size and pattern of any deviations from the SM discovered using EFT methods can indicate whether a HEFT or a SMEFT description is appropriate. Any deviation from the SMEFT expectation that follows from the exact H doublet structure can carry significant information about the possible UV physics matched onto the lower energy EFT description. Theoretical tools that can consistently account for the possibility that the Higgs boson may not be part of an exact SU_L(2) multiplet with the π^I, and to probe experimentally this hypothesis as a part of a wider precision measurements program are essential. The experimental collaborations have already started to address this issue, providing independent measurements of several Higgs couplings and testing for the presence of anomalous Lorentz structures (see e.g. Ref. <cit.>). At the present moment, all the observations are compatible with the SM, and the SMEFT extension, but the evidence in favor of this option is still not compelling. The presence of large uncertainties (roughly of order 10–20% as discussed in Section <ref>), together with the fact that some decay and production channel are still not accessible, allows for significant deviations.§.§.§ Topology of the scalar manifold In parallel to the HEFT formalism being continually rediscovered over the years, there has been a continual rediscovery of the resistance to this idea in the theoretical community. This resistance is usually due to the field reparameterization equivalence theorem <cit.> of S matrix elements. A consequence of this theorem is that a coordinate choice for a scalar manifold does not matter for S matrix elements. In the SM, the HEFT or the SMEFT, the scalar manifold of interest is ℳ(π^I,h). A coordinate choice for this manifold has no physical effect.The HEFT literature is not due to a misunderstanding of this point. This EFT is designed to describe a set of possible low energy IR limits that are not consistent with a predictive version of the SMEFT (i.e. when the operator expansion in the EFT converges). Specific examples are given by the Dilaton constructions of Refs. <cit.>, which have long been known to be represented by the HEFT. Nevertheless, a formulation of the HEFT/SMEFT discriminant in terms of scalar manifold topology was not present in the literature until recently.It is not surprising that this discriminant is topological in nature. Manifold topology plays an important role in theories of symmetry breaking such as in the SM, or in the Landau-Ginzburg effective theory of superconductivity (see Section <ref>). Topological distinctions of vacuum states are common in field theory, and have long been understood to underlie the properties of the scattering of pions<cit.> based on the curvature of the scalar manifold describing the Goldstone bosons of chiral perturbation theory. The heuristic embedding of possible deviations from SM expectations given by SM ⊂ SMEFT ⊂ HEFT is also well known for many years, but the precise theoretical statement is given as* The SM has a flat scalar manifold ℳ and an O(4) fixed point which is the unbroken global custodial group O(4) ∼ SU_L(2) × SU_R(2). This group is respected by a subset of the SM Lagrangian <cit.>. When ⟨ H^† H ⟩ = v^2/2 a SU_L+R(2) = SU_c(2) subgroup (also generally referred to the custodial group) is unbroken that leads to the prediction m_W = m_Z cosθ_W. The linearization lemma <cit.> then allows for a local linear transformation of the scalar manifold coordinates, which results in the embedding of the scalar fields into a linear multiplet H. This is the group theory equivalent of assuming a H field in the SM construction. * The SMEFT has a curved scalar manifold <cit.> due to the presence of two derivative Higgs operators. In the Warsaw basis these operators are Q_HD,Q_H. The SMEFT also has a O(4) fixed point which allows the EFT to be constructed with the linear multiplet H. The presence of this scalar curvature is a key point of the SMEFT construction and is the reason that one cannot remove all the two derivative Higgs operators from the SMEFT basis with gauge independent field redefinitions. * HEFT has a curved scalar manifold ℳ and does not contain aO(4) fixed point <cit.>. This distinction between the SM, SMEFT and HEFT is field redefinition invariant.[Note that in Refs. <cit.> and some other works this field redefinition invariant distinction between the HEFT and SMEFT is obscured and Ref. <cit.> is incorrectly credited for the introduction of a nonlinear realization.]§.§.§ UV embeddings of HEFT The HEFT is a general field theory that can describe a wide variety of scenarios, including composite Higgs models <cit.> and Dilaton constructions <cit.> while reproducing the SM in a specific limit of parameter space. The HEFT formalism is of most interest, if it captures the IR limit of a UV completion of the SM. Determining the necessary and sufficient conditions on UV dynamics that leads to the HEFT construction at low energies is an unsolved problem. The naive assumption that the distinction between SMEFT and HEFT is a statement of the presence of a linear multiplet in the UV sector integrated out of the theory is not correct. The assumption is an IR assumption on the nature of the EFT that corresponds to an (unknown) set of criteria on UV dynamics.This can be directly seen in the non-minimal coupling to gravity of the Higgs doublet in a UV sector. In this case the classical background field scalar manifold also plays a central role. This Lagrangian is given as a tower of higher dimensional operators as ℒ_H,inf= √(- ĝ)(ℒ_SM - M_p^2R̂/2 - ξ H^† HR̂).This Lagrangian is known as the Higgs-Inflation Lagrangian, and was popularized in recent literature in Ref. <cit.>. Here ĝ is the determinant of the Jordan frame metric ĝ_μν, M_p= 2.44 × 10^18GeV is the reduced Planck mass, R̂ is the Ricci scalar and ξ is a dimensionless coupling. When ⟨ H^† H ⟩≡v̂^2/2 ≫ v^2/2 this Lagrangian leads to a flat Higgs potential due to mixing between the scalar state h and a scalar component of the graviton.[The scalar degree of freedom of gravity can only be removed in the usual manner with diffeomorphism invariance in Minkowski space when this mixing is not present, which requires the theory is first canonically normalized.] This can be seen expanding the metric about Minkowski space as ĝ_μν = η_μν + h_μν/M_p giving δℒ = ξ/M_p (h + v̂)^2η^μ ν∂^2 h_μ ν.This mixing was analyzed in a complete fashion in Ref. <cit.> for the first time and becomes significant when √(⟨ H^† H ⟩)∼ M_p/ξ. At this scale the canonical normalization of the h field and the graviton leads to significant non-linearities introduced into an EFT description of the propagating degrees of freedom –resulting in the HEFT. This explains the scattering results found related to the cut-off scale behavior of this theory <cit.>. Although the UV dynamics includes a H doublet, the IR EFT construction useful for scattering calculations around background fields √(⟨ H^† H ⟩)≳ M_p/ξ is a version of the HEFT. As this operator is induced renormalizing the SM in curved space, this physics is formally always present and the (small) non-linearities due to gravity are present in EFT descriptions of Higgs physics. Of course these effects are below the current (and most likely future!) experimental resolution.The relevant point for LHC phenomenology is the possibility that dynamics present in a UV sector that explains EWSB leads to similar IR effects, that can be accommodated in the HEFT. In composite Higgs theories, some new physics states may mix with the h field, thus weakening the unitarity constraint as captured by the HEFT formalism, and permitting deviations from an exact doublet structure. Due to our inability to calculate the low energy limit of all possible strongly interacting sectors to understand the precise conditions on dynamics that can lead to such IR effects, this formalism should not be casually dismissed. For recent discussions on IR limits associated with TeV scale physics that require the HEFT, see Refs. <cit.>.§.§.§ The HEFT Lagrangian, preliminariesIn analogy with the formalism employed in chiral perturbation theory (χPT) for the pions of QCD, in HEFT the three Goldstone bosons π^I are embedded into a dimensionless unitary matrix= exp(i τ^I π^I/v) ,↦ LR^† ,which transforms as a bi-doublet under global SU_L(2)× SU_R(2) transformations. Inside the exponential, the π^I fields appear suppressed by the scale v rather than by a heavier cutoff Λ. This is a reasonable choice, as v can be regarded as an order parameter of EWSB and the characteristic scale of the π^I. This implies that, unlike in the SMEFT case, π^I insertions are not suppressed in general. Additional suppressions may be induced if the HEFT is matched to specific UV models that associate these degrees of freedom with a heavier scale. This is the case of Composite Higgs models, in which both the π^I and h fields arise weighted by a scale f that satisfies the relation 4π f ≥Λ, with Λ being the cutoff of the EFT <cit.>. In this case, the scalar fields are always accompanied in the EFT by factors of ξ = (v/f)^2<1. In the EW vacuum ⟨⟩=1 the global chiral symmetry is spontaneously broken down to the custodial group SU_c(2)=SU_L+R(2). In addition, the SU_R(2) component is explicitly broken in the EW sector by the Yukawa couplings and the gauged U(1) subgroup with coupling g_1 generated by τ^3. It is convenient then to define objects that transform in the adjoint of SU_L(2), to be employed as building blocks of a SU_c(3)× SU_L(2)× U_Y(1) invariant Lagrangian. It is possible to construct a scalar and a vector as follows:= τ^3 ^†↦ LL^† _μ = (D_μ) ^† _μ ↦ L _μ L^† ,where the covariant derivative of thefield isD_μ=_μ+ig_2/2 W_μ^I τ^I- ig_1/2 B_μτ^3 .Note that the fieldis invariant under hypercharge transformations but not under the global SU_R(2) group so it can be treated as a custodial symmetry breaking spurion.The physical Higgs scalar is introduced as a gauge singlet h. This choice ensures the most general approach to the physics of this state and makes the HEFT a versatile tool which includes the SU_L(2) doublet case as a particular limit. Specifically, the SM Higgs doublet H can be written as a fixed combination of the fields h andaccording to:(H̃H) = v+h/√(2).Being a singlet, the h field has completely arbitrary couplings, that are customarily encoded in generic functions <cit.> _i(h) = 1+ 2a_ih/v+b_ih^2/v^2+…that constitute another building block for the HEFT Lagrangian. Note that here the polynomial structure should be interpreted as the result of a Taylor expansion in (h/v), which may include an infinite number of terms.§.§.§ The HEFT Lagrangian The HEFT Lagrangian is composed of the gauge and fermion fields in Section <ref>, and the scalar fields , h defined in the previous section. Its construction historically builds upon that of the EW chiral Lagrangian <cit.>, in which only the three π^I fields were retained in the spectrum, while the physical Higgs was assumed to be sufficiently heavy to be integrated out. Extensions with addition of extra (pseudo-)scalar fields were also explored more recently <cit.>. The HEFT Lagrangian can be defined asℒ_HEFT = ℒ_0+Δℒ+… ,where ℒ_0 contains the leading order terms and Δℒ includes first order deviations. Unlike the SMEFT, it is not possible to classify the HEFT invariants based on just the canonical dimension. This is due to the fact that the HEFT is a fusion of χPT (in the scalar sector) with the SMEFT (in the fermions and gauge sector). Because these two theories follow different counting rules, the structure of the mixed expansion is complex. A rigorous and self-contained method to determine the expected suppression of a given HEFT invariant represents a non-trivial and subtle task that has been intensely debated in the literature. The discussion of this technical point is postponed to Section <ref>.The leading Lagrangian ℒ_0can be written as[Due to the lack of a unique criterion for the classification of the invariants, there is some freedom in the definition of the LO itself. To define one universal rule that uniquely determines the order a certain HEFT operator should be assigned, in a UV independent manner, is the essential challenge. This difficulty has been overcome in the literature by identifying a set of heuristic criteria, which essentially follow from internal consistency requirements and some common sense considerations. Unfortunately, some UV dependence is generally also present. Nevertheless, there is substantial agreement in the community with respect to the organization of the HEFT expansion which we stress here, despite disagreements on the definition of counting rules. For definiteness, here we use the conventions of Ref. <cit.>. We stress this is not a value judgment, simply a convention. The same LO Lagrangian has been previously adopted in Refs. <cit.>, although in a slightly different notation. Different conventional choices were made in Refs. <cit.>.] ℒ_0=-14 G_μν^A G^A μν -1/4 W_μν^I W^I μν-14 B_μνB^μν+∑_ψψ̅iDψ ++1/2_μ h ^μ h-v^24(_μ^μ)_C(h)-V(h)+-v/√(2)(q̅ 𝒴_Q(h) q_R+) -v/√(2)(ℓ̅ 𝒴_Q(h) ℓ_R +) ,where the right-handed fermions have been collected into doublets of the global SU_R(2) symmetryq_R = [ u_R; d_R ],ℓ_R = [ 0; e_R ]and the Yukawa couplings include a dependence on h:𝒴_Q(h) = diag(∑_n Y_u^(n)h^n/v^n, ∑_n Y_d^nh^n/v^n),𝒴_L(h) = diag(0, ∑_n Y_e^(n)h^n/v^n) .The first term in the sum (n=0) generates fermion masses, while higher orders describe the couplings with an arbitrary number of h insertions. The term (_μ^μ) in the second line of Eq. <ref> contains the kinetic terms of the π^I and the mass terms of the gauge bosons. Because the matrixis adimensional, this invariant has canonical dimension 2 and therefore appears in the Lagrangian multiplied by a factor v^2. The reason why it is the scale v, and not the cutoff Λ, that multiplies this term is that this choice ensures a canonically normalized kinetic term for the π^I, given the definition ofin Eq. <ref>. The same principle applies to the Yukawa couplings.The Lagrangian ℒ_0 is equivalent to the SM Lagrangian up to the presence of an arbitrarily large number of Higgs and π^I insertions and to the fact that the Higgs couplings (which includes the scalar potential) are parameterized by independent coefficients rather than being fixed by the doublet structure. Despite being allowed by symmetry, Higgs couplings to kinetic term structures are absent in ℒ_0. In the case of the fermion and h kinetic terms, this is because any extra (h) factor can be removed via field redefinitions <cit.> and reabsorbed into the coefficients that parameterize the Higgs interactions. This can be understood to follow directly from the general arguments on constructing on-shell EFTs given in Refs. <cit.>. In the case of gauge kinetic terms such couplings are chosen to be retained in the basis, but they are customarily classified as higher order effects due to a suppressed Wilson coefficient, under the assumption that the transverse components of the gauge bosons do not couple strongly to the Higgs sector. The same choice has been made for the suppression of the Wilson coefficient of the operator (_μ)^2, which would belong to ℒ_0 according to the derivative expansion of χPT, but that must carry an implicit suppression in its matching if the custodial symmetry is assumed to be only weakly broken, as suggested by experimental observations. The Lagrangian Δℒ contains the leading deviations from ℒ_0. Again, it is not possible to infer in a rigorous and universal way which classes of operators belong to this order. The relative impact of two given invariants depends in general on the kinematic regime considered (see Section <ref> for further details). It is possible to identify a set of invariants that can be responsible for the largest deviations at least at energies up to a few TeV. They amount to 148 independent invariants in total <cit.>. A complete basis of operators encompassing both bosonic and fermionic sector has been first proposed in Ref. <cit.> and an alternative set has been derived independently inRef. <cit.>. The two works agree on the classes of operators, although there are differences in the choice of individual terms retained. We describe the operators present sector by sector as follows. Bosonic operators: This must include chiral invariants with four derivatives, which are next-to-leading terms in the chiral expansion and can be reducedto an independent set with the structures[Not all of the Lorentz indices in these expressions match. This indicates the possibility of multiple Lorentz contractions.] (_μ)^4, (_μ)^3 _ν_i(h),(_μ)^2(_ν_j(h))^2,(_μ_k(h))^4X_μν (_ρ)^2, X_μν_ρ_σ_l(h), (X_μν)^2_m(h) . Note that operators in this category are not suppressed by any powers of Λ in the Lagrangian, as they have formally canonical dimension 4. The two-derivative operator v^2 (_μ)^2/4 also belongs to Δℒ as explained above. Finally, operators with the structure (X_μν)^3 can be included at this order despite containing 6 derivatives. This can be justified assuming that, being composed of only transverse gauge boson, these operators follow the ordering rules of the SMEFT (see Section <ref>).A complete set of interactions for the bosonic sector, constructed avoiding reduction via the EOM, was presented and studied phenomenologically in Ref. <cit.> (CP conserving terms) and Ref. <cit.> (CP odd terms). Operators with fermions:Δℒ contains operators with one fermionic current and up to two derivatives, as at least a subset of this category of invariants is required as counter-terms in the one-loop renormalization of ℒ_0. Schematically, they can be reduced to the set of independent structures(ψ̅γ^μψ) _μ,(ψ̅ψ)(_μ)^2, (ψ̅ψ)_μ^μ_n(h), (ψ̅σ^μνψ)(_μ)_ν_o(h),(ψ̅σ^μνψ)(X_μν) .Here operators with one derivative are unsuppressed, while operators with two derivatives are multiplied by Λ^-1, having canonical dimension 5. Four-fermion operators appear in Δℒ as well, because a subset of terms in this class is required for the renormalization of ℒ_0. These considerations lead to the construction of a consistent basis of independent operators in the HEFT, reported by multiple groups, which is closed under the EOM relations. §.§ SMEFT vs. HEFT S matrix elements and relations between S matrix elements constructed in the SMEFT and the HEFT expansions are potentially distinguishable, as the theories are distinct in a field redefinition invariant manner <cit.>. The ordering of the theories in their IR assumptions is given by SM (H,Λ→∞) ⊂ SMEFT (H,Λ≠∞) ⊂ HEFT (h,Λ≠∞).Early statements on the need to experimentally check the assumption of a H field in the EFT used, appeared in the literature before the turn on of LHC <cit.>. Developing a precise statement on what exact experimental result would decide between these two frameworks conclusively is an ongoing challenge. The differences between SMEFT and HEFT identified to date, and existing proposals to take advantage of these differences are as follows.§.§.§ Higgs/Triple Gauge Couplings While in SMEFT the Higgs couplings follow a polynomial dependence in (v+h)^ndue to the H doublet, in the HEFT they do not. Current LHC results are compatible with H in the spectrum, but with large uncertainties that allow for significant deviations. A fundamental probe of this point would be the observation of double-Higgs production, which is challenging at the LHC. A discussion of this process in relation to the HEFT can be found in Refs. <cit.>.Another distinctive prediction of the HEFT is the decorrelation of triple gauge couplings and Higgs-gauge bosons interactions, due to the fact that in this EFT the covariant derivative D_μ H ∼_μ h+(v+h)D_μ is formally split into two independent pieces. The possibility of isolating this effect experimentally has been studied in Refs. <cit.>.The appearance of Hermitian derivative terms acting on Higgs fields in the SMEFT of the form H^†iD_β H and H^†iD_β^I H in ℒ_6 relates anomalous Z couplings in ψ̅ψ→ Z →ψ̅ψ to contact operator contributions to p p → h → Z Z^⋆→ψ̅ψ ψ̅ψ. This correlation can be probed as a consistency test of the SMEFT accommodating any discovered deviations in the kinematic spectra of h → Zψ̅ψ and h → Z Z^⋆→ψ̅ψ ψ̅ψ determined in the narrow width approximation <cit.>. For recent analysis of this spectra in the SMEFT/HEFT, see Refs. <cit.>.§.§.§ Organization of the expansion As detailed in Section <ref>, the HEFT Lagrangian is not organized as an expansion in canonical dimensionsas in the SMEFT, but has a more complex structure. This is because the physical Higgs h and the π^I containing matrixare independent in HEFT. A consequence is the re-shuffling the orders at which interactions appear in the expansion.Insertions of longitudinal gauge bosons are less suppressed in HEFT. Couplings of these fields or, equivalently, of the π^I can be probed in high energy scattering. These processes are also potentially sensitive to the presence of strong interactions in the EWSB sector. Several channels have been analyzed in HEFT, including V_L V_L → V'_L V'_L (V,V'=Z,W^±) <cit.>, V_L V_L→ hh <cit.>, V_L V_L→ tt̅ <cit.> andγγ→ V_LV_L <cit.>. The latter is more promising, having been already observed at the LHC <cit.>.Some interesting differences in the allowed interactions in HEFT comparedto the SMEFT at fixed order in the power counting of each theory have been identified.For example, the operator ϵ_μνρ(^μ)(^ν W^ρ)(h), introduces triple and quartic gauge couplings with an anomalous antisymmetricLorentz structure in HEFT.[This operator violates explicitly the custodial symmetry group (SU_c(2))Therefore it may be suppressed in scenarios where this symmetry is broken only through standard effects (non-homogeneous Yukawas and g'≠0).].A phenomenological study of this term has been carried out in Ref. <cit.>.§.§.§ Number of operators/operator structureHEFT contains a larger number of invariants compared to the SMEFT order by order in the expansions of each theory. In the flavour-blind limit, the complete HEFT basis Δℒ contains 148+h.c operators <cit.>, compared to the 76 parameters when n_f = 1 in the SMEFT at ℒ^(6) <cit.>. Most of the operators appearing in HEFT contribute to the same interaction vertices in the SMEFTdue to ℒ^(6). When this is the case, the larger number of invariantsis then present in the corrections to predicted S matrix elementswith the same interaction vertices.This does not lead to a measurable difference in a single observable as multiple observables are requiredto fix the free parameters. Potentially, patterns of allowed deviations in multiple processescould be more easily accommodated in HEFT than the SMEFT. §.§.§ Tails vs polesDifferences between SMEFT and HEFT identified to date exist in interactions involving at least three boson fields: hhh, hhhh, hVV,VVV. In each EFT, one can work in the canonically normalized basis of fields. For pole observables, using the different EFTs label different unknown parameters in Vψ̅ψ and the mass terms VV. Studying ψ̅ψ→ V →ψ̅ψ observables the power counting of the corrections in each EFT scales as C_i v^2/Λ^2. At present, no SM-resonant exchange (i.e. pole process) measured at LEP, the Tevatron, or LHC has deviated in a statistically significant manner from the SM expectation, see Section <ref>.Future Run II data, the High Luminosity phase of the LHC, and future facilities will offer more precise experimental results on pole observables, and S matrix elements that receive contributions from hhh, hhhh, hVV, VVV. These off-shell vertices have a non-trivial relation to LHC observables, with other possible deviations simultaneously present at each order in the EFT expansions in vertex corrections. When studying these processes, two expansions are present {v^2/Λ^2, p^2/Λ^2} where p^2 stands for a general kinematic invariant, and in general p^2/Λ^2 → 1 in the tail of a distribution.It is an unsolved problem to develop a method for consistent SMEFT/HEFT analysis of the tails of kinematic distributions and to project the global constraints of each EFT into these distributions.[For recent results aimed at this problem, see Refs. <cit.>.] Defining the interference of the SM for an observable O with a tail expansion parameter factored out with HEFT or SMEFT as ⟨⟩_O, to distinguish these theories and conclude that the SMEFT cannot accommodate a discovered deviation from the SM, while the HEFT can accommodate such a deviation, requires Δ⟨ℒ_6 -(ℒ_0+Δℒ)⟩_O p^2/Λ^2> ⟨ℒ^8 ⟩_O(p^2)^2/Λ^4.Due to the large multiplicity of operators appearing in the SMEFT expansion at ℒ_8<cit.>, examining the distinguishability of these theories in the limit p^2/Λ^2 ≲ 1 (as opposed to p^2/Λ^2 << 1) is difficult. An additional challenge is the suppression of the interference terms due to ℒ_6 in several tail observables of this form following from the Helicity non-interference arguments of Ref. <cit.>. It remains to be conclusively shown if HEFT and SMEFT can be functionally distinguished in the tails of distributions. Determining the global constraints on these theories from the LEP and LHC data sets is an essential step to examine this question quantitatively.§ POWER COUNTINGThe SMEFT and the HEFT are constructed out of an infinite series of operators based on a separation of scales. It is required that these theoretical descriptions are consistent, and they are of interest if they are predictive. For this reason, it is important to establish:* How a Lagrangian term scales in a consistent manner with the dimensionful quantities in the theory, i.e. the normalization of terms. * A prescription for ordering such invariants within the EFT expansion to estimate the relative physical impact of any given Lagrangian term on a measurement, so that theapproximate theoretical precision of the EFT can be well defined. The procedure of establishing the consistent scaling/ordering of the terms in the Lagrangian is usually called “power counting”. There is a canonical interpretation of this concept that runs through Refs. <cit.> and many other works. We adopt this interpretation in this review. A different usage of “power counting” also exists in the literature, as we stress below. §.§ Counting rules for the SMEFT The SMEFT is built out of the SM fields and couplings that have the well defined canonical mass dimensions [..]. Defining the total mass dimensions of the [ℒ_SM] = d, then [ψ] = d-1/2, [Y_i] = 4-d/2, [V] = d-2/2, [g_i] = 4-d/2, [H] = d-2/2,[λ] = 4-d.A power counting in canonical mass dimension leads to an operator built out of field insertions with total mass dimension d multiplied by a factor Λ^4-d. According to its canonical dimension, operators with d ≤ 4 (the SM Lagrangian) are leading terms (LT), operators with d=6 are next-to-leading terms (NLT) in the SMEFT and so on. This power counting rule is an ordering and normalization statement on the EFT.For the EFT to be predictive it is also required that the expansion parameters are perturbative C_i v^2/Λ^2 <1,C_i p^2/Λ^2 <1.The accompanying Wilson coefficients are UV dependent. This condition translates into a condition on the UV completion by insisting that the IR EFT construction is predictive. A power counting in Naive Mass Dimension allows an estimate of the neglected higher order terms in a calculation by varying the unknown C^d_i v^d-4/Λ^d-4,C^d_i |p|^d-4/Λ^d-4 over a declared range of values, allowing an interpretation of the data in the SMEFT.§.§.§ NDA countingOne can determine a consistent set of power counting rules that are a property of the SMEFT (and other field theories) that are distinct from just using the canonical mass dimension estimate. This alternative approach is known as Naive Dimensional Analysis (NDA) <cit.>. NDA normalization has a physical intuition underlying it. One can determine the generation of operators Q_i from operators Q_j using topological relations due to the full set of connected diagrams in the EFT. Applying this idea to the SMEFT, one can assume roughly homogeneous matching C_i ∼ C_j for operators in ℒ_6, and demand that parameter tuning is avoided. Then one expects |C_i| ≳ |Δ C_i| where |Δ C_i| is the absolute value of the induced Wilson coefficient for C_i due to ∫ℒ_SM× C_j Q_j from the allowed connected diagrams.The generalized version of NDA <cit.> of this form, which builds upon the work in Refs. <cit.> gives a normalization of Lagrangian terms asΛ^4/16π^2(_μ/Λ)^D (4π V_μ/Λ)^A(4πψ/Λ^3/2)^F (4π H/Λ)^S(g/4π)^N_g(y/4π)^N_y(λ/16π^2)^N_λ,for an interaction term with D derivatives, A gauge fields, F fermion insertions and S scalar fields, accompanied by N_g gauge coupling constants, N_y Yukawas and N_λ quartic scalar couplings.NDA was first determined in the context of the chiral quark <cit.> model, but its approach has been successfully applied to many other EFT descriptions. This is because it is a consistency condition on a EFT construction. NDA does not change the Lagrangian. Itmakes manifest an ordering of the Wilson coefficients that corresponds to a lack of tuning of parameters, consistent with its assumptions in the EFT. This normalization is a property of the EFT and the EOM generated by varying the action of the EFT (see Section <ref>) obeys this power counting as a result.NDA supplies a reasonableestimate of the normalization of a Lagrangian term, which still multiplies an unknown Wilson coefficient. Eq. <ref> characterizes the scales in a UV sector as ∼Λ. If a distinct set of scales exists in matching onto a UV sector, this can lead to the Wilson coefficients having to differ significantly if a NDA normalization is used. Applying a NDA normalization to global fits in the SMEFT the Wilson coefficients should be treated as free parameters for this reason.§.§.§ UV matching scenariosA distinct approach to ordering parameters in the Lagrangian also goes under the name power counting in the literature. To clarify this difference we willrefer to this approach as “Meta-matching”. The idea is to make more assumptions on the UV sector, that leads to broad conditions on the Wilson coefficients. The difficulty is in making UV assumptions in a well defined, precise, and consistent manner. Meta-matching is closely related to power counting in that it defines a normalization of Lagrangian terms in the EFT, this is why it is also referred to as power counting in some literature. It is also a distinct UV assumption heavy approach that blurs the strict separation of scales defining an EFT construction. Despite this, it can still be a useful tool.Topological tree/loop Meta-matching: The most commonly used version of Meta-matching is the “tree-loop” classification scheme of Artz, Einhorn and Wudka <cit.>. The idea is to study topologically all of the field content of weakly coupled renormalizable models that can couple at tree level to the SM states exhaustively, and to determine which ℒ_6 operators can be generated at tree level in matching to a basis. For the Warsaw basis, the operators classified as “tree” in this manner for ℒ_6 are those in Classes {2,3,5,7,8} in Table <ref>. As the operators in Classes {1,4,6} contain field strengths, this classification scheme could have been misunderstood as being related to a “minimal coupling” condition in matching results. Artz, Einhorn and Wudka <cit.> do not assert this.[For further discussion on minimal coupling, see Refs. <cit.>.] The analysis in Ref. <cit.> discussed the possibility of kinetic mixing of the form V_μ ν F^μ ν, for F a beyond the SM U(1) vector boson only briefly. This interaction is redundant, so this does not limit the conclusions of Ref. <cit.>.This scheme is an assertion of UV assumptions, not a consistency condition of the EFT, which is the point of NDA. The results of Ref. <cit.> are limited to weakly coupled and renormalizable UV scenarios, they do not apply if the UV contains an EFT or a strongly interacting theory <cit.>. The classification scheme of Ref. <cit.> also cannot capture the low energy effects of the multi-pole expansion discussed in Section <ref>. Furthermore, ℒ_6 operators equated by the EOM have Wilson coefficients that differ in this scheme, which can cause some confusion. An example is given in the EOM operator relation 𝒫_HB = 1/2 g_1^2 _H Q_H +2g_1^2 _H Q_H D - 1/2_H g_1^2 Q_H B - 1/4 g_1g_2Q_H WB,+ 1/2g_1^2[_l Q_H l tt^(1) +_e Q_H e tt + _q Q_H q tt^(1)+_u Q_H u tt+ _d Q_H d tt].Here the Wilson coefficients of the operators Q_H B,Q_H WB are considered to be “loop level”, and the remaining operators are considered to have Wilson coefficients that are tree level in this scheme. Recall that the operator forms are equated as a result of the freedom to redefine the SM fields by 𝒪(1/Λ^2) corrections without changing the S matrix. It is important to understand the distinction between such IR relations that are statements of consistency conditions in the SMEFT (in this case the freedom to change variables in a path integral without physical effect) and the UV assumptions that can break the corresponding relations that follow between Wilson coefficients. Unfortunately, this distinction is frequently lost in the literature where references to “tree” and “loop” operators abound, which blurs the distinction between the IR operators and the UV matching coefficients that follows from the separation of scales defining the EFT. One scale, one coupling Meta-matching: It was observed in Ref. <cit.> that the NDA results of Ref. <cit.> could follow from the assumption of NDA being respected by QCD as an assumed high energy theory, and then considered to be matched to χPT with g_3 ≃ 4π. This point was not stressed, as it is well known that when QCD is strongly interacting a useful EFT description transitions to χPT treated as a bottom up EFT, as an essential singularity is present. In the case of QCD with m_q→ 0, the beta function leads to the relation( Λ_qcd/μ)^b_0 = e^-8π^2/[ ħg_3^2(μ)],with the factors of ħ explicit <cit.>. Λ_qcd is not due to summing the g_3^2 expansion to all orders with a choice g_3 ≃ 4π due to the essential singularity at g_3^2ħ=0 in the g_3^2 and ħ expansions. No useful perturbative matching can be performed in this limit.Ref. <cit.> observed that if some of the vector resonances of χPT were considered lighter than the remaining states, an alternate normalization of the Lagrangian terms of χPT under this assumption could be constructed.A highly influential version of Meta-matching was put forth in Ref. <cit.> based on arguing these two observations hold in a useful manner for general scenarios of strongly interacting physics with a light pseudo-Goldstone Higgs. The modern version of the Lagrangian normalization for a composite sector that results <cit.> is given by m_⋆^4/g_⋆^2(∂/m_⋆)^A (g_⋆Π/m_⋆)^B (g_⋆σ/m_⋆)^C (g_⋆Ψ/m_⋆^3/2)^D.Here the σ model scale is given by f = m_⋆/g_⋆where m_⋆ and g_⋆ are the corresponding one scale and one coupling assumed in the composite sector. The Ψ, Π and σ are assumed states in the composite sector with the latter two associated with the σ model scale f. Further assuming that g_⋆ < 4 π one can (at least hypothetically) hope to avoid the equivalent challenge of an essential singularity being present in perturbative expansions in the strong coupling limit, and obtain a result consistent with the NDA result of Ref. <cit.> (with 4π replaced by g_⋆), assuming a matching onto the SM states making up ℒ_6. It is however necessary to stress that the presence of an essential singularity in the strong coupling limit is the basic reason that any such approach taking the g_⋆→ 4 π (for example in Eq. <ref>) in a naive analytic continuation is not expected to be useful or informative in general, and rather unwise on very basic grounds. For a strongly coupled UV theory in general, there is no reason to expect such a simplistic counting to hold for this reason, and also for the reasons discussed in Sections <ref>, <ref>. As initially formulated in Ref. <cit.>, this approach to Meta-matching had several other essential inconsistencies, see discussion in Refs. <cit.>. Alternative reasoning <cit.> has since been advanced to justify some of the results of Ref. <cit.> and other assumptions of this work have been effectively abandoned <cit.>.Universal theories: An oblique, or universal theory assumption <cit.> is another example of Meta-matching. This idea asserts that the dominant effects of physics beyond the SM can be sequestered into modifying the two point functions of the gauge bosons Π_WW,ZZ,γ Z. The idea of oblique, or universal theories, is motivated out of past approaches to EWPD, where the global S,T EWPD fit assumes that vertex corrections due to physics beyond the SM are neglected - giving the “oblique" qualifier <cit.>. It is problematic that the oblique, or universal, assumption is not field redefinition invariant,[A corollary to the point that Lagrangian parameters are not physical (see Section <ref>) is that field redefinitions cannot be used to satisfy defining physical conditions on the EFT, such as the oblique, or universal, assumption. For discussion on this point see Ref. <cit.>.] see Section <ref>.Oblique, or universal, assumptions were tolerable in the 1990's and early 2000's as the SM Higgs couples in a dominant fashion to Π_WW,ZZ when generating the mass of the W,Z bosons, and has small couplings to the light fermions, satisfying the assumption in a particular UV model that was not directly experimentally supported.LHC results now indicate the W,Z bosons obtain their mass in a manner that is associated with the Higgs-like scalar. Corrections to Π_WW,ZZ can be included for the SM, or more generally <cit.> due to the discovered scalar. There is no strong theoretical support to maintain an oblique, or universal, assumption to constrain the further perturbations due to new physics scenarios in the SMEFT with EWPD. It is also unclear if this idea is even a consistent theoretical concept.The reason is that a fully defined mechanism of dynamical mass generation in a UV sector has never been demonstrated to be consistent with this assumption, to define a consistent IR limit. Furthermore, assumed oblique, or universal, theories generate non-universal theories <cit.> using the renormalization group to run the operators matched onto from Λ→m̂_z. For these reasons, this assumption has largely been abandoned in consistent SMEFT analyses in recent years.[For more discussion on the limitations of a universal theories assumption, see Refs. <cit.>.]§.§.§ Meta-matching vs power countingEFT formalizes the separation of scales in a problem so that Meta-matching assumptions are avoidable. An analogy to the SM is useful. The SM predictions depend on the numerical values of the gauge couplings {g_1,g_2,g_3}. Different values of the gauge couplings in the SM lead to different predictions of S matrix elements and relations between S matrix elements. The SM is studied without invoking as necessary a reference to a grand unified theory (GUT) embedding, even though such an embedding would form a UV boundary condition that dictates the values of {g_1,g_2,g_3} at lower scales. The same point holds in the SMEFT and HEFT, for different values of the Wilson coefficients. It is largely a sterile and unproductive debate to dispute assumptions about the size of {g_1,g_2,g_3} or C_i. Measuring parameters is more productive. Making Meta-matching assumptions in an EFT framework causes confusion and is the source of much conflict in the literature. The separation of scales fundamental to the EFT construction can be violated, the assumptions are frequently assigned to the IR operator forms, not the UV dependent Wilson coefficients, and yet EFT language and concepts are used. The conclusions derived are then not model independent EFT statements, although they are usually argued to be “broadly applicable" conclusions as a subjective assertion.On the other hand, due to the large number of parameters present in the HEFT and the SMEFT the reduction in the parameter space seems required. We stress that it is universally accepted to use theIR assumptions of experimentally motivated flavour symmetries being present in the operator basis, which leads directly to a sufficient reduction in Wilson coefficient parameters to enable a global constraint program.At times extra assumptions are still employed which can lead to much stronger conclusions studying the data. Such conclusions must be defined in a consistent IR limit of a UV sector to be meaningful and, even if this is the case, are limited by model dependence. Such results can be of great value if they are properly qualified and defined. §.§ Counting rules for the chiral LagrangiansPower counting in HEFT has also been intensely debated recently in Refs. <cit.>. In HEFT, it is not possible to define an ordering criterion among the invariants that holds in any kinematic regime. Furthermore, assumptions made on the size of the Wilson coefficients can have significant phenomenological consequences. In this HEFT discussion, LT will indicate ℒ_0 while NLT is Δℒ.§.§.§ χPTChiral perturbation theoryThe HEFT is a fusion of the SMEFT with χPT. These two EFTs follow different power countings. One of the main applications of χPT is the description of the octet of light QCD mesons (π^0,±, K^0,±, η) at E≪Λ_ QCD. These particles are the Goldstone bosons of the spontaneous breaking of the chiral symmetry SU_L(3)× SU_R(3) into the diagonal SU_L+R(3) and they are collectively described by the fieldU = exp(2iΦ/f),Φ = ϕ^A T^A ,that transforms as U↦ LUR^† under the chiral group. Here f is the meson decay constant[At this level all the eight mesons are degenerate and share the same decay constant. The inclusion of higher-order terms breaks the SU_L+R(3) symmetry leading to the splitting f_π≃93 MeV, f_K≃113 MeV. ], that satisfies the relation 4π f ≥Λ, where Λ is the cutoff of the effective description <cit.>. Omitting the electromagnetic interactions, and in the limit of massless quarks, the leading order Lagrangian isℒ_2 = f^2/4(_μ U^†^μ U),where the presence of f^2 ensures canonical kinetic terms for the ϕ^A fields upon expanding the exponential. Because the matrix U is adimensional, this EFT is organized as an expansion in derivatives: ℒ_χ PT=ℒ_2 + ℒ_4 + ℒ_6 +… ,where ℒ_d contains invariants with d derivatives. This is Weinberg's power counting approach, which is supported by a renormalization argument <cit.>. The Lagrangian ℒ_2 contains an infinite series in powers of the meson fields ϕ^A. The counterterms required to reabsorb the one-loop divergences appearing at this level correspond to interaction terms with four derivatives. This can be seen with a topological analysis: an amplitude with L loops and containing N_d vertices with d derivatives must scale with D powers of momentum, where <cit.> D = 2L+2 + ∑_d (d-2)N_d .Computing at one loop (L=1) with d=2 vertices gives only amplitudes with D=4, i.e. four derivatives. Computing at two loops with L_2 or at one loop with one insertion of a 4-derivative interaction plus an arbitrary number of d=2 vertices, gives instead 4-derivative terms and so on. This is in contrast with the SMEFT case, where loops containing one insertion of a d-dimensional operators generates divergences of order ≤ d.[See Ref. <cit.> for the complete set of these terms.] Eq. <ref> implies that χPT is renormalizable at fixed order in the momentum expansion, which provides a solid iterative method for organizing the effective series: each order of the EFT must contain at least all the operators that are required as counterterms for the one-loop renormalization of the preceding order. An alternative counting prescription for χPT is that of NDA <cit.>. This provides a rule for establishing the normalization of a given operator and it is constructed so that the scaling assigned to a given invariant is independent of the loop order at which it is generated in the EFT. Restricted to our simple case, it states that the overall coefficient of a generic interaction with D derivatives and S pion fields is estimated by the formulaf^2Λ^2 (_μ/Λ)^D(ϕ/f)^S.This rule assigns the correct coefficient f^2 to the Goldstones kinetic term (Eq. <ref>). The suppression in powers of Λ assigned by the NDA rule is equivalent to the derivative counting. In the NDA notation, the χPT Lagrangian can be writtenℒ_χ PT=f^2/4((_μ U^†^μ U) + 1/Λ^2ℒ_4 + 1/Λ^4ℒ_6 +…) .Requiring that operators with 4 derivatives must be weighted by a factor at least as large as a loop suppression one obtains the constraint f^2/Λ^2≥ 1/(4π)^2, i.e. Λ≤ 4π f. χPT without electromagnetic interactions and in the limit of vanishing quark masses, is a simple example, in which the three counting methods (derivative, loops, NDA) all lead to the same result. The scenario becomes more complicated as soon as dimensionful quantities or fields, such as photons or mass terms, are incorporated in the EFT. The explicit chiral symmetry breaking termℒ_χ =f^2/4(U^†χ + χ^† U) ,has χ as a quantity proportional to the quarks masses χ∼ diag(M_u,M_d,M_s) that transforms as χ→ Lχ R^†. The derivative counting can be extended to this object assuming that, being dimensionful, χ shall scale as p. In this way the expansion in p/Λ is formally maintained, but upgraded to the nomenclature of “chiral dimension”, which counts both thenumber of derivatives and χ insertions. The term ℒ_χ has chiral dimension 2 and therefore is a leading term, that should be included in ℒ_2.The scaling χ∼ p is formal and represents an approximation valid only in the regime p≃ M_π,K. The chiral dimension gives a correct estimate of the impact of a given operator only for processes with on-shell mesons, while it loses physical meaning as one moves outside this kinematic regime. This is because the relative importance of two given operators (such as the kinetic term and ℒ_χ) can be different in S matrix elements measured at distinct energies.§.§.§ HEFT countingHEFT is complex as an EFT due to the simultaneous presence of a scalar sector embedded in a χPT-like construction and of fermions and longitudinal gauge bosons, whose interactions are organized according to their canonical dimension.Following the two-step procedure of power counting presented at the beginning of this section, one first has to address the question of assigning weights to each Lagrangian term. This can be done in a self-consistent EFT approach adopting the generalized version of NDA <cit.>, which builds upon the work in Refs. <cit.>. For the HEFT case, this modern NDA assigns the scalingΛ^4/16π^2(_μ/Λ)^D (4π A_μ/Λ)^A (4πψ/Λ^3/2)^F (4πϕ/Λ)^S(g/4π)^N_g(y/4π)^N_y,to an interaction with D derivatives, A gauge fields, F fermion insertions and S scalar fields (either the Goldstones or the physical Higgs), accompanied by N_g gauge coupling constants and N_y Yukawas.Determining the scaling of a given term does not ensure the existence of a unique ordering of effects in the EFT that holds at any energy regime and that is also independent of any UV assumptions. There are several different elements that coexist in the HEFT: one may choose to organize the expansion in loops, i.e. according to the requirement of an order by order renormalization, or rather to order the operators by their number of derivatives, chiral dimensions, inverse powers of Λ or of 4π. These rules are not generally consistent with each other and they all operate simultaneously. Such a choice can be made upon restricting to a particular kinematic regime or assuming a particular class of UV completions. It is not possible, to our knowledge, to select one of these elements as the unique dominating rationale to order terms in importance for all S matrix elements within the predictive regime of validity of this EFT, and for all possible values of the Wilson coefficients.To see the contradictions that may emerge between different countings, consider the operator ϵ_IJKW^I_μνW^JνρW^Kμ_ρ. This term is not required to reabsorb one-loop divergences of ℒ_0 and it has chiral dimension 6, so it is a NNLT according to Weinberg's counting (ℒ_6 in the χPT series).[This is the case both with the classical notion of chiral dimension in χPT, defined in the previous paragraphs, and with the chiral dimension generalized to the HEFT case, defined inRef. <cit.>. Note that the latter requires the definition of the operator to include a factor g_2^3.] It may be argued that, containing only transverse gauge fields, this operator should not follow the chiral counting, but rather the SMEFT classification, that sets it as a NLT. An alternative observation is that as this operator contains two derivatives more than the gauge kinetic term, which belongs to ℒ_0, this operator can be a NLT even in a chiral approach. Finally, NDA assigns to this term a coefficient 4π/Λ^2, which has both a (4π) enhancement and a Λ^2 suppression. The different counting rules give contradictory estimates in this case because each rule stems from a different assumption.Depending on both the kinematic regime of interest and particular values of Wilson coefficients considered, any one of these arguments may drive the organization of the EFT: for instance in the limit p/Λ≪ v/Λ the derivative expansion dominates over other criteria, but this is not the case in general.It is necessary, whatever power counting is followed, that inconsistencies in the theory are avoided. Classifying as NLT the operators appearing as counterterms for the renormalization of ℒ_0 is required. Even within this category, the physical impact of the invariants can be different and dependent on the kinematic regime to which the HEFT is applied. Two such structures are for instance Q̅_L_μ^μ Q_R and (Q̅_L Q_R)^2. The former has one fermionic current and two derivatives, that correspond to gauge bosons insertions in unitary gauge, the latter is a four-fermion operator. The NDA weight for the single-current term is 1/Λ, and the one for the four-fermion term is (4π)^2/Λ^2. So the two can have a comparable impact in a specific kinematic region, but this isn't the case in general for all possible S matrix elements. There are many intrinsic challenges to a systematic classification of HEFT operators into well defined orders. Some of those are present in ℒ_0 where one can select whether couplings of the physical Higgs to the gauge kinetic terms should be included in ℒ_0 or not. The choice made in Eq. <ref> to define these terms as NLT is phenomenologically sound, but it requires a specific assumption that the transverse gauge fields not being directly coupled to the scalar sector in the UV, leading to a suppressed Wilson coefficient. Once the leading Lagrangian is fixed, the next order Δℒ can be identified following:* All terms required for the one-loop renormalization on ℒ_0 must be included in Δℒ.To this end, the one-loop renormalization of the scalar sector of the HEFT has been worked outin Refs. <cit.>.These results build upon previous results obtained in the absence of the h degree offreedom <cit.>. * Structures that are not strictly required as counterterms, but that receive finite loop contributions from ℒ_0 and/or have a similar field composition or NDA suppression as the counterterms, should also be retained. An example corresponding to finite loops of ℒ_0 is dipole operators, that have the structure ψ̅_L^μνψ_R X_μν. An example of inclusion by similarity in the field composition are four-fermion operators with vector/axial currents (schematically ψ̅γ_μψ ψ̅'γ^μψ'), which are included by analogy with the four-fermion operators with scalar currents (ψ̅_Lψ_R ψ̅'_Lψ'_R) that are required for the renormalization of ℒ_0. * Operator categories that are classified as NLT in at least one of the countingprinciples described above should be included, e.g. the term ϵ_IJKW^I_μνW^JνρW^Kμ_ρ. This point ensures that all the operators that can be relevant in at least one specific scenario are included, preserving the generality of the HEFT description. * The set of operators that form the Δℒ basis should closeunder the use of the EOMs. These rules lead to the construction of the bases of Refs. <cit.>. The characterization of higher orders in the HEFT expansion poses further difficulties due to the mismatch among the expansions in different parameters.§ THE SS-MATRIX, LAGRANGIAN PARAMETERS AND MEASUREMENTSSubtleties that can complicate the interpretation of experimental data in EFTs are clarified by a clear definition of S matrix elements and their distinction from Lagrangian parameters. In this section we review these two concepts for later use and emphasize the distinctions between them and their relation to measurements. The aim of this section is to discuss the physical nature of S matrix elements, the unphysical nature of Lagrangian terms, and to then build up to how the pseudo-observable concept is introduced to attempt to bridge between full fledged S matrix elements and common gauge invariant quantities that appear in many measurements. §.§ SS-matrix elementsConsider a set of interpolating spin-{0,1/2,1 } fields, that are representations ofSO^+(3,1), which we denote schematically as θ = {ϕ,ψ,V^μ}. These fields are used to construct asymptotic perturbative expansions that satisfy a set of global symmetries (G).[The global symmetry is emphasized as the local gauge redundancy does not lead to relations between S matrix elements. Gauging a symmetry does not give any additional conserved charges (and corresponding S matrix relationships) beyond those of the corresponding global symmetry, see Refs. <cit.> for discussion.] We denote the Lagrangian composed of θ, constructed to manifestly preserve G, as ℒ(θ), and the sources of the fields as J_θ. The generating functional W[J_θ] of the connected Green's functions is then defined ase^i W[J_θ] = ∫𝒟θexp[i ∫ d^4 x (ℒ(θ) + θJ_θ) ].Here 𝒟 is the measure of the functional integral. Functional differentiation of W[J_θ] with respect to n fields then defines an n-point time ordered correlation function (or Green's function) between ground states (Ω) of the interacting theory ⟨Ω | T θ(x_1)θ(x_2) ⋯θ(x_n)| Ω⟩ = (-i)^n δ^n W/δ J_θ(x_1) ⋯δ J_θ(x_n).The LSZ reduction formula <cit.> defines the n-point S matrix elements related to these correlation functions as ∏_1^a ∫ d^4 x_i e^i p_i · x_i∏_1^b ∫ d^4 y_i e^i k_i · y_i⟨Ω | T θ(x_1)θ(x_2) ⋯θ(x_a)θ(y_1)θ(y_2) ⋯θ(y_b)| Ω⟩,≃ (∏_i =1^a i √(Z_i)/p_i^2 - m^2_i+ i ϵ) (∏_j =1^b i √(Z_j)/k_j^2 - m_j^2+ i ϵ) ⟨ p_1 ⋯ p_a|S|k_1 ⋯ k_b ⟩.Here n = a+b, and p_i,k_j are four momenta associated via the Fourier transform of each field, which is also associated with a mass term m_i,j and renormalization factor Z_i,j. The ≃ is due to the requirement to isolate each single particle pole experimentally so that each p_i^0 → E_p_i, k_j^0 → E_k_j when treating the particles as asymptotic states.S matrix elements, unlike Lagrangian parameters, directly define the measurable quantities in the theory: the scattering and decay observables.[The S matrix was first introduced in Ref. <cit.>.]S matrix elements conserve overall four momentum and are unitary. The decomposition of an S matrix element into an Lorentz invariant amplitude ℳ is given as ⟨ p_1 ⋯ p_a|S|k_1 ⋯ k_b ⟩ = I + i(2 π)^4 δ^4(p_i - k_j)ℳ/√(2 E_i)√(2 E_j),with implicit sum over i,j. For most scattering observables, 2 → n processes are sufficient to consider, which are given asd σ = (2π)^4 |ℳ|^2/4 √((p_1 · p_2)^2 - m_1^2 m_2^2) δ^4 (p_1 + p_2 - ∑_j k_j) ∏_j=1^n d^3 k_j/(2 π)^3 E_j.Similarly, particle decays of an initial state with momentum (P) and mass (M) into ∑_j k_j final states are given byd Γ = (2π)^4/2 M|ℳ|^2δ^4 (P - ∑_j k_j) ∏_j=1^n d^3 k_j/(2 π)^3 E_j.Measured event rates correspond to d σ or d Γ integrated over a measured phase space volume. Such phase space integrations can be highly non-trivial, see Refs. <cit.> for excellent discussions on overcoming this technical hurdle. The relation between the differential scattering and decay observables and S matrix elements is direct. Intuitively, changing field variables on the path integral defining the generating functional in Eq. <ref> can be expected to have no physical effect on the S matrix elements derived from it, as the field variables are essentially dummy variables used to define the G preserving perturbative expansions of the correlation functions in the path integral formulation of the QFT. This intuition is correct, and not violated by quantum corrections. Such variable changes do modify the Lagrangian terms and the source terms in a correlated manner, which can be used to arrange the Lagrangian into a particular form. This understanding of field redefinitions is formalized in what is known as the Equivalence theorem <cit.>, which is a precise formulation of the invariance of S matrix elements under G preserving field redefinitions for renormalized quantities. See Refs. <cit.> for related discussion. These formal developments are all focused on gauge independent, and G preserving, field redefinitions. §.§ Lagrangian parametersUnlike S matrix elements, individual Lagrangian terms are not invariant under field redefinitions <cit.>. Lagrangian parameters are neither directly, nor trivially related to S matrix elements, or measured observables. In modern times this understanding has been advanced to a level where efforts are underway to avoid Lagrangian formulations completely in highly symmetric field theories.[See Refs. <cit.> for discussion on this approach.]Several historical examples of the equivalence of Lagrangians of a distinct interaction and field variable form indicate that a particular interaction term in a Lagrangian should not be mistaken for a physical S matrix element. The famous equivalence of the Thirring <cit.> and sine-Gordon models under bosonization proven by Coleman <cit.> (see also Ref. <cit.>) illustrates the equivalence of a particular quantum theory with purely interacting fermions and bosons in each case. The Lagrangians are given by ℒ_T = ψ̅ (i d - m) ψ - g/2 (ψ̅γ^μψ)^2, ℒ_sG = 1/2∂^μϕ∂_μϕ + α/β^2cos (βϕ).The couplings are identified as β^2/4 π = 1/(1+ g/π). These Lagrangians describe the same physics despite the drastically different field content and interaction terms <cit.>.The practical consequences of Lagrangian terms being distinct from physical S matrix elements is drastically different in the SM and in the SMEFT/HEFT field theories. In the case of an operator basis choice in the SMEFT or HEFT, the general non-physical nature of individual Lagrangian terms manifests itself in the freedom to pick different sets of Lagrangian terms to represent the same physics. This occurs via (1/Λ^2) small field redefinitions in the SMEFT[Again, only those that are gauge independent.] and the EOM play an essential role in equating Lagrangian terms that are naively distinct.In the SM, Λ→∞, so there is no direct equivalent of these small field redefinitions. The same point on the nature of Lagrangian terms is made by the use of auxiliary fields. When performing gauge fixing, the BRST formalism <cit.> introduces the non-propagating Lautrup-Nakanishi <cit.> auxiliary field B^a into a non-abelian Lagrangian as ℒ_BRST = -1/4 F_a^μ νF^a_μ ν + ψ̅ (i D - m) ψ + ξ/2 (B^a)^2 + B^a ∂^μ A_μ^a + c̅ (- ∂^μ D_μ^ac)c^c.Performing the Gaussian functional integral over the B^a field variable returns ℒ_FP = -1/4 F_a^μ νF^a_μ ν + ψ̅ (i D - m) ψ - 1/2ξ(∂^μ A_μ^a)^2 + c̅ (- ∂^μ D_μ^ac)c^c.The two Lagrangians give equivalent S matrix elements despite having a distinct form and naively differing field variables. Another example is supplied by the Hubbard–Stratonovich transformation <cit.> where a non-dynamical scalar field is “integrated in”. In this case the curvature R^2 action of a gravitational theory (i.e. Starobinsky inflation <cit.>) ℒ_inf = √(ĝ)[-M_p^2/2R̂ + ζR̂^2 ],is equivalent to the transformed Lagrangian with the auxiliary scalar field ℒ'_inf = √(ĝ)[-M_p^2/2R̂ - 2α Φ^2R̂ - Φ^4 ].This can be seen by performing a Gaussian functional integral with ζ = α^2. Performing the Gaussian integrals modifies the field variables to a reduced (less redundant) set that still produce the same S matrix. In these cases the auxiliary field variables are non-propagating and do not have Fock spaces associated with them. So the difference in the Lagrangians is a vanishing difference for the external states labeling S matrix elements. This is the analogy to the vanishing of the difference between operators equivalent by use of the on-shell EOM (as in Eq. <ref>) essential to choosing an operator basis in the EFT.[We stress that the point here is on the non-physical nature of Lagrangian parameters in general not that field redefinitions leave the Lagrangian alone invariant. In general this is not the case as discussed in Section <ref>.]The fact that Lagrangian parameters are not physical quantities is not changed when working with mass eigenstate fields, or unitary gauge, in any way. §.§ Scattering measurements and renormalizable/non-renormalizable theories The distinction between Lagrangian terms and S matrix elements is crucial in the SMEFT and HEFT. It is also the case that using naive physical intuition to assign SM Lagrangian terms a naive physical meaning classically, although formally incorrect, is a reasonable rough approximation for a subset of parameters in the SM. Essentially the naive physical intuition at work is accidentally supported by the renormalizable nature of the SM Lagrangian, the small perturbative corrections in the EW sector of the SM, and the related fact that the SM unstable states have small widths compared to their masses. We discuss each of these points in turn and compare the differences that appear when extending SM studies to the SMEFT and HEFT.§.§.§ The SM case Extracting the SM Lagrangian parameters {v, g_1, g_2, g_3, Y_i} from measurements of S matrix elements at tree level in the SM is generally done in the presence of significant experimental correlations and a degeneracy of parameters present in the mapping to the Lagrangian. This degeneracy is first an issue at the level of perturbative and non-perturbative corrections to the Lagrangian due to a quirk of renormalizable theories. In the renormalizable SM on-shell vertices are unique in the interaction Lagrangian, and off-shell gauge coupling vertices are related by global symmetries to on-shell coupling parameters. This is no longer trivially the case when one transitions to EFT extensions of the SM.An unusual case of a Lagrangian parameter extraction in the SM, is the extraction of v from μ^- → e^- + ν̅_e+ ν_μ. The physics at work is due to the SM being a weakly coupled renormalizable theory describing a Higgsing of SU_L(2) × U_Y(1) → U(1)_em, in conjunction with neutrinos being weakly interacting particles. Extracting v in the SM is unusual as it can be extracted from a ψ̅_Lψ_L →ψ̅_Lψ_L process at low energies where an effective Lagrangian of the form ℒ_GF = - 4 G_F/√(2) V_ijV_kl^⋆ (ψ̅_i γ^μ P_L ψ_j)(ψ̅_k γ^μ P_L ψ_l), 4 G_F/√(2) = g_2^2/2M_w^2 = 2/v^2.is used.[Remarkably, interest in Fermi theory has been acute in recent years in the literature, see Refs. <cit.>. Our discussion is most consistent with Refs. <cit.> due to the important role that the CKM and PMNS-matrix plays, and the standard understanding of EFT adopted in this review.] Here V_ij is the CKM or PMNS-matrix in the case of quarks or leptons. The gauge couplingg_2 cancels as a direct consequence that the SM has a Higgs mechanism generating the gauge boson masses. This cancellation is not generic in EFTs but is a rather unique result of the SM. Furthermore, as individual asymptotic eigenstates of neutrinos are not experimentally identified in this decay process, practical measurements of μ^- → e^- + ν̅_e+ ν_μ sum over all neutrino species. Unitarity of the PMNS-matrix then leads to a direct extraction of v from this decay, which is another unique feature in extracting v from μ^- → e^- + ν̅_e+ ν_μ.Subsequently when using the input parameter set {Ĝ_F,M̂_W,M̂_Z} mapping these measured quantities to the Lagrangian parameters g_1,g_2 can be done with a m_Z pole scan as performed at LEP <cit.>, and a study of transverse W mass variables as performed at the Tevatron <cit.> or similar studies that have begun at LHC <cit.>. This set of Lagrangian parameter extractions is afflicted with significant experimental correlations.* In the case of the extraction of m̂_Z at LEP, despite the presence of a resonance peak, the Lagrangian parameters are extracted from a fit that simultaneously defines the pseudo-observables {m̂_Z,Γ_Z, σ_had^0,R_e^0,R_μ^0,R_τ^0 }<cit.>. See Section <ref>. * In the case of m̂_W, due the calibration of the electromagnetic calorimeter to Z decays, the experimental extraction is effectively an extraction of the ratio m̂_W/m̂_Z<cit.>. These extractions of Lagrangian parameters from S matrix elements take place in the perturbative expansion of the EW theory. The precision of these measurements require that higher order corrections be included. It is only because the EW interactions are perturbative with typical leading loop corrections ≲𝒪(1%) that the resulting parameter degeneracy introduced leads to small perturbations of the highly correlated central values extracted in a naive tree level analysis of {v,g_1,g_2}.If an input set of the form {Ĝ_F,,M̂_Z} is used then the challenge of parameter degeneracy also appears due to the requirement to run the extracted Lagrangian parameterthrough the hadronic resonance region. Using an input parameteractually corresponds to a simultaneous input set of {,∇α} as extractions ofare dominated by q^2 → 0 measurements determined by probing the Coulomb potential of a charged particle. The low scale measurement extracts this parameter with the mapping to the two point functions Π^ab - i[4π α̂(q^2)/q^2]_q^2 → 0≡- ie̅_0/q^2[1 +ReΣ^AA(m_Z^2)/m_Z^2 - ∇α].The finite terms in the low scale matching that are the largest effect are due to the vacuum polarization of the photon in the q^2 → 0 limit. This unknown term is given by the last two terms on the right hand side of Eq. <ref>, with the notation ∇α = [ ReΣ^AA(m_Z^2)/m_Z^2- [Σ^AA(q^2)/q^2]_q^2 → 0].This quantity parameterizes contributions to the two point function that have to be simultaneously determined to define . For further discussion see Refs. <cit.>. The uncertainty on ∇α due to this parameter degeneracy completely dominates the uncertainty onwhen it is used in LHC applications.The requirement of simultaneous extractions and highly correlated Lagrangian parameters from S matrix elements in conjunction with other non-perturbative unknown parameters also returns in the cases of {g_3,Y_i}:* Extractions of g_3 from studies of e^+ e^- event shapes simultaneously extract α_s and the leading non-perturbative parameter from thrust distributions <cit.>. Similarly the extraction of g_3 from the event shape C parameter is a simultaneous extraction of α_s and a leading non-perturbative parameter <cit.>. * The extraction of m_b from inclusive B̅→ X_c ν̅ℓ decays using HQET is a simultaneous extraction of m̂_b and the leading non-perturbative corrections <cit.>. * The extraction of light quark masses {m̂_u,m̂_d,m̂_s,m̂_c} using Lattice QCD occurs with a simultaneous lattice cut-off parameter and is related to MS masses in perturbation theory <cit.>. The same point holds for extractions of m̂_c<cit.>. Precise extractions of quark mass parameters {m̂_u,m̂_d,m̂_s} in χPT are extractions of quark mass ratios <cit.>, not single Lagrangian terms. Degeneracy and correlations in the space of Lagrangian parameters that are extracted from S matrix elements is essentially unavoidable. This is a reflection of the fact that Lagrangian parameters are not physical. This remains the case despite a protection of the SM against parameter degeneracy due to the nature of the Higgs mechanism in the SMEFT. As ⟨ H^† H ⟩ = v^2/2, interaction terms that differ by dimension d=2 can lead to a degenerate interaction term of the remaining fields when expanding around the vacuum expectation value. As ℒ_SM has d ≤ 4 this degeneracy is avoided in the SM, as an accidental simplification due to renormalizability.§.§.§ The EFT case The challenge of correlations and parameter degeneracy in relating Lagrangian terms to S matrix elements is serious in the SM, and this challenge is even more acute in EFTs. The new parameters introduced in the Taylor series defining the EFTs are local (see Eq. <ref>), and several parameters appear simultaneously in the power counting expansions. The presence of a Higgsed phase in an EFT now acts to increase parameter degeneracy.Examining the field redefinition freedom that exists in defining a ℒ_6 basis further reinforces the gap between the S matrix elements and the EFT parameters. Generally, when constructing a ℒ_6 basis, a version of an on-shell effective field theory is constructed <cit.>. In this approach derivative terms are systematically removed (if possible) and kinetic terms are canonically normalized. This shifts EFT corrections to vertices, which increases parameter degeneracy in how ℒ_6 corrections modify SM predictions. The natural expectation is a highly correlated fit space, and this is indeed frequently found <cit.>.[See also Refs. <cit.>.]To expand on the example in the previous section in the context of an EFT extension of the SM, the generic conclusion is that the fit spaces of Lagrangian parameters are even more correlated. The accidental nature of the extraction of v in the SM now projects onto a three-fold Lagrangian parameter degeneracy in the U(3)^5 limit asℒ_G_F ≡-4𝒢_F/√(2) (ν̅_μ γ^μ P_L μ) (e̅ γ_μ P_L ν_e), -4𝒢_F/√(2) =-2/v̅_T^2 - 4Ĝ_Fδ G_F,where the leading order shift result is <cit.>δ G_F=-1/4Ĝ_F (C_llμ ee μ +C_ll e μμ e) + 1/2Ĝ_F(C^(3)_Hl ee+C^(3)_Hlμμ).Using a {, m̂_Z, Ĝ_F} input parameter set one finds the results listed in the Appendix for Z,W vertex terms in the SMEFT. The Wilson coefficients that now appear simultaneously in ℒ_6 in ψ̅ψ→ψ̅ψ scattering in the SMEFT (in addition to the SM couplings) are[The U(3)^5 limit used here treats the two flavour contractions (C_llδ_mn δ_op + C'_llδ_mp δ_no)(l̅_m γ_μ l_n)(l̅_o γ^μ l_p) as independent <cit.>.] C̃_i ≡v̅_T^2/Λ^2{C_He, C_Hu, C_Hd, , , , , C_HWB, C_HD, C_ll, C_ll'}.Furthermore, beginning at ℒ_6, an EOM degeneracy appears between these parameters in ψ̅ψ→ψ̅ψ and a subset of ψ̅ψ→ (ψ̅ψ)^n amplitudes due to ⟨ H^† H ⟩ being dimension two, and the SMEFT having a Higgs mechanism. See Section <ref> for further discussion on this resulting reparameterization invariance.[The first appearance of a bilinear in H outside the Higgs potential in the SM appears at dimension five, see Eqn. <ref>. This Lagrangian term does not have a direct degeneracy with parameters already present up to ℒ_4 in the SM, due to the interplay of the global symmetry structure of the SMEFT operator expansion with operator dimension <cit.>.] Extracting {g_3, m̂_b } from lower energy data introduces an even larger set of ℒ_6 parameters.Highly correlated Wilson coefficient fit spaces are a central feature of the SMEFT. Even mild assumptions about parameter correlations (or lack thereof) have a significant impact on the constrained Wilson coefficient space as a result. §.§ The idea of pseudo-observables The previous sections can be summarized as: S matrix elements directly correspond to physical observables, and Lagrangian parameters do not. Common Lagrangian parameters do feed into many observables in a manner that is sometimes consistent with naive classical reasoning, despite classical intuition being misplaced in a QFT. A common reason that this dichotomy between formally correct field theory and naive classical reasoning persists for collider physics studies is frequently the fact that the unstable states in the SM arenarrow, i.e have the property that Γ/m≪ 1.The ratios of the widths (Γ_V) of the unstable V= {W,Z} bosons to their masses (m_V) are <cit.>Γ̂_Z/m̂_Z = 2.4952/91.1876∼ 0.03,Γ̂_W/m̂_W = 2.085/80.385∼ 0.03.The width of the Higgs is bounded (in the SM) to be <cit.>Γ̂_h/m̂_h≲0.013/125.09∼ 10^-4.The constraint on the Higgs width is model dependent <cit.> and corresponds to a bound on Γ̂_h in conjunction with κ_g and κ_t. Assuming that the width of the Higgs is only perturbed from its SM value in the SMEFT, one can expand in Γ̂_h/m̂_h.Narrow widths effectively factorize a full amplitude contributing to an S matrix element up into sub-blocks. Frequently, a naive classical intuition is also assigned to these sub-blocks, at least in a tree level calculation.[See Ref. <cit.> for a discussion on the explicit and implicit assumptions in the narrow width approximation.] An unfortunate side effect of the highly correlated nature of the Wilson coefficient space in SMEFT studies is that such factorizing up of observables with the narrowness of the SM states, can break flat directions in the Wilson coefficient space in an inconsistent manner, and bias conclusions by orders of magnitude <cit.>.The phase space of the scattering events is populated in a manner that usually is dominated by the impact of the narrow resonances, unless selection cuts are applied to remove all resonant contributions. This remains true studying EFT extensions of the SM in colliders. The narrowness of the SM states is actually fortunate for EFT studies at LHC, as while simultaneously avoiding the large non-resonant QCD backgrounds, EFT approaches can exploit this fact and gain a significant benefit – so long as the exploitation of the narrow width effect occurs in a well defined manner. §.§.§ The naive narrow width expansionThe most naive attempt to use the narrowness of the SM states to expand is the naive narrow width approximation. Consider an s-channel scattering ψ̅ψ→ S(s)^⋆→ψ̅ψ where S ={h,W,Z}. Expanding around the limit Γ_S/m → 0 is subtle. It corresponds to a series around the kinematic point in the amplitude where some propagators vanish due to an intermediate state going “on-shell” at p^2 =m^2. It is not a trivial expansion in the Γ_S/m ratio at the amplitude level, as this limit must be taken in the sense of a distribution over the phase space.Due to the Feynman propagator prescription <cit.> defining the Green's function when intermediate states go on-shell corresponds to a discontinuity in the imaginary part of the Feynman diagram. Cutkosky <cit.> developed formally the proof that the discontinuity across a branch cut when a stable intermediate state goes on-shell in a Feynman diagram is given by the replacement of the propagator dp^2/p^2 - m^2 +iϵ→ - 2πi δ(p^2 - m^2) dp^2.This approach can be extended to the case where the intermediate state is unstable, leading to a generalization of the optical theorem. The most naive version of the resulting reasoning is the narrow width approximation. Define the particle mass by the conditionm^2 - m_0^2 -Re Σ(m_0^2) = 0,where m_0 is the bare mass parameter. The propagator pole is then shifted off of the real axis if the intermediate state decays, in a manner that approximates a Breit-Wigner distribution formula ∝ 1/(p^2 - m_0^2 + i m_0 Γ_S(p^2)). If Γ_S/m ≪ 1, it can be approximated as a constant Γ_S(p^2) ≃Γ_S for the phase space events populated by scattering through the resonance peak. Then, in a distribution sense where the phase space is integrated over a sufficiently inclusive region, the appearance of a propagator squared in a full cross section can be simplified. Shifting the zero point of the symmetric d p^2 distribution, and performing the resulting integral gives ∫_- ∞^∞d p^2/(p^2 - m_0^2)^2 + m^2_0Γ_S^2= ∫_- ∞^∞d p^2/(p^2)^2 + m^2_0Γ_S^2 = 1/m_0Γ_S∫_-∞^∞dx/1+ x^2 = π/m_0Γ_S.To utilize this expansion for an on-shell intermediate state contributing to an S matrix element we replace d p^2/(p^2 - m_0^2)^2 + m^2_0Γ_S^2→π/m_0Γ_S δ(p^2 - m^2)dp^2.For a total ψ̅_k ψ_l → S(s)^⋆→ψ̅_i ψ_j cross section one obtains σ = 1/32π s ∫d^3 k_i/(2 π)^3 E_jd^3 k_j/(2 π)^3 E_j |𝒜(ψ̅_k ψ_l→ S)|^2 |𝒜(S →ψ̅_i ψ_j)|^2 π/mΓ_S δ(s - m^2),= σ(ψ̅_k ψ_l→ S)Br(S →ψ̅_i ψ_j).This is the narrow width approximation. Using this approximation for on-shell production corresponds to naive classical intuition and factorizes up phase space in a manner that is a significant numerical benefit in evaluating cross sections. The presence of narrow widths of the SM unstable states should be exploited to aid in studying the SMEFT and the HEFT at LHC. Attempting to utilize the naive narrow width approximation to achieve this end, is problematic. Narrow width approximations are subject to severe limitations, as should be evident from the derivation. The limitations include* The derivation is limited to on-shell kinematics and a tree level exchange, and not performed in the presence of experimental cuts limiting the phase space. * Higher order corrections and renormalization is not obviously consistent with the derivation, it is also not obvious that the approach is gauge invariant at higher orders. * The factorization of the cross section into sub-blocks for the limited tree level exchange does not change the Hilbert space of the theory, and the excited unstable S state is still not an external particle. Technically the unstable particles do not allow a plane wave expansion as asymptotic states; their energies are imaginary and the asymptotic plane wave expansions either diverge or vanish. * It is unclear how to formally justify using replacements such as in Eq. <ref> for other processes, even ψ̅ψ→ψ̅ψ ψ̅ψ where one S state is on-shell while a second S state is off-shell for the same region of phase space. The formal developments of Cutkosky <cit.> for stable intermediate states do not directly justify replacing with Eq. <ref> in arbitrary Feynman diagrams for unstable intermediate states, and all points in phase space. * In the SMEFT or HEFT cases, the corrections to the narrow width approximations are comparable to the order of the SMEFT corrections to the SM. One expects in many cases Γ_W,Z/M_W/Z∼v̅_T^2/Λ^2, p^2/Λ^2. If a narrow width prescription adopted is ambiguous or essentially an arbitrary scheme choice this can significantly impact the allowed parameter space of the Wilson coefficients in the EFT using off-shell data, biasing global fit conclusions. *Formally, the narrow width expansion and SMEFT expansion do not commute when the {,m̂_Z, Ĝ_F} input parameter scheme is used. To avoid an ambiguity, the ordering of the expansions should be defined, see Ref. <cit.>. It is known that the ambiguity is experimentally constrained to be small, even in EFT extensions of the SM <cit.>. This affects the results for the top width given in the Appendix (Eq. <ref>).The challenges of unstable states in field theory are well known, see Refs. <cit.> for formal developments. The modern approach of utilizing the narrowness of the SM unstable states is still challenged by these issues, but some progress has been made. Modern efforts to decompose a cross section into gauge invariant sub-blocks exploiting the narrowness of the SM states avoid the most naive narrow width approximation.§.§.§ Double pole expansionsConsider the process with the next level of complexity compared to ψ̅ ψ→ S^⋆→ψ̅ ψ. When two propagators are present one has to determine an expansion exploiting narrow width enhancements for the process ψ̅ ψ→ S^⋆(s_12) S^⋆(s_34) →ψ̅ ψψ̅ ψ.An extension of the naive narrow width approach can be developed directly by expanding the amplitude result around the two S poles, assuming the intermediate states are stable, giving the decomposition 𝒜(s_12,s_34)= 1/s_12 - m̅_W^21/s_34 - m̅_W^2 DR[s_12,s_34, d Ω] + 1/s_12 - m̅_W^2 SR_1[s_12,s_34,d Ω], + 1/s_34 - m̅_W^2 SR_2[s_12,s_34,d Ω] +NR[s_12,s_34,d Ω].Here DR, SR_1,2 and NR refer to the doubly resonant, singly resonant and non-resonant contributions to the amplitude, respectively, and Ω refers to all angular dependence defined in an s_12,s_34 independent manner. This expansion is defined as in Refs. <cit.> and is not a trivial Feynman diagram decomposition for the off-shell phase space, but a reorganization of the full amplitude result around the physical poles[One can understand that the situation is more subtle when considering off-shell production as double resonant diagrams contributing to Eq. <ref> are not gauge invariant as a subset of the full amplitude. The difference in axial and 't Hooft-Feynman gauge expressions for the doubly resonant diagrams generates a single-resonant diagram process<cit.>.] in a Laurent expansion. The residues of the poles are gauge invariant as they can be experimentally measured (at least in principle). This factorization of the process into gauge invariant sub-blocks can be considered an example of a pseudo-observable decomposition, with the individual terms in Eq. <ref> being pseudo-observables.Adding the width of the unstable S state into these pole expressions can be performed after the residues are determined. This approach, with perturbative corrections, underlies the SM prediction of the ψ̅ ψ→ W^⋆(s_12) W^⋆(s_34) →ψ̅ ψψ̅ ψ process in Refs. <cit.>. The systematic perturbative improvement of a double-pole decomposition with higher order radiative corrections and higher order terms in the Γ/m expansion is technically challenging. The challenge arises from the need to calculate soft photon radiative corrections that are non-factorizable as well as factorizable in the sense of the pole decomposition.In the case of ψ̅ ψ→ W^⋆(s_12) W^⋆(s_34) →ψ̅ ψψ̅ ψ the non-factorizable corrections were small compared to the experimental uncertainty at LEP <cit.>. This conclusion does not directly translate to the LHC experimental environment, or future colliders where pseudo-observable approaches face significant challenges from the need to characterize radiative corrections.§.§.§ Complex mass schemeProbing for small SMEFT or HEFT corrections to the SM predictions argues that improvements beyond a naive narrow width or tree level double pole approximation could be required in the long term LHC program. A systematically improvable theoretical framework is required to determine such corrections. A preferred approach is to expand around unstable particle poles relying[This expansion is also known in some literature as a multi-pole expansion. This expansion is distinct from the expansion discussed in Section <ref>.] on the generalization of the idea of particle mass introduced in Refs. <cit.> and developed for EW applications in Refs. <cit.>. The key observation is that an unstable particle corresponds to a pole on the second Riemann sheet of an analytic continuation of the S matrix.[For a detailed discussion on the formal development of this analytical continuation for LHC processes, see Ref. <cit.>.] The complex mass of a state S is the solution ofs - m_S^2 + Σ_SS(s) = 0,with renormalized mass m_S and self energy Σ_SS and the negative imaginary solution is taken <cit.>.[A drawback when considering the SMEFT is that this scheme is tied to on-shell renormalization schemes, while EFT studies generally use MS subtraction.] Use of the Nielsen identities <cit.> establishes that the position of the pole is gauge parameter independent in the SM and the SMEFT. The decomposition of a propagator square in the cross section can be augmented from Eq. <ref> to be d p^2/(p^2 - m_0^2)^2 + m^2_0Γ^2→π/m_0Γ δ(p^2 - m^2) +PV[1/(p^2 - m_0^2)^2] dp^2,where PV indicates a principal value. Simultaneously factorizing up the phase space in a manner consistent with this pole decomposition allows a systematic analytical continuation of amplitudes into the complex mass scheme at tree level. In the SM this approach was also pushed to report full one loop corrections to ψ̅ ψ→ψ̅ ψψ̅ ψ, see Ref. <cit.>.The complex mass scheme treats resonant and non-resonant regions of phase space in a unified manner and one loop calculations have a difficulty similar to one loop calculations in the SM. It is reasonable to expect that this approach could be extended to one loop results in the SMEFT, but such full calculations have never been carried out to date in any LHC process involving the Higgs boson. As no explicit expansion in Γ/m need be performed using the complex mass scheme, the application of this approach to SMEFT studies is favored to systematically develop pseudo-observables <cit.>.An alternative approach is to use unstable particle EFT that was developed in Refs. <cit.>. It is unclear if these results can be extended to a SMEFT study. It is also unclear if the comparative (one loop) benefits of the complex mass scheme over the unstable particle EFT present in the SMwill persist in SMEFT studies. For more discussion comparing these approaches in the SM, see Ref. <cit.>. §.§ Basics of EFT studies at colliders Practically implementing EFT studies using data from modern colliders is challenging on several fronts. A large number of free parameters is characteristic of the EFTs. A significant parameter degeneracy (until measurements are combined) is also a fundamental feature. Utilizing a well defined naive narrow width approximation is an essential step to factorizing up S matrix elements and reducing the complexity of the analysis to a manageable level. In addition, one can simultaneously exploit the relative scaling of leading corrections in the EFTs in how phase space is populated, IR symmetry assumptions in the EFT, and the fact that parameters that violate symmetries approximately preserved in the SM interfere in a numerically suppressed fashion. The resulting reduction in parameters enables a systematic EFT program using LHC, and lower energy, datato be practically carried out.The relative population of phase space due to resonance enhancements or suppressions is important at Hadron colliders, where an experimental measurement is always a combination of signal and background processes. Exploiting this effect in EFT studies was recently discussed more systematically in Ref. <cit.>, and we largely reproduce this discussion here.A general scattering amplitude is depicted in Fig. <ref>, and shows the decomposition around the physical poles of the narrow propagating bosons B of the SM 𝒜 = 𝒜_a(p_1^2, ⋯ p_M^2)/(p_1^2 - m_B_1^2 + i Γ_B_1 m_B_1) ⋯ (p_N^2 - m_B_N^2 + i Γ_B_N m_B_N),+ 𝒜_b(p_1^2, ⋯ p_M^2)/(p_1^2 - m_B_1^2 + i Γ_B_1 m_B_1) ⋯ (p_N-1^2 - m_B_N-1^2 + i Γ_B_N-1 m_B_N-1), + ⋯ + 𝒜_j(p_1^2, ⋯ p_M^2).If experimental selection cuts are made so that the process is numerically dominated by a set of leading pole contributions of narrow bosons B, then this phase space volumeΩis (d σ/Ω̣)_pole ≃ (d σ_SM/Ω̣)^1[1+ 𝒪(C_iv̅_T^2/g_SMΛ^2) + 𝒪(C_jv̅_T^2 m_B/Λ^2Γ_B)], + (d σ_SM/Ω̣)^2[1+ 𝒪(C_k p_i^2/g_SMΛ^2)].The differential cross sections (d σ_SM/Ω̣)^1,2 are distinct in each case, see Ref. <cit.> for further details. The interference with a complex Wilson coefficient denoted C, that occurs when a resonance exchange is not present compared to the leading resonant SM signal result (shown in Fig. <ref> d)) scales as|𝒜|^2∝ (g_SM^2/(p_i^2 - m_B^2 + iΓ(p) m_B) + C/Λ^2) (g_SM^2/(p_i^2 - m_B^2 + iΓ(p) m_B) + C/Λ^2)^⋆⋯∝ [g_SM^2/(p_i^2 - m_B^2)^2 + Γ_B^2 m_B^2 + (p_i^2 - m_B^2) (C/Λ^2 + C^⋆/Λ^2) - i Γ_B m_B (C^⋆/Λ^2 - C/Λ^2)/(p_i^2 - m_B^2)^2 + Γ_B^2 m_B^2] ⋯ In the near on-shell region of phase space (√(p_i^2) - m_B ∼Γ_B), the SMEFT then has the additional numerically subleading corrections(d σ_SM/Ω̣)^1𝒪(Γ_B m_B{ Re(C), Im(C)}/g_SM^2Λ^2) + (d σ_SM/Ω̣)^2𝒪(Γ_B m_B{ Re(C), Im(C)}/g_SM^2Λ^2) ⋯ For this reason, the numerical effect of the parameters not resonantly enhanced are relatively suppressed by (Γ_B m_B/v̅_T^2) { Re(C), Im(C)}/g_SMC_i, (Γ_B m_B/p_i^2) { Re(C), Im(C)}/g_SM C_k,This relative numerical suppression occurs in addition to the power counting in the SMEFT and the combination of these two suppressions is what is experimentally relevant.In addition to maximally exploiting resonance enhancements or suppressions of parameters other IR assumptions can be directly made. Examples of such assumptions are* Symmetry assumptions made on the SMEFT or the HEFT operators. Typically these are global symmetry assumptions on Baryon or Lepton number conservation, U(3)^5 flavour symmetry or a flavour subgroup. Assumed IR symmetries lead to relations between scattering amplitudes in the EFT, and hence lead to constraints even if the symmetry is spontaneously broken. Weinberg refers to such assumptions as “algebraic symmetries”<cit.>, as they lead to algebraic relations between S matrix elements. Note that the IR limit of the full theory by definition is reproduced in the EFT. For this reason, the assumption of an algebraic symmetry in an EFT can directly enforce a UV class of theories which is required for matching. The distinction of this assumption being an IR constraint on the EFT itself, is an important conceptual point. * Taking the Y_u,d,c,s→ 0 limit in a process.[The dependence on these parameters referred to here is due to ℒ_SM and not a normalization choice on operator Wilson coefficients.] * Neglecting higher order SMEFT loops, i.e. those involving the SM fields and ℒ_6 that are perturbative corrections determined in the IR EFT construction.Finally, we note that when a parameter is retained in the EFT that violates a symmetry approximately preserved in the SM, the interference term is still numerically suppressed. This fact can also be used to neglect parameters that are numerically small in contributions to measured observables. Such numerical suppressions affect a large number of parameters in Class 5, 6, 7 (in Table <ref>) for flavour changing neutral current contributions that interfere with small GIM suppressed processes in the SM. Numerical suppressions of this form are also present for interference between the SM and the operators of Class 5, 6, 7 that introduce CP violation, see Refs. <cit.> for recent studies. Operators that introduce chirality flips of the light SM fermions are also suppressed when interfering with the SM, leading to the possible neglect of some parameters in dipole operators in Class 6, and contributions from right handed currents induced bu 𝒬_Hud in Class 7<cit.>. Taking all of this into account reduced sets of parameters can be well motivated in global fits in the SMEFT, see Ref. <cit.> for more discussion.IR assumptions, or simplifications of this form due to kinematics in scattering processes suppressing dependence on the C_i, can be made so long as a theoretical error for the SMEFT is introduced to accommodate the neglected higher order terms that violate the assumption/simplification. In contrast, UV assumptions (setting C_i = 0, dropping operators and using an incomplete basis, or assuming C_i/C_j ∼ 16 π^2 etc.) are dangerous. The EOM makes it non-trivial and non-intuitive to determine how such assumptions modify the SMEFT framework into an alternate consistent field theory formulation. § THE LEP EXAMPLEWe now turn to the interpretation of LEP data in EFT extensions of the SM, primarily focused on a SMEFT interpretation, utilizing the results of the previous sections to frame this discussion. Interpreting LEP data in EFT extensions of the SM is illustrative of the challenges that the LHC experimental program will face long term in enabling or disabling model independent re-interpretations of its results. It is also informative as to how a program of decomposing experimental data into pseudo-observables (PO) and mapping to EFT extensions of the SM has comparative advantages and disadvantages if LHC data reporting attempts to follow the same path as LEP, despite the very different collider environment. Here we illustrate these challenges by comparing EFT and PO interpretations of the legacy LEPI-II results. §.§ LEPI pseudo-observables and interpretation. The LEPI pseudo-observables are inferred from a data set that is a scan through the Z pole including 40 pb^-1 of off-peak data with 155 pb^-1 of on-peak data. The narrowness of the Z is directly exploited in the LEPI PO set definitions, as summarized in the Appendix. The LEPI PO results is an ideal case of model independent reporting of experimental results with numerous consistency checks on SM assumptions used to extract and define the PO. This fortunate result was enabled by the clean LEPI collision environment, with known e^+ e^- initial states scattering through a Z resonance at relatively low and fixed energies. The resulting legacy data reporting still enables EFT interpretations to be revisited and systematically improved over the years. §.§.§ Checking the SM-like QED radiation field assumption at LEPLEPI data is summarized in Ref. <cit.>. The LEPI pseudo-observables sets include flavour universal and flavour non-universal experimental results. Leading order contributions to the resonant exchanges and the photon emissions are shown in Fig. <ref>. We will restrict our detailed discussion to the flavour universal PO results. The raw experimental results of the LEP pole scan through the Z resonance are deconvolved with initial and final state photon emissions being subtracted out, under a SM-like QED radiator function H^tot_QED assumption. The radiator function is calculated to third order in the SM in QED and the data is deconvolved using this SM function. The measured pole scan is treated as a convolution of a electroweak kernel cross section σ_ew(s) with H^tot_QED as <cit.>σ(s) = ∫^1_4 m_f^2/sdz H^tot_QED(z,s)σ_ew(z s).Radiator functions are also applied to deconvolute the forward and backward cross sections σ_F/B used to define the A_FB pseudo-observable extractions. The effect of this deconvolution is very dramatic. The peak is modified by 36 % and its central value is shifted by 100 MeV, which is fifty times larger than the quoted error on m̂_Z<cit.>. The deconvolved cross section is then mapped to the PO set {m̂_Z,Γ_Z, σ_had^0,R_e^0,R_μ^0,R_τ^0 } given in Table <ref>, and a subsequent mapping from PO to EFT parameters at tree or loop level is then made. An immediate challenge to this procedure is the possibility that the QED radiator function H^tot_QED is modified transitioning to the SMEFT in a manner that dramatically biases the results. A modification of the current ℒ_A,eff=√(4 πα̂)[ Q_xJ_μ^A, x] A^μ,for x = {ℓ,u,d} due to SMEFT corrections at tree level is not present ifis used as an input parameter in a global fit. The modification of the dipole moment of the electron still leads to a potential bias to H^tot_QED. The SMEFT electron dipole moment is given by <cit.> ℒ =ev/√(2)(1/g_1 C_e B rs - 1/g_2 C_e W rs ) e_rσ^μν P_R e_sF_μν+h.c.where r and s are flavour indices. Under a U(3)^5 assumption dominantly broken by the SM Yukawas C_e B,C_e W∝ Y_e, yielding an effective suppression by a small fermion mass for the LEP events. Such dipole insertions contribute to the S matrix element of e^+ e^- → e^- e^+ through a direct contribution to a γ exchange between the initial and final state, and also through modifying the external legs of the process in the LSZ formula in a disconnected contribution, similar to the case of photon emissions in the effective Hamiltonian for inclusive B̅→ X_sγ<cit.>. The SMEFT result of this form for has recently been calculated and is reported in Ref. <cit.>.Despite these facts supplying reasonable arguments that assuming H^tot_QED is SM like in the deconvolution procedure in the SMEFT introduces only a small theoretical bias, it was still checked at LEP what the impact is of large anomalous γ-Z interference effects on the reported PO. Using a general S matrix parameterization of this interference <cit.> the possibility of anomalous γ-Z interference does change the inferred pseudo-observable results, primarily the inferred value of m̂_Z by increasing the error by a factor of three compared to the quoted error in Table <ref>. The remaining pseudo-observables are modified by 20 % of their quoted errors, when a 50 % correction to the SM value is introduced that is far in excess of a SMEFT correction expected to be ∼ few %<cit.> ifis not used as an input and Λ∼ fewTeV.A benefit of the γ-Z interference cross check being performed, is that even whenis not used as an input parameter, deviations from the SM expectation in γ-Z interference can be understood to only perturb EFT conclusions extracted from the LEP PO. The simple addition of an extra theoretical error in interpreting the PO when a SMEFT interpretation is used andis not an input is then justified.A secondary benefit of the S matrix parameterization check of LEP data is that it constrains another potential bias. Due to potential interference with ψ^4 operators that are present in the SMEFT and feed into LEP data due to the presence of off the Z pole scattering events <cit.>, a potential bias in the LEP PO scales as ∼ (m_ZΓ_Z/v^2) times a function of this ratio of off/on peak data <cit.>. The corresponding uncertainty does not disable using EWPD to obtain ∼% level constraints on the C^6_i, as an anomalously large effect would also have shown up in the S matrix cross check reported in Refs. <cit.>.LEPI results and cross checks establish that the assumption of a SM-like QED radiator function, and neglect of ψ^4 interference effects, is validated for the Z pole scan data set. A slight increase in the errors quoted on the pseudo-observable extractions is sufficient and appropriate ifis not used as an input to use the PO in the SMEFT. We stress that the LEP PO approach did not simply assume only SM like radiative corrections, it checked that a SM like QED radiation field was present for the processes of interest. An assumption of a SM like radiation field for pseudo-observables, is essentially an assumption that the multi-pole expansion that appears in the SMEFT derivative expansion directly does not lead to a significant perturbation of the SM radiation field. This is a strong condition on UV dynamics that should be avoided to maintain model independence.An essential challenge for the PO program at LHC is to address the challenge of radiative corrections to Po's with suitable rigor to enable a precision PO program to characterize the properties of the Higgs-like scalar in a model independent fashion. To date this challenge has not been met, but some initial studies in the direction of characterizing such corrections have appeared, for example in Ref. <cit.>. We return to this point below.§.§.§ LEPI interpretationsMost interpretations of EWPD that go beyond the PO level and make contact with specific models use the S,T formalism (or related approaches <cit.>). The S,T oblique formalism parameterizes a few common corrections to the two point functions (Π_WW,ZZ,γ Z) that feed into the extracted PO in the standard form <cit.>α̂(m_Z)/4ŝ_Z^2ĉ_Z^2S≡ Π_ZZ^new(m_Z^2) - Π_ZZ^new(0)/m_Z^2- ĉ_Z^2-ŝ_Z^2/ĉ_Zŝ_ZΠ^new_Zγ(m_Z^2)/m_Z^2 -Π^new_γ γ(m_Z^2)/m_Z^2,α̂ T≡ Π_WW^new(0)/m_W^2-Π_ZZ^new(0)/m_Z^2.One calculates Π_WW,ZZ,γ Z in a model and uses global fit results on EWPD with S,T corrections to constrain the model. See Ref. <cit.> for recent results of such fits. This can be done if the conditions on the global S,T EWPD fit are satisfied; namely that vertex corrections due to physics beyond the SM are neglected, which is the origin of the “oblique" qualifier of EWPD <cit.>. The SM Higgs couples in a dominant fashion to Π_WW,ZZ when generating the mass of the W,Z bosons, and has small couplings to the light fermions due to the small Yukawa couplings, so it satisfies the oblique assumption. This is the reason why this assumption was usually adopted before the Higgs-like scalar discovery. In general, this assumption is very problematic as it is a UV condition, not an IR assumption in defining an EFT, as the vertex correction operators are present in general. Furthermore, the oblique assumption is not field redefinition invariant,[Attempts to translate the oblique condition into a requirement to use a particular operator basis by using these EOM relations are best ignored. As such attempts are based on a misconception of the nature of field redefinitions (assuming that they can satisfy physical conditions), see Ref. <cit.> for related discussion. It has also been argued that the oblique requirement can be interpreted as a condition defining a class of UV theories known as universal theories <cit.>. See Section <ref> for more discussion on this idea.] as can be seen by inspecting the EOM that result from SM field redefinitions, given in Section <ref>.LHC results indicate the W,Z bosons obtain their mass in a manner that is associated with the Higgs-like scalar, see Section <ref>. Corrections to Π_WW,ZZ can be included for the SM, or in extensions due to this scalar, see Ref. <cit.>. Once this is done, there is no strong theoretical support for an oblique assumption to be invoked on the remaining perturbations to EWPD. Dropping this problematic assumption leads to a SMEFT analysis which has several benefits. For example, a SMEFT analysis permits the determination of higher order corrections when interpreting EWPD, see Ref. <cit.>.In a SMEFT analysis of EWPD a model is mapped to the SMEFT Wilson coefficients in a tree or loop level matching calculation and model independent global fit results are used to constrain the Wilson coefficients in a global χ^2 fit.Initial works pioneering this approach are Refs. <cit.>. The analysis of Ref. <cit.> identified unconstrained directions in the EWPD set and correctly found a highly correlated Wilson coefficient space in the SMEFT. Recent analyses that do not break these correlations by assumption (or a chosen marginalization procedure) still find that the EWPD Wilson coefficient space is highly correlated, see Refs. <cit.>. In determining the constraints on the Wilson coefficients of the SMEFT, one chooses an input parameter set, and predicts the EWPD PO. In the Z,W pole results of Refs. <cit.> the mapping is {m̂_Z,m̂_h,m̂_t,Ĝ_F,,α̂_s,Δα̂}→{m_W,σ_h^0,Γ_Z,R_ℓ^0,R_b^0,R_c^0,A_FB^ℓ,A_FB^c,A_FB^b},through ℒ_SMEFT. Here the hat superscript indicates an input parameter. In more detail, a SMEFT fit procedure is as follows:A set of observables is denoted O_i, O̅^LO_i, Ô_i for the SM prediction, the SMEFT prediction to first order in the C^(6), and the experimental value of the extracted PO respectively. The measured value Ô_i is assumed to be a Gaussian variable centered about O̅_i and the likelihood function (L(C)) and χ^2 are defined asL(C) = 1/√((2 π)^n |V|)exp(-1/2( Ô - O̅^LO)^T V^-1( Ô - O̅^LO)), χ^2 = - 2 Log[L(C)],where V_ij = Δ^exp_i ρ^exp_ijΔ^exp_j + Δ^th_i ρ^th_ijΔ^th_j is the covariance matrix with determinant |V|. ρ^exp/ρ^th are the experimental/theoretical correlation matrices and Δ^exp/Δ^th the experimental/theoretical error of the observable O_i. This approach is an approximation, with neglected effects introducing a theoretical error <cit.>. The theoretical error Δ_i^th for an observable O_i is defined as Δ^th_i = √(Δ_i,SM^2 + (Δ_i,SMEFT× O_i)^2), where Δ_i,SM, Δ_i,SMEFT correspond to the absolute SM theoretical, and the multiplicative SMEFT theory error. The χ^2 is χ^2_C_i^6 = χ^2_C_i^6, min + (C_i^6 - C_i,min^6 )^Tℐ(C_i^6 - C_i,min^6),where C_i,min^6 corresponds to the Wilson coefficients vector minimizing the χ^2 and ℐ is the Fisher information matrix. Recent results <cit.> using this methodology are shown in Fig. <ref>. The effect of modifying the input parameters: {,Δα̂}→m̂_W was been recently examined <cit.>, which does not change this conclusion. The Fisher matrices of the SMEFT fit space allow the construction of the SMEFT χ^2 function. These matrices were developed in a fit using 177 observables <cit.> and are available upon request of the authors of Ref. <cit.>. §.§.§ Loop corrections to ZZ decayThe SMEFT and HEFT formalisms allows one to combine data sets into a global constraint picture in a consistent fashion and these EFTs also allow the systematic determination of perturbative corrections. One loop calculations in these theories are absolutely required to gain the full constraining power of the most precisely measured observables, such as the LEPI PO.This is not surprising; far more startling is that the complete one loop results of this form remain undetermined decades after the LEPI data set was reported! In the case of 𝒪(y_t^2,λ) corrections to {Γ_Z, Γ_Z →ψ̅ψ̅,Γ_Z^had,R_ℓ^0,R_b^0}, about thirty loops were recently determined in Ref. <cit.> mapping the input parameters to these observables, while retaining the m̂_t,m̂_h mass scales in the calculation. The renormalization of the ℒ_6 operators in the Warsaw basis <cit.> is used in this result, which simultaneously provides a check of the terms ∝ y_t^2,λ that appear in these observables <cit.>. The results define a perturbative expansion for the LEPI PO O̅_i = O̅^LO_i(C_i^6) + 1/16π^2(F_1[C_j^6] + F_2[λ,C_k^6]logμ^2/m̂_h^2 + F_3[y_t^2,C_l^6]logμ^2/m̂_t^2) + ⋯ The LO results depend on ten Wilson coefficients in the Warsaw basis, defining the {C_i}, and dim({C_j})≠ dim({C_k}) ≠ dim({C_l}) > dim({C_i}) holds in general, where ({C}) here denotes the dimension of a set of coefficients {C}. At one loop, considering 𝒪(y_t^2,λ) corrections the new SMEFT parameters that appear are <cit.>{C_qq^(1), C_qq^(3), C_qu^(1),C_uu, C_qd^(1),C_ud^(1),C_ℓ q^(1),C_ℓ q^(3), C_ℓ u,C_qe,C_eu,C_Hu, C_HB+ C_HW,C_uB,C_uW,C_uH}. It has been shown in Ref. <cit.> in this manner that the number of parameters exceeds the number of precise LEPI PO measurements when one loop corrections are calculated. The LEPI PO are very important to project into the SMEFT, as for a few observables Δ^exp_j ∼ 0.1 %. When 1/16π^2(F_1[C_j^6] + F_2[λ,C_k^6]logμ^2/m̂_h^2 + F_3[y_t^2,C_l^6]logμ^2/m̂_t^2) + ⋯≳Δ^exp_j Ô_i,it is clear that these corrections can have a significant effect on the interpretation of the LEPI PO for the exact same reason. If this is the case depends on the values of the UV dependent Wilson coefficients and the global constraint picture, which is unknown. However, we note that recent global fit results indicate directly that some of the four fermion operators that feed in at one loop are very weakly constrained by lower energy data <cit.>. For the near Z pole observables, one can fix μ = m̂_Z, but the new weakly constrained parameters are still present. Although EFT techniques can sum all of the logs that appear relating various scales, the extraction and prediction of the LEPI PO is a complex multi-scale problem with the scales0 ≪m̂_μ^2 ≪m̂_Z^2 < m̂_h^2 < m̂_t^2.The required calculations to sum all the logs are not available to date.[Ideally these results would have been determined in the decadesbefore the turn on of LHC.] These results already establish that LEPI PO data does not constrain the SMEFT parameters appearing at tree level to the per-mille level in a model independent fashion. This is very good news for hopes of the indirect techniques discussed in this review discovering evidence for physics beyond the SM at LHC.Using the determined SMEFT constraints that result from EWPD to study LHC data, one must also run the determined constraints on C_i,j,k,l(m_Z) to the LHC measurement scales. This also acts to reduce the power of constraints when mapping between the data sets by renormalization group equation (RGE) running. It is simply not advisable to setC_i^6(μ) = 0 in LHC analyses to attempt to incorporate EWPD constraints for all these reasons. The challenge of combining LEPI PO consistently with Higgs data requires further theoretical development of the SMEFT.§.§.§ SMEFT reparameterization invarianceA central feature of interpreting LEP data in the SMEFT is the highly correlated Wilson coefficient fit space. This results from the unphysical nature of Lagrangian parameters and the fact that several parameters in ℒ_6 appear at the same order in the power counting of the theory simultaneously (see Section <ref>). A further wrinkle is that unconstrained directions due to LEPI data in this Wilson coefficient space are manifest in the Warsaw basis but not in other formalisms. As a result, these unconstrained directions have caused enormous confusion over the years.[For discussion on these unconstrained directions in the Wilson coefficient space, see Refs. <cit.>.] This has lead to some misplaced intuition that the number of SMEFT parameters is too large to do a consistent analysis of the global data set. It has also lead to claims that some operator bases are better related to experimental measurements than others. The logical extension of this thinking has lead to ad-hoc phenomenological parameterizations being promoted for the LHC experimental program, which are also argued to be better related to experimental measurements. (See Section <ref> for more discussion.)The physics of these unconstrained directions is now understood in an operator basis independent manner <cit.>. To understand this result the unphysical nature of Lagrangian parameters is an essential feature, and these flat directions follow from a scaling argument that is a property of ψ̅ψ→ψ̅ψ data.The scaling argument underlying the reparameterization invariance is simple. A vector boson can always be transformed between canonical and non-canonical form in its kinetic term by a field redefinition without physical effect, due to a corresponding correction in the LSZ formula. Such a shift can be canceled by a corresponding shift in the V ψ̅ψ coupling. The same set of physical scatterings can then be parameterized by an equivalence class of fields and coupling parameters in the SMEFT as a result <cit.>(V,g ) ↔(V' (1+ ϵ), g' (1- ϵ) ), whereϵ∼𝒪(v̅_T^2/Λ^2). This is the SMEFT reparameterization invariance identified in Ref. <cit.>. Denoting ⟨⋯⟩_S_R as the class of ψ̅ψ→ V →ψ̅ψ matrix elements, the following operator relations follow from the SM EOM given in Section <ref>⟨ y_h g_1^2 Q_HB⟩_S_R = ⟨∑_ψ y_k g_1^2ψ_κ γ_βψ_κ(H^†iD_β H) + 2 g_1^2Q_HD - 1/2 g_1 g_2 Q_HWB⟩_S_R,⟨g_2^2 Q_HW⟩_S_R = ⟨g_2^2(q τ^I γ_βq + l τ^I γ_βl ) (H^†iD_β^I H) - 2 g_1 g_2 y_h Q_HWB⟩_S_R. Because of the SMEFT reparameterization invariance, a Wilson coefficient multiplying the left hand side of these equations is not observable in ψ̅ψ→ψ̅ψ scatterings. The invariance of S matrix elements under field configurations equivalent by use of the EOMmeans then, that the corresponding fixed linear combinations of Wilson coefficients that appear on the right-hand sides of these equations are also not observable in the S_R matrix elements. These combinations are EOM equivalent to physical effects that cancel out due to the reparameterization invariance.The S_R class of data is simultaneously invariant under the two independent reparameterizations (defining w_B,W) that leave the products (g_1 B_μ) and (g_2W^i_μ) unchanged. The unconstrained directions in the global fit are found to bew_1= v^2/Λ^2(C_Hd/3-2C_HD+ C_He+/2 -/6-2 C_Hu/3 -1.29 (+ )+ 1.64 C_HWB), w_2=v^2/Λ^2(C_Hd/3-2C_HD+ C_He+/2 -/6-2 C_Hu/3 + 2.16 (+ )- 0.16 C_HWB).These unconstrained directions can be projected into the vector space defined by w_B,W as <cit.>w_1 = -w_B - 2.59 w_W, andw_2= -w_B +4.31 w_W.We stress that it is important to understand that the existence of these flat directions should not be considered a sign of the SMEFT having too many parameters to interface with the data. Conversely, a correct interpretation of this physics is that a consistent EFT formalism retaining all parameters can indicate that hidden structures such as the reparameterization invariance are present.It is required to include data from ψ̅ψ→ψ̅ψψ̅ψ scattering to lift the flat directions <cit.>. This is understood to be the case when considering Fig. <ref> (a) that contributes to ψ̅ψ→ψ̅ψψ̅ψ scattering which is not invariant under Eq. <ref>. Fig. <ref> (a) contains a TGC vertex, which is the reason the reparameterization invariance is broken at an operator level. We emphasize that there is a distinction between the scaling argument in Eq. <ref> that applies to S matrix elements in a basis independent manner and the presence (or not) of an operator contributing to an anomalous TGC vertex. The latter depends upon the operator basis chosen and unphysical field redefinitions.A recent approach of using mass eigenstate (unitary gauge) coupling parameters to characterize deviations from the Standard Model makes the presence of these unconstrained directions even harder to uncover in data analyses. The reason is that EOM relations key to understanding the reparameterization invariance do not have a (manifest) equivalent in the parameterization chosen, although the fact that there remain un-probed aspects of the Z boson phenomenology in ψ̅ψ→ψ̅ψ scatterings is directly acknowledged in Refs. <cit.>. Defining correlations for mass eigenstate parameters in a form that manifestly preserves the consequences of the reparameterization invariance remains an unsolved problem, and assuming no correlations between these parameters can bias results by breaking the reparameterization invariance.As ψ̅ψ→ψ̅ψψ̅ψoccurs through narrow W^± states, the requirement to utilize the narrowness of the W^± boson in a consistent theoretical approach is now front and center.§.§ LEPII pseudo-observables and interpretation.The LEPII data and analysis summary is reported in Ref. <cit.>. A major goal of the LEPII run was to exploit the sensitivity of scattering σ_2ψ^4ψ≡σ(e^+ e^- → W^+ W^- → f_1 f̅_2 f_3 f̅_4) to CP even Triple Gauge Couplings (TGCs) vertices to test the non-abelian structure of the SM. For this reason, measurements of σ_2ψ^4ψ were stepped up to a running energy of √(s) = 206.5 GeV through the W^± pair production threshold.The SM prediction of this process was developed in Refs. <cit.>. The direct calculation of the process σ_2ψ^4ψ, and related differential distributions, is sufficiently complex that it benefits from using spinor-helicity results developed in Refs. <cit.>. See the Appendix for the results broken into Helicity eigenstates in the SMEFT. To define radiative corrections to this process in the SM, the narrowness of the W^± bosons is exploited in a double pole expansion <cit.>. Functionally the program RacoonWW is used <cit.> which utilizes this expansion. Due to the reparameterization invariance present in e^+ e^- → f_1 f̅_2 scattering, this data set is critical to lifting the flat directions in the Wilson coefficient space, but this was not the motivation for extending the LEPI data set to measure σ_2ψ^4ψ. Only retrospectively was it realized how this extension of the LEP data is critical to globally constrain the SMEFT Wilson coefficient space in a consistent SMEFT analysis <cit.>. Unfortunately, the pseudo-observables reported for LEPII were subject to less cross checks on SM-like assumptions used in extracting the PO, and are less directly connected to S matrix elements. The LEPI PO set is focused on the observable cross sections given in Table <ref> while LEPII PO generally refer to the reported values of some Lagrangian parameters treated as extracted pseudo-observable quantities.[In this latter case a label of pseudo-nonobservable is perhaps more appropriate, as the distinction between Lagrangian parameters and S matrix elements has been obscured. We use this tongue-in-cheek nomenclature in what follows to emphasize this point. Note that the neutral gauge boson parameters h_i^Z,γ,f_i^Z,γ also are in this class of pseudo-nonobservable.] This distinction is important for the LHC program, where proposals exist that are closely related to the LEPII approach to PO as opposed to the LEPI PO. For this reason, we review what has been understood about the LEPII case using the SMEFT in recent years. §.§.§ TGC pseudo-nonobservables and effective pseudo-observablesSeveral studies of σ_2ψ^4ψ in an EFT context developed the formalism used in LEPII results <cit.>. The most general C,P conserving TGCs is introduced as <cit.>- ℒ_TGC/g_VWW=i g̅_1^V( W_μν^+ W^- μ- W_μν^- W^+ μ)V^ν + i κ̅_VW^+_μW^-_νV^μν + i λ̅_V/m̂^2_WV^μν W^+ ρ_νW^-_ρμ,where V={Z,A}, W^±_μν=∂_μ W^±_ν - ∂_νW^±_μ and similarly V_μν = ∂_μ V_ν - ∂_ν V_μ. At tree level in the SMg_AWW= e, g_ZWW= e θ, g_1^V=κ_V=1,λ_V=0.As it stands, this parameterization of SM deviations is not based on a linearly realized operator formalism <cit.>. It can be considered to be an example of a non-linearly realized SU_L(2) × U_Y(1) theory; in modern EFT Language this requires an embedding in the HEFT. A complete HEFT description contains more interaction terms, see Ref. <cit.> for more discussion. This ad-hoc construction can also be embedded into the SMEFT by relating it to gauge invariant operators as shown in Ref. <cit.>. An up-to-date embedding of this form in the Warsaw basis is given in the Appendix <ref>. This embedding is straightforward as this parameterization does not have any gauge dependent defining conditions.Traditional interpretations of LEPII pseudo-nonobservables, reported as effective bounds on the TGC parameters shifting the SM predictions δ g_1^V, δκ_V,δλ_V are problematic. In many studies, including Refs. <cit.>, the constraints on δ g_1^V,δλ_V reported by the LEPII experiments are used as observables. These parameters cannot be treated directly as physical observables to constrain the SMEFT parameter space consistently <cit.>.[ See Section <ref> for a general discussion on the observable/Lagrangian parameter distinction.] Losing this distinction in SMEFT studies of σ_2ψ^4ψ related observables leads to spurious results. The reason is that constraints on the non-physical TGC parameters are generally developed using Monte-Carlo tools where vertex corrections of the massive gauge boson couplings are assumed to be “SM-like” for all fermion species. This assumption when interpreted as an operator basis independent constraint on the SMEFT, implies a “SM-like” TGC vertex is also required in the SMEFT <cit.> due to the EOM relations linking the relevant Lagrangian parameters.[Ref. <cit.> pointed out that the conclusion does not hold in the case of universal theories. On the other hand, see Section <ref> for a discussion on universal theories.] Attempts to then develop “effective TGC parameters" were subsequently reported in Ref. <cit.>.[Ref. <cit.> does not specify the treatment of the expansion used due to the narrowness of the SM states to define the observables. It seems apparent that a narrow width approximation is implicitly used to define the σ_2ψ^4ψ observable. This result is then combined with SM predictions defined using the double pole expansion results of Ref. <cit.>. The inconsistencies so introduced in such a procedure are not suppressed by Γ_W/m_W.] Ref. <cit.> showed that to define a PO set of“effective TGC parameters" for the SMEFT also requires that the shift to the W^± mass (δ m_W^2) and width (δΓ_W^2) must be taken into account in addition to the gauge boson vertex corrections, to accommodate using a double pole expansion to define the observables. Taking these corrections into account gives(δ g_1^V)^eff = δ g_1^V - 2 δ D^W(m_W^2) - δ D^V(m_V^2), (δκ_V)^eff = δκ_V - 2 δ D^W(m_W^2)- δ D^V(m_V^2).We have approximated the dependence in the residue in the effective TGC as s_ij = m_W^2, s = m_V^2, consistent with a double pole expansion. δ D^W, δ D^V are not relatively suppressed by an extra factor of Γ_W/m_W or Γ_V/m_V as might be expected to result from the narrow width expansions of the SM gauge bosons. Results of this formwere reported in Ref. <cit.>, as detailed in the Appendix. §.§.§ LEP II boundsThe LEP experiments during the LEPII run extracted limits on the effective parameters in Eq. <ref>, both in the individual experiments and in combination. This was a significant focus of experimental efforts. The focus of the theoretical community leading up to the LEPII data reporting was to move beyond the naive narrow width approximation in σ_2ψ^4ψ and to define SM radiative corrections to hypothesis test the SM at LEPII. A good summary of the theoretical issues that were priorities going into LEPII is reported inRef. <cit.>. A self-consistent SMEFT approach was not a theoretical priority as the SM had not yet been validated in a test of the non-abelian nature of the massive SM gauge boson interactions, and no Higgs-like scalar was known to be found experimentally. The situation has now changed due to LHC and LEP results.LEPII data has been reexamined in recent years to develop a consistent SMEFT interpretation in Refs. <cit.>. The conclusions found are generally consistent but the detailed numerical results are subject to significant uncertainties. This is illustrated in Table <ref> where the quoted results for bounds on the parameters δ g_1^V,δκ_V,δλ_V that have been produced from the experiments, and an external group, are listed. These LEPII bounds were reported by varying one parameter at a time while assuming the other TGC parameters vanish in Ref. <cit.>.[Ref <cit.> does not specify this is the procedure it follows, but we assume this is the practice as no correlation matrix is reported for the results quoted.] All of these results are produced under the assumption of a “SM-like” coupling of the massive gauge bosons to all fermions in the Monte-Carlo modeling, despite the fact that imposing such a constraint in a basis independent manner in the general SMEFT renders this approach functionally redundant <cit.>. Comparing the quoted results in Ref, <cit.> shows the bounds on anomalous TGC parameters are sensitive to the inclusion of quartic terms in the likelihood. Expanding a general χ^2 function as defined in Eq. <ref> in the correction to the observables one obtains <cit.> + 2 ∑_i=1^n∑_k,l=1^q∑1/Δ_i^2[ζ_i,k,lC_k^6C_l^6] ( Ô - O)_i + 2 ∑_i=1^n∑_k=1^r1/Δ_i^2γ_i,k C_k^8( Ô - O)_i ,when neglecting correlations between the different observables. These effects are numerically suppressed relative to the terms ∼∑_i=1^n∑_k=1^q_i∑_l=1^q_iC_i,k^6C_i,l^6/(Δ_i)^2.This is due to the fact that when ( Ô - O)_i ∼Δ_i a relative suppression of C_k^8 terms by Δ_i is numerically present.[This does not correspond to a power counting suppression as there is no evidence of beyond the SM (BSM) physics.] Including C_i,k^6C_i,l^6 effects and neglecting C_k^8 corrections in the χ^2 function is most justified if Δ_i << 1. In the case of TGC parameters treated as observables, the results in Table <ref> have Δ_i ∼ 1-10 % at one sigma, while σ_2ψ^4ψ based results, that only use actual observables, have Δ_i ∼ 10-50 %<cit.>. Retaining C_i,k^6C_i,l^6 terms while neglecting C_k^8 corrections relies on other numerical effects not overwhelming this relative numerical enhancement in either case. Unfortunately, it is known that* The number of operators dramatically increases order by order in the SMEFT expansion. The increase is exponential <cit.>, leading to the expectation of a large multiplicity of C_k^8 parameters compared to the number of C_i,k^6 parameters. * There are numerical suppressions of the linear interference terms due to C_i,l^6 following from the Helicity arguments of Refs. <cit.>, introducing more numerical sensitivity to C_k^8 corrections.σ_2ψ^4ψ results projected into the SMEFT retaining C_i,k^6C_i,l^6 terms in the likelihood are subject to substantial theoretical uncertainties for all these reasons. The approach of Ref. <cit.> is to assign and vary a theoretical error due to neglected higher order terms in the SMEFT when using LEPII data, and to only use the total and differential σ_2ψ^4ψ results with identified final states, avoiding the use of reported LEPII pseudo-nonobservables. It was noted in Ref. <cit.> that when numerical behavior indicates that the neglect or inclusion of quartic terms have a small effect on the likelihood, without simultaneously changing the theory error in the fit, can lead to the wrong conclusion on the sensitivity of the fit to higher order effects. The substantial theoretical uncertainties present when projecting LEPII results into the SMEFT are acknowledged in Ref. <cit.>. This work also argued that a likelihood that combines Higgs data and TGC data has numerical behavior that indicates that these higher order terms have a small effect on the likelihood. §.§ LEP SMEFT summaryThe SMEFT interpretation of LEP data demonstrates the challenge of consistently combining the data sets in this EFT can be overcome. This requires a careful separation of IR and UV assumptions and a consistent SMEFT analysis.Inconsistent treatments of the LEPII results treat Lagrangian parameters as directly observable, even though many of the corresponding vertices are off-shell, and formalisms used have been subject to gauge dependence. LEPII quantities are extracted with other SMEFT parameters being set to zero in reported results. This introduces non-intuitive consequences in the SMEFT, and inconsistencies due to the EOM relations between ℒ_6 operators. A consistent approach to LEPII results can be developed using the double pole expansion to exploit the narrowness of the SM massive gauge bosons, and only using the experimental σ_2ψ^4ψ total and differential observables. LEPI PO and interpretations are on a much firmer footing. They are more directly related to measured quantities and extractions involved cross checks of assumptions of a QED like radiation field with S matrix techniques. As a direct result, the LEPI PO are not subject to the degree of misinterpretation that has plagued LEPII interpretations.The numerical differences between results developed using inconsistent methodology and more consistent SMEFT interpretations is small, as can be seen comparing results in Refs. <cit.>. The experimental uncertainties at LEPII are significant, and no evidence of physics beyond the SM emerged from the LEP data sets. This fact does not validate, and should not encourage, using inconsistent results and methods to interpret LHC data. The inconsistencies that can be introduced in SMEFT studies illustrated with LEP data here can tragically obscure the meaning of a real deviation being discovered using EFT techniques, if the inconsistency is numerically larger than the experimental errors. In the presence of unknown UV dependent Wilson coefficients, numerically estimating the size of the inconsistencies introduced is a severe challenge. This point holds for LHC studies of the Higgs-like boson using EFT methods.§ THE HIGGS-LIKE SCALARThe experimental determination of the couplings of the newly discovered J^P = 0^+ scalar is essential. It is important to study these couplings in a consistent framework and to upgrade this approach to a full EFT interpretation. The SMEFT and HEFT approach are now fully defined at leading order and available to be used, but transitioning from the currently used formalism known as the "κ formalism" to these consistent field theory frameworks is still a work in progress for the LHC experiments. The challenges to performing this task in the LHC collider environment are profound, but it is important to emphasize that these challenges can be overcome in a consistent program of applying EFT methods to collider studies, including the discovered scalar's couplings, benefiting from the lessons learned interpreting LEP data. In this section, we review the current dominant paradigm for reporting constraints on Higgs properties, known as the κ formalism, due to the notation of Ref. <cit.>. §.§ The κkappa formalismThe κ formalism is not an EFT approach to Higgs data as it was set up in Ref. <cit.>, but is the fusion of two approaches. The idea to reweigh the SM couplings and extract and limit deviations in the partial and total widths of a discovered scalar was laid out in Ref. <cit.>. This approach is an ad-hoc rescaling of couplings in the SM without a field theory embedding. See Fig. <ref> for Feynman diagrams of some of the production and decay modes rescaled.Some of the rescaled couplings appear in loop diagrams, which mediate the leading production gg → h and decay mode h →γγ used to probe the properties of this state. That these modes appear first at the one loop level in the SM is due to the fact that the SM is renormalizable. Introducing such ad-hoc rescalings into the SM parameters is not a small perturbation for such one loop processes, as the counterterm structure is changed and the theory being used is no longer the SM. The κ formalism can thus only make sense when it is embedded into a consistent field theory framework that can be renormalized. Refs. <cit.> did not perform this embedding, although this challenge was understood to be present.A rescaling approach to the parameters of the SM can be interpreted in principle in the SMEFT or the HEFT. Systematic field theory embeddings of a rescaling of the Higgs-like scalar couplings, to interpret the emergence of a signal for this state, appeared in the influential Refs. <cit.>.[These works themselves were also influenced by Ref. <cit.>.] Ref. <cit.> closely follows in its proposed methodology these works, as it directly notes in its introduction. The κ formalism made a series of assumptions in its detailed implementation that are not IR assumptions consistent with either field theory embedding. As a result, in both the HEFT and the SMEFT, a direct relation between the κ approach and a general EFT framework is not present. In both cases, when further UV assumptions are made a mapping can be performed. For example, Refs. <cit.> are formulated with a general HEFT approach in mind, while Ref. <cit.> is constructed in a linear SMEFT formalism in unitary gauge. The κ formalism can accommodate large corrections to the properties of the Higgs-like scalar, and relations due to SU_L(2) × U_Y(1) linear ℒ_6 operators are not directly imposed. The HEFT also contains parameters whose behavior is analogous to that of the κ's, for example a_C plays the same role as κ_V. For these reasons, the rescaling approach, of which the κ formalism is a particular example, has been widely considered to be a restricted version of the HEFT.[Works based on this understanding include Refs. <cit.>. Perhaps the most comprehensive discussion of this embedding is given in Refs. <cit.>.] The κ formalism assumes the existence of a CP even scalar resonance with m̂_h ∼ 125 GeV, whose couplings have the same Lorentz structure as those of the SM Higgs boson and whose width is Γ∼Γ^SM_h, i.e. narrower than the energy resolution of the LHC experiments. The assumption of a narrow resonance even in the presence of SM perturbations is used in Ref. <cit.> to factorize the total event rate into a production cross section and a partial decay width. Although the exact expansion used for the narrow width is not specified in Ref. <cit.>, a naive narrow width assumption is functionally present (and explicitly mentioned in Ref. <cit.> in this context).A set of scale factors κ_i are defined, such that each Higgs production cross section and decay channel is formally rescaled by a corresponding κ_i^2. For instance, in the case of the process gg→ H →γγ one has the parameterizationσ(gg→ H)· BR(H →γγ) =κ_g^2κ_γ^2/κ_H^2 σ(gg→ H)_ SM· BR(H →γγ)_ SM ,where κ_H rescales the Higgs total width. In the limit κ_i≡ 1 the SM is recovered, while values κ_i≠ 1 indicate deviations from the SM. A list of relevant κ factors is defined as <cit.>:σ_WH/σ_WH^ SM =κ^2_W σ_ZH/σ_ZH^ SM =κ^2_Z σ_VBF/σ_VBF^ SM =κ^2_VBF σ_ggH/σ_ggH^ SM = κ^2_g σ_ttH/σ_ttH^ SM =κ^2_tΓ_WW^*/Γ_WW^*^ SM = κ^2_W Γ_ZZ^*/Γ_ZZ^*^ SM = κ^2_Z Γ_γγ/Γ_γγ^ SM = κ^2_γ Γ_Zγ/Γ_Zγ^ SM = κ^2_Zγ Γ_ff/Γ_ff^ SM = κ^2_f .The κ's are in a sense pseudo-observables, as they are defined as a rescaling of SM observables in many cases, or inferred quantities while using the narrowness of the SM gauge bosons to factorize a scattering amplitude. The assumption of SM like soft radiation effects is present, and no detailed procedure for a correction factor is introduced to remove these corrections. This pseudo-observable understanding is not developed in great detail in the specific κ proposal, and projecting constraints on these parameters onto the Higgs couplings requires further information and assumptions.[More explicit PO interpretations of these parameters have since been advanced in Refs. <cit.>.] It deserves to be emphasized that many of these limitations and subtleties are well understood and very clearly stated in Ref. <cit.>.An example of an assumption adopted in a κ fit is the case where the scaling factors for VH production and H→ VV^* decay have been defined to be the same. This approximation holds only for tree level computations[In this section, any reference to perturbative orders is implicitly referred to the EW sector. Radiative QCD corrections can be factorized due to the narrow SM widths, with some exception, in the κ's definitions (see Eqn <ref>).] and under the assumption that deviations in these observables can be interpreted in terms of an anomalous hZZ coupling only, with corrections to e.g. Zqq vertices being neglected.[This procedure is beset with inconsistencies that are discussed in Section <ref>.] Within these assumptions, the prescriptions of the κ formalism are equivalent to the use of the phenomenological ad-hoc Lagrangianℒ_κ = -∑_ψκ_ψ√(2) M_ψ/v̂ψ̅ψ h + κ_Z M_Z^2/v̂ Z_μ Z^μ h+ κ_W 2M_W^2/v̂ W^+_μ W^-μ h,+ κ_g,cg_3^2/16π^2 v̂ G_μν G^μν h+ κ_γ,ce^2/16π^2 v̂ F_μν F^μν h+ κ_Zγ,ce^2/16π^2 c_θ̂v̂ Z_μν F^μν h,where h denotes the physical Higgs boson and F_μν is the photon field strength. For this reason, the κ's are often referred to as “coupling modifiers”. Those in the second line of Eq. <ref> have been defined with an additional “c” to underline that they are associated to effective contact interactions. To match the κ's definitions above (with in particular κ_i,c=κ_i for i=g,γ,Zγ), the Lagrangian ℒ_κ must be strictly used for tree-level computations only. We stress that this parameterization of Lagrangian terms should not be interpreted to give a naive physical meaning to the κ_i. The distinction between observables and Lagrangian parameters discussed in Section <ref> applies, and formulating the κ_i in terms of mass eigenstate shifts does not change this fact.For processes that are produced at one-loop already in the SM, it is possible to employ the Lagrangian above at NLO, with the caveat that in this case the parameters κ_g,c, κ_γ,c and κ_Zγ,c appearing in Eq. <ref> do not coincide with the κ's defined in Eq. <ref> and in particular their SM value is κ_i,c=0. Consider the one loop SM process σ(gg → h), the leading contribution is generated radiatively with a top or bottom quark running in the loop. In this case, the amplitude can be formally split separating the various contributions as𝒜_ggH = κ_t 𝒜^t_ggH + κ_b 𝒜^b_ggH + κ_g,c,where 𝒜_ggH^t(b) represents the contribution from the top (bottom) loop.Although the most general approach is that of retaining κ_g,c as an independent parameter that can capture contributions from BSM diagrams, it is possible to consider the approximation κ_g,c=0. This corresponds to the implicit assumption that the ggH vertex does not receive direct new physics contributions from e.g. a heavy state running in the loop.As a consequence the factor κ_g (see Eq. <ref>) can be expressed as a function of κ_t and κ_b according toκ_g^2(κ_t,κ_b) =κ_t^2σ^tt_ggH +κ_b^2 σ^bb_ggH +κ_t κ_b σ^tb_ggH/σ^tt_ggH+σ^bb_ggH+σ^tb_ggH .Here σ^tt(bb)_ggH denotes the contributionof the top (bottom) loop to the ggH cross section, while σ^tb_ggH stands for the interference term.Similar considerations hold for the γγ and Zγ decays, for the tree-level case of vector boson fusion (VBF) production, where the SM diagrams are modified with κ_Z,W, and for the total Γ_h, whose corresponding modifier κ_H receives contributions from κ_f,W,Z,γ,g in addition to a BSM component that can be denoted κ_H, BSM.The experimental collaborations have provided limits on the κ parameters, that are extracted from a global fit to Higgs production and decay measurements <cit.>.Both benchmarks described above in the ggH example have been considered: Figure <ref> shows the most recent constraints <cit.> obtained including κ_γ,c, κ_g,c, κ_H, BSM (the latter is denoted B_BSM in the plot) as free parameters, while Figure <ref> shows the results for κ_γ,c=κ_g,c=κ_H, BSM=0. In the first case (Figure <ref>) the system is under-constrained, and therefore one additional condition is required to constrain all the parameters. Two alternative choices were considered: either imposing |κ_V|=|κ_Z|,|κ_W|≤ 1 while allowing κ_H, BSM≠ 0 (left panel), or fixing κ_H, BSM= 0 (right panel).The results generically show that κ_t is sensitive to which of the two scenarios is assumed, as expected considering that it gives the dominant contribution to the radiative decay and production channels. The other parameters do not show a dramatic variation among the different setups. It also worth noting that, as anticipated in Section <ref> the current uncertainty in the determination of the Higgs couplings is10 – 20% on average.[Note that the degeneracy of the κ_t sign can be lifted using the method discussed in Refs. <cit.>.]Notably, as the experimental accuracy drops below the 10%, EW radiative corrections become significant. As a consequence, it is not appropriate to use the κ framework to project directly the signal-strength measurements into constraints on the Higgs couplings except in some limited applications.The κ-formalism was constructed as a first probe of the Higgs boson's properties and constitutes a reasonable framework for the interpretation of theHiggs dataset collected so far at the LHC. The key strength of this approach is not that it is an EFT, but that it allows a series of hypothesis tests addressing the question on the consistency of the properties of the discovered scalar with the SM Higgs. Perhaps the most elegant of these tests is the two dimensional test with only a universal (κ_F,κ_V).A comparison of results of this form produced at the time of discovery in 2012 in Ref. <cit.> in Fig.<ref> (left) and the combined ATLAS+CMS results inFig.<ref> (right) demonstrates the degree to which the Run I data set increasingly supported the hypothesis that the discovered scalar is the Higgs boson.The κ formalism does not constitute a suitable tool for a consistent analysis of the Higgs properties going forward, as the experimental data improves. Figure <ref> shows the projections for these measurements at the CMS experiment at 14 TeV and for integrated luminosities of 300 fb^-1 (left panel) and 3000 fb^-1 (right panel) <cit.>. Similar results are expected at ATLAS <cit.>. The plot shows that the sensitivity will approximately reach the 5 – 10 % level, with a significant dependence on the scaling of experimental errors assumed. The green lines correspond to a quite conservative case (Scenario 1) in which all the systematics are assumed to be the same as in the 2012 performance. The red lines (Scenario 2), instead, are derived rescaling the theoretical uncertainties by a factor 1/2 and the other systematics by the square root of the luminosity. Although the projections are extremely uncertain, and subject to a number of unverified assumptions it is clear that the properties of the Higgs-like scalar will be increasingly resolved experimentally in Run II and beyond. The κ formalism is not the right tool for this era of increasing experimental precision. Some of its main limitations are*Theκ formalism is not an EFT as formulated in Ref. <cit.>. As the κ formalism can only be related to EFT constructions with a set of further UV assumptions, it is not guaranteed that it captures a consistent IR limit of an underlying new physics sector. If κ fits show deviations from the SM constructing a consistent inverse map to the UV sector is not guaranteed to be possible as a result.*The κ formalism is not systematically improvable with perturbative corrections. This is a fatal flawthat introduces a multitude of difficulties. These difficulties always appear in any ad-hoc construction. Any replacement formalism must be able to systematically determine perturbative corrections without assuming the SM to address this central flaw. The only known way to accomplish this is with a well defined effective field theory embedding. As the accuracy of the data descends below the ∼ 10 % range, higher order calculations simply become indispensable in important channels sensitive to Higgs properties.*The rescalings of parameters that are off-shell vertices, in particular h → V V^⋆ is ambiguous as the off-shell massive gauge boson is not an external state and has no precise definition without a field theory embedding.*The κ formalismcannot be consistently used to interface Higgs data with LEP data, due to the different scales involved in the measurements. Again this requires a field theory embedding to relate the Wilson coefficient constraints. Similarly, theκ formalism is not a useful tool to interface with even lower energy measurements.*The formulation is intrinsically non-gauge invariant, and at best an example of a non-linear realization of the gauge symmetry of the SM, i.e. a restricted version of the HEFT. The couplings of the scalar to fermions and gauge bosons are left arbitrary. It is also the case that amplitudes computed with the Lagrangian given in Eq. <ref> generically lead to non-unitary S-matrix elements. This is not a concern if the κformalism is embedded in an EFT extension of the SM, as such a theory need only be unitary to the cut off scale. As the κ formalism is not so embedded in an EFT without further assumptions, its lack of unitarity at high energies renders it obviously inconsistent, and unsuitable for studies of differential distributions at high energies in particular.* The construction of theκ formalism is not a truly general set of deviations from the SM, but a biased constructioninformed by experimental constraints in a fairly haphazard fashion. For example, having two independent parameters κ_Z≠κ_W introduces a hard breaking of the custodial symmetry, but this is avoided in many κ fits due to experimental constraints (see the discussion in Ref. <cit.>). On the other hand, these constraints are not consistently determined in theκ formalism itself, due to its inability to relate LEP and LHC data which requires a field theory embedding.*The κ formalism includes only couplings with standard Lorentz structures in a renormalizable field theory. As such, it can only capture deviations in total production/decay rates, and this is another reason it cannot be consistently used for the analysis of kinematic distributions. Despite all of these flaws, and the clear need to go beyond the κ framework, we wish to emphasize that the κ framework and Run I results reported in it were a profound and important achievement. The theoretical framework to interface with LHC data in a consistent EFT extension of the SM was simply not available in Run I. As such, the κ framework, despite all its flaws, was a sensible and insightful choice to project the raw experimental data into a useful and informative form. It is clear that the use of the κ frameworkto hypothesis test the SM was an informative application. This data reporting formalism should be maintained into Run II and beyond despite all its limitations for this application. Nevertheless, it is time to go beyond the κ framework. The HEFT and the SMEFT have now been developed to a sufficient degree that they can be systematically used for this task going forward. §.§ Relation of the κkappa formalism to SMEFT and HEFTIn order to overcome the κ framework limitations listed above, it is necessary to switch from the κ-parameterization to one given in terms of Wilson coefficients of a non-redundant EFT basis (see e.g.<cit.>). Once such a basis has been chosen, it is possible to identify a one way mapping between the {κ_i} and {C_i}.[ This does not turn theκ's into a basis, as no gauge invariant field redefinitions result in the mapping.] In general, each κ_i can be decomposed asκ_i^2 = 1 + Δκ_i,with Δκ_i a linear combination of EFT parameters, whose numerical coefficients are computed calculating the relevant σ_i or Γ_i at a given order in the EFT. We stress that translating the κ framework into an EFT form does not just represent a bare reparameterization, but actually improves the theoretical consistency of the description. The EFT embedding makes manifest the presence of correlations among different observables that are required by gauge invariance or other imposed symmetries and structures such as the SMEFT reparameterization invariance due to the EOM. The generic result, once again, is a correlated fit space of unphysical Lagrangian parameters, as in the LEP case represented in Fig. <ref>.To illustrate how the procedure is carried out in the SMEFT, consider for instance the decay h→b̅b. A partial NLO calculation of this process in the SMEFT has been presented in Refs. <cit.>. For illustrative purposes, here we report only the tree-level result computed in the Warsaw basis, which givesκ_b^2=𝒜^2(h→b̅b)_ SMEFT/𝒜^2(h→b̅b)_ SM =1+ Δκ_b , Δκ_b= 2v̅_T^2(C_H□-C_HD/4-C_Hl^(3)+C_ll'/2-C_dH/[Y_d]_33) .The terms C_H□,C_HD come from normalizing the Higgs' kinetic term to the canonical form, while C_Hl^(3),C_ll' appear due to the shift between the true vev v̅_T and the value inferred from the measurement of G_F. Finally, C_dH represents the only direct d=6 contribution, which is due to the operator 𝒬_dH, that perturbs the Yukawa coupling. See Appendix <ref> for details on shift parameters.Another important example is that of h→γγ. In this case it is necessary to carry out the computation at one-loop, which givesκ_γ^2 = 𝒜^2(h→γγ)_ SMEFT/𝒜^2(h→γγ)_ SM= 1 +Δκ_γ^ LO + Δκ_γ^ NLOwhere Δκ_γ^ LO and Δκ_γ^ NLO include the contributions computed respectively at tree-level and at one-loop in the SMEFT, both normalized to the SM amplitude calculated at a specific order in perturbation theory. Here we normalize by the one loop SM amplitude. The tree-level term is easily derived in the Warsaw basis. The SMEFT contributions from CP even operators at LO reads <cit.> Δκ_γ^ LO = -16π^2/e^2 ℐ^γ v_T^2(C_HWB c_θ̂ -C_HW^2-C_HBc^2_θ̂),where ℐ^γ encodes the adimensional SM amplitude, whose expression can be found in Ref. <cit.>. It is interesting to notice that Δκ_γ^ LO carries an enhancement of 8π^2/e^2 compared e.g. to Δκ_b. This could give 𝒪(1) deviations for C_i ∼ 1 and Λ as low as about 4 TeV (barring cancellations among the coefficients), which simply reflects the fact that processes that are radiatively suppressed in the SM are a priori more sensitive to the presence of new physics as the SMEFT has a multi-pole expansion in general. This was emphasized long ago in Ref. <cit.>. The one-loop term Δκ_γ^ NLO has a much more complex structure and is not so easy to derive. It contains a large number of ℒ_6 Wilson coefficients, feeding in relatively suppressed by 16 π^2 compared to the LO results. Many of these Wilson coefficients do not appear at tree level in the SMEFT, see the results in Refs. <cit.>. Note also that an explicit matching of the κ's into the Warsaw basis, extended to all the Higgs two-body decay channels, has been partially given in Ref. <cit.>.The mapping procedure illustrated above for the SMEFT case can be done with the HEFT. In this case, the κ's are mapped to combinations of parameters belonging both to the leading and next-to-leading order Lagrangian. Consider the tree-level expression of Δκ_b. Using the basis of Ref. <cit.> one hasΔκ_b = 32π^2v_T^2/Λ^2(r_2^l-r_5^l)+[Y_d^(1)]_33/[Y_d]_33 .The coefficients r^l_2,5 are associated to four-fermion operators that enter through δ v_T ∼δ G_F (analog to C_ll' in <ref>) and belong to the NLO Lagrangian[Compared to Eq. <ref>, here there is no equivalent of the terms C_H□ and C_HD because there is no operator in the basis of Ref. <cit.> that modifies the Higgs kinetic term. The operator corresponding to 𝒬_Hl^(3) was not retained either.]Δℒ. In the last term, instead, Y_d^(1) belongs to ℒ_0: itis the Yukawa matrix appearing in the second term of the expansion of the functional 𝒴_Q(h), defined in Eq. <ref>. The impact of this term is analogous to that of the operator 𝒬_dH in the SMEFT, with the difference that in the HEFT case anomalous Higgs couplings appear already at LO due to the singlet nature of the h field. This is true for the interaction terms with d≤ 4, while it is still necessary to include Δℒ terms to match Hγγ, HGG, HZγ couplings. Namely, κ_W,Z and κ_f receive leading contributions respectively from a_C and Y_f^(1), while κ_γ, κ_g, κ_Zγ are mapped to a combination of higher order Wilson coefficients. For instance,[This result can be compared with Eq. <ref>. There is a close correspondence between the coefficients C_HWB→ a_1, C_HB→ a_B, C_HW→ a_W, while the term a_12 in the HEFT comes from the custodial breaking operator (W_μν )^2 that has an equivalent only at d = 8 in the SMEFT.] Δκ_γ^ LO =8π^2/e^2 ℐ^γ( -4c_θ̂ã_1 + c_θ̂^2ã_B + ^2(ã_W-4ã_12)) ,where the shorthand notation ã_i = C_ia_i stands for the product of C_i with the coefficient of the linear term in the function ℱ_i(h)=1+2a_i h/v+…Restricted versions of the SMEFT and the HEFT can be used in this manner to develop one way mappings to the κ's. This can be done as no defining conditions in the κ approach are fundamentally gauge dependent.As the assumptions of the κ formalism itself starts to fail at the experimental precision where this mapping becomes of interest, this task is not a high priority. Directly developing the corresponding results in the SMEFT and HEFT to interface with past data and future LHC results at leading, and next to leading order, is ongoing in a manner that is essentially bypassing the κ formalism. § SMEFT DEVELOPMENTS IN THE TOP SECTOR Being the heaviest known particle, and the one with the largest Yukawa coupling, the top quark is possibly the SM state that is closest to new physics sectors. In particular, it represents a sensitive probe of new physics driving the EW symmetry breaking (EWSB): for instance, its mass plays a fundamental role in determining the RG evolution and stability of the Higgs potential in the UV. At the same time, the top is typically expected to exhibit the largest mixing with exotic states in scenarios with non-linear EWSB sectors, such as composite Higgs models or models with warped extra dimensions.Studying the properties and couplings of the top quark can thus give a unique insight into new physics, which is complementary to that offered by the Higgs boson. Top physics also benefits from a significantly larger dataset compared to Higgs physics, as top quarks are abundantly produced at high energy hadron colliders such as Tevatron and LHC. This has motivated several analyses of the top sector based on the SMEFT approach. Early studies explored the possibility of constraining its interactions at e^+e^- colliders <cit.> (for recent analyses at future lepton colliders see e.g. <cit.>) and in γγ collisions <cit.>, which constitute a particularly suitable environment to probe CP violating couplings. Here we give an overview of SMEFT studies of the top sector, focusing on the processes relevant for top physics at the Tevatron and LHC.In the Warsaw basis there are 28 operators that directly involve the top quark at ℒ^(6) in unitary gauge (see Table <ref> for the operators definitions)[Prior to the construction of the Warsaw basis, a systematic parameterization of d=6 effects in the top sector was proposed in <cit.>.]:_uH, _Hu, _Hq^(1),(3), _Hud, _uW, _uB, _uG, _dW, _qq^(1),(3), _lq^(1),(3),_uu, _ud^(1),(8), _eu, _lu, _qe, _qu^(1),(8), _qd^(1),(8), _ledq, _quqd^(1),(8), _lequ^(1),(3).In addition to these, other operators can be relevant for a global analysis of the top sector, either because they enter the top couplings due to input parameter definitions (see Appendix <ref>) or because they modify other interactions entering top production processes. The first class includes_H□, _HD, _HWB, _ll, _Hl^(3),while the operators fulfilling the latter condition are_G, _G̃, _HG, _HG̃.Considering a general flavor scenario and retaining only the index contractions that select a top quark, the overall number of independent parameters is 1179 (622 absolute values + 557 complex phases). The picture is remarkably simplified in the approximation in which the quarks of the first two generations respect a U(2)^3 symmetry (a U(2) for each field q, u, d) and, simultaneously, flavor universality is assumed in the lepton sector. In this case, that represents the customary set of assumptions for global top analyses, the number of independent parameters is reduced to 85, corresponding to 68 absolute values and 17 phases[The number of independent parameters is further reduced in the presence of a full U(3)^5 flavor symmetry, that gives 46 independent quantities (38 absolute values + 8 phases). However, this option is rarely considered in top analyses, as they implicitly assume that new physics effects may impact top physics more significantly compared to processes involving the first two generations.].This number is still too large for a complete analysis to be performed, but a rich variety of studies has been carried out in the literature, focusing on specific subsets of the parameter space.In particular, the operators in Eq. <ref> are usually neglected in top physics analyses, under the assumption that they can be constrained in other classes of measurements. An exception are fits to EWPD, where C_WB and C_HD are typically retained. These studies allow to constrain top operators of classes 6 and 7, via loop contributions to the gauge bosons self energies <cit.> and four fermion operators arising either at tree level or at 1-loop due to RG mixing <cit.>.CP violating contributions are also negligible for a large number of top measurements at the LHC, because the spin-averaged SM amplitudes for these processes are dominantly CP-conserving, implying that the interference term 𝒜_SM𝒜^*_d=6 is typically suppressed. Bounds on the CP-odd parameters are rather inferred from measurements of top polarizations and t-t̅ spin correlations <cit.> or from lower energy experiments, such as EDM measurements <cit.> and B meson decays <cit.>.The first global fit to top results from Tevatron and LHC has been presented by the TopFitter collaboration in <cit.> (see also <cit.>), where 12 independent combinations of 14 Wilson coefficients were constrained using both differential and inclusive measurements. The relevant processes for top physics measurements at the LHC are the following:*Top pair production p p → t t̅. In the SM, this process is dominated by QCD contributions: both qq̅ and gg initiated diagrams contribute, as shown in Figure <ref> (a)-(c).In the SMEFT, diagram (a) is corrected only by contributions proportional to (C_uG)_33, modifying the G tt̅ coupling. This is the coefficient with the largest impact on top pair production. Diagrams obtained inserting (C_uG)_ii with i={1,2} in the initial G qq̅ vertex are negligible because their interference with the SM amplitude is proportional to m_u or m_d, see Section <ref>.In addition, qq̅ initiated top pair production receives tree-level SMEFT contributions from the 6 four-quark operators _qq^(1), _qq^(3), _uu, _ud^(8), _qu^(8), _qd^(8) <cit.>, whose impact can be expressed in terms of only four combinations of Wilson coefficients <cit.>:C_u^1= (C_qq^(1))_1331 +(C_qq^(3))_1331+(C_uu)_1331, C_d^1= 4 (C_qq^(3))_1133 + (C_ud^(8))_3311, C_u^2= (C_qu^(8))_1133 + (C_qu^(8))_3311, C_d^2= (C_qu^(8))_1133 + (C_qd^(8))_3311. Diagrams induced by _qu^(1), _qd^(1), having color-singlet contractions in the fermion currents, do not interfere with the QCD SM diagrams, but only with the EW production. Their contributions to the total cross section are roughly an order of magnitude smaller that those of the six four-quark operators listed above[Note that the interference of _qq^(1),(3) and _uu contributions with SM QCD production is non-vanishing, as particular linear combinations of these operators are equivalent to the color octet contractions (q̅_μ T^Aq)^2, (u̅_μ T^Au)^2 via Fierz transformations and the SU(3) completeness relation T^A_BCT^A_DE=2 _̣BE_̣DC-2/3 _̣BC_̣DE.Among the 3 remaining operators admitting interactions with two top quarks, _ud^(1) gives contributions that only interfere with the EW SM diagrams. Q_quqd^(1),(8) contain currents with a (L̅R) chiral structure, so they give only diagrams whose interference with the SM is suppressed by the mass of the initial state quarks.].For the gg initiated channel, that dominates at high energy, the SM diagrams in Fig. <ref> (b) and (c) can be dressed with d=6 contributions from _uG in the Gtt̅ vertices.This operator contributes, in addition, through a GGtt̅ four-point interaction._uG is the operator with the most significant impact on tt̅ production, and it has been extensively studied in the literature, see e.g. <cit.>.Diagram (b) can also be corrected with an insertion of _G (or _G̃) in the GGG vertex. Due to the helicity structure of the external gluons, this term interferes only with the diagram in Figure <ref> (c) proportionally to m_t^2, and thus yields a smaller correction compared to _uG. The operators _HG, _HG̃ can also contribute inducing a tree gg→ h→ t t̅ diagram. However, this term is suppressed by the Higgs propagator being always largely off-shell.1em The main observables in pp → tt̅ is the total cross section, that is currently measured at the LHC with an uncertainty ≲ 5% <cit.>.Differential cross sections are also important for constraining the relevant Wilson coefficients. In particular, the analysis of the m_tt̅ spectrum is useful to target and disentangle four-fermion operators <cit.>. Further, charge asymmetries received much attention in the past, mainly due to a discrepancy with the SM expectation registered by the CDF experiment at Tevatron <cit.>, in the forward-backward asymmetryA_FB = N(Δ y > 0)- N(Δ y < 0)/N(Δ y > 0) + N(Δ y < 0),Δ y = y_t-y_t̅,where y_f is the rapidity of the fermion f in the laboratory frame. The excess was subsequently reduced in analyses with higher statistics, and the most recent combination of the Tevatron measurements finds agreement with the SM expectation at NNLO QCD + NLO EW within 1.6σ <cit.>. Besides refined experimental techniques, an important role in reducing the anomaly has been played by higher order calculations in the SM <cit.> showing that A_FB receives large radiative corrections. At the LHC, the forward-backward asymmetry is washed out by the huge symmetric gg→ t t̅ contributions and by the fact that the initial pp state is forward-backward symmetric. A better observable for LHC measurements is the central charge asymmetry A_C, which is correlated with A_FB and defined as <cit.> A_C = N(Δ |y| > 0)- N(Δ |y| < 0)/N(Δ |y| > 0) + N(Δ |y| < 0).The A_C asymmetry has been measured by the ATLAS and CMS collaborations in pp collisions at 7 and 8 TeV (see <cit.> and references therein), finding agreement with the SM expectation. Interestingly, the two asymmetries A_FB and A_C are different linear functions of the parameters in Eq. <ref>. Therefore the four combinations of Wilson coefficients can be constrained combining Tevatron measurements of A_FB and LHC measurements of A_C <cit.>.The first systematic EFT studies of tt̅ production were carried out in Refs. <cit.>, including a phenomenological analysis considering all the observables mentioned above. The role of angular observables in the top decay products has been explored, for instance, in <cit.>.Higher order results are also available: the SMEFT amplitude has been computed at NLO QCD including the contribution of a real (C_uG)_33 <cit.>; effects induced via RG mixing were also considered within the class of four-fermion operators <cit.>.*Single top production p p → q t (q≠ t), pp → t W. Single top production is usually classified into s-channel, t-channel and tW production. Focusing on the p p → q t modes, the s-channel is characterized by a b in the final state (Fig. <ref> (d)) while the t-channel has typically a light quark in the final state (Fig. <ref> (e)).The SM diagrams receive SMEFT corrections from operators entering the Wtb vertex <cit.>, namely _uW, _Hq^(3) and the operators in Eq. <ref> (all but _H□, that only corrects the top Yukawa coupling), or modifying the W coupling to a light quark pair. Among the latter, only _Hq^(3) and the terms in Eq. <ref> give relevant contributions, while the interference terms for _uW, _dW, _Hud are always suppressed by the mass of one of the quarks entering the vertex.Among four-fermion operators, _qq^(1),(3) have the largest impact: due to the chiral structure of the SM diagrams, the interference terms of invariants with at least one right-handed fermion are proportional to the mass of one of the external quarks <cit.>. The pp→ tW mode is mostly produced from the gb partonic initial state and and is sensitive to (C_uG)_33 in addition to the coefficients of the operators affecting pp→ q t <cit.>.1emThe total cross section for mono-top production is dominated by the t-channel contribution, that has been measured at the LHCwith a precision of ∼ 10% <cit.>. The cross section for the s-channel is ∼ 20 times smaller at the LHC, due to the presence of an anti-quark in the initial state <cit.>.ATLAS and CMS have reported evidence for this process with LHC-Run I data, although with a quite low significance of 3.2 and 2.5 σ respectively <cit.>(see <cit.> for a recent experimental review).The SM cross section of pp → tW is also sizable at the LHC (although about 3–4 times smaller than the t-channel cross section). Evidence for this process at the LHC has been found already at √(s)=7 TeV <cit.>. Recently, the ATLAS and CMS collaborations reported measurements of the total and differential rates at √(s)=13 TeV <cit.>, with a maximum precision of ∼ 10% reached by CMS using the full 35.9 fb^-1 dataset collected in 2016. Despite the accessible cross sections, measurements of tW production are challenging at the LHC, mainly because the process interferes with tt̅ production beyond LO in QCD. Disentangling the two signals is a quite complex task <cit.>. Measurements of the top polarizations in mono-top processes can play an important role in this context, helping to constrain anomalous Wtb interactions <cit.> and four-fermion operators <cit.>.Comprehensive EFT analyses of mono-top can be found in <cit.>.NLO QCD corrections to the SMEFT amplitude have been computed for all three channels <cit.> finding, in particular, that they have a non-trivial impact on differential distributions.*Top pair production in association with a neutral gauge boson pp → tt̅ V, V={Z,γ}. The production of a top in association with a neutral gauge boson Z/γ can be both qq̅ and gg-initiated: the relevant SM diagrams have the same structure as those in Figure <ref> (a)-(c), with a gauge boson emitted from any of the fermion lines.In addition to the set of operators that modify tt̅ production, these channels give a unique access to the Ztt and tt couplings <cit.>, that are corrected by _Hq^(1),(3), _Hu, _uW, _uB, _dW, _dB and the operators in Eq. <ref> (except _H□).Four-fermion operators also contribute to the qq̅-initiated channel. The corresponding corrections behave as in the pp→ tt̅ case. The total cross section for both tt̅ Z, t t̅γ have been measured at the LHC with an accuracy of ∼15-20% <cit.>. Their constraining power is quite low at the moment <cit.>, but shall become significant with higher statistics.It has been pointed out in Ref. <cit.> that cross section ratios of tt̅ V/tt̅ are also convenient observables, that allow to isolate and constrain the top dipole moments induced by (C_uW)_33, (C_dW)_33, (C_uB)_33.The SMEFT contributions from class 6 and 7 operators to the total cross section and differential distributions of pp→ tt̅ Z/γ have been computed at NLO QCD accuracy <cit.>.*Top pair production in association with a Higgs boson pp → t t̅ h. In the SM pp→ tt̅ h takes place mainly through diagrams analogous to Figs. <ref> (a)-(c), with a Higgs radiated from one of the top propagators. SMEFT corrections to these diagrams are therefore analogous to those for pp→ tt̅, with the addition of possible insertions of the coefficients C_H□, C_HD and (C_uH)_33 in the tth vertex.The operator _HG (_HG̃) also contributes, inducing new diagrams in which the Higgs is radiated from a gluon line rather than from a top, and a gg-initiated diagram containing a GGGh four-point interaction. This process represents therefore an interesting bridge between top and Higgs global analyses, providing information complementary to that extracted from Higgs production and decay <cit.>.In particular, the interplay of the tt̅ h channel with h, hh and h+j production at the LHC has been explored with the inclusion of NLO QCD corrections to contributions from (C_uH)_33, (C_uG)_33, C_HG <cit.>.The analysis of angular observables of the decay products of the tops is also interesting in this context, as it would allow to probe the imaginary part of (C_uH)_33, testing the CP nature of the top Yukawa vertex <cit.>. Due to the low cross section and the presence of large irreducible backgrounds, significant constraints from pp→ tt̅h are likely to be extracted only at the high luminosity phase of the LHC. At present, the ATLAS and CMS collaborations have reported the observation of this process at √(s)=13 TeV. The uncertainties on the measured cross sections are of the order of 15–20% <cit.>. *Top decays. In the SM, top quarks decay nearly 100% of the time to b W. As such, a study of the properties of the top decay products can probe the Wtb interaction to a good accuracy. In the Warsaw basis, this vertex receives tree-level corrections from_Hq^(3) and the operators in Eq. <ref> (except _H□), that preserve the Lorentz structure of the SM interaction[The operators in Eq. <ref> enter the vertex due to the parameter shifts determined with the choice of the input quantities. In particular, (C_ll)_1221 and (C_Hl^(3))_11,22 are present due to Ĝ_F being chosen as an input. On the other hand C_HD and C_HWB enter when choosing the set {,m̂_Z,Ĝ_F} as inputs but do not contribute ifis replaced with m̂_W.], from _Hud that introduces a right-handed coupling, and from (C_uW)_33 and (C_dW)_33, with a dipole contraction. The presence of _Hq^(3), _ll, _Hl^(3), _HD or _HWB therefore determines a rescaling of the total decay rate, while _Hud, _uW, _dW impact the kinematic properties of the decay products.The four fermion operators _qq^(3), _lq^(3) also contribute to this decay for the hadronic and leptonic final states of the W respectively <cit.>. The total width of the top has been measured both at the Tevatron <cit.> and at the LHC <cit.> with quite large uncertainties.More precise measurements (with a 3–4% accuracy) are available for the helicity fractions of the W boson <cit.>, that are among the most promising observables. They are predicted to the permille accuracy in the SM <cit.> and they are modified only by[There are in principle corrections proportional to (C_Hud,uW,dW)_ii, i=1,2 and (C_eW)_jj that enter the W decay vertex. These, however, interfere with the SM amplitude proportionally to the mass of one of the W decay products, which is always negligible.](C_Hud)_33, (C_uW)_33 and (C_dW)_33. Measurements of these quantities (possibly in combination with other angular observables) allow therefore to derive significant constraints on these coefficients, see e.g. <cit.>. The decay t→ Wb in the presence of anomalous couplings has been computed analytically up to NLO QCD <cit.>.In particular, Ref. <cit.> also explored the impact of four-fermion operators.Finally, upper limits on the observation of exotic top decays through FCNC can be used to set bounds on the flavor off-diagonal entries of some Wilson coefficients. Relevant processes in this sense include t→ u_iX, with u_i={u,c} and X={Z,γ,g, h} <cit.>. A full list of the operators contributing (up to NLO QCD accuracy) can be found in <cit.>.Contributions from class 6 and 7 operators to t→ u_i Z/γ have been computed to NLO QCD <cit.>. More recently, the calculation has been extended to the t→ u_i h channel <cit.> and with the inclusion of parton shower effects <cit.>.Ref. <cit.> included these observables in a global analysis of flavor-changing top couplings.Operators giving flavor-changing charged currents and flavor-changing four-fermion interactions were considered in Ref. <cit.>, that also performed a global analysis including constraints from single top production and B and Z decays (see also <cit.>). In a complementary approach, Ref. <cit.>, considered lepton flavor violating top decays t→ q ℓ^+ ℓ^'-, ℓ≠ℓ^' induced by four-fermion interactions, showing that their measurement at the LHC can give bounds comparable to those obtained from flavor physics at HERA.*Other processes. Other processes can be relevant for constraining the couplings of the top at the LHC. For instance, pp → tt̅ tt̅ and pp → tt̅ bb̅ would give access to the (3333) flavor contraction of the operators with four quarks, that do not affect significantly any other process above <cit.>. Single top production in association with a neutral gauge boson pp → t Z/γis also of interest, as it can give relevant constraints on FCNC top interactions, complementary to those from decays <cit.>. Single top production in association with a Higgs boson pp→ t h can instead help setting constraints on (C_uH)_33, as this process is sensitive to the sign (an therefore the complex phase) of the top Yukawa coupling<cit.>, and on (C_uH)_3i, as to the presence of flavor-changing twh couplings <cit.>.§ PROGRESS AND CHALLENGES FOR LHC PSEUDO-OBSERVABLES, HEFT AND SMEFTThe key problem of the κ formalism is that it is not a systematically improvable framework. It does not have well defined perturbative corrections, and its results cannot be combined in a predictive fashion with different data sets. In short, the κ framework is not an EFT. To go beyond the κ framework there are currently two main approaches being developed for LHC applications, the pseudo-observables (PO) approach, and the systematic development of EFT extensions to the SM. Both of these approaches face significant challenges and are underdeveloped. In the previous sections we have reviewed the extensive development of EFT extensions to the SM studying LEPI, LEPII and top physics. In this section we summarize the overall state of affairs partway through LHC Run II. §.§ Challenges for pseudo-observablesRecently the paradigm of pseudo-observables (PO) has been reinvigorated at the LHC. Initial work in this direction was reported in Ref. <cit.>, and further precursor studies <cit.> have been developed into a theoretical paradigm in Refs. <cit.>. These developments are a welcome advance over the κ formalism. They are theoretically grounded in formal expansions <cit.> around the poles of the narrow SM states, factorizing observables into gauge invariant sub-blocks (see Section <ref>). This transitions a κ approach to a firmer theoretical footing. The evolution of the κ approach to the results of Ref. <cit.> for inclusive Higgs decays is rather direct. A key strength of the PO approach is that it is defined as a gauge invariant decomposition around the physical poles in the process that is disconnected from an underlying assumed Lagrangian. This approach directly exploits the narrowness of the unstable massive states known to exist, and can be mapped to both the HEFT and the SMEFT in principle as the underlying Lagrangian field theory is not fixed. This approach is also based on the correct understanding of the distinction between observable S matrix elements and unphysical Lagrangian parameters (see Section <ref>). For this reason, a PO decomposition, at least in inclusive Higgs decays, can form a sensible bridge (at tree level) to the underlying EFTs in data reporting. PO decompositions also have some predictive power. By exploiting crossing symmetry, relations between different classes of observables can be determined in the PO approach, such as h → V ℱ and ℱ→ h V<cit.> or between h → f_1 f_2 f_3 f_4 and f_1 f_2 → h f_3 f_4<cit.>.The challenges to the successful development of a model independent PO program at LHC are also clear. Decomposing all observable amplitudes into a PO set is challenging at the LHC compared to LEP for a simple reason, the LEP initial state was well defined, while the LHC initial state is an overlap of various partonic processes convoluted with parton distribution functions. This challenge can be avoided if a narrow width expansion is employed to factorize up an observable, and this issue is not present for characterizing inclusive Higgs decays, where PO approaches have been well developed <cit.> and are clearly applicable.A further important issue is related to the core model independent strength of the pseudo-observable approach, namely the lack of a fixed embedding in an explicit EFT Lagrangian that extends the SM. As a direct result, some of the limitations of the κ approach remain. Without mapping to a particular field theory, relations between observables (unrelated by crossing symmetry) are absent, and radiative emission is ill-defined in general. The idea to accommodate soft radiation has been to use universal radiator functions, mimicking the approach at LEP, to dress amplitudes. However, unlike at LEP there is currently no feasible proposal to check the SM-like radiator functions assumption in the LHC environment. Explicit assumptions of no effects of physics beyond the SM in radiative emissions dressing the PO have been invoked, but these are UV assumptions, not IR assumptions, and they have no known interpretation as a precise condition on UV dynamics. It is possible these challenges can be overcome to enable a precision pseudo-observable program at LHC that extends beyond characterizing inclusive Higgs decays. Assuming away these pressing issues with UV assumptions reduces the model independence of the PO approach, and should be avoided if at all possible. §.§ Challenges for SMEFT/HEFTAny formalism that seeks to improve upon theκ framework must address its core defects comprehensively, consistently and without invoking UV assumptions if it seeks to maintain model independence. The new framework must be able to capture the IR limit of physics beyond the SM, without assuming that the physics beyond the SM is already known, and allow an inverse map to the underlying theory if deviations are discovered at LHC, or in future facilities where LHC data is also used as legacy information. It is fortunate that EFT is constructed and defined to exactly meet these demands, that have essentially resulted from the LHC data set not indicating non-SM resonances around the EW scale. The powerful constraints of* Lorentz invariance and the global symmetry constraints due to the Higgsing of SU_L(2) × U_Y(1) → U_em(1) in the case of the SMEFT, * local analytic operators extending the SM due to the assumption of a degree of decoupling v̅_T ≪Λ, (see Section <ref>), leads to a predictive and well defined extension of the SM, that can be systematically constrained experimentally. The EFT approach allows a well defined characterization of higher order perturbative and non-perturbative neglected effects defining theapproximate theoretical precision in an analysis, as is a fundamental part of the EFTdescription. Such an approximate precision can be characterized as a theoretical error in global data analysis that is varied to represent various cases of the size of the neglected higher order terms. This error can be continually and appropriately reduced with further development of this theoretical paradigm. This approach stops one from overinterpreting the data set and being too aggressive on the constraints found in the EFT framework, considering the limited theoretical precision of the EFT description. The HEFT has less constraints due to the presence of a singlet scalar in the spectrum, but still carries powerful constraints due to a local analytic operator expansion and can accommodate assumed global symmetries asIR assumptions reducing its complexity. A core challenge to SMEFT/HEFT is – which EFT should be used? The existence of two self-consistent constructions must be seriously considered. It is not appropriate to casually dismiss the HEFT construction just because the Higgs-like scalar has converged on the properties of the SM Higgs to date. The key distinction between the HEFT and the SMEFT is an IR assumption about the states in the spectrum in the presence of SU_L(2) × U_Y(1) → U_em(1). Considering the viewpoint laid out in Section <ref>, on the Higgs potential being an effective parameterization of the true dynamical mechanism underlying SU_L(2) × U_Y(1) → U_em(1), this EFT choice essentially corresponds to the assumption of one low energy parameterization of such physics being preferred over the other. SU_L(2) × U_Y(1) → U_em(1) can occur due to weakly coupled or strongly coupled dynamics in the UV sector. The core problem with making this choice outright is that no good understanding of the low energy limit of all possible strongly interacting sectors exists, due to the difficulties in calculating non-perturbative physics. On the other hand, the fact that the HEFT and the SMEFT seem to be functionally indistinguishable on SM pole processes requiring studies of tails of distributions to seek out differences (see Section <ref>) indicates that a dedicated pole constraint program formulated in one of these theories can be mapped to the alternate EFT construction directly.The efficient way to develop the constraint picture for each EFT is fairly clear and is undergoing a rapid development in the community. First, a consolidation/data mining phase that distills past LEPI/LEPII/Tevatron/LHC Run I and lower energy results into constraints on the Wilson coefficients is developed. The first priority for this effort is to map the constraints on pole processes (scattering events where p^2 ∼ m^2 for a SM intermediate state) to the SMEFT. This restriction to single pole resonantly enhanced processes allows the narrow width factorization of the dependence on Wilson coefficients in ℒ_6 into those that are resonantly enhanced, and those suppressed by an additional factor of Γ/m. As the data set of such pole processes is limited, it is appropriate to invoke flavour symmetries when pursuing such bounds, at least initially.This initial analysis phase is extended with data from the tails of distributions and low energy observables. The much larger data sets present in these cases allow the flavour symmetries to be relaxed. Tails of distributions are an important source of information on Wilson coefficients, but it cannot be avoided that when the EFT expansion is breaking down, predictivity is lost. This can be the case in the tails of distributions. It is also clear that when examining tails of distributions the choice between the HEFT and the SMEFT as a field theory approach is more pressing. Furthermore, in either EFT, the number of parameters grows dramatically as the IR effect of class 8 ψ^4 operators being further suppressed is absent. These challenges can all be overcome without invoking UV assumptions so long as an appropriate theoretical error is assigned to EFT studies in such tails of distributions. A related question is how PDF uncertainties impact the extraction of constraints from LHC measurements of high-energy tails. PDF uncertainties in the large-x region strongly depend on the partons considered, being largest for gluons and antiquarks. At the LHC with √(s)=13 TeV,for a partonic c.o.m. energy of √(ŝ)∼2-3 TeV, they are typically in the ranges (-30%,+10%) and (-15%, +10%) for gg and qq̅-initiated processes respectively, but they can grow up to 100% at √(ŝ)∼5 TeV.[Additionally, the gg, qg and qq̅ PDF luminosities are sensitive to some flavor assumptions and parameterization choices that lead to energy-dependent discrepancies of up to 20–40% between predictions of different groups. See Ref. <cit.> for a recent detailed discussion.] Precision measurements for this class of observables are then particularly challenging and generally possible only for E≲2 TeV.Very different is the situation for dominantly qq or qg-initiated processes, that typically carry PDF uncertainties of the order 10-20% or smaller on the entire spectrum <cit.>. Around √(ŝ)∼3 TeV, EFT effects can exceed the PDF uncertainty band for Λ≲10 TeV, thus allowing the extraction of constraints within the region of validity of the EFT expansion.A more accurate comparison of PDF uncertainties with possible BSM signals can be done only on a case by case basis, as both quantities are strongly process-dependent. The reduction of PDF uncertainties at large-x, that is crucial for EFT analysis, will be possible in the near future, thanks to the inclusion of high-energy LHC data in PDF global fits (see e.g. <cit.> for the impact of top quark pair production measurements). To combinedata sets measured at disparate energies, it is required to develop the SMEFT and HEFT to the order of one loop calculations for the most precise observables. It is also clear that the challenge of statistical estimates of constraints in multi-dimensional Wilson coefficient spaces are underdeveloped. One loop results are not available in almost every process of phenomenological interest, although it has been shown they can have a remarkable impact on standard interpretations of precise measurements such as the LEP EWPD PO <cit.>. It is unclear if a systematic one loop SMEFT and HEFT paradigm can be developed in time to have maximum impact on LHC studies.A non-physics challenge to the HEFT and the SMEFT is the literature is manifestly conflicted. There is little agreement on EFT conventions, basis choice (the definition of a basis), the meaning of power counting, the degree of constraint on Wilson coefficients, the possibility of doing model independent EFT studies or not, and other issues. The great interest in developing EFT extensions of the SM, in response to the discovery of a Higgs-like scalar and no other resonances in the LHC data set, has resulted in a significant disarray in the rapidly advancing literature. This makes this fascinating and important area of research incomprehensible when comparing various parts of the literature, and effectively unapproachable for the next generation of students. We hope this review will have a positive and clarifying impact by removing some of these barriers and resolving some of these issues. We hope it will encourage students to work in this area.[To this end, in the Appendix we have also included some results on LO corrections to a number of processes with a unified and common notation, in the Warsaw basis, for the SMEFT.] There is an enormous amount of important work to do to develop and use the SMEFT and the HEFT to gain the most out of the unprecedented LHC data set, that is soon to arrive.MT and IB thank the Villum Foundation, NBIA and the Discovery Centre at Copenhagen University for support. MT thanks the many experts in EFT that he has learned from over the years, and acknowledges informative and revealing conversations on this subject with Cliff Burgess, Ben Grinstein, Gino Isidori, Mike Luke, Aneesh Manohar, Giampiero Passarino, Maxim Pospelov and Mark Wise. MT also thanks past collaborators and members of the Higgs Cross Section Working Group, in particular André Mendes and Michael Dührssen, for discussions and motivation. IB acknowledges instructive conversations with past collaborators and thanks in particular Belén Gavela and Luca Merlo for many teachings and formative discussions. We thank Poul Damgaard, Andreas Helset, Yun Jiang, André Mendes, Subodh Patil, Duccio Pappadopulo, William Shepherd, Michael Spira, Anagha Vasudeven and Jordy de Vries for comments on the review. We welcome comments on the review and encourage feedback and corrections. § CROSS SECTIONS/DECAY WIDTHS OF SELECTED PROCESSES IN THE SMEFTIn this Appendix we report the analytic expression of the cross sections for selected EW processes in the SMEFT including leading order shifts due to ℒ_6. All the observables are computed at tree-level in the Warsaw basis with the operators defined as in Ref. <cit.>. We choose the set , m̂_Z, Ĝ_F,m̂_h} as input parameters[The subscript inwill be dropped in the following.] and adopt the notation and assumptions of Ref. <cit.>. Many of these results were already reported in the literature in various bases. A unified presentation with common notational conventions is lacking, so we have included this summary of known results here, and simultaneously extended the known literature.We adopt theIR assumption that light quark masses are neglected both in the SM predictions and in the SMEFT corrections. Unless otherwise specified, the SMEFT Lagrangian is assumed to respect an approximate U(3)^5 flavour symmetry as a furtherIR assumption, which is only violated through insertions of the Yukawa couplings. In this scenario, the Wilson coefficients of operators containing chirality-flipping fermion currents have the[At lowest order in the linear MFV expansion <cit.>.] flavour structureC_f Hrs, C_f Wrs, C_f Brs, C_f Grs ∝[Y^†_f]_rs, C_H ud rs∝[Y_u Y_d^†]_rs, C_ledqrs∝[Y^†_e Y_d]_rs,C_quqdrs^(1),(8) ∝[Y^†_u Y^†_d]_rs,C_lequrs^(1),(3)∝[Y^†_e Y^†_u]_rs.The terms that give very suppressed contributions to interactions involving light fermions proportional to the light quark masses are neglected here. Furthermore, the Wilson coefficients have been redefined so as to absorb the factor Λ^-2: we use the dimensionful parameters C_i' = C_i / Λ^2 with the prime implicitly dropped. We use the indices p, r, s… for the flavour space and I, J, K… for SU_L(2) consistent with the notational conventions in Section <ref>. §.§ Core shifts in the {,Ĝ_F,m̂_Z}{alpha, GF, mZ} input schemeThe SM Lagrangian is written in terms of several internal parameters whose values can be determined via the measurement of a few input quantities. For the EW sector it is customary to adopt the inputs set {, Ĝ_F, m̂_Z,m̂_h, ⋯}, whereis the electromagnetic structure constant extracted from Thomson scattering, Ĝ_F is the Fermi constant extracted from muon decay, and m̂_Z, m̂_h are respectively the measured masses of the Z and Higgs bosons.[Arguably a transition to the {m̂_W, Ĝ_F, m̂_Z,m̂_h, ⋯} scheme is favoured for several reasons as discussed in Ref. <cit.>. We do not fight the tide of historical convention here but note that an m̂_W input scheme is available to be used in the SMEFTsim package <cit.>.] The numerical values of some of the inputs are reported in Table <ref>.The other relevant Lagrangian parameters are fixed by the following definitions:^2= 1/2[1-√(1-4π/√(2)Ĝ_F m̂_Z^2)], ê =√(4π), ĝ_1= ê/, ĝ_2= ê/, v̂_T= 1/2^1/4√(Ĝ_F), m̂_W^2= m̂_Z^2^2.When applied in the SMEFT, this procedure introduces a mismatch between the quantities determined from the input measurements and the parameters defined in the canonically normalized Lagrangian consistent with an on-shell EFT construction. Denoting the former quantities with a hat and the latter with a bar, a generic parameter κ receives a shift from its SM value given byκ̣= κ̅- κ̂ . In the SM limit (C_i→ 0) hatted and bar quantities coincide. It is convenient to define the quantitiesG̣_F= 1/√(2) Ĝ_F(√(2)C^(3)_Hl - C_ll'/√(2)),ṃ_Z^2= 1/2√(2) m̂_Z^2/Ĝ_F C_HD + 2^1/4√(πα̂) m̂_Z/Ĝ_F^3/2 C_HWB, ṃ_h^2=m̂_h^2/√(2)Ĝ_F(-3C_H /2λ+2 C_H□-C_HD/2),ṃ_W^2= m̂_W^2 (√(2)G̣_F + 2 g̣_2/ĝ_2), =-v̅_T^2/4( / C_HD +/(4-2C_ll') +4C_HWB),Using this notation, related results are[To define the SMEFT in R^ξ gauge the approach used here to define the diagonalization of the mass eigenstate fields is advantageous, see Refs. <cit.>.] ṿ_T^2 =v̅_T^2 - v̂_T^2 = G̣_F/Ĝ_F g̣_1 =g̅_1-ĝ_1 =ĝ_1/2[^2(√(2)G̣_F+ṃ_Z^2/m̂_Z^2)+^2 v̅_T^2 C_HWB]=ĝ_1/v̅_T^2/4[^2(C_HD+4-2C_ll')+2 C_HWB],g̣_2 =g̅_2 - ĝ_2 =-ĝ_2/2[(√(2)G̣_F+ṃ_Z^2/m̂_Z^2)+^2 v̅_T^2 C_HWB]=-ĝ_2/v̅_T^2/4[^2(C_HD+4-2C_ll')+2 C_HWB] ^2 = s_θ̅^2- ^2= 2^2^2(g̣_1/ĝ_1-g̣_2/ĝ_2) +v̅_T^2/2 C_HWB=/8√(2)Ĝ_F[(C_HD+4-2C_ll')+4 C_HWB] §.§.§ Couplings of the gauge bosons to fermionsThe photon couplings do not receive corrections in this input parameter set, due to the fact thatis an input and, at the same time, the U_ em(1) gauge symmetry is preserved in the SM. The relevant Lagrangian term isℒ_A,eff = -2√(π) Q_ψJ^ψ, em_μA^μ ,where J^ψ,em_ν is the electromagnetic current with the fermion ψ={ℓ,u,d}. On the other hand, the Z and W couplings to fermions are modified. Using the notation of Refs <cit.>, the former can be parameterized asℒ_Z, eff=ĝ_Z (J_μ^Z ℓ Z^μ + J_μ^Z ν Z^μ + J_μ^Z u Z^μ +J_μ^Z d Z^μ),where ĝ_Z = -ĝ_2/= - 2 2^1/4 √(Ĝ_F) m̂_Z, (J_μ^Z ψ)^pr = ψ̅_pγ_μ[(g̅^ψ_V)_pr- (g̅^ψ_A)_pr γ_5 ] ψ_r for ψ = {u,d,ℓ,ν}. The couplings' normalization is such thatψV = T_3/2 - Q_ψs_θ̂^2,ψA = T_3/2with T_3 = ± 1/2 and Q_ψ = {-1,2/3,-1/3 } for ψ = {ℓ,u,d}. The couplings deviate from the SM expressions asδ (g^ψ_V,A)_pr = (g̅^ψ_V,A)_pr - (ψV,A)_pr,with F[C_1,C_2,C_3 + ⋯]_pr = (C_1 pr + C_2 pr + C_3 pr + ⋯)/(4 √(2)Ĝ_F) δ (g^ℓ_V)_pr =δg̅_Z (ℓV)_pr - F[C_H e,C^(1)_H ℓ,C^(3)_H ℓ]_pr + δ s_θ^2 δ_pr, δ(g^ℓ_A)_pr =δg̅_Z (ℓA)_pr + F[C_H e,-C^(1)_H ℓ,-C^(3)_H ℓ]_pr,δ(g^ν_A/V)_pr =δg̅_Z (νA/V)_pr - F[C^(1)_H ℓ,-C^(3)_H ℓ]_pr, δ (g^u_V)_pr =δg̅_Z (uV)_pr+ F[-C_H q,C^(3)_H q,-C_H u]_pr - 2/3δ^2δ_pr,δ(g^u_A)_pr =δg̅_Z (uA)_pr -F[C_H q^(1),-C^(3)_H q,-C_H u]_pr,δ (g^d_V)_pr =δg̅_Z(dV)_pr -F[C_H q^(1),C^(3)_H q,C_H d]_pr +1/3δ^2δ_pr, δ(g^d_A)_pr =δg̅_Z(dA)_pr +F[-C_H q^(1),-C^(3)_H q,C_H d]_pr,andδg̅_Z =- δ G_F/√(2) - δ m_Z^2/2m̂_Z^2 + s_θ̂c_θ̂/√(2)Ĝ_FC_HWB. For the charged currentsℒ_W,eff = - √(2π )/[(J_μ^W_±, ℓ)_pr W_±^μ + (J_μ^W_±, q)_pr W_±^μ],with(J_μ^W_+, ℓ)_pr =ν̅_pγ^μ (g̅^W_+,ℓ_V - g̅^W_+,ℓ_A γ_5 ) ℓ_r,(J_μ^W_-, ℓ)_pr =ℓ̅_pγ^μ (g̅^W_-,ℓ_V - g̅^W_-,ℓ_A γ_5) ν_r,and analogously for quarks. In the SM(g̅_V^W_±,ℓ)^SM_pr= (g̅_A^W_±,ℓ)^SM_pr = (U_PMNS^†)_pr/2, (g̅_V^W_±,q)^SM_pr=(g̅_A^W_±,q)^SM_pr = V_pr/2,where V is the CKM matrix. In the SMEFT g̅_V/A^W_±,ψ = (g̅_V/A^W_±,ψ)^SM + ̣̅g_V/A^W_±,ψ where, for the flavour diagonal component:δ(g^W_±,ℓ_V)_rr = δ(g^W_±,ℓ_A)_rr = 1/2√(2)Ĝ_F(C^(3)_H ℓrr + 1/2/C_HWB) - 1/4δ s_θ^2/s^2_θ̂, δ(g^W_±,q_V)_rr = δ(g^W_±,q_A)_rr =1/2√(2)Ĝ_F(C^(3)_H q rr + 1/2/C_HWB) - 1/4δ s_θ^2/s^2_θ̂.the off diagonal components are the obvious generalization of this result. §.§.§ Triple Gauge CouplingsGoing from the SM to the SMEFT, the TGC couplings get redefined by a subset of ℒ_6 operators, so that g̅_1^V = g_1^V+ δ g_1^V, κ̅_V= κ_V + δκ_V, λ̅_V = λ_V+δλ_V withg̣_1^A = 0, g̣_1^Z=1/ 2 √(2)Ĝ_F(/+/) C_HWB - 1/2^2(1/^2+1/^2),κ̣_A= 1/√(2)Ĝ_F/ C_HWB,κ̣_Z=1/ 2 √(2)Ĝ_F(- /+/)C_HWB -1/2^2(1/^2+1/^2), λ̣_A =6 m̂^2_W/g_AWWC_W, λ̣_Z =6 m̂^2_W/g_ZWWC_W.Notice that three relations between parameters in this input scheme (at the level of ℒ_6<cit.>) hold in the SMEFT: δκ_Z = δ g_1^Z - t_^2 δκ_A, δλ_A = δλ_Z and δ g_1^A = 0. §.§.§ Fermion masses and Yukawa couplingsIf the assumption of massless fermions is relaxed, the measured masses of quarks and leptons can be incorporated in the set of input parameters and they allow to determine the Yukawa couplings through the definitionŶ_f = 2^3/4m̂_f √(Ĝ_F) .In the SM the coupling of the Higgs boson to fermions is then g_hf̅f^SM =Ŷ_f/√(2). In the SMEFT it is shifted as g̅_hf̅f = g_hf̅f^SM + g̣_hf̅f where <cit.> g̣_hf̅f =Ŷ_f/√(2)[v̅_T^2(C_H□-C_HD/4) -G̣_F/√(2)]-v̅_T^2/√(2)C^*_fH .§.§ Generic 2→22->2 scattering processes via gauge boson exchange §.§.§Scattering ℓ^+ℓ^-→f̅ f , f={ℓ'≠ℓ,u,c,b,d,s}The general s channel differential cross section dσ(ℓ^+ℓ^-→ ff̅)/d, valid on and off resonance scattering, has been computed in the SMEFT inRef. <cit.>. The result includes the contributions from Z and γ exchange, the effect of ψ^4 operators and the interference of all of these terms, up to leading order in the interference of the ψ^4 operators with the SM amplitude. Initial and final state radiation (including possible α_s corrections to final state fermions) have been neglected, together with fermion masses. Consistently with the other results reported here, a U(3)^5 flavour symmetry is assumed, which allows to neglect interference effects with operators of the form LRRL, LRLR, that are proportional to SM Yukawas. Finally, the initial e^+,e^- are taken to be unpolarized. The final expression, in Feynman gauge, reads [Here we correct a factor of 1/π in the first line compared to Ref. <cit.>, which has a typo.] 1/N_cdσ/d= Ĝ_F^2 m̂_Z^4/π χ̅(s) [(|g̅^ℓ_V|^2 + |g̅^ℓ_A|^2)(|g̅^f_V|^2 + |g̅^f_A|^2) (1+ c_θ^2 ) - 8 Re[g̅^ℓ_Ag̅^ℓ,⋆_V ] Re[g̅^f_A g̅^f, ⋆_V ] c_θ],+ |α̂|^2 |Q_ℓ|^2 |Q_f|^2π/2 s(1+ c_θ^2 ) + Ĝ_F m̂_Z^2 Q_ℓQ_f/√(2) [α^⋆g̅^ℓ_Vg̅^f_V (1+ c_θ^2 ) + 2 c_θ g̅^ℓ_Ag̅^f_A/s - m̅_Z^2 + iw̅(s) +h.c.],+ Q_ℓQ_f/32 [α^⋆C_LL,RR^ℓ, f(1+ c_θ)^2+h.c.] + Q_ℓQ_f/32 [α^⋆C_LR^ℓ, f(1- c_θ)^2+h.c.], + (Ĝ_F m̂_Z^2/16√(2) π)[ (s /s - m̅_Z^2 + iw̅(s)) C_LL,RR,LR^ℓ, f,⋆ (g̅^ℓ_V ±g̅^ℓ_A)(g̅^f_V ±g̅^f_A) (1+ c_θ^2 ) +h.c.], + (Ĝ_F m̂_Z^2/16√(2) π) [ (s/s - m̅_Z^2 + iw̅(s))C_LL,RR,LR^ℓ, f,⋆(g̅^ℓ_A ±g̅^ℓ_V)(g̅^f_A ±g̅^f_V) 2 c_θ +h.c.].where in the last two lines the couplings combinations with signs (++, –, +-) are associated to the LL, RR and LR operators respectively. We have also definedχ̅(s) = s/(s - m̅_Z^2)^2 + | w̅(s)|^2 ,where w̅(s) represents the Breit-Wigner distribution <cit.>, that can be either expressed as an s dependent width (w̅(s) = sΓ̅_Z /m̅_Z, which is the approach used at LEP) or alternatively using directly the real part of the complex pole w̅(s) =Γ̅_Zm̅_Z. The parameteris the cosine of the angle between the incoming ℓ^- and the outgoing f̅, and s = (p_ℓ^+ + p_ℓ^-)^2. N_C is the dimension of the SU(3) group of the produced fermion f.Flavour indices on the ψ^4 operator Wilson coefficients and the effective gauge couplingshave been suppressed. Reintroducing them, one has C^⋆→ C^⋆_ℓ ℓf f, C^⋆_ℓf fℓ, C^⋆_fℓ ℓf for C^⋆_LL,RR. For the LR operators C^⋆→ C^⋆_ℓ ℓf f is as in the previous chirality cases, while the cases C^⋆_ℓf fℓ,C^⋆_fℓ ℓf vanish.Expanding linearly in the Wilson coefficient, the differential cross section is shifted compared to the SM prediction by[ Here we correct a 1/π compared to the result in the first three lines that is derived from Ref. <cit.>.] 1/N_c(dσ/d )=Ĝ_F^2 m̂_Z^4/πχ(s) [2 Re[(ℓV)^* g̣^ℓ_V +(ℓA)^* g̣^ℓ_A] (|fV|^2 + |fA|^2) (1+ ^2 ) +(ℓ↔ f ) ], - 8 Ĝ_F^2 m̂_Z^4/πχ(s) [ Re[g̣^ℓ_A(ℓV)^* + (ℓA)^*g̣^ℓ,⋆_V ]Re[fA (fV)^* ]+(ℓ↔ f ) ], + Ĝ_F^2 m̂_Z^4/πχ̣(s) [(|ℓV|^2 + |ℓA|^2) (|fV|^2 + |fA|^2) (1+ ^2 )- 8Re[ℓA(ℓV)^* ]Re[fA (fV)^*] ], + Ĝ_F m̂_Z^2 Q_ℓ Q_f/√(2)[α^⋆χ_2(s) (g̣^ℓ_V fV+ ℓVg̣^f_V) (1+ ^2 ) + 2 (g̣^ℓ_A fA +ℓAg̣^f_A) /s +h.c.], + Ĝ_F m̂_Z^2 Q_ℓ Q_f/√(2)[α^⋆χ̣_2(s) ℓVfV(1+ ^2 ) + 2 ℓAfA/s +h.c.],+ Q_ℓ Q_f/32[α^⋆ C_LL,RR^ℓ, f (1+ )^2+h.c.] + Q_ℓ Q_f/32[α^⋆ C_LR^ℓ, f (1- )^2+h.c.], +(Ĝ_F m̂_Z^2/16 √(2)π)[ χ_2(s) C_LL,RR,LR^ℓ, f,⋆ (ℓV±ℓA)(fV±fA) (1+ ^2 ) +h.c.], +(Ĝ_F m̂_Z^2/16 √(2)π) [χ_2(s) C_LL,RR,LR^ℓ, f,⋆ (ℓA±ℓV)(g^f,SM_A ±fV) 2+h.c.].Hereχ(s)= |Ξ(s)|^2/s,χ̣(s)= 1/s[Ξ(s)Ξ̣^⋆(s) + Ξ̣(s)Ξ^⋆(s)], χ_2(s)=Ξ(s),χ̣_2(s)= Ξ(s),withΞ(s)= s/s -m̂_Z^2 + i (w(s))_SM,Ξ̣(s)= s/[s -m̂_Z^2 + i (w(s))_SM]^2[ - i ẉ(s) ].The shift in the Breit-Wigner distribution depends on the specific form assumed for w(s):w̅(s)= s Γ̅_Z/m̅_Z →ẉ(s)= sΓ̣_Z/m̂_Z,w̅(s)=Γ̅_Z m̅_Z →ẉ(s)=m̂_ZΓ̣_Z.The expression for the Z width correction Γ̣_Z is given in Eq. <ref>. §.§.§Scattering f̅f→f̅ fHere we consider the particular case where the initial and final states fermion f are identical. Two kinematic channels, s and t, are present in this process. Adopting the same set of approximations and assumptions as above, the differential cross section for Bhabha scattering (e^+ e^- → e^+ e^-) in the SMEFT is given by <cit.> d σ/d c_θ = 2Ĝ_F^2 m̂_Z^4/π s[ ( |g̅^ℓ_V|^2 + |g̅^ℓ_A|^2 )^2 (u^2 + s^2/(t-m̅_Z^2)^2 +χ̅(s)/s(u^2 + t^2 ) +2χ̅(s) u^2 (1 - m̅_Z^2/s)/t - m̅_Z^2), .. - 4 Re[g̅^ℓ *_V g̅^ℓ_A]^2 (s^2 - u^2/(t-m̅_Z^2)^2 +χ̅(s)/s(u^2 - t^2 ) - 2χ̅(s) u^2 (1 - m̅_Z^2/s)/t - m̅_Z^2) ], + √(2) Ĝ_F m̂_Z^2/s[α̂^* (g̅^ℓ_V)^2 (u^2 + t^2) + (g̅^ℓ_A)^2 (u^2-t^2)/s (s - m̅_Z^2 + i w̅(s) ) + α̂^* (g̅^ℓ_V)^2 (u^2 + s^2) + (g̅^ℓ_A)^2 (u^2-s^2) /t (t - m̅_Z^2 ) + h.c. ],+ √(2) Ĝ_F m̂_Z^2 u^2/s[α̂^*/t(g̅^ℓ_V)^2 + (g̅^ℓ_A)^2/(s - m̅_Z^2 + i w̅(s) ) + α̂/s(g̅^ℓ,⋆_V)^2 + (g̅^ℓ,⋆_A)^2/(t - m̅_Z^2 )],+2π α̂^2/s[u^2 + s^2/t^2+ u^2+t^2/s^2 + 2 u^2/ts]+ α̂/4 s[ 2(u^2/s + u^2/t)C_LL,RR^⋆ + (t^2/s+ s^2/t)C^⋆_LR + h.c ], + Ĝ_F m̂_Z^2/4 √(2)π s[4 u^2 (g̅^ℓ_A ±g̅^ℓ_V)^2 C^⋆_LL,RR+ 2 t^2((g̅^ℓ_V)^2 - (g̅^ℓ_A)^2)C^⋆_LR/s-m̅_Z^2 +i w(s) + h.c ],+ Ĝ_F m̂_Z^2/4 √(2)π s[4 u^2 (g̅^ℓ_A ±g̅^ℓ_V)^2 C^⋆_LL,RR+ 2 s^2((g̅^ℓ_V)^2 - (g̅^ℓ_A)^2)C^⋆_LR/t-m̅_Z^2 + h.c ].The shift from the SM result is <cit.> δ(d σ_e^+e^- → e^+ e^-/d cos(θ)) = 2Ĝ_F^2 /π s[ u^2 F_3^+ + s^2F_3^-/P(t)^2 + u^2 F_3^-+ t^2 F_3^+/P(s)^2 + 2u^2 F_3^+/P(s)P(t)],+ 2√(2)Ĝ_F α̂/s[u^2 F_7^+ + t^2 F_7^-/s P(s) + u^2 F_7^+ + s^2 F_7^-/t P(t) +u^2 F_7^+/t P(s) + u^2 F_7^+/s P(t)],+2 Ĝ_F/π s[ F_4 u^2 ( 1/P(s)+ 1/P(t)) + F_5(t^2/P(s) + s^2/P(t)) ],+ α̂/2 s[ 2 (u^2/s+u^2/t)C_LL/RR + (t^2/s +s^2/t)C_LR].where P(x) = x/m̂_Z^2 - 1. The factors F_i are defined as follows[Note that in the U(3)^5 limit and for F=ℓ, the operator _ll admits two independent flavor contractions and both contribute to C_LL, so that C_LL = (C_ll + C_ll'). This also applies to _qq^(1), _qq^(3), contributing to C_LL for F=q.]:F_3^± = 4 (N_VA^ℓ)^3 G_VA^ℓδ G_VAAV^ℓ± 8(N_VA^ℓ)^2 δ G_VVAA^ℓ, F_4= (ℓV±ℓA)^2/√(2)C_LL/RR, F_5=(ℓV)^2-(ℓA)^2/2 √(2)C_LR, F_7^± = 2 ℓVδ g_V^ℓ± 2 ℓAδ g_A^ℓ,whereN_VA^ℓ = ℓVℓA,G_VA^ℓ =(ℓV)^2 + (ℓA)^2/(ℓVℓA)^2,δ G_ijkl^ℓ = δ g_i^ℓ/ℓj+δ g_k^ℓ/ℓl.§.§ Electroweak observables near the Z poleAnalytic expressions for the electroweak precision observables in the SMEFT can be extracted from the general parameterization of 2→2 scattering given in the previous section. This section summarizes the results, using the notation of Ref <cit.>. §.§.§ Partial and total Z widthsIn the SMEFT, at tree level, one hasΓ̅(Z → f f̅)=√(2) Ĝ_F m̂_Z^3 N_c/3 π( |g̅^f_V|^2 + |g̅^f_A|^2 ),Γ̅(Z → Had)= 2Γ̅(Z → u u̅)+ 3 Γ̅(Z → d d̅).These expressions can be written down separating the SM contribution from the SMEFT correction:Γ̅(Z → f f̅)= Γ_Z → f f̅^SM + δΓ_Z → f f̅for each fermion f. The same kind of relation holds for the total width Γ̅_Z. Specifically, the shifts in each channel read:δΓ_Z →ℓℓ̅= √(2)Ĝ_F m̂_Z^3/6 π[ -g̣^ℓ_A + (-1 + 4 ^2 ) g̣^ℓ_V ] + δΓ_Z →ℓ̅ℓ, ψ^4, Γ̣_Z →νν̅= √(2)Ĝ_F m̂_Z^3/6 π[ g̣^ν_A +g̣^ν_V ] + Γ̣_Z →νν̅,ψ^4 , Γ̣_Z → Had=2 Γ̣_Z u̅ u + 3 Γ̣_Z d̅ d, = √(2)Ĝ_F m̂_Z^3/π[ g̣^u_A - 1/3(- 3 + 8 s_^2 ) g̣^u_V - 3/2g̣ ^d_A + 1/2(- 3 + 4 s_^2 ) g̣^d_V]+ Γ̣_Z → Had, ψ^4, Γ̣_Z =3Γ̣_Z →ℓℓ̅ + 3 Γ̣_Z →νν̅ +Γ̣_Z→ Had, = √(2)Ĝ_F m̂_Z^3/2π[ g̣^ν_A +g̣^ν_V -g̣^ℓ_A + (-1 + 4 ^2 ) g̣^ℓ_V . . + 2 g̣^u_A - 2/3(- 3 + 8 s_^2 ) g̣^u_V - 3 g̣ ^d_A + (- 3 + 4 s_^2 ) g̣^d_V ] +Γ̣_Z → Had, ψ^4 + 3 Γ̣_Z →ℓℓ̅, ψ^4 + 3 Γ̣_Z →νν̅,ψ^4.The corrections due to four-fermion operators have been generically denoted by Γ̣_Z→ f f̅, ψ^4 and can be derived directly from Eq. <ref>. Their expressions are not given here as they are suppressed by v̅_TΓ_Z/m_Z^2 beyond the power counting suppression, see Ref <cit.> for details.The ratios of decay rates are defined in the SM as R^0_f=Γ_Z→ Had/Γ_Z →f̅ f where f can be a charged lepton ℓ or a neutrino. These are shifted byR̅^0_f= R^0_f + Ṛ^0_f withṚ^0_f=1/(Γ_Z → f f̅^SM)^2[ Γ̣_Z → HadΓ_Z → f f̅^SM - Γ̣_Z → f f̅Γ_Z → Had^SM].When f is an identified quark, the ratioR^0_q is defined as the inverse of the lepton case. §.§.§ Forward-backward asymmetriesThe forward backward asymmetry for the scattering ℓ^+ℓ^-→ ff̅ is defined asA_FB = σ_F - σ_B/σ_F + σ_B,where σ_F is defined by θ∈[0,π/2] and σ_B by θ∈[π/2, π] with θ the angle between the incoming ℓ^- and the outgoing f̅. In the SM:A_FB^0, f= 3/4 A_ℓ A_f, withA_ℓ= 2 g^ℓ_V g^ℓ_A/ (g^ℓ_V)^2 + (g^ℓ_A)^2,A_f= 2 g^f_V g^f_A/ (g^f_V)^2 + (g^f_A)^2.In the SMEFT A̅_f can be written asA̅_f = 2 r̅_f/1 + r̅_f^2,r̅_f = g̅^f_V/g̅^f_Aand it is shifted due to modifications of the Z couplings as A̅_f = (A_f)^SM+Ạ_f withẠ_f = (A_f)^SM( 1 - 2 (r_f^2)^SM/1+ (r_f^2)^SM) ṛ_fandr_f = (r_f)^SM( 1 + ṛ_f) ,ṛ_f =g̣^f_V/fV - g̣^f_A/fA .The corresponding correction to A_FB^0,f isẠ_FB^0,f =3/4[ Ạ_ℓ (A_f)^SM + (A_ℓ)^SM Ạ_f].Corrections due to four-fermion operators are negligible, as the forward backward asymmetry measurements are direct cross section measurements extracted near the Z pole. §.§ Properties of the W^±W boson §.§.§ W widthThe partial W^± widths in the SMEFT read <cit.> Γ̅_W →f̅_p f_r = Γ_W →f̅_p f_r^SM + δΓ_W →f̅_p f_r, Γ_W →f̅_p f_r^SM = N_C |V^f_pr|^2 √(2)Ĝ_Fm̂_W^3/12 π, δΓ_W →f̅_p f_r = N_C |V^f_pr|^2 √(2)Ĝ_Fm̂_W^3/12 π( 4 δ g_V/A^W, f + 1/2δ m_W^2/m̂_W^2).As above, N_C depends on the color representation of final state fermions and V^f corresponds to the CKM (f=q) or PMNS (f=ℓ) matrix. In the lepton case, as the neutrino flavour of the decay of a W^± boson is not identified, the sum over the neutrino species gives ∑_r |V^ℓ_pr|^2 = 1. As a result, the total width is Γ̅_W = Γ_W^SM + δΓ_W withΓ^SM_W= 3√(2)Ĝ_F m̂_W^3/4π, Γ̣_W= Γ^SM_W(4/3δ g^ℓ_W + 8/3δ g^q_W +δ m_W^2/2 m̂_W^2).Here m̂_W is the standard model value of the W-mass at tree level in terms of the input parameters, m̂_W = c_ m̂_Z. §.§ Scattering ℓ^+ ℓ^- → 4 fll->4f through W^±W currentsThe doubly resonant contribution to the ℓ^+ ℓ^- → f_1 f̅_2 f_3 f̅_4 scattering via W^± currents was computed in the SMEFT in Ref. <cit.>. There are two relevant diagrams contributing for each fixed final state, as illustrated in Figure <ref>. The total amplitude can be written as 𝒜 = 𝒜_V + 𝒜_ν with𝒜_V= 𝒜_ℓℓ→ WW,V^λ_12λ_23λ_+λ_- D^W(s_12)D^W(s_23) 𝒜_W^+→ f_1f̅_2^λ_12𝒜_W^-→ f_3f̅_4^λ_34 , 𝒜_ν = 𝒜_ℓℓ→ WW,ν^λ_12λ_23λ_+λ_- D^W(s_12)D^W(s_23) 𝒜_W^+→ f_1f̅_2^λ_12𝒜_W^-→ f_3f̅_4^λ_34 .Here λ_± is the helicity of the initial ℓ^± andλ_12,λ_34 = {0,+,-,L} are the helicities of the W^+ and W^- boson respectively (see Refs <cit.> for further details on this spinor helicity formalism). The contribution of the longitudinal helicity (L) vanishes in the limit of massless fermions and therefore it can be neglected. Each amplitude has been decomposed as the product of three helicity-dependent sub-amplitudes (for WW production via V={Z,γ} or ν exchange and for the decay of each W) and of the W^± propagators, parameterized as:D^W(s_ij) = 1/s_ij-m̅_W^2+iΓ̅_W m̅_W+iϵ ,with s_ij = s_12 for the W^+ and s_ij=s_34 for the W^-. In the SMEFT and with the {,Ĝ_F, m̂_Z} input scheme both the W pole mass and width are shifted compared to the SM prediction. The propagators the need to be expanded up to linear order in the Wilson coefficients as <cit.> D^W(s_ij)= 1/s_ij-m̂_W^2+iΓ̂_W m̂_W+iϵ[1+Ḍ^W(s_ij)] , Ḍ^W(s_ij)= 1/s_ij-m̂_W^2+iΓ̂_W m̂_W[(1-iΓ̂_W/2m̂_W) ṃ_W^2-im̂_WΓ̣_W] .The expressions of 𝒜_ℓℓ→ WW,V^λ_12λ_23λ_+λ_- , 𝒜_ℓℓ→ WW,ν^λ_12λ_23λ_+λ_- in the SMEFT for each λ_ij assignment are listed in Tables <ref>, <ref> and <ref>. Those for 𝒜_W^+→ f_1f̅_2^λ_12, 𝒜_W^-→ f_3f̅_4^λ_34 are instead in Table <ref>. The tables use the notationD^V(s)= 1/s-m̂_V^2+i Γ̂_V m̂_V+iϵ, F_1^Z= -ĝ_Z,effg_ZWW g̅_L^e,F_2^Z= -ĝ_Zg_ZWW g̅_R^e, F_1^γ = F̅_2^γ =√(4πα̂)g_AWW,for the V couplings and propagators, and λ^1/2(x,y,z) is the square root of the Källén functionλ(x,y,z) = [x^2+y^2+z^2-2 xy-2xz -2 yz] .The quantities g̅_1^V, κ̅_V, λ̅_V are the triple gauge couplings in the parameterization of Eq. <ref>. The phase space is parameterized as follows: θ is the angle between the momenta of the incoming e^- and the outgoing W^+ in the center-of-mass frame; ϕ̃_12(34) and θ̃_12(34) are the azimuthal and polar angle of the momentum of the final state fermion f_1(3) in the rest frame of the W^+(-) boson.The total spin averaged cross section isσ̅(s)=∫∑ |𝒜|^2/8 sd s_12 d s_34/(2π)^2[ β̅_12/8 πd cosθ̃_12/2d ϕ̃_12/2 π][ β̅_34/8 πd cosθ̃_34/2d ϕ̃_34/2 π] [ β̅/8 πd cosθ/2d ϕ/2 π],where, for ℓ=e ∑ |𝒜|^2= | D^W(s_12)D^W(s_34)|^2 ∑_λ_12,λ_12'∑_λ_34,λ_34'( 𝒜^λ_12_W^+→ f_1f̅_2)( 𝒜^λ_12'_W^+→ f_1 f̅_2)^*( 𝒜^λ_34_W^-→ f_3 f̅_4)(𝒜^λ_34'_W^-→ f_3 f̅_4)^* ×∑_λ_+∑_λ_-( 𝒜_ee → WW^λ_12λ_34,λ_+,λ_-)( 𝒜_ee → WW^λ_12' λ_34',λ_+,λ_-)^*,and the WW production amplitudes contain both the V and ν exchange contributions.The β-factors areβ̅=√(1-2(s_12+s_34)/s+(s_12-s_34)^2/s^2),β̅_ij=1and the integration over the parameters can be performed numerically in the regionsϕ̃_ij ∈ [0,2π],s_34 ∈[0,(√(s)-√(s)_12)^2], cosθ, cosθ̃_ij ∈ [-1,1], s_12 ∈[0,s]. §.§.§ Universality in β decaysIt is possible to place bounds on combinations of four fermion operators and W^± vertex corrections by comparing the extraction of G_F from μ^- → e^- + ν̅_e + ν_μ decays to the value determined in semileptonic β decays <cit.>. Assuming U(3)^5 universality in the SMEFT, this represents a constraint on the unitarity of the CKM matrix and it translates into a bound on the followingcombination of operators|̣V_CKM|^2 = √(2)/Ĝ_F(- C_l q^(3)+C_l l' + C_Hq^(3) - C_H l^(3)).§.§ Higgs production and decayIn the following we list the expressions of the partonic cross sections for the main Higgs production processes and the partial width for the relevant Higgs decay channels, computed at tree level in the SMEFT.§.§.§ gg→ h and h→ ggThe dominant Higgs production mechanism at the LHC is via gluon fusion. In the SM this process is generated at one loop, as the SM is renormalizable. The leading contribution comes from a top quark loop and it can be computed in the m_t→∞ approximation[In this approximation, the gluon-fusion cross section is now known at N3LO in QCD corrections <cit.>.], where the contact interaction h G^A_μνG^Aμν is present. In the SMEFT, this coupling receives a contribution from the operator _G. In addition, the operator _G̃ introduces a CP violating Lorentz structure that does not interfere with the SM amplitude.The leading SM contribution (at tree-level in the EFT obtained integrating out the top quark) gives the partonic cross section <cit.> ^SM(gg→ h) = G_F_s^2/32√(2)πI^g^2where I^g is a Feynman integral that accounts for the top-quark loop contribution and the result is understood in a distribution sense multiplying a suppressed delta function. Including QCD corrections up to NLO[The normalization is such that I^g=1/3 if QCD corrections are omitted and in the limit m_h/m_t→ 0.] <cit.>:I^g = (1+11/4_s/π) ∫^1_0 dx∫^1-x_0 dy 1-4 x y/1- (m_h^2/m_t^2) x y≃0.375.Compared to the SM, the total cross section in the SMEFT is rescaled by <cit.> σ(gg → h)/σ^SM(gg → h)≃1+ 16 π^2 v̅_T^2/g̅_3^2 I^gC_HG^2+16 π^2 v̅_T^2/g̅_3^2 I^g C_H̃G̃^2 .The decay h→ gg (which is not observable at the LHC) proceeds through the same diagrams if initial and final state gluon emission is neglected andtherefore (for this limited case) it is modified by the same relative correction:Γ(h → gg )/Γ^SM(h → gg)≃σ(gg → h)/σ^SM(gg → h). §.§.§ hV associated productionThe amplitude for Vh associated production can be decomposed as <cit.>[See also Ref. <cit.>.] 𝒜_hV = iN_V g_V^2 m̂_V/q^2-m̂_V^2+iΓ̂_V m̂_V J^V,ψ_ν ϵ^*_μT_V^μνwhere g_V = {g̅_2, g̅_2/} and N_V={1/√(2),1} for V={W,Z} respectively and ϵ^*_μ denotes the polarization vector of the V boson.[The massive vector boson is not an external state and does not appear in the Hilbert space of the SMEFT. Here we are considering the approximate experimental reconstruction of the massive vector boson V with kinematics being dominated by the approximate on-shell region of phase space.]The fermionic currents are defined as in Eqs. <ref>, <ref>. The tensor T_V^μν can be decomposed in the sum of four independent Lorentz structures and form factors <cit.> T_V^μν = f_1^V(q^2) g^μν+f_2^V(q^2)q^μ q^ν+f_3^V(q^2)(p· q g^μν-q^μ p^ν) + f_4^V(q^2)ϵ^μνρ p_ρq_ ,where q denotes the four-momentum of the fermion pair (q^2=s) and p is the four-momentum of the outgoing V boson. In the SM, at tree level and in unitary gauge: f_1^V,SM(q^2)=-m_V^2 f_2^V,SM(q^2)≡ 1, f_3,4^V,SM(q^2)≡ 0.In the SMEFT, the amplitude receives corrections that can be decomposed as followsA_hV = iN_V g_V^2 m̂_V/q^2-m̂_V^2 + iΓ̂_V m̂_V ϵ^*_μ [J̣_ν^V,ψ (T_V^μν)^SM + (J_ν^V,ψ)^SMṬ_V^μν] + A_hV,where the first term contains corrections to the SM diagram,while A_hV stands for the corrections from extra diagrams that are present only in the SMEFT case. In particular J̣_ν^V,ψ contains the shift in the fermion couplings to the V boson that can be inferred from the results in Section <ref> andṬ_V^μν=f̣_1^V(q^2) g^μν+f̣_2^V(q^2)q^μ q^ν+f̣_3^V(q^2)(p· q g^μν-q^μ p^ν) + f̣_4^V(q^2)ϵ^μνρ p_ρq_accounts for modifications to the VVh interaction and the V propagator (in the W case). For hW production, A_hW corresponds to a contribution from the 4-point interaction udhW, while for hZ production A_hZ contains both the contribution from the 4-point interaction ψ̅ψ hZ and that stemming from the diagram with photon exchange in the s-channel (see Figure <ref>).In the SMEFT the corrections f̣_i^V(q^2) and the amplitude shifts A_hV can be expressed as linear functions in the Wilson coefficients. For the case V=Z, with the Warsaw basis and in the U(3)^5 limit <cit.>[Here we correct factors of 2 compared to results quoted in Ref. <cit.> for only a subset of SMEFT operators.] f̣_1^Z(q^2)= Ḍ^Z(q^2)+v̅_T^2(C_H□ +C_HD/4-+C_ll'/2) , f̣_2^Z(q^2)=-1/m̂_Z^2f̣_1^Z(q^2) , f̣_3^Z(q^2)= 2v̅_T^2/m̂_Z^2[^2C_HW+^2 C_HB+ C_HWB] , f̣_4^Z(q^2)=-2v̅_T^2/m̂_Z^2(^2C_HW̃+^2 C_H B̃+ C_HW̃B) , A_hZ =2ie̅v̅_T/q^2Q_ψ J_ν^ψ,em ϵ^*_μ[((C_HW-C_HB)- C_HWB)(p· qg^μν-q^μ p^ν)++((C_HB̃-C_HW̃)+ C_HW̃B)ϵ^μνρp_ρ q_σ]++ 2 im̂_Z ϵ^*_μ ψ̅_s^μ(C_HψsrP_R + (±)_srP_L)ψ_r.In <ref>, Ḍ^Z(q^2) denotes the correction due to the modified Z-width in the propagator: choosing the Breit-Wigner distribution to be Γ̅_Z m̅_Z, it is given byḌ^Z(q^2)= -im̂_ZΓ̣_Z/q^2-m̂_Z^2+iΓ̂_Z m̂_Z . The first two lines of <ref> contain the contribution from the s-channel photon exchange and J^em,ψ_ν =ψ̅_νψ is the electromagnetic current. The last line accounts for the 4-point interaction ψ̅ψ hZ with ψ a quark (the expression for a lepton current is analogous). HereP_L,R=(1∓_5)/2 are the left and right chirality projectors and s,r are flavour indices. The left-handed couplings is ∼ (-) for ψ=u and (+) for ψ=d. For the charged current case V=W^+:f̣_1^W(q^2)=Ḍ^W(q^2) +v̅_T^2(C_H□ -(2+1/)C_HD/4+C_ll'-2/2-/C_HWB) , f̣_2^W(q^2)=-1/m̂_Z^2f̣_1^W(q^2) , f̣_3^W(q^2)= 2v̅_T^2/m̂_W^2 C_HW , f̣_4^W(q^2)=-2v̅_T^2/m̂_W^2 C_HW̃ , A_hW =-2√(2)im̂_W ϵ^*_μ u̅_L,a^μ (V_CKM)_srd_L,r.Here Ḍ^W(s) is the correction to the W propagator due to the shift in the W pole mass and width defined in <ref>.The partonic cross sections are completely determined by the amplitude structure given above. Their analytic expressions are simple in the case in which only the V={W,Z} mediated diagrams are retained. This approximation is justified as these diagrams are assumed to largely dominate the process in the kinematic region selected for the experimental measurement. With this simplification, the SM partonic cross sections at fixed q^2 are <cit.> σ ( ψ̅ψ→ Z h)^ SM = 2 πα^2 [(ψV)^2 + (ψA)^2]/3N_c^4 ^4|p⃗_h|/√(q^2) |p⃗_h|^2 + 3 m̂_Z^2/(q^2- m̂_Z^2+iΓ̂_Z m̂_Z)^2 , σ ( ψ̅_s ψ_r → W h)^ SM =πα^2 |V_rs|^2/18 ^4|p⃗_h|/√(q^2) |p⃗_h|^2 + 3 m̂_W^2/(q^2- m̂_W^2+iΓ̂_W m̂_W)^2 ,where |p⃗_h| = [λ(m_h^2,m_V^2,q^2)/(4 q^2)]^1/2 is the center of mass momentum of the Higgs-like boson and V_rs denotes CKM matrix elements. The function λ(x,y,z) was defined in <ref>. In the SMEFT these expressions are modified according to[Partial results for this expression were reported in Ref. <cit.>.] σ^ BSM (ψ ψ̅→ V h)/σ^ SM (ψ ψ̅→ V h)= | f^V_1 (q^2) |^2 +3 Re[f^V_1(q^2)f^V*_3(q^2)]m̂_V^2 (q^2 + m̂_V^2 -m̂_h^2)/ |p⃗_h|^2 +3 m̂_V^2+ m̂_V^2 q^2/ |p⃗_h|^2 +3 m̂_V^2 [ | f^V_3(q^2) |^2(3 m̂_V^2 + 2|p⃗_h|^2) + 2|p⃗_h|^2| f^V_4(q^2)|^2 ]+2/(g_V,SM^V,ψ)^2 + (g_A,SM^V,ψ)^2[ g_V,SM^V,ψ g̣_V^V,ψ + g_A,SM^V,ψ g̣_A^V,ψ],where f_i^V(q^2)=f_i^V,SM(q^2)+f̣_i^V(q^2) and the notation g_χ,SM^V,ψ denotes the SM coupling of the fermion ψ with chiral structure χ={V,A} to the V boson.§.§.§ VBF production Adopting a formalism similar to that employed for Vh associated production, the generic amplitude for Higgs production via VV={ZZ,W^+W^-} fusion can be written as𝒜_VBF,VV = iN_V^2g_V^3 m̂_V/(q_1^2-m̂_V^2+iΓ̂_V m̂_V)(q_2^2-m̂_V^2+iΓ̂_V m̂_V)J^V,ψ_1_μ J^V,ψ_2_ν T^μν_V,where the currents and momenta are labeled as in Figure <ref> and, in the limit of massless fermions, the tensor T_V^μν can be decomposed into three Lorentz structures[An alternative, equivalent decomposition was used in <cit.>.]:T_V^μν = f_1^V(q_1^2,q_2^2) g^μν+f_3^V(q_1^2,q_2^2)(q_1· q_2 g^μν-q_1^ν q_2^μ) + f_4^V(q_1^2,q_2^2)ϵ^μνρ q_1ρq_2 .For the ZZ fusion case f_1^Z,SM(q_1^2,q_2^2) ≡ 1, f_3,4^Z,SM(q_1^2,q_2^2)≡0, while in the SMEFTf_1^Z(q_1^2,q_2^2)=1+Ḍ^Z(q_1^2)+Ḍ^Z(q_2^2)+v̅_T^2(C_H□ +C_HD/4-+C_ll'/2) , f_3^Z(q_1^2,q_2^2)=-2v̅_T^2/m̂_Z^2[^2C_HW+^2 C_HB+ C_HWB] , f_4^Z(q_1^2,q_2^2)= 2v̅_T^2/m̂_Z^2(^2C_HW̃+^2 C_H B̃+ C_HW̃B) .In the SMEFT the neutral current process receives additional contributions from diagrams with Z andfusion, whose amplitudes are𝒜_VBF,Zγ =2ie̅ g_Zv̅_T (Q_ψ_2 J_μ^Z,ψ_1J_ν^ψ_2,em/(q_1^2-m̂_Z^2+iΓ̂_Z m̂_Z)q_2^2+ Q_ψ_1J_μ^ψ_1,emJ_ν^V,ψ_2,/(q_2^2-m̂_Z^2+iΓ̂_Z m̂_Z)q_1^2) ×[((C_HB-C_HW)+ C_HWB)(q_1· q_2g^μν-q_2^μ q_1^ν)++((C_HW̃-C_HB̃)- C_HW̃B)ϵ^μνρq_1ρ q_2σ] , 𝒜_VBF,γγ =4ie̅^2v̅_T/q_1^2q_2^2Q_ψ_1Q_ψ_2 J_μ^ψ_1,emJ_ν^ψ_2,em [(^2C_HW+^2C_HB- C_HWB)(q_1· q_2g^μν-q_2^μ q_1^ν)++(^2 C_HW̃+^2C_HB̃- C_HW̃B)ϵ^μνρq_1ρ q_2σ] .The total shift in the neutral-current VBF production can then be written asA_VBF, n.c. = ig̅_2^3 m̂_Z/^3(q_1^2-m̂_Z^2+iΓ̂_Z m̂_Z)(q_2^2-m̂_Z^2+iΓ̂_Z m̂_Z)·[ J̣_μ^Z,ψ_1 (J_ν^Z,ψ_2T_Z^μν)^SM + J̣_ν^Z,ψ_2 (J_μ^Z,ψ_1T_Z^μν)^SM+ (J_μ^Z,ψ_1J_ν^Z,ψ_2)^SMṬ_Z^μν]++𝒜_VBF,Zγ+𝒜_VBF,γγ .For the charged current casef_1^W,SM(q_1^2,q_2^2) ≡ 1, f_3,4^W,SM(q_1^2,q_2^2)≡0 and in the SMEFTf_1^W(q_1^2,q_2^2)= 1+Ḍ^W(q_1^2) +Ḍ^W(q_2^2)++v̅_T^2(C_H□ -(2+1/)C_HD/4+C_ll'-2/2-/C_HWB) , f_3^W(q_1^2,q_2^2)=-2v̅_T^2/m̂_W^2 C_HW , f_4^W(q_1^2,q_2^2)= 2v̅_T^2/m̂_W^2 C_HW̃ .with Ḍ^W(s) defined in Eq. <ref>. Since there are no additional contributions from different diagrams, in this case the overall shift in the amplitude, compared to the SM isA_VBF, c.c. = ig̅_2^3 m_W/2(q_1^2-m_W^2)(q_2^2-m_W^2)[ J̣_μ^W_+,q (J_ν^W_-,qT_W^μν)^SM + J̣_ν^W_-,q (J_μ^W_+,qT_W^μν)^SM++(J_μ^W_+,qJ_ν^W_-,q)^SMṬ_W^μν],where the expression has been specialized to the quark current case.§.§.§ h→f̅fThe decay width of the Higgs boson into a fermion pair is given byΓ(h →f̅f) =(g̅_hff̅)^2m̂_h N_c/8π(1 - 4 m_f^2/m̂_h^2)^3/2,where N_c is the number of colors of the final state fermion f. The generic coupling is <cit.> [g̅_hff̅]_rs= 1/√(2)[Ŷ_f]_rs[ 1+ v̅_T^2(C_H□-C_HD/4)-G̣_F/√(2)] - v̅_T^2/√(2) C^*_f H rs ,f=u,d,e with [Ŷ_f]_rs = 2^3/4√(Ĝ_F) [m_f]_rr _̣rs as defined in Eq. <ref>.§.§.§ h→ Vf̅f'The decay of the Higgs into an experimentally reconstructed nearly on-shell vector boson V={W^±,Z} and a fermion pair f̅f' proceeds through the same diagrams that give Vh associated production. The amplitude of this process can then be decomposed in the same way <cit.> exploiting the narrow width of the intermediate vector boson.𝒜_h→ Vψ̅ψ'= iN_V g_V^2m_V/q^2-m_V^2J^V,ψ_ν ϵ^*_μT^μν_Vwhere q is the four-momentum of the fermion pair in the final state and the momentum p appearing in the decomposition of T_V^μν (Eq. <ref>) is again that of the outgoing V boson.Because the two processes have the same diagram forms, the relative correction to the partial width is also analogous to that of <ref>. The contributions from the 4-point contact interactions and from a -mediated diagram in Z ff̅ production:dΓ(h→ V ψψ̅)/dq^2/dΓ(h→ V ψψ̅)^SM/dq^2 = | f^V_1 (q^2) |^2 -3 Re[f^V_1(q^2)f^V*_3(q^2)] m_V^2 (q^2 + m_V^2 - m_h^2)/ |p⃗|^2 +3 m_V^2+m_V^2 q^2/ |p⃗|^2 +3 m_V^2 [ | f^V_3(q^2) |^2(3m_V^2 + 2|p⃗|^2) + 2|p⃗|^2| f^V_4(q^2)|^2 ]+2/(g_V,SM^V,ψ)^2 + (g_A,SM^V,ψ)^2[ g_V,SM^V,ψ g̣_V^V,ψ + g_A,SM^V,ψ g̣_A^V,ψ],where |p⃗|=[λ(m_h^2,m_V^2,q^2)/(4q^2)]^1/2 is the momentum of the V boson in the final state and the differential SM width is <cit.>[ Ref. <cit.> corrects an overall factor of 1/2 compared to the result in Ref. <cit.>.] dΓ(h→ V ψψ̅)^SM/dq^2= N_V^2g_V^4[(g_V,SM^V,ψ)^2+g_A,SM^V,ψ)^2]/96π^3m_h^33 m_V^2+ |p⃗|^2/(q^2-m_V^2)^2 (q^2)^3/2 |p⃗|= N_V^2g_V^4[(g_V,SM^V,ψ)^2+g_A,SM^V,ψ)^2]/768π^3m_h^312 q^2 m_V^2+ λ(m_h^2,m_V^2,q^2)/(q^2-m_V^2)^2λ^1/2(m_h^2,m_V^2,q^2).The form factors read[Note the difference in sign in f_3(q^2) and f_4(q^2) w.r.t. the Vh production case, which is due to the fact that here p and q are both outgoing momenta, while in the production q is incoming and p is outgoing. The form factors for h→ V ff̅ decay coincide with those for VBF production in the limit of massless fermions.] f_1^Z(q^2)= 1+Ḍ^Z(q^2)+v̅_T^2(C_H□ +C_HD/4-+C_ll'/2) , f_2^Z(q^2)=-1/m̂_Z^2f̣_1^Z(q^2) , f_3^Z(q^2)=-2v̅_T^2/m̂_Z^2[^2C_HW+^2 C_HB+ C_HWB] ,f_4^Z(q^2)= 2v̅_T^2/m̂_Z^2(^2C_HW̃+^2 C_H B̃+ C_HW̃B),f_1^W(q^2)=1+ Ḍ^W(q^2) +v̅_T^2(C_H□ -(2+1/)C_HD/4+C_ll'-2/2-/C_HWB) , f_2^W(q^2)=-1/m̂_Z^2f̣_1^W(q^2) , f_3^W(q^2)=-2v̅_T^2/m̂_W^2 C_HW , f_4^W(q^2)= 2v̅_T^2/m̂_W^2 C_HW̃ .The contributions from the diagram h→ Z γ^*→ Z ff̅ and from the 4-point contact interactions ψ̅ψ hZ, ψ̅ψ hW have the same form, with an opposite sign, as those given for hV associated production in Eqs. <ref>, <ref>. §.§.§ h→γγAt one loop in the SM, the partial width of the Higgs decay into photons is <cit.> Γ^SM(h→) = √(2)Ĝ_F ^2 m̂_h^3/16 π^3I^^2where I^γ is a Feynman parameter integral including both contributions from the top quark (with leading QCD corrections) and W-boson loops. Its analytic expression is given in Ref <cit.> and numerically it is I^γ≈ -1.65 <cit.>.At tree level in the SMEFT, the decay rate is modified as <cit.> Γ(h→γγ) /Γ^SM(h →γγ)≃1+8 π^2 v̅_T^2/I^γ𝒞_γγ^2+8 π^2 v̅_T^2/I^γ𝒞_γγ^2where𝒞_γγ = 1/g̅_2^2C_HW+1/g̅_1^2C_HB-1/g̅_1 g̅_2C_HWB 𝒞̃_γγ = 1/g̅_2^2C_HW̃+1/g̅_1^2C_HB̃-1/g̅_1 g̅_2C_HW̃B. §.§.§ h→ Z γThe one-loop rate for the Higgs decay into Z in the SM reads <cit.> Γ^SM(h→ Z) = √(2) G_F ^2 m̂_h^3/8π^3(1-m̂_Z^2/m̂_h^2)^3I^Z^2,where, including both top and W loop contributions,I^Z≈ -2.87 <cit.>. The modification of the decay rate in the SMEFT has the form <cit.> Γ(h→γ Z) /Γ^SM(h →γ Z)≃1+8 π^2 v̅_T^2/I^Z𝒞_γ Z^2+8 π^2 v̅_T^2/I^Z𝒞_γ Z^2with𝒞_γ Z = 1/g̅_1 g̅_2C_HW- 1/g̅_1 g̅_2C_HB -(1/2 g̅_1^2-1/2 g̅_2^2) C_HWB 𝒞̃_γ Z = 1/g̅_1 g̅_2C_HW̃- 1/g̅_1 g̅_2C_HB̃ -(1/2 g̅_1^2-1/2 g̅_2^2) C_HW̃B.Note the inverse gauge coupling dependence that follows from the choice to not scale the Wilson coefficients by a gauge coupling. §.§ Top quark properties §.§.§ Top width The width of the top quark can be computed at tree-level in the SMEFT using the narrow width approximation for the W boson:Γ̅(t→ b f̅_p f_r) = Γ̅(t→ b W^+)Br(W^+→f̅_p f_r) .The narrow width approximation and the SMEFT approximation do not commute. We perform first the narrow width approximation and then the SMEFT expansion, finding, in the notation of Sec. <ref> Γ̅(t→ b f̅_p f_r) =Γ(t → b W^+)^SM Γ_W→f̅_p f_r^SM/Γ_W^SM + Γ̣(t → b W^+) Γ_W→f̅_p f_r^SM/Γ_W^SM + Γ(t → b W^+)^SM Γ̣_W→f̅_p f_r/Γ_W^SM -Γ(t → b W^+)^SM Γ_W→f̅_p f_r^SM/Γ_W^SM Γ̣_W/Γ_W^SM .Summing over the polarizations of the W boson one hasΓ(t→ b W^+)^SM = g̅_2^2/64πm̂_t/m̂_W^2λ^1/2(m̂_t^2,m̂_b^2,m̂_W^2)|V_tb|^2(1+x_W^2-2x_b^2-2x_W^4+x_W^2 x_b^2+x_b^4),Γ̣(t→ b W^+) = g̅_2^2/64πm̂_t/m̂_W^2λ^1/2(m̂_t^2,m̂_b^2,m̂_W^2)[ 2[V_tb^* ((̣g_V^W,q)_33+(̣g_A^W,q)_33)](1+x_W^2-2x_b^2-2x_W^4+x_W^2 x_b^2+x_b^4)- 12 x_W^2 x_b[V_tb^* ((̣g_V^W,q)_33-(̣g_A^W,q)_33)] +6 /Ĝ_F x_W (1-x_W^2-x_b^2) ((V_tb C_uW 33^*)-(V_tb C_dW 33)x_b)],where x_i= m̂_i/m̂_t, λ(x,y,z) is defined in Eq. <ref> and g̣_V,A^W,q are the shifts for the W couplings to quarks defined in Eq. <ref>. These results are in agreement with previous calculations, see e.g. <cit.>. Note that if _Hud is included, it also contributes to the latter as(̣g_V^W,q)_rr = -v̅_T^2/4 C_Hud rr^*+…,(̣g_A^W,q)_rr = v̅_T^2/4 C_Hud rr+…,making these quantities complex.§ LOW ENERGY PRECISION MEASUREMENTS IN LEFT §.§.§ ν–lepton scatteringThe scattering process νe^±→νe^± can be described by the following Effective Lagrangian <cit.> ℒ_ν e= - Ĝ_F/√(2)[ e̅γ^μ( (g̅^ν e_V) - (g̅^ν e_A) γ^5 ) e ] [ ν̅γ_μ( 1-γ^5 ) ν].the shifts are then g̅^ν e_V = g^ν e_V + g̣^ν e_V, g̅^ν e_A = g^ν e_A + g̣^ν e_A where(̣g^ν e_V) = 2 (g̣^ℓ_V + 2 g̣^ℓ, W_±_V )+ 4 g̣^ν_V (- 1/2 + 2 s_^2) - 1/2 √(2)Ĝ_F(2C_l l + 2 C_ll' + C_ l e) -ṃ_W^2/m_W^2, (̣g^ν e_A)= 2(g̣^ℓ_A + 2g̣^ℓ, W_±_A ) - 2g̣^ν_V- 1/2 √(2)Ĝ_F(2C_l l + 2C_ll' - C_l e)- ṃ_W^2/m_W^2.these shifts add the contributions of W and Z exchange. Depending on the neutrino flavour some terms are absent. The shift that is relevant for g_A,V^ν_μ e does not have a Ṃ_W^2 or g̣^ℓ, W_±_V,A contribution, whereas a shift for g_A,V^ν_μμ has both contributions. §.§.§ ν–Nucleon scatteringThe scattering process νN →νX via Z exchange can be described by the following Effective Lagrangian <cit.> ℒ^NC_ν q=-Ĝ_F/√(2)[ν̅γ^μ(1-γ^5 )ν] [ϵ̅_L^q q̅γ_μ(1-γ^5)q +ϵ̅_R^q q̅γ_μ(1+γ^5)q ].where q={u,d}. At tree level in the SM: (ϵ_L^q)^SM = qV + qA and (ϵ_R^q)_SM = qV - qA. The redefinition of the Z couplings and the corrections due to ψ^4 operators lead to a shift in ϵ_L^q and ϵ_R^q of the formϵ̅_L/R^q = ϵ_L/R^q + ϵ̣_L/R^q withϵ̣_L^u= -1/2 √(2)Ĝ_F(C_l q^(1) + C_l q^(3)) + g̣_V^u + g̣_A^u + 4 g̣_V^ν(ϵ_L^u)^SM, ϵ̣_L^d= -1/2 √(2)Ĝ_F(C_l q^(1) - C_l q^(3)) + g̣_V^d + g̣_A^d + 4 g̣_V^ν(ϵ_L^d)^SM, ϵ̣_R^u=-1/2 √(2)Ĝ_F C_l u + g̣_V^u - g̣_A^u + 4 g̣_V^ν (ϵ_R^u)^SM,ϵ̣_R^d= - 1/2 √(2)Ĝ_F C_l u + g̣_V^u - g̣_A^u + 4 g̣_V^ν (ϵ_R^u)^SM.The scattering νN →ℓX and the inverse process take place through W exchange and can be described byℒ=-Ĝ_F/√(2)[ℓ̅γ^μ(1-γ^5 )ν] [Σ̅^ij_L u̅_iγ_μ(1-γ^5)d_j] +,where at tree level in the SM (Σ_L^ij)_SM = V_CKM^ij. This coupling receives corrections from redefinitions of the W mass and couplings, so that Σ̅^ij_L= (Σ_L^ij)_SM + Σ̣_L^ij withΣ̣_L^ij =[- ṃ_W^2/m̂_W^2+ 2g̣_V^q,W + 2g̣_V^ℓ, W- 1/√(2)Ĝ_FC_l q^(3)] V_CKM^ij.The inclusion of a right-handed coupling Σ̅_R^ij is not forbidden in principle in the SMEFT, it can be generated by the operator _ℓ e d q. This contribution has been neglected here because such a correction is proportional to the Yukawa matrices in the U(3)^5 limit assumed, and therefore vanishes in the limit of massless fermions.Charged and neutral current process are related by <cit.> d^2σ(ν N →ν X)/d x d y = g_L,eff^2d^2σ(ν N →μ^- X)/d x d y + g_R,eff^2d^2σ(ν̅ N →μ^+ X)/d x d y.for the scattering variables x = - q^2 / (2p_N· q), y = (p_N · q)/(p_N·p_ν), where q^2 is the momentum transfer and p_N, p_ν are respectively the nucleon and neutrino momenta. The effective couplings g_L/R,eff receive corrections in the SMEFT so that g̅_L/R,eff^2=g_L/R,eff^2+ g̣_L/R,eff^2 and they can be expressed in terms of the ϵ_L/R^q parameters asg̅_L/R,eff^2= ∑_i,j[|ϵ̅_L/R^u^i|^2 + |ϵ̅_L/R^d^j|^2 ]|(Σ̅_L^ij)|^-2.Relevant quantities for these processes are the ratios of cross sectionsR^ν = σ( ν N →ν X)/σ(ν N →ℓ^- X) = g_L,eff^2 + rg_R,eff^2,R^ν̅ = σ(ν̅ N →ν̅ X)/σ(ν̅ N →ℓ^+ X) = g_L,eff^2 + g_R,eff^2/r,where the factor r in an ideal experiment with full acceptance (in the absence of sea quarks) is given by r=1/3.SMEFT contributions to the r parameter can be neglected as long as dimension-6 operators have a negligible impact on the parton distributions of nucleons. This condition is plausibly verified, as such corrections are expected to scale as Λ_ QCD^2/Λ^2.Finally, the parameter κ defined byκ = 1.7897 g_L,eff^2+ 1.1479 g_R,eff^2 - 0.0916 h_L,eff^2 - 0.0782 h_R,eff^2has been used to report data, e.g. by the CCFR collaboration <cit.>. Here g_L/R,eff are the couplings introduced in Eq. <ref> andh̅_L/R,eff^2 =∑_i,j[|ϵ̅_L/R^u^i|^2 - |ϵ̅_L/R^d^j|^2 ]|(Σ̅_L^ij)|^-2. §.§.§ Neutrino Trident ProductionNeutrino trident production is the pair production of leptons from the scattering of a neutrino off the Coulomb field of a nucleus, ν N →νNℓ^+ℓ^-. The SM calculation of this process is well known, see Refs.<cit.>. Using the notation of the effective Lagrangian in Eq.<ref>, the constraint on the SMEFT is through the ratio of the partonic cross sectionsσ̅_SMEFT/σ_SM = (g̅^ν_e e_V)^2 + (g̅^ν_e e_A)^2/(ν_e eV)^2 + (ν_e eA)^2. §.§.§ Atomic Parity ViolationFor Atomic Parity Violation (APV) the standard Effective Lagrangian is given by <cit.> ℒ_eq= Ĝ_F/√(2) [ ∑_q g̅^eq_AV(e̅γ_μγ^5 e ) ( q̅γ^μ q) + g̅^eq_VA( e̅γ_μ e ) ( q̅γ^μγ^5 q ) ].In the SM:e qAV = 8qV ℓA and e qVA = 8qA ℓV and the relevant couplings are for q = u,d. The effective shifts areg̣^eu_AV = 1/2√(2)Ĝ_F( - C_l q^(1) +C_l q^(3) - C_l u + C_e u + C_q e) + 2 (1- 8/3s_^2) g̣^ℓ_A- 2 g̣^u_V,g̣^eu_VA = 1/2√(2)Ĝ_F( - C_l q^(1) +C_l q^(3) + C_l u + C_e u - C_q e) +2 g̣^u_A (-1 + 4 s_^2 )+ 2 g̣^ℓ_V,g̣^ed_AV = 1/2√(2)Ĝ_F( - C_l q^(1) -C_l q^(3) - C_l d + C_e d + C_q e) + 2 (-1 + 4/3s_^2)g̣^ℓ_A - 2 g̣^d_V,g̣^ed_VA = 1/2√(2)Ĝ_F( - C_l q^(1) -C_l q^(3) + C_l d + C_e d - C_q e) + 2 g̣^d_A (-1 + 4 s_^2 ) - 2 g̣^ℓ_V.It is convenient to define the four combinationsg̅^ep_AV/VA = 2 g̅^eu_AV/VA + g̅^ed_AV/VA, g̅^en_AV = g̅^eu_AV/VA + 2 g̅^ed_AV/VA,that are shifted from their SM values by g̣^ep_AV/VA =2 g̣^eu_AV/VA + g̣^ed_AV/VA,g̣^en_AV/VA = g̣^eu_AV/VA + 2 g̣^ed_AV/VA.The weak charge Q_W^Z,N of an element X^A_Z defined by <cit.> Q_W^Z,N = - 2 [ Z (g^ep_AV + 0.00005 ) + N ( g^en_AV + 0.00006 )](1- α̅/2 π),is measured very precisely for thallium <cit.> and cesium <cit.> and in the SMEFT it is shifted byQ̣_W^Z,N = - 2 [Z g̣_AV^ep + N g̣_AV^en](1- α̂/2 π)compared to the SM value. §.§.§ Parity Violating Asymmetry in eDIS For inelastic polarized electron scattering e_L,R N → e X the right-left asymmetry A is defined as<cit.>:A = σ_R - σ_L/σ_R + σ_L,where A/Q^2 =a_1 + a_2 1-(1-y)^2/1+(1-y)^2a_1= 3Ĝ_F/5 √(2)πα̂(g_AV^eu - 1/2 g_AV^ed),a_2= 3Ĝ_F/5 √(2)πα̂(g_VA^eu - 1/2 g_VA^ed).Here Q^2 ≥ 0 is the momentum transfer and y is the fractional energy transfer in the scattering y ≃ Q^2/s. In the SMEFT g̅_AV/VA^eq = g_AV/VA^eq + g̣_AV/VA^eq so that a_1 and a_2 receive the corrections <cit.>ạ_1= 3Ĝ_F/5 √(2)πα̂(g̣_AV^eu - 1/2g̣_AV^ed),ạ_2= 3Ĝ_F/5 √(2)πα̂(g̣_VA^eu - 1/2g̣_VA^ed). §.§.§ Mø ller scatteringParity Violation Asymmetry (A_PV) in Mø ller scattering can be parameterized with the standard Effective Lagrangianℒ_ee= Ĝ_F /√(2) g_AV^ee(e̅γ^μγ^5 e )(e̅γ_μ e).In the SM g_AV^ee = 8 ℓVℓA = 1/2(1 - 4 ^2). In the SMEFT we have the correction <cit.> g̣_AV^ee = 1/√(2)Ĝ_F(-C_l l-C_ll' + C_ee)- 2 g̣_V^ℓ - 2 (1 - 4 ^2) g̣_A^ℓ,so that the parity violating asymmetry A_PV is expressed asA_PV/Q^2 = - 2 g_AV^eeĜ_F/√(2)πα̂1-y/1+y^4+(1-y)^4. Q^2 and y are defined as above.science_arxiv | http://arxiv.org/abs/1706.08945v3 | {
"authors": [
"Ilaria Brivio",
"Michael Trott"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170627171752",
"title": "The Standard Model as an Effective Field Theory"
} |
basicstyle=, columns=fullflexible, showstringspaces=false, keywordstyle= ,commentstyle=, frame=shadowbox, rulesepcolor=Roaming across the Castle Tunnels: an Empirical Study of Inter-App Navigation Behaviors of Android Users Ziniu Hu, Yun Ma, Qiaozhu Mei, Jian Tang December 30, 2023 ======================================================================================================== Mobile applications (a.k.a., apps), which facilitate a large variety of tasks on mobile devices, have become indispensable in our everyday lives. Accomplishing a task may require the user to navigate among various apps. Unlike Web pages that are inherently interconnected through hyperlinks, mobile apps are usually isolated building blocks, and the lack of direct links between apps has largely compromised the efficiency of task completion. In this paper, we present the first in-depth empirical study of inter-app navigation behaviors of smartphone users based on a comprehensive dataset collected through a sizable user study over three months. We propose a model to distinguish informational pages and transitional pages, based on which a large number of inter-app navigation are identified. We reveal that developing “tunnels” between of “isolated” apps has a huge potential to reduce the cost of navigation.Our analysis provides various practical implications on how to improve app-navigation experiences from both the operating system's perspective and the developer's perspective.§ INTRODUCTIONAlice was wandering through restaurant reviews in the Yelp app. It mentioned that the restaurant reminded people of a classical scene in the movie “Pretty Woman,” which made her very eager to watch the clip. All she had to do was to go back to the OS home screen and then launch the Youtube app. When the landing page of the app was loaded, she looked for the search bar, typed in a query, and navigated through a few results before finding the clip of the scene. And the moment was gone.Mobile applications (a.k.a., apps) are already indispensable in our everyday lives. It is reported that the traffic from mobile devices has already surpassed that from PCs and apps have become the major entrance to the Internet <cit.>. Mobile users usually need to “navigate” among a set of apps to complete a specific task <cit.>. For example,one may receive a piece of news in the email app, read it in the newsreader app, and share it to the social networking app. Such a process is quite similar to browsing through Web pages. However, compared to the Web users who can easily navigate through the hyperlinks, app users like Alice often have to go through a frustrating procedure to manually switch from one app to another. This frustration is amplified when there are more and more apps installed on a device and when the user needs to switch back and forth between various apps.The inter-app navigation is indeed non-trivial for user interaction and has been drawing a lot of attention.Some solutions have been proposed to help bridge the “isolated”apps <cit.>. In particular, the recent concept of “deep links” has been proposed to facilitate the navigation from one app to another. Essentially, deep links are the URLs that point to specific locations inside an app page, which launches the app if it has been already installed on the device. Today, all major mobile platforms, including Android, iOS, and Windows, have supported deep links, and have been encouraging developers to implement and define the deep links to their apps <cit.>. So far, the deep link has been reported to have a poor coverage <cit.> - only approximately 25% of apps provide deep links and only a small number of pages within an app, as predefined by the developers, can be directly accessed via deep links.Deep link is a desirable concept, but what is holding it back? While there are historical reasons such as the fear of “stolen page views”,the most straightforward reason is that it takes non-trivial manual effort to do so. Indeed, unlike hyperlinks which are standardized and facilitated by the HTTP, there is no gold standard for deep links. The app developers, the platforms, and the apps stores all have to spend substantial effort to define, implement, standardize, and maintain them. Without teleportation spell, tunneling through the castles means tedious human work.Is it that bad? Perhaps not really. In practice, navigation between apps is actually performed between the “pages” inside the apps. Analogical to a website, an app also contains various pages, e.g., the landing page, advertisement pages, content pages, etc. Not all apps need to be deeply linked, and for those that do, not every single page needs to be linked from outside the apps.If one can distinguish those “important” pages from the rest and understand their relationship in navigation, the effort can be significantly reduced.Developers may focus their effort on creating deep links for these pages; they may also target the “partner” apps/pages.Furthermore, if the next page that a user is likely to visit can be predicted and a method that candynamically generate deep links is available <cit.>, runtime facilities such as Avitate <cit.> and FALCON <cit.> can be leveraged to navigate users to the destination page more efficiently.At the minimum, even if either of the wishes comes true, a simple estimation of the potential benefit of deep links could make the “deep link advocacy” more persuasive. Answers to these questions exactly requires an in-depth analysis of the inter-app navigation behavior, which unfortunately does not exist in literature. This paper makes the first empirical study of the page-level inter-app navigation behavior of Android users. We present an in-depth analysis of inter-app navigation based on a three-month behavioral data set collected from 64 users in 389 Android apps, consisting of about 0.89 million records of app-page level navigation. Inspired by the Web search and browsing, we classify the informational pages where users tend to stay and interact for long at an app, and the transitional pages where users only stay awhile or tend to skip when they navigate among informational pages. We summarize various patterns of inter-app navigation which can imply some routines of users. We believe that our empirical study results can provide useful insights to various stakeholders in the app-centric ecosystem.The main contributions of this paper are:* We propose a classification model to distinguish pages in an app into informational pages and transitional pages. We study the distribution of users' staying time of these two kinds of pages. The staying time in transitional pages looks like a Gaussian distribution with a mean of 4.3 seconds and a small value of variance, while the staying time of informational pages follows a log-normal distribution with the mean staying time of 29.4 seconds and a large variance, indicating that the time spent on this type of pages can differ dramatically. * We then demonstrate the inefficiency caused by transitional pages in inter-app navigation. The average time cost of inter-app navigation is around 13 seconds when navigating among different apps, in which the transitional pages account for 28.2%. Such an overhead of inter-app navigation is non-trivial for mobile users. * We explore two frequent patterns in inter-app navigations by analyzing which apps/pages are more likely to be linked during inter-app navigation, and under what contexts such navigation would happen. The task-specific pattern indicates that apps can be classified into clusters under which the apps cooperate with each other to accomplish specific tasks. The contextual pattern indicates that the inter-app navigation is related to context information such as network type, time, location, etc. * We propose some practical implications based on the findings, to facilitate app developers, OS vendors, and end-users. For example, we make a proof-of-concept demonstration by employing a machine learning based approach to accurately predicting the next informational page from current state and thus recommend a navigation path between two pages (i.e., a potential deep link) to reduce unnecessary transitional pages. To the best of our knowledge, this is the first empirical study on inter-app navigation behavior at a fine-grained level, i.e., page level rather than app level. The rest of this paper is organized as follows. Section <ref> presents the background of Android apps and formulate the inter-app navigation at the page level. Section <ref> presents our behavioral data set and how it was collected. Section <ref> describes an empirical study of inter-app navigation at page level and characterizes the navigation patterns. Section <ref> discusses some practical and potential useful implications that can be explored based on our empirical study. Section <ref> discusses the limitations of our work. Section <ref> relates our work to existing literature and Section <ref> concludes the paper. § INTER-APP NAVIGATION: IN A NUTSHELLAn Android app <cit.>, identified by its , usually consists of multiplethat are related to each other. Anis a component that provides an interface for users to interact with, such as dialing phones, watching videos, reading news, or viewing maps. Each activity has a uniqueand is assigned a window to draw its graphical user interface.For ease of understanding, we can draw an analogy between Android apps and the Web, as compared in Table <ref>. An Android app can be regarded as a website where the package name of the app is similar to the domain of the website. An activity can be regarded as a template of Web pages and an instance of an activity is like a Web page instance. Different Web pages of the same template differ in the values of parameters in their URLs. For example, URLs of different video pages on the Youtube website have the format of <https://www.youtube.com/watch?v=[xxx]>. This template can be regarded as the VideoActivity in the Youtube app. Without loss of generality, in this paper, we use the term page or app page to represent the activity that has a UI for users to interact with on their smartphones.In order to accomplish a task, users usually have to navigate through many pages. Among them, different pages serve different roles. Inspired by studies on the Web <cit.>, we classify the pages of mobile apps into two categories:* informational page: These pages serve to provide content/information, such as news article pages, video pages, mail editing page or chatting page, etc. Users accomplish their desired intention of one app in its informational pages. * transitional page: These pages are the intermediate pages along the way to reach the informational pages. Some of the pages serve to narrow the search space of possible informational pages and direct the users to them, including the system transitional pages, such as Launcher, where users can select the desired app that may contain the potential informational pages, and in-app transitional pages, such as ListPage in a news app, where users can choose a topic to filter the recommended articles. Other pages are somehow less helpful for navigation from the users' perspective, such as the advertisement pages and the transition splash pages. Based on our definition, the app navigation is the process of reaching the target informational page from the source informational page, via multiple transitioanl pages. Specifically, inter-app navigation is the navigation whose source and target informational pages belong to different apps.To illustrate the navigation process, a simple scenario is drawn in Fig <ref> . Suppose a user has found a ramen restaurant in the Yelp, and wants to read detailed comments about this restaurant in Reddit. To accomplish this task, the user has to first quit Yelp app to the OS launcher, click the app menu and then opens the Reddit app. Afterwards, the user has to start from the home page of Reddit app, walking through a series of transitional pages such as Search page and List page, and finally reaches the post page to read the comment about this restaurant. In this scenario, the true intention of the user is to be directly navigated to the target informational page, i.e., the post page, to read the comment. However, the user needs various transitional pages before landing on the final target page.Intuitively, users would like to spend more time on informational pages and avoid transitional pages. In view of that, some popular apps adopt flat UI design and carefully organize its functionalities and information display in order to reduce the number of transitional pages for reaching the target informational page. However, due to the fact that most of the apps provide only dedicated functionalities, users usually need to switch among multiple apps to achieve one task, resulting in notable time cost on inter-app navigations.We formally define a page-level app navigation as a triple <s, t, ψ>, where s is the source informational page, t is the target informational page, and ψ={p_i} is a sequence of transitional pages that represent a navigation path from s to t. We ensure that the screen state of the smartphone is always On in a whole navigation process. ψ can be empty, indicating that there is a direct link from s to t without having to pass through any transitional pages. By the definition, the inter-app navigation is a special type of app navigation where s and t belong to different apps. § DATA COLLECTION To study the inter-app navigation on Android smartphones, we conduct a field study by collecting behavioral data from real-world users. In this section, we present the design of our data collection tool and the description of the data set.As mentioned above, we focus on the page-level inter-app navigation. To this end, we develop a tool to monitor the system events representing such behaviors, as shown in Fig <ref>. The tool is a monitoring app running at the background of Android platforms. It consists of four modules, Activity Monitor, Context Extractor, Daemon, and Uploader. The activity monitor reads the top of the activity stack from the system every one second and produces a record entry whenever the top activity changes, indicating that a transition between app pages occurs. In the meantime, the monitor invokes the context extractor to collect the user's context information, including network type (cellular/Wi-Fi/Off), local time and screen status (ON/OFF). The records are stored in a local database, and the uploader will upload the records to our server once a day at night and under Wi-Fi network condition. Therefore, the tool does not influence the normal usage of mobile devices. To keep our tool from being killed by the system, the daemon module periodically checks the status of the tool and re-launches it if necessary. To protect user privacy, we anonymize the device ID with a hash string.We recruited student volunteers for the data collection via an internal social network site in Peking University. We got 64 on-campus student volunteers who fully agreed with the collection statements, and we installed the tool on their Android smartphones. The data collection lasted for three months, and we finally collected 894,542 records, containing 3,527 activities from 389 apps[The data collection and analysisprocess was conducted with IRB approval from the Research Ethic Committee of Institute of Software, Peking University. We plan to release the collected data once the work is published.]. Table <ref> provides some illustrating examples: denotes the unique and anonymized identifier of the user;anddenote which app and page (i.e., activity) that the user is interacting with, respectively.refer to the local Beijing time when the page is visited.indicates the network type when the page is visited, i.e., cellular, Wi-Fi, or offline;denotes the status of the device screen, i.e., ON and OFF.§ EMPIRICAL ANALYSISAccording to the definition of inter-app navigation, the most important issue is to identify the informational pages from transitional pages. In this section, we present an empirical study on our collected user behavioral data, based on which we can identify the informational pages and transitional pages through a clustering approach. We then explore the characteristics and find some interesting patterns of inter-app navigations. §.§ Clustering the App Pages According to the definition in Section <ref>, to identify the inter-app navigation, we should distinguish informational pages from transitional pages in our collected records of page usage. Just like the similar experiences on the Web pages <cit.>, a simple but intuitive measure is based on the length of staying time spent on the pages. Intuitively, users are likely to spend longer time on the informational pages, while the transitional pages are only for navigation purpose and thus users are likely to spend rather short time on them. Since our data records the sequence of pages, we are able to calculate the time interval for every single page by the timestamp when this page is visited and the timestamp when its subsequent page is visited. As each page could appear several times, we can obtain the distribution of the staying time for each page.Table <ref> lists example pages with the longest and shortest average staying time. From the name of these pages, we can speculate that the pages with longer staying time are more likely to be the pages that provide substantial information and available services. While the pages with shorter staying time have names containing “list” or “launcher”, indicating that these pages are more likely to be transitional pages. These examples imply that the staying time can be a vital clue to identify transitional pages.Inspired by the work by Van et al. <cit.>, which propose a classification approach to determine the seesion time threshold of user interaction, we propose to use an unsupervised clustering approach to separate the pages according to the users' staying time on the pages. A simple approach may be employing the average staying time by all users as the threshold. However, simply using the average staying time cannot reflect the overall staying time distribution of mobile users. Thus, we represent each page with the probability distribution of all visits' staying time.To measure the distance between the staying time distribution of pages, we may use the classic Kullback-Leibleer divergence (KLD) <cit.> and its symmetric version, the Jensen-Shannon divergence (JSD) <cit.>. In this paper, we choose the JSD distance as the distance between the distribution of every pairs of pages because its symmetric property makes it more appropriate for cluster. Then, we use the spectral clustering method <cit.> to cluster the pages.More specifically, such a process first learns a low-dimensional representation of each page according to the distance matrix between the pages, and then deploys the K-means algorithm to cluster the pages based on the low-dimensional representations. We assign different numbers of clusters in the K-means algorithm and select the optimal number of clusters through the Silhouette score <cit.>. Fig. <ref> visualizes the clustering results with different numbers of clusters using Fruchterman-Reingold force-directed algorithm <cit.>. The best clustering results are obtained when the number of clusters equals to 2, which well verifies our intuition that the pages can be classified into two categories.To further validate the distribution of staying time of these two types of pages, for each page, we calculate the median of its staying time and then depict the Probability Density Function (PDF) of all the pages in Fig <ref>. The blue curve indicates the PDF of staying time for informational pages while the red curve indicates transitional pages. The distribution of the transitional pages's staying time looks like a Gaussian distribution with a mean of 4.3 seconds and a small value of variance. The distribution of informational pages' staying time follows a log-normal distribution with the mean staying time of 29.4 seconds and a large variance, indicating that the time spent on this type of pages can differ dramatically.To evaluate the authenticity of our cluster, we randomly select 100 activities, and manually classify these activities by their name, staying time and screenshots. It turns out that this result is completely accorded with our cluster labels, verifying the reliability of our result.After identifying informational and transitional pages, we extract all the navigations from the records. Since we focus on only the navigation between different apps, we select all the inter-app navigation. The amount of inter-app navigation records is 41,619, which accounts for 26.9% of the total navigations. Our following analysis is conducted on these inter-app navigation records. §.§ Characterizing Inter-App Navigation Ideally, users prefer to be directly navigated between two informational pages, without any transitional pages on the navigation path.To explore whether the current inter-app navigation performs in such user-desired fashion, we use two metrics to quantitatively measure: 1) time cost, which is the aggregated value of staying time for all the transitional pages between the source informational page and target informational page; 2) step cost, which is the number of transitional pages on a navigation path. The distribution of time cost and step cost for all the inter-app navigations are illustrated as the red curve and labeled as total in Fig <ref>. The average time cost is 13.01s, and the average step cost is 2.69, indicating thatusers have to transit about 3 pages and spend 13 seconds when navigating among different apps. Additionally, we find that the time cost on the transitional pages takes up to 28.2% of the total time in the entire inter-app navigation, which is the total time cost of the source informational page, the subsequent transitional pages and the target informational page during a single navigation process. Therefore,the overhead of inter-app navigation is non-trivial for mobile users.We observe that most of the inter-app navigations do not jump directly from the source app to the target app. Instead, a large number of inter-app navigations involve pages from the Android system such as , which is the system home screen, or , which is the system list page showing all the recently used apps. We call these navigations as indirect navigations. However, a small number of inter-app navigations do not contain these system pages, but consist of transitional pages only from the source and target apps, e.g., the landing page of the target app. We call this kind of navigation as direct navigation between apps.Comparing the distribution of time cost and step cost between direct navigations and indirect navigations in Fig <ref>, we have the following observations:* The average time cost of direct and indirect navigations are 8.97 seconds and 24.43 seconds respectively; the average step cost of direct and indirect navigation are 0.97 and 3.86 respectively. This result indicates that the overhead of direct navigations is much smaller than that of indirect navigations.* All the indirect navigations have at least one intermediate step, i.e., the transitional page belongs to the Android system. In contrast, 66% of direct navigation have no such cost, indicating no navigation overhead. Therefore, the direct navigation is more appreciated.We further explore how direct navigations could happen. Some links navigate to the system-wide apps such as Call, Maps, Camera, and so on, which are pre-installed in the OS and have special APIs to invoke. Other links navigate the users directly from the source page to the target page of third-party apps without passing through system navigational pages, such as Launcher. There links are likely to be implemented by the emerging popular deep link <cit.>. Similar to hyperlinks of Web pages, deep links also employ URI to locate pages and can be executed to directly open the target page from the source page. For example, when a user reads an interesting article and wants to share it to facebook, he/she can click a share button, invoking a deep link that directly navigate he/she to the facebook interface without walking through multiple transitional pages.We are interested in whether the direct navigation is achieved by means of deep link. To this end, we first label the category information of each app[In this paper, we simply use the categorization system from a leading Android market, called Wandoujia <http://www.wandoujia.com>.], and filter out the system-wide apps. The remained third-party apps accounts for 82.7% of the total apps. Next, we check the manifest file of the third-party apps that contain the target pages to ensure they provide deep link interfaces. Finally, we employ a popular Android analysis tool IC3 <cit.> to check whether the source page actually invokes the target page's deep link interfaces.Given two apps, IC3 can compute all the inter-component communications (ICC) between these apps. If there exists an ICC between the activities of the source page and target page, then we can confirm that the source page has an entry to invoke the target page's deep link interfaces. Results show that within the direct navigation pairs, all the target pages provide deep link interfaces and all the source pages have an entry to invoke the corresponding interfaces, indicating that these direct navigations are highly likely to be implemented by deep links. §.§ Patterns of Inter-App Navigation Next, we explore whether there exist any patterns of inter-app navigations. By analyzing which apps/pages are more likely to be linked during inter-app navigation, and under which contexts such navigation would happen, we find two kinds of navigation patterns: task-specific pattern that the inter-app navigation is related to a specific type of task, which is inferred by page property, and contextual pattern that the inter-app navigation is related to context information such as network type, time, location, etc, which is the characteristic feature of mobile usage.§.§.§ Task-Specific Navigation Pattern We draw a graph representing all inter-app navigations by removing those transitional pages. As is illustrated in Fig <ref>, each node corresponds to an independent informational page, and is labeled with the category information of the app that the page belongs to. We assign an edge between two nodes from different apps if there exists a navigation between them. We then measure the distance by the static transition probability between every single informational page, which is calculated by the frequency of a particular navigation divided by the total navigation times. In this way, we get the distance matrix of the graph and visualize it with the classic Fruchterman-Reingold force-directed algorithm <cit.>, where each node size is determined by its out-degree.Interestingly, there are some significant clusters. In particular, there is a “core” cluster, in which most of the pages belong to the SOCIAL, SYSTEM, and MEDIA apps. Other clusters around the core include smaller nodes from different categories. From the distance matrix, we find that those small clusters around the core are mutually isolated with each other, while the nodes in these clusters are strongly cohesive. Additionally, all these small clusters have a strong correlation with the core cluster. We also find that apps consisting these small clusters belong to different categories.For example, Fig <ref> shows the pages of one cluster that is zoomed in. We can see that these pages come from a photo beautification app called(mtxx for short), a gallery app called ,a video app called , system camera and gallery, along with the directed links among them. Such observations may reflect the following scenario: a user captures a screenshot from a video page ofor takes a photo using the camera, then he/she navigates to the edit page in theand decorates this image. Eventually he/she saves the image into the . Such observations indicate that the inter-app navigation can be intuitively classified into some clusters. Each cluster could represent a specific user-task scenario, where different pages may cooperate with one another to accomplish this task.Indeed, the current results can only imply that such patterns may consist of a task. In our future work, we plan to combine session-level analysis to comprehensively specify the task characteristics <cit.>. §.§.§ Contextual Navigation Pattern Next, we are interested if any contextual patterns exist, i.e., whether some navigations are more likely to occur under specific contexts. From our collected data, we focus on two types of contexts, i.e., network and time.We first analyze whether the type of network affects user's preference of inter-app navigation. We classify all navigations into three clusters based on their network type. Similar to the task-specific pattern, for each cluster, we calculate the static transition probability between each pair of informational pages. Then we plot a heatmap for each network-type cluster, where pages are denoted by their categories. The source pages are plotted on the Y-axis while target pages are plotted on the X-axis. The results are presented in Fig <ref>.It is observed that only theis significant when the users are offline, while all the other categories are rather shallow. It indicates that users may mainly switch betweenapps such as SMS, Dialing, Camera, etc., when they are offline. When the network is under Wi-Fi and cellular, the inter-app navigations are more likely to happen, andapps are usually involved. However, the frequent patterns can vary between these two network types. When users are under Wi-Fi,apps are more likely to be linked with other apps. Sinceapps such as video players may require high-bandwidth and stable connection, it makes sense that such a navigation pattern exists. In a sense, one can infer which pages/apps that are more likely to be visited given the network type.Then we investigate whether some navigations are time sensitive. We compute the visit frequency of a page during a specific period of time, with a metric namely Usage Time Gap (UTG), which is defined as the interval between current time when a page is visited and the time when this page was latest visited. We draw the boxplot of the UTG for each categories in Fig <ref>. The UTG varies among different categories. Forapps, the median value and variance of UTG is the smallest, indicating that users may frequently visit the pages of these apps in a short time interval. For theandapps, their median UTG are much larger than the others, indicating that these apps are not “daily routines" for users. In other words, when used once, these apps are not likely to be revisited by users in a short time interval. Intuitively, with the metric such as UTG, one can infer which pages are more likely to be visited. Combined with the sequential information between two informational pages, it would be possible to prefetch these pages according to the patterns captured with UTG. § IMPLICATIONSThe preceding results have illustrated that the inter-app navigation has some patterns. Thus, the most intuitive implication is whether the navigation could be predicted, so that the transitional pages can be reduced or even be eliminated. Based on the prediction, our study would be useful for stakeholders including end-users, app developers, and OS vendors. §.§ Predicting Page Navigation We first investigate whether the inter-app navigation could be predicted, i.e., given a source informational page s, which target informational page t a user is likely to s navigate to? In fact, we can predict such behaviors based on our previously derived patterns. To make a proof-of-concept demonstration, we use a machine learning technique to conduct the prediction. Intuitively, such a prediction problem can be treated as a classification problem. We can treat a pair of app page navigation (s, t) as an instance. Each pair of navigation (s, t) can be represented with different types of features such as the features for source informational page s and target informational page t, as well as the contextual features when the navigation occurs.With the actual inter-app navigation behaviors in our collected data, we can observe many positive instances of inter-app navigation pairs. However, given a source informational page s, it isunknown which target information pages t will never be navigated to. Theoretically, this is known as a typical one-class classification problem <cit.> in literature. Here we adopt a start-of-the-art algorithm PU proposed by Liu et al. <cit.> <cit.> to solve the problem. The basic idea of the algorithm is to use the positive data to identify a set of informative negative samples from the unlabelled data (Here, each possible pair of source and target informational pages can be treated as an unlabeled instance.) In our experiments, we compare different variants of the algorithms, including S-EM, Spy+SVM, Spy+SVM-I, NB+EM, NB+SVM, and NB+SVM-I, and a naive solution which simply samples some random target pages for each source page and treats them as the negative samples.The overall classification process can be simply described in Algorithm <ref>. In the beginning, the algorithm generates an unlabelled data set U by randomly sampling some target pages t for each source page s. Then the algorithm identifies informative negative samples according to the positive training data. For prediction, given a source page s, we can rank the candidate target page t according to the probability of the pair (s,t) belonging to the positive class.In Table <ref>, we illustrate two types of features to represent each navigation pair (s, t),including the task-specific features of the pages and the contextual features associated with the pairs. For the task-specific features, three different features are identified including the name of the pages, the name of the apps which the pages belong to, and the name of the category that the page belongs to. The contextual features include the network type and UTG, representing the elapsed time of the target app since it was used last time until now. Note that all the features except UTG are quantitative variables, so we process them in one-hot encoding.We report the prediction results with different combinations of these features to illustrate their importance. As a ranking problem, we evaluate the prediction result according to a well adopted measure for ranking, called mean reciprocal rank (MRR) metric <cit.>. The MRR of a recommendation list is the multiplicative inverse of the rank for the correct answer. Obviously, the MRR score and the recommendation performance is positively correlated. Indeed, there are different variants of the PU algorithm. Therefore, we first randomly filter 30% of the data reserved for subsequent evaluation, and then conduct a five-fold cross validation to compare different variants of the algorithms. The results show that S-EM algorithm yields the better results.We compare the proposed approach with two other straightforward algorithms including popularity approach and Markov chain based approach. Popularity approach ranks the pages according to their usage frequency, andMarkov chain based approach is ranked by the static transition probability for one page to be navigated from the given source page. To evaluate our approach, we use the 30% data as test set which is previously reserved for evaluation and the remaining 70% data as training set.Fig <ref> presents the prediction results with different algorithms and features. First, the classification algorithm with all the features outperform the naive Popularity and Markov Chain methods, with its MRR score of 0.328, implying that the actual page is on average ranked at the third place of our recommendation list. This is because the classification algorithm can effectively integrate the task-specific features of pages and contextual features for prediction, which have been proved to be very important according to the exploratory analysis in previous section.Second, as for the importance of different features for prediction, we tried different combinations of features for prediction. We first treat the page name as the basic feature. Adding the name of the app or the category of the app only slightly improves the performance. This may be because all the three features are about the task-specifics of the pages, which are strongly correlated, and the name of the pages already well represent the task-specifics of the pages. Adding either of the contextual features significantly improve the result, showing that the contexts are indeed very important factors to affect the inter-app navigation. By combing all the features, the best-of-breed prediction result is obtained.§.§ Further Potential Application Scenarios The preceding proof-of-concept demonstration indicates that the inter-app navigation could be predicted to some extent, based on which we can further explore some potential applications for different stakeholders, including app developers, OS vendors, and end-users.Based on the information of inter-app navigation, app developers can more efficiently identify the upstream partner apps from which users can navigate to the current app, anddownstream partner apps to which users can navigate from the current app. On one hand, app developers could expose “deep links” to upstream apps in order to attract more user visits from upstream apps. On the other hand, app developers could integrate deep links from downstream apps in order to precisely introduce users for downstream apps. Similar to the hyperlinks of Web pages, deep links can introduce more “page visits” of an app. In particular, the in-app ads are the major revenue channels for app developers, hereby establishing the deep links can potentially increase the possibility of ads clicking.For current system-wide smart assistant such as Aviate <cit.>, our analysis of navigation patterns can provide insights to further improve the user experiences. Aviate predicts the next app that is likely to be used by users and displays the shortcut of apps on system home screen. However, the prediction is performed at app level, thus users still have to manually locate the desired page by several tedious steps. Combined with our analysis, the prediction can be performed at a fine-grained level with less “transitional” overhead but navigate users directly to the page that they are really interested in. In addition, the predicted page can be more accurate to contexts (e.g., the current network condition), and can be prefetched into RAM to enable fast launch.Finally, the ultimate goal of our study is to build a recommendation assistant that can predict which app pages a user desires to access, and can dynamically generate deep links for such desired pages. Thus, the user can pass through a few transitional pages in the recommendation assistant, rather than the time-consuming navigation steps. Since previous research efforts such as uLink <cit.> have already proposed practical approaches to dynamically generating deep links for app pages, the key point of such a recommendation assistant is to accurately predict which pages should be navigated. Our prediction can provide which kind of informational pages the user is likely to visit. Based on the prediction, the assistant can recommend the possible pages and generate a deep link for each page, enabling the user to choose the desired page and navigate directly to that page.For example, if we predict that a user is going to watch a movie, the system can offer a top recommended movie list for the user to select. Our findings can integrate the recent efforts on in-app semantic analysis <cit.>, by which the system can more precisely predict which app page the user desires. § DISCUSSION This paper does make the first-step empirical study to derive preliminary knowledge of inter-app navigation. Our results come from the field study conducted on 64 student volunteers' devices for three months, with 0.89 million records of user behaviors. Indeed, such a scale is a bitsmall, and the selected user group may not be comprehensive enough, e.g., reflecting the preferences of only on-campus students. In this way, the empirical results and derived knowledge may not generalize to users from other groups. However, our analysis approach and prediction techniques can still be generalized. Indeed, predicting the inter-app navigation behavior is quite meaningful in various aspects, e.g., traffic accounting and in-app ads.Our data collection tool introduces very little additional overhead. Based on our previous industrial experiences of large-scale user study collaborated with leading app store operators <cit.> and input methodapps <cit.>, we plan to evaluate and integrate our data collection tool in these platforms, so that we can learn more comprehensive knowledge of inter-app navigation. In addition, with the access to a large number of user profile, we can add features of user modeling to enrich the analysis of navigation patterns.Another limitation of our empirical study is that our data collection is focused at page (activity) level. However, modern Android apps make use of fragment, which is a portion of user interfaces (e.g., tabs) in an activity and can be roughly regarded as dynamic sub-pages. Ideally, it is more comprehensive to measure the session time the user spends on each fragment and on each activity, respectively. We plan to consider conducting this measurement study in our future work.This paper demonstrates the potential overhead caused by those transitional pages. From the user's perspective, we argue that such overhead could compromise user experiences and shall be avoided. However, from the developer's perspective, some of the transitional pages are still meaningful, e.g., containing some in-app ads to increase developers' revenues. Hence, it may not be reasonable or realistic to remove all transitional pages. In practice, we need to carefully justify whether a transitional page should be contained or not on the navigation path, e.g., allowing the developers to configure when releasing deep links, or the end users to decide by their own preferences. § RELATED WORK In this section, we discuss the related existing literature studies and compare with our work.§.§ Field Study of Smartphone UsageThere have been some field studies to investigate user behaviors on smartphones. Ravindranath et al. <cit.> developed AppInsight to automatically identify and characterize the critical paths in user transactions, and they conducted a field trial with 30 users for over 4 months to study the app performance. Rahmati et al. <cit.> designed LiveLab, a methodology to measure real-world smartphone usage and wireless networks with a reprogrammable in-device logger designed for long-term user studies. They conducted a user experiment on iPhone and analyzed how users use the network on their smartphones. In their following work <cit.>, they conducted a study involving 34 iPhone 3GS users, reporting how users with different economic background use smartphones differently. Mathur et al. <cit.> carried out experiments with 10 users to model user engagement on mobile devices and tested their model with smartphone usage logs from 130 users. Ferreira et al. <cit.> collected smartphone application usage patterns from 21 participants and participants context to study how they manage their time interacting with the device. Our field study collected user behavior data from 64 volunteers for 3 months, of which the quantity is comparable with existed studies. §.§ Mining Navigation PatternsA lot of research efforts have been made on mining the navigation patterns on the Web. Some existing literatures leveraged graph models. Borges et al. <cit.> proposed an N-gram model to exploit user navigation patterns and they used entropy as an estimator of the user sessions' statistical property. Anderson et al. <cit.> proposed a Relational Markov Model(RMM) to model the behavior of Web users for personalizing websites. Liu et al. <cit.> proposed a method of computing page importance by user browsing graph rather than the traditional way of analyzing link graph. Chierichetti et al. <cit.> studied the extent to which the Markovian assumption is invalid for Web users. Other literatures use sequential pattern mining to exploit association rules. Fu et al. <cit.> designed a system which actively monitors and tracks a user's navigation, and applied A-priori algorithm to discover hidden patterns. Wang et al. <cit.> divided navigation sessions into frames based on a specific time internal, and proposed a personalized recommendation method by integrating user clustering and association-mining techniques. West et al. <cit.> studied how Web users navigate among Wikipedia with hyperlinks. Our work is inspired by these previous efforts on the Web and we focus on how users navigate among apps on mobile system. Similar to our work, Srinivasan et al. <cit.> developed a service called MobileMiner that runs on the phone to collect user usage information. They conducted a user experiment and use a sequential mining algorithm to find user behavior patterns. Jones et al. <cit.> present a revisitation analysis of smartphone use to investigate whether smartphones induce usage habbits. They distinguish the pattern granularity into macro and micro level, and find unique usage characteristics on micro level. Both of them dig into app-level usage patterns, but our work focuses on the much deeper page-level navigation patterns rather than the app level. §.§ Prediction of App UsagePredicting the apps to be used not only facilitates mobile users to target the following apps, but also can be leveraged by mobile systems to improve the performance. Abhinav et al. <cit.> proposed a method similar to a text compression algorithm that regards the usage history as sequential patterns and uses the preceding usage sequence to compute the conditional probability distribution for the next app. Yan et al. <cit.> proposed an algorithm for predicting next app to be used based on user contexts such as location and temporal access patterns. They built an app FALCON that can pop the predicted app to home screen for fast launching. Richardo et al. <cit.> proposed Parallel Tree Augmented Naive Bayesian Network (PTAN) as the prediction model, and used a large scale of data from Aviate log dataset to train their model, and achieve high precision result.Our work differs from previous efforts much in several aspects. On one hand, the granularity of our research is on the page level, which is deeper than the app level. As a result, we cannot exploit the existing dataset that mainly records the user behaviors at the app level. On the other hand, we reveal that the inter-app navigation has many transitional pages, which is a tremendous noise for the purpose of predicting informational pages. Thus, we propose a clustering approach to filter out the transitional pages, and enable more precise prediction. § CONCLUSION In this paper, we have presented a quantitative study of the inter-app navigation behavior of Android users. The analysis was based on behavioral data collected from a real user study, which contains nearly a million records of page transitions. We found that unnecessary transitional pages visited between two inter-app informational pages take up to 28.2% of the time in the entire inter-app navigation, a cost that can be significantly reduced through building direct links between the informational pages.An in-depth analysis reveals clear clustering patterns of inter-app pages, and a machine learning algorithm effectively predicts the next informational page a user is navigating into. Our results provide actionable insights to app developers and OS vendors. While we have made the preliminary results of demonstrating the feasibility of predicting the next page using a standard algorithm, it is not our intent to optimize the prediction performance in this study. It is a meaningful future direction to build advanced prediction models, especially to consider the previous navigation sequences in the same session and to personalize the prediction. It is also intriguing to enlarge the scale of the user study to cover users with diverse demographics. abbrv | http://arxiv.org/abs/1706.08274v1 | {
"authors": [
"Ziniu Hu",
"Yun Ma",
"Qiaozhu Mei",
"Jian Tang"
],
"categories": [
"cs.HC"
],
"primary_category": "cs.HC",
"published": "20170626082421",
"title": "Roaming across the Castle Tunnels: an Empirical Study of Inter-App Navigation Behaviors of Android Users"
} |
]Roberto AmadiniDepartment of Computing and Information Systems, The University of Melbourne, Australia.Maurizio GabbrielliDISI, University of Bologna, Italy /FOCUS Research Team, INRIA, France.Jacopo MauroDepartment of Informatics, University of Oslo, Norway.SUNNY-CP and the MiniZinc Challenge[ December 30, 2023 ===================================== In Constraint Programming (CP) a portfolio solver combines avariety of different constraint solvers for solving a given problem.This fairly recent approach enables to significantly boost the performanceof single solvers, especially when multicore architectures are exploited. In this work we give a brief overview of theportfolio solver , and we discuss its performance in theMiniZinc Challenge—the annual internationalcompetition for CP solvers—where it won two gold medals in 2015 and 2016.Under consideration in Theory and Practice of Logic Programming (TPLP) § INTRODUCTION In Constraint Programming (CP) the goal is to model and solveConstraint Satisfaction Problems (CSPs) andConstraint Optimisation Problems (COPs) <cit.>. Solving a CSP means finding a solution that satisfies all the constraintsof the problem, while for COPs the goal is to find an optimal solution, whichminimises (or maximises) an objective function.A fairly recent trend to solve combinatorial problems, based on the well-knownAlgorithm Selection problem <cit.>, consists of building portfoliosolvers <cit.>.A portfolio solver is a meta-solver that exploits a collectionof n > 1 constituent solvers s_1, …, s_n. When a new, unseen problemcomes, the portfolio solver seeks to predict and run its best solver(s)s_i_1, …, s_i_k (with 1 ≤ k ≤ n) for solving the problem.Despite that plenty of Algorithm Selectionapproaches have been proposed <cit.>, a relatively small number of portfolio solvers havebeenpractically adopted <cit.>.In particular, only few portfolio solvers participated in CP solverscompetitions. The first one (for solving CSPsonly) was CPHydra <cit.> that in 2008 won the International CSP SolverCompetition.[The International CSP SolverCompetition ended in 2009.] In 2013 a portfolio solver based onNumberjack <cit.> attended the MiniZinc Challenge(MZNC) <cit.>,nowadays the only surviving international competition for CPsolvers.Between 2014 and 2016, was the only portfolio solver thatjoined the MZNC. Its first, sequential version had appreciable results in the MZNC 2014 but remained off the podium. In MZNC2015 and 2016 its enhanced, parallel version <cit.> demonstratedits effectiveness by winning the gold medal in the Open category of thechallenge.In this paper, after a brief overview of ,wediscuss the performance it achieved in the MiniZinc Challenges2014—2016 and we propose directions for future works. The lessons we learned are: * a portfolio solver is robust even in prohibitive scenarios, like the MiniZinc Challenge, characterised by small-size test sets and unreliable solvers; * in a multicore setting, a parallel portfolio ofsequential solvers appears to be more fruitful than a single, parallelsolver; * can be a useful baselineto improve the state-of-the-art for (not only) the CP field,where dealing with non-reliable solvers must be properly addressed. § SUNNY AND SUNNY-CP In this section we provide a high-level description of , referring theinterested reader to Amadini et al.for a moredetailedpresentation.is an open-source CP portfolio solver.Its firstimplementationwas sequential <cit.>, while the current version exploits multicorearchitectures to run more solvers in parallel and to enabletheir cooperation via bounds sharing and restarting policies.To the best of our knowledge, it is currently the only parallelportfolio solver able to solve generic CP problems encoded in MiniZinclanguage <cit.>. is built on top of SUNNY algorithm <cit.>. Given aset of known problems, asolving timeout T and a portfolio Π, SUNNY uses the k-Nearest Neighbours (k-NN) algorithm to produce a sequential scheduleσ = [(s_1, t_1), …, (s_k, t_n)] where solver s_i ∈Π has to be run fort_i seconds and ∑_i = 1^n t_i = T.The time slots t_i and the ordering of solvers s_i are definedaccordingto the average performance of the solvers of Π on the k traininginstancescloser tothe problem to be solved. For each problem p,a feature vector is computed and theEuclidean distance is used to retrieve the k instances in the training setcloser top. In a nutshell, a feature vector is a collection of numerical attributes thatcharacterise a given problem instance. uses several features,e.g., statistics over the variables, the (global) constraints,the objective function (when applicable), the search heuristics. In total,uses 95 features.[The firstversion of also used graph features and dynamic features, afterwardsremoved for the sake of efficiency and portability. For more details about features, please seemzn2feat and <https://github.com/CP-Unibo/mzn2feat>.]The sequential schedule σis then parallelisedon the c ≥ 1 available cores by running the first and most promisingc-1 solvers in the k-neighbourhood on the first c-1 cores, while theremaining solvers (if any) are assigned to the last available core by linearlywidening their allocated times to cover the time window [0, T]. The notion of “promising solver” depends on the context. For CSPs, theperformance is measured only in termsof number of solved instances and average solving time. For COPs, also the quality of the solutions is taken into account <cit.>. We might say that uses a conservative policy: first, it skims the originalportfolio by selecting a promising subset of its solvers; second, it allocatesto each of these solvers an amount of time proportional to theirsupposed effectiveness.Solvers are run in parallel and a “bound-and-restart” mechanism is usedfor enabling the bounds sharing between the running COPsolvers <cit.>.This allows one to use the (sub-optimal) solutions found by a solver to narrow the search space ofthe other scheduled solvers. If there are fewer solvers than cores, simply allocates a solver percore.Since treats solvers as black boxes, it can not support thesharing of the bounds knowledge without the solvers interruption. Forthis reason,a restarting threshold T_r is used to decide when to stop a solverandrestart it with a new bound. A running solver is stopped andrestarted when: (i) it hasnot found a solution in the last T_r seconds; (ii) its current bestbound is obsolete w.r.t. the overall best bound found by another scheduled solver.Table <ref> summarises the solvers used by in the MZNCs 2014–2016.For more details about these solvers,seechoco,minisatid, picat-sat, haifacsp, mzn, jacop, mistral,ortools, chuffed, opturion, izplus. § SUNNY-CP AND THE MINIZINC CHALLENGEThe MiniZinc Challenge(MZNC) <cit.> is the annualinternational competition for CP solvers. Portfolio solvers compete in the “Open” class of MZNC,where all solvers are free to use multiple threads or cores,and no search strategy is imposed.The scoring system of the MZNC is based on a Bordacount <cit.>where a solver s is compared against each other solver s'over 100 problem instances—belonging to different classes—defined in theMiniZinc language. If s gives a betteranswer than s' then it scores 1 point, if it gives a worse solution it scores0 points.If s and s' give indistinguishable answers the scoring is based on thesolving time.[Please refer to<http://www.minizinc.org/challenge.html> for further details.]Until MZNC 2014, the solving timeout was 15 minutes and did not include the MiniZinc-to-FlatZinc conversion time. Starting from the MZNC 2015 thistime has been included, and the timeout has been extended to 20 minutes. §.§ MiniZinc Challenges 2013–2014Table <ref> summarizes the Open class results in the MZNCs2013–2014. The first portfolio solver that attended a MiniZinc Challenge in 2013 (see Table <ref>)was based on Numberjack platform <cit.>. In the following years, was unfortunately the only portfolio solver that entered thecompetition. In 2014, was a sequential solver running just one solver at time. We willdenote it with to distinguish such version fromthe current parallel one. came with two versions:the default one and a version withpre-solving denoted in Table <ref> as. In the latter astaticselection of solvers was run for a short time, before executingthe default version in the remaining time. Both of the two versions usedthe same portfolio of 7 solvers, viz. Chuffed, CPX, G12/FD, G12/LazyFD, Gecode, MinisatID, MZN/Gurobi. For more details, we refer the reader to sunnycp.improved on Numberjack and obtained respectable results: the two variants ranked4th and 7th out of 18.had to compete also with parallel solvers and all its solvers exceptMinisatID and MZN/Gurobi adopted the “fixed” strategy, i.e., they used thesearch heuristic defined in the problems instead of their preferredstrategy. As described by sunny_rcpsp, we realisedafterward that this choice is often not rewarding.To give a measure of comparison, as shown in Table<ref>, in the “Fixed”[According toMZNC rules, each solver in the Fixed category that has not a Free versionis automatically promoted in the Free category (analogously, solvers inthe Free category can be entered in the Parallel category, and thenin turn in the Open category). ]category—where sequential solvers mustfollow the search heuristic defined in the model—would have been ranked1st and 3rd. Moreover, unlike other competitors, the results of were computed by including also the MiniZinc-to-FlatZinc[MZNC uses the MiniZinc language to specify the problems, while the solvers use the lower level specification language FlatZinc, whichis obtained by compilation from MiniZinc models.] conversion timesince, by its nature, can not be a FlatZinc solver (seesunnycpfor more details). This penalised especially for the easier instances.§.§ MiniZinc Challenge 2015Several enhancements of were implemented after the MZNC 2014: (i) became parallel, enabling the simultaneous execution of its solverswhile retaining the bounds communication for COPs; (ii) newstate-of-the-art solvers were incorporated in its portfolio; (iii)became more stable, usable, configurable and flexible.These improvements, detailed by sunnycp2 where has been tested on largebenchmarks, have been reflected in itsperformance in the MZNC 2015. participated in the competition with two versions: a defaultone and an “eligible” one, denoted in Table <ref>. Thedifference is that did not includesolvers developed by the organisers of the challenge, and thereforewas eligible for prizes.usedChoco, Gecode, HaifaCSP, iZplus, MinisatID, Opturion CPX and OR-Tools solvers,while used also Chuffed, MZN/Gurobi, G12/FD and G12/LazyFD. Since theavailability of eight logical cores, performed algorithm selection forcomputing and distributing the SUNNY sequential schedule,while launched all its solvers in parallel. Table <ref> shows that is theoverall best solver while won thegold medal since Chuffed—the best sequential solver—was not eligible forprizes.The column “Incomplete” refers to the MZNCscore computed without giving any point for proving optimality orinfeasibility. This score, meant to evaluate local search solvers, only takesinto account the quality of a solution. Note that with this metricalso overcomes Chuffed, without having it in the portfolio.Several reasons justify the success of in MZNC 2015. Surely theparallelisation on multiple cores of state-of-the-art solvers was decisive,especially because it was cooperative thanks tobounds sharing mechanism.Moreover, differently from MZNC 2014, all thesolvers were run with their free version instead of the fixed one. Furthermore, the MZNC rules were less penalising for portfolio solvers since forthe first time in the history of the MZNCs the total solving time included also the MiniZinc-to-FlatZinc conversion time.We underline that the constituent solvers of do not exploit multi-threading.Hence, the parallel solvers marked with * in Table <ref>are not the constituent solvers of but their (new) parallelvariants. The overall best single solver is Chuffed, whichis sequential. Having it in theportfolio is clearly a benefit for. However, even without Chuffed, is able to provide solutions ofhigh quality (see “Incomplete” column ofTable <ref>) proving that also the other solvers are important forthe success of .We remark that—as pointed out also by sunnycp—when compared to the best solver for a given problem, a portfolio solveralways has additional overheads(e.g., due to feature extraction or memory contention issues) that penaliseits score. The 100 problems of MZNC 2015 are divided into 20different problem classes, each of whichconsisting of 5 instances: in total,10 CSPs and 90 COPs. was the best solver for only two classes:and . Interestingly, for the wholeproblem class, scored 0 points because it always provided anunsound answer due to a buggy solver.This is a sensitive issue that should not be overlooked. On the one hand, abuggy solverinevitably affects the whole portfolio making it buggy as well. On the otherhand, not using an unstable solver may penalize the global performance sinceexperimental solvers like Chuffed and iZpluscan be very efficient even if not yet in astable stage.As we shall see also in Section <ref>, unlike SAT but similarly to SMT field, most CP solversare not fully reliable (e.g., in MZNC 2014 one third of the solvers providedat least an unsound answer). When unreliable solvers are used, a possible way to mitigate the problemisto verify a posteriori the solution. For instance, another constituentsolver can be used for double-checking a solution. Obviously, checking all thesolutions of all the solvers implies a slowdown in the solving time. Note that the biggestproblems arise when the solver does not produce a solution or when it declaresa sub-optimal solution as optimal. In the first case, since solversusually do not present a proof of the unsatisfiability, checking thecorrectness of the answer requires solving the same problem from scratch. Inthe second case, the presence of a solution may simplify the check of theanswer, but checking if a solution is optimal is still an NP-hard problem.In MZNC 2015 checkedHaifaCSP, since its author warned us about its unreliability.This allowed to detect 21 incorrect answers. Without this check its performance would have beendramatically worse: would have scored 87.5 points less—thusresultingworse than Chuffed—while would have scored 206.84 points less, passingfrom the goldmedal to no medal. However, this check was not enough: due to bugs in other constituentsolvers provided 5 wronganswers, while provided 7 wrong answers. §.§ MiniZinc Challenge 2016In the MiniZinc Challenge 2016 we enrolled three versions, namely:, , and .was not eligible for prizes and added to the portfolio of MZNC 2015three new solvers: JaCoP, Mistral, and Picat-SAT.contained only the eligible solvers of , i.e.,Choco, Gecode, HaifaCSP, JaCoP, iZplus, MinisatID, Mistral, Opturion CPX, OR-Tools,Picat.[We did not have an updated version of Choco and Opturionsolvers, so we used their2015 version.]contained only the solvers of that never won a medalin the Free category of the last three challenges, i.e., Gecode, HaifaCSP, JaCoP, MinisatID, Mistral, Picat. Ideally, we aimed to measure the contribution of the supposedly best solvers of . Conversely, to and , does not need to schedule itssolvers, having fewer solvers thanavailable cores. It just launches all its solvers in parallel.Table <ref> shows the Open category ranking of the MZNC 2016.These results are somehow unexpected if compared with thoseof the previous editions. For the first time, Chuffed has been outperformed by asequential solver, i.e., the new, experimental LCG-Glucose—a lazy clause generation solverbased on Glucose SAT solver. Surprisingly, solvers like OR-Tools, iZplus, Choco had subduedperformance. Conversely, HaifaCSP and Picat-SAT performed very well. The sharp improvementof the solvers based on Gurobi and CPLEX is also clear, arguably due to a better linearisation of theMiniZinc models <cit.>. Local search solvers still appear immature.The results of are definitely unexpected. In particular, it appears quite strange thatperformed far worse than although having more, and ideally better, solvers. We then thoroughly investigated this anomaly since, as alsoshown in Amadini et al. , the dynamic schedulingof the availablesolvers is normally more fruitful than statically running an arbitrarily good subset of themover the available cores.Firstly, we note that for the easier instancesis inherently faster than and because it does not need to scheduleits solvers, and therefore it skips the feature extraction and the algorithm selection phasesof the SUNNY algorithm <cit.>. Nevertheless, most of the MZNC 2016instances are noteasy to solve.Another reason is that always runs HaifaCSP and Picat-SAT,two solvers that performed better than expected, while executes Picat-SAT onlyfor 37 problems. Nonetheless, always executes HaifaCSP so also this explanationcan not fully explain the performance difference.The actual reason behind the performance gap relies on some buggy solvers which belongs to but not to.[With the term “buggy solver” we not necessarily mean thatthe solver itself isactually buggy. The problems may arise due to a misinterpretation of theFlatZincinstances or to the wrongdecomposition of global constraints <cit.>.]In our pre-challenge tests we did not notice inconsistencies in anyof the solvers, except for Choco. So we decided to check the solutions only for Choco andHaifaCSP (the latter because of the unreliability shown in the MZNC 2015, see Section <ref>). However, none of these solvers gave an unsound outcome in the MZNC 2016.Conversely, Opturion and OR-Tools solvers provided a lot of incorrect, andunfortunately unchecked, answers. We also noticed that for some instances our version ofMistral failed when restarted with a new bound, while on the same instances the Free version ofMistral provided a sound outcome. In total, gave 24 wronganswers,[Namely, all the 5 instances of , , andclasses; 4 instances ofclass;1 instance of , , ,,classes.] meaning that itcompeted only on the 76% of the problems of MZNC 2016. failed instead on 5 instances.Table <ref> shows the results without the 24 instancesfor which gave an incorrect answer.We underline that this table has a purely indicative value: for a more comprehensivecomparison, also the instances where other solvers provided an incorrect answershould be removed. On these 76 problemsovercomes LCG-Glucose, while impressivelygains 7 positions and becomes gold medallist being the first of the eligible solvers. however behaves well (silver medallist), being overtaken by only.Note that the results of are good also in the original ranking of Table <ref> since, being this version not eligible for prizes, the organisers enabled the solutions checking ofG12/LazyFD, HaifaCSP, Mistral, Opturion, OR-Tools. This allowed to detect 19 incorrectanswers. An interesting insight is given by the Incomplete score, which does not give any benefitwhen a solver concludes the search (i.e., when optimality or unsatisfiability is proven).As observed also in Section <ref>, with this metriccan significantlyovercome a solver that has a greater score (e.g., see the Incompletein Table <ref>). This confirms the attitude of in findinggood solutions even when it does not conclude the search. § CONCLUSIONSWe presented an overview of , a fairly recent CP portfolio solverrelying on the SUNNY algorithm, and we discussed its performance inthe MiniZinc Challenge—the annual international competition forCP solvers. In the MiniZinc Challenge 2014 received an honourable mention,in 2015 it has been the first portfolio solver to wina (gold) medal, and in 2016—despite several issues with buggy solvers—itconfirmed the first position.For the future of CP portfolio solvers, it would be interestinghaving more portfolio competitors to improve the state of the art in this field. Different portfolio approaches have been already compared w.r.t. and itsversions <cit.>. The Algorithm Selection approaches of the ICONChallenge 2015 <cit.> might be adapted to deal with generic CPproblems.The SUNNY algorithm itself, which is competitive in the CP scenarios ofsunny,paper_amai,[ We submitted such scenarios, namelyand, to the Algorithm Selection Library <cit.>. ] provided very poorperformance in the SAT scenarios of the ICON Challenge andsunny_plus show that it can be strongly improved with a propertraining phase. runs inparallel different single-threaded solvers. This choice so far has proved to bemorefruitful than parallelising the search of a single solver. However, thepossibility of using multi-threaded solvers may have some benefits when solvinghard problems as shown by 3s-par for SAT problems.The multi-threaded execution also enables search splitting strategies. It is not clear to us if the use of all the available cores, as done by, is the best possible strategy. As shown by parallelSatInsights it is possible that running in parallel all thesolvers on the same multicore machine slows down the execution of theindividual solvers.Therefore, it may be more convenient to leave free one or more coresand run just the most promising solvers. Unfortunately, it is hard to extrapolate a general pattern to understandthe interplay between the solvers and their resource consumption.One direction for further investigations, clearly emerged from the challengeoutcomes, concerns how to deal with unstable solvers. Under these circumstances it is important to find a trade-offbetween reliability and performance. Developing an automated way of checking aCP solver outcome when the answer is “unsatisfiable problem” or“optimal solution” is not a trivial challenge: we can not merely do a solutioncheck, but we have to know and check the actual explanation for which thesolver provided such an outcome.A major advancement for CP portfolio solvers would be having API for injectingconstraints at runtime, without stopping a running solver. Indeed,interrupting a solver means losing all the knowledge it has collected so far.This is particularly bad for Lazy Clause Generation solvers, and in general forevery solver relying on no-good learning.Another interesting direction for further studies is to consider the impact ofthe global constraints <cit.> on the performances of the portfoliosolver. It is well-known that the propagation algorithms and thedecompositions used for global constraints are the keys of solverseffectiveness. We believe that the use of solvers supporting different globalconstraint decompositions may be beneficial.We underline that—even if focused on Constraint Programming—this work can beextended to other fields, e.g., Constraint Logic Programming, Answer-Set Programmingor Planning, where portfolio solving has been used only marginally.To conclude, in order to follow the good practiceof making the tools publicly available and easy toinstall and use, we stress that ispublicly available at <https://github.com/CP-Unibo/sunny-cp> and can beeasily installed, possibly relying on the Docker container technology for avoidingthe installation of its constituent solvers. All the resultsof this paper can be reproduced and verified by using the web interface of<http://www.minizinc.org/challenge.html>. § ACKNOWLEDGEMENTSWe are grateful to all the authors and developers of the constituent solvers of ,for providing us the tools and the instructions to use the solvers. Wethank all the MiniZinc Challenge staff, and in particular Andreas Schutt, for theavailability and technical support.This work was supported by the EU project FP7-644298 HyVar: Scalable Hybrid Variability for Distributed, Evolving Software Systems.acmtrans | http://arxiv.org/abs/1706.08627v3 | {
"authors": [
"Roberto Amadini",
"Maurizio Gabbrielli",
"Jacopo Mauro"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170626234814",
"title": "SUNNY-CP and the MiniZinc Challenge"
} |
[email protected] [email protected] 1CAS Key Laboratory of Geospace Environment, Department of Geophysics and Planetary Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, China 2School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai, Guangdong, 519000, China3Collaborative Innovation Center of Astronautical Science and Technology, China 4Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China5Mengcheng National Geophysical Observatory, University of Science and Technology of China 6Institute of Physics/IGAM, University of Graz, Universitätsplatz 5/II, 8010 Graz, Austria In this paper, we identified the magnetic source locations of 142 quasi-homologous (QH) coronal mass ejections (CMEs), of which 121 are from solar cycle (SC) 23 and 21 from SC 24. Among those CMEs, 63% originated from the same source location as their predecessor (defined as S-type), while 37% originated from a different location within the same active region as their predecessor (defined as D-type). Their distinctly different waiting time distribution, peaking around 7.5 and 1.5 hours for S- and D-type CMEs,suggests that they might involve different physical mechanisms with different characteristic time scales.Through detailed analysis based on non-linear force free (NLFF) coronal magnetic field modeling of two exemplary cases, we propose that theS-type QH CMES might involve a recurring energy release process from the same source location (by magnetic free energy replenishment),whereas the D-type QH CMEs can happen when a flux tube systemdisturbed by a nearby CME. § INTRODUCTIONCoronal mass ejections, huge expulsions of plasma and magnetic fields from the solar corona, are among the drivers of hazardous space weather.Besides the knowledge on the propagation of a CME in interplanetary space, a successful space weather forecast also requires a precise understanding of the physical mechanisms behind CMEs, as well as their relation to other phenomena in the solar atmosphere. CMEs may originate from either active regions (ARs)or quiescent filament regions <cit.>. Statistical studies suggestthat about two thirds of CMEs originate from ARs, although the percentages vary from63% to 85%in different studied samples <cit.>.The flare and CME productivity of different ARs varies <cit.>. Some ARs barely produce an eruption, some produce numerous subsequent flares without accompanying CME <cit.>, and some others can generate many flare-associated CMEs within a short duration. It appears that ARs which accumulate large amounts of magnetic free energy tend to produce a larger number and more powerful flares and CMEs than ARs with a small magnetic free energy budget <cit.>. Additionally, the larger a flare, the more likely it is accompanied by a CME <cit.>. The triggering mechanism of a CME itself, however, is most likely determined by the involved magnetic field topology, both, of the unstable CME structure and its AR environment. CMEs are termed “homologous” when they originate from the same region within an AR and exhibit a close morphological resemblance in coronal and coronagraphic observation <cit.>.However, CMEs may originate from different parts of an AR, and/or even have different appearances.Following <cit.>, we use the term “quasi-homologous" CMEs, to denote subsequent CMEs that originate fromthe same AR, but disregarding their detailed magnetic source locations and appearances.Statistical analysis of the waiting times of QH CMEs has been performed by<cit.> and <cit.> in order to explore the physical nature of their initiation. The waiting time is defined as the time interval between the first appearance of a CME and that of its immediate predecessor in coronagraphic images. The waiting time distribution for QH CMEs observed during1997-1998 consists of two components separated by 15 hours, where only the first componentclearly exhibits the shape of a Gaussian, peaking around 8 hours <cit.>. This is significantly different from thewaiting times of CMEs in general, appearing in the form of a Poisson distribution <cit.>.When only considering the QH CMEs that originated from the super ARs in solar cycle 23, theseparation between the two components increases to about 18 hours, while the peak of the first component shifts to 7 hours<cit.>.CMEs with waiting times less than 18 hours, i.e. the ones which contribute to the Gaussian component, are thought to have a close physical connection.In addition, numerical simulations reveal that successive eruptions from a single AR may be driven by continuous shearing motions on the photosphere, the emergence of twisted magnetic flux tubes, reconnection between emerging and pre-existing flux systems, or perturbations induced by a preceding eruption <cit.>. Most CME-productive ARs exhibit a complex photospheric magnetic field configuration, consisting of a mix of flux concentrations.Adjacent flux concentrations with opposite polarities, which may hold a flux tube, are separated by a polarity inversion line (PIL).Depending on the polarity pairsbeing present within an AR, a number of PILs (of different length and shape) may be present. Note that in some conditions, more than one polarity pairs are closely located in the vicinity of each other, with same polarity placed at the same side, forming a long PIL; i.e., a long PIL may be spanned by more than one flux tubes , thus, be divided into different parts. Based on this, <cit.> envisaged three possible scenarios for QH CMEs to occur: successive CMEs may originate (i) from exactly the same part of a PIL, (ii) from different parts of the same PIL, (iii) from different PILs within the same AR.The first scenario has been envisaged as the recurring release of quickly replenished magnetic energy/helicity. The other two have been regarded as scenarios where neighbouring flux tubes, either spanning different parts of a common long PIL or spanning distinctly different PILs, are disturbed, become unstable and erupt. Since the peak value of the waiting time distribution may represent the characteristic time scale of the most probable involved physicalprocess (either recurring release of the magnetic free energy or destabilization), we further explore the database of <cit.> in this work, in order to depict the most probable scenarios for QH CMEs to occur. § IDENTIFICATION AND CLASSIFICATION OF QH CMES §.§ Event sampleThe event sample of <cit.> consists of 281 QH CMEs that originated from 28 super ARs in SC 23.The CMEs are all listed in the SOHO/LASCO CME catalog[<http://cdaw.gsfc.nasa.gov/CME_list/>] <cit.>, and their source ARs have been determined[<http://space.ustc.edu.cn/dreams/quasi-homologous_cmes/>] following the process described in <cit.>.It is based on a combination of flares and EUV dimmings or waves, as they are strong evidence for the presence of CMEs.In particular, in the present work, we use localized flare-associated features, such as flare kernels, flare ribbons, and post-flare loops in order to determine the (portions of the) PIL relevant to the individual CME. Another two well-studied CME-rich ARs,NOAA AR 11158 and 11429, are added into the sample for detailed case study, as they were observed during the SDO <cit.> era, allowing an in-depth study of the associated flare emission using coronal imagery from AIA <cit.> and the involved coronal magnetic field structure and evolution based on vector magnetic field measurements from HMI <cit.>. Out of all of the events, 188 QH CMEs exhibit a waiting time of less than 18 hours, thus we assume them to be physically connected. Due to limitations in the observational data, not all of the 188 QH CME events could be successfully assigned to one of the three categories introduced above, i.e., whether to originate, from the exactly same portion of a PIL,from different portions of the same PIL, or a different PILwithin the same AR as their predecessor.The CME assigned to the first category (the latter two categories) are defined as S-type (D-type) QH CMEs.Note that QH CMEs were assigned to the second category, only when they originated from totally different portions of a long PIL (with non-overlapped post-flare loops, ribbons, etc.).In total, we were able to clearly identify the magnetic sources of 142QH CMEs.Among them, 90 are classified as S-type,accounting for 63%; 52 are classified as D-type, accounting for 37%.Selected QH CMEs are discussed in detail in the following two subsections, in order to demonstrate the identification process. The preceding CME is referred to as CME1, and the following CME is referred to as CME2. The associated flares are accordingly referred to as flare1 and flare2.§.§ Examples of S-type QH CMEsS-type QH CME from AR NOAA 9026AR NOAA 9026, observed in the form of a large bipolar sunspot region with a δ-spot, (Fig. <ref>(a)), was a highly CME-productive AR that launched at least 12 CMEs during its disk passage. Note that the strong positive polarity at the [-300,320] in Fig. <ref>(a) belongs to AR 9030.Fig. <ref> shows the magnetic source location, morphology and the time evolution of an S-type CME and its predecessor that both originated from the main PIL,located within the yellow box L1in Fig. <ref>(a).Fig. <ref>(b) - (d) show the evolution of the CME1-associated M7.1 flare1,as observed by TRACE <cit.> at 1600 Å, while the white-light appearance of CME1 in LASCO/C2 <cit.>is shown in Fig. <ref>(e).Fig. <ref>(f) - (i) show the corresponding features of CME2 and its associated X2.3 flare2. From Fig. <ref> it is evident that the chromospheric ribbons of both, flare1 and flare2, appear and evolve along the same part of the main PIL of the AR. Thus, CME2, with a waiting time of one hour, is classified as an S-type CME.S-type QH CME from AR NOAA 9236AR NOAA 9236produced more than 15 CMEs during its disk passage. The AR hosted a δ-spot of positive polarity surrounded by scattered elements ofnegative polarity (see Fig. <ref>(a)).The PIL of interest is located within the yellow box L1.The two CMEs (see Fig. <ref>(e) and <ref>(i)) wereassociated with an X2.3 and an X1.8 flare, respectively. The according TRACE 1600 Å observations (Fig. <ref>(b) - (d) and Fig <ref>(f) - (h), respectively) reveal thatthe ribbons of the twoflares appeared at the same location. CME2 had a waiting time of 7 hours and is thus classified as an S-type event. Note that these two CMEs were also classified as homologous events in <cit.> and <cit.>. S-type QH CME from AR NOAA 11158AR NOAA 11158 was the first super AR in SC 24 and produced more than 10 CMEs during disk passage.A pair of opposite polarities in the quadrupolar AR (outlined by the yellow box L1 in Fig. <ref>(a)) produced a number of CMEs within two days.Most of the CMEs were front-side, narrow events and missed by LASCO. However, they were all well captured by STEREO/COR1 <cit.>.The pair of CMEs shown in Fig. <ref>(e) and <ref>(i) were associated with an M2.2 and a C6.6 flare, respectively (see Fig. <ref>(b) - (d) and Fig.～<ref>(f) - (h)).The mass ejections (marked by the white arrows in Fig. <ref>(d) and (h)) shared the same source location. CME2, with a waiting time of 2.2 hours, is thus classified as an S-type QH CME. The cyan curve A1 in Fig. <ref>(a) indicates the projection of the flux rope axis along therelated PIL at Time1, i.e., before the occurrence of CME1. The pink curve A2 indicates the flux rope axis position at Time2, i.e., at a time instance after CME1 happend but before CME2 was launched. The lines C1 and C2mark the position of two vertical cuts that will be used to derive some flux rope parameters at the two time instances. For details see Sec. <ref>. §.§ Examples of D-type QH CMEsD-type QH CME from AR NOAA 10030AR NOAA 10030 adhered to a quadrupolarconfiguration (see Fig. <ref>(a)) and produced at least 8 CMEs during disk passage. A CME and its QH predecessor are shown in Fig. <ref>(i) and <ref>(e).The yellow boxes L1 and L2 in Fig. <ref>(a) enclose the pairsof opposite polarities, relevant to the respective CMEs, CME1 and CME2, and defining the accordingly relevant PILs (PIL1 and PIL2, respectively). CME1 was accompanied by a X3.0 flare(see Fig. <ref>(b) - (d)).Though an extra ribbon appeared in the positive polarity in L2 in Fig. <ref>(b),the helical structure marked by the white arrow in Fig. <ref>(b), and the observed chromospheric ribbonssupport that CME1 originated from L1.Fig. <ref>(f) - (h) show the time evolution of the chromospheric ribbons of theCME2-associated M1.8 flare2, clearly aligned with PIL2.CME2, with a waiting time of 1 hour, thus is classified as a D-type CME.Already <cit.> demonstrated that the two CMEs should have originated from two different magnetic flux tube systems, and further argued that the observational signatures matched a breakout scenario.D-type QH CME from AR NOAA 10696AR NOAA 10696, similar to NOAA 9236,consisted of a concentrated negative polarity region surrounded by scattered small positive polarity patches (see Fig. <ref>(a)). It produced more than 12 CMEs. The yellow boxes L1 and L2 in Fig. <ref>(a) mark thesource locations of CME1 and CME2, respectively. Fig. <ref>(b) - (d) and <ref>(f) - (h) show the evolution of the associated M5.0 and M1.0 flare, respectively. Fig. <ref>(e) and <ref>(i) show the appearance of the CMEs in LASCO/C2. The white arrows in Fig. <ref>(d) and <ref>(h) mark the post-flare loops associated with the two CMEs,further supporting that they originated from different flux tubesystems. CME2 had a waiting time of 2.8 hours and is therefore classified as a D-type QH CME.D-type QH CME from AR NOAA 11429AR NOAA 11429, a super active AR in SC 24, produced more than 12 CMEs during disk passage. The AR exhibited a complicated topology with a δ-spot. The two yellow boxes L2 and L1 in Fig. <ref>(a) mark the magnetic source locations of a CME and its QH predecessor. The cyan curve A1 indicates the projection of the flux rope axis along PIL2 at Time1, i.e., before the occurrence of CME1. The cyan line C1 mark the position of a vertical plane that perpendicular to A1 at Time1. The pink curves A2 and C2 are corresponding axis and plane for PIL2 at Time2, i.e., a time instance after CME1 happened but before CME2 was launched. See more details in Sec. <ref>.The time evolution of the flares that accompanied the two CMEs, an X5.4 and an X1.3 flare,is shown in Fig. <ref>(b) - (d) and <ref>(f) - (h), respectively.The white arrowin Fig. <ref>(h) marks the post-flare loops ofCME2, while the black arrows in Fig. <ref>(f) - (h) mark the post-flare loops of CME1.CME2, with a waiting time of one hour, is classified as a D-type CME, in agreement with its classification by <cit.>. §.§ Waiting-time DistributionThe waiting time distribution of the 188 CMEs (with waiting times < 18 hours) is shown as a black curve in Fig. <ref>, exhibiting a Gaussian-like distribution with a peak at about 7.5 hours, suggesting that they are physically related. The distributions of precisely located S- and D-type QH CMEs,are shown as a blue curve and a red curve in Fig. <ref>, respectively. The two are distinctly different from each other: the former peaks at 7.5 hours while the latter peaks at 1.5 hours, strongly supporting that these two types of QH CMEs may be involved into different physical mechanisms. Another slightly lower peak appears around 9.5 hours in the waiting time distribution of D-type QH CMEs. One possible reason is that in some cases, a CME triggers a D-type QH CME in a short interval of around 1.5 hours, after which the first CME's source region undergoes a energy replenishment and produces another QH CME with a interval around 7.5 hours. However, the third CME would be classified as a D-type, as it originates from a different source location from its predecessor, with a waiting time of around 6 hours. Considering the 3 hours bin size of the distribution, a peak around 9 hours may be reasonable. Another possible reason is that those D-type QH CMEs with waiting times around 9.5 hours may follow a different mechanism from the ones with short waiting times (around 1.5 hours). The work aims to find the most possible (but not only) scenario for the two types of QH CMEs.In order to explore the different underlying mechanisms,the aforementioned S-type CME in AR 11158 and D-type CME in AR 11429 are analyzed in details in the next section.These two cases were observed during the SDO era, allowing for sophisticated modeling of the three dimensional (3D) coronal magnetic field, based on the measurements of the photospheric magnetic field vector at a high spatial resolution from SDO/HMI. § CORONAL MAGNETIC FIELD TOPOLOGY OF S- AND D-TYPE CMES§.§ Method It is widely accepted that the expulsion of a CME is determined by the inner driving force (associated to, e.g., an erupting flux rope) and the external confining force (exerted by the large-scale, surrounding coronal magnetic field) <cit.>.In order to investigate the involved mechanisms, the knowledge of the 3D coronal magnetic fieldis necessary.A method developed by Wiegelmann <cit.> is employed tothe two selected cases, to reconstruct the 3D potential (current-free) and nonlinear force-free (NLFF) fields in the corona, based on the surface magnetic vector field measurements from HMI.A magnetic flux rope, characterized by magnetic fields twisted about a common axis, may become unstable and act as a driver for an eruption <cit.>. A flux rope can be identifiedusing a combination of topological measures deduced from the employed NLFF models, e.g.,in the form of the twist number T_wand the squashing factor Q <cit.>. T_w gives the number of turns by which two infinitely approaching field lines, i.e., two neighbouring field lines whose separation could be arbitrarily small, wind around each other, and is computed by T_w = 1/4π∫_Lα dlwhere α is the force-free parameter, dl is the length increment along a magnetic field line, L is the length of the field line <cit.>. Q is a measure of the local gradients in magnetic connectivity; regions with high values of Q are referred to as Quasi-separatrix Layers (QSLs) <cit.>.The cross section of a flux ropewith twisted field lines treading the plane, would be visible asa region of strong T_w enclosed by a surface of high Q valuesthat separatingthe magnetic fields of the flux rope from its magnetic environment.The location of the local extremum T_w in the cross section of a coherent flux rope is a reliable proxy of the location of its central axis. Additionally,a cross section perpendicular to the axis of the flux rope (e.g., the section at the apex point of the flux rope axis) would allow the axis run through the plane horizontally, so thatthe in-plane vector field will show a clear rotational pattern around the axis, which is represented by the point where T_w is maximal.The external confining force can be measured by the decay indexn=- dln B_ex(h)/ dln hwhere h is the radial height from the solar surface, B_ex is the horizontal component of the strapping potential field above the AR. Basically, n measures the run of the strapping field's confinement with height.Theoretical works predict the onset of torus instability when n is in the range of [1.5,2.0]<cit.>, while observations of eruptive prominences suggest a critical value n∼ 1 <cit.>.It is suggestedthat the former value is representative for the top of the flux rope axis, while the latter value is typical for the location of magnetic dipsthat hold the prominence material <cit.>. Therefore, n=[1,1.5] are used as critical decay index values for our analysis.Torus instability sets in once the axis of the flux rope reaches a height in the corona at which the strapping potential fields decrease fast enough <cit.>, thus the vertical distribution of n, along the axis of the flux rope, will hint at its instability.Since a physical relation is assumed to exist between the QH CMEs (CME2 and its predecessor, CME1), we may expecta change in themagnetic field configuration of the CME2's source location after CME1,detectable in the form of a change of the related parameters defined above (T_w, Q and n). Therefore, we deduce these parameters from the NLFF models (for T_w and Q) and potential models (for n) of the pre-CME1, and post-CME1 (i.e., pre-CME2) corona as follows: * Locate the axis of the flux rope using the method of<cit.>, which calculates the twist maps in many vertical planes at first, and traces the field line running through the peak T_w point at each map. All traced field lines should be coinciding with each other if a coherent flux rope is present. The line is then considered as to represent the flux rope axis.* Calculate T_w and Q in a vertical plane perpendicular to the flux rope axis. The in-plane vector field, B⃗_∥, can provide additional evidence of the presence of a flux rope in the form of a clear rotational pattern, centered on the flux rope axis' position. * Calculate the decay index n in a vertical plane, aligned with the flux rope axis and extending from the flux rope axis upwards, as a function of height in the corona. Using the above introduced models and concepts, we investigate the pre-CME1 and post-CME1 (pre-CME2)coronal magnetic field configuration of the mentioned two cases in ARs NOAA 11158 (Sect. <ref>) and 11429 (Sect. <ref>) in great detail. The quality of all the NLFF extrapolation in this paper is shown in Appendix <ref>. §.§ S-type QH CME from AR NOAA 11158As demonstrated in Sec. <ref>, the S-type CME and its predecessor originated from the same PIL within NOAA 11158.We study the magnetic parameters at the CMEs' source location (L1)at two time instances: once before CME1, at 2011-02-14T17:10:12 UT (Time1), and once after CME1 but before CME2 at 2011-02-14T18:10:12 UT (Time2).At both times, we find a flux rope structure from the constructed corona field (see Fig. <ref>(g) and (h)).The magnetic properties of the pre- and post-CME1 flux rope in a vertical plane perpendicular to its axis are shown in Fig. <ref>(a) - (c) and <ref>(d) - (f) (from left to right: Q, T_w, and B⃗_∥), respectively. The footprints of the vertical planes at the two timesare marked as C1 and C2 in Fig. <ref>(a).Their vertical extensions are indicated by the yellow lines in Fig. <ref>(g) and (h).At Time1 (pre-CME1), a region of strong twist(Fig. <ref>(b)) is surrounded by a pronounced Q-surface(Fig. <ref>(a)).The diamond symbols in Fig. <ref>(a) - (c)mark the location where T_w is strongest, at T=-1.94, and are assumed to represent the 3D location of the flux rope axis, at a coronal height of h≳2 Mm. The in-plane vector magnetic fields, B⃗_∥ (Fig. <ref>(c)),show a clear rotational pattern, centered around theflux rope axis, suggesting a left-handedness of the flux rope, since the blue arrows indicate the vector fields with the normal components going into the plane. The field lines passing through thestrong twisted region are shown in Fig. <ref>(g) in cyan,even adhering to a Bald Patch (BP) <cit.>. A representative field line in the BP is plotted as a white line, which is determined by the criteria introduced in <cit.>. At Time2 (post-CME1), the highest value of twist in the vertical plane perpendicular to the flux rope axis is found as T_w=-2.11, marked by the diamond symbols in Fig. <ref> (d) - (f). Again, a region of strong twist (Fig. <ref>(e))is surrounded by a pronounced Q-surface (Fig. <ref>(d)), but located lower in the model corona (height of the flux rope axis h≲2 Mm). The fields traced from the high-T_w region areshown in Fig. <ref>(h) as pink curves. For comparison, the outline of the flux rope at Time1 is shown again as cyan curves. The more potential arcade fields (white lines) are traced at Time2 but from the coordinates of the top of the flux rope at Time1. The direct comparison between the pre- and post-CME1 model magnetic field configuration suggests that the upper part of the flux rope might erupt during CME1, while the lower-lying part of the flux rope seems to remain. In order to check the conjecture, we further trace the field lines within the pre-CME1 corona from exactly the same starting locations used for tracing the post-CME1 flux rope (i.e., the high-T_w region enclosed by the high-Q boundary at Time2; see Fig. <ref>(d) and <ref>(e)). The traced pre-CME1 field configuration (red lines in Fig. <ref>(h)) clearly differs from the post-CME1 field structure (pink lines in Fig. <ref>(h)), which may suggest two possibilities: (i) the flux rope totally erupted during CME1, after which a new one emerged, or reformed; (ii) the flux rope underwent a topology change that part (not simple the upper part) of it was expelled during CME1, while the other part was left, being responsible for CME2. See Appendix <ref> for some details for CME2.We also calculated the unsigned vertical magnetic flux from the strong T_w region in the aforementioned planes. No strong twist region exists outside of the flux rope, thus, instead of doing a image-based flux rope recognition, we directly select the regions with |T_w|≳ 1.25. |T_w|=1.25 isa threshold value for kink instability <cit.>. The flux is calculated by Φ_1.25 = ∫_A_1.25|B⃗_⊥| dAin which B⃗_⊥ is the magnetic fields perpendicular to the vertical plane, dA is the element area. The planes are perpendicular to the axes of the pre- and post-CME1 flux ropes, thus the vertical magnetic flux can represent the axial flux of the flux rope. The unsigned vertical magnetic flux (given at the the header of Fig. <ref>(a) and <ref>(d)) decreased from 4.52× 10^19 Mx at Time1 to 3.10× 10^19 Mx at Time2, which can be due to either ejection or simple redistribution of twisted field lines, since twist is not supposed to be conserved during the flux rope evolution. However, CME1 has been confirmed to be related with source location L1 based on observation, as discussed in Sec. <ref>, the decrease here is more likely to support a twist release through eruption rather than redistribution.We can not make a definite conclusion on whether the flux rope at Time2 is a partial eruption remnant, or is a newly emerged/reformed one. However, the pre-CME1 flux rope has a BP, and the post-CME1 rope has some nearly-potential loops right above it. Thus,we prefer a partial expulsion model<cit.>, consisting of a coherent flux rope with a BP, to explain the eruption process: the field lines in the BP are not free to escape so that during the writhing and upward expansion of the ends of the field lines, a vertical current sheet may form, along which internal reconnection may occur and finally split the flux rope into two parts. The white arcades in Fig. <ref>(h) could be the post-eruption loops, which may also support that part of the flux rope erupted with CME1.Fig. <ref>(a) and (b) shows the distribution of the decay index n as a function of height above the flux rope axis, for Time1 and Time2, respectively. The projections of the flux rope axis at the two timesare indicated by the curves A1 (for Time1) and A2 (for Time2) in Fig. <ref>(a). The solid lines in Fig. <ref> indicate the height where n=1 and n=1.5.It is evident that, for both time instances, the vertical run of n varies strongly along the flux rope, with the n=1.5 level being located at a height above 48 Mm at one end, and around 16 Mm at the other end of the flux rope. The height where n=1 varies less dramatically along the flux rope, and is located at the height around 10 Mm. Comparison of the n=1.5 level at Time1 and Time2 (represented by the dotted and solid curves in Fig. <ref>(b), respectively) suggests that the critical height at the south-eastern end (x=0 Mm in Fig <ref>) is lowered by about 8 Mm. In the remaining part of the flux rope, no significant change was detected, which indicates that the external confining force was not lowered significantly by the first eruption. The critical height, both before and after CME1, were located relatively low in the solar atmosphere (e.g., n = 1 at h≈10 Mm), but still far above the height of the flux rope axis (red lines in Fig. <ref>(a) and (b)) located below 3 Mm at both times. The maximal n at the flux rope axis reaches 0.80 at Time1, and0.44 at Time2,which are both lower than the critical n=1.5 for torus instability. The results argue against torus instability in triggering the two QH CMEs. <cit.> studied the long-term evolution of AR NOAA 11158 and showed that the fast emergence and continuous shear of a bipolar photospheric magnetic field (L1 in Fig. <ref>) accumulated a large amount of magnetic free energy before the onset of a series of QH CMEs. They showed that the emerging fields reconnected with pre-existing fields, which finally led to the eruptions. Together with our analysis, their results hint at a multi-stage energy release process during which the magnetic free energy is released due to the successive eruptions from the same bipolar region (L1 in Fig. <ref>). Meanwhile, the energy was replenished through the shearing motion and ongoing flux emergence. We also calculate the magnetic free energy in the entire extrapolation volume at the two time instances (shown as E_F in Fig. <ref> (b) and (e)) by E_F=∫_V B^2_N/8πdV-∫_V B^2_P/8πdVB_N is the NLFF filed, B_P is the potential field and dV is the element volume. E_F shows a slight increase by 5% from Time1 (2.06 × 10^32 erg) to Time2 (2.17 × 10^32 erg), which is against the expectation that the magnetic free energy would decrease after CME1, since CME1 should have taken part of the free energy during the multi-stage energy release process. The slight increase could be due to the small fraction of the big, fast evolving AR that the erupting bipolar system account for, and/or the fast accumulation of the magnetic free energy byflux emergence and shear motions. Besides, the free energy calculated from the model coronal field has an uncertainty of around 10%<cit.>, so that no definite conclusion on the loss of free energy during CME1 could be made here.§.§ The D-type QH CMEs from AR NOAA 11429As discussed in Sec. <ref>, a D-type CME and its predecessor originated from two different locations within NOAA 11429, separated by a waiting time of just 1 hour. A physical relation is assumed to exist between the two QH CMEs, thus, a change at the source location of CME2 after CME1, is expected (see Sec. <ref>). Therefore, we study the magnetic parameters at the source location (L2) of CME2 at two time instances in the following. Once before CME1, at 2012-03-06 23:46:14 UT (Time1), and once after CME1 but before CME2 at 2012-03-07 00:58:14 UT (Time2). Fig. <ref> shows the T_w, Q, in-plane vector fields (B⃗_∥) maps and the traced flux ropes for AR 11429. Through checking the T_w and Q maps in many vertical cuts across PIL2, we found three possible flux ropes at Time1.The peak T_w point resides in the middle structure, thus, we again identified the axis of the middle rope with the peak T_w point and then place a plane perpendicular to the flux rope axis. The plane's footprint is marked as C1 in Fig. <ref>(a) and its vertical extent is marked by the yellow vertical line in Fig. <ref>(g). Fig. <ref>(a) - (c) show the distribution of Q, T_w and B⃗_∥ calculated in the plane. The axis of the middle flux rope, with a peak value T_w=1.86, is indicated by diamond symbols. The in-plane vector field, B⃗_∥, displays three clearly rotational patterns with opposite handedness, alternately. This supports that there were three flux ropes present along PIL2 at Time1. A configuration with two vertically arranged flux ropes,i.e., a so-called double-decker flux rope, has been studied <cit.>. However, a similar configuration, with three flux ropes presented here, is barely reported to our knowledge, we name it a triple-decker flux rope, analogically. The blue arrows indicate the vector magnetic fields with vertical component going into the plane, thus the upper one and the lower one (FR^3_2 and FR^1_2 in Fig. <ref>) is left-handed (i.e., the in-plane vector field exhibits a counter-clockwise sense of rotation), while the middle one (FR^2_2) is right-handed.The square and triangle symbols in Fig. <ref> (a), (b) mark the position of the axes of FR^3_2 and FR^1_2 , with local peak values T_w=-1.82 and T_w=-1.49, respective. The plane is not perpendicular to the axes of FR^3_2 and FR^1_2, positions of which are not well corresponding with the rotational centers of the ropes' in-plane fields, thus the symbols are not marked in Fig. <ref>(c). Fig. <ref>(g) depicts the structure of the flux ropes, FR^3_2 in blue, FR^2_2 in orange and FR^1_2 in cyan.A longer, strongly twisted rope (marked as FR_1 in Fig. <ref>(g)–(h)), is aligned with PIL1, and is resulted in CME1. The white lines in Fig. <ref>(g) represent some nearly-potential arcades above the flux ropes. Note that the south-western end of FR_1 was located closely to the triple-decker flux rope along PIL2, and part of the arcade field was overlying both, the south-west end of the CME1-associated flux rope and the eastern part of the tripple-decker flux rope. Therefore, we may assume that the eruption of FR_1 easily affected the triple-decker flux rope through various ways, e.g., by removing the common overlying arcades, disturbance, compressing the neighbouring fields through expansion of the post-eruption loop system below the erupted flux rope, even reconnecting with the neighbour fields during expansion. At Time2 (see Fig. <ref>(d)–(f)), the upper two flux ropes along PIL2 evidently disappeared from the extrapolated domain, while the lower one was now located higher, with a peak value T_w=-1.81 (indicated by triangles) located at h∼6 Mm. The whole structure also appears expanded compared to that at Time1. The in-plane vector field, B⃗_∥, exhibits a rotational pattern around the maximum value of T_w, which is evidence for the presence of a flux rope (Fig. <ref>(f)). The footprint of the vertical plane is marked as C2 in Fig. <ref>(a) and its vertical extent is marked as a yellow line in Fig. <ref>(h). Field lines traced from the strong T_w region at Time2 are shown in pink in Fig. <ref>(h). For comparison, the flux ropes which was present at Time1 is shown as cyan lines. Comparison of FR_2^1 at Time1 and Time2 reveals that it elevated and expanded, as well as gained internal twist. The vertical magnetic fluxes calculated by Equ. <ref> from the strongT_w region (|T_w| ≳ 1.25) of the lowermost structure of the triple-decker flux rope (h ≲ 5 Mm at Time1 and h ≲ 8 Mm at Time2) ,i.e., the representation of the axial magnetic flux of the lower flux rope (shown at the headers of Fig. <ref>(a) and (d)) indicate an increase by 2.48 times (from 2.28× 10^19 Mx at Time1 to 5.66× 10^19 Mx at Time2), supporting the enhancement of the twist. The upper two flux ropes, with opposite handedness, clearly disappeared from the system with almost no remnant left behind. A QSL exists between the two ropes (strong Q line at around 8.5 Mm in Fig. <ref>(a)). Thus, we prefer annihilation due to local reconnection started from the QSL, rather than expulsion, to be account for the absence of them at Time2. Annihilation of the ropes would cause decrease of the local magnetic pressure, which is likely to allow FR^1_2 to rise, expand and finally erupt, giving rise to the faint CME2.Further support for this scenario is given by the evolution of the observed chromospheric ribbons as shown in Fig. <ref>. At the beginning of flare1, two ribbons, labeled R_1^1 and R_1^2 in Fig. <ref>(a), expand on both sides of PIL1. While R_1^2 grew southward in time (Fig. <ref>(b)), two more faint and small ribbons, R_1^3 and R_1^4, became visible along PIL2 (Fig. <ref>(c)).Comparison to the flux ropes shown in Fig. <ref>(g) and (h), this pair of ribbons indicate the involvement of FR_2^2 and FR_2^3 in the magnetic process. The two ribbons showed no clear sign of development that departed from, or along the PIL, which may be evidence for a local, small scale reconnection process. FR_2^2 andFR_2^3 should have reconnected and annihilated during the first eruption.After flare1/CME1, the lower flux rope became unstable as well and erupted, giving rise to a further pair of flare ribbons, R_2^1 and R_2^2, at the beginning of flare2.Note that there still existed a flux rope at PIL1 after CME1, though we cannot determine whether it's a remnant or a newly emerged/reformed one. A similar analysis is performed across PIL1.See Appendix <ref> for details.The magnetic free energy in the extrapolated pre- and post-CME1 corona volume (shown as E_F in Fig. <ref> (g) and (h)) shows a decrease of 25% (from 10.61 × 10^32 erg at Time1 to 8.01 × 10^32 erg at Time2), which is beyond the uncertainty (10%), implying a clear energy release with CME1. Fig. <ref>(a) and (b) shows the distribution of the decay index n as a function of height above the axis of the lower flux rope at PIL2, for Time1 and Time2, respectively. The projection of the flux rope axis at the two times is indicated by the curves A1 and A2 in Fig. <ref>(a).The solid curves mark the height where n=1 and n=1.5. The height at which n=1.5 variesbetween h=30 Mm and 50 Mm along the flux rope axis, while the height at which n=1 shows a similar trend but at lower heights (about 15 Mm lower). The dotted lines in Fig. <ref>(b) are critical heights at Time1 for comparison. The red lines indicate the height of the flux rope axis, that both are lower than 6 Mm at the two time instances. No significant change is found, suggesting that CME1 may not significantly lower the constraining force of the overlying field. At both times, the predicted critical height for the onset of torus instability (n=1.5) is located much higher in the corona than the axis of the flux rope. Also the observation-based critical height (where n = 1) is located clearly above the flux rope. The maximal n at the flux rope axis is 0.59 at Time1, 0.53 at Time2, respectively, both lower than the critical value n=1.5, also suggests that torus instability may not have been the direct trigger for the two CMEs. We conclude for the D-type CME and its predecessor from AR NOAA 11429, their magnetic source regions were located very close to each other, and bridged by the same large-scale potential field arcade. The first occurring CME1 (associated to the flux rope along PIL1) destabilized the magnetic environment of the nearby flux tube system (above PIL2), leading to the reconnecting annihilation of the upper two flux ropes along PIL2, which decreased the local magnetic pressure, led the lower flux rope along PIL2 to rise and expand, and to finally erupt as well (during flare2 and causing the associated CME2).See Appendix <ref> for some details of CME2. § SUMMARY AND DISCUSSIONS In this paper, we analyze 188 quasi-homologous CMEs with waiting times less than 18 hours, and find that the waiting times show a Gaussian distribution peaking at about 7.5 hours. Thus, the CMEs are believed to be physical related in the statistical sense.A classification based on the precise source locations has been performed:QH CMEs that sharing the source locations with their predecessors are defined as S-type, and the ones having different source locations from their predecessors are defined as D-type. Same source location means the involvement of the same part of a PIL and different source locations mean different parts of one PIL or different PILs in an AR. In total, we classified 90 S-type QH CMEs, and 52 D-type ones. Six cases, three of D-type and three of S-type, are discussed in Sec. <ref> to show theprocess of detailed identification,basically based on the corresponding localized flaring signatures such as ribbons and post-flare loops across the PILs.The waiting time distributions of the two types of QH CMEs are significantly different: the distribution of the S-type CMEs peaks at around 7.5 hours while the distribution of the D-type CMEs peaks at around 1.5 hours,suggesting that the major mechanisms of the two types of QH CMEs are probably different.In order to picture the differences in the possibly underlying mechanisms, one of S type and one of D type cases, are analysed in detail. The S-type CME and its predecessor (i.e., CME2 and CME1) originated from the same location with a waiting time of 2.2 hours in the quadrupolar AR 11158.Three parameters: the squashing factor Q and the twist number T_w that can locate the inner flux rope, the decay index n that measures the external confining force, are investigated at the erupting region atTime1 (the time instance before the CME1) and Time2 (the time instance after CME1 but before CME2). The decay index above the erupting region shows no significant change,supporting that CME1 did not weaken the external confinement significantly. Note, the coronal magnetic field is extrapolated using the photospheric magnetograms as boundaries.It is possible that the change of the magnetic field in the corona cannot feed back to the photosphere within a short duration due to the high plasma β (ratio of gas pressure to magnetic pressure) and the long response times of the photosphere relative to the corona, thus, the decay index remains unchanged. At both time instances, the height where decay index reaches the critical value for torus instability is much higher than the height of the flux rope axis, which suggests that torus instability may not be the direct causes for the two CMEs.The differences between the flux rope field lines that traced from the same starting coordinates in the pre- and post-CME1 corona indicates a topological change during flare1/CME1; while the reduction of the representation of the flux rope axial magnetic flux from Time1 to Time2 evidence an eruption; presence of a BP and post-eruption loop at the position of the upper part of the flux rope at Time1 is more likely to support a partial expulsion process : part of the flux rope erupted as CME1, while the other part may survive, erupting later as CME2, which fits into a free energy multi-stage release process. However, the magnetic free energy in the extrapolation volume almost remains unchanged, which may be due to three reasons:(i) the small extent of the CME-involved corona, small compared to the entire AR for which the energy budget was estimated, (ii) on-going free energy replenishment, (iii) the uncertainty of the free energy estimate itself. Besides the scenario of the S-type case in AR 11158, the eruptions from the same location can also be in a energy consuming and replenishment processas studied in <cit.>. Two CMEs with a waiting time of 13 hours originated from the main PIL of a bipolar AR, AR 11817. The first one erupted and took the majority of the twist of the flux rope structure <cit.>.A very weakly twisted structure still existed after the eruption,and gained the twist through continuous shear motion on the photosphere <cit.>, and finally grew into a highly twisted seed flux rope for the next eruption.In this case, CME1 consumed most of the free energy at the erupting location, and the energy for CME2 was refilled after CME1. In the case of AR 11158, CME1 may only consumed part of the free energy, and the energy regain was ongoing before and after CME1 through the shear motion and flux emergence at the PIL <cit.>.Although the amount of the consumed energy for CME1 may be different, they both are due to continuous energy input, fitting into the energy regain scenario. The BP of the flux rope in AR 11158 is probably the reason for preventing the flux ropefrom a full eruption whereas therebuilding of magnetic free energy, e.g., flux emergence and shear motions, should be the main reason for the S-type eruptions.Detailed study of another CME-rich AR, AR 9236 that produced more than 10 S-type CMEs with a mean waiting time around 7 hours, also suggests that those S-type CMEs were caused by continuously emerging flux, supporting the free energy regain scenario <cit.>.The peak value around 7.5 hours of the S-type QH CMEs waiting time distribution could be a characteristic time scale of the free energy replenishment process.The D-type eruption and its predecessor originated from two different locations in AR 11429 with a waiting time of 1 hour. No significant change is found in the decay index, like that in AR 11158. Again, the heights where decay index reaches the critical value for torus instability are much higher than the heights of the flux rope axes at both time instances, arguing against torus instability in triggering the two CMEs. However, the seed flux rope for CME2, i.e., the lower flux rope at PIL2 shows a stronger twist, clear rising and expansion after CME1, which are favourable for its eruption. The most possible reason for the change of the flux rope is that CME1 influence the magnetic environment on PIL2 thatmake the upper two flux ropesdisappear, lead to decrease of the local magnetic pressureand allow the lower one to erupt. In post-CME1 model corona, the upper two flux ropes totally disappeared from the domain. During flare1, a pair of ribbons ignited along PIL2 after the brightening of the ribbons along PIL1, with no development departing from the PIL, supporting a local reconnecting annihilation between the upper two flux ropes, rather than expulsion of them. The details about how the eruption of the flux rope along PIL1 resulted in the reconnection of the upper two flux ropes along PIL2 remains unclear, though the observation data has been analysed. The first CME can remove the common overlying arcades, cause disturbance, compress the fields in neighbour system,even reconnect with neighbour fields. Somehow the equilibrium of the triple-decker flux rope is broken, and the upper two flux ropes reconnect.The key reason for the D-type eruption studied here is that the two flux rope systems are close enough that CME1 can impact on the pre-eruptive structure of CME2. It should be noted that the triple-decker flux rope presented here delivers a quiet uncommon configuration, of which equilibrium and evolution is worth to be studied in the future. A well-studied D-type QH CME from AR 11402, with a waiting time of 48 minutes, also suggests that the CME was initiatedby its predecessor<cit.>. The first CME may have opened some overlying arcade, allowed the neighbouring fields to expand and lowered the downward magnetic tension above the neighbouring flux rope, leading to the second CME. The scenario, that one eruption weakens the magnetic confinement of another flux tube system and promotes other eruptions, has been demonstrated in simulations <cit.>. The configuration in <cit.>contains a pseudo-streamer (PS), with two flux ropes located in the PS and one flux rope located next to the PS.The flux rope outsideexpands and erupts as the first CME, causing a breakout reconnection above one of the flux ropes in the PS, resulting the second CME; the current sheet formed below the second erupted flux rope causes reconnection at the overlying arcades of the other flux rope in the PS, leading to the third CME. The latter two CMEs can happen in a more generic configuration, without a flux rope outside the PS to eurpt at first to trigger them,although the underlying evolution is the same <cit.>. The model of <cit.> or <cit.> is applicable in a PS configuration.More generally, it is applicable ina configuration witha closed flux system containing a flux rope located nearby the erupting flux rope, e.g., a quadrupolar configuration, as the D-type CME and it's preceding one from AR 10030 shown in Fig. <ref>.The CME had a waiting time of 1 hour, following a process similar as the second and third CMEs in <cit.>, or the two CMEs in <cit.>, according to <cit.>: the core flux rope of the first CME was released from one flux tube system in a quadrupolar region by a breakout reconnection at the X point above the region;the neighbouring flux rope started to expand and finally erupted out due to the decrease of the overlying magnetic tension, which was caused by the reconnection at the current sheet formed below the first erupted flux rope.More generally, in an AR with multiple flux tube systems, one eruption causes destabilizations that promote other eruptions could be described as a “domino effect" scenario <cit.>.The peak value of the waiting time distribution of the D-type QH CMEs, around 1.5 hour, could be the characteristic time scale of the growth of distablizationcaused by their predecessors.This kind of consecutive CMEswith extremely short waiting time are sometimes called as “twin-CMEs” or “sympathetic-CMEs", although they are not necessarily produced from the same AR<cit.>.The source locations of a D-type QH CME and its predecessorare expected to be located close to each other, or have some magnetic connection that one eruption can induce the other one. Note, there is another slightly lower peak around 9.5 hours in thewaiting time distribution of D-types, may be due to the method of classification, or even different mechanism from the one for those with waiting time around 1.5 hours. In conclusion, through the two cases studied in depth, we propose possible mechanisms for most of the two types of QH CMEs, i.e., the ones located around the peak of the waiting time distribution: S-type QH CME can occur in a recurring energy release process by free energy regain, while D-type QH CME can happenwhen disturbed by its preceding one.The different peak values of the waiting time distributions: 7.5 hours for S-type and 1.5 hour for D-type QH CMEs might be the characteristic time scales of the two different scenarios.The classification is only based on the source PILs. S-type QH CME may also happen when disturbed by its predecessor, following a process as similar to the D-type. For example, in a configuration with more than one flux ropes vertically located above the same PIL, like the ones in AR 11429, in which change (reconnection, expulsion, etc.) of the upper flux ropes caused the eruption of the lower one.More cases with high spatial and temporal resolution data (e.g., data from SDO) are worth to be studied to discover more scenarios. We thank our anonymous referee for his/her constructive comments that significantly improved the manuscript. We acknowledge the use of the data from HMI and AIA instruments onboard Solar Dynamics Observatory (SDO), EIT, MDI and LASCO instruments onboard Solar and Heliospheric Observatory (SOHO), and Transition Region and Coronal Explorer (TRACE).This work is supported by the grants from NSFC (41131065, 41574165, 41421063, 41274173, 41474151) CAS (Key Research Program KZZD-EW-01-4), MOEC (20113402110001) and the fundamental research funds for the central universities.§ APPENDIX§ QUALITY OF NLFFF EXTRAPOLATIONLorentz force (J×B, where J is the current density) and the divergence of the magnetic field (∇·B) shouldbe as small as possible to meet force-free and divergence-free condition in the NLFF coronal fields.We follow <cit.>, using two parameters: θ (the angle between B and J) and ⟨|f_i|⟩ (fractional flux increase), to measure the quality of the model fields: σ_J=(∑_i=1^n|J×B|_i/B_i)/∑_i=1^nJ_i θ=sin^-1σ_J ⟨|f_i|⟩=1/n∑_i=1^n|∇·B|_iΔ V_i/B_i·Δ S_in is the number of the grid points, Δ V_i and Δ S_i is the volume and surface area of the i_th cell, respective. σ_J gives average sinθ weighted by J.See Table. <ref> for θ and ⟨|f_i|⟩ in the aforementioned (and aftermetioned in the next three sections) model NLFF fields, which all meet force-free and divergence-free conditions.§ CHANGE OF MAGNETIC PARAMETERS DURING CME2 IN AR 11158 In Sec. <ref>, the magnetic parameters at the source location (L1) are studied in pre-CME1 (at Time1) and post-CME1 but pre-CME2 (at Time2) corona. In this section, we perform a similar analysis in a plane perpendicular to the flux rope axis along PIL1 in the post-CME2 corona (2011-02-14T19:46:20 UT, defined as Time3), as shown in Fig. <ref> (d), (e), and (f) (Q, T_w and B⃗_∥, respective). The parameters at Time2 are shown in Fig. <ref> (a) - (c) for comparison. The triangles mark the peak T_w position, i.e., the position where the flux rope axis threading the plane. At Time3, the pronounced Q boundary, strong T_w region and the rotational structure around the peak T_w point in the in-plane vector fields evidence a flux rope. However, the vertical magnetic flux from the strong T_w region (|T_w|≳1.25) calculated by Equ. <ref> is reduced by 77% after CME2 (from 3.10× 10^19 Mx at Time2 to 0.70× 10^19 Mx at Time3, shown at the header of Fig. <ref>(a) and (d)).The magnetic free energy still shows a slight increase of 5.5% (from 2.17 × 10^32 erg at Time2 to 2.29 × 10^32 erg Time3, as shown at the header of Fig. <ref>(b) and (e)), which is below the uncertainty. CME2 that is confirmed to be correlated to the source location based on observation, and decrease of twist of the rope, all evidence that the flux rope is involved into the eruption. However, the information is not enough for distinguishing whether the flux rope at Time3 is a remnant of the previous flux rope which may undergo a partial eruption accompanied by topology reconfiguration during CME2, or is a newly emerged/reformed one after CME2. Study of the CME2's eruption detail is beyond this paper's scope.§ CHANGE OF MAGNETIC PARAMETERS DURING CME1 IN AR 11429 In Sec. <ref>, the magnetic parameters at the source location of CME2 (L2) are studied in pre-CME1 (at Time1) and post-CME1 but pre-CME2 (at Time2) corona to see the possible influence from CME1 to CME2. In this section, we perform a similar analysis at the source location of CME1 (L1) to see what happened during CME1. Q, T_w and B⃗_∥ are calculated in a plane perpendicular to the flux rope axis along PIL1 at Time1 (Fig. <ref> (a) - (c)) and Time2 (Fig. <ref> (d) - (f)), respective. Flux rope is found at PIL1 both before CME1 and after CME1. The vertical magnetic flux from the strong T_w region, with a threshold of 1.25 turns (|T_w|≳ 1.25) , shows no significant change. However, when changing the threshold to 1.6 turns (|T_w|≳ 1.6), the vertical magnetic flux shows a significant reduction of 47% (from 3.23× 10^19 Mx at Time1 to 1.70× 10^19 Mx at Time2, shown at the header of Fig. <ref>(a) and (c)). CME1 has been confirmed to be related to the source location L1 based on observation, as discussed in Sec. <ref>, thus, the flux rope should be responsible to the eruption. It's representative axial flux with |T_w|≳ 1.6 decreased, at the mean time, the flux with |T_w|≳ 1.25 almost kept constant. Partial expulsion of the flux rope, accompanied by replenishment of twist through shear motion or reconnection, can explain the phenomenon.§ CHANGE OF MAGNETIC PARAMETERS DURING CME2 IN AR 11429In this section, we perform a similar analysis as in Sec. <ref> in a plane perpendicular to the flux rope axis along PIL2 in the post-CME2 corona (2012-03-07T01:10:12 UT, defined as Time3), as shown in Fig. <ref> (d), (e), and (f) (Q, T_w and B⃗_∥, respective), to see the eruption detail during CME2. The parameters at Time2 are shown in Fig. <ref> (a) - (c) for comparison.After CME2, there still existed a flux rope along PIL2, showing a significant topology change compared to that at Time2. The vertical magnetic flux from the strong T_w region (|T_w|≳1.25) calculated by Equ. <ref> decreased by 38% after CME2 (from 5.66× 10^19 Mx at Time2 to 3.51× 10^19 Mx at Time3, shown at the header of Fig. <ref>(a) and (d)).The magnetic free energy also shows a slight decrease of 1.5% (from 8.01 × 10^32 erg at Time2 to 7.89 × 10^32 erg at Time3, as shown at the header of Fig. <ref>(b) and (e)), which is far below the uncertainty. The flux ropes traced by the model method in our cases, and two eruptive events in <cit.> all show twist remnant after the eruption. We come up two possible explanation: it is due to a partial expulsion process, or quick replenishment of twist through emergence/reformation after the eruption. The phenomenon is worth to be studied in the future.aasjournal | http://arxiv.org/abs/1706.08878v1 | {
"authors": [
"Lijuan Liu",
"Yuming Wang",
"Rui Liu",
"Zhenjun Zhou",
"M. Temmer",
"J. K. Thalmann",
"Jiajia Liu",
"Kai Liu",
"Chenglong Shen",
"Quanhao Zhang",
"A. M. Veronig"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170627142050",
"title": "The causes of quasi-homologous CMEs"
} |
1Union College, Department of Physics & Astronomy, 807 Union Street, Schenectady NY 12308; [email protected], [email protected] 2Washington & Jefferson College, Department of Computing and Information Studies, 60 S Lincoln Street, Washington PA, 15301. 3Cornell Center for Astrophysics and Planetary Science (CCAPS), Space Sciences Building, Cornell University, Ithaca, NY 14853; [email protected], [email protected], [email protected] 4CCPP, New York University, 4 Washington Place, New York, NY 10003; [email protected] 5Kapteyn Astronomical Institute, University of Groningen, Landleven 12, Groningen NL-9747AD, The Netherlands; [email protected] We investigate a sample of 3 dwarf elliptical galaxies in the Virgo Cluster which have significant reservoirs of . We present deep optical imaging (from CFHT and KPNO),spectra (Arecibo) and resolvedimaging (VLA) of this sample. These observations confirm theircontent and optical morphologies, and indicate that the gas is unlikely to be recently accreted. The sample has more in common with dwarf transitionals, although dwarf transitionals are generally lower in stellar mass and gas fraction. VCC 190 has antidal tail from a recent encounter with the massive spiral galaxy NGC 4224. In VCC 611, blue star-forming features are observed which were unseen by shallower SDSS imaging. § INTRODUCTIONThe relationship between gas-rich late-type dwarfs—irregulars and blue compact dwarfs (BCDs)—and early-type dwarfs—dwarf ellipticals and spheroidals—is unclear. Are late-type dwarfs the progenitors of early-types, or are the two populations largely distinct, with late-types evolving in the field and dwarf ellipticals in cluster environments? On one hand, many early-type dwarfs show late-type features: faint disk-like structures (; ), rotational support (; ; ), and lingering central star formation (; ). In turn, the underlying old stellar population of late-type dwarfs resemble dwarf ellipticals and spheroidals (). On the other hand, simulations (; ; ) and stellar population modeling (; ; ; ; ; ) suggest that early-type dwarfs essentially halted star formation at z∼1, and so late-type dwarfs at z=0 are unlike their progenitors.Essentially all dwarfs in the field are star-forming or in a starburst phase (; ) while quiescent dwarf ellipticals predominate in the cluster (; ). If these populations are related, then it is assumed that the cluster environment is responsible for this transformation. As gas-rich late-type galaxies fall onto the cluster, their gas is removed via processes such as ram-pressure stripping (; ) or galaxy harassment (; ). This process is relatively rapid, less than 100 Myr (). By the time a dwarf galaxy has crossed through the cluster core, they are almost entirely gas free (>99% gas removed). Without gas, star formation ceases, the galaxy's colors redden, and it loses any irregular or spiral features, becoming a smooth dwarf elliptical.Given that gas removal precedes morphological changes, there should be few, if any dwarf elliptical galaxies which still have a detectable reservoir of . And indeed there are only a few, about 2%, which do (e.g. ; ; ). In the work of <cit.>, hereafter H12, we identified a sample of 7 gas-bearing dwarf elliptical galaxies in the Virgo Cluster. The star formation in this sample is clearly suppressed: the galaxies are as red as typical dwarf ellipticals in the g-i, NUV-r, and FUV-r bands. Additionally, their gas fractions (GF≡ M_HI/M_*) are typical for unstripped dwarfs both in Virgo and the field <cit.>. Based on this, we argued that the gas has been recently accreted.How plausible is this assertion, given their evidence? Most dwarf ellipticals have cluster orbits which are highly radial <cit.>. This means that they will spend most of their time near the cluster edge—which is where the sample preferentially lies—where they can encounter clouds of neutral gas falling onto the cluster. At their present positions outside the x-ray emitting region of Virgo, accretion is possible, as evaporation of clouds of neutraldue to the hot intracluster medium is relatively slow compared with the time it would take to accrete a cloud of(H12).The purpose of this work is to test the re-accretion hypothesis of H12, using new deep optical andobservations. In <ref>, we review and update the sample of H12, removing galaxies whosewas not confirmed. In <ref>, we present the results of new optical andobservations of the sample. We weigh the evidence for morphological transformation of the galaxies in <ref>, for recent accretion in <ref>, and for quenched star formation without gas removal in <ref>. In <ref>, we argue that VCC 190 has had gas removed via tidal stripping.§ SAMPLE SELECTION AND UPDATES Our sample selection is outlined in Table 1, as well as below. In H12, we defined a sample of dwarf elliptical galaxies in the Virgo cluster. These galaxies were identified using the morphology and subcluster assignmentsfrom the Virgo Cluster Catalog (VCC; ), updates from <cit.>, and our own internal assignments. lr 2 0ptSample CountsSelection Criteria Galaxies Binggeli et al 1985 (VCC) Galaxies 2096Dwarfs Ellipticals in Virgo 365Red dEs withDetections (H12)7 Confirmed by Follow-Up 5Removal of VCC 956 and VCC 1993 (see text)3Number of galaxies in sample after each cut was applied. This work is concerned only with the 3 which remain after all cuts have been applied.Of these 365 dwarf ellipticals, 7 both had enoughsuch that they were detected in the Arecibo Legacy Fast ALFA survey (ALFALFA; ; ) and clearly lay along the red sequence (SDSS g-r≥0.45). §.§ ALFALFA Follow-UpThedetections of this sample were often near the detection limit of ALFALFA's, with signal-to-noise ratios of 3<SN<10. So, we re-observed the each object as part of a larger ALFALFA campaign to confirm thein unusual and low signal-to-noise galaxies.We performed a 3 minute ON/OFF observation of each galaxy using the single pixel L-Band Wide (LBW) receiver. These pointed observations have an rms of 1.1 mJy in 10wide channels, and are thus twice as sensitive as ALFALFA, which has a typical rms of 2.2 mJy when smoothed to the same width.Out of the seven, theemission in five of the galaxies was confirmed. The two galaxies with the lowest signal-to-noise (VCC 421 and VCC 1649) were not confirmed, and so we remove them from our sample.§.§ The Trouble With VCC 956 and VCC 1993 <cit.> mapped antail caused by ram-pressure stripping off of NGC 4388, which ranges in velocity from 2000<v<2600 . The sky position of VCC 956 coincides with the end of the tail, as shown in Figure <ref>. Our nominaldetection has a velocity of v=2200 , which perfectly coincides with the velocity of NGC 4388's tail at that position (see , Figures 1 and 2). There are no independent redshift measurements of VCC 956. It is thus likely that our nominalmeasurement of VCC 956 is better identified with the tail of NGC 4388, and we remove it from our sample.The surface brightness profile of VCC 1993 indicates that it is not a dwarf elliptical, but a low mass elliptical. It also has the highest M_* in the sample by a factor of 10. So, we remove it from present consideration and focus on the lowest mass, truly dwarf, galaxies.The remaining three galaxies, VCC 190, VCC 611, and VCC 1533 are the focus of this work.§ OBSERVATIONS AND RESULTS§.§ Summary of Prior SDSS, GALEX, and ALFALFA Observations Figure <ref> (left) shows an NUV-r color-magnitude diagram of the dwarf sample of H12. Late-type dwarfs withare crosses, while early type dwarfs undetected in ALFALFA are gray dots. The three gas-rich dwarf ellipticals in our sample are plotted as red squares, and labelled with their VCC numbers. With a few exceptions, the late-type and early type dwarfs separate themselves neatly. From our sample, VCC 190 is unambiguously red in NUV-r color. VCC 611 and VCC 1533 lie in the green valley; while they are redder than almost all of the late-type dwarfs, they are also bluer than most dwarf ellipticals. The galaxies show a similar segregation in FUV-r and g-r colors (see H12; Figure 6).Figure <ref> (right) shows thegas fraction as a function of stellar mass of late type dwarfs (crosses) along with a best-fit trend. The -bearing late-type dwarfs lie along the same line as -bearing dwarfs outside of the Virgo cluster (H12; ). This suggests that such galaxies have yet to undergo any significant stripping processes. Red boxes with labels are our sample. All three galaxies lie at or below the line defined by the HI-bearing late types, but none significantly so. Arrows are 3σ upper limits on the gas fractions of dwarf ellipticals which were not detected in ALFALFA derived from spectral stacking. The Arecibo Galaxy Environment Survey (AGES; ) found similar upper limits over a smaller region of the cluster (). The three dwarf ellipticals in our sample thus fall into an unusual class: from their optical and ultraviolet magnitudes alone, they appear to fit in with the dwarf elliptical population by being either “red and dead” (VCC 190), or nearly so (VCC 611 and 1533). However, from theirgas fractions, they appear to be unstripped late-type dwarfs, and several orders of magnitude more gas-rich than other dwarf ellipticals. §.§ Optical Observations We obtained optical r-band imaging at the WIYN 0.9m telescope with the HDI camera at the Kitt Peak National Observatory (KPNO). VCC 190 and VCC 611 were observed for 30 and 36 minutes, respectively, with seeing 1^''.6. In addition, all three galaxies lay within the footprint of the Next Generation Virgo Survey (NGVS; ), performed with the CFHT. All three galaxies were observed for between 30 minutes and 1 hour in the u, g, and i bands.Optical NGVS images of all three galaxies in our sample are shown in Figure <ref> (top panels). The three galaxies all appear roughly elliptical but with some peculiar morphology. The highest optical surface density in VCC 190 (top left) is offset from the center, with possible tails to the west and south. The shape of VCC 1533 (right) is boxy, with a nucleated center. Figure <ref> (second row) shows r-band (VCC 190 and 611) and i-band (VCC 1533) surface brightness profiles. All three galaxies are well-fit with an exponential profile for r > 5^'' (dashed line), consistent with the standard dwarf elliptical profile. At smaller radii, the surface brightness profiles of VCC 1533 is contaminated by what is likely a foreground star.VCC 611 (top center) appears like a smooth elliptical, likely due to its relatively higher stellar mass and surface brightness compared with the other two galaxies in our sample. However, it clearly has a very blue center indicative of star formation, which can be seen in Figure <ref>. The left panel presents an i-band image, which shows a smooth old stellar population. The right panel is a g-i color map of the galaxy, which shows several star-forming knots in the galaxy's center. VCC 611 was observed as part of the SDSS spectroscopic survey and hasin emission with an equivalent width (EW) of 25 Å (NASA Sloan Atlas[www.nsatlas.org]). <cit.> and <cit.> previously performed unsharp masking on all three of these galaxies using shallower SDSS images, but did not detect the underlying structure or the blue center in VCC 611. §.§ HI Observations The HI content of the three dwarf elliptical galaxies in our sample were observed at high spectral resolution (native 0.65channel width) using the Arecibo Observatory. Observations were performed with the LBW receiver in a series of 5 minute ON/OFF pairs. Results are summarized in Table 2. The ALFALFA data cubes indicate that there are no sources within 10^' which could be contaminating the sources by being detected in Arecibo's sidelobes above the 5% level.Thespectra of the three galaxies are shown in Figure <ref> (third row). VCC 611 and VCC 1533 (center and right) are in good agreement with a gaussian profile: the normalized residuals (bottom center and right) show low scatter and non-significant trends. In addition, we fit 3rd order gauss-hermite polynomials, to both spectra to look for asymmetric components. For VCC 611, we find no significant asymmetries, while VCC 1533'sprofile is asymmetric at a 3σ level. The velocity widths of both galaxies are very narrow: W_50 = 19 and 28 , respectively). Fit parameters are presented in Table 3. lccccc 5 0ptSample HI PropertiesGalaxy S_HI log M_HI V_sys W_50Jy log M_⊙(1) (2)(3) (4) VCC 190 A0.18 ± 0.01 7.12345.4 ± 0.214.9 ± 0.6VCC 190 B0.29 ± 0.02 7.32371 ± 246 ±2VCC 6110.12 ± 0.01 6.91304.5 ± 0.619 ± 1VCC 1533 0.45 ± 0.01 7.5 648 ± 127.7 ± 0.6 Results of single pixelspectral observations. VCC 190 is split into a relatively blueshifted (A) and redshifted (B) peak. Column 1: integrated flux density; Column 2:mass, assuming a distance of 16.7 Mpc; Column 3: heliocentric optical velocity; Column 4: Full-width half maximum of gaussian fit. For VCC 190, the fit is two simultaneous gaussian fits, one to the narrower, low velocity piece, and to the wider, higher velocity tail.Theprofile of VCC 190 (Figure <ref>; third row left) is neither gaussian nor does it have a typical symmetric two-horn profile. Attempting to fit a single, asymmetric profile to the galaxy produces a rather poor fit overall (χ^2_ν = 3.0). Such asymmetries in the globalprofile are typical in galaxies for which ongoing ram-pressure stripping is observed (; ; ; ), though stripping is not the only possible explanation. Instead, guided by our VLA observations (see <ref>), we fit two gaussians, one to the lower redshift peak, and one to the higher redshift peak. The combined model fits the data well (χ^2_ν=1.5), with no trend in the residuals (bottom left). The lower redshift peak has a very narrow W_50 = 15 , while the higher redshift peak has W_50 = 46 .§.§ VLA Observations of VCC 190 We observed VCC 190 using the VLA in the D and C configurations for 2 hours and 4 hours respectively. The cubes were produced in CASA using multiscale clean, with a Briggs robustness weighting of 1.0. Continuum subtraction was performed in the image plane with the imcontsub task using the line-free channels. This yielded an rms of 1.0 mJyin a 37^''× 31^'' beam, with7wide channels. The moment 0 maps were produced by first smoothing the data cubes to half their original spatial resolution and calculating a mask at 3 σ (3.0 mJy ). This mask was then applied to the original data cube.Figure <ref> (left) shows a moment 0 image overlaid on a KPNO r-band optical image. Contours begin at 3σ and increase by 3σ at each additional contour. The gas is not localized directly on the optical, but shows a tail to the southeast. The total flux recovered by the VLA is 0.44±0.01 Jy km s^-1, which agrees with the Arecibo flux of 0.47 ± 0.02 Jy km s^-1. In Figure <ref> (right), we overlay two sets of contours, corresponding to the velocities of the two peaks observed at Arecibo. The blue contours, corresponding to the narrow, lower velocity peak (2320-2343 ) VCC 190 A roughly coincide with the location of the optical galaxy, with an offset of 12^''=1 kpc. The red dashed contours correspond to the wider, high velocity peak (2343-2374 ) VCC 190 B correspond solely to the tail.§ GALAXY MORPHOLOGYIn H12, we argued that this sample is morphologically well-described as dwarf ellipticals, based on optical morphology and color from relatively shallow imaging. All three galaxies in our refined sample were classified as dwarf ellipticals by <cit.>, are redder than almost all gas-rich late-type dwarfs in the cluster, and no faint substructure was recovered in the galaxies by either <cit.> or <cit.> using stacked SDSS imaging. Do the new, significantly deeper, optical andobservations support this assessment?In the CFHT images, we observe that only VCC 611 appears by eye to be a `typical' dwarf elliptical: it is smooth and clearly elliptically symmetric. VCC 190 and 1533 are a bit peculiar (see Figure <ref>, first row): VCC 190 is asymmetric and VCC 1533 is boxy in shape. Such peculiar features are, however, not unusual. Massive elliptical galaxies in the field which are observed to have detectable quantities ofoften have peculiar optical morphology, appearing blue, distorted, or with faint disk-like features (e.g. ; ; ; ). It is possible that VCC 611 is a massive elliptical galaxy, but appears to be a dwarf because it is behind the cluster. This can be tested using surface brightness profiles: all dwarf galaxies (early- and late-type) are expected to have exponential profiles, as compared with the de Vaucouleurs profile of more massive ellipticals.Using r- and i-band imaging from KPNO and the CFHT, we observe that exponential profiles do fit all three galaxies very well at radii larger than 5^'' (see Figure <ref>, second row). This measure does not, unfortunately, allow us to differentiate between dwarf ellipticals and irregulars. The bulk motion of stars and gas in dwarf ellipticals is generally not rotational, but pressure-supported. Typical values of V_rot≲σ, where σ is the dispersion velocity. For the stellar component, typical values are 0 <V_rot<30 , while 20 < σ_*<40(e.g. ; ; ). This relationship only seems to break down for the most extreme of the so-called fast-rotating and rotationally supported dwarf ellipticals ().In agreement with stellar observations of other dwarf ellipticals, all three galaxies in our sample haveprofiles which show little sign of rotation. All three can be very well fit with a gaussian profile, with widths of 14 < W_50 < 28(see Figure <ref>, bottom two rows). We can estimate a sky-projected rotation velocity via: V_rot=√(W_50^2-σ_HI^2)/2 where σ_HI = 11is our assumed velocity dispersion, and the factor of two accounts for the integrated spectrum containing both the approaching and receding half of the rotation curve. We then obtain approximaterotational velocities of 4-13 , or V_rot≲σ_HI.By comparison, the estimated rotational velocities for other late-type dwarfs is much higher. For late-types dwarfs in the Virgo Cluster observed by ALFALFA, the average rotational velocity is 31 , with an interquartile range (IQR) of 18 < V_rot < 40 , several times σ_HI. The FIGGS sample of field irregulars <cit.> are typically narrower, with an average V_rot of 39 (IQR of 11 < V_rot < 24 . While the rotational velocities of the FIGGS sample just overlaps with our sample, the FIGGS galaxies are also typically of lower mass. Finally, we note that the spectra of most late-type dwarfs in ALFALFA and FIGGS also have a two-horned profile or otherwise show clear features of rotational broadening.§ DISCUSSIONThus, our previous assessment that these galaxies are morphologically dwarf ellipticals appears to hold, regardless of whether we consider surface brightness profiles, visual appearance, or velocity widths. There are, however, a few peculiarities which must be explained, primarily their largereservoirs in comparison with other dwarf ellipticals. We now explore two hypotheses about thecontent of the galaxies. First, we reconsider the hypothesis of H12 that the gas has been recently accreted. Second, we consider whether these galaxies are a part of the irregular or BCD dwarf population, but are between starburst phases, and so only appear optically red and dead. §.§ Gas Accretion Approximately half of giant elliptical galaxies in the field have been observed to bear(; ; ; ; ). In most cases, these authors attributed the presence of gas to accretion—whether of gas-rich satellites, the cooling of ionized gas, or cold-mode accretion from the intergalactic medium (e.g. ; ; ; ). However, in clusters, ellipticals do not appear to be accreting <cit.>. <cit.>, <cit.>, and <cit.> observed thein a combined 54 gas-bearing giant ellipticals in the field and the Virgo cluster. These authors found a wide range ofmorphologies, from scattered clouds and tails to regularly rotatingdisks. They infer recent or ongoing accretion as the origin of the gas when theis in the form of clouds, tails, and warped and disturbed disks. <cit.> also observed that star formation is generally observed in galaxies where accretion is recent or ongoing. When accretion is not recent, star formation is lacking, even when a galaxy is gas-rich.However, the most appropriate population to compare our sample to is other low-mass early-type galaxies, such as NGC 404 and FCC046 (; ).has been observed in both galaxies, and attributed by the authors to gas accretion in the last Gyr. Like the more massive ellipticals, the gas in NGC 404 and FCC046 is rotating in a disk. The rotational structure of the gas is clear even in an integrated spectrum: both galaxies have a classic two-horned profile, with a velocity widths of 65<cit.> and 67<cit.>.[We have re-fit the spectrum of FCC046, as the W_50 = 34<cit.> report is from a gaussian, not two-horned, fit to the spectrum.] We note that the true rotational velocity for NGC 404 must be quite large, as itsdisk is observed nearly face-on. For FCC046, the misalignment between the angular momentum of the accreted gas and the stellar component of the galaxy is very clear: the gas is in a polar ring. Like more massive ellipticals, accretion coincides with recent star formation for these dwarfs (; ).Whether comparing with massive or dwarf ellipticals, it is clear that accreted gas is associated with two features: themorphology is in the form of gas clouds, streams, or warped or disturbed disks, and the presence of star formation. For our sample—where our primary observations are single dish spectra—the direct observable ofin streams or disks would be a large velocity width (≳ 50 ) or a two-horned profile as a signature of a rotating disk. We note however that the spectral features are not a perfect analogue to resolvedmorphology and are insufficient to conclusively point to recent accretion as the source of the gas. Most notably, we would be unable to discern the existence of a warp in an otherwise smoothly rotating disk. Instead, we can argue that the lack of such features weakens the case for accretion. Neither wide velocity widths nor two-horned profiles are observed for any of the three galaxies. We do, however, observe evidence of recent star formation in VCC 611 and VCC 1533. For VCC 190, where theis partially resolved, we do observe a gas tail, but we argue that gas removal is a more plausible explanation (see <ref>).Finally, we consider the properties of freeclouds in the Virgo cluster. Such clouds would be the source of the accretedin our sample. Here we run into a signficant problem. If the accretion is ongoing or recent, then a tidal interaction between the clouds and the galaxies would increase the velocity widths of the clouds'profiles. However, the velocity profile of thein each of the three galaxies is much narrower than that of any of the clouds in Virgo, which have W_50≳ 50(; ). §.§ Between Bursts of Star Formation The high gas fractions in our sample (see Figure <ref>, right) and their location at the edge of the cluster (; ; ) suggest that these galaxies are new arrivals to the cluster. However, essentially all dwarf galaxies in the field show signs of recent star formation, to varying degrees (; ). If these galaxies were quenched prior to arriving in the cluster, then approximately 50% of their gas reservoirs could have been removed, consistent withobservations (; ) and simulations (; ) of galaxy groups.In groups, dwarfs with morphological features which are somewhere between irregulars and spheroidal dwarfs are observed (; ). We observe such features in our sample: VCC 190 and VCC 1533 show deviations from a purely elliptical shape, and VCC 611 has some star-forming features at its center. Called dwarf transitionals, such galaxies have little ongoing star formation, and often have detectable reservoirs of . The star formation history of such galaxies is very similar to irregulars and spheroidals until roughly 1 Gyr ago (; ).The general interpretation is that dwarf transitionals are either between major episodes of star formation, or have simply permanently stopped forming stars (; ).The typical dwarf transitional has a lower M_* and much lower GF than our sample (; ). Our sample could be the gas-rich and more massive tail of the dwarf transitional population, holding on to more of their gas while in the group environment by virtue of their higher total mass. Indeed, our most gas-poor galaxy, VCC 611, appears in the dwarf transitional sample of <cit.>.§ THE TAIL OF VCC 190 Of the three galaxies in our sample, only VCC 190 has clear ongoing gas removal, as indicated by both its unusual HI spectrum and tail. The two processes most likely to produce such a tail are ram pressure stripping and a tidal interaction with a larger galaxy or galaxies. Which is it?Evidence for ram pressure stripping is fairly weak. VCC 190's projected position is beyond the 3σ detection of the ROSAT satellite <cit.>, where the effects of the ICM should be minimal. In addition, massive galaxies at this distance from the cluster center are notdeficient, and rarely show ram pressure stripping tails <cit.>. However, VCC 190 is very low mass, and may be susceptible to ram pressure stripping even when it is too weak for the more massive galaxies.The tail geometry is also not right for ram pressure stripping. In general, tails point opposite the motion of the galaxy, which generally means they point roughly away from the subcluster center (e.g. ; ; ). In this case, the tail should point away from either M87 or M49, to the northwest and west, respectively, but VCC 190's tail points southwest. In addition, tails caused by stripping generally point towards the mean redshift of the cluster, as the removed gas is decelerated relative to the ICM, but VCC 190's tail is blueshifted towards higher velocity.Instead, the tidal interaction hypothesis is stronger. First, we note that the distribution of stars in VCC 190 is not a smooth ellipse, but asymmetric; the top left portion of the galaxy has a higher surface brightness than the bottom right. Ram pressure stripping would only disturb the gas in the galaxy, while a strong tidal encounter, or multiple weaker encounters can disturb the stars as well.Second, it is clear which galaxies collided with VCC 190. With the exception of a few dwarf galaxies, three galaxies are within 100 kpc of VCC 190 and at a similar redshift, all to the southwest. Figure <ref> shows an optical image with ALFALFA contours overlaid of these galaxies. NGC 4224 and AGC 221988 are interacting, with a tidal bridge visible between them (see Figure <ref>). <cit.> found stellar tidal streams coming off of NGC 4224 pointing to the north, for which they could not find an obvious interaction partner; we suggest that AGC 221988 is the cause. An optical spectrum indicates that NGC 4233 is at the same redshift, and so may be interacting with the other galaxies. But, its lack ofmeans that such an interaction unseen in the image.The tidal tail of VCC 190 points toward NGC 4224, suggesting an encounter between the two. In addition, NGC 4224 lies at a slightly higher redshift (v = 2606 ) than VCC 190 (v = 2345 ). As VCC 190 passed by NGC 4224, the removed gas in VCC 190's tail would be perturbed to a higher redshift, as is observed. We performed additional observations at Arecibo between VCC 190 and NGC 4224 (Figure <ref>; red crosses) to look for a tidal bridge between the two galaxies. No gas was detected, with a 3σ upper limit of 10^6.3 M_⊙ (0.10 Jy ), assuming a 30wide tail, or approximately 5% of themass of VCC 190.Based on the disturbed and asymmetric stellar light distribution in the galaxy, the presence of a narrow and undisturbed (W_50 = 14 ) component of HI centered on the optical galaxy, and finally NGC 4224 as a clear collision partner, this tail is likely to be tidal in nature.§ SUMMARY We revisited sample of dwarf elliptical galaxies in the Virgo Cluster identified by H12. Despite being `red and dead,' all had substantial reservoirs ofas detected in ALFALFA. We found that the two lowest signal-to-noisedetections reported in H12 and <cit.>, VCC 421 and VCC 1649, were spurious. We further eliminated two more galaxies from the sample of H12: VCC 956 appears at the same sky location as a longtidal tail off of NGC 4388, and VCC 1993 is not a dwarf galaxy. Our primary sample of interest was thus narrowed to three galaxies: VCC 190, VCC 611, and VCC 1533.New radio and optical observations. High resolution (channel width of 0.65 ) Arecibo observations reveal that VCC 611 and 1533 have narrow (W_50<30 ) velocity widths and gaussian profiles. VCC 190 has two gaussian peaks, one narrow and one wide (W_50=45 ). VLA observations of VCC 190 show antail pointing toward NGC 4224 to the south, and the tail corresponds to the wide velocity feature observed at Arecibo. The ALFALFA data cubes show that NGC 4224 is interacting with one of its neighbors, AGC 221988.Our KPNO observations and CFHT archival data from the NGVS show that all three galaxies have exponential profiles and lack widespread star-forming features. They do have some unusual features, however: the surface brightness of VCC 190 is asymmetric, VCC 611 has a star-forming feature at its center, and VCC 1533 is boxy in shape.Elliptical morphology without significant removal of gas is observed for all three galaxies, even though gas removal in clusters should be very rapid compared with morphological change. This is supported both by their optical (exponential profiles and a lack of widespread star-forming features) and(narrow, undisturbed velocity profiles) features. Together, these features suggest that the galaxies are relatively new arrivals in the cluster which have not undergone significant gas stripping. We observe ongoing gas removal in VCC 190 (see below), but the current tidal interaction is too recent to be responsible for its optical properties.These galaxies are similar to dwarf transitionals—galaxies with optical morphologies between irregulars and spheroidals with little sign of star formation. All three galaxies show some deviation from a smooth elliptical shape; VCC 190 is asymmetrical, VCC 611 has ongoing central star formation, and VCC 1533 is box-like in shape. Transition-type dwarfs may be between bursts of star formation; our sample additionally may have been weakly stripped (<50% gas removal) while still in a group environment. Our sample is more gas rich and has a higher M_* than is typical for dwarf transitionals (; ), and so we could be observing the most extreme galaxies in the transition population. Ambiguous evidence for gas accretion. In elliptical galaxies with accreted gas, recent or ongoing star formation is generally observed, and theis in the form of clouds, streams, or a perturbed rotating disk. For two galaxies in our sample, we do observe star formation: the SDSS spectrum of VCC 611 shows strong centrally-located star formation, and the overall color of VCC 1533 is at the edge of what is expected for star-forming dwarfs (NUV-r=3). With single-dish spectra, we can only imperfectly probe themorphology, but andisk will have a clear two-horned profile, and both disks and streams have wide velocity widths even in dwarfs (W_50≳50 ). Thelines of VCC 611 and VCC 1533 galaxies are both narrow (W_50<30 ) and lack evidence of rotation as both are well-fit by a gaussian profile. VCC 190 is not star-forming, and theat its optical position is similarly non-rotating. A tail is evident inimaging of VCC 190, but is more consistent with gas removal than accretion.Evidence of tidal gas removal in VCC 190. An observedtail in VCC 190 points towards NGC 4224, while the gas coincident with the optical galaxy is practically undisturbed (fit well by a gaussian with W_50 = 15 ). The tail is redshifted relative to thein the optical galaxy, as would be expected from an encounter with NGC 4224, which is at a higher redshift than VCC 190. These all suggest that the tail is a tidal feature from an encounter approximately 1 Gy ago. § ACKNOWLEDGEMENTSWe are extremely grateful for the observations performed by the faculty and students of the Undergraduate ALFALFA Team (UAT), without which this work would not be possible.They performed the observations at Arecibo to confirm each galaxy's , and performed the optical observations at KPNO of VCC 190 and VCC 611. The UAT is supported by NSF grant AST-1211005.ALFALFA has been supported by NSF grant AST-1107390, and grants from the Brinson Foundation.This work is based in part on observations made with the Karl G. Jansky Very Large Array, a facility of the National Radio Astronomy Observatory (NRAO). The NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.This work is based in part on observations made with the Arecibo Observatory. The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968), and in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities Space Research Association.This work is based in part on observations made with the WIYN 0.9m telescope operated by WIYN Inc. on behalf of a consortium of nine partner Universities and Organizations.[Aguerri & González-García(2009)]AguerriGonzalez2009 Aguerri, J. A. L., & González-García, A. C. 2009, , 494, 891[Auld et al.(2006)]Auld2006 Auld, R., Minchin, R. F., Davies, J. I., et al. 2006, , 371, 1617[Beasley et al.(2009)]Beasley2009 Beasley, M. A., Cenarro, A. J., Strader, J., & Brodie, J. P. 2009, , 137, 5146[Begum et al.(2008)]Begum2008 Begum, A., Chengalur, J. N., Karachentsev, I. D., Sharina, M. E., & Kaisin, S. S. 2008, , 386, 1667[Bekki(2009)]Bekki2009 Bekki, K. 2009, , 399, 2221[Binggeli et al.(1985)]Binggeli1985 Binggeli, B., Sandage, A., & Tammann, G. A. 1985, , 90, 1681 [Binggeli et al.(1987)]Binggeli1987 Binggeli, B., Tammann, G. A., & Sandage, A. 1987, , 94, 251[Binggeli et al.(1993)]Binggeli1993 Binggeli, B., Popescu, C. C., & Tammann, G. A. 1993, , 98, 275 [Blanton et al. (2011)]Blanton2011 Blanton, M. R., Kazin, E., Muna, D., Weaver, B. A., & Price-Whelan, A. 2011, , 142, 31 [Böhringer et al.(1994)]Bohringer1994 Böhringer, H., Briel, U. G., Schwarz, R. A., et al. 1994, , 368, 828[Boselli et al.(2008)]Boselli2008 Boselli, A., Boissier, S., Cortese, L., & Gavazzi, G. 2008, , 674, 742-767 [Boselli et al.(2016)]Boselli2016 Boselli, A., Roehlly, Y., Fossati, M., et al. 2016, , 596, A11[Burstein et al.(1987)]Burstein1987 Burstein, D., Krumm, N., & Salpeter, E. E. 1987, , 94, 883[Catinella et al.(2010)]Catinella2010 Catinella, B., Schiminovich, D., Kauffmann, G., et al. 2010, , 403, 683[Cen et al.(2014)]Cen2014 Cen, R., Roxana Pop, A., & Bahcall, N. A. 2014, Proceedings of the National Academy of Science, 111, 7914[Chung et al.(2007)]Chung2007 Chung, A., van Gorkom, J. H., Kenney, J. D. P., & Vollmer, B. 2007, , 659, L115 [Conselice et al.(2001)]Conselice2001 Conselice, C. J., Gallagher, J. S., III, & Wyse, R. F. G. 2001, , 559, 791[Davis et al.(2012)]Davis2011 Davis, T. A., Alatalo, K., Sarzi, M., et al. 2011, , 417, 882[del Río et al.(2004)]delRio2004 del Río, M. S., Brinks, E., & Cepa, J. 2004, , 128, 89[De Looze et al.(2013)]DeLooze2013 De Looze, I., Baes, M., Boselli, A., et al. 2013, , 436, 1057[De Rijcke et al.(2013)]DeRijcke2013 De Rijcke, S., Buyle, P., & Koleva, M. 2013, , 770, L26 [di Serego Alighieri et al.(2007)]diSeregoAlighieri2007 di Serego Alighieri, S., Gavazzi, G., Giovanardi, C., et al. 2007, , 474, 851[Dressler(1980)]Dressler1980 Dressler, A. 1980, , 236, 351[Duprie & Schneider(1996)]DuprieSchneider1996 Duprie, K., & Schneider, S. E. 1996, , 112, 937[Ferrarese et al.(2012)]Ferrarese2012 Ferrarese, L., Côté, P., Cuillandre, J.-C., et al. 2012, , 200, 4 [Geha et al.(2012)]Geha2012 Geha, M., Blanton, M. R., Yan, R., & Tinker, J. L. 2012, , 757, 85[Giovanelli et al.(2005)]Giovanelli2005 Giovanelli, R., Haynes, M. P., Kent, B. R., et al. 2005, , 130, 2598[Grebel et al.(2003)]Grebel2003 Grebel, E. K., Gallagher, J. S., III, & Harbeck, D. 2003, , 125, 1926[Grossi et al.(2009)]Grossi2009 Grossi, M., di Serego Alighieri, S., Giovanardi, C., et al. 2009, , 498, 407[Guérou et al.(2015)]Guerou2015 Guérou, A., Emsellem, E., McDermid, R. M., et al. 2015, , 804, 70[Hallenbeck et al.(2012)]Hallenbeck2012 Hallenbeck, G., Papastergis, E., Huang, S., et al. 2012, , 144, 87[Hau et al.(2008)]Hau2008 Hau, G. K. T., Bower, R. G., Kilborn, V., et al. 2008, , 385, 1965[Haynes et al.(2007)]Haynes2007 Haynes, M. P., Giovanelli, R., & Kent, B. R. 2007, , 665, L19[Haynes et al.(2011)]Haynes2011 Haynes, M. P., Giovanelli, R., Martin, A. M., et al. 2011, , 142, 170[Hess & Wilcots(2013)]Hess2013 Hess, K. M., & Wilcots, E. M. 2013, , 146, 124[Huang et al.(2012a)]Huang2012pop Huang, S., Haynes, M. P., Giovanelli, R., & Brinchmann, J. 2012, , 756, 113[Huang et al.(2012b)]Huang2012 Huang, S., Haynes, M. P., Giovanelli, R., et al. 2012, , 143, 133 [Huchtmeier & Richter(1986)]HuchtmeierRichter1986 Huchtmeier, W. K., & Richter, O.-G. 1986, , 64, 111[Jaffé et al.(2016)]Jaffe2016 Jaffé, Y. L., Verheijen, M. A. W., Haines, C. P., et al. 2016, , 461, 1202[Koopmann & Kenney(2004)]KoopmannKenney2004 Koopmann, R. A., & Kenney, J. D. P. 2004, , 613, 866[Kenney et al.(2014)]Kenney2014 Kenney, J. D. P., Geha, M., Jáchym, P., et al. 2014, , 780, 119[Kent et al.(2007)]Kent2007 Kent, B. R., Giovanelli, R., Haynes, M. P., et al. 2007, , 665, L15 [Kent et al.(2009)]Kent2009 Kent, B. R., Spekkens, K., Giovanelli, R., et al. 2009, , 691, 1595[Koleva et al.(2009)]Koleva2009 Koleva, M., de Rijcke, S., Prugniel, P., Zeilinger, W. W., & Michielsen, D. 2009, , 396, 2133[Lee et al.(2009)]Lee2009 Lee, J. C., Kennicutt, R. C., Jr., Funes, S. J. J. G., Sakai, S., & Akiyama, S. 2009, , 692, 1305-1320[Lisker et al.(2006a)]Lisker2006bc Lisker, T., Glatt, K., Westera, P., & Grebel, E. K. 2006, , 132, 2432 [Lisker et al.(2006b)]Lisker2006disk Lisker, T., Grebel, E. K., & Binggeli, B. 2006, , 132, 497[Lisker et al.(2013)]Lisker2013 Lisker, T., Weinmann, S. M., Janz, J., & Meyer, H. T. 2013, , 432, 1162[Mayer et al.(2001)]Mayer2001 Mayer, L., Governato, F., Colpi, M., et al. 2001, , 559, 754[Meyer et al.(2014)]Meyer2014 Meyer, H. T., Lisker, T., Janz, J., & Papaderos, P. 2014, , 562, A49[Mateo(1998)]Mateo1998 Mateo, M. L. 1998, , 36, 435 [Mei et al.(2007)]Mei2007 Mei, S., Blakeslee, J. P., Côté, P., et al. 2007, , 655, 144[Miskolczi et al.(2011)]Miskolczi2011 Miskolczi, A., Bomans, D. J., & Dettmar, R.-J. 2011, , 536, A66 [Mistani et al.(2016)]Mistani2016 Mistani, P. A., Sales, L. V., Pillepich, A., et al. 2016, , 455, 2323[Morganti et al.(2006)]Morganti2006 Morganti, R., de Zeeuw, P. T., Oosterloo, T. A., et al. 2006, , 371, 157 [Odekon et al.(2016)]Odekon2016 Odekon, M. C., Koopmann, R. A., Haynes, M. P., et al. 2016, , 824, 110 [Oosterloo & van Gorkom(2005)]OosterloovanGorkom2005 Oosterloo, T., & van Gorkom, J. 2005, , 437, L19 [Oosterloo et al.(2010)]Oosterloo2010 Oosterloo, T., Morganti, R., Crocker, A., et al. 2010, , 409, 500[Paudel et al.(2010)]Paudel2010 Paudel, S., Lisker, T., Kuntschner, H., Grebel, E. K., & Glatt, K. 2010, , 405, 800[Penny et al.(2014)]Penny2014 Penny, S. J., Forbes, D. A., Pimbblet, K. A., & Floyd, D. J. E. 2014, , 443, 3381[Rakos & Schombert(2004)]Rakos2004 Rakos, K., & Schombert, J. 2004, , 127, 1502[Roediger & Brüggen(2008)]Roediger2008 Roediger, E., & Brüggen, M. 2008, , 388, 465[Roediger et al.(2011)]Roediger2011 Roediger, J. C., Courteau, S., MacArthur, L. A., & McDonald, M. 2011, , 416, 1996[Ryś et al.(2013)]Rys2013 Ryś, A., Falcón-Barroso, J., & van de Ven, G. 2013, , 428, 2980[Sandage & Hoffman(1991)]SandageHoffman1991 Sandage, A., & Hoffman, G. L. 1991, , 379, L45[Serra et al.(2012)]Serra2012 Serra, P., Oosterloo, T., Morganti, R., et al. 2012, , 422, 1835 [Skillman et al.(2003)]Skillman2003 Skillman, E. D., Côté, S., & Miller, B. W. 2003, , 125, 593[Struve et al.(2010)]Struve2010 Struve, C., Oosterloo, T., Sancisi, R., Morganti, R., & Emonts, B. H. C. 2010, , 523, A75[Taylor et al.(2012)]Taylor2012 Taylor, R., Davies, J. I., Auld, R., & Minchin, R. F. 2012, , 423, 787[Taranu et al.(2014)]Taranu2014 Taranu, D. S., Hudson, M. J., Balogh, M. L., et al. 2014, , 440, 1934[Thilker et al.(2010)]Thilker2010 Thilker, D. A., Bianchi, L., Schiminovich, D., et al. 2010, , 714, L171[Toloba et al.(2011)]Toloba2011 Toloba, E., Boselli, A., Cenarro, A. J., et al. 2011, , 526, A114[Toloba et al.(2014)]Toloba2014 Toloba, E., Guhathakurta, P., Peletier, R. F., et al. 2014, , 215, 17[Toloba et al.(2016)]Toloba2016 Toloba, E., Li, B., Guhathakurta, P., et al. 2016, , 822, 51[van Zee et al.(2004)]vanZee2004 van Zee, L., Skillman, E. D., & Haynes, M. P. 2004, , 128, 121[Weinmann et al.(2011)]Weinmann2011 Weinmann, S. M., Lisker, T., Guo, Q., Meyer, H. T., & Janz, J. 2011, , 416, 1197[Weisz et al.(2011a)]Weisz2011a Weisz, D. R., Dalcanton, J. J., Williams, B. F., et al. 2011, , 739, 5[Weisz et al.(2011b)]Weisz2011b Weisz, D. R., Dolphin, A. E., Dalcanton, J. J., et al. 2011, , 743, 8[Weisz et al.(2014)]Weisz2014 Weisz, D. R., Dolphin, A. E., Skillman, E. D., et al. 2014, , 789, 147[Yıldız et al.(2017)]Yildiz2017 Yıldız, M. K., Serra, P., Peletier, R. F., Oosterloo, T. A., & Duc, P.-A. 2017, , 464, 329 | http://arxiv.org/abs/1706.08555v1 | {
"authors": [
"Gregory Hallenbeck",
"Rebecca Koopmann",
"Riccardo Giovanelli",
"Martha P. Haynes",
"Shan Huang",
"Lukas Leisman",
"Emmanouil Papastergis"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170626184004",
"title": "HI in Virgo's \"Red and Dead\" Dwarf Ellipticals - A Tidal Tail and Central Star Formation"
} |
Beijing Computational Science Research Center, Beijing 100084, ChinaSchool of Physical Sciences and Technology, Shenzhen University, Shenzhen 518060, ChinaDepartment of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China [email protected] of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, ChinaWe theoretically study artificial light harvesting by a dimerized Möbius ring. When the donors in the ring are dimerized, the energies of the donor ring are splitted into two sub-bands. Because of the nontrivial Möbius boundary condition, both the photon and acceptor are coupled to all collective-excitation modes in the donor ring. Therefore, the quantum dynamics in the light harvesting are subtly influenced by the dimerization in the Möbius ring. It is discovered that energy transfer is more efficient in a dimerized ring than that in an equally-spaced ring. This discovery is also confirmed by the calculation with the perturbation theory, which is equivalent to the Wigner-Weisskopf approximation. Our findings may be beneficial to the optimal design of artificial light harvesting.Artificial Light Harvesting by Dimerized Möbius Ring Qing Ai ... ====================================================§ INTRODUCTION Photosynthesis is the main resource of energy supply for living beings on earth. Therein, efficient light harvesting process, which delivers the captured photon energy to the reaction center, plays a crucial role in natural photosynthesis <cit.>. Because a series of experimental and theoretical explorations <cit.> have demonstrated that quantum coherent phenomena might exist and even optimize the natural photosynthesis, much effort has been made to reveal the effect of quantum coherence on efficient light harvesting <cit.>.The researches on optimal geometries for efficient energy transfer are rare <cit.>. Wu et al. demonstrated the trapping-free mechanism for efficient light harvesting in a star-like artificial system <cit.>. Hoyer et al. showed ratchet effect for quantum coherent energy transfer in a one-dimensional system <cit.>. In 2013, one of the authors Q.A. and his collaborators elucidated that clustered geometries utilize exciton delocalization and energy matching condition to optimize energy transfer in a generic tetramer model <cit.> as well as in Fenna-Matthews-Olson complex <cit.>. However, photosynthesis with ring-shape geometry is more frequently observed in natural photosynthetic complexes, e.g. LH1 and LH2 <cit.>. This observation inspired Yang et al. to prove that symmetry breaking in B850 ring of LH2 complex boosts efficient inter-complex energy transfer <cit.>. Dong et al. further proposed that a perfect donor ring for artificial light harvesting makes full use of collective excitation and dark state to enhance the energy transfer efficiency <cit.>.Mathematically speaking, the rings as in LH1 and LH2 are topologically trivial, since both the photon and acceptor are only coupled to the zero-momentum collective-excitation mode regardless of the dimerization as shown in Appendix <ref>. On the other hand, Möbius strips <cit.> manifest novel physical properties and can be used to fabricate novel devices and materials <cit.>, e.g. topological insulators and negative-index metamaterials <cit.>. In these Möbius strips, the electrons in the ring experience different local effective fields at different positions due to the topologically non-trivial boundary condition. This observation enlightens us on the investigation of the Möbius strips in artificial light harvesting. When the donors in the ring are dimerized, there are two energy sub-bands for collective excitation modes. Due to the Möbius boundary condition, both the photon and acceptor interact with all collective excitations in the ring. This is in remarkable contrast to the case in Ref. <cit.>, where they are only coupled to single collective excitation mode.This paper is organized as follows: the donor ring with Möbius boundary condition in introduced for light harvesting in the next section. Then, in Sec. <ref>, the energy transfer efficiency is numerically simulated for various of parameter regimes. Finally, the main results are summarized in the Conclusion part. In Appendix <ref>, a brief description of diagonalizing a dimerized ring with Möbius boundary condition is given. In Appendix <ref>, we prove that both the photon and acceptor only interact with one of the collective excitation modes in the ring with periodical boundary condition, no matter whether the dimerization exists in the ring or not. In Appendix <ref>, we present a detailed perturbation theory for describing energy transfer in a ring with Möbius boundary condition.§ DIMERIZED RING WITH MÖBIUS BOUNDARY CONDITION In this paper, we consider the light harvesting in a Peierls distorted chain with Möbius boundary condition <cit.>. The quantum dynamics of the whole system is governed by the HamiltonianH =ω b^†b+ε_AA^†A+H_PM+∑_j=1^N(ξ d_j^†A+Jd_j^†b)+h.c.,where b^†(b) is the creation (annihilation) operator of photon with frequency ω, A^†(A) is the creation (annihilation) operator of excitation at the acceptor with site energy ε_A, d_j^†(d_j) is the creation (annihilation) operator of an excitation at jth donor, and the Peierls distorted ring with Möbius boundary condition is described byH_PM=∑_j=1^N-1g[1-(-1)^jδ]d_j+1^†d_j-g[1-(-1)^Nδ]d_1^†d_N+h.c.with g being coupling constant between nearest neighbors and δ being dimerization constant. Notice that the minus sign before the second term on the r.h.s. indicates the Möbius boundary condition <cit.>, and the site energies of donors are homogeneous and chosen as the zero point of energy. Here we assume that the photon and acceptor are coupled to all donors with equal coupling strength J and ξ, respectively.By the diagonalization method of H_PM in Appendix <ref>, the total Hamiltonian is rewritten asH =ω b^†b+ε_AA^†A+∑_kε_k(A_k^†A_k-B_k^†B_k)+H_1.There are two energy bands in the donor ring denoted by the annihilation (creation) operatorsA_k=∑_j=1^N/2e^-ikj/√(N)(e^i(2j-1)π/Nd_2j-1+e^i(θ_k+2jπ/N)d_2j),B_k=∑_j=1^N/2e^-ikj/√(N)(e^i(2j-1)π/Nd_2j-1-e^i(θ_k+2jπ/N)d_2j), (A_k^† and B_k^†) with the eigen energies being ±ε_k, whereε_k= 2g√(cos^2(k/2-π/N)+δ^2sin^2(k/2-π/N)),and the momentumk=4π/N(0,1,2,⋯N/2-1)-π+2π/N.The interaction Hamiltonian among the photon and acceptor and donor ring isH_1=∑_k(ξ_AkA_k^†+ξ_BkB_k^†)A+(J_AkA_k^†+J_BkB_k^†)b+h.c. =∑_k(h_AkA_k^†+h_BkB_k^†)(ξ A+Jb)+h.c.with the k-dependent factorsh_Ak=1/√(N)∑_j=1^N/2e^-ikje^-i(2j-1)π/N(1+e^iθ_ke^iπ/N),h_Bk=1/√(N)∑_j=1^N/2e^-ikje^-i(2j-1)π/N(1-e^iθ_ke^iπ/N),e^iθ_k=g/ε_k[(1+δ)e^-iπ/N+(1-δ)e^-i(k-π/N)]. Before investigating the quantum dynamics of light harvesting, we shall analyze the energy spectrum of Möbius ring for different dimerization δ. As shown in Fig. <ref>, the characteristics of the energy spectrum vary remarkably in response to the change of δ. When the donors in the ring are equally distributed, i.e. δ=0, the energy spectrum ε_k=2gcos(k/2) changes over a range 2g. For other parameters, i.e. 0<|δ|<1, the energy spectrum ε_k, which lies in the range [2g|δ|,2g], shrinks as |δ| approaches unity. There is an energy gap between the two sub-bands 4g|δ|.Previously, Möbius strip <cit.> was proposed to fabricate novel devices and materials <cit.>, e.g. topological insulators and negative-index metamaterials <cit.>. Although it seems that the ring with Möbius boundary condition in this paper is somewhat different from Möbius strip in Refs. <cit.>, we remark that the ring with Möbius boundary condition can be considered as Möbius strip in the Hilbert space. Further comparison shows that the ring with Möbius boundary condition is not evenly twisted as the previous Möbius strip. As a result of Möbius boundary condition, the photon and acceptor are coupled to all the collective modes in the ring, and thus leads to different quantum dynamics as compared to that for a ring with periodical boundary condition.§ NUMERICAL RESULTS In this section, we consider the quantum dynamics of light harvesting by a Peierls distorted chain with Möbius boundary condition. For an initial state |ψ(0)⟩=|1_b⟩≡|1_b,0_A,0_Ak,0_Bk⟩, the state of the total system at any time t|ψ(t)⟩=α_b|1_b⟩+α_A|1_A⟩+∑_kβ_Ak|1_Ak⟩+∑_kβ_Bk|1_Bk⟩,with |1_A⟩≡|0_b,1_A,0_Ak,0_Bk⟩, |1_Ak⟩≡|0_b,0_A,1_Ak,0_Bk⟩, |1_Bk⟩≡|0_b,0_A,0_Ak,1_Bk⟩, is governed by the Schrödinger equationi∂_t|ψ(t)⟩=H|ψ(t)⟩,if the system does not interact with the environment. However, an open quantum system inevitably suffers from decoherence due to the couplings to the environment. Generally speaking, the quantum dynamics in the presence of decoherence is described by the master equation instead of Schrödinger equation. Despite this, it has been shown that the quantum dynamics for light harvesting can be well simulated by the quantum jump approach with a non-Hermitian Hamiltonian where the imaginary parts in the diagonal terms represent the decoherence processes <cit.>. In this case, the Hamiltonian H in Eq. (<ref>) is obtained by replacing the energies of the acceptor and collective-excitation modes in the following way, i.e.A:ε_A → ε_A^'=ε_A-iΓ,A_k:ε_k → ε_k^-=ε_k- iκ,B_k:-ε_k →-ε_k^+=-(ε_k+ iκ),where κ and Γ are respectively the fluorescence rate at the donors and the charge separation rate at the acceptor.It was shown <cit.> that in a composite system including a photon, and an acceptor, and a perfect ring with periodical boundary condition, the quantum dynamics of the total system can be effectively modeled as the interaction of photon and acceptor with the donor's single collective-excitation mode. Furthermore, as proven in Appendix <ref>, both the photon and acceptor are still coupled to the same collective mode of donor ring even when the ring is dimerized. However, the situation is different when we explore the coherent energy transfer in a dimerized Möbius ring. In Fig. <ref>, we plot population of acceptor P_a(t)=|α_A(t)|^2 vs time for three cases with different δ. Clearly, the quantum dynamics in Möbius ring is subtly influenced by the dimerization. In all cases, the populations quickly rise to a maximum and then it is followed by damped oscillations. After t≃10, all three curves converge to a similar same exponential decay as the charge separation rates are the same for all three cases.To quantitatively characterize the energy transfer, there is the transfer efficiency η defined as <cit.>η= 2Γ∫_0^∞|α_A(t)|^2dt.In Fig. <ref>(a), we investigate the dependence of transfer efficiency on the detuning Δ=ω-ε_A for a fast fluorescence decay κ/ξ=1. For an evenly-distributed donor ring, i.e. δ=0, the transfer efficiency reaches maximum near the resonance, i.e. Δ=0, which is slightly different from that discovered in Ref. <cit.>. As the photon frequency deviates from the resonance, the transfer efficiency quickly drops. When we further investigate the case for a dimerized ring with δ=0.6, the transfer efficiency has been increased over the whole range of photon frequency, especially in the blue-detuned regime. The same characteristics has also been observed in the case with a slow fluorescence decay κ/ξ=0.1 as shown in Fig. <ref>(b).To validate the above numerical simulation, we also calculate the quantum dynamics and light-harvesting efficiency by the perturbation theory. The details are presented in Appendix <ref>. In Ref. <cit.>, it has been proven that the Wigner-Weisskopf approximation is equivalent to the perturbation theory in the study of an excited state coupled to a continuum. Here we generalize the perturbation theory to the investigation of coherent energy transfer between few bodies via a continuum. In Fig. <ref>, the energy transfer efficiency of a dimerized ring is generally larger than that of an equally-spaced ring. This result is qualitatively consistent with Fig. <ref>(a).In order to explore the underlying physical mechanism, we turn to coupling constants |h_Ak| and |h_Bk| as shown in Fig. <ref>. Although the coupling constants |h_Ak| for δ=0 almost coincide with the ones for δ=0.6, the coupling constants |h_Bk| are significantly suppressed when the dimerization occurs. Furthermore, because |h_Ak| are generally larger than |h_Bk| by an order, the transfer efficiency is slightly tuned by the dimerization. § CONCLUSION In this paper, we investigate the quantum dynamics of light harvesting in a Peierls distorted ring with Möbius boundary condition. Due to the nontrivial Möbius topology, there are two energy bands for the excitation in the ring when the donors in the ring are dimerized. Because both the photon and acceptor interact with all collective-excitation modes in the Möbius ring, the quantum dynamics and thus the efficiency for light harvesting is effectively influenced by the presence of dimerization. By numerical simulations, we show that when the donors in the Möbius ring are dimerized, the energy transfer is generally optimal for a wide range of photon frequencies. Our discoveries together with previous findings <cit.> may be beneficial for the future design of optimal artificial light harvesting.In addition, we also remark that in fact change of protein environment inevitably leads to static and dynamic disorders in the transition energies of chlorophylls in natural photosynthesis <cit.>. In Ref. <cit.>, assuming that all site energies of donors are homogeneous, the absorbed photon energy utilizes dark-state mechanism to effectively avoid fluorescence loss via the donors. However, this mechanism might not work well when the static and dynamic disorders take place.We thank stimulating discussions with C. P. Sun and H. Dong. This work was supported by the National Natural Science Foundation of China (Grant No. 11121403 and No. 11534002 and No. 11505007), the National 973 program (Grant No. 2014CB921403 and No. 2012CB922104), and the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University Grant No. KF201502. § DIAGONALIZATION OF MÖBIUS HAMILTONIAN The dimerized ring with Möbius boundary condition is described byH_PM=∑_j=1^N-1g[1-(-1)^jδ]d_j+1^†d_j-g[1-(-1)^Nδ]d_1^†d_N+h.c.,where |δ|≤1 is a dimensionless dimerization constant. By applying a unitary transformationU=∑_je^ijπ/Nd_j^†d_j,the Hamiltonian readsUH_PMU^†=∑_j=1^N-1g[1-(-1)^jδ]e^iπ/Nd_j+1^†d_j -g[1-(-1)^Nδ]e^-i(N-1)π/Nd_1^†d_N+h.c.=∑_j=1^Ng_0[1-(-1)^jδ]d_j+1^†d_j+h.c.=∑_j=1^N/2[g_0(1-δ)d_2j+1^†d_2j+g_0(1+δ)d_2j^†d_2j-1]+h.c.,whereg_0=ge^iπ/N.After the unitary transformation, the Möbius boundary condition has been canceled.By defining fermion operators <cit.>α_k=1/√(N)∑_j=1^N/2e^-ikj(d_2j-1+e^iθ_kd_2j), β_k=1/√(N)∑_j=1^N/2e^-ikj(d_2j-1-e^iθ_kd_2j), with inverse transformationd_2j-1=1/√(N)∑_j=1^N/2e^ikj(α_k+β_k),d_2j=1/√(N)∑_j=1^N/2e^ikj-iθ_k(α_k-β_k), where the momentum isk =4π/N(0,1,2,⋯N/2-1)-π+2π/N,e^iθ_k=g/ε_k[(1+δ)e^-iπ/N+(1-δ)e^-i(k-π/N)], ε_k= 2g√(cos^2(k/2-π/N)+δ^2sin^2(k/2-π/N))is the eigen energy, N is an even number,UH_PMU^†=∑_kε_k(α_k^†α_k-β_k^†β_k).Here the momentum is chosen in such way that cos(k/2-π/N)≥0 for δ=0 and thus the eigen energies ε_k=2gcos(k/2-π/N). Therefore, the original Hamiltonian can be diagonalized asH_PM=∑_kε_k(A_k^†A_k-B_k^†B_k),where the annihilation operators of collective excitation modes of the two bands areA_k=α_kU =∑_j=1^N/2e^-ikj/√(N)(e^i(2j-1)π/Nd_2j-1+e^i(θ_k+2jπ/N)d_2j),B_k=β_kU =∑_j=1^N/2e^-ikj/√(N)(e^i(2j-1)π/Nd_2j-1-e^i(θ_k+2jπ/N)d_2j). According to Eq. (<ref>), there are two sub-bands in the dimerized chain, i.e. ε_k and -ε_k. For the upper band, there is a minimum 2| gδ| at k=π+2π/N, while for the lower band, there is a maximum -2| gδ| also at k=π+2π/N. Therefore, there is an energy gap between the two sub-bands 4| gδ| as long as the ring is dimerized.§ LIGHT HARVESTING BY RING WITH PERIODICAL BOUNDARY CONDITION The Hamiltonian for a dimerized ring with periodical boundary condition readsH =ω b^†b+ε_AA^†A+∑_jg[1-(-1)^jδ]d_j^†d_j+1+∑_j(Jd_j^†b+ξ d_j^†A)+h.c.We define two sets of collective-excitation operators α_k=1/√(N)∑_j=1^N/2e^-ikj(d_2j-1-e^iθ_kd_2j), β_k=1/√(N)∑_j=1^N/2e^-ikj(d_2j-1+e^iθ_kd_2j), and the reverse transformation isd_2j-1=1/√(N)∑_j=1^N/2e^-ikj(α_k+β_k),d_2j=1/√(N)∑_j=1^N/2e^-ikj-iθ_k(β_k-α_k), where k=4π(n-1)/N, n=1,2,...,N/2, ande^iθ_k= [(1+δ)+(1-δ)e^-ik]g/ε_k, ε_k= 2g√(cos^2(k/2)+δ^2sin^2(k/2)). Thus, the Hamiltonian is transformed asH =ω b^†b+ε_AA^†A+∑_kε_k(β_k^†β_k-α_k^†α_k)+√(N)β_0^†(Jb+ξ A)+h.c.Since only the zero-momentum modeβ_0=1/√(N)∑_j=1^Nd_j with eigen energy ε_0=2g is coupled to the photon and acceptor, the effective Hamiltonian is further simplified asH_eff=ω b^†b+2gβ_0^†β_0+ε_AA^†A+√(N)β_0^†(Jb+ξ A)+h.c.In conclusion, for a ring with periodical boundary condition, both the photon and acceptor are coupled to the same collective-excitation mode irrespective of the dimerization in the ring. In other words, we have proven that the quantum dynamics of light harvesting by a ring with periodical boundary condition is not affected by the dimerization.§ EQUIVALENCE OF PRESENT THEORY AND WIGNER-WEISSKOPF APPROXIMATION The quantum dynamics in light harvesting is governed by the Schrödinger equation, which is equivalent to a set of coupled differential equations. Formally, it can be solved by Laplace transformation. However, due to the presence of branch cut, the exact solution may not be easily obtained. Generally speaking, it can be solved by the Wigner-Weisskopf approximation <cit.>. In Ref. <cit.>, it has been proven that the Wigner-Weisskopf approximation is equivalent to the perturbation theory in the study of excited state of a few-body system coupled to a continuum. Therefore, we make use of the perturbation theory to investigate the quantum dynamics of light-harvesting in a dimerized Möbius chain.For an initial state |ψ(0)⟩=|1_b⟩≡|1_b,0_A,0_Ak,0_Bk⟩, the state of system at any time t|ψ(t)⟩=α_b(t)|1_b⟩+α_A(t)|1_A⟩+∑_kβ_Ak(t)|1_Ak⟩+∑_kβ_Bk(t)|1_Bk⟩,with |1_A⟩≡|0_b,1_A,0_Ak,0_Bk⟩, |1_Ak⟩≡|0_b,0_A,1_Ak,0_Bk⟩, |1_Bk⟩≡|0_b,0_A,0_Ak,1_Bk⟩, is governed by the Schrödinger equationi∂_t|ψ(t)⟩=H|ψ(t)⟩.We obtain a set of equations for the coefficients, i.e.iα̇_b=ωα_b+ϵ∑_kJ_Ak^*β_Ak+ϵ∑_kJ_Bk^*β_Bk,iα̇_A=ε_A^'α_A+ϵ∑_kξ_Ak^*β_Ak+ϵ∑_kξ_Bk^*β_Bk,iβ̇_Ak=ε_k^-β_Ak+ϵξ_Akα_A+ϵ J_Akα_b,iβ̇_Bk= -ε_k^+β_Bk+ϵξ_Bkα_A+ϵ J_Bkα_b,where the parameter ϵ is introduced to keep track of the orders of perturbation.By introducing renormalized frequencies, i.e.Ω_b=ω+ϵΩ_b^(1)+ϵ^2Ω_b^(2)+ϵ^3Ω_b^(3)+…, Ω_A=ε_A^'+ϵΩ_A^(1)+ϵ^2Ω_A^(2)+ϵ^3Ω_A^(3)+…, Ω_Ak=ε_k^-+ϵΩ_Ak^(1)+ϵ^2Ω_Ak^(2)+ϵ^3Ω_Ak^(3)+…, Ω_Bk= -ε_k^++ϵΩ_Bk^(1)+ϵ^2Ω_Bk^(2)+ϵ^3Ω_Bk^(3)+…,we can define dimensionless timesτ_b=Ω_bt, τ_A=Ω_At, τ_Ak=Ω_Akt, τ_Bk=Ω_Bkt,and re-express the above differential equations asiΩ_b∂α_b/∂τ_b=ωα_b+ϵ∑_kJ_Ak^*β_Ak+ϵ∑_kJ_Bk^*β_Bk,iΩ_A∂α_A/∂τ_A=ε_A^'α_A+ϵ∑_kξ_Ak^*β_Ak+ϵ∑_kξ_Bk^*β_Bk,iΩ_Ak∂β_Ak/∂τ_Ak=ε_k^-β_Ak+ϵξ_Akα_A+ϵ J_Akα_b,iΩ_Bk∂β_Bk/∂τ_Bk= -ε_k^+β_Bk+ϵξ_Bkα_A+ϵ J_Bkα_b.By inserting Eq. (<ref>) into Eq. (<ref>), and expanding the coefficients asα_b=α_b^(0)+ϵα_b^(1)+ϵ^2α_b^(2)+ϵ^3α_b^(3)+…, α_A=α_A^(0)+ϵα_A^(1)+ϵ^2α_A^(2)+ϵ^3α_A^(3)+…, β_Ak=β_Ak^(0)+ϵβ_Ak^(1)+ϵ^2β_Ak^(2)+ϵ^3β_Ak^(3)+…, β_Bk=β_Bk^(0)+ϵβ_Bk^(1)+ϵ^2β_Bk^(2)+ϵ^3β_Bk^(3)+…,we could obtain a set of equations for different orders of coefficients,i(ω+ϵΩ_b^(1)+ϵ^2Ω_b^(2)+ϵ^3Ω_b^(3)…)∂(α_b^(0)+ϵα_b^(1)+ϵ^2α_b^(2)+ϵ^3α_b^(3)…)/∂τ_b=ω(α_b^(0)+ϵα_b^(1)+ϵ^2α_b^(2)+ϵ^3α_b^(3)…)+ϵ∑_kJ_Ak^*β_Ak+ϵ∑_kJ_Bk^*β_Bk, i(ε_A^'+ϵΩ_A^(1)+ϵ^2Ω_A^(2)+ϵ^3Ω_A^(3)…)∂(α_A^(0)+ϵα_A^(1)+ϵ^2α_A^(2)+ϵ^3α_A^(3)…)/∂τ_A=ε_A^'(α_A^(0)+ϵα_A^(1)+ϵ^2α_A^(2)+ϵ^3α_A^(3)…)+ϵ∑_kξ_Ak^*β_Ak+ϵ∑_kξ_Bk^*β_Bk, i(ε_k^-+ϵΩ_Ak^(1)+ϵ^2Ω_Ak^(2)+ϵ^3Ω_Ak^(3)…)∂(β_Ak^(0)+ϵβ_Ak^(1)+ϵ^2β_Ak^(2)+ϵ^3β_Ak^(3)…)/∂τ_Ak=ε_k^-(β_Ak^(0)+ϵβ_Ak^(1)+ϵ^2β_Ak^(2)+ϵ^3β_Ak^(3)…)+ϵξ_Akα_A+ϵ J_Akα_b, i(-ε_k^++ϵΩ_Bk^(1)+ϵ^2Ω_Bk^(2)+ϵ^3Ω_Bk^(3)…)∂(β_Bk^(0)+ϵβ_Bk^(1)+ϵ^2β_Bk^(2)+ϵ^3β_Bk^(3)…)/∂τ_Bk= -ε_k^+(β_Bk^(0)+ϵβ_Bk^(1)+ϵ^2β_Bk^(2)+ϵ^3β_Bk^(3)…)+ϵξ_Bkα_A+ϵ J_Bkα_b.For the zeroth-order coefficients, we haveω(i∂α_b^(0)/∂τ_b-α_b^(0)) = 0, ε_A^'(i∂α_A^(0)/∂τ_A-α_A^(0)) = 0, ε_k^-(i∂β_Ak^(0)/∂τ_Ak-β_Ak^(0)) = 0, ε_k^+(i∂β_Bk^(0)/∂τ_Bk-β_Bk^(0)) = 0.The solutions to the above equations areα_b^(0)= A_be^-iτ_b, α_A^(0)= A_Ae^-iτ_A, β_Ak^(0)= B_Ake^-iτ_Ak, β_Bk^(0)= B_Bke^-iτ_Bk,where the constants A_u (u=b,A) and B_vk (v=A,B) will be determined by the initial condition later.For the first-order coefficients, we haveω(i∂α_b^(1)/∂τ_b-α_b^(1)) = -iΩ_b^(1)∂α_b^(0)/∂τ_b+∑_kJ_Ak^*β_Ak^(0)+∑_kJ_Bk^*β_Bk^(0), ε_A^'(i∂α_A^(1)/∂τ_A-α_A^(1)) = -iΩ_A^(1)∂α_A^(0)/∂τ_A+∑_kξ_Ak^*β_Ak^(0)+∑_kξ_Bk^*β_Bk^(0), ε_k^-(i∂β_Ak^(1)/∂τ_Ak-β_Ak^(1)) = -iΩ_Ak^(1)∂β_Ak^(0)/∂τ_Ak+ξ_Akα_A^(0)+J_Akα_b^(0),-ε_k^+(i∂β_Bk^(1)/∂τ_Bk-β_Bk^(1)) = -iΩ_Bk^(1)∂β_Bk^(0)/∂τ_Bk+ξ_Bkα_A^(0)+J_Bkα_b^(0).The first terms on the r.h.s will lead to divergence in the long-time limit, because they are resonant driving. As a result, the first-order renormalizations to energies vanish, i.e.Ω_b^(1)=Ω_A^(1)=Ω_Ak^(1)=Ω_Bk^(1)=0.By further inserting Eq. (<ref>) into Eq. (<ref>):ω(i∂α_b^(1)/∂τ_b-α_b^(1)) =∑_kJ_Ak^*B_Ake^-iτ_Ak+∑_kJ_Bk^*B_Bke^-iτ_Bk, ε_A^'(i∂α_A^(1)/∂τ_A-α_A^(1)) =∑_kξ_Ak^*B_Ake^-iτ_Ak+∑_kξ_Bk^*B_Bke^-iτ_Bk, ε_k^-(i∂β_Ak^(1)/∂τ_Ak-β_Ak^(1)) =ξ_AkA_Ae^-iτ_A+J_AkA_be^-iτ_b,-ε_k^+(i∂β_Bk^(1)/∂τ_Bk-β_Bk^(1)) =ξ_BkA_Ae^-iτ_A+J_BkA_be^-iτ_b,we obtain the first-order terms asα_b^(1)=∑_kB_AkΩ_bJ_Ak^*/ω(Ω_Ak-Ω_b)e^-iτ_Ak+∑_kB_BkΩ_bJ_Bk^*/ω(Ω_Bk-Ω_b)e^-iτ_Bk, α_A^(1)=∑_kB_AkΩ_Aξ_Ak^*/ε_A^'(Ω_Ak-Ω_A)e^-iτ_Ak+∑_kB_BkΩ_Aξ_Bk^*/ε_A^'(Ω_Bk-Ω_A)e^-iτ_Bk, β_Ak^(1)= A_AΩ_Akξ_Ak/ε_k^-(Ω_A-Ω_Ak)e^-iτ_A+A_bΩ_AkJ_Ak/ε_k^-(Ω_b-Ω_Ak)e^-iτ_b, β_Bk^(1)= A_AΩ_Bkξ_Bk/-ε_k^+(Ω_A-Ω_Bk)e^-iτ_A+A_bΩ_BkJ_Bk/-ε_k^+(Ω_b-Ω_Bk)e^-iτ_b.In the same way, we can obtain the equations for the second-order coefficients asω(i∂α_b^(2)/∂τ_b-α_b^(2)) = -iΩ_b^(2)∂α_b^(0)/∂τ_b+∑_kJ_Ak^*(A_AΩ_Akξ_Ak/ε_k^-(Ω_A-Ω_Ak)e^-iτ_A+A_bΩ_AkJ_Ak/ε_k^-(Ω_b-Ω_Ak)e^-iτ_b]+∑_kJ_Bk^*[A_AΩ_Bkξ_Bk/-ε_k^+(Ω_A-Ω_Bk)e^-iτ_A+A_bΩ_BkJ_Bk/-ε_k^+(Ω_b-Ω_Bk)e^-iτ_b], ε_A^'(i∂α_A^(2)/∂τ_A-α_A^(2)) = -iΩ_A^(2)∂α_A^(0)/∂τ_A+∑_kξ_Ak^*[A_AΩ_Akξ_Ak/ε_k^-(Ω_A-Ω_Ak)e^-iτ_A+A_bΩ_AkJ_Ak/ε_k^-(Ω_b-Ω_Ak)e^-iτ_b]+∑_kξ_Bk^*[A_AΩ_Bkξ_Bk/-ε_k^+(Ω_A-Ω_Bk)e^-iτ_A+A_bΩ_BkJ_Bk/-ε_k^+(Ω_b-Ω_Bk)e^-iτ_b], ε_k^-(i∂β_Ak^(2)/∂τ_Ak-β_Ak^(2)) = -iΩ_Ak^(2)∂β_Ak^(0)/∂τ_Ak+ξ_Ak[∑_k^'B_Ak^'Ω_Aξ_Ak^'^*/ε_A^'(Ω_Ak^'-Ω_A)e^-iτ_Ak^'+∑_k^'B_Bk^'Ω_Aξ_Bk^'^*/ε_A^'(Ω_Bk^'-Ω_A)e^-iτ_Bk^']+J_Ak[∑_k^'B_Ak^'Ω_bJ_Ak^'^*/ω(Ω_Ak^'-Ω_b)e^-iτ_Ak^'+∑_k^'B_Bk^'Ω_bJ_Bk^'^*/ω(Ω_Bk^'-Ω_b)e^-iτ_Bk^'],-ε_k^+(i∂β_Bk^(2)/∂τ_Bk-β_Bk^(2)) = -iΩ_Bk^(2)∂β_Bk^(0)/∂τ_Bk+ξ_Bk[∑_k^'B_Ak^'Ω_Aξ_Ak^'^*/ε_A^'(Ω_Ak^'-Ω_A)e^-iτ_Ak^'+∑_k^'B_Bk^'Ω_Aξ_Bk^'^*/ε_A^'(Ω_Bk^'-Ω_A)e^-iτ_Bk^']+J_Bk[∑_k^'B_Ak^'Ω_bJ_Ak^'^*/ω(Ω_Ak^'-Ω_b)e^-iτ_Ak^'+∑_k^'B_Bk^'Ω_bJ_Bk^'^*/ω(Ω_Bk^'-Ω_b)e^-iτ_Bk^'].Because the resonant-driving terms will result in divergence, the second-order energies are thusΩ_b^(2)=∑_k[Ω_Ak|J_Ak|^2/ε_k^-(Ω_b-Ω_Ak)+Ω_Bk|J_Bk|^2/-ε_k^+(Ω_b-Ω_Bk)], Ω_A^(2)=∑_k[Ω_Ak|ξ_Ak|^2/ε_k^-(Ω_A-Ω_Ak)+Ω_Bk|ξ_Bk|^2/-ε_k^+(Ω_A-Ω_Bk)], Ω_Ak^(2)=Ω_A|ξ_Ak|^2/ε_A^'(Ω_Ak-Ω_A)+Ω_b|J_Ak|^2/ω(Ω_Ak-Ω_b), Ω_Bk^(2)=Ω_A|ξ_Bk|^2/ε_A^'(Ω_Bk-Ω_A)+Ω_b|J_Bk|^2/ω(Ω_Bk-Ω_b).And the corresponding second-order coefficients are α_b^(2)=∑_kA_A[J_Ak^*ξ_AkΩ_Ak/ε_k^-(Ω_A-Ω_Ak)+J_Bk^*ξ_BkΩ_Bk/-ε_k^+(Ω_A-Ω_Bk)]Ω_be^-iτ_A/ω(Ω_A-Ω_b), α_A^(2)=∑_kA_b[J_Akξ_Ak^*Ω_Ak/ε_k^-(Ω_b-Ω_Ak)+J_Bkξ_Bk^*Ω_Bk/-ε_k^+(Ω_b-Ω_Bk)]Ω_Ae^-iτ_b/ε_A^'(Ω_b-Ω_A), β_Ak^(2)=∑_k^'^'B_Ak^'[Ω_Aξ_Akξ_Ak^'^*/ε_A^'(Ω_Ak^'-Ω_A)+Ω_bJ_AkJ_Ak^'^*/ω(Ω_Ak^'-Ω_b)]Ω_Ake^-iτ_Ak^'/ε_k^-(Ω_Ak^'-Ω_Ak)+∑_k^'B_Bk^'[Ω_Aξ_Akξ_Bk^'^*/ε_A^'(Ω_Bk^'-Ω_A)+Ω_bJ_AkJ_Bk^'^*/ω(Ω_Bk^'-Ω_b)]Ω_Ake^-iτ_Bk^'/ε_k^-(Ω_Bk^'-Ω_Ak), β_Bk^(2)=∑_k^'^'B_Bk^'[Ω_Aξ_Bkξ_Bk^'^*/ε_A^'(Ω_Bk^'-Ω_A)+Ω_bJ_BkJ_Bk^'^*/ω(Ω_Bk^'-Ω_b)]Ω_Bke^-iτ_Bk^'/-ε_k^+(Ω_Bk^'-Ω_Bk)+∑_k^'B_Ak^'[Ω_Aξ_Bkξ_Ak^'^*/ε_A^'(Ω_Ak^'-Ω_A)+Ω_bJ_BkJ_Ak^'^*/ω(Ω_Ak^'-Ω_b)]Ω_Bke^-iτ_Ak^'/-ε_k^+(Ω_Ak^'-Ω_Bk),where the prime over summation ∑_k^'^' indicates that the term related to k^'=k is removed from the summation.For the third-order coefficients, we have iω∂α_b^(3)/∂τ_b-ωα_b^(3)= -iΩ_b^(2)∂α_b^(1)/∂τ_b-iΩ_b^(3)∂α_b^(0)/∂τ_b+∑_kJ_Ak^*β_Ak^(2)+∑_kJ_Bk^*β_Bk^(2),iε_A^'∂α_A^(3)/∂τ_A-ε_A^'α_A^(3)= -iΩ_A^(2)∂α_A^(1)/∂τ_A-iΩ_A^(3)∂α_A^(0)/∂τ_A+∑_kξ_Ak^*β_Ak^(2)+∑_kξ_Bk^*β_Bk^(2),iε_k^-∂β_Ak^(3)/∂τ_Ak-ε_k^-β_Ak^(3)= -iΩ_Ak^(2)∂β_Ak^(1)/∂τ_Ak-iΩ_Ak^(3)∂β_Ak^(0)/∂τ_Ak+ξ_Ak∑_kA_b[J_Akξ_Ak^*Ω_Ak/ε_k^-(Ω_b-Ω_Ak)+J_Bkξ_Bk^*Ω_Bk/-ε_k^+(Ω_b-Ω_Bk)]Ω_Ae^-iτ_b/ε_A^'(Ω_b-Ω_A)+J_Ak∑_kA_A[J_Ak^*ξ_AkΩ_Ak/ε_k^-(Ω_A-Ω_Ak)+J_Bk^*ξ_BkΩ_Bk/-ε_k^+(Ω_A-Ω_Bk)]Ω_be^-iτ_A/ω(Ω_A-Ω_b),-iε_k^+∂β_Bk^(3)/∂τ_Bk+ε_k^+β_Bk^(3)= -iΩ_Bk^(2)∂β_Bk^(1)/∂τ_Bk-iΩ_Bk^(3)∂β_Bk^(0)/∂τ_Bk+ξ_Bk∑_kA_b[J_Akξ_Ak^*Ω_Ak/ε_k^-(Ω_b-Ω_Ak)+J_Bkξ_Bk^*Ω_Bk/-ε_k^+(Ω_b-Ω_Bk)]Ω_Ae^-iτ_b/ε_A^'(Ω_b-Ω_A)+J_Bk∑_kA_A[J_Ak^*ξ_AkΩ_Ak/ε_k^-(Ω_A-Ω_Ak)+J_Bk^*ξ_BkΩ_Bk/-ε_k^+(Ω_A-Ω_Bk)]Ω_be^-iτ_A/ω(Ω_A-Ω_b).The coefficients of resonant terms are zero and thus lead toΩ_b^(3)=Ω_A^(3)=Ω_Ak^(3)=Ω_Bk^(3)=0. In conclusion, to the order of ϵ^3, the renormalized energies areΩ_b=ω+ϵ^2∑_k(|J_Ak|^2/ω-ε_k^-+|J_Bk|^2/ω+ε_k^+), Ω_A=ε_A^'+ϵ^2∑_k(|ξ_Ak|^2/ε_A^'-ε_k^-+|ξ_Bk|^2/ε_A^'+ε_k^+), Ω_Ak=ε_k^-+ϵ^2(|ξ_Ak|^2/ε_k^--ε_A^'+|J_Ak|^2/ε_k^--ω), Ω_Bk= -ε_k^++ϵ^2(|ξ_Bk|^2/-ε_k^+-ε_A^'+|J_Bk|^2/-ε_k^+-ω).To the order of ϵ^2, the coefficients are α_b(t) = A_be^-iτ_b+ϵ∑_k[B_AkΩ_bJ_Ak^*/ω(Ω_Ak-Ω_b)e^-iτ_Ak+B_BkΩ_bJ_Bk^*/ω(Ω_Bk-Ω_b)e^-iτ_Bk]+ϵ^2∑_kA_A[J_Ak^*ξ_AkΩ_Ak/ε_k^-(Ω_A-Ω_Ak)+J_Bk^*ξ_BkΩ_Bk/-ε_k^+(Ω_A-Ω_Bk)]Ω_be^-iτ_A/ω(Ω_A-Ω_b), α_A(t) = A_Ae^-iτ_A+ϵ∑_k[B_AkΩ_Aξ_Ak^*/ε_A^'(Ω_Ak-Ω_A)e^-iτ_Ak+B_BkΩ_Aξ_Bk^*/ε_A^'(Ω_Bk-Ω_A)e^-iτ_Bk]+ϵ^2∑_kA_b[J_Akξ_Ak^*Ω_Ak/ε_k^-(Ω_b-Ω_Ak)+J_Bkξ_Bk^*Ω_Bk/-ε_k^+(Ω_b-Ω_Bk)]Ω_Ae^-iτ_b/ε_A^'(Ω_b-Ω_A), β_Ak(t) = B_Ake^-iτ_Ak+ϵ[A_AΩ_Akξ_Ak/ε_k^-(Ω_A-Ω_Ak)e^-iτ_A+A_bΩ_AkJ_Ak/ε_k^-(Ω_b-Ω_Ak)e^-iτ_b]+ϵ^2{∑_k^'^'B_Ak^'[Ω_Aξ_Akξ_Ak^'^*/ε_A^'(Ω_Ak^'-Ω_A)+Ω_bJ_AkJ_Ak^'^*/ω(Ω_Ak^'-Ω_b)]Ω_Ake^-iτ_Ak^'/ε_k^-(Ω_Ak^'-Ω_Ak)+∑_k^'B_Bk^'[Ω_Aξ_Akξ_Bk^'^*/ε_A^'(Ω_Bk^'-Ω_A)+Ω_bJ_AkJ_Bk^'^*/ω(Ω_Bk^'-Ω_b)]Ω_Ake^-iτ_Bk^'/ε_k^-(Ω_Bk^'-Ω_Ak)}, β_Bk(t) = B_Bke^-iτ_Bk+ϵ[A_AΩ_Bkξ_Bk/-ε_k^+(Ω_A-Ω_Bk)e^-iτ_A+A_bΩ_BkJ_Bk/-ε_k^+(Ω_b-Ω_Bk)e^-iτ_b]+ϵ^2{∑_k^'^'B_Bk^'[Ω_Aξ_Bkξ_Bk^'^*/ε_A^'(Ω_Bk^'-Ω_A)+Ω_bJ_BkJ_Bk^'^*/ω(Ω_Bk^'-Ω_b)]Ω_Bke^-iτ_Bk^'/-ε_k^+(Ω_Bk^'-Ω_Bk)+∑_k^'B_Ak^'[Ω_Aξ_Bkξ_Ak^'^*/ε_A^'(Ω_Ak^'-Ω_A)+Ω_bJ_BkJ_Ak^'^*/ω(Ω_Ak^'-Ω_b)]Ω_Bke^-iτ_Ak^'/-ε_k^+(Ω_Ak^'-Ω_Bk)}.By using the initial conditionα_b(0) = 1, α_A(0)=β_Ak(0)=β_Bk(0)=0,we can obtain the constants as A_b= 1-ϵ^2∑_k[|J_Ak|^2/(Ω_b-Ω_Ak)^2+|J_Bk|^2/(Ω_b-Ω_Bk)^2] = 1-∑_k[|J_Ak|^2/(ω-ε_k^-)^2+|J_Bk|^2/(ω+ε_k^+)^2],A_A= -ϵ^2∑_k[J_Ak/(Ω_b-Ω_Ak)ξ_Ak^*/(Ω_A-Ω_Ak)+J_Bk/(Ω_b-Ω_Bk)ξ_Bk^*/(Ω_A-Ω_Bk)]-ϵ^2∑_k(J_Akξ_Ak^*/Ω_b-Ω_Ak+J_Bkξ_Bk^*/Ω_b-Ω_Bk)1/Ω_b-Ω_A = -∑_k[J_Ak/(ω-ε_k^-)ξ_Ak^*/(ε_A^'-ε_k^-)+J_Bk/(ω+ε_k^+)ξ_Bk^*/(ε_A^'+ε_k^+)]-∑_k(J_Akξ_Ak^*/ω-ε_k^-+J_Bkξ_Bk^*/ω+ε_k^+)1/ω-ε_A^',B_Ak= -ϵJ_Ak/Ω_b-Ω_Ak = -J_Ak/ω-ε_k^-,B_Bk= -ϵJ_Bk/Ω_b-Ω_Bk = -J_Bk/ω+ε_k^+.To the order of ϵ^2 , the probability amplitudes are α_b(t) = A_be^-iτ_b+∑_k[B_AkΩ_bJ_Ak^*/ω(Ω_Ak-Ω_b)e^-iτ_Ak+B_BkΩ_bJ_Bk^*/ω(Ω_Bk-Ω_b)e^-iτ_Bk] = A_be^-iτ_b+∑_k(B_AkJ_Ak^*/ε_k^--ωe^-iτ_Ak+B_BkJ_Bk^*/-ε_k^+-ωe^-iτ_Bk), α_A(t) = A_Ae^-iτ_A+∑_k[B_AkΩ_Aξ_Ak^*/ε_A^'(Ω_Ak-Ω_A)e^-iτ_Ak+B_BkΩ_Aξ_Bk^*/ε_A^'(Ω_Bk-Ω_A)e^-iτ_Bk]+∑_kA_b[J_Akξ_Ak^*Ω_Ak/ε_k^-(Ω_b-Ω_Ak)+J_Bkξ_Bk^*Ω_Bk/-ε_k^+(Ω_b-Ω_Bk)]Ω_Ae^-iτ_b/ε_A^'(Ω_b-Ω_A) = A_Ae^-iτ_A+∑_k(B_Akξ_Ak^*/ε_k^--ε_A^'e^-iτ_Ak+B_Bkξ_Bk^*/-ε_k^+-ε_A^'e^-iτ_Bk)+A_be^-iτ_b/ω-ε_A^'∑_k(J_Akξ_Ak^*/ω-ε_k^-+J_Bkξ_Bk^*/ω+ε_k^+), β_Ak(t) = B_Ake^-iτ_Ak+A_bΩ_AkJ_Ak/ε_k^-(Ω_b-Ω_Ak)e^-iτ_b = B_Ake^-iτ_Ak+A_bJ_Ak/ω-ε_k^-e^-iτ_b, β_Bk(t) = B_Bke^-iτ_Bk+A_bΩ_BkJ_Bk/-ε_k^+(Ω_b-Ω_Bk)e^-iτ_b = B_Bke^-iτ_Bk+A_bJ_Bk/ω+ε_k^+e^-iτ_b,where we have taken ϵ=1.To the lowest order of ϵ,α_b(t) =A_be^-iτ_b =exp[-iω t-it∑_k(|J_Ak|^2/ω-ε_k^-+|J_Bk|^2/ω+ε_k^+)].Regardless of fluorescence in the donor ring, and assuming there is a small imaginary part in ω, we recover the result under Wigner-Weisskopf approximation, i.e.α_b(t) =A_be^-iτ_b =exp[-iω t-it∑_k(|J_Ak|^2/ω-ε_k+i0^++|J_Bk|^2/ω+ε_k+i0^+)] =exp[-i(ω+Δ^')t-γ t],whereΔ^'=∑_k℘(|J_Ak|^2/ω-ε_k+|J_Bk|^2/ω+ε_k), γ=π∑_k[J_Ak|^2δ(ω-ε_k)+|J_Bk|^2δ(ω+ε_k)]. 10 Blankenship02R. E. Blankenship, Molecular Mechanisms of Photosynthesis (Blackwell Science, Oxford, 2002).May04V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, 2nd ed. (Wiley-VCH Verlag, Berlin, 2004).Amerongen00H. van Amerongen, L. Valkunas, and R. van Grondelle, Photosynthetic Excitons (Singapore, World Scientific, 2000).Fleming08G. R. Fleming and M. A. Ratner, Phys. Today 617, 28 (2008).Engel07G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mancal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, Nature(London) 446, 782 (2007).Lee07H. Lee, Y.-C. Cheng, and G. R. Fleming, Science 316, 1462 (2007).Collini10E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature(London) 463, 644 (2010).Hildner13 R. Hildner, D. Brinks, J. B. Nieder, R. J. Cogdell, and N. F. van Hulst, Science 340, 1448 (2013).Ishizaki09A. Ishizaki and G. R. Fleming, Proc. Natl. Acad. Sci. USA 106, 17255 (2009).Cheng06Y.-C. Cheng and R. J. Silbey, Phys. Rev. Lett. 96, 028103 (2006).Chin13A.W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, Nat. Phys. 9, 113 (2013).Ai13Q. Ai, T.-C. Yen, B.-Y. Jin, and Y.-C. Cheng, J. Phys. Chem. Lett. 4, 2577 (2013).Sarovar10M. Sarovar, A. Ishizaki, G. R. Fleming, and K. B. Whaley, Nat. Phys. 6, 462 (2010).Liao10J.-Q. Liao, J.-F. Huang, L.-M. Kuang, and C. P. Sun, Phys. Rev. A 82, 052109 (2010).Mohseni08 M. Mohseni, P. Rebentrost, S. Lloyd, A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008).Qin14M. Qin, H. Z. Shen, X. L. Zhao, and X. X. Yi, Phys. Rev. E 90, 042140 (2014).Tao16M.-J. Tao, Q. Ai, F.-G. Deng, and Y.-C. Cheng, Sci. Rep. 6, 27535 (2016).Wu13J. L. Wu, R. J. Silbey, and J. S. Cao, Phys. Rev. Lett. 110, 200402 (2013).Hoyer12S. Hoyer, A. Ishizaki, and K. B. Whaley, Phys. Rev. E 86, 041911 (2012).Yang10S. Yang, D. Z. Xu, Z. Song, and C. P. Sun, J. Chem. Phys. 132, 234501 (2010).Dong12H. Dong, D. Z. Xu, J. F. Huang, and C. P. Sun, Light: Sci. Appl. 1, e2 (2012).Sarovar11M. Sarovar, Y.-C. Cheng, and K. B. Whaley, Phys. Rev. E 83, 011906 (2011).Nalbach11P. Nalbach, D. Braun, and M. Thorwart, Phys. Rev. E 84, 041926 (2011).Olaya-Castro08A. Olaya-Castro, C. F. Lee, F. F. Olsen, and N. F. Johnson, Phys. Rev. B 78, 085115 (2008).Heilbronner64E. Heilbronner, Tetrahedron Lett. 5, 1923 (1964).Ajami03D. Ajami, O. Oeckler, A. Simon, and R. Herges, Nature 426, 819 (2003).Yoneda14T. Yoneda, Y. M. Sung, J. M. Lim, D. Kim, and A. Osuka, Angew. Chem. 126, 13385 (2014).Zhao09N. Zhao, H. Dong, S. Yang, and C. P. Sun, Phys. Rev. B 79, 125440 (2009).Balzani08V. Balzani, A. Credi, and M. Venturi, Molecular Devices and Machines: Concepts and Perspectives for the Nanoworld (VCH-Wiley, Weinheim, 2008).Guo09Z. L. Guo, Z. R. Gong, H. Dong, and C. P. Sun, Phys. Rev. B 80, 195310 (2009).Chang10C. W. Chang, M. Liu, S. Nam, S. Zhang, Y. Liu, G. Bartal, and X. Zhang, Phys. Rev. Lett. 105, 235501 (2010).Fang16Y. N. Fang, Y. Shen, Q. Ai, and C. P. Sun, Phys. Rev. A 94, 043805 (2016).Peierls55R. E. Peierls, Quantum Theory of Solids (Oxford University Press, London, 1955).Shen13Y. Shen and B.-Y. Jin, J. Phys. Chem. A 117, 12540 (2013).Ai14Q. Ai, Y.-J. Fan, B.-Y. Jin, and Y.-C. Cheng, New J. Phys. 16, 053033 (2014).Caruso09F. Caruso, A.W. Chin, A. Datta, S. F. Huelga, and M. B. Plenio, J. Chem. Phys. 131, 105106 (2009).Fassioli10F. Fassioli and A. Olaya-Castro, New J. Phys. 12, 085006 (2010).BqAi12B.-q. Ai and S.-L. Zhu, Phys. Rev. E 86, 061917 (2012).Wang74Y. K. Wang and I. C. Khoo, Opt. Commun. 11, 323 (1974).Rey13M. del Rey, A. W. Chin, S. F. Huelga, M. B. Plenio, J. Phys. Chem. Lett. 4, 903 (2013).Dong16H. Dong, S.-W. Li, Z. H. Yi, G. S. Agarwal, and M. O. Scully, arXiv:1608.04364 (2016).Pajusalu15M. Pajusalu, R. Kunz, M. Ratsep, K. Timpmann, J. Kohler, and A. Freiberg, Phys. Rev. E 92, 052709 (2015).Huo08M. X. Huo, Y. Li, Z. Song, and C. P. Sun, Europhys. Lett. 84, 30004 (2008).Weisskopf30W. Weisskopf and E. Wigner, Z. Physik 63, 54 (1930).Knight72P. L. Knight and L. Allen, Phys. Lett. 38A, 99 (1972).Lee73P. S. Lee, Y. C. Lee, and C. T. Chang, Phys. Rev. A 8, 1722 (1973). | http://arxiv.org/abs/1706.08297v2 | {
"authors": [
"Lei Xu",
"Z. R. Gong",
"Ming-Jie Tao",
"Qing Ai"
],
"categories": [
"quant-ph",
"cond-mat.mtrl-sci",
"physics.atom-ph",
"physics.chem-ph",
"physics.optics"
],
"primary_category": "quant-ph",
"published": "20170626091723",
"title": "Artificial Light Harvesting by Dimerized Mobius Ring"
} |
Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USATU Dresden, Department of German, Applied Linguistics, D-01062 Dresden, Germany Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany School of Mathematics and Statistics, University of Sydney, 2006 NSW, AustraliaWe use an information-theoretic measure of linguistic similarity to investigate the organization and evolution of scientific fields.An analysis of almost 20M papers from the past three decades reveals that the linguistic similarity is related but different from experts and citation-based classifications, leading to an improved view on the organization of science. A temporal analysis of the similarity of fields shows that some fields (e.g., computer science) are becoming increasingly central, but that on average the similarity between pairs has not changed in the last decades. This suggests that tendencies of convergence (e.g., multi-disciplinarity) and divergence(e.g., specialization) of disciplines are in balance. Using text analysis to quantify the similarity and evolution of scientific disciplines Eduardo G. Altmann December 30, 2023 ====================================================================================== § INTRODUCTIONThe digitization of scientific production opens new possibilities for quantitative studies on scientometrics and science of science <cit.>, bringing new insights into questions such as how knowledge is organized (maps of science) <cit.>, how impact evolves over time (bibliometrics) <cit.>, or how to measure the degree of interdisciplinarity <cit.>.At the heart of these questions lies the problems of identifying scientific fields and how they relate to each other. The difficulty of these problems, and the inadequacy of a purely essentialist approach, was clear to K. R. Popper already in the 1950's <cit.>: “The belief that there is such a thing as physics, or biology, or archaeology, and that these 'studies' or 'disciplines' are distinguishable by the subject matter which they investigate, appears to me to be a residue from the time when one believed that a theory had to proceed from a definition of its own subject matter. But subject matter, or kinds of things, do not, I hold, constitute a basis for distinguishing disciplines.” <cit.>. Instead, he argued that disciplines have a cognitive and a social dimension <cit.>, i.e. they “are distinguished partly for historical reasons and reasons of administrative convenience (such as the organization of teaching and of appointments), and partly because the theories which we construct to solve our problems have a tendency to grow into unified systems.” <cit.>.On the one hand, the social dimension of scientific fields can be defined in terms of different institutions establishing stable recurring patterns of behavior <cit.>:producing and reproducing institutions such as research institutes and universities, communicative institutions such as scientific societies, journals or conferences, collecting institutions (journals, libraries), as well as directing institutions (ministries, scientific advisory boards), etc. All these institutions contribute to the formation, stabilization, and reproduction of a discipline as well as its distinction from others. On the other hand, the cognitive dimension has been specifiedin Ref. <cit.> as a number of fundamental invariants in the procedural knowledge, which lead to the categorical construction of scientific knowledge.If this process causes a change in the cognitive realm for an object of knowledge, it constitutes a certain discipline. The brief discussion above is sufficient to show that both the definition and relation between scientific fields depend on multiple dimensions (e.g., essentialist, social, and cognitive). Traditional (expert) classifications are mostly motivated by the ”subject matters” under investigation and can be associated to an essentialist view. The empirical analysis of citation networks, an approach with a long tradition in scientometry <cit.>, can be regarded as capturing the social dimension (i.e. collecting institutions in the form of journals).While citations offer valuable insights into the structure and dynamics of science, they thus reflect only one particular dimension of the relationship between publications (or scientists) largely ignoring the actual content of the scientific articles.In contrast, the cognitive dimension can be operationalized with the help of linguistic features (e.g., keywords as indicators for conceptual imprints of disciplines). The increasing availability of full text of scientific articles (e.g. of Open Access journals) provides new opportunities to study the latter aspect in the form of written language. Examples include i) the tracking of the spread of individual words (memes) <cit.> or ideas <cit.>, ii) quantifying differences in the scientific discourse between subdomains in biomedical literature <cit.> or “hard” and “soft” science <cit.>, or iii) efforts to combine citation and textual information <cit.>.In this work we advance the idea that the organization and evolution of science should be studied through different, complementary, dimensions.We add a new methodology that provides a meaningful, language-based, organization of scientific disciplines based on written text, we study how it compares to classifications obtained from experts as well as citations, and we study the temporal evolution in the relation between different scientific disciplines. More specifically, we introduce an unsupervised methodology to analyze the text of scientific articles. Our methodology is based on an information-theoretic dissimilarity measure we proposed recently <cit.> (more technically, it is a generalized and normalized Jensen-Shannon divergence between two corpora). The main advantage of this measure is that it has an absolute meaning (i.e., it is not based on relative comparisons) and it is statistically more robust than traditional approaches <cit.>, e.g. with respect to the detection of spurious trends due to rare words and increasing corpus sizes. We measure the similarity between scientific fields based on ≈ 10^7 abstracts from the last 3 decades (Web of Science database).Comparing our language analysis to a citation analysis and an experts classification, we find that the language and citation are more similar to each other but the language is even more distinct from the experts than the citation analysis. Following the relation between scientific fields over time, our language analysis reveals the scientific fields that are becomingmore central in science.However, overall (averaged over all pairs of disciplines) we find that the similarity between the language of different fields is not increasing. § DISSIMILARITY MEASURES OF SCIENTIFIC FIELDS We are interested in the general problem <cit.> of quantifying the relationship between two scientificfields i,j through the computation of dissimilarity measures D(i,j), i.e., a quantification of how different i and j are. Dissimilarity measures are symmetric D(i,j)=D(j,i), non-negative D(i,j)≥ 0, and D(i,i)=0 <cit.>.Each scientific field is defined by (at least hundreds of) papers classified by Web of Science as belonging to the same category (see Methods Sec. <ref> for details on the data). We consider dissimilarities computed based on the following three different information.§.§ ExpertsThe classification of disciplines by their relationship is as old as science itself.The most used structure is a strict hierarchical tree, as seen in the traditional departmental division of Universities.The collection of papers used here, provided by ISI Web Of Science <cit.>, provides a classification of papers according to the OECD classification of fields of science and technology <cit.>.This scheme is a hierarchical tree with scientific fields defined at 3 levels (domains, disciplines, and specialties). For instance, Applied Mathematics (a specialty) is part of Mathematics (a discipline) which is part of Natural Sciences (a domain). The natural dissimilarity measure D_exp(i,j) between two fields in this structure is the number of links needed to reach a common ancestor of i and j.For instance, considering i,j at the specialty level, D_exp can assume three different values: D_exp=1 for specialties belonging to the same discipline (e.g., Applied Mathematics and Statistics & Probability), D_exp=2 for specialties belonging to the samedomain (e..g, Applied Mathematics and Condensed Matter Physics), and D_exp=3 for the other pairs of specialties (e.g., Applied Mathematics and Linguistics). While researchers have pointed out potential issues with classification into categories of ISI Web Of Science <cit.>, it offers the most extensively available classification and remains widely used to relate articles and journals to disciplines <cit.>.§.§ Citations Another popular approach is to consider that fields i and j are more similar if there are citations from (to) papers in i to (from) papers in j <cit.>. Here we consider a dissimilarity measure D_cite(i,j) which decreases for every citation between papers in i and j, increases with every citation from ithat is not to j (and vice-versa), but that remains unchanged by the number of citations that do not involve neither i nor j.These requirements are achieved using (for ij) a symmetrized Jaccard-like dissimilarity <cit.>D_cite(i,j) = 1/2(C_i,j̅+C_i̅,j/c_i,j+C_i,j̅+C_i̅,j +C_j,i̅+C_j̅,i/c_j,i+C_j,i̅+C_j̅,i) where c_i,j are the number of citations from i to j, C_a,b̅ = ∑_t=1,t≠ b^N c_a,t, and C_a̅,b = ∑_t=1,t a^N c_t,b[Each of the two terms in Eq. (<ref>) can be interpreted as a directed Jaccard distance i → j (j → i) in the sense that we divide the number of edges that are out-links of field i (j) and in-links of field j (i) by the number of edges that are out-links of field i (j) or in-links of field j (i).].§.§ LanguageWe compare the language of fields i and j based on the frequency of words in each field using methods from Information Theory.Measuring the frequency p(w) of word w, for each field i we obtain a vector of frequencies 𝐩_i ≡ p_i(w) for w=1, …, V, where V is the size of the vocabulary (i.e. number of different words). From this, following Ref. <cit.>, the dissimilarity between two fields i and j is D_lang(i,j) = 2 H_2(𝐩_i+𝐩_j/2) - H_2(𝐩_i) - H_2(𝐩_j)1/2(2- H_2(𝐩_i) - H_2(𝐩_j) ), where H_2 (𝐩_i) = 1 -∑_w p_i(w)^2 is the generalized entropy of order 2 and the denominator ensures normalization (i.e., 0 ≤ D_lang(i,j)≤ 1).In order to increase the discrimination power and to avoid statistical biases in our estimation, we removed a list of stop wordsand included only the V=20,000 most frequent words (seeMethods Sec. <ref> for a justification). The dissimilarity (<ref>) corresponds to a generalized (and normalized) Jensen-Shannon divergencewhich yields statistically robust estimations in texts <cit.> (for details and motivation, seeMethods Sec. <ref>). The advantages of Eq. (<ref>) are twofold. On the one hand, it is well-founded in Information Theory and its statistical properties (in terms of systematic and statistical errors) are well understood <cit.> distinguishing it from other heuristic approaches. On the other hand, it has convenient properties: i) 0 ≤ D_lang(i,j) ≤ 1; ii) it depends only on the papers contained in fields i and j; and iii) it does not require training corpora.As a result, the measured distance between two fields, D_lang(i,j), has an absolute meaning. This is in contrast to alternative similarity measures <cit.>, including machine-learning approaches (e.g., topic models <cit.>) based on (un-) supervised classification of documents into coherent subgroups. Here, the main limitations stem from the fact that either i) the division into subgroups is typically based on statistically significant differences in the usage of words between the different subgroups independent of the actual effect size, or ii) the resulting distance between two fields depends on all other fields as well (e.g. the distance between 'Physics' and 'Chemistry' depends on whether one includes articles about 'Anthropology' in the classification). § RESULTS We now present and interpret results obtained computing the three dissimilarity measures (D_exp, D_cite, and D_lang) reported above for scientific fields i,j defined by papers published in different time intervals and categorized (by Web of Science) as belonging to the same specialty (e.g., Applied Mathematics), discipline, (e.g., Mathematics) or domain (e.g., Natural Sciences). §.§ Comparison of dissimilarity measuresFigure <ref> shows the three D(i,j) at the level of specialties (i,j) for the complete time interval 1991-2014. The concentration of low D(i,j) close to the diagonal shows that both the citations and language of scientific papers partially reflect the disciplinary classification done by the experts. However, visual inspection already reveals that citations and our language analysis show relationships not present in the expert classification, e.g., the low dissimilarity between Engineering and Natural Sciences (most clearly between Electrical Engineering and Physical Sciences) and between Agriculture and Biological Sciences.We start by quantifying the relationship between the three different dissimilarity measures, i.e. (D_exp, D_cite, and D_lang), across all pairs of specialties (i,j). In Tab. <ref>we report the rank-correlation between the three measures, which we obtain from ranking for each dissimilarity the pairs of (i,j) according to D(i,j). The choice of this non-parametric correlation is motivated by the fact that the range of the three measures differs dramatically (e.g. D_exp∈{0,1,2,3} and D_lang∈ [0,1]). The positive statistically-significant correlation between all pairs of D(i,j)'s confirms the visual impression described above.The correlation between citations and language is higher than the correlation with the experts classification. Remarkably, language and citations show a very similar correlation with experts but language is systematically less correlated than citations (p -value=1.8 × 10^-5 for Spearman-ρ andp-value=2.2 × 10^-5 for Kendall-τ [Obtained from 10^3 bootstrapping samples of each joint distribution P(D_exp,D_cite) and P(D_exp,D_lang), i.e. comparison of 10^6 pairs of correlation values]).We conclude that the language dissimilarity D_lang introduced here is able to retrieve the well-known relationships between disciplines in a similar extent that the (well-studied) citation analysis.We now explore how the relationship between the different dimensions depends on the different scientific fields. The results in Fig. <ref> confirm the conclusions of the aggregated analysis but shows further interesting features. First, the correlation in (D_exp,D_lang) is smaller than (D_exp,D_cite) mainly in the natural sciences. Second, while the correlation between citations and language remains largely constant, large fluctuations in the correlations between expert and citations (as well as expert and language) exist. This is seen both as the strong downward spikes and also in themanifested dependence on disciplines and domains.The titles of the specialties at the low peaks already suggest that these are specialties with interdisciplinary connections.For instance, Chemistry, Medicinal is a specialty that (according to the experts classification) belongs to the discipline Basic Medicine and to the domain Medical Science.Therefore D_exp=3 between Chemistry, Medicinal and all specialties of the Natural Sciences (in particular, for all specialties from the discipline Chemical Sciences). Instead, the dissimilarity measured by citations D_cite and language D_lang yield much smaller values revealing the proximity of Chemistry, Medicinal to the Natural Sciences thus explaining the low correlation in (D_exp,D_cite) and in (D_exp,D_lang). The central role of the natural sciences in other disciplines explains also the other spikes: computing for a list of selected specialties i=i^spikes the pairs (i,j) which suffered the largest rank change we find that 9 from the the top 10 specialties which increased most in ranks (comparing D_exp with D_lang) were from the domain Natural Sciences(5 of them from the discipline Chemical sciences, including the top 2 specialties). §.§ Hierarchical ClusteringA strict hierarchical classification of scientific fields is both aesthetically appealing and of practical use in bibliographical and document classification tasks.It also allows us to further highlight the differences in the relationship between scientific fields revealed by the different dissimilarity measures (in particular by D_lang). While D_exp is precisely based on one such hierarchical classifications, D_cite and D_lang are not.In Fig. <ref> we show the hierarchical classifications inducedby D_cite and D_lang through the computation of a simple clustering method at the level of domains and disciplines. At the top level of the 6 domains (top row in Fig. <ref>), the clustering obtained from citations and from language are very similar.In particular, both identify Engineering-NaturalSciences andHumanities-Social Science as clusters that separate from the other domains in a similar fashion.The only difference is that, based on citations, Agriculture appears more isolated while based on language this happens for Medical Science.A more detailed picture of the differences between language and citation is revealed at the level of disciplines (bottom row in Fig. <ref>). While at the first division, both citations and language create a cluster in which all disciplines of the domains Humanitiesand Social Sciences appear, further divisions show more subtle differences between the two dissimilarity measures.Remarkably, the hierarchy obtained from language creates a cluster containing all and only Humanities disciplines. In contrast, the hierarchy based on citations creates one clustering with three of the five Humanities disciplines (Lang. and Literature, Arts, and Other Humanitieswhile the two remaining ones (History & Archaeology and Philosophy, ethics, religion) are clustered together in the middle of a cluster of disciplines in Social Science.Another interesting difference between the clusterings is revealed looking at 3 disciplines of the domain Medicine: In the analysis based on Citations the minimum cluster that includes the three disciplines includes Biological sciences and Other natural sciences,while in thelanguage analysis this cluster includes additionally three related Engineering disciplines(Medical eng., Ind. biotechnology, and Envir. biotechnology).Probably the most remarkable feature of the clustering obtained by, both, citations and language is that it repeatedly clusters together related disciplines from Natural Sciences with disciplines from Engineering and Medicine (e.g., Chemical Sciences and Materials Science).This clustering, not present in the experts classification, suggests that the distinction between fundamental and applied sciences present in the expert classification has no strong effect on citations and the language of the publications. Instead, in this specific case, the citation and language analysis seem to be capturing a connection between “subject matters” that was necessarily absent from the strict hierarchical expert classification. §.§ Temporal evolutionWhile in the previous sections we looked at a static snapshot of the relation between disciplines, here we are interested in how the linguistic relationship D_lang(i,j) between pairs (i,j) of disciplines evolved over the last three decades [We work at the level of disciplines because most specialties fail to have enough publications in a single year.]. In Figure <ref> we show the temporal evolution for five out of 703 pairs (i,j), with focus on the discipline Physical Sciences, illustrating different types of dynamic patterns. On the one hand, the dissimilarity to Chemical Sciences (its most similar discipline) and Mathematics stay roughly constant over time. On the other hand, we also observe systematic trends of disciplines becoming more or less similar over time. While the proximity to Biological Sciences and Computer and information Science has steadily increased (decreased dissimilarity D_lang(i,j)) after the year 2000, the opposite trend is seen forElectrical, electronical, and information Engineering. These observations are consistent with the increasing number of biological and computational-related publications in Physics, and with a departure from the historical connections to Engineering.The observations reported above raise the question whether scientific disciplines are showing an overall tendency to become more similar to each other. In a more general context, this amounts to the question whether the purported increase in interdisciplinarity leads to a larger overlap in the language used by different disciplines.We address this question by computing, for each pair of disciplines, the mean yearly variation ν(i,j)= 1Δ t∑_t ∈Δ tD_lang^(t)(i,j)- D_lang^(t-1)(i,j) = 1Δ t(D_lang^(t_f)(i,j) - D_lang^(t_0-1)(i,j)), where the time interval Δ t≡ t_f-t_0 was usually from t_0=1991 to t_f=2014.The distribution of values of ν for all disciplines pairs (i,j) is shown at the (rightmost) box plot in the right panel of Fig. <ref>. We see that there are both positive and negative variations, consistent with our qualitative observations in the example of Physical Sciences in left panel of the Fig. <ref>. However, the average variation ⟨ν⟩≈ -0.00025 over all pairs of disciplines (i,j) is not distinguishable from zero (the null hypothesis of ⟨ν⟩=0 has a p-value=0.07 in the T-test for the mean of one sample and a p-value = 0.21 in thenon-parametric Wilcoxon test), i.e.the typical dissimilarity remains unchanged. This result suggests that, while there are systematic trends for individual pairs of disciplines, on average there is no significant increase or decrease in the interdisciplinarity for the science as a whole in the last 3 decades as measured by the language. On a more fine-grained level, however, we observe systematic trends that suggest that individual disciplines tend to become more (less) central. For this, we focus on the discipline pairs (i,j) which experienced the most extreme variation in the last decade (one standard deviation away from ⟨ν⟩). These pairs have typically |ν|⪆ 0.003 meaning that their (normalized) dissimilarity changes roughly 3% in a decade.The three disciplines that are most frequently seen in the left tail (ν < 0) are: 1-02 Computer and information sciences, 2-08 Environmental biotechnology, and 3-01 Basic medicine.The language of these disciplines became significantly more similar to the language of other disciplines in the last 3 decades, suggesting that these disciplines became more central. In contrast, the three disciplines that experienced most strongly the opposite effect (most frequently seen in the right tail, ν > 0) are: 5-01 Psychology, 2-05 Materialsengineering, and2-02 Electrical engineering, electronic engineering, informationengineering. § DISCUSSIONWe investigated the similarity between scientific fields from different perspectives: an expert classification, a citation analysis, and a newly proposed measure of linguistic similarity. We found that these different dimensions are related yet different, yielding thus new insights on the relationship between disciplines, their hierarchical organization, and their temporal evolution. Our first main finding is that the language and citation relationships between disciplines are similar and substantially different from the expert classification. This is consistent with the motivation exposed in our introduction which associated the expert classification to the (largely idealized) essentialist view of scientific disciplines, while the citation (social) and language (cognitive) were closer to dimensions that play a more important role in the relationship between fields. Interestingly, our results indicate that the language-relation of fields is more distinct from the expert classification than the citation-relation is, specially in the natural sciences.Our second main finding is that in the last 30 years the language of different scientific fields remain, on average, at the same distance from all other fields.While individual disciplines show clear trends of increasing (or decreasing) centrality, this suggests that, overall, diverging tendencies in science (e.g., specialization) are in balance with converging tendencies (e.g., multidisciplinarism).This is a remarkable quantitative finding because of the substantial changes observed in this period.The latter result demonstrates that our textual measure is of practical relevance for the study of interdisciplinarity.In recent years, interdisciplinary research achieved a central position <cit.> due to its broader relation to the concept of diversity <cit.> and its effect on impact <cit.> and performance of teams <cit.> as well as its implications for policy making, e.g. in terms of funding <cit.>. Is it just a fashion or science is really getting more and more interdisciplinary?A usual way to assess interdisciplinarity is based on citation networks using heuristic approaches <cit.> or methods from complex networks <cit.>. In line with the arguments exposed in the introduction, interdisciplinarity can be viewed through different dimensions and the cognitive dimension would be best measured using textual data. However, there are only very few works <cit.> relating textual measures with interdisciplinarity, despite the increasing availability of the text of scientific articles. In this view, the significance of our approach is that it provides a measure of interdisciplinarity based on how much the usage of words in different disciplines overlap.Finally, we hope our results and methodology will stimulate a multiple-dimensional approach in other problems related to the study of sciences, profiting from the modern availability of large (textual) databases of scientific publications that allow us to go beyond traditional bibliometric analysis <cit.>.These include, but are not limited to, the formulation of more meaningful bibliometric indicators <cit.>, the identification and prediction of influential papers and disciplines <cit.>,or the inclusion of textual information in recommending related scientific papers <cit.>. § MATERIALS AND METHODS §.§ Data and grouping of corporaWe use the Web of Science database <cit.> and explore the following information available for individual articles: citations, title, abstract, and the classification in one scientific specialty (per OECD classification <cit.>). We use all papers published between 1991 and 2014 because the number of articles with text in the abstract is substantial only after 1991 and because at the time we started our analysis 2014 was the last complete year available to us. The text of an article was built concatenation its title and abstract. The corpus representing a specialty in a given year is obtained from the concatenation of the text of all articles for that specialty in that year. The corpus for one discipline (or domain) concatenates all articles in all specialties belonging to that discipline (or domain).Our analysis is based on 19,589,166 articles for each the textual and classification information were available (92% of all articles indexed in Web of Science between 1991-2014). In our analysis we considered only citations from and to the papers in our list because only for these papers we had a reliable classification of specialties. These citations corresponded to roughly half of the ≈ 625M citations associated with these papers.§.§ Data processingFor each article in our database we performed the following steps to process the textual information:* The copyright information contained in the abstract was removed.* Title and abstract were concatenated.* The text was converted to lowercase.* Contractions were replaced by their non-contracted form. * The text was tokenized, and the nouns and verbs were lemmatized using the Natural Language Toolkit <cit.>.* Symbols (except hyphen, to avoid remove significant compound modifiers) inside tokens were replaced by white space, therefore generatingtwo or more distinct tokens.* Tokens composed by numbers or single letter were removed.* Tokens belonging to a preset stop-word list were discarded. §.§ Minimum corpus sizeWe computed D_lang using only the 20,000 most frequent word types, disregarding the scientific fields for which there was not enough data to achieve this cut-off. This choice is motivated by the slow convergence of entropy estimations (and thus D_lang) <cit.>. By choosing a fixed number of word types we reduce the effect of the remaining bias (in the estimation of D_lang) on our comparative analysis of textual dissimilarity between pairs of fields. This happens because the residual bias acts as an off-set in all cases (when a fixed cut-off is chosen) instead of affecting differently each case (as obtained if the maximum amount of data is used in each case). The bias decays with the number of word types used because the more frequent types are responsible for almost all the dissimilarity, specially for α=2 <cit.>. Using 10,000 types as a cut-off, we estimated the textual dissimilarity relative standard deviation, computed over multiple samples of the same scientific field, to be σ̂(D_lang)/D_lang≈ 1%. Our cut-off of 20,000 types is a conservative choice to ensure that σ̂(D_lang)/D_lang < 1%.§.§ Generalized Jensen-Shannon Divergence Given two texts (indexed by p and q), we define the probability distributions over all words w=1,…,Vas 𝐩 = (p_w) and 𝐪 = (q_w). An Information-theoretic measure to quantify their similarity is the generalized Jensen-Shannon divergenceD_α(𝐩, 𝐪) = H_α(𝐩+𝐪/2) - 1/2H_α(𝐩) - 1/2H_α(𝐪),based on the generalized entropy of order α (∈ℝ), where H_α(𝐩) = 1/1-α( ∑_w p_w^α - 1 ). Here, we consider a normalized similarity <cit.>D̃_α(𝐩, 𝐪) = D_α(𝐩, 𝐪)/D_α^max(𝐩, 𝐪)such that D̃_α∈ [0,1] where D_α^max(𝐩, 𝐪) = 2^1-α - 1/2( H_α(𝐩) + H_α(𝐪) + 2/1-α) is the maximum possible D_α between 𝐩 and 𝐪 assuming that the the set of symbols in each distribution (i.e., the support of 𝐩 and 𝐪) are disjoint.Note that for α = 1, Eq. (<ref>) yields the Shannon-entropy <cit.>, i.e. H_α=1(𝐩) = -∑_w p_wlog p_w, and D_α=1 is the well-known Jensen-Shannon divergence <cit.>. Ref. <cit.> shows that α=2 provides the most robust statistical measure of similarity of texts. L.D. received financial support from CNPq/Brazil through the program “Science without Borders”. We thank M. Palzenberger and the Max Planck Digital Library for providing access to the data, M. de Domenico for insightful discussions, and S. Haan and the Centre for Translational Data Science (University of Sydney) for helping with Figs. <ref> and <ref>. 60evans.2011 J. A. Evans, J. G. Foster, Science 331, 721 (2011).boerner.2003 K. Börner, C. Chen, K. W. Boyack, Annual review of information science and technology 37, 179 (2003).Shiffrin2004 R. M. Shiffrin, K. Borner, Proceedings of the National Academy of Sciences 101, 5183 (2004).boyack.2005 K. W. Boyack, R. Klavans, K. Börner, Scientometrics 64, 351 (2005).Rosvall2008 M. Rosvall, C. T. Bergstrom, Proceedings of the National Academy of Sciences 105, 1118 (2008).glaser.2017 J. Gläser, W. Glänzel, A. Scharnhorst, Scientometrics 111, 979 (2017).Wang2013 D. Wang, C. Song, A.-L. Barabási, Science 342, 127 (2013).moreira.2015 J. A. G. Moreira, X. H. T. Zeng, L. A. N. Amaral, PLoS ONE 10, e0143108 (2015).lariviere.book2014 V. Larivière, Y. Gingras, Beyond Bibliometrics (MIT Press, 2014).Noorden2015 R. V. Noorden, Nature 525, 306 (2015).popper.1952 K. R. Popper, The British Journal for the Philosophy of Science 3, 124 (1952).balsiger.book2005 P. W. Balsiger, Transdisziplinarität : systematisch-vergleichende Untersuchung disziplinenübergreifender Wissenschaftspraxis (Fink, 2005).guntau.book1991 M. Guntau, H. Laitko, World Views and Scientific Discipline Formation, R. W. Woodward, R. S. Cohen, eds. (Springer Netherlands, 1991).Garfield1964 E. Garfield, I. H. Sher, R. J. Torpie, The use of citation data in writing the history of science (Institute for Scientific Information, Philadelphia, 1964).DeSollaPrice1965 D. J. de Solla Price, Science 149, 510 (1965).kuhn.2014 T. Kuhn, M. Perc, D. Helbing, Physical Review X 4, 041036 (2014).chavalarias.2013 D. Chavalarias, J.-P. Cointet, PLoS ONE 8, e54847 (2013).lippincott.2011 T. Lippincott, D. Ó. Séaghdha, A. Korhonen, BMC bioinformatics 12, 212 (2011).evans.2016a E. Evans, C. Gomez, D. McFarland, Sociological Science 3, 757 (2016).braam.1991 R. R. Braam, H. F. Moed, A. F. J. van Raan, Journal of the American Society for Information Science 42, 233 (1991).vilhena.2014 D. Vilhena, et al., Sociological Science 1, 221 (2014).silva.2016 F. N. Silva, D. R. Amancio, M. Bardosova, L. d. F. Costa, O. N. Oliveira, Journal of Informetrics 10, 487 (2016).sienkiewicz.2016 J. Sienkiewicz, E. G. Altmann, Royal Society Open Science 3, 160140 (2016).Gerlach2016 M. Gerlach, F. Font-Clos, E. G. Altmann, Physical Review X 6, 021009 (2016).Altmann2017 E. G. Altmann, L. Dias, M. Gerlach, Journal of Statistical Mechanics: Theory and Experiment 2017, 014002 (2017).webb.2002 A. Webb, Statistical Pattern Recognition (Wiley, 2002).webOfScience http://apps.webofknowledge.comWeb of Science is a product of http://thomsonreuters.com/en.htmlThomson Reuters.bibliographicWoSClassification Working Party of National Experts on Science and Technology, OECD (2006) available at <http://www.oecd.org/science/inno/38235147.pdf>.porter.2009 A. L. Porter, I. Rafols, Scientometrics 81, 719 (2009).leydesdorff.2008 L. Leydesdorff, Journal of the American Society for Information Science and Technology 59, 77 (2008).Note1 Each of the two terms in Eq. (<ref>) can be interpreted as a directed Jaccard distance i → j (j → i) in the sense that we divide the number of edges that are out-links of field i (j) and in-links of field j (i) by the number of edges that are out-links of field i (j) or in-links of field j (i).grosse.2002 I. Grosse, et al., Physical Review E 65, 041905 (2002).Landauer2004 T. K. Landauer, D. Laham, M. Derr, Proceedings of the National Academy of Sciences 101 Suppl, 5214 (2004).Boyack2011 K. W. Boyack, et al., PLoS One 6, e18029 (2011).Note2 Obtained from 10^3 bootstrapping samples of each joint distribution P(D_exp,D_cite) and P(D_exp,D_lang), i.e. comparison of 10^6 pairs of correlation values.Sokal1958 R. Sokal, C. Michener, University of Kansas Science Bulletin 38, 1409 (1958).Note3 We work at the level of disciplines because most specialties fail to have enough publications in a single year.stirling.2007 A. Stirling, Journal of The Royal Society Interface 4, 707 (2007).uzzi.2013 B. Uzzi, S. Mukherjee, M. Stringer, B. Jones, Science (New York, N.Y.) 342, 468 (2013).wang.2015 J. Wang, B. Thijs, W. Glänzel, PLoS ONE 10, e0127298 (2015).lungeanu.2014 A. Lungeanu, Y. Huang, N. S. Contractor, Journal of Informetrics 8, 59 (2014).academy.book2004 Committee on Facilitating Interdisciplinary Research; Committee on Science, Engineering, P. P. I. of Medicine; Policy, G. A. N. A. of Sciences; National Academy of Engineering, Facilitating Interdisciplinary Research (National Academies Press, 2004).wagner.2011 C. S. Wagner, et al., Journal of Informetrics 5, 14 (2011).pan.2012 R. K. Pan, S. Sinha, K. Kaski, J. Saramäki, Scientific Reports 2, 1 (2012).sayama.2012 H. Sayama, J. Akaishi, PLoS ONE 7, e38747 (2012).sinatra.2015 R. Sinatra, P. Deville, M. Szell, D. Wang, A.-L. Barabási, Nature Physics 11, 791 (2015).omodei.2016 E. Omodei, M. D. Domenico, A. Arenas, Network Science pp. 1–12 (2016).bache.2013 K. Bache, D. Newman, P. Smyth, Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '13 (ACM Press, 2013).nichols.2014 L. G. Nichols, Scientometrics 100, 741 (2014).evans.2016 E. D. Evans, Socius: Sociological Research for a Dynamic World 2 (2016).mann.2006 G. S. Mann, D. Mimno, A. McCallum, Proceedings of the 6th ACM/IEEE-CS joint conference on Digital libraries - JCDL '06 (ACM Press, 2006).gerrish.2010 S. Gerrish, D. M. Blei, Proceedings of the 27th International Conference on Machine Learning (ICML-10), June 21-24, 2010, Haifa, Israel (2010), pp. 375–382.foulds.2013 J. Foulds, P. Smyth, Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing pp. 113–123 (2013).whalen.2015 R. Whalen, Y. Huang, A. Sawant, B. Uzzi, N. Contractor, Quantifying and Analysing Scholarly Communication on the Web (ASCW'15) (2015).achakulvisut.2016 T. Achakulvisut, D. E. Acuna, T. Ruangrong, K. Kording, PLoS ONE 11, e0158423 (2016).nltk Natural language toolkit, <http://www.nltk.org/>.cover2006 T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 2006).lin1991 J. Lin, IEEE Transactions on Information Theory 37, 145 (1991). | http://arxiv.org/abs/1706.08671v1 | {
"authors": [
"Laercio Dias",
"Martin Gerlach",
"Joachim Scharloth",
"Eduardo G. Altmann"
],
"categories": [
"cs.DL",
"physics.soc-ph"
],
"primary_category": "cs.DL",
"published": "20170627051209",
"title": "Using text analysis to quantify the similarity and evolution of scientific disciplines"
} |
[email protected] Institute for Quantum Science and Engineering and Department of Physics, South University of Science and Technology of China, Shenzhen 518055, China [email protected] for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, ChinaAlthough the emergence of a fully-functional quantum computer may still be far away from today, in the near future, it is possible to have medium-size, special-purpose, quantum devices that can perform computational tasks not efficiently simulable with any classical computer. This status is known as quantum supremacy (or quantum advantage), where one of the promising approaches is through the sampling of chaotic quantum circuits. Sampling of ideal chaotic quantum circuits has been argued to require an exponential time for classical devices. A major question is whether quantum supremacy can be maintained under noise without error correction, as the implementation of fault-tolerance would cost lots of extra qubits and quantum gates. Here we show that, for a family of chaotic quantum circuits subject to Pauli errors, there exists an non-exponential classical algorithm capable of simulating the noisy chaotic quantum circuits with bounded errors. This result represents a serious challenge to a previous result in the literature suggesting the failure of classical devices in simulating noisy chaotic circuits with about 48 qubits and depth 25. Moreover, even though our model does not cover all types of experimental errors, from a practical point view, our result describes a well-defined setting where quantum supremacy can be tested against the challenges of classical rivals, in the context of chaotic quantum circuits.Can Chaotic Quantum Circuits Maintain Quantum Supremacy under Noise? Xun Gao December 30, 2023 ====================================================================Introduction. A major challenge in the field of quantum computation is related to the question, what exactly are the computational problems quantum computers can solve but classical computers cannot? More precisely, in terms of the language of computational complexity <cit.>, is the class BQP (bounded-error quantum polynomial time) a proper superset of BPP (bounded-error probabilistic polynomial time) or P (polynomial time)? This question is related to the goal of disproving a fundamental assumption in computer science, known as the extended Church-Turing thesis (ECT) <cit.>. The ECT thesis, relevant in the asymptotic limit, asserts that, with a polynomial overhead, every physical process (classical or quantum) can be simulated by a classical probabilistic Turing machine. Although from the point of view of simulating the dynamics of quantum systems, a quantum computer seems to be capable of exhibiting computational advantages <cit.> over classical devices, there is still no rigorous proof excluding the existence an efficient classical algorithm. Moreover, it is known <cit.> that not all quantum computational models are hard with classical computation. For example, quantum circuits with only Clifford gates <cit.>, with sparse distributions <cit.>,fermionic <cit.> (matchgates), andbosonic <cit.> (linear optics with finite number of bosons) quantum computation, are classically simulable.As a step towards this goal, it is important to understand the conditions required to achieve the status of “quantum supremacy" <cit.>, where a quantum device can efficiently perform well-defined tasks beyond the capacities of state-of-the-art of classical computers, even in the absence of quantum error correction. In other words, one has to determine the minimal complexity, in terms of the number of qubits and gates, of a quantum computation model that are not achievable by a classical device. Additionally, one also needs to be sure the status of quantum supremacy of the circuit is robust against the perturbation of experimental noise; in practice, a fault-tolerant implementation of a generic quantum circuit requires an enormous amount of extra resources. By definition, quantum supremacy is a time-dependent concept, depending on the progress of high-performance computing. Practically, the status of quantum supremacy would also built on multiple complexity assumptions or conjectures. To achieve quantum supremacy, the computational tasks in question are usually special-purpose, or non-universal, minimizing the technological requirements for a realization. Notably, there are three intermediate models of quantum computation proposed for demonstrating quantum supremacy, including boson sampling <cit.>, one-clean-qubit model (or DQC1) <cit.>, and IQP (instantaneous quantum polynomial-time) <cit.> and it variants <cit.>. It has been shown <cit.> that a fault-tolerant implementation of boson sampling or IQP sampling is robust against noise in maintaining a quantum advantage. However, without error correction, the distribution of a typical quantum computation can deviate significantly from the ideal implementation, when independent noise is applied to each qubit <cit.>.Recently, the Google team proposed <cit.> the use of cross entropy to characterize quantum supremacy. They focused on a computation model based on chaotic quantum circuits, where random quantum gates are applied. There, the authors argued that quantum supremacy is preserved when a certain amount of noise is included. However, it has been shown <cit.> that IQP circuits, when subject to experimental noise, may become classically simulable and hence lose the status of quantum supremacy. Furthermore, a threshold theorem for quantum supremacy has been established <cit.>, showing that the threshold value for quantum supremacy is significantly higher than that of universal fault-tolerant quantum computation. This result is consistent with that of Ref. <cit.>, which showed that a classical algorithm simulating noisy IQP circuits fails after error correction is implemented. However, the applicability of the threshold theorem relies on the assumption ofthe ability of performing quantum error correction.The central question of interest is “can chaotic quantum circuits maintain quantum supremacy under noise?", even in the absence of quantum error correction. A major assumption made in Ref. <cit.> is that, in terms of the cross entropy, the correlation between a family of chaotic quantum circuits and polynomial-time classical algorithms is negligible. The authors in Ref. <cit.> managed to find a bayesian-based classical algorithm exhibiting the desired behavior, i.e., with an exponential runtime. However, as we shall argue, there exist non-exponential classical algorithms for a family of noisy chaotic circuits. Therefore, the proposal of Ref. <cit.> suggesting the demonstration of quantum supremacy with 48 qubits becomes questionable in our setting.To address the open question, here we consider a family of chaotic quantum circuits, where noisy Pauli-X and Pauli-Z gates are probabilistically applied to simulate experimental noises. The goal is to look for non-exponential classical algorithms for simulating the outputs of these chaotic circuits. Our results indicate that (i) in order the bound the average l_1 norm of these chaotic circuits, a classical algorithm can run in a polynomial time only. (ii) In order to bound the cross entropy, the classical algorithm runs at most in quasi-polynomial time. Our results are established under the same assumption made in Ref. <cit.>, namely the outputs of the n-qubit chaotic quantum circuits obey the Porter-Thomas distribution <cit.>, which is a characteristic of quantum chaos. In fact, we can relax the condition to that the second-moment of the output distribution scales as O(1/2^n). A summary of our main results is included after we introduce all the necessary background. However, we do not claim that all chaotic quantum circuits can be simulated efficiently with the proposed classical algorithm. In addition to the need of justifying the assumption mentioned, the family of chaotic circuits considered in this work is a sub-class of chaotic quantum circuits, and the type of experimental noise considered is a toy model. Furthermore, in order to achieve the bounds for a given noise strength ε, there is a factor of n^O(1/ε) in the runtime, which could become a very large number when ε→ 0.Nevertheless, our model can be taken as a well-defined platform for benchmarking the performance of an experimental implementation of chaotic quantum circuits against the challenge of the non-exponential classical algorithms. At the time of writing, it is possible that as many as 40 to 50 qubits might be already available in research centers from the industry. We propose to test these qubits with random samples of the quantum circuits defined in this work, and compare it with the classical algorithm described below using a reasonably-fast classical workstation. On the other hand, we do not claim our classical algorithm to be optimal; there are still rooms to improve the classical algorithm, e.g., by combining compressed sensing or machine learning methods to optimize the random-sampling subroutine. An optimization of our algorithm becomes interesting when its performance can be directly compared with a realistic quantum device.Finally, perhaps the existence of non-exponential classical algorithms for simulating quantum chaotic processes under noise is by itself an interesting direction for a further investigation, given that classical chaotic systems are notoriously difficult to simulate. Quantum chaotic behaviors include a rapid delocalization of quantum states. Consequently, compared with the overlap of a pair of quantum states generated by Hamiltonians deviated from each other, it decreases exponentially. Therefore, our results suggest that in the presence of noise, the corresponding overlap may not decrease exponentially anymore. This result is consistent in the asymptotic limit where the noise is so strong that the outcomes of all quantum circuits, chaotic or not, become identical to the uniform distribution. §—PART I: SETTING THE STAGE—Three different sources of bit-strings. In this work, we are considering outputs in the form of n-bit strings of 0's and 1's, i.e., x ∈{ 0,1}^n, coming from three different sources, namely (i) ideal quantum circuits, (ii) noisy quantum circuits modeling experimental implementations, and (iii) randomized classical algorithms. We denote the corresponding probability distributions as, respectively, (i) P_ qc( x ) = | ⟨ x |U| ψ _ in⟩|^2,(ii) P_exp( x ) = ⟨ x |ℰ( | ψ _ in⟩⟨ψ _ in|)| x ⟩,(iii) P_ cl( x ).Here the quantum state | ψ_ in⟩ is the initial state of the quantum circuit, the operator U is the unitary transformation generated by the quantum circuit, and the super-operator ℰ(·) represent some noisy channel, modeling an experimental implementation. We shall describe the classical algorithm in a later section (see Eq. (<ref>)). In fact, instead of a single circuit, we shall consider an ensemble of quantum circuits, where each circuit is labeled by a string x', i.e., U_x'. Therefore, it is necessary to compare the corresponding (conditional) probabilities for each circuit, i.e., P_qc( x|U_x'),P_exp( x|U_x'),P_cl( x|U_x') .Later, we shall distinguish two types of classical algorithms. The first type aims to obtain the numerical values of P_cl( x|U_x'). In the second type, we aim to produce a classical distribution described by P_cl( x|U_x'). The relationship between them is rather subtle.Vector norm. In order to characterize the performance of simulation from one source to another, it is common (see e.g. Refs. <cit.>) to consider the l_1-norm between two distributions, e.g., Λ≡P_exp - P_cl_1 = ∑_x ∈{0,1}^n| P_exp( x ) - P_cl( x )| ,In particular, subject to a couple of reasonable conjectures, if there exists a classical algorithm that can sample any IQP (instantaneous quantum polynomial time) circuit to a small constant in the l_1 norm, then the polynomial hierarchy collapses to the third level <cit.>, which is widely believed to be implausible.Cross entropy. Alternatively, it has been suggested <cit.> that one may consider the cross entropy S_c between two distributions, e.g., S_c( P_exp,P_qc) ≡- ∑_x ∈{0,1}^nP_exp( x )log P_qc( x ).The operational meaning of the cross entropy is as follows: suppose we generate a sequence of many, m ≫ 1, independent bit-strings x's from each source, e.g. from an ideal quantum circuit s_ qc={ x_1^ qc,x_2^ qc,...,x_m^ qc}, and experiment s_ exp={ x_1^ exp,x_2^ exp,...,x_m^ exp}. The probability of obtaining the sequence s_ qc is given by the product, ( s_qc) = ∏_i = 1^m P_qc( x_i^qc) . Now, we can also ask the question: how likely does the same quantum circuit generate the sequence s_ exp in the experiment? The corresponding probability is given by ( s_exp) = ∏_i = 1^m P_qc( x_i^exp) . (Note that we have to use P_qc instead of P_exp.) After applying the central-limit theorem (see e.g. Ref. <cit.>), the probability ( s_exp) can be determined by the cross entropy in Eq. (<ref>) through, ( s_exp) = e^ - mS_c( P_exp,P_qc) . Shannon versus cross entropy. In the noise-free (or ideal) limit, we expect that the two probabilities become the same, i.e., ( s_exp)→( s_qc), which means that the cross entropy S_c becomes identical to the Shannon entropy, S( P_qc) ≡- ∑_x ∈{0,1}^nP_qc( x )log P_qc( x ),i.e., S_c( P_exp,P_qc)→S( P_qc). More generally, one may quantify the performance of the experiment by the difference of the two quantities, Δ_ exp≡| S( P_qc) - S_c( P_exp,P_qc)|.In a similar way, we may consider quantifying the performance of a classical algorithm byΔ _cl≡| S( P_qc) - S_c( P_cl,P_qc)|. Note that for the purpose of demonstrating quantum supremacy, it is not necessary for Δ_ exp to be small. Instead, one should make sure Δ _ exp to be significantly less than Δ _ cl, for all possible classical algorithms.Constraint of quantum supremacy. Now, applying the triangle inequality to Δ _ cl, we have, Δ _ cl ⩽ Δ_ exp+ Δ_S ,where Δ_S≡| S_c( P_exp,P_qc) - S_c( P_cl,P_qc)|is the main quantity we shall focus on. This inequality implies that if there exists an efficient classical algorithm such that Δ _S can be bounded by a small constant, then the performance of the classical algorithm, in terms of simulating the ideal quantum circuit, cannot be much worse than that of the experiment. In other words, for the cases where the values of Δ_S are small, the value of cross entropy cannot be used to justify the status of quantum supremacy in an experimental implementation. Of course, a similar argument is applicable to the case using the l_1 norm (see Eq. (<ref>)). Summary of main results.In this work, we ask the following question: given an ensemble of chaotic quantum circuits, how well can classical algorithms approximateexperimental implementations of the circuits? In Ref. <cit.>, the performance of a classical algorithm simulating the chaotic quantum circuit was exponentially close to the performance ofusing a simple uniform distribution. The result gives a lower bound on the performance of classical algorithms, but it does not exclude the possibility of the existence of better classical algorithms, which is the main task of this work.More precisely, the authors of Ref. <cit.> employed a measure defined by Δ _H( P_cl) ≡S_0 - S_c( P_cl,P_qc),where S_0 is the cross entropy when a uniform distribution is compared with the quantum circuit. It was shown that if a classical algorithm can perfectly reproduce the distribution of the quantum circuit, i.e., P_cl→P_qc, then Δ_H ( P_qc) becomes unity, i.e., Δ _H( P_qc) → 1. However, it approaches zero Δ _H→ 0, if we replace it with a uniform distribution, i.e., P_cl( x ) → 1/N.The central idea of Ref. <cit.> is that if it was true that for a class of quantum circuits, the value of Δ _H( P_qc) for all polynomial-time classical algorithms is significantly smaller than that the values Δ _H( P_exp) resulting from experimental implementations, then the status of quantum supremacy can be achieved. In our context, one needs to show that Δ _S = | Δ _H( P_exp) - Δ _H( P_cl)|,to be very different from zero, comparing with experimental implementation and the best classical algorithm.Here we consider a class of chaotic quantum circuits, where the imperfection in their experimental implementations are modeled by random bit-flip errors and depolarizing noise. Our goal in this work is to show that there exists (quasi) polynomial-time classical algorithms that can simulate a class of chaotic quantum circuits under noise. In other words, our results indicate that in some situations, it can happen that a polynomial-time classical algorithm leads to a small Δ _S.More precisely, our classical algorithm can output the values of the classical probabilites such that, on average, the l_1 norm with the experimental simulation P_exp - P_cl_1 is bounded by a given small constant δ. The runtime of the classical algorithm scales as O( (n + m)^L/δ ^2 ), where n is the number of qubits in the quantum circuits, m=poly(n) is the number of ancilla qubits. Here L = O(log (α / δ ^2 )/ε ), where ε is the strength of the noise and α =O(1).The classical algorithm can then be employed to produce a classical distribution of bit strings, approximating those from the experimental simulation. In particular, if we randomly choose a quantum circuit in our ensemble, with a probability at least 1-2/k^2 for any k>1, the l_1 norm between the probability distribution generated by the classical algorithm (denoted as “Alg") and that of experimental simulation is bounded by Alg - P_exp_1⩽ 4kδ /( 1 - kδ). Furthermore, on average, the difference in cross entropy in Eq. (<ref>) is bound by O( δ√(( nlog 2 + γ)^2 + π ^2/6)), where γ = 0.57721 ... is the Euler constant. In Ref. <cit.>, n=48 is taken to be the “supremacy frontier".The assumption involved in this work is the chaotic nature of the ensemble of the quantum circuits. Starting from a chaotic quantum circuit of n qubits (to be defined later), the other members in the ensemble are generated by applying single-qubit Pauli-Xgates to each of the local gates in the original circuit. We assume these quantum circuits remains chaotic.Overview on technical details. In the remaining of this work, we shall first define chaotic quantum circuits, which involves an requirement that the second moment of the probability distribution be bounded by O(1/2^n). The main idea is inspired by the techniques in Ref. <cit.> and Ref. <cit.>. In addition, we start with a circuit decomposition of a chaotic quantum circuit based in the universal set {H,Z( α),CZ} .In this way, we are able to encode the original circuit as one of the branches in a IQP circuit using the idea in Ref. <cit.> which maps an ensemble of random circuit to an IQP circuit. Each circuit in the other branches represent a variant of the original circuit subject to some additional Pauli-X gates.To model noise, a depolarizing channel is applied to each of the qubits in the IQP circuit. In this way, we show that effectively each of the local gates are applied with a bit-flip error in the chaotic quantum circuits. Further analysis are performed by including several results in Ref. <cit.>, which proposed the use of Fourier components in IQP circuits to simplify calculations of probability distributions. In particular, it is shown that the effect of including depolarizing noise on IQP circuits is to reduce the size of the Fourier components, which makes it possible to approximate the probability distribution through truncation of the small Fourier components. Finally, we found that the chaotic nature of the quantum circuits implies that the difference in the cross entropy can be bounded in a similar way as the l_1 norm. However, bounding the cross entropy difference to a small constant is more demanding than that for the l_1 norm. The former case requires a quasi-polynomial time algorithm. Before we get started on the technical details, we remark that most of the technical work presented here is devoted to provide a rigorous proof on the lower bound of the performance of the proposed classical algorithms. In practice, we expect the average performance should be better than the lower bound presented here.Furthermore, one may consider performing additional (including heuristic) optimization through otherclassical methods to further improve the efficiency. For example,one may apply the technique of compressed sensing to reduce the number of sampling on the Fourier components to reduce computational costs. Therefore, the classical algorithms outlined in this work provide a framework for comparing the performance of near-future experimental demonstrations of quantum supremacy based on chaotic quantum circuits. § —PART II: CHAOTIC QUANTUM CIRCUITS—Chaotic quantum circuits. In this work, we are interested in some chaotic quantum circuit U_0 applied to a fixed initial state, namely |ψ_ in⟩≡|+⟩ ^⊗ n = H^⊗ n| 000...0⟩ , of n qubits, where H ≡|+⟩⟨ 0 | + |-⟩⟨ 1 | is the Hadamard gate and |±⟩≡( | 0 ⟩±| 1 ⟩)/√(2). Here chaotic quantum circuits <cit.> correspond to the class of quantum circuits where the values of P_ qc(x | U_0)≡| ⟨ x |U_0| ψ _in⟩|^2 obey the Porter-Thomas distribution (i.e., the distribution of the distribution), N^2 exp(- NP) for N=2^n. In fact, we will only need a weaker condition that the second moment is O(1/N), i.e., R_0≡∑_x ∈{0,1}^nP_qc( x|U_0) ^2 ⩽ α _0/2^n ,for some constant α_0. Note that for quantum circuits obeying strictly the Porter-Thomas distribution, ⟨P^2⟩≡ N^2∫_0^∞e^ - NPP^2dP =2/N ,implying that Q_0= 2/2^n.Circuit decomposition. In addition, we consider the chaotic quantum circuit U_0described by a universal gate set {H,Z( α),CZ}, where Z( α) ≡| 0 ⟩⟨ 0 | + e^iα| 1 ⟩⟨ 1 | covers all possible values of α, and CZ ≡ I - 2| 11⟩⟨11|. We further require that each single-qubit gate is always realized by the productJ( α) ≡ HZ( α),which is possible because any single qubit unitary gate U_qubit can always be decomposed into at most four applications of J(α), i.e., U_qubit = e^iξJ( 0 )J( α)J( β)J( γ) = e^iξZ( α)X( β)Z( γ). Consequently, such a chaotic quantum circuit can be generated by a post-selection in the setting of measurement-based quantum computation. In particular, for a general qubit state | ψ⟩ and another qubit in state |+⟩, when we apply a CZ gate to correlate them followed by applying J( α) ≡ HZ( α) to the first qubit, we obtain the following state before measurement (after a swapping the two qubits for clarity),| ψ _J⟩ = 1/√(2)( J( α)| ψ⟩⊗| 0 ⟩+ XJ( α)| ψ⟩⊗| 1 ⟩),where | ψ _J⟩≡ (J( α)⊗ I)CZ( | ψ⟩|+⟩) and X ≡| 1 ⟩⟨ 0 | + | 0 ⟩⟨ 1 |. Hence, for example, the J(α) gate is applied if the state |0⟩ is obtained in a measurement. Ensemble from chaotic quantum circuits. However, we are not interested in physically generating such a quantum circuit by post-selection, which is highly inefficient when scaled up. Instead, the purpose here is to take the chaotic quantum circuit U_0 as the “seed" for describing an ensemble of quantum circuits generated by applying bit-flip gates after each J-gate in the original quantum circuit U_0. For example in Eq. (<ref>), thequantum circuit obtained by the result |1⟩ is the one with a Puali-X gate (or bit-flip) applied after the J(α) gate. Generally, for a quantum circuit U_0 with m single-qubit J gates, there are exactly 2^m different quantum circuits in the ensemble. We label each quantum circuit U_x' by a m-bit string x'=x'_1 x'_2 x'_3⋯ x'_m .The original one is always x'=000 ⋯ 0, i.e., U_0 = U_000 ⋯ 0. Furthermore, if the j-th bit (x'_j) is non-zero, it means that a Pauli-X gate is applied to the j-th J gate in U_0.Robustness of chaotic circuits. The only assumption behind this work is thatan application of these Pauli-X gates does not change the chaotic nature of the original quantum circuit U_0, at least on average. More specifically, given the condition in Eq. (<ref>), we assume it still holds that for some constant α,1/2^m∑_x'∈{0,1}^mR_x'⩽α/2^n ,where R_x'≡∑_x ∈{0,1}^nP_qc( x|U_x')^2 is the second moment of circuit U_x'. In fact, we shall see that such an assumption is equivalent to the assumption made in Ref. <cit.> for the case of random IQP circuit. Furthermore, a similar but different approach of generating random ensemble of quantum circuits have been proposed <cit.>, where the last set of Hamdamard gates are not applied to the ancilla qubits before measurement. There, numerical evidence was provided for showing the convergence of the probability distribution towards the Porter-Thomas distribution. For the sake of argument, here we keep the condition in Eq. (<ref>) as an assumption.§ —PART III: UNIVERSAL IQP CIRCUITS — Encoding unitaries in IQP circuit.Now, we can extend the argument in Eq. (<ref>) to the whole quantum circuit of U_0, in the usual way described in measurement-based quantum computation. However, the difference is that it is sufficient to consider the procedure in an non-adaptive manner as follows. Suppose there are m+n qubits initialized in the |0⟩ state. Then, we apply Hadamard gates to all the qubits, i.e., H^⊗ (m+n )| 0^m+n⟩. Next, we apply a diagonal gate D, which contains three sets of commuting gates: (i) all the CZ gates in the original quantum circuit U_0 and (ii) the CZ gates with the m ancilla qubits generating the J gates, (iii) the Z gates associated with each J gate. For example, the case of Eq. (<ref>) corresponds to the case D = ( Z(α) ⊗ I )CZ, with | ψ⟩= | 0 ⟩.Finally, when this unitary gate, U_ IQP≡ H^⊗ (m+n )D H^⊗ (m+n ) .is applied to the initial state | 0^n + m⟩ (without loss of generality), we have U_IQP| 0^n + m⟩= 1/2^m/2∑_x' ∈{0,1}^mU_x'| ψ_ in⟩⊗| x' ⟩ ,where | ψ _in⟩ is defined in Eq. (<ref>), and U_x' is the Pauli-X variants of the original quantum circuit U_0 discussed previously. It is now evident that any member U_x' in the ensemble of chaotic quantum circuits is an instance after a partial measurement of a IQP circuit. Of course, one may deterministically pick any choice of the quantum circuit U_x' for an experimental implementation. Note that we have the following relation: P_IQP( x,x') = P_qc( x|U_x')P( x'),or explicitly,P_IQP( x,x') = 1/2^m| ⟨ x |U_x'| ψ _in⟩|^2 = 1/2^mP_qc( x|U_x'),since P( x') = 1/2^m and P_qc( x|U_x') = | ⟨ x |U_x'| ψ _in⟩|^2.Hadamard transform and IQP circuits. Note that the elements of the diagonal gate D can be sorted out readily as a function,f( x,x') ≡⟨x,x'|D| x,x'⟩ ,of the system and ancilla qubits. Given the initial state | 0^n + m⟩, the Hamdamard gates produce a uniform superposition of quantum states, which means that DH^⊗ (n + m)| 0^n + m⟩= 2^ - (n + m)/2∑_x,x'f( x,x')| x,x'⟩. Applying to any computational basis, H^⊗ (n + m)| x,x'⟩= ∑_y,y'(- 1)^x · y + x' · y'/2^(n + m)/2| y,y'⟩,which implies that the probability,P_IQP( x,x') ≡| ⟨x,x'|U_IQP| 0^n + m⟩|^2 ,from the IQP circuit is given by,P_IQP( x,x') = 1/2^n + m| ∑_y,y'f( y,y')(- 1)^xx' · yy'|^2 ,where y ∈{ 0,1}^n, y' ∈{ 0,1 } ^m, and xx' · yy' ≡ x · y + x' · y'.Adding noises to quantum circuits. Let us now turn our attention to the problem of modeling the noise in experimental implementations of these chaotic quantum circuits. For the sake of argument, we consider adding two types of noises to the quantum circuits U_x', namely (i) bit-flip noise X to each single-qubit gate,ℰ_bf( ρ) = ( 1 - ε /2)ρ+ ( ε /2)Xρ X,and (ii) a depolarizing channel, ℰ_dp( ρ) = ( 1 - ε)ρ+ ε I/2,for each of the system qubits just before a final quantum measurement.In fact, applying the bit-flip errors for each single-qubit gate in the ensemble is equivalent to applying a depolarizing channel before the measurement in the corresponding IQP circuit. To justify this point, let us look at Eq. (<ref>) again. If a depolarizing channel is applied to the ancilla qubit before measurement, the final state after measurement is of the form (apart from a normalization factor):[ ( 1 - ε/2)ρ _0 + ε/2ρ _1]Π _0 + [ ( 1 - ε/2)ρ _1 + ε/2ρ _0]Π _1 ,where Π _0≡| 0 ⟩⟨ 0 | and Π _1≡| 1 ⟩⟨ 1 | are projectors, ρ _0 = J( α)| ψ_ini⟩⟨ψ_ini|J( α)^†, and ρ _1 = Xρ _0X. Alternatively, we can also express it as,ℰ_bf(ρ _0) ⊗Π _0 + ℰ_bf(ρ _1) ⊗Π _1 . Therefore, we have mapped the ensemble of noisy problem of chaotic quantum circuits to the problem of an IQP circuit associated with depolarizing noise applied to each qubit before measurement. We denote the corresponding probability of obtaining the string x byP_exp( x|U_x') ≡⟨ x |ℰ_x'( ρ _ini)| x ⟩ ,where ℰ_x' denotes the channel describing the noisy quantum circuit labeled by x'. See appendix for an example. As a result, the probability P_IQP^ε( x,x') of getting the strings x and x' on a noisy IQP circuit is given by, P_IQP^ε( x,x') = 1/2^mP_exp( x|U_x'),which is reduced to Eq. (<ref>) when ε= 0, i.e., the noise-free limit.§ —PART IV: FOURIER ANALYSIS— Fourier representation. We shall discuss how to evaluate the probabilities, P_IQP(x,x') ≡| ⟨x,x'|U_IQP| 0^n + m⟩|^2 ,of an IQP circuit classically. Here x∈{0,1}^n is an n-bit string associated with the system qubits, and x'∈{0,1}^m is for the m ancilla qubits. For this purpose, we need to apply the Fourier analysis to the function f(x,x') in the IQP circuit. Let us define χ _s,s'( x,x') ≡(- 1)^s s' · x x' .Any function f(x,x') can be expanded by χ _s,s' by f( x,x') = ∑_s,s'f̂( s,s')χ _s,s'( x,x'), where f̂( s,s') ≡1/2^n + m∑_x,x'f( x,x')(- 1)^s s' · x x'is called the Fourier coefficient of f(x,x'). As a special case, f̂( 0,0) = ∑_x,x'f( x,x') /2^n + mis simply the average value of f(x,x'). Comparing Eq. (<ref>) and (<ref>), we may also interpret the probability as given by the absolute square of the Fourier coefficient, i.e., P_IQP( x,x') = | f̂( x,x') |^2 . Classical approximation of Fourier coefficients. In the same way, the probabilities of the IQP circuit can also be expanded by its Fourier coefficients,P̂_IQP( s,s') = 1/2^n + m∑_x,x'P_IQP( x,x')(- 1)^s s' · x x' .Now, from Eq. (<ref>), we can choose to expand only one of the Fourier coefficients for P_IQP( x,x'), i.e., P_IQP( x,x') = 2^ - (n + m)∑_yy'f^*( y,y')f̂( x,x')(- 1)^xx' · yy', which gives P̂_IQP( s,s') = 1/2^2(n + m)∑_y,y' f_y,y'^*f_y + s,y' + s' ,where f_x,x'≡ f( x,x'). Consequently, one can approximate the value of the Fourier coefficient P̂_IQP( s,s') by uniformly sampling the product of the functions f_y,y'^*f_y + s,y' + s' (and taking the real part at the end). From the standard Chernoff bound, for any η>0, if we take an average value from T_run=O(1/η^2),independent trials, the approximating value, denoted by Q̂_cl( s,s'), is exponentially accurate with a high probability, i.e., | Q̂_cl( s,s') - P̂_IQP( s,s') | ⩽η 2^ - ( n + m) .Effect of noise on Fourier coefficients. In the following, we shall show that the effect of applying the depolarizing channel in Eq. (<ref>) on all qubits in the IQP circuit before measurement is to change each of the Fourier coefficient by some factor, i.e., P̂_IQP( s,s') →P̂_IQP^ε ( s,s' )≡( 1 - ε)^| ss'| P̂_IQP( s,s'), where | ss'| is the Hamming weight of the string ss', i.e., the number of 1's. This result was stated without proof in Ref. <cit.>; the following discussion provides a physical picture of this result and can be potentially extended for a generalization for further applications. As a result, the probability distribution of the IQP circuit under noise is given by,P_IQP^ε( x,x') = ∑_s,s'( 1 - ε)^|ss'|P̂( s,s')(- 1)^ss' · xx' .First, for any given density matrix ρ, the operation of measurement ℳ and depolarizing channel ℰ_dp commute, i.e.,ℳ( ℰ_dp( ρ)) = ℰ_dp( ℳ( ρ)) = ( 1 - ε)ℳ( ρ) + ε I/2. Let us now consider a general n-qubit quantum state ρ after a quantum measurement given by the following form, ℳ( ρ) = ∑_x ∈{0,1}^np( x )| x ⟩⟨ x |,where p( x ) = ⟨ x |ℳ( ρ)| x ⟩. The same quantity can be written as,ℳ( ρ) = ∑_x ∈{0,1}^np( x ) Π_x,where Π_x ≡Π _x_1Π _x_2⋯Π _x_n, and Π _x_k≡| x_k⟩⟨x_k| for x ∈{ 0 ,1 } is a projector for the k-th qubit, a notation already introduced in Eq. (<ref>).Next, we consider the expansion in Eq. (<ref>) as a vector space, and introduce a Hadamard matrix ℋ in the same space, i.e., for k∈{ 0,1}, ℋ Π _k = ( Π _0 + (- 1)^k Π _1)/√(2).Note that we have the relationship,ℋ^⊗ n Π _x = 1/√(2^n)∑_s ∈{0,1}^n(- 1)^s · x Π _s ,which is similar to the standard Hadamard matrix in normal quantum circuits (see Eq. (<ref>)).Furthermore, it is also true that, ℋ^2 = ℐ (identity). Consequently, from the fact that, ℳ( ρ) = ℋ^⊗ n ℋ^⊗ nℳ( ρ), we haveℳ( ρ) = √(2^n)∑_s ∈{0,1}^nP̂( s ) ℋ^⊗ nΠ _s ,where ℋ^⊗ nΠ _s = ℋΠ _s_1⊗ℋΠ _s_2... ⊗ℋΠ _s_n.Let us consider applying the depolarizing channel to one of the qubits, ℰ_dp( ℳ( ρ)). One can readily show that ℰ_dp( ℋΠ _0) = ℋΠ _0 and ℰ_dp( ℋΠ _1) = ( 1 - ε)ℋΠ _1. Therefore, for each string s=s_1s_2 ⋯ s_n, we obtain a factor of ( 1 - ε)^s_1( 1 - ε)^s_2⋯( 1 - ε)^s_n≡( 1 - ε)^|s| for each Fourier coefficient P̂( s ), which completes the proof.§ —PART V: CLASSICAL APPROXIMATIONS— Approximating the noisy circuits. We are now ready to approximate the noisy quantum circuits based on the family of chaotic quantum circuits encoded in the IQP circuit(see Eq. (<ref>)), in terms of the l_1 norm. In Eq. (<ref>), we have seen that one can approximate each Fourier component P̂_IQP( s,s') of the IQP circuit with a high probability. The question is how well can we approximate the probabilities of individual quantum circuit P_qc( x|U_x') or the experimental implementation P_exp( x|U_x')? The starting point is to calculate the quantities classically, P_cl( x,x') ≡∑_|ss'| ⩽ LQ̂^ε_cl( s,s')(- 1)^ss' · xx' ,which contains Fourier coefficients,Q̂_cl^ε( s,s') ≡( 1 - ε)^|ss'| Q̂_cl( s,s'),that is associated with a Hamming bound |ss'| less than a given constant L. The number of terms in P_cl( x,x') is bounded by ∑_k = 0^L n+mk⩽(n+m)^L + 1.Here each term Q̂_cl( s,s') is obtained by the classical sampling method discussed below Eq. (<ref>).In the light of the relation shown in Eq. (<ref>), we define P_cl( x|U_x') ≡2^mP_cl( x,x'),to be the classical approximation for the experimental implementation of U_x' under our noise model. For a given quantum circuit U_x', the deviation between the classical algorithm and experimental implementation can be quantified by the l_1 norm, i.e., P_cl( x|U_x') - P_exp( x|U_x')_1 = ∑_x ∈{0,1}^n| P_cl( x|U_x') - P_exp( x|U_x')|. We are interested in bounding the average performance of the classical algorithm in terms of the mean value of the l_1 norm over all of the quantum circuits in the ensemble, Λ _av≡ (1/2^m)∑_x' ∈{ 0,1}^mΛ _x' ,where Λ _x'≡P_cl( x|U_x') - P_exp( x|U_x')_1 . Bounding the average of the distributions.Using Eq. (<ref>), the average value Λ _av can be expressed as the l_1 norm between difference between the probabilities of the IQP circuit under noise P_IQP^ε( x,x') (see Eq. (<ref>)) and the classical algorithm P_cl( x,x') approximating it, i.e., Λ_av = P_cl( x,x') - P_IQP^ε( x,x') .The goal of this section is to show there exists a polynomial-time classical algorithm such that Λ_av can be bounded by a small constant δ, i.e., Λ _av⩽ cδ ,for some constant c.For any vector, x=(x_1,x_2,..,x_n)^T, the l_1 norm, 𝐱_1 = ∑_i = 1^n | x_i|, can always be bounded by the l_2 norm, 𝐱_2 = (∑_i = 1^n |x_i| ^2)^1/2, i.e., 𝐱_1⩽√(n)𝐱_2. Therefore, we can write, Λ _av^2 ⩽2^n + m∑_x,x'( P_cl( x,x') - P_IQP^ε( x,x'))^2 .The right-hand side can be replaced by the corresponding Fourier coefficients using Parseval's identity, which gives Λ _av^2 ⩽2^2(n + m)∑_s,s'( P̂_cl( s,s') - P̂_IQP^ε( s,s') )^2 .Note that there is an extra factor of 2^n+m. Recall in Eq. (<ref>) that P̂_cl( s,s') = Q̂_cl^ε( s,s') for the Hamming distance of the string ss' to be less than a constant L; otherwise P̂_cl( s,s') = 0. Recall also in Eq. (<ref>) that there are O(n+m)^L non-zero terms. Therefore, we divide the summation into two parts, i.e.,Λ _av^2⩽ Ω _1 + Ω _2 ,where the first term is given by, Ω _1≡2^2(n + m)∑_|ss'| ⩽ L( P̂_cl( s,s') - P̂_IQP^ε( s,s') )^2 ,and the second term can be written as,Ω _2≡2^2(n + m)∑_|ss'| > LP̂_IQP^ε( s,s')^2 , which (using Eq. (<ref>)) can be bounded by Ω _2⩽2^2(n + m)( 1 - ε)^2L∑_s,s'P̂_IQP( s,s')^2 . Let us further investigate the two Ω terms separately; the following analysis is similar to the one performed in Ref. <cit.>; we provide the details in terms of our notations. For the term Ω_1, since ( 1 - ε)^|ss'|⩽ 1, we have | P̂_cl( s,s') - P̂_IQP^ε( s,s') | ⩽ | Q̂_cl( s,s') - P̂_IQP( s,s') |.Together withEq. (<ref>), we conclude that, Ω _1≤η ^2((n+m)^L+1) .In order to have it bounded by a small constant, e.g., Ω _1≤δ ^2, we need to make, η= O(δ /(n + m)^L/2),which requires the runtime of the Monte Carlo algorithm to scales as (see Eq. (<ref>))T_run = O( (n + m)^L/δ ^2 ).The value of L is determined by the second term.For the second term, we can now go back to the standard basis, i.e., Ω _2⩽2^n + m( 1 - ε)^2L∑_x,x'P_IQP( x,x')^2 .Using Eq. (<ref>), we have Ω _2⩽( 1 - ε)^2L2^n - m∑_x,x'P_qc( x|U_x')^2 , which implies that,Ω _2⩽( 1 - ε)^2L2^n - m∑_x'R_x'⩽α( 1 - ε)^2L ,with the use of the assumption made in Eq. (<ref>). Unless α is of order δ^2 or smaller, we need to make L to be sufficiently large, so that α( 1 - ε)^2L⩽αe^ - 2 ε L = δ ^2, and hence L = O(log (α/δ ^2)/ε ).As a result, the average l_1 norm Λ _av can be bounded by the classical polynomial-time algorithm to a small constant, i.e., ∑_x,x'( P_cl( x,x') - P_IQP^ε( x,x'))^2⩽c^2 δ^2 /2^n + m ,and henceΛ _av⩽ cδ from Eq. (<ref>).Bounding the norm for each circuit. We have shown that the average value of the l_1 norms, Λ_av, can be bounded by a small constant by a polynomial-time classical algorithm. Next, we can further ask the following question: if we randomly pick one of the quantum circuits, U_x', what is the probability that the classical algorithm fails to maintain anl_1 norm close to the average value, i.e., within a constant multiple of δ; the answer to this question depends on the variance of the distribution of Λ_x' defined in Eq. (<ref>).Recall that for a random variable X, the Chebyshev inequality states that ( | X - μ| ⩾λ) ⩽Var( X )/λ ^2 ,where μ≡∑_a a ( X = a) is the mean value, and Var( X ) = ⟨| X |^2⟩- | μ|^2 is the variance. For our case, we set X=Λ_x', and hence μ= Λ _av. Furthermore, we have Var( X ) ⩽⟨X^2⟩= 2^ - m∑_x'Λ _x'^2 , where Λ _x'^2 ⩽2^n∑_x (P_cl( x|U_x') - P_exp( x|U_x'))^2 ,using again 𝐱_1⩽√(n)𝐱_2. From Eq. (<ref>) and (<ref>), we have ⟨X^2⟩⩽2^n + m∑_x,x'(P_cl( x,x') - P_IQP^ε( x,x'))^2 , which is exactly the right-hand side of Eq. (<ref>). Therefore, we have ⟨ X^2 ⟩≤ 2δ^2.Now, if we set λ = kδ, then the probability for the l_1 norm Λ_x' to be deviated from the mean value by an amount of k δ is less than 2/k^2, i.e., ( | Λ _x' - Λ _av| ⩾ kδ) ⩽ 2/k^2 ,which can become very small, e.g. by making k=10.Classical sampling algorithm. So far, we have explained how to obtain an numerical approximation of the probability P_exp( x|U_x') for a given x and U_x'. In order to produce a “distribution" in practice, we need a sampling algorithm that can be justified to be an accurate approximation of the noisy quantum circuits. The sampling algorithm we shall be discussing is a modified version of the one described in Ref. <cit.>. However, we shall present the algorithm in an alternative, and from our point of view, more direct approach.Our goal is to show how to produce classically the distribution described by any U_x' in the quantum ensemble under noise, i.e., bit strings following closely the distribution described by P_exp( x|U_x'). The challenge is that there is no guarantee that the classical approximation of the probabilities are necessarily positive numbers, through the truncation of the Fourier coefficients Eq. (<ref>). Fortunately, as we shall show that such a difficulty can be overcome.The classical sampling algorithm proposed in Ref. <cit.> is quite simple; it is a random walk, involving an application of two repeating steps. Each time the walker has to decide randomly to take the next step to be `0' or `1'. The position of the walk is described by a binary string of variable length. To describe the algorithm in detail, suppose after k steps, the walker stopped at a position labeled by z_k ≡ x_1 x_2 ⋯ x_k. Step 1: calculate the pseudo-probability (omitting the label of x' for simplicity), p( z_k) ≡∑_x,z_kP_cl( x) ,for any given partial string, which is the sum of all calculated approximation of the probabilities subject to the constraint that the first k bits are fixed to be z_k. Note that when k=n, p( z_n) = P_cl( z_n) = P_cl( x_1x_2⋯x_n). Similarly, the values of p( z_k0) and p( z_k1) are also needed, where p( z_k) = p( z_k0) + p( z_k1), by definition. These values can be determined by evaluating the Fourier coefficients (see Ref. <cit.>). Note that the values of p( z_k0) and p( z_k1) may be negative, but not both, i.e., p( z_k) is necessarily positive, from the construction of the next step. Step 2: check if [case 1:] both p(z_k0) and p(z_k1) are positive, or [case 2:] one of them is negative. If both values are positive, then the walker takes `0' or `1' based on the following probabilities, ( 0 ) = p( z_k0)/p( z_k),( 1 ) = p( z_k1)/p( z_k) . However, if one of them is negative, e.g., p( z_k0) > 0 and p( z_k1) < 0, then the walker chooses the positive side with deterministically, i.e., ( 0 ) = 1, ( 1 ) = 0. Performance and the sampling algorithm. To analyze the performance of the sampling algorithm, we suppose, after k-1 steps, the distribution generated by the algorithm is given by,Alg_k - 1 = a_0Π _0 + a_1Π _1 + ... + a_mΠ _m ,where Π_i labels one of the projectors, and for all a_i > 0, we have a_0 + a_1 + ... + a_m = 1.We shall compare the distribution with a mathematical vector calculated classically, P_math^( k )≡( p_0Π _0 + p_1Π _1 + ... + p_2^k - 1Π _2^k - 1)/S,where value of the p_i is taken to be one of the p(z_k) in Eq. (<ref>), and S = p_0 + p_1 + ... + p_2^k - 1is a normalization factor. Note that the value of S is in fact independent of k, as S = ∑_x P_cl( x ).Furthermore, from the discussion around Eq. (<ref>), there is a high probability to find a quantum circuit U_x', such that the classically-calculated norm is bounded by k δ, i.e., ∑_x | P_exp( x ) - P_cl( x )|⩽ k δ .The left-hand side is larger than the following: |∑_x (P_exp( x ) - P_cl( x )) | = | 1 - S|,which means that | 1 - S| ⩽ kδ, or equivalently, 1 + kδ⩾ S ⩾ 1 - kδ .Practically, we would need to choose k δ≪ 1.Now, we shall show that for each term, a_x≤p_x/S ,which means that the algorithm cannot generate a probability larger than the calculated value. The proof can be achieved by induction.Suppose, at some point, it is true that a_x = g_xp_x/S for some g_x, where 0 < g_x⩽ 1. In the next step, we consider the values of p_x0 and p_x1, which are the calculated values of the next step, i.e., p( z_k0) and p( z_k1) in Eq. (<ref>). According to the mechanism of the sampling algorithm, if both terms are positive, i.e., p( z_k0) > 0 and p( z_k1) > 0. Then we set c_xΠ _x→g_x( p_x0/S)Π _x0 + g_x( p_x1/S)Π _x1. Compared with the vector in Eq. (<ref>), we have ( p_x0/S) - g_x( p_x0/S) ⩾ 0 and ( p_x1/S) - g_x( p_x1/S) ⩾ 0. Now, if one of them is negative (recall that it is impossible to have both terms negative), e.g., p_x0 > 0 and p_x1 < 0. We then have c_xΠ _x→g_x( p_x/S)Π _x0 = g_x( p_x/p_x0)( p_x0/S)Π _x0. Note that p_x = p_x0 + p_x1 < p_x0. Therefore, the value g_x( p_x/p_x0)( p_x0/S) is also smaller than that ( p_x0/S) of P_math^( k ). At the end, i.e., when k=n, we have p_x = P_cl(x). From the result of Eq. (<ref>), the l_1 norm between the sampling algorithm and the calculation can be written as, P_math - Alg_1 = ∑_x,p_x > 0( p_x/S - a_x)+ ∑_x,p_x < 0| p_x|/S ,where the first summation contains all positive terms. The second summation contains the negative terms. Now, we can use Eq. (<ref>) to make ∑_x,p_x > 0a_x= 1 = S/S = 1/S∑_x p_x .Note that the summation on the right includes all possible values of x. Consequently, with a high probability, we have P_math - Alg_1 = 2/S∑_x,p_x < 0| p_x|⩽2kδ/1 - kδ , where we used Eq. (<ref>), and Eq. (<ref>) in the following way: ∑_x,p_x < 0| p_x|⩽∑_x,p_x < 0| P_exp( x ) - P_cl( x )|⩽ kδ . Finally, we can ask how good is the sampling algorithm compared with the experimental implementation. We first consider the l_1 norm, i.e., Alg - P_exp_1⩽P_math - Alg_1 + P_math - P_exp_1 ,using the triangle inequality, where P_math - P_exp_1 = ∑_x | P_cl( x )/S - P_exp( x )| .Applying again the triangle inequality, we have P_math - P_exp_1⩽1/SP_cl - P_exp_1 + | 1/S - 1|P_exp_1, which implies that P_math - P_exp_1⩽kδ/S + | 1 - S|/S⩽2kδ/1 - kδ ,from Eq. (<ref>) and Eq. (<ref>). Consequently, we haveAlg - P_exp_1⩽4 kδ/1 - kδ . Bounding the cross entropy. Now, we are ready to consider the cross entropy. In Eq. (<ref>), we argued that the main quantity of interest is the difference in cross entropy, Δ_S≡| S_c( P_exp,P_qc) - S_c( P_cl,P_qc)|, which is bounded by the following:Δ _S⩽∑_x ∈{0,1}^n| P_exp( x ) - P_cl( x )| ·| logP_qc( x )| ,which can be extended to any circuit U_x' in the ensemble, e.g., P_exp( x ) →P_exp( x|U_x'). Let us now consider an ensemble of this quantity, E_Δ≡1/2^m∑_x'Δ _S,x' ,where Δ _S,x' denotes the Δ _S associated with one of the quantum circuits U_x' in the ensemble. Using Eq. (<ref>) and Eq. (<ref>), the average value is bounded by the following: E_Δ⩽∑_x,x'A( x,x')B( x,x') ,where A( x,x') ≡| P_IQP^ε( x,x') - P_cl( x,x')| and B( x,x') ≡| logP_qc( x|U_x')|. Applying the Cauchy-Schwarz inequality to the upper bound of E_Δ, we have E_Δ ^2 ⩽∑_x,x'A( x,x')^2∑_x,x'B( x,x')^2. According to Eq. (<ref>), ∑_x,x'A( x,x')^2⩽c^2δ ^2/2^n + m .On the other hand, assuming the Porter-Thomas distribution for the quantum circuits, we have (see appendix)∑_x,x'B( x,x')^2= 2^n + m[ ( nlog 2 + γ)^2 + π ^2/6].As a result, we haveE_Δ = O( δ√(( nlog 2 + γ)^2 + π ^2/6)).Therefore, the average value can be bounded as above. To keep it a constant, it is sufficient to require δ to scale as 1/n. This completes our analysis.§ CONCLUSIONIn this work, we consider the problem of demonstrating the quantum supremacy for afamily of chaotic quantum circuits, subject to noise. For chaotic circuits obeying the Porter-Thomas distribution, we found that there exist a polynomial-time classical algorithm that can simulate the average distribution within a constant l_1 norm. Furthermore, the classical algorithm becomes quasi-polynomial if we further require it to bound the difference in cross entropy within a small constant.Going back to the original question, can chaotic quantum circuits maintain quantum supremacy under noise? Our results suggest that it really depends on our knowledge on the noise and the strength ε of the noise. In the extreme case, where the quantum circuits are subject to very strong depolarizing noise, the final state becomes very close to a completely-mixed state, and therefore becomes simulable classically. Indeed, our bound is very sensitive to the value of ε. The runtime of the classical algorithm scales as n^1/ε; it becomes a very expensive polynomial when ε approaches zero. Therefore, the answer to the question depends how much can we improve the classical algorithm, or how small the ε is. The bottom line is that our results highlight the challengesfor an experimental demonstration of quantum supremacy. We believe that a better theoretical understanding of the problem is needed before one can design a “loop-hole free" experimental demonstrating on quantum supremacy.Acknowledgments— M.-H.Y. acknowledges the support by the National Natural Science Foundation of China under Grants No. 11405093, and Guangdong Innovative and Entrepreneurial Research Team Program (No.2016ZT06D348). X. G. acknowledges the support by the National key Research and Development Program of China. apsrev4-1 § APPENDIX: CROSS ENTROPYHere we review the emergence of cross entropy. Let | ψ⟩= U| ψ _0⟩ ,be the output of a given random quantum circuit, x_j^qc be a bit string obtained from a quantum measurement on the final state in the computational basis {| x_j⟩},s_qc = {x_1^qc,x_2^qc,...,x_m^qc} be a sequence obtained by m quantum measurements. The probability of obtaining the sequence s_ qc is given by the product, ( s_qc) = ∏_i = 1^m P_qc( x_i^qc) ,where P_qc( x)≡| ⟨ x |. ψ⟩|^2. Let us now consider (removing the superscript for simplicity),log( s_qc) = m ×1/m∑_x_j∈ s_qclog P_qc( x_j).If we uniformly and randomly pick the x_j's, then from the central limit theorem, log( s_qc) =- mS ( P_qc) + O( m^1/2),whereS ( P_qc) ≡- ∑_j = 1^N P_qc( x_j)log P_qc( x_j),is the Shanon entropy.Now, let s_cl = {x_1^cl,x_2^cl,...,x_m^cl} be the sequence of bit strings generated by a classical algorithm. Let us consider ( s_cl) = ∏_x_j∈s_clP_qc( x_j^cl) ,the joint probability the quantum circuit would produce the same sequence s_cl. Taking the logarithm, we have log( s_cl) = ∑_x_j∈s_cllogP_qc( x_j) . Following the procedure in equation (<ref>), where the central limit theorem was applied, we havelog( s_cl) =- mS_c ( P_cl,P_qc) + O( m^1/2),where S_c ( P_cl,P_qc) ≡- ∑_j = 1^N P_cl(x_j)logP_qc( x_j),is the cross entropy between the strings generated by the classical algorithm and quantum circuit.Similarly, for a sequence of strings s_exp produced by an experiment, we havelog( s_exp) =- mS_c ( P_exp,P_qc) + O( m^1/2). § APPENDIX: PORTER-THOMAS AND THE SECOND MOMENT Consider the second moment of the probabilities P(x) in sampling the bit-strings x∈{0,1}^n of a general quantum circuit:∑_x ∈{0,1}^nP( x )^2= ∑_i = 1^N P( x_i)^2≡∑_i = 1^N P_i^2 ,where we labelled the probabilities as P_i≡ P( x_i) for simplicity, and set N=2^n. Of course, we can also express it as follows: ∑_i = 1^N P_i^2= ∫_0^∞f( P ) P^2 dP,where the distribution function is given by,f( P ) ≡∑_i = 1^N δ( P - P_i) . Note that the normalization of f(P) has to be N instead of 1, as ∫_0^∞f( P ) dP = ∫_0^∞∑_i = 1^N δ( P - P_i) dP = N .Therefore, for the Porter-Thomas distribution, we have to make f( P ) = N^2e^ - NP ,which gives the second moment,⟨P^2⟩≡∫_0^∞f( P )P^2dP = 2/N.In general, the k-th moment of the Porter-Thomas distribution is given by, ⟨P^k⟩= N^ - k + 1k!,which can be obtained by a similar argument.§ APPENDIX: NOISE CONVERSION FOR TWO QUBITS Let us consider an extension of the Eq. (<ref>) for the case of two qubits for a further elaboration of how depolarizing noise can be converted into Pauli noise, | ψ _J⟩= 1/2∑_x' ∈{0,1}^2U_x'| ψ_ ini⟩⊗| x'⟩ ,where U_00 = J_2( β)J_1( α), U_01 = J_2 ( β) X J_1( α), U_10 = X J_2( β) J_1( α), andU_11 = X J_2( β)XJ_1( α).Suppose a single-qubit depolarizing channel is applied to the first ancilla qubits, after the measurement, we have the resulting state,∑_x,y ∈{0,1}( ( 1 - ε/2)ρ _xy + ε/2ρ _x̅y) ⊗Π _xy ,where x̅ represents the complement of x, and ρ _xy≡U_xy ρ _ini U_xy^† .This expression can be viewed as an application of the bit-flipping channel to the first qubit, i.e., ∑_x,y ∈{ 0,1}( ℰ_bf⊗ I)(U_xyρ _iniU_xy^† ) ⊗Π _xy . Let us apply the depolarizing channel to the second ancilla qubit, giving ∑_x,y( ( 1 - ε/2)( ℰ_bf⊗ I)ρ _xy + ε/2( ℰ_bf⊗ I)ρ _xy̅) ⊗Π _xy .For example, let us focus on the Π_00 term, which is associated with the state,p_ε ^2ρ _00 + p_εq_ε ρ _10 + q_εp_ε ρ _01 + q_ε ^2ρ _11 ,wherep_ε≡ 1 - ε /2 and q_ε≡ε /2. Therefore, if we measure the ancilla qubits and obtain the outcome 00, then it is equivalent to the case where a quantum circuit U_00 is implemented under the bit-flip noises after applying each J gate. In general, given x' from the ancilla qubits, the probability of getting a bit-string x is given by P_exp( x|U_x') ≡⟨ x |ℰ_x'( ρ _ini)| x ⟩ ,where ℰ_x' labels the quantum channel of the noisy quantum circuit. For example, if x'=00, we have ℰ_00( ρ) = ( I ⊗ℰ_bf)(J_2( β)( ℰ_bf⊗ I)(J_1( α)ρJ_1( α)^†)J_2( β)^†). Note that in the main text, the quantity ℰ_x' also includes a series of depolarizing channel applied to the system qubits.§ APPENDIX: INTEGRALS AND EULER CONSTANT The integral representation of the Euler constant is given by γ=- ∫_0^∞e^ - xlog x dx = 0.57721 ... .A related integral is given by∫_0^∞e^ - x(log x)^2 dx = γ ^2 + π ^2/6 .Recall that ∑_x,x'B( x,x')^2= ∑_x,x'| logP_qc( x|U_x')| ^2 .To have an estimation of the value of left-hand side, we follow a similar procedure as in Eq. (<ref>) and write,∑_x | logP_qc( x|U_x')| ^2 = ∫_0^∞f( P )( log P)^2dP .We further assume that each of the quantum circuit U_x' obeys Porter-Thomas distribution (see Eq. (<ref>)), which means that the right-hand side becomesN^2∫_0^∞e^ - NP( log P)^2dP .Let us now consider the following integral, I_0 ≡∫_0^∞e^ - NP( log NP)^2dNP = γ ^2 + π ^2/6 ,which is equal toI_0 = ∫_0^∞e^ - NP( log N + log P)^2dNP .This integral can be decomposed into three terms,I_0 = I_1 + I_2 +I_3,where the first term on the right is given by,I_1≡(log N)^2∫_0^∞e^ - NPdNP= (log N)^2 ,and the second term is given by I_2≡ 2log N∫_0^∞e^ - NPlog P dNP .To obtain I_2, let us now write the Euler constant as- γ= ∫_0^∞e^ - NP(log NP) dNP ,where the right hand side can be written as log N∫_0^∞e^ - NPdNP+ ∫_0^∞e^ - NP(log P) dNP .Consequently, we have - ∫_0^∞e^ - NP(log P) dNP= log N + γ ,and hence I_2 =- 2log N( log N + γ).Lastly, the third term on the right of Eq. (<ref>) is given byI_3≡∫_0^∞e^ - NP(log P)^2dNP .Putting these together, I_3 = I_0 - I_1 - I_2, and is given byI_3 = ( log N + γ)^2 + π ^2/6 . Note that we can also write∑_x | logP_qc( x|U_x')|^2= NI_3 ,which gives the values for the bound for quantum circuits with Porter-Thomas distribution. | http://arxiv.org/abs/1706.08913v1 | {
"authors": [
"Man-Hong Yung",
"Xun Gao"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170627154425",
"title": "Can Chaotic Quantum Circuits Maintain Quantum Supremacy under Noise?"
} |
40pt New Large Volume Solutions Ross Altman^a, Yang-Hui He^b, Vishnu Jejjala^c, and Brent D. Nelson^a========================================================================= ^aDepartment of Physics, Northeastern University, Boston, MA 02115, USA ^b Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK; School of Physics, NanKai University, Tianjin, 300071, P.R. China; and Merton College, University of Oxford, OX1 4JD, UK ^c Mandelstam Institute for Theoretical Physics, National Institute for Theoretical Physics, CoE-MASS, and School of Physics, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa [email protected], [email protected], [email protected], [email protected] previous work, we have commenced the task of unpacking the 473,800,776 reflexive polyhedra by Kreuzer and Skarke into a database of Calabi–Yau threefolds <cit.> (see <www.rossealtman.com>). In this paper, following a pedagogical introduction, we present a new algorithm to isolate Swiss cheese solutions characterized by “holes,” or small 4-cycles, descending from the toric divisors inherent to the original four dimensional reflexive polyhedra. Implementing these methods, we find 2,268 explicit Swiss cheese manifolds, over half of which have h^1,1=6. Many of our solutions have multiple large cycles. Such Swiss cheese geometries facilitate moduli stabilization in string compactifications and provide flat directions for cosmological inflation. empty equationsection .equation § INTRODUCTION Kreuzer and Skarke have exhaustively classified the 473,800,776 reflexive polyhedra in four dimensions <cit.>. Each of these reflexive polyhedra gives rise to a four dimensional toric variety in which the anticanonical hypersurface is a singular Calabi–Yau threefold <cit.>. Moreover, each of these singular hypersurfaces admits at least one, but potentially many maximal projective crepant partial (MPCP) desingularizations, some of which represent adjacent regions in the moduli space of the same manifold, and some of which are entirely independent. This leaves us with an indeterminate, but undeniably large class of Calabi–Yau threefolds, well in excess of the half billion reflexive polytopes. In a previous work <cit.>, we have started to compile a catalog of Calabi–Yau threefolds extracted from the Kreuzer–Skarke dataset <cit.> into a new database indexed by the topological and geometric properties of the threefolds (see <www.rossealtman.com> <cit.>). As important features of geometries for compactification are readily available in a format that can be queried or scanned in batch, our database provides an efficient and useful resource for the string phenomenology and string cosmology communities.With the enormous number of candidate Calabi–Yau compactifications in hand, model builders are confronted with the challenge of isolating the set of constructions which might potentially replicate physics in the real world. In type IIB string theory, the particularly difficult problem of moduli stabilization can be avoided via flux considerations in one of two prevailing Calabi–Yau threefold compactification paradigms: KKLT <cit.> or the large volume scenario <cit.>. A particularly interesting subset of the latter are the so-called “Swiss cheese” compactifications. The name derives from the fact that a subset of the Kähler moduli are large and control the overall volume of the manifold, while the the rest of the Kähler moduli remain small and control the volumes of the “holes” at which non-perturbative contributions to the superpotential, such as E3-instantons, are localized. In this paper, we consider a special subclass of Swiss cheese compactifications characterized by large and small cycles that descend directly from the toric divisors of the Calabi–Yau threefold and are therefore directly encoded in the four dimensional reflexive polyhedra of Kreuzer and Skarke. We detail an algorithm for identifying such geometries.Implementing this algorithm, we have conducted a first scan of the current database of Calabi–Yau threefolds (h^1,1≤ 6) for the existence of the special class of Swiss cheese geometries, which we refer to as the toric Swiss cheese solutions. When we find a solution of this type, we compute the rotation matrices from the given basis of 2-cycle and 4-cycle volumes (represented by t^i and τ_i, respectively) into the bases where the large and small cycles are manifest. Our main result is to report the data for 2,268 of these toric Swiss cheese Calabi–Yau geometries, over half of which have h^1,1=6. Of these, 70 have two or more large cycles. The full details are available in the database of toric Calabi–Yau threefolds located at <www.rossealtman.com>. The number of large cycles in these geometries range from 1 to h^1,1(X)-1.The organization of the paper is as follows. In Section <ref>, we outline the conditions for the possible existence of a large volume solution in the language of toric geometry. This allows us to set our notation and conventions. The conditions in the general Swiss cheese case are summarized in Subsection <ref>, while the particular case of toric Swiss cheese is presented in Subsection <ref>. Section <ref> contains a schematic of the algorithm used into detect and compute toric Swiss cheese solutions for the various Calabi–Yau threefold vacua. In preparation for presenting our results, we establish in Section <ref> some terminology on classifying large volume solutions on the basis of the form of the Calabi–Yau volume. As terminology is used by various groups in slightly different contexts, we hope that this classification helps disambiguate language often used in the literature. In Section <ref>, we show an explicit example of a Swiss cheese manifold with Hodge numbers (h^1,1,h^2,1) = (4,94) and two large cycles, and perform a minimization of the potential. We then present our results and discuss some implications in Section <ref>. Finally, Appendices <ref> and <ref> provide a self-contained pedagogical review on the background of the Large Volume Scenario (LVS).§ METHODS: DETECTING TORIC SWISS CHEESE SOLUTIONS Consider a Calabi-Yau threefold hypersurface X in an ambient 4-dimensional toric variety 𝒜 with k toric coordinates x_1,...,x_k, each corresponding to a divisor D_i={x_i=0} on X. The Kähler moduli space is given by H^1,1(X)∩ H^2(X;ℤ) with dimension h^1,1(X)=dim H^1,1(X); we shall be largely concerned with the so-called favorable manifolds where all divisor classes on X descend from that of the ambient A, so that h:=h^1,1(X)=h^1,1(𝒜). A ℤ-basis of 2-form classes, corresponding to 4-cycles in homology via Poincaré duality, can be chosen as {J_1,...,J_h}∈ H^1,1(X)∩ H^2(X;ℤ) spanning the space. Because a Calabi-Yau manifold is Kähler, it is naturally equipped with a characteristic Kähler 2-form class J∈ H^1,1(X)∩ H^2(X;ℤ). Expanding the Kähler form in a basis, we find J=t^iJ_i, with Kähler parameters t^i∈ℤ. However, the Kähler form itself is basis independent, and we can therefore choose any basis[Note: the superscript Latin characters A, B, C,… are labels rather than indices, and will not obey the Einstein summation convention.] {J^A_1,...,J^A_h}∈ H^1,1(X)∩ H^2(X;ℚ), where, for the sake of computational efficiency, we have relaxed the requirement of ℤ-valued coefficients to the more general case of ℚ-valued ones. We can then expand the Kähler form in this new A-basis as follows J=t^AiJ^A_i. Note that because J is basis-independent, we can easily do this as many times as we want with new bases B, C, etc. The cohomology or Chow ring structure on X, however, is basis-dependent. In this chapter, we wish to identify a “Swiss cheese” basis in which the large, volume-modulating 4-cycles are manifestly separated from the small, blowup 4-cycles, which are phenomenologically useful in achieving moduli stabilization. But since we have no natural choice of basis to work with, finding one which satisfies the Swiss cheese condition <cit.> must involve an arbitrary basis change with many unconstrained degrees of freedom. It is therefore an extremely computationally expensive undertaking, especially when faced with higher dimensional moduli spaces.Therefore, in order to work around this bottleneck, the only options remaining are to narrow the scope of the search to a special case or to find a particularly natural basis to work with. Later, we will outline a technique that is a combination of these two approaches.We now consider only the class of smooth toric Calabi-Yau threefolds <cit.>, i.e. those obtained as the anticanonical hypersurface in a 4-dimensional toric variety with no worse than terminal singularities. A database<cit.> of these Calabi-Yau threefolds is available through a robust search engine at <www.rossealtman.com>. The topological and geometric information for these manifolds is presented in an arbitrary ℤ-basis {J_1,...,J_h}.We can define the A-basis of the Kähler class as a linear transformation of the original basis J_i. This transformation should be invertible, so we define the transformation matrix 𝐓^A∈ GL_h(ℚ) by J^A_i=(T^A)_i^ jJ_j. In the same manner, we may introduce matrices 𝐓^B, 𝐓^C, etc. for the B-, C-, etc. basis representations of the Kähler class. §.§ Volume, Large Cycle, and Small Cycle Conditions The complex subvarieties of X can be written in terms of 2-cycle curves 𝒞^i, 4-cycle divisors J_i, and the compact Calabi-Yau 6-cycle X. Curves are dual to divisors, and can be expressed in a basis 𝒞^1,...,𝒞^h∈ℳ(𝒜) of linear functionals on the space of divisors, 𝒞^i: H^1,1(𝒜)→ℤ, where ℳ(𝒜) is called the Mori cone, or cone of curves. The Kähler class J acts as a calibration 2-form on these 2n-cycles on X, fixing their volumes according to[We have slightly abused notation by writing J_i for both the divisor cohomology class and its Poincaré dual in homology.] vol(𝒞^i) =1/1!∫_𝒞^iJ=1/1!∫_𝒞^it^jJ_j=t^jδ^i_ j=t^i, vol(J_i) =1/2!∫_J_iJ∧ J=1/2!∫_XJ_i∧ t^jJ_j∧ t^kJ_k=1/2t^jt^kκ_ijk:=τ_i, vol(X) =1/3!∫_XJ∧ J∧ J=1/3!∫_Xt^iJ_i∧ t^jJ_j∧ t^kJ_k=1/6t^it^jt^kκ_ijk:=𝒱,where κ_ijk=∫_XJ_i∧ J_j∧ J_k is the triple intersection tensor corresponding to the Chow ring structure of the Calabi-Yau threefold X. We can expand the volume 𝒱 in complete generality by assuming that each of the three copies of J in the integral is written in a different basis 𝒱 =1/3!∫_XJ∧ J∧ J=1/3!t^Ait^Bjt^Ck∫_XJ^A_i∧ J^B_j∧ J^C_k=1/3!t^Ait^Bjt^Ck∫_X((T^A)_i^ rJ_r)∧((T^B)_j^ sJ_s)∧((T^C)_k^ tJ_t)=1/3!t^Ait^Bjt^Ck(T^A)_i^ r(T^B)_j^ s(T^C)_k^ t∫_XJ_r∧ J_s∧ J_t=1/3!t^Ait^Bjt^Ck(T^A)_i^ r(T^B)_j^ s(T^C)_k^ tκ_rst. The volume of each of the 4-cycles τ_i can then be written as the derivative of the total volume with respect to each of the 2-cycle volumes t^i τ^A_i =d𝒱/d t^Ai=d/d t^Ai[1/3!t^A'i't^B'j't^C'k'∫_XJ^A'_i'∧ J^B'_j'∧ J^C'_k']=1/2t^Bjt^Ck∫_XJ^A_i∧ J^B_j∧ J^C_k=1/2t^Bjt^Ck(T^A)_i^r(T^B)_j^ s(T^C)_k^ t∫_XJ_r∧ J_s∧ J_t=1/2t^Bjt^Ck(T^A)_i^ r(T^B)_j^ s(T^C)_k^ tκ_rst. In a generic basis J^A_i, the Kähler moduli may be arbitrarily large or small. When looking at phenomenological models in the large Volume Scenario (LVS), however, we wish to choose a basis in which some set of cycles can shrink to zero size (i.e. small), while the remaining cycles must be left non-zero (i.e. large). Thus, in the following formulation, the number of large and small cycles will be labeled N_L and N_S, respectively, such that h=N_L+N_S. For compactness of notation and in analogy to computational pseudocode, we define the following index intervals I^Toric =[1,k] (Toric divisors)I =[1,h] (Original basis)I^A =[1,h], I^A_L=[1,N_L], and I^A_S=[N_L+1,h] (A-basis)I^B =[1,h], I^B_L=[1,N_L], and I^B_S=[N_L+1,h] (B-basis) where k is the total number of toric divisors on the resolved Calabi-Yau threefold[An n-dimensional toric variety 𝒜 constructed from an n-dimensional reflexive lattice polytope M obeys the short exact sequence0→ M→⊕_i=1^kℤD_i→Pic(𝒜)≅ H^1,1(𝒜)∩ H^2(𝒜;ℤ)→ 0where the D_i are toric divisor classes. Therefore, k=h^1,1(𝒜)+dim(M)=h^1,1(𝒜)+dim_ℂ(𝒜). So, when the codimension 1 hypersurface X⊂𝒜 is favorable, we have k=h^1,1(X)+dim_ℂ(X)+1. In the case of a Calabi-Yau threefold, k=h^1,1(X)+4 specifically.] X. We will assume that there is a specific basis {J^A_i} such thatt^Ai=Large, i∈ I^A_LSmall, i∈ I^A_S and a specific basis {J^B_i} such that τ^B_i=Large, i∈ I^B_LSmall, i∈ I^B_S, and we will work only in these two bases for the remainder of this work.We then see that from Equations (<ref>) and (<ref>)-(<ref>) that if we want the total volume 𝒱 to be large, then ∃ (i,j,k)∈ I^A_L× I^A× I^A: (T^A)_i^ r(T^A)_j^ s(T^A)_k^ tκ_rst≠ 0 . We also see from Equations (<ref>) and (<ref>)-(<ref>) that ∃(j,k)∈ I^A_L× I^A: (T^B)_i^ r(T^A)_j^ s(T^A)_k^ tκ_rst≠ 0,∀ i∈ I^B_L(T^B)_i^ r(T^A)_j^ s(T^A)_k^ tκ_rst=0,∀ (i,j,k)∈ I^B_S× I^A_L× I^A. But, if {J^A_i} is a basis, then 𝐓^A must be full rank. This implies that we can write Equation (<ref>) as (T^B)_i^ r(T^A)_j^ sκ_rst=0,∀ (i,j,t)∈ I^B_S× I^A_L× I .§.§ Kähler Cone Condition§.§.§ The Mori and Kähler Cones A Kähler manifold is defined as a symplectic manifold with a closed symplectic 2-form J, which is simultaneously consistent with an almost complex and Riemannian structure. The former imposes the constraint that J is in fact a (1,1)-form, while the latter requires J to be locally positive definite. This is directly related to the fact that the volume of an effective curve vol(𝒞)=∫_𝒞J>0. By expanding J=t^iJ_i in a basis {J_i}∈ H^1,1(X), we ensure that it is indeed a closed (1,1)-form, however we must also constrain it to be positive definite. To make this explicit, we check that J has positive intersection with every subvariety of complementary codimension, i.e. curves 𝒞 𝒦(𝒜)={J∈ H^1,1(X)| vol(𝒞)=∫_𝒞J>0.}. Thus, the allowed values of J form a convex cone in the Kähler moduli space. The curves 𝒞 then form a dual cone, known as the Mori cone ℳ(𝒜)⊂Hom(H^1,1(𝒜),ℚ)≅ℚ^h which is generated[Again, for the sake of computational efficiency, we have relaxed the requirement of J∈ H^1,1(𝒜)∩ H^2(𝒜;ℤ) to J∈ H^1,1(𝒜)∩ H^2(𝒜;ℚ).] by a set of extremal rays 𝒞^1,...,𝒞^r such that ℳ(𝒜)={∑_i=1^ra_i𝒞^i| a_i∈ℝ_≥ 0}. These extremal rays can be regarded as linear functionals on the divisors, and can therefore easily be computed in terms of the toric divisors from symplectic moment polytope information provided in the Kreuzer-Skarke database. Then, given our original basis of divisor classes {J_i}_i∈ I of H^1,1(X)≅ H^1,1(𝒜) for favorable geometries, we can define the r× h Kähler cone matrix of intersection numbers between the generating curves 𝒞^i and the basis divisor classes 𝐊^i_ j=∫_𝒞^iJ_j, whose rows represent the generating curves, or equivalently, rays of the Mori cone ℳ(𝒜). Using this Kähler cone matrix, and referring to Equations (<ref>) and (<ref>), we see that ∫_𝒞^iJ=t^Aj∫_𝒞^iJ^A_j=t^Aj(T^A)_j^ k∫_𝒞^iJ_k=t^Aj(T^A)_j^ kK^i_ k, where J∈ H^1,1(𝒜) is the Kähler form on 𝒜. If we want J∈𝒦(𝒜), then we must satisfy ∫_𝒞J>0,∀𝒞∈ℳ(𝒜). This is equivalent to 0<∫_𝒞J=∫_∑_ia_i𝒞^iJ=∑_ia_i∫_𝒞^iJ,witha_i∈ℝ_>0, ∀ i∈ I. Since this must be true for arbitrary a_i, then each term of the sum must satisfy the inequality independently 0<∫_𝒞^iJ=t^Aj(T^A)_j^ kK^i_ k=(K^A)^i_ jt^Aj,∀ i∈ I . This, then, is the set of conditions which must be satisfied in order for the Kähler form J to lie within the Kähler cone. Unfortunately, this procedure only tells us the Kähler cone of the ambient toric variety 𝒜, while that of the Calabi-Yau hypersurface may be larger. It is still, however, a sufficient condition.In order to approximate better the full Kähler cone of the hypersurface, we have implemented the procedure, as described in our previous work <cit.>, of gluing together the Kähler cones of all resolutions of 𝒜 that are related by flops, and between which the hypersurface X continues smoothly. It has been shown <cit.> that in some cases, this procedure still results in a subcone of the full hypersurface Kähler cone. With some knowledge of the divisor structure, it can be further refined <cit.>, however, we will leave this to future work. §.§.§ Large and Small Cycle Kähler Cone Conditions Without loss of generality, we can always rearrange the rows of 𝐊^A=𝐊(𝐓^A)^T to put as many zero entries as possible in the lower left quadrant 𝐊^A=ccc|c(ccc|c)𝐩^A_1 … 𝐩^A_N_L ⋱ 1-4 0[ (𝐪^A_1)^T; ⋮; (𝐪^A_m)^T ] 3c 1c 3cN_L 1ch-N_L [ 2r-m; 4m; ; ;],0≤ m≤ r . Then, in the large volume limit where (t^A)^1,…,(t^A)^N_L→±∞, the Kähler cone condition of Equation (<ref>) becomes ∑_i=1^N_Llim_t^Ai→±∞𝐩^A_it^Ai >0𝐪^A_j·[[ (t^A)^N_L+1; ⋮; (t^A)^h ]] >0,∀ j∈ [1,m] . For the first expression to be well-defined, each term must either be satisfied independently or be identically zero, so that lim_t^Ai→±∞𝐩^A_it^Ai≥ 0,∀ i∈ I^A_L ⇒ 𝐩^A_i≥0or𝐩^A_i≤0,∀ i∈ I^A_L ⇒ ±𝐩^A_i≥0,∀ i∈ I^A_L.In order to satisfy Equation (<ref>), we first recognize that the rows of 𝐊^A are just the generating rays of the Mori cone ℳ(𝒜)⊂ℚ^h, as expressed in the A-basis of H^1,1(X). Then, we see that cone(𝐪^A_1,…,𝐪^A_m) must be a convex subcone of at most dimension h-N_L. Then, defining the dual σ^∨ to a d-dimensional convex cone σ by σ^∨={𝐧∈ℚ^d|⟨𝐦,𝐧⟩≥ 0,∀𝐦∈σ⊂ℚ^d.}, we see that the solution space of Equation (<ref>) is just the relative interior of the dual cone, where the inequality is strict [[ (t^A)^N_L+1; ⋮; (t^A)^h^1,1 ]]∈relint(cone(𝐪^A_1,…,𝐪^A_m)^∨) . Thus, a solution exists if and only if dim[relint(cone(𝐪^A_1,…,𝐪^A_m)^∨)]>0.§.§ Homogeneity Condition The effective potential in the low-energy supergravity limit of a type IIB theory in the Large Volume Scenario (LVS) has exponential factors involving small cycle moduli that are proportional to 𝒱, and can often be volatile unless the terms are carefully balanced. More specifically, to have a finite minimum, each term must be of the same order in 𝒱^-1. We refer to this property as homogeneity of the terms in the effective potential. This leads to a restrictive requirement on the Kähler potential, and in turn on the Kähler metric. Because the 4-cycle volumes obey the ordering τ^B_i≫τ^B_j,∀(i,j)∈ I^B_L× I^B_S, all terms in the effective potential involving τ^B_i are exponentially suppressed for each i∈ I^B_L. The requirement on the Kähler metric can then be expressed as (K^-1)_ii∼𝒱h_i^1/2({τ^B_k}_k∈ I^B_S),∀ i∈ I^B_S, where the {h_i^1/2}_i∈ I^B_S are h-N_L functions of degree-1/2 in the small 4-cycles {τ^B_k}_k∈ I^B_S. Now, we consider the expansion of the Kähler metric (K^-1)_ij in 𝒱^-1 (see Appendix <ref> for details) <cit.> (K^-1)_ij =-4𝒱(∫_XJ^B_i∧ J^B_j∧ J)+4τ^B_iτ^B_j+𝒪(𝒱^-1)=-4𝒱t^Ak(∫_XJ^B_i∧ J^B_j∧ J^A_k)+4τ^B_iτ^B_j+𝒪(𝒱^-1)=-4𝒱t^Ak(T^B)_i^ r(T^B)_j^ s(T^A)_k^ t(∫_XJ_r∧ J_s∧ J_t)+4τ^B_iτ^B_j+𝒪(𝒱^-1)=-4𝒱t^Ak(T^B)_i^ r(T^B)_j^ s(T^A)_k^ tκ_rst+4τ^B_iτ^B_j+𝒪(𝒱^-1). The diagonal elements of (K^-1)_ij have the form (K^-1)_ii/𝒱 =-4t^Aj(T^B)_i^ r(T^B)_i^ s(T^A)_j^ tκ_rst+4(τ^B_i)^2/𝒱+𝒪(𝒱^-1). But, but by definition, τ^B_i≪𝒱,∀ i∈ I^B_S, so (K^-1)_ii/𝒱 ≈ -4t^Aj(T^B)_i^ r(T^B)_i^ s(T^A)_j^ tκ_rst,∀ i∈ I^B_S. Then, because the 4-cycle volumes are quadratic in the 2-cycle volumes, we have found our degree-1/2 functions {h_i^1/2}_i∈ I^B_S from Equation (<ref>) h_i^1/2({τ^B_j}_j∈ I^B_S)=-4t^Aj(T^B)_i^ r(T^B)_i^ s(T^A)_j^ tκ_rst,∀ i∈ I^B_S. By inspecting Equations (<ref>) and (<ref>), we find the following (T^B)_i^ r(T^B)_i^ s(T^A)_j^ tκ_rst=0,∀ (i,j)∈ I^B_S× I^A_L ∃ j∈ I^A_S: (T^B)_i^ r(T^B)_i^ s(T^A)_j^ tκ_rst≠ 0,∀ i∈ I^B_S. Note that because κ_rst is a symmetric tensor, Equation (<ref>) is implied by Equation (<ref>) and therefore redundant.This homogeneity condition is critically important for finding a Swiss cheese solution with N_S=1. However, when N_S>1, the exponential factors in the effective potential have more degrees of freedom, and the necessity of this condition is loosened. However, it remains a sufficient condition in most circumstances, and we simply flag these cases in our scan when we encounter them, rather than constraining the search parameters. §.§ General List of Conditions In this section, we have compiled all the conditions necessary for X to have a Swiss cheese solution in the large volume scenario. For ease of notation, we make the following definitions κ^AAA_ijk =(T^A)_i^ r(T^A)_j^ s(T^A)_k^ tκ_rst κ^BAA_ijk =(T^B)_i^ r(T^A)_j^ s(T^A)_k^ tκ_rst κ^BA0_ijk =(T^B)_i^ r(T^A)_j^ sκ_rsk κ^BBA_ijk =(T^B)_i^ r(T^B)_j^ s(T^A)_j^ tκ_rst. Then, in order for a Swiss cheese solution to exist with N_L large 4-cycles, there must exist invertible[𝐓^A,𝐓^B∈ GL_h(ℚ) implies that both are invertible: Det(𝐓^A)≠ 0 and Det(𝐓^B)≠ 0.] A- and B-bases such that 1. (Equation (<ref>): Volume) ∃ (i,j,k)∈ I^A_L× I^A× I^A: κ^AAA_ijk≠ 02. (Equation (<ref>): Large Cycle) ∃(j,k)∈ I^A_L× I^A: κ^BAA_ijk≠ 0,∀ i∈ I^B_L3. (Equation (<ref>): Small Cycle) κ^BA0_ijk=0,∀ (i,j,k)∈ I^B_S× I^A_L× I4. (Equation (<ref>): Homogeneity) ∃ j∈ I^A_S: κ^BBA_iij≠ 0,∀ i∈ I^B_S5. (Equation (<ref>): Kähler Cone (L)) ±𝐩^A_i≥0,∀ i∈ I^A_L6. (Equation (<ref>): Kähler Cone (S)) dim[relint(cone(𝐪^A_1,…,𝐪^A_m)^∨)]>0where 𝐊^A=ccc|c(ccc|c)𝐩^A_1 … 𝐩^A_N_L ⋱ 1-4 0[ (𝐪^A_1)^T; ⋮; (𝐪^A_m)^T ] 3c 1c 3cN_L 1ch-N_L [ 2r-m; 4m; ; ;],0≤ m≤ r , is the Kähler cone matrix after rotation into the A-basis. §.§ Special Case: Toric Swiss Cheese Each favorable toric Calabi-Yau threefold is endowed with a set of special 2-form toric classes {D_i}_i∈ I^Toric dual to the 4-cycle toric divisors, which descend directly from the ambient space 𝒜. The Kähler moduli space H^1,1(X) is always spanned by these toric 2-forms, up to some redundancy. We know, therefore, than any basis expansion of a point in the moduli space may be equivalently described as a linear combination of the toric 2-forms, though it will not be unique. Practically speaking, however, this form is advantageous for a scan since the ring structure of H^1,1(X) has already been computed for these directly via toric methods and will not cost us anything. In addition, the redundancy of the toric divisors allows us to scan over multiple choices of basis (in particular, many will naturally be ℤ-bases) simply by sampling subsets of the toric divisors. So, while there is no natural basis for our calculations, the toric divisor classes {D_i}_i∈ I^Toric form a natural “pseudo-basis” in spite of their redundancy.A basis formed by a pure subset of the toric 2-forms will not always have a Swiss cheese solution. Even if such a solution exists, an arbitrary rotation may still be required. However, if we limit ourselves to the case in which some subset of the toric 2-forms is already a Swiss cheese basis (i.e. {J^A_i}_i∈ I^A,{J^B_i}_i∈ I^B⊂{D_i}_i∈ I^Toric), then our problem is reduced to a relatively simple combinatorical one. In order to see this, we define the injective maps α :I^A↪ I^Toric β :I^B↪ I^ToricJ^A_i↦ D_α(i)J^B_i↦ D_β(i). We also define the toric triple intersection tensor and the Mori cone matrix[The Mori cone matrix is essentially the same as the Kähler cone matrix, but expanded in the toric divisors rather than a basis. We give it this name because it is the object that is directly computed from torus invariant curves viewed as linear functionals relating the toric divisors.] d_ijk =∫_XD_i∧ D_j∧ D_k 𝐌^i_ j =∫_C^iD_j. Then, we can rewrite κ^AAA_ijk =d_α(i)α(j)α(k) κ^BAA_ijk =d_β(i)α(j)α(k) κ^BA0_ijk =d_β(i)α(j)k κ^BBA_ijk =d_β(i)β(j)α(k) (𝐊^A)^i_ j =𝐌^i_α(j). It is clear, then, that the conditions in Section <ref> become purely combinatoric in nature and take the form 1. (Equation (<ref>): Volume) ∃ (i,j,k)∈α(I^A_L)×α(I^A)×α(I^A):d_ijk≠ 02. (Equation (<ref>): Large Cycle) ∃(j,k)∈α(I^A_L)×α(I^A):d_ijk≠ 0,∀ i∈β(I^B_L)3. (Equation (<ref>): Small Cycle) d_ijk=0,∀ (i,j,k)∈β(I^B_S)×α(I^A_L)× I^Toric4. (Equation (<ref>): Homogeneity) ∃ j∈α(I^A_S):d_iij≠ 0,∀ i∈β(I^B_S)5. (Equation (<ref>): Kähler Cone (L)) ±𝐩^A_i≥0,∀ i∈α(I^A_L)6. (Equation (<ref>): Kähler Cone (S)) dim[relint(cone(𝐪^A_1,…,𝐪^A_m)^∨)]>0where 𝐌^A=ccc|c(ccc|c)𝐩^A_α(1) … 𝐩^A_α(N_L) ⋱ 1-4 0[ (𝐪^A_1)^T; ⋮; (𝐪^A_m)^T ] 3c 1c 3cN_L 1ch-N_L [ 2r-m; 4m; ; ;],0≤ m≤ r . Now, instead of solving a complex linear system for two arbitrary rotation matrices 𝐓^A,𝐓^B∈ GL_h(ℚ), we simply need to choose two subsets α(I^A),β(I^B)⊂ I^Toric. Since the toric triple intersection tensor d_ijk and the Mori cone matrix 𝐌^i_ j are basis-independent, it is a simple combinatoric matter to search d_ijk for subtensors that meet these constraints. If one is found, then we are done and the sets α(I^A) and β(I^B) determine the bases[Again, even if the original basis {J_i}_i∈ I is a ℤ-basis, it is not guaranteed in our analysis that {J^A_i}_i∈ I^A and {J^B_i}_i∈ I^B are as well.] {J^A_i}_i∈ I^A={D_j}_j∈α(I^A) and {J^B_i}_i∈ I^B={D_j}_j∈β(I^B) for which there exists such a Swiss cheese solution. From toric methods, we can easily obtain the rectangular transformation matrix 𝐑 given by D_i=R_i^ jJ_j. Then, the rotation matrices 𝐓^A and 𝐓^B are simply defined by (T^A)_i^ j=R_α(i)^ jand(T^B)_i^ j=R_β(i)^ j.§ IMPLEMENTING TORIC SWISS CHEESE DETECTION Given the combinatorial conditions set forward in the previous section, it is fairly straightforward to scan the database of toric Calabi-Yau threefolds<cit.> for Swiss cheese solutions. The procedure we use is as follows, but there are many variations.* From the database, we can readily obtain the toric triple intersection tensor d_ijk, the Mori cone matrix 𝐌, and the weight matrix 𝐖. The latter is defined by the conditions ∑_ρ=1^kW_r^ρn_ρ=0 and W≥ 0, where the k 4-dimensional vectors {n_1,...,n_k} are the vertices of the dual polytope. * The Small Cycle condition reads d_ijk=0,∀ (i,j,k)∈β(I^B_S)×α(I^A_L)× I^Toric. This tells us that we can search for any row of any submatrix of d_ijk that contains all zeroes, and the indices of those rows and submatrices give us all possible combinations of α(I^A_L) and β(I^B_S).* We then assemble all possible complementary sets of indices α(I^A_S) and β(I^B_L) from among the k toric divisor indices to get the full sets α(I^A) and β(I^B), each of which contain h total indices.* We construct the submatrices W_α(i)^ α(j) and W_β(i)^ β(j) of the weight matrix and check that both are full rank, otherwise we have chosen redundant toric divisors.* We then check the Volume condition ∃ (i,j,k)∈α(I^A_L)×α(I^A)×α(I^A):d_ijk≠ 0 .* Given the Mori cone matrix 𝐌 and the set of indices α(I^A), we construct the submatrix 𝐌^i_α(j) and reorder the rows until it takes the form 0𝐩^A_α(1) … 𝐩^A_α(N_L)1(𝐪^A_1)^T ⋮ (𝐪^A_m)^T𝐌^i_α(j)=[[0 [1]⋱; ; [1]01 ]],0≤ m≤ h .* Next, we check the Large Cycle Kähler cone condition ±𝐩^A_i≥0,∀ i∈α(I^A_L) ,* Then the Small Cycle Kähler cone condition dim[relint(cone(𝐪^A_1,…,𝐪^A_m)^∨)]>0 ,* The Large Cycle condition ∃(j,k)∈α(I^A_L)×α(I^A):d_ijk≠ 0,∀ i∈β(I^B_L) ,* And finally, if we choose to, we can also check the Homogeneity condition ∃ j∈α(I^A_S):d_iij≠ 0,∀ i∈β(I^B_S).* If all the conditions in Section <ref> are satisfied, then the sets of indices α(I^A) and β(I^B) are converted into rotation matrices(T^A)_i^ j=R_α(i)^ jand(T^B)_i^ j=R_β(i)^ j where D_i=R_i^ jJ_j.* We also check whether the A- and B-bases are ℤ-bases. This is the case if and only if the remaining redundant toric divisors all intersect each other smoothly at a point on the desingularized ambient toric variety 𝒜, up to an action of the fundamental group. * We repeat this procedure for N_L=1,…,h-1, so that at least one 4-cycle is always large and at least one 4-cycle is always small. The results are recorded in the database<cit.> as well. * Finally, we can take multiple passes at the dataset, beginning with a randomly chosen GL_k(ℤ) transformation on the toric divisors {D_1,...,D_k} each time. The full Swiss cheese solution set should begin to converge after many iterations, but it is unclear how slow that convergence should be. This is still a significant improvement over the method of solving the linear system for 𝐓^A and 𝐓^B, as each loop will uncover a handful of solutions with purely combinatorial efficiency. We save this larger scan for a later work.§ SWISS CHEESE CLASSIFICATION In previous studies, the majority of Swiss cheese geometries have been constructed explicitly using a top down approach. Here, working from a vast database of known candidate geometries <cit.>, we attack the problem from the bottom up with the hope of identifying as many viable Swiss cheese vacua as possible. Toward this end, in this section we lay out a scheme for categorizing Swiss cheese geometries with varying degrees of generality.The Kähler moduli t^i are the natural geometrical parameters on the Calabi-Yau threefold X, and it is a simple matter to write the volume form in terms of these as 𝒱=1/3!t^it^jt^kκ_ijk, where the intersection tensor κ_ijk encodes the Chow ring structure on X. In the low energy ten dimensional supergravity limit, the relevant field parameters descending from X are the complexified[The b_i are axionic partners of the τ_i 4-cycle volumes.] 4-cycle volumes T_i=τ_i+ib_i. There is a natural injective map from the 2-cycles to the 4-cycles via t^i↦τ_i≡∂𝒱/∂ t^i. Depending on the Chow ring structure hidden in 𝒱, it may be possible to choose a basis in which the map is invertible, at least on some subset of t^i. If so, then it is possible to write 𝒱 explicitly in terms of the 4-cycle volumes (at least partially). In this case, we say that X is explicitly Swiss cheese.In addition, the Swiss cheese condition requires that some set of large 4-cycles determine the scale of the overall volume 𝒱, while the remaining small 4-cycles determine the scale of the missing “holes”. This can be observed directly from the form of 𝒱 when each 4-cycle volume τ_i contributes independently as its own term, such that 𝒱=∑_i=1^h^1,1λ_iτ_i^3/2. When this is the case, we say that the volume is diagonalized, as there are no mixed terms. The conditions set forward in Section <ref> guarantee that X obeys the Swiss cheese condition, but even when X is explicit, it is not always possible to find a basis that makes Equation (<ref>) manifest. When it is possible, though, we say that X is diagonal.Finally, a Swiss cheese geometry X that is both maximally explicit and maximally diagonal has special properties, and we refer to it as a strong Swiss cheese geometry. In any other case, X is said to be weak.In order to give a more thorough classification, we first define the monomial functions* f^d≡ f^d(t^1,…,t^h): a degree d monomial in the 2-cycle volumes.* g^d≡ g^d(τ_1,…,τ_h): a degree d monomial in the 4-cycle volumes. With this notation, Table <ref> enumerates the monomial forms M_i that can appear in the expression for the overall volume 𝒱=∑_iλ_iM_i. Note that a special case of g^d occurs when the case in question is a function of only one 4-cycle volume. In these cases we have replaced g^d with τ^d in Table <ref>.The volume is naturally expressed in terms of 2-cycles as a sum of monomials of the form f^3, as in Equation (<ref>). If none of the maps in Equation (<ref>) are invertible, then this form of the volume must remain implicit only. If one or more of the maps can be inverted, then mixed terms involving momomials from the various terms g^d are possible. Thus, a partially explicit case may contain terms that remain in the form f^3, but must contain terms involving g^d. A completely explicit case involves a sum of monomials from the set g^3/2 only. We note that a fully explicit form for the volume may not always display the geometrical properties of the Calabi-Yau manifold most clearly. For example, a K3-fibration is often evidenced by terms of the form fτ in the volume <cit.>.We now focus our attention on the lower-right quadrant of Table <ref>. These are the cases that are designated as partially, or fully, diagonal. When restricted to these cases, we can write the most general volume form in a new basis C as 𝒱=f^3+f^2g^1/2(τ^C_1,...,τ^C_h)+fg(τ^C_1,...,τ^C_h)+g^3/2(τ^C_1,...,τ^C_h)+∑_i=1^nλ^C_i(τ^C_i)^3/2, with n≤ h. Thus, a partially diagonal volume form may contain factors from the general terms f^d and g^d, but must contain at least one term of the form τ^3/2. A circumstance such as Equation (<ref>) is both fully explicit (no terms involving 2-cycle volumes) and fully diagonal. In contrast, consider the general form of the volume in terms of the Kähler moduli given by Equation (<ref>): 𝒱 =1/3!t^Cit^Cjt^Ck(T^C)_i^ r(T^C)_j^ s(T^C)_k^ tκ_rst=1/3!t^Cit^Cjt^Ckκ^CCC_ijk, where κ^CCC_ijk=(T^C)_i^ r(T^C)_j^ s(T^C)_k^ tκ_rst. Then, we can recover the 4-cycle volumes τ^C_i =∂𝒱/∂ t^Ci=1/2t^Cjt^Ckκ^CCC_ijk and rewrite the volume as 𝒱 =1/3t^Ciτ^C_i . We can then scan the database of Calabi-Yau vacua for cases in which the Chow ring structure allows for the identification τ^D_i=(T^D)_i^ jτ_j=(T^D(T^C)^-1)_i^jτ^C_j=1/9(λ_i^C)^2t^Cit^Ci,∀ i≤ n, When this is the case, the volume takes the explicit form 𝒱 =∑_i=1^h±λ_i^Cτ^C_i√(τ^D_i) where the sign of each coefficient λ_i^C can be fixed by Kähler cone and non-negative volume considerations. Furthermore, when the C- and D-bases coincide, the volume can be written in the diagonal form 𝒱 =∑_i=1^h±λ_i^C(τ^C_i)^3/2 . In this case, the LVS vacuum takes the form of a “strong” Swiss cheese compactification, in which terms with negative sign punch out “holes” in an overall volume. Comparing Equations (<ref>) and (<ref>), we see that κ^CCC_ijk={[ 2/9(λ^C_i)^2,i=j=k≤ n;; 0 ,i,j,k≤ n;; Undetermined, otherwise ]. Therefore, we see that in the C-basis, κ^CCC_ijk=(T^C)_i^r(T^C)_j^s(T^C)_k^tκ_rst is a partially-diagonal, rank three tensor. In fact, if n=h, then κ^CCC_ijk is fully diagonal, and X is a strong Swiss cheese geometry. This result is derived via similar methods in <cit.>.It must be noted carefully, however, that the above procedure is not exhaustive in identifying explicit Swiss cheese cases. It is clear from Equation (<ref>) that in this case, the maps of Equation (<ref>) can be inverted with the specific form τ^C_i↦ t^Ci=√(∑_j=1^ha_i^ jτ^C_j). This is, in fact, a very restrictive condition and will fail to detect many potentially interesting solutions. In Section <ref>, we showcase a specific example that is both an explicit and a toric Swiss cheese manifold. While the toric methods of Section <ref> allowed us to identify it as a solution, the methods of this section were insufficient to detect it as an explicit case. As a result, a more robust algorithm for inverting the maps of Equation (<ref>) is currently in development. § EXAMPLE MODULI STABILIZATION: TORIC SWISS CHEESE WITH N_L=N_S=2We choose an example from our database at <www.rossealtman.com> with h^1,1(X)=4,h^2,1(X)=94, χ(X)=-180 and database indexes [ Polytope ID Geometry ID;1145 1; ] The intersection numbers and Kähler cone matrix in the original bases are given by I_3= J_1^2 J_2-3 J_1 J_2^2-9 J_2^3+2 J_1^2 J_3+6 J_1 J_2 J_3+6 J_1 J_3^2+18 J_2 J_3^2+18 J_3^3+J_1^2 J_4-3 J_1 J_4^2+9 J_4^3 , 𝐊= ([0101;100 -3;000 -1;0 -110;]) . This Calabi-Yau geometry was found, as a result of our toric Swiss cheese scan presented in Section <ref>, to have a Swiss cheese solution with N_L=N_S=2, with original basis, A-basis, and B-basis given by J_1=D_3 ,J_2=D_6 , J_3=D_7 , J_4=D_8, J^A_1=D_5 ,J^A_2=D_7 ,J^A_3=D_1 ,J^A_4=D_4, J^B_1=D_1 ,J^B_2=D_5 ,J^B_3=D_4 ,J^B_4=D_8. The toric divisors have independent Hodge numbers h^∙={h^0,0,h^0,1,h^0,2,h^1,1} given by[In order to determine the Hodge number of an individual divisor on the Calabi-Yau threefold, we use the Koszul extension to thepackage <cit.> with themodule.] h^∙(D_1)=h^∙(D_2)=h^∙(D_3) ={1,0,2,30}h^∙(D_4)=h^∙(D_8) ={1,0,0,1}h^∙(D_5) ={1,0,1,20}h^∙(D_6) ={1,0,0,19}h^∙(D_7) ={1,0,10,92}. Since the B-basis separates 4-cycles into large and small volumes given by τ^B_i=1/2!∫_J^B_iJ∧ J, this tells us immediately that the two small volume divisors J^B_3 and J^B_4 are both dP_0 blowup cycles, while J^B_2 is a K3 fiber. Then, J^B_3 and J^B_4 are precisely the divisors desired to host the non-perturbative contributions to the superpotential (due to E3-instantons or gaugino condensation on a stack of D7 branes) required to stabilize some of the Kähler moduli using the LVS prescription.Using Equation (<ref>) and the relations between toric divisors, we find the rotation matrices 𝐓^A=([ -311 -1;0010;1000; -301 -1 ])and𝐓^B=([1000; -311 -1; -301 -1;0001 ]) . We can use these to rotate the intersection tensor into the AAA, BAA, and BBA configurations κ^AAA_ijk =(T^A)_i^ r(T^A)_j^ s(T^A)_j^ tκ_rst κ^BAA_ijk =(T^B)_i^ r(T^A)_j^ s(T^A)_j^ tκ_rst κ^BBA_ijk =(T^B)_i^ r(T^B)_j^ s(T^A)_j^ tκ_rst. The intersection numbers in this configuration are given by I^AAA_3= 18 J_1^A (J_2^A)^2+18 (J_2^A)^3+6 J_1^A J_2^A J_3^A+6 (J_2^A)^2 J_3^A+2 J_1^A (J_3^A)^2+2 J_2^A (J_3^A)^2+(J_3^A)^2 J_4^A-3 J_3^A (J_4^A)^2+9 (J_4^A)^3,I^BAA_3= 6 J_1^A J_1^B J_2^A+6 J_1^B (J_2^A)^2+18 (J_2^A)^2 J_2^B+2 J_1^A J_1^B J_3^A+2 J_1^B J_2^A J_3^A+6 J_2^A J_2^B J_3^A+2 J_2^B (J_3^A)^2+(J_3^A)^2 J_3^B+J_1^B J_3^A J_4^A-3 J_3^A J_3^B J_4^A-3 J_1^B (J_4^A)^2+9 J_3^B (J_4^A)^2+(J_3^A)^2 J_4^B, I^BBA_3= 2 J_1^A (J_1^B)^2+2 (J_1^B)^2 J_2^A+6 J_1^B J_2^A J_2^B+2 J_1^B J_2^B J_3^A+J_1^B J_3^A J_3^B-3 J_3^A (J_3^B)^2+(J_1^B)^2 J_4^A-3 J_1^B J_3^B J_4^A+9 (J_3^B)^2 J_4^A+J_1^B J_3^A J_4^B-3 J_3^A (J_4^B)^2. We can then write out the τ^B_i in terms of the t^Ai using τ^B_i=1/2!t^Ajt^Akκ^BAA_ijk and we get τ^B_1 =3(t^A2)^2+2t^A2t^A3+2t^A1(3t^A2+t^A3)+t^A3t^A4-3/2(t^A4)^2 τ^B_2 =(3t^A2+t^A3)^2 τ^B_3 =1/2(t^A3-3t^A4)^2 τ^B_4 =1/2(t^A3)^2. Thus, we see that the B-basis is at least partially explicit. We can invert the perfect squares to get 3t^A2+t^A3 =±√(τ^B_2) 1/√(2)(t^A3-3t^A4) =±√(τ^B_3) t^A3/√(2) =±√(τ^B_4). We can fix the signs on the right hand side by computing the Kähler cone in the A-basis 𝐊^A=𝐊(𝐓^A)^T=( [000 -1;0010;1001;0101;]), with (K^A)_i^ jt^Aj>0, so that t^A4<0,t^A3>0,t^A1+t^A4>0,t^A2+t^A4>0. This fixes the signs in Equation (<ref>) to be (+,+,+). Solving the rest of Equation (<ref>), we get the rather messy result t^A_1 =1/6√(τ^B_2)(3τ^B_1-τ^B_2+τ^B_3+τ^B_4)t^A_2 =1/3(√(τ^B_2)-√(2τ^B_4))t^A_3 =√(2τ^B_4)t^A_4 =√(2)/3(√(τ^B_4)-√(τ^B_3)). Substituting these into the expression for volume, we get 𝒱 =1/3!t^Ait^Ajt^Akκ^AAA_ijk=1/18[9τ^B_1√(τ^B_2)+3√(τ^B_2)(τ^B_3+τ^B_4)-(τ^B_2)^3/2 -2√(2)((τ^B_3)^3/2+(τ^B_4)^3/2)]. Thus, we have determined that this is an explicit and partially diagonal Swiss cheese solution. In order to stabilize the Kähler moduli, we must write down the effective potential and find a stable AdS minimum, which can later be uplifted. To find the form of the potential, we need to know the inverse Kähler metric. This is, to leading order in 𝒱^-1 (see[Note that in Appendix <ref>, the inverse Kähler metric for the Kähler moduli is denoted (_T^-1)_ij̅, while here we refer to it simply at (^-1)_ij̅.] Equations <ref> and <ref>), given by (𝒦^-1)_ij̅ =-4𝒱κ^BBA_ijkt^Ak=4√(2)𝒱( [ (√(τ^B_3)+√(τ^B_4))/3 -(3 τ^B_1+τ^B_2+τ^B_3+τ^B_4)/3 √(2τ^B_2) -√(2τ^B_2)-√(τ^B_3)-√(τ^B_4); -√(2τ^B_2)000;-√(τ^B_3)0 3 √(τ^B_3)0;-√(τ^B_4)00 3 √(τ^B_4);]). From this form of the inverse Kähler metric, the effective potential takes the form V(𝒱,τ^B_3,τ^B_4) =a_3^2 |A_3|^2 (𝒦^-1)_33 e^-2 a_3τ^B_3/2𝒱^2+a_4^2 |A_4|^2 (𝒦^-1)_44 e^-2 a_4τ^B_4/2𝒱^2+2 a_3a_4 |A_3A_4| (𝒦^-1)_34 e^-(a_3τ^B_3+a_4τ^B_4)/2𝒱^2-2 a_3 |A_3W_GVW| τ^B_3 e^-a_3τ^B_3/𝒱^2-2 a_4 |A_4W_GVW| τ^B_4 e^-a_4τ^B_4/𝒱^2+3 ξ|W_GVW|^2/8 𝒱^3=6√(2)a_3^2 |A_3|^2 √(τ^B_3) e^-2 a_3τ^B_3/𝒱+6√(2) a_4^2 |A_4|^2 √(τ^B_4) e^-2 a_4τ^B_4/𝒱-2 a_3 |A_3W_GVW| τ^B_3 e^-a_3τ^B_3/𝒱^2-2 a_4 |A_4W_GVW| τ^B_4 e^-a_4τ^B_4/𝒱^2+3 ξ|W_GVW|^2/8 𝒱^3. We attempt to plot ln V(τ^B_3,τ^B_4), using an estimate of ⟨𝒱⟩∼ 10^32, and reasonable values for a_3=a_4=2π, A_1=A_2=1, W_GVW=1, and ξ =-χ (X)ζ (3)/2, in Figure <ref>.Where the logarithm gets cut off, the potential has gone negative. This gives us an AdS minimum. We notice that the potential is symmetric in τ^B_3 and τ^B_4, so we can choose the direction where they are equal, i.e. τ^B_s:τ^B_3=τ^B_4. Then we can plot the potential in terms of τ^B_s and 𝒱, as in Figure <ref>.Again, we see the AdS minimum. With τ^B_3 and τ^B_4 identified, the potential takes the form V(𝒱,τ_s) =12 √(2) a_s^2 |A_s|^2 √(τ^B_s) e^-2 a_sτ^B_s/𝒱-4 a_s |A_sW_GVW| τ^B_s e^-a_sτ^B_s/𝒱^2+3 ξ|W_GVW|^2/8 𝒱^3. In fact, this is exactly the form of Equation (<ref>), the potential for N_S=1 with A_s↦ 2A_s and c=3/√(2). Using Equations (<ref>) and (<ref>) to find the minima, we arrive at ⟨τ^B_s⟩ ≃1/4(3cχ (X)ζ (3)/4)^2/3≃ 12.3 ,⟨⟩ ≃| W_GVW|/2ca_s| A_s|√(τ^B_s)e^a_sτ^B_s≃ 2.12× 10^32. Thus, we do indeed get a large volume solution. And finally, we notice that using this minimum, we can find a flat direction for the other two large 4-cycle volumes τ^B_1 and τ^B_2. τ^B_1=(τ^B_2)^3/2-6 τ^B_s√(τ^B_2)+4 √(2)(τ^B_s)^3/2+18 𝒱/9 √(τ^B_2). This kind of feature will be particular interesting in the context of fiber modulus mediated inflation. Recall that J^B_2 is, in fact, a K3 fiber in this case. Work on these direct cosmological consequences is already being pursued and will be presented in future manuscripts. § RESULTS AND DISCUSSION In this section we present the results of our search for toric Swiss cheese manifolds within the Kreuzer–Skarke database, for those polytopes with h^1,1≤ 6. As indicated in Table <ref>, this represents 23,573 reflexive polytopes, giving rise to 101,681 unique Calabi-Yau threefolds, of which 100,368 are favorable. The first stage of the analysis was to scan these ∼ 10^5 favorable geometries, implementing the search strategy outlined in Section <ref>. This was performed using resources at the Massachusetts Green High Performance Computing Center. The computations were performed on dual Intel E5 2650 CPUs with 128GB of RAM per node, and the total time consumed for this stage in the analysis was 4,930 core-hours. It is clear from these results that there is a scarcity of Calabi-Yau threefolds X, whose volumes can be made explicit at higher values of h^1,1(X). Recall from Section <ref> that this does not mean that we cannot ever write the 6-cycle volume of these manifolds in terms of 4-cycle volumes, but merely that to do so might involve a linear combination of arbitrary square roots, for which it is difficult to scan.We also see from Table <ref> that there are few toric Swiss cheese manifolds X at higher h^1,1(X) with many large 4-cycles. Unfortunately, when N_S=h^1,1(X)-N_L>2, it becomes difficult to stabilize the axion component of the complexified 4-cycle moduli <cit.>. However, we see that there are still 18 cases in h^1,1(X)=4 for which the Kähler moduli can still be explicitly stabilized through the LVS. We demonstrate an example of this in Section <ref>. We note in Section <ref> that these results can be expanded by running this scan iteratively, each time with an arbitrary rotation of the toric intersection tensor.Finally, we notice that a large number of the toric Swiss cheese solutions satisfy the homogeneity condition of Section <ref>. This condition ensures that the effective potential contains terms with the correct order in 𝒱^-1 for a minimum to exist. It is not always necessary to achieve such a minimum when N_S>1, but it greatly simplifies the minimization procedure <cit.>.In total, 2,268 toric Swiss cheese manifolds were identified, 2,055 of which satisfied the homogeneity condition. These solutions are distributed as shown in the penultimate section of Table <ref>. While these numbers represent only a subset of all the possible Swiss cheese manifolds that may exist in this dataset, they represent an ample starting point for phenomenological investigation with nearly 160 explicit examples of guaranteed moduli stabilization using the techniques of <cit.>. A concrete example of this was explored in Section <ref>.More general techniques that do not rely on the ability to trivialize the unknown basis change (see Equation (<ref>)), such as those employed in <cit.> are far more computationally expensive. Nevertheless, we expect such cases to form the majority of all manifolds which admit a large volume limit, particularly for higher values of h^1,1 and/or greater numbers of large cycles. A full analysis, extending the results of <cit.>, is currently underway.The 𝒪(2,000) cases with h^1,1>4 are, to our knowledge, unknown before now. Such cases were not approachable using thesoftware <cit.>, and required the redesigned techniques described in <cit.>. These high Picard number cases include 29 cases with two or more large cycles. These are of particular interest for the possibility of identifying flat directions which may be relevant for inflationary cosmology <cit.>.The second stage of the analysis involved studying the 100,368 favorable Calabi-Yau geometries and identifying directly when a basis of divisor classes can be found for which the volume can be written explicitly in terms of 4-cycle volumes. Given the difficulty of studying the dynamics of Kähler moduli in the 4-dimensional effective supergravity theory when the Calabi-Yau volume cannot be put into a fully-explicit or (preferably) strong form, we also flag those cases for which a suitable basis exists, and provide the necessary rotation matrix (𝐓^C in Section <ref>). We expect these results to be expanded significantly in the future as the more general case is completed. While this brute force attempt was performed over the entire dataset of Calabi-Yau threefolds, the identification of the components of the basis matrix involves solving a large system of polynomials using Groebner basis techniques. As such, this stage is computationally expensive, and both the CPU time and physical memory required to find a solution for any given example can be hard to predict. The results of the second stage in the analysis are found in the final section of Table <ref>, and we find a total of 109 usable cases in h^1,1≤ 3. http://rossealtman.com/mongosearch.php?mongocode=%7B%22SWISSCHEESE%22%3A%7B%22%24ne%22%3A%7B%7D%7D%7D onoffswitch=on polyprops%5B0%5D=POLYID polyprops%5B2%5D=H11 geomprops%5B0%5D=GEOMN geomprops%5B8%5D=SWISSCHEESE count=NONE limit=0 matches=GEOM submit=Search%21this link inAll 2,268 toric Swiss cheese manifolds can be queried immediately viathe on-line database at <www.rossealtman.com>, allowing for quick access to this subset for the purposes of model building or further phenomenological study[For further convenience, we also flag whether or not our solutions of the A- and B-bases are integer-valued.]. Given the difficulty of studying the dynamics of Kähler moduli in the 4-dimensional effective supergravity theory when the Calabi-Yau volume cannot be put into a fully-explicit or (preferably) strong form, we also flag those cases for which a suitable basis exists, and provide the necessary rotation matrix 𝐓^C. We expect these results to be expanded significantly in the future as the more general case is completed.§ ACKNOWLEDGEMENTS We thank James Gray for discussions. The work of RA and BDN is supported in part by the National Science Foundation, under grant PHY-1620575. VJ is supported by the South African Research Chairs Initiative and the National Research Foundation. YHH would like to thank the Science and Technology Facilities Council, UK, for grant ST/J00037X/1, the Chinese Ministry of Education, for a Chang-Jiang Chair Professorship at NanKai University as well as the City of Tian-Jin for a Qian-Ren Scholarship, and Merton College, Oxford, for her enduring support.§ LARGE VOLUME SCENARIO IDENTITIESThe following two appendices provide a start-to-finish, self-contained derivation of all of the relevant expressions for moduli stabilization in the Large Volume Scenario, as well as key expressions from toric geometry which expedite the final result. While much of this material is present in the papers cited in the reference section of this work, we find that many of the intermediate steps are elided in the literature, and thus a more complete, pedagogical treatment is warranted.For a given Calabi-Yau threefold X, the volume form is given by =1/3!κ_ijkt^it^jt^k. From this, we can derive the volume of the i^th 4-cycle divisor τ_a =t^a=1/3!κ_ijk(t^it^at^jt^k+t^it^jt^at^k+t^it^jt^kt^a)=1/3!κ_ijk(δ^i_at^jt^k+t^iδ^j_at^k+t^it^jδ^k_a)=1/2κ_ajkt^jt^k. For ease of computation, we will also derive the following two relations involving derivatives of 2-cycle divisor volumesδ^a_b =τ_bτ_a=τ_a(1/2κ_bjkt^jt^k)=1/2κ_bjk(t^jτ_at^k+t^jt^kτ_a)=κ_bjkt^kt^jτ_a,0=δ^a_bτ_c=κ_bjk(t^kτ_ct^jτ_a+t^kt^jτ_cτ_a) ⇒κ_bjkt^kt^jτ_cτ_a=-κ_bjkt^kτ_ct^jτ_a , . Using Equation (<ref>), we now compute the first derivative of the volume form ^a =τ_a=1/3!κ_ijk(t^iτ_at^jt^k+t^it^jτ_at^k+t^it^jt^kτ_a)=1/2t^iτ_aκ_ijkt^jt^k=1/2δ^a_kt^k=1/2t^a, and using Equation (<ref>), we compute the second derivative ^ab =^aτ_b=1/2κ_ijk(t^iτ_bτ_at^jt^k+t^iτ_at^jτ_bt^k+t^iτ_at^jt^kτ_b)=1/2κ_ijk(-t^kτ_bt^iτ_at^j+t^iτ_at^jτ_bt^k+t^iτ_at^jt^kτ_b)=1/2δ^b_it^iτ_a=1/2t^bτ_a. Again using Equation (<ref>), we can then derive the inverse of the second derivative of the volume form δ^a_b=κ_bjkt^kt^jτ_a=2κ_bjkt^k^aj ⇒_jb=2κ_jbkt^k. We now derive the following three relations, which will be useful in computing the inverse Kähler metric ^ab_bc =(1/2t^bτ_a)(2κ_bckt^k)=δ^a_c,^a_ab =(1/2t^a)(2κ_abkt^k)=2τ_b,^a_ab^b =(1/2t^a)(2κ_abkt^k)(1/2t^b)=3.§ KÄHLER MODULI STABILIZATION The bosonic field content in the string frame is as follows * R-R sector: A 0-form potential C_0, a 2-form potential C_2, and a 4-form potential C_4.* NS-NS sector: The 0-form dilaton ϕ, the 2-form 10D graviton g_μν, and the antisymmetric Kalb-Ramond 2-form B_2.* Scalar moduli: Kähler moduli (τ_i) and complex structure moduli (U_i). A constant variation in the dilaton can be shown to produce a corresponding variation in the string coupling according to δ g_s/g_s=δϕ, so that we may express the coupling as g_s∼ e^ϕ. Furthermore, in the Einstein frame, the 10D string-frame graviton is rescaled by g^(s)_μν→ g^(E)_μν=g_s^-1/2g^(s)_μν. This results in the rescaling of each 2-cycle volume as t^i→^i=g_s^-1/2t^i, and therefore τ_i→_i=g_s^-1τ_i and →=g_s^-3/2.It is convenient to make the following field redefinitions * The axion-dilaton S=g_s^-1+iC_0, which corresponds to the complex structure of the elliptic fiber in the F-theory generalization.* The complexified Kähler moduli T_i=∫_J_i(J∧ J+iC_4)=_i+ib_i. In this section, we will show that at tree level, the scalar potential of the 4D effective supergravity theory exhibits a “no-scale” structure, at which only the axion-dilaton and complex structure moduli are stabilized. We further show that in order to break the “no-scale” structure and stabilize the volume modulus, we must include the leading α ' correction to the volume in the Kähler potential. And finally, we show that to stabilize the remaining Kähler blowup modes, we must consider non-perturbative corrections to the superpotential resulting from the structure on the blowup cycles. In the end, we will write down the corrected scalar potential [V= V_tree+V_α '+ V_non-perturbative ] which can be minimized to stabilize the Kähler blowup moduli, as well as the volume modulus, which gets fixed exponentially large with respect to the blowup moduli. The presence of a flat direction in the moduli space at this minimum leaves the door open for Kähler blowup moduli-mediated inflation. We will discuss this in the next section.In the absence of flat directions, the blowup moduli are all stabilized at small values. However, this still leaves any non-blowup Kähler moduli (which correspond instead to fibration modes and typically have large values) unstabilized. This “extended no-scale” structure can only be broken by adding subleading string loop corrections. The fibration moduli can then be stabilized by minimizing the further corrected potential. §.§ V_tree At tree level in α ', the Kähler potential for X can be expressed in the separated form[Technically, the Einstein frame volume =g_s^-3/2 depends on the axion-dilaton S through g_s∼S+S̅/2. However, we can disregard this, since we will be differentiating with respect to the Einstein frame Kähler moduli , which have a complementary dependence on S.] [ =_S +_T +_U; = -ln(S+S̅) + -2 ln() + -ln(-i∫_XΩ∧Ω̅), ] where Ω is the unique, holomorphic (3,0)-form on X which contains the dependence on the complex structure moduli U_i,i=1,...,h^2,1. If we turn on the non-trivial RR and NS-NS gauge fluxes F_3=dC_2 and H_3=dB_2, and construct the complexified flux G_3=F_3+iSH_3, then the superpotential can be written as W=W_Gukov-Vafa-Witten=∫_XΩ∧ G_3. In order to obtain a warped 10D background, the Bianchi identity constrains G_3 to be imaginary self-dual (i.e. *_6G_3=iG_3). It can be shown that this is equivalent to the constraints on the GVW superpotential D^AW=0,Φ_A∈{S,U_i}. The full scalar potential in the 4D effective supergravity theory then has the form V=e^[_ab̅D^aWD^b̅W-3|W|^2],Φ_a,Φ_b∈{S,T_1,...,T_h^1,1,U_1,...,U_h^2,1} where the inverse Kähler metric is defined as _ab̅≡(∂/∂Φ_a∂̅Φ̅_̅b̅)^-1 and the gauge connection for the covariant derivative is given by ∂_a≡∂/∂Φ_a so that D^aW=∂^aW+W∂^a. Becauseis separated into _S, _T, and _U, the inverse Kähler metric has a block diagonal form _ab̅=([ (_S^-1) 0 0; 0_ij̅ 0; 0 0 (_U^-1)_AB̅; ]) where Φ_i,Φ_j∈{T_1,...,T_h^1,1} and Φ_A,Φ_B∈{U_1,...,U_h^2,1}. Then, the scalar potential separates [V=V_S+V_T+V_U+ -3e^K|W|^2; =0+V_T+0+ -3e^K|W|^2 ]. where the second equality follows from the constraints in Equation (<ref>). Then, focusing on V_T, we have V_T=e^[_ij̅D^iWD^j̅W] The first derivative of K_T is given by _T^i =∂^i_T=∂/∂ T_i_T=-2^-1∂/∂ T_i=-2^-11/2(∂/∂_i-i∂/∂ b_i)=-^-1^i. Then, the second derivative is given by _T^ij̅ =∂^i∂^j̅_T=-(∂/∂ T_i^-1)^j-^-1(∂/∂ T_i^j)=^-21/2(∂/∂_i-i∂/∂ b_i)^j-^-11/2(∂^j/∂_i-i∂^j/∂ b_i)=1/2(1/^i^j-^ij). Now, we assume that _ij̅ is of the form _ij̅ =u_ij+v^k_ki^l_lj+𝒪(higher order in ^-1). Then, we have _T^ij̅_j̅k =1/2[1/^i^j-^ij][u_jk+v^l_lj^m_mk]=1/2[u^i^j_jk+v/^i^l_lj^j^m_mk-u^ij_jk..-v^ij_jl^l^m_mk]=1/2[u^i^j_jk+v/^i(3)^m_mk-uδ^i_k-vδ^i_l^l^m_mk]=1/2[u^i^j_jk+3v^i^j_jk-uδ^i_k-v^i^j_jk]=1/2[(u+2v)^i^j_jk-uδ^i_k]=(u/2+v)1/(1/2^i)(2_k)-u/2δ^i_k=(u/2+v)^i_k/-u/2δ^i_k. In order for this result to be consistent and general, we must have u=-2andv=1 so that _ij̅ =-2_ij+^k_ki^l_lj+𝒪(higher order in ^-1). We also know that since W=W_GVW is independent of the Kähler moduli, we can write D^iW =W∂^i_T=-W^-1^i. Then, expanding Equation (<ref>), we have V_T =e^[_ij̅D^iWD^j̅W]=e^|W|^2[-2_ij+^k_ki^l_lj]^-2^i^j=e^|W|^2^-2[-2^i_ij^j+^k_ki^i^l_lj^j]=e^|W|^2^-2[-2(3)+(3)(3)]=3e^|W|^2, and the scalar potential in Equation (<ref>) reduces to the “no-scale” form with V_tree=0. Thus, at tree level, this potential cannot be minimized, and so both the volume and the individual Kähler moduli are left unstabilized. §.§ V_α ' In order to stabilize the volume , we must break the “no-scale” structure. We do this by considering the leading (α ')^3 correction to , given by ξ/2, where[χ (X) is the Euler characteristic of X, and ζ (3)≈ 1.20206 is the Riemann zeta function, evaluated at 3.] ξ =-χ (X)ζ (3)/2. With this correction, the Einstein frame volume becomes =g_s^-3/2→ g_s^-3/2( + ξ/2)= + ξ/2(S+S̅/2)^3/2. Then, the Kähler potential for X can be expressed in a partially separated form as [ =_S/T +_U; = -ln(S+S̅)-2 ln( +ξ/2(S+S̅/2)^3/2) + -ln(-i∫_XΩ∧Ω̅). ] For ease of notation, we also define_T=-2 ln( +ξ/2(S+S̅/2)^3/2)=-2 ln( +ξ̂/2). This allows us to write K^ij̅=∂^i∂^j̅K in block diagonal form ^a̅b =([(_S/T)^00(_S/T^T)^0j0; ; (K_S/T)^i̅0̅ (_T)^i̅j0; ;00 (_U)^A̅B ]) =([ a 𝐛^T; 𝐛 𝐂 ]), where Φ_i,Φ_j∈{T_1,...,T_h^1,1} and Φ_A,Φ_B∈{U_1,...,U_h^2,1}. The inverse is given by _ab̅=1/d([1 -𝐛^T𝐂^-1; -𝐂^-1𝐛 d𝐂^-1+𝐂^-1𝐛𝐛^T𝐂^-1 ]), where d=a-𝐛^T𝐂^-1𝐛. In addition, our matrix 𝐂 is diagonal, so 𝐂^-1 =([ (^-1_T)_ij̅ 0; 0 (^-1_U)_AB̅ ]) and Equation (<ref>) reduces to _ab̅=1/d×([1-(_S/T^T)^0k(^-1_T)_kj̅0; ; -(^-1_T)_ik̅(_S/T)^k̅0 d(^-1_T)_ij̅+(^-1_T)_ik̅(^T_S/T)^k̅0(_S/T)^0l(^-1_T)_lj̅0; ;00 d(^-1_U)_AB̅ ]), where d=(_S/T)^00-(_S/T^T)^0i(^-1_T)_ij̅(_S/T)^j̅0.The full scalar potential in the 4D effective supergravity theory again has the form V=e^[_ab̅D^aWD^b̅W-3|W|^2],Φ_a,Φ_b∈{S,T_1,...,T_h^1,1,U_1,...,U_h^2,1}. Due to the superpotential constraints (see Equation (<ref>)) that stabilize the axion-dilaton and complex structure moduli at tree level, any terms proportional to D^SW or D^U_iW vanish, and we need only consider the center block of the Equation (<ref>). This gives us the corrected inverse Kähler metric<cit.> (_T^-1)_ij̅=_ij̅+1/d_ik̅(^T_S/T)^k̅0(_S/T)^0l_lj̅ where, from Equation (<ref>) and Equation (<ref>) with →+ξ̂/2 and ξ̂=g_s^-3/2ξ, we find that _ij̅ =-(2+ξ̂)_ij+2(2+ξ̂/4-ξ̂)^k_ki^l_lj, (_S/T)^0i =3ξ̂g_s/4(2+ξ̂)^2^i,(_S/T)^i =-2/2+ξ̂^i, and d =g_s^2(-ξ̂)/4(4-ξ̂). The scalar potential then takes the form V=e^[(^-1_T)_ij̅D^iWD^j̅W-3|W|^2]. Recall that the superpotential is independent of the Kähler moduli, so that D^iW=W∂^i=W(_S/T)^i. Then, the scalar potential reduces to [V= V_tree+V_α '; ; =0+ 3e^|W|^2ξ̂(ξ̂^2+7ξ̂+^2)/(-ξ̂)(2+ξ̂)^2. ] We can now find a stable minimum for the volume modulus. §.§ V_non-perturbative In the previous subsection, we were able to use the leading α ' correction to the volume in order to break the “no-scale” structure of the potential and stabilize the volume modulus, but this still did not give us a mechanism for stabilizing the remaining Kähler moduli. In order to find such a mechanism, we must further consider the effect of non-perturbative features on the superpotential. The superpotential then takes the form [W=W_GVW+ W_non-perturbative; =∫_XΩ∧ G_3+ A_ie^-â_iT_i ] where the scalar constants â_i depend on non-perturbative effects such as D brane instantons (â_i=2π/g_s) or gaugino condensation (â_i=2π/g_sN) on the corresponding 4-cycle J_i∈ H_4(X;ℤ), and the complex constants A_i encode threshold effects depending implicitly on the complex structure and D3 brane positions.The scalar potential still takes the form in Equation (<ref>), except that D^iW=∂^iW+W∂^i=-â_iA_ie^-â_iT_i+W(_ad/K)^i. Plugging this in, we find that V=e^ [(^-1_T)_ij̅â_iâ_jA_iA̅_je^-(â_iT_i+â_jT̅_j)..-(^-1_T)_ij̅(â_iA_iW̅e^-â_iT_i(_S/T)^j̅+(_S/T)^iâ_jA̅_jWe^-â_jT̅_j).+.(^-1_T)_ij̅|W|^2(_S/T)^i(_S/T)^j̅-3|W|^2], where the last line is just V_α '. This then reduces to [V= V_tree+V_np1+V_np2+V_α ';] with V_np1 =e^(^-1_T)_ij̅â_iâ_jA_iA̅_je^-(â_iT_i+â_jT̅_j)=e^â_iâ_jA_iA̅_je^-(â_iT_i+â_jT̅_j)(2+ξ̂)[-_ij+(2/4-ξ̂)^k_ki^l_lj]=2e^â_iâ_j| A_iA_j| e^-(â_i_i+â_j_j)e^i(θ_i-θ_j-â_ib_i+â_jb_j)(2+ξ̂)[-κ_ijk^k+(4/4-ξ̂)_i_j] and V_np2 =-e^(^-1_T)_ij̅(â_iA_iW̅e^-â_iT_i(_S/T)^j̅+(_S/T)^iâ_jA̅_jWe^-â_jT̅_j)=e^â_i4ξ̂^2+ξ̂ +4^2/( -ξ̂)(ξ̂+2)(A_i_iW̅e^-â_iT_i+A̅_i_iWe^-â_iT̅_i)=2e^â_i| A_iW|_ie^-â_i_i4ξ̂^2+ξ̂ +4^2/( -ξ̂)(2+ξ̂)cos(θ_i-ϕ -â_ib_i), where A_i=| A_i| e^iθ_i and W=|W|e^iϕ.The axionic part b_i of the complexified Kähler moduli can be decoupled and stabilized independently. The result, however, depends heavily on the topology of X as encoded in the triple intersection tensor κ_ijk, and therefore its complexity also scales rapidly with increasing numbers of blowup moduli. In their appendix, the authors of <cit.> did an excellent job of classifying the resulting axion-stabilized scalar potential for various forms of κ_ijk in the large volume limit for up to two blowup moduli, and the reader is encouraged to refer there for more detail.In this appendix, we consider only the “Swiss cheese” case in which the small 4-cycle blowup moduli can be explicitly separated from the large moduli which control the volume and flbration structure. In addition, for the sake of simplicity, we turn our attention only to cases with one small blowup modulus _s, while the rest are sent large. In this case, it is a relatively simple matter to stabilize the single axion b_s, as its contribution cancels in V_np1. We find that V_np1 =2e^â_s^2| A_s|^2e^-2â_s_s(2+ξ̂)(-κ_ssi^i+(4/4-ξ̂)_s_s) V_np2 =-2e^â_s| A_sW| e^-â_s_s_s4ξ̂^2+ξ̂ +4^2/( -ξ̂)(ξ̂+2)=-2e^â_s| A_s| e^-â_s_s(| W_GVW|-| A_s| e^-â_s_s)_s4ξ̂^2+ξ̂ +4^2/( -ξ̂)(2+ξ̂). Furthermore, <cit.> shows that in the case of a single small blowup modulus, there will only be a large volume AdS minimum when an additional so-called “homogeneity condition” κ_ssi^i≃ -c√(_s), c>0 is satisfied[The minus sign in Equation (<ref>) originates from the fact that inside the Kähler cone ∫_C^iJ>0, the Kähler metric must be positive definite.]. Then, in the large volume limit we have e^→∞=g_se^_cs/2^2, and to leading order in each term V_np1 →∞=g_se^_cs2câ_s^2| A_s|^2e^-2â_s_s√(_s)/,V_np2 →∞=-g_se^_cs2â_s| A_sW_GVW| e^-â_s_s_s/^2, V_α ' →∞=g_se^_cs3| W_GVW|^2ξ̂/8^3, and we obtain the full potential V(_s,) =V_tree+V_np1+V_np2+V_α '=g_se^_cs[2câ_s^2| A_s|^2e^-2â_s_s√(_s)/-2â_s| A_sW_GVW| e^-â_s_s_s/^2+3| W_GVW|^2ξ̂/8^3]. We can now stabilize both the blowup modulus and the volume by finding a local minimum of the potential where ∂ V/∂_s=∂ V/∂=0. Following the work of <cit.> and taking â_s_s∼ln ≫ 1 in order to cut off higher instanton corrections, one obtains the simple result ⟨_s⟩≃(3cξ̂/16)^2/3and⟨⟩≃| W_GVW|/2câ_s| A_s|√(_s)e^â_s_s. Finally, we convert back to the string frame using the transformations =g_s^-3/2, _i=g_s^-1τ_i, â_i=g_sa_i, and ξ̂=g_s^-3/2ξ =-χ (X)ζ (3)/2g_s^3/2. We find that[For a small number h^1,1(X) of Kähler moduli relative to complex structure moduli h^2,1(X), the Euler number will have a negative value χ (X)=2(h^1,1(X)-h^2,1(X))<0. Thus, the volume >0.] ⟨τ_s⟩ ≃1/4(3cχ (X)ζ (3)/4)^2/3,⟨⟩ ≃| W_GVW|/2ca_s| A_s|√(τ_s)e^a_sτ_s.99 Altman:2014bfaR. Altman, J. Gray, Y. H. He, V. Jejjala and B. D. Nelson, “A Calabi-Yau Database: Threefolds Constructed from the Kreuzer–Skarke List,” [arXiv:http://arxiv.org/abs/1411.14181411.1418 [hep-th]]. Kreuzer:2000xyM. Kreuzer and H. Skarke, “Complete classification of reflexive polyhedra in four-dimensions,” Adv. Theor. Math. Phys.4, 1209 (2002) [http://arxiv.org/abs/hep-th/0002240hep-th/0002240].Batyrev:1994hmV. V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,” J. Alg. Geom.3, 493 (1994) [http://arxiv.org/abs/alg-geom/9310003alg-geom/9310003].Kreuzera M. Kreuzer and H. Skarke, Calabi-Yau data, <http://hep.itp.tuwien.ac.at/ kreuzer/CY/>Altmana R. Altman, Toric Calabi-Yau Threefold Database, <http://www.rossealtman.com>kklt S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, “De Sitter vacua in string theory,” Phys. Rev. D 68, 046005 (2003). [hep-th/https://arxiv.org/pdf/hep-th/0301240.pdf0301240].Bobkov:2004cyK. Bobkov, “Volume stabilization via alpha-prime corrections in type IIB theory with fluxes,” JHEP 0505, 010 (2005) doi:10.1088/1126-6708/2005/05/010 [hep-th/https://arxiv.org/pdf/hep-th/0412239.pdf0412239].bbcq V. Balasubramanian, P. Berglund, J. P. Conlon and F. Quevedo, “Systematics of moduli stabilisation in Calabi-Yau flux compactifications,” JHEP 0503, 007 (2005). [hep-th/https://arxiv.org/pdf/hep-th/0502058.pdf0502058]. Cicoli:2008vaM. Cicoli, J. P. Conlon and F. Quevedo, “General Analysis of large Volume Scenarios with String Loop Moduli Stabilisation,” JHEP 0810, 105 (2008) [arXiv:http://arxiv.org/abs/0805.10290805.1029 [hep-th]]. Gray:2012jyJ. Gray, Y. H. He, V. Jejjala, B. Jurke, B. D. Nelson and J. Simon, “Calabi-Yau Manifolds with Large Volume Vacua,” Phys. Rev. D 86, 101901 (2012) [arXiv:http://arxiv.org/abs/1207.58011207.5801 [hep-th]].Szendroi2001B. Szendroi, “On a conjecture of Cox and Katz,” Math. Z 240, 233 (2002) [arXiv:math/https://arxiv.org/pdf/math/0110166.pdf0110166]. Cicoli:2012vwM. Cicoli, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, “D-Branes at del Pezzo Singularities: Global Embedding and Moduli Stabilisation,” JHEP 1209, 019 (2012) doi:10.1007/JHEP09(2012)019 [arXiv:https://arxiv.org/pdf/1206.5237.pdf1206.5237 [hep-th]]. Cicoli:2011itM. Cicoli, M. Kreuzer and C. Mayrhofer, “Toric K3-Fibred Calabi-Yau Manifolds with del Pezzo Divisors for String Compactifications,” JHEP 1202, 002 (2012) [arXiv:http://arxiv.org/abs/1107.03831107.0383 [hep-th]].Blumenhagen:2010pv R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of Line Bundles: A Computational Algorithm,” J. Math. Phys. 51 (2010) 103525, http://www.arXiv.org/abs/1003.52171003.5217.cohomCalg:Implementation “cohomCalg package.” Download link: <http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg/>, 2010. High-performance line bundle cohomology computation based on <cit.>.Braun:2012vhA. P. Braun, J. Knapp, E. Scheidegger, H. Skarke and N. O. Walliser, “PALP - a User Manual,” [arXiV:http://arxiv.org/abs/1205.41471205.4147 [math.AG]].Conlon:2005jmJ. P. Conlon and F. Quevedo, “Kahler moduli inflation,” JHEP 0601, 146 (2006) doi:10.1088/1126-6708/2006/01/146 [hep-th/https://arxiv.org/pdf/hep-th/0509012.pdf0509012].Cicoli:2008gpM. Cicoli, C. P. Burgess and F. Quevedo, “Fibre Inflation: Observable Gravity Waves from IIB String Compactifications,” JCAP 0903, 013 (2009) doi:10.1088/1475-7516/2009/03/013 [arXiv:https://arxiv.org/pdf/0808.0691.pdf0808.0691 [hep-th]].Burgess:2016owbC. P. Burgess, M. Cicoli, S. de Alwis and F. Quevedo, “Robust Inflation from Fibrous Strings,” JCAP 1605, no. 05, 032 (2016) doi:10.1088/1475-7516/2016/05/032 [arXiv:https://arxiv.org/pdf/1603.06789.pdf1603.06789 [hep-th]].Cicoli:2016chbM. Cicoli, D. Ciupke, S. de Alwis and F. Muia, “α' Inflation: Moduli Stabilisation and Observable Tensors from Higher Derivatives,” [arXiv:https://arxiv.org/pdf/1607.01395.pdf1607.01395 [hep-th]].Becker:2002nn K. Becker, M. Becker, M. Haack and J. Louis, “Supersymmetry breaking and alpha-prime corrections to flux induced potentials,” JHEP 0206 (2002) 060 doi:10.1088/1126-6708/2002/06/060 [hep-th/https://arxiv.org/pdf/hep-th/0204254.pdf0204254].Altman R. Altman, J. Gray, Y. He, V. Jejjala, and B. D. Nelson. Manuscript in preparation. | http://arxiv.org/abs/1706.09070v1 | {
"authors": [
"Ross Altman",
"Yang-Hui He",
"Vishnu Jejjala",
"Brent D. Nelson"
],
"categories": [
"hep-th",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170627230612",
"title": "New Large Volume Solutions"
} |
Web Science Group Heinrich-Heine-University Duesseldorf [email protected] Science Group Heinrich-Heine-University Duesseldorf [email protected] this paper we consider the modern theory of the Bayesian brain from cognitive neurosciences in the light ofrecommender systems and expose potentials for our community.In particular, we elaborate on noisy user feedback and the thus resulting multicomponent user models, which have indeed a biological origin. In real user experiments we observe the impact of both factors directly in a repeated rating task along with recommendation.As a consequence, this contribution supports the plausibility of contemporary theories of mind in the context of recommender systems and can be understood as a solicitation to integrate ideas of cognitive neurosciences into our systems in order to further improve the prediction of human behaviour.Bayesian Brain meets Bayesian RecommenderTowards Systems with Empathy for the Human Nature Sergej Sizov December 30, 2023 ============================================================================================§ INTRODUCTIONIn our community of recommender systems, there are continual efforts to make predictions more precise and systems more efficient and user-friendly. In doing so, the classic approach is to model the relationship between a user and items in terms of optimising a target function in order to predict future user decisions based on training data. However, there are two major problems that are caused by human nature. First of all, many studies prove that users are not entirely certain about a decision so that a given rating may fluctuate when the rating task is repeated (noisy user feedback) <cit.>. Second, optimising a target function might not sufficiently account for dynamic changes in behaviour (need for multicomponent user models) which again can be proven in systematic user experiments<cit.>.The theories of cognitive neurosciences know these phenomena and describe these aspects of human cognition by means of probabilistic models. For example, the origin of volatile decision-making and noisy user feedback is due to the irregular transmission of informations through the synaptic cleft. A Bayesian formulation of this effect leads directly to multicomponent models as an explicit consequence of user noise. Both of these factors, noisy user feedback and multicomponent models, have recently been proven to have a significant impact on prediction quality in recommender systems <cit.>. Therefore, this contributions seeks the benefits of implementing neuroscientific models and will give experimental indication for possible deficiencies within the field of recommender research in case of omission.§ THEORY AND EXPERIMENTSThe Bayesian brain theory is a composition of Bayesian inference and theoretical neurology. We will first demonstrate similarity to Bayesian recommender systems and then discuss the modelling of noisy data as well as multicomponent user models.§.§ Bayesian Learning BasicsFrom the perspective of cognitive neuroscience, it all starts with the brain observing sensory input Y (visual, auditory, etc.) and making an estimate of the state of the world X <cit.>. To continiously improve this estimation - or subjective beliefs to be more precisely -the brain has to learn by comparing reality against predictions based on these beliefs <cit.>. This makes the human brain a highly sophisticated recommender system itself. The Bayes Theorem provides the basis for the processing of beliefs along with real world evidence. Those confirmed or modified beliefs are thereby brought to ever new situations. Mathematically spoken, the posterior serves as prior in subsequent cognitions. For multiple independent sensory observations y=(y_1,…,y_n) we yieldP(X| y)∝ P(X| y_1)·…· P(X| y_t-1)· P(X| y_t)where ∏_i=1^t-1P(X| y_i) is the posterior of X given sensory data until time t-1 and serves as prior for time t (learning from the past). When the world state itself changes while making observations, we need to consider a transition probabilities P(X_t| X_t-1)P(X| y)∝P(X_t| X_t-1) ·∏_i=1^t-1 P(X_t-1| y_i) · P(X| y_t)which is the basis of learning by iteration. Here, we can already see the similarity to Bayesian recommender systems clearly <cit.>. But how does the brain actually model prior probabilities? Mathematically, this question is pointless when hierarchical networks are used since these models optimise the priors themselves through mutual back-propagation of predictions and forward-feeding of prediction errors <cit.> as to see in Figure <ref>. The optimisation task itself is done by neuron clusters (agents) via minimising the so called free energy <cit.>. In consequence, beliefs do exist in the form of probability densities, from which particular draws are made for decision making. We will see later that these distributions indeed exist in the case of product ratings.§.§ Neural Noise and Decision-Making Message passing works by relaying electrical signals from one neuron to the next through the synaptic gap by means of neurotransmitters. However, same signals never result into emitting the same amount of transmitters (neural noise) <cit.>. This noise may raise too weak (or sufficient) signals above (or below) a certain threshold, causing a neuron to inhibit (or to fire) <cit.>. In fact, for the firing of a neuron, we can only specify a probability <cit.>. In a recommender's language, biological irregularity implies that every time a decision-making is repeated, other prior probabilities might be used and thus the resulting belief is never quite the same as before. In a systematic experiment, real users repeatedly rated theatrical trailers on a 5-star scale. It turns out that only 35% of all users show constant rating behaviour, whereas about 50% use two different answer categories and 15% of all users make use of three or more categories. Figure <ref> is a characteristic example for these results. This sows that individuals are not able to perfectly reproduce their decision-making.These fluctuations can be explained by the theory of neural noise, and have a direct effect on recommender systems. Assuming the model based prediction to be π=3, a user rating r=4 can not be seen as a deviation according to Figure <ref>. Furthermore, by gathering information about temporary belief posteriors <cit.>, it can be proven that all user responses (aggregated for each item) hold the same expectation. This is an indication for a common source of noise with constant magnitude, i.e. the manifestation of neural noise. §.§ Modelling User Preferences When it comes to a repeated product rating where the participant does not remember his previous response, as induced in our controlled experiment, the process of decision making is restarted. Accordingly, the participant receives a new and slightly different belief distribution for each rating trial. This has been mathematically explained in <cit.> and implies the need for multicomponent user models, which have recently been adressed in recommender systems research <cit.>.Figure <ref> shows the RMSEs of three different systems utilised in our experiment, based on one-component models along with the scores each system has achieved in each trial. It is apparent that some draws (scores) can not be drawn from the corresponding distribution. This indicates that users had changed their rating behaviour and sampled from different distributions for different trials. It can be proven via hypothesis testing that rating behaviour of trial 1 and 5 significantly deviates from trials 2 to 4.This can be explained by memory effects: In trial 1, decision-making was initialised for the first time. After the trial 2, participants got aware that the experiment was about repetition and so started to remember their ratings. Therefore, belief distributions remained more or less the same. After the trial 4 plus constant addition of new distractors, short-time memory was not able to keep all previous information and further decision-making produced different belief distributions again. These findings within simple recommendation scenarios can be entirely described by the Bayesian brain theory and may help systems to learn human behaviour more naturally. § CONCLUSION We have shown that the Bayesian brain theory uses the same models as Bayesian recommender systems. In addition, we have shown that the Bayesian brain theory is already in a position to model quantities whose impact can beseen in recommendation scenarios. We have demonstrated this by the example of noisy user feedback and multicomponent models. The impact of these factors has been discussed briefly, i.e. the interpretation of correct and false predictions has to be considered more differentiated whereas the choice of user models holds a strong dependency on time. We therefore recommend to adopt corresponding models from neurosciences in order to optimise recommender systems in terms of imitating human behaviour. ACM-Reference-Format | http://arxiv.org/abs/1706.08319v2 | {
"authors": [
"Kevin Jasberg",
"Sergej Sizov"
],
"categories": [
"cs.HC"
],
"primary_category": "cs.HC",
"published": "20170626110234",
"title": "Bayesian Brain meets Bayesian Recommender - Towards Systems with Empathy for the Human Nature"
} |
Large Noether number]Finite groups with large Noether number are almost cyclicThe research was partly supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K115799. The second and third authors were also funded by the National Research, Development and Innovation Office (NKFIH) Grant No. ERC_HU_15 118286. Their work on the project leading to this application has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 741420). The second author received funding from ERC 648017 and was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Department of MathematicsCentral European UniversityNádor utca 9H-1051 Budapest, Hungary [email protected] Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesReáltanoda utca 13-15H-1053, Budapest, Hungary [email protected] Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesReáltanoda utca 13-15H-1053, Budapest, Hungary [email protected] polynomes invariants, majorant de Noether, groupes simples de type de Lie[2010]13A50, 20D06, (20D08, 20D99)Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order |G| of a finite group G, then the polynomial invariants of G are generated by polynomials of degrees at most |G|. Let β(G) denote the largest indispensable degree in such generating sets. Cziszter and Domokos recently described finite groups G with |G|/β(G) at most 2. We prove an asymptotic extension of their result. Namely, |G|/β(G) is bounded for a finite group G if and only if G has a characteristic cyclic subgroup of bounded index. In the course of the proof we obtain the following surprising result. If S is a finite simple group of Lie type or a sporadic group then we have β(S) ≤|S|^39/40. We ask a number of questions motivated by our results. Noether, Fleischmann et Fogarty ont montré que si le caractéristique du corps sous-jacent ne divise pas l'ordre |G| d'un groupe fini, alors l'anneau de polynomes invariants de G est engendré par des polynomes de degré au plus égal à |G|. Notons par β(G) le plus haut degré indispensable en un tel système de générateurs. Cziszter et Domokos ont récemment décrit les groupes finis G tels que |G|/β(G) est au plus égal à 2.Nous démontrons une extension asymptotique de leur résultat, à savoir que |G|/β(G) est borné pour un groupe fini G si et seulement s'il admet un sous-groupe caractéristique cyclique d'indice borné. Durant la démonstration nous trouvons le résultat surprenant suivant : si S est un groupe fini simple de type de Lie ou l'un des groupes sporadiques alors on a β(S) ≤|S|^39/40. Nous posons égalament quelques questions motivées par nos résultats. [ László Pyber December 30, 2023 =====================§ INTRODUCTION Let G be a finite group and V an FG-module of finite dimension over a field F. By a classical theorem of Noether <cit.>, the algebra of polynomial invariants on V, denoted by F[V]^G, is finitely generated. Define β(G, V) to be the smallest integer d such that F[V]^G is generated by elements of degrees at most d. In case the characteristic of F does not divide |G|, the numbers β(G,V) have a largest value as V ranges over the finite dimensional FG-modules. This number is called the Noether number and is denoted by β(G). The notation β(G) suppresses the dependence on the field but it should not cause misunderstanding. In fact, for fields of the same characteristic the Noether number is the same and we may assume that F is algebraically closed. See <cit.> for details.Noether <cit.> also proved that β(G)≤ |G| over fields of characteristic 0. This bound was verified independently by Fleischmann <cit.> and Fogarty <cit.> to hold also in positive characteristics not dividing |G|. For characteristics dividing |G|, a deep result of Symonds <cit.> states that β(G,V)≤(V)(|G|-1).From now on throughout the whole paper, except in Question <ref>, we assume that the characteristic of the field F is 0 or is coprime to the order of G.Schmid <cit.> proved that over the field of complex numbers β(G)=|G| holds only when G is cyclic. This was sharpened by Domokos and Hegedűs <cit.> (and later by Sezer <cit.> in positive coprime characteristic) to β(G)≤3/4|G| unless G is cyclic.An important ingredient in Schmid's argument was to show that β(G)≥β(H) holds for any subgroup H≤ G. In particular, β(G) is bounded from below by the maximal order of the elements in G, that is, the Noether index n(G) = |G|/β(G) of a finite group G is at most the minimal index of a cyclic subgroup in G.Recently Cziszter and Domokos <cit.> described finite groups G with n(G) at most 2. Their deep result <cit.> states that for a finite group G (with order not divisible by the characteristic of F) we have n(G) ≤ 2 if and only if G has a cyclic subgroup of index at most 2, or G is isomorphic to Z_3× Z_3, Z_2× Z_2× Z_2, the alternating group A_4, or the binary tetrahedral group A_4. In particular, the inequality n(G) ≤ 2 implies that G has a cyclic subgroup of index at most 4.Our main result is as follows.Let G be a finite group with Noether index n(G). Then G has a characteristic cyclic subgroup of index at most n(G)^10 log_2k where k denotes the maximum of 2^10 and the largest degree of a non-Abelian alternating composition factor of G, if such exists. Furthermore if G is solvable, then G has a characteristic cyclic subgroup of index at most n(G)^10. In view of Theorem <ref> and Section <ref>, the bound n(G)^10 holds even for a large class of non-solvable groups.Theorem <ref> has a consequence which can be viewed as an asymptotic version of the afore-mentioned result of Cziszter and Domokos.Let G be a finite group with Noether index n(G). If G is nonsol-vable, then n(G)>2.7 and G has a characteristic cyclic subgroup of index at most n(G)^100 + 10 log_2log_2 n(G). If G is solvable then G contains a characteristic cyclic subgroup of index at most n(G)^10. It is an open question whether there exists a polynomial bound in n(G) for the index of a characteristic cyclic subgroup in an arbitrary finite group G. Theorem <ref> is a major step in answering this question.As a step in our proofs we obtain a result which may be of independent interest.Let S be a finite simple group of Lie type or a sporadic simple group. Then n(S) ≥ |S|^1/40. It would be interesting to know if the bound in Theorem <ref> holds for alternating groups of arbitrarily large degrees. Our methods are sufficient only for degrees up to 17. For degrees no greater than 17 (but at least 5) the claim follows from the remark after Lemma <ref>.Assume that, for some fixed constant ϵ > 0, we have n(S) ≥|S|^ϵ for every alternating group S of degree at least 5. Then our proofs show that, for some other (computable) fixed constant ϵ' > 0 with ϵ' ≤ 0.1, any finite group G has a characteristic cyclic subgroup of index at most n(G)^1/ϵ'.§ AFFINE GROUPSOur main aim in the present section is to give upper bounds on β(G) for the Frobenius group G≅ Z_p⋊ Z_n, where p is a prime and n| p-1.It is an open conjecture of Pawale <cit.> that β(Z_p⋊ Z_q)=p+q-1 for a prime q. This is verified for q=2 <cit.> (where β(D_2n)=n+1 is shown for composite n, as well) and for q=3 <cit.>. Cziszter and Domokos obtain an upper bound which we extend to a more general one if q is not a prime. See Lemma <ref>, Theorem <ref> and Corollary <ref>.In this section we rely heavily on the techniques developed by Cziszter and Domokos. For convenience and completeness we include here those that we need. However, we try to simplify and not include them in full generality.Let G be the Frobenius group of order pn with Z_p≤ G≤_p. Then every G-module has a Z_p-eigenbasis permuted up to scalars by G.The regular module is relevant because every irreducible Z_p-character occurs in it. For every Z_p-module V the polynomial invariants are linear combinations of Z_p-invariant monomials. The Z_p-invariant monomials correspond to 0-sum sequences of irreducible Z_p-characters. These motivate all the definitions below.Let𝒴={y_1,…,y_p} be the set of variables from F[Z_p] that are Z_p-eigenvectors and y_1 is Z_p-invariant. For a monomial f=∏_i=1^p y_i^a_i let us define b(f)=∏_a_i>0y_i. Let g_1=b(f) and construct recursively the finite list of monomials g_1, g_2,… in such a way that g_k+1=b(f/∏_j=1^k g_j) for every k, stopping if f=∏ g_j.We call this list the row decomposition of f. (In <cit.> the corresponding list of irreducible Z_p-characters is considered and called the row decomposition.) This list consists of monomials each dividing the previous one and the exponent of every variable y_i is at most 1.Let l be a positive integer. Suppose a set of variables {x_1,…, x_l} consists of Z_p-eigenvectors on which G/Z_p acts by permutation, but not necessarily transitively.For each x_i there is a corresponding unique y_i̅∈𝒴 having the same Z_p-action on them.This defines a map m↦ f_m from the monomials in {x_1,…, x_l} to the monomials in 𝒴 by m=∏ x_i^a_i↦ m_f=∏ y_i̅^a_i. This map is G/Z_p-equivariant. Moreover, the Z_p-action on m is the same as on f_m, so m is Z_p-invariant if and only if f_m is.Given a monomial m we determine the row decomposition g_1,…, g_h of f_m. Suppose that for every G-orbit 𝒪⊆𝒴 and every index i<h the following holds. If g_i involves some variables from𝒪, but not all then g_i+1 involves fewer variables than g_i does. Such a monomial m is called gapless in <cit.>. If g_i=g_i+1 for a gapless monomial m then g_i is G/Z_p-invariant. In particular, as nontrivial G/Z_p-orbits on 𝒴 are of length n,if y_1∤ g_i and (g_i)<n then (g_i+1)<(g_i). Let M =⊕_d=0^∞ M_d be a graded module over a commutative graded F-algebra R=⊕_d=0^∞ R_d. We also assume that R_0 = F when 1∈ R and R_0 = 0 otherwise.Define M_≤ s =⊕_d=0^s M_d, a subspace of M, and R_+ =⊕_d=1^∞ R_d◃ R a maximal ideal. The subalgebra of R generated by R_≤ sis denoted by F[R_≤ s]. Define β(M, R)=min{s| M=⟨ M_≤ s⟩_R_+}, the highest degree needed for an R_+-generating set of M. In other words, it is the highest degree of nonzero components of M/MR_+ (the factor space M/MR_+ inherits the grading).The following three propositions from <cit.> will be used in the proof of Theorem <ref>. They are paraphrased and not stated in their full generality.<cit.> Let G be the Frobenius group of order pn with Z_p≤ G≤_p. Let V be an FG-module, L=F[V] the polynomial algebra, R=L^G its invariants. Suppose the variables of L are permuted by G up to non-zero scalar multiples. Then the vector space L_+/L_+R_+ is spanned by monomials of the form b_1⋯ b_r m, where the b_i are Z_p-invariant of degree 1 or of prime degree q_i|n and m has a gapless divisor of degree at least (m)-(p-1). (Note that the so-called bricks mentioned in the original version of Proposition <ref> are Z_p-invariant.)<cit.>Let G be the Frobenius group of order pn with Z_p≤ G≤_p. Let V be an FG-module, L=F[V] the polynomial algebra, R=L^G and I=L^Z_p its invariants. Then for every s≥ 1 the following bound is valid:β(L_+,R)≤ (n-1)s+max{β(L_+/L_+R_+,I),β(L_+/L_+R_+,F[I_≤ s])-s}.(The original version of Proposition <ref>holds for the generalized Noether numbers β_r, however we only use the case r=1.)<cit.> Let S be a sequence over Z_p with maximal multiplicity h. If |S|≥ p then S has a zero-sum subsequence T⊆ S of length |T|≤ h. The following proposition is a simple corollary.Suppose f is a monomial in 𝒴 of degree at least p such that the exponent of each y_i∈𝒴 is at most h. Then f has a Z_p-invariant submonomial f^' such that (f^')≤ h.Let f=∏ y_i^a_i. Fix a generator element z∈ Z_p and a primitive p-th root of unity, μ∈ F. Define S to be the sequence over ℤ/ℤ_p consisting of a_i copies of the exponent of μ as the eigenvalue of z on y_i. This satisfies the assumptions of the previous lemma. Let then f^' be the product of the elements of T, it is a submonomial of degree |T|≤ h. That T is zero-sum means exactly that f^' is Z_p-invariant. The following upper bound is used frequently.Let E = (Z_p)^k be a non-cyclic elementary Abelian p-group for some prime p. Then β(E) = kp - k + 1. Thus β(E) < |E|^0.8. Furthermore if |E| ≠ 2^2, 3^2, 5^2, then β(E) < |E|^0.67.The first statement is the combination of Olson's Theorem <cit.> and a “folklore result” of invariant theory <cit.>. We have β(E) < |E|^0.8 since k ≥ 2. The other statement follows from an easy calculation. We reformulate the result of <cit.> for affine groups in a form that can be applied in inductive arguments. For our purposes the following lemma is sufficient. However, as the proof shows, β(G)≤ (1+ε)p√(q) is true for fixed ε > 0 and for p, q large enough.Let q| p-1 for primes p,q and let G≤_p be of order pq. Then β(G)≤ pq^0.8.If q=2, then β(G)=p+1<p2^0.8 (see <cit.> and <cit.>). Let q>2. By <cit.> we have β(G)≤3/2(p+q(q-2))-2< 3p-2 if p>q(q-2). If here q≥ 5 then 3p-2<pq^0.8. If q=3 then β(G) is at most 3/2(p+q(q-2))-2=3/2p+2.5<p3^0.8, as required.So let p<q(q-2), in particular q>3. In this case <cit.> concludes β(G)≤ 2p+(q-2)q-2 and β(G)≤ 2p+(q-2)(c-1)-2 if there exists c≤ q such that c(c-1)<2p<c(c+1). Note that if q(q-1)<2p then q<√(2p) and if q(q-1)>2p then there exists c≤ q such that c(c-1)<2p<c(c+1) and c-1<√(2p). So in both cases β(G)≤ 2p+(q-2)√(2p)-2. If q=5 then p<15 and 5| p-1 imply p=11. We have β(G)≤ 22+3√(22)-2<11·5^0.8.Finally let q≥ 7. Using q-2<√(q)√(p/2) we getβ(G)< p(2+√(q)√(p/2)√(2p)/p)=p(2+√(q)).As q^0.8-q^0.5 is increasing and 7^0.8-7^0.5>2 we get the claimed bound. Let G be the Frobenius group of order pn with Z_p≤ G≤_p. Suppose that n ≥ 6 has no prime divisor larger than p/√(n). Then β(G)< 2p√(n).Let Vbe an arbitrary FG-module, L=F[V] the polynomial algebra and R=L^G and I=L^Z_p the respective group invariants. Put s=[p/√(n)]. As β(Z_p)=p we have β(L_+/L_+R_+,I)≤ p. Hence by Proposition <ref>,β(G,V)≤ (n-1)s+max{p,β(L_+/L_+R_+,F[I_≤ s])-s}.The first term of this sum is smaller than p√(n) so it is enough to prove thatβ(L_+/L_+R_+,F[I_≤ s])≤ p√(n)+s.We assume that the basis of the dual module V^* is a Z_p-eigenbasis {x_1,x_2,…,x_l} permuted by G/Z_p.Now apply Proposition <ref>. The space L_+/L_+R_+ is spanned by monomials m that either have a Z_p-invariant divisor of degree at most s or have a gapless monomial divisor of degree at least (m)-(p-1). The former kind are in F[I_≤ s] so we need an upper bound for the degrees of the latter kind. More precisely, we have that if m^' is the largest degree gapless monomial with no Z_p-invariant divisor of degree at most s thenβ(L_+/L_+R_+,F[I_≤ s])≤ p-1+(m^'). Consider now the row decomposition g_1,…,g_h of f_m^'. In the submonomial f=g_1+g_2+⋯+g_s of f_m^' all the exponents are at most s, so by Proposition <ref>, f≤ p-1. This implies that (g_s)≤ (p-1)/s. It is below √(n)+1 because if s= (p/√(n))-ε then(p/√(n)-ε)(√(n)+1)=p+p/√(n)-ε√(n)-ε>p-1. So (g_s)≤√(n)+1. In particular, (g_s)<n and by (<ref>), (g_i+1)<(g_i) for i≥ s. Hence we have the following bound on the degree.(m^')=∑_i=1^s(g_i)+∑_i=s+1^h (g_i)<p-1+1/2√(n)(√(n)+1)=p-1+n+√(n)/2. Now (<ref>) and2+n/2(p-1)≤ 2.5<√(n)+1/√(n) (as n>5) imply thatβ(L_+/L_+R_+,F[I_≤ s]) ≤ p-1+(m^')≤ 2(p-1)+n+√(n)/2 ==(p-1)(2+n/2(p-1))+√(n)/2<<(p-1)(√(n)+1/√(n))+√(n)-1< p√(n)+s,which is exactly (<ref>). We continue with a useful tool.[Schmid <cit.> and Sezer <cit.>]Let H be a subgroup and N a normal subgroup in a finite group G. Then β(G) ≤β(N) β(G/N) and β(G) ≤|G:H| β(H).See Schmid <cit.> and Sezer <cit.>. Let N be a normal subgroup of prime order p in a finite group G. Assume that N=C_G(N) and that G/N is cyclic of order m prime to p. Then β(G) ≤ pm^0.9.The group G is an affine Frobenius group. If m is prime, then the claim follows from Lemma <ref>. For m=4 we have β(G)≤ p+6<4^0.9p by <cit.>. If m has a prime divisor q > p/√(m) then first, m<p<q√(m) implies q>√(m). Second, Z_p⋊ Z_q≤ G, so by Lemma <ref> and Lemma <ref>, β(G)≤m/qpq^0.8=mpq^-0.2<pm^0.9. Finally, if m ≥ 6 has no prime divisor larger than p/√(m) then by Theorem <ref> we have β(G)≤ 2p√(m)≤ pm^0.9. § SOLVABLE GROUPSIn this section we will give a general upper bound for β(G) in case G is a finite solvable group.Let C be a characteristic cyclic subgroup of maximal order in a finite nilpotent group G. Then β(G) ≤|C|^0.2|G|^0.8.Suppose that G is a counterexample with |G| minimal. By the afore-mentioned result of Noether <cit.>, Fleischmann <cit.> and Fogarty <cit.>, G must be non-cyclic. By Lemma <ref>, G must also be a p-group for some prime p. Then G/Φ(G) must be a non-cyclic elementary Abelian p-group where Φ(G) denotes the Frattini subgroup of G. By Lemma <ref>, β(G/Φ(G)) < |G/Φ(G)|^0.8. By minimality, there exists a characteristic cyclic subgroup C in Φ(G), characteristic in G, such that β(Φ(G)) ≤|C|^0.2|Φ(G)|^0.8. We get a contradiction using Lemma <ref>. We repeat the following result from the Introduction.[Domokos and Hegedűs <cit.> and Sezer <cit.>]For any non-cyclic finite group G we have β(G) ≤3/4|G|. The next bound holds for every finite solvable group, but it is slightly weaker than the one in Proposition <ref>.Let C be a characteristic cyclic subgroup of maximal order in a finite solvable group G. Then β(G) ≤|C|^0.1|G|^0.9.By Proposition <ref>, we may assume that G is not nilpotent. Consider the Fitting subgroup F(G) and the Frattini subgroup Φ(G) of G. Since F(G) is normal in G, we have, by <cit.>, that Φ(F(G)) ≤Φ(G) ≤ F(G). Thus F(G)/Φ(G) is a product of elementary Abelian groups. The socle of the group G/Φ(G) is F(G)/Φ(G) on which G/F(G) acts completely reducibly (in possibly mixed characteristic) and faithfully (see <cit.>).Let N be the product of O_p(G) ∩Φ(F(G)) for all primes p for which O_p(G) is cyclic, together with the subgroups O_p(G) ∩Φ(F(G)) for all primes p for which p divides |F(G)/Φ(G)| but p^2 does not, together with O_p(G) ∩Φ(G) for all primes p for which p^2 divides |F(G)/Φ(G)|. Clearly, F(G)/N is a faithful G/F(G)-module (of possibly mixed characteristic) with a completely reducible, faithful quotient.We claim that the bound in the statement of the theorem holds when C is taken to be the product of the (direct) product of all cyclic Sylow subgroups of F(G) and a characteristic cyclic subgroup of maximal order in N. By our choice of C and Proposition <ref>, we have β(N) ≤(|C|/s)^0.1|N|^0.9, where s denotes the product of the primes for which O_p(G) is cyclic. In order to finish the proof of the theorem, it is sufficient to show that β(G/N) ≤ s^0.1|G/N|^0.9.This latter bound will follow from the following claim. Let H be a finite solvable group with a normal subgroup V that is the direct product of elementary Abelian normal subgroups of H. Let π be the set of prime divisors of |V| and write V in the form ×_p ∈π O_p(V). Assume that V is self-centralizing in H and that the H/V-module V has a completely reducible, faithful quotient module. We claim that β(H) ≤ s^0.1|H|^0.9 where s denotes the product of all primes p for which |O_p(V)| = p.We prove the claim by induction on |π|. Let p ∈π. Assume that |π| = 1. If |V| = p then Corollary <ref> gives the claim. Assume that |V| ≥ p^2. By a result of Pálfy <cit.> and Wolf <cit.>, |H/V| < |V|^2.3. First assume that |V| is different from 2^2, 3^2, 5^2. By Lemma <ref> and Lemma <ref>,β(H) < |V|^0.67|H/V| < |H|^0.9.Thus assume that |V| = 2^2, 3^2, or 5^2. If |H| < |V|^2, thenβ(H) < |V|^0.8|H/V| < |H|^0.9,again by Lemmas <ref> and <ref>. So assume also that |H| ≥|V|^2, in particular that H/V is not cyclic. By Theorem <ref>, we have β(H) < 3/4|V|^0.8|H/V| ≤|H|^0.9, since H is solvable.Assume that |π| > 1. The group H can be viewed as a subdirect product in Y = Y_p× Y_p' where Y_p and Y_p' are solvable groups with the following properties. There is an elementary Abelian normal p-subgroup V_p in Y_p and a direct product V_p' of elementary Abelian normal p'-subgroups in Y_p' such that both the Y_p/V_p-module V_p and the Y_p'/V_p'-module V_p' have a completely reducible, faithful quotient module. Let N be the kernel of the projection of H onto Y_p. Clearly, N satisfies the inductive hypothesis with the set π∖{ p } of primes. Thus Lemma <ref> gives the bound of the claim. § FINITE SIMPLE GROUPS OF LIE TYPE The following is inherent in <cit.> without being explicitly stated. We reproduce their argument with a slight modification.If G is a nonsolvable finite group then n(G)> 2.7.By Lemma <ref>, it is enough to prove this for minimal non-Abelian simple groups. By a theorem of Thompson <cit.> these are (3,3), Suzuki groups (2^p), for p>2 prime and (2,q), where q=2^p, 3^p (p a prime, p>2 for q=3^p) or q>3 is a prime such that q ≡± 2 5.If G≅(2^p) or G≅(2,2^p), for p>2 then G has an elementary Abelian subgroup H≅ Z_2^3 of index k=|G:H|≥ 63. So n(G)≥8k/2k+3 = 4 - 12/2k+3 > 3.9. (See the proof of <cit.>.)If G≅(3,3) or G≅(2,3^p), for p>2 then G has an elementary Abelian subgroup H≅ Z_3^2 of index k=|G:H|≥ 624. So n(G)≥9k/3k+2=3-6/3k+2>2.9. (See the proof of <cit.>.)If G≅(2,4)≅ A_5 or G≅(2,p) then G contains a subgroup H≅ A_4 of index k=|G:H|≥ 5. So n(G)≥6k/2k+1=3-3/2k+1>2.7. (See the proof of <cit.>.) This implies that if G is a nonsolvable group with order less than 2.7^40 then β(G)< |G|/2.7 <|G|^39/40. The following theorem claims this bound for every finite simple group of Lie type.Let S be a finite simple group of Lie type. Then β(S)≤ |S|^39/40, in other words, n(S)≥ |S|^1/40.The proof requires a case by case check of the 16 families of simple groups of Lie type. In each case we find a subgroup E≤ S with Noether index n(E) relatively large, more precisely n(E)≥ |S|^1/40 and hence n(S)≥ n(E)≥ |S|^1/40 as required.If the rank of the group is at least 2 then we find a non-cyclic elementary Abelian p-subgroup E in the defining characteristic p satisfying |E|^8>|S|. The relevant data can be found for example in <cit.> which we summarise below. By Lemma <ref> we have n(S)≥ n(E)>|S|^1/40 which implies our statement in this case. However Table <ref> gives the best bounds for each type that can be obtained this way. (For notational ease C_2(2^a) is used instead of B_2(2^a) below. The Tits group is not in the list, but using a Sylow 2-subgroup we can easily obtain n(S)>|S|^0.2 for that S.)So this method gives a better bound log_|S|n(E)≥ 0.051> 1/20, the worst group being S≅(3,2), with |E|=4. The rank 1 case remains. First let p>3 be a prime and S=(2,p). Then S contains a Frobenius subgroup H≅ Z_p⋊ Z_(p-1)/2 of index |S:H|=p+1. By Corollary <ref>, we have the bound β(H)≤ p (p-1/2)^0.9. It follows by Lemma <ref> that β(S)≤ (p+1)β(H)≤ (p+1)p(p-1/2)^0.9. This implies β(S)<|S|^1-1/40 for p≥ 13.For S≅(2,p) with p=5, 7, 11 the order of the group S is less than 2.7^40, so the theorem holds by the remark after Lemma <ref>.Finally let S=(2,q) where q=p^f, p a prime and f>1. Then S=(2,q) contains an elementary Abelian subgroup E of order p^f for which, by Lemma <ref>, β(E)=(p-1)f+1<p^0.8f. Since |S|<q^3=p^3f, we haven(E)=p^f/(p-1)f+1>p^0.2f>|S|^1/15.This finishes the proof. § A REDUCTION TO ALMOST SIMPLE GROUPSWe will proceed to prove the following result.Let G be a finite group and C a characteristic cyclic subgroup in G of largest size. Then β(G) ≤|C|^ϵ|G|^1-ϵ with ϵ = (10 log_2 k)^-1, where k denotes the maximum of 2^10 and the largest degree of a non-Abelian alternating composition factor of G, if such exists. If G is solvable, then β(G) ≤|C|^0.1|G|^0.9. The second statement of Theorem <ref> is Theorem <ref>. The following result reduces the proof of Theorem <ref> to a question on almost simple groups.Let G be a finite group. Let ϵ be a constant with 0 < ϵ≤ 0.1 such that β(H) ≤ 2^-ϵ|H|^1-ϵ for any (if any) almost simple group H whose socle is a composition factor of G. Let C be a characteristic cyclic subgroup of maximal order in G. Then β(G) ≤|C|^ϵ|G|^1-ϵ. Note that for any finite group G the ϵ in Theorem <ref> can be taken to be positive by Theorem <ref>. Let G be a counterexample to the statement of Theorem <ref> with |G| minimal. By Theorem <ref>, G cannot be solvable. Let R be the solvable radical of G. By Theorem <ref> there exists a characteristic cyclic subgroup C of R (which is also characteristic in G) such that β(R) ≤|C|^ϵ|R|^1 - ϵ. If R ≠ 1, then, by minimality, β(G/R) ≤|G/R|^1 - ϵ, and so Lemma <ref> gives a contradiction. Thus R = 1.Let S be the socle of G. This is a direct product of, say r ≥ 1, non-Abelian simple groups. Let K be the kernel of the action of G on S. By our hypothesis on almost simple groups and by Lemma <ref>, β(K) ≤|K|^1-ϵ/2^ϵ· r.Let T = G/K. We claim that β(T) ≤ 2^ϵ (r-1)|T|^1 - ϵ. By Lemma <ref> this would yield β(G) ≤|G|^1 - ϵ, giving us a contradiction.To prove our claim we will show that if P is a permutation group of degree n such that |P| ≤ |T|, n ≤ r, and every non-Abelian composition factor (if any) of P is also a composition factor of T, then β(P) ≤ 2^ϵ (n-1)|P|^1 - ϵ. Suppose that P acts on a set Ω of size n. Let P be a counterexample to the bound of this latter claim with n minimal. Then n > 1. Suppose that P is not transitive. Then P has an orbit Δ of size, say k, with k < n. Let B be the kernel of the action of P on Δ. Then β(P/B) ≤ 2^ϵ (k-1)|P/B|^1 - ϵ and β(B) ≤ 2^ϵ ((n-k)-1) |B|^1 - ϵ. We get a contradiction using Lemma <ref>. So P must be transitive. Suppose that P acts imprimitively on Ω. Let Σ be a (non-trivial) system of blocks with each block of size k with 1 < k < n. Let B be the kernel of the action of P on Σ. By minimality, β(P/B) ≤ 2^ϵ ((n/k)-1)|P/B|^1 - ϵ. By minimality and Lemma <ref>, we also have β(B) ≤ 2^ϵ (k-1)(n/k)|B|^1 - ϵ. Again, Lemma <ref> gives a contradiction. Thus P must be primitive. If the solvable radical of P is trivial, we get β(P) ≤|P|^1- ϵ by |P| < |G|. In fact, the same conclusion holds unless n is prime and P is meta-cyclic. In this latter case Corollary <ref> gives β(P) ≤ n^ϵ|P|^1 - ϵ. We get a contradiction by n ≤ 2^n-1. § ALMOST SIMPLE GROUPSLet H be an almost simple group. In view of Theorem <ref> in this section we will give a bound for β(H) of the form 2^-ϵ|H|^1-ϵ where ϵ is such that 0 < ϵ≤ 0.1. Let S be the socle of H. §.§ The case when S is a finite simple group of Lie type We first show that we may take ϵ = 0.01. By Theorem <ref>, β(S) ≤|S|^39/40. By this and Lemma <ref>, we getβ(H) ≤ |H:S| ·|S|^39/40 = |H:S|^0.01·|S|^0.01 - (1/40)·|H|^0.99.Thus it is sufficient to see that |H:S|^0.01·|S|^0.01 - (1/40)≤ 2^-0.01. But this is clear since |H:S| ≤ |Out(S)| < |S|^1.5/2.For the remainder of this subsection set ϵ = 0.1. In order to prove the bound for this ϵ, by the previous argument, it would be sufficient to show that β(S) ≤|S|^0.8. We claim that this holds once the Lie rank m of S is sufficiently large. Let E be an elementary Abelian subgroup in S of maximal size. By Lemma <ref> and by Table <ref>, if m →∞, we have log_2|E|/ log_2β(E) →∞. Again by Table <ref>, log_2|S|/log_2|E| = 4 + o(1) as m →∞. Thus we havelog_2β(S) ≤log_2β(E) - log_2|E| + log_2|S| = (-1 + o(1)) log_2|E| + log_2|S| = = (-(1/4) + o(1)) log_2|S| + log_2|S| = ((3/4)+o(1))log_2|S| < 0.8 log_2|S|,as m →∞.Let p be a defining characteristic for S and let q = p^f be the size of the field of definition. Unfortunately we cannot prove the bound β(H) ≤ 2^-0.1|H|^0.9 for all groups H with q large enough, but we can establish this bound in case f is sufficiently large. By Table <ref>, if the Lie rank m is at least 2 then S contains an elementary Abelian p-subgroup E such that |E|^8 > |S|. Notice that this bound also holds for m = 1, at least for sufficiently large groups S. Thus log_2|S|/log_2|E| < 8. If f →∞, then log_2|E|/ log_2β(E) →∞. In a similar way as in the previous paragraph, we obtain log_2β(S) < ((7/8)+o(1)) log_2|S|, that is, β(S) < |S|^0.89, for sufficiently large S. Since |H:S| is at most a universal constant multiple of f, we certainly have |H:S| < |S|^o(1), as f →∞. The claim follows by Lemma <ref>. §.§ The case when S is a sporadic simple group or the Tits group In this subsection we set ϵ = 0.1 and try to establish the proposed bound in as many cases as possible. Here we also complete the proof of Theorem <ref>.In this paragraph for a prime p and a positive integer k let p^k denote the elementary Abelian p-group of rank k and let 2^1+4 denote a group of order 2^5 with center of size 2. By the Atlas <cit.>, the groups S = 4 and S = 1 contain a section isomorphic to 2^12. Furthermore the groups S = 2, 3, , 22, 23, 24,andcontain a section isomorphic to 2^10, 3^5, 3^4:10, 2^10, 2^10, 2^12, 2^22, and 2^24 respectively and the group S = contains a subgroup isomorphic to 3^4:2^1+4. If S is any of these previously listed groups, we may use Lemmas <ref> and <ref> together with the estimate β(10)/|10| ≤ 3/4 in one case (see Theorem <ref>) to obtain the bound β(H) ≤ 2^-ϵ|H|^1-ϵ with ϵ = 0.1. The same estimate holds in case S is the Tits group, as shown in the proof of Theorem <ref>.If S is not a group treated in the previous paragraph, then |H| < 2.7^40. Thus, by the remark after Lemma <ref>, we have β(H) < |H|/2.7 < |H|^39/40. This and Theorem <ref> complete the proof of Theorem <ref>. Notice also that for ϵ = 0.01 we have |H|^39/40 < 2^-ϵ|H|^1-ϵ. §.§ The case when S is an alternating group Let S = A_k be the alternating group of degree k at least 5.Assume first that k > 10. Put s=[k/4]≥ 2. There exists an elementary Abelian 2-subgroup P ≤ A_k of rank 2s. By Lemma <ref>, we have β(P)=2s+1. By Lemma <ref>, this gives n(S) ≥ n(P) = 2^2s/(2s+1). Thus log_2(n(S)) > k log_21.11 > k/10. This gives β(H) < |H|/2^k/10. Thus ifϵ = k/10 + 10 log_2 |H| > 1/(10/k) + 10 (log_2(k) - 1) > 1/10 log_2k,then β(H) < 2^-ϵ|H|^1-ϵ.Now let k ≤ 10. Then |H| < 2.7^16. By the remark after Lemma <ref> we have β(H) < |H|/2.7 < |H|^15/16. This is certainly less than 2^-ϵ|H|^1-ϵ for ϵ = 0.01.§ PROOFS OF THE THREE MAIN RESULTSLet G be a finite group. By Theorem <ref>, we may assume that G is nonsolvable. Let H be an almost simple group whose socle S is a composition factor of G. By Sections <ref>, <ref>, and <ref>, we see that β(H) ≤ 2^-0.01|H|^0.99 provided that S is not an alternating group of degree at least 2^10. If S is an alterna-ting group of degree k at least 2^10, then β(H) ≤ 2^-ϵ|H|^1-ϵ with ϵ = (10 log_2k)^-1. The result now follows from Theorem <ref>.Let G be a finite group with Noether index n(G). Let k denote the maximum of 2^10 and the largest degree of a non-Abelian alternating composition factor of G, if such exists. Let C be a characteristic cyclic subgroup in G of largest possible size. Put f = |G:C|. By Theorem <ref>, β(G) ≤|C|^ϵ|G|^1-ϵ with ϵ = (10 log_2 k)^-1. In other words, n(G) ≥ f^ϵ. Thus G has a characteristic cyclic subgroup of index at most n(G)^10 log_2 k. If G is solvable, then β(G) ≤|C|^0.1|G|^0.9 by Theorem <ref>. In other words, n(G) ≥ f^0.1 and so f ≤n(G)^10.Let G be a finite group with Noether index n(G). By Theorem <ref> we may assume that G is nonsolvable. Thus n(G) > 2.7 by Lemma <ref>. By Theorem <ref> we may also assume that G has an alternating composition factor A_k with k ≥ 2^10. From Section <ref> we have k < 10 log_2(n(A_k)). Since n(A_k) ≤ n(G) by Lemma <ref>, we get 10 ≤log_2 k < log_210 + log_2log_2(n(G)). The result now follows from Theorem <ref>. § QUESTIONS We close with three questions which suggest another connection between the Noether number of a group and the Noether numbers of its special subgroups. Is it true that β(S)≤max{o(g)^2|g∈ S} for a finite simple group S?Is it true that β(G) ≤max{β(A)^100 | A ≤ G,AAbelian} for a finite group G? Let V be a finite dimensional FG-module for a field F and finite group G. Is it true that β(G,V) ≤(V) |G:H| β(H,V) for every subgroup H of G? AcknowledgementsThe authors are grateful to Mátyás Domokos for comments on an earlier version of the paper.99 ATLAS Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. Atlas of finite groups. Maximal subgroups andordinary characters for simple groups. Oxford UniversityPress, Eynsham, (1985). cz3p Cziszter, Kálmán. The Noether number of the non-Abelian group of order 3p. Period. Math. Hungar. 68 (2014), no. 2, 150–159. CzD Cziszter, Kálmán and Domokos, Mátyás. Groups with large Noether bound. Ann. Inst. Fourier (Grenoble) 64 (2014), no. 3, 909–944.DH Domokos, Mátyás and Hegedűs, Pál. Noether's bound for polynomial invariants of finite groups.Arch. Math. (Basel) 74 (2000), no. 3, 161–167. fleischmann Fleischmann, Peter. The Noether bound in invariant theory of finite groups. Adv. Math. 156 (2000), no. 1, 23–32. fogarty Fogarty, John. On Noether's bound for polynomial invariants of a finite group. Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 5–7 (electronic).GLyS Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald. The classification of the finite simple groups. Number 5. Part III. Chapters 1–6. The generic case, stages 1–3a. Mathematical Surveys and Monographs, 40.5. American Mathematical Society, Providence, RI, 2002. Huppert Huppert, B. Endliche Gruppen. I.Die Grundlehren der Mathematischen Wissenschaften, Band 134 Springer-Verlag, Berlin-New York, 1967. knop Knop, Friedrich. On Noether's and Weyl's bound in positive characteristic. Invariant theory in all characteristics, 175–188,CRM Proc. Lecture Notes, 35, Amer. Math. Soc., Providence, RI, 2004. noether Noether, Emmy. Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann. 77 (1915), no. 1, 89–92. olson Olson, John E. A combinatorial problem on finite Abelian groups. I. J. Number Theory 1 (1969) 8–10. Palfy Pálfy, P. P. A polynomial bound for the ordersof primitive solvable groups. J. Algebra 77 (1982), 127–137. pawale Pawale, Vivek M. Invariants of semidirect product of cyclic groups. Ph.D. Thesis, Brandeis University. 1999. schmid Schmid, Barbara J. Finite groups and invariant theory. Topics in invariant theory (Paris, 1989/1990), 35–66, Lecture Notes in Math., 1478, Springer, Berlin, 1991. sezer Sezer, Müfit. Sharpening the generalized Noether bound in the invariant theory of finite groups. J. Algebra 254 (2002), no. 2, 252–263. symonds Symonds, Peter. On the Castelnuovo-Mumford regularity of rings of polynomial invariants. Ann. of Math. (2) 174 (2011), no. 1, 499–517. thompson Thompson, John G. Nonsolvable finite groups all of whose local subgroups are solvable. Bull. Amer. Math. Soc. 74 (1968) 383–437. Wolf Wolf, Thomas R. Solvable and nilpotent subgroups ofGL(n,q^m). Canad. J. Math. 34 (1982), 1097–1111. | http://arxiv.org/abs/1706.08290v3 | {
"authors": [
"Pál Hegedűs",
"Attila Maróti",
"László Pyber"
],
"categories": [
"math.GR",
"math.AC",
"13A50, 20D06, (20D08, 20D99)"
],
"primary_category": "math.GR",
"published": "20170626090839",
"title": "Finite groups with large Noether number are almost cyclic"
} |
MatlabΓ <cit.>𝐤·𝐩 L>l<defnDefinition*defn*DefinitionconditionspropPropositioncorCorollaryexplExampletheoremTheoremDepartment of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, SpainPrinceton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USADepartment of Physics, Princeton University, Princeton, New Jersey 08544, USADonostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, SpainDepartment of Applied Physics II, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, SpainMax Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany.Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USAMax Planck Institute for Chemical Physics of Solids, 01187 Dresden, GermanyDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USADonostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, SpainLaboratoire Pierre Aigrain, Ecole Normale Supérieure-PSL Research University, CNRS, Université Pierre et Marie Curie-Sorbonne Universités, Université Paris Diderot-Sorbonne Paris Cité, 24 rue Lhomond, 75231 Paris Cedex 05, FranceSorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, FranceDepartment of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, SpainDepartment of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, SpainDepartment of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations ( DGENPOS) and of irreducible representations of the double (point and space) groups ( REPRESENTATIONS DPG andREPRESENTATIONS DSG). The toolDCOMPREL provides compatibility relations between the irreducible representations of double space groups at different 𝐤-vectors of the Brillouin zone when there is a group-subgroup relation between the corresponding little groups. The programDSITESYM implements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the programBANDREP calculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results ofBANDREP have been extensively applied in the description and the search of topological insulators. Double crystallographic groups and their representations on the Bilbao Crystallographic Server Mois I. Aroyo December 30, 2023 ==============================================================================================§ INTRODUCTION The Bilbao Crystallographic Server (http://www.cryst.ehu.es) website offers crystallographic databases and programs <cit.>. It can be used free of charge from any computer with a web browser via Internet.The applications on the server are organized into different sections depending on their degree of complexity, in such a way that the most complex tools make use of the results obtained by the simpler ones. The server is built on a core of databases that includes data from the International Tables for Crystallography, Vol. A: Space- group symmetry (ita, henceforth abbreviated as ), Vol. A1: Symmetry Relations between Space Groups<cit.> and Vol. E: Subperiodic groups<cit.>. A k-vector database with Brillouin-zone figures and classification tables of all the wave vectors for all 230 space groups is also available. Databases of magnetic space groups and magnetic structures have recently been implemented, alongside with a set of computationaltools that facilitate the systematic application of symmetry arguments in the study of magnetic materials <cit.>.The database of incommensurate structures, hosted by the server, contains both single-modulated structures and composites. Parallel to the databases and the crystallographic software there are a number of programs facilitating the study of specific problems related to solid-state physics, structural chemistry and crystallography involving crystallographic groups, their group-subgroup relations and irreducible representations.In a number of physical applications it is necessary to include spin-dependent terms in the Hamiltonian of a crystal system: for example, in taking into account relativistic effects in band structure calculations or spin-orbit coupling in crystal-field theory. Then, instead of the crystallographic groups and their representations, it is necessary to consider the so called double crystallographic groups and the related single-valued or vector and double-valuedor spinor representations. Since their introduction during the first half of the last century <cit.>, the double groups and their representations have been discussed in detail in the literature (see for example, the account given by bradley1972 and the references therein). Several reference compilations of the character tables of double point groups (cf.altmann1994), and of the irreducible representations of double space groups <cit.>, abbreviated as CDML exist. However, we are not aware of any online available databases of double crystallographic groups nor of any computing tools for the calculation of their representations. Here, we report on the recently completed development of such programs and their implementation on the Bilbao Crystallographic Server (BCS). For what follows, it will be convenient for us to recall briefly some essential features of the double crystallographic groups. Suppose that G is a point group which consists of pure rotations, i.e. a subgroup of the group SO(3). The two-to-one homomorphism φ between the group SU(2) of all (2×2) unitary unimodular matrices onto SO(3) can be used for the formal definition of the double groups ^dG<cit.>: the double group ^dG of a group G of order |G| which is a subgroup of the group SO(3), is the abstract group of order 2|G| having the same multiplication table as the 2|G| matrices of SU(2) which correspond under φ to the elements of the group G. The assignment of theSU(2) matrices to the crystallographic symmetry operations used in the databases and programs of the server follows the choice by altmann1994.The kernel of the homomorphism φ:SU(2)⟶ SO(3) consists of two elements: {1=[ 1 0; 0 1 ], ^d1=[ -10;0 -1 ]}. As a result, the preimage of each element R of G consist of two elements of ^dG, namely R and ^dR=^d1R.Formally, one can write^dG={R}∪{^dR}.It is important to note that the subset of elements {R} does not form a subgroup of ^dG as it is not closed under the binary group operation. For instance, if R represents a 2-fold rotation or a mirror plane, R^2= ^d1. In that sense it is wrong to refer to eq. (<ref>) as a coset decomposition of ^dG with respect to G. Physically, it is considered that the SU(2) matrices act on the spinors of a 1/2 spin space. Then, the operation ^d1 is of order 2 and it is often interpreted as a 2π rotation. The above statements about double point groups of pure rotations can be generalized in a straightforward way to the case of double point groups that contain improper rotations. For this purpose it is sufficient to indicate that the two SU(2) matrices 1=[ 1 0; 0 1 ] and ^d1=^d11=[ -10;0 -1 ] are assigned to the symmetry operation of inversion.Consider a space group G and its decomposition G:T into cosets with respect to its translation subgroup T, i.e. G=T∪T{R_2|𝐯_2}∪…∪T{R_n|𝐯_n}. In a similar way, the symmetry operations of the double space group^dG can be conveniently represented using its coset decomposition with respect to T as:^dG=T∪T{R_2|𝐯_2}∪…∪T{R_n|𝐯_n}∪T{^d1|𝐨}∪⋯∪T{^dR_n|𝐯_n}, where {R_1|𝐯_1}={1|𝐨} is omitted. Here, R_i and ^dR_i are the elements of the double point group ^dG of ^dG. Consequently, there are two elements {R_i|𝐯_i} and {^dR_i|𝐯_i} of the double space group ^dG, that correspond to every element of the space group G. The translation subgroup T is an invariant subgroup of the double space group ^dG.In the following, we shall discuss the development and implementation of databases and programs involving the double crystallographic groups for the BCS. We start with the presentation of the double space group database and the retrieval tools that access the stored crystallographic symmetry information (Section <ref>). The introduction to the basic programs available on BCS for the computation of representations of double crystallographic groups is given in Section <ref>. The last sections of the article are devoted to the presentation of the accompanying applications of the representations of double crystallographic groups, such as their compatibility relations (Section <ref>), the site-symmetry approach (Section <ref>), and finally, the determination of the band representations and the elementary band representations (Section <ref>). In the Appendix we briefly describe the normal-subgroup induction procedure. § DOUBLE CRYSTALLOGRAPHIC GROUPS The double space groups are infinite groups, i.e. they contain an infinite number of symmetry operations generated by the set of all translations of the space group. As already noted, a practical way to represent the symmetry operations of the double space group ^dG is based on the coset decomposition of ^dGwith respect to its translation subgroup T, cf. eq.(<ref>). The set of coset representatives {{R_i|𝐯_i},{^dR_i|𝐯_i}, i=1,…, n} of the decomposition ^dG:T (often referred to as General positions of ^dG) represents, in a clear and compact way, the infinite number of symmetry operations of the double space group ^dG. The infinite symmetry operations in a coset have the same linear part R_i (or ^dR_i) while their translation parts differ by lattice translations.The translations {1|𝐭}∈T form the first coset with the identity {1|𝐨} as a coset representative. The number of cosets is always finite and is equal to the order of the double point group ^dG of the double space group ^dG. The database of double space groups, available on BCS, includes the lists of the representatives of the general positions of each double space group ^dG. These data can be retrieved using theDGENPOS tool (http://www.cryst.ehu.es/cryst/dgenpos), by specifying the sequential number or the Hermann-Mauguin symbol of the space group G corresponding to ^dG. The symmetry operations are specified by their matrix presentations, i.e by the (3×4) matrix-column pairs and the (2×2) matrices of SU(2), and by Seitz symbols.*Matrix presentations With reference to a coordinate system (O, 𝐚_1,𝐚_2, 𝐚_3), consisting of an origin O and a basis𝐚_k, the symmetry operations of the space group ^dG are described by matrix-columnpairs (W, w). The set of translations are represented by the matrix-column pairs (I, t_i), where I is the (3× 3) unit matrix and t_i is the column of coefficients of the translation vector 𝐭_i that belongs to thevector lattice𝐋 of ^dG. The programs and databases of the double space groups, as well the rest of the computer tools on BCS, use specific settings of space groups (hereafter referred to as standard, or default, settings) that coincide with the conventional space-group descriptions found in . For space groups with more than one description in , the following settings are chosen as standard: unique axis b setting, cell choice 1 for monoclinic groups, hexagonal axes setting for rhombohedral groups, and origin choice 2 (origin on 1 ) for the centrosymmetric groups listed with respect to two origins in .The pair of symmetry operations {{R_i|𝐯_i},{^dR_i|𝐯_i}} related by {^d1|𝐨} have the same matrix-column presentation (W_i, w_i) while their (2×2) matrices differ by a sign. As already noted, the chosen correspondence scheme between the rotations and the (2×2) matrices follows closely the choice made by altmann1994. The shorthand descriptions of the (3×4) matrix-column pairs and of the (2×2) matrices of the symmetry operations are also shown in the general-position table. *Seitz symbolsThe Seitz symbols {R_i|𝐯_i} of space-group symmetry operations consist of two parts: a rotation (or linear) part R and a translation part 𝐯<cit.>. The Seitz symbol is specified between braces and the rotational and the translational parts are separated by a vertical line. The translation parts v correspond exactly to the columns w of the matrix-column presentation (W,w) of the symmetry operations. The rotation parts R consist of symbols that specify (i) the type and the order of the symmetry operation, and (ii) the orientation of the corresponding symmetry element with respect to the space-group basis. The symbols 1 and 1are used for the identity and the inversion, m for reflections, the symbols 2, 3, 4 and 6 denote rotations and 3 , 4and 6 rotoinversions. For rotations and rotoinversions of order higher than 2, a superscript + or - is used to indicate the sense of the rotation. The orientation of the symmetry element is denoted by the direction of the axis for rotations or rotoinversions, or by the direction of the normal to reflection planes. Subscripts of the symbols specify the characteristic direction of the operation: for example, the subscripts 100, 010 and 11 0 refer to the directions [100], [010] and [11 0], respectively.The symmetry operations of the double space groups are denoted by the modified Seitz symbols. The modified Seitz symbols include a superscript d added to the rotational part of the symmetry operationsto distinguish between the symmetry operations {R|𝐯} and those obtained by their combinations with{^d1|𝐨}.As an example, consider the general-position table of the double space group P2_12_12_1 (No. 19) shown in Fig. <ref>. The listed eight symmetry operations are the chosen coset representatives of the decomposition of P2_12_12_1 with respect to its translation subgroup. The pairs of elements related by {^d1|𝐨} are clearly distinguished by their Seitz symbols {R_i|𝐯_i} and {^dR_i|𝐯_i}, e.g.(2) {2_001|1/2,0,1/2} and (6) {^d2_001|1/2,0,1/2}. As explained, the pairs of symmetry operations {{R_i|𝐯_i}, {^dR_i|𝐯_i}} are represented by the same 3×4 matrix-column pair while their 2×2 matrices differ by a sign.§ REPRESENTATIONS OF THE DOUBLE CRYSTALLOGRAPHIC GROUPS§.§ The problemThere are two programs on the Bilbao Crystallographic Server that compute the irreducible representations (irreps) of space groups explicitly, namelyREPRES (http://www.cryst.ehu.es/cryst/repres) andRepresentations SG(http://www.cryst.ehu.es/cryst/ representationsSG). Given a space group G and ak-vector, both programs calculate the irreps of space groups following the algorithm based on a normal-subgroup induction method, i.e. the irreps of a group G are calculated starting from those of a normal subgroup HG. The main steps of the procedure involve the construction of all irreps of H and their distribution into orbits under G, determination of the corresponding little groups and the allowed (small) irreps and finally, construction of the irreps of G by induction from the allowed irreps. The labels assigned to the irreps calculated byREPRES andRepresentations SG correspond to those used by CDML. The correct assignment of the labels to the irreps calculated byREPRESis achieved with the help of the database of space-group irreps recently developed by stokes2013. Here we report on the development of the programRepresentations DSG (http://www.cryst.ehu.es/cryst/ representationsDSG), available on BCS, for the computation of the irreps of double crystallographic groups. The method for calculating of the irreps of double crystallographic groups is based on a generalization of the normal-subgroup induction procedure for the calculation of space-group irreps implemented in the programsREPRES andRepresentations SG.To make the exposition self consistent, we include in the Appendix the main concepts and statements of the normal-subgroup induction procedure (for a detailed presentation, cf. aroyo2006). §.§ The methodThe normal-subgroup induction procedure for the calculation of space-group representations (see the Appendix) can be generalized in a straightforward way for the calculations of the representations of the double space groups (cf. Miller & Love, 1967). Consider the coset decomposition of the double group ^dG with respect to its translation subgroup T (cf. eq. (<ref>)): ^dG=⋃_iTq_i ∪T ^dq_i.It will be convenient to rewrite the coset decomposition of ^dG with respect to the group ^dT=T⊗{{1|000},{^d1|000}}:^dG=^dT∪^dTq_2 ∪⋯∪^dTq_m.The group ^dT is a trivial central extension of T by the group generated by{^d1|000}; it is an abelian group, and it is a normal subgroup of ^dG, i.e.^dG▹^dT. Each irrep Γ^k of T generates two irreps Γ^𝐤 and Γ^𝐤 of ^dT: Γ^𝐤({1|𝐭})=exp(-ik·t), Γ^𝐤({^d1|𝐭})=exp(-ik·t), Γ^𝐤({1|𝐭})=exp(-ik·t), Γ^𝐤({^d1|𝐭})=-exp(-ik·t). The irrep Γ^𝐤 is known as a single-valued irrep: the same (one-dimensional) matrix represents the elements {1|𝐭} and {^d1|𝐭}. On the contrary, the (one-dimensional) matrices of the elements {1|𝐭} and {^d1|𝐭} differ by a sign in the double-valued irrep Γ^𝐤.Note that the term single-valued (double-valued) representation is kept for any representation 𝐃^𝐤 (𝐃^𝐤) induced from Γ^𝐤 (Γ^𝐤) as all such representations are characterized by the same relationship between the matrices representing the elements related by the operation ^d1, namely:𝐃^𝐤({R|𝐭})=𝐃^𝐤({^dR|𝐭}), 𝐃^𝐤({R|𝐭})=-𝐃^𝐤({^dR|𝐭}). As the wave vector is left invariant under the action of ^d1, the double little cogroup ^dG ^𝐤 of the wave vector 𝐤 consists of the sets of elements of {R^𝐤}and {^dR^𝐤} that correspond to the elements of G ^𝐤, cf. eq. (<ref>):^dG ^𝐤={R}∪{^dR}.The group ^dG ^𝐤 is a subgroup of the double point group ^dG of the space group ^dG. The index |^dG/^dG ^𝐤| equals the index G ^𝐤 in G, and for the coset decomposition of ^dG with respect to ^dG ^𝐤 one can choose the same set of coset representatives as of the decomposition of G with respect to G ^𝐤. As a result, the star *𝐤 of the wave vector 𝐤 in ^dG coincides with *𝐤 in G. In analogy to the relationship between the little co-group G ^𝐤 and the little group G ^𝐤 (cf. eq. (<ref>)), the little group ^dG ^𝐤 can be defined as:^dG^𝐤={{R^𝐤|𝐯^𝐤}∈G|R^𝐤∈ ^dG ^𝐤}. The allowed single-valued 𝐃^𝐤,i and double-valued irreps 𝐃^𝐤,j of ^dG ^𝐤 can be determined from the allowed single-valued Γ^𝐤 and double-valued irreps Γ^𝐤 of ^dT by stepwise induction along the composition series of ^dG ^𝐤:^dG^𝐤^dH_1^𝐤…^dH_m-1^𝐤^dH_m^𝐤…^dH_n^𝐤 =^dT where |^dH_m-1^𝐤/^dH_m^𝐤|=23.The full single-valued irreps 𝐃^*𝐤,i and full double-valued irreps 𝐃^*𝐤,j of the double space group ^dG are induced from the allowed single-valued and double-valued irreps of the little group ^dG^𝐤. The full irreps are of dimension r× s where s is the number of arms of the star *𝐤 and r is the dimension of the corresponding allowed irrep of ^dG^𝐤.The matrices of the full irreps have a block structure with s× s blocks, each of dimension r× r.§.§ Complex conjugation For applications involving time-reversal symmetry, it is necessary to recall briefly the classification of irreps with respect to complex conjugation or, as it is commonly referred to, with respect to their reality (for more details, cf.bradley1972). Note that the concepts and results formulated in the following for ordinary representations of groups, are equally valid for single- and double-valued representations of double groups.An irrep 𝐃 of the group G is of:(i) the first kind (or real) if 𝐃 is equivalent to a group of real matrices;(ii) the second kind (or pseudoreal) if 𝐃 is equivalent to 𝐃^* but not to any group of real matrices;(iii) the third kind (or complex) if 𝐃 is not equivalent to 𝐃^*.One can show that <cit.> 1/| G|∑_j=1^|G|χ(g_j^2)={[1iff 𝐃 is of the first kind,; -1 iff 𝐃 is of the second kind,;0iff 𝐃 is of the third kind,;]. where χ is the character of the irrep 𝐃 of G. The obvious difficulties in the direct application of the reality test (eq. <ref>) to space-group irreps can be overcome ifthe sum over all operations of G can be replaced by a sum over a relatively small number of space-group elements. This can be achieved using the relationship between the space-group irreps and the allowed irreps of the little group from which they are induced. As a result, the reality test for space group irreps can be written in the following form (see bradley1972 for details of the proof): the irrep 𝐃^*𝐤,j of the space group G induced from the allowed irrep 𝐃^𝐤,j of the little group G^𝐤, 𝐃^*𝐤,j=𝐃^𝐤,j↑G, is of the first, second or third kind according to:1/|𝐇_𝐤|∑_{R_s|𝐯_𝐬}∈𝐇_𝐤χ^𝐤,j({R_s|𝐯_𝐬}^2)=1, -1, or 0,where χ^𝐤,j is the character of the allowed irrep 𝐃^𝐤,j of the little group G^𝐤 and the sum is restricted to the set 𝐇_𝐤 of coset representatives {R_s|𝐯_𝐬} of G with respect to T whose rotation parts transform 𝐤 into a vector equivalent to -𝐤. Obviously, the element {R_s|𝐯_𝐬}^2 leaves 𝐤 invariant and therefore it belongs to G^𝐤.The irreps of the first and of the second kind are also known as self-conjugate while the irreps of the third kindform pairs of conjugated irreps (𝐃, 𝐃^*) which, in general, may be induced from allowed irreps of wave vectors belonging to different stars.The concepts of physically irreducible representations or time-reversal invariant representations used in some physical applications are closely related to the above-discussed reality of the representations. Once the reality of a space-group irrep has been determined and, for the complex irreps, the pairs of conjugated irreps have been identified, the time-reversal (TR)-invariant irreps (single- and double-valued irreps of the double space groups) can be constructed according to:*If the irrep 𝐃 is (a) single-valued and real, or (b) double-valued and pseudo-real, it is TR-invariant.*If the irrep 𝐃 is (a) single-valued and pseudo-real or (b) double-valued and real, the TR-invariant irrep is the direct sum of 𝐃 with itself. The dimension of the TR-invariant irrep doubles the dimension of 𝐃. The label of the TR-invariant irrep consists of two copies of the label of 𝐃.*If 𝐃_1 and 𝐃_2 form a pair of mutually conjugated irreps, the direct sum of both irreps is TR-invariant. The label of the TR-invariant representation is the union of the labels of the two irreps.§.§ The programRepresentations DSG An algorithm based on the normal-subgroup procedure for the calculation of the irreps of the double space groups (Section <ref>) is implemented in the programRepresentations DSG. As Input, the program needs the specification ofthe double space group ^dG by the sequentialITA number of G. (As already noted, the programs of BCS use the standardsettings for the description of the space groups.) Thek-vector data can be introduced by choosing thek-vector directly from a table provided by the program, where CDML labels are used to designate the symmetryk-vector types. The listedk-vector coefficients (called conventionalk-vector coefficients) refer to a basis 𝐚^*,𝐛^*,𝐜^* which is dual to the conventional ITA settings of the space groups.The Output of the program consists of two tables showing the matrices of the little-group irreps and the matrices of the irreps of the space group (i.e the full representations). In detail, the program produces:*The table of little-group representations (illustrated by the allowed irreps of the point X:(0,1/2,0) of the double space group P4/ncc (No. 130) shown in Fig. <ref>). The rows of the table are labeled by the symmetry operations of the little group while the columns are specified by the allowed little-group irreps. The symmetry operations are represented by the (3× 4) matrix-column pairs and (2× 2)SU(2) matrices, and by the (modified) Seitz symbol. The first row corresponds to a general translation {1|𝐭}, while the subsequent rows show the data of the representatives {R|𝐯} of the coset decomposition of the little group with respect to its translation subgroup. (From the fact that anyelement of the group {R|𝐯+𝐭} can be expressed as a combination of a translation {1|𝐭} and a coset representative {R|𝐯} follows that the irrep matrices of {R|𝐯+𝐭} are equal to products of the type 𝐃^𝐤({R|𝐯+𝐭})=𝐃^𝐤({1|𝐭})𝐃^𝐤({R|𝐯}).) The columns of single-valued irreps are followed by those of the double-valued irreps that are ordered according to their dimension. The labels of the single-valued irreps follow the notation of CDML: the labels consist of a 𝐤-vector letter(s) and a sequential index. The labels of the double-valued irreps are constructed in a similar way: a bar over the 𝐤-vector letter(s) distinguishes the double-valued irreps from the single-valued ones. In the example of the space group P4/ncc (No. 130) and 𝐤 vector X, 𝐤=(0,1/2,0) (Fig. <ref>), the single-valued irreps have labels X_1 and X_2, and the two double-valued irreps are labelled X_3 and X_4.The irrep matrices are shown explicitly if the dimension of the irrep is ≤ 4. When the dimension of the irrep is larger than 4, the output shows only the non-zero elements of the matrix, in the format: (i;j):r, signifying that the (i,j) element of the matrix has the value r. * The full-group irreps (illustrated by the full-group irreps of the double space group P4/ncc at the point X:(0,1/2,0) shown in Fig. <ref>).The indication of the arms of the star *𝐤 precedes the table of the matrices of the full-group irreps 𝐃^*k,j of ^dG induced from the allowed irreps 𝐃^𝐤,j of the little group ^dG^𝐤. The structure and the organization of the data of the full-group irrep table is very similar to that of the little-group irreps: the coset representatives of the decomposition ^dG:T of the double space group with respect to the translation subgroup specify the rows of the table, while the columns correspond to the full space-group irreps. The symmetry operations are described by their matrix-column pairs, SU(2) matrices and Seitz symbols. The labels of the space-group irrep follow the labels of the allowed little-group irreps from which they are induced: e.g. as in the case of little-group irreps, a bar over the 𝐤 vector letter indicates a double-valued irrep.The number in brackets after the irrep label specifies the reality of the irrep per eq. (<ref>): (1) indicates an irrep of the first kind,i.e. real; (-1) - an irrep of the second kind, orpseudoreal;and (0) - an irrep of the third kind, orcomplex. The pairs of the complex conjugated irreps are also indicated in the output. For example, the double-valued irreps^*X_3 and ^*X_4 form a complex-conjugated pair as indicated immediately before the table of full irreps (Fig. <ref>).As another example, the double-valued irrep P_7 in the cubic double space group Ia3 (No. 206) is real, so that it doubles when time-reversal is considered. On the contrary, the double-valued irrep P_7 in the cubic double space group I4_132 (No. 214) is pseudoreal and it does not double with time-reversal. §.§ Representations of the double point groupsThe crystallographic double point groups and their representations have been extensively discussed in the literature. Sets of irrep compilations can be found, for example, inkoster1963, bradley1972 and altmann1994.The data on the irreps of the crystallographic double pointgroups are now also online accessible via BCS. The irrep data was obtained by the programRepresentations DSG (cf. Section <ref>) for the particular case of 𝐤 = Γ (0,0,0). For each of the 32 crystallographic double point groups, specified by their international (Hermann-Mauguin)and Schoenflies symbols, the retrieval toolRepresentations DPG (http://www.cryst.ehu.es/cryst/representationsDPG) displays the following set of tables: * Character table As usual, the table entries correspond to characters of the irreps (rows) for the conjugacy classes of symmetry operations(columns) of the chosen double point group. The irreps are labelled according to the notation of mulliken1933, by the Γ labels given by koster1963 and by the labels generated by the programRepresentations DSG.A bar over the irrep label distinguishes the double-valued (known also asspinor) representations. The distribution into conjugacy classes of the symmetry operations of the double point group (designated by (modified) Seitz symbols), is also indicated.* Table of irrep matricesThe matrices of the irreps, as calculated byRepresentations DSG for the particular case of 𝐤=Γ (0,0,0) are also provided byRepresentations DPG. The matrices are explicitly listed for each symmetry operation (rows) and irrep (columns). The symmetry operations of the double point group are specified by the pair of (3× 3) rotation and (2× 2)SU(2) matrices, and by the (modified) Seitz symbol. The classification of irreps with respect to complex conjugation is revealed by the number in brackets after the irrep label: as already noted, (1) indicates an irrep of the first kind,i.e. real; (-1) - an irrep of the second kind, or orpseudoreal;and (0) - an irrep of the third kind, orcomplex. The pairs of complex conjugated irreps are also listed. Screenshots of the character table and the table of irrep matrices for the double point group mm2 (C_2v) are shown in Fig. <ref> and Fig. <ref>.§ COMPATIBILITY RELATIONS§.§ The problemThe so-called compatibility relations have different applications in solid-state physics as, for example, in the analysis of the electronic band structures or phonon dispersion curves.The compatibility relations are essential in the study of connectivity of energy bands as we move in a continuous way from one k-vector point to a neighbouring one with a different symmetry, or in crystal-field splitting problems that arise when a high-symmetry crystal field is perturbed by a field of lower symmetry.From a group-theoretical point of view, the compatibility relations correspond to the so-called subduction relationships betweenthe little-group representations of different k-vectors of the same space group G whose little groups form a group-subgroup pair. Let G and H be two groups with group-subgroup relation G > H with n = | G | / | H | being the index of H in G. Consider an irrep𝐃_β= {𝐃_β(g), g∈G} of a group G: The subduction of 𝐃_β to the subgroup H results in a representation of the subgroup, the so-called subduced representation, formed by the matrices of those elements of G that also belong to the subgroup H, that is, 𝐃_β(h) with h∈H<G. This subduced representation (𝐃_β↓H)=𝐃^S is in general reducible, and is decomposable into irreps 𝐝_α of H: (𝐃_β↓H)(h) = 𝐃^S(h) ∼⊕_α n_α^(β)𝐝_α(h), h∈H. The multiplicities n_α^(β) of the irreps𝐝_α of H in the subduced representation 𝐃_β↓H can be calculated by the reduction formula (known also as the Schur orthogonality relation or `magic' formula): n_α^(β) = 1/|H|∑_hχ^S(h)χ_α(h)^*,where χ^S(h) is the character of the subduced representation and χ_α(h) is the character of the irrep 𝐝_α for the same element h ∈H. Consider two k-vectors k_1 and k_2 where k_2 = k_1 + κ with κ - an infinitesimal k-vector; e.g.k_1 could be the wave vector of a symmetry point (line) 'sitting' on a symmetry line (plane) k_2. The little group G^k_2 is in general a subgroup of the little group G^k_1 and the compatibility relations between the irreps at k_1 and k_2 (in the limitκ→ 0) can be established by the subduction of the representations D^k_1,i of the little group G^k_1 onto the little subgroup G^k_2. In the following we say that these two 𝐤-vectors are connected.Tables of compatibility relations of double space groups can be found in Miller & Love (1967) but the compiled tablesprovide only the relationships between the irreps of k-vector points and lines; neither the relations between the k-vector lines and planes nor the relations between the k-vector points and planes are made available. The programDCOMPREL(http://www.cryst.ehu.es/cryst/dcomprel) calculates the compatibility relations between the little-group irreps of the double space groups for any pair of symmetry-related wave vectors.Compatibility relations between little-group irreps of ordinary space groups can be calculated by the programCOMPATIBILITY RELATIONS, also accessible in BCS (http://www.cryst.ehu.es/rep/rep_correlation_relations). §.§ The methodEssentially, the same algorithm is implemented in the programsCOMPATIBILITY RELATIONS andDCOMPREL: it follows closely the subduction procedure explained above applied to the special case of little-group representations. Consider two symmetry-related wave vectors k_1 and k_2, where k_2 = k_1 + κ with G^k_2<G^k_1. The matrices of an irrep of the little group G^k_1 associated to the symmetry operations that belong to G^k_2 form a representation of the little group of 𝐤_2, which in general, is reducible. The compatibility relations are extracted from the decomposition of the subduced representation G^k_1↓G^k_2 into irreps of G^k_2: 𝐃^k_1,i↓G^k_2∼⊕_j=1^s m_k_2,j𝐃^k_2,j, where m_k_2,j represents the multiplicity of the irrep 𝐃^k_2,j in the subduced representation. Given the characters χ_𝐃^k_1,i of 𝐃^k_1,i↓G^k_2 and χ_𝐃^k_2,j of 𝐃^k_2,j for all g∈G^k_2 (e.g. calculated by the programRepresentations DSG), the multiplicities can be obtained by a variation of the reduction formula: m_k_2,j = 1/|G^k_2|∑_gχ_𝐃^k_2,j(g) χ_𝐃^k_1,i(g)^*, ∀ g ∈G^k_2.The expression (<ref>) is not convenient for practical calculations due to the infinite order of the little group G^k_2. However, the summation in eq. (<ref>) can be first performed over all translations in G^k_2 which will reduce the infinite sum to a finite one over the set 𝐆^k_2_o of the representatives g_k of the decomposition of G^k_2 in cosets with respect to its translation subgroup: m_k_2,j = 1/n∑_k=1^n χ_𝐃^k_2,j(g_k) χ_𝐃^k_1,i(g_k)^*; ∀ g_k ∈𝐆^k_2_o.The number n of coset representatives g_k is equal to the order of the little co-group G^k_2.§.§ The programDCOMPRELTheprocedure described above for the calculation of the compatibility relations between the little-group irreps of space groups is implemented in the program . The program is available on BCS, and for a number of cases the results have been successfully checked against the compatibility-relation data listed in miller1967. Input: The specification of the double space group by thenumber leads to a menu with the list of the different symmetry 𝐤-vectors for the group chosen. The choice of a 𝐤-vector, 𝐤_1, produces a second output with the list of all 𝐤-vectorsthat can be connected to it. The user can then ask for the compatibility relations between the chosen 𝐤_1-vector and a single 𝐤-vector or between 𝐤_1 and all the 𝐤-vectors in the list.Output: After a summary of the input data, a table with the compatibility relations between the little-group irreps of the selected k-vectors is shown.Fig. <ref> shows a partial output of the programDCOMPREL of the compatibility relations of the 𝐤-vector line W(0,1/2,w) with all symmetry-related wave vectors in the double space group P4/ncc. Note that in some cases, W is the high-symmetry point of the pair (as in W→ B,F,GP) while in others W is the point of lower symmetry (as in R→ W). The number in parenthesis indicates the dimension of the representation. §.§ Example: electronic bands of germaniumAs an application of the use of the representations of the double space groups and the compatibility relations we consider the electronic bands of germanium, with space group Fd3̅m (No. 227) and occupied atomic position 8a(1/8,1/8,1/8). Figure <ref> reproduces the figures 12.10 and 14.41 in dresselhaus2008. The labels of the irreducible representations have been changed to the labels used inREPRESENTATIONS DSG. Figure <ref> (a) shows the band structure when the spin-orbit coupling is not considered and the relevant irreps are the single-valued irreducible representations, and Figure <ref> (b) reproduces the band structure when the spin-orbit coupling is considered and the relevant irreps are the double-valued irreducible representations. Figure <ref> shows the compatibility relations between the Γ(0,0,0) point and the Δ(0,v,0) line and between the X(0,1/2,0) point and the Δ line in the double space group Fd3̅m.It can be checked that these compatibility relations agree with the paths Γ↔Δ↔ X in Figures <ref>(a) and <ref>(b). In some cases, these relations could be useful to identify the irreducible representations at the k⃗ points of high symmetry. The compatibility relations give the connectivities between the branches of the electronic band along the Brillouin zone and the dimensions of the irreps at each point give the degeneracies of the energy.§ SITE-SYMMETRY APPROACH§.§ The problem The symmetry of the extended states (phonons, soft modes, electronic bands, etc.) of crystal structures (described by the irreps of the space group of the crystal) over the entire Brillouin zone is related to the symmetry of localized states (local atomic displacements, atomic orbitals, etc.) of the constituent structural units (described by the irreps of the local symmetry groups). A procedure for the determination of such a relationship is very useful, as it allows the prediction of the symmetry of the possible extended states starting from the crystal structural data. In particular, it is useful in distinguishing topological materials <cit.>. Formally, the procedure relating localized and extended crystalline states can be described by induction of a representation of a space group G from a finite subgroup H, followed by a reduction into irreps. In other words, the induction method permits the calculation of the symmetry of the compatible extended states transforming according to irreps of crystal space group G induced by a localized state described by an irrep of the local or site symmetry group H=S. However, the calculation of the space-group irreps induced by the irreps of the site-symmetry group is not an easy task: the fact that the site-symmetry group S (isomorphic to a point group) is a subgroup of infinite index of G implies that the representation of G induced by an irrep of S must be of infinite dimension, and therefore difficult to calculate directly. The so-called site-symmetry approach resolves this problem by applying the Frobenius reciprocity theorem, which states that the multiplicities of the irreps of a group G in the induced representation from an irrep of a subgroup H of G can be determined from the multiplicities of the irreps of H in the representations subduced from G to H.(For a detailed presentation of the method, its discussion and applications, cf.evarestov1997). In this way, the knowledge of the subduced representation of the irreps of the space group onto the site-symmetry group, which are relatively easy to compute once the space group irreps are known, is enough to obtain all the necessary information about the representations induced by the irreps of the site-symmetry group into the space group.The site-symmetry method is implemented in two programs of BCS: the programSITESYM for ordinary representations of space groups, and the programDSITESYM for the double space groups. The following explanations of the site-symmetry method are equally valid for the cases of space groups and double space groups.§.§ The method The objective of the site-symmetry programs is to find the symmetry of the crystal extended states induced by localized states of some of the constituent structural units. Examples of the application of this method can be found in the supplementary material of NaturePaper. In group-theoretical terms, this task requires the derivation of the irreps of a space group G at any point in the reciprocal space (which classify the extended states of the structure) induced by the irreps of the site-symmetry group of a Wyckoff position (according to which localized states are classified). Two basic concepts of representation theory, namely subduction and induction are essential for the site-symmetry approach. As already explained, the subduction coefficients specify the decomposition of a representation of a group G into irreps of one of its subgroups H< G whilethe induction procedure permits the construction of a representation of G starting from a representation of H. Consider the decomposition of G in cosets with respect to H with coset representatives {q_m, m=1,…,n}, cf. eq. (<ref>). If 𝐝_α={𝐝_α(h),h∈H} is an irrep of H < G, then the matrices of the induced representation (𝐝_α↑G) = 𝐃^I of G are constructed as follows: 𝐃^I(g)_kt,js ={[ 𝐝_α(q_k^-1 g q_j)_ts if q_k^-1 g q_j ∈H;0pt4ex0ifq_k^-1 g q_j ∉H ].,where k,j = 1,…,n and t,s=1,…,m with m the dimension of the irrep 𝐝_α of H. Its characters are given by:χ^I_γ(g) =∑_j χ^α(q_j^-1g q_j), where χ^α(q_j^-1g q_j) is the trace of the j-th diagonal block of 𝐃^I(g). In general, the induced representations are reducible, and as such it is possible to decompose them into irreps 𝐃_β of G: (𝐝_α↑G) = 𝐃^I ∼⊕_β n^(α)_β𝐃_β,where n^(α)_β are the multiplicities of the irreps of G in the induced representation, and it is possible to calculate them applying the reduction formula (<ref>): n^(α)_β = 1/|G|∑_gχ^I(g)χ^*_β(g), where χ_β(g) is the character of the element g of the irrep 𝐃_β of G.The dimension of the induced representation can be directly read off the equation for its construction, eq.(<ref>): dim(𝐝_α↑G)= dim (𝐝_α) |G|/|H|.This result points to the difficulties for the direct calculation of a representation of a space group G induced from an irrep of a finite subgroup H of G: due to the infinite index of H in G, the dimension of the representation of G induced from an irrep of H is infinite. By means of the site-symmetry approach it is possible to determine the multiplicities n^(α)_β of an irrep of G in the induced representation without the necessity of constructing the infinite-dimensional representation. As stated above, the method is based on the Frobenius reciprocity theorem, according to which the multiplicity of an irreducible irrep 𝐃_β(g) of G in a representation 𝐝_α↑G of G induced by an irrep 𝐝_α of H<G is equal to the multiplicity of the irrep 𝐝_α of H in the representation 𝐃_β↓H subduced by 𝐃_β of G to H,i.e.n_α^(β)=n_β^(α). In other words, in order to calculate the frequencies n_β^(α), eq.(<ref>), it is sufficient to calculate the multiplicities n_α^(β) of 𝐝_α in the subduced representation 𝐃_β↓H∼⊕_αn_α^(β)d_α.As already discussed, the irreps of a space group G are classified according to the reciprocal space wave vectors and for each vector k there is a finite set of irreps of G. The main idea of the site-symmetry approach is that although the representation induced by an irrep of a site-symmetry group in a space group G has infinite dimension, it is possible to know the part of this induced representation that corresponds to any set (finite) of irreps of G for a given wave vector k (calculated in advance). As in most applications only the irreps of G related to a few wave vectors are of interest, this partial knowledge of the induced representation proves to be sufficient.The algorithmic procedure, based on the site-symmetry approach and implemented in the site-symmetry programsSITESYMandDSITESYM, is the following:*Consider an occupied Wyckoff position and determine its site-symmetry group S. Note that it is not sufficient to determine the point group isomorphic to the site-symmetry group: it is necessary to obtain the set of space-group symmetry operations, with their rotation and translations parts, that belong to the site-symmetry group, since the representation matrices of symmetry operations related by translations are, in general, different.*Calculate the irreps of the space group G for the wave vectors k of interest. The BCS programREPRES for ordinary space-group representations (orRepresentations DSG for single- and double-valued irreps of double space groups) provides the irrep matrices for any element of the space group (or of the double space group).*From the obtained space-group irreps, calculate the representations subduced to the site-symmetry group S obtained in the first step.*From the irreps of the site-symmetry group (tabulated by the BCS programsPOINT<cit.> for ordinary point-group representations orRepresentations DPG for single- and double-valued irreps of double point groups), and making use of the reduction formula (<ref>), calculate the multiplicities of the site-symmetry irreps in the subduced representations.*Apply the Frobenius reciprocity theorem to obtain the multiplicities of the irreps in the induced representation of G from the multiplicities of irreps of S.§.§ The programsSITESYM andDSITESYMThis algorithm for the calculation of the multiplicities of space-group irreps in the representations induced by the irreps of a site-symmetry group (S<G) has been implemented in the site-symmetry programs of BCS. The description that follows refers to the programDSITESYM (http://www.cryst.ehu.es/cryst/dsitesym) but similar explanations are also applicable to the programSITESYM (http://www.cryst.ehu.es/cryst/sitesym). The site symmetry approach allows the determination of the symmetry relationships between the extended (Bloch) and localised (Wannier-type) electronic states in crystals. As an example of the utility ofDSITESYM we will consider the determination of the symmetry of electronic states in (GaAs)_m(AlAs)_n semiconductor superlattices grown along the [001] direction, a problem discussed in detail in kitaev1997. The output of the program will be illustrated by the specific calculations of the symmetry relationshipsbetween the electronic band states at the point k⃗=M(1/2,1/2,0) of the double space group P4m2 (No. 115) (the structure-symmetry group of (GaAs)_m(AlAs)_n) and the localised states of atomic orbitals at the Wyckoff position 2g(0,1/2,z). In other words, the multiplicities of the irreps of the double space group P4m2 at the point k⃗=M(1/2,1/2,0) shown in the output ofDSITESYM, describe the transformation properties of the extended electronic states induced by the irreps of the double site-symmetry group of the Wyckoff position 2g(0,1/2,z) according to which the localised one-electron wave functions transform. Input: The required input data include the specification of the double space group, of the occupied Wyckoff position and of the wave-vector label of the space-group irreps whose induced-representation multiplicities are to be calculated. The information is entered using three forms: in the first one, the double space group is specified by itssequential number; the wave vector of interest and the occupied Wyckoff position can be selected from the corresponding lists produced by the program.Output: After a header that reproduces the input data, the program displays the following tables with results of the intermediate steps of the procedure (the induction table, that lists the final results of the site-symmetry calculations, is at the bottom of thescreen (v)):(i)List of operations of the double site-symmetry group S: each of the symmetry operations of the double space group that leaves the Wyckoff position representative point invariant, is represented in a x,y,z and in a matrix notation. Labels (g_1,...,g_n) necessary for later referencing are assigned to each element of S.The double site-symmetry group of the position 2g(0,1/2,z) of the double space group P4̅m2 isformed by 8 symmetry operations as shown in the screenshot (Fig. <ref>) of the programDSITESYM. (ii)Character table of the double point group. The table reproduces the character table of the irreps of the double point group isomorphic to the site-symmetry group, as provided by the programRepresentations DPG of BCS. The labels of the irreps are given in notations by mulliken1933 and as generated by the program.The site symmetry group of (0,1/2,z) is isomorphic to the double point group mm2 and has five irreps. Its character table is shown in the Fig.<ref>.(iii)Table of characters of the subduced representations. The programRepresentations DSG calculates the characters of the elements of the site-symmetry group S (obtained in the first step) for each of the irreps 𝐃_β of ^dG of the selected wave vector. In this way, we obtain the characters of the subduced representations 𝐃_β↓S of S.The double space group P4̅m2 has seven irreps for the wave-vector M which subduce seven representations of the double site-symmetry group S=mm2 with the characters shown in Fig. <ref>.(iv)Table of the decompositions of the subduced representations. The multiplicities n_α^(β) of the double point-group irreps 𝐝_α of S in the subduced representations 𝐃_β↓S are obtained by the application of the reduction formula, cf. eq. (<ref>). In the example, the decompositions of the representations ^*M_i ↓ mm2, i=1,…,7, into irreps of mm2 are shown in Fig. <ref>:(v)Table of induced representations. According to Frobenius reciprocity theorem, the multiplicities n_β^(α) of the irreps of ^dG for a given k-vector in the representations 𝐝_α↑^dG (induced from the irreps 𝐝_α of the site-symmetry group S) are obtained by "transposing" the table of the decompositions of the subduced representations 𝐃_β↓S. The table of representations of the double space group P4̅m2 at the point M induced by the irreps of the site-symmetry group mm2 of the Wyckoff position 2g is shown in Fig. <ref>. The rows of the table correspond to the irreps 𝐝_α of the site-symmetry group mm2 (cf. Fig. <ref>); the entries in each row indicate the multiplicities of the M-irreps ofP4̅m2 in the (infinite) induced representation 𝐝_α↑P4̅m2: A_1 ↑P4̅m2∼M_5⊕⋯ A_2 ↑P4̅m2∼M_5⊕⋯ B_2 ↑P4̅m2∼M_1⊕M_2⊕⋯ B_1 ↑P4̅m2∼M_3⊕M_4⊕⋯ E↑P4̅m2∼M_6⊕M_7⊕⋯ The obtained results coincide exactly with the corresponding data of Table 2 of ref. kitaev1997. § BAND REPRESENTATIONS AND ELEMENTARY BAND REPRESENTATIONS§.§ The problemThe concept of a band representation (BR) was introduced by zak1982, as a set of extended energy states over the entire reciprocal space, E_n(𝐤), related to the symmetry of (exponentially) localized states (Wannier orbitals). The basic structure description of a crystal includes the assignation of a space group and the determination of the lattice parameters and of the atomic coordinates (occupied Wyckoff positions) of a minimal set of atoms that belong to the asymmetric unit (known also as the set of independent atoms). The atomic positions of all the atoms in the crystal are obtained through the action of the symmetry operations of the space group onto the coordinates of the independent atoms. As explained in Section <ref>, the finite set of symmetry operations of the space group that keeps a point of a Wyckoff position invariant is its site-symmetry group S, and it is isomorphic to a point group. It is important to note that the site-symmetry groups S_q of all points q belonging to a Wyckoff position 𝐐={q} are isomorphic to the same point-group type, and in that sense one speaks of a site-symmetry group of a Wyckoff position S_𝐐. The localized states(the atomic orbitals, for example) of an atom that occupy a given Wyckoff position transform according to a representation 𝐝_α of its site-symmetry group. When the spin-orbit coupling is also considered, the localized states transform according to a representation of the double site-symmetry group, isomorphic to a double point group. A band representation can be defined as the induced representation 𝐝_α↑𝒢, being a particular case of the general description of the site-symmetry approach discussed in Section <ref> If the representation 𝐝_α of the site-symmetry group S is reducible, then it decomposes into irreps of the site-symmetry group,𝐝_α= ⊕_β n_β^(α)𝐝_β,and the induced BR decomposes into BRs induced from the irreducible representations 𝐝_β 𝐝_α↑𝒢= ⊕_βn_β^(α)(𝐝_β↑𝒢).Therefore, in order to have a complete information of the BRs induced from a given Wyckoff position, it is necessary to calculate only the BRs induced from the irreps of its site-symmetry group. The BRs induced from different irreps of the same site-symmetry group are not equivalent, but BRs induced from different Wyckoff positions could be equivalent. The definition of equivalence of BRs is different from the definition of equivalence of representations, where we say that two representations are equivalent if all the multiplicities of their decomposition into irreps are the same. However, for certain applications, as for example, the study of topological phases, we need a stronger form of equivalence <cit.>. Definition 1: Two band representations ρ_G^𝐤 and σ_G^𝐤 are equivalent iff there exists a unitary matrix-valued function S(𝐤,s,g) smooth in 𝐤 and continuous in s such that for all g∈𝒢 1.S(𝐤,s,g) is a band representation for all s∈[0,1],2.S(𝐤,0,g)=ρ_G^𝐤(g) and3.S(𝐤,1,g)=σ_G^𝐤(g) This definition implies that the BRs ρ_G^𝐤(g) and σ_G^𝐤(g) subduce into the same little group representations at all points in the Brillouin zone. This definition also implies the following: consider two Wyckoff positions 𝐐 and 𝐐 ' and let ρ_G^𝐤(g) and σ_G^𝐤(g) be two BRs induced from an irrep ρ of the site-symmetry group 𝒮_𝐐 of 𝐐 and an irrep σ of the site-symmetry group 𝒮_𝐐 ' of 𝐐 '. The intersection 𝒮_𝐐_0=𝒮_𝐐∩𝒮_𝐐 ' of the two site-symmetry groups is the site-symmetry group of another Wyckoff position 𝐐_0 that can be identified relatively easy: some of the point coordinates of 𝐐_0 should be represented by variable parameters that interpolate between the point coordinates of the Wyckoff positions 𝐐 and 𝐐 '. If for a given irrep ρ_0 of 𝒮_𝐐_0, the induced representations into the site-symmetry groups of 𝐐 and 𝐐 ' satisfy, ρ_0↑ S_𝐐=ρ and ρ_0↑ S_𝐐 '=σ,then the two BRs (ρ↑ G)↓ G^𝐤=ρ_G^𝐤 and (σ↑ G)↓ G^𝐤=σ_G^𝐤 are equivalent.Once a criterion of equivalence of BR is established, we define:Definition 2: A band representation is called composite if it is equivalent to the direct sum of other band representations. A band representation that is not composite is called elementary.All the elementary band representations (EBR) are induced from the so-called Wyckoff positions of maximal symmetry (cf. Section <ref> for the definition of the concept) but the opposite is not true. This fact was already pointed out for single-valued BRs in bacry1988, where also the complete list of exceptions was given. The list of exceptions for BRs induced from double-valued irreducible representations of the double space groups can be found in the supplementary material of NaturePaper.The generalization of the procedure for the calculation of EBRs for systems with time-reversal symmetry is straightforward. In this case, it is enough to consider the BRsinduced from the physically irreducible representations of the site-symmetry groups of the Wyckoff positions of maximal symmetry. The physically irreducible or TR-invariant representations of the site-symmetry groups can be constructed following the steps explained in Section <ref>. The TR-invariant irreps induce TR-invariant BRs in 𝒢, and using the definitions 1 and 2 above the TR-invariant EBRs can be calculated. Not all the single-valued TR-invariant BRs induced from the double site-symmetry groups of Wyckoff positions are elementary: the list of exceptions is given in <cit.>. However, all the double-valued TR-invariant BRs induced from the double site-symmetry groups of Wyckoff positions of maximal symmetry are elementary, as a consequence of Kramer's theorem <cit.>.In a series of concomitant articles <cit.>, in relation to the problem of topological insulators, we have studied the connectivities of energy bands of the EBRs. The energy bands are continuous functions defined in the Brillouin zone. The possible different ways in which the Bloch states at bands of 𝐤-vectors of maximal symmetry (cf. Section <ref>) are connected through intermediate bands of 𝐤-vectors of lower symmetry, are restricted by the corresponding compatibility relations (cf. Section <ref>). In most cases the EBRs are fully connected, which means that we can continuously 'travel' along all the points that form the energy band. However, in some cases, depending on the specific values of the electronic band energies at each 𝐤-vector, the compatibility relations allow the decomposition of the whole group of bands into disconnected branches, separated by an energy gap. Exactly such EBRs describe topological insulators. In a companion paper <cit.> we describe in detail the algorithms used to determine if an EBR (with and without TR) is decomposable or not.§.§ The methodThe algorithms implemented in the programBANDREP to calculate the BRs follows the site-symmetry procedure explained in Section <ref> and makes use of different tools of the BCS. Given a space group G, a BR is fully identified by a Wyckoff position, an irrep of its site-symmetry group and the irreps of the little groups of any 𝐤-vector of the reciprocal space. In fact, it is only necessary to consider the 𝐤-vectors of maximal symmetry (cf. Section <ref>), because the multiplicities of the irreps of the little groups of any other 𝐤-vector can be calculated from the former ones using the corresponding compatibility relations described in Section <ref>. The multiplicities of the irreps of the little groups of any 𝐤-vector in BRs are determined applying the site-symmetry approach.Given a space group 𝒢 and a Wyckoff position 𝐐={q}, first the program identifies the symmetry elements included in the site-symmetry group S_q: for each of the coset representatives {R|𝐯} of G:T the program calculates 𝐭 that satisfies the equation,𝐭=Rq+𝐯-q.If 𝐭 belongs to the set of lattice translations of 𝒢, then {R|𝐯-𝐭} belongs to the site-symmetry group 𝒮_q. Once all elements of 𝒮_q are determined, the toolIDENTIFY GROUP (www.cryst.ehu.es/cryst/identify_group) of the BCS is used to identify the point (or double point) group type isomorphic to 𝒮_q and to establish the corresponding isomorphic mapping between the group elements. As a next step, for each element g∈𝒮_q, we calculate: (i) the characters χ_α(g) of each irrep 𝐝_α of the double point group (usingRepresentations DPG), and (ii) the characters χ_β^^*𝐤(g) of the irreps D_β^*𝐤 of 𝒢 for each of the tabulated 𝐤-vectors in the Brillouin zone, induced from the allowed irreps (or TR-invariant irreps) 𝐝_β^𝐤 of the little group 𝒢^𝐤 of 𝐤 (applying the toolRepresentations DSG). The decomposition of the subduced representations 𝐃_β^*𝐤↓𝒮_q into irreps 𝐝_α of the site-symmetry group S_q is performed using eqs. (<ref>) and (<ref>): 𝐃_β^*𝐤↓𝒮_q∼⊕ n_α^𝐤,(β)𝐝_α,where the multiplicities are,n_α^𝐤,(β)=1/|𝒮_q|∑_g∈𝒮_qχ_α(g)^*χ_β^*𝐤(g). As explained in Section <ref>, according to the Frobenius reciprocity theorem, these multiplicities coincide with the multiplicities of the irreps of the little group 𝒢^𝐤 in the BR induced from the irrep 𝐝_α of the double point group 𝒮_q: (𝐝_α↑𝒢)↓𝒢^𝐤∼⊕ n_β^𝐤,(α)𝐝_β^𝐤. We thus fully identify the BR induced from the irrep 𝐝_α of the site-symmetry group of the Wyckoff position 𝐐 giving the tabulated irreps 𝐝_β^𝐤 of the little group for every 𝐤 of the space group 𝒢 (obtained byREPRESENTATIONS DSG) and calculate the multiplicities n_α^𝐤,(β).Finally, we check if the BR induced from the Wyckoff position 𝐐 is elementary or not. As all EBRs of a space group are induced from Wyckoff positions of maximal symmetry, the check is reduced to BRs induced from such Wyckoff positions. Each BR is characterized by a set of multiplicities of the little-group irreps but only for 𝐤-vectors of maximal symmetry. Using these lists of multiplicities we can easily check if the given BR decomposes at every k as the direct sum of two or more irreps. If such decompositions exist, the BR is a candidate to be a composite BR. However, as explained in Section <ref>, this condition is not sufficient to consider the BR as equivalent to the direct sum of BRs induced from a different Wyckoff position 𝐐 '. In addition, it is necessary to calculate the intersection of the site-symmetry groups S_Q and S_Q ' of 𝐐 and 𝐐 ', respectively, and identify the Wyckoff position 𝐐_0 (of non-maximal symmetry) which has this intersection S_Q_0 as its site-symmetry group. Then, we induce representations of S_Q and S_Q ' from every irrep of S_Q_0. If for some irrep of S_Q_0 we get ρ, the irrep of S_Q that induces the candidate BR to be composite, and a reducible representation σ of S_Q ', that induces a composite BR that decomposes at every 𝐤 as ρ↑G does, the candidate BR is composite. Otherwise, it is elementary. §.§ Wyckoff positions of maximal symmetryThe so-called Wyckoff positions of maximal symmetry are of importance for certain applications, and in particular, for the calculation of the EBRs. In the following, we comment briefly on the definition of Wyckoff positions of maximal symmetry The intersection 𝒮_𝐐_0=𝒮_𝐐∩𝒮_𝐐 ' of the two site-symmetry groups is the site-symmetry group of another Wyckoff position 𝐐_0 that can be identified relatively easy: some of the point coordinates of 𝐐_0 should be represented by variable parameters that interpolate between the point coordinates of the Wyckoff positions 𝐐 and 𝐐 '. If for a given irrep ρ_0 of 𝒮_𝐐_0, the induced representations into the site-symmetry groups of 𝐐 and 𝐐 ' satisfy, ρ_0↑ S_𝐐=ρ and ρ_0↑ S_𝐐 '=σ, then the two BRs (ρ↑ G)↓ G^𝐤=ρ_G^𝐤 and (σ↑ G)↓ G^𝐤=σ_G^𝐤 are equivalent.and on the procedure we used for their calculation. Note that the sets of Wyckoff positions of double space groups and those of the corresponding space groups are closely related. In fact, the essential difference concerns the site-symmetry groups which, in the former case, are isomorphic to double point groups, while in the latter, only to point groups. As a consequence, practically the same procedure for the determination of the Wyckoff positions of maximal symmetry can be used for space groups and for double space groups.It is common to describe a Wyckoff position by its multiplicity, Wyckoff letter, symbol of the site-symmetry group and a set of coordinate triplets of the points in the unit cell that belong to the Wyckoff position, possibly depending on one or two variable parameters (three for the general position) (for a detailed introduction to Wyckoff positions of space groups, cf.). A Wyckoff position Q with a site-symmetry group S_Q has maximal symmetry if it is not connected to another Wyckoff position Q ' whose site-symmetry group S_Q ' is a supergroup of S_Q. We say that two Wyckoff positions are connected if (i) the coordinate triplet of at least one of them depends on one or more variable parameters, and (ii) if for specific values of the variableparameters the coordinate triplets of the two Wyckoff positions coincide. For instance, in the space group P2/m (No. 10), the site-symmetry group of the Wyckoff position 2i: (0,y,0) is isomorphic to the point group 2. It is not a Wyckoff position of maximal symmetry because it is connected to the Wyckoff position 2a: (0,0,0):the coordinate-triplet description of 2a is obtained from that of 2i by the substitution y=0, and the site-symmetry group of the position 2a is isomorphic to 2/m which is supergroup of 2.The algorithm to identify the Wyckoff positions of maximal symmetry is straightforward. In the space groups with no points in a special position (the so-called fixed-point-free or Bieberbach groups) the general Wyckoff position is the only Wyckoff position of maximal symmetry. For the rest of space groups, we distribute the special Wyckoff positions into three subsets, according to the number (0,1,2) of variable parameters of their coordinate triplets. All Wyckoff positions with 0 variable parameters are Wyckoff positions of maximal symmetry. Those Wyckoff positions of the subset with one variable parameter which are connected to at least one Wyckoff position of the subset with no variable parameter, in the sense explained above, are not maximal. The rest of Wyckoff positions with one variable parameter are maximal. Finally, we repeat the check for the subset of Wyckoff positions with two variable parameters, trying to find if they are connected to at least one of the Wyckoff positions of the subsets with 0 and 1 variable parameter. Those that have no connection are Wyckoff positions of maximal symmetry.As an example, consider the Wyckoff positions of the (double) space group P4/ncc (No. 130) shown in Table <ref>.The Wyckoff positions 4a,4b, 4c and 8dare of maximal symmetry while 8e,8f and 16g are not. If a Wyckoff position is not maximal, then the corresponding Wyckoff position of maximal symmetry (i.e. the one to which it is connected) together with the specific values of the variable parameters for which the two coordinate-triplet descriptions coincide, are indicated in the last column of the table. §.§ 𝐤-vectors of maximal symmetryIn analogy to the Wyckoff positions of maximal symmetry introduced in the previous section, for each space group we can define a set of 𝐤-vectors of maximal symmetry. These vectors play also an important role, for example, in the analysis of the connectivities of the BRs (cf.<cit.>). Similar to the distribution of points of direct space into Wyckoff positions, the set of all 𝐤-vectors can be distributed into the so-called 𝐤-vector types. A 𝐤-vector type consists of complete orbits of 𝐤-vectors and thus of full stars of 𝐤-vectors. The 𝐤-vectors belonging to a 𝐤-vector type are represented by a 𝐤-vector letter (here we follow CDML notation), by the point-group type of the little co-groups of 𝐤-vectors, and by a set of 𝐤-vectors coefficients: the zero, one or two variable parameters in the 𝐤-vector coefficients correspond to special𝐤-vector types, i.e., they define symmetry points, symmetry lines or symmetry planes in the Brillouin zone. Three variable parameters indicate a general𝐤-vector type.We say that a 𝐤-vector type (or just a 𝐤-vector, for short) is of maximal symmetry if its little co-group is not a subgroup of the little co-group of another 𝐤 '-vector type connected to 𝐤 in the same sense as the connected Wyckoff positions discussed in Section <ref>. The procedure to identify the 𝐤-vectors of maximal symmetry is analogous to the procedure for the determination of the Wyckoff positions of maximal symmetry. In general, the set of 𝐤-vectors of maximal symmetry for non-centrosymmetric space groups (i.e space groups G whose point groups G do not include the operation of (space) inversion 1) could be modified when time-reversal symmetry is taken into account. One can show that in such cases wave vectors of the same 𝐤-vector type (with respect to the spacial symmetry) could behave differently under the action of time reversal: some wave vectors are time-reversal invariant, the so-called Time Reversal Invariant Momentum (TRIM) points, while others are not. In other words, when TR symmetry is taken into account, the TRIM points are 𝐤-vectors of maximal symmetry and the corresponding physically irreducible representationsdetermine the transformation properties of the eigenfunctions of the Hamiltonian of the system. As an example, consider the polar space group P4 (No. 75) and the symmetry line Λ (0,0,w) which is a 𝐤-vector line of maximal symmetry as its little co-group is the point group 4 (C_4). There are 4 one-dimensional single-valued irreps at the Λ (0,0,w) point: Λ_i, i=1,…,4. If however, time-reversal symmetry is taken into account, the 𝐤-vector type Λ (0,0,w) 'splits' into three 𝐤-vector types: the points Γ(0,0,0) and Z (0,0,1/2) are TRIM points and become 𝐤-vector types of maximal symmetry while the rest of the points of the line Λ form a 𝐤-vector type of non-maximal symmetry. Two of the four single-valued irreducible representations at these points, Γ_3, Γ_4 and Z_3, Z_4, form a pair of complex conjugated irreps and become doubly-degenerate.Tables <ref> and <ref> show the lists of 𝐤-vectors of maximal symmetry of P4 (No. 75), without and with TR, respectively.The above example indicates the important consequences of the time-reversal symmetry in the derivation of the compatibility relations, and in particular, in the study of EBR connectivities between pairs of 𝐤-vectors of maximal symmetry. §.§ The programBANDREPThe programBANDREP, recently added to BCS, applies the method explained in Section <ref> to calculate the BRs of any of the 230 double space groups. The fact that the list of single-valued BRs identified as elementary by the program, coincides exactly with the list obtained by bacry1988, alongside with the many checks performed and the agreement between the program and the results of NaturePaper, can be considered a proof test of the program. Input: The main page of the input requires the specification of double space group (by itssequential number) and offers four options: calculation either of EBRs of the double space group with or without TR symmetry, or of the BRs with or without TR symmetry. For the last two options the program produces the list of Wyckoff positions of the selected space group, separated into those of maximal and non-maximal symmetry. Figures <ref> and <ref> show screenshots of the input pages.Output: The option selected by the user determines the specific output produced by the program:*Option: Elementary band representations without TR symmetry A screenshot of the table output of the programBANDREP for the double space group P4/ncc (No. 130) is shown on Figure <ref>. The EBRs are listed in columns specified by (i) the Wyckoff positions of P4/ncc (No. 130) followed by the symbol of the double point group S_𝐐 isomorphic to the site-symmetry group (first row of the header), and (ii) the irrep 𝐝_α of the site-symmetry group from which theEBR 𝐝_α↑G is induced (second row of the header); the dimension of 𝐝_α↑G is shown in brackets after the EBR symbol. The entries of the third row of the header indicate if the EBR is decomposable or not (cf. Section <ref>). The entries of the output table show the decompositions of EBRs into irreps of the little groups of the 𝐤-vectors of maximal symmetry which denote the rows of the table. The dimensions of the little-group irreps are given in brackets after their symbols. Clicking on the button "Show all types of 𝐤-vectors" generates a table with the decompositions of each EBR into irreps of the little groups of maximal and non-maximal 𝐤-vectors.Clicking on the button "Minimal set of paths and compatibility relations to analyse the connectivity", produces the set of paths between pairs of 𝐤-vectors of maximal symmetry to be considered in the analysis of the connectivities of the EBRs in the given space group, once the known redundancies have been removed <cit.>, and the corresponding independent compatibility relations. Figure <ref> shows the screenshot of the minimal set of paths and compatibility relations for the space group P4/ncc. For more details about the problem of connectivity of EBRs see GraphDataPaper.*Option: Elementary band representations with TR symmetryThe essential difference in comparison with the output of Option 1 is that the shown results refer to physically-irreducible representations (for a discussion on physically-irreducible representations, cf. Section <ref>).*Option: Band representations from a Wyckoff position without TR symmetryUnder this option the program shows the band representations for a specific Wyckoff position chosen from a list of Wyckoff positions of maximal or of non-maximal symmetry. Fig. <ref> shows the output produced by the program for the case of the double space group P4/ncc, and the Wyckoff position of maximal symmetry 8d. The structure of the output table is similar to those of elementary bands: the essential difference is that the program shows not only the elementary but also the composite band representations (specified in the second row of the table header). Whether the EBRs are decomposable or not is also indicated. In the partial output shown in Fig. <ref> all the EBRs are decomposable. By clicking on the button "Decomposable", we obtain all different ways to decompose the EBR into disconnected branches, which correspond to topological insulators. The four possible ways to decompose the EBR A_g↑G induced from the A_g irreducible representation of the site-symmetry group of the 8d Wyckoff position of the P4/nccspace group are shown in Fig. <ref>. Each column gives the arrangements of the set of the little-group irreps at each 𝐤-vector of maximal symmetry in a branch. The band-graph (not yet available in the BCS) that illustrates a disconnected EBR is shown in Fig. <ref>. The arrangements of irreps at each 𝐤-vector in Fig. <ref> correspond to the decomposition shown on the first row of Fig. <ref>.*Option: Band representations from a Wyckoff position with TR symmetry. This provides similar information as Option 3 but for the physically-irreducible representations. § CONCLUSIONS The group-theoretical description of physical systems where the Hamiltonian depends on spin components, would require the use of the so-called double crystallographic groups and their single- and double-valued representations. In this paper, we describe a set of databases and programs of double crystallographic groups that recently have been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). As the rest of the programs on BCS, the new tools are freely available and can be accessed via user-friendly web interfaces. Some of the algorithms applied in the new programs are extensions of the algorithms used in the BCS for ordinary space groups.The toolDGENPOS provides in different formats the symmetry operations of the 230 double space groups in the standard or conventional setting. The programREPRESENTATIONS DPG gives access to the irreducible representations of the 32 crystallographic double point groups whileREPRESENTATIONS DSG calculates the irreducible representations of the double space groups and analyses their reality indicating also the pairs of conjugated irreducible representations. The programDSITESYM applies the site-symmetry approach to the double space groups. It can be considered as a bridge between a local description of the atomic orbitals on-site and a global description through extended states along the Brillouin zone in a structure. The programDCOMPREL calculates the compatibility relations between the irreducible representations of double space groups at high- and low-symmetry points in the Brillouin zone, necessary in the analysis of the connectivity of the functions defined in the reciprocal space (e.g. in the analysis of the structure of the electronic bands and their connectivity through the Brillouin zone). As an application of the site-symmetry approach, we have developed the programBANDREP that provides the band representations and the subset of elementary band representations induced from any Wyckoff position of any double space group. The tool also identifies the subset of decomposable elementary band representations and the different ways of their decomposition, together with large amount of additional data necessary for their study. Concomitantly with the current paper, the results provided byBANDREP have been successfully applied in a novel method for the description, search and prediction of topological insulators <cit.>.§ NORMAL-SUBGROUP INDUCTION PROCEDUREThe irreps of a space group G are obtained by induction from the irreps of its translation group T. Assuming the Born-von Karman (periodic) boundary conditions (I, t_i)^N_i = (I, o) to hold, where t_i = (1,0,0), (0,1,0) or (0,0,1) and N_i is a large integer for i = 1, 2 or 3, respectively. Then, the irreps Γ^𝐤 of the translation group T are given by:Γ^k [ (I,t)] = exp(-ik·t). There are N_1N_2N_3 different irreps of T which aredistinguished by the wave vectors:𝐤 = ∑_i=1^3 k_ia_i^*,where k_i = q_i/N_i; q_i=0, 1, 2, … , N_i-1. The basis a_1^*, a_2^*, a_3^*is called the basis of the reciprocal latticeL^* andit is the dual basis of 𝐚_1, a_2,a_3 of L. The vectors 𝐚_i^*are defined by the relations 𝐚_i·𝐚_j^*=2πδ_ijwhere δ_ij is the Kronecker symbol.The wave vectors k and k' = k + K, where K is a vector of the reciprocal lattice L^*, describe the same irreps of T. Therefore, in order to determine all the irreps of T, it is necessary to consider only the k-vectors of the first Brillouin-zone.The little co-group of k is the point group consisting of all the rotational parts R^k of the symmetry operations of the space group G that either leave the k-vector invariant, or map it to an equivalent vector, i. e.:k =kW^𝐤 +K, K∈ L^*. Here, W^𝐤 is the (3×3) matrix representation of R^k. The little co-group G^k is a subgroup of the point group G of the space group G. The vector k is called a generalkvector if the little co-group contains the identity operation only, i.e.G ^𝐤 = {I}; otherwise G ^𝐤 > {I}, and k is called a specialkvector. Consider the coset decomposition of G relative to G ^𝐤. If {R_m} is the corresponding set of coset representatives, then the set *𝐤={ kW_m+𝐊}, with 𝐊∈L^*, is called the star ofk and the vectors 𝐤W_m+𝐊 are called the arms of*𝐤.An orbit of Γ^𝐤(T) under conjugation by G comprisesall irreps Γ^𝐤'(T) with 𝐤'belonging to *𝐤. Irreps of T belonging to the same orbit give rise to equivalent irreps of G, i.e. in order to obtain each irrep of G exactly once it is necessary to consider one k vector per star. (A simply connected subset of the Brillouin zone which containsexactly one k vector per*𝐤, is called arepresentation domain.)Given a space group G, its translation subgroup T, and an irrepΓ^𝐤(T), one can define the littlegroupG^𝐤 of the wave vector k: it is a space groupthat consists of all those elements of G whose rotation partsR^𝐤 leave either k unchanged ortransform it into an equivalent vector, G^𝐤={{R^𝐤|𝐯^𝐤}∈G|R^𝐤∈G ^𝐤}.The irreps of space groups are obtained by induction from the so-called allowed irreps of the little groups G^𝐤 of k. IfD^𝐤,i is anallowed irrep of G^𝐤, then D^𝐤,i ({1| 𝐭})=exp(-ik· t) I.(The matrix I is the identity matrix withI=D^𝐤, i). The allowed irreps of the little group G^𝐤 are determined by an induction procedure <cit.> which is based on the fact that the little groups (as all crystallographic groups) are solvable groups, i.e. for each group G^𝐤 there exists a series of subgroups H_i^𝐤 (the so-called composition series), such that: G^𝐤H_1^𝐤…H_m-1^𝐤H_m^𝐤…H_n^𝐤=Tand that the factor groups H_m-1^𝐤/H_m^𝐤 are cyclic groups of prime order. The (allowed) irreps of G^𝐤 can be obtained from the (allowed) irreps ofT by applying several times the general induction procedure'climbing up' the chain of normal subgroups (eq. <ref>). Important for the irrep calculation is the observation that the factor groups in the composition series of crystallographic groups have orders 2 or 3 which simplifies considerably the induction procedure. The corresponding induction formulae and a detailed example of application of the induction method in the case of the space group P4bm and 𝐤=𝐗(0,1/2,0) can be found, for example, in Aroyo et al. (2006).Finally, following the normal-subgroup induction procedure, the irreps of a space group G (the so-called full irreps) for a given k vector are obtainedby induction from the allowed irreps 𝐃^𝐤, i ofthe corresponding little group G^𝐤. Let the elementsq_m={R_m| 𝐯_m},m=1, … ,s be the representatives of the cosets in the decomposition of G relative toG^𝐤: G = G^𝐤∪q_2G^𝐤∪…∪q_sG^𝐤.If dim D^𝐤, i=r,and if s is the number of arms in (the order of) the star of k,then the induced irrep D^*𝐤, i(G) has the dimension r× s and its matrices can be written in the form: D^*𝐤, i({R| 𝐯}_mp,nq)= M({R| 𝐯}_m,n)D^𝐤, i ({R^𝐤| 𝐯^𝐤}_p,q),where {R^𝐤| 𝐯^𝐤}= (q_m)^-1 {R| 𝐯} q_n is an element of the little group G^𝐤, with n,m=1,…,s. Because the s × sinduction matrixM({R| 𝐯}) is a monomial matrix,the matricesD^*𝐤, i({R| 𝐯}) have a block structure with exactly onenon-zero (r × r) block in every column and every row; the block is thematrix D^𝐤, i({R^𝐤| 𝐯^𝐤}), and{R^𝐤| 𝐯^𝐤} is fixed by the condition {R^𝐤| 𝐯^𝐤}=(q_m)^-1 {R| 𝐯} q_n ∈G^𝐤. The work of LE, GF and MIA was supported by the Government of the Basque Country (project IT779-13) and the Spanish Ministry of Economy and Competitiveness and FEDER funds (project MAT2015-66441-P). The work of MVG was supported by FIS2016- 75862-P and FIS2013-48286-C2-1-P national projects of the Spanish MINECO. ZW and BAB, as well as part of the development of the initial theory and further ab-initio work, were supported by the NSF EAGER Grant No. DMR-1643312, ONR - N00014-14-1-0330, ARO MURI W911NF-12-1-0461, and NSF-MRSEC DMR-1420541. The development of the practical part of the theory, tables, some of the code development, and ab-initio work was funded by Department of Energy de-sc0016239, Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. BB, JC, ZW, and BAB acknowledge the hospitality of the Donostia International Physics Center, where parts of this work were carried out. JC acknowledges the hospitality of the Kavli Institute for Theoretical Physics, and BAB acknowledges the hospitality and support of the École Normale Supérieure and Laboratoire de Physique Théorique et Hautes Energies. | http://arxiv.org/abs/1706.09272v2 | {
"authors": [
"Luis Elcoro",
"Barry Bradlyn",
"Zhijun Wang",
"Maia G. Vergniory",
"Jennifer Cano",
"Claudia Felser",
"B. Andrei Bernevig",
"Danel Orobengoa",
"Gemma de la Flor",
"Mois I. Aroyo"
],
"categories": [
"cond-mat.mes-hall",
"physics.data-an"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170626134423",
"title": "Double crystallographic groups and their representations on the Bilbao Crystallographic Server"
} |
a4papershowonlyrefs=true *§ * §.§ *§.§.§ ** teoTheorem[section] lema[teo]Lemma cor[teo]Corollary prop[teo]Proposition defi[teo]Definition mytheoremstyle. .5emmytheoremstyle notaRemark[section] mytheoremstyle exemploExample[section] mytheoremstyle *notacaoNotation equationsection | http://arxiv.org/abs/1706.08475v1 | {
"authors": [
"Simão Correia"
],
"categories": [
"math.AP",
"35Q55, 35A01"
],
"primary_category": "math.AP",
"published": "20170626164726",
"title": "Local Cauchy theory for the nonlinear Schrödinger equation in spaces of infinite mass"
} |
Branching-point selection for trilinear monomialsE. Speakman Dept. of Industrial and Operations Engineering, University of Michigan, Ann Arbor. [email protected]. Lee Dept. of Industrial and Operations Engineering, University of Michigan, Ann Arbor. [email protected] On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation[This work extends and presents parts of the first author's doctoral dissertation<cit.>, and itcorrects results first announced in the short abstract <cit.>.]Emily Speakman Jon Lee============================================================================================================================================================================================================================================================================ In Speakman and Lee (2017), we analytically developed the idea of using volume as a measure for comparing relaxations in the context of spatial branch-and-bound. Specifically, for trilinear monomials, we analytically compared the three possible“double-McCormick relaxations” with the tight convex-hull relaxation. Here, again using volume as a measure, for the convex-hull relaxation of trilinear monomials, we establish simple rules for determining the optimal branching variable and optimal branching point. Additionally, we compare our results with current software practice. § INTRODUCTION In this article, we consider the spatial branch-and-bound (sBB) family of algorithms (see, for example, <cit.>,<cit.>,<cit.>, building on <cit.>) which aim to find globally-optimal solutions of factorable mathematical-optimization formulations via a divide-and-conquer approach (building on the branch-and-bound approach for discrete optimization, see <cit.> and <cit.>).Implementations of these sBB algorithms for factorable formulations work by introducing auxiliary variables in such a way as to decompose every function of the original formulation which we can then view as a labeled directed graph (DAG). Leaves correspond to original model variables, and we assume that the domain of each such model variable is a finite interval.We have a library of basic functions, including `linear combination' of an arbitrary number variables, and other simple functions of a small number of variables.The out-degree of each internal node, labeled by a library function f∈ℱ that is not`linear combination'is typically small (say d_f≤ 3, for all f∈ℱ). We assume that we have methods for convexifying each low-dimensional library function f on an arbitrary box domain in ℝ^d_f. From these DAGs, relaxations are composed and refined (see <cit.>, for example). For a given function f, the associated DAG can be constructed in more than one way, and therefore sBB has choices to make inthis regard.Such choices can have a strong impact on the quality of the convex relaxation obtained from the formulation. Because sBB algorithms obtain bounds from these convex relaxations, these choices can have a significant impact on the performance of the algorithm.There has been substantial research on how to obtain good-quality convex relaxations of graphs of low-dimensional nonlinear functions on various domains (see, for example, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>), and some consideration has been given to constructing DAGs in a favorable way.In particular, in <cit.>, weobtained analytic results regarding the convexifications obtained from different ways of treating trilinear monomials, f=x_1 x_2 x_3, on non-negative box domains {x∈^3 : x_i∈[a_i,b_i], i=1,2,3}.We computed both the extreme point and inequality representations of the alternative relaxations (derived from iterating McCormick inequalities) and calculated their 4-dimensional volumes (in the space of {(f,x_i,x_j,x_k)∈ℝ^4}) as a comparison measure.Using volume as a measure gives a way to analytically compare formulations and corresponds to a uniform distribution of the optimal solution across a relaxation; when concerned with non-linear optimization, such a uniform distribution is quite natural.This is in contrast to linear optimization, where an optimal solution can always be found at an extreme point, and therefore, the distribution of the optimal solution (or optimal solutions) is clearly not uniform across the feasible region. Experimental corroboration for using volume as a measure of the quality of relaxations for trilinear monomials appears in <cit.> (also see <cit.>, concerning quadrilinear monomials).Along with utilizing good convex relaxations, other important issues in the effective implementation of sBB for factorable formulations are: (i) the choice of branching variable, and (ii) the selection of the branching point.Software developers have tuned their choice of branching point using extensive problem test beds.It is common practice for solvers to branch on the value of the variable at the current solution, adjusted using some method to ensure that the branching point is not too close to either of the interval endpoints.Often this is done by weighting the interval midpoint and the variable at the optimal solution of the current relaxation, and/or restricting the branching choice to a central part of the interval.For example, in <cit.> (also see <cit.>), they suggest branching at the current relaxation point when it is in the middle 60% of the interval and failing that, branch at the midpoint. Thesoftware,see <cit.>, uses the solution of an upper-bounding problem as a reference solution, if such a solution is found; otherwise the solution of the lower-bounding relaxation is used as a reference solution.then identifies the non-convex term with the greatest separation distance with respect to its convex relaxation. The branchingvariable is then chosen as the variable whose value at the reference solution is nearest to the midpoint of its range. But it is not clear howthen chooses the branching point. <cit.> describe a typical way to avoid the interval endpoints by choosing the branching point asmax{a_i+β(a_i-b_i), min{b_i-β(b_i-a_i), αx̂_i+(1-α)(a_i+b_i)/2}},where x̂_i is the value of the branching variable x_i at the current solution.The constants α∈ [0,1] and β∈ [0, 1/2] are algorithm parameters. So, the branching point is the closest point in the interval[a_i+β(a_i-b_i), b_i-β(b_i-a_i)]to the weighted combination αx̂_i+(1-α)(a_i+b_i)/2 (of the x-value in the current optimal solution and the interval midpoint), thus explicitly ruling outbranching in the bottom and top β fraction of the interval. Note that ifβ≤ (1-α)/2, then there is no such explicit restriction, because already the weighted combination αx̂_i+(1-α)(a_i+b_i)/2 precludes branching in the bottom and top (1-α)/2 fraction of the interval.Current available software use a variety of values for the parameters α and β. The method (mostly) employed by(see <cit.>, <cit.> and the open-source code itself) is to select the branching point as the closest point in themiddle 60% of the interval to the variable value x̂_i.This is equivalent to setting α=1 and β=0.2 and gives an explicit restriction via the choice of β.The current default settings of [Private communication with Ruth Misener] (<cit.> and <cit.>), [Private communication with Nick Sahinidis] (<cit.>) and(see <cit.> and the open-source code itself) all have β≤ (1-α)/2, and so the default branching pointis simply the weighted combination αx̂_i+(1-α)(a_i+b_i)/2; see Table <ref>.The different choices are based on combinations of intuition and substantial empirical evidence gathered by the software developers. We note that there is considerable variation in the settings of these parameters, across the various software packages. Furthermore, there are other factors (especially in )that sometimes supersede selecting a branching point according to formula (<ref>); in particular, functional forms involved, the solution of the current relaxation, available incumbent solutions, complementarity considerations, etc. Our work is based solely on analyzing a single trilinear monomial, afterbranching on a variable in that trilinear monomial, with the goal of helping to guide, and in some cases mathematically support, the choice of a branching point. Of course variables often appear in multiple functions.So, when deciding on a branching variable or a branching point, we may obtain conflicting guidance.But this is an issue with most branching rules, including those developed empirically, and it is always a challenge to find good ways to combine local information to make algorithmic decisions (see <cit.>).We hope that our results can help influence such decisions. For example, taking weighted averages of scores based on our metric would be a reasonable way to proceed.§ PRELIMINARIESIn this work, we focus on trilinear monomials; that is, functions of the form f=x_1x_2x_3.This is an important class of functions for sBB algorithms, because such monomials may also involve auxiliary variables.This means that whenever a formulation contains the product of three (or more) expressions (possibly complicated themselves), our results apply.Following <cit.>, for the variables x_i ∈ [a_i,b_i], i=1,2,3, throughout this paper we assume the following conditions hold:.0 ≤ a_i < b_ifori=1,2,3, anda_1b_2b_3+b_1a_2a_3 ≤ b_1a_2b_3 + a_1b_2a_3 ≤ b_1b_2a_3 + a_1a_2b_3. }Ω To see that the latter two inequalities are without loss of generality, let 𝒪_ia_i(b_jb_k) + b_i(a_ja_k), for i=1,2,3.Then we can construct a labeling such that 𝒪_1 ≤𝒪_2 ≤𝒪_3.Note that because we are only considering non-negative bounds, the latter part of this condition is equivalent to: a_1/b_1≤a_2/b_2≤a_3/b_3.This follows from Lemma <ref> (in the Appendix), and that b_i>0, i=1,2,3.Also, it is very important to note that once we have labeled our variables to satisfy <ref>, our trilinear monomial cannot be treated assymmetric across variables. This condition also arises in the complete characterization of the inequality description for the (polyhedral) convex hull of the graph of the trilinear monomial f:=x_1x_2x_3 (in ^4) (see <cit.> and <cit.>).We introduce the following notation for the convex hull of the graph of f:= x_1x_2x_3 on a box domain:_hconv ({(f,x_1,x_2,x_3) ∈^4 : f= x_1x_2x_3,x_i ∈ [a_i,b_i], i=1,2,3 }). Instead of referring to convex lower envelopes and concave upper envelopes, we take the view that any given monomial is likely to be composed in many different ways in a complicated formulation, and so we are agnostic about focusing on only one of convex lower envelopes and concave upper envelopes, and rather we look at the convex hull of the graph of the function on the domain of interest (and it's total volume; not just the volume below or above the graph).The extreme points of _h are the eight points that correspond to the 2^3=8 choices of each x-variable at its upper or lower bound (see <cit.>). We label these eight points (all of the form [x_1x_2x_3, x_1, x_2, x_3]^T) as follows: v^1:=[[ b_1a_2a_3; b_1; a_2; a_3 ]] ,v^2:=[[ a_1a_2a_3; a_1; a_2; a_3 ]],v^3:=[[ a_1a_2b_3; a_1; a_2; b_3 ]],v^4:=[[ a_1b_2a_3; a_1; b_2; a_3 ]], v^5:=[[ a_1b_2b_3; a_1; b_2; b_3 ]],v^6:=[[ b_1b_2b_3; b_1; b_2; b_3 ]],v^7:= [[ b_1b_2a_3; b_1; b_2; a_3 ]],v^8:=[[ b_1a_2b_3; b_1; a_2; b_3 ]]. The (complicated) inequality description of the convex hull (see <cit.> and <cit.>) is directly used by some global-optimization software (e.g.,and ).However, other software packages (e.g.,and )instead use McCormick inequalities iteratively to obtain a (simpler) convex relaxation fortrilinear monomials.These alternative approaches reflect the tradeoff between using a more complicated but stronger convexification and a simpler but weaker one, especially in the context of global optimization (see <cit.>, for example). From <cit.>, we have a formula for the volume of the convex-hull relaxation (additionally, for the various double-McCormick relaxations), parameterized in terms of the upper and lower variable bounds.Under <ref>, we havevol(_h) = (b_1-a_1)(b_2-a_2)(b_3-a_3)× (b_1(5b_2b_3-a_2b_3-b_2a_3-3a_2a_3) + a_1(5a_2a_3-b_2a_3-a_2b_3-3b_2b_3))/24. Note that due to the asymmetry introduced by <ref>, the formula does not treat all variables in the same manner. In particular, the role of x_1 is quite different than the roles of x_2 and x_3 (which can be interchanged). This observation is very important in our analysis that follows.In the context of branching within sBB, let c_i ∈ [a_i,b_i] be the branching point of variable x_i. We obtain two children. By substituting a_i=c_i and b_i=c_i (respectively) for a given variable x_i into the appropriate formula (i.e., Theorem <ref> after a possible relabeling of the variables), and summing the results, we obtain the total resulting volume of the relaxations at the two child nodes, given that we branch on variable x_i at point c_i. It is important to realize that the volume formula only holds when the labeling <ref> is respected.Given when we branch, the bounds in our problem change, we must be careful to ensure that we always use the formula correctly (i.e. if necessary, we relabel the variables to ensure that <ref> holds). In <ref>, we present our results analyzing optimal branching-point selection for x_1. Then, in <ref>, we present the analysis forx_2 and x_3.Due to the special role of x_1, we will see that the analysis (and even the result) is significantly simpler forx_2 and x_3 than for x_1.In <ref>, we analyze branching-variable selection, and in particular, we demonstrate that it is always best to branch on x_1. In<ref>, wemake some concluding remarks. In the Appendix, we provide proofs of various technical results that we utilize.§ BRANCHING ON X_1First, we define the following quantities (note that because we assume b_i>a_i,i=1,2,3, the denominators will not be zero for any valid parameter choice): q_1 3a_1a_2a_3+a_1a_2b_3-a_1b_2a_3-3a_1b_2b_3+4b_1a_2a_3-4b_1b_2b_3/2(3a_2a_3+a_2b_3-4b_2b_3); q_2 a_1+b_1/2;q_3 4a_1a_2a_3-4a_1b_2b_3+3b_1a_2a_3+b_1a_2b_3-b_1b_2a_3-3b_1b_2b_3/2(4a_2a_3-b_2a_3-3b_2b_3).Next, we refer to Procedure <ref> which depicts a procedure for choosing a branching point when branching on variable x_1. Note that q_1 is not used in the procedure, but it is used in the analysis of the procedure.[h!] < g r a p h i c s >Output is the optimal branching point when branching on variable x_1 First, consider what happens when we pick a branching variable x_i, and branch at a given point c_i: we obtain two children, now with different bounds on the branching variable.The upper bound of the branching variable in the left child becomes the value of the branching point, as does the lower bound of the branching variable in the right child. That is, the domain of x_i for the left child is [a_i,c_i], and the domain of x_i for the right child is [c_i,b_i].We reconvexify the two children using our chosen method of convexification (i.e., the convex hull), and we can sum the volumes from both children to obtain the total volume when branching at that given point.We are interested in finding the branching point that leads to the least total volume.For an example of this principle in a lower dimension, see the diagram of Figure <ref> which illustrates reconvexifying after branching in sBB.Here, because we have a one dimensional function, the graph of the function is a set in ^2.Therefore, in the context of this diagram, we wish to find the branching point that minimizes the sum of the areas of the two diagonally striped (green) regions.Clearly this depends on the choice of convexification method.We can compute the volume of the relaxation for each of the children using Theorem <ref> (i.e., Theorem 4.1 from <cit.>). To ensure that we compute the appropriate volumes, we need to check that as the bounds on the branching variable change, we still respect the labeling <ref>.To illustrate this, consider the left child obtained by branching on variable x_1 at some point c_1 ∈ [a_1,b_1].For this left child, the lower bound on the branching variable remains the same and the new upper bound is c_1.We can see that if c_1 is `close enough' to b_1, then <ref> willremain satisfied, however as c_1 decreases, there comes a point wherethe labeling must change.By simple algebra, we calculate that this critical point is when a_1/c_1 = a_2/b_2⇔ c_1 = a_1b_2/a_2 (assuming for now that a_2>0).We can consider the right child in the same manner.On the right, the upper bound on the branching variable remains the same, and the new lower bound is c_1.When c_1 is close to a_1, <ref> willremain satisfied; however, as c_1 becomes larger, eventually the labeling must change.This critical point for the right child is at c_1/b_1 = a_2/b_2⇔ c_1 = b_1a_2/b_2.We note that because of the structure of the volume function of the convex hull, (see Theorem <ref>), the second and third variables are interchangeable.This means that we do not need to consider what happens when the bounds vary enough for x_1 to be relabeled as x_3.Before stating Theorem <ref>, we need to clarify some definitions and state four technical lemmas that will be needed in the proof.The proofs of these four lemmas can be found in the Appendix.We first defineV(l_1,u_1,l_2,u_2,l_3,u_3)(u_1-l_1)(u_2-l_2)(u_3-l_3)×(u_1(5u_2u_3 - l_2u_3 - u_2l_3 - 3l_2l_3) + l_1(5l_2l_3 - u_2l_3 - l_2u_3 -3u_2u_3))/24,to be the volume of the convex hull with variable lower bounds l_i and upper bounds, u_i, for i=1,2,3.Then, we define the following parameterized function: TV(c_1)V_1(c_1),a_1 ≤ c_1 ≤b_1a_2/b_2; V_2(c_1), b_1a_2/b_2 < c_1 < a_1b_2/a_2; V_3(c_1), a_1b_2/a_2≤ c_1 ≤ b_1,where:V_1(c_1) V(a_2,b_2,a_1,c_1,a_3,b_3) + V(c_1,b_1,a_2,b_2,a_3,b_3), V_2(c_1) V(a_2,b_2,a_1,c_1,a_3,b_3) + V(a_2,b_2,c_1,b_1,a_3,b_3),V_3(c_1) V(a_1,c_1,a_2,b_2,a_3,b_3) + V(a_2,b_2,c_1,b_1,a_3,b_3). And finally the second parameterized function: TV(c_1)V_1(c_1)a_1 ≤ c_1 ≤b_1a_2/b_2; V_4(c_1) b_1a_2/b_2 < c_1 < a_1b_2/a_2; V_3(c_1) a_1b_2/a_2≤ c_1 ≤ b_1, where V_1(c_1) and V_3(c_1) are defined as before and:V_4(c_1)V(a_1,c_1,a_2,b_2,a_3,b_3) + V(c_1,b_1,a_2,b_2,a_3,b_3). Both TV(c_1) and TV(c_1) are piecewise-quadratic functions in c_1. We can easily observe this by noticing that V is the product of a pair of multilinear functions in the parameters. Given that the upper- and lower-bound parameters respect the labeling <ref>, and b_1a_2/b_2≤a_1b_2/a_2,V_1(b_1a_2/b_2)=V_2(b_1a_2/b_2) ≥ V_2(a_1b_2/a_2) = V_3(a_1b_2/a_2).Given that the upper- and lower-bound parameters respect the labeling <ref>, and b_1a_2/b_2 > a_1b_2/a_2,V_1(a_1b_2/a_2)=V_4(a_1b_2/a_2) ≥ V_4(b_1a_2/b_2) = V_3(b_1a_2/b_2).Given that the parameters satisfy the conditions <ref>, and furthermore, b_1a_2/b_2≤a_1b_2/a_2, we haveq_1 ≥b_1a_2/b_2. Given that the parameters satisfy the conditions <ref>, and furthermore, b_1a_2/b_2≥a_1b_2/a_2, we haveq_1 ≥a_1b_2/a_2.We are now ready to state the theorem.Assume initial bounds a_1,b_1,a_2,b_2,a_3,b_3 that satisfy <ref> and that we branch on x_1.Furthermore, assume that q_2 and q_3 are defined as in Equation <ref> and Equation <ref>.Procedure <ref> gives the optimal branching point with respect to minimizing the sum of the volumes of the two convex-hull relaxations of the children.Given our earlier discussion, it is natural to think about three cases.First, when a_2=0 (we refer to this as Case 0). Second (Case 1), whena_2 ≠ 0 andb_1a_2/b_2≤a_1b_2/a_2⟺a_2^2/b_2^2≤a_1/b_1⟺ b_1a_2^2 ≤ a_1b_2^2 ,and third (Case 2), whena_2 ≠ 0 andb_1a_2/b_2 > a_1b_2/a_2⟺a_2^2/b_2^2 > a_1/b_1⟺ b_1a_2^2 > a_1b_2^2 .The case of equality, i.e., b_1a_2/b_2 = a_1b_2/a_2, is arbitrarily included with Case 1.In fact, when equality holds, the analysis that follows is simplified, and it could be contained in either of the cases.Depending on the case, the necessary relabeling to ensure Ω remains satisfied is different, and the functions we defined as V_i(c_1), i=1…4 reflect these different relabelings.For an illustration of when the variable labeling must change on the left child, the right child, or on both children to ensure that <ref> remains satisfied (as the branching point varies), see Figure <ref>.Case 0: a_2=0. From the condition <ref>, we know that a_2=0 ⇒ a_1=0.In this special case, the labeling for the left child does not change no matter how small the upper bound becomes.Conversely, the labeling for the right child changes as soon as the lower bound becomes positive.We therefore have the picture shown in Figure <ref>, and the function (i.e. V_3(c_1)) describing the sum of the volumes of the two child relaxations over the entire domain, [a_1,b_1], does not need to be defined in a piecewise manner.As we will see shortly, this function is a convex quadratic, and therefore it is easy to check (by calculating where the derivative is zero) that in this special case the minimizer of this function is q_3 (defined above) and this is the minimizer of the total volume of the two children.Furthermore, when a_2=0 (and therefore a_1=0), this minimizer simplifies to b_1/2=a_1+b_1/2=q_2, the midpoint of the interval. Case 1: b_1a_2/b_2≤a_1b_2/a_2. As illustrated in Figure <ref>, in Case 1, the function describing the sum of the volumes of the child relaxations is TV(c_1) (Equation <ref>).It is straightforward to check that the function TV(c_1) is continuous over its domain.Furthermore, by observing that the leading coefficient of each piece (V_i(c_1), i=1,2,3) is positive for all parameter values satisfying <ref>, we conclude that each piece is strictly convex.We are able to claim strict convexity because we assume b_i > a_i for all i. Using this fact, for each coefficient below we observe that each multiplicand in the numerator is strictly positive and therefore each leading coefficient is strictly positive. The coefficient of c_1^2 in the quadratic function V_1(c_1) is: (b_3-a_3)(b_2-a_2)(6(b_2b_3-a_2a_3) + 2b_3(b_2-a_2))/24 > 0. The coefficient of c_1^2 in the quadratic function V_2(c_1) is: (b_3-a_3)(b_2-a_2)(4(b_2b_3-a_2a_3)+2(b_3+a_3)(b_2-a_2))/24 > 0. The coefficient of c_1^2 in the quadratic function V_3(c_1) is: (b_3-a_3)(b_2-a_2)(6(b_2b_3-a_2a_3)+2a_3(b_2-a_2))/24 > 0. Figure <ref> gives some idea of what this function could look like. The example depicts a globally convex function and we are yet to prove that this will always be the case.However, in later analysis (Theorem <ref> in <ref>) we will demonstrate that global convexity always holds.Now that we know that TV(c_1) has this structure, to find the global minimizer over the domain [a_1,b_1], we can simply find the local minimizer on each of the three pieces and pick the point with the least function value.Because we have convex functions, the local minimum of a given piece will either occur at the global minimizer of V_i(c_1) (if this occurs over the appropriate subdomain), or at one of the end points of the subdomain.Therefore, to find the local minimizer for a given segment, we first find the global minimizer of V_i(c_1) over the entire real line and check if it occurs in the interval; if so, it is the local minimizer, if not, we examine the interval end points to locate the local minimizer.We can then compare the function value of the local minimizer of each of the three pieces to find the global minimizer of TV(c_1), i.e., the branching point that obtains the least total volume.Given that each V_i is a parameterised convex-quadratic function in c_1, it is easy to use a computer algebra system to calculate the following:The minimum of V_1(c_1) occurs at:c_1=3a_1a_2a_3+a_1a_2b_3-a_1b_2a_3-3a_1b_2b_3+4b_1a_2a_3-4b_1b_2b_3/2(3a_2a_3+a_2b_3-4b_2b_3)=q_1.The minimum of V_2(c_1) occurs at:c_1=a_1+b_1/2=q_2.The minimum of V_3(c_1) occurs at:c_1=4a_1a_2a_3-4a_1b_2b_3+3b_1a_2a_3+b_1a_2b_3-b_1b_2a_3-3b_1b_2b_3/2(4a_2a_3-b_2a_3-3b_2b_3)=q_3. Therefore, the candidate points for the minimizer are a_1, b_1a_2/b_2, a_1b_2/a_2, b_1, q_1, q_2 and q_3.We can immediately discard a_1 and b_1 because these are both equivalent to not branching. By branching and reconvexifying over the two children, we can never do worse with regard to volume.Therefore, we have five points to consider.For a given set of parameters, it is straightforward to evaluate and check which of these five points is the minimizer.However, making use of the following observations, we can further reduce the possibilities.If q_1 were to be the global minimizer, then it must fall in the appropriate subdomain; i.e., it must be that q_1 ≤b_1a_2/b_2.However, by Lemma <ref>, in Case 1 we always have q_1 ≥b_1a_2/b_2. Therefore, we can discard q_1 as a candidate point for the minimizer because for it to be the minimizer, this quantity would have to be exactly equal to b_1a_2/b_2, which is already on the list of candidate points.Now, consider the quantities:q_1 - q_2 = (b_3-a_3)(b_1a_2-a_1b_2)/2(4b_2b_3-a_2b_3-3a_2a_3)≥ 0,andq_3 - q_2 = (a_3-b_3)(b_1a_2-a_1b_2)/2(3b_2b_3+b_2a_3-4a_2a_3)≤ 0. The inequalities follow from b_i > a_i, i=1,2,3, and Lemma <ref> in the Appendix.We therefore have:q_1 ≥ q_2 = a_1+b_1/2≥ q_3. From this, we can observe that if q_3 ≥a_1b_2/a_2, then q_2 ≥ q_3 ≥a_1b_2/a_2, and therefore q_3 is the minimizer.This is because neither q_1 nor q_2 fall in their key intervals (i.e. in the appropriate subdomain); furthermore, by the definition of q_3 as the minimizer of V_3, we must have that V_3(q_3) ≤ V_3(a_1b_2/a_2), and by Lemma <ref>, we have that V_3(a_1b_2/a_2) ≤ V_2(b_1a_2/b_2).If this does not occur, i.e. q_3 < a_1b_2/a_2, then if b_1a_2/b_2≤a_1+b_1/2≤a_1b_2/a_2, the midpoint q_2 is the minimizer.This is because under these conditions, q_2 is the only minimizer that occurs in the `correct' function piece, and by definition of q_2 as the minimizer of V_2, the function value is not more than at either of the end points.Otherwise, if none of the above occurs (i.e., none of the intervals contain their function global minimizer), we have that a_1b_2/a_2 is the minimizer by Lemma <ref>. Case 2: b_1a_2/b_2 > a_1b_2/a_2. In this second case, for a given problem with initial upper and lower bounds (a_1, b_1, a_2, b_2, a_3, b_3), the sum of the volumes of the two child relaxations after branching at point c_1, is given by the function TV(c_1) (Equation <ref> and illustrated in Figure <ref>).This is similar, but distinct, from the function in Case 1.Recall that this is a piecewise-quadratic function in c_1, and, as before, it is simple to check that the function is continuous over its domain.Furthermore, by observing that the leading coefficient of each piece is positive for all parameter values satisfying <ref>, we know that each piece is strictly convex.Strict convexity comes from the knowledge b_i>a_i, i=1,2,3. The coefficient of c_1^2 in the quadratic function V_4(c_1) is:8(b_3-a_3)(b_2-a_2)(b_2b_3-a_2a_3)/24 > 0. Therefore, we can take the same approach as before to find the global minimizer: first find the local minimizer for each segment.We do this by finding the global minimizer for the appropriate function (V_i(c_1)), over the whole real line and checking if it occurs in the segment.If it does, we have found the minimizer for that segment, if not, we examine the interval end points.We then compare the minimum in each of the three segments to find the branching point that obtains the least total volume.From our analysis of Case 1, we know that the minimums of V_1(c_1) and V_3(c_1) occur at q_1 and q_3 respectively.We compute that the minimum of V_4(c_1) occurs at the midpoint of the whole interval, i.e., atc_1=a_1+b_1/2 = q_2. As before, the candidate points for the minimizer are b_1a_2/b_2, a_1b_2/a_2, q_1, q_2 and q_3. However, by making the following observations we can further reduce the points we need to examine.If q_1 were to be the global minimizer, then it must fall in the appropriate subdomain, i.e., it must be that q_1 ≤a_1b_2/a_2.However, by Lemma <ref>, in Case 2 we always have q_1 ≥a_1b_2/a_2. Therefore, we can discard q_1 as a candidate point for the minimizer because for it to be the minimizer it would have to be exactly equal to a_1b_2/a_2, which is already on the list of candidate points.If q_3 ≥b_1a_2/b_2, then q_2 ≥ q_3 ≥b_1a_2/b_2, and therefore q_3 is the minimizer.This is because neither q_1 nor q_2 fall in their key intervals; furthermore, by definition of q_3 as the minimizer of V_3, we must have that V_3(q_3) ≤ V_3(b_1a_2/b_2), and by Lemma <ref> we know that V_3(b_1a_2/b_2) ≤ V_1(a_1b_2/a_2).If this does not occur, i.e. q_3 < b_1a_2/b_2, then if a_1b_2/a_2≤a_1+b_1/2≤b_1a_2/b_2, the midpoint q_2 is the minimizer.This is because under these conditions, q_2 is the only minimizer that occurs in the `correct' function piece, and by definition of q_2 as the minimizer of V_4, the function value is no more than at either of the end points.Otherwise, we have that b_1a_2/b_2 is the minimizer by Lemma <ref>.Therefore Procedure <ref> is correct. §.§ Side note For completeness, and as an interesting side point, wenote that in Case 1, if it were possible to have q_1 ≤b_1a_2/b_2, then q_3 ≤ q_2 ≤ q_1 ≤b_1a_2/b_2, and therefore q_1 would be the minimizer.This is because neither q_2 nor q_3 would fall in their key intervals; furthermore, by the definition of q_1 as the minimizer of V_1, we have that V_1(q_1) ≤ V_1(b_1a_2/b_2), and by Proposition <ref>(see the Appendix), we know that V_1(q_1) ≤ V_2(a_1b_2/a_2).However, by Lemma <ref> we have already discarded this case.As another interesting side point, we also note that in Case 2, if it were possible to have q_1 ≤a_1b_2/a_2, then q_3 ≤ q_2 ≤ q_1 ≤a_1b_2/a_2, and q_1 would be the minimizer.This is because neither q_2 nor q_3 would fall in their key intervals.Furthermore, by definition of q_1 as the minimizer of V_1, we must have that V_1(q_1) ≤ V_1(a_1b_2/a_2), and by Proposition <ref> (see the Appendix), we know that V_1(q_1) ≤ V_4(b_1a_2/b_2).However, by Lemma <ref> we have already discarded this case.§.§ Some examples We can illustrate these piecewise-quadratic functions for the possible outcomes of Procedure 1. In this illustration, we focus on Case 1, and therefore Figure <ref> shows the function TV(c_1) over the domain [a_1,b_1].The (orange) dashed curve illustrates an example where the minimizer of V_3(c_1), (i.e. q_3), falls in the relevant interval, and therefore is the minimizer over our whole domain.The (purple) solid curve illustrates an example where q_3 does not fall in this interval, however the midpoint, q_2, falls in between the quantities b_1a_2/b_2 and a_1b_2/a_2 and is therefore the required minimizer.The (green) dotted curve illustrates an example where neither of the above happens, and therefore the breakpoint between the function V_2(c_1) and the function V_3(c_1) is the minimizer.In this example we are in Case 1, and therefore this point is a_1b_2/a_2. It is important to note that each of the cases in Procedure 1 actually can occur.It is easy to check the following:* An example of a dashed curve (minimum occurs at q_3) is (a_1=1, b_1=35, a_2=2,b_2=12, a_3=12, b_3=35). * An example of a solid curve (minimum occurs at q_2) is (a_1=1, b_1=34, a_2=2, b_2=36, a_3=12, b_3=35). * An example of a dotted curve (minimum occurs at a_1b_2/a_2) is (a_1=1, b_1=8, a_2=5, b_2=22, a_3=1, b_3=4). Unfortunately, the plots of the actual functions do not display the key details as clearly as our illustration, so we do not include them here.Furthermore, an example of Case 2, where the minimum occurs at the breakpoint between the function V_4 and the function V_3, i.e. the point b_1a_2/b_2 is (a_1=1, b_1=13, a_2=1, b_2=2, a_3=2, b_3=4).Finally, a simple example of Case 0, is the special case (a_1=0, b_1=1, a_2=0, b_2=1, a_3=0, b_3=1).In Figure <ref> we can see the plot of this function and the minimum, which falls at the midpoint.In Case 0 we always have q_1=q_2=q_3=a_1+b_1/2=b_1/2.§.§ Global convexity of our piecewise-quadratic function over its domain We have seen that each piece of TV(c_1) and TV(c_1) is a convex quadratic function.However, this does not imply that the functions are convex over the whole domain, [a_1,b_1].Nevertheless, as we show in the following theorem, with a bit more work, we are able to demonstrate that TV(c_1) and TV(c_1) are convex over the domain, [a_1,b_1].It is very useful that these functions are globally convex; if a variable appears in many trilinear terms, it is quite reasonable to combine volumes in a reasonable manner (see <cit.>). For example, we can take a weighted average (of the sum of the two volumes for each term) as a measure for deciding on a branching point.A weighted average (assuming positive weights) of convex functions is convex, and therefore, the global-convexity property of these functions allows us to find the optimal branching point (defined as the minimum of the weighted-average function) by a simple bisection search.However, it is important to note that we are not advocating a bisection search if there is only one term being considered.In this case, Procedure (<ref>) is more efficient. Given that the upper- and lower-bound parameters respect the labeling <ref>, the functions TV(c_1) and TV(c_1) are globally-convex functions in the branching point c_1 over the domain [a_1,b_1].To demonstrate the global convexity of TV(c_1), we will establish that it is the pointwise maximum of the convex functions V_1(c_1), V_2(c_1) and V_3(c_1).Similarly, to demonstrate the global convexity of TV(c_1), we will establish that it is the pointwise maximum of the convex functions V_1(c_1), V_4(c_1) and V_3(c_1). Global convexity of TV(c_1):Consider the difference of V_1(c_1) and V_2(c_1): V_1(c_1) - V_2(c_1) = (b_3-a_3)^2(b_1-c_1)(b_2-a_2)(b_1a_2-c_1b_2)/12. Note that for all parameter values such that <ref> is satisfied and a_1 < c_1 < b_1, we have that V_1(c_1) > V_2(c_1) if and only if c_1 < b_1a_2/b_2 and conversely V_1(c_1) > V_2(c_1) if and only if c_1 > b_1a_2/b_2.They are equal when c_1 = b_1a_2/b_2.Now consider the difference of V_3(c_1) and V_2(c_1): V_3(c_1) - V_2(c_1) = (b_3-a_3)^2(c_1-a_1)(b_2-a_2)(c_1a_2-a_1b_2)/12. Again, note that for all parameter values such that <ref> is satisfied and a_1 < c_1 < b_1, we have that V_3(c_1) > V_2(c_1) if and only if c_1 > a_1b_2/a_2 and conversely V_3(c_1) < V_2(c_1) if and only if c_1 < a_1b_2/a_2.They are equal when c_1 = a_1b_2/a_2.Also recall that in the definition of TV(c_1), we implicitly assume b_1a_2/b_2≥a_1b_2/a_2.We can make the following observations.On the interval c_1 ∈(a_1,b_1a_2/b_2) we have V_1(c_1)>V_2(c_1)>V_3(c_1), at c_1=b_1a_2/b_2 we have V_1(c_1)=V_2(c_1)>V_3(c_1), on the interval c_1 ∈(b_1a_2/b_2, a_1b_2/a_2) we have V_2(c_1)>V_1(c_1) and V_2(c_1)>V_3(c_1). At c_1=a_1b_2/a_2 we have V_3(c_1)=V_2(c_1)>V_1(c_1) and on the interval c_1 ∈(a_1b_2/a_2, b_1) we have V_3(c_1)>V_2(c_1)>V_1(c_1).Furthermore, when c_1=a_1, we have V_1(c_1)>V_2(c_1)=V_3(c_1) and when c_1=b_1 we have V_3(c_1)>V_2(c_1)=V_1(c_1).From these observations, it is clear that TV(c_1), is the pointwise maximum of the convex functions V_1(c_1), V_2(c_1) and V_3(c_1) over the domain [a_1,b_1] therefore we observe that TV(c_1) is globally convex over the domain [a_1,b_1]. Global convexity of TV(c_1):Consider the difference of V_1(c_1) and V_4(c_1): V_1(c_1) - V_4(c_1) = (b_3-a_3)^2(c_1-a_1)(b_2-a_2)(a_1b_2-c_1a_2)/12. Note that for all parameter values such that <ref> is satisfied and a_1 < c_1 < b_1, we have that V_1(c_1) > V_4(c_1) if and only if c_1 < a_1b_2/a_2 and conversely V_1(c_1) > V_2(c_1) if and only if c_1 > a_1b_2/a_2.They are equal when c_1 = a_1b_2/a_2.Now consider the difference of V_3(c_1) and V_4(c_1): V_3(c_1) - V_4(c_1) = (b_3-a_3)^2(b_1-c_1)(b_2-a_2)(c_1b_2-b_1a_2)/12. Again, note that for all parameter values such that <ref> is satisfied and a_1 < c_1 < b_1, we have that V_3(c_1) > V_4(c_1) if and only if c_1 > b_1a_2/b_2 and conversely V_3(c_1) < V_2(c_1) if and only if c_1 < b_1a_2/b_2.They are equal when c_1 = b_1a_2/b_2.Now recall that TV(c_1) is defined with the assumption b_1a_2/b_2 < a_1b_2/a_2.We can make almost identical observations to those above to see that TV(c_1)is the pointwise maximum of the convex functions V_1(c_1), V_4(c_1) and V_3(c_1) and therefore is also globally convex over the domain [a_1,b_1].§.§ Bounds on where the optimal branching point can occur We have seen in <ref> that software employ methods to avoid selecting a branching point that falls too close to either endpoint of the interval.Therefore, a natural issue to consider is whether our minimizer can fall close to either of the endpoints.We want to know how likely it is that solvers are routinely precluding our optimal branching point.The following theorems give some insight on this issue and show that, in fact, software is unlikely to be cutting off our optimal branching point. The branching point for variable x_1 that obtains the least total volume, never occurs at a point in the interval greater than the midpoint. If a_2=0, then we are in Case 0, and the minimizer is at the midpoint, which is clearly no greater than the midpoint. If a_1b_2/a_2≥b_1a_2/b_2, then we are in Case 1.If q_3 ≥a_1b_2/a_2, then q_3 is the minimizer, but we know that q_3 ≤a_1+b_1/2 (see Equation <ref>).If q_2 = a_1+b_1/2 falls in the interval [b_1a_2/b_2,a_1b_2/a_2], then the midpoint is the minimizer.If it does not, then (i) a_1b_2/a_2 is the minimizer, and (ii) it must be that either that a_1+b_1/2 > a_1b_2/a_2, in which case our claim is valid, ora_1+b_1/2 < b_1a_2/b_2≤a_1b_2/a_2.We will show by contradiction that this cannot be the case.Toward this end, assume that:a_1+b_1/2 < b_1a_2/b_2anda_1+b_1/2 < a_1b_2/a_2.This implies:2b_1a_2 -b_1b_2 - a_1b_2 = b_1(a_2-b_2) + (b_1a_2-a_1b_2) > 0,and2a_1b_2-a_1a_2-b_1a_2 = a_1(b_2-a_2) +(a_1b_2-b_1a_2) > 0.Now let X b_2-a_2 and Y b_1a_2-a_1b_2 (note that both X and Y are non-negative: Lemma <ref>).Therefore we can write our assumption as:b_1(-X) + Y > 0 and a_1(X) + (-Y) > 0,which impliesY > b_1X and Y < a_1X,a contradiction.Therefore, in Case 1 the minimizer must be no larger than the midpoint.We make a similar argument for Case 2.Herea_1b_2/a_2<b_1a_2/b_2.If q_3 ≥b_1a_2/b_2, then q_3 is the minimizer, but we know that q_3 ≤a_1+b_1/2 (see Equation <ref>).If q_2 = a_1+b_1/2 falls in the interval [a_1b_2/a_2,b_1a_2/b_2], then the midpoint is the minimizer.If it does not, then (i) b_1a_2/b_2 is the minimizer, and (ii) it must be that either that a_1+b_1/2 > b_1a_2/b_2, in which case our claim is valid, ora_1+b_1/2 < a_1b_2/a_2 < b_1a_2/b_2.However, we have just shown by contradiction that this cannot be the case.Therefore, in Case 2 the minimizer must be no larger than the midpoint.This theorem gives an upper bound on the fraction through the interval the minimizer can fall (namely 1/2).Furthermore, this bound is sharp (i.e. it is obtained and therefore cannot be strengthened) because we know examples when the minimizer is exactly at the midpoint.It would be nice to also obtain a sharp lower bound on this fraction.By demonstrating that the minimizer cannot fall too close to the end points of the interval, we are providing mathematical evidence to justify the current choices of branching point in software, as discussed in <ref>.The following theorem gives a lower bound on this fraction when a_2 ≠ 0, (when a_2 =0, we know that the minimizer will be exactly at the midpoint).We note that because of the condition <ref>, the problem is no longer symmetric and therefore knowledge about the upper bound does not allow us to draw conclusions about the lower bound.Given upper- and lower-bound parameters (a_1, b_1, a_2, b_2, a_3, b_3) satisfying <ref>, and a_2 ≠ 0.The branching point for variable x_1 that obtains the least total volume, never occurs at a point in the interval less thanmin{max{a_1(b_2-a_2)/a_2(b_1-a_1),b_1a_2-a_1b_2/b_1b_2-a_1b_2}, 1/2}of the way through the interval. There are four candidate points where the minimizer can occur.Namely, q_2=a_1+b_1/2, q_3, a_1b_2/a_2, and b_1a_2/b_2.Therefore min{a_1+b_1/2, q_3, a_1b_2/a_2, b_1a_2/b_2},is a trivial lower bound on this minimizer.We know that if q_3 is the minimizer, then we must have q_3 ≥a_1b_2/a_2 (Case 1), or q_3 ≥b_1a_2/b_2 (Case 2), so we can discard this point.Additionally, we know that if a_1b_2/a_2 is the minimizer, then we have a_1b_2/a_2≥b_1a_2/b_2 (Case 1), and if b_1a_2/b_2 is the minimizer, then we have b_1a_2/b_2 > a_1b_2/a_2 (Case 2).Therefore we have that a lower bound on the minimizer is: min{max{a_1b_2/a_2,b_1a_2/b_2}, a_1+b_1/2}.Moreover, a lower bound for the fraction of the interval where this point can fall is: min{max{a_1b_2/a_2 - a_1/b_1-a_1,b_1a_2/b_2-a_1/b_1-a_1}, a_1+b_1/2-a_1/b_1-a_1}=min{max{a_1(b_2-a_2)/a_2(b_1-a_1),b_1a_2-a_1b_2/b_1b_2-a_1b_2}, 1/2}.We note that this lower bound is unlikely to be sharp.Consider the case where a_1=0, a_2= ϵ>0 and b_2=1.This bound becomes ϵ, and is therefore not particularly informative, given that we can make ϵ as close to zero as we wish.However, we have computationally checked many examples, and we have yet to find an example where the minimizer occurs less than ∼ 0.45 of the way through the interval.It would be nice to sharpen this bound, and our computations indicate that this should be possible.§ BRANCHING ON X_2 AND X_3We noted in <ref> that because of the structure of the volume function of the convex hull, the second and third variables are interchangeable.Therefore, the branching-point analyses for these variables will be equivalent.To see how the results in this case are less complex than in the x_1 case, recall the condition <ref>, which due to our non-negativity assumption can be written as a_1/b_1≤a_2/b_2≤a_3/b_3. Now consider what happens to the quantity a_2/b_2 when we branch on x_2.In the left interval, a_2 remains constant, and b_2 becomes the branching point, c_2 < b_2.Therefore, a_2/b_2 cannot decrease further.In the right interval b_2 remains constant and a_2 becomes the branching point, c_2 > a_2.Therefore, again, a_2/b_2 cannot decrease further.Because of this, the labeling for x_1 and x_2 will not have to be switched to ensure <ref> remains satisfied.Furthermore, x_2 and x_3 are interchangeable in the formula, so we do not need to consider what happens when the ratios change such that a_2/b_2>a_3/b_3.The case of x_2 and x_3 therefore both require the analysis of only one convex quadratic function.This is formalized in the following theorem.Let c_i ∈ [a_i,b_i] be the branching point for x_i, i=2,3.With the convex-hull relaxation, the least total volume after branching is obtained when c_i=(a_i+b_i)/2, i.e., branching at the midpoint is optimal. We first consider branching on x_2.Consider the sum of the two resulting volumes, given by the following function:TV_2(c_2)=V(a_1,b_1,c_2,b_2,a_3,b_3)+V(a_1,b_1,a_2,c_2,a_3,b_3),which is quadratic in c_2.The leading coefficient (i.e. second derivative) isTV_2(c_2)=1/12(b_1-a_1)(b_3-a_3)(3(b_1b_3-a_1a_3)+(b_1a_3-a_1b_3)),which is greater than or equal to zero for all parameters satisfying <ref> and hence all c_2 ∈ [a_2,b_2] (Lemma <ref>).Therefore this function is convex. Setting the first derivative equal to zero and solving for c_2, we obtain that the minimum occurs at c_2=(a_2+b_2)/2.Similar analysis can be completed for i=3 to obtain the result. § THE OPTIMAL BRANCHING VARIABLENow that we have established the optimal branching point for each variable in all cases, it is interesting to compare the total volumes obtained when branching at the optimal point for each variable.In this section we establish the optimal branching variable.Given that the upper- and lower-bound parameters respect the labeling <ref>, if we assume optimal branching-point selection, then branching on x_1 obtains the least total volume, and branching on x_3 obtains the greatest total volume. Additionally, even if we branch at the midpoint for x_1 (which may not be optimal), this is at least as good as doing optimal branching-point selection (i.e., midpoint branching) on either x_2 or x_3. First, we establish that branching optimally (at the midpoint) on variable x_2 obtains a lower total volume than branching optimally (at the midpoint) on variable x_3.The optimal total volume when branching on variable x_3 is:(b_3-a_3)(b_2-a_2)(b_1-a_1)/48× (7a_1a_2a_3+a_1a_2b_3-3a_1a_3b_2-5a_1b_2b_3-5a_2a_3b_1-3a_2b_1b_3+a_3b_1b_2+7b_1b_2b_3). The optimal total volume when branching on variable x_2 is:(b_3-a_3)(b_2-a_2)(b_1-a_1)/48×(7a_1a_2a_3-3a_1a_2b_3+a_1a_3b_2-5a_1b_2b_3-5a_2a_3b_1+a_2b_1b_3-3a_3b_1b_2+7b_1b_2b_3). Therefore, the difference in total volume from branching on x_3 compared with x_2 is:(b_3-a_3)(b_2-a_2)(b_1-a_1)^2(b_2a_3-a_2b_3)/12,which is greater than or equal to zero by Lemma <ref>.Therefore, if we assume optimal branching, branching on x_3 always results in a greater volume than branching on x_2.Now let us consider the optimal total volume when branching on x_1, this quantity must always be less than or equal to the total volume when branching at the midpoint of the interval (it will be equal exactly when the midpoint is the optimal branching point).Therefore, if we can establish that branching on variable x_1 at the midpoint always obtains a lesser total volume than branching on variable x_2 at the midpoint, we will have shown our claim.Recall Figure <ref>.We know from the proof of Theorem <ref>, that the midpoint can never be less than: min{a_1b_2/a_2, b_1a_2/b_2}.Therefore, in every case, the midpoint must fall in a subdomain where: (i) the labeling for left interval stays the same, and the labeling for the right changes; (ii) the labeling changes for both intervals; or, (iii) the labeling remains the same for both intervals.This means that we are interested in the function value (total volume) at the midpoint for the functions V_2(c_1), V_3(c_1) and V_4(c_1).The total volume of branching (on variable x_1) at the midpoint if it occurs in the subdomain corresponding to V_2 is:(b_3-a_3)(b_2-a_2)(b_1-a_1)/48×(7a_1a_2a_3-3a_1a_2b_3-5a_1a_3b_2+a_1b_2b_3+a_2a_3b_1-5a_2b_1b_3-3a_3b_1b_2+7b_1b_2b_3).Therefore, the difference in total volume from branching on x_2 compared with this quantity is:(b_3-a_3)^2(b_2-a_2)(b_1-a_1)(b_1a_2-a_1b_2)/8,which is greater than or equal to zero by Lemma <ref>.The total volume of branching (on variable x_1) at the midpoint if it occurs in the subdomain corresponding to V_3 is:(b_3-a_3)(b_2-a_2)(b_1-a_1)/48×(6a_1a_2a_3-2a_1a_2b_3-3a_1a_3b_2-a_1b_2b_3-4a_2b_1b_3-3a_3b_1b_2+7b_1b_2b_3).Therefore, the difference in total volume from branching on x_2 compared with this quantity is:(b_3-a_3)^2(b_2-a_2)(b_1-a_1)(4(b_1a_2-a_1b_2)+a_2(b_1-a_1))/48,which is greater than or equal to zero by Lemma <ref>. The total volume of branching (on variable x_1) at the midpoint if it occurs in the subdomain corresponding to V_4 is:(b_3-a_3)(b_2-a_2)(b_1-a_1)/24×(3a_1a_2a_3-a_1a_2b_3-a_1a_3b_2-a_1b_2b_3-a_2a_3b_1-a_2b_1b_3-a_3b_1b_2+3b_1b_2b_3).Therefore, the difference in total volume from branching on x_2 compared with this quantity is:(b_3-a_3)^2(b_2-a_2)(b_1-a_1)(b_1b_2-a_1a_2+3(b_1a_2-a_1b_2))/48,which is greater than or equal to zero by Lemma <ref>.Therefore, for each one of these possible scenarios, optimally branching on x_2 results in a greater volume than branching on x_1 at the midpoint. And so we can conclude that given optimal branching, branching on x_1 obtains the least total volume, and branching on x_3 obtains the greatest total volume. § CONCLUDING REMARKS AND FUTURE WORKWe have presented some analytic results on branching variable and branching-point selection in the context of sBB applied to models having functions involving the multiplication of three or more terms.In particular, for trilinear monomials f=x_1x_2x_3 on a box domain satisfying <ref>,we have shown that when the convex-hull relaxation is used, and the branching variable is x_2 or x_3, branching at the commonly-used midpoint results in the least total volume.We have presented a simple procedure for obtaining the optimal branching point when using the convex-hull relaxation and branching on variable x_1.We have provided a sharp upper bound on where in the interval the minimizer can occur, and we have also obtained a lower bound for this fraction.By computationally checking many examples, we have evidence to suggest that this lower bound can be sharpened, thus providing analysis that backs up software's current choice of branching point.Furthermore, we have shown that the piecewise-quadratic functions we have been considering are globally convex over their entire domain.Given that we branch at an optimal branching point, we have also compared the choice of branching variable.We demonstrate that branching on x_1 gives the least total volume.We are in the process ofcarrying out a similar analysis to what we have done here, but for the best of the double-McCormick convexifications rather than for the convex-hull relaxation. However, due to the structure of the volume formula for the best double-McCormick convexification (see <cit.>), our task is significantly more complex.Finally, we hope that our mathematical results can be used as some guidance toward justifying, developing and refining practical branching rules. We believe that our work is just a first step in this direction. In this regard, we hope to further extend our mathematical analysis to directly deal with variables appearing in multiple non-linear terms.This work was supported in part by ONR grants N00014-14-1-0315 and N00014-17-1-2296. The authors gratefully acknowledge conversations with Ruth Misener and Nick Sahinidis concerning how branching points are selected in and . spmpsciAppendix: technical propositions and lemmas In this section, we provide the technical propositions and lemmas used for our analysis.Given that the upper- and lower-bound parameters respect the labeling <ref>, and b_1a_2/b_2≤a_1b_2/a_2,V_1(q_1) ≤ V_2(a_1b_2/a_2) = V_3(a_1b_2/a_2). It is easy to check that V_2(a_1b_2/a_2) = V_3(a_1b_2/a_2). V_2(a_1b_2/a_2) - V_1(q_1) = (b_3-a_3)(b_2-a_2)/48(4b_2b_3-a_2b_3-3a_2a_3)a_2^2× (pa_1^2 + qa_1 + r ), wherep =(-3a_2a_3-a_2b_3+b_2a_3+3b_2b_3)×(-3a_2^3a_3-a_2^3b_3+13a_2^2b_2a_3+7a_2^2b_2b_3-12a_2b_2^2a_3-20a_2b_2^2b_3+16b_2^3b_3)=(3(b_2b_3-a_2a_3)+b_2a_3-a_2b_3) ×((-3a_2^3+13a_2^2b_2-12a_2b_2^2)a_3 + (-a_2^3+7a_2^2b_2-20a_2b_2^2+16b_2^3)b_3),q =4a_2b_1(2a_2^2a_3-3a_2b_2a_3-3a_2b_2b_3+4b_2^2b_3) × (3a_2a_3+a_2b_3-b_2a_3-3b_2b_3),r =4a_2^2b_1^2(a_2a_3+a_2b_3-2b_2b_3)^2. To show that V_2(a_1b_2/a_2) - V_1(q_1) is non-negative for all parameters satisfying <ref>, we will show that pa_1^2 + qa_1 + r ≥ 0 for all parameters satisfying <ref>.We observe:((-a_2^3+7a_2^2b_2-20a_2b_2^2+16b_2^3)b_3 + (-3a_2^3+13a_2^2b_2-12a_2b_2^2)a_3)b_3Y + a_3Z,whereY + Z = 4(b_2-a_2)(2b_2-a_2)^2 ≥ 0,andY = (b_2-a_2)(4b_2(b_2-a_2)+12b_2^2+a_2^2) + 2a_2^2b_2 ≥ 0. Therefore, by Lemma <ref> we have that b_3Y + a_3Z is non-negative and so p is non-negative (Lemma <ref>).From this we know that pa_1^2 + qa_1 + r is a convex function in a_1 and we can find the minimizer by setting the derivative to zero and solving for a_1.The minimum occurs ata_1 = 2b_1a_2(2a_2^2a_3-3a_2b_2a_3-3a_2b_2b_3+4b_2^2b_3)/(-3a_2^3a_3-a_2^3b_3+13a_2^2b_2a_3+7a_2^2b_2b_3-12a_2b_2^2a_3-20a_2b_2^2b_3+16b_2^3b_3). Substituting this in to pa_1^2 + qa_1 + r, we obtain that the minimum value of this quadratic is:4a_2^2b_1^2(b_3-a_3)(b_2-a_2)^3(3a_2a_3+a_2b_3-4b_2b_3)^2/(-3a_2^3a_3-a_2^3b_3+13a_2^2b_2a_3+7a_2^2b_2b_3-12a_2b_2^2a_3-20a_2b_2^2b_3+16b_2^3b_3). In demonstrating the non-negativity of p, we have already shown that the denominator is non-negative, and it is easy to see that the numerator is non-negative for all values of the parameters satisfying <ref>.Therefore pa_1^2 + qa_1 + r ≥ 0, and consequently, V_2(a_1b_2/a_2) - V_1(q_1) ≥ 0 as required.Given that the upper- and lower-bound parameters respect the labeling <ref>, and b_1a_2/b_2≤a_1b_2/a_2,V_1(b_1a_2/b_2)=V_2(b_1a_2/b_2) ≥ V_2(a_1b_2/a_2) = V_3(a_1b_2/a_2) It is easy to check that V_1(b_1a_2/b_2)=V_2(b_1a_2/b_2) and V_2(a_1b_2/a_2) = V_3(a_1b_2/a_2).Furthermore,V_2(b_1a_2/b_2) - V_2(a_1b_2/a_2) = (b_3-a_3)(b_2-a_2)^2(b_1a_2-a_1b_2)(a_1b_2^2-a_2^2b_1)(3(b_2b_3-a_2a_3)+b_2a_3-a_2b_3)/12a_2^2b_2^2≥ 0,as required.Given that the upper- and lower-bound parameters respect the labeling <ref>, and b_1a_2/b_2 > a_1b_2/a_2,V_1(q_1) ≤ V_4(b_1a_2/b_2) = V_3(b_1a_2/b_2). It is easy to check that V_4(b_1a_2/b_2) = V_3(b_1a_2/b_2). V_4(b_1a_2/b_2) - V_1(q_1) = (b_3-a_3)(b_2-a_2)/48(4b_2b_3-a_2b_3-3a_2a_3)b_2^2× (pa_1^2 + qa_1 + r ), wherep =b_2^2(5b_2b_3-b_2a_3-a_2b_3-3a_2a_3)^2,q =8b_1b_2(6a_2^2a_3+2a_2^2b_3-3a_2b_2a_3-9a_2b_2b_3+b_2^2a_3+3b_2^2b_3)(b_2b_3-a_2a_3),r =16b_1^2(-3a_2^3a_3-a_2^3b_3+3a_2^2b_2a_3+5a_2^2b_2b_3-a_2b_2^2a_3-4a_2b_2^2b_3+b_2^3b_3)×(b_2b_3-a_2a_3). To show this is non-negative for all parameters satisfying <ref>, we will show pa_1^2 + qa_1 + r ≥ 0 for all parameters satisfying <ref>.Firstly, we observe thatp=b_2^2(5b_2b_3-b_2a_3-a_2b_3-3a_2a_3)^2 ≥ 0.From this we know that pa_1^2 + qa_1 + r is a convex function in a_1, and we can find the minimizer by setting the derivative to zero and solving for a_1.The minimum occurs ata_1 = 4b_1(6a_2^2a_3+2a_2^2b_3-3a_2b_2a_3-9a_2b_2b_3+b_2^2a_3+3b_2^2b_3)(a_2a_3-b_2b_3)/b_2(3a_2a_3+a_2b_3+b_2a_3-5b_2b_3)^2. Substituting this in to pa_1^2 + qa_1 + r, we obtain that the minimum value of this quadratic is:16b_1^2(b_3-a_3)(b_2-a_2)^3(b_2b_3-a_2a_3)(3a_2a_3+a_2b_3-4b_2b_3)^2/(3a_2a_3+a_2b_3+b_2a_3-5b_2b_3)^2, which is non-negative for all parameters satisfying <ref>.Therefore pa_1^2 + qa_1 + r ≥ 0, and consequently, V_4(b_1a_2/b_2) - V_1(q_1) ≥ 0, as required. Given that the upper- and lower-bound parameters respect the labeling <ref>, and b_1a_2/b_2 > a_1b_2/a_2,V_1(a_1b_2/a_2)=V_4(a_1b_2/a_2) ≥ V_4(b_1a_2/b_2) = V_3(b_1a_2/b_2). It is easy to check that V_1(a_1b_2/a_2)=V_4(a_1b_2/a_2) and V_4(b_1a_2/b_2) = V_3(b_1a_2/b_2).Furthermore,V_4(a_1b_2/a_2)-V_4(b_1a_2/b_2) = (b_3-a_3)(b_2-a_2)^2(b_1a_2^2-a_1b_2^2)(b_1a_2-a_1b_2)(b_2b_3-a_2a_3)/3a_2^2b_2^2≥ 0,as required.Given that the parameters satisfy the conditions <ref>, and furthermore, b_1a_2/b_2≤a_1b_2/a_2, we haveq_1 ≥b_1a_2/b_2. From the proof of Theorem <ref>, we know that the midpoint, q_2, cannot be less than both a_1b_2/b_1 and b_1a_2/b_2.Therefore we have:q_2 ≥min{a_1b_2/b_1,b_1a_2/b_2},and because we saw in <ref> that q_1 ≥ q_2 we also haveq_1 ≥min{a_1b_2/b_1,b_1a_2/b_2}. Therefore, under the conditions of the lemma, q_1 ≥b_1a_2/b_2 as required. Given that the parameters satisfy the conditions <ref>, and furthermore, b_1a_2/b_2≥a_1b_2/a_2, we haveq_1 ≥a_1b_2/a_2. We saw in the proof of Lemma <ref> thatq_1 ≥min{a_1b_2/b_1,b_1a_2/b_2}. Therefore, under the conditions of the lemma, q_1 ≥a_1b_2/a_2 as required.For completeness, we state and give proofs of two very simple lemmas (from <cit.>) which we used several times.For all choices ofparameters 0≤ a_i < b_i satisfying <ref>, we have: b_1a_2-a_1b_2≥ 0, b_1a_3-a_1b_3 ≥ 0 and b_2a_3-a_2b_3 ≥ 0. (b_3-a_3)(b_1a_2-a_1b_2) = b_1a_2b_3 + a_1b_2a_3 - a_1b_2b_3 -b_1a_2a_3 ≥ 0by <ref>. This implies b_1a_2-a_1b_2 ≥ 0, because b_3-a_3 > 0. b_1a_3-a_1b_3 ≥ 0 and b_2a_3-a_2b_3 ≥ 0 follow from <ref> in a similar way. Let A,B,C,D ∈ with A ≥ B≥ 0, C+D ≥ 0, C ≥ 0.Then AC+BD ≥0.AC+BD ≥ B(C+D) ≥ 0. | http://arxiv.org/abs/1706.08438v2 | {
"authors": [
"Emily Speakman",
"Jon Lee"
],
"categories": [
"math.OC",
"90C26"
],
"primary_category": "math.OC",
"published": "20170626153107",
"title": "On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation"
} |
Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, [email protected] Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology, Beijing 100081, China [email protected] Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, SingaporeDirac points in two-dimensional (2D) materials have been a fascinating subject of research, with graphene as the most prominent example. However, the Dirac points in existing 2D materials, including graphene, are vulnerable against spin-orbit coupling (SOC). Here, based on first-principles calculations and theoretical analysis, we propose a new family of stable 2D materials, the HfGeTe-family monolayers, which represent the first example to host so-called spin-orbit Dirac points (SDPs) close to the Fermi level. These Dirac points are special in that they are formed only under significant SOC, hence they are intrinsically robust against SOC. We show that the existence of a pair of SDPs are dictated by the nonsymmorphic space group symmetry of the system, which are very robust under various types of lattice strains. The energy, the dispersion, and the valley occupation around the Dirac points can be effectively tuned by strain. We construct a low-energy effective model to characterize the Dirac fermions around the SDPs.Furthermore, we find that the material is simultaneously a 2D ℤ_2 topological metal, which possesses nontrivial ℤ_2 invariant in the bulk and spin-helical edge states on the boundary. From the calculated exfoliation energies and mechanical properties, we show that these materials can be readily obtained in experiment from the existing bulk materials. Our result reveals HfGeTe-family monolayers as a promising platform for exploring spin-orbit Dirac fermions and novel topological phases in two-dimensions. Two-dimensional Spin-Orbit Dirac Point in Monolayer HfGeTe Shengyuan A. Yang December 30, 2023 ==========================================================§ INTRODUCTIONGraphene, as the most prominent example of two-dimensional (2D) materials <cit.>, has been attracting tremendous interest in the past decade. Many of the excellent properties of graphene can be attributed to its Dirac-cone-type band structure <cit.>. The conduction and valence bands in graphene touch with linear dispersion at discrete Dirac points on the Fermi level, around which the low-energy electrons behave like relativistic massless Dirac fermions in 2D, exhibiting properties distinct from the usual Schrödinger fermions. Inspired by graphene, much effort has been devoted to the search for other 2D materials which host Dirac/Weyl points, and a number of candidates have been proposed <cit.>, such as silicene <cit.>, germanene <cit.>, graphyne <cit.>, 2D carbon and boron allotropes <cit.>, group-VA phosphorene structures <cit.>, and 5d transition metal trichloride <cit.>. The Dirac points in all these materials (including graphene) are protected by symmetry, but only in the absence of spin-orbit coupling (SOC). When SOC is included, a gap will be opened at the Dirac point, so strictly speaking, graphene is formally a 2D topological insulator (also known as quantum spin Hall insulator) <cit.>, although the gap size is very small <cit.>.Is it possible to have 2D Dirac points that are robust against SOC? This question has been theoretically addressed by Young and Kane <cit.>. Via symmetry analysis and tight-binding model studies, they showed that certain nonsymmorphic space group symmetries, i.e., symmetries involving fractional lattice translations, can stabilize a new kind of 2D Dirac points that are robust against SOC. Besides the stability under SOC as their defining signature, such Dirac points, termed as 2D spin-orbit Dirac points (SDPs), also exhibit the following features that are different from the previously studied (SOC-vulnerable) Dirac points: (i) their presence is solely dictated by the specific space group symmetry and does not require band inversion; (ii) they must locate at time-reversal-invariant momenta at the Brillouin zone (BZ) boundary; and (iii) their appearance close to Fermi level typically requires partial filling of multiple bands.Despite the recent exciting advance in theory <cit.> and in finding analogous 3D SDPs in several bulk materials <cit.>, the search for realistic 2D materials that possess 2D SDPs at low energy is still challenging. The crucial issue is regarding the structural stability. Stable 2D materials with nonsymmorphic space group symmetries are typically insulators. For example, group-VA 2D materials with phosphorene structure <cit.> or 2D materials with α-SnO structure <cit.> do possess SDPs in their band structures, but these SDPs are away from the Fermi level (usually by energy >0.5 eV), so they will hardly manifest in electronic properties. On the other hand, if one tries to expose these SDPs to Fermi level by replacing the elements with other species of different valence, the resulting 2D materials are usually found to be structurally unstable. So far, a stable 2D material hosting SDPs close to its Fermi level has not been found yet, and how to realize such a material remains an open problem.In this work, we reveal a family of monolayer materials as the first example of 2D SDP materials, which realize the 2D SDPs proposed by Young and Kane. Using first-principles calculations and theoretical analysis, we show that monolayer HfGeTe-family materials host a pair of 2D SDPs close to the Fermi level. We demonstrate the stability of these materials in the monolayer form, and more importantly, for several of the member materials (including HfGeTe), the corresponding three-dimensional bulk materials already exist and the calculated exfoliation energy is relatively low, so that the monolayer can be more readily obtained, e.g., by mechanical exfoliation from the bulk. From symmetry analysis, we show that the two SDPs are dictated by the nonsymmorphic space group symmetry to appear at high symmetry points on the BZ boundary. An effective model is constructed to characterize the low-energy Dirac fermions. We further show that the SDPs are not only robust against SOC, they also survive under a variety of lattice strains: biaxial, uniaxial, and shear strains all preserve the SDPs; and they serve as effective means to tune the Dirac dispersion as well as the energy of the SDPs relative to the Fermi level. In addition, we find that these materials carry a well-defined ℤ_2 invariant despite a vanishing global bandgap, corresponding to a 2D ℤ_2 topological metal. Consequently, they possess topological edge states at the system boundary, which is indeed confirmed by our calculation. Our finding opens the door to the exploration of 2D spin-orbit Dirac materials, and provides a realistic material platform for the fundamental research as well as promising nanoscale applications. § COMPUTATIONAL DETAILSOur first-principles calculations are based on the density functional theory (DFT), using the projector augmented wave method <cit.> as implemented in the Vienna ab-initio Simulation Package <cit.>. The generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) realization <cit.> is used for the exchange-correlation functional. Our main results are also verified by the hybrid functional approach (HSE06) <cit.>. The plane-wave energy cutoff is set to be 330 eV. Monkhorst-Pack k-point mesh with size of 30×30×1 is applied for the Brillouin zone sampling. A vacuum layer of 13 Åthickness is added to avoid artificial interactions between periodic images. All lattice structures are fully relaxed until energy and force are converged with accuracy of 10^-6 eV and 0.005 eV/Å, respectively. Van der Waals interaction is taken into account by using the approach of Dion et al <cit.>.The phonon spectrum is calculated using the PHONOPY code through the DFPT approach <cit.>, with a 6× 6× 1 supercell and a 4× 4× 1 q-grid (such that the total energy converges with an accuracy of 10^-8 eV). The transition metal d orbitals may have important correlation effects, so we also validate our results by using the DFT+U method following the approach of Dudarev et al <cit.>. Several on-site Hubbard U parameters (U =1.0, 1.5, 2.0 eV) are tested for Hf(5d) orbitals, which yield almost the same results as that from the GGA calculations (see Supporting Information). To study the topological edge states, we construct the maximally localized Wannier functions using the Wannier90 code <cit.>, and then calculate the edge states using the iterative Green's function method as implemented in the Wannier_tools package <cit.>.§ RESULTSIn their three-dimensional (3D) bulk form, the HfGeTe-family materials take the PbFCl-type structure with space group No. 129 (P4/nmm) (see Fig. <ref>(a)) <cit.>. In fact, this structure is shared by a large group of (more than 200) existing compounds with chemical formula WHM, with W a transition metal or rare earth element, and H/M two main group elements <cit.>. Some of these materials have been gaining interest in recent research. For example, Dirac lines have been investigated by several works in the 3D bulk ZrSiX (X= S, Se, Te) <cit.> and HfSiS materials <cit.>, and some materials like ZrSiO have been predicted to be topological insulators in the monolayer form <cit.>.In this work, we shall consider the family with W=Hf, M=Te, and H being an element from the carbon group (group IVA). These materials all share similar electronic properties, so we shall mainly focus on HfGeTe as a representative in the following discussion.As shown in Fig. <ref>(a), the 3D bulk HfGeTe crystal has a layered structure with each HfGeTe layer consisting of five atomic layers in the sequence of Te-Hf-Ge-Hf-Te, and each atomic layer is of a square lattice. The lattice parameters obtained from DFT calculations are given by a = b = 3.885 Åand c = 8.464 Å, which is in good agreement with experimental values (a = b = 3.87 Åand c = 8.50 Å) <cit.>.In this work, our focus is on the monolayer HfGeTe (ML-HfGeTe). The corresponding lattice structure is shown in Fig. <ref>(b) and <ref>(c). The monolayer possesses the same space group symmetry as the bulk. The fully relaxed structure has lattice parameters a=b=3.727 Å, which are slightly decreased from the bulk values. To check the structural stability of ML-HfGeTe,we have calculated the phonon spectrum. As shown in Fig. <ref>(d), there is no imaginary frequency (soft mode) in the phonon spectrum throughout the Brillouin zone, indicating that the material is dynamically stable. One also notes that at low frequencies near Γ point, apart from the linearly dispersing in-plane transverse acoustic modes, there is also the parabolic out-of-plane acoustic (ZA) branch, which is a characteristic feature of 2D materials <cit.>.We further estimate the capability of ML-HfGeTe to form a freestanding membrane. The elastic strength of a 2D material can be characterized by its in-plane stiffness constant C=(1/A_0)(∂^2 E_S/∂ε^2)|_ε=0, where A_0 is the equilibrium area, E_S is the strain energy given by the energy difference between the strained and unstrained systems, and ε is the applied uniaxial strain. From DFT calculations, we find that C≈ 0.69 eV/Å^2 for ML-HfGeTe. Consider the deformation of a freestanding ML-HfGeTe flake under gravity, from the elastic theory by balancing gravity and 2D strain energy <cit.>, we find the ratio between the out-of-plane deformation and the dimension of the flake to be as small as 10^-3 to 10^-4 even for large flakes with a size about 10^4 μm^2. This result suggests that ML-HfGeTe could be strong enough to form a freestanding 2D structure even without support of a substrate.Having confirmed the stability of ML-HfGeTe, we then check the possibility to obtain ML-HfGeTe from its bulk samples via mechanical exfoliation. For bulk samples, the binding between the HfGeTe layer is relatively weak. In Fig. <ref>, we plot the energy variation (E_ex) when a ML-HfGeTe is separated from the bulk by a distance d, simulating the exfoliation process. With increasing d, the energy quickly saturates to a value corresponding to the exfoliation energy about 0.98 J/m^2. This value is comparable to that of graphene (∼0.37 J/m^2) <cit.> and MoS_2 (∼ 0.41 J/m^2) and is less than that of Ca_2N (∼ 1.14 J/m^2) <cit.>, suggesting the feasibility to obtain ML-HfGeTe by mechanical exfoliation from the bulk. In Fig. <ref>, we also show the exfoliation strength σ, which is obtained as the maximum derivative of E_ex with respect to the separation distance d. The calculated value of σ is about 4.2 GPa, also similar to values for typical 2D materials, such as graphene (∼ 2.1 GPa). In fact, mechanical exfoliation of ultrathin layers of the closely related materials ZrSiSe and ZrSiTe have already been demonstrated in experiment <cit.>.Next, we come to the electronic properties of ML-HfGeTe. Figure <ref>(a) and <ref>(b) show the electronic band structures of ML-HfGeTe without and with SOC, respectively. In the absence of SOC, the system exhibits a metallic state, with several band-crossings observed around X point and along the Γ-M path. By projecting the states onto atomic orbitals, we find that the low-energy states near the Fermi level are mainly from the Hf-5d orbitals, which should have a large SOC effect. Indeed, after including SOC, the low-energy bands undergo noticeable changes. First, SOC opens gap at the original band-crossings around the X point and along the Γ-M path. However, a global bandgap is not opened, so the state remains a metal, with electron pocket near X point and hole pocket on Γ-M. Second, the original band-degeneracy along X-M path is lifted. Importantly, one observes Dirac-like dispersion with the Dirac point exactly located at X point and close to the Fermi level (at -38 meV). In addition, one notes that the structure preserves inversion symmetry, thus each band is at least two-fold degenerate. Hence the crossing point at X is four-fold degenerate, conforming with the definition of a Dirac point.To confirm the linear dispersion around this point, in Fig. <ref>(d), we plot the 2D energy surface around X point for the two crossing bands. One indeed observes Dirac-type linear dispersion along all directions from the point. The obtained Fermi velocities are anisotropic due to the absence of four-fold rotational symmetry at X, with v_1=6.64× 10^5 m/s along Γ-X direction, and v_2=3.86× 10^5 m/s along X-M direction. This Dirac point exhibits the unusual feature that it is robust under strong SOC (actually it only appears when SOC is included), and it is located at the time-reversal-invariant momentum (TRIM) point on the BZ boundary. These observations suggest that it is a SDP.To fully demonstrate its identity as a 2D SDP, we need to clarify the mechanism that protects the Dirac point against SOC. In the following analysis, the electron spin and the SOC are included. We shall show that the presence of SDP at X point is solely dictated by the following symmetries of the system: glide mirror plane ℳ_z:(x,y,z)→ (x+1/2,y+1/2,-z), inversion 𝒫, and time reversal symmetry 𝒯. Here ℳ_z is a nonsymmorphic symmetry, which involves translation of half the lattice parameter along both x and y directions. Since every k-point in BZ is invariant under ℳ_z operation, each Bloch state can be chosen as eigenstate of ℳ_z. To find the possible ℳ_z eigenvalues, we note that(ℳ_z)^2=T_110E=-e^-ik_x-ik_y,where T_110 denotes the translation by one unit cell along both x and y directions, and E is a 2π rotation on spin which leads to a (-1) factor (here the wave-vectors k_x and k_y are measured in unit of 1/a). Hence the eigenvalues of ℳ_z are given byg_z=± i e^-ik_x/2-ik_y/2.It is important to note that the inversion does not commute with the glide mirror operation, instead, one findsℳ_z𝒫=T_110𝒫ℳ_z=e^-ik_x-ik_y𝒫ℳ_z.As a result, for an eigenstate |g_z⟩ of ℳ_z with eigenvalue g_z, the following relation holds:ℳ_z(𝒫𝒯|g_z⟩)=-g_z(𝒫𝒯|g_z⟩).As we have mentioned, due to the presence of both 𝒫 and 𝒯 symmetries, with SOC, each band here is (at least) two-fold degenerate with the pair of states |g_z⟩ and 𝒫𝒯|g_z⟩ at each k-point. Now Eq. (<ref>) shows that the two states |g_z⟩ and 𝒫𝒯|g_z⟩ have opposite ℳ_z eigenvalues. At the special point X with k_x=π and k_y=0, the relations in Eqs. (<ref>) and (<ref>) are reduced to g_z=± 1 and {ℳ_z,𝒫}=0. Further note that X is a TRIM point, hence any state |g_z⟩ at X has another degenerate Kramers partner 𝒯|g_z⟩ with the same eigenvalue g_z (because g_z=± 1 is real at X). This ensures four-fold degeneracy at X point, with the following linearly independent states { |g_z⟩, 𝒯|g_z⟩, 𝒫𝒯|g_z⟩, 𝒫|g_z⟩}.Note that for a generic k-point deviating from X, it will not be invariant under 𝒯 operation, hence the four-fold degeneracy will generally split into two doubly-degenerate bands away from X point. Thus, an isolated four-fold band-crossing point must appear at X. From the band structure in Fig. <ref>(b), one indeed observes that the states at X are all of this type, such as the crossing-point below the local gap. In the symmetry analysis, spin is explicitly considered, so the resulting degeneracy is robust against SOC. In the group-theory language, the SDPs here derive from the existence of four-fold irreducible representations of the nonsymmorphic space group at X point. For k-points on the BZ boundary, these representations are obtained from projective representations of the associated crystal point group (using Herring's method), which can be related to the regular representations of a larger group known as the central extension group <cit.>. This typically gives rise to higher dimensional irreducible representations, leading to band degeneracy points on the BZ boundary.The SDP and the Dirac fermions around it are characterized by the effective k· p Hamiltonian. This model can be obtained from the symmetry constraints that the Hamiltonian ℋ is invariant under the symmetry operations at X <cit.>. The symmetry operations in the little group at X include 𝒯, 𝒫, ℳ_z, and additionally ℳ_y: (x,y,z)→ (x,-y+1/2,z), which is an (off-centered) mirror plane perpendicular to y.The matrix representations of these operations can be found in the standard reference <cit.>, with 𝒯=-iσ_y⊗τ_0 K, 𝒫=σ_0⊗τ_x, ℳ_z=σ_0⊗τ_z, ℳ_y=-iσ_y⊗τ_x. Here K is the complex conjugation, σ_i and τ_i (i=x,y,z) are the Pauli matrices representing the degree of freedom with the four degenerate states at X, σ_0 and τ_0 are the 2× 2 identity matrix. Constrained by these symmetries, the effective Hamiltonian expanded around the 2D SDP can be expressed asℋ( k)=v_1 k_x(cosθσ_z⊗τ_z+sinθσ_x⊗τ_z)+v_2 k_yσ_y⊗τ_z,where the wave-vector k and the energy are measured from the SDP, and the model parameters v_1, v_2, and θ are real and depend on the microscopic details. The obtained energy dispersion is given by E=±√(v_1^2k_x^2+v_2^2 k_y^2),where each eigenvalue is doubly degenerate due to the combined 𝒫𝒯 symmetry, consistent with the anisotropic Dirac-cone spectrum that one observes in Fig. <ref>(d).Up to this point, we have established the presence of SDP in ML-HfGeTe. It should be pointed out that Y point at (0,π) is connected to X point by a four-fold rotational symmetry (see Fig. <ref>(c)), hence there is also a SDP at Y with its Dirac-cone rotated by π/2 compared to X. Thus, ML-HfGeTe is a 2D spin-orbit Dirac material with a pair of SDPs close to its Fermi level.Since the SDP in ML-HfGeTe is protected by the 𝒯, 𝒫, and ℳ_z symmetries, the Dirac point cannot be destroyed as long as these symmetries are preserved. We find that these symmetries are quite robust: they survive under a variety of strains, such as in-plane biaxial, uniaxial, and shear strains. In Fig. <ref>, we plot the calculated band structures under several different strains. One indeed observes that the SDP is maintained for all these cases, only the energy of the Dirac point and the dispersion are changed by strain. The strain-stress curves in Fig. <ref>(e) show that ML-HfGeTe also has excellent mechanical properties. It exhibits a linear elastic region up to 8% strain, and the critical strain is beyond 20%. These suggest that strain can be employed as an effective way to tune the properties of spin-orbit Dirac fermions in ML-HfGeTe.For example, the Fermi velocities can be tuned by strain. As plotted in Fig. <ref>(a), under biaxial strain, the two Fermi velocities v_1 and v_2 can be changed on the order of 5× 10^5 m/s in the range between -5% to +5% strains. The case with uniaxial strain is even more interesting, because in this case, the four-fold rotational symmetry that connects X and Y points are broken. Consequently, the two valleys at X and Y become independent. In Fig. <ref>(c), one observes that the two SDPs are shifted in opposite directions along energy axis. This behavior can be understood from the different bonding features at X and Y valleys. In Fig. <ref>(c) and <ref>(d), we plot the charge distribution for states at X and Y valleys. One observes that the states at X valley shows a bonding character along x, whereas the states at Y valley shows a bonding character along y (as dictated by the four-fold rotational symmetry). Therefore, when stretched along x, the X valley will be pushed up in energy, while the Y valley will be shifted down due to contraction along y. This is consistent with the result in Fig. <ref>(c). The variation of SDP energiesversus uniaxial strain along x-direction is shown in Fig. <ref>(b). The energy separation between the two SDPs can be up to 0.6 eV at 5% strain. We find that above ∼6% uniaxial strain, the valley at X is above the Fermi level and becomes unoccupied, resulting in a large valley polarization with Dirac fermions all in the Y valley.It is known that a ℤ_2-classification applies for 2D insulators with preserved time-reversal symmetry, where a nontrivial ℤ_2 invariant indicates a 2D topological insulator phase <cit.>. Interestingly, we find that ML-HfGeTe also possesses a nontrivial ℤ_2 invariant. Here, although ML-HfGeTe does not have a global bandgap, one notes that its bandgap is closed indirectly, i.e., there is no direct bandgap closing at any k-point [see Fig. <ref>(b)]. Thus, the band structure can be adiabatically connected to a fully-gapped insulating phase without any band crossing in the process (e.g., by raising the conduction bands at X point and by lowering the valence bands along the Γ-M path), such that it is also characterized with a ℤ_2 invariant. Here, the ℤ_2 invariant is defined for the bands below the local gap. Such an idea was theoretically proposed before in Ref. Pan2014 based on a model study, in which the proposed 2D metallic phase with a nontrivial ℤ_2 invariant was termed as a 2D ℤ_2 topological metal.We have rigorously evaluated the ℤ_2 invariant using the formula derived by Fu and Kane for centrosymmetric systems <cit.>. In this approach, one analyzes the parity eigenvalues at the four TRIM points (Γ, X, Y, and M). At each TRIM point, we calculate the quantity δ_i=∏_m=1^N ξ_2m(i), where i∈{Γ, X, Y, M}, ξ_2m(i)=± 1 is the parity eigenvalue of the 2mth band at i, which shares the same eigenvalue ξ_2m=ξ_2m-1 with its Kramers degenerate partner, and m runs through the 2N bands below the local gap. Then the ℤ_2 invariant ν=0,1 is obtained from the product of the four δ_i's through <cit.>(-1)^ν=∏_i δ_i.We have calculated the δ_i's using our DFT result, and their values are listed in Table <ref> [also see Fig. <ref>(a)]. We indeed find that the ℤ_2 invariant ν=1 is nontrivial for ML-HfGeTe. Thus, ML-HfGeTe also serves as the first realistic example that realizes the 2D ℤ_2 topological metal phase proposed in Ref. Pan2014.The nontrivial ℤ_2 invariant dictates the presence of topological edge states at the sample boundaries <cit.>. We consider a semi-infinite system of ML-HfGeTe with a boundary along y-direction. The edge spectrum is calculated and plotted in Fig. <ref>(b). One indeed observes a pair of topological edge states around the edge-projected X point (at k_y=0), similar to those for 2D topological insulators. These states are spin-helical, i.e., the states (at the same energy) propagating along opposite directions are with opposite spin polarizations. To further confirm the localization of these states around the edge, we directly calculate the charge density distribution of the edge state for a ML-HfGeTe nano-ribbon. The result in Fig. <ref>(c) shows that the edge state is indeed confined at the sample edge. The existence of these topological edge states manifests the nontrivial topology of the bulk band structure of ML-HfGeTe.§ DISCUSSION We emphasize that the existence of SDPs here is solely dictated by the nonsymmorphic space group symmetry. Hence they also appear in other member materials of the HfGeTe-family which share the same crystal symmetry, such as ML-HfSnTe and ML-HfSiTe etc. (see Supporting Information) and also materials with Te substituted by other chalcogen elements. However, symmetry cannot constrain the energy of the SDP. Fortunately, we find that for all the members of this family, the SDPs appear close to the Fermi level, hence qualifying them as 2D SDP materials. This provides a number of candidate materials that can be used to explore 2D SDP fermions.We mentioned that these proposed materials represent the first realistic material platform that realizes the 2D SDPs proposed by Young and Kane <cit.>. In this family of materials, there are two symmetry equivalent SDPs (at X and Y) connected by the four-fold rotation, which corresponds to Case I discussed in Ref. Young2015a. When the rotational symmetry is broken by the lattice strain [as in Fig. <ref>(c)], the two SDPs become inequivalent, which then corresponds to Case II in Ref. Young2015a.Several experimental and computational works have found Dirac lines in 3D bulk materials ZrSiX (X=S, Se, Te) and HfSiS <cit.>. The Dirac lines there are also derived from nonsymmorphic symmetries, similar to our case. But there are important distinctions. (i) Those works are on 3D bulk materials, whereas our work focuses on 2D materials. Note that this dimensionality difference is crucial for the stabilization of Dirac points, as we mentioned in the Introduction. (ii) The materials studied in those works have relatively weak SOC, such that the bands along certain paths (X-M and R-A in the 3D BZ) become nearly degenerate, forming the Dirac lines. In comparison, the ML-HfGeTe studied here has stronger SOC, such that the bands well split except at the X and Y points, leading to well-defined 2D SDPs.As we have mentioned, the 3D bulk material of HfGeTe (and of HfSiTe) already exist. This would greatly facilitate the realization of 2D monolayers, e.g., by using the mechanical exfoliation method. Experimentally, it has been demonstrated that for two closely related materials ZrSiSe and ZrSiTe, ultrathin layers with thickness less than 10 nm can be readily obtained by mechanical exfoliation <cit.>. Therefore, we expect that ML-HfGeTe (and other materials in the family) could also be fabricated in the near future. Once realized, the Dirac dispersion can be directly probed via angle-resolved photoemission spectroscopy (ARPES). The Dirac fermion character typically would tend to enhance the carrier mobility. The topological edge states can be detected via ARPES or scanning tunneling spectroscopy at the sample edge.In this work, we focus on the Dirac points which have linear energy dispersions. There could exist other kinds of protected band degeneracy points withquadratic or higher-order dispersions. For example, 2D quadratic band-touching points have been found in bilayer graphene <cit.> and in blue phosphorene oxide <cit.>. It will be interesting to explore these exotic band degeneracy points and their novel physics in future studies.Finally, we point out that the HfGeTe-family is still not ideal in terms of manifesting the Dirac physics, because besides the SDP, there are also extraneous non-Dirac bands passing the Fermi level. They will lead to additional contributions in electronic properties. Thus, in future research, it will be desirable to explore new materials with clean Dirac band structures, and/or explore methods to engineer the band structure of existing materials to get rid of the extraneous bands. Nevertheless, our current work serves an important first step towards this goal. § CONCLUSIONIn conclusion, based on first-principles calculation and theoretical analysis, we have proposed the first realistic material that realizes 2D SDPs close to its Fermi level. This material, ML-HfGeTe, is shown to be stable in monolayer form, and may be easily exfoliated from the corresponding bulk material. There exists a pair of SDPs in ML-HfGeTe. From symmetry analysis, we identify the nature and protection of the SDPs, demonstrating that they are intrinsically robust against SOC. We construct an effective k· p model for characterizing the low-energy fermions around the SDP. It is shown that these SDPs are quite robust against lattice deformations, and various strains can be used as powerful means to tune the SDPs. Furthermore, we find that ML-HfGeTe also represents the first example of the previously proposed 2D ℤ_2 topological metal. It possesses a nontrivial ℤ_2 invariant in the bulk, and a pair of topological edge states on the boundary. The above features are shared by a number of 2D materials in the HfGeTe-family. Our findings thus provide a promising platform to explore the intriguing physics of 2D spin-orbit Dirac fermions and the associated nanoscale applications.The authors thank D.L. Deng for helpful discussions. This work is supported by the MOST Project of China (Grants No.2016YFA0300603 and No.2014CB920903), the National Natural Science Foundation of China (Grants No.11574029), the Singapore Ministry of Education Academic Research Fund Tier 2 (MOE2015-T2-2-144) and Tier 1 (SUTD-T1-2015004).We acknowledge computational support from Texas Advanced Computing Center. 57 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Novoselov et al.(2004)Novoselov, Geim, Morozov, Jiang, Zhang, Dubonos, Grigorieva, and Firsov]Novoselov2004 author author K. S. Novoselov, author A. K. Geim, author S. V. Morozov, author D. Jiang, author Y. Zhang, author S. V. Dubonos, author I. V.Grigorieva,and author A. A.Firsov,journal journal Science volume 306, pages 666 (year 2004)NoStop [Castro Neto et al.(2009)Castro Neto, Guinea, Peres, Novoselov, and Geim]CastroNeto2009 author author A. H. Castro Neto, author F. Guinea, author N. M. R. Peres, author K. S. Novoselov,and author A. K. Geim, journal journal Rev. Mod. Phys. volume 81, pages 109 (year 2009)NoStop [Wang et al.(2015)Wang, Deng, Liu, and Liu]Wang2015 author author J. Wang, author S. Deng, author Z. Liu,and author Z. Liu,journaljournal National Science Review volume 2, pages 22 (year 2015)NoStop [Cahangirov et al.(2009)Cahangirov, Topsakal, Aktürk, Şahin, andCiraci]Cahangirov2009 author author S. Cahangirov, author M. Topsakal, author E. Aktürk, author H. Şahin,and author S. Ciraci,journal journal Phys. Rev. Lett. volume 102, pages 236804 (year 2009)NoStop [Liu et al.(2011)Liu, Feng, and Yao]Liu2011 author author C.-C. Liu, author W. Feng,andauthor Y. Yao,journal journal Phys. Rev. Lett. volume 107, pages 076802 (year 2011)NoStop [Liu et al.(2015)Liu, Liu, Xu, Ouyang, Song, Guan, and Yang]Liu2015 author author G. Liu, author S. B. Liu, author B. Xu, author C. Y. Ouyang, author H. Y. Song, author S. Guan,and author S. A. Yang, booktitle booktitle The Journal of Physical Chemistry Letters,journal journal J. Phys. Chem. Lett. volume 6, pages 4936 (year 2015)NoStop [Malko et al.(2012)Malko, Neiss, Viñes, and Görling]Malko2012 author author D. Malko, author C. Neiss, author F. Viñes,andauthor A. Görling,journal journal Phys. Rev. Lett. volume 108, pages 086804 (year 2012)NoStop [Xu et al.(2014)Xu, Wang, Miao, Wei, Chen, Yan, Lau, Liu, and Ma]Xu2014 author author L.-C. Xu, author R.-Z. Wang, author M.-S. Miao, author X.-L. Wei, author Y.-P. Chen, author H. Yan, author W.-M. Lau, author L.-M.Liu,and author Y.-M.Ma,journal journal Nanoscale volume 6, pages 1113 (year 2014)NoStop [Zhou et al.(2014)Zhou, Dong, Oganov, Zhu, Tian, and Wang]Zhou2014 author author X.-F. Zhou, author X. Dong, author A. R. Oganov, author Q. Zhu, author Y. Tian,and author H.-T.Wang,journal journal Phys. Rev. Lett. volume 112, pages 085502 (year 2014)NoStop [Ma et al.(2016)Ma, Jiao, Gao, Gu, Bilic, Chen, and Du]Ma2016 author author F. Ma, author Y. Jiao, author G. Gao, author Y. Gu, author A. Bilic, author Z. Chen,and author A. Du, booktitle booktitle Nano Letters,journal journal Nano Lett. volume 16, pages 3022 (year 2016)NoStop [Jiao et al.(2016)Jiao, Ma, Bell, Bilic, andDu]Jiao2016 author author Y. Jiao, author F. Ma, author J. Bell, author A. Bilic,and author A. Du,journal journal Angew. Chem. volume 128, pages 10448 (year 2016)NoStop [Feng et al.(2017)Feng, Sugino, Liu, Zhang, Yukawa, Kawamura, Iimori, Kim, Hasegawa, Li, Chen, Wu, Kumigashira, Komori, Chiang, Meng, andMatsuda]Feng2017 author author B. Feng, author O. Sugino, author R.-Y. Liu, author J. Zhang, author R. Yukawa, author M. Kawamura, author T. Iimori, author H. Kim, author Y. Hasegawa, author H. Li, author L. Chen, author K. Wu, author H. Kumigashira, author F. Komori, author T.-C.Chiang, author S. Meng,and author I. Matsuda,journal journal Phys. Rev. Lett. volume 118, pages 096401 (year 2017)NoStop [Kim et al.(2015)Kim, Baik, Ryu, Sohn, Park, Park, Denlinger, Yi, Choi, and Kim]Kim2015 author author J. Kim, author S. S. Baik, author S. H. Ryu, author Y. Sohn, author S. Park, author B.-G. Park, author J. Denlinger, author Y. Yi, author H. J.Choi,and author K. S.Kim,journal journal Science volume 349, pages 723 (year 2015)NoStop [Lu et al.(2016)Lu, Zhou, Chang, Guan, Chen, Jiang, Jiang, Wang, Yang, Feng, Kawazoe, and Lin]Lu2016 author author Y. Lu, author D. Zhou, author G. Chang, author S. Guan, author W. Chen, author Y. Jiang, author J. Jiang, author X.-S. Wang, author S. A. Yang, author Y. P. Feng, author Y. Kawazoe,and author H. Lin,journal journal npj Computational Materials volume 2, pages 16011 (year 2016)NoStop [Sheng and Nikolić(2017)]Sheng2017 author author X. L. Sheng and author B. K. Nikolić, journal journal Phys. Rev. B volume 95, pages201402 (year 2017)NoStop [Kane and Mele(2005a)]Kane2005 author author C. L. Kane and author E. J. Mele,journal journal Phys. Rev. Lett. volume 95, pages 226801 (year 2005a)NoStop [Min et al.(2006)Min, Hill, Sinitsyn, Sahu, Kleinman, and MacDonald]Min2006 author author H. Min, author J. E. Hill, author N. A. Sinitsyn, author B. R. Sahu, author L. Kleinman,and author A. H. MacDonald,journal journal Phys. Rev. B volume 74, pages 165310 (year 2006)NoStop [Yao et al.(2007)Yao, Ye, Qi, Zhang, andFang]Yao2007 author author Y. Yao, author F. Ye, author X.-L. Qi, author S.-C. Zhang,and author Z. Fang,journal journal Phys. Rev. B volume 75, pages 041401 (year 2007)NoStop [Young and Kane(2015)]Young2015a author author S. M. Young and author C. L. Kane,journal journal Phys. Rev. Lett. volume 115, pages 126803 (year 2015)NoStop [Yang and Nagaosa(2014)]Yang2014 author author B.-J. Yang and author N. Nagaosa,@noopjournal journal Nat Communvolume 5, pages 4898 (year 2014)NoStop [Wieder and Kane(2016)]Wieder2016 author author B. J. Wieder and author C. L. Kane, @noopjournal journal Phys. Rev. B volume 94, pages 155108 (year 2016)NoStop [Young et al.(2012)Young, Zaheer, Teo, Kane, Mele, and Rappe]Young2012 author author S. M. Young, author S. Zaheer, author J. C. Y. Teo, author C. L. Kane, author E. J. Mele,and author A. M. Rappe, @noopjournal journal Phys. Rev. Lett. volume 108, pages 140405 (year 2012)NoStop [Steinberg et al.(2014)Steinberg, Young, Zaheer, Kane, Mele, and Rappe]Steinberg2014 author author J. A. Steinberg, author S. M. Young, author S. Zaheer, author C. L. Kane, author E. J. Mele,and author A. M. Rappe, @noopjournal journal Phys. Rev. Lett. volume 112, pages 036403 (year 2014)NoStop [Singh and Hennig(2014)]Singh2014 author author A. K. Singh and author R. G. Hennig, booktitle booktitle Applied Physics Letters,journal journal Appl. Phys. Lett. volume 105, pages 042103 (year 2014)NoStop [Blöchl(1994)]Bloechl1994 author author P. E. Blöchl, @noopjournal journal Phys. Rev. B volume 50, pages 17953 (year 1994)NoStop [Kresse and Hafner(1993)]Kresse1993 author author G. Kresse and author J. Hafner,journal journal Phys. Rev. B volume 47,pages 558 (year 1993)NoStop [Kresse and Furthmüller(1996)]Kresse1996 author author G. Kresse and author J. Furthmüller, @noopjournal journal Phys. Rev. B volume 54, pages 11169 (year 1996)NoStop [Perdew et al.(1996)Perdew, Burke, and Ernzerhof]PBE author author J. P. Perdew, author K. Burke, and author M. Ernzerhof,@noopjournal journal Phys. Rev. Lett.volume 77, pages 3865 (year 1996)NoStop [Heyd et al.(2003)Heyd, Scuseria, and Ernzerhof]Heyd2003 author author J. Heyd, author G. E. Scuseria,and author M. Ernzerhof,booktitle booktitle The Journal of Chemical Physics,journaljournal The Journal of Chemical Physics volume 118, pages 8207 (year 2003)NoStop [Dion et al.(2004)Dion, Rydberg, Schröder, Langreth, and Lundqvist]Dion2004 author author M. Dion, author H. Rydberg, author E. Schröder, author D. C. Langreth,and author B. I. Lundqvist,journal journal Phys. Rev. Lett. volume 92, pages 246401 (year 2004)NoStop [Togo and Tanaka(2015)]Togo2015 author author A. Togo and author I. Tanaka,journal journal Scripta Materialia volume 108, pages 1 (year 2015)NoStop [Dudarev et al.(1998)Dudarev, Botton, Savrasov, Humphreys, and Sutton]Dudarev1998 author author S. L. Dudarev, author G. A. Botton, author S. Y. Savrasov, author C. J. Humphreys,and author A. P. Sutton, @noopjournal journal Phys. Rev. B volume 57, pages 1505 (year 1998)NoStop [Marzari and Vanderbilt(1997)]Marzari1997 author author N. Marzari and author D. Vanderbilt, @noopjournal journal Phys. Rev. B volume 56, pages 12847 (year 1997)NoStop [Souza et al.(2001)Souza, Marzari, and Vanderbilt]Souza2001 author author I. Souza, author N. Marzari, and author D. Vanderbilt,@noopjournal journal Phys. Rev. Bvolume 65, pages 035109 (year 2001)NoStop [Wu_()]Wu_tool @noopnote Q. S. Wu, S. N. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, arXiv:1703.07789.Stop [Onken et al.(1964)Onken, Vierheilig, and Hahn]Onken1964 author author H. Onken, author K. Vierheilig, and author H. Hahn,journal journal Z. anorg. allg. Chem. volume 333,pages 267 (year 1964)NoStop [Wang and Hughbanks(1995)]Wang1995 author author C. Wang and author T. Hughbanks, booktitle booktitle Inorganic Chemistry,journal journal Inorg. Chem. volume 34, pages 5524 (year 1995)NoStop [Schoop et al.(2016)Schoop, Ali, Straßer, Topp, Varykhalov, Marchenko, Duppel, Parkin, Lotsch, and Ast]Schoop2016 author author L. M. Schoop, author M. N. Ali, author C. Straßer, author A. Topp, author A. Varykhalov, author D. Marchenko, author V. Duppel, author S. S. P. Parkin, author B. V.Lotsch,and author C. R.Ast, journal journal Nature Communications volume 7, pages 11696 (year 2016)NoStop [Hu et al.(2016)Hu, Tang, Liu, Liu, Zhu, Graf, Myhro, Tran, Lau, Wei, and Mao]Hu2016 author author J. Hu, author Z. Tang, author J. Liu, author X. Liu, author Y. Zhu, author D. Graf, author K. Myhro, author S. Tran, author C. N. Lau, author J. Wei,and author Z. Mao,journal journal Phys. Rev. Lett. volume 117, pages 016602 (year 2016)NoStop [Neupane et al.(2016)Neupane, Belopolski, Hosen, Sanchez, Sankar, Szlawska, Xu, Dimitri, Dhakal, Maldonado, Oppeneer, Kaczorowski, Chou, Hasan, and Durakiewicz]Neupane2016 author author M. Neupane, author I. Belopolski, author M. M. Hosen, author D. S. Sanchez, author R. Sankar, author M. Szlawska, author S.-Y.Xu, author K. Dimitri, author N. Dhakal, author P. Maldonado, author P. M. Oppeneer, author D. Kaczorowski, author F. Chou, author M. Z. Hasan,and author T. Durakiewicz,journal journal Phys. Rev. B volume 93, pages 201104 (year 2016)NoStop [Topp et al.(2016)Topp, Lippmann, Varykhalov, Duppel, Lotsch, Ast, and Schoop]Topp2016 author author A. Topp, author J. M. Lippmann, author A. Varykhalov, author V. Duppel, author B. V. Lotsch, author C. R. Ast,and author L. M. Schoop,journal journal New Journal of Physics volume 18, pages 125014 (year 2016)NoStop [Hosen et al.(2017)Hosen, Dimitri, Belopolski, Maldonado, Sankar, Dhakal, Dhakal, Cole, Oppeneer, Kaczorowski, Chou, Hasan, Durakiewicz, and Neupane]Hosen2017 author author M. M. Hosen, author K. Dimitri, author I. Belopolski, author P. Maldonado, author R. Sankar, author N. Dhakal, author G. Dhakal, author T. Cole, author P. M.Oppeneer, author D. Kaczorowski, author F. Chou, author M. Z. Hasan, author T. Durakiewicz,and author M. Neupane,journal journal Phys. Rev. B volume 95, pages 161101 (year 2017)NoStop [Singha et al.(2017)Singha, Pariari, Satpati, and Mandal]Singha2017 author author R. Singha, author A. K. Pariari, author B. Satpati,andauthor P. Mandal,journal journal Proceedings of the National Academy of Sciencesvolume 114, pages 2468 (year 2017)NoStop [Takane et al.(2016)Takane, Wang, Souma, Nakayama, Trang, Sato, Takahashi, andAndo]Takane2016 author author D. Takane, author Z. Wang, author S. Souma, author K. Nakayama, author C. X. Trang, author T. Sato, author T. Takahashi,and author Y. Ando, journal journal Phys. Rev. B volume 94, pages 121108 (year 2016)NoStop [Chen et al.(2017)Chen, Xu, Jiang, Wu, Qi, Yang, Wang, Sun, Schröter, Yang, Schoop, Lv, Zhou, Chen, Yao, Lu, Chen, Felser, Yan, Liu, and Chen]Chen2017 author author C. Chen, author X. Xu, author J. Jiang, author S.-C. Wu, author Y. P. Qi, author L. X.Yang, author M. X. Wang, author Y. Sun, author N. B. M. Schröter, author H. F. Yang, author L. M. Schoop, author Y. Y. Lv, author J. Zhou, author Y. B.Chen, author S. H. Yao, author M. H. Lu, author Y. F. Chen, author C. Felser, author B. H. Yan, author Z. K. Liu,and author Y. L. Chen,journal journal Phys. Rev. B volume 95, pages 125126 (year 2017)NoStop [Xu et al.(2015)Xu, Song, Nie, Weng, Fang, and Dai]Xu2015 author author Q. Xu, author Z. Song, author S. Nie, author H. Weng, author Z. Fang,and author X. Dai,journal journal Phys. Rev. B volume 92, pages 205310 (year 2015)NoStop [Liu et al.(2007)Liu, Ming, and Li]Liu2007 author author F. Liu, author P. Ming,andauthor J. Li,journal journal Phys. Rev. B volume 76, pages 064120 (year 2007)NoStop [Zhu et al.(2014)Zhu, Zhang, and Li]Zhu2014 author author L. Zhu, author G. Zhang,andauthor B. Li,journal journal Phys. Rev. B volume 90, pages 214302 (year 2014)NoStop [Booth et al.(2008)Booth, Blake, Nair, Jiang, Hill, Bangert, Bleloch, Gass, Novoselov, Katsnelson,and Geim]Booth2008 author author T. J. Booth, author P. Blake, author R. R. Nair, author D. Jiang, author E. W. Hill, author U. Bangert, author A. Bleloch, author M. Gass, author K. S.Novoselov, author M. I.Katsnelson,and author A. K.Geim, booktitle booktitle Nano Letters,journaljournal Nano Lett. volume 8,pages 2442 (year 2008)NoStop [Zhao et al.(2014)Zhao, Li, and Yang]Zhao2014 author author S. Zhao, author Z. Li,andauthor J. Yang, booktitle booktitle Journal of the American Chemical Society,journal journal J. Am. Chem. Soc. volume 136, pages 13313 (year 2014)NoStop [Zacharia et al.(2004)Zacharia, Ulbricht, and Hertel]Zacharia2004 author author R. Zacharia, author H. Ulbricht,and author T. Hertel,journal journal Phys. Rev. B volume 69, pages 155406 (year 2004)NoStop [Guan et al.(2015)Guan, Yang, Zhu, Hu, andYao]Guan2015 author author S. Guan, author S. A. Yang, author L. Zhu, author J. Hu,and author Y. Yao,journal journal Sci. Rep. volume 5, pages 12285 (year 2015)NoStop [Bradley and Cracknell(1972)]Bradley1972 author author C. J. Bradley and author A. P. Cracknell, @nooptitle The Mathematical Theory of Symmetry in Solids (publisher Clarendon Press,address Oxford, year 1972)NoStop [Kane and Mele(2005b)]Kane2005a author author C. L. Kane and author E. J. Mele,journal journal Phys. Rev. Lett. volume 95, pages 146802 (year 2005b)NoStop [Pan et al.(2014)Pan, Li, Qiao, Liu, Yao, and Yang]Pan2014 author author H. Pan, author X. Li, author Z. Qiao, author C.-C. Liu, author Y. Yao,and author S. A.Yang,journal journal New Journal of Physics volume 16, pages 123015 (year 2014)NoStop [Fu and Kane(2007)]Fu2007 author author L. Fu and author C. L. Kane, journal journal Phys. Rev. B volume 76, pages 045302 (year 2007)NoStop [Hasan and Kane(2010)]Hasan2010 author author M. Z. Hasan and author C. L. Kane,journaljournal Rev. Mod. Phys. volume 82,pages 3045 (year 2010)NoStop [Qi and Zhang(2011)]Qi2011 author author X.-L. Qi and author S.-C. Zhang, journal journal Rev. Mod. Phys. volume 83, pages 1057 (year 2011)NoStop [McCann and Fal'ko(2006)]McCann2006 author author E. McCann and author V.I. Fal'ko, journal journal Phys. Rev. Lett. volume 96, pages 086805 (year 2006)NoStop [Zhu et al.(2016)Zhu, Wang, Guan, Liu, Zhang, Chen, and Yang]Zhu2016 author author L. Y. Zhu, author S. S. Wang, author S. Guan, author Y. Liu, author T. T. Zhang, author G. B. Chen,and author S. A.Yang,journal journal Nano Lett. volume 16, pages 6548 (year 2016)NoStop | http://arxiv.org/abs/1706.08692v2 | {
"authors": [
"Shan Guan",
"Ying Liu",
"Zhi-Ming Yu",
"Shan-Shan Wang",
"Yugui Yao",
"Shengyuan A. Yang"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170627070804",
"title": "Two-dimensional Spin-Orbit Dirac Point in Monolayer HfGeTe"
} |
Deviation inequalities for convex functions]Deviation inequalities for convex functions motivated by the Talagrandconjecture Supported by the grants ANR 2011 BS01 007 01, ANR 10 LABX-58, ANR11-LBX-0023-01Université Paris Descartes - MAP 5 (UMR CNRS 8145), 45 rue des Saints-Pères 75270Paris cedex 6, France. University of Delaware, Department of Mathematical Sciences, 501 Ewing Hall, Newark DE 19716, USA. Université Paris Ouest Nanterre La Défense - Modal'X, 200 avenue de la République 92000 Nanterre, France Université Paris Est Marne la Vallée - Laboratoire d'Analyse et de Mathématiques Appliquées (UMR CNRS 8050), 5 bd Descartes, 77454 Marne la Vallée Cedex 2, [email protected], [email protected], [email protected], [email protected] 60E15, 32F32 and 26D10[ Nathael Gozlan, Mokshay Madiman, Cyril Roberto, Paul-Marie Samson December 30, 2023 ===================================================================== Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on integrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure. § INTRODUCTIONIn the late eighties, Talagrand conjectured that the “convolution by a biased coin”, on the hypercube {-1,1}^n, satisfies some refined hypercontractivity property. We refer to Problems 1 and 2 in <cit.> for precise statements. A continuous version of Talagrand's conjecture for the Ornstein-Uhlenbeck operator has recently attracted some attention <cit.>;in particular, it was resolved by<cit.> by first proving a deviation inequality for log-semiconvex functions above their means under Gaussian measure. In this paper, we discuss a simpler approach to proving this deviation inequality for the special case of log-convex functions (which is already of interest). Let us start by presenting the continuous version of Talagrand's conjecture and the history of its resolution. Denote by γ_n the standard Gaussian (probability) measure in dimension n, with density x ↦ (2 π)^-n/2exp{ -|x|^2/2}(where |x| denotes the standard Euclidean norm of x∈^n) and, for p ≥ 1, by 𝕃^p(γ_n) the set of measurable functions f ℝ^n →ℝ such that |f|^p is integrable with respect to γ_n. Then, given g ∈𝕃^1(γ_n), the Ornstein-Ulhenbeck semi-group is defined asP_tg(x) := ∫ g(e^-tx + √(1-e^-2t)y) dγ_n(y)x ∈ℝ^n , t ≥ 0. It is well known that the family (P_t)_t ≥ 0 enjoys the so-called hypercontractivity property <cit.> which asserts that, for any p>1, any t>0 and q ≤ 1 + (p-1)e^2t, P_t g is more regular than g in the sense that, if g ∈𝕃^p(γ_n) then P_tg ∈𝕃^q(γ_n) and moreoverP_t g_q ≤g_p.However this property is empty when one only assumes that g ∈𝕃^1(γ_n). A natural question is therefore to ask if the semi-group has anyway some regularization effect also in this case. Given g : ℝ^n →ℝ non-negativewith ∫ gdγ_n =1, by Markov's inequality and the fact that ∫ P_s g dγ_n=1 we haveγ_n({P_sg ≥ t }) ≤1/t∀ t>0 .The continuous version of Talagrand's conjecture (adapted from <cit.>) states that as soon as s>0,lim_t →∞sup_g ≥ 0, ∫ g dγ_n=1t γ_n({P_s g ≥ t }) = 0. The most recent paper dealing with this conjecture is due to Lehec <cit.> who proved that, for any s >0 there exists a constant α_s ∈ (0, ∞) (depending only on s and not on the dimension n) such that for any non-negative function g : ℝ^n →ℝ^+ with ∫ gdγ_n =1,γ_n({P_s g ≥ t }) ≤α_s/t √(log t)∀ t>1and this bound is optimal in the sense that the factor √(log t) cannot be improved. In the first paper dealing with this question <cit.>, Ball, Barthe, Bednorz, Oleszkiewicz and Wolff already obtained a similar bound but with a constant α_s depending heavily on the dimension n plus some extra loglog t factor in the numerator. Later Eldan and Lee <cit.> proved that the above bound holds with a constant α_s independent on n but again with the extra loglog t factor in the numerator. Finally the conjecture was fully proved by Lehec removing the loglog t factor <cit.> and giving an explicit bound on α_s, namely that α_s := αmax(1,1/2s) for some numerical constant α.In both Eldan-Lee and Lehec's papers, the two key ingredients are the following:(1) for any s>0, the Ornstein-Uhlenbeck semi-group satisfies, for all non-negative function g ∈𝕃^1(γ_n),Hess (log P_s g) ≥ -1/2sId,where Hess denotes the Hessian matrix and Id the identity matrix of ℝ^n. This is a somehow standard property easy to prove thanks to the kernel representation (<ref>);(2) for any positive function g with Hess (log g) ≥ -βId, for some β>0, and ∫ gdγ_n =1, it holdsγ_n({g ≥ t }) ≤C_β/t √(log t)∀ t>1,with C_β = αmax(1, β). It will be more convenient to deal with g=e^f in the sequel so we move to this setting now. The last inequality can be reformulated as follows:for any f : ℝ^n →ℝ with ∫ e^fdγ_n =1 and Hess (f) ≥ -βId, it holdsγ_n ({ f ≥t }) ≤ C_βe^-t/√(t)∀ t >0 .We now describe the two main contributions of this note (whichwere independently obtained by Ramon van Handel).First, as a warm up, we give in Section <ref> a short proof of (<ref>) in dimension 1. The main argument of thisproof is that due to the semi-convexity of f, the condition (2π)^-1/2∫ e^f-1/2|x|^2 dγ =1implies a pointwise comparison between f and the function |x|^2/2, which then can be turned into a tail comparison. Then, in dimension n, we give in Section <ref> a sharp version of the upper bound (<ref>) for convex functions. Our main result states: Suppose that f: ^n → is a convex function such that ∫ e^f dγ_n=1, thenγ_n (f ≥ t) ≤Φ(√(2t)),∀ t≥0,where Φ(t) = 1/√(2π)∫_t^+∞ e^-u^2/2 du, t∈. Let us make a few comments on this result. First, using the following classical bound (which is asymptotically optimal)Φ(s) = 1/√(2π)∫_s^∞ e^-x^2/2dx≤1/√(2π)∫_s^∞x/s e^-x^2/2 dx = e^-s^2/2/√(2π)s, ∀ s>0,one immediately recovers (<ref>) with the constant C_0' = 1/(2√(π)). Furthermore, the bound (<ref>) is sharp.Indeed, for a given value of t≥ 0, Inequality (<ref>) becomes an equality for the function f_t(x) = √(2t)x_1 -t, x = (x_1,…,x_n) ∈^n.Finally, since the Ornstein-Uhlenbeck semigroup preserves log-convexity (this follows from the fact that any positive combination oflog-convex functions remains log-convex, see e.g <cit.> p. 649), Theorem <ref> immediately implies the following corollary. Let g be a log-convex function such that ∫ g dγ_n=1, then for any s≥ 0,γ_n(P_sg ≥ t) ≤Φ(√(2log(t))),∀ t≥1.In the special case when g is log-convex, Corollary <ref> is a sharp improvement of Lehec's result (<ref>).Note that for log-convex g, the constant α_s can be taken independent of s unlike in (<ref>), but this already followed from Lehec's inequality (<ref>) combined with the preservation of log-convexity by the Ornstein-Uhlenbeck semigroup.Another consequence of Theorem <ref> is that a deviation inequality for structured functions also follows for other measures that can be obtained by “nice” pushforwards of Gaussian measure.Indeed, observe that for any coordinate-wise non-decreasing, convex function f on ^n, and any convex functions g_1,…, g_n:^N, the composition f(g_1(x), …, g_n(x)) is convex on ^N. Hence we immediately have the following corollary. For a standard Gaussian random vector Z in ^N, let the probability measure μ on ^n be the joint distribution of (g_1(Z), …, g_n(Z)), where g_1,…,g_n : ^N are convex functions. Suppose that f: ^n → is a coordinate-wise non-decreasing, convex function such that ∫_^n e^f dμ=1. Thenμ (f ≥ t) ≤Φ(√(2t)),∀ t≥0,For example, consider the exponential distribution, whose density is e^-x on _+=(0,∞) and which can be realized as Z_1^2+Z_2^2/2 with Z_1, Z_2 i.i.d. standard Gaussian. Clearly a product of exponential distributions on the line is an instance covered by Corollary <ref>, since we can take N=2n and g_i(x)=x_i^2+x_i+1^2/2. More generally, Corollary <ref> applies toa product of χ^2 distributions with arbitrary degrees of freedom, and also to some cases with correlation (consider for example N=3, g_1(x)=x_1^2+x_2^2/2 and g_2(x)=x_2^2+x_3^2/2).The proof of Theorem <ref> is given in Section <ref>. It relies on the Ehrhard inequality, which we recall now: according to <cit.>, if A,B ⊂^n are two convex sets, then Φ^-1 (γ_n(λ A + (1-λ) B)) ≥λΦ^-1 (γ_n(A)) + (1-λ) Φ^-1 (γ_n(B)),∀λ∈ [0,1],where λ A + (1-λ) B := {λ a + (1-λ)b : a ∈ A, b ∈ B} denotes the usual Minkowski sum and Φ^-1 is the inverse of the cumulative distribution function Φ of γ_1:Φ(t) = 1/√(2π)∫_-∞^t e^-u^2/2 du, t ∈.After Ehrhard's pioneer work, Inequality (<ref>) was shown to be true if only one set is assumed to be convex by Latała <cit.> and finally to arbitrary measurable sets by Borell <cit.>. See also <cit.> and the references therein for recent developments on this inequality. Inequality (<ref>) (for arbitrary sets A,B) is a very strong statement in the hierarchy of Gaussian geometric and functional inequalities. For instance, it gives back the celebrated Gaussian isoperimetric result of Sudakov-Tsirelson <cit.> and Borell <cit.>. Another elegant consequence of (<ref>) due to Kwapień is that if f is a convex function on ^n which is integrable with respect to γ_n, then the median of f is always less than or equal to the mean of f under γ_n. The key ingredient in Kwapień's proof is the observation that the function α(t) = Φ^-1(γ_n(f ≤ t)), t ∈is concave over ; this observation (already made in Ehrhard's original paper) also plays a key role in our proof of Theorem <ref>. After the completion of this work, we learned that Paouris and Valettas <cit.> developed in a recent paper similar ideas to derive from (<ref>) deviation inequalities for convex functions under their mean.In Section <ref>, we give a second proof of Theorem <ref>, and also discuss (following an observation of R. van Handel) the difficulty of its extension to the log-semiconvex case.Acknowledgement. The results of this note were independently obtained by Ramon van Handela few months before us, as we learnt after a version of this note was circulated. Although he chose not to publish them, these observations should be considered as due to him. We are also grateful to him for numerous comments on earlier drafts of this note.§ THE CONTINUOUS TALAGRAND CONJECTURE IN DIMENSION 1 In the next lemma we take advantage of the semi-convexity property Hess (f) ≥ -βId to derive information on f.More precisely we may compare f to x ↦ |x|^2/2. The result holds in any dimension, and we give two proofs for completeness. Let f ℝ^n →ℝ and β≥ 0 be such that ∫ e^f dγ_n =1, f is smooth and Hess (f) ≥ -βId. Then,f(x) ≤n/2ln(1+β) +1/2|x|^2 , ∀ x ∈ℝ^n . [First proof of Lemma <ref>] Let h(x)=f(x)+ β/2|x|^2. By assumption on f, the function h is convex on ^n and hence h(x)= sup_t ∈ℝ^n{⟨ x,t⟩ - h^*(t) },∀ x ∈^n, where h^*(t):= sup_x ∈ℝ^n{⟨ t,x⟩ - h(x) }, t ∈^nis the Legendre transform of h.Now, we have for all t ∈ℝ^n1=∫ e^fdγ_n = ∫exp{ h(x) - β/2|x|^2 }dγ_n(x) ≥(2 π)^-n/2 e^-h^*(t)∫exp{⟨ x,t⟩ - 1+β/2 |x|^2 }dx= (1+β)^-n/2exp{ -h^*(t) + 1/2(1+β)|t|^2 } .Therefore, for all t ∈ℝ^n it holdsh^*(t) ≥ - n/2ln(1+β) + 1/2(1+β)|t|^2 .In turnh(x)=sup_t {⟨ x,t⟩ - h^*(t) }≤n/2ln(1+β) + sup_t {⟨ x,t⟩ - 1/2(1+β)|t|^2 } = 1/2(n ln(1+β) + (1+β) |x|^2 )which leads to the desired conclusion. [Second proof of Lemma <ref>] Define h̃(x) = h(x) + β/2|x|^2, x ∈^n and let γ_n,β be the gaussian measure 𝒩(0, 1/1+βI), then it holds 1=∫ e^h(x) dγ_n(x) = (1+β)^-n/2∫ e^h̃(x) dγ_n,β(x)For all a ∈^n, the change of variable formula then gives 1= (1+β)^-n/2e^-(1+β)/2|a|^2∫ e^h̃(y+a)-(1+β)y· a dγ_n,β(dy).The function y↦h̃(y+a)-(1+β)y· a is convex and the function x ↦ e^x is convex and increasing so the function y↦exp(h̃(y+a)-(1+β)y· a) is also convex. So applying Jensen inequality yields to1≥ (1+β)^-n/2e^-(1+β)/2|a|^2exp(h̃(a+∫ y dγ_n,β(y)) - (1+β)∫ y· a dγ_n,β(y)) = e^-(1+β)/2|a|^2+ h̃(a)and so h(a) ≤ |a|^2/2 + n/2log(1+β).The β=0 case of Lemma <ref> (i.e., for convex functions f, which is the essential case) is contained in Graczyk et al. <cit.> (curiously it does not appear in the published version <cit.> of the paper), and in fact was proved in the more general setting of subharmonic functions. The second proof given above is theirs and works for the more general setting. Also note that neither proof requires smoothness of f, which however is sufficient for our purposes.In principle, one would hope to already get some deviation bound from the above lemma. More precisely, given f as in Lemma <ref>, we haveγ_n ( { f ≥ t })≤γ_n ( { |x|^2 ≥ 2t - n ln(1+β) }) ,thanks to Lemma <ref>, and we are left with a tail estimate for a ^2 distribution with n degrees of freedom.In dimension n=1, the tail of the ^2 distribution behaves like e^-t/√(t). Therefore, the above simple argument already gives backthe estimate (<ref>) and thus provides a quick proof of the continuous Talagrand's conjecture for n=1, moreover with clean dependence on β, as detailed below.If f ℝ→ℝ is smooth and β≥ 0 are such that ∫ e^fdγ_1 =1 and f”≥ -β pointwise, thenγ_1 ({ f ≥t }) ≤1+β/√(2)e^-t/√(t)∀ t ≥1 . Assume first that t ≥ (1+β)ln(1+β)/(2β). Using Inequality (<ref>), we get from Lemma <ref>γ_1 ( { f ≥ t }) ≤γ_1 ( { |x| ≥√(2t - ln(1+β))})≤2(2 π)^-1/2exp{ -t + 1/2ln(1+β)}/√(2t-ln(1+β))=√(1+β/π)e^-t/√(t)1/√(1-(ln(1+β)/(2t)))≤√(1+β/π)e^-t/√(t)1/√(1-(β/(1+β))) = 1+β/√(π)e^-t/√(t) .Now assume that t ≤ (1+β)ln(1+β)/(2β).Thanks to Markov's inequality, we haveγ_1 ({ f > t }) ≤e^-t≤ √((1+β)ln(1+β) /(2β))e^-t/√(t)≤√(1+β)/√(2)e^-t/√(t)≤1+β/√(2)e^-t/√(t)where, in the third inequality, we used that ln(1+β) ≤β. The result follows. Unfortunately this naive approach of using the pointwise bound from Lemma <ref> is specific to dimension 1,since in higher dimension the tail of the^2 distribution does not have the correct behavior. It should be noticed that Ball et al. <cit.> also have a quick direct proof of the Talagrand conjecture for n=1that also uses a similar tail comparison with the ^2 distribution, and also noticed that such a tail is not of the correct order for n ≥ 2. § THE DEVIATION INEQUALITY FOR LOG-CONVEX FUNCTIONS Throughout this section f ℝ^n →ℝ is a convex function satisfying ∫ e^fdγ_n =1 where γ_n is the standard Gaussian measure on ℝ^n. Given s ∈ℝ, let A_s:={f ≤ s } and φ(s):=Φ^-1( γ_n(A_s) ),whereΦ^-1 is the inverse of the Gaussian cumulative function Φ defined by (<ref>).The key ingredient in the proof of Theorem <ref> is the concavity of the function φ that, as we shall see in the proof of the next lemma, is a direct consequence of Ehrhard's inequality (<ref>).Let f and φ be defined as above. Then φ is concave, non-decreasing, lim_s →∞φ(s)=+ ∞ and lim_s → -∞φ(s)=- ∞.The concavity of φ was first observed by Ehrhard in <cit.>. Below we recall the proof for the reader's convenience. That φ is non-decreasing and satisfies lim_s →∞φ(s)=+ ∞ and lim_s → -∞φ(s)=- ∞ is a direct and obvious consequence of the definition. Now we prove that φ is concave, using Ehrhard's inequality. Given λ∈ [0,1] and s_1, s_2 ∈ℝ, we have, by convexity of f,A_λ s_1 + (1-λ)s_2⊃λ A_s_1 + (1-λ)A_s_2.Hence, by monotonicity of Φ^-1, it holdsφ(λ s_1 + (1-λ)s_2) ≥Φ^-1( γ_n( λ A_s_1 + (1-λ)A_s_2) ) .Then, Ehrhard's inequality (<ref>) implies thatΦ^-1( γ_n( λ A_s_1 + (1-λ)A_s_2) ) ≥λΦ^-1( γ_n(A_s_1 ) ) + (1-λ) Φ^-1( γ_n( A_s_2) )=λφ(s_1) + (1-λ)φ(s_2)from which the concavity of φ follows. [Proof of Theorem <ref>] Let f and φ be defined as above. Then, it is enough to show thatφ(u) ≥√(2u),∀ u ≥ 0.Since -φ : →∪{+∞} is convex by Lemma <ref> and lower-semicontinuous, the Fenchel-Moreau Theorem applies and guarantees that-φ(u)= sup_t ∈ℝ{ ut - ψ(t)},∀ u ∈, where ψ(t)=(-φ)^*(t):=sup_u ∈ℝ{ ut + φ(u) }is the Fenchel-Legendre transform of -φ. Also we observe that, since lim_u →∞φ(u)=+∞, necessarily ψ(t)=+∞ for all t >0 so that φ(u)= - sup_t ≤ 0{ ut - ψ(t)} = inf_t ≤ 0{ -ut + ψ(t)} .Now observe that1 = ∫ e^fdγ_n = ∫_-∞^∞ e^u γ_n(f ≥ u)du = ∫_-∞^∞ e^u (1-Φ(φ(u)) du = ∫_-∞^∞ e^u Φ(φ(u)) duwhere we recall that Φ=1-Φ. Using integration by parts and the fact Φ is decreasing, we have for all t ≤ 01= ∫_-∞^∞ e^u Φ(φ(u))du≥∫_-∞^∞ e^u Φ(-ut+ψ(t))du = (-t)e^ψ(t)/t∫_-∞^+∞e^-v/tΦ(v) dv =e^ψ(t)/t1/√(2π)∫_-∞^+∞ e^-v/t e^-v^2/2 dv = exp{ψ(t)/t + 1/2t^2}.Therefore, for all t ≤ 0 it holdsψ(t) ≥ -1/2t .In turn,φ(u)= inf_t ≤ 0{ -ut + ψ(t)}≥inf_t ≤ 0{ -ut - 1/2t} = √(2u)as expected.§ REVISITING THE DEVIATION INEQUALITY, WITH A DISCUSSION OF THE SEMI-CONVEX CASE Suppose that f : ^n → is a function such that ∫ e^f dγ_n = 1. Define μ_f the distribution of f under γ_n, that it to sayμ_f(A) := γ_n ({x ∈^n : f(x) ∈ A}),∀ BorelA ⊂.Consider the monotone rearrangement transport map T_f sending γ_1 onto μ_f. It is defined byT_f(u) = F_f^-1∘Φ(u),∀ u ∈,where F_f(t) = μ_f((-∞,t]), t ∈, denotes the cumulative distribution function of μ_f and F_f^-1(s) = inf{ t : F_f(t)≥ s}, s∈ (0,1)its generalized inverse.The following proposition will yield to a slightly different proof of Theorem <ref>.With the notation above, if T_f is κ-semiconvex, for some κ≥0 i.eT_f((1-t)x+ty) ≤ (1-t)T_f(x) + t T_f(y) + κ/2 t(1-t) |x-y|^2,∀ x,y ∈,∀ t ∈ [0,1],then γ_n({f>u}) ≤Φ(√(2u-log(1+κ))),∀ u≥1/2log(1+κ). The κ-semiconvexity condition is equivalent to the convexity of the function x↦ T_f(x)+κx^2/2. Now observe that1=∫ e^fdγ_n = ∫ e^ydμ_f(y) = ∫ e^T_f(x) dγ_1(x).Applying Lemma <ref> to the function T_f in dimension 1, one concludes that T_f(x) ≤1/2x^2 + 1/2log(1+κ),∀ x ∈.This is equivalent toΦ(x) ≤ F_f(1/2x^2 + 1/2log(1+κ))and thusF_f(u) ≥Φ(√(2u-log(1+κ))),∀ u≥1/2log(1+κ)or in other words,γ_n({f>u}) ≤Φ(√(2u-log(1+κ))),∀ u≥1/2log(1+κ)[Second proof of Theorem <ref>] Suppose that f : ^n → is convex and such that ∫ e^f dγ_n=1. Then according to Lemma <ref>, the function Φ^-1∘ F_f = T_f^-1 is concave. Being also non-decreasing, its inverse T_f is convex. Applying Proposition <ref> with κ=0 completes the proof.Inview of Proposition <ref>, a natural conjecture would be the following:Conjecture. There exists a function κ : [0,∞) → [0,∞) such that if f: ^n → is a smooth function such that Hess f ≥ -βId, β≥0, then the map T_f is κ(β)-semiconvex on .If this conjecture was true, then one would recover completely Eldan-Lee-Lehec result (<ref>). Besides the convex case, let us observe that the conjecture is obviously true in dimension 1 for non-decreasing functions f. Indeed, f is clearly a transport map between γ_1 and μ_f.Being non-decreasing, f is necessarily the monotone rearrangement map, that is to say : f = T_f. Since f is κ-semiconvex, then so is T_f. Unfortunately, this probably too naive conjecture turns out to be false in general. As explained to us by R. van Handel, the presence of local minimizers for f breaks down the semi-convexity of T_f. Let us illustrate this in dimension 1. Consider a function f:→ of class 𝒞^1 such that f'(x) vanishes only at a finite number of points and such that there is some point x_o ∈ and η >0 such that f'(x_o)=0, f'(x)<0 on [x_o-η,x_o[ and f'(x_o)>0 on ]x_o,x_o+η]. Denoting by t_o = f(x_o), we assume that inf_ f < t_o, that is to say, f only presents a local minimizer at x_o. Let us further assume that there are some α_o,β_o>0 and some positive integer N such that, for all t_o-α_o≤ t<t_o, Card{ x ∈ : f(x)=t}≤ Nand |f'(x)|≥β_o for all x such that t_o-α_o ≤ f(x)<t_o.Claim. There is no λ≥ 0 for which the map T:=T_f is λ-semi-convex.It is not difficult to exhibit semi-convex functions f enjying the assumptions above, which disclaim the conjecture.[Proof of the Claim.] First let us remark that if T was λ-semi-convex for some λ≥ 0, then the map x↦ T(x) + λ/2x^2 would be convex, and so would admit finite left and right derivatives everywhere. Moreover for a convex function the left derivative at some point is always less than or equal to the right derivative at this same point. So the λ-semi-convexity of T would in particular imply thatT_-'(x) ≤ T'_+(x),∀ x ∈.We are going to show that T_-'(u_o) > T'_+(u_o) for some u_o ∈ which will prove the claim. Since, denoting F:=F_f, T'_±(u) = φ(u)/F'_±∘ T(u),at every point u∈ where the derivative exists, one conludes that it is enough to show that F'_-(t_o)<F'_+(t_o)to have the desired inequality at u_o=T^-1(t_o). Note that |T^-1(t_o)| <∞ because μ_f((t_o,+∞))=γ_1((T^-1(t_o),+∞))>0 and μ_f((∞,t_o))=γ_1((-∞,T^-1(t_o)))>0, as easily follows from our assumptions.According to the one dimensional general change of variable formula, the probability measure μ_f admits the following densityh(t) = ∑_x ∈{f=t}φ(x)/|f'(x)|,t ∈,where φ(x) = 1/√(2π) e^-x^2/2, x ∈. Define ε_o = max_[x_o-η, x_o+η]f-t_o>0 ; then, for h< ε_o, it holds F(t_o+h)-F(t_o) = ∫_t_o^t_o+h h(t) dt≥ hm/M(h),where m = inf_[x_o-η, x_o+η]φ and M(h) = sup{ |f'(x)| : x ∈ [x_o-η, x_o+η], f(x) ∈ [t_o,t_o+h]}.It is easily seen that M(h)→ 0 as h to 0^+, which implies that F'_+(t_o)=+∞. Now let us consider the left derivative. Let us note that one can assume without loss of generality that the left derivative exists at t_o, since otherwise the function T would clearly not be semi-convex.For any h>0, it holdsF(t_o)-F(t_o-h) = ∫_t_o-h^t_o h(t) dt ≤ hN/√(2π)β_oand so F'_-(t_o) < +∞, which completes the proof of the claim.plain10BBBOW13 K. Ball, F. Barthe, W. Bednorz, K. Oleszkiewicz, and P. Wolff. L^1-smoothing for the Ornstein-Uhlenbeck semigroup. Mathematika, 59(1):160–168, 2013.BH09 F. Barthe and N. Huet. On Gaussian Brunn–Minkowski inequalities. Studia Math., 191(3):283–304, 2009.Bor75b C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. Math., 30(2):207–216, 1975.Bor03 C. Borell. The Ehrhard inequality. C. R. Math. Acad. Sci. Paris, 337(10):663–666, 2003.Ehr83 A. Ehrhard. Symétrisation dans l'espace de Gauss. Math. Scand., 53(2):281–301, 1983.EL14:2 R. Eldan and J. R. Lee. Regularization under diffusion and anti-concentration of temperature. Preprint, arXiv:1410.3887, 2014.GKLZ08 P. Graczyk, T. Kemp, J.-J. Loeb, and T. Zak. Hypercontractivity for log-subharmonic functions. Preprint, arXiv:0802.4260v2, 2008.GKL10 P. Graczyk, T. Kemp, and J.-J. Loeb. Hypercontractivity for log-subharmonic functions. J. Funct. Anal., 258(6):1785–1805, 2010.Gro75 L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math., 97(4):1061–1083, 1975.Lat96 R. Latała. A note on the Ehrhard inequality. Studia Math., 118(2):169–174, 1996.Leh16 J. Lehec. Regularization in L_1 for the Ornstein-Uhlenbeck semigroup. Ann. Fac. Sci. Toulouse Math. (6), 25(1):191–204, 2016.MOA11:book A. W. Marshall, I. Olkin, and B. C. Arnold. Inequalities: theory of majorization and its applications. Springer Series in Statistics. Springer, New York, second edition, 2011.Nel66 E. Nelson. A quartic interaction in two dimensions. In Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965), pages 69–73. M.I.T. Press, Cambridge, Mass., 1966.Nel73 E. Nelson. The free Markoff field. J. Functional Analysis, 12:211–227, 1973.PV16 G. Paouris and P. Valettas. A small deviation inequality for convex functions. Preprint, Available online at arXiv:1611.01723, 2016.ST74 V.N. Sudakov and B.S. Tsirel'son. Extremal properties of half-spaces for spherically invariant measures. Zap. Nauch. Sem. L.O.M.I., 41:14–24, translated in J. Soviet Math. 9, 9–18 (1978) 1974.Tal89:1 M. Talagrand. A conjecture on convolution operators, and a non-Dunford-Pettis operator on L^1. Israel J. Math., 68(1):82–88, 1989.Han16 R. van Handel. The Borell-Ehrhard game. Preprint, Available online at arXiv:1605.00285, 2016. | http://arxiv.org/abs/1706.08688v1 | {
"authors": [
"Nathael Gozlan",
"Mokshay Madiman",
"Cyril Roberto",
"Paul-Marie Samson"
],
"categories": [
"math.PR",
"math.FA"
],
"primary_category": "math.PR",
"published": "20170627065158",
"title": "Deviation inequalities for convex functions motivated by the Talagrand conjecture"
} |
Skeleton-based human action recognition has attracted a lot of research attention during the past few years. Recent works attempted to utilize recurrent neural networks to model the temporal dependencies between the 3D positional configurations of human body joints for better analysis of human activities in the skeletal data. The proposed work extends this idea to spatial domain as well as temporal domain to better analyze the hidden sources of action-related information within the human skeleton sequences in both of these domains simultaneously. Based on the pictorial structure of Kinect's skeletal data, an effective tree-structure based traversal framework is also proposed. In order to deal with the noise in the skeletal data, a new gating mechanism within LSTM module is introduced, with which the network can learn the reliability of the sequential data and accordingly adjust the effect of the input data on the updating procedure of the long-term context representation stored in the unit's memory cell. Moreover, we introduce a novel multi-modal feature fusion strategy within the LSTM unit in this paper. The comprehensive experimental results on seven challenging benchmark datasets for human action recognition demonstrate the effectiveness of the proposed method.Action Recognition, Recurrent Neural Networks, Long Short-Term Memory, Spatio-Temporal Analysis, Tree Traversal, Trust Gate, Skeleton Sequence. Skeleton-Based Action Recognition Using Spatio-Temporal LSTM Network with Trust Gates Jun Liu, Amir Shahroudy, Dong Xu, Alex C. Kot, and Gang Wang J. Liu, A. Shahroudy, and A. C. Kot are with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore.E-mail: {jliu029, amir3, eackot}@ntu.edu.sg. G. Wang is with Alibaba Group, Hangzhou, 310052, China.E-mail: [email protected]. D. Xu is with School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2006, Australia.E-mail: [email protected]. Manuscript received August, 2016; revised June, 2017.========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTIONHuman action recognition is a fast developing research area due to its wide applications in intelligent surveillance, human-computer interaction, robotics, and so on. In recent years, human activity analysis based on human skeletal data has attracted a lot of attention,and various methods for feature extraction and classifier learning have been developed for skeleton-based action recognition <cit.>. A hidden Markov model (HMM) is utilized by Xia <cit.> to model the temporal dynamics over a histogram-based representation of joint positions for action recognition. The static postures and dynamics of the motion patterns are represented via eigenjoints by Yang and Tian <cit.>. A Naive-Bayes-Nearest-Neighbor classifier learning approach is also used by <cit.>. Vemulapalli <cit.> represent the skeleton configurations and action patterns as points and curves in a Lie group, and then a SVM classifier is adopted to classify the actions. Evangelidis <cit.> propose to learn a GMM over the Fisher kernel representation of the skeletal quads feature. An angular body configuration representation over the tree-structured set of joints is proposed in <cit.>.A skeleton-based dictionary learning method using geometry constraint and group sparsity is also introduced in <cit.>.Recently, recurrent neural networks (RNNs) which can handle the sequential data with variable lengths <cit.>, have shown their strength in language modeling <cit.>, image captioning <cit.>, video analysis <cit.>,and RGB-based activity recognition <cit.>. Applications of these networks have also shown promising achievements in skeleton-based action recognition <cit.>.In the current skeleton-based action recognition literature, RNNs are mainly used to model the long-term context information across the temporal dimension by representing motion-based dynamics. However, there is often strong dependency relations among the skeletal joints in spatial domain also, and the spatial dependency structure is usually discriminative for action classification. To model the dynamics and dependency relations in both temporal and spatial domains, we propose a spatio-temporal long short-term memory (ST-LSTM) network in this paper.In our ST-LSTM network, each joint can receive context information from its stored data from previous frames and also from the neighboring joints at the same time frame to represent its incoming spatio-temporal context. Feeding a simple chain of joints to a sequence learner limits the performance of the network, as the human skeletal joints are not semantically arranged as a chain. Instead, the adjacency configuration of the joints in the skeletal data can be better represented by a tree structure. Consequently, we propose a traversal procedure by following the tree structure of the skeleton to exploit the kinematic relationship among the body joints for better modeling spatial dependencies.Since the 3D positions of skeletal joints provided by depth sensors are not always very accurate, we further introduce a new gating framework, so called “trust gate”, for our ST-LSTM network to analyze the reliability of the input data at each spatio-temporal step. The proposed trust gate gives better insight to the ST-LSTM network about when and how to update, forget, or remember the internal memory content as the representation of the long-term context information.In addition, we introduce a feature fusion method within the ST-LSTM unit to better exploit the multi-modal features extracted for each joint.We summarize the main contributions of this paper as follows.(1) A novel spatio-temporal LSTM (ST-LSTM) network for skeleton-based action recognition is designed. (2) A tree traversal technique is proposed to feed the structured human skeletal data into a sequential LSTM network. (3) The functionality of the ST-LSTM framework is further extended by adding the proposed “trust gate”. (4) A multi-modal feature fusion strategy within the ST-LSTM unit is introduced. (5) The proposed method achieves state-of-the-art performance on seven benchmark datasets. The remainder of this paper is organized as follows. In section <ref>, we introduce the related works on skeleton-based action recognition, which used recurrent neural networks to model the temporal dynamics. In section <ref>, we introduce our end-to-end trainable spatio-temporal recurrent neural network for action recognition. The experiments are presented in section <ref>. Finally, the paper is concluded in section <ref>.§ RELATED WORKSkeleton-based action recognition has been explored in different aspects during recent years <cit.>. In this section, we limit our reviewto more recent approaches which use RNNs or LSTMs for human activity analysis.Du <cit.> proposed a Hierarchical RNN network by utilizing multiple bidirectional RNNs in a novel hierarchical fashion. The human skeletal structure was divided to five major joint groups. Then each group was fed into the corresponding bidirectional RNN. The outputs of the RNNs were concatenated to represent the upper body and lower body, then each was further fed into another set of RNNs. By concatenating the outputs of two RNNs, the global body representation was obtained, which was fed to the next RNN layer. Finally, a softmax classifier was used in <cit.> to perform action classification.Veeriah <cit.> proposed to add a new gating mechanism for LSTM to model the derivatives of the memory states and explore the salient action patterns. In this method, all of the input features were concatenated at each frame and were fed to the differential LSTM at each step.Zhu <cit.> introduced a regularization term to the objective function of the LSTM network to push the entire framework towards learning co-occurrence relations among the joints for action recognition. An internal dropout <cit.> technique within the LSTM unit was also introduced in <cit.>.Shahroudy <cit.> proposed to split the LSTM's memory cell to sub-cells to push the network towards learning the context representations for each body part separately. The output of the network was learned by concatenating the multiple memory sub-cells.Harvey and Pal <cit.> adopted an encoder-decoder recurrent network to reconstruct the skeleton sequence and perform action classification at the same time. Their model showed promising results on motion capture sequences.Mahasseni and Todorovic <cit.> proposed to use LSTM to encode a skeleton sequence as a feature vector. At each step, the input of the LSTM consists of the concatenation of the skeletal joints' 3D locations in a frame. They further constructed a feature manifold by using a set of encoded feature vectors. Finally, the manifold was used to assist and regularize the supervised learning of another LSTM for RGB video based action recognition.Different from the aforementioned works, our proposed method does not simply concatenate the joint-based input features to build the body-level feature representation. Instead, the dependencies between the skeletal joints are explicitly modeled by applying recurrent analysis over temporal and spatial dimensions concurrently. Furthermore, a novel trust gate is introduced to make our ST-LSTM network more reliable against the noisy input data. This paper is an extension of our preliminary conference version <cit.>. In <cit.>, we validated the effectiveness of our model on four benchmark datasets. In this paper, we extensively evaluate our model on seven challenging datasets. Besides, we further propose an effective feature fusion strategy inside the ST-LSTM unit. In order to improve the learning ability of our ST-LSTM network, a last-to-first link scheme is also introduced. In addition, we provide more empirical analysis of the proposed framework. § SPATIO-TEMPORAL RECURRENT NETWORKS In a generic skeleton-based action recognition problem, the input observations are limited to the 3D locations of the major body joints at each frame. Recurrent neural networks have been successfully applied tothis problem recently <cit.>. LSTM networks <cit.> are among the most successful extensions of recurrent neural networks. A gating mechanism controlling the contents of an internal memory cell is adopted by the LSTM model to learn a better and more complex representation of long-term dependencies in the input sequential data. Consequently, LSTM networks are very suitable for feature learning over time series data (such as human skeletal sequences over time). We will briefly review the original LSTM model in this section, and then introduce our ST-LSTM network and the tree-structure based traversal approach. We will also introduce a new gating mechanism for ST-LSTM to handle the noisy measurements in the input data for better action recognition. Finally, an internal feature fusion strategy for ST-LSTM will be proposed. §.§ Temporal Modeling with LSTMIn the standard LSTM model, each recurrent unit contains an input gate i_t, a forget gate f_t, an output gate o_t, and an internal memory cell state c_t, together with a hidden state h_t. The input gate i_t controls the contributions of the newly arrived input data at time step t for updating the memory cell, while the forget gate f_t determines how much the contents of the previous state (c_t-1) contribute to deriving the current state (c_t).The output gate o_t learns how the output of the LSTM unit at current time step should be derived from the current state of the internal memory cell.These gates and states can be obtained as follows: ([ i_t; f_t; o_t; u_t; ])= ([σ;σ;σ; tanh;]) (M([ x_t; h_t-1; ]) )c_t =i_t⊙ u_t+ f_t⊙c_t-1h_t =o_t⊙tanh( c_t)where x_t is the input at time step t, u_t is the modulated input, ⊙ denotes the element-wise product, and M: ℝ^D+d→ℝ^4d is an affine transformation. d is the size of the internal memory cell, and D is the dimension of x_t. §.§ Spatio-Temporal LSTMRNNs have already shown their strengths in modeling the complex dynamics of human activities as time series data, and achieved promising performance in skeleton-based human action recognition <cit.>.In the existing literature, RNNs are mainly utilized in temporal domain to discover the discriminative dynamics and motion patterns for action recognition. However, there is also discriminative spatial information encoded in the joints' locations and posture configurations at each video frame, and the sequential nature of the body joints makes it possible to apply RNN-based modeling to spatial domain as well.Different from the existing methods which concatenate the joints' information as the entire body's representation, we extend the recurrent analysis to spatial domain by discovering the spatial dependency patterns among different body joints. We propose a spatio-temporal LSTM (ST-LSTM) network to simultaneously model the temporal dependencies among different frames and also the spatial dependencies of different joints at the same frame. Each ST-LSTM unit, which corresponds to one of the body joints, receives the hidden representation of its own joint from the previous time step and also the hidden representation of its previous joint at the current frame. A schema of this model is illustrated in<ref>.In this section, we assume the joints are arranged in a simple chain sequence, and the order is depicted in<ref>(a). In section <ref>, we will introduce a more advanced traversal scheme to take advantage of the adjacency structure among the skeletal joints.We use j and t to respectively denote the indices of joints and frames, where j ∈{1,...,J} and t ∈{1,...,T}. Each ST-LSTM unit is fed with the input (x_j, t, the information of the corresponding joint at current time step), the hidden representation of the previous joint at current time step (h_j-1,t), and the hidden representation of the same joint at the previous time step (h_j,t-1).As depicted in<ref>, each unit also has two forget gates, f_j, t^T and f_j, t^S, to handle the two sources of context information in temporal and spatial dimensions, respectively.The transition equations of ST-LSTM are formulated as follows:([ i_j, t; f_j, t^S; f_j, t^T; o_j, t; u_j, t;])= ([σ;σ;σ;σ; tanh;]) (M([ x_j, t; h_j-1, t; h_j, t-1;]) ) c_j, t = i_j, t⊙ u_j, t + f_j, t^S⊙c_j-1, t + f_j, t^T⊙c_j, t-1h_j, t =o_j, t⊙tanh( c_j, t)§.§ Tree-Structure Based Traversal Arranging the skeletal joints in a simple chain order ignores the kinematic interdependencies among the body joints. Moreover, several semantically false connections between the joints, which are not strongly related, are added.The body joints are popularly represented as a tree-based pictorial structure <cit.> in human parsing, as shown in<ref>(b). It is beneficial to utilize the known interdependency relations between various sets of body joints as an adjacency tree structure inside our ST-LSTM network as well. For instance, the hidden representation of the neck joint (joint 2 in<ref>(a)) is often more informative for the right hand joints (7, 8, and 9) compared to the joint 6, which lies before them in the numerically ordered chain-like model. Although using a tree structure for the skeletal data sounds more reasonable here, tree structures cannot be directly fed into our current form of the proposed ST-LSTM network.In order to mitigate the aforementioned limitation, a bidirectional tree traversal scheme is proposed. In this scheme, the joints are visited in a sequence, while the adjacency information in the skeletal tree structure will be maintained. At the first spatial step, the root node (central spine joint in<ref>(c)) is fed to our network. Then the network follows the depth-first traversal order in the spatial (skeleton tree) domain.Upon reaching a leaf node, the traversal backtracks in the tree. Finally, the traversal goes back to the root node.In our traversal scheme, each connection in the tree is met twice, thus it guarantees the transmission of the context data in both top-down and bottom-up directions within the adjacency tree structure. In other words, each node (joint) can obtain the context information from both its ancestors and descendants in the hierarchy defined by the tree structure. Compared to the simple joint chain order described in section <ref>, this tree traversal strategy, which takes advantage of the joints' adjacency structure, can discover stronger long-term spatial dependency patterns in the skeleton sequence. Our framework's representation capacity can be further improved by stacking multiple layers of the tree-structured ST-LSTMs and making the network deeper, as shown in<ref>.It is worth noting that at each step of our ST-LSTM framework, the input is limited to the information of a single joint at a time step, and its dimension is much smaller compared to the concatenated input features used by other existing methods. Therefore, our network has much fewer learning parameters. This can be regarded as a weight sharing regularization for our learning model, which leads to better generalization in the scenarios with relatively small sets of training samples. This is an important advantage for skeleton-based action recognition, since the numbers of training samples in most existing datasets are limited.§.§ Spatio-Temporal LSTM with Trust Gates In our proposed tree-structured ST-LSTM network, the inputs are the positions of body joints provided by depth sensors (such as Kinect), which are not always accurate because of noisy measurements and occlusion. The unreliable inputs can degrade the performance of the network.To circumvent this difficulty, we propose to add a novel additional gate to our ST-LSTM network to analyze the reliability of the input measurements based on the derived estimations of the input from the available context information at each spatio-temporal step.Our gating scheme is inspired by the works in natural language processing <cit.>, which use the LSTM representation of previous words at each step to predict the next coming word. As there are often high dependency relations among the words in a sentence, this idea works decently. Similarly, in a skeletal sequence, the neighboring body joints often move together, and this articulated motion follows common yet complex patterns, thus the input data x_j,t is expected to be predictable by using the contextual information (h_j-1,t and h_j,t-1) at each spatio-temporal step.Inspired by this predictability concept, we add a new mechanism to our ST-LSTM calculating a prediction of the input at each step and comparing it with the actual input. The amount of estimation error is then used to learn a new “trust gate”. The activation of this new gate can be used to assist the ST-LSTM network to learn better decisions about when and how to remember or forget the contents in the memory cell. For instance, if the trust gate learns that the current joint has wrong measurements, then this gate can block the input gate and prevent the memory cell from being altered by the current unreliable input data. Concretely, we introduce a function to produce a prediction of the input at step (j,t) based on the available context information as:p_j, t = tanh(M_p([ h_j-1, t; h_j, t-1;]) )where M_p is an affine transformation mapping the data from ℝ^2d to ℝ^d, thus the dimension of p_j,t is d.Note that the context information at each step does not only contain the representation of the previous temporal step, but also the hidden state of the previous spatial step. This indicates that the long-term context information of both the same joint at previous frames and the other visited joints at the current frame are seamlessly incorporated. Thus this function is expected to be capable of generating reasonable predictions.In our proposed network, the activation of trust gate is a vector in ℝ^d (similar to the activation of input gate and forget gate). The trust gate τ_j, t is calculated as follows:x'_j, t = tanh(M_x( x_j, t) )τ_j, t =G (p_j, t - x'_j, t) where M_x: ℝ^D→ℝ^d is an affine transformation. The activation function G(·) is an element-wise operation calculated as G(z) = exp(-λ z^2), for which λ is a parameter to control the bandwidth of Gaussian function (λ > 0). G(z) produces a small response if z has a large absolute value and a large response when z is close to zero.Adding the proposed trust gate, the cell state of ST-LSTM will be updated as:c_j, t = τ_j, t⊙ i_j, t⊙ u_j, t + (1 - τ_j, t) ⊙ f_j, t^S⊙c_j-1, t + (1 - τ_j, t) ⊙ f_j, t^T⊙c_j, t-1 This equation can be explained as follows: (1) if the input x_j,t is not trusted (due to the noise or occlusion), then our network relies more on its history information, and tries to block the new input at this step; (2) on the contrary, if the input is reliable, then our learning algorithm updates the memory cell regarding the input data.The proposed ST-LSTM unit equipped with trust gate is illustrated in<ref>. The concept of the proposed trust gate technique is theoretically generic and can be used in other domains to handle noisy input information for recurrent network models.§.§ Feature Fusion within ST-LSTM UnitAs mentioned above, at each spatio-temporal step, the positional information of the corresponding joint at the current frame is fed to our ST-LSTM network. Here we call joint position-based feature as a geometric feature. Beside utilizing the joint position (3D coordinates), we can also extract visual texture and motion features (HOG, HOF <cit.>, or ConvNet-based features <cit.>) from the RGB frames, around each body joint as the complementary information. This is intuitively effective for better human action representation, especially in the human-object interaction scenarios. A naive way for combining geometric and visual features for each joint is to concatenate them in the feature leveland feed them to the corresponding ST-LSTM unit as network's input data. However, the dimension of the geometric feature is very low intrinsically, while the visual features are often in relatively higher dimensions. Due to this inconsistency, simple concatenation of these two types of features in the input stage of the network causes degradation in the final performance of the entire model.The work in <cit.> feeds different body parts into the Part-aware LSTM <cit.> separately, and then assembles them inside the LSTM unit. Inspired by this work, we propose to fuse the two types of features inside the ST-LSTM unit, rather than simply concatenating them at the input level. We use x_j,t^ℱ (ℱ∈{1,2}) to denote the geometric feature and visual feature for a joint at the t-th time step. As illustrated in<ref>, at step (j,t), the two features (x_j,t^1 and x_j,t^2) are fed to the ST-LSTM unit separately as the new input structure. Inside the recurrent unit, we deploy two sets of gates, input gates (i_j,t^ℱ), forget gates with respect to time (f_j,t^T, ℱ) and space (f_j,t^S, ℱ), and also trust gates (τ_j, t^ℱ), to deal with the two heterogeneous sets of modality features. We put the two cell representations (c_j,t^ℱ) together to build up the multimodal context information of the two sets of modality features. Finally, the output of each ST-LSTM unit is calculated based on the multimodal context representations, and controlled by the output gate (o_j,t) which is shared for the two sets of features.For the features of each modality, it is efficient and intuitive to model their context information independently. However, we argue that the representation ability of each modality-based sets of features can be strengthened by borrowing information from the other set of features. Thus, the proposed structure does not completely separate the modeling of multimodal features.Let us take the geometric feature as an example. Its input gate, forget gates, and trust gate are all calculated from the new input (x_j,t^1) and hidden representations (h_j,t-1 and h_j-1,t), whereas each hidden representation is an associate representation of two features' context information from previous steps. Assisted by visual features' context information, the input gate, forget gates, and also trust gate for geometric feature can effectively learn how to update its current cell state (c_j,t^1). Specifically, for the new geometric feature input (x_j,t^1), we expect the network to produce a better prediction when it is not only based on the context of the geometric features, but also assisted by the context of visual features. Therefore, the trust gate (τ_j, t^1) will have stronger ability to assess the reliability of the new input data (x_j,t^1).The proposed ST-LSTM with integrated multimodal feature fusion is formulated as: ([ i_j, t^ℱ; f_j, t^S,ℱ; f_j, t^T,ℱ; u_j, t^ℱ;])= ([σ;σ;σ; tanh;]) (M^ℱ([ x_j, t^ℱ; h_j-1, t; h_j, t-1;]) ) p_j, t^ℱ = tanh(M_p^ℱ([ h_j-1, t; h_j, t-1;]) )x'_j, t^ℱ = tanh(M_x^ℱ([ x_j, t^ℱ;]) )τ_j, t^ℱ =G (x'_j, t^ℱ - p_j, t^ℱ) c_j, t^ℱ = τ_j, t^ℱ⊙ i_j, t^ℱ⊙ u_j, t^ℱ + (1 - τ_j, t^ℱ) ⊙ f_j, t^S,ℱ⊙c_j-1, t^ℱ + (1 - τ_j, t^ℱ) ⊙ f_j, t^T,ℱ⊙c_j, t-1^ℱo_j, t = σ( M_o([ x_j, t^1; x_j, t^2; h_j-1, t; h_j, t-1;]) ) h_j, t =o_j, t⊙tanh([ c_j, t^1; c_j, t^2;]) §.§ Learning the Classifier As the labels are given at video level, we feed them as the training outputs of our network at each spatio-temporal step. A softmax layer is used by the network to predict the action class ŷ among the given class set Y. The prediction of the whole video can be obtained by averaging the prediction scores of all steps.The objective function of our ST-LSTM network is as follows:ℒ = ∑_j=1^J ∑_t=1^T l(ŷ_j,t, y)where l(ŷ_j,t, y) is the negative log-likelihood loss <cit.> that measures the difference between the prediction result ŷ_j,t at step (j,t) and the true label y. The back-propagation through time (BPTT) algorithm <cit.> is often effective for minimizing the objective function for the RNN/LSTM models. As our ST-LSTM model involves both spatial and temporal steps, we adopt a modified version of BPTT for training. The back-propagation runs over spatial and temporal steps simultaneously by starting at the last joint at the last frame. To clarify the error accumulation in this procedure, we use e_j,t^T and e_j,t^S to denote the error back-propagated from step (j,t+1) to (j,t) and the error back-propagated from step (j+1,t) to (j,t), respectively. Then the errors accumulated at step (j,t) can be calculated as e_j,t^T+e_j,t^S. Consequently, before back-propagating the error at each step, we should guarantee both its subsequent joint step and subsequent time step have already been computed. The left-most units in our ST-LSTM network do not have preceding spatial units, as shown in<ref>. To update the cell states of these units in the feed-forward stage, a popular strategy is to input zero values into these nodes to substitute the hidden representations from the preceding nodes. In our implementation, we link the last unit at the last time step to the first unit at the current time step.We call the new connection as last-to-first link. In the tree traversal, the first and last nodes refer to the same joint (root node of the tree), however the last node contains holistic information of the human skeleton in the corresponding frame. Linking the last node to the starting node at the next time step provides the starting node with the whole body structure configuration, rather than initializing it with less effective zero values. Thus, the network has better ability to learn the action patterns in the skeleton sequence.§ EXPERIMENTSThe proposed method is evaluated and empirically analyzed on seven benchmark datasets for which the coordinates of skeletal joints are provided. These datasets are NTU RGB+D, UT-Kinect, SBU Interaction, SYSU-3D, ChaLearn Gesture, MSR Action3D, and Berkeley MHAD.We conduct extensive experiments with different models to verify the effectiveness of individual technical contributions proposed, as follows: (1) “ST-LSTM (Joint Chain)”. In this model, the joints are visited in a simple chain order, as shown in<ref>(a);(2) “ST-LSTM (Tree)”. In this model, the tree traversal scheme illustrated in<ref>(c) is used to take advantage of the tree-based spatial structure of skeletal joints;(3) “ST-LSTM (Tree) + Trust Gate”. This model uses the trust gate to handle the noisy input. The input to every unit of of our network at each spatio-temporal step is the location of the corresponding skeletal joint (i.e., geometric features) at the current time step. We also use two of the datasets (NTU RGB+D dataset and UT-Kinect dataset) as examples to evaluate the performance of our fusion model within the ST-LSTM unit by fusing the geometric and visual features. These two datasets include human-object interactions (such as making a phone call and picking up something) and the visual information around the major joints can be complementary to the geometric features for action recognition. §.§ Evaluation DatasetsNTU RGB+D dataset <cit.> was captured with Kinect (v2). It is currently the largest publicly available dataset for depth-based action recognition, which contains more than 56,000 video sequences and 4 million video frames. The samples in this dataset were collected from 80 distinct viewpoints. A total of 60 action classes (including daily actions, medical conditions, and pair actions) were performed by 40 different persons aged between 10 and 35. This dataset is very challenging due to the large intra-class and viewpoint variations. With a large number of samples, this dataset is highly suitable for deep learning based activity analysis. The parameters learned on this dataset can also be used to initialize the models for smaller datasets to improve and speed up the training process of the network. The 3D coordinates of 25 body joints are provided in this dataset.UT-Kinect dataset <cit.> was captured with a stationary Kinect sensor. It contains 10 action classes.Each action was performed twice by every subject. The 3D locations of 20 skeletal joints are provided. The significant intra-class and viewpoint variations make this dataset very challenging.SBU Interaction dataset <cit.> was collected with Kinect. It contains 8 classes of two-person interactions, and includes 282 skeleton sequences with 6822 frames. Each body skeleton consists of 15 joints. The major challenges of this dataset are: (1) in most interactions, one subject is acting, while the other subject is reacting; and (2) the 3D measurement accuracies of the joint coordinates are low in many sequences.SYSU-3D dataset <cit.> contains 480 sequences and was collected with Kinect. In this dataset, 12 different activities were performed by 40 persons. The 3D coordinates of 20 joints are provided in this dataset. The SYSU-3D dataset is a very challenging benchmark because: (1) the motion patterns are highly similar among different activities, and (2) there are various viewpoints in this dataset.ChaLearn Gesture dataset <cit.> consists of 23 hours of videos captured with Kinect. A total of 20 Italian gestures were performed by 27 different subjects.This dataset contains 955 long-duration videos and has predefined splits of samples as training, validation and testing sets. Each skeleton in this dataset has 20 joints.MSR Action3D dataset <cit.> is widely used for depth-based action recognition. It contains a total of 10 subjects and 20 actions.Each action was performed by the same subject two or three times. Each frame in this dataset contains 20 skeletal joints.Berkeley MHAD dataset <cit.> was collected by using a motion capture network of sensors. It contains 659 sequences and about 82 minutes of recording time. Eleven action classes were performed by five female and seven male subjects. The 3D coordinates of 35 skeletal joints are provided in each frame. §.§ Implementation Details In our experiments, each video sequence is divided to T sub-sequences with the same length, and one frame is randomly selected from each sub-sequence. This sampling strategy has the following advantages: (1) Randomly selecting a frame from each sub-sequence can add variation to the input data, and improves the generalization strengths of our trained network. (2) Assume each sub-sequence contains n frames, so we have n choices to sample a frame from each sub-sequence. Accordingly, for the whole video, we can obtain a total number of n^T sampling combinations. This indicates that the training data can be greatly augmented. We use different frame sampling combinations for each video over different training epochs. This strategy is useful for handling the over-fitting issues, as most datasets have limited numbers of training samples. We observe this strategy achieves better performance in contrast with uniformly sampling frames. We cross-validated the performance based on the leave-one-subject-out protocol on the large scale NTU RGB+D dataset, and found T=20 as the optimum value.We use Torch7 <cit.> as the deep learning platform to perform our experiments. We train the network with stochastic gradient descent, and set the learning rate, momentum, and decay rate to 2×10^-3, 0.9, and 0.95, respectively. We set the unit size d to 128, and the parameter λ used in G(·) to 0.5. Two ST-LSTM layers are used in our stacked network. Although there are variations in terms of joint number, sequence length, and data acquisition equipment for different datasets, we adopt the same parameter settings mentioned above for all datasets. Our method achieves promising results on all the benchmark datasets with these parameter settings untouched, which shows the robustness of our method.An NVIDIA TitanX GPU is used to perform our experiments. We evaluate the computational efficiency of our method on the NTU RGB+D dataset and set the batch size to 100. On average, within one second, 210, 100, and 70 videos can be processed by using “ST-LSTM (Joint Chain)”, “ST-LSTM (Tree)”, and “ST-LSTM (Tree) + Trust Gate”, respectively.§.§ Experiments on the NTU RGB+D DatasetThe NTU RGB+D dataset has two standard evaluation protocols <cit.>. The first protocol is the cross-subject (X-Subject) evaluation protocol, in which half of the subjects are used for training and the remaining subjects are kept for testing. The second is the cross-view (X-View) evaluation protocol, in which 2/3 of the viewpoints are used for training, and 1/3 unseen viewpoints are left out for testing. We evaluate the performance of our method on both of these protocols. The results are shown in<ref>.In<ref>, the deep RNN model concatenates the joint features at each frame and then feeds them to the network to model the temporal kinetics, and ignores the spatial dynamics. As can be seen, both “ST-LSTM (Joint Chain)” and “ST-LSTM (Tree)” models outperform this method by a notable margin. It can also be observed that our approach utilizing the trust gate brings significant performance improvement, because the data provided by Kinect is often noisy and multiple joints are frequently occluded in this dataset. Note that our proposed models (such as “ST-LSTM (Tree) + Trust Gate”) reported in this table only use skeletal data as input.We compare the class specific recognition accuracies of “ST-LSTM (Tree)” and “ST-LSTM (Tree) + Trust Gate”, as shown in<ref>. We observe that “ST-LSTM (Tree) + Trust Gate” significantly outperforms “ST-LSTM (Tree)” for most of the action classes, which demonstrates our proposed trust gate can effectively improve the human action recognition accuracy by learning the degrees of reliability over the input data at each time step.As shown in<ref>, a notable portion of videos in the NTU RGB+D dataset were collected in side views. Due to the design of Kinect's body tracking mechanism, skeletal data is less accurate in side view compared to the front view. To further investigate the effectiveness of the proposed trust gate, we analyze the performance of the network by feeding the side views samples only. The accuracy of “ST-LSTM (Tree)” is 76.5%, while “ST-LSTM (Tree) + Trust Gate” yields 81.6%. This shows how trust gate can effectively deal with the noise in the input data.To verify the performance boost by stacking layers, we limit the depth of the network by using only one ST-LSTM layer, and the accuracies drop to 65.5% and 77.0% based on the cross-subject and cross-view protocol, respectively. This indicates our two-layer stacked network has better representation power than the single-layer network. To evaluate the performance of our feature fusion scheme, we extract visual features from several regions based on the joint positions and use them in addition to the geometric features (3D coordinates of the joints). We extract HOG and HOF <cit.> features from a 80×80 RGB patch centered at each joint location. For each joint, this produces a 300D visual descriptor, and we apply PCA to reduce the dimension to 20. The results are shown in<ref>. We observe that our method using the visual features together with the joint positions improves the performance. Besides, we compare our newly proposed feature fusion strategy within the ST-LSTM unit with two other feature fusion methods: (1) early fusion which simply concatenates two types of features as the input of the ST-LSTM unit; (2) late fusion which uses two ST-LSTMs to deal with two types of features respectively, then concatenates the outputs of the two ST-LSTMs at each step, and feeds the concatenated result to a softmax classifier. We observe that our proposed feature fusion strategy is superior to other baselines.We also evaluate the sensitivity of the proposed network with respect to the variation of neuron unit size and λ values. The results are shown in<ref>. When trust gate is added, our network obtains better performance for all the λ values compared to the network without the trust gate. Finally, we investigate the recognition performance with early stopping conditions by feeding the first p portion of the testing video to the trained network based on the cross-subject protocol (p ∈{0.1, 0.2, ..., 1.0}). The results are shown in<ref>. We can observe that the results are improved when a larger portion of the video is fed to our network. §.§ Experiments on the UT-Kinect DatasetThere are two evaluation protocols for the UT-Kinect dataset in the literature. The first is the leave-one-out-cross-validation (LOOCV) protocol <cit.>. The second protocol is suggested by <cit.>, for which half of the subjects are used for training, and the remaining are used for testing. We evaluate our approach using both protocols on this dataset.Using the LOOCV protocol, our method achieves better performance than other skeleton-based methods, as shown in<ref>. Using the second protocol (see<ref>), our method achieves competitive result (95.0%) to the Elastic functional coding method <cit.> (94.9%), which is an extension of the Lie Group model <cit.>.Some actions in the UT-Kinect dataset involve human-object interactions, thus appearance based features representing visual information of the objects can be complementary to the geometric features. Thus we can evaluate our proposed feature fusion approach within the ST-LSTM unit on this dataset. The results are shown in<ref>. Using geometric features only, the accuracy is 97%. By simply concatenating the geometric and visual features, the accuracy improves slightly. However, the accuracy of our approach can reach 98% when the proposed feature fusion method is adopted. §.§ Experiments on the SBU Interaction DatasetWe follow the standard evaluation protocol in <cit.> and perform 5-fold cross validation on the SBU Interaction dataset. As two human skeletons are provided in each frame of this dataset, our traversal scheme visits the joints throughout the two skeletons over the spatial steps.We report the results in terms of average classification accuracy in<ref>. The methods in <cit.> and <cit.> are both LSTM-based approaches, which are more relevant to our method.The results show that the proposed “ST-LSTM (Tree) + Trust Gate” model outperforms all other skeleton-based methods. “ST-LSTM (Tree)” achieves higher accuracy than “ST-LSTM (Joint Chain)”, as the latter adds some false links between less related joints.Both Co-occurrence LSTM <cit.> and Hierarchical RNN <cit.> adopt the Svaitzky-Golay filter <cit.> in the temporal domain to smooth the skeletal joint positions and reduce the influence of noise in the data collected by Kinect.The proposed “ST-LSTM (Tree)” model without the trust gate mechanism outperforms Hierarchical RNN, and achieves comparable result (88.6%) to Co-occurrence LSTM. When the trust gate is used, the accuracy of our method jumps to 93.3%.§.§ Experiments on the SYSU-3D Dataset We follow the standard evaluation protocol in <cit.> on the SYSU-3D dataset. The samples from 20 subjects are used to train the model parameters, and the samples of the remaining 20 subjects are used for testing. We perform 30-fold cross validation and report the mean accuracy in <ref>.The results in <ref> show that our proposed “ST-LSTM (Tree) + Trust Gate” method outperforms all the baseline methods on this dataset. We can also find that the tree traversal strategy can help to improve the classification accuracy of our model. As the skeletal joints provided by Kinect are noisy in this dataset, the trust gate, which aims at handling noisy data, brings significant performance improvement (about 3% improvement).There are large viewpoint variations in this dataset. To make our model reliable against viewpoint variations, we adopt a similar skeleton normalization procedure as suggested by <cit.> on this dataset. In this preprocessing step, we perform a rotation transformation on each skeleton, such that all the normalized skeletons face to the same direction. Specifically, after rotation, the 3D vector from “right shoulder” to “left shoulder” will be parallel to the X axis, and the vector from “hip center” to “spine” will be aligned to the Y axis (please see <cit.> for more details about the normalization procedure).We evaluate our “ST-LSTM (Tree) + Trust Gate” method by respectively using the original skeletons without rotation and the transformed skeletons, and report the results in <ref>. The results show that it is beneficial to use the transformed skeletons as the input for action recognition. §.§ Experiments on the ChaLearn Gesture DatasetWe follow the evaluation protocol adopted in <cit.> and report the F1-score measures on the validation set of the ChaLearn Gesture dataset.As shown in <ref>,our method surpasses the state-of-the-art methods <cit.>, which demonstrates the effectiveness of our method in dealing with skeleton-based action recognition problem.Compared to other methods, our method focuses on modeling both temporal and spatial dependency patterns in skeleton sequences. Moreover, the proposed trust gate is also incorporated to our method to handle the noisy skeleton data captured by Kinect, which can further improve the results. §.§ Experiments on the MSR Action3D Dataset We follow the experimental protocol in <cit.> on the MSR Action3D dataset, and show the results in <ref>.On the MSR Action3D dataset, our proposed method, “ST-LSTM (Tree) + Trust Gate”, achieves 94.8% of classification accuracy, which is superior to the Hierarchical RNN model <cit.> and other baseline methods. §.§ Experiments on the Berkeley MHAD DatasetWe adopt the experimental protocol in <cit.> on the Berkeley MHAD dataset. 384 video sequences corresponding to the first seven persons are used for training, and the 275 sequences of the remaining five persons are held out for testing. The experimental results in<ref> show that our method achieves very high accuracy (100%) on this dataset. Unlike <cit.> and <cit.>, our method does not use any preliminary manual smoothing procedures.§.§ Visualization of Trust GatesIn this section, to better investigate the effectiveness of the proposed trust gate scheme, we study the behavior of the proposed framework against the presence of noise in skeletal data from the MSR Action3D dataset. We manually rectify some noisy joints of the samples by referring to the corresponding depth images. We then compare the activations of trust gates on the noisy and rectified inputs. As illustrated in<ref>(a), the magnitude of trust gate's output (l_2 norm of the activations of the trust gate) is smaller when a noisy joint is fed, compared to the corresponding rectified joint. This demonstrates how the network controls the impact of noisy input on its stored representation of the observed data.In our next experiment, we manually add noise to one joint for all testing samples on the Berkeley MHAD dataset, in order to further analyze the behavior of our proposed trust gate. Note that the Berkeley MHAD dataset was collected with motion capture system, thus the skeletal joint coordinates in this dataset are much more accurate than those captured with Kinect sensors. We add noise to the right foot joint by moving the joint away from its original location. The direction of the translation vector is randomly chosen and the norm is a random value around 30cm, which is a significant noise in the scale of human body.We measure the difference in the magnitudes of trust gates' activations between the noisy data and the original ones. For all testing samples, we carry out the same operations and then calculate the average difference. The results in<ref>(b) show that the magnitude of trust gate is reduced when the noisy data is fed to the network. This shows that our network tries to block the flow of noisy input and stop it from affecting the memory. We also observe that the overall accuracy of our network does not drop after adding the above-mentioned noise to the input data. §.§ Evaluation of Different Spatial Joint Sequence ModelsThe previous experiments showed how “ST-LSTM (Tree)” outperforms “ST-LSTM (Joint Chain)”, because “ST-LSTM (Tree)” models the kinematic dependency structures of human skeletal sequences. In this section, we further analyze the effectiveness of our “ST-LSTM (Tree)” model and compare it with a “ST-LSTM (Double Joint Chain)” model.The “ST-LSTM (Joint Chain)” has fewer steps in the spatial dimension than the “ST-LSTM (Tree)”. One question that may rise here is if the advantage of “ST-LSTM (Tree)” model could be only due to the higher length and redundant sequence of the joints fed to the network, and not because of the proposed semantic relations between the joints. To answer this question, we evaluate the effect of using a double chain scheme to increase the spatial steps of the “ST-LSTM (Joint Chain)” model. Specifically, we use the joint visiting order of 1-2-3-...-16-1-2-3-...-16, and we call this model as “ST-LSTM (Double Joint Chain)”. The results in <ref> show that the performance of “ST-LSTM (Double Joint Chain)” is better than “ST-LSTM (Joint Chain)”, yet inferior to “ST-LSTM (Tree)”. This experiment indicates that it is beneficial to introduce more passes in the spatial dimension to the ST-LSTM for performance improvement. A possible explanation is that the units visited in the second round can obtain the global level context representation from the previous pass, thus they can generate better representations of the action patterns by using the context information.However, the performance of “ST-LSTM (Double Joint Chain)” is still weaker than “ST-LSTM (Tree)”, though the numbers of their spatial steps are almost equal.The proposed tree traversal scheme is superior because it connects the most semantically related joints and avoids false connections between the less-related joints (unlike the other two compared models). §.§ Evaluation of Temporal Average, LSTM and ST-LSTMTo further investigate the effect of simultaneous modeling of dependencies in spatial and temporal domains, in this experiment, we replace our ST-LSTM with the original LSTM which only models the temporal dynamics among the frames without explicitly considering spatial dependencies.We report the results of this experiment in<ref>. As can be seen, our “ST-LSTM + Trust Gate” significantly outperforms “LSTM + Trust Gate”. This demonstrates that the proposed modeling of the dependencies in both temporal and spatial dimensions provides much richer representations than the original LSTM.The second observation of this experiment is that if we add our trust gate to the original LSTM, the performance of LSTM can also be improved, but its performance gain is less than the performance gain on ST-LSTM. A possible explanation is that we have both spatial and temporal context information at each step of ST-LSTM to generate a good prediction of the input at the current step ((see Eq. (<ref>)), thus our trust gate can achieve a good estimation of the reliability of the input at each step by using the prediction (see Eq. (<ref>)). However, in the original LSTM, the available context at each step is from the previous temporal step, i.e., the prediction can only be based on the context in the temporal dimension, thus the effectiveness of the trust gate is limited when it is added to the original LSTM. This further demonstrates the effectiveness of our ST-LSTM framework for spatio-temporal modeling of the skeleton sequences.In addition, we investigate the effectiveness of the LSTM structure for handling the sequential data. We evaluate a baseline method (called “Temporal Average”) by averaging the features from all frames instead of using LSTM. Specifically, the geometric features are averaged over all the frames of the input sequence (i.e., the temporal ordering information in the sequence is ignored), and then the resultant averaged feature is fed to a two-layer network, followed by a softmax classifier. The performance of this scheme is much weaker than our proposed ST-LSTM with trust gate, and also weaker than the original LSTM, as shown in <ref>. The results demonstrate the representation strengths of the LSTM networks for modeling the dependencies and dynamics in sequential data, when compared to traditional temporal aggregation methods of input sequences. §.§ Evaluation of the Last-to-first Link SchemeIn this section, we evaluate the effectiveness of the last-to-first link in our model (see section <ref>). The results in<ref> show the advantages of using the last-to-first link in improving the final action recognition performance. § CONCLUSION In this paper, we have extended the RNN-based action recognition method to both spatial and temporal domains. Specifically, we have proposed a novel ST-LSTM network which analyzes the 3D locations of skeletal joints at each frame and at each processing step.A skeleton tree traversal method based on the adjacency graph of body joints is also proposed to better represent the structure of the input sequences and to improve the performance of our network by connecting the most related joints together in the input sequence.In addition, a new gating mechanism is introduced to improve the robustness of our network against the noise in input sequences. A multi-modal feature fusion method is also proposed for our ST-LSTM framework. The experimental results have validated the contributions and demonstrated the effectiveness of our approach which achieves better performance over the existing state-of-the-art methods on seven challenging benchmark datasets.§ ACKNOWLEDGEMENT This work was carried out at Rapid-Rich Object Search (ROSE) Lab, Nanyang Technological University. ROSE Lab is supported by the National Research Foundation, Singapore, under its IDM Strategic Research Programme.IEEEtran 10 url@samestyle zhu2016handcrafted F. Zhu, L. Shao, J. Xie, and Y. Fang, “From handcrafted to learned representations for human action recognition: a survey,” Image and Vision Computing, 2016. presti20163d L. L. Presti and M. La Cascia, “3d skeleton-based human action classification: A survey,” Pattern Recognition, 2016. han2016review F. Han, B. Reily, W. Hoff, and H. Zhang, “Space-time representation of people based on 3d skeletal data: a review,” arXiv, 2016. HOJ3D L. Xia, C. Chen, and J. Aggarwal, “View invariant human action recognition using histograms of 3d joints,” in CVPRW, 2012. eigenjointsJournal X. Yang and Y. Tian, “Effective 3d action recognition using eigenjoints,” Journal of Visual Communication and Image Representation, 2014. vemulapalli2014liegroup R. Vemulapalli, F. Arrate, and R. Chellappa, “Human action recognition by representing 3d skeletons as points in a lie group,” in CVPR, 2014. skeletalQuads G. Evangelidis, G. Singh, and R. Horaud, “Skeletal quads: Human action recognition using joint quadruples,” in ICPR, 2014. hog2-ohnbar E. Ohn-Bar and M. Trivedi, “Joint angles similarities and hog^2 for action recognition,” in CVPRW, 2013. Luo_2013_ICCV J. Luo, W. Wang, and H. Qi, “Group sparsity and geometry constrained dictionary learning for action recognition from depth maps,” in ICCV, 2013. graves2013speechICASSP A. Graves, A.-r. Mohamed, and G. Hinton, “Speech recognition with deep recurrent neural networks,” in ICASSP, 2013. sutskever2014sequence I. Sutskever, O. Vinyals, and Q. V. Le, “Sequence to sequence learning with neural networks,” in NIPS, 2014. mikolov2011extensions T. Mikolov, S. Kombrink, L. Burget, J. H. Černockỳ, and S. Khudanpur, “Extensions of recurrent neural network language model,” in ICASSP, 2011. sundermeyer2012lstm M. Sundermeyer, R. Schlüter, and H. Ney, “Lstm neural networks for language modeling,” in INTERSPEECH, 2012. mesnil2013investigation G. Mesnil, X. He, L. Deng, and Y. Bengio, “Investigation of recurrent-neural-network architectures and learning methods for spoken language understanding.” in INTERSPEECH, 2013. vinyals2015show O. Vinyals, A. Toshev, S. Bengio, and D. Erhan, “Show and tell: A neural image caption generator,” in CVPR, 2015. xu2015show K. Xu, J. Ba, R. Kiros, K. Cho, A. Courville, R. Salakhudinov, R. Zemel, and Y. Bengio, “Show, attend and tell: Neural image caption generation with visual attention,” in ICML, 2015. srivastava2015unsupervised N. Srivastava, E. Mansimov, and R. Salakhudinov, “Unsupervised learning of video representations using lstms,” in ICML, 2015. Singh_2016_CVPR B. Singh, T. K. Marks, M. Jones, O. Tuzel, and M. Shao, “A multi-stream bi-directional recurrent neural network for fine-grained action detection,” in CVPR, 2016. Jain_2016_CVPR A. Jain, A. R. Zamir, S. Savarese, and A. Saxena, “Structural-rnn: Deep learning on spatio-temporal graphs,” in CVPR, 2016. Alahi_2016_CVPR A. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and S. Savarese, “Social lstm: Human trajectory prediction in crowded spaces,” in CVPR, 2016. Deng_2016_CVPR Z. Deng, A. Vahdat, H. Hu, and G. Mori, “Structure inference machines: Recurrent neural networks for analyzing relations in group activity recognition,” in CVPR, 2016. Ibrahim_2016_CVPR M. S. Ibrahim, S. Muralidharan, Z. Deng, A. Vahdat, and G. Mori, “A hierarchical deep temporal model for group activity recognition,” in CVPR, 2016. Ma_2016_CVPR S. Ma, L. Sigal, and S. Sclaroff, “Learning activity progression in lstms for activity detection and early detection,” in CVPR, 2016. Ni_2016_CVPR B. Ni, X. Yang, and S. Gao, “Progressively parsing interactional objects for fine grained action detection,” in CVPR, 2016. li2016online Y. Li, C. Lan, J. Xing, W. Zeng, C. Yuan, and J. Liu, “Online human action detection using joint classification-regression recurrent neural networks,” arXiv, 2016. yue2015beyond J. Yue-Hei Ng, M. Hausknecht, S. Vijayanarasimhan, O. Vinyals, R. Monga, and G. Toderici, “Beyond short snippets: Deep networks for video classification,” in CVPR, 2015. donahue2015long J. Donahue, L. Anne Hendricks, S. Guadarrama, M. Rohrbach, S. Venugopalan, K. Saenko, and T. Darrell, “Long-term recurrent convolutional networks for visual recognition and description,” in CVPR, 2015. li2016action Q. Li, Z. Qiu, T. Yao, T. Mei, Y. Rui, and J. Luo, “Action recognition by learning deep multi-granular spatio-temporal video representation,” in ICMR, 2016. wu2015ACMMM Z. Wu, X. Wang, Y.-G. Jiang, H. Ye, and X. Xue, “Modeling spatial-temporal clues in a hybrid deep learning framework for video classification,” in ACM MM, 2015. du2015hierarchical Y. Du, W. Wang, and L. Wang, “Hierarchical recurrent neural network for skeleton based action recognition,” in CVPR, 2015. veeriah2015differential V. Veeriah, N. Zhuang, and G.-J. Qi, “Differential recurrent neural networks for action recognition,” in ICCV, 2015. nturgbd A. Shahroudy, J. Liu, T.-T. Ng, and G. Wang, “Ntu rgb+d: A large scale dataset for 3d human activity analysis,” in CVPR, 2016. 7284883 M. Meng, H. Drira, M. Daoudi, and J. Boonaert, “Human-object interaction recognition by learning the distances between the object and the skeleton joints,” in FG, 2015. actionletPAMI J. Wang, Z. Liu, Y. Wu, and J. Yuan, “Learning actionlet ensemble for 3d human action recognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2014. MMMP_PAMI A. Shahroudy, T. T. Ng, Q. Yang, and G. Wang, “Multimodal multipart learning for action recognition in depth videos,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2016. MMTW J. Wang and Y. Wu, “Learning maximum margin temporal warping for action recognition,” in ICCV, 2013. Vemulapalli_2016_CVPR R. Vemulapalli and R. Chellapa, “Rolling rotations for recognizing human actions from 3d skeletal data,” in CVPR, 2016. rahmani2014real H. Rahmani, A. Mahmood, D. Q. Huynh, and A. Mian, “Real time action recognition using histograms of depth gradients and random decision forests,” in WACV, 2014. shahroudy2014multi A. Shahroudy, G. Wang, and T.-T. Ng, “Multi-modal feature fusion for action recognition in rgb-d sequences,” in ISCCSP, 2014. rahmani2015learning H. Rahmani and A. Mian, “Learning a non-linear knowledge transfer model for cross-view action recognition,” in CVPR, 2015. lillo2014discriminative I. Lillo, A. Soto, and J. Carlos Niebles, “Discriminative hierarchical modeling of spatio-temporally composable human activities,” in CVPR, 2014. jhuang2013towards H. Jhuang, J. Gall, S. Zuffi, C. Schmid, and M. J. Black, “Towards understanding action recognition,” in ICCV, 2013. chen_2016_icassp C. Chen, R. Jafari, and N. Kehtarnavaz, “Fusion of depth, skeleton, and inertial data for human action recognition,” in ICASSP, 2016. liu2016IVC Z. Liu, C. Zhang, and Y. Tian, “3d-based deep convolutional neural network for action recognition with depth sequences,” Image and Vision Computing, 2016. cai2016TMM X. Cai, W. Zhou, L. Wu, J. Luo, and H. Li, “Effective active skeleton representation for low latency human action recognition,” IEEE Transactions on Multimedia, 2016. al2016PRL A. S. Al Alwani and Y. Chahir, “Spatiotemporal representation of 3d skeleton joints-based action recognition using modified spherical harmonics,” Pattern Recognition Letters, 2016. Tao_2015_ICCV_Workshops L. Tao and R. Vidal, “Moving poselets: A discriminative and interpretable skeletal motion representation for action recognition,” in ICCVW, 2015. zhu2016co W. Zhu, C. Lan, J. Xing, W. Zeng, Y. Li, L. Shen, and X. Xie, “Co-occurrence feature learning for skeleton based action recognition using regularized deep lstm networks,” in AAAI, 2016. dropout N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: A simple way to prevent neural networks from overfitting,” Journal of Machine Learning Research, 2014. harvey2015semi F. G. Harvey and C. Pal, “Semi-supervised learning with encoder-decoder recurrent neural networks: Experiments with motion capture sequences,” arXiv, 2016. mahasseni2016regularizing B. Mahasseni and S. Todorovic, “Regularizing long short term memory with 3d human-skeleton sequences for action recognition,” in CVPR, 2016. liu2016spatio J. Liu, A. Shahroudy, D. Xu, and G. Wang, “Spatio-temporal lstm with trust gates for 3d human action recognition,” in ECCV, 2016. lstm S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural Computation, 1997. zou2009automatic B. Zou, S. Chen, C. Shi, and U. M. Providence, “Automatic reconstruction of 3d human motion pose from uncalibrated monocular video sequences based on markerless human motion tracking,” Pattern Recognition, 2009. yang2011articulated Y. Yang and D. Ramanan, “Articulated pose estimation with flexible mixtures-of-parts,” in CVPR, 2011. dalal2006human N. Dalal, B. Triggs, and C. Schmid, “Human detection using oriented histograms of flow and appearance,” in ECCV, 2006. wang2011action H. Wang, A. Kläser, C. Schmid, and C.-L. Liu, “Action recognition by dense trajectories,” in CVPR, 2011. simonyan2014very K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” in ICLR, 2015. cheron2015p G. Chéron, I. Laptev, and C. Schmid, “P-cnn: Pose-based cnn features for action recognition,” in ICCV, 2015. graves2012supervised A. Graves, Supervised Sequence Labelling with Recurrent Neural Networks.1em plus 0.5em minus 0.4emSpringer, 2012. yun2012two K. Yun, J. Honorio, D. Chattopadhyay, T. L. Berg, and D. Samaras, “Two-person interaction detection using body-pose features and multiple instance learning,” in CVPRW, 2012. jianfang_CVPR15 J.-F. Hu, W. Zheng, J. Lai, and J. Zhang, “Jointly learning heterogeneous features for rgb-d activity recognition,” in CVPR, 2015. escalera2013multi S. Escalera, J. Gonzàlez, X. Baró, M. Reyes, O. Lopes, I. Guyon, V. Athitsos, and H. Escalante, “Multi-modal gesture recognition challenge 2013: Dataset and results,” in ICMI, 2013. li2010action W. Li, Z. Zhang, and Z. Liu, “Action recognition based on a bag of 3d points,” in CVPRW, 2010. ofli2013berkeley F. Ofli, R. Chaudhry, G. Kurillo, R. Vidal, and R. Bajcsy, “Berkeley mhad: A comprehensive multimodal human action database,” in WACV, 2013. collobert2011torch7 R. Collobert, K. Kavukcuoglu, and C. Farabet, “Torch7: A matlab-like environment for machine learning,” in NIPS Workshop, 2011. cippitelli2016evaluation E. Cippitelli, E. Gambi, S. Spinsante, and F. Flórez-Revuelta, “Evaluation of a skeleton-based method for human activity recognition on a large-scale rgb-d dataset,” in TechAAL, 2016. rahmani20163d H. Rahmani and A. Mian, “3d action recognition from novel viewpoints,” in CVPR, 2016. zhu2013fusing Y. Zhu, W. Chen, and G. Guo, “Fusing spatiotemporal features and joints for 3d action recognition,” in CVPRW, 2013. anirudh2015elastic R. Anirudh, P. Turaga, J. Su, and A. Srivastava, “Elastic functional coding of human actions: from vector-fields to latent variables,” in CVPR, 2015. slama2015accurate R. Slama, H. Wannous, M. Daoudi, and A. Srivastava, “Accurate 3d action recognition using learning on the grassmann manifold,” Pattern Recognition, 2015. jetley20143d S. Jetley and F. Cuzzolin, “3d activity recognition using motion history and binary shape templates,” in ACCVW, 2014. devanne2013space M. Devanne, H. Wannous, S. Berretti, P. Pala, M. Daoudi, and A. Del Bimbo, “Space-time pose representation for 3d human action recognition,” in ICIAP, 2013. devanne20153d M. Devanne, H. Wannous, S. Berretti, P. Pala, M. Daoudi, and A. Bimbo, “3-d human action recognition by shape analysis of motion trajectories on riemannian manifold,” IEEE Transactions on Cybernetics, 2015. jiang2015informative M. Jiang, J. Kong, G. Bebis, and H. Huo, “Informative joints based human action recognition using skeleton contexts,” Signal Processing: Image Communication, 2015. chrungoo2014activity A. Chrungoo, S. Manimaran, and B. Ravindran, “Activity recognition for natural human robot interaction,” in ICSR, 2014. Wang_2016_CVPR_Mining C. Wang, Y. Wang, and A. L. Yuille, “Mining 3d key-pose-motifs for action recognition,” in CVPR, 2016. ji2014interactive Y. Ji, G. Ye, and H. Cheng, “Interactive body part contrast mining for human interaction recognition,” in ICMEW, 2014. li2015category W. Li, L. Wen, M. Choo Chuah, and S. Lyu, “Category-blind human action recognition: a practical recognition system,” in ICCV, 2015. savitzky1964smoothing A. Savitzky and M. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Analytical chemistry, 1964. hu2016ECCV J.-F. Hu, W.-S. Zheng, L. Ma, G. Wang, and J. Lai, “Real-time rgb-d activity prediction by soft regression,” in ECCV, 2016. wang2015hierarchical H. Wang, W. Wang, and L. Wang, “Hierarchical motion evolution for action recognition,” in ACPR, 2015. fernando2015modeling B. Fernando, E. Gavves, J. Oramas, A. Ghodrati, and T. Tuytelaars, “Modeling video evolution for action recognition,” in CVPR, 2015. yao2014gesture A. Yao, L. Van Gool, and P. Kohli, “Gesture recognition portfolios for personalization,” in CVPR, 2014. wu2013fusing J. Wu, J. Cheng, C. Zhao, and H. Lu, “Fusing multi-modal features for gesture recognition,” in ICMI, 2013. pfister2014domain T. Pfister, J. Charles, and A. Zisserman, “Domain-adaptive discriminative one-shot learning of gestures,” in ECCV, 2014. gowayyed2013histogram M. A. Gowayyed, M. Torki, M. E. Hussein, and M. El-Saban, “Histogram of oriented displacements (hod): Describing trajectories of human joints for action recognition,” in IJCAI, 2013. lillo2016hierarchical I. Lillo, J. Carlos Niebles, and A. Soto, “A hierarchical pose-based approach to complex action understanding using dictionaries of actionlets and motion poselets,” in CVPR, 2016. Ofli2014jvci F. Ofli, R. Chaudhry, G. Kurillo, R. Vidal, and R. Bajcsy, “Sequence of the most informative joints (smij): A new representation for human skeletal action recognition,” Journal of Visual Communication and Image Representation, 2014. vantigodi2013real S. Vantigodi and R. V. Babu, “Real-time human action recognition from motion capture data,” in NCVPRIPG, 2013. vantigodi2014action S. Vantigodi and V. B. Radhakrishnan, “Action recognition from motion capture data using meta-cognitive rbf network classifier,” in ISSNIP, 2014. kapsouras2014action I. Kapsouras and N. Nikolaidis, “Action recognition on motion capture data using a dynemes and forward differences representation,” Journal of Visual Communication and Image Representation, 2014. | http://arxiv.org/abs/1706.08276v1 | {
"authors": [
"Jun Liu",
"Amir Shahroudy",
"Dong Xu",
"Alex C. Kot",
"Gang Wang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170626083545",
"title": "Skeleton-Based Action Recognition Using Spatio-Temporal LSTM Network with Trust Gates"
} |
=116.2 cm 22.75 cm -1.25 cm -0.0 cmA B C D E S F G H I J K L M N O P Q R V W T tr α̇hep-ph/***DFPD-2017/TH/091.5cmbold Are neutrino masses modular forms? normal .3cm 0.5cm Ferruccio Feruglio.7cmDipartimento di Fisica e Astronomia `G. Galilei', Università di Padova INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy .1cm.5cm [l].9 E-mail: [email protected] 1cmWe explore a new class of supersymmetric models for lepton masses and mixing angles where the role of flavour symmetry is played by modular invariance. The building blocks are modular forms of level N and matter supermultiplets, both transforming in representations of a finite discrete group Γ_N.In the simplest version of these models, Yukawa couplings are just modular forms and the only source of flavour symmetry breaking is the vacuum expectation value of a single complex field, the modulus. In the special case where modular forms are constant functions the whole construction collapses to a supersymmetric flavour model invariant under Γ_N, the case treated so far in the literature. The framework has a number of appealing features. Flavon fields other than the modulus might not be needed.Neutrino masses andmixing angles are simultaneously constrained by the modular symmetry. As long as supersymmetry is exact, modular invariance determines all higher-dimensional operators in the superpotential.We discuss the general framework and we provide complete examplesof the new construction. The most economical model predicts neutrino mass ratios, lepton mixing angles, Dirac and Majorana phases uniquely in terms of the modulus vacuum expectation value, with all the parameters except one within the experimentally allowed range.As a byproduct of the general formalism we extend the notion of non-linearly realised symmetries to the discrete case.emptyParma, circa 2006 2.cm.5cmA contribution to: “From my vast repertoire:the legacy of Guido Altarelli"S. Forte, A. Levy and G. Ridolfi, eds.§ INTRODUCTIONA considerable interest in discrete flavour symmetries <cit.> has been fostered by early models of quark masses and mixing angles <cit.> and, more recently, by the discovery of neutrino oscillations. Early data were well-compatible with a highly symmetric lepton mixing pattern, the tri-bimaximal one <cit.>, which could be derived from small non-abelian discrete symmetry groups such asA_4<cit.>. Other discrete groups like S_4 and A_5 produced interesting alternative mixing patterns, which could be adopted as zeroth-order approximation to the data. Today this approach is facing several difficulties. The formidable recent experimental progress has sharpened the neutrino oscillation parameters, revealing many details that require a precise description, such as the non-vanishing value of the reactor angle, the deviation of the atmospheric angle from the maximal value and a non-trivial Dirac CP-violating phase. Inclusion of these features in a realistic model based on discrete symmetries requires departure from minimality. Large corrections to the zeroth-order approximation can be introduced at the price of spoiling predictability, due to the ignorance about the non-negligible higher-order contributions. Alternatively, groups of large dimensionality can be invoked to correctly fit the data <cit.>. Discrete flavour symmetries can also be combined with CP invariance in predictive models <cit.>. Apart from the loss of minimality, there are several drawbacks in this program. The breaking of flavour symmetries typically relies on a generous set of scalar multiplets, the so-called flavons,and the Yukawa interactions generally include non-renormalizable operators with flavon insertions. Higher-dimensional operators with multiple flavon insertions come with unknown coefficients that affect the model predictions. Moreover the flavon energy density has to be cleverly designed to get the correct vacuum alignment. The approach is mainly focused onlepton mixing angles while neutrino masses are reproduced by tuning the available parameters. Finally, it is not straightforward to extend the construction to the quark sector that seems not to like discrete symmetries. In view of these disadvantages, anarchy <cit.>and its generalizations have gained considerable momentum. Anarchy in the neutrino sector can arise in a variety of different frameworks providing acommon description to both quark and lepton mass/mixing parameters,also in the context of grand unifiedtheories <cit.>. However in the anarchy paradigm the observed lepton mixing angles are regarded as environmental quantities <cit.> and cannot be accurately predicted. For their intrinsic nature models based on anarchy essentially escape experimental tests aiming at an accuracythat matches the experimental precision.In this wavering between order and anarchy we feel encouraged to investigate new directions. Aim of the present work is to explore a new class of models generalizing the current approach based on discrete symmetry groups.These models are required to be invariant under transformations of the modular group, acting on the complex modulus τ (Im(τ)>0) as linear fractional transformations:τ→a τ+b/c τ+d , (a,b,c,d integers , ad-bc=1) . In a supersymmetric theory these transformations naturally induce transformations of the matter multiplets according to representations of Γ_N,the so-called finite modular groups. Moreover there are holomorphic combinations of the modulus τ, themodular forms of level N, that also transform in representations of Γ_N. We can thus exploit group theoretical properties of Γ_N to build supersymmetric, modular-invariant models of fermion masses, where matter fields and modular forms are used as building blocks. A particular case of modular forms is that of constant functions. In this case the whole construction collapses to the casethat have been considered in the literature so far when dealing with discrete flavour symmetries. On the other hand a new, presently unexplored, class of models arises when modular forms carry a non-trivial dependence on τ. For small N the finite modular groups are permutation groups and the new models are generalizations of well-known existing models. In this paper we illustrate how the new construction can be realized, by discussing the general formalism and by illustrating it with simple examples. In the minimal examples, see section 3.1.4, there are no flavonsand the only source of flavour symmetry breaking is the vacuum expectation value (VEV) of the modulus τ. A truly remarkable feature of these examples is that all higher-dimensional operators in the superpotential are unambiguously determined in the limit of unbroken supersymmetry. The possible corrections to the theory predictions can only come from the Kahler potential or from supersymmetry breaking contributions. Moreover the modular symmetry constrains lepton mixing angles and neutrino masses at the same time.In the most economical examples, discussed in section 3.1.2 and 3.1.3, neutrino mass ratios, lepton mixing angles, Dirac and Majorana phasesdo not depend on any Lagrangian parameter, but only on the modulus VEV. In the non-trivial task of reproducing five known observables in terms of a single complex parameter, the model of section 3.1.2 scores relatively well: only sin^2θ_13, predicted to be about 0.045, is out of the allowed experimental range. The model predicts inverted neutrino mass ordering and determines completely the Majorana phases. The model of section 3.1.3 predicts normal neutrino mass ordering and is almost equally well-performing.An unexpected byproduct of the new framework is a new concept of non-linearly realized symmetry, applied to discrete groups. This is a built-in feature of modular forms of level N but, in the second part of this work, we will analyze it in a more general framework. In the context of relativistic quantum field theories, non-linear realizations of a continuous symmetry are synonymous of symmetry breaking and ofthe systematic low-energy expansion describing the massless modes predicted by the Goldstone theorem. The applications of these concepts are ubiquitous,ranging from the description of fundamental particle interactions to the theory of phase transitions.Extension of this formalism to the case of discrete symmetries may look unmanageable, especially when thinking to the Callan-Coleman-Wess-Zumino construction <cit.>,which is tightly connected to the Goldstone theorem.Moving from the properties of level-3 modular forms, we show how non-linear realizations of discrete symmetries can be defined and how they are related to the symmetry breaking pattern ofa discrete group. In the continuous case non-linear realizations can be defined also by imposing invariant constraints on the field space, such as for instance φ^T φ=M^2, the multiplet φ transforming in the fundamental of SO(N).In the discrete case this approach benefits from an important generalization. While a (connected, semisimple) Lie group has only the trivial one-dimensional representation, a discrete group can possess a number of non-invariant singlets χ_i. Therefore the invariant conditions χ_i=0 have no counterpart in(connected, semisimple) Lie groups. If the product of a set of multiplets φ contains one ore more non-invariant singlets χ_i(φ), the conditionsχ_i(φ)=0 may define a non-linear realization of the discrete group. This is precisely what happens with modular forms of level 3. Non-linear realizations of a discrete symmetry do not define low-energy effectivetheories. They should rather be view as consistent truncations of some ultraviolet completion. As we will see, in general, the requirementχ_i(φ)=0 leaves no non-trivialresidual symmetry. We identify the cases in which a non-trivial residual symmetry group survives and we discuss the relation of such event with the properties of the so-called orbit space, the space spanned by the invariants of the group.This last discussion is only very tentative,but we hope that in the future it may be better clarified, offering further possibilities in model building.§ MODULAR FORMS AND MODULAR INVARIANT THEORIESIn this section we briefly recap definitions and properties of modular forms. We also review the framework of modular invariant supersymmetric theories. The material of this Section is quite known, but will be presented here from a different perspective, emphasizing the role of finite discrete symmetries in modular invariant theories.Modular invariance has a long history both in string and in field theories. Target space modular invariance has been investigated soon after the discovery that the spectrum of a closed string,when compactified on a circle of radius R, is invariant under the duality transformation R→ 1/2R, which corresponds to the linear fractional transformation τ→ -1/τ restricted tothe imaginary part of τ (τ=2 i R^2). Modular invariance controls orbifold compactifications of the heterotic string and requires that the couplings among twisted states are modular forms <cit.>.This holds in particular for realistic Yukawa couplings <cit.>. In orientifold compactifications of Type II strings the Yukawa couplings are functions with specific modular properties <cit.>. A similar feature is found also in magnetised extra dimensions <cit.>. Modular invariant supersymmetric field theories have been analyzed in the late 80s <cit.>, both for global and local supersymmetry. Modular invariance in field theory constructions has been invoked while addressing several aspects of the flavour problem in model building <cit.>. Duality and modular invariance have beensuggested as underlying properties of the quantum Hall effect <cit.>. §.§ Modular forms of level NWe consider the series of groups Γ(N)(N=1,2,3,....) defined by: Γ(N)={([ a b; c d ])∈ SL(2,Z), ([ a b; c d ])=([ 1 0; 0 1 ]) ( mod N)} ,where Γ≡ SL(2,Z) is the group of two by two matrices with integer entries and determinant equal to one, also called homogeneous modular group. We have Γ=Γ(1) and the groups Γ(N)(N≥ 2)are infinite normal subgroups of Γ, called principal congruence subgroups. The group Γ(N) acts on the complex variable τ, varying in the upper-half complex plane H= Im(τ)>0, as the linear fractional transformation γτ=a τ+b/c τ+d .The quotient space H/Γ(N) can be compactified by adding special points called cusps. They coincide with i∞ or with rational real numbers. While it is very convenient to deal with two by two matrices, the groups Γ(N) of linear fractional transformations are slightly different from the groups Γ(N). Since γ and -γ induce the same linear fractional transformation, for N=1 and N=2, the transformations (<ref>) are in a one-to-one correspondence with the elements of the group Γ≡Γ/{±1} and Γ(2)≡Γ(2)/{±1}, respectively.Γ is called inomhogeneous modular group, or simply modular group. When N>2 the element -1 does not belong to Γ(N) and we have Γ(N)≡Γ(N). The group Γ is generated by two elements S and T satisfying:S^2=1 , (ST)^3=1 .Theycan be represented bythe SL(2,Z) matrices:S=( [01; -10 ]) T=( [ 1 1; 0 1 ]) ,corresponding to the transformationsS: τ→ -1/τ , T: τ→τ+1 .The quotient groupsΓ_N≡Γ/Γ(N) are called finite modular groups. Some of their properties can be found in ref. <cit.>.Modular forms f(τ) of weight 2k and level N are holomorphic functions of the complex variable τwith well-defined transformation properties under the group Γ(N)<cit.>:f(γτ)=(c τ+d)^2k f(τ) γ=([ a b; c d ])∈Γ(N) ,where k≥ 0 is an integer [Following <cit.>, here we only consider modular forms of even weight.]. The function f(τ) is required to be holomorphic in H and at all the cusps. In the special case when it vanishes at all the cusp, f(τ) is called a cusp form. For Γ(N)(N≥ 2), N is the level of the group and T^N belongs to Γ(N). It follows that f(τ+N)=f(τ) and we have the Fourier expansion (or simply q-expansion):f(τ)=∑_i=0^∞ a_n q_N^n q_N=e^i 2πτ/N .For k<0 there are no modular forms. If k=0 the only possible modular form is a constant. Modular forms of weight 2k and level N form a linear space M_2k(Γ(N)) of finite dimension d_2k(Γ(N)). The dimensions d_2k(Γ(N)) for few levels N are shown in table 1, derivedfrom ref. <cit.>. General formulas are reported in Appendix A.The product of two modular forms of level N and weights 2k, 2k' is a modular form of level N and weight 2(k+k') and the set M(Γ(N)) of all modular forms of level N is a ringM(Γ(N))=⊕_k=0^∞ M_2k(Γ(N)) ,generated by few elements. For instance M(Γ) is generated by two modular forms E_4(τ) and E_6(τ) of weight 4 and 6 respectively, so that each modular form in M_2k(Γ) can be written as a polynomial ∑_ij c_ij E_4(τ)^n_i E_6(τ)^n_j, with powers satisfying 2k=4 n_i+6 n_j.Central to the construction illustrated in the next Sections is the following result, explicitly proved in Appendix B. Modular forms of weight 2k and level N≥ 2 are invariant, up to the factor (cτ+d)^2k, under Γ(N) but they transform under the quotient group Γ_N≡Γ/Γ(N). As show in Appendix B, it is possible to choose a basis in M_2k(Γ(N)) such that this transformation is described by a unitary representation ρ of Γ_N:f_i(γτ)=(c τ+d)^k ρ(γ)_ijf_j(τ) ,where γ=([ a b; c d ])stands for a representative element of Γ_N. In practise, it is sufficient to determine the representation ρ for the two elements S and T that generate the entire modular group and, for N≥ 2, are not contained in Γ(N). While the groups Γ(N) are infinite, the quotient groups Γ_N are finite. In table <ref> we show few groups Γ_N and their dimensions. §.§ Modular-invariant supersymmetric theoriesHere we briefly review the formalism <cit.>, starting from the case of N=1 global supersymmetry, where the action takes the general form[We focus on Yukawa interactions and we turn off the gauge interactions.]:S=∫ d^4 x d^2θ d^2θ̅ K(Φ,Φ̅)+∫ d^4 x d^2θ w(Φ)+h.c.where K(Φ,Φ̅), the Kahler potential, is a real gauge-invariant function of the chiral superfields Φ and their conjugates and w(Φ), the superpotential, is a holomorphic gauge-invariant function of the chiral superfields Φ. With Φ=(τ,φ) we denote collectively all chiral supermultiplets of the theory, including the modulus [In the literature it is common to make use of the field T=-iτ. The modulus τ describes a dimensionless chiral supermultiplet, depending on both space-time and Grassmann coordinates. By introducing a fundamental scale Λ, a chiral supermultiplet σ=Λτ with mass dimension one can be defined. In the limit of unbroken supersymmetry the results presented in this paper depend on σ only through the τ combination.]τ plus a number of additional chiral supermultiplets φ separated into sectors φ^(I). In general each sector I contains several chiral supermultiplets φ^(I)_i_I but, to keep our notation more concise, we will not write the additional index i_I. We ask for invariance under transformations of the modular group Γ, under which the supermultiplets φ^(I) of each sector I are assumed to transform in a representation ρ^(I) of a quotient group Γ_N,kept fixed in the construction, with a weight -k_I: {[τ→a τ+b/c τ+d; φ^(I)→ (cτ+d)^-k_Iρ^(I)(γ) φ^(I) ] . .The supermultiplets φ^(I) are not modular forms and there are no restrictions on the possible value of k_I, a priori. The invariance of the action S under (<ref>)requires the invariance of the superpotential w(Φ) and the invariance of the Kahler potential up to a Kahler transformation: {[ w(Φ)→ w(Φ); K(Φ,Φ̅)→ K(Φ,Φ̅)+f(Φ)+f(Φ) ]. .The requirement of invariance of the Kahler potential can be easily satisfied. An example of Kahler potential invariant under (<ref>) up to Kahler transformations isK(Φ,Φ̅)=-h log(-iτ+iτ̅)+ ∑_I (-iτ+iτ̅)^-k_I |φ^(I)|^2 ,where h is a positive constant. Since (τ-τ̅)→ |c τ+d|^-2(τ-τ̅), we have:K(Φ,Φ̅)→ K(Φ,Φ̅)+h log(c τ+d)+h log(cτ̅+d) .Assuming that only the modulus τ acquires a vacuum expectation value (VEV), this Kahler potential gives rise to kinetic terms for the scalar components of the supermultiplets τ and φ^(I) of the kind h/⟨-iτ+iτ̅⟩^2∂_μτ̅∂^μτ+∑_I ∂_μφ^(I)∂^μφ^(I)/⟨-iτ+iτ̅⟩^k_I .On the contrary, the invariance of the superpotential w(Φ) under the modular group provides a strong restriction on the theory. Consider the expansion of w(Φ) in power series of the supermultiplets φ^(I):w(Φ)=∑_n Y_I_1...I_n(τ) φ^(I_1)... φ^(I_n) .For the n-th order term to be modular invariant the functions Y_I_1...I_n(τ) should be modular forms of weight k_Y(n) transforming in the representation ρ of Γ_N:Y_I_1...I_n(γτ)=(cτ+d)^k_Y(n)ρ(γ) Y_I_1...I_n(τ) ,with k_Y(n) and ρ such that:1. The weight k_Y(n) should compensate the overall weight of the product φ^(I_1)... φ^(I_n):k_Y(n)=k_I_1+....+k_I_n .2. The product ρ×ρ^I_1× ... ×ρ^I_n contains an invariant singlet.When we have k_I=0 in all sectors I of the theory, we get k_Y(n)=0 for all n and the functions Y_I_1...I_n(τ) are τ-independent constants since, as we have seen, there are no non-trivial modular forms of weight zero. The product ρ^I_1× ... ×ρ^I_n should contain an invariant singlet and Y_I_1...I_n is an invariant tensor under the group Γ_N. This particular limit reproduces the well-studied case of a supersymmetric theory invariant under the discrete symmetry Γ_N. Some of the fields φ^(I) can be gauge singlets,i.e. flavons, whose vacuum expectation values break Γ_N in the appropriate direction.This scheme applies to most of the models of fermion masses based on discrete symmetries that have been proposed so far. However this is just a particular case of the more general setup considered here. When k_I 0 in some sector I, the Yukawa couplings Y_I_1...I_n(τ) should carry a non-trivial τ dependence. The whole modular group acts on the field space and Y_I_1...I_n(τ) are strictly constrained by the relatively small number of possible modular forms that match the above requirements. This setup can be easily extended to the case of N=1 local supersymmetry where Kahler potential and superpotential are not independent functions since the theory depends on the combinationG(Φ,Φ̅)=K(Φ,Φ̅)+log w(Φ)+log w(Φ̅) .The modular invariance of the theory can be realized in two ways. Either K(Φ,Φ̅) and w(Φ) are separately modular invariant or the transformation of K(Φ,Φ̅) under the modular group is compensated by that of w(Φ). An example of this second possibility is given by the Kahler potential of eq. (<ref>), with the superpotential w(Φ) transforming asw(Φ)→ e^iα(γ) (cτ+d)^-h w(Φ) In the expansion (<ref>) the Yukawa couplings Y_I_1...I_n(τ) should now transform as Y_I_1...I_n(γτ)= e^iα(γ) (cτ+d)^k_Y(n)ρ(γ) Y_I_1...I_n(τ) ,with k_Y(n)=k_I_1+....+k_I_n-h and the representation ρ subject to the requirement 2. When we have k_I_1+....+k_I_n=h, we get k_Y(n)=0 and the functions Y_I_1...I_n(τ) are τ-independent constants. This occurs for supermultiplets belonging to the untwisted sector in the orbifold compactification of the heterotic string.The above framework is known and we have reported it here for completeness. However there are two aspects that we have emphasized in this presentation. First of all, the close relationship between supersymmetric modular-invariant theories and finite discrete symmetries. As stressed above the whole class of models invariant under discrete groups of the type Γ_N can be regarded as a particular case of the wider class of modular invariant theories. The second aspect is that this formulation is particular suitable to a bottom-up approach. Namely, by explicitly constructing the whole ring M(Γ(N)) of modular forms of level N, one can systematically explore all possible Yukawa couplings Y_I_1...I_n(τ) which can occur in the superpotential w(Φ). As we shall see in a specific example, the limited number of generators of M(Γ(N)) severely constrain the candidate couplings and lead to a new class of predictive models of lepton masses. § MODELS OF LEPTON MASSES In this section we apply the formalism outlined above to models of lepton masses and mixings. Here we give the general rules to build modular invariant N=1 supersymmetric models with matter supermultiplets transforming according to representations of a finite modular group Γ_N, with a fixed integer N. The relevant matter fields and their transformation properties are collected in table <ref>.The field content includes a chiral supermultiplet forthe modulus τ, transforming as in (<ref>) under modular transformations and singlet under gauge transformations.There is a minimal class of models where the only source of breaking of the modular symmetry is the VEV of τ. In this class of models we do not introduce any flavon multiplet. The role of flavons is replaced by the modular forms Y(τ) of level N. In a more general context, we can allow for the existence of a set of gauge-singlet flavons φ, which transform non-trivially under Γ_N and contribute to the breaking of the discrete symmetry through their VEVs. Without loosing generality the flavon fields can be assumed dimensionless, as the modulus τ. As a preliminary step, we need to identify the modular forms Y(τ) and their transformation properties under the modular group,i.e. their representations ρ under Γ_N. We assume a Kahler potential of the form (<ref>). For the superpotential, we distinguish two cases.1. When neutrino masses originate from a type I see-saw mechanism,the superpotential in the lepton sector reads:w=α (E^c H_d L f_E(Y,φ))_1+g(N^c H_u L f_N(Y,φ))_1+Λ (N^c N^c f_M(Y,φ))_1 ,where f_I(Y,φ)(I=E,N,M) denotes the most general combination ofmodular forms Y(τ) and flavon fields φ such that the total weight of each term in (<ref>) is zero. Moreover (...)_1 stands for the invariant singlet of Γ_N. It can occur that there are several independent invariant singlets.In this case a sum over all contributions with arbitrary coefficients is understood and in w there are more parameters than those explicitly indicated.2. When neutrino masses directly originate from the Weinberg operator,the superpotential in the lepton sector reads:w=α (E^c H_d L f_E(Y,φ))_1+1/Λ(H_u H_u L L f_W(Y,φ))_1 .The supermultiplet τ and the modular forms Y(τ) do not carry lepton number. In the above superpotential the lepton number of the matter multiplets is assigned in the usual way and the total lepton number is effectively broken either by the VEV of some flavon field or by the parameter Λ, that can be regarded as a spurion. We will not address here the vacuum alignment problem. We will not attempt to build the most general supersymmetric and modular invariant scalar potential for τ and φ. To explore the ability of this class of models in reproducing the existing data, both τ and φ will be treated as spurions whose valued will be varied when scanning the parameter space of the model. In principle, for each value of τ the kinetic terms of the chiral multiplets in table <ref> have to be rescaled to match their canonical form. In practice, in the concrete models we will explore, this effect can be absorbed into the unknown parameters of the superpotential. By scanning the parameter space of the model (α,g,Λ) and the VEVs (⟨τ⟩,⟨φ⟩) we can compute lepton masses, mixing angles and phases and compare them to the data. §.§ Models based on Γ_3To assess the viability of the above formalism, we build in this section models of lepton masses based on modular forms of level 3. The group Γ_3 is isomorphic to A_4, see Table 1, and we will be able to compare the present construction with a class of models widely discussed in the literature, especially in connection to the tribimaximal mixing. Though the explicit example will refer to the case N=3, it is prettyclear from the previous section that there are no conceptual obstacle to develop a similar construction for any N. A comprehensive analysis of all models that can be constructed along these lines is beyond the scope of the present work. The examples discussed here, admittedly not fully realistic, are meant to illustrate how this new type of construction can be brought to completion and compared with the data.§.§.§ Modular Forms of Level 3 We focus on modular forms of level 3, satisfyingf(γτ)=(c τ+d)^2k f(τ) γ=([ a b; c d ])∈Γ(3) ,with Γ(3)={([ a b; c d ])∈ SL(2,Z), ([ a b; c d ])=([ 1 0; 0 1 ]) ( mod 3)} .The quotient space H/Γ(3) can be described by a fundamental domain F for Γ(3), that is a connected region of H such that each point of H can be mapped into F by a Γ(3) transformation, but no two points in the interior of F are related under Γ(3). The space H/Γ(3) is simply F with certain boundary points identified. A fundamental domain for Γ(3) is shown in fig. 1. H/Γ(3) can be made compact by adding the points i∞, -1, 0 and +1, which are the cusps. The compactified space H/Γ(3) has genus zero and can be thought of as a tetrahedronwhose vertices are the cusps. Indeed the cuspsare related by transformations of Γ_3=Γ̅/Γ(3), which is the symmetry group of a regular tetrahedron, given the isomorphism between Γ_3 and A_4. The group A_4 is generated by the elements S and T, satisfying the relations:S^2=T^3=(ST)^3=1 .Basic properties of A_4 are summarized in Appendix C.Modular forms of weight 2k and level 3 transform according to unitary representations of A_4. In the final part of this work we will see that they provide a non-linear representation of A_4. Modular forms of level 3 can be constructed starting from those of lower weight, k=1. From table 1 we see that there are three linearly independent such forms, which we call Y_i(τ). Three linearly independent weight 2 and level-3 forms are constructed in the Appendix C. They read:Y_1(τ) = i/2 π[η'(τ/3)/η(τ/3)+η'(τ+1/3)/η(τ+1/3)+η'(τ +2/3)/η(τ +2/3)-27 η'(3 τ )/η (3 τ)]Y_2(τ) = -i/π[η'(τ/3)/η(τ/3) +ω^2 η'(τ+1/3)/η(τ+1/3)+ω η'(τ +2/3)/η(τ +2/3)]Y_2(τ) = -i/π[η'(τ/3)/η(τ/3) +ω η'(τ+1/3)/η(τ+1/3)+ω^2 η'(τ +2/3)/η(τ +2/3)] . where η(τ) is the Dedekind eta-function, defined in the upper complex plane: η(τ)=q^1/24∏_n=1^∞(1-q^n ) q≡ e^i 2 πτ .They transform in the three-dimensional representation of A_4. In a vector notation where Y^T=(Y_1,Y_2,Y_3) we haveY(-1/τ)=τ^2 ρ(S) Y(τ) , Y(τ+1)=ρ(T) Y(τ) ,with unitary matrices ρ(S) and ρ(T) 0.1cm ρ(S)=1/3( [ -122;2 -12;22 -1 ]) , ρ(T)= ( [ 1 0 0; 0 ω 0; 0 0 ω^2 ]) , ω=-1/2+√(3)/2i .0.1cmThe q-expansion of Y_i(τ) reads:Y_1(τ) = 1+12q+36q^2+12q^3+...Y_2(τ) = -6q^1/3(1+7q+8q^2+...)Y_3(τ) = -18q^2/3(1+2q+5q^2+...) . From the q-expansion we see that the functions Y_i(τ) are regular at the cusps. Moreover Y_i(τ) satisfy the constraint:Y_2^2+2 Y_1 Y_3=0 .As discussed explicitly in Appendix D, the constraint (<ref>) is essential to recover the correct dimension of the linear space M_2k(Γ(3)). On the one side from table 1 we see that this space has dimension 2k+1. On the other hand the number of independent homogeneous polynomial Y_i_1Y_i_2··· Y_i_k of degreek that we can form with Y_i is (k+1)(k+2)/2. These polynomials are modular forms of weight 2k and, to match the correct dimension, k(k-1)/2 among themshould vanish. Indeed this happens as a consequence of eq. (<ref>). Therefore the ring M(Γ(3)) is generated by the modular forms Y_i(τ)(i=1,2,3).There are two special sets of VEV for τ that preserve a subgroup of the modular group (and of Γ_3).The subgroup generated by T is preserved by ⟨τ⟩=i∞[This can be seen more clearly by going from H to the unit disk |w|<1, via w=(τ-1)/(τ+1). The transformation T acts on w as w→(1+w)/(3-w), having a unique fixed point at w=1, belonging to the boundary of the unit disk and corresponding to τ=i∞.]. This induces the well-known VEV for the triplet Y_i:(Y_1,Y_2,Y_3)|_τ=i∞=(1,0,0) .The subgroup generated by S is preserved by ⟨τ⟩=i, which results into:(Y_1,Y_2,Y_3)|_τ=i=Y_1(i)(1,1-√(3),-2+√(3))This VEV pattern is completely different from (1,1,1), the eigenvector of the matrix S corresponding to the eigenvalue +1. Indeed we should satisfy:Y(-1/τ)|_τ=i=-ρ(S) Y(τ)|_τ=i .For this reason Y(τ)|_τ=i must be an eigenvector of the matrix S with eigenvalue -1. There are two such eigenvectors and the one in (<ref>) satisfies the relation (<ref>). Notice that the configuration (Y_1,Y_2,Y_3)=(1,1,1), widely used in model building, cannot be realised since it violates the constraint (<ref>).Of course, values of ⟨τ⟩ related to i by a modular transformation, have little groups isomorphic to Z_2, the subgroup generated by S. A similar consideration applies to the vacuum configurations related to ⟨τ⟩=i∞ by modular transformations.§.§.§ Example 1: neutrino masses from the Weinberg operatorWe consider an example where the neutrino masses and mixing angles originate directly from the Weinberg operator. In this model we have no contribution to the mixing from the charged lepton sector. This can be achieved bychoosing the field content of table 3. In particular we need a flavon φ_T transforming as a triplet of Γ_3 and developing a VEV of the type: ⟨φ_T⟩=(u,0,0) .Such a VEV breaks Γ_3 down to the Z_3 subgroup generated by T.We choose the weights k_E_i, k_L, k_d and k_φ such thatk_E_i+k_L+k_d+k_φ=0. Moreover, to forbid a dependence of the charged lepton masses on Y(τ) (and a dependence of the Weinberg operator on φ_T), we take, for instance, k_φ=-3. The superpotential for the charged lepton sector reads:w_e=α E_1^c H_d (L φ_T)_1+β E_2^c H_d (L φ_T)_1'+γ E_3^c H_d (L φ_T)_1” .The VEV of eq. (<ref>) leads to a diagonal mass matrix for the charged leptons:m_e= diag(α,β,γ)u v_d .The charged lepton masses can be reproduced by adjusting the parameters α, β and γ, with an ambiguity related to the freedom of permuting the eigenvalues. As a result, the lepton mixing matrix U_PMNS is determined up to a permutation of the rows. Finally, by choosing k_L=+1 and k_u=0, we uniquely determine the form of the Weinberg operator: w_ν=1/Λ(H_u H_u L L Y)_1 The superpotential w=w_e+w_ν depends on the four parameters α,β,γ,Λ.The charged lepton masses m_e, m_μ and m_τ are in a one-to-one correspondence with α, β and γ, which can be taken real without loosing generality. The neutrino mass matrix is given by:m_ν=([2 Y_1 -Y_3 -Y_2; -Y_32 Y_2 -Y_1; -Y_2- Y_1 2Y_3 ])v_u^2/Λ We see that the fourth parameter, Λ, controls the absolute scale of neutrino masses.A remarkable feature of this model is that neutrino mass ratios, lepton mixing angles, Dirac and Majorana phases are completely determined bythe modulus τ. We have eight dimensionless physical quantities that do not depend on any coupling constant.Assuming the VEV of eq. (<ref>) for the flavon φ_T, they all uniquely depend on τ. Four of these parameters,r=Δ m^2_sol/|Δ m^2_atm|, sin^2θ_12, sin^2θ_13 and sin^2θ_23, have been measured with good precision. A fifth one, δ_CP, starts to be constrained by the present data, see table 4. It is a significant challenge to reproduce all of them by varying a single complex parameter. By scanning a portion of the upper complex plane where τ varies, wefound that the agreement between predictions and data is optimized by the choice: τ=0.0111+0.9946 i ,giving rise to an inverted neutrino mass ordering. Assuming that the charged lepton sector induces a permutation between the second and the third rows, the neutrino mass/mixing parameters are predicted to be [Dirac and Majorana phases are in the PDG convention.][ Δ m^2_sol/|Δ m^2_atm|=0.0292;sin^2θ_12=0.295 sin^2θ_13=0.0447sin^2θ_23=0.651;δ_CP/π=1.55α_21/π=0.22α_31/π=1.80 . ] Also the absolute scale of neutrino masses is determined, since v_u^2/Λ can be fixed by requiring that the individual square mass differences |Δ m^2_atm| and Δ m^2_sol are reproduced. We find the central values:m_1=4.998× 10^-2 eV m_2=5.071× 10^-2 eV m_3=7.338× 10^-4 eV .Some comments are in order.∙ The value of τ that minimises the χ^2 is intriguingly close to the self-dual point τ=i where S is unbroken. It is easy to verify that the self-dual point leads to an inverted-ordering spectrum with Δ m^2_sol=0 and relative Majorana phase π between the first two neutrino levels. For τ=i the CP symmetry is unbroken and the non-trivial phases in eq. (<ref>) are entirely generated by the departure of τ from i[When τ=i the mixing matrix U is real and, from eq. (<ref>), we find sin^2θ_12=(11-6√(3))/26≈ 0.02,sin^2θ_23=(16-4√(3))/26≈0.35 and |U_e3|=(3-√(3))/6≈ 0.21.]. ∙ Three observables, r=Δ m^2_sol/|Δ m^2_atm|, sin^2θ_12 and δ_CP are within the 1σ experimental range. A fourth one, sin^2θ_23, is very close to the 3σ range <cit.> for the inverted ordering.∙ The minimum χ^2 is completely dominated by sin^2θ_13, which is many σs away from the allowed range. Nevertheless θ_13 is predicted to be the smallest angle, in qualitative agreement with the data.∙ The observed value of δ_CP was not included in the data set used to minimise the χ^2, but it turns out to be in very good agreement with the current determination. ∙ The two Majorana phases are predicted and the parameter |m_ee| of neutrinoless double beta decay is completely determined.∙ Had we assumed no permutation between the second and third row of the mixing matrix, we would have obtained sin^2θ_23=0.349 and δ/π=-1.55, with no change for the other predictions.∙ Notice that modular symmetry and supersymmetry completely determine the superpotential in the lepton sector in terms of the free parameters α, β, γ and Λ.There are no extra contribution we can add to w that cannot be absorbed in a redefinitions of those parameters. In particular the couplings of the modulus τ to the matter supermultiplets are fixed to any order in the τ power expansion and all higher-dimensional operators involving τ are known as a function of a finite set of parameters. This remarkable feature has no counterpart in the known models based on discrete symmetries where, in general, to fix all higher-dimensional operators involving the relevant flavon fields, an infinite number of parameters is needed. As a consequence, corrections to the above predictions can only come from few sources. One is the vacuum alignment. In this example we have kept fixed the VEV of φ_T. Additional, small contributions to the mixing can be expected if we relax this assumption.Other sources of corrections can be either supersymmetry breaking contributions or corrections to the Kahler potential. A candidate modification of the Kahler potential (<ref>) able to induce corrections to the above resultsis an additive contribution depending explicitly on both the matter supermultiplets and on the modular forms Y_i(τ). The problem of accounting for these additional terms will be analyzed elsewhere. The properties discussed here are not specific of the example under examination, but rather general features of the setup proposed here. A small correction to the model, coming for instance from the charged lepton sector, might bring the model to agreement with θ_13, perhaps without spoiling the predictions for the remaining observables. A more quantitative analysis will be pursued elsewhere. We consider the above zeroth-order approximation as an excellent starting point.§.§.§ Example 2: neutrino masses from the see-saw mechanismIn this example neutrino masses and mixing angles come from the see-saw mechanism. The field content is the one of table 5 and includes a triplet of chiral supermultiplets N^c describing right-handed neutrinos.The charged lepton sector is exactly the same as in the previous example. The relevant superpotential w_e is the one of eq. (<ref>).As before the weights k_E_i, k_L, k_d and k_φ satisfy k_E_i+k_L+k_d+k_φ=0 and we choose k_φ=-3. We assume that the flavon φ_T, transforming as a triplet of Γ_3, develops the VEV in eq. (<ref>), which guarantees a diagonal mass matrix for the charged leptons, as in eq. (<ref>). There is no contribution to the mixing from w_e and the charged lepton masses are in a one-to-one correspondence with the parameters α, β and γ. Also in this case there is an ambiguity related to the freedom of permuting the eigenvalues of m_e and the lepton mixing matrix U_PMNS is determined up to a permutation of the rows.By choosing the weights k_N+k_u+k_L=0 and k_N=+1, the superpotential of the neutrino sector is given by:w_ν=g (N^c H_u L)_1+Λ (N^c N^c Y)_1With the above choice of weights this the only gauge-invariant holomorphic polynomial singlet under Γ_3 and satisfying the constraint of eq. (<ref>). Using a vector notation (E^c^T=(E_1^c,E_2^c,E_3^c),...), we can write the superpotential w=w_e+w_ν as: [ w=E^c^TY_e H_d L+N^c^TY_ν H_u L+ N^c^TM_R N^c;Y_e=([ α φ_T1 α φ_T3 α φ_T2; β φ_T2 β φ_T1 β φ_T3; γ φ_T3 γ φ_T2 γ φ_T1 ]);Y_ν=g([ 1 0 0; 0 0 1; 0 1 0 ]); M_R=([2 Y_1 -Y_3 -Y_2; -Y_32 Y_2 -Y_1; -Y_2- Y_1 2Y_3 ])Λ ] The light neutrino mass matrix m_ν derived from the see-saw and the charged lepton mass matrix m_e read [ m_ν= -Y_ν^T M_R^-1 Y_ν⟨ H_u⟩^2; m_e=Y_e ⟨ H_d⟩ . ] The number of independent low-energy parameters is the same as in the previous example, since g only enters in the combination g^2/Λ[The phases of g and Λ are unobservable.]. A good agreement between data and predictions is obtained with the choice: τ=-0.195 + 1.0636 igiving rise to a normal neutrino mass ordering. Assuming that the charged lepton sector induces a permutation between the second and the third rows, the neutrino mass/mixing parameters are predicted to be: [ Δ m^2_sol/|Δ m^2_atm|=0.0280;sin^2θ_12=0.291 sin^2θ_13=0.0486sin^2θ_23=0.331;δ_CP/π=1.47α_21/π=1.83α_31/π=1.26 . ] Also the absolute scale of neutrino masses is determined, since g^2 v_u^2/Λ can be fixed by requiring that the individual square mass differences |Δ m^2_atm| and Δ m^2_sol are reproduced. We find the central values:m_1=1.096× 10^-2 eV m_2=1.387× 10^-2 eV m_3=5.231× 10^-2 eV .Most of the comments on the previous model apply to the present example as well. The distinctive feature of the model is the normal ordering of neutrino masses. Beyond the prediction of θ_13, which deviates by many standard deviations from the experimental range, also the value of sin^2θ_23 is slightly below the 3σ allowed range for normal ordering.Nevertheless θ_13 is the smallest angle and θ_23 the largest one, in qualitative agreement with the data. Corrections coming either from the Kahler potential or from the VEV of the flavon φ_T might restore the full agreement between data and predictions.§.§.§ Example 3: no flavonsThe following example can be regarded as minimal, in that it makes no use of any flavon field other than the modulus τ. We chose for lepton and Higgs supermultiplets the transformations of table <ref>. We can exploit the fact that the whole ring M(Γ(3)) is generated by the A_4 triplet Y_i(τ)(i=1,2,3), to cast the general superpotential of eq. (<ref>) in the form: [w=w_e+w_ν; w_e=α E_1^c H_d (L Y^a_1)_1+β E_2^c H_d (L Y^a_2)_1'+γ E_3^c H_d (L Y^a_3)_1” ,;w_ν=g(N^c H_u L Y^b)_1+Λ (N^c N^c Y^c)_1 ] where Y^a denotes a insertions of the basic modular forms Y_i(τ) and (...)_r(r=1,1',1”,3) stands for the r representation of A_4. It can occur that there are several independent combinations of the type (...)_r. In this case a sum over all contributions with arbitrary coefficients is understood. The invariance of w under modular transformations implies the following relations for the weights: {[ 2 a_i=k_E_i+k_d+k_L (i=1,2,3); 2 b=k_N+k_u+k_L;2c=2 k_N; ].To get canonical kinetic terms we have to rescale the supermultiplets of the theory. The effect of such rescaling can be absorbed into the arbitrary coefficients α, β, γ, g and Λ. We now proceed by selecting one example with specific choices of the weights k_I. By choosing k_u,d=0 and k_E_i=k_N=k_L=1, we have a_i=b=c=1 and all couplings are linear in Y_i: [ w_e=α E_1^c H_d (L Y)_1+β E_2^c H_d (L Y)_1'+γ E_3^c H_d (L Y)_1” ,;w_ν=g(N^c H_u L Y)_1+Λ (N^c N^c Y)_1 ] Using a vector notation (E^c^T=(E_1^c,E_2^c,E_3^c),...), we can write: [w=E^c^TY_e H_d L+N^c^TY_ν H_u L+ N^c^TM_R N^c;Y_e=([ α Y_1 α Y_3 α Y_2; β Y_2 β Y_1 β Y_3; γ Y_3 γ Y_2 γ Y_1 ]); Y_ν=([2 g_1 Y_1(-g_1+g_2)Y_3 (-g_1-g_2) Y_2;(-g_1-g_2)Y_32 g_1 Y_2 (-g_1+g_2) Y_1;(-g_1+g_2)Y_2 (-g_1-g_2) Y_12 g_1 Y_3 ]);M_R=([2 Y_1 -Y_3 -Y_2; -Y_32 Y_2 -Y_1; -Y_2- Y_1 2Y_3 ])Λ ] Notice that the contribution g(N^c H_u L Y)_1 depends on two parameters g_1 and g_2 corresponding to the two independent invariant singlets that we can form out of N^c, L and Y. The light neutrino mass matrix m_ν derived from the see-saw and the charged lepton mass matrix m_e read [ m_ν= -Y_ν^T M_R^-1 Y_ν⟨ H_u⟩^2; m_e=Y_e ⟨ H_d⟩ . ] From the matrices m_e and m_νwe can compute lepton masses, mixing angles and phases in terms of the effective parameters: α, β, γ, g_1,2 and τ. We focus on mass ratios and,without loss of generality, we can set the spurion Λ to one. Moreover, by exploiting field redefinitions, α, β, γ and g_1 can be taken real. Notice that the light neutrino mass matrix m_ν depends on a single complex parameter g_2/g_1, up to an overall factor that does not affect mass ratios and mixing angles. Mass ratios among charged leptons depend on α/γ and β/γ. By scanning the parameter space of the model we could identify the point α/γ=15.3, β/γ=0.054, g_2/g_1=0.029 i, τ=0.008+ 0.98 i which better approaches the data, collected in table 4, though not providing a realistic set of observables. We find that neutrino masses have inverted ordering. Mass ratios are in good agreement with the data: Δ m^2_sol/|Δ m^2_atm|=0.0292 , m_e/m_μ=0.0048 , m_μ/m_τ=0.0567 .The mixing angles are: sin^2θ_12=0.459 , sin^2θ_13=0.001 , sin^2θ_23=0.749 .An appealing feature of this pattern is that it predicts large solar and atmospheric mixing angles and a small reactor angle in qualitative agreement with the observations. Nevertheless, given the present experimental accuracy this set of angles is clearly excluded by many standard deviations. For completeness we also list the Dirac and Majorana phases in the PDG convention: δ_CP/π=1.25 , α_21/π=1.04 , α_31/π=1.02 . The observed value of δ_CP was not included in the data used to search for the best region of the parameter space and it turns out to be in good agreement with the current determination. As in the previous examples, modular symmetry and supersymmetry completely determine the superpotential in the lepton sector in terms of the free parameters α, β, γ, g_1,2 and Λ. In this case the model does not include flavons and corrections to the above predictions can only come from two sources. Either from supersymmetry breaking contributions or from corrections to the Kahler potential. §.§ Symmetry RealizationsConsider the minimal theory defined by eqs. (<ref>), (<ref>) and (<ref>) in the absence of flavon multiplets φ. It is reminiscent of a non-linear sigma model andit is interesting to better understand how the underlying symmetries are effectively realized. If we set the superpotential w(Φ) to zero, the action becomes invariant under the full SL(2,R) continuous group. Indeed the actionS=∫ d^4x d^2θ d^2θ̅ K(Φ,Φ̅)whereK(Φ,Φ̅)=-h log(-iτ+iτ̅)+ ∑_I (-iτ+iτ̅)^-k_I |φ^(I)|^2 ,is invariant under SL(2,R), with no restriction on the real parameters a, b, c, d other than ad-bc=1. Starting from a generic τ( Imτ>0), it is always possible to reach i by a transformation of SL(2,R) and we can choose τ=i as representative of the vacuum configuration. Such configuration has an SO(2) invariance, since cosα i+sinα/-sinα i+ cosα=i .Hence the modulus τ parametrizes the coset space SL(2,R)/SO(2). In this limit SL(2,R) is non-linearly realized and the theory describes the spontaneous breaking of SL(2,R) down to the subgroup SO(2), τ describing the corresponding massless Goldstone bosons. In the full theory, which includes a non-vanishing superpotential, the continuous symmetry SL(2,R) is explicitly broken down to SL(2,Z). Here the group Γ_N comes into play. Level-N modular forms and matter supermultiplets transform in representations of the groups Γ_N. Modular forms depend on the modulus τ and, in the examples considered so far, the symmetry Γ_N is always in the broken phase, as in the case of non-linearly realised continuous symmetries. The analogy can be pushed further. In the continuous case non-linear realizations can be defined by imposing invariant constraints on the field space of the theory, such as for instance φ^T φ=M^2(M^2> 0), the multiplet φ transforming in the fundamental of SO(N).This constraint induces the breaking of SO(N) down to SO(N-1). Consider the case of a supersymmetric modular-invariant theory with matter multiplets in representations of Γ_3, as in Section 3.1. We have seen that the three modular forms of level 3 and weight 2 obey the relation:(YY)_1”=Y_2^2+2 Y_1Y_3=0 .The left-hand side is covariant, since (YY)_1” transforms with a phase factor under A_4. Thus the constraint itselfis left invariant by A_4 transformations and induces a restriction on the field space similar to that induced by φ^T φ=M^2 in the case of SO(N).For generic values of Y(τ) satisfying the relation (<ref>), the symmetry Γ_3 is completely broken. Only for the special values related to τ=i or τ=i∞ by a Γ_3 transformation, a subgroup of Γ_3 is left unbroken. Therefore, in some sense, Γ_3 is non-linearly realised in the models considered here.Of course there are several differences with the continuous case. In the discrete case there are no Goldstone bosons and the modulus τ is expected to be massive.Non-linear covariant constraints of the type (<ref>) do not define low-energy effective theories in the discrete case. These theories should rather be viewed as consistent truncations of some ultraviolet completion.In general, as in the case of the models under discussions, covariant constraints leaves no non-trivial residual symmetry.We can nevertheless identify the cases in which a non-trivial residual symmetry group survives, by discussing the properties ofthe so-called orbit space, the space spanned by the invariants of the group.We do this in the remaining part of this work.§ NON-LINEAR REALISATIONS OF DISCRETE SYMMETRIESConsider a Lagrangian L(φ,ψ) invariant under the action of a discrete group G_d, depending on a set of scalar fields φ responsible for the spontaneous breaking of G_d, plus additional fields ψ that do not play a role in the symmetry breaking. As in the case of a continuous symmetry, we can restrict the field space M to which φ belong by means of constraints invariant under the action of G_d.We classify these constraints into two types:1. constraints of the type I, I standing forinvariant, are relations such as:I_i(φ)=0 (i=1,...,n) [type I] where I_i(φ) are invariants under G_d. 2. constraints of the type C, C standing forcovariant, are defined by the conditions:C_i(φ)=0 (i=1,...,m) [type C] where C_i(φ) are covariant, but not invariant, combinations of φ. Here we focus on thosetransforming with a phase factor under G_d C_i(φ')=e^i α_iC_i(φ) ,where α_i depend on the G_d transformation.We will consider polynomial invariants, that is I_i(φ) are polynomials in the components of the multiplet φ. The ring of invariant polynomials is infinite, but it is generated by a finite number of invariants γ_α(φ), which means that any invariant polynomial can be written as a polynomial in γ_α. The invariants γ_α might be related by a number of algebraic relations, or syzygies, Z_s(γ)=0.Also in the case of C constraints we will consider only polynomial expressions. Constraints of type I are those usually considered to define non-linear realisations of continuous symmetries. In the discrete case there are other possibilities to restrict the field space M in an invariant way. While a (connected, semisimple) Lie group has only the trivial one-dimensional representation, a discrete group can possess a number of non-invariant one-dimensional representations, or singlets, χ_i.Therefore the invariant conditions χ_i=0 have no counterpart in (connected, semisimple) Lie groups and give rise of constraints of type C. Notice that both conditions of type I and type C are left invariant bytransformations of G_d.In this Section we would like to analyse the pattern of symmetry breaking induced by the above constraints. To this purpose it is useful to inspect the orbits of the group,i.e. the set of points in the field space M that are related by group transformations. Each point of a given orbit has the same little group, up to a conjugation. The union of orbits having isomorphic little groups forms a stratum. The full field space M is partitioned into several strata. For instance the origin of M is the stratum of type G_d, since for φ=0 the symmetry is unbroken. Most of the field space M is made of orbits having trivial little group,i.e. the symmetry G_d is completely broken. This subset of M is called principal stratum. To analyse the symmetry breaking pattern induced by a constraint, it is sufficient to determine the orbit type of the strata affected by the constraint. An even more effective picture is obtained by moving to the orbit space M_I, spanned by the invariants γ_α of the theory.A whole orbit of M is mapped into a single point of M_I. The crucial property of M_I is that while M has no boundaries, M_I has boundaries that describe the possible breaking chains of the group. The tools that allow to characterise the orbit space M_I are the Jacobian matrix <cit.> J≡∂γ/∂φ ,and the so-called P-matrix <cit.> P=JJ^T .The manifold M_I is identified by the requirement that the matrix P is positive semidefinite, resulting in a set of inequalities involving the invariants γ_α. The boundaries of M_I can be found by studying the rank of J. In the interior of M_I the matrix J has maximum rank r_max. This region corresponds to the principal stratum where G_d is completely broken. On the boundaries P has some vanishing eigenvalue and the rank of J is reduced.If the dimension of M_I is d, in general we have (d-1)-dimensional boundaries where rank(J)=r_max-1. They corresponding to strata of type G'_d, withG'_d⊂ G_d. These boundaries meet along (d-2)-dimensional spaces, where rank(J)=r_max-2. They correspond to strata of type G”_d, with G'_d⊂ G”_d⊂ G_d. And so on, until the 1-dimensional boundaries meet in a point corresponding to the stratum of type G_d. Thus the symmetry breaking induced by a constraint can be immediately visualised by identifying the region of M_I involved by the constraint. Here we provide some examples, by discussing the case of the groups S_3 and A_4. §.§ Covariant constraints in S_3S_3 is the group of permutations of three objects. It is also the symmetry group of an equilateral triangle, generated by a reflection a with respect to an axis crossing a vertex and orthogonal to the opposite side and a 2π/3 rotation ab around the center. In terms of the two generators a and b its six elements are given byS_3={e,a,b,ab,ba,bab} .The elements (ab,ba) are of order three and the elements (a,b,bab) are of order two. They fall into three distinct conjugacy classes C_1={e}, C_2={ab,ba}, C_3={a,b,bab} and the three irreducible representations of the group are an invariant singlet 1, a singlet 1' and a doublet 2. A real basis for the generators a and b is given by: [1: a=1 b=1; 1':a=-1b=-1;2: a=( [10;0 -1 ]) b=( [-1/2 -√(3)/2; -√(3)/2 1/2 ]) . ] Given two doublets φ=(φ_1,φ_2) and ψ=(ψ_1,ψ_2), we have(φψ)_1 = φ_1ψ_1+φ_2ψ_2(φψ)_1' = φ_1ψ_2-φ_2ψ_1(φψ)_2 = ( [ -φ_1ψ_1+φ_2ψ_2;φ_1ψ_2+φ_2ψ_1 ]) . We consider a theory invariant under S_3 and depending on a single real doublet φ(x) in the scalar sector.The ring of invariant polynomials is generated by the two basic invariants [The Molien function of S_3 in the doublet representation is M(x)=1/(1-x^2)(1-x^3). It follows that there are two independent invariants, one quadratic and one cubic <cit.>.]: γ_1=(φφ)_1 , γ_2=(φφφ)_1 .In our real basis these read: γ_1=φ_1^2+φ_2^2 , γ_2=φ_1^3-3φ_1φ_2^2 .The invariants γ_1,2 are independent: there are no algebraic relations among them. Examples of possible constraints of type I are φ_1^2+φ_2^2=M_1^2 , φ_1^3-3φ_1φ_2^2=M_2^3 .We notice that the constraint (<ref>) is invariant under the full orthogonal group O(2) and is analogue to the constraint defining a nonlinear sigma model G/H, G=O(2) and H={e}. In general, imposing these constraints breaks completely the group S_3, in the sense that a generic pair (φ_1,φ_2) satisfying (<ref>) or (<ref>) or both is not left invariant by any subgroup of S_3. These constraints might still be useful in the model building, since they limit the possible set of vacuum configuration of the theory. The group S_3 has a unique one-dimensional non invariant representation, the singlet 1'. One can verify that the most general combination of (φ_1,φ_2) transforming as a 1' can be written asI(φ)χ(φ) ,where I(φ) is an S_3 invariant while χ(φ) is given by χ(φ)=(φφφ)_1'=3 φ_1^2φ_2-φ_2^3 ,where in the last equality we display the explicit expression in the real basis. Thus the unique non-trivial example of type C constraint for S_3 with a real doublet φ is3 φ_1^2φ_2-φ_2^3=0 .This has no analogue in O(2). It defines a domain D, contained in the full field space M≡ R^2, with the property of being invariant under the action of S_3. It is satisfiedby points belonging to the orbits O_ξ=((ξ,0),(-ξ/2,-√(3)/2ξ),(-ξ/2,+√(3)/2ξ)) -∞≤ξ≤ +∞ .For ξ 0 these orbits have little group isomorphic to Z_2 while for ξ=0 the little group coincides with S_3. When ξ varies from -∞ to +∞, O_ξ describes a domain D consisting of three straight lines that are permuted under the action of S_3. The domain D is the union of the stratum of type Z_2 and the stratum of type S_3. The orbits O_ξ are not generic ones. Generic orbits of S_3 consist of six points in the field space M, see Figure 2.In generic orbits the symmetry S_3 is completely broken. The full field space M is partitioned into three strata: the origin, the stratum of type Z_2 and the principal stratum, having trivial little group, see Figure 2.Restricting the field space of φ through the covariant constraint (<ref>) amounts to describe the breaking of S_3 down to Z_2, but for the particular case where φ=0, which leaves S_3 unbroken. The condition (<ref>) is invariant under S_3 transformations and fields φ=(φ_1,φ_2) obeying such conditions do not abandon the domain D when undergoing an S_3 transformation. We can say that the discrete symmetry S_3 is non-linearly realised, reflecting the fact that the domain D cannot be globally parametrized in terms of a single real field variable. The LagrangianL=1/2∂_μφ_1∂^μφ_1+1/2∂_μφ_2∂^μφ_2-V(γ_1,γ_2)+δ L(φ,ψ) φ∈ D where δ L(φ,ψ) represents S_3-invariant terms depending on possible additional fields ψ, describes an S_3-invariant theory, where S_3 is unbroken or spontaneously broken down to Z_2. At variance with non-linear realizations of a continuous group, the scalar field ξ that locally parametrizes the domain D is not a Goldstone boson. Moreover it is expected to have a non-vanishing mass, since it parametrizes a continuous direction linking different group orbits, albeit of the same orbit type.In general, we cannot interpret the theory restricted to D as a low-energy effective description obtained by integrating out the heavy degrees of freedom included in φ=(φ_1,φ_2), unless appropriate assumptions on the parameters of the scalar potential V(γ_1,γ_2) are made. Thus the theory described by the Lagrangian (<ref>) should be viewed as a consistent truncation of the full theory, rather than a low-energy approximation.The orbit space M_I is spanned by (γ_1,γ_2). A whole orbit of M is mapped into a single point of M_I, see Figure 3. The Jacobian matrix J and the P matrix readJ≡∂γ/∂φ= ( [2φ_12φ_2; 3φ_1^2-3φ_2^2-6φ_1φ_2 ]) , P=JJ^T= ( [ 4γ_1 6γ_2; 6γ_2 9γ_1^2 ]) .The manifold M_I is identified by requiring that the matrix P is positive semidefinite, which defines the region: γ_1≥ 0 , |γ_2|≤√(γ_1^3) .We have: [ rank(J)=1 ↔ 3φ_1^2 φ_2-φ_2^3=0 (φ 0); rank(J)=0 ↔φ= 0 ] The 2-dimensional orbit space M_I is shown in fig. 3. The conditions 3φ_1^2 φ_2-φ_2^3=0(φ 0)) and rank(J)=1 are equivalent and identify the 1-dimensional boundary, corresponding to the stratum of type Z_2. The rank of J vanishes for φ=0, where γ_1=γ_2=0. Thus the origin ofM_I represents the stratum of type S_3. In the interior of M_I, the principalstratum, we have rank(J)=2 and S_3 is completely broken. Even when we do not make use of constraints to restrict the field space of the theory,the above conditions can still be very useful to identify extrema of the scalar potential V(γ_1,γ_2) in the unrestricted field space M. Indeed the boundary of M_I provides an important tool to characterize the extrema of V<cit.>. The extremum condition reads: δ V=∑_i ∂ V/∂γ_i∂γ_i/∂φ_αδφ_α=∑_i ∂ V/∂γ_iJ_iα δφ_α=0 .One can prove that extrema of V with respect to the points of the boundary of M_I are also extrema of V. Moreover extrema on the boundary of M_I are more natural than extrema in the interior of M_I, in the sense that they require the vanishing of a smaller number of derivatives of V,since on the boundary the rank of the Jacobian matrix J is reduced. Therefore the condition χ(φ)=3φ_1^2 φ_2-φ_2^3=0 parametrizes possible natural extrema of the scalar potential. Additional conditions are in general required to guarantee the existence of an extremum on the boundary of M_I<cit.>. Invariants of continuous groups relevant to particle physics have been extensively used in the literature <cit.>.The previous example shows a first relation between constraints of type C and the breaking of a discrete symmetry.The covariant constraint 3 φ_1^2φ_2-φ_2^3=0 enforces the breaking of S_3 down to a Z_2 subgroup. This is nota general result, but rather a special outcome related to the fact that with a single real doublet there is only one independent constraint of type C that can be constructed. To analyse a more general case we move to the group A_4. §.§ Covariant constraints in A_4We replicate the previous analysis in the case of A_4, isomorphic to Γ_3 and of direct interest to modular forms of level 3. The basic properties of A_4 are summarised in Appendix C.We consider a theory invariant under A_4, whose scalar sector depends on a single real triplet φ(x).As we did in the example of the S_3 group, we restrict the field space M=R^3 by means of constraints. Also in this case we can formulate two types of constraints. Constraints of type I are of the form I_i(φ)=0 as in eq. (<ref>), where I_i(φ) are polynomial invariants. With a single real triplet the ring of invariant polynomials is generated by the four invariants γ_i(i=1,...,4) listed in table <ref>. These invariants are not algebraically independent. They satisfy the relation, or Syzygy <cit.>:Z(γ)=4γ_4^2-2γ_3^3+108 γ_2^4+γ_1^6+36 γ_3γ_2^2γ_1-20 γ_2^2γ_1^3+5γ_3^2γ_1^2-4γ_3γ_1^4=0 .As we discussed in the case of S_3, imposing a generic constraint I_i(φ)=0 of type I breaks completely the A_4 symmetry.We turn to constraints of type C, which we formulate by using the non-invariant singlets χ_i(i=1,...,4) shown in table <ref>.By working in the real basis, we start by examining the constraint: χ_1=φ_1^2+ω^2φ_2^2+ωφ_3^2=0 .By taking the real and the imaginary parts of eq. (<ref>) we get φ_1^2=φ_2^2=φ_3^2 ,which impliesalso χ_2= 0.The constraint is satisfiedby points belonging to the orbits O_2ξ≡((ξ,ξ,ξ),(-ξ,ξ,-ξ),(-ξ,-ξ,ξ),(ξ,-ξ,-ξ)) -∞≤ξ≤ +∞ .For ξ 0 these orbits have little group isomorphic to Z_3 while for ξ=0 the little group coincides with A_4. When ξ varies from -∞ to +∞, O_2ξ describes a domain D_2 consisting of four straight lines that are permuted under the action of A_4. The domain D_2 is the union of the stratum of type Z_3 and the stratum of type A_4. As a whole, the set of four lines is left invariant under A_4.To complete the partition of the field space M into strata, we analyze the the constraint [The conditions χ_3=0 and χ_4=0have the same solutions, for a real triplet.]χ_3=φ_2^2φ_3^2+ω^2φ_3^2φ_1^2+ωφ_1^2φ_2^2=0 .This constraint is solved by all the points in D_2 and also by points belonging to the orbitsO_4ξ≡((ξ,0,0),(-ξ,0,0),(0,ξ,0),(0,-ξ,0),(0,0,ξ),(0,0,-ξ)) 0≤ξ≤ +∞} .For ξ 0 these orbits have little group isomorphic to Z_2 while for ξ=0 the little group coincides with A_4. When ξ varies from -∞ to +∞, O_4ξ describes a domain D_4 consisting of three straight lines that are permuted under the action of A_4. Excluding the origin, these lines form a stratum of type Z_2.Thus the two covariant constraints (<ref>,<ref>) provide a partition of the field space M into strata. The domains D_2 and D_4 describe, for ξ 0, strata of type Z_3 and Z_2 respectively. The origin is the stratum of type A_4 and the rest is the principal stratum.Moving to the space of invariants M_I, it is easy to check that both the strata D_2 and D_4 are mapped into boundaries of M_I. Indeed along the orbits O_2ξ and O_4ξ the rank of the Jacobian matrixJ^T≡(∂γ/∂φ)^T= ( [ 2φ_1 φ_2φ_3 4φ_1^3 2φ_1(φ_2^2-φ_3^2)(φ_2^2-2 φ_1^2+φ_3^2); 2φ_2 φ_3φ_1 4φ_2^3 2φ_2(φ_3^2-φ_1^2)(φ_3^2-2 φ_2^2+φ_1^2); 2φ_3 φ_1φ_2 4φ_3^3 2φ_3(φ_1^2-φ_2^2)(φ_1^2-2 φ_3^2+φ_2^2) ]) ,is equal to one for ξ 0 and zero when ξ=0, while for generic orbits the rank of J is equal to three. The space of invariants M_I is more complicated than the one discussed in the S_3 case, due to the presence of the Syzygy Z(γ)=0. M_I lies on the three-dimensional hyper-surface defined by Z(γ)=0 and it is identified by the requirement that the matrix P=JJ^T is positive semidefinite.There are no two-dimensional boundaries, as it might be expected in a smooth three-dimensional manifold, and M_I is a sort of hyperconical surface.Concerning the applications to fermion masses, the models that we can build using the constraints (<ref>) and (<ref>) are still far from realistic. In most of the known models specific vacuum alignments among several multiplets of a discrete group are required. For instance, in the lepton sector, typically we ask distinct residual symmetries in the neutrino and in the charged lepton sectors. If the model contains just a single real triplet of A_4, the above constraints only allow to break A_4 down to a Z_2 or a Z_3 subgroup.Nevertheless this approach can be easily generalized to models with a larger scalar sector and may help exploring different paths in model building. An interesting case occurs when a covariant constraint does not correspond to a non-trivial residual symmetry, as it happens in the class of supersymmetric modular-invariant modelsbuilt by using modular forms Y_i(τ) of level 3. We consider an A_4-invariant theory depending on a complex scalar triplet φ=(φ_1,φ_2,φ_3) and for convenience we adopt the complex basis for the three-dimensional representation. We restrict to holomorphic constraints, having in mind the case of supersymmetric theories where φ represents a triplet of chiral supermultiplets of N=1 supersymmetry. We can read the relevant invariant and covariant combinations of φ in tables <ref> and <ref>, second columns. In particular we focus on the covariant constraint(φφ)_1”≡φ_2^2+2 φ_1 φ_3=0 .This is precisely the algebraic relation met when examining modular forms of level 3 and weight 2. We see that this constraint does not imply any more (φφ)_1'=0, due to the complex nature of the φ_i components.Therefore imposing (<ref>) does not necessarily enforce the breaking of A_4 into Z_3, as in the case of a real triplet. Indeed this is what we found when exploring the modelsof Section 3.1. Generic orbits of the field space satisfying this constraint belong to the principal stratum and are mapped into the interior of the orbit space, where thesymmetry is completely broken. There can be special orbits satisfying the constraint and belonging to strata with a non-trivial little group, but these orbits do not represent generic solutions of the constraint. The region defined by the constraint is invariant under A_4 transformations and the theory can be regarded as a non-linear realisation of the discrete symmetry A_4.§ DISCUSSIONWe have proposed a new bottom-up approach to the problem of lepton masses and mixing angles based on supersymmetric modular invariant theories. By recalling general properties of such theories we have seen how they naturally involve the finite modular groups Γ_N. When chiral multiplets transform, up to an overall factor, in unitary representations of Γ_N, the Yukawa couplings have to be modular forms of level N. Modular invariance plays the role of flavour symmetry. Flavons field might not be needed and the flavour symmetry can be entirely broken by the VEV of the modulus. The superpotential of the theory is very restricted and modular invariance constrains both neutrino masses and mixing angles. As long as supersymmetry is exact all higher-dimensional operators in the superpotential are completely determined by modular invariance. Possible sources of corrections are supersymmetry-breaking contributionsand modifications of the Kahler potential. When the weights of all matter supermultiplets vanish, the theory collapse to a supersymmeric Γ_N-invariant model, of the type that has been extensively studied in the past. We have provided several complete examples based on Γ_3, by explicitly constructing the ring of modular forms of level 3.In the most economical example, discussed in section 3.1.2, neutrino mass ratios, lepton mixing angles, Dirac and Majorana phasesdo not depend on any Lagrangian parameter, but only on the modulus VEV. In the difficult challengeof reproducing five known observables in terms of a single complex parameter, this model scores relatively well. Only sin^2θ_13, predicted to be about 0.045, is out of the allowed experimental range, while providing a reasonable zeroth-order approximation. The model predicts inverted neutrino mass ordering. The example presented in section 3.1.3 predicts normal neutrino mass ordering and is almost equally well-performing. The last example, discussed in section 3.1.4, is minimal in the sense that there are no additional fields breaking the flavour symmetry other than the modulus. Lepton mass ratios and the CP-violating phase are in agreement with the observations, but the mixing angles are only qualitatively reproduced. Solar and atmospheric angles are large and the reactor angle is small, but they are out of range by many standard deviations. Nevertheless the example provides an additional proof that this type of constructions can be carried out and that they are worth exploring. When discussing modular forms of level 3 we came across a new feature, allowing to extend the notion of non-linearly realised symmetry to the discrete case. While in the continuous case non-linear realizations can be defined by imposing invariant constraints on the field space, in the discrete case we can also consider covariant constraints of the type χ_i(φ)=0, χ_i denoting non-invariant singlets of the group, a possibility that is precluded to (connected, semisimple) Lie groups. These conditions can be used to define non-linear realizations of a discrete group and we have explicitly analyzed how they can arise for the groups S_3 and A_4.Non-linear realizations of a discrete symmetry do not define low-energy effective theories. They should rather be view as consistent truncations of some ultraviolet completion. In general covariant constraints leaves no non-trivial residual symmetry. We identify the cases in which a non-trivial residual symmetry group survives, by discussing the properties ofthe so-called orbit space, the space spanned by the invariants of the group. The discussion given here aboutnon-linearly realized discrete symmetries is very provisional and has been motivated by a puzzle about the dimension of the linear spaces of modular forms. Nevertheless we hope that in the future it may turn into a useful tool when looking for new possibilities in model building.§ AKNOWLEDGEMENTSI am grateful to Stefano Forte, Aharon Levy and Giovanni Ridolfifor giving me the chance to honour the memory of Guido Altarelli with this contribution. I thank Franc Cameron for useful correspondence about modular forms and Gianguido Dall'Agata and Roberto Volpato for comments on the manuscript. ThisworkwassupportedinpartbytheMIUR-PRINproject2015P5SBHT 003“Search for the Fundamental Laws and Constituents” andbythe EuropeanUnion network FP10ITN ELUSIVESand INVISIBLES-PLUS (H2020-MSCA- ITN- 2015-674896 and H2020- MSCA- RISE- 2015- 690575). § APPENDIX AWe list here general formulas for the dimensions d_2k(Γ(N)) of the linear space M_2k(Γ(N)) of modular forms of level N and weight 2k<cit.>. The dimensions readd_2k(Γ)= {[[k/6] k=1 ( mod 6);[k/6]+1 k 1 ( mod 6) ].where [x] is the integer part of x, and {[d_2k(Γ(2))=k+1; d_2k(Γ(N))=(2k-1)N+6/24N^2∏_p|N (1-1/p^2) N>2 ].where the product is over the prime divisors p of N. § APPENDIX BWe show that modular forms f_i(τ) of weight 2k and level N≥ 2 transform under Γ_N asf_i(γτ)=(c' τ+d')^2kρ(γ)_ijf_j(τ)where γ=([ a' b'; c' d' ])is representative of an element in Γ_N and ρ(γ) is a unitary representation of Γ_N. It is sufficient to show that (<ref>) holds for γ equal to S and T, which generate the entire modular group Γ and are not elements of Γ(N) for N≥ 2. When γ is equal to S and Teq. (<ref>) becomes {[ (Sτ)^2k f_i(S τ)=ρ(S)_ijf_j(τ); f_i(T τ)=ρ(T)_ijf_j(τ) ]. .We start by observing that the holomorphic functions F_Ti(τ)=f_i(T τ) and F_Si(τ)=(Sτ)^2kf_i(S τ) are modular forms of weight 2k and level N. Let h h=([ a b; c d ])be a generic element of Γ(N).Making use of the fact that Γ(N) are normal subgroups of Γ, we have:F_Ti(hτ)=f_i(ThT^-1 Tτ)=(c τ+d)^2k F_Ti(τ)and [ F_Si(hτ)=(Shτ)^2k f_i(ShS^-1Sτ)= (cτ+d)^2k(Sτ)^2k f_i(Sτ)=(cτ+d)^2k F_Si(τ) ] .Thus f_i(T τ) and (Sτ)^2kf_i(S τ) are in M_2k(Γ(N)) and can be written as linear combinations of f_j(τ), like in eq. (<ref>). The fact that the coefficients ρ_ij(S) and ρ_ij(T) define representations of Γ_N follows from the algebraic relations satisfied by S and T. For instance, the generators S and T of Γ_3 satisfy the relations:S^2=T^3=(ST)^3=1 .It follows that: [ ρ_ik(S)ρ_kj(S)=ρ_ik(T)ρ_km(T)ρ_mj(T)=δ_ij; ρ_ik(S)ρ_km(T)ρ_mp(S)ρ_pq(T)ρ_qr(S)ρ_rj(T)=δ_ij , ] and ρ_ij(S) and ρ_ij(T) are linear representations of the generators of Γ_3. The presentations of several groups Γ_N in terms of the elements S and T can be found in ref. <cit.>.Finally, the unitarity of the representation stems from the finite dimensionality of Γ_N.§ APPENDIX CA_4 is the group of even permutations of four objects. It is also the symmetry group of a regular tetrahedron. It has 12 elements and two generators, S and T, satisfyingS^2=T^3=(ST)^3=1 .The 12 elements fall into four conjugacy classes C_1={e}, C_2={T,ST,TS,STS}, C_3={T^2,ST^2,T^2S,TST}, C_4={S,T^2ST,TST^2}. The group has four irreducible representations: an invariant singlet 1, two non-invariant singlets 1',1” and a triplet 3. The elements S and T in the irreducible representations of A_4 are shown in table <ref>. The one-dimensional representations are determined non-ambiguously by the conditions (<ref>), while the three-dimensional representation is determined up to a unitary transformation, representing a change of basis. The last two columns of table 9 display S and T in two convenient basis, the real one and the complex one, respectively.Given two triplets φ=(φ_1,φ_2,φ_3) and ψ=(ψ_1,ψ_2,ψ_3), their product decomposes in the sum 1+1'+1”+3_S+3_A, where 3_S(A) denotes the symmetric(antisymmetric) combination. We show the result in table <ref>, both in the real and in the complex basis.§ APPENDIX DWe explicitly construct a basis of weight 2 modular forms for Γ(3). The dimension of the space M_2(Γ(3)) of modular forms of weight 2 for Γ(3) has dimension 3 and we look for three linearly elements. We start by observing that if f(τ) transforms asf(τ)→ e^iα (cτ+d)^k f(τ) ,then d/dτlog f(τ)→ (cτ+d)^2 d/dτlog f(τ)+ k c(cτ+d) .The inhomogeneous term can be removed if we combine several f_i(τ) with weights k_id/dτ∑_ilog f_i(τ)→ (cτ+d)^2 d/dτ∑_ilog f_i(τ)+ (∑_i k_i) c(cτ+d) ,provided the sum of the weights vanishes. Consider the Dedekind eta-function η(τ), defined in the upper complex plane: η(τ)=q^1/24∏_n=1^∞(1-q^n ) q≡ e^i 2 πτ .It satisfies η(-1/τ)=√(-i τ) η(τ) , η(τ+1)=e^i π/12 η(τ) .So η(τ)^24 is a modular form of weight 12 under the full modular group, actually a cusp form.We further observe that the set composed by η(3τ), η(τ/3), η((τ+1)/3) and η((τ+2)/3) is closed under the modular group. Under T we have η(3τ) →e^ i π/4 η(3τ) η(τ/3) → η(τ+1/3) η(τ+1/3) → η(τ+2/3) η(τ+2/3) → e^ i π/12 η(τ/3) . Under S we have η(3τ) → √(1/3) √(-i τ) η(τ/3)η(τ/3) → √(3) √(-i τ) η(3τ) η(τ+1/3) → e^- i π/12 √(-i τ) η(τ+2/3) η(τ+2/3) → e^ i π/12 √(-i τ) η(τ+1/3) . A candidate weight 2 form isY(α,β,γ,δ|τ)=d/dτ[αlogη(τ/3)+βlogη(τ+1/3)+γlogη(τ+2/3)+δlogη(3τ) ] .with α+β+γ+δ=0 to eliminate the inhomogeneous term. Under S and T we haveY(α,β,γ,δ|τ)τ^2 Y(δ,γ,β,α|τ) , Y(α,β,γ,δ|τ) Y(γ,α,β,δ|τ) .We search for three independent forms Y_i(τ) transforming in the three-dimensional representation of A_4(in a vector notation where Y^T=(Y_1,Y_2,Y_3))Y(-1/τ)=τ^2 ρ(S) Y(τ) , Y(τ+1)=ρ(T) Y(τ) ,with unitary matrices ρ(S) and ρ(T) 0.1cm ρ(S)=1/3( [ -122;2 -12;22 -1 ]) , ρ(T)= ( [ 1 0 0; 0 ω 0; 0 0 ω^2 ]) , ω=-1/2+√(3)/2i .0.1cmThetransformation T requires:Y_1(τ)=c_1 Y(1,1,1,-3|τ) , Y_2(τ)=c_2 Y(1,ω^2,ω,0|τ) , Y_3(τ)=c_3 Y(1,ω,ω^2,0|τ) ,while the transformation S fixes the coefficients c_i up to an overall factor:c_1=3 c , c_2=-6 c , c_3=-6 c .For convenience we choose c=i/2π and explicitly we haveY_1(τ) = i/2 π[η'(τ/3)/η(τ/3)+η'(τ+1/3)/η(τ+1/3)+η'(τ +2/3)/η(τ +2/3)-27 η'(3 τ )/η (3 τ)]Y_2(τ) = -i/π[η'(τ/3)/η(τ/3) +ω^2 η'(τ+1/3)/η(τ+1/3)+ω η'(τ +2/3)/η(τ +2/3)]Y_2(τ) = -i/π[η'(τ/3)/η(τ/3) +ω η'(τ+1/3)/η(τ+1/3)+ω^2 η'(τ +2/3)/η(τ +2/3)] . The q-expansion of Y_i(τ) reads:Y_1(τ) = 1+12q+36q^2+12q^3+...Y_2(τ) = -6q^1/3(1+7q+8q^2+...)Y_3(τ) = -18q^2/3(1+2q+5q^2+...) . It agrees, up to an overall factor, with the q expansion derived in ref. <cit.>, where the functions Y_i(τ) are expressed in terms of hypergeometric series.From the q-expansion we see that the modular forms Y_i(τ) satisfy the constraint:(YY)_1”≡ Y_2^2+2 Y_1 Y_3=0 .Notice that this constraint is left invariant by A_4 transformations, since the combination (YY)_1” is invariant up to a phase factor. This constraint has a direct consequence on the number of non-vanishing multilinear combinations of Y_i. If we do not require eq. (<ref>), there are (k+1)(k+2)/2 independent k-linear combinations of the typeY_i_1Y_i_2··· Y_i_k (i_1,i_2,i_k=1,2,3) .For instance, for k=2, we have the 6 combinations Y_1 Y_1, Y_2 Y_1, Y_3 Y_1, Y_2 Y_2, Y_3 Y_2, Y_3 Y_3. These combinations can be arranged into irreducible representations of A_4. For k=2 the six independent Y_iY_j decompose as 3+1+1'+1” under A_4:Y^(4)_3 = (Y_1^2-Y_2 Y_3,Y_3^2-Y_1 Y_2,Y_2^2-Y_1 Y_3)Y^(4)_1 = Y_1^2+2 Y_2 Y_3Y^(4)_1' = Y_3^2+2 Y_1 Y_2Y^(4)_1” = Y_2^2+2 Y_1 Y_3 , where, in the left-hand side, the lower index denotes the representation and the upper index stands for 2 k.For k=3, we have the 10 independent combinations Y_i Y_j Y_l. It is not difficult to see that they should decompose as 3+3+3+1 under A_4. They areY^(6)_1 = Y_1^3+Y_2^3+Y_3^3-3 Y_1 Y_2 Y_3Y^(6)_3,1 = (Y_1^3+2 Y_1Y_2Y_3,Y_1^2Y_2+2Y_2^2Y_3,Y_1^2Y_3+2Y_3^2Y_2)Y^(6)_3,2 = (Y_3^3+2 Y_1Y_2Y_3,Y_3^2Y_1+2Y_1^2Y_2,Y_3^2Y_2+2Y_2^2Y_1)Y^(6)_3,3 = (Y_2^3+2 Y_1Y_2Y_3,Y_2^2Y_3+2Y_3^2Y_1,Y_2^2Y_1+2Y_1^2Y_3) As a consequence of the constraint in eq. (<ref>) we see that Y^(4)_1”=0 and Y^(6)_3,3=0. In general, k(k-1)/2 among the Y_i_1Y_i_2··· Y_i_k are equal to zero and we are left with 2k+1 nonvanishing combinations. These should still decompose into A_4 representations, since the constraint (<ref>) is A_4-invariant.This explains why the linear space of modular forms of level 3 and weight 2k has dimension 2k+1, as it should, and not (k+1)(k+2)/2 as naively derived by counting the homogeneous polynomials of degree k in Y_i(τ). Thus a modular invariant supersymmetric theory using as building blocks modular forms of level 3, as the one we have explicitly illustrated as a candidate model for lepton masses, can be regarded as a sort of non-linear realisation of the discrete symmetry A_4. 00 Altarelli:2010gt G. Altarelli and F. Feruglio,Rev. Mod. Phys.82 (2010) 2701 [arXiv:1002.0211 [hep-ph]].Ishimori:2010au H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto,Prog. Theor. Phys. Suppl.183 (2010) 1[arXiv:1003.3552 [hep-th]]. Hernandez:2012ra D. Hernandez and A. Y. Smirnov,Phys. Rev. D86 (2012) 053014[arXiv:1204.0445 [hep-ph]].King:2013eh S. F. King and C. Luhn,Rept. Prog. Phys.76 (2013) 056201[arXiv:1301.1340 [hep-ph]]. King:2014nza S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto,New J. Phys.16 (2014) 045018 [arXiv:1402.4271 [hep-ph]]. King:2017guk S. F. King,Prog. Part. Nucl. Phys.94 (2017) 217[arXiv:1701.04413 [hep-ph]]. Hagedorn:2017zks C. Hagedorn,arXiv:1705.00684 [hep-ph].Pakvasa:1977in S. Pakvasa and H. Sugawara,Phys. Lett.73B (1978) 61. Wilczek:1977uh F. Wilczek and A. Zee,Phys. Lett.70B (1977) 418Erratum: [Phys. Lett.72B (1978) 504].Harrison:2002er P. F. Harrison, D. H. Perkins and W. G. Scott,Phys. Lett. B530 (2002) 167[hep-ph/0202074].Ma:2001dn E. Ma and G. Rajasekaran,Phys. Rev. D64 (2001) 113012[hep-ph/0106291]. Babu:2002dz K. S. Babu, E. Ma and J. W. F. Valle,Phys. Lett. B552 (2003) 207[hep-ph/0206292]. Altarelli:2005yp G. Altarelli and F. Feruglio,Nucl. Phys. B720 (2005) 64[hep-ph/0504165].Toorop:2011jnR. d. A. Toorop, F. Feruglio and C. Hagedorn,Phys. Lett. B703, 447 (2011) [arXiv:1107.3486 [hep-ph]]. deAdelhartToorop:2011reR. de Adelhart Toorop, F. Feruglio and C. Hagedorn,Nucl. Phys. B858, 437 (2012) [arXiv:1112.1340 [hep-ph]]. Holthausen:2012wtM. Holthausen, K. S. Lim and M. Lindner,Phys. Lett. B721, 61 (2013) [arXiv:1212.2411 [hep-ph]].King:2013vnaS. F. King, T. Neder and A. J. Stuart,Phys. Lett. B726, 312 (2013) [arXiv:1305.3200 [hep-ph]].Hagedorn:2013nra C. Hagedorn, A. Meroni and L. Vitale,J. Phys. A47 (2014) 055201 [arXiv:1307.5308 [hep-ph]].Fonseca:2014koa R. M. Fonseca and W. Grimus,JHEP1409 (2014) 033[arXiv:1405.3678 [hep-ph]].Feruglio:2012cw F. Feruglio, C. Hagedorn and R. Ziegler,JHEP1307 (2013) 027[arXiv:1211.5560 [hep-ph]].Holthausen:2012dk M. Holthausen, M. Lindner and M. A. Schmidt,JHEP1304 (2013) 122[arXiv:1211.6953 [hep-ph]]. Hall:1999snL. J. Hall, H. Murayama and N. Weiner,Phys. Rev. Lett.84, 2572 (2000) [hep-ph/9911341].Haba:2000be N. Haba and H. Murayama,Phys. Rev. D63 (2001) 053010 [hep-ph/0009174].deGouvea:2003xe A. de Gouvea and H. Murayama,Phys. Lett. B573 (2003) 94 [hep-ph/0301050].Espinosa:2003qz J. R. Espinosa,hep-ph/0306019.deGouvea:2012ac A. de Gouvea and H. Murayama,arXiv:1204.1249 [hep-ph].Feruglio:2015jfa F. Feruglio,Eur. Phys. J. C75 (2015) no.8,373[arXiv:1503.04071 [hep-ph]].Schellekens:2013bpa A. N. Schellekens,Rev. Mod. Phys.85 (2013) no.4,1491[arXiv:1306.5083 [hep-ph]]. Coleman:1969sm S. R. Coleman, J. Wess and B. Zumino,Phys. Rev.177 (1969) 2239.Callan:1969sn C. G. Callan, Jr., S. R. Coleman, J. Wess and B. Zumino,Phys. Rev.177 (1969) 2247.Hamidi:1986vh S. Hamidi and C. Vafa,Nucl. Phys. B279 (1987) 465.Dixon:1986qv L. J. Dixon, D. Friedan, E. J. Martinec and S. H. Shenker,Nucl. Phys. B282 (1987) 13.Lauer:1989ax J. Lauer, J. Mas and H. P. Nilles,Phys. Lett. B226 (1989) 251. Lauer:1990tm J. Lauer, J. Mas and H. P. Nilles,Nucl. Phys. B351 (1991) 353. doi:10.1016/0550-3213(91)90095-F Erler:1991nr J. Erler, D. Jungnickel and J. Lauer,Phys. Rev. D45 (1992) 3651.Ibanez:1986ka L. E. Ibanez,Phys. Lett. B181 (1986) 269.Casas:1991ac J. A. Casas, F. Gomez and C. Munoz,Int. J. Mod. Phys. A8 (1993) 455[hep-th/9110060]. Lebedev:2001qg O. Lebedev,Phys. Lett. B521 (2001) 71[hep-th/0108218]. Kobayashi:2003vi T. Kobayashi and O. Lebedev,Phys. Lett. B566 (2003) 164[hep-th/0303009].Cremades:2003qj D. Cremades, L. E. Ibanez and F. Marchesano,JHEP0307 (2003) 038[hep-th/0302105]. Blumenhagen:2005mu R. Blumenhagen, M. Cvetic, P. Langacker and G. Shiu,Ann. Rev. Nucl. Part. Sci.55 (2005) 71[hep-th/0502005]. Abel:2006yk S. A. Abel and M. D. Goodsell,JHEP0710 (2007) 034[hep-th/0612110]. Blumenhagen:2006ci R. Blumenhagen, B. Kors, D. Lust and S. Stieberger,Phys. Rept.445 (2007) 1[hep-th/0610327]. Marchesano:2007de F. Marchesano,Fortsch. Phys.55 (2007) 491[hep-th/0702094 [HEP-TH]]. Antoniadis:2009bg I. Antoniadis, A. Kumar and B. Panda,Nucl. Phys. B823 (2009) 116[arXiv:0904.0910 [hep-th]].Kobayashi:2016ovu T. Kobayashi, S. Nagamoto and S. Uemura,PTEP2017 (2017) no.2,023B02[arXiv:1608.06129 [hep-th]].Cremades:2004wa D. Cremades, L. E. Ibanez and F. Marchesano,JHEP0405 (2004) 079[hep-th/0404229]. Abe:2009vi H. Abe, K. S. Choi, T. Kobayashi and H. Ohki,Nucl. Phys. B820 (2009) 317[arXiv:0904.2631 [hep-ph]]. Ferrara:1989bc S. Ferrara, D. Lust, A. D. Shapere and S. Theisen,Phys. Lett. B225 (1989) 363. Ferrara:1989qb S. Ferrara, .D. Lust and S. Theisen,Phys. Lett. B233 (1989) 147. Brax:1994kv P. Brax and M. Chemtob,Phys. Rev. D51 (1995) 6550[hep-th/9411022].Binetruy:1995nt P. Binetruy and E. Dudas,Nucl. Phys. B451 (1995) 31[hep-ph/9505295]. Dudas:1995eq E. Dudas, S. Pokorski and C. A. Savoy,Phys. Lett. B369 (1996) 255[hep-ph/9509410].Dudas:1996aa E. Dudas,hep-ph/9602231. Leontaris:1997vw G. K. Leontaris and N. D. Tracas,Phys. Lett. B419 (1998) 206[hep-ph/9709510].Dent:2001cc T. Dent,Phys. Rev. D64 (2001) 056005[hep-ph/0105285].Dent:2001mn T. Dent,JHEP0112 (2001) 028[hep-th/0111024]. Lutken:1991jk C. A. Lutken and G. G. Ross,Phys. Rev. B45 (1992) no.20,11837.Cappelli:1996np A. Cappelli and G. R. Zemba,Nucl. Phys. B490 (1997) 595[hep-th/9605127]. Dolan:1998vs B. P. Dolan,Nucl. Phys. B554 (1999) 487[cond-mat/9809294 [cond-mat.mes-hall]]. Dolan:1998vr B. P. Dolan,J. Phys. A32 (1999) L243[cond-mat/9805171]. Burgess:2000kj C. P. Burgess and B. P. Dolan,Phys. Rev. B63 (2001) 155309[hep-th/0010246]. Burgess:2001sy C. P. Burgess and B. P. Dolan,Phys. Rev. B65 (2002) 155323[cond-mat/0105621]. Burgess:2006fw C. P. Burgess and B. P. Dolan,Phys. Rev. B76 (2007) 113406[cond-mat/0612269 [cond-mat.mes-hall]]. Lutken:2007zz C. A. Lutken and G. G. Ross,Phys. Lett. B653 (2007) 363.Lutken:2011zz C. A. Lutken and G. G. Ross,Nucl. Phys. B850 (2011) 321. gunning R. C. Gunning, Lectures on Modular Forms, Princeton, New Jersey USA, Princeton University Press 1962.Capozzi:2017ipn F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri and A. Palazzo,Phys. Rev. D95 (2017) no.9,096014[arXiv:1703.04471 [hep-ph]]. Ross:2007az G. Ross and M. Serna,Phys. Lett. B664 (2008) 97[arXiv:0704.1248 [hep-ph]]. RamondP. Ramond, Group Theory: a Physicist's Survey, Cambridge University Press, New York 2010. Cabibbo:1970rza N. Cabibbo and L. Maiani,in Evolution of particle physics, Academic Press (1970), 50, App. I.Talamini:2006wd V. Talamini,J. Phys. Conf. Ser.30 (2006) 30 [hep-th/0607165].Michel:1970mua L. Michel and L. A. Radicati,Proc. of the FifthCoral Gables Conference on Symmetry principles at High Energy, ed. by B. Kursunogluet al.,W. H. Benjamin, Inc. New York (1965);Annals Phys.66(1971)758. Jenkins:2009dy E. E. Jenkins and A. V. Manohar,JHEP0910 (2009) 094 [arXiv:0907.4763 [hep-ph]]and references within; A. Hanany, E. E. Jenkins, A. V. Manohar and G. Torri,JHEP1103 (2011) 096 [arXiv:1010.3161 [hep-ph]] Degee:2012sk A. Degee, I. P. Ivanov and V. Keus,JHEP1302 (2013) 125[arXiv:1211.4989 [hep-ph]].Espinosa:2012uu J. R. Espinosa, C. S. Fong and E. Nardi,JHEP1302 (2013) 137[arXiv:1211.6428 [hep-ph]]. Alonso:2013nca R. Alonso, M. B. Gavela, G. Isidori and L. Maiani,JHEP1311 (2013) 187[arXiv:1306.5927 [hep-ph]]. Fong:2013dnk C. S. Fong and E. Nardi,Phys. Rev. D89 (2014) no.3,036008[arXiv:1307.4412 [hep-ph]]. francFranc, C. and Mason, G.The Ramanujan Journal", 41 (2016) 233, arXiv:1503.05519. | http://arxiv.org/abs/1706.08749v2 | {
"authors": [
"Ferruccio Feruglio"
],
"categories": [
"hep-ph",
"hep-th"
],
"primary_category": "hep-ph",
"published": "20170627092624",
"title": "Are neutrino masses modular forms?"
} |
Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, JapanTIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, JapanWe study a general phase transition between spinless topological nodal-line semimetal and Weyl semimetal phases. We classify topological nodallines into two types based on their positions and shapes, and their phase transitions depends on their types.We show that a topological nodal-line semimetal becomes the Weyl semimetal by breaking time-reversal symmetry when the nodal lines enclose time-reversal invariant momenta (type-A nodal lines).We also discuss an effect of crystallographic symmetries determining the band structure of the topological nodal-line semimetals. Thanks to protection by the symmetries, the topological nodal-line semimetals can transition into spinless Weyl semimetalsor maintain the nodal lines in many crystals after inversion symmetry is broken.73.20.At, 73.43.Nq Universal phase transition and band structures for spinless nodal-line and Weyl semimetals Shuichi Murakami December 30, 2023 ============================================================================================ § INTRODUCTIONMany topological semimetals are realized by strong spin-orbit interactions leading to the gap closing. One example of the topological semimetals is a Weyl semimetal (WSM) <cit.>. The WSMs have three-dimensional nondegenerate Dirac cones. The gapless points called Weyl nodes are protected by the topology in the momentum space, and necessarily appear in pairs. The WSMs also show topological surface states called Fermi arcs <cit.>. As another example, topological Dirac semimetals have been also investigated <cit.>. The topological Dirac semimetals can be obtained in systems with time-reversal (TR) and inversion (I) symmetries, and have Dirac nodes with fourfold degeneracy.Moreover, recent works propose new topological semimetals whose band crossings lie on symmetry points with high-dimensional irreducible representations <cit.>.On the other hand, a novel topological semimetal appears in spinless systems with TR- and I symmetries.It is called a topological nodal line (TNL) semimetal (SM).The TNLSMs have twofold degenerate nodal lines on general points in the three-dimensional Brillouin zone. (If spin degeneracy is considered, the TNLs are fourfold degenerate.) The line degeneracy is acccidental, and characterized by a quantized Berry phase equal to π <cit.>. Hence, the nodal lines are protected topologically. However, existence of characteristic surface states called drumhead surface states is not necessarily guaranteed<cit.>. The TNLSMs have been predicted theoretically in various materials with negligible spin-orbit interaction. The candidates are carbon allotropes <cit.>, Cu_3(Pd, Zn)N <cit.>, Ca_3P_2 <cit.>, LaN <cit.>,compressed black phosphorus <cit.>, alkaline-earth metals <cit.>, BaSn_2 <cit.>, and CaP_3 family <cit.>. DC conductivities are calculated in hyperhoneycomb lattices with the TNLs <cit.>. Recently, ZrSiS has been observed as a TNLSM experimentally <cit.>.Meanwhile, one can find another type ofspinless nodal lines protected by mirror or glide symmetry but not topology. The nodal lines lie on the mirror/glide plane. The spinless nodal-line semimetals have been reported in CaAgP and CaAgAs, which are noncentrosymmetric <cit.>. In general, the protection by the mirror symmetry can coexist with the topological protection. Actually, the nodal lines in ZrSiS are protected also by the glide symmetry <cit.>.We can also realize the WSM phase in spinless systems when either TR or I symmetry is absent <cit.>.The spinless WSM phase has been realized experimentally in a photonic crystal <cit.>. Some spinless WSM phases appear between topologically trivial and nontrivial insulator phases characterized by some crystal symmetries <cit.>. In some models,the spinless WSM is expected to be driven from the nodal-line semimetal by a circularly polarized light <cit.>. Yet the purely spinless TNLSMs and the spinless WSMs have not been discovered experimentally in three-dimensional electronic systems. Additionally, the topological nodal-bands are suggested in non-electronic systems which do not have spin-orbit interactions <cit.>. Therefore, it is important to give a general framework of a phase transition between the spinless topological semimetal phases, as studied in spinful systems <cit.>. Moreover, it is recently shown that band evolutions of spinful WSMs are determined by crystallographic symmetries,although the Weyl nodes may arise at generic points <cit.>. Extending this theory to the TNLSMs helps us to understand the band structures easily.In this work, we study a generic topological phase transition between the spinless semimetal phases.To elucidate the phase transition, we classify TNLs into two types, type-A and type-B.The type-A and type-B TNLs are distinguished by their locations and shapes, which roughly corresponds towhether or not the TNLs enclose a time-reversal invariant momentum. We show that depending on the type of the TNL, its topological nature and its evolution under symmetry-breaking perturbations are quite varied.It is shown that the type-A TNLSM phase always becomes the spinless WSM phase when the TR symmetry is broken. Furthermore, we show how other crystallographic symmetries constrain positions of the TNLs in the type-A TNLSMs. As a result, even if the I symmetry is broken, the system remains in a nontrivial topological semimetal phase by the crystal symmetries in many cases.We also demonstrate the phase transition between the TNLSM phase and the WSM phase by using a lattice model to confirm our theory.This paper is organized as follows. We classify the topological nodal line into the type-A and the type-B TNLSMs andshow corresponding effective models in Sec. <ref>. In Sec. <ref>, we show general phase transitions in TNLSMs when the TR or I symmetryis broken.We elucidate effects of other crystal symmetries on band structures and nodal lines of the topological semimetals in Sec. <ref>. In Sec. <ref>, we discuss phase transitions in TNLSMs with additional crystallographic symmetries, when thethe TR or I symmetryis broken. Our results are summarized in Sec. <ref>. § CLASSIFICATION OF TNLSMS INTO TWO TYPESIn this section we classify TNLs into two types: type-A and type-B. We consider systems with I and TR symmetries, whoseoperators are denoted by P and Θ, respectively. Θ is a complex conjugation operator K. These symmetries give constraints H(-k)=PH(k)P^-1=Θ H(k)Θ ^-1, where H(k) is the Hamiltonian. Because of these symmetries, it is important to describe behaviors of the energy bands at a time-reversal invariant momentum (TRIM). We classify TNLs based on their shapes around one of the TRIM. Because of the TR symmetry, TNLs appear symmetrically with respect to TRIM. When there are more than one TNL in the Brillouin zone,we consider each TNL separately. It may sometimes happen that a single TNL is not time-reversal invariant in itself, i.e.it is not symmetric withrespect to the TRIM considered;in such cases, we consider instead a pair of TNLs which is symmetric withrespect to the TRIM, as shown in Fig. <ref>(b). Obviously, this pairing of TNLs is independent of thechoice of the TRIM.It may also happen that some TNLs may traverse across the Brillouin zone, like an “open orbit”of an electron under a magnetic field within semiclassical theory. Our theory also works in such cases. To classify individualTNLs, we first define a TR-invariant plane, as a plane in k space containing the TRIM considered.This plane is invariant under the TR symmetry.Since the TNL are symmetric with respect to the TRIM, the TNL always intersects with the TR-invariant plane 2(2N+1) or 4N times, where N is a non-negative integer (Fig. <ref>). If the TNL intersects with the TR-invariant plane 2(2N+1) times, the TNL encloses the TRIM as shown in Fig. <ref>(a), andwe call the TNL a type-A TNL. On the other hand, if the number of the intersection points is 4N, we call the TNL a type-B TNLas shown in Fig. <ref>(b). If the TNLs are tangential to the TR-invariant plane, we slightly move the TR-invariant planeto eliminate the points of tangency, and count the number of intersections.This classification is independent of the choice of the TR-invariant plane for the fixed choice of theTRIM. Furthermore, it is also independent of the choice of the TRIM, which can be directly shown by considering a TR-invariant plane containing more than one TRIM.In the following, we construct a two-band effective Hamiltonian consisting of the conduction and the valence bands around the TRIM,in order to facilitate our understanding of the behaviors of the TNLs. To construct an effective Hamiltonian we assume that each TNL is isolated, meaning that we can take a vicinity of the TRIM which contain only one TNL. First, we consider the case where the parity eigenvalues of the conduction and the valence bands are different at the TRIM, and are inverted fromthe other TRIM. As we see later, this corresponds to the type-A TNL. Then the I symmetry is given by P=±σ _z,where σ _i=x,y,z denote Pauli matrices acting on the space spanned by the conduction and the valence bands. Then, from the TR- and I symmetries, the effective Hamiltonian isH_TNL(q)=a_y(q)σ _y+a_z(q)σ _z,where q is a wavevector measured from the TRIM, a_y(-q)=-a_y(q), and a_z(-q)=a_z(q). Therefore, the TNL is represented by a_y(q)=0 and a_z(q)=0.Here, we are considering the casewhere the parities of the bands at the TRIM are inverted fromthose at other TRIM. Therefore, the coefficient a_z(q) changes sign as we go away from the TRIM (q=0). Hence, theequation a_z(q)=0 definesa closed surface encircling the TRIM, and together with the other condition a_y(q)=0, it indeed defines a TNL enclosing the TRIM, corresponding to the type-A TNL.In this case, a sign of a parameter mdefined by m≡ a_z(q=0) describes whether the bands are inverted or not. Suppose we start from the TNLSM phase and change this parameter m across zero. As m approaches zero, the nodal line shrinks. At m=0 the gap closes at the TRIM(q=0), and then the gap opens. In addition, some of the type-B TNLs can also be described by Eq. (<ref>). It happens when the sign of a_z(q) at q=0 and that away from q=0 are the same, whereasa_z(q) vanishes at some q. This corresponds to the type-B TNL, by countingthe number of intersections between the TNL and the TR-invariant plane.Second,when the parity eigenvalues of the conduction and the valence bands are identical at the TRIM,P=±σ _0 and the effective Hamiltonian isH_TNL(q)=a_x(q)σ _x+a_z(q)σ _z,where a_x(-q)=a_x(q) and a_z(-q)=a_z(q). TNLs exist if a_x(q)=0 and a_z(q)=0. It is straightforward to see thatthe number of intersections between the TNL and the TR-invariant planeis 4N (N: integer), meaning that this TNL is of type B. Unlike Eq. (<ref>), the gap closingat the TRIM is prohibited by level repulsion.Meanwhile, as we explained later, the TNL can be annhilated without crossing the TRIM.In some cases, there are more than one TNLs in the Brillouin zone.Ca (calcium) has four type-A TNLs and and Yb (ytterbium) without the spin-orbit coupling has six pairs oftype-B TNLs <cit.>.Let n_ A and n_ B denote the number of type-A TNLs and that of type-B TNLs, respectively.Then one can relate these numbers withthe ℤ_2 topological invariants ν_i (i=0,1,2,3) introduced in Ref. Kim15.The topological invariants are defined as(-1)^ν _0=∏ _n_j=0,1∏ _m^occ.ξ _m(Γ _n_1n_2n_3), (-1)^ν _i=∏ _n_i=1,n_j≠ i=0,1∏ _m^occ.ξ _m(Γ _n_1n_2n_3),where ξ _m(Γ _n_1n_2n_3) is a parity eigenvalue of the m-th occupied bandat a TRIM Γ _n_1n_2n_3 =(n_1G_1+n_2G_2+n_3G_3)/2, n_i=0,1.G_i=1,2,3 are reciprocal vectors. These topological invariants determine whether the number of intersections between the TNLs and a half of an arbitrary plane including four TRIM is even or odd <cit.>. In particular, it directly follows from Ref. Kim15 thatν_0≡ n_ A ( mod 2). For example, both in Ca and Yb (without the the spin-orbit coupling),the ℤ_2 topological numbers are trivial, i.e. (ν _0; ν_1 ν_2 ν_3)=(0;000) <cit.>, and it agrees with the number of TNLs, (n_ A,n_ B)=(4,0) in Ca and (n_ A,n_ B)=(0,6) in Yb. Next we consider an evolution of a TNL under continuous deformation of the system.A TNL may change its shape under the deformation, and sometimes the number of intersections witha TR-invariant plane may change. We first note that as long as the TNL does not go across the TRIM,the number of intersections between a TNL and a TR-invariant plane can changeonly by an integer multiple of four, because the TNL remains symmetric with respect to the TRIM.Therefore, a type-B TNL can shrink and be annihilated without crossing the TRIM, because 4N≡ 0 (mod 4). On the other hand, to annihilate a type-A TNL, it should go across the TRIM, and thereby the gap closes at the TRIM.From the argument of the ℤ_2 topological numbers,in order to annihilate a type-A TNL, the Z_2 topological number should change, and thus this gap closingshould necessarily accompany an exchange of the parity eigenvalues at the TRIM between the valence and the conduction bands. This agrees with the argument in Eq. (<ref>). If the two bands forming the TNL have the same parity eigenvalues,the gap closing at the TRIM is not allowed because of the level repulsion. Meanwhile, when the bands have opposite parity eigenvalues, there are no constraints for gap-closing points. We remark that one can change the numbers of type-A TNLs and type-B TNLs under continuous deformation of the system without changing the topological invariant ν_0.For example, one can continuously deform from the TNLs in Ca to those in Yb via Lifshitz transitions,without changing ν_0.§ PHASE TRANSITION INVOLVING TNLSMSTo elucidate phase transitions involving the TNLSM phase,we add symmetry-breaking perturbations to the system.We use the effective Hamiltonians for the TNLSMs described by Eqs. (<ref>) and (<ref>). We assume that the TNLsare realized before breaking the symmetry, and let ℓ denote the TNL.On the TNLs, a_y(q)=a_z(q)=0 in Eq. (<ref>) or a_x(q)=a_z(q)=0 in Eq. (<ref>) holds. In this section, we ignore crystallographic symmetries other than I symmetry. §.§ Type-A TNLSMs with TR breakingFirstly, we break the TR symmetry in type-A TNLSMs. The allowed perturbation term is a_x(q)σ _x which satisfies a_x(-q)=-a_x(q) because of the I symmetry. We can assume that the perturbation is so small that the coefficients of Eq. (<ref>) remain zero on ℓ after the TR breaking.Thus, the gap closes when a_x(q)=0,^∃q⊂ℓ in the presence of the small TR breaking term. In fact, such wavevectors satisfying a_x(q)=0 always exist somewhere on ℓ because a_x(q) is an odd function of q and the type-A TNL ℓ encloses the TRIM represented by q=0.The emergent gapless points are Weyl nodes [Fig. <ref> (a)]. The Weyl nodes appear symmetrically with respect to the TRIM,and the minimal number of Weyl nodes is two. The two Weyl nodes are related by the I symmetry, and thus have opposite monopole charges. Hence, when the TR symmetry is broken, the system changes from the type-A TNLSM phase to the spinless WSM phase.We also show another proof of the appearance of the spinless WSM phase by breaking the TR symmetry based on a topological description. We assume that a type-A TNL encloses a TRIM Γ, and that the energy bands are gapless only on the TNL. We consider a TR-invariant plane P_Γ which includes Γ. The TR-invariant plane has 2(2N+1) intersection points ±k_i (i=1, ⋯ ,2N+1) with the type-A TNL. We focus on pairs of the gapless points on P_Γ, which arerelated by the I symmetry. Because the closings of the gap at these gapless pointsare protected topologically by the TR- and I symmetries,the bands generally become gapped at these points when we weakly break the TR symmetry. The perturbation terms obtained in each pair ±k_i have opposite signs, because of the I symmetry.Thus, the bands at each pair of wavevectors ±k_i contribute by +1 or -1 to the Chern number defined on the planeP_Γ <cit.>. By summing over all the (2N+1) pairs, the Chern number on the planeP_Γ is nonzero. On the other hand, we introduce another plane P_Γ∥ which is parallel to P_Γ, but does not intersect nodal lines [Fig. <ref> (b)]. By assumption, the Chern number defined on P_Γ∥ is zero before introducing the perturbation.As long as the perturbation is small, the band gap does not close on the plane P_Γ∥, and the Chern number remains zero on P_Γ∥ after the TR symmetry is broken. Therefore, the Chern numbers are different between P_Γ and P_Γ∥, and it means that between P_Γ and P_Γ∥ the energy bands should have gapless points, i.e. Weyl nodes. As a consequence, the WSM phase necessarily emerges from the type-A TNLSM phase by breaking the TR symmetry. §.§ Type-A TNLSMs with I breakingSecondly, we introduce a term which weakly breaks the I symmetry but preserves the TR symmetry in type-A TNLSMs. The allowed term is described by a_x(q) which satisfies a_x(-q)=a_x(q). Then, a_x(q) can be nonzero on the whole loop ℓ since a_x(q) is an even function of q. Therefore, the energy bands can become gapped. It is natural from the viewpoint of topology; because the perturbation terms obtained in each pair ±k_i have the same signs, the Chern number on the plane P_Γ is zero, implying thatthere appear no gapless points in general.§.§ Type-B TNLSMs with TR or I breakingNext, we study a phase transtion of the type-B TNLSM phase by breaking the TR- or I symmetry.The additional perturbation term for Eq. (<ref>) and (<ref>) for breaking either of the TR- or I symmetriesis a_x(q)σ _x and a_y(q)σ _y, respectively.Now the perturbation is generally nonzero everywhere on ℓ, whichever symmetry is broken. Even if the perturbation term is an odd function of q, it can be nonzero on ℓ because the type-B TNLs do not enclose the TRIM, unlike the type-A TNLs. Therefore, in general, by breaking the TR- or I symmetry, a gap opens, and the WSM phase does not appear from the type-B TNLSM phase. §.§ Phase transition in a lattice modelIn this subsection, we see a phase transition from the type-A TNLSM phase to the spinless WSM phase by using a lattice model,and we see agreement with the discussion in Sec. <ref>A.We use a model on a diamond lattice given byH=∑ _<ij>t_ijc^†_ic_j +∑ _≪ ij≫t'_ijc^†_ic_j.The first term represents nearest-neighbor hoppings between the sublattices A and B. Here, we denote the three translation vectors by t_1=a/2(0,1,1),t_2=a/2(1,0,1), and t_3=a/2(1,1,0), where a is a lattice constant. Then, the four nearest-neighbor bonds are τ=a/4(1,1,1), and δ_i=1,2,3=τ-t_i=1,2,3.We express the hoppings in the direction of δ as subscripts. For example, the hoppings in the direction of τ and δ _i=1,2,3 are written by t_τ and t_δ _i=1,2,3, respectively.The second term represents the next nearest-neighbor hoppings. The twelve next nearest-neighbor bonds are represented by ±t_i=1,2,3, and ±u_i=1,2,3, where u_1=t_3-t_2, u_2=t_3-t_1, and u_3=t_1-t_2. In addition, we denote the next nearest-neighbor hoppings between the same sublattices A(B) by t'^A(B)_δ. When the system is I-symmetric, t_τ and t_δ _i=1,2,3 are real, and t'^A_δ=(t'^B_δ)^∗. The Hamiltonian in the momentum space is H(k)= [2∑_dRe[t'^A_d]cosk·d] σ _0+ [ t_τ+∑ _it_δ_icosk·t_i] σ _x+ [∑ _i t_δ_isink·t_i] σ _y+ [ 2∑_dIm[t'^A_d]sink·d] σ _z, where d in the sum runs over t_i=1,2,3 and u_i=1,2,3. The Pauli matrices σ _i=0,x,y,z act on the sublattice degree of freedom. In this model, the parity operator is represented by P=σ _x. Then, the parity eigenvalues ξ of the occupied bands at the TRIM Γ_n_1n_2n_3 are given byξ (Γ_n_1n_2n_3)=-sgn[ t_τ+∑ _it_δ_i(-1)^n_i] .The topological invariant ν _0 is obtained from (-1)^ν _0=∏ _n_j=0,1ξ (Γ_n_1n_2n_3).When t'^A(B)_δ are real i.e. Im[t'^A_δ]= 0, the model has TR symmetry, which case has been studied in Ref. Takahashi13. In Ref. Takahashi13, it is shown that the energy bands can have a type-A TNL around the TRIM Γ_111=L=π/a(1,1,1). The type-A TNL exists when the parity eigenvalue ξ (Γ_111) is opposite from those at the other TRIM. To realize it, we set t_τ/t_δ _1=1.4, t_δ_2/t_δ _1=1.1, and t_δ_3/t_δ _1=0.9. We also assume that all the second nearest neighbor hopping are identical, having the values t'^A_δ/t_δ _1=0.1. Since all the nearest-neighbor hoppings are different and real, the model has only TR- and I symmetries. Then, the type-A TNL appears around the L point as seen in Fig. <ref> (c), whichis as expected from our argument in Sec. <ref>A.Next we break the TR symmetry by adding finite imaginary parts of t'^A_δ. For example, this TR breaking can be included as Peierls phases from magnetization. We put t'^A_d=te^iϕ forthe next nearest-neighbor hoppingsrepresented by d=t_i=1,2,3 and u_i=1,2,3, where t and ϕ are real constants.Then, t'^A_d=te^-iϕ when d =-t_i=1,2,3 and d=-u_i=1,2,3.In order to break the TR symmetry, we put t/t_δ _1=0.1 and ϕ =0.1 for instance. Consequently, we find that a topological phase transition occurs from the type-A TNLSM phase to the spinless WSM phase. Figure <ref> (c) shows the two Weyl nodes which emerge from the type-A TNL.Instead of the TR-breaking, we can break the I symmetry by adding an on-site staggered potential given by H_IB=M_s∑ _iλ _ic_i^†c_i, where λ _i takes values +1 for the A sublattices and -1 for the B sublattices. Then, we can directly see that the type-A TNL becomes gapped.§ CRYSTAL SYMMETRIES AND BAND STRUCTURES OF TYPE-A TNLSMSIn Sec. <ref>, we have discussed the TNLSMs by considering only TR- and I symmetries. In this section, we also take account into twofold rotational (C_2) and mirror (M) symmetries, because C_2M is equal to the space inversion.Particularly, we show how the two symmetries, C_2 and M, constrain band structures having type-A TNLs. §.§ Band structures of the type-A TNLSMsNow, we classify the type-A TNLs into two cases according to whether or not the TRIM which the TNLs enclose is invariant under C_2 and M symmetries. Because P=C_2M, in I-symmetric systems, a little group of the TRIM often contains C_2 and M symmetries in pairs.If the TRIM is not invariant under the two symmetries, we call this case (I). When the TRIM considered is invariant under C_2 and M, we call the case (II).Here, twofold screw symmetries and glide symmetries can be treated similarly toC_2 and M symmetries, respectively, and the systems with these symmetries can be included in the case (II), except for some special cases at theBrillouin zone boundary for nonsymmorphic space groups (see the Appendix <ref>). Actually, the case (II)is moreimportant for application to real materials because 89 space groups of all the 92 space groups with I symmetry have the two symmetries <cit.>.In fact,the band structures and the phase transition of the type-A TNLSMs for (I) have alreadybeen studied in Sec. <ref> and <ref>,because there is no additional symmetry which further constrains the phase transition.For example, CaP_3 <cit.> is included in the case (I).In the case (II), the two symmetries C_2 and M give some constraints to the effective Hamiltonian described by Eq. (<ref>).Since the parity eigenvalues are different for the conduction and the valence bands of the type-A TNLs, either of the C_2 or the M symmetry has different eigenvalues for the conduction and the valence bands. In spinless systems, eigenvalues of the C_2 and the M symmetries take values ± 1. Then, because P=MC_2, there are two cases for combinations of eigenvalues C_2 and M at the TRIM; (II)-(i) eigenvalues of M are the same and those of C_2 are different, and(II)-(ii) eigenvalues of M are different and those of C_2 are the same. They correspond to two different matrix representations: (II)-(i) M=±σ _0 and C_2=±σ _z, and (II)-(ii) M=±σ _z and C_2=±σ _0. We can calculate the band structures for these cases, and the details are shown in the Appendix <ref>. The resulting positions of the nodal lines are shown inFig. <ref>, where we set the twofold rotational axis and the mirror plane to be the z axis and xy plane, respectively. For (II)-(i), as seen in Fig. <ref> (a), the type-A TNL encircles the TRIM and intersects the C_2-invariant axis. The TNL is symmetric with respect to the mirror plane q_z=0. For (II)-(ii), the type-A TNL appears on the mirror-invariant plane as shown in Fig. <ref> (b). §.§ Applications ofthe theory of the band structures to the candidate materials of TNLSMsThe results in the previous subsection can be easily generalized to little groups with many different pairs of C_2 and M symmetries. We apply the theory to several candidates of the type-A TNLSMs. First, we consider fcc Ca, whose space group is No. 225 <cit.>. Ca have four type-A TNLs near each of the four TRIM L. The little group at the L points is D_3d which contains three C_2-rotational operations. The two bands forming the TNLs belong to A_1g and A_2u states at the point L, and they have different C_2 eigenvalues. Therefore, the TNLs in Ca intersect the L-W lines, which are the C_2 invariant axesbut do not lie on mirror planes, in accordance with our theory.Next, we apply this theory to Cu_3ZnN <cit.>. The space group of Cu_3ZnN is No. 221. The energy bands have type-A TNLs around the three TRIM X, whose little groups are D_4h. The type-A TNLs are formed by A_2u and A_1g states at the X points. The D_4h group contains a fourfold-rotational (C_4) operation, four C_2 operations whose rotational axes are normal to the principal axis, and the corresponding five mirror operations. Then, the A_2u and A_1g states have opposite eigenvalues of the four C_2 symmetries. Thus, the TNLs cross the X-M lines and X-R lines, which are the C_2-invariant axes. Meanwhile, the two states have the same eigenvalues of the C_4 symmetry. Hence, C_4=σ _0 leads to different eigenvalues of the mirror symmetry M=P(C_4)^2=σ _z . Therefore, the TNLs also appear on the mirror-invariant plane normal to the C_4-invariant axis. As a result, the type-A TNLs in Cu_3ZnN not only cross the C_2-invariant axes but also exist on the mirror plane.Last, we remark that the type-A TNLs are predicted to appear on the mirror planes in many candidates such as Cu_3(Pd, Zn)N <cit.>, Ca_3P_2 <cit.>, LaN <cit.>, and compressed black phosphorus <cit.>. They belong to (II)-(ii) andthe existence ofTNLs is understood from the difference in eigenvalues of M between the conduction and valence bands. § PHASE TRANSITIONS OF TYPE-A TNLSMS AND CRYSTAL SYMMETRIESIn this section, we show that for type-A TNLSMs in the case (II), the presence of C_2 and M symmetries changes phase transitions when we break TR- or I symmetry.§.§ Type-A TNLs protected by crystal symmetries with TR breakingHere, we break the TR symmetry in the type-A TNL. When energy bands cross on high-symmetry lines or planes, and have different eigenvalues of crystal symmetries, the band crossing is protected by the symmetries. Therefore, such degeneracyremains on high-symmetry lines or planes, even when the TR symmetry is broken.Therefore, in the case (II)-(i), where the type-A TNL crosses the C_2-invariant axis, the TR breaking creates Weyl nodes on the C_2-invariant axis as shown in Fig. <ref>. In the effective model, the protection originates from the fact that the perturbation a_x(0,0,q_z) always vanishes on the C_2 axis. Next, in the case (II)-(ii), where the type-A TNL is always on the mirror-invariant plane,the nodal line remains on the mirror plane even without the TR symmetry.In the effective 2× 2 model, it is seen from the fact that the perturbationa_x(q_x,q_y,0) vanishes on the mirror plane. In particular, one needs to break the M symmetry in order to realize the WSM phase. §.§ Type-A TNLs protected by crystal symmetries with I breakingWe study effects of the I breaking for the case (II) in this subsection.In the case (II), where the system has C_2 and M symmetries,violation of the I symmetry is equivalent to breaking either C_2 or M symmetry because P=MC_2. Therefore, the topological semimetal phases may survive in a different way between (II)-(i) and (II)-(ii).First, we consider the case (II)-(i). When the type-A TNL intersects C_2-invariant axes, the system becomes a spinless WSM phase by breaking the I symmetry while retaining the C_2 symmetries. Then, we obtain Weyl nodes not only on the C_2 invariant axes but also on theΘ C_2-invariant plane (q_z=0),because of the symmetry protection. In the effective model,Θ C_2=Kσ _z symmetry leads toa_x(q_x,q_y,0)=0, meaning that the perturbation is absent on this Θ C_2-invariantplane. Here, within each pair of nodes related by the TR symmetry, monopole charges are the same. The four nodes correspond to the minimal number of Weyl nodes in TR-invariant WSMs<cit.>.In fact, the appearance of four Weyl nodes can be understood by expanding the I-breaking perturbation term proportional to σ _x. The term expanded near the TRIM to the lowest order is a_x(q)=(α q_x+β q_y)q_z.Therefore, we can see that the Weyl nodes appear when either q_z=0 or q_x=q_y=0is satisfied, giving the four Weyl nodes. On the other hand, for the case (II)-(ii) of the type-A TNLs on the mirror plane. if we leave the M symmetry and break the C_2 symmetry, the nodal line survives because of the M symmetry. The mirror symmetry protects the nodal lines regardless of existence of the I symmetry.Hence, even if the I symmetry is broken,the nodal line remains unless the mirror symmetry is broken.§ CONCLUSION AND DISCUSSIONIn the present paper, we study phase transitions and band evolutions of topological nodal-line semimetals. We classified topological nodal-line semimetals into type-A and type-B in order to describe general phase transitions by breaking time-reversal or inversion symmetry. This classification is based on the geometrical positions of the nodal lines, andwe give effective Hamiltonians for each case for analysis of symmetry breaking.The results show that the topological nodal lines enclosing a TRIM(type-A topological nodal lines) always become Weyl nodes when the time-reversal symmetry is broken. However, breaking of inversion symmetry opens a band gap in the type-A topological nodal-line semimetals, andit is confirmed by our calculation on the lattice model. On the other hand, it is shown that the type-B topological nodal lines, which do not enclose a TRIM,become gapped by breaking time-reversal symmetry. The two types are distinguishable from the shapes of the topological nodal lines.We also showed how band structures of type-A topological nodal lines are determined by the little group at the TRIM. When the topological nodal line encircles the TRIM, which is invariant under C_2 and M symmetries of the system, they cross the C_2-invariant axis and/or appear on the mirror-invariant plane, and consequently are protected by the symmetries. Therefore, the nodal lines or points survive in some cases, even when the time-reversal or the inversion symmetries is broken. The revealed properties are also useful to search spinless topological semimetals in many materials because many space groups with I symmetry have various C_2 and M symmetries. As a result, the spinless WSMs can be predicted in many candidates of the topological nodal-line semimetals protected by the C_2 symmetries when we break the I symmetry.Our study tells us how to realize a spinless WSM phase. In electronic systems, the spinless WSM phase appear from the type-A topological nodal-line semimetal phasesnot only by a circularly polarized light <cit.> but also by magnetic ordering, an external electric field, structural transition, and so on. Moreover, our theory can be applied to spinless fermions in cold atoms and bosonic bands. The experiments have potential in bosonic metamaterials of photons and phonons where lattice structure and its symmetry are flexibly controllable.The TNLs may cross each other, and our classification into type-A and type-B still works for the TNLs with mutual crossings. Meanwhile, in the presence of crossings,the effective models become different from those discussed in our paper, which is beyond the scope of this paper. In this context, classification ofpossible patterns of their crossingsand their evolution underthe spin-orbit coupling was studied recently in Ref. Kobayashi17.In Ref. Kobayashi17 only the TNLs based on the mirror symmetry are discussed,and the main focus is on the crossings of the TNLs. Meanwhile our paper includes both the TNLs from the mirror symmetry and those from the π Berry phase, and therefore the target of our research is different from that of Ref. Kobayashi17.This work was supported by JSPS KAKENHI Grant Numbers 16J08552 and 26287062 and by MEXT Elements Strategy Initiative to Form Core Research Center (TIES).§ CLASSIFICATION OF THE TYPE-A TNLS AND THEIR BAND STRUCTURESWe have classified type-A TNLs into the two cases (I) and (II) in Sec. <ref>. The classification provides information on band evolutions and phase transitions involving type-A TNLs. In this appendix, we explain band structures of type-A TNLs in the two cases (I) and (II).The type-A TNLs are formed by two bands with opposite parity eigenvalues at the TRIM.and they can be descrived by the two-band effective Hamiltonian by Eq. (<ref>). Here, to describe the TNLs by the two-band effective Hamiltonian, we assume that the TNLs are formed only by two nondegenerate states. In fact, a similar two-band effective Hamiltonian has been used in spinful WSMs in order to describe band evolutions in Ref. Murakami17, and therefore, here we can extend the analysis in Ref. Murakami17 to some of the spinless TNLSMs as well. In Ref. Murakami17, it is shown that when two bands touch each other on high-symmetry lines or planes, emergent gapless nodes evolve along the lines and the planes where the two bands have the different eigenvalues. By using these results, we can understand band structures of type-A TNLs.For example, in Sec. <ref> we introduced two cases(II)-(i) and (ii) for type-A TNLs,formed by two bands with C_2 and M symmetry.These two casesare classified according to the C_2 and M eigenvalues at the TRIM, andthe effective Hamiltonian for the two cases are constrained by these two symmetries. For simplicity, we set the C_2 axis and the M plane to be the z axis and xy plane, respectively. From the constraints, we obtain σ _z H_TNL(-q_x,-q_y,q_z)σ _z=H_TNL(q_x,q_y,q_z) in the case (II)-(i). Meanwhile, in the case (II)-(ii), we obtain σ _z H_TNL(q_x,q_y,-q_z)σ _z=H_TNL(q_x,q_y,q_z). Although eigenvalues of the C_2 and M symmetries are different in spinless and spinful systems, the expressions for these constraints are the same both in spinless and in spinful cases <cit.>. As a result, the type-A TNL of the case (II)-(i) crosses the C_2 axis while the type-A TNL of the case (II)-(ii) appears on the M plane, both in spinless and in spinful systems. In some cases, several type-A TNLs can enclose the same TRIM if a little group at the TRIM contains some C_2 and M symmetries.There are various options for the little group at the TRIM, which affects the position of thetype-A TNLs.(I) refers to the case where the TRIM is neither C_2- nor M-symmetric. Therefore, the little group is C_i or C_3i. In this case (I), the type-A TNL does not necessarily cross high-symmetry lines. For example, if the two bands have the same C_3 eigenvalues at the TRIM whose little group is C_3i, the type-A TNL lies at a general position.(II) refers to the case where the TRIM is C_2- and M-symmetric. In this case, as we have shown in Sec. <ref>,the type-A TNLs necessarily cross the high-symmetry lines or appear on the mirror-invariant planes, thanks to symmetry protection. The little groups can also have rotational symmetries besides the C_2 symmetry. When the conduction and the valence bands belong to different subspaces of the C_n-rotational symmetries, the type-A TNLs can intersect the high-symmetry lines. On the other hand, when the two bands belong to different subspaces of the mirror symmetry, the type-A TNLs are on the mirror-invariant plane. In particular, if the eigenvalues of the C_4 or C_6 symmetry are the same, the type-A TNL exists on the mirror-invariant plane perpendicular to the C_4- or C_6-invariant axes because (C_4)^2=C_2 and (C_6)^3=C_2.Here we comment on TNLs in systems with a nonsymmorphic space group having twofold screw (S_2) symmetries or glide (G) symmetries. Inside the Brillouin zone, the TNLs are similar to those with a symmorphic space group, because there is no extra degeneracy due to nonsymmorphic symmetry together with TR symmetry.Meanwhile, S_2 and G symmetries may give rise to extra degeneracy on the Brillouin zone boundary by TR symmetry, if the square of the symmetry operations becomes -1. For example, by combination of several G symmetries and the TR symmetry, spinless nodal lines can contain fourfold-degenerate points at theTRIM on the surface of the Brillouin zone <cit.>. As another example, a nodal surface appears on the Θ S_2-invariant plane on the surface of the Brillouin zone <cit.>. Such cases are beyond the scope of this paper because the band crossing cannot be described by the two-band effective Hamiltonian.apsrev4-1 | http://arxiv.org/abs/1706.08551v2 | {
"authors": [
"Ryo Okugawa",
"Shuichi Murakami"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170626182144",
"title": "Universal phase transition and band structures for spinless nodal-line and Weyl semimetals"
} |
Full dataset and evaluation routines available at https://jasbergk.wixsite.com/research Web Science Group Heinrich-Heine-University Duesseldorf [email protected] Science Group Heinrich-Heine-University Duesseldorf [email protected] this paper, we examine the statistical soundness of comparative assessments within the field of recommender systems in terms of reliability and human uncertainty. From a controlled experiment, we get the insight that users provide different ratings on same items when repeatedly asked. This volatility of user ratings justifies the assumption of using probability densities instead of single rating scores. As a consequence, the well-known accuracy metrics (e.g. MAE, MSE, RMSE) yield a density themselves that emerges from convolution of all rating densities. When two different systems produce different RMSE distributions with significant intersection, then there exists a probability of error for each possible ranking. As an application, we examine possible ranking errors of the Netflix Prize. We are able to show that all top rankings are more or less subject to high probabilities of error and that some rankings may be deemed to be caused by mere chance rather than system quality.Re-Evaluating the Netflix Prize - Human Uncertainty and its Impact on Reliability Sergej Sizov October 24, 2017 ===================================================================================§ INTRODUCTIONRecommender systems play a central role nowadays and their sound evaluation is crucial. For this purpose, a variety of quality metrics have been developed <cit.>, such as the RMSE which has been used in one of the largest recommender competitions, the Netflix Prize. In this contribution we draw attention to possible inaccuracies within recommender assessment caused by uncertain user feedback, exemplary in the evaluation of the Netflix Prize.In a systematic experiment, we required participants to (re-)rate theatrical trailers several times. Our results reveal that users are not able reproduce their own decisions, i.e. given ratings fluctuate around a central tendency. This result is consistent with other studies <cit.> and theoretical models of the human mind <cit.>. Based on our experiment and in accordance to the Netflix Prize, one may compute the RMSE for different recommender systems for each of the rating trials. Figure <ref> shows a histogram of these different RMSE outcomes for three sample recommender systems (defined by their predictors π). It is apparent that the RMSE itself yields a particular degree of uncertainty, due to uncertain user feedback. When ranking these recommender systems, Figure <ref> allows for a variety of possible orders that emerge with different frequencies. The problem is most obvious for recommender R2 (green) as it could be both, the best or the worst recommender, although it operates on the same users rating the same items. Thus, the question for a comparison changes, namely from “Is R1 better than R2?” to “How likely is it that R1 is better than R2?”. Vice versa, no matter what ranking we finally opt for, there is always a certain chance of error for this decision.The impact of uncertain user feedback and possible ranking errors is in the main focus of this paper and will be exemplified using the Netflix Prize. The central research question is thus: How reliable is the Netflix Prize (as an example for evaluations in general) when considering human uncertainty? § RELATED WORKThe observation of uncertain user feedback in product evaluations was been made before in <cit.>.The concept of this study has been combined with modern methods of experimental psychology <cit.> to conduct out our own study. Latest neuroscience research considers action-coordinating cognitions to be based on perceptions in the form of distributions which are constantly updated by a complicated generative process within the human cortex <cit.>. Decision making thus yields a specific volatility, which we denote human uncertainty in our context. This uncertainty can be explained by the irregular release of neuromodulators like dopamine and acetylcholine <cit.>. These findings support our idea of modelling user feedback as individual distributions. The handling of uncertainty has a long tradition in the field of physics and metrology <cit.>. In particular, <cit.> describes the propagation of uncertain quantities when new ones are calculated therefrom. This model of uncertainty is used to calculate the distributions of the RMSE. With this collection of methods, we are able to determine the human uncertainty experimentally, to investigate their propagation in the RMSE, and to uncover possible ranking errors in the Netflix Prize. § CASE STUDY Let X_ν∼𝒩(μ_ν,σ_ν^2)be a family of n random variables (representing user ratings) which are assumed to be normally distributed in accordance to <cit.>. The RMSE thus becomes a random variable itself. The distribution emerges as a convolution of n density functions with respect to the mathematical model RMSE = √(1/n∑_ν (X_ν- π_ν )^2).Using the Gaussian Error Propagation <cit.> and the Central Limit Theorem, the RMSE∼𝒩(μ,σ^2)yields a normal distribution withμ≈√(1/n∑_νσ_ν^2+Δ_ν^2)andσ^2≈∑_νσ_ν^4+2σ_ν^2Δ_ν^2/2n·∑_νσ_ν^2+Δ_ν^2 .with the substitution Δ_ν=μ_ν-π_ν. Let now Z_1∼𝒩(μ_1,σ_1^2) and Z_2∼𝒩(μ_2,σ_2^2) be two RMSE random quantities that correspond to different recommender systems. Assuming μ_1< μ_2, we would consider system 1 to be better than system 2. However, this decision may be subject to an error which occurs with a probability of P(Z_1≥ Z_2)= Φ ( (σ_1^2+σ_2^2)^-1/2(μ_1-μ_2)).where Φ is the standard-normal cumulative distribution function. With this framework we are able to elaborate the reliability of the Netflix Prize. At this point, it appears to be challenging that Netflix did not collect any information about human uncertainty. However, for the size of Netflix's test record (n=2.8· 10^6), this is not a problem at all since the RMSE's variance scales with 1/2n. This is illustrated in Figure <ref>. It is apparent that the true extent of human uncertainty no longer influences the variance significantly when one has to deal with big data. In fact, we estimated the uncertainty for the Netflix Prize in three different ways:Approach A) ML-fitting of human uncertainty based on our experiment provided a density from which random draws were made to be associated to each rating of the Netflix record.Approach B) Human uncertainty was randomly sampled from different distributions (e.g. uniform, triangular, beta) and associated to each rating of the Netflix record.Approach C) Having a 5-star scale, human uncertainty yields certain limitations. Association of minimum and maximum uncertainty to each Netflix rating produces an interval in which the RMSE's variance is located. With <ref> we can then transform each RMSE score in the Netflix leaderboard into a random quantity Z∼𝒩(score, σ_score^2). In doing so, methods A and B always provide the same value σ_score^2= 0.0006. For method C, there are intervals whose mean exactly corresponds to the result of method A and B.This empirically shows that the extent of human uncertainty for each individual rating no longer contributes to the variance of the RMSE since only the size of the data record is decisive here. With <ref> we can then estimate the error probabilities that correspond to each pair-wise ranking. The results are listed in the Table below. R_i represents the recommender system with leaderboard placing i. The entry p_ij is the error probability of the ranking R_i < R_j. For example, the error probability of placing 3 being better than placing 4 is nearly 25%, i.e. these systems would swap placings on the leaderboard in one of four repeated evaluations. Especially for the last placings there is a disillusioning message: Placings 9, 10, 11 and 12 hold nearly 50% probability of error. Thus, the entry into the top 10 of the Netflix Price might be based on mere chance rather than system quality. R_1/2 R_3 R_4R_5R_6R_7R_8R_9R_10R_11R_12 R_1/2 .50 .04 .01 .00 .00 .00 .00 .00 .00.00.00R_3 .50 .24 .14 .08 .01 .00 .00 .00.00.00R_4.50.36 .24.06 .00 .00 .00.00.00R_5 .50 .36 .12 .01 .00 .00.00.00R_6.50 .20 .02 .00 .00.00.00R_7 .50 .10.01 .00.00.00R_8.50.12.10.10.08 R_9 .50 .45.45.41 R_10.50.50.45 R_11 .50 .45 R_12 .50This example encourages to consider evaluations based on user feedback more carefully, i.e. not to search for the only true ranking, but to weigh all possibilities against each other on the basis of their probabilities.§ CONCLUSION AND FUTURE WORKHuman uncertainty strongly influences the evaluation of recommender systems. Hence, it is crucial to continue investigating this impact in our systems and evaluation processes. In particular, this contribution is an opportunity to rethink about statistical soundness of even more modern and sophisticated quality measures than the RMSE. Future research may focus on the impact on other forms of recommender assessment and on developing new metrics that explicitly take human uncertainty into account. ACM-Reference-Format | http://arxiv.org/abs/1706.08866v1 | {
"authors": [
"Kevin Jasberg",
"Sergej Sizov"
],
"categories": [
"cs.HC"
],
"primary_category": "cs.HC",
"published": "20170627140906",
"title": "Re-Evaluating the Netflix Prize - Human Uncertainty and its Impact on Reliability"
} |
arabic | http://arxiv.org/abs/1706.08860v2 | {
"authors": [
"Teppei Okumura",
"Takahiro Nishimichi",
"Keiichi Umetsu",
"Ken Osato"
],
"categories": [
"astro-ph.CO",
"astro-ph.GA"
],
"primary_category": "astro-ph.CO",
"published": "20170627135941",
"title": "Intrinsic Alignments and Splashback Radius of Dark Matter Halos from Cosmic Density and Velocity Fields"
} |
24pt Multi-polaron solutions, nonlocal effectsand internal modes in a nonlinear chain M. Pereiro December 30, 2023 ================================================================================== 24pt The beta family owes its privileged status within unit interval distributions to several relevant features such as, for example, easyness of interpretation and versatility in modeling different types of data. However, its flexibility at the unit interval endpoints is poor enough to prevent from properly modeling the portions of data having values next to zero and one. Such a drawback can be overcome by resorting to the class of the non-central beta distributions. Indeed, the latter allows the density to take on arbitrary positive and finite limits which have a really simple form. That said, new insights into such class are provided in this paper. In particular, new representations and moments expressions are derived. Moreover, its potential with respect to alternative models is highlighted through applications to real data. Keywords: generalizations of beta distribution, unit interval limits, non-centrality. § INTRODUCTIONThe beta distribution plays a prominent role in the analysis of random phenomena which take on values with lower and upper bounds. Indeed, allowing its probability density function to have a great variety of shapes, such a distribution is versatile enough to model data arisen from a wide range of fields. In this regard, for example see <cit.>, <cit.>, <cit.>, <cit.>.However, the density of the latter shows poor flexibility at the unit interval endpoints. In fact, its limiting values are equal to one if the shape parameters are unitary (in this case it reduces to the uniform one) and are equal to zero or infinity otherwise. As a consequence of this, the beta distribution prevents from properly modeling the portions of data having values next to zero and one.In this regard, in the literature there exists some generalizations of the beta model that enable to overcome this limitation thanks to a richer parametrization. For instance, we recall the Libby and Novick's generalized beta <cit.>, the Gauss hypergeometric <cit.> and the confluent hypergeometric <cit.> distributions. Indeed, the densities of the aforementioned models can take on positive and finite values at zero and one when the shape parameters are unitary. See for example <cit.> to get an overview of such distributions. That said, the present paper aims at getting an insight into the class of the non-central beta distributions. The latter is another extension of the beta model that exhibits the aforementioned peculiarity. Indeed, its density shows positive and finite limits that, interestingly, have a really simple form <cit.>. Hence, such class is considered to be worthy of further investigating. More specifically, our intent is to provide a valid point of reference for the study of such distributions. In this regard, we are supported by the fact that in recent years the non-central beta distributions have attracted many applications. For example, <cit.> pointed out that the semblance of a single wave propagating across a receiver array with added Gaussian noise is distributed according to a special case of non-central beta, called type 1. In the setting of magnetic resonance image reconstruction, <cit.> introduced a new estimating method for coil sensitivity profiles that uses spatial smoothing and additional body coil data for phase normalization. Upon providing detailed information on the statistical distribution of this estimator, they showed that the square of the random variable ℛ_k(), which plays a relevant role in the definition of such a method, follows a doubly non-central beta distribution, the latter being the most general non-central extension of the beta one. In order to analyze the bias and the variance of this estimator, the calculation of the first two raw moments of the aforementioned distribution was needed.Finally, the present paper is organized as follows. In Section <ref>, in order to go into the matter of interest in due depth, we shall focus on the non-central chi-squared distribution. Indeed, the latter covers a crucial role in the study of the family of generalizations of the beta distribution we are interested in. More precisely, its definition and some useful properties are briefly recalled and a new general expression for its moments about zero is derived. In Section <ref> the definition and various representations of the doubly non-central beta distribution are provided. In particular, a new representation of a random variable distributed as previously said is here obtained in terms of a convex linear combination of a central component and a purely non-central one. In Section <ref> some significant plots of the density are shown. In this regard, a special focus is given to the case in which both the shape parameters are unitary; in fact, in this case the density shows the attractive feature of taking on arbitrary finite and positive limits at zero and one. Section <ref> presents how a simple approximation for the doubly non-central beta distribution can be determined by applying the Patnaik's approximation for the non-central chi-squared one <cit.>. Section <ref> sheds new light on the issue of moments expression. As a matter of fact, a more straightforward general formula for the moments about zero of such distribution is derived. Last but not the least interesting, in Section <ref> the potential of the doubly non-central beta distribution is highlighted through applications to real data with respect to the above mentioned alternative models on the real interval (0,1). The issue of the parameters estimation is here addressed using both the moments and the maximum-likelihood methods. Some concluding remarks are provided in Section <ref>.For clarity of exposition, the proofs of all the following results are given in the Appendix <ref>, while in the Appendix <ref> the implementation of the major issues dealt with in this paper is provided inprogramming language.§ PRELIMINARIES ON THE NON-CENTRAL CHI-SQUARED DISTRIBUTION §.§ Definition, representations and propertiesIn this Section we shall recall the definition and some useful properties of the non-central chi-squared distribution. The latter represents the main ingredient for the study of the class of distributions on the real interval (0,1) we are interested in. In fact, some results included in the remainder of this paper, such as Propositions <ref> and <ref>, ensue from analogous results regarding the present distribution (Properties <ref> and <ref>, respectively), while others, such as Propositions <ref> and <ref>, are strongly implied by some of its properties (Properties <ref> and <ref>, <ref>, respectively).That said, the non-central extension of the chi-squared distribution is defined as follows.Let W_k, k=1,…,g, be independent and normally distributed random variables with expectations μ_k and unitary variances. Then, a random variable is said to have a non-central chi-squared distribution with g>0 degrees of freedom and non-centrality parameter λ=∑_k=1^gμ^2_k≥ 0, denoted by χ'^ 2_g (λ), if it is distributed as Y'=∑_k=1^gW^2_k <cit.>. The case λ=0 clearly corresponds to the χ^2_g distribution.The density function f_Y' of Y' ∼χ'^ 2_g (λ) can be expressed as:f_Y'(y;g,λ)=∑_i=0^+∞e^-λ/2(λ/2)^i/i!y^g+2i/2-1e^-y/2/Γ(g+2i/2)2^g+2i/2,y>0,i.e. as the series of the χ^2_g+2i densities, i ∈ℕ∪{0}, weighted by the probabilities of a Poisson random variable with mean λ/2, λ≥0 (the case λ=0 corresponding to a random variable degenerate at zero).In view of Eq. (<ref>), the χ'^ 2_g (λ) distribution admits the following mixture representation.[Mixture representation of χ'^ 2_g (λ)]Let Y' have a χ'^ 2_g (λ) distribution and M be a Poisson random variable with mean λ/2. Then, Y' admits the following representation:Y' | M ∼χ^2_g+2M.Interestingly, a non-central chi-squared random variable with g degrees of freedom and non-centrality parameter λ can be additively decomposed into two components, a central one with g degrees of freedom and a purely non-central one with non-centrality parameter λ <cit.>. The latter can be easily obtained from Property <ref> by making use of the reproductive property of the chi-squared distribution with respect to degrees of freedom. Such representation turns out to be as follows. [Sum of a central part and a purely non-central part]Let Y' ∼χ'^ 2_g (λ). Then:Y'=Y+∑_j=1^MF_j,where: i) Y, M, {F_j} are mutually independent,ii) Y ∼χ^2_g, M ∼(λ/2) and {F_j} is a sequence of independent random variables with χ^2_2 distribution.In the notation of Property <ref>, the random variable Y'_pnc=∑_j=1^MF_j is said to have a purely non-central chi-squared distribution with non-centrality parameter λ. Indeed, it is denoted by χ'^ 2_0 (λ), the degrees of freedom being equal to zero.The case g=2 is of prominent interest in the present setting; in fact, in such case the limit at 0 of the non-central chi-squared density is decreasing in λ.[Limit at 0 of the χ'^ 2_g (λ) density when g=2]Let Y' be a χ'^ 2_2 (λ) random variable and f_Y'(y; 2,λ) denote its density function. Then lim_y → 0^+f_Y'(y; 2,λ)=1/2e^-λ/2. Plots of the latter are displayed in Figure <ref> for selected values of the non-centrality parameter.The non-central chi-squared distribution is reproductive with respect to both degrees of freedom and non-centrality parameter. The latter can be easily derived from its characteristic function <cit.>. [Reproductive property of χ'^ 2_g (λ)]If Y'_j, j=1,…,m, are independent with χ'^2_g_j(λ_j) distributions, then Y'^+=∑_j=1^m Y'_j ∼χ'^2_g^+(λ^+), with g^+=∑_j=1^m g_j and λ^+=∑_j=1^m λ_j.Finally, we recall the simple approximation for the non-central chi-squared distribution suggested by Patnaik <cit.>. [Patnaik's approximation for χ'^ 2_g (λ)]Let Y' have a χ'^2_g(λ) distribution with g>0 and λ>0 and Y have a χ^2_ν distribution with ν=(g+λ)^2/g+2λ. Furthermore, let Y'_P=ρY ∼(ν/2,1/2 ρ), with ρ=g+2λ/g+λ. Then, one can approximate Y' d≈ Y'_P. In the notation of Property <ref>, observe that as λ tends to 0^+, ν tends to g and ρ tends to 1; therefore, the distributions of both Y' and Y'_P tend to the χ^2_g one. §.§ A note on the moments about zeroThe r-th moment about zero of Y' ∼χ'^ 2_g (λ), g>0, can be evaluated according to the following formula set out by <cit.>:𝔼[(Y' )^r ]=2^rΓ(r+g/2) ∑_j=0^rrj(λ/2)^j/Γ(j+g/2). A new moment formula for the non-central chi-squared distribution can be derived regardless of Eq. (<ref>) by means of the following simple expansion of the ascending factorial of a binomial, which, as far as we know, has never been discussed in the literature.In this regard, we recall that:(a)_0=1, (a)_l=a(a+1)…(a+l-1),l ∈ℕis the ascending factorial or Pochhammer's symbol of a ∈ℝ <cit.>. Observe that for every a ∈ℝ-{0} Eq. (<ref>) is tantamount to:(a)_l=Γ(a+l)/Γ(a),l ∈ℕ∪{0}.Furthermore, in light of Eq. (<ref>), one has:(a)_l+m=Γ(a+l+m)/Γ(a)={[ Γ(a+l)/Γ(a) Γ(a+l+m)/Γ(a+l)=(a)_l(a+l)_m; ; Γ(a+m)/Γ(a) Γ(a+m+l)/Γ(a+m)=(a)_m(a+m)_l ].for every l,m ∈ℕ∪{0}.That said, the aforementioned expansion follows.Let a, b ∈ℝ-{0}. Then, for every l ∈ℕ∪{0}:(a+b)_l=∑_i=0^l1/i![d^i/d a^i(a)_l] b^i,where d^i f/d a^i denotes the i-th derivative of f with respect to a (the case i=0 corresponding to f) and (a)_l is defined as in Eq. (<ref>). For the proof see <ref> in the Appendix.The latter result and the mixture representation in Eq. (<ref>) lead to the following new general formula for the moments of the non-central chi-squared distribution. Let Y' have a χ'^ 2_g (λ) distribution with g>0. Then, for every r ∈ℕ, the r-th moment about zero of Y' can be written as:𝔼[(Y' )^r ]=2^r∑_i=0^r∑_j=0^i𝒮(i,j) 1/i![d^i/d h^i(h)_r] (λ/2)^j,where 𝒮(i,j) is a Stirling number of the second kind, h=g/2 and (h)_r is defined as in Eq. (<ref>). For the proof see <ref> in the Appendix. However, neither the moments formula available in the literature nor the one herein derived apply in case of zero degrees of freedom. As far as the computation of the r-th moment about zero of the purely non-central chi-squared distribution is concerned, the following formula can be used. Let Y'_pnc have a χ'^ 2_0 (λ) distribution. Then, for every r ∈ℕ, the r-th moment about zero of Y'_pnc can be written as:𝔼[(Y'_pnc)^r ]=2^r∑_i=0^r∑_j=0^i|s(r,i)|𝒮(i,j) (λ/2)^j,where |s(r,i)| is an unsigned Stirling number of the first kind and 𝒮(i,j) is a Stirling number of the second kind. For the proof see <ref> in the Appendix. Finally, the comparison between Eq. (<ref>) and Eq. (<ref>) leads to the following identity. The latter will be used in Section <ref> in order to derive a new general formula for the moments about zero of the non-central beta distributions. Let r ∈ℕ and h ∈ℝ - {0}. Then:rj(h)_r/(h)_j=∑_i=j^r𝒮(i,j) 1/i![d^i/d h^i(h)_r], ∀ j=0,…,r,where (h)_r is defined as in Eq. (<ref>) and 𝒮(i,j) is a Stirling number of the second kind. For the proof see <ref> in the Appendix. § THE NON-CENTRAL BETA DISTRIBUTIONS§.§ Definitions and representationsIt is well known that if Y_i, i=1,2, are independent chi-squared random variables with 2 α_i>0 degrees of freedom, then the random variable:X=Y_1/Y_1+Y_2has a beta distribution with shape parameters α_1, α_2, denoted by Beta(α_1, α_2). We point out that a (α_1,0) random variable with α_1>0 is degenerate at one: in fact, the chi-squared random variable present only at denominator in Eq. (<ref>) is degenerate at zero. Similarly, a (0,α_2) random variable with α_2>0 is degenerate at zero: in fact, the chi-squared random variable present at both numerator and denominator in Eq. (<ref>) is degenerate at zero, too. Then, we recall that the beta density function takes the following form:(x;α_1,α_2)=x^α_1-1 (1-x)^α_2-1/B(α_1,α_2),0<x<1 . That said, in order to go into the matter of interest in due depth, we recall herein a characterizing property of independent chi-squared (and, more generally, gamma) random variables. The latter is a matter of great consequence for our interests; in fact, it will be largely used in the derivation of all the results proved in the sequel. Thus, it is given a special reference. [Characterizing property of independent χ^2 random variables]Y_i, i=1,2, are independent chi-squared random variables if and only if the compositional ratio X=Y_1/(Y_1+Y_2) is independent of Y_1+Y_2. By replacing the two chi-squared random variables involved in Eq. (<ref>) with two independent non-central ones, we obtain the definition of the “doubly” non-central beta distribution, that is the most general non-central extension of the beta one. The latter is defined as follows.Let Y'_i, i=1,2, be independent χ'^ 2_2α_i(λ_i) random variables. Then, a random variable is said to have a doubly non-central beta distribution with shape parameters α_1,α_2 and non-centrality parameters λ_1,λ_2, denoted by ”(α_1,α_2,λ_1,λ_2), if it is distributed asX'=Y'_1/Y'_1+Y'_2<cit.>. The case λ_1=λ_2=0 clearly corresponds to the beta distribution. Moreover, by taking α_1=α_2=0 in Eq. (<ref>), the latter degenerates into the compositional ratio X'_pnc of two purely non-central chi-squared independent random variables with non-centrality parameters λ_1, λ_2. Its distribution is denoted by ”(0,0,λ_1,λ_2). The ” density can be easily derived by using the mixture representation of the non-central chi-squared distribution. Specifically, let M_i, i=1,2, be independent Poisson random variables with means λ_i/2. Conditionally on (M_1,M_2), X' has a Beta(α_1+M_1, α_2 + M_2) distribution, Y'_i|(M_1,M_2) being independent with distributions χ^2_2α_i + 2 M_i, i=1,2. Therefore, the density function f_X' of X' ∼”(α_1,α_2,λ_1,λ_2) can be stated as:f_X'(x;α_1,α_2,λ_1,λ_2)= =∑_j=0^+∞∑_k=0^+∞e^-λ_1/2(λ_1/2)^j/j!e^-λ_2/2(λ_2/2)^k/k!x^α_1+j-1(1-x)^α_2+k-1/B(α_1+j,α_2+k), 0<x<1,i.e. as the double series of the Beta(α_1+j,α_2+k) densities, j,k ∈ℕ∪{0}, weighted by the joint probabilities of the bivariate random variable (M_1,M_2), where M_i, i=1,2, are independent with (λ_i/2) distributions.By analogy with the density, the distribution function F_X' of X' ∼”(α_1,α_2,λ_1,λ_2) can be stated as:F_X'(x;α_1,α_2,λ_1,λ_2)= =∑_j=0^+∞∑_k=0^+∞e^-λ_1/2(λ_1/2)^j/j!e^-λ_2/2(λ_2/2)^k/k!B(x;α_1+j,α_2+k)/B(α_1+j,α_2+k), 0<x<1,i.e. as the double series of the Beta(α_1+j,α_2+k) distribution functions, j,k ∈ℕ∪{0}, weighted by the joint probabilities of the bivariate random variable (M_1,M_2), where M_i, i=1,2, are independent with (λ_i/2) distributions. We recall that B(x; a,b)=∫_0^x t^a-1 (1-t)^b-1dt is the incomplete beta function. An implementation of Eq. (<ref>) inlanguage is proposed in <ref>, <ref>.The above discussion directly leads to the following mixture representation.[Mixture representation of B”]Let X' have a ”(α_1,α_2,λ_1,λ_2) distribution and M_i, i=1,2, be independent Poisson random variables with means λ_i/2. Then, X' admits the following representation:X' |(M_1, M_2) ∼(α_1+M_1, α_2+M_2) .In view of the foregoing arguments, it's clear that Property <ref> is no longer valid in the non-central setting. However, an interesting generalization of the latter holds true. As a matter of fact, a doubly non-central beta random variable is herein proved to be independent of the sum of the two non-central chi-squared random variables involved in its definition in a suitable conditional form. More precisely, in the notation of Eq. (<ref>), the latter occurs conditionally on the sum M^+ of the two Poisson random variables on which both X' and Y'_1+Y'_2 depend. As a side effect, the distribution of X' given M^+ is also obtained.Let X' ∼”(α_1,α_2,λ_1,λ_2) and Y'_i, i=1,2, be independent χ'^ 2_2α_i(λ_i) random variables, with Y'^+=Y'_1+Y'_2. Furthermore, let M_i, i=1,2, be independent Poisson random variables with means λ_i/2 and M^+=M_1+M_2. Then, X' and Y'^+ are conditionally independent given M^+ and the density of X' given M^+ is:f_.X'|M^+(x)=∑_i=0^M^+(i;M^+,λ_1/λ^+) ·(x;α_1+i,α_2+M^+-i),where:(i;M^+,λ_1/λ^+)=M^+i(λ_1/λ^+)^i (1-λ_1/λ^+)^M^+-i,i=0,…,M^+.For the proof see <ref> in the Appendix.The doubly non-central beta density can be equivalently written as a perturbation of the corresponding central case, i.e. the beta one, as follows.Let X' have a ” distribution with shape parameters α_1, α_2 and non-centrality parameters λ_1, λ_2. Then, the density f_X' of X' can be written as:f_X'(x;α_1,α_2,λ_1,λ_2)= (x; α_1,α_2 ) · e^-λ^+/2 Ψ_2[α^+;α_1,α_2;λ_1/2x,λ_2 /2(1-x)] ,where (x;α_1,α_2) is defined as in Eq. (<ref>), α^+=α_1+α_2, λ^+=λ_1+λ_2 andΨ_2[α;γ,γ';x,y]=∑_j=0^+∞∑_k=0^+∞(α)_j+k/(γ)_j (γ')_kx^j/j!y^k/k!,x,y ≥ 0is the Humbert's confluent hypergeometric function <cit.>. For the proof see <ref> in the Appendix. Unfortunately, the perturbation representation of the doubly non-central beta density is not so easily tractable and interpretable. Indeed, Eq. (<ref>) shows that, unless a constant term, the beta density is perturbed by a function in two variables given by the sum of the double power series reported in Eq. (<ref>). The latter has not a simple behavior and, to our knowledge, is not reducible into a more tractable analytical form. Therefore, the effect of such perturbation is not easy to understand. However, it can be clearly seen when α_1=α_2=1, because in this case the beta density reduces to the uniform one (see Section <ref>). In this regard, note that, in light of Eq. (<ref>), one obtains:Ψ_2[α^+;α_1,α_2;λ_1/2x,λ_2 /2(1-x)]= =∑_j=0^+∞∑_k=0^+∞(α^+)_j+k/(α_1)_j (α_2)_k(λ_1/2x)^j/j![λ_2/2(1-x)]^k/k!==∑_j=0^+∞(α^+)_j/(α_1)_j(λ_1/2x)^j/j!∑_k=0^+∞(α^++j)_k/(α_2)_k[λ_2/2(1-x)]^k/k!==∑_j=0^+∞(α^+)_j/(α_1)_j(λ_1/2x)^j/j!_1 F_1[α^++j;α_2;λ_2/2 (1-x)],where _1F_1(a;b;x)=∑_k=0^+∞(a)_k/(b)_kx^k/k! is the Kummer's confluent hypergeometric function <cit.>. From Eq. (<ref>) it's immediate to see that the Ψ_2 function can be equivalently expressed as a series of weighted Kummer's confluent hypergeometric functions. Therefore, this formula can be usefully adopted as a natural basis for implementing such a function in any statistical package where the _1F_1 function is already implemented, for instance theprogramming environment. In this regard, an implementation of Eq. (<ref>) inlanguage is proposed in <ref>.A new representation of X'∼”(α_1,α_2,λ_1,λ_2) is now introduced. According to the latter, a doubly non-central beta random variable can be expressed in terms of a convex linear combination of a central component and a purely non-central one. These two additive components are given random weights that can be fully understood by recalling the type 1 and the type 2 non-central beta distributions, namely two special cases of the doubly non-central one. The latter are now briefly recalled but for more details the reader can refer for example to <cit.>.If two random variables Y'_1 and Y_2 are independently distributed according to χ'^ 2_2α_1(λ) and χ^2_2α_2, respectively, then the random variable:X'_1=Y'_1/Y'_1+Y_2is said to have a type 1 non-central beta distribution, denoted by '_1(α_1,α_2,λ). The density function f_X'_1 of X'_1 ∼'_1(α_1,α_2,λ) can be derived by means of a reasoning analogous to the one leading to Eq. (<ref>) and it is given by:f_X'_1(x_1;α_1,α_2,λ)=∑_j=0^+∞e^-λ/2(λ/2)^j/j!x_1^α_1+j-1(1-x_1)^α_2-1/B(α_1+j,α_2), 0<x_1<1,i.e. the series of the Beta(α_1+j,α_2) densities, j ∈ℕ∪{0}, weighted by the probabilities of M ∼(λ/2). Roughly speaking, Eq. (<ref>) can be intuitively established by taking λ_2=0 and renaming λ_1 with λ in Eq. (<ref>). Such a distribution admits the following mixture and perturbation representations. [Mixture representation of '_1]Let X'_1 have a '_1(α_1,α_2,λ) distribution and M be a Poisson random variable with mean λ/2. Then, X'_1 admits the following representation: X'_1| M ∼(α_1+M, α_2).Let X'_1∼'_1(α_1,α_2,λ) and α^+=α_1+α_2. Then, the density f_X'_1 of X'_1 can be written as:f_X'_1(x_1;α_1,α_2,λ)= (x_1; α_1,α_2 ) · e^-λ/2_1 F_1(α^+;α_1;λ/2x_1).The proof follows the same lines as the proof of Proposition <ref>. By integrating Eq. (<ref>) or, roughly speaking, by taking λ_2=0 and renaming λ_1 with λ in Eq. (<ref>), the distribution function F_X'_1 of X'_1 ∼'_1(α_1,α_2,λ) can be obtained as follows:F_X'_1(x_1;α_1,α_2,λ)=∑_j=0^+∞e^-λ/2(λ/2)^j/j!B(x_1;α_1+j,α_2)/B(α_1+j,α_2), 0<x_1<1,i.e., by analogy with the '_1 density, as the series of the Beta(α_1+j,α_2) distribution functions, j ∈ℕ∪{0}, weighted by the probabilities of M ∼(λ/2). As previously said, the case of α_1=α_2=1 is hugely important in this context. In the latter, the '_1 density becomes significantly easier. Indeed, by considering Eq. (<ref>) and letting a=2, z=λ/2x in the following formula (http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/03/01/02/link):_1F_1(a;a-1;z)=e^z (1+z/a-1),we obtain:f_X'_1(x_1;1,1,λ)=e^-λ/2_1F_1(2;1;λ/2x_1)=e^-λ/2(1-x_1)(1+λ/2x_1),0<x_1<1.Hence, by integrating Eq. (<ref>), it's easy to see that the '_1 distribution function turns out to be:F_X'_1(x_1;1,1,λ)=x_1 e^-λ/2(1-x_1),0<x_1<1.Finally, the type 2 non-central beta, denoted by '_2(α_1,α_2,λ), is the distribution of the random variable:X'_2=Y_1/Y_1+Y'_2,where Y_1 and Y'_2 are independently distributed according to χ^2_2α_1 and χ'^ 2_2α_2(λ), respectively.The type 1 and the type 2 non-central beta random variables are connected by the following relationship. [Relationship between '_1 and '_2]Let X'_2 ∼'_2(α_1,α_2,λ) and X'_1 ∼'_1(α_2,α_1,λ). Then:X'_1=1-X'_2.For the proof see <ref> in the Appendix. Hence, the density function f_X'_2 of X'_2 ∼'_2(α_1,α_2,λ) can be easily derived by making use of the transformation of variable in Eq. (<ref>) and it is given by:f_X'_2(x_2;α_1,α_2,λ)=∑_k=0^+∞e^-λ/2(λ/2)^k/k!x_2^α_1-1(1-x_2)^α_2+k-1/B(α_1,α_2+k), 0<x_2<1,i.e. the series of the Beta(α_1,α_2+k) densities, k ∈ℕ∪{0}, weighted by the probabilities of M ∼(λ/2). Roughly speaking, Eq. (<ref>) can be intuitively established by taking λ_1=0 and renaming λ_2 with λ in Eq. (<ref>). Such a distribution admits the following mixture and perturbation representations. [Mixture representation of '_2]Let X'_2 have a '_2(α_1,α_2,λ) distribution and M be a Poisson random variable with mean λ/2. Then, X'_2 admits the following representation:X'_2| M ∼(α_1, α_2+M). Let X'_2∼'_2(α_1,α_2,λ). Then, the density f_X'_2 of X'_2 can be written as:f_X'_2(x_2;α_1,α_2,λ)= (x_2; α_1,α_2 ) · e^-λ/2_1 F_1[α^+;α_2;λ/2(1-x_2)] .The proof follows the same lines as the proof of Proposition <ref>. By integrating Eq. (<ref>) or, roughly speaking, by taking λ_1=0 and renaming λ_2 with λ in Eq. (<ref>), the distribution function F_X'_2 of X'_2 ∼'_2(α_1,α_2,λ) can be obtained as follows:F_X'_2(x_2;α_1,α_2,λ)=∑_k=0^+∞e^-λ/2(λ/2)^k/k!B(x_2;α_1,α_2+k)/B(α_1,α_2+k), 0<x_2<1,i.e., by analogy with the '_2 density, as the series of the Beta(α_1,α_2+k) distribution functions, k ∈ℕ∪{0}, weighted by the probabilities of M ∼(λ/2). Finally, in view of Property <ref>, for α_1=α_2=1 we have that the '_2 density takes on the following simple form:f_X'_2(x_2;1,1,λ)=e^-λ/2 x_2[1+λ/2(1-x_2)],0<x_2<1;moreover, the following holds true for the '_2 distribution function:F_X'_2(x_2;1,1,λ)=1- e^-λ/2x_2(1-x_2),0<x_2<1.That said, we are now ready to establish the aforementioned representation of a ” random variable. Let X' ∼”(α_1,α_2,λ_1,λ_2) and α^+=α_1+α_2, λ^+=λ_1+λ_2. Furthermore, let M_r, r=1,2, be independent Poisson random variables with means λ_r/2 and M^+=M_1+M_2. Then:X'=X'_2 X +(1-X'_2)X'_pnc,where: i) X and (X'_2,X'_pnc) are mutually independent and X ∼(α_1,α_2), ii) (X'_2, X'_pnc) are conditionally independent given M^+ with:.X'_2|M^+ ∼(α^+,M^+), .X'_pnc|M^+ ∼∑_i=0^M^+(i;M^+,λ_1/λ^+) ·(x;i,M^+-i), iii) X'_2 ∼'_2(α^+,0,λ^+) and X'_pnc∼”(0,0,λ_1,λ_2). For the proof see <ref> in the Appendix. It's apparent that the doubly non-central beta model can be easily simulated in different ways. Until now we have seen that this can be done by means of its definition, its mixture representation and its conditional distribution given M^+ in Eq. (<ref>). However, such issue can be alternatively addressed by resorting to the above proved representation.To this end, in the notation of Proposition <ref>, it's necessary to generate the random variables X, M^+ and simulate accordingly from the distributions of X'_2|M^+ and X'_pnc|M^+. Firstly, it's to be noted that X'_2|(M^+=0) is degenerate at one and X'_pnc|(M^+=0) is degenerate at zero. Secondly, in case of M^+ ≠ 0, the distribution of X'_pnc|M^+ is given by a mixture of M^++1 beta distributions, two of which have one shape parameter equal to zero. To sample from such mixture, one chooses an index i^* from {0,…,M^+} according to the probabilities of the binomial distribution referred to hereinabove and then simulates a value from the corresponding (i^*,M^+-i^*) distribution. An implementation of this algorithm inlanguage is proposed in <ref>. That said, Figures <ref>, <ref>, <ref> show the results of the simulation from the ” model for selected values of the shape and the non-centrality parameters. The generating process of the ” random variate was carried out by means of two algorithms: the former is based on the definition while the latter on the new representation we have just introduced. In all the cases considered, the histogram of the simulated values was plotted together with the true density, thus anticipating the matters that will be discussed in the subsequent Section relating to the variety of shapes taken on by it. The graphs show that the results of the two approaches are indeed comparable.Finally, in Section <ref> the latter representation will be used in the derivation of an interesting expression for the mean of the doubly non-central beta distribution in terms of a convex linear combination of the mean of the beta distribution and a compositional ratio of the non-centrality parameters.§.§ Density plotsA key feature of the ” distribution over the beta one lies in the much larger variety of shapes reachable by its density on varying the non-centrality parameters.In this regard, it's worth recalling that by reversing both the shape and the non-centrality parameters the ” density turns out to be symmetrical with respect to the midpoint of the interval (0,1). Let X' ∼”(α_1,α_2,λ_1,λ_2). Then 1-X' ∼”(α_2,α_1,λ_2,λ_1). For the proof see <ref> in the Appendix.As a special case of Property <ref>, the ” density with α_1=α_2 and λ_1=λ_2 is symmetrical with respect to x=1/2.Some significant plots of the B” density are displayed in the following Figures <ref>, <ref> for selected values of the shape and the non-centrality parameters.In this regard, we recall that the limits at 0 and 1 of the beta density are equal to 0 or +∞ if α_i ≠ 1 and are equal to 1 if α_i=1, i=1,2. In the latter case the beta density reduces to the uniform one. When α_i ≠ 1 the ” density shows the same limiting characteristics as the beta model ones. On the contrary, when α_i=1 the doubly non-central beta density shows a more flexible behavior at the unit interval endpoints by taking on arbitrary finite and positive limits at 0 and 1. Some examples of this particularly relevant feature of the ” density are shown in Figures <ref> for selected values of the non-centrality parameters λ_1, λ_2. Such peculiarity follows from Remark <ref> and was set out in <cit.>.More specifically, the limits of the ” density have the following expressions, that, interestingly, are really simple. Let X' ∼ ”(α_1,α_2,λ_1,λ_2). Then, the limits at 0 and 1 of the density f_X' of X' when α_1=α_2=1 are: lim_x → 0^+f_X'(x; 1,1,λ_1,λ_2)=e^-λ_1/2(λ_2/2+1), lim_x → 1^-f_X'(x; 1,1,λ_1,λ_2)=e^-λ_2/2(λ_1/2+1).For the proof see <ref> in the Appendix. Thanks to this essential characteristic, the ” distribution enables to properly model the portions of data having values next to the endpoints of the real interval (0,1). In this regard, the ” applicative potential will be highlighted in Section <ref> through the analysis of real data.Finally, by carrying out the same lines as the proof of Proposition <ref> or, roughly speaking, by taking λ_2=0 and renaming λ_1 with λ in Eqs. (<ref>), (<ref>), one can obtain the limits of the '_1 density when α_1=α_2=1. In view of Property <ref>, the limits of the '_2 density can be simply stated by reversing the '_1 ones. Following are their expressions. Let X'_1 ∼ '_1(α_1,α_2,λ) and X'_2 ∼ '_2(α_1,α_2,λ). Then, the limits at 0 and 1 of the density f_X'_1 of X'_1 when α_1=α_2=1 are: lim_x_1 → 0^+f_X'_1(x_1; 1,1,λ)=e^-λ/2, lim_x_1 → 1^-f_X'_1(x_1; 1,1,λ)=λ/2+1,while the limits at 0 and 1 of the density f_X'_2 of X'_2 when α_1=α_2=1 are: lim_x_2 → 0^+f_X'_2(x_2; 1,1,λ)=λ/2+1, lim_x_2 → 1^-f_X'_2(x_2; 1,1,λ)=e^-λ/2.§.§ Patnaik's approximationA simple and reliable approximation for the doubly non-central beta distribution can be easily derived by applying the Patnaik's approximation for the non-central chi-squared one <cit.>.Indeed, hereafter we prove that the ” model can be approximated by the three-parameter generalization of the beta one introduced by Libby and Novick <cit.>. Let X' have a ” distribution with shape parameters α_r and non-centrality parameters λ_r, r=1,2. Furthermore, let Y_r be independent χ^2_ν_r random variables, with:ν_r=(2α_r+λ_r)^2/2(α_r+λ_r).By taking:β_r=ν_r/2, ρ_r=2(α_r+λ_r)/2α_r+λ_r,r=1,2, γ=ρ_2/ρ_1,one can approximate X' d≈ X'_P, where X'_P=ρ_1 Y_1/ρ_1 Y_1+ρ_2 Y_2∼(β_1,β_2,γ) and:(x';β_1,β_2,γ)=(x';β_1,β_2)γ^β_1/[1-(1-γ)x']^β_1+β_2,0<x'<1is the probability density function of the Libby and Novick's generalized beta distribution <cit.>. For the proof see <ref> in the Appendix. Observe that as λ_r tends to 0^+, r=1,2, ν_r tends to 2 α_r and ρ_r tends to 1; therefore, the distributions of both X' and X'_P tend to the (α_1,α_2) one.A graphic comparison between the ”(α_1,α_2,λ_1,λ_2) density and its approximation herein derived is shown in Figures <ref>, <ref>, <ref> for selected values of the shape and the non-centrality parameters. Note that in all the cases depicted the plots of the two densities are very similar, except for more or less slight differences on the tails. In this regard, the approximation results particularly unsatisfactory on the tails for α_1=α_2=1, due to its inability to replicate the behavior of the ” density at the unit interval endpoints (see the right-hand panel of Figure <ref>).A three-parameter generalized beta random variable, thanks to its relationship with the beta, has distribution function that takes a really simple form. Indeed, from Eq. (<ref>) in the proof of Proposition <ref>, it's immediate to see that if X'_P ∼(β_1,β_2,γ) then X=γ X'_P/γ X'_P+1-X'_P∼(β_1,β_2). By exploiting the latter and by denoting the distribution functions of X'_P and X with F_X'_P and F_X respectively, for every x' ∈ (0,1) we have accordingly:F_X'_P(x') =(X'_P ≤ x')=(X/X+γ (1-X)≤ x')==(X ≤γ x'/γ x'+1-x')=F_X(γ x'/γ x'+1-x')=B(γx'/γ x'+1-x';β_1,β_2)/B(β_1,β_2). In view of the foregoing arguments, the latter can be used to approximate the ” distribution function. In this regard, a graphic analysis was performed in order to investigate the goodness of approximation of Eq. (<ref>). The results are displayed in Figures <ref>, <ref>, <ref>, that show a deep reliability of the G3B distribution function as an approximation of the ” one.codes for the density and the distribution function of the Libby and Novick's generalized beta model are proposed in <ref>, <ref>. §.§ MomentsBy analogy with the form of the ” density in Eq. (<ref>) and the ” distribution function in Eq. (<ref>), the r-th moment about zero of the doubly non-central beta distribution can be stated as the double series of the r-th moments about zero of the Beta(α_1+j,α_2+k) distributions, j,k ∈ℕ∪{0}, weighted by the joint probabilities of (M_1,M_2), where M_i, i=1,2, are independent with (λ_i/2) distributions. As far as we know, the latter is the only analytical form available in the literature for the moments of the ” distribution.That said, in the present Section a new general formula for the moments of such distribution is provided. This formula allows the computation of moments to be reduced from a double series to a single one. According to the latter, in fact, the r-th moment can be evaluated in terms of a perturbation of the corresponding moment of the beta distribution through a weighted sum of Kummer's confluent hypergeometric functions. More specifically, the present result extends and completes Proposition 7 in <cit.> and concludes that the r-th moment of the ” distribution can be expressed as follows. Let X' ∼”(α_1,α_2,λ_1,λ_2) and α^+=α_1+α_2, λ^+=λ_1+λ_2. Let M_j, j=1,2, be independent Poisson random variables with means λ_j/2 and M^+=M_1+M_2. Then, for every r ∈ℕ, the r-th moment about zero of X' admits the following expression:𝔼[(X')^r]=(α_1)_r/(α^+)_re^-λ^+/2∑_i=0^rri(α^+)_i(λ_1/2)^i/(α_1)_i (α^++r)_i_1F_1(α^++i;α^++r+i;λ^+/2).For the proof see <ref> in the Appendix. The first two moments of the ” distribution can thus be computed as special cases of Eq. (<ref>) by taking r=1 and r=2 as follows:𝔼(X')=α_1/α^+e^-λ^+/2[_1F_1(α^+;α^++1;λ^+/2)+α^+λ_1/2/α_1 (α^++1)_1F_1(α^++1;α^++2;λ^+/2)], 𝔼[(X')^2]=(α_1)_2/(α^+)_2e^-λ^+/2[_1F_1(α^+;α^++2;λ^+/2)+α^+λ_1/α_1 (α^++2)·.· . _1F_1(α^++1;α^++3;λ^+/2)+(α^+)_2(λ_1/2)^2/(α_1)_2 (α^++2)_2 _1F_1(α^++2;α^++4;λ^+/2)].An implementation inlanguage of the moments formula in Eq. (<ref>) is proposed in <ref>.Now we should like to make a few comments on the moments of the type 1 and the type 2 non-central beta distributions. More specifically, by making use of the definition of the r-th moment about zero of a random variable, the following formula holds for the moments of X'_1 ∼ '_1(α_1,α_2,λ):𝔼[(X'_1)^r]=(α_1)_r/(α^+)_re^-λ/2_2F_2(α_1+r,α^+;α_1, α^++r;λ/2),where _2F_2(a_1,a_2;b_1,b_2;x)=∑_k=0^+∞(a_1)_k (a_2)_k/(b_1)_k (b_2)_kx^k/k! is the generalized hypergeometric function _pF_q with p=2 and q=2 coefficients respectively at numerator and denominator <cit.>.That said, a new general formula for the moments about zero of the '_1 distribution can be derived regardless of Eq. (<ref>) in light of Eq. (<ref>). Indeed, by taking λ_2=0 and renaming λ_1 with λ in Eq. (<ref>), the following holds true. Let X'_1 ∼ '_1(α_1,α_2,λ) and α^+=α_1+α_2. Then, for every r ∈ℕ, the r-th moment about zero of X_1' admits the following expression:𝔼[(X'_1)^r]=(α_1)_r/(α^+)_re^-λ/2∑_i=0^rri(α^+)_i(λ/2)^i/(α_1)_i (α^++r)_i_1F_1(α^++i;α^++r+i;λ/2).As a side effect, by comparing Eqs. (<ref>), (<ref>), the following identity between the aforementioned hypergeometric functions holds true:_2F_2(α_1+r,α^+;α_1, α^++r;λ/2)=∑_i=0^rri(α^+)_i(λ/2)^i/(α_1)_i (α^++r)_i_1F_1(α^++i;α^++r+i;λ/2). Finally, the general formula for the moments about zero of the type 2 non-central beta distribution can be stated by taking λ_1=0 and renaming λ_2 with λ in Eq. (<ref>). The latter can be also derived by making use of the definition of the r-th moment about zero of a random variable, obtaining as follows:𝔼[(X'_2)^r]=(α_1)_r/(α^+)_re^-λ/2_1F_1(α^+;α^++r;λ/2). An interesting relationship applies among the means of the three non-central beta distributions recalled herein. More precisely, the mean of the ” distribution with shape parameters α_1, α_2 and non-centrality parameters λ_1, λ_2 can be expressed as a convex linear combination of the means of the '_1 and '_2 distributions with shape parameters α_1, α_2 and non-centrality parameter λ^+=λ_1+λ_2.Let X' ∼ ” (α_1,α_2,λ_1,λ_2), X'_1 ∼ '_1(α_1,α_2,λ^+) and X'_2 ∼ '_2(α_1,α_2,λ^+), where λ^+=λ_1+λ_2. Then:𝔼(X')=λ_1/λ^+ 𝔼(X'_1)+λ_2/λ^+ 𝔼(X'_2).For the proof see <ref> in the Appendix. Moreover, by resorting to Proposition <ref>, we can obtain an alternative and interesting expression for the mean of the doubly non-central beta distribution in terms of a convex linear combination of the mean of the beta distribution and a compositional ratio of the non-centrality parameters.Let X' ∼ ” (α_1,α_2,λ_1,λ_2) and α^+=α_1+α_2, λ^+=λ_1+λ_2. Then:𝔼(X')=α_1/α^+[e^-λ^+/2_1F_1(α^+;α^++1;λ^+/2)]+λ_1/λ^+[1-e^-λ^+/2_1F_1(α^+;α^++1;λ^+/2)].For the proof see <ref> in the Appendix. We conclude the present Section by further investigating the moments of the ” distribution when α_1=α_2=1. In the latter case, the mean and the variance interestingly take on the following simple forms. Let X' ∼ ” (α_1,α_2,λ_1,λ_2) with α_1=α_2=1. Then:𝔼(X')=1/2+λ_1-λ_2/2(λ^+)^3[(λ^+)^2-4λ^++8-8 e^-λ^+/2], (X') =4/(λ^+)^2+4λ_1λ_2/(λ^+)^5[(λ^+-2)(λ^+-8)-2 e^-λ^+/2(λ^++8)]+ +8 (λ_1-λ_2)^2/(λ^+)^6[(λ^+)^2 e^-λ^+/2 -2(1-e^-λ^+/2)(λ^++1-e^-λ^+/2)].For the proof see <ref> in the Appendix. Observe that 𝔼(X')>1/2 when λ_1>λ_2; in fact, in view of Eq. (<ref>), (λ^+)^2-4λ^++8>8 e^-λ^+/2 for every λ^+>0. Note that Eqs. (<ref>), (<ref>) become considerably simplified by assuming that the non-centrality parameters are equal. Indeed, by taking λ_1=λ_2=λ we have accordingly:𝔼(X')=1/2, (X')=1/λ^2+1/2λ^3[(λ-1)(λ-4)-(λ+4) e^-λ]. Finally, by carrying out the same lines as the proof of Proposition <ref> or, roughly speaking, by taking λ_2=0 and renaming λ_1 with λ in Eqs. (<ref>), (<ref>), one can obtain simple expressions for the mean and the variance of the '_1 distribution when α_1=α_2=1. Similarly, simple expressions can be derived also for the mean and the variance of the '_2 distribution when α_1=α_2=1 by taking λ_1=0 and renaming λ_2 with λ in Eqs. (<ref>), (<ref>). Following are their expressions. Let X'_1 ∼'_1(α_1,α_2,λ) and X'_2 ∼'_2(α_1,α_2,λ) with α_1=α_2=1. Then:𝔼(X'_1)=1/2+1/2λ^2(λ^2-4λ+8-8 e^-λ/2), 𝔼(X'_2)=1/2-1/2λ^2(λ^2-4λ+8-8 e^-λ/2), (X'_1)=(X'_2)=4/λ^2+8/λ^4[λ^2 e^-λ/2 -2(1-e^-λ/2)(λ+1-e^-λ/2)].In view of the foregoing arguments, note that 𝔼(X'_1)>1/2 and 𝔼(X'_2)<1/2 for every λ>0.§.§ ApplicationsThe applicative potential of the ” model is now highlighted through the analysis of real data. To this end, we first turned our attention to three significant examples arisen respectively from the sectors of geology, economics and psychology.More specifically, we focused on the proportion of sand in 21 sediment specimens, the proportion of males involved in agriculture as occupation for 47 French-speaking provinces of Switzerland at about 1888 and the subjective diagnostic probability of calculus deficiency assigned by 15 statisticians. The first data are available in <cit.> (p. 380) and details about the geologic interpretation can be found in <cit.>. The second data are taken from the “swiss” data set, which is included in the“datasets” package (http://stat.ethz.ch/R-manual/R-devel/library/datasets/html/swiss.htmllink); details about the reference frame are available in theon-line Documentation and in the references quoted therein. Finally, the third data are again available in <cit.> (p. 375).In a comparative perspective, five distributions were fitted to the above mentioned three data sets. The first is the standard beta, that is one of the most frequently employed to model proportions. The second is the doubly non-central beta, which is the subject of interest in the present work.The third model is the three-parameter generalization of the beta distribution proposed by Libby and Novick <cit.>, the density function of which has been previously reported. In the notation of Eq. (<ref>), the parameter γ>0 allows the G3B density to take a much wider variety of shapes than the beta one. In particular, when β_1=β_2=1, the latter shows a more flexible behavior at the unit interval endpoints than the beta. In fact, its limits at 0 and 1 have the following expressions:lim_x → 0^+(x;1,1,γ)=γ, lim_x → 1^-(x;1,1,γ)=1/γ,which are clearly subject to the strong constraint of being mutual to each other. Furthermore, we recall that the r-th moment about zero of X ∼(β_1,β_2,γ) is:𝔼(X)^r=(β_1)_r/(β_1+β_2)_r γ^β_1_2F_1(β_1+r,β_1+β_2;β_1+β_2+r;1-γ),where _2F_1=∑_k=0^+∞(a)_k (b)_k/(c)_kx^k/k!, |x|<1 is the Gauss hypergeometric function. Notice that the _2F_1 function in Eq. (<ref>), despite its representation in terms of infinite series converges only for |1-γ|<1, can be computed for any γ>0 by using suitably one of the Euler transformation formulas <cit.>:_2F_1(a,b;c;x) =(1-x)^-a_2F_1(a,c-b;c;x/x-1) =(1-x)^-b_2F_1(c-a,b;c;x/x-1) =(1-x)^c-a-b_2F_1(c-a,c-b;c;x),that enable to rewrite such function to have absolute values of the argument less than one.Then, we considered the Gauss hypergeometric model <cit.>. A random variable X is said to have a Gauss hypergeometric distribution with shape parameters a>0, b>0 and additional parameters λ∈ℝ, z>-1, denoted by (a,b,λ,z), if its probability density function is:(x;a,b,λ,z)=(x;a,b)/(1+z x)^λ_2F_1(λ,a;a+b;-z) ,0<x<1.Note that the case z=0 corresponds to the beta distribution. When a=b=1 its limiting values are given by the following functions of λ and z:lim_x → 0^+(x;1,1,λ,z)=1/_2F_1(λ,1;2;-z), lim_x → 1^-(x;1,1,λ,z)=1/(1+z)^λ_2F_1(λ,1;2;-z);the latter are analitically hard, not so easily interpretable and not particularly simplified by using any of the transformation formulas of the _2F_1 function that are valid in case of specific values for its arguments. In this regard, see <cit.>. Moreover, the r-th moment about zero of X ∼(a,b,λ,z) is:𝔼(X)^r=(a)_r/(a+b)_r_2F_1(λ,a+r;a+b+r;-z)/_2F_1(λ,a;a+b;-z). Finally, we used the confluent hypergeometric model, proposed by Gordy <cit.>, who applied it to the auction theory. A random variable X is said to have a confluent hypergeometric distribution with shape parameters c>0, d>0 and additional parameter δ∈ℝ, denoted by (c,d,δ), if its probability density function is:(x;c,d,δ)=(x;c,d)e^-δx/ _1F_1(c;c+d;-δ) ,0<x<1.Note that the case δ=0 corresponds to the beta distribution. In this regard, by taking a=1 and z=-δ in the following formula:_1F_1(a;a+1;z)=a(-z)^-a[Γ(a)-Γ(a,-z)](http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/03/01/02/link), where Γ(a,-z)=∫_-z^+∞t^a-1e^-tdt is the incomplete gamma function, we obtain:_1F_1(1;2;-δ)=1/δ(1-∫_δ^+∞e^-tdt)=e^δ-1/δe^δ.Hence, when c=d=1, the CH density function takes on the following form:(x;1,1,δ)=δe^δ(1-x)/e^δ-1,0<x<1;moreover, as x tends to the endpoints of (0,1), the latter tends to:lim_x → 0^+(x;1,1,δ)=δe^δ/e^δ-1, lim_x → 1^-(x;1,1,δ)=δ/e^δ-1.The r-th moment about zero of X ∼(c,d,δ) is:𝔼(X)^r=(c)_r/(c+d)_r _1F_1(c+r;c+d+r;-δ)/_1F_1(c;c+d;-δ).The latter can be easily obtained by using the definition of r-th moment of a random variable and the integral representation of the Kummer's confluent hypergeometric function <cit.>, that is:Γ(b-a)Γ(a)/Γ(b)_1F_1(a;b;z)=∫_0^1t^a-1(1-t)^b-1 e^z tdt,b>a>0. That said, the method of moments was applied in order to obtain the estimates for the parameters of each model. The shape parameters were assigned unitary values in all the models except obviously for the beta one. In case of two parameters to be estimated, the mean and the variance of the model were set simultaneously equal to the values of the corresponding sample statistics; in case of one parameter, instead, only the mean was considered. In particular, the formulas in Eqs. (<ref>), (<ref>) were used as the expressions for the mean and the variance of the ” model. Clearly, none of the estimates of interest admits an explicit expression (except for the beta one), due to the hard analytical formulas of the moments. Therefore, the aforementioned systems of equations were solved numerically by means of the built-in function “FindRoot” of Mathematica language.The data histogram together with the estimated densities of the five models considered are shown in Figure <ref> for the first analysis setting, in Figure <ref> for the second one and in Figure <ref> for the third one.Notice that the beta and the doubly non-central beta produce a fairly accurate fit for all the proportions/probabilities considered (left-hand panels), exhibiting a substantially better fit than the G3B, the GH and the CH, the inadequate performances of which are evident (right-hand panels).In particular, it is remarkable that the ” distribution allows for the tails of the data histograms to be captured and modeled and appears to be more helpful in data interpretation, as it recognizes the presence of values next to zero and one by showing positive and finite limits. On the contrary, the beta distribution cannot display such ability. Furthermore, none of the three alternative models considered enables to capture this peculiarity of the data pattern. Therefore, for such models, good fitting and having positive and finite limits would sound irreconcilable features.Finally, it should be stressed that there might be situations in which the method of moments results in negative estimates for one of the two non-centrality parameters of the ” distribution. This may be explained by incompatibilities between the data concerned and the range of shapes the ” density can take on for α_1=α_2=1 by varying λ_1, λ_2. To avoid this inconvenience, the method of moments could be adopted to estimate all four parameters of the ” model without fixing the shape parameters to one. However, in this case the moments formulas to be used are special cases of Eq. (<ref>) and therefore are computationally heavy. Moreover, no real solutions might exist. Then, one can fall back to using the type 1 or the type 2 non-central beta distributions. More specifically, in the case of negative estimates for λ_1 and λ_2, the type 2 and the type 1 models should be respectively fitted to data.By way of example, we considered the proportion by weight of cornite in 25 specimens of kongite. The latter data are once again available in <cit.> (p. 356).In order to derive the method of moments estimates for the ”(1,1,λ_1,λ_2) distribution, the mean and the variance of such model were set simultaneously equal to the values of the corresponding sample statistics, obtaining the following solutions λ̃_1=-0.530932, λ̃_2=7.36794 for the non-centrality parameters. As λ̃_1<0, we are lead to believe that the present situation is not well suited for being modeled by the doubly non-central beta distribution while the type 2 model may be more appropriate in this respect. The latter was thus fitted to the aforementioned data in place of the ” and the simple formula in Eq. (<ref>) was used as the expression for the mean.That said, Figure <ref> shows the data histogram and the estimated densities of the five models considered.In the previous examples the differences in fitting between the ” and the three alternative models were unquestionably clear. In the present case, instead, their performances are indeed comparable and all satisfactory, with the small exception of the G3B model, the density of which shows a much higher limiting value at zero (right-hand panel), due to the strong constraint existing between its limits. This means that, contrarily to the ” distribution, the potential of the '_2 in capturing the observations with low and high values does not exceed the one of the other three models and similar conclusions might be drawn with reference to the '_1 where appropriate. The foregoing conclusions lead us to compare the performances of the beta and the doubly non-central beta distributions more deeply. For this purpose, we resorted to the Akaike information criterion <cit.>, given by =-2 l(θ̂)+2 p, where l=l(θ) is the log-likelihood function for the p-dimensional vector θ of the model parameters and θ̂ is the maximum-likelihood estimate of θ. The distribution with the smallest value for this criterion is taken as the one that gives the best description of the data. In this regard, let X_1,…,X_n be independent random variables with identical distribution depending on an unknown parameter vector θ to be estimated.Suppose first that X_i ∼(α_1,α_2), i=1,…,n. It is well known that, given the observed sample x=(x_1,…,x_n), the log-likelihood function for the vector (α_1,α_2) ∈ℝ^2+ of the shape parameters is:l(α_1,α_2;x)=(α_1-1) ∑_i=1^nlog x_i+(α_2-1)∑_i=1^nlog(1-x_i)-nlog[B(α_1,α_2)].Secondly, let X_i ∼”(1,1,λ_1,λ_2), i=1,…,n. Then, by resorting to Eq. (<ref>), it's easy to see that the log-likelihood function for the vector (λ_1,λ_2) ∈ℝ^2+ of the non-centrality parameters is given by:l(λ_1,λ_2;x)=-n/2(λ_1+λ_2) +∑_i=1^nlogΨ_2[2;1,1;λ_1/2x_i,λ_2 /2(1-x_i)]. The number of parameters of both the above models is p=2; therefore, in view of the AIC definition, the maximized value of the log-likelihood is the only discriminating criterion between them.In the present setup the log-likelihoods are to be maximized numerically. This procedure can be easily accomplished by using for example the “optim” built-in-function from thestatistical package or, alternatively, the “FindMaximum” one from Mathematica software. These routines are able to locate the maximum of the log-likelihood surface for a wide range of starting values. However, to ease computations, it is useful to have reasonable starting values, such as, for example, the method of moments estimates.That said, the aforementioned algorithms were applied on the four data sets subject to the analyses previously carried out in this Section.The standard errors of such estimates can be evaluated by recalling that, under suitable regularity conditions, the maximum-likelihood estimator Θ̂ of θ is asymptotically distributed according to a multivariate normal with mean vector θ and asymptotic covariance matrix that can be approximated by the inverse of the observed information matrix I(θ̂;x)={-∂^2l(θ;x)/∂θ ∂θ^T}_θ=θ̂. The required second-order derivatives can be computed numerically by means of the“optim” function.Table <ref> lists the maximum-likelihood estimates, their standard errors and the AIC statistics of the two models of interest for each of the above cases, labelled as “sand”, “male”, “calculus” and “cornite”. By the comparison of the present results with the first ones, it's immediate to see that the parameters estimates are very similar for both methods. In particular, it's to be noted that the considerations previously drawn with regards to the “cornite” data are now confirmed by applying the maximum-likelihood approach. Indeed, as λ̂_1=0, we are inclined to think that the ” distribution results overparametrized to model such data while its special case '_2 is enough to this end. Furthermore, the results indicate that the ” model has the smallest value for the AIC statistic in half the cases (in bold in Table <ref>). So, in these cases the latter could be chosen as the most suitable model.Before concluding, we want to further illustrate the flexibility of the ” distribution. To this end, we used four data sets taken from <cit.> and related to the subjects of petrology and geology. The first data set consists of the proportions of magnesium oxide in 23 specimens of aphyric Skye lavas (p. 360); more details for a petrological interpretation of the latter can be found in <cit.>. The second one deals with the proportions by weight of albite in 25 specimens of kongite (p. 356). Finally, we considered the proportion of clay in 39 sediment samples at different water depths in an Arctic Lake (p. 359), adapted from <cit.> (Table 1) and the proportion of abies in 30 specimens of fossil pollen from three different locations (p. 389). As it did before, the fit of the ” distribution with unitary shape parameters was compared with the beta one for each of the above cases. As criteria for comparing the fits, we used the AIC statistic, based on the maximum-likelihood estimates for the model parameters. For both distributions, the latter were evalutated numerically by means of thefunction “optim” using the method of moments estimates as starting values. The results of fitting are shown in Table <ref>, where the aforementioned data are labelled as “oxide”, “albite”, “clay” and “abies”. Moreover, Figures <ref>, <ref> show the data histograms superimposed with the fitted probability density functions of the two models.Note that in all these cases the ” model has the lowest AIC and so it could be chosen as the best one. Moreover, the present analysis shows that the higher are the non-centrality parameters of the doubly non-central beta distribution, the better is the ability of the latter to model even data with no values next to zero and one (right-hand panels). § CONCLUSIONSNew insights into the class of the non-central beta distributions were provided in this paper. More specifically, new representations and moments expressions were derived for the doubly non-central beta distribution despite its uneasy analytical tractability. A particularly relevant advantage of this model over alternative ones on the real interval (0,1), such as the beta, the Libby and Novick's generalized beta, the Gauss hypergeometric and the confluent hypergeometric models, is its ability to properly capture the tails of data by allowing its density to take on finite and positive limits. Indeed, various applications using real data proved the superior performance of the doubly non-central beta distribution over the others in terms of fitting. In particular, in many cases the doubly non-central beta showed lower values of the Akaike information criterion than the beta one, which is well known to be the most frequently employed to model proportions. That is why we hope this model may attract wider applications in statistics.An investigation of its multivariate generalization is clearly needed. However, the poor analytical simplicity characterizing the unidimensional case can only get worse as the dimensionality increases.In this regard, a first analysis of a more easily tractable and interpretable analogue of the doubly non-central beta distribution was carried out in <cit.>. An in-depth study of such distribution seems to be desirable; in fact, the latter preserves the applicative potential of the standard doubly non-central beta model and its straightforward form can make possible its extension to the multidimensional setting in a simple manner. Ultimately, it seems worthwhile to continue developing this research line by tackling it in future work. 99. Ait03 Aitchison, J.: The Statistical Analysis of Compositional Data. The Annals of Probability, The Blackburn Press, London (2003)Aka74 Akaike, H.: A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723 (1974)ArmBay94 Armero, C., Bayarri, M.J.: Prior assessments for prediction in queues. The Statistician, 43, 139–153 (1994)CoaRus68 Coakley, J.P., Rust, B.R.: Sedimentation in an Arctic Lake. Journal of Sedimentary Petrology, 38, 1290–1300 (1968)EttMef02 Ettoumi, F.Y., Mefti, A., Adane, A., Bouroubi, M.Y.: Statistical analysis of solar measurements in Algeria using beta distributions. Reneweable Energy, 26, 47–67 (2002)Gor98 Gordy, M.B.: Computationally convenient distributional assumptions for common-value auctions. Computational Economics, 12, 61–78 (1998)HasPac95 Haskett, J.D., Pachepsky, Y.A., Acock, B.: Use of the beta distribution for parameterizing variability of soil properties at the regional level for crop yield estimation. Agricultural Systems, 48, 73–86 (1995)Hjo88 Hjort, N.L.: The eccentric part of the non-central chi-square distribution. The American Statistician, 42, 130–132 (1988)JohKemKot05 Johnson, N.L., Kemp, A.W., Kotz, S.: Univariate Discrete Distributions, 3rd edition. John Wiley & Sons, Hoboken, New Jersey (2005)JohKotBal94 Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions Vol. 1, 2nd edition. John Wiley & Sons, New York (1994)JohKotBal95 Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions Vol. 2, 2nd edition. John Wiley & Sons, New York (1995)KimSch98 Kimball, C.V., Scheibner, D.J.: Error bars for sonic slowness measurements. Geophysics, 63, 345–353, (1998)LibNov82 Libby, D.L., Novick, M.R.: Multivariate generalized beta distributions with applications to utility assessment. Journal of Educational Statistics, 7, 271–294 (1982)MalRos59 Malcolm, D.G., Roseboom, J.H., Clark, C.E., Fazar, W.: Application of a technique for research and development program evaluation. Operations Research, 7, 646–669 (1959)McC75 McCammon, R.B.: Concepts in geostatistics. John Wiley & Sons, New York (1975)NadGup04 Nadarajah, S., Gupta, A. K.: Generalizations and Related Univariate Distributions. In Handbook of Beta Distribution and Its Applications, eds. Gupta, A.K., Nadarajah, S., Marcel Dekker, Inc., New York, pp. 97–163 (2004)OngOrs15 Ongaro, A., Orsi, C.: Some results on non-central beta distributions. Statistica, 75, 85–100 (2015)Pat49 Patnaik, P.B.: The non-central χ^2 and F-distributions and their applications. Biometrika, 36, 202–232 (1949)SriKar85 Srivastava, H.M., Karlsson, W.: Multiple Gaussian Hypergeometric Series. Ellis Horwood, Chichester (1985) Sta15 Stamm, A., Singh, J., Afacan, O., Warfield, S. K.: Analytic quantification of bias and variance of coil sensitivity profile estimators for improved image reconstruction inmri. Medical Image Computing and Computer-Assisted Intervention MICCAI 2015, 684–691 (2015)ThoEssDun72 Thompson, R.N., Esson, J., Duncan, A. C.: Major element chemical variation in the Eocene lavas of the Isle of Skye, Scotland. Journal of Petrology, 13, 219–253 (1972)WilHer89 Wiley, J.A., Herschkorn, S.J., Padian, N.S.: Heterogeneity in the probability of HIV transmission per sexual contact: the case of male-to-female transmission in penile-vaginal intercourse. Statistics in Medicine, 8, 93–102 (1989) § APPENDIX. PROOFS [Proposition <ref>]The proof of Eq. (<ref>) follows from Eq. (<ref>) by noting that l=i+(l-i), ∀ i=0,…,l and by doing multiplications in a way such that the left-hand side of Eq. (<ref>) can be written in the following form:(a+b)_l=∑_i=0^l[P_i(a)] b^i,where P_i(a) is a polynomial in the variable a with degree equal to l-i, ∀ i=0,…,l. It's easy to see that P_i(a) can be written in the same form as in the right-hand side of Eq. (<ref>).[Proposition <ref>]By virtue of the law of iterated expectations, one has 𝔼[(Y' )^r ]=𝔼_M{𝔼[.(Y' )^r|M ]}, where, in the notation of Property <ref>, M is a Poisson random variable with mean λ/2 and, conditionally on M, Y' has a χ^2_g+2M distribution. In view of the general formula for the moments about zero of the gamma distribution <cit.>, one obtains 𝔼[.(Y' )^r|M ]=2^r (h+M)_r, where h=g/2; therefore:𝔼[(Y' )^r ]=2^r𝔼[(h+M)_r].By virtue of Proposition <ref>, Eq. (<ref>) can be restated as follows: 𝔼[(Y' )^r ]=2^r∑_i=0^r1/i![d^i/d h^i(h)_r] 𝔼(M^i), where, in view of the general formula for the moments about zero of Poisson distribution <cit.>:𝔼(M^i)=∑_j=0^i𝒮(i,j) (λ/2)^j,i ∈ℕ,𝒮(i,j) being a Stirling number of the second kind. Thus, Eq. (<ref>) is established.[Proposition <ref>]By taking h=0 in Eq. (<ref>), one has 𝔼[(Y'_pnc)^r ]=2^r𝔼[(M)_r], where M ∼(λ/2). By bearing in mind that (M)_r=∑_i=0^r |s(r,i)| M^i and by virtue of Eq. (<ref>), Eq. (<ref>) is established. [Proposition <ref>]By virtue of Eq. (<ref>), one has Γ(j+h)=Γ(h) ·(h)_j, ∀ j=0,…,r, so that Eq. (<ref>) can be rewritten as follows:𝔼[(Y' )^r ]=2^r ∑_j=0^rrj(h)_r/(h)_j(λ/2)^j,where h=g/2. Furthermore, by noting that ∑_i=0^r∑_j=0^ia_ij=∑_j=0^r∑_i=j^ra_ij, Eq. (<ref>) can be restated according to the following form:𝔼[(Y' )^r ]=2^r∑_j=0^r(λ/2)^j ∑_i=j^r𝒮(i,j) 1/i![d^i/d h^i(h)_r].Hence, by equating Eq. (<ref>) and Eq. (<ref>), Eq. (<ref>) is established. [Proposition <ref>]Observe that Eq. (<ref>) can be restated as follows:.X'|(M_1,M^+) ∼(α_1+M_1,α_2+M^+-M_1).In view of Property <ref>, Y'^+ has a χ'^2_2α^+(λ^+) distribution with α^+=α_1+α_2 and λ^+=λ_1+λ_2; moreover, by virtue of Property <ref>, one has that:.Y'^+|(M_1,M^+) d=.Y'^+|M^+ ∼χ^2_2α^++2M^+,where d= stands for “equal in distribution”. By Property <ref>, X' and Y'^+ are conditionally independent given (M_1,M^+). Hence, conditionally on (M_1,M^+), the joint distribution of (X',Y'^+) factorizes into the marginal distributions of X' and Y'^+.That said, the proof follows by noting that the joint density function of .(X',Y'^+)|M^+ turns out to factorize into the marginal density functions of X'|M^+ and Y'^+|M^+. Indeed, under Eq. (<ref>) one can obtain:f_.(X',Y'^+)|M^+(x,y) =∑_i=0^M^+(.M_1=i|M^+) · f_.(X',Y'^+)|(M_1,M^+)(x,y)= = f_.Y'^+|M^+(y) · f_.X'|M^+(x),where, under Eq. (<ref>) and by bearing in mind that M_1|M^+ ∼(M^+,λ_1/λ^+), the density f_.X'|M^+ of X' given M^+ is of the same form as in Eq. (<ref>).[Proposition <ref>]In light of Eq. (<ref>) and in the notation of Eq. (<ref>), it follows that:B(α_1+j,α_2+k) =Γ(α_1+j)Γ(α_2+k)/Γ(α^++j+k)=Γ(α_1)(α_1)_jΓ(α_2)(α_2)_k/Γ(α^+)(α^+)_j+k= = B(α_1,α_2)(α_1)_j(α_2)_k/(α^+)_j+k.Hence, Eq. (<ref>) can be obtained from Eq. (<ref>) with simple computations by making use of the latter result and of Eq. (<ref>); indeed we have:f_X'(x;α_1,α_2,λ_1,λ_2 )=(x;α_1,α_2) · e^-λ^+/2∑_j=0^+∞∑_k=0^+∞(α^+)_j+k/(α_1)_j (α_2)_k(λ_1/2x)^j/j![λ_2/2(1-x)]^k/k!. [Property <ref>]By virtue of Eq. (<ref>) one has:X'_2=Y_1/Y_1+Y'_2=1-Y'_2/Y'_2+Y_1 ⇔ Y'_2/Y'_2+Y_1=1-X'_2,where Y_1 ∼χ^2_2α_1 and Y'_2 ∼χ'^ 2_2α_2(λ) independently. In view of Eq. (<ref>), Eq. (<ref>) is established.[Proposition <ref>]Let Y_r, r=1,2, be independent χ^2_2α_r random variables and Y^+=Y_1+Y_2 ∼χ^2_2α^+, with α^+=α_1+α_2. In light of Eqs. (<ref>), (<ref>) one has: X'=Y'_1/Y'_1+Y'_2=Y_1+∑_j=1^M_1F_j/Y^++∑_j=1^M^+F_j=Y_1/Y^++∑_j=1^M^+F_j+∑_j=1^M_1F_j/Y^++∑_j=1^M^+F_j.Observe that the first term on the right-hand side of Eq. (<ref>) can be restated as:Y_1/Y^++∑_j=1^M^+F_j=Y^+/Y^++∑_j=1^M^+F_j·Y_1/Y^+;similarly, with respect to the second term, we have:∑_j=1^M_1F_j/Y^++∑_j=1^M^+F_j= ∑_j=1^M^+F_j/Y^++∑_j=1^M^+ F_j·∑_j=1^M_1F_j/∑_j=1^M^+ F_j,which is meaningful provided we set that:∑_j=1^M_1F_j/∑_j=1^M^+ F_j=0 M_r=0, ∀ r=1,2.Finally, by setting:X'_2=Y^+/Y^++∑_j=1^M^+F_j,X=Y_1/Y^+,X'_pnc=∑_j=1^M_1F_j/∑_j=1^M^+ F_j,the decomposition in Eq. (<ref>) is established.Now consider the random vector (X,X'_2,X'_pnc). In light of Eq. (<ref>), the marginal random vector (X'_2,X'_pnc) is a function of (Y^+,M_1,M_2,{F_j}); moreover, the latter is independent of X: in fact, Y^+ is independent of X by virtue of Property <ref>. Finally, X and (X'_2,X'_pnc) are mutually independent and, in view of Eq. (<ref>), X ∼(α_1,α_2): result i) is thus proved.In order to prove result ii), observe first that:.X'_2|(M_1,M_2)=.Y^+/Y^++∑_j=1^M^+F_j|(M_1,M_2) ∼(α^+,M^+)and:.X'_pnc|(M_1,M_2)=.∑_j=1^M_1F_j/∑_j=1^M^+ F_j|(M_1,M_2) ∼(M_1,M_2);moreover, X'_2 and X'_pnc are conditionally independent given (M_1,M_2). That said, the proof of ii) follows by noting that the joint density function of (X'_2,X'_pnc) |M^+ turns out to factorize into the marginal distributions of .X'_2|M^+ and .X'_pnc|M^+. Indeed, by bearing in mind that .M_1|M^+ ∼(M^+,λ_1/λ^+), one can obtain:f_.(X'_2,X'_pnc)|M^+(x_2,x)= =∑_i=0^M^+ f_.(X'_2,X'_pnc)|(M_1,M_2)(x_2,x) ·(.M_1=i|M^+)= =∑_i=0^M^+ f_.X'_2|(M_1,M_2)(x_2) · f_.X'_pnc|(M_1,M_2)(x) ·(.M_1=i|M^+)= =∑_i=0^M^+(x_2;α^+,M^+) ·(x;i,M^+-i) ·(.M_1=i|M^+)= =(x_2;α^+,M^+) ·∑_i=0^M^+(.M_1=i|M^+) ·(x;i,M^+-i)= = f_.X'_2|M^+(x_2) · f_.X'_pnc|M^+(x).Finally, result iii) follows from Eq. (<ref>) and Eq. (<ref>).[Property <ref>]Let Y'_r, r=1,2, be independent χ'^ 2_2α_r(λ_r) random variables. The proof follows from Eq. (<ref>) by noting that:X'=Y'_1/Y'_1+Y'_2=1-Y'_2/Y'_2+Y'_1 ⇔1-X'=Y'_2/Y'_2+Y'_1∼”(α_2,α_1,λ_2,λ_1). [Proposition <ref>]By taking α_1=α_2=1 in Eq. (<ref>), one has:f_X'(x;1,1,λ_1,λ_2)=∑_j=0^+∞∑_k=0^+∞e^-λ_1/2(λ_1/2)^j/j!e^-λ_2/2(λ_2/2)^k/k!x^j (1-x)^k/B(1+j,1+k), 0<x<1.Hence, by taking the limit of both sides of Eq. (<ref>) as x tends to 0^+, the outcome is:lim_x → 0^+ f_X'(x;1,1,λ_1,λ_2)= = e^-λ_1/2 ∑_k=0^+∞e^-λ_2/2(λ_2/2)^k/k!1/B(1,1+k)=e^-λ_1/2 ∑_k=0^+∞ (k+1)e^-λ_2/2(λ_2/2)^k/k!= = e^-λ_1/2[∑_k=0^+∞ ke^-λ_2/2(λ_2/2)^k/k!+∑_k=0^+∞e^-λ_2/2(λ_2/2)^k/k!]=e^-λ_1/2(λ_2/2 +1),that is Eq. (<ref>).Similarly the limit at 1 of Eq. (<ref>) turns out to be:lim_x → 1^- f_X'(x;1,1,λ_1,λ_2)= = e^-λ_2/2 ∑_j=0^+∞e^-λ_1/2(λ_1/2)^j/j!1/B(1+j,1)=e^-λ_2/2 ∑_j=0^+∞ (j+1)e^-λ_1/2(λ_1/2)^j/j!= = e^-λ_2/2[∑_j=0^+∞ je^-λ_1/2(λ_1/2)^j/j!+∑_j=0^+∞e^-λ_1/2(λ_1/2)^j/j!]=e^-λ_2/2(λ_1/2 +1),that is Eq. (<ref>).[Proposition <ref>]By virtue of Eq. (<ref>) and Property <ref>, we have:X'=Y'_1/Y'_1+Y'_2 d≈X'_P=ρ_1 Y_1/ρ_1 Y_1+ρ_2 Y_2,where ρ_r=2(α_r+λ_r)/2α_r+λ_r, r=1,2 and Y_r are independent χ^2_ν_r random variables with ν_r=(2α_r+λ_r)^2/2(α_r+λ_r); moreover, Eq. (<ref>) is tantamount to:X' d≈1/ρ_1 Y_1+ρ_2 Y_2/ρ_1 Y_1=1/1+ρ_2/ρ_1Y_2/Y_1.Now let X have a (ν_1/2,ν_2/2) distribution. Therefore, we have:X=Y_1/Y_1+Y_2 ⇔ Y_2/Y_1=1-X/X;in light of Eq. (<ref>), Eq. (<ref>) can be thus restated as follows:X' d≈1/1+ρ_2/ρ_11-X/X=ρ_1 X/ρ_1 X+ρ_2(1-X)=f(X)=X'_P.By noting that X=f^-1(X'_P)=ρ_2 X'_P/ρ_1 (1-X'_P)+ρ_2 X'_P, dx/dx'=ρ_1ρ_2/[ρ_1(1-x')+ρ_2 x']^2, by taking β_r=ν_r/2, r=1,2 and by denoting the densities of X and X'_P with f_X and f_X'_P, respectively, the proof is straightforward once we observe that:f_X'_P(x')=f_X(ρ_2 x'/ρ_1 (1-x')+ρ_2 x') dx/dx'=(ρ_2/ρ_1)^β_1/B(β_1,β_2)x'^β_1-1 (1-x')^β_2-1/[1-(1-ρ_2/ρ_1) x']^β_1+β_2,with x' ∈ (0,1).[Proposition <ref>]In the notation of Eq. (<ref>) and by virtue of Proposition <ref>, one has: 𝔼[.(X')^r|M^+]=𝔼[.(Y'_1)^r|M^+]/𝔼[.(Y'^+)^r|M^+]; moreover, in view of the general formula for the moments of the gamma distribution <cit.>, one has:𝔼[.(Y'_1)^r|M^+]=𝔼{.𝔼[.(Y'_1)^r |M_1,M^+] |M^+}=𝔼{.𝔼[.(Y'_1)^r |M_1]|M^+}= =𝔼[.2^r(α_1+M_1)_r|M^+]=2^r ∑_i=0^M^+(α_1+i)_r M^+i(λ_1/λ^+)^i (1-λ_1/λ^+)^M^+-iand 𝔼[.(Y'^+)^r|M^+]=2^r (α^++M^+)_r. Therefore:𝔼[.(X')^r|M^+]=1/(α^++M^+)_r∑_i=0^M^+(α_1+i)_r M^+i(λ_1/λ^+)^i (1-λ_1/λ^+)^M^+-i.By letting L ∼(M^+,λ_1/λ^+), Eq. (<ref>) can be restated as follows:𝔼[.(X')^r|M^+]=𝔼[(α_1+L)_r]/(α^++M^+)_r.In this regard, by replacing a and b with, respectively, α_1 and L in Eq. (<ref>), one has:𝔼[(α_1+L)_r]=∑_i=0^r1/i![d^i/dα_1^i(α_1)_r ]𝔼(L^i),where, ∀ i ∈ℕ∪{0}:(L^i)=∑_k=0^i𝒮(i,k) M^+!/(M^+-k)!(λ_1/λ^+)^k<cit.>, 𝒮(i,k) being a Stirling number of the second kind. By making use of M^+!/(M^+-k)!=∑_j=0^k s(k,j) ·(M^+)^j <cit.>, s(k,j) being a Stirling number of the first kind, Eq. (<ref>) can be rewritten as follows:𝔼(L^i)=∑_k=0^i 𝒮(i,k) [∑_j=0^k s (k,j) ·(M^+)^j](λ_1/λ^+)^k.By noting that ∑_k=0^i∑_j=0^ka_kj=∑_j=0^i∑_k=j^ia_kj and by letting θ_1=λ_1/λ^+, Eq. (<ref>) turns out to be tantamount to:𝔼(L^i)=∑_j=0^i[∑_k=j^i𝒮(i,k) s(k,j)θ_1^k](M^+)^j;under Eq. (<ref>), Eq. (<ref>) can be written accordingly in the form of:𝔼[(α_1+L)_r]=∑_i=0^r1/i![d^i/d α_1^i(α_1)_r ] ·∑_j=0^i[∑_k=j^i𝒮(i,k) s(k,j)θ_1^k](M^+)^jand finally, under Eq. (<ref>), Eq. (<ref>) can be stated in the following form:𝔼[(X')^r|M^+]=∑_i=0^r1/i![d^i/d α_1^i(α_1)_r ] ·∑_j=0^i[∑_k=j^i𝒮(i,k) s(k,j)θ_1^k](M^+)^j/(α^++M^+)_r.Therefore, by virtue of the law of iterated expectations, since M^+ ∼(λ^+/2), the r-th moment about zero of the doubly non-central beta distribution turns out to have the following expression:𝔼[(X')^r]=e^-λ^+/2∑_i=0^r1/i![d^i/d α_1^i(α_1)_r ] ·∑_j=0^i[∑_k=j^i𝒮(i,k) s(k,j)θ_1^k] ·∑_l=0^+∞l^j/(α^++l)_r(λ^+/2)^l/l!.Observe that, in view of Eq. (<ref>), the following holds:(α^+)_l (α^++l)_r=(α^+)_r (α^++r)_l ⇔1/(α^++l)_r=(α^+)_l/(α^++r)_l1/(α^+)_rand Eq. (<ref>) can be rewritten accordingly as follows:𝔼[(X')^r]=e^-λ^+/2/(α^+)_r∑_i=0^r1/i![d^i/d α_1^i(α_1)_r ] ·∑_j=0^i[∑_k=j^iS(i,k) s(k,j)θ_1^k] ·∑_l=0^+∞l^j(α^+)_l/(α^++r)_l(λ^+/2)^l/l!. Now let M^+_* be a random variable on the non-negative integers such that:(M^+_*=l)=(α^+)_l/(α^++r)_l(λ^+/2)^l/l!/_1F_1(α^+;α^++r;λ^+/2), ∀ l ∈ℕ∪{0},so that, for every j ∈ℕ∪{0}:𝔼(M^+_*)^j=1/_1F_1(α^+;α^++r;λ^+/2)∑_l=0^+∞l^j(α^+)_l/(α^++r)_l(λ^+/2)^l/l!and:𝔼[(X')^r]=e^-λ^+/2/(α^+)_r_1F_1(α^+;α^++r;λ^+/2) ·· ∑_i=0^r1/i![d^i/d α_1^i(α_1)_r ] ·∑_j=0^i[∑_k=j^iS(i,k) s(k,j)θ_1^k] ·𝔼(M^+_*)^j.Note that the generating function of the descending factorial moments of M^+_* has the following expression:𝔼[(1+t)^M^+*]=_1F_1(α^+;α^++r;(1+t)λ^+/2)/_1F_1(α^+;α^++r;λ^+/2),t ∈ℝand its derivative of order m ∈ℕ is:d^m/dt^m𝔼[(1+t)^M^+*]=(λ^+/2)^m (α^+)_m/(α^++r)_m_1F_1(α^++m;α^++r+m;(1+t)λ^+/2)/_1F_1(α^+;α^++r;λ^+/2);hence, by taking t=0 in Eq. (<ref>), it follows that the m-th descending factorial moment of M^+_* turns out to be:𝔼(M^+_*)_[m]=(λ^+/2)^m(α^+)_m/(α^++r)_m_1F_1(α^++m;α^++r+m;λ^+/2)/_1F_1(α^+;α^++r;λ^+/2).By bearing in mind that 𝔼(M^+_*)^j=∑_m=0^j𝒮(j,m)𝔼(M^+_*)_[m] <cit.> and in light of Eq. (<ref>), Eq. (<ref>) can be rewritten as follows:[(X')^r]= =e^-λ^+/2/(α^+)_r∑_i=0^r1/i![d^i/d α_1^i(α_1)_r ] ·∑_j=0^i[∑_k=j^i𝒮(i,k) s(k,j)θ_1^k] ·· ∑_m=0^j𝒮(j,m) (λ^+/2)^m(α^+)_m/(α^++r)_m_1F_1(α^++m;α^++r+m;λ^+/2).Furthermore, by virtue of the following properties of the Stirling numbers of the first and the second kinds <cit.>:s(a,0)=s(0,a)=𝒮(a,0)=𝒮(0,a)=0, ∀ a>0, ∑_j=m^n𝒮(n,j)s(j,m)=∑_j=m^ns(n,j)𝒮(j,m)={[ 1 m=n; 0 ]., one has:[(X')^r]=e^-λ^+/2/(α^+)_r∑_i=0^r[∑_j=i^r1/j!d^j/dα_1^j(α_1)_r 𝒮(j,i)] ··s(i,i)𝒮(i,i)(α^+)_i (θ_1 λ^+/2)^i /(α^++r)_i_1F_1(α^++i;α^++r+i;λ^+/2);finally, in view of Eq. (<ref>) and by bearing in mind that s(a,a)=𝒮(a,a)=1, for every a ≥ 0, Eq. (<ref>) is established.[Proposition <ref>]Observe that Eq. (<ref>) can be rewritten as follows:𝔼(X')= =α_1/α^+e^-λ^+/2_1F_1(α^+;α^++1;λ^+/2)+e^-λ^+/2λ_1/2/α^++1_1F_1(α^++1;α^++2;λ^+/2)= =λ_1/λ^+[α_1/α^+e^-λ^+/2_1F_1(α^+;α^++1;λ^+/2)+e^-λ^+/2λ^+/2/α^++1_1F_1(α^++1;α^++2;λ^+/2)]+ +λ_2/λ^+[α_1/α^+e^-λ^+/2_1F_1(α^+;α^++1;λ^+/2)];Eq. (<ref>) is thus established.[Proposition <ref>]In the notation of Proposition <ref>, one has:𝔼(X')=𝔼[X'_2 X +(1-X'_2)X'_pnc];in view of result i) of the aforementioned Proposition, Eq. (<ref>) can be rewritten as 𝔼(X')=α_1/α^+ 𝔼(X'_2)+ 𝔼[(1-X'_2)X'_pnc]. As X'_2 ∼'_2(α^+,0,λ^+) in light of Eq. (<ref>), one has: 𝔼(X'_2)=e^-λ^+/2_1F_1(α^+;α^++1;λ^+/2). By virtue of the law of iterated expectations and in view of result ii) of Proposition <ref>, the following holds true:𝔼[(1-X'_2)X'_pnc]= =𝔼_M^+{𝔼[.(1-X'_2)X'_pnc|M^+]}=𝔼_M^+{𝔼[.(1-X'_2)|M^+]𝔼(.X'_pnc|M^+)},where:𝔼(.X'_pnc|M^+) =∑_i=0^M^+M^+i(λ_1/λ^+)^i (1-λ_1/λ^+)^M^+-i 𝔼[(x;i,M^+-i)]= =∑_i=0^M^+M^+i(λ_1/λ^+)^i (1-λ_1/λ^+)^M^+-i i/M^+= =1/M^+ ∑_i=0^M^+ iM^+i(λ_1/λ^+)^i (1-λ_1/λ^+)^M^+-i= =1/M^+· M^+λ_1/λ^+=λ_1/λ^+.Therefore, Eq. (<ref>) can be restated as follows:𝔼[(1-X'_2)X'_pnc]=λ_1/λ^+ 𝔼_M^+{𝔼[.(1-X'_2)|M^+] }= =λ_1/λ^+ 𝔼(1-X'_2)=λ_1/λ^+[1- 𝔼(X'_2)]=λ_1/λ^+[1- e^-λ^+/2_1F_1(α^+;α^++1;λ^+/2)]and Eq. (<ref>) is established.[Proposition <ref>]By taking α_1=α_2=1 in Eq. (<ref>), we have:𝔼(X')=λ_1/λ^++(1/2-λ_1/λ^+) e^-λ^+/2_1F_1(2;3;λ^+/2).Observe that by taking a=2 and z=λ^+/2 in Eq. (<ref>), one has:_1F_1(2;3;λ^+/2) =2(-λ^+/2)^-2[Γ(2)-Γ(2,-λ^+/2)]=8/(λ^+)^2[1-∫_-λ^+/2^+∞t e^-tdt]= =8/(λ^+)^2[1+(λ^+/2-1)e^λ^+/2];therefore:𝔼(X') =λ_1/λ^++8/(λ^+)^2(1/2-λ_1/λ^+)(e^-λ^+/2+λ^+/2-1)= =λ_1/λ^+ -4λ_1/(λ^+)^2+2/λ^++8λ_1/(λ^+)^3-4/(λ^+)^2+4/(λ^+)^2e^-λ^+/2-8λ_1/(λ^+)^3e^-λ^+/2.Eq. (<ref>) can be thus obtained with simple computations by noting that:λ_1/λ^+=1/2+λ_1-λ_2/2λ^+,-4λ_1/(λ^+)^2+2/λ^+=-2 (λ_1-λ_2)/(λ^+)^2, 8λ_1/(λ^+)^3-4/(λ^+)^2=4 (λ_1-λ_2)/(λ^+)^3 , 4/(λ^+)^2e^-λ^+/2-8λ_1/(λ^+)^3e^-λ^+/2=-4 (λ_1-λ_2)/(λ^+)^3e^-λ^+/2. We now come to the proof of Eq. (<ref>). When α_1=α_2=1, Eq. (<ref>) can be rewritten as follows:𝔼[(X')^2]= =1/3e^-λ^+/2_1F_1(2;4;λ^+/2)+λ_1/6e^-λ^+/2_1F_1(3;5;λ^+/2)+λ_1^2/80e^-λ^+/2_1F_1(4;6;λ^+/2).Observe that by taking a=2, 3, 4 and z=λ^+/2 in the following formula (http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/03/01/02/link):_1F_1(a;a+2;z)=(-z)^-a/z·· {Γ(a) a^3+(za+a+z)Γ(a+1)-(a+1) [e^z (-z)^a+1+(a+z)Γ(a+1,-z)]}one has respectively:_1F_1(2;4;λ^+/2)= =8/(λ^+)^3[12+3 λ^++3(λ^+)^3/8e^λ^+/2-3(2+λ^+/2)∫_-λ^+/2^+∞t^2 e^-tdt]= =96/(λ^+)^3+24/(λ^+)^2 -96/(λ^+)^3e^λ^+/2+24/(λ^+)^2 e^λ^+/2, _1F_1(3;5;λ^+/2)= = -16/(λ^+)^4[72+12 λ^+-(λ^+)^4/4e^λ^+/2-4(3+λ^+/2)∫_-λ^+/2^+∞t^3 e^-tdt]= = -1152/(λ^+)^4-192/(λ^+)^3 +1152/(λ^+)^4e^λ^+/2-384/(λ^+)^3 e^λ^+/2+48/(λ^+)^2e^λ^+/2, _1F_1(4;6;λ^+/2)= =32/(λ^+)^5[480+60 λ^++5(λ^+)^5/32e^λ^+/2-5(4+λ^+/2)∫_-λ^+/2^+∞t^4 e^-tdt]= =15360/(λ^+)^5+1920/(λ^+)^4 -15360/(λ^+)^5e^λ^+/2+5760/(λ^+)^4 e^λ^+/2-960/(λ^+)^3e^λ^+/2+80/(λ^+)^2e^λ^+/2.Therefore, one has:𝔼[(X')^2]= = -32/(λ^+)^3+8/(λ^+)^2 +32/(λ^+)^3e^-λ^+/2+8/(λ^+)^2 e^-λ^+/2+ +λ_1 [192/(λ^+)^4-64/(λ^+)^3 +8/(λ^+)^2- 192/(λ^+)^4e^-λ^+/2-32/(λ^+)^3 e^-λ^+/2]+ +λ_1^2 [-192/(λ^+)^5+72/(λ^+)^4 -12/(λ^+)^3+1/(λ^+)^2+ 192/(λ^+)^5e^-λ^+/2+24/(λ^+)^4 e^-λ^+/2].Moreover:𝔼[(X')]^2= =16/(λ^+)^4-16/(λ^+)^3 +4/(λ^+)^2-32/(λ^+)^4e^-λ^+/2+16/(λ^+)^3 e^-λ^+/2+16/(λ^+)^4 e^-λ^++ +λ_1 [-64/(λ^+)^5+64/(λ^+)^4 -24/(λ^+)^3+ 4/(λ^+)^2+128/(λ^+)^5e^-λ^+/2-64/(λ^+)^4 e^-λ^+/2+ .+. 8/(λ^+)^3 e^-λ^+/2-64/(λ^+)^5 e^-λ^+]+ +λ_1^2 [64/(λ^+)^6-64/(λ^+)^5 +32/(λ^+)^4-8/(λ^+)^3+ 1/(λ^+)^2-128/(λ^+)^6e^-λ^+/2+64/(λ^+)^5 e^-λ^+/2+ . -.16/(λ^+)^4 e^-λ^+/2+64/(λ^+)^6 e^-λ^+].Hence:(X')=𝔼[(X')^2]-𝔼[(X')]^2= = -16/(λ^+)^4-16/(λ^+)^3 +4/(λ^+)^2+32/(λ^+)^4e^-λ^+/2+16/(λ^+)^3 e^-λ^+/2+8/(λ^+)^2 e^-λ^+/2+ -16/(λ^+)^4 e^-λ^++ +λ_1 [64/(λ^+)^5+128/(λ^+)^4 -40/(λ^+)^3+ 4/(λ^+)^2-128/(λ^+)^5e^-λ^+/2-128/(λ^+)^4 e^-λ^+/2+ .-. 40/(λ^+)^3 e^-λ^+/2+64/(λ^+)^5 e^-λ^+]+ +λ_1^2 [-64/(λ^+)^6-128/(λ^+)^5 +40/(λ^+)^4-4/(λ^+)^3+ 128/(λ^+)^6e^-λ^+/2+128/(λ^+)^5e^-λ^+/2+ . +. 40/(λ^+)^4 e^-λ^+/2 - 64/(λ^+)^6 e^-λ^+].At this point, the proof of the variance formula follows from the mere application of some tedious algebra. Upon noting that:-16/(λ^+)^4 e^-λ^++32/(λ^+)^4e^-λ^+/2-16/(λ^+)^4= -16/(λ^+)^4(1-e^-λ^+/2)^2, λ_1 [64/(λ^+)^5e^-λ^+-128/(λ^+)^5e^-λ^+/2 +64/(λ^+)^5] =64λ_1/(λ^+)^5 (1-e^-λ^+/2)^2, λ_1^2 [-64/(λ^+)^6 e^-λ^++ 128/(λ^+)^6e^-λ^+/2-64/(λ^+)^6] = -64λ_1^2/(λ^+)^6 (1-e^-λ^+/2)^2,we have:-16 (λ_1-λ_2)^2/(λ^+)^6 (1-e^-λ^+/2)^2= = -16/(λ^+)^4(1-e^-λ^+/2)^2+64λ_1/(λ^+)^5 (1-e^-λ^+/2)^2-64λ_1^2/(λ^+)^6 (1-e^-λ^+/2)^2;moreover:λ_1λ_2/λ^+[128/(λ^+)^4-40/(λ^+)^3+4/(λ^+)^2-128/(λ^+)^4 e^-λ^+/2-40/(λ^+)^3 e^-λ^+/2]= =λ_1[128/(λ^+)^4 -40/(λ^+)^3+ 4/(λ^+)^2-128/(λ^+)^4 e^-λ^+/2-40/(λ^+)^3 e^-λ^+/2]+ +λ_1^2 [-128/(λ^+)^5 +40/(λ^+)^4-4/(λ^+)^3+ 128/(λ^+)^5e^-λ^+/2+40/(λ^+)^4 e^-λ^+/2],so that:(X') = -16 (λ_1-λ_2)^2/(λ^+)^6 (1-e^-λ^+/2)^2+ +λ_1λ_2/λ^+[128/(λ^+)^4-40/(λ^+)^3+4/(λ^+)^2-128/(λ^+)^4 e^-λ^+/2-40/(λ^+)^3 e^-λ^+/2]+ -16/(λ^+)^3 +4/(λ^+)^2+16/(λ^+)^3 e^-λ^+/2+8/(λ^+)^2 e^-λ^+/2.By observing that:16/(λ^+)^5(1-e^-λ^+/2) [4λ_1λ_2- (λ_1-λ_2)^2]= =λ_1λ_2/λ^+[128/(λ^+)^4-128/(λ^+)^4 e^-λ^+/2]-16/(λ^+)^3 +16/(λ^+)^3 e^-λ^+/2,we have:(X') = -16 (λ_1-λ_2)^2/(λ^+)^6 (1-e^-λ^+/2)(λ^++1-e^-λ^+/2)+ +λ_1λ_2/λ^+[64/(λ^+)^4-40/(λ^+)^3+4/(λ^+)^2-64/(λ^+)^4 e^-λ^+/2-40/(λ^+)^3 e^-λ^+/2]+ +8/(λ^+)^2 e^-λ^+/2 +4/(λ^+)^2.Note that:λ_1λ_2/λ^+[64/(λ^+)^4-40/(λ^+)^3+4/(λ^+)^2]=4λ_1λ_2/(λ^+)^5(λ^+-2)(λ^+-8)and:-40λ_1λ_2/(λ^+)^4 e^-λ^+/2 +8/(λ^+)^2 e^-λ^+/2=8/(λ^+)^4 e^-λ^+/2[(λ_1-λ_2)^2-λ_1λ_2],so that:8/(λ^+)^4 e^-λ^+/2(λ_1-λ_2)^2-16 (λ_1-λ_2)^2/(λ^+)^6 (1-e^-λ^+/2)(λ^++1-e^-λ^+/2)= =8(λ_1-λ_2)^2/(λ^+)^6[(λ^+)^2e^-λ^+/2 -2 (1-e^-λ^+/2)(λ^++1-e^-λ^+/2) ].Finally:-8λ_1λ_2/(λ^+)^4e^-λ^+/2-64λ_1λ_2/(λ^+)^5e^-λ^+/2=-8λ_1λ_2/(λ^+)^5(λ^++8) e^-λ^+/2and:4λ_1λ_2/(λ^+)^5(λ^+-2)(λ^+-8)-8λ_1λ_2/(λ^+)^5(λ^++8) e^-λ^+/2= =4λ_1λ_2/(λ^+)^5[(λ^+-2)(λ^+-8)-2e^-λ^+/2(λ^++8) ].Eq. (<ref>) is thus established.§ APPENDIX. R FUNCTIONS [Perturbation factor of the beta density in the perturbation representation of the doubly non-central beta density in Eq. (<ref>)] Arguments: * x: vector of quantiles* shape1, shape2: shape parameters of the doubly non-central beta distribution* ncp1, ncp2: non-centrality parameters of the doubly non-central beta distribution* tol: tolerance with zero meaning to iterate until additional terms to not change the partial sum* maxiter: maximum number of iterations to perform* debug: Boolean, with TRUE meaning to return debugging information and FALSE meaning to return just the evaluate [Perturbation representation of the doubly non-central beta density in Eq. (<ref>)] Arguments: * x: vector of quantiles* shape1, shape2: shape parameters of the doubly non-central beta distribution* ncp1, ncp2: non-centrality parameters of the doubly non-central beta distribution The cases =0 and =0 correspond respectively to the densities of the type 2 and the type 1 non-central beta distributions.[Internal series of the doubly non-central beta distribution function in Eq. (<ref>) for any fixed value of the index of the external one] Arguments: * x: vector of quantiles* first: value of the index of the external series* shape2: second shape parameter of the doubly non-central beta distribution* ncp2: second non-centrality parameter of the doubly non-central beta distribution* tol: tolerance with zero meaning to iterate until additional terms to not change the partial sum* maxiter: maximum number of iterations to perform* debug: Boolean, with TRUE meaning to return debugging information and FALSE meaning to return just the evaluate[Doubly non-central beta distribution function] Arguments: * x: vector of quantiles* shape1, shape2: shape parameters of the doubly non-central beta distribution* ncp1, ncp2: non-centrality parameters of the doubly non-central beta distribution* lower.tail: logical, if TRUE, probabilities are (X ≤ x), otherwise, (X > x).* tol: tolerance with zero meaning to iterate until additional terms to not change the partial sum* maxiter: maximum number of iterations to perform* debug: Boolean, with TRUE meaning to return debugging information and FALSE meaning to return just the evaluateThe cases =0 and =0 correspond respectively to the type 2 and the type 1 non-central beta distribution functions. [Generating a doubly non-central beta random variable by means of its definition] Arguments: * n: number of determinations * shape1, shape2: shape parameters of the doubly non-central beta distribution* ncp1, ncp2: non-centrality parameters of the doubly non-central beta distribution[Generating a doubly non-central beta random variable by means of its representation as a convex linear combination in Eq. (<ref>)] Arguments: * n: number of determinations * shape1, shape2: shape parameters of the doubly non-central beta distribution* ncp1, ncp2: non-centrality parameters of the doubly non-central beta distribution[Libby and Novick's generalized beta density in Eq. (<ref>)] Arguments: * x: vector of quantiles * shape1, shape2: shape parameters of the Libby and Novick's generalized beta distribution* gamma: additional parameter of the Libby and Novick's generalized beta distribution [Libby and Novick's generalized beta distribution function in Eq. (<ref>)] Arguments: * x: vector of quantiles * shape1, shape2: shape parameters of the Libby and Novick's generalized beta distribution* gamma: additional parameter of the Libby and Novick's generalized beta distribution* lower.tail: logical, if TRUE, probabilities are (X ≤ x), otherwise, (X > x).[Moments formula of the doubly non-central beta distribution in Eq. (<ref>)] Arguments: * order: vector of integers * shape1, shape2: shape parameters of the doubly non-central beta distribution* ncp1, ncp2: non-centrality parameters of the doubly non-central beta distribution The cases =0 and =0 correspond respectively to the moments formulas of the type 2 and the type 1 non-central beta distributions. | http://arxiv.org/abs/1706.08557v1 | {
"authors": [
"Carlo Orsi"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170626184113",
"title": "New insights into non-central beta distributions"
} |
=-.31truein=-.48truein landscape=1000=2ptcm`=11@FiNeD#1#2#3#4#4#1#2#3#4#4bb(This will come out unreduced. =1200=16pt plus 2pt minus 1pt ==(This will be reduced. ł@r=L =1000=16pt plus 2pt minus 1pt =7truein =8truein=4.75truein=10truein===0 to#1Lł@r 1=1 [0.1] =#1 ł@r=R1=2to#1ł@r=L=by -2000 #1#2HUTP-/A#1#2 #1#2=#1 1in#2.5in=0#1#1==page DRAFTMODE preliminary draft: // ==20pt plus 2pt minus 2pt255=255 by 60 255 255 by-60255 by:255<10 0255##1##1##1 ##1=` .03in##1 ##1-1.05in##1 ##1##1=0 =1@csym #1by1 0#1(. 0)=0@t@csym@msection. #1 section. #1@t.=1 @t=0 #1by1 0#1(@csym. 0) >9000subsection@m. @m.. #1subsection@m.@m.. #1#1#2=1=0#1. Appendix appendix#1 #1. #20(#1. #2)0 @csym#1.@m#1 appendix#1Appendix #1. #2 @de#1#2@rst##1##1equation #1#2equation#1#2#1#1#1(@de @csym) #1#1#1by1@rst#1@̧t#1a@ark@̧t#1#2@ark#1 #1#1#1{} #1##1(@de @csym@rst##1##1) #11#11by1#1#2#1 #1(equation@csym ) #2(equation@csym)#1#1#1by1@n@n(#1)#1 #1@n#1 #1(@ns #1) @ns#1@FiNeD#1eqnlabel #1 is undefined. #1(?.?)=#1 @rk#1#1@xt#1@xt#1#1=@ns#1#1#1#1#1#1 #1@FiNeD#116*** WARNING: the label #1 is already defined *** 14pt plus 1pt minus 1pt@t#1#1foot @@tfoot =height9.5pt depth4.5pt width0pt=0by1@rgfootnote @rgfootnote[^@rg]by1=footnote^ =1=.fts * footnote f:=20pt Footnotes.fts=1 <ref> #1#1 #1[reference] #1#1 =1=.refs= * [reference]#1.31inby1#1#=`^̂M@rg @rg#1@line#1^^M^^M @line#1^^M0@x #1 0@rk=0=@line @x#1 @rk@f @f#1#2#1#1#1=@FiNeD<ref>1#2; #1 * #1 =References=20pt =` .refs#1=.refs=#1@f@f[#1]#1 #1255=1[@̊fs #1] @̊fs#1@FiNeD#1reflabel #1 is undefined. #1need to supply reference #1.=#1 @rk #1255#1255=0#1255=1=@̊fsFigure Captions=40pt ##1##2 * Fig. figure##1##1. ##2 =1fig. figure #1#1 #1fig. figure #1fig. #1 =1=.figs== * Fig.figure.#1.55inby140pt 14ptFigure Captions=` .figs@g@g fig.M#1figs. @gs #1 @gs#1=#1@rk#1#1#1=@gs @FiNeD (NO epsf.tex, FIGURES WILL BE IGNORED) ##12in##1.5in (FIGURES WILL BE INCLUDED) ##1##1##1##1 #1#1#2#3#1#1fig. figure #1fig. #1 #312pt by -1truein#1=#1Fig. figure: #2 by1-1%{ }# =.defs ##1 ===##1 #1#1#1 #1section #1#1#1=#1 #1#1 #1subsection@m. @m.#1#1#1=#1 #1#1 #1appendix@m@m #1#1#1=#1#1 to 1em.=.toc##1##1page @ckfile @ckfile=.toc @ckfileno file .toc, no table of contents this pass @ckfileContents=12pt=0pt`=11.toc `=12`=12scaled3scaled4 =cmr10 =cmr7 =cmr5 =cmmi10 =cmmi7 =cmmi5 =cmsy10 =cmsy7 =cmsy5 =cmti10 ='177 ='177 ='177 ='60 ='60 ='600 0=0=0=1=1=1=2=2=2===cmcsc10 scaled1=cmti10 scaled 1 =cmsl10 scaled 1 =cmr10 scaled1 =cmr7 scaled1 =cmr5 scaled1 =cmmi10 scaled1 =cmmi7 scaled1 =cmmi5 scaled1 =cmsy10 scaled1 =cmsy7 scaled1 =cmsy5 scaled1 =cmbx10 scaled1 ='177 ='177 ='177 ='60 ='60 ='600 0=0=0=1=1=1=2=2=2== = =0 0=0=0=1=1=1=2=2=2== ==cmr9 =cmr6 =cmmi9 =cmmi6 =cmsy9 =cmsy6 =cmbx9 =cmti9 =cmsl9 ='177 ='177 ='60 ='600 0=0=0=1=1=1=2=2=2== ==0pt anom-aly anom-alies coun-ter-term coun-ter-terms^.15ex--.05em 1 ^.15ex/-13.5mu D.15ex/-.57em∂ ∂=cmti10 scaled1$ -.2in #13pt3pt #13pt3pt 1#2height#2in #1in#1e^^#1 #1 ∇_#1#1#2 ∇_#1∇_#2φ ψ #1#2ω^#1_#2 ⊔⊓ #11.5ex↔-16.5mu #1 $12 #1.3ex#1-.75em1ex∼mssym psfZ-.4em ZZ-.4em Z .9ptZ-.4em Z 1.2ptZ-.4em ZZ-.4em Z#1 #2(#3), #1 #2 (19#3)#1 #2(#3)#1 #2 (19#3)i.e. e.g. c.f. et.al. etc α' #1#1 #/1#2#1#2 #1#2#1#2height1.5ex width.4pt depth0pt -.3em C I-.18em R I-.18em P 1-3pt l n^ th#1#2#3Nucl. Phys. B#1 (#2) #3 #1#2#3Phys. Lett. #1B (#2) #3 #1#2#3Phys. Lett. #1B (#2) #3 #1#2#3Phys. Rev. Lett. #1 (#2) #3 #1#2#3Phys. Rev. D#1 (#2) #3 #1#2#3Phys. Rev. D#1 (#2) #3#1#2#3Phys. Rep. #1 (#2) #3 #1#2#3Rev. Mod. Phys. #1 (#2) #3 #1#2#3Comm. Math. Phys. #1 (#2) #3 #1#2#3Class. Quant. Grav. #1 (#2) #3 #1#2#3Mod. Phys. Lett. #1 (#2) #3`=11 #1@̧ncel#1 =0pt ÅA B C D E F G H I J K L M N O P Q R S T U V W X Y S Z Z R λ ϵ ε #1#2@#1_#2 <∼ ⪅<∼ `=12 2ptheight 4.1pt depth -.3pt width .25pt -2pt× mod sinhsinh coshcosh sgn expexp sinh cosh NS^' (2,1) (2,0) (1,1) (0,1) 1+1 2+2 2+1 ȷj x̅ z̅ θ̅ ψ̅ ϕ̅ α̅ β̅ γ̅ θ̅ a̅ b̅ i̅ j̅ k̅ ℓ̅ ṃ̅̅S ijij̅ Kähler ϑ ϑ̅ AdS_3 SL(2) U(1) /J G Lϵ [ ]#1#12ex1em1em1exNOTE: #11ex1em#12ex1em1em1exInternal Note: #11ex1em#1(∙#1∙)L M N 5mm 2mm ,5mm ^-1 ∂ → ⇒ * * ∙* ∙∙ ∂ ã c̃ ã^el ã^m x̃ m̃z̃ tr → β̱ ỹ N CS_grav g Z C H W T=msbm10 =msbm7 =msbm5 1=1=1=⟨#|1⟨#1| |#⟩1| #1⟩ #1⟨#1⟩ ⟨#|1⟩#2#1 #2#1#̱1̱^2.5pt2 #1#2#̱1̱^2.5pt#2 #1#2#1^2.5pt#2 #1#2#2 ^#1 #1↔#1 Re Im det codim coker Tr tr Res sgn × 2√(2) ∂̣∇ mod diff Diff #1|#1| #1#1 #1#⃗1⃗ →α̇ β̇ γ̇ δ̇ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z1 R C H̋ Z N P S Tε #1#1_C ∪_1 ∪_2 S2πi 1-.26em I 0-.30em lΛ_NP M_Pl g tmod I-.18em L I-.18em H I-.18em R -.3em C Z-.4em ZZ-.4em Z .9ptZ-.4em Z 1.2ptZ-.4em ZZ-.4em Z M κ_CFT -̸2̸p̸t̸ ̸∂̸ O codim cok coker P P ch Hom diff Diffg^-1 g i̅ z̅ w̅ q̅ =manfnt 3.5pt127 #1#1-1.5ptZ-.4em ZZ-.4em Z .9ptZ-.4em Z 1.2ptZ-.4em ZZ-.4em Z 122ptheight 4.1pt depth -.3pt width .25pt -2pt×bat. Bye. Ram#1 * #1 ∂̅ β̇ α̇ Log ψ_ϕ ∂∘∘2√(2) 21√(2) √(3) √(3)2 #10=#1 0=0 1=/ 1=1 0>1 to 0/ #1to 1#1/gcd lcm#1#2#3#4_#1#2#3#4#1#2#1#1Figure figure #1Figure #1 Figure figure:#2 #1=#1 by1A. G. Abanov and P. B. Wiegmann, “Theta terms in nonlinear sigma models,” Nucl. Phys. B 570, 685 (2000). [hep-th/9911025].I. Affleck, J. A. Harvey and E. Witten, “Instantons and (Super)Symmetry Breaking in (2+1)-Dimensions,” Nucl. Phys. B 206, 413 (1982).O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg and M. J. Strassler, “Aspects of N=2 supersymmetric gauge theories in three-dimensions,” Nucl. Phys. B 499, 67 (1997). [hep-th/9703110]. O. Aharony, “IR duality in d = 3 N=2 supersymmetric USp(2N_c) and U(N_c) gauge theories,” Phys. Lett. B 404, 71 (1997). [hep-th/9703215].O. Aharony and I. Shamir, “On O(N_c) d=3 N=2 supersymmetric QCD Theories,” JHEP 1112, 043 (2011). [arXiv:1109.5081 [hep-th]].O. Aharony, G. Gur-Ari and R. Yacoby, “d=3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories,” JHEP 1203, 037 (2012). [arXiv:1110.4382 [hep-th]].O. Aharony, G. Gur-Ari and R. Yacoby, “Correlation Functions of Large N Chern-Simons-Matter Theories and Bosonization in Three Dimensions,” JHEP 1212, 028 (2012). [arXiv:1207.4593 [hep-th]].O. Aharony, S. Giombi, G. Gur-Ari, J. Maldacena and R. Yacoby, “The Thermal Free Energy in Large N Chern-Simons-Matter Theories,” JHEP 1303, 121 (2013). [arXiv:1211.4843 [hep-th]].O. Aharony, N. Seiberg and Y. Tachikawa, “Reading between the lines of four-dimensional gauge theories,” JHEP 1308, 115 (2013). [arXiv:1305.0318 [hep-th]].O. Aharony, S. S. Razamat, N. Seiberg and B. Willett, “3d dualities from 4d dualities,” JHEP 1307, 149 (2013). [arXiv:1305.3924 [hep-th]].O. Aharony, S. S. Razamat, N. Seiberg and B. Willett, “3d dualities from 4d dualities for orthogonal groups,” JHEP 1308, 099 (2013) [arXiv:1307.0511 [hep-th]].O. Aharony, “Baryons, monopoles and dualities in Chern-Simons-matter theories,” JHEP 1602, 093 (2016). [arXiv:1512.00161 [hep-th]].O. Aharony, F. Benini, P. S. Hsin and N. Seiberg, “Chern-Simons-matter dualities with SO and USp gauge groups,” JHEP 1702, 072 (2017). [arXiv:1611.07874 [cond-mat.str-el]].C. Wang, A. Nahum, M. A. Metlitski, C. Xu and T. Senthil, “Deconfined quantum critical points: symmetries and dualities,” [arXiv:1703.02426 [cond-mat.str-el]]. L. Alvarez-Gaume, S. Della Pietra and G. W. Moore, “Anomalies and Odd Dimensions,” Annals Phys.163, 288 (1985).D. Anninos, T. Hartman and A. Strominger, “Higher Spin Realization of the dS/CFT Correspondence,” [arXiv:1108.5735 [hep-th]].D. Anninos, R. Mahajan, D�. Radicevic and E. Shaghoulian, “Chern-Simons-Ghost Theories and de Sitter Space,” JHEP 1501, 074 (2015). [arXiv:1405.1424 [hep-th]].M. F. Atiyah, V. K. Patodi and I. M. Singer, “Spectral Asymmetry in Riemannian Geometry, I,” Math. Proc. Camb. Phil. Soc. 77 (1975) 43–69. T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and Gravity,” Phys. Rev. D 83, 084019 (2011). [arXiv:1011.5120 [hep-th]].M. Barkeshli and J. McGreevy, “Continuous transition between fractional quantum Hall and superfluid states,” Phys. Rev. B 89, 235116 (2014). C. Beem, T. Dimofte and S. Pasquetti, “Holomorphic Blocks in Three Dimensions,” [arXiv:1211.1986 [hep-th]]. C. Vafa and E. Witten, “Parity Conservation in QCD,” Phys. Rev. Lett.53, 535 (1984). F. Benini, C. Closset and S. Cremonesi, “Comments on 3d Seiberg-like dualities,” JHEP 1110, 075 (2011). [arXiv:1108.5373 [hep-th]].D. Bernard, “String Characters From Kac-Moody Automorphisms,” Nucl. Phys. B 288, 628 (1987). J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, “Indices for Superconformal Field Theories in 3,5 and 6 Dimensions,” JHEP 0802, 064 (2008). [arXiv:0801.1435 [hep-th]].J. de Boer, K. Hori, H. Ooguri and Y. Oz, “Mirror symmetry in three-dimensional gauge theories, quivers and D-branes,” Nucl. Phys. B 493, 101 (1997). [hep-th/9611063].J. de Boer, K. Hori, Y. Oz and Z. Yin, “Branes and mirror symmetry in N=2 supersymmetric gauge theories in three-dimensions,” Nucl. Phys. B 502, 107 (1997). [hep-th/9702154].F. Wilczek and A. Zee, “Linking Numbers, Spin, and Statistics of Solitons,” Phys. Rev. Lett.51, 2250 (1983).P. Bonderson, C. Nayak and X. L. Qi, “A time-reversal invariant topological phase at the surface of a 3D topological insulator,” J. Stat. Mech.2013, P09016 (2013). V. Borokhov, A. Kapustin and X. k. Wu, “Topological disorder operators in three-dimensional conformal field theory,” JHEP 0211, 049 (2002). [hep-th/0206054]. E. Witten, “Global Aspects of Current Algebra,” Nucl. Phys. B 223, 422 (1983). E. Witten, “Current Algebra, Baryons, and Quark Confinement,” Nucl. Phys. B 223, 433 (1983).D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, “Theta, Time Reversal, and Temperature,” [arXiv:1703.00501 [hep-th]].V. Borokhov, A. Kapustin and X. k. Wu, “Monopole operators and mirror symmetry in three-dimensions,” JHEP 0212, 044 (2002). [hep-th/0207074].Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, “Comments on Abelian Higgs Models and Persistent Order,” [arXiv:1705.04786 [hep-th]].Z. Komargodski, T. Sulejmanpasic and M. Unsal, “Walls, Anomalies, and (De) Confinement in Quantum Anti-Ferromagnets,” [arXiv:1706.05731 [cond-mat.str-el]].W. Browder and E. Thomas, “Axioms for the generalized Pontryagin cohomology operations,” Quart. J. Math. Oxford 13, 55–60 (1962).F. van de Bult, “Hyperbolic Hypergeometric Functions,” University of Amsterdam Ph.D. thesis M. Camperi, F. Levstein and G. Zemba, “The Large N Limit Of Chern-simons Gauge Theory,” Phys. Lett. B 247 (1990) 549.W. Chen, M. P. A. Fisher and Y. S. Wu, “Mott transition in an anyon gas,” Phys. Rev. B 48, 13749 (1993). [cond-mat/9301037].E. Witten, “Supersymmetric index of three-dimensional gauge theory,” In *Shifman, M.A. (ed.): The many faces of the superworld* 156-184. [hep-th/9903005]. X. Chen, L. Fidkowski and A. Vishwanath, “Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator,” Phys. Rev. B 89, no. 16, 165132 (2014). [arXiv:1306.3250 [cond-mat.str-el]].K. Intriligator and N. Seiberg, “Aspects of 3d N=2 Chern-Simons-Matter Theories,” JHEP 1307, 079 (2013). [arXiv:1305.1633 [hep-th]].M. Cheng and C. Xu, “Series of (2+1)-dimensional stable self-dual interacting conformal field theories,” Phys. Rev. B 94, 214415 (2016).[arXiv:1609.02560 [cond-mat.str-el]].C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, “Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories,” JHEP 1210, 053 (2012). [arXiv:1205.4142 [hep-th]].C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, “Comments on Chern-Simons Contact Terms in Three Dimensions,” JHEP 1209, 091 (2012). [arXiv:1206.5218 [hep-th]]. C. Closset, T. T. Dumitrescu, G. Festuccia and Z. Komargodski, “Supersymmetric Field Theories on Three-Manifolds,” JHEP 1305, 017 (2013). [arXiv:1212.3388 [hep-th]].M. Cvetic, T. W. Grimm and D. Klevers, “Anomaly Cancellation And Abelian Gauge Symmetries In F-theory,” JHEP 1302, 101 (2013). [arXiv:1210.6034 [hep-th]].X. z. Dai and D. S. Freed, “eta invariants and determinant lines,” J. Math. Phys.35, 5155 (1994), Erratum: [J. Math. Phys.42, 2343 (2001)]. [hep-th/9405012].C. Dasgupta and B. I. Halperin, “Phase Transition in a Lattice Model of Superconductivity,” Phys. Rev. Lett.47, 1556 (1981). N. M. Davies, T. J. Hollowood, V. V. Khoze and M. P. Mattis, “Gluino condensate and magnetic monopoles in supersymmetric gluodynamics,” Nucl. Phys. B 559, 123 (1999). [hep-th/9905015].N. M. Davies, T. J. Hollowood and V. V. Khoze, “Monopoles, affine algebras and the gluino condensate,” J. Math. Phys.44, 3640 (2003). [hep-th/0006011]. T. Dimofte, D. Gaiotto and S. Gukov, “3-Manifolds and 3d Indices,” [arXiv:1112.5179 [hep-th]].F. A. Dolan and H. Osborn, “Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories,” Nucl. Phys. B 818, 137 (2009). [arXiv:0801.4947 [hep-th]]. F. A. H. Dolan, V. P. Spiridonov and G. S. Vartanov, “From 4d superconformal indices to 3d partition functions,” Phys. Lett. B 704, 234 (2011). [arXiv:1104.1787 [hep-th]].M. R. Douglas, “Chern-Simons-Witten theory as a topological Fermi liquid,” [hep-th/9403119]. R. Eager, J. Schmude and Y. Tachikawa, “Superconformal Indices, Sasaki-Einstein Manifolds, and Cyclic Homologies,” [arXiv:1207.0573 [hep-th]].S. Elitzur, A. Giveon and D. Kutasov, “Branes and N=1 duality in string theory,” Phys. Lett. B 400, 269 (1997). [hep-th/9702014].A. Kapustin and R. Thorngren, “Anomalies of discrete symmetries in three dimensions and group cohomology,” Phys. Rev. Lett.112, no. 23, 231602 (2014). [arXiv:1403.0617 [hep-th]]. A. Kapustin and R. Thorngren, “Anomalies of discrete symmetries in various dimensions and group cohomology,” [arXiv:1404.3230 [hep-th]].S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and A. Schwimmer, “Brane dynamics and N=1 supersymmetric gauge theory,” Nucl. Phys. B 505, 202 (1997). [hep-th/9704104].A. M. Essin, J. E. Moore and D. Vanderbilt, “Magnetoelectric polarizability and axion electrodynamics in crystalline insulators,” Phys. Rev. Lett.102, 146805 (2009). [arXiv:0810.2998 [cond-mat.mes-hall]]. J. Felder, A. Varchenko, “The elliptic gamma function and SL(3,Z) × Z^3,”[arXiv:math/0001184].G. Festuccia and N. Seiberg, “Rigid Supersymmetric Theories in Curved Superspace,” JHEP 1106, 114 (2011). [arXiv:1105.0689 [hep-th]].L. Fidkowski, X. Chen and A. Vishwanath, “Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model,” Phys. Rev. X 3, no. 4, 041016 (2013). [arXiv:1305.5851 [cond-mat.str-el]].E. H. Fradkin and F. A. Schaposnik, “The Fermion - boson mapping in three-dimensional quantum field theory,” Phys. Lett. B 338, 253 (1994). [hep-th/9407182]. A. Gadde, L. Rastelli, S. S. Razamat and W. Yan, “On the Superconformal Index of N=1 IR Fixed Points: A Holographic Check,” JHEP 1103, 041 (2011). [arXiv:1011.5278 [hep-th]].D. K. Hong and H. U. Yee, “Holographic aspects of three dimensional QCD from string theory,” JHEP 1005, 036 (2010), Erratum: [JHEP 1008, 120 (2010)]. [arXiv:1003.1306 [hep-th]].A. Gadde and W. Yan, “Reducing the 4d Index to the S^3 Partition Function,” JHEP 1212, 003 (2012). [arXiv:1104.2592 [hep-th]].A. Gadde and S. Gukov, “2d Index and Surface operators,” [arXiv:1305.0266 [hep-th]].C. Vafa and E. Witten, “Eigenvalue Inequalities for Fermions in Gauge Theories,” Commun. Math. Phys.95, 257 (1984).D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,” Adv. Theor. Math. Phys.13, no. 3, 721 (2009). [arXiv:0807.3720 [hep-th]].D. Gaiotto, G. W. Moore and A. Neitzke, “Framed BPS States,” [arXiv:1006.0146 [hep-th]].D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,” JHEP 1502, 172 (2015). [arXiv:1412.5148 [hep-th]].S. D. Geraedts, M. P. Zaletel, R. S. K. Mong, M. A. Metlitski, A. Vishwanath and O. I. Motrunich, “The half-filled Landau level: the case for Dirac composite fermions,” Science 352, 197 (2016). [arXiv:1508.04140 [cond-mat.str-el]].S. Giombi and X. Yin, “On Higher Spin Gauge Theory and the Critical O(N) Model,” Phys. Rev. D 85, 086005 (2012). [arXiv:1105.4011 [hep-th]].S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia and X. Yin, “Chern-Simons Theory with Vector Fermion Matter,” Eur. Phys. J. C 72, 2112 (2012). [arXiv:1110.4386 [hep-th]].S. Giombi and X. Yin, “The Higher Spin/Vector Model Duality,” J. Phys. A 46, 214003 (2013). [arXiv:1208.4036 [hep-th]].S. Giombi, V. Gurucharan, V. Kirilin, S. Prakash and E. Skvortsov, “On the Higher-Spin Spectrum in Large N Chern-Simons Vector Models,” JHEP 1701, 058 (2017). [arXiv:1610.08472 [hep-th]].A. Giveon and D. Kutasov, “Brane dynamics and gauge theory,” Rev. Mod. Phys.71, 983 (1999). [hep-th/9802067].A. Giveon and D. Kutasov, “Seiberg Duality in Chern-Simons Theory,” Nucl. Phys. B 812, 1 (2009). [arXiv:0808.0360 [hep-th]].P. Goddard, J. Nuyts and D. I. Olive, “Gauge Theories and Magnetic Charge,” Nucl. Phys. B 125, 1 (1977).P. Goddard, A. Kent and D. I. Olive, “Virasoro Algebras and Coset Space Models,” Phys. Lett. B 152, 88 (1985). D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, “Exactly Marginal Deformations and Global Symmetries,” JHEP 1006, 106 (2010). [arXiv:1005.3546 [hep-th]].G. Gur-Ari and R. Yacoby, “Three Dimensional Bosonization From Supersymmetry,” JHEP 1511, 013 (2015). [arXiv:1507.04378 [hep-th]].G. Gur-Ari, S. A. Hartnoll and R. Mahajan, “Transport in Chern-Simons-Matter Theories,” JHEP 1607, 090 (2016). [arXiv:1605.01122 [hep-th]].B. I. Halperin, P. A. Lee and N. Read, “Theory of the half filled Landau level,” Phys. Rev. B 47, 7312 (1993). N. Hama, K. Hosomichi and S. Lee, “SUSY Gauge Theories on Squashed Three-Spheres,” JHEP 1105, 014 (2011). [arXiv:1102.4716 [hep-th]]. K. Hasegawa, “Spin Module Versions of Weyl's Reciprocity Theorem for Classical Kac-Moody Lie Algebras - An Application to Branching Rule Duality,” Publ. Res. Inst. Math. Sci. 25, 741-828 (1989).K. Hori and D. Tong, “Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2) Theories,” JHEP 0705, 079 (2007). [hep-th/0609032]. K. Hori, “Duality In Two-Dimensional (2,2) Supersymmetric Non-Abelian Gauge Theories,” [arXiv:1104.2853 [hep-th]].P. S. Hsin and N. Seiberg, “Level/rank Duality and Chern-Simons-Matter Theories,” JHEP 1609, 095 (2016). [arXiv:1607.07457 [hep-th]].C. M. Hull and B. J. Spence, “The Geometry of the gauged sigma model with Wess-Zumino term,” Nucl. Phys. B 353, 379 (1991). C. Hwang, H. Kim, K. -J. Park and J. Park, “Index computation for 3d Chern-Simons matter theory: test of Seiberg-like duality,” JHEP 1109, 037 (2011). [arXiv:1107.4942 [hep-th]].C. Hwang, K. -J. Park and J. Park, “Evidence for Aharony duality for orthogonal gauge groups,” JHEP 1111, 011 (2011). [arXiv:1109.2828 [hep-th]]. C. Hwang, H. -C. Kim and J. Park, “Factorization of the 3d superconformal index,” [arXiv:1211.6023 [hep-th]]. Y. Imamura and S. Yokoyama, “Index for three dimensional superconformal field theories with general R-charge assignments,” JHEP 1104, 007 (2011). [arXiv:1101.0557 [hep-th]]. Y. Imamura,“Relation between the 4d superconformal index and the S^3 partition function,” JHEP 1109, 133 (2011). [arXiv:1104.4482 [hep-th]]. Y. Imamura and D. Yokoyama,“N=2 supersymmetric theories on squashed three-sphere,” Phys. Rev. D 85, 025015 (2012). [arXiv:1109.4734 [hep-th]]. Y. Imamura and D. Yokoyama,“S^3/Z_n partition function and dualities,” JHEP 1211, 122 (2012). [arXiv:1208.1404 [hep-th]].K. Inbasekar, S. Jain, S. Mazumdar, S. Minwalla, V. Umesh and S. Yokoyama, “Unitarity, crossing symmetry and duality in the scattering of N=1 SUSY matter Chern-Simons theories,” JHEP 1510, 176 (2015). [arXiv:1505.06571 [hep-th]].K. A. Intriligator and N. Seiberg, “Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(N(c)) gauge theories,” Nucl. Phys. B 444, 125 (1995). [hep-th/9503179].K. A. Intriligator and P. Pouliot, “Exact superpotentials, quantum vacua and duality in supersymmetric SP(N(c)) gauge theories,” Phys. Lett. B 353, 471 (1995). [hep-th/9505006].K. A. Intriligator and N. Seiberg, “Phases of N=1 supersymmetric gauge theories and electric - magnetic triality,” In *Los Angeles 1995, Future perspectives in string theory* 270-282. [hep-th/9506084].K. A. Intriligator and N. Seiberg, “Lectures on supersymmetric gauge theories and electric - magnetic duality,” Nucl. Phys. Proc. Suppl.45BC, 1 (1996). [hep-th/9509066].K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories,” Phys. Lett. B 387, 513 (1996). [hep-th/9607207].K. Intriligator and N. Seiberg, “Aspects of 3d N=2 Chern-Simons-Matter Theories,” JHEP 1307, 079 (2013). [arXiv:1305.1633 [hep-th]]. E. A. Ivanov,“Chern-Simons matter systems with manifest N=2 supersymmetry,” Phys. Lett. B 268, 203 (1991). D. L. Jafferis, “The Exact Superconformal R-Symmetry Extremizes Z,” JHEP 1205, 159 (2012). [arXiv:1012.3210 [hep-th]].D. L. Jafferis, I. R. Klebanov, S. S. Pufu and B. R. Safdi, “Towards the F-Theorem: N=2 Field Theories on the Three-Sphere,” JHEP 1106, 102 (2011). [arXiv:1103.1181 [hep-th]].J. K. Jain, “Composite fermion approach for the fractional quantum Hall effect,” Phys. Rev. Lett.63, 199 (1989). S. Jain, S. Minwalla, T. Sharma, T. Takimi, S. R. Wadia and S. Yokoyama, “Phases of large N vector Chern-Simons theories on S^2 × S^1,” JHEP 1309, 009 (2013). [arXiv:1301.6169 [hep-th]].S. Jain, S. Minwalla and S. Yokoyama, “Chern Simons duality with a fundamental boson and fermion,” JHEP 1311, 037 (2013). [arXiv:1305.7235 [hep-th]].S. Jain, M. Mandlik, S. Minwalla, T. Takimi, S. R. Wadia and S. Yokoyama, “Unitarity, Crossing Symmetry and Duality of the S-matrix in large N Chern-Simons theories with fundamental matter,” JHEP 1504, 129 (2015). [arXiv:1404.6373 [hep-th]].S. Kachru, M. Mulligan, G. Torroba and H. Wang, “Mirror symmetry and the half-filled Landau level,” Phys. Rev. B 92, 235105 (2015). [arXiv:1506.01376 [cond-mat.str-el]].S. Kachru, M. Mulligan, G. Torroba and H. Wang, “Bosonization and Mirror Symmetry,” Phys. Rev. D 94, no. 8, 085009 (2016). [arXiv:1608.05077 [hep-th]].S. Kachru, M. Mulligan, G. Torroba and H. Wang, “Nonsupersymmetric dualities from mirror symmetry,” Phys. Rev. Lett.118, 011602 (2017). [arXiv:1609.02149 [hep-th]].K. Kajantie, M. Laine, T. Neuhaus, A. Rajantie and K. Rummukainen, “Duality and scaling in three-dimensional scalar electrodynamics,” Nucl. Phys. B 699, 632 (2004). [hep-lat/0402021].A. Kapustin and M. J. Strassler, “On mirror symmetry in three-dimensional Abelian gauge theories,” JHEP 9904, 021 (1999). [hep-th/9902033].A. Kapustin, “Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality,” Phys. Rev. D 74, 025005 (2006). [hep-th/0501015]. A. Kapustin, B. Willett and I. Yaakov, “Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,” JHEP 1003, 089 (2010). [arXiv:0909.4559 [hep-th]]. A. Kapustin,2010 Simons Workshop talk, a video of this talk can be found at http://media.scgp.stonybrook.edu/video/video.php?f=20110810_1_qtp.mp4A. Kapustin, B. Willett and I. Yaakov, “Nonperturbative Tests of Three-Dimensional Dualities,” JHEP 1010, 013 (2010). [arXiv:1003.5694 [hep-th]].A. Kapustin, “Seiberg-like duality in three dimensions for orthogonal gauge groups,” arXiv:1104.0466 [hep-th]. A. Kapustin and B. Willett, “Generalized Superconformal Index for Three Dimensional Field Theories,” [arXiv:1106.2484 [hep-th]].A. Kapustin, H. Kim and J. Park, “Dualities for 3d Theories with Tensor Matter,” JHEP 1112, 087 (2011). [arXiv:1110.2547 [hep-th]].A. Kapustin and N. Seiberg, “Coupling a QFT to a TQFT and Duality,” JHEP 1404, 001 (2014). [arXiv:1401.0740 [hep-th]].A. Karch, “Seiberg duality in three-dimensions,” Phys. Lett. B 405, 79 (1997). [hep-th/9703172].A. Karch and D. Tong, “Particle-Vortex Duality from 3d Bosonization,” Phys. Rev. X 6, 031043 (2016).[arXiv:1606.01893 [hep-th]].A. Karch, B. Robinson and D. Tong, “More Abelian Dualities in 2+1 Dimensions,” JHEP 1701, 017 (2017). [arXiv:1609.04012 [hep-th]]. S. Kim, “The Complete superconformal index for N=6 Chern-Simons theory,” Nucl. Phys. B 821, 241 (2009), [Erratum-ibid. B 864, 884 (2012)]. [arXiv:0903.4172 [hep-th]].H. Kim and J. Park, “Aharony Dualities for 3d Theories with Adjoint Matter,” [arXiv:1302.3645 [hep-th]]. J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju, “An Index for 4 dimensional super conformal theories,” Commun. Math. Phys.275, 209 (2007). [hep-th/0510251].I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical O(N) vector model,” Phys. Lett. B 550, 213 (2002). [hep-th/0210114]. C. Krattenthaler, V. P. Spiridonov, G. S. Vartanov, “Superconformal indices of three-dimensional theories related by mirror symmetry,” JHEP 1106, 008 (2011). [arXiv:1103.4075 [hep-th]]. S. M. Kravec, J. McGreevy, and B. Swingle, “All-Fermion Electrodynamics And Fermion Number Anomaly Inflow,” arXiv:1409.8339. D. Kutasov, “A Comment on duality in N=1 supersymmetric nonAbelian gauge theories,” Phys. Lett. B 351, 230 (1995). [hep-th/9503086].D. Kutasov and A. Schwimmer, “On duality in supersymmetric Yang-Mills theory,” Phys. Lett. B 354, 315 (1995). [hep-th/9505004].D. Kutasov, A. Schwimmer and N. Seiberg, “Chiral rings, singularity theory and electric - magnetic duality,” Nucl. Phys. B 459, 455 (1996). [hep-th/9510222].K. -M. Lee and P. Yi, “Monopoles and instantons on partially compactified D-branes,” Phys. Rev. D 56, 3711 (1997). [hep-th/9702107].K. -M. Lee, “Instantons and magnetic monopoles on R**3 x S**1 with arbitrary simple gauge groups,” Phys. Lett. B 426, 323 (1998). [hep-th/9802012].J. M. Maldacena, G. W. Moore and N. Seiberg, “D-brane charges in five-brane backgrounds,” JHEP 0110, 005 (2001). [hep-th/0108152].J. Maldacena and A. Zhiboedov, “Constraining Conformal Field Theories with A Higher Spin Symmetry,” J. Phys. A 46, 214011 (2013). [arXiv:1112.1016 [hep-th]].M. A. Metlitski and A. Vishwanath, “Particle-vortex duality of 2d Dirac fermion from electric-magnetic duality of 3d topological insulators,” [arXiv:1505.05142 [cond-mat.str-el]].M. A. Metlitski, C. L. Kane and M. P. A. Fisher, “Symmetry-respecting topologically ordered surface phase of three-dimensional electron topological insulators,” Phys. Rev. B 92, no. 12, 125111 (2015). M. A. Metlitski, “S-duality of u(1) gauge theory with θ =π on non-orientable manifolds: Applications to topological insulators and superconductors,” [arXiv:1510.05663 [hep-th]].M. A. Metlitski, A. Vishwanath and C. Xu, “Duality and bosonization of (2+1)d Majorana fermions,” arXiv:1611.05049 [cond-mat.str-el].S. Minwalla and S. Yokoyama, “Chern Simons Bosonization along RG Flows,” JHEP 1602, 103 (2016). [arXiv:1507.04546 [hep-th]].E. J. Mlawer, S. G. Naculich, H. A. Riggs and H. J. Schnitzer, “Group level duality of WZW fusion coefficients and Chern-Simons link observables,” Nucl. Phys. B 352, 863 (1991). G. W. Moore and N. Seiberg, “Naturality in Conformal Field Theory,” Nucl. Phys. B 313, 16 (1989).G. W. Moore and N. Seiberg, “Taming the Conformal Zoo,” Phys. Lett. B 220, 422 (1989).T. Morita and V. Niarchos, “F-theorem, duality and SUSY breaking in one-adjoint Chern-Simons-Matter theories,” Nucl. Phys. B 858, 84 (2012). [arXiv:1108.4963 [hep-th]].D. F. Mross, J. Alicea and O. I. Motrunich, “Explicit derivation of duality between a free Dirac cone and quantum electrodynamics in (2+1) dimensions,” [arXiv:1510.08455 [cond-mat.str-el]].M. Mulligan, S. Raghu and M. P. A. Fisher, “Emergent particle-hole symmetry in the half-filled Landau level,” [arXiv:1603.05656 [cond-mat.str-el]].J. Murugan and H. Nastase, “Particle-vortex duality in topological insulators and superconductors,” [arXiv:1606.01912 [hep-th]].S. G. Naculich, H. A. Riggs and H. J. Schnitzer, “Group Level Duality in WZW Models and Chern-Simons Theory,” Phys. Lett. B 246, 417 (1990).S. G. Naculich and H. J. Schnitzer, “Level-rank duality of the U(N) WZW model, Chern-Simons theory, and 2-D qYM theory,” JHEP 0706, 023 (2007). [hep-th/0703089 [HEP-TH]].M. Nakahara, “Geometry, topology and physics,” Boca Raton, USA: Taylor and Francis (2003) 573 p. T. Nakanishi and A. Tsuchiya, “Level rank duality of WZW models in conformal field theory,” Commun. Math. Phys.144, 351 (1992).A. K. Nguyen and A. Sudbo, “Topological phase fluctuations, amplitude fluctuations, and criticality in extreme type II superconductors,” Phys. Rev. B 60, 15307 (1999). [cond-mat/9907385].V. Niarchos, “Seiberg Duality in Chern-Simons Theories with Fundamental and Adjoint Matter,” JHEP 0811, 001 (2008). [arXiv:0808.2771 [hep-th]].V. Niarchos, “R-charges, Chiral Rings and RG Flows in Supersymmetric Chern-Simons-Matter Theories,” JHEP 0905, 054 (2009).= [arXiv:0903.0435 [hep-th]]. V. Niarchos, “Seiberg dualities and the 3d/4d connection,” JHEP 1207, 075 (2012). [arXiv:1205.2086 [hep-th]].A. J. Niemi and G. W. Semenoff, “Axial Anomaly Induced Fermion Fractionization and Effective Gauge Theory Actions in Odd Dimensional Space-Times,” Phys. Rev. Lett.51, 2077 (1983). V. Ostrik and M. Sun, “Level-Rank Duality Via Tensor Categories,” Comm. Math. Phys. 326 (2014) 49-61. [arXiv:1208.5131 [math-ph]]. J. Park and K. J. Park, “Seiberg-like Dualities for 3d N=2 Theories with SU(N) gauge group,” JHEP 1310, 198 (2013). [arXiv:1305.6280 [hep-th]].C. Pauly, “Strange duality revisited,” Math. Res. Lett.21, 1353 (2014). M. E. Peskin, “Mandelstam 't Hooft Duality in Abelian Lattice Models,” Annals Phys.113, 122 (1978).A. M. Polyakov, “Quark Confinement and Topology of Gauge Groups,” Nucl. Phys. B 120, 429 (1977).A. M. Polyakov, “Fermi-Bose Transmutations Induced by Gauge Fields,” Mod. Phys. Lett. A 3, 325 (1988).A. C. Potter, M. Serbyn and A. Vishwanath, “Thermoelectric transport signatures of Dirac composite fermions in the half-filled Landau level,” Phys. Rev. X 6, 031026 (2016). [arXiv:1512.06852 [cond-mat.str-el]].X. L. Qi, T. Hughes and S. C. Zhang, “Topological Field Theory of Time-Reversal Invariant Insulators,” Phys. Rev. B 78, 195424 (2008). [arXiv:0802.3537 [cond-mat.mes-hall]].C. Cordova, P. S. Hsin and N. Seiberg, “Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups,” [arXiv:1711.10008 [hep-th]].E. Rabinovici, A. Schwimmer and S. Yankielowicz, “Quantization in the Presence of Wess-Zumino Terms,” Nucl. Phys. B 248, 523 (1984).D. Radicevic, “Disorder Operators in Chern-Simons-Fermion Theories,” JHEP 1603, 131 (2016). [arXiv:1511.01902 [hep-th]].D. Radicevic, D. Tong and C. Turner, “Non-Abelian 3d Bosonization and Quantum Hall States,” JHEP 1612, 067 (2016). [arXiv:1608.04732 [hep-th]]. S. S. Razamat, “On a modular property of N=2 superconformal theories in four dimensions,” JHEP 1210, 191 (2012). [arXiv:1208.5056 [hep-th]].A. N. Redlich, “Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions,” Phys. Rev. D 29, 2366 (1984). K.-H. Rehren, “Algebraic Conformal QFT”, 3rd Meeting of the French-Italian Research Team on Noncommutative Geometry and Quantum Physics Vietri sul Mare, 2009.C. Romelsberger, “Counting chiral primaries in N = 1, d=4 superconformal field theories,” Nucl. Phys. B 747, 329 (2006). [hep-th/0510060].D. Roscher, E. Torres and P. Strack, “Dual QED_3 at "N_F = 1/2" is an interacting CFT in the infrared,” [arXiv:1605.05347 [cond-mat.str-el]].B. R. Safdi, I. R. Klebanov and J. Lee, “A Crack in the Conformal Window,” [arXiv:1212.4502 [hep-th]].N. Seiberg, “Exact results on the space of vacua of four-dimensional SUSY gauge theories,” Phys. Rev. D 49, 6857 (1994). [hep-th/9402044].N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge theories,” Nucl. Phys. B 435, 129 (1995). [hep-th/9411149].N. Seiberg and E. Witten, “Gauge dynamics and compactification to three-dimensions,” In *Saclay 1996, The mathematical beauty of physics* 333-366. [hep-th/9607163].N. Seiberg, “Modifying the Sum Over Topological Sectors and Constraints on Supergravity,” JHEP 1007, 070 (2010). [arXiv:1005.0002 [hep-th]].N. Seiberg and E. Witten, “Gapped Boundary Phases of Topological Insulators via Weak Coupling,” PTEP 2016, 12C101 (2016).[arXiv:1602.04251 [cond-mat.str-el]].N. Seiberg, T. Senthil, C. Wang and E. Witten, “A Duality Web in 2+1 Dimensions and Condensed Matter Physics,” Annals Phys.374, 395 (2016). [arXiv:1606.01989 [hep-th]].T. Senthil and M. P. A. Fisher, “Competing orders, non-linear sigma models, and topological terms in quantum magnets,” Phys. Rev. B 74, 064405 (2006). [cond-mat/0510459].E. Sezgin and P. Sundell, “Massless higher spins and holography,” Nucl. Phys. B 644, 303 (2002), Erratum: [Nucl. Phys. B 660, 403 (2003)]. [hep-th/0205131].E. Sezgin and P. Sundell, “Holography in 4D (super) higher spin theories and a test via cubic scalar couplings,” JHEP 0507, 044 (2005). [hep-th/0305040].N. Shaji, R. Shankar and M. Sivakumar, “On Bose-fermi Equivalence in a U(1) Gauge Theory With Chern-Simons Action,” Mod. Phys. Lett. A 5, 593 (1990). I. Shamir, “Aspects of three dimensional Seiberg duality,” M. Sc. thesis submitted to the Weizmann Institute of Science, April 2010. S. H. Shenker and X. Yin, “Vector Models in the Singlet Sector at Finite Temperature,” [arXiv:1109.3519 [hep-th]].D. T. Son, “Is the Composite Fermion a Dirac Particle?,” Phys. Rev. X 5, 031027 (2015). [arXiv:1502.03446 [cond-mat.mes-hall]].V. P. Spiridonov and G. S. Vartanov, “Superconformal indices for N = 1 theories with multiple duals,” Nucl. Phys. B 824, 192 (2010). [arXiv:0811.1909 [hep-th]]. V. P. Spiridonov and G. S. Vartanov, “Elliptic Hypergeometry of Supersymmetric Dualities,” Commun. Math. Phys.304, 797 (2011). [arXiv:0910.5944 [hep-th]]. V. P. Spiridonov and G. S. Vartanov, “Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices,” [arXiv:1107.5788 [hep-th]]. V. P. Spiridonov and G. S. Vartanov, “Elliptic hypergeometric integrals and 't Hooft anomaly matching conditions,” JHEP 1206, 016 (2012). [arXiv:1203.5677 [hep-th]].M. J. Strassler, “Duality, phases, spinors and monopoles in SO(N) and spin(N) gauge theories,” JHEP 9809, 017 (1998). [hep-th/9709081].M. A. Vasiliev, “Holography, Unfolding and Higher-Spin Theory,” J. Phys. A 46, 214013 (2013). [arXiv:1203.5554 [hep-th]].D. Verstegen, “Conformal embeddings, rank-level duality and exceptional modular invariants,” Commun. Math. Phys.137, 567 (1991).C. Wang, A. C. Potter and T. Senthil, “Gapped symmetry preserving surface state for the electron topological insulator,” Phys. Rev. B 88, no. 11, 115137 (2013). [arXiv:1306.3223 [cond-mat.str-el]]. C. Wang, A. C. Potter, and T. Senthil, “Classification Of Interacting Electronic Topological Insulators In Three Dimensions,” Science 343 (2014) 629, [arXiv:1306.3238]. C. Wang and T. Senthil, “Interacting fermionic topological insulators/superconductors in three dimensions,” Phys. Rev. B 89, no. 19, 195124 (2014), Erratum: [Phys. Rev. B 91, no. 23, 239902 (2015)]. [arXiv:1401.1142 [cond-mat.str-el]].C. Wang and T. Senthil, “Dual Dirac Liquid on the Surface of the Electron Topological Insulator,” Phys. Rev. X 5, no. 4, 041031 (2015). [arXiv:1505.05141 [cond-mat.str-el]]. C. Wang and T. Senthil, “Half-filled Landau level, topological insulator surfaces, and three-dimensional quantum spin liquids,” Phys. Rev. B 93, no. 8, 085110 (2016). [arXiv:1507.08290 [cond-mat.st-el]]. C. Wang and T. Senthil, “Composite fermi liquids in the lowest Landau level,” Phys. Rev. B 94, 245107 (2016).[arXiv:1604.06807 [cond-mat.str-el]].C. Wang and T. Senthil, “Time-Reversal Symmetric U(1) Quantum Spin Liquids,” Phys. Rev. X 6, no. 1, 011034 (2016).J. H. C. Whitehead, “On simply connected, 4-dimensional polyhedra,” Comm. Math. Helv. 22 (1949) 48. F. Wilczek, “Magnetic Flux, Angular Momentum, and Statistics,” Phys. Rev. Lett.48, 1144 (1982).F. Wilczek and A. Zee, “Linking Numbers, Spin, and Statistics of Solitons,” Phys. Rev. Lett.51, 2250 (1983).J. Gasser and H. Leutwyler, “Chiral Perturbation Theory to One Loop,” Annals Phys.158, 142 (1984). B. Willett and I. Yaakov, “N=2 Dualities and Z-extremization in Three Dimensions,” arXiv:1104.0487 [hep-th].E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math. Phys.121, 351 (1989).C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, “Comments on Chern-Simons Contact Terms in Three Dimensions,” JHEP 1209, 091 (2012). [arXiv:1206.5218 [hep-th]].E. Witten, “The Verlinde algebra and the cohomology of the Grassmannian,” In *Cambridge 1993, Geometry, topology, and physics* 357-422. [hep-th/9312104].E. Witten, “On S duality in Abelian gauge theory,” Selecta Math.1, 383 (1995). [hep-th/9505186].E. Witten, “Supersymmetric index of three-dimensional gauge theory,” In *Shifman, M.A. (ed.): The many faces of the superworld* 156-184. [hep-th/9903005].L. Di Pietro and E. Stamou, “Scaling dimensions in QED_3 from the ϵ-expansion,” [arXiv:1708.03740 [hep-th]].L. Di Pietro and E. Stamou, “Operator mixing in ϵ-expansion: scheme and evanescent (in)dependence,” [arXiv:1708.03739 [hep-th]].L. Di Pietro, Z. Komargodski, I. Shamir and E. Stamou, “The ϵ -expansion for QED,” Int. J. Mod. Phys. A 31, no. 28&29, 1645039 (2016)..E. Witten, “SL(2,Z) action on three-dimensional conformal field theories with Abelian symmetry,” In *Shifman, M. (ed.) et al.: From fields to strings, vol. 2* 1173-1200. [hep-th/0307041].E. Witten, “Fermion Path Integrals And Topological Phases,” [arXiv:1508.04715 [cond-mat.mes-hall]].T. T. Wu and C. N. Yang, “Dirac Monopole Without Strings: Monopole Harmonics,” Nucl. Phys. B 107, 365 (1976).F. Xu, “Algebraic coset conformal field theories,” Commun. Math. Phys.211, 1 (2000). [math/9810035].C. Vafa and E. Witten, “Restrictions on Symmetry Breaking in Vector-Like Gauge Theories,” Nucl. Phys. B 234, 173 (1984).F. Benini, P. S. Hsin and N. Seiberg, “Comments on global symmetries, anomalies, and duality in (2 + 1)d,” JHEP 1704, 135 (2017). [arXiv:1702.07035 [cond-mat.str-el]].A. N. Redlich, “Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions,” Phys. Rev. D 29, 2366 (1984).C. Xu and Y. Z. You, “Self-dual Quantum Electrodynamics as Boundary State of the three dimensional Bosonic Topological Insulator,” Phys. Rev. B 92, 220416 (2015).[arXiv:1510.06032 [cond-mat.str-el]].D. Radicevic, “Disorder Operators in Chern-Simons-Fermion Theories,” JHEP 1603, 131 (2016). [arXiv:1511.01902 [hep-th]].O. Aharony, “Baryons, monopoles and dualities in Chern-Simons-matter theories,” JHEP 1602, 093 (2016). [arXiv:1512.00161 [hep-th]]. B. M. Zupnik and D. G. Pak,“Topologically Massive Gauge Theories In Superspace,” Sov. Phys. J.31, 962 (1988). T. Appelquist and D. Nash, “Critical Behavior in (2+1)-dimensional QCD,” Phys. Rev. Lett.64, 721 (1990).D. S. Freed, “Pions and Generalized Cohomology,” J. Diff. Geom.80, no. 1, 45 (2008). [hep-th/0607134]. B. I. Zwiebel, “Charging the Superconformal Index,” JHEP 1201, 116 (2012). [arXiv:1111.1773 [hep-th]].Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, “Comments on Abelian Higgs Models and Persistent Order,” [arXiv:1705.04786 [hep-th]].D. Gaiotto, Z. Komargodski and N. Seiberg, “Time-Reversal Breaking in QCD_4, Walls, and Dualities in 2+1 Dimensions,” [arXiv:1708.06806 [hep-th]].D. S. Freed, Z. Komargodski and N. Seiberg, “The Sum Over Topological Sectors and θ in the 2+1-Dimensional ^1 σ-Model,” [arXiv:1707.05448 [cond-mat.str-el]]. -60pt A Symmetry Breaking Scenario for QCD_3 -15ptZohar Komargodski^1,2 and Nathan Seiberg^3 15pt ^1 Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Israel^2 Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY ^3 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA25pt We consider the dynamics of 2+1 dimensional SU(N) gauge theory with Chern-Simons level k and N_f fundamental fermions. By requiring consistency with previously suggested dualities for N_f≤ 2k as well as the dynamics at k=0 we propose that the theory with N_f> 2k breaks the U(N_f) global symmetry spontaneously toU(N_f/2+k)× U(N_f/2-k). In contrast to the 3+1 dimensional case, the symmetry breaking takes place in a range of quark masses and not just at one point. The target space never becomes parametrically large and the Nambu-Goldstone bosons are therefore not visible semi-classically. Such symmetry breaking is argued to take place in some intermediate range of the number of flavors, 2k< N_f< N_*(N,k), with the upper limit N_* obeying various constraints. The Lagrangian for the Nambu-Goldstone bosons has to be supplemented by nontrivial Wess-Zumino terms that are necessary for the consistency of the picture, even at k=0. Furthermore, we suggest two scalar dual theories in this range of N_f.A similar picture is developed for SO(N) and Sp(N) gauge theories.It sheds new light on monopole condensation and confinement in the SO(N) &Spin(N) theories.June 2017 = =cmbxti10 #1#1#1 IntroductionIn this paper we study QCD in 2+1 dimensions, namely non-Abelian 2+1 dimensional gauge theories coupled to fermionic matter fields.We will focus on SU(N), SO(N), and Sp(N) gauge theories coupled to N_f fermion flavors in the fundamental representation.In SU(N) each fermion flavor is in an N dimensional complex representation, in SO(N) each flavor is in an N dimensional real representation, and in Sp(N) each flavor is in a 2N dimensional pseudo-real representation. These are the 2+1 dimensional analogs of Quantum Chromodynamics. While there is a rather concrete picture for the dynamics of 3+1 dimensional QCD, the 2+1 dimensional counterpart is still poorly understood in part because these theories depend not just on the number of flavors and the gauge group but also on the Chern-Simons level. Our main goal is to propose a scenario for the phases of these theories as a function of the gauge group, Chern-Simons level k, and the number of flavors N_f.We will follow the notation and conventions of -, which we summarize now. Let us start from a single Dirac fermion of charge 1 coupled to a gauge field A, L = i ψ^†D_A ψ . The effective action for the background field A can be computed by performing the path integral over the fermion. The effective action for A is however ambiguous, since we could add various counter-terms.More precisely, we could add to the action k_bare4π∫A∧dA , with k_bare∈. We will fix this ambiguity by choosing the following convention for the effective action of the free fermion. If we add to the Lagrangian a positive mass term for the fermion, then our IR effective action has no Chern-Simons term for the background field. If, on the other hand, we add a negative fermion mass, then the IR effective action is -1 4π∫ A∧ dA, i.e. the Chern-Simons level in the infrared is -1. This convention for the effective action breaks time reversal symmetry, but this is unavoidable in 2+1 dimensions . Our specific convention amounts (up to a sign) to identifying the phase of the fermion path integral with the eta invariant (see for a recent detailed discussion). Our convention for fermions coupling to non-Abelian gauge fields is analogous.With this convention, let us suppose that we start from N_f fermions coupled to a U(1) gauge field with Chern-Simons level k_bare∈. Then, for positive mass the low energy effective theory contains a Chern-Simons term of level k_bare and with negative mass the infrared Chern-Simons level is k_bare-N_f. We could label this theory by its bare Chern-Simons term and the number of matter particles. In order to conform with the common notation used in the literature wedefine k=k_bare-N_f/2, in terms of which the infrared levels are k± N_f/2 for positive and negative mass, respectively. This choice is more symmetric. Note that while k_bare was an integer, k may be a half integer, and it is a half integer if and only if the number of fermions is odd. Henceforth we label this theory by U(1)_k+N_f fermions. Our labelling of theories with non-Abelian gauge groups is analogous.In all the theories that we will discuss here, the bare Chern-Simons term must always be an integer.Above, to define k, we shiftedthe bare Chern-Simons term by -N_f/2. We can thinkIn the general case of an interacting conformal field theory, the “quantum” Chern-Simons term was defined in ,.It can be an arbitrary real number. about this shift as k_quantum and then k=k_bare +k_quantum . In our conventions, k_quantum=-N_f/2 in the SU(N), SO(N), and Sp(N) theories, and, for example, k_quantum=-h/2 in the =1 supersymmetric theories with h the dual Coxeter number (N in SU(N)).When the fermions have masses and they are integrated out the Chern-Simons level in the low energy theory is k_IR=k -(m) k_quantum , which is a properly normalized Chern-Simons level because k is a half integer if and only if k_quantum is a half integer. Note that the theories with k=0 are distinguished, since they have manifest time reversal invariance at m=0.Note also that weakly coupled bosons have k_quantum=0 in all cases.Before we proceed, we would like to make a general comment about the effect of the global aspects of the gauge group on the dynamics of 2+1 dimensional systems.Often our gauge symmetry G is connected but not simply connected and then the system has a magnetic global symmetry H.The charged objects under that symmetry are local monopole operators.One typical example is G=U(1), where the magnetic symmetry is H=U(1), and another example, which we will discuss in section 3, is G=SO(N), whose magnetic symmetry is H=_2.Suppose we take a subgroup H̃⊂H where H̃ is isomorphic to some _n. We couple H̃ to a dynamical _n gauge field. This has the effect of changing the gauge group G to a multiple cover of it G̃ such that G̃/ H̃ = G.Note that the magnetic global symmetry group H is always Abelian and here we gauge only a discrete subgroup of it, H̃.If we gauge a continuous subgroup of H, then there are new local operators, monopole operators, and the discussion is more complicated. For example, starting with G=U(1) and H̃=_n the new gauge group is G̃=U(1), but all the particle charges are multiplied by n.Similarly, for G=SO(N)with H̃ = _2 the new gauge group is G̃= Spin(N).How does the dynamics of the G̃ theory differ from that of the original G gauge theory?Since all we did to the G gauge theory is to gauge a discrete cyclic global symmetry, the resulting G̃ gauge theory is an H̃ orbifold of the original theory.It is important that in 2+1d such orbifolding does not change the phase diagram.The phase transitions in the G gauge theory are the same as in the G̃ theory.The only difference between the two theories is that the G̃ theory has fewer local monopole operators and it has more line operators from the twisted sector. If the G theory has a phase with some nonlinear sigma model (i.e. with a moduli space of vacua) on which H̃ acts, then the G̃ theory has the same phase and the target space of the nonlinear sigma model is the quotient of the original target space by H̃. We therefore see that the space of vacua could depend on global aspects of the gauge group. (This phenomenon does not occur in four dimensions.) This will be important in section 3.As a simple example, consider the U(1)_k gauge theory with N_f fermions all with charge one.This theory has a magnetic H=U(1) global symmetry.Let us quotient the theory by H̃=_n⊂ U(1).This is easily done in the Lagrangian by replacing the gauge field a by na; i.e. all the fermions now have charge n and the Chern-Simons level of the new theory is kn^2.The reasoning above shows that the phase diagram and the phase transitions between them are independent of n.In particular, the phase diagram of the U(1)_0 theory is the same as that of thegauge theory without a Chern-Simons term . In section 3 we will consider a similar example with SO(N) and Spin(N) gauge theories.The dynamics of 2+1 dimensional non-Abelian gauge theories has been recently revitalised by the proposal of boson/fermion dualities -,,,-.They were motivated by ideas in 2+1d field theory -, supersymmetric quantum field theory -, and string theory -.Roughly speaking, these dualities shed light on theories with small N_f not larger than the Chern-Simons level in the fermionic theory.As stated, these dualities do not apply in the time-reversal invariant case k=0.Below we will clarify the role of duality in this case. The dualities can be summarized asU(k+N_f/2)_Nwith N_f scalars⟷SU(N)_-kwith N_f fermions SO(k+N_f/2)_Nwith N_f real scalars⟷SO(N)_-kwith N_f real fermions Sp(k+N_f/2)_Nwith N_f scalars⟷Sp(N)_-kwith N_f fermions and other dualities that follow from them.(Our notation is U(k)_N ≡ U(k)_N,N.)The unitary dualities were conjectured to hold for N_f ≤ k, the symplectic dualities for N_f ≤ k, and the orthogonal dualities for N_f ≤ k-2 if N=1, N_f ≤ k -1 if N=2, and N_f ≤ k if N>2. We will also need their time-reversed versionsU(k+N_f/2)_-Nwith N_f scalars⟷SU(N)_kwith N_f fermions SO(k+N_f/ 2)_-Nwith N_f real scalars⟷SO(N)_kwith N_f real fermions Sp(k+N_f/ 2)_-Nwith N_f scalars⟷Sp(N)_kwith N_f fermionsIn all these cases the scalars have quartic interactions.We flow to the IR and allow to tune the fermion and boson masses to a critical point, if it exists.Except for some special cases, we cannot prove that such a critical point exists. Many of the statements that we will make about the phases of these three-dimensional theories are independent of whether the corresponding transitions are second order or not.As we said, the dualities above , were conjectured to hold for values of N_f that are bounded from above by, loosely speaking, the Chern-Simons level of the fermionic theory. The central point of this note is to extend this conjecture about the behavior of these theories to larger values of N_f. It is logically possible that the previous conjectures are true, buttheir newer extended version is not.Of course such proposals are subject to stringent consistency checks from symmetries, anomalies, mass deformations that allow to decrease N_f, and constraints from the large N_f limits of 2+1 dimensional gauge theories .Our conjecture is further motivated by the study of domain walls in 3+1-dimensional QCD. This will be developed in great detail in . Briefly, pure Yang-Mills theory in 3+1 dimensions at θ=π has a first order transition associated with the spontaneous breaking of time-reversal.Because of anomaly considerations the domain wall at this transition carries a nontrivial TQFT – SU(N)_1 Chern-Simons theory . Suppose we now add to the 3+1-dimensional theory heavy quarks with real positive masses, such that time reversal symmetry is preserved by the Lagrangian but broken in the vacuum. For large masses the domain wall is still described by an SU(N)_1 Chern-Simons theory, since it is hard to make a topological theory disappear. As we lower the quarks masses, the theory on the domain wall undergos a phase transition. Our scenarios below about the dynamics of the 2+1-dimensional theory are consistent by the transitions on these domain walls .The phase diagram of SU(N)_k for N_f≤ 2k.We define the transition point to be at m=0.The gapped phases have topological field theories, SU(N)_k±N_f/ 2, which are related by level-rank duality of spin theories to U(k±N_f/ 2) .For k=N_f/ 2 the phase for negative m is trivial.4inSUlargek.epsSummary of our proposed scenario Let us summarize our conjecture for SU(N)_k+N_f fermions, starting from the case N_f≤ 2k, which is covered by the dualities ,. The theory has two phases, depending on the mass of the quarks. One phase is SU(N)_k+N_f/2 pure Chern-Simons theory and the other phase is SU(N)_k-N_f/2 pure Chern-Simons theory. The transition could be first or second order. The global symmetry is never broken. The theory is dual to U(k+N_f/2)_-N+N_f bosons, which has exactly these two pure Chern-Simons phases (this can be seen by level-rank duality). For N_f=2k this description still makes sense; on one of the sides of the transition the ground state is trivial. This summarizes the first line in . We also summarize this situation in .For larger N_f this picture breaks down because the bosonic theory has a sigma model at low energies for negative mass squared (for N_f≤ 2k the bosonic model does not have a sigma model phase in its semi-classical regimes and instead it flows to a Chern-Simons TQFT as depicted in ). By taking the mass squared of the scalars to be large and negative we can make the sigma model as weakly coupled as we like (i.e. the radius of the target space can be as large as we like). No such weakly coupled sigma model with a large target space exists in the fermionic theory and therefore the duality fails . Indeed let us turn on equal masses m for all the quarks.Let us analyze it for |m|≫ g^2 (where g is the gauge coupling).Then we can integrate out the fermions and useto find that the low energy theory is gapped and it is described by a pure SU(N)_k_IR Chern-Simons theory with level k_IR=k+(m)N_f/2 . This leaves the interesting question of what the theory does for small values of m, of order g^2. The phase diagram of SU(N)_k for 2|k|<N_f<N_*.The gapped phases have topological field theories, SU(N)_ k±N_f/ 2, which are related by level-rank duality of spin theories to U( N_f/ 2± k)_∓ N.The middle gapless phase is a nonlinear sigma model on a Grassmannian with a Wess-Zumino term Γ, whose coefficient is N. For N=2 the Grassmannian is replaced by Sp(N_f)/( Sp(N_f/2 + k)× Sp(N_f/2 -k)).4inSUsmallk.epsOur conjecture is that there exists a window N_*(N,k)>N_f>2k where N_* is an unknown function of the parameters of the theory such that the U(N_f) symmetry that rotates the various flavorsMore precisely, only a certain quotient of U(N_f) acts faithfully(see also for a similar discussion in a slightly different context) and the system also has charge conjugation symmetry.These will not play an important role in our discussion. is spontaneously broken leading to the coset (N_f,k)=U(N_f)U(N_f/2+k)×U(N_f/2-k) . Our proposal is that this is a purely quantum phase, not visible in the semi-classical limits of the theory and the target space of this sigma model (i.e. the complex Grassmannian ) never becomes large. It is important to mention that we need to add to the sigma model Lagrangian a certain Wess-Zumino term, which we will discuss in detail soon.In the special casek=N_f/2-1 this result is consistent with the study of domain walls in four dimensions . This proposal for N_*(N,k)>N_f>2k implies the dualities , for lower values of N_f and shifted values of k, as we explain in the bulk of the paper.In addition, one can “derive” this picture by starting fromSU(N)_0+N_f fermions (with even N_f).It was suggested in - that in some range 0<N_f<N_*(N,0) this theory breaks its global symmetry leading to a nonlinear sigma model with target space (N_f,k=0)=U(N_f)U(N_f/2)×U(N_f/2) . The case k=0 is therefore an important consistency check of our scenario. But it is not just a consistency check – we can reverse the logic and assume this scenario at k=0 and then derive all the rest. By deforming the nonlinear sigma model with masses that correspond to fermion masses in the ultraviolet theory, one can see that this breaking must take place in a whole region of parameter space (as we vary the symmetry-preserving mass deformation) and not just at one point, and, in addition, one can derive all the other symmetry breaking patterns . In particular, as we will explain, if the symmetry breaking does not occur, then our conjecture is necessarily wrong.In addition, we stress again that we need to add to this discussion the Wess-Zumino term. There are various ways to see that this Wess-Zumino term is necessary, for example, by considering the quantum numbers of Skyrmions (baryons). This derivation also leads to interesting constraints on N_*(N,k), which we discuss in section 2.The sigma model arises from a condensation of the quark bilinear. Indeed, consider the condensate ψψ^†=diag(x,...,x,y,...,y) , with x appearing N_f/2+k times and y appearing N_f/2-k times (y≠ x). It breaks the symmetry to U(N_f/2+k)× U(N_f/2-k). Therefore, we obtain the coset . In the interesting special case of k=0 the model has time reversal symmetry at m=0. The symmetry breaking phase , if it exists for the given value of N_f, must contain the point where the quarks are massless.The fermionic theory therefore has two special points (see ), one where it makes a transition from the semi-classical phase SU(N)_k+N_f/2 to the phase and the other is where it makes a transition from to the second semi-classical phase, SU(N)_k-N_f/2. We propose that the former is dual to scalar theory U(N_f/2+k)_-N+N_f scalars and the latter to the theory U(N_f/2-k)_N+N_f scalars. As a simple consistency check, note that both of these scalar theories lead to the coset . In other words, the fermionic theory covers the whole parameter space, while each of its bosonic duals describes only a patch of the parameter space – the patch around each transition.This is consistent with the fact that all three theories have the same global symmetry and the same 't Hooft anomalies . Again, we do not have a clear picture of whether these phase transitions are first or second order, but the symmetry breaking phase should be robust and independent of the question of the order of the transition. Thisproposal for the dynamics of QCD in 2+1 dimensions for N_*(N,k)> N_f>2k is summarized in . The phases of the theory that are described in the deep infrared by pure Chern-Simons theory are intuitively not confining as the low energy observer can see the Wilson lines of quarks. The symmetry breaking phase at small |m| should be viewed as a confining phase. Therefore, confinement takes place for N_*(N,k)> N_f>2k, but it does not take place for N_f≤ 2k.There is no rigorous order parameter for confinement here, so this discussion is only at the intuitive level. In the case of SO(N) gauge theory (more precisely, Spin(N)) there is a rigorous notion of confinement and what we said in the text is rigorously true for the analogous phases there.See section 3. Finally, it remains to discuss what happens for N_f≥ N_*. We know that for sufficiently large N_f the theory has to be conformal, with only Chern-Simons phases (as in ) on both sides of the transition .We therefore assume that this is the behavior of the theory for all N_f≥ N_*. No scalar dual is known in this regime. N_* is therefore the critical number below which (and above 2k) the symmetry breaks in an intermediate, quantum phase, that is invisible semi-classically.Our proposal for the phases of SO and Sp gauge theories is rather similar in spirit, with an intermediate symmetry breaking phase for sufficiently large (but not too large) values of N_f.In section 2 we discuss some additional details about SU(N)_k with N_f fermions in the fundamental representation. In section 3 we discuss the dynamics and phases of the SO and Sp theories, where there are a few novelties. More on SU(N)_k with N_f Fermion FlavorsIn this section we discuss a number of topics related to the proposed Grassmannian phase of SU(N)_k with N_f fermion that have not been described in detail in the introduction.Studying the sigma modelWe will study some properties of the sigma model with target space(N_f,k)=U(N_f)U(N_f/2+k)×U(N_f/2-k) . One way to think about it is intrinsic to this sigma model and uses properties of this space.This approach is standard in the literature. Alternatively, we find it convenient to view this theory as the low energy approximation of a gauged linear sigma model.In the special case of (N_f,k=N_f/2-1) =^N_f-1 this approach is also standard in the literature.Specifically, we can think about as the low energy theory of the gauge theory U(N_f/2-k)+N_fscalars Φ^i , where we add large negative mass squared for the scalars, such that the model is weakly coupled and we can treat it classically. Of course, as we explained in the introduction, this model, when appropriately modified by Chern-Simons terms, plays a more fundamental role in the story than just an auxiliary linear sigma model. However, here we will use it to study some properties of the Grassmannian per se without referring to the more general role this model plays (upon modifying it with Chern-Simons terms and allowing it to be strongly coupled).Later we will add the Chern-Simons term and show that it leads to a Wess-Zumino term in the non-linear model.Following the standard treatment in the ^N_f-1 model, it is straightforward to write down explicitly the two-derivative nonlinear sigma model Lagrangian with target space . The fundamental degree of freedom is a bi-fundamental U(N_f/2+k)× U(N_f/2-k) scalar π and the Lagrangian is L_kinetic∼Tr(ππ^†(1+ππ^†)^-1 -ππ^†(1+ππ^†)^-1ππ^†(1+ππ^†)^-1) . This metric can be derived from the Kähler potential K = Trlog(1+ππ^†) .Instead, we will continue to analyze the gauged linear sigma model with large, degenerate negative mass squared for the scalars, such that the model flows to .Wess-Zumino Terms and Skyrmions Here we describe the Wess-Zumino terms that must be added to the nonlinear sigma model Lagrangians.We start from the simplest Grassmannians, namely the ^N_f-1 manifolds, and then briefly describe the more general story.As above, we describe it in terms of a U(1) gauge theory with a gauge field b with N_f scalars.In this theory we can add the Chern-Simons terms N4π bdb +12π db B for N∈2 N4π bdb +12π db A for N∈2+1 , where A is a classical background spin_c connection and B is a classical background U(1) gauge field.For even N each term inis separately meaningful.The same is true for odd N on a spin manifold.But on a non-spin manifold with odd N only the sum of the two terms inis meaningful.(See - for more details.) The equations of motion of b set db = w+ … where the ellipses denote higher order termsThe low energy nonlinear model is characterized by a scale that originates from the expectation value of the scalars in the linear model.It appears in front of the kinetic term in the nonlinear model.These higher order terms are suppressed by inverse powers of that scale. and w is the pull back of the Kähler form of the ^N_f-1 manifold to spacetime, normalized such that its integrals over two-cycles Σ_2 are ∫_Σ_2 w∈2π . For N_f>2 we can substituteback in the Lagrangian and find N4π∫_M_4ww +12π ∫_M_3w B for N∈2 N4π∫_M_4 ww +12π∫_M_3 w A for N∈2+1 , where M_3 is our spacetime and M_4 is a four manifold whose boundary is M_3.The first term can be interpreted as a Wess-Zumino term in the nonlinear model. We see that on a non-spin manifold N should be even.If N is odd, we need to have a spin_c structure with a connection A. In terms of the massless pions the Wess-Zumino term induces interactions such as ∼N ∫d^3 x ϵ^μνρ (_μπ^†·π)(_νπ^†·_ρπ)+⋯ , where the ellipses stand for higher order interaction terms (with three derivatives) that are necessary for SU(N_f) invariance.The background fields A and B couple into monopole operators of the linear model.The spin of the monopole is N/2 and hence, for even N it is a boson, which couples to an ordinary background gauge field B and for odd N it is a fermion, which couples to a spin_c connection A.In the nonlinear model these background fields couple to configurations with nonzero ∫_Σ_2 w.If Σ_2 is our space, then these charged objects are Skyrmions.Therefore, the Skyrmions of the model are bosons or fermions depending whether N is even or odd.Let us consider the baryons of the fermionic SU(N) theory. They are created by acting with the baryon operator on the vacuum. In terms of the bosonic dual, we can act on the vacuum with the (appropriately dressed) monopole operator. Since the monopole operator is dual to the baryon operator, we get dual excitations in the Hilbert space ,. In our case we can further observe that the monopole operators flows to the Skyrmion operator in the sigma model phase. Therefore, the baryon particles can be identified with the Skyrmions of the non-linear sigma model.See a related discussion in . As a check, the quantum numbers of these baryon operators match with those of the monopole operators of the bosonic theory and with the Skyrmion operators of the low energy theory.The SU(N_f) classical gauge fields already couple to the two-derivative Lagrangian, which is the reason we do not discuss them here. By contrast, the baryon symmetry does not act onthe two-derivative Lagrangian. For completeness it is however worth mentioning that the SU(N_f) gauge fields do couple to Skyrmions, since the baryon carries an SU(N_f) representation that is necessarily nontrivial, i.e. with nonzero N_f-ality, if gcd(N_f,N_c)≠ 1.This identification of the Skyrmions with the microscopic baryons and the way their quantum numbers are determined by N through the Wess-Zumino term are exactly as in the similar four-dimensional theory ,. In addition, as in four dimensions, the Wess-Zumino term breaks a symmetry that exists accidentally in the nonlinear-sigma model. For k≠ 0 it is time-reversal symmetry and for k=0 it is another discrete symmetry (exchanging the two U(N_f/2) factors, which acts as transposition on the matrix of pions).A very important exception occurs when N_f=2. Clearly, the term ww in vanishes then.There is however still a discrete Wess-Zumino action, which can be further extended to non-spin manifolds by allowing A to be a spin_c connection. The construction of this discrete term will be discussed in . Note that in a non-local modification of the ^1 action was described and it was shown to affect the quantum numbers of the Skyrmions. The relation of this to the discrete invariant of will be described there.Now let us discuss briefly the general case of (N_f,k)=U(N_f) U(N_f/2+k)× U(N_f/2-k), which arises in SU(N)_k gauge theory with N_f fermions. The main novelty of the general case compared to^N_f-1 is that the H^4 cohomologyFor a more precise treatment of this Wess-Zumino term see .is generated by two four-forms and not just one. Therefore, there are a priori two Wess-Zumino terms. It is easiest to describe these two coefficients using the linear sigma model description, where they are the standard U(N_f/2-k) invariants, ∫_M_4 Tr F∧ Tr F and ∫_M_4 Tr F∧ F. In our proposal we take only the latter one with coefficient proportional to N.Let us finish this discussion with a comment about strings in the sigma model. Such strings exist, if the fundamental group of the target space is non-trivial.Then consider a configuration that is independent of one direction, say x^1, and wraps the nontrivial cycle in x^2.The energy density is localized in a small region in x^2 and therefore such a configuration is a strings in ^2.This fundamental group and the corresponding strings are associated with an unbroken one-form global symmetry ,. In the SU(N)_k+N_f fermions theory there is no such one-form symmetry at short distancesand it is therefore nice to find out that indeed π_1((N_f,k)) vanishes. Flowing down in N_fWe now study some mass perturbations of our model. Instead of perturbing the nonlinear model we will perturb the gauged linear model by a small mass term δ m^2_ij̅Φ^iΦ̅^j̅.One simple perturbation δ m^2_ij̅∼δ_ij̅ preserves the SU(N_f) global symmetry.It just shifts the mass squared of the scalars, and in the semi-classical regime they would still condense and lead to the Grassmannian , albeit with a different size. The Grassmannian should therefore remain when a mass term proportional to the identity matrix is added to the Lagrangian. This fact has some ramifications for the mapping of parameters between the Grassmannian Lagrangian, the gauged linear sigma model, and the fermionic theory. In essence, that means that we can mix general mass perturbations, which are in the adjoint representation, with the singlet representation when we map the parameters.Next, we analyze two mass perturbations that decrease N_f to N_f-1. First, consider a mass deformation with the only nonzero entry being δ m^2_11̅>0. Then the N_f-1 scalars with degenerate masses would still condense. The scalar Φ^1 is now massive around this configuration. Therefore we obtain the coset (N_f-1,k-1/2)=U(N_f-1)U(N_f/2+k-1)×U(N_f/2-k) .Second, let δ m^2_11̅<0 and small. Then we condense first Φ^1 and obtain a U(N_f/2-k-1) gauge theory coupled to N_f-1 scalars. Then, condensing them we obtain the coset (N_f-1,k+1/2)=U(N_f-1)U(N_f/2+k)×U(N_f/2-k-1) . Clearly, if we started with N_f/2-k=1 this resulting theory is trivial.Here we have treated the linear sigma model as a proxy for the nonlinear sigma model Lagrangian with target space .Clearly, instead of analyzing these deformations in the linear model, we could have analyzed them directly in the nonlinear model.That would have led directly to .Returning to our general proposal, the two mass deformations (N,N_f,k)→(N,N_f-1,k±1/2) show that if our proposal about the phase diagram is right for (N,N_f,k), it must also be right for (N,N_f-1,k± 1/2).This fact has two implications that we discuss now.Constraints on N_*(N,k) We can use in order to obtain interesting constraints on N_*(N,k).First, using the same argument as in , it is clear that for large enough N_f with or without large k the system has only two phases separated by a second order point.For N_f≤ 2k in this range we have a proposed bosonic dual description of this phase transition.Next, we consider finite N_f and look for the Grassmannian phase with 2k<N_f<N_*.Assuming we found such a point, requiring that the deformations of the Grassmannian theory by a small mass always agree (as above) in the infrared with deformation of the full theory by a large mass parameter, we get a nontrivial constraint on N_*(N,k). We see that if N_f<N_*(N,k), then it must also be true that N_f-1<N_*(N,k±), and therefore, for all N,k (with small enough k) N_*(N,k)-1≤N_*(N,k±) .Interestingly, the constraint implies that the size of the window for the Grassmannian phase, namely, N_*-2k, is necessarily maximized at k=0 (i.e. the size of the window can only decrease as k is increased).In particular, if there is no symmetry breaking at k=0 according to U(N_f/2)→ U(N_f/2)× U(N_f/2), then there cannot be symmetry breaking of the sort that we discuss at any k. Furthermore, N_* cannot decrease too fast (since N_*(N,k+1/2)≤ N_*(N,k)+1, the average derivative with which it can decrease is not bigger than 2 in absolute value). Going up in kNext, we useto offer an alternative point of view on our proposal. Let us start with the SU(N)_0+N_f fermions theory with an even number of fermions.This theory is time-reversal invariant for m=0.It has been suggested that its global U(N_f) symmetry is spontaneously broken to U(N_f/2)× U(N_f/2).One argument for it is that we can give the fermions a time-reversal invariant mass preserving this unbroken symmetry.We do that by giving N_f/2 of them mass +m and N_f/2 of them mass -m.This way we can gap the system in a smooth way.Such a deformation was used in,, to argue that time reversal and the U(N_f/2)× U(N_f/2) symmetry should remain unbroken.Of course, his does not mean that U(N_f) must be broken.A more abstract point uses the fact that this theory is time-reversal invariant as well as U(N_f) invariant. However, there is a mixed 't Hooft anomaly between these two symmetries. This is just a variant of the standard mixed anomaly between time reversal and fermion number symmetry .For N_f≥ N_* this anomaly is represented in the IR theory by a nontrivial conformal field theory. At N_f<N_* the anomaly can be saturated by breaking U(N_f) → U(N_f/2)× U(N_f/2). It does not mean that the range N_f<N_* existsIn principle, there could be other ways to match the anomaly in the infrared, e.g. with a nontrivial TQFT and an invariant vacuum or with a nontrivial CFT. (i.e. it could be that N_*=0), but let us assume it does. Hence, this symmetry breaking pattern must exist in a range of the U(N_f)-preserving mass parameter, and not just at m=0 as in four dimensions.Next, we can deform the model as in and arrive at the other Grassmannians with other values of N_f and k.More precisely, this way we can get only to N_f<N_*(N,0)-2k. We conclude that assuming the U(N_f) → U(N_f/2)× U(N_f/2) breaking in the k=0 theory, we derive our more general proposal about the intermediate phase. A central nontrivial point is that already the k=0 model needs to be supplemented with the Wess-Zumino term we described above. Then, its existence for the other values of k follows. SO(N)_k and Sp(N)_k with N_f Fermion FlavorsThe situation for SO(N) and Sp(N) gauge theories is similar to the SU(N) situation described above.Here we useSO(N)_kwith N_f fermions⟷SO(k+N_f2)_-Nwith N_f scalarsSO(N)_kwith N_f fermions⟷SO(-k+N_f2)_Nwith N_f scalars Sp(N)_kwith N_f fermions⟷Sp(k+N_f2)_-Nwith N_f scalarsSp(N)_kwith N_f fermions⟷Sp(-k+N_f2)_Nwith N_f scalarsThe phase diagram of SO(N)_k forN_f≤ 2k.4inSOlargek.epsThe phase diagram of SO(N)_k for 2|k|<N_f.4inSOsmallk.epsThe phase diagram of Sp(N)_k for N_f≤ 2k.4inSplargek.epsThe phase diagram of Sp(N)_k for 2|k|< N_f<N_*.4inSpsmallk.epsAgain, we expect some N_*(N,k) such that for N_f≥ N_* there are only two non-confining phases with a second order point between them.The value of N_*(N,k) can depend on the gauge group.We use the same notation as in the SU(N) discussion above and suppress the dependence on the gauge group. For 2k<N_f<N_* we expect three phases with a gapless middle phase describing the spontaneous breaking of the global symmetry.And for N_f≤ 2k the intermediate phase is absent.For SO(N) it is depicted inand , and for Sp(N) it is depicted inand .All the comments we made in the previous section still apply, but there are a few additional noteworthy facts.The SO(N) fermionic theory has a _2 magnetic global 0-form symmetry.Under the duality it is mapped to a charge conjugation symmetry – the symmetry that extends the SO(N_f 2± k)_N gauge symmetry to O(N_f 2± k) .In the topological phases with SO(N)_k± N_f/2 this symmetry is unbroken.However, in the Grassmannian phase the magnetic _2 symmetry is spontaneously broken.This can be seen most easily using the scalar dual theories, where the Grassmannian can be seen at tree level. Without loss of generality consider the scalar theory with gauge group SO(N_f 2- k). In the Grassmannian phase the scalars have an expectation value, which, up to symmetry transformations, is of the form ϕ_ia = x δ_ia where i labels the colors and a labels the flavors, and x is nonzero. This breaks the O(N_f) global symmetry to SO(N_f 2- k) × O(N_f 2+k), where the first factor is a diagonal subgroup between a subgroup of the global symmetry group and the gauge group.This leads to the vacuum manifold =SO(N_f)SO(N_f2+k)×SO(N_f2-k) , as in .The point ϕ_ia = x δ_ia and the point where the first entry, ϕ_11, flips sign are related by a broken global symmetry transformation reflecting the spontaneously broken charge conjugation symmetry of the bosonic theory. Alternatively, we can parameterize the vacuum manifold by a gauge invariant order parameter of the form ϕϕ.It lives in =SO(N_f)S[O(N_f2+k)×O(N_f2-k)] . The true target spaceis a double cover of this space because every point incorresponds to two different points in , which differ by the expectation value of a gauge invariant baryon operator of the form ϕϕ...ϕ, where the color indices are contracted using an epsilon symbol.This order parameter is odd under the charge conjugation symmetry, thus establishing that this symmetry is spontaneously broken.Using the duality it is mapped to a monopole operator of the underlying SO(N) gauge theory.Its expectation value shows that the magnetic symmetry of the fermionic SU(N) theory is spontaneously broken.We see that the magnetic symmetry of the fermionic theory is unbroken in the topological phase and it is broken in the Grassmannian phase.This is consistent with the intuitive picture that the topological phases are not confining, but the Grassmannian phase is confining (we typically think of a phase with broken magnetic symmetry as confining).A more precise way to see that uses one-form global symmetries ,.None of our theories have such a symmetry.This follows from the fact that the gauge group acts faithfully on the matter fields. In particular, the space is simply connected, which is encouraging, since that means that there are no stable strings in the macroscopic theory.However, we can change the SO(N) theories and replace them by Spin(N) theories (by gauging the discrete 0-form _2 magnetic symmetry, as described in the introduction) and then these theories have a “bonus” _2 one-form global symmetry.The charged objects under this one-form symmetry are Wilson lines in the spinor representation of Spin(N).They can be used to diagnose confinement rigorously.In order to change the gauge group to Spin(N) we turn the global _2 magnetic symmetry of the SO(N) theories to a gauge symmetry. Then the topological phases are Spin(N)_k± N/2 and the existence of nontrivial spinor Wilson lines in these phases confirms our assertion that these phases are not confining.The Grassmannian phase is more interesting.Again, we analyze it using the two dual scalar theories.Here the _2 magnetic symmetry of the fermionic theory acts as charge conjugationand correspondingly the gauge group is O(N_f/2 ± k).For a more detailed discussion see . The effect of this extension of the gauge group on the Grassmannian is to mod it out by _2 turning it fromofto =/_2 of .The resulting space is not simply connected π_1()=_2. The fact that the moduli space of vacua in the SO(N), Spin(N), and O(N) theories are related by certain quotients and its relation to duality has already been noted in supersymmetric theories in ,. We see that the underlying _2 one-from global symmetry, which is associated with the center of Spin(N), is realized as a one-form global symmetry in the macroscopic theory.Because of this nontrivial π_1 the macroscopic theory has stable _2 strings.These are configurations of the scalars that are independent of one spatial direction and the other spatial direction winds around the non-contractible cycle in the target space.These charged strings are interpreted as the _2 confining strings of the microscopic theory.We see that in the Grassmannian phase the one-form global symmetry is unbroken and as described in , this means that the theory is confining. A similar comment in a closely related four-dimensional context was made in ,.In general, when we gauge an electric one-form symmetry we obtain in the new gauge theory a bonus magnetic zero-form symmetry.The reverse process was described in the introduction. If the electric one-form symmetry is unbroken, the theory confines.In this case the bonus zero-form magnetic symmetry of the new theory is expected to be broken.Similarly, if the electric one-form symmetry is broken and hence the theory does not confine, the bonus magnetic symmetry of the new theory is unbroken. This is the standard relation between spontaneously broken magnetic symmetry and confinement.The original SO(N) gauge theory (or the Spin(N) theory we have just discussed) has a _2 charge conjugation symmetry that extends SO(N) to O(N).It is mapped under the duality to the _2 magnetic symmetry of the bosonic gauge theory .This symmetry is realized in the macroscopic Grassmannian phase as associated with a non-trivial π_2 of the target space.The charged operators under this symmetry are baryon operators in the fermionic gauge theory.They are mapped to monopole operators in the bosonic gauge theory.And in the low energy Grassmannian sigma model they are mapped to operators creating nontrivial π_2.This π_2 of the Grassmannian leads to Skyrmions, which are odd under this _2 symmetry.They are identified with the baryons of the underlying fermionic gauge theory. An interesting exception occurs when one or both of the factors in the denominator ofis SO(2).In that case π_2()has one or more factors of .For example, for N_f=4, k=0 we have π_2(SO(4) SO(2)× SO(2))=×.In this case the Grassmannian theory should be supplemented with Skyrmion operators that allow Skyrmions to decay, making the global Skyrmion number only _2.In the case where the bosonic dual has a SO(2)=U(1) gauge symmetry, this enhancedπ_2 is associated with an enhanced U(1)_T magnetic symmetry.In this case this symmetry should be explicitly broken to _2by adding monopole operators of charge two to the Lagrangian in order to preserve the duality (as was done recently in other models in ,). These operators have a negligible effect in the Higgs phase, except that they allow Skyrmions to decay, while preserving the remaining _2 charge. However, these operators can become important, if their dimension becomes relevant in the IR.In summary, in the generic case π_2()=_2 is nicely consistent with the expectations from the fermionic theory. In the special cases where N_f/2-k=2 or N_f/2+k=2 we have to add to the bosonic description even monopole operators. Analogously, the nonlinear sigma model Lagrangian needs to be modified by adding monopole-like operators, which would allow even Skyrmions to decay to the vacuum.Finally, as in the SU(N) theories, also here the Grassmannian has to be accompanied by certain Wess-Zumino terms that are responsible for the quantum numbers of Skyrmions. We do not describe them in detail here, except to note that they follow from the Chern-Simons terms in the bosonic gauged linear models.A Comment about SO(2) In the discussion of we restricted ourselves to N>2. In fact, our results nicely apply to N=2 when properly interpreted. Here we would like to explain how this comes about.As we will see, not only is our general picture correct also for N=2, this case is actually on stronger footing than the more general case and thus supports our general picture.First, let us explain why we restricted our discussion to N>2. In the case of SO(2)≅ U(1) gauge theory with N_f flavors, the symmetry of the massless theory is locally SU(N_f)× U(1), where the first factor is the flavor symmetry and the second is the magnetic symmetry. This differs from the SO(N) series with N>2, where the magnetic symmetry is discrete (it is _2) and the flavor symmetry is SO(N_f).For example, let us consider an SO(2)_0≅ U(1)_0 gauge theory with N_f=2 fermions in the fundamental representation, i.e. two fermions of charge 1, ψ^i. It was suggested in and clarified in that this theory enjoys self-duality; i.e. it is dual to another U(1)_0 theory with two fermions χ^I. Unlike the original fermions ψ^i, they are labelled by an upper case I, because they are acted upon by a different SU(2) global symmetry.The precise global symmetry, its 't Hooft anomaly, and the deformations of the theory were analyzed in detail in . We will summarize below the facts that are necessary for our discussion.When the fermions are massless (hence time reversal symmetry is not explicitly broken) and no monopole operators are added to the Lagrangian the model flows to a conformal field theory with an enhanced O(4) symmetry. This conformal field theory has several notable operators.First, an SU(2) invariant fermion bilinear in the UV _1=ψ^iψ_i^† (with i=1,2 labels the two flavors) flows to an SO(4) singlet in the IR.Second, the basic monopole operator in the UV ^i is in an SU(2) doublet.^i and its conjugate ^i form an SO(4) vector in the IR; i.e. the magnetic U(1) is enhanced in the IR to SU(2).Finally, consider an SU(2) triplet fermion bi-linear _3^a,3=ψ^iσ_i^ajψ_j^† (the superscript 3 will be explained shortly).In the IR it combines with a double-monopole and its conjugate ^iσ_ij^a^j=(_3^a,1+i_3^a,2)^iσ_ij^a^j=(_3^a,1-i_3^a,2) , which are SU(2) triplets, to form a traceless symmetric tensor of SO(4) in the IR.In terms of SU(2)× SU(2) representation it transforms as ( 3,3) and hence the notation with the two superscripts in _3^a,A in and .The superscript A is distinguished from the superscript a as it denotes a triplet of another SU(2). In the dual description we have two fermions χ^I and the flavor SU(2) that mixes them is identified in the IR with the enhanced SU(2) of the original theory.The singletis identified as χ^Iχ_I^†, the monopole and its conjugate are identified with the SO(4) vector.And most interestingly, the double-monopole and the fermion bilinear are mapped nontrivially _3^3,A= χ^Iσ_I^AJχ_J^† .Our analysis of the phases of the general SO(N) theories can be naturally continued all the way to SO(2) if we properly deform the SO(2) theory such that the UV symmetries are as in the general case. This can be achieved by adding the charge-two monopole operator and its conjugate, say ^iσ_ij^3^j + c.c.=_3^3,1to the Lagrangian.(The superscripts 3,1 where chosen without loss of generality.)This has the effect of explicitly breaking the flavor SU(2) to SO(2) and the magnetic U(1) to _2, such that the UV symmetries are as in a generic SO(N) theory in .In the UV this operator has a very large dimension, but it becomes relevant in the IR.The duality allows us to analyze its effect in the IR.This is achieved because in the dual description this operator is a fermion bilinear.Up to an SU(2) flavor rotation of the dual theory it is _3^0,3 =χ^1χ_1^†- χ^2χ_2^† .Before adding the monopole operator to the Lagrangian the UV U(2) global symmetry was enhanced in the IR to SO(4) (we suppress discrete factors).After adding the monopole operator to the Lagrangian the UV symmetry is explicitly broken to U(1) ×_2 ⊂ U(2) and it is enhanced in the IR to U(1)× U(1) (and we again suppress some discrete factors) .Adding the perturbationin the dual theory we see that the two fermions are massive and their mass has opposite sign. As a result, when they are integrated out we remain with a U(1) gauge field and no Chern-Simons term. It is dual to a compact scalar and represents the spontaneous breaking of the magnetic U(1) in the dual theory. Using the duality map this corresponds to the spontaneous breaking of the original SO(2) flavor symmetry that remains in the presence of the monopole operator .This is precisely as implied by upon setting N=2 and N_f=2.We can further consider the phase transitions in . The horizontal axis corresponds to a deformation by the singlet mass term , which is identified as χ^Iχ_I^†. We should add the operator in conjunction with .This means that for some value of the singlet mass χ^1 is massless and at another value χ^2 is massless.Around these two points we have U(1)_ with N_f=1, which is dual to the O(2) Wilson-Fisher theory.This is exactly as predicted by the dualities in .This conclusion is quite satisfying because of the following reason.As explained in , the self-duality of U(1)_0 with N_f=2 follows from the web of dualities of , which is supported by a lot of evidence.So this is not an additional assumption.Then, the analysis of the deformation by the double-monopole operator, which proceeds via this duality also follows from the same assumption.This means that at least for SO(2) with N_f=2 the intermediate phase scenario that we have been advocating throughout this paper follows logically from the web of dualities of without further assumptions!A similar treatment applies to the U(1)_ with a N_f=1 theory.This model again has a U(1) magnetic symmetry, which is not present in the general SO(N) case. We thus interpret our phase diagram as being the result of adding a double-monopole to the Lagrangian. Again, we analyze the effect of the double-monopole operator using duality.The model SO(2)_1/2+ψ without monopole operators in the Lagrangian is dual to theO(2) Wilson-Fisher fixed point with a complex scalar Φ.The double-monopole perturbation is translated in this theory to Φ^2+Φ̅^2.Now, as we dial the coefficient of the invariant operator |Φ|^2 we encounter one Ising point (where one scalar is massless and the other has positive mass squared), exactly as in if one substitutes N=2, k=, N_f=1.We expect that for large enough N_f this double-monopole operator is irrelevant in the IR fixed point and therefore it cannot split it as above.Again, this is consistent with our general picture.Conversely, a microscopic lattice model of these systems might not have the global magnetic U(1) symmetry.For large enough N_f we recover the U(1) magnetic symmetry in the IR and the IR fixed point is not split.But for smaller values of N_f, where this interaction is relevant in the IR, the fixed point is split and we find some global symmetry breaking there.This could explain why people who studied microscopic lattice models of this system, which implicitly include the double-monopole operator in the Lagrangian found global symmetry breaking for low values of N_f. Acknowledgments We would like to thank O. Aharony, A. Armoni, F. Benini, D. Freed, D. Gaiotto, S. Snigerov, and E. Witten for useful discussions. Z.K. is supported in part by an Israel Science Foundation center for excellence grant and by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12). Z.K. is also supported by the ERC STG grant 335182 and by the Simons Foundation grant 488657 (Simons Collaboration on the Non-Perturbative Bootstrap). N.S. was supported in part by DOE grant DE-SC0009988. \end | http://arxiv.org/abs/1706.08755v2 | {
"authors": [
"Zohar Komargodski",
"Nathan Seiberg"
],
"categories": [
"hep-th",
"cond-mat.str-el"
],
"primary_category": "hep-th",
"published": "20170627094529",
"title": "A Symmetry Breaking Scenario for QCD$_3$"
} |
Magnetic fluctuations and superconducting properties of CaKFe_4As_4 studied by ^75As NMR Y. Furukawa, December 30, 2023 ========================================================================================§ ABSTRACTThe ROS navigation stack is powerful for mobile robots to move from place to place reliably. The job of navigation stack is to produce a safe path for the robot to execute, by processing data from odometry, sensors and environment map. Maximizing the performance of this navigation stack requires some fine tuning of parameters, and this is not as simple as it looks. One who is sophomoric about the concepts and reasoning may try things randomly, and wastes a lot of time.This article intends to guide the reader through the process of fine tuning navigation parameters. It is the reference when someone need to know the "how" and "why" when setting the value of key parameters. This guide assumes that the reader has already set up the navigation stack and ready to optimize it. This is also a summary of my work with the ROS navigation stack. 2 [ § TOPICS] * Velocity and Acceleration * Global Planner* Global Planner Selection* Global Planner Parameters * Local Planner* Local Planner Selection* DWA Local Planner * DWA algorithm* DWA forward simulation* DWA trajectory scoring* Other DWA parameters* Costmap Parameters* AMCL* Recovery Behavior* Dynamic Reconfigure* Problems § VELOCITY AND ACCELERATIONThis section concerns with synchro-drive robots. The dynamics (e.g. velocity and acceleration of the robot) of the robot is essential for local planners including dynamic window approach (DWA) and timed elastic band (TEB). In ROS navigation stack, local planner takes in odometry messages ("odom" topic) and outputs velocity commands ("cmd_vel" topic) that controls the robot's motion.Max/min velocity and acceleration are two basic parameters for the mobile base. Setting them correctly is very helpful for optimal local planner behavior. In ROS navigation, we need to know translational and rotational velocity and acceleration. §.§ To obtain maximum velocityUsually you can refer to your mobile base's manual. For example, SCITOS G5 has maximum velocity 1.4 m/s[This information is obtained from http://www.metralabs.com/en/researchMetraLabs's website.]. In ROS, you can also subscribe to thetopic to obtain the current odometry information. If you can control your robot manually (e.g. with a joystick), you can try to run it forward until its speed reaches constant, and then echo the odometry data.Translational velocity (m/s) is the velocity when robot is moving in a straight line. Its max value is the same as the maximum velocity we obtained above. Rotational velocity (rad/s) is equivalent as angular velocity; its maximum value is the angular velocity of the robot when it is rotating in place. To obtain maximum rotational velocity, we can control the robot by a joystick and rotate the robot 360 degrees after the robot's speed reaches constant, and time this movement. For safety, we prefer to set maximum translational and rotational velocities to be lower than their actual maximum values. §.§ To obtain maximum accelerationThere are many ways to measure maximum acceleration of your mobile base, if your manual does not tell you directly. In ROS, again we can echo odometry data which include time stamps, and them see how long it took the robot to reach constant maximum translational velocity (t_i). Then we use the position and velocity information from odometry (nav_msgs/Odometry message) to compute the acceleration in this process. Do several trails and take the average. Use t_t, t_r to denote the time used to reach translationand and rotational maximum velocity from static, respectively. The maximum translational acceleration a_t,max=max dv / dt≈ v_max/t_t. Likewise, rotational acceleration can be computed by a_r,max=max dω / dt≈ω_max/t_r. §.§ Setting minimum valuesSetting minimum velocity is not as formulaic as above. For minimum translational velocity, we want to set it to a large negative value because this enables the robot to back off when it needs to unstuck itself, but it should prefer moving forward in most cases. For minimum rotational velocity, we also want to set it to negative (if the parameter allows) so that the robot can rotate in either directions. Notice that DWA Local Planner takes the absolute value of robot's minimum rotational velocity. §.§ Velocity in x, y direction x velocity means the velocity in the direction parallel to robot's straight movement. It is the same as translational velocity. y velocity is the velocity in the direction perpendicular to that straight movement. It is called "strafing velocity" in . y velocity should be set to zero for non-holonomic robot (such as differential wheeled robots). § GLOBAL PLANNER§.§ Global Planner Selection To use thenode in navigation stack, we need to have a global planner and a local planner. There are three global planners that adhere tointerface: ,and .§.§.§ carrot_planner This is the simplest one. It checks if the given goal is an obstacle, and if so it picks an alternative goal close to the original one, by moving back along the vector between the robot and the goal point. Eventually it passes this valid goal as a plan to the local planner or controller (internally). Therefore, this planner does not do any global path planning. It is helpful if you require your robot to move close to the given goal even if the goal is unreachable. In complicated indoor environments, this planner is not very practical.§.§.§ navfn and global_planneruses Dijkstra's algorithm to find a global path with minimum cost between start point and end point.is built as a more flexible replacement ofwith more options. These options include (1) support for A*, (2) toggling quadratic approximation, (3) toggling grid path. Bothand global planner are based on the work by <cit.>: §.§ Global Planner ParametersSinceis generally the one that we prefer, let us look at some of its key parameters. Note: not all of these parameters are listed on ROS's website, but you can see them if you run the rqt dynamic reconfigure program: with We can leave (true), (true), (true), (false), (false) to their default values. Setting (false) to true is helpful when we would like to visualize the potential map in RVIZ. Besides these parameters, there are three other unlisted parameters that actually determine the quality of the planned global path. They are , , . Actually, these parameters also present in . The source code[<https://github.com/ros-planning/navigation/blob/indigo-devel/navfn/include/navfn/navfn.h>] has one paragraph explaining howcomputes cost values.cost values are set toIncoming costmap cost values are in the range 0 to 252. The comment also says:Withof 50, theneeds to be about 0.8 to ensure the input values are spread evenly over the output range, 50 to 253.Ifis higher, cost values will have a plateau around obstacles and the planner will then treat (for example) the whole width of a narrow hallway as equally undesirable and thus will not plan paths down the center.Experiment observations Experiments have confirmed this explanation. Settingto too low or too high lowers the quality of the paths. These paths do not go through the middle of obstacles on each side and have relatively flat curvature. Extremevalues have the same effect. For , setting it to a low value may result in failure to produce any path, even when a feasible path is obvious. Figures 5-10 show the effect ofandon global path planning. The green line is the global path produced by .After a few experiments we observed that when= 0.55, , and , the global path is quite desirable.§ LOCAL PLANNER SELECTION Local planners that adhere tointerface are ,and . They use different algorithms to generate velocity commands. Usuallyis the go-to choice. We will discuss it in detail. More information on other planners will be provided later. §.§ DWA Local Planner§.§.§ DWA algorithm See next page. uses Dynamic Window Approach (DWA) algorithm. ROS Wiki provides a summary of its implementation of this algorithm: 33 em * Discretely sample in the robot's control space (dx,dy,dtheta)* For each sampled velocity, perform forward simulation from the robot's current state to predict what would happen if the sampled velocity were applied for some (short) period of time.* Evaluate (score) each trajectory resulting from the forward simulation, using a metric that incorporates characteristics such as: proximity to obstacles, proximity to the goal, proximity to the global path, and speed. Discard illegal trajectories (those that collide with obstacles).* Pick the highest-scoring trajectory and send the associated velocity to the mobile base.* Rinse and repeat.DWA is proposed by <cit.>. According to this paper, the goal of DWA is to produce a (v,ω) pair which represents a circular trajectory that is optimal for robot's local condition. DWA reaches this goal by searching the velocity space in the next time interval. The velocities in this space are restricted to be admissible, which means the robot must be able to stop before reaching the closest obstacle on the circular trajectory dictated by these admissible velocities. Also, DWA will only consider velocities within a dynamic window, which is defined to be the set of velocity pairs that is reachable within the next time interval given the current translational and rotational velocities and accelerations. DWA maximizes an objective function that depends on (1) the progress to the target, (2) clearance from obstacles, and (3) forward velocity to produce the optimal velocity pair.Now, let us look at the algorithm summary on ROS Wiki. The first step is to sample velocity pairs (v_x, v_y, ω) in the velocity space within the dynamic window. The second step is basically obliterating velocities (i.e. kill off bad trajectories) that are not admissible. The third step is to evaluate the velocity pairs using the objective function, which outputs trajectory score. The fourth and fifth steps are easy to understand: take the current best velocity option and recompute. This DWA planner depends on the local costmap which provides obstacle information. Therefore, tuning the parameters for the local costmap is crucial for optimal behavior of DWA local planner. Next, we will look at parameters in forward simulation, trajectory scoring, costmap, and so on.§.§.§ DWA Local Planner : Forward Simulation Forward simulation is the second step of the DWA algorithm. In this step, the local planner takes the velocity samples in robot's control space, and examine the circular trajectories represented by those velocity samples, and finally eliminate bad velocities (ones whose trajectory intersects with an obstacle). Each velocity sample is simulated as if it is applied to the robot for a set time interval, controlled by (s) parameter. We can think ofas the time allowed for the robot to move with the sampled velocities.Through experiments, we observed that the longer the value of , the heavier the computation load becomes. Also, whengets longer, the path produced by the local planner is longer as well, which is reasonable. Here are some suggestions on how to tune thisparameter. How to tuneSettingto a very low value (<= 2.0) will result in limited performance, especially when the robot needs to pass a narrow doorway, or gap between furnitures, because there is insufficient time to obtain the optimal trajectory that actually goes through the narrow passway. On the other hand, since with DWA Local Planner, all trajectories are simple arcs, setting theto a very high value (>= 5.0) will result in long curves that are not very flexible. This problem is not that unavoidable, because the planner actively replans after each time interval (controlled by (Hz)), which leaves room for small adjustments. A value of 4.0 seconds should be enough even for high performance computers.Besides , there are several other parameters that worth our attention.Velocity samples Among other parameters, ,determine how many translational velocity samples to take in x, y direction.controls the number of rotational velocities samples. The number of samples you would like to take depends on how much computation power you have. In most cases we prefer to setto be higher than translational velocity samples, because turning is generally a more complicated condition than moving straight ahead. If you setto be zero, there is no need to have velocity samples in y direction since there will be no usable samples. We picked= 20, and= 40. Simulation granularityis the step size to take between points on a trajectory. It basically means how frequent should the points on this trajectory be examined (test if they intersect with any obstacle or not). A lower value means higher frequency, which requires more computation power. The default value of 0.025 is generally enough for turtlebot-sized mobile base.§.§.§ DWA Local Planner : Trajactory Scoring As we mentioned above, DWA Local Planner maximizes an objective function to obtain optimal velocity pairs. In its paper, the value of this objective function relies on three components: progress to goal, clearance from obstacles and forward velocity. In ROS's implementation, the cost of the objective function is calculated like this: cost=* (distance(m) to path from the endpoint of the trajectory)+* (distance(m) to local goal from the endpoint of the trajectory)+* (maximum obstacle cost along the trajectory in obstacle cost (0-254)) The objective is to get the lowest cost.is the weight for how much the local planner should stay close to the global path <cit.>. A high value will make the local planner prefer trajectories on global path.is the weight for how much the robot should attempt to reach the local goal, with whatever path. Experiments show that increasing this parameter enables the robot to be less attached to the global path. is the weight for how much the robot should attempt to avoid obstacles. A high value for this parameter results in indecisive robot that stucks in place. Currently for SCITOS G5, we setto 32.0,to 20.0,to 0.02. They work well in simulation.§.§.§ DWA Local Planner : Other ParametersGoal distance tolerance These parameters are straightforward to understand. Here we will list their description shown on ROS Wiki:*(double, default: 0.05) The tolerance in radians for the controller in yaw/rotation when achieving its goal.*(double, default: 0.10) The tolerance in meters for the controller in the x & y distance when achieving a goal.*(bool, default: false) If goal tolerance is latched, if the robot ever reaches the goal xy location it will simply rotate in place, even if it ends up outside the goal tolerance while it is doing so.Oscilation reset In situations such as passing a doorway, the robot may oscilate back and forth because its local planner is producing paths leading to two opposite directions. If the robot keeps oscilating, the navigation stack will let the robot try its recovery behaviors. *(double, default: 0.05) How far the robot must travel in meters before oscillation flags are reset. § COSTMAP PARAMETERS As mentioned above, costmap parameters tuning is essential for the success of local planners (not only for DWA). In ROS, costmap is composed of static map layer, obstacle map layer and inflation layer. Static map layer directly interprets the given static SLAM map provided to the navigation stack. Obstacle map layer includes 2D obstacles and 3D obstacles (voxel layer). Inflation layer is where obstacles are inflated to calculate cost for each 2D costmap cell.Besides, there is a global costmap, as well as a local costmap. Global costmap is generated by inflating the obstacles on the map provided to the navigation stack. Local costmap is generated by inflating obstacles detected by the robot's sensors in real time.There are a number of important parameters that should be set as good as possible. §.§ footprint Footprint is the contour of the mobile base. In ROS, it is represented by a two dimensional array of the form [x_0, y_0],[x_1,y_1],[x_2,y_2],...], no need to repeat the first coordinate. This footprint will be used to compute the radius of inscribed circle and circumscribed circle, which are used to inflate obstacles in a way that fits this robot. Usually for safety, we want to have the footprint to be slightly larger than the robot's real contour.To determine the footprint of a robot, the most straightforward way is to refer to the drawings of your robot. Besides, you can manually take a picture of the top view of its base. Then use CAD software (such as Solidworks) to scale the image appropriately and move your mouse around the contour of the base and read its coordinate. The origin of the coordinates should be the center of the robot. Or, you can move your robot on a piece of large paper, then draw the contour of the base. Then pick some vertices and use rulers to figure out their coordinates. §.§ inflation Inflation layer is consisted of cells with cost ranging from 0 to 255. Each cell is either occupied, free of obstacles, or unknown. Figure 13 shows a diagram [Diagram is from <http://wiki.ros.org/costmap_2d>] illustrating how inflation decay curve is computed. andare the parameters that determine the inflation.controls how far away the zero cost point is from the obstacle.is inversely proportional to the cost of a cell. Setting it higher will make the decay curve more steep.Dr. Pronobis sugggests the optimal costmap decay curve is one that has relatively low slope, so that the best path is as far as possible from the obstacles on each side. The advantage is that the robot would prefer to move in the middle of obstacles.As shown in Figure 8 and 9, with the same starting point and goal, when costmap curve is steep, the robot tends to be close to obstacles. In Figure 14,= 0.55,= 5.0; In Figure 15,= 1.75,= 2.58Based on the decay curve diagram, we want to set these two parameters such that the inflation radius almost covers the corriders, and the decay of cost value is moderate, which means decrease the value of. §.§ costmap resolution This parameter can be set separately for local costmap and global costmap. They affect computation load and path planning. With low resolution (>=0.05), in narrow passways, the obstacle region may overlap and thus the local planner will not be able to find a path through.For global costmap resolution, it is enough to keep it the same as the resolution of the map provided to navigation stack. If you have more than enough computation power, you should take a look at the resolution of your laser scanner, because when creating the map using gmapping, if the laser scanner has lower resolution than your desired map resolution, there will be a lot of small "unknown dots" because the laser scanner cannot cover that area, as in Figure 10. For example, Hokuyo URG-04LX-UG01 laser scanner has metric resolution of 0.01mm[data from <https://www.hokuyo-aut.jp/02sensor/07scanner/download/pdf/URG-04LX_UG01_spec_en.pdf>]. Therefore, scanning a map with resolution <=0.01 will require the robot to rotate several times in order to clear unknown dots. We found 0.02 to be a sufficient resolution to use.§.§ obstacle layer and voxel layerThese two layers are responsible for marking obstacles on the costmap. They can be called altogether as obstacle layer. According to ROS wiki, the obstacle layer tracks in two dimensions, whereas the voxel layer tracks in three. Obstacles are marked (detected) or cleared (removed) based on data from robot's sensors, which has topics for costmap to subscribe to.In ROS implementation, the voxel layer inherits from obstacle layer, and they both obtain obstacles information by interpreting laser scans or data sent withortype messages. Besides, the voxel layer requires depth sensors such as Microsoft Kinect or ASUS Xtion. 3D obstacles are eventually projected down to the 2D costmap for inflation. How voxel layer works Voxels are 3D volumetric cubes (think 3D pixel) which has certain relative position in space. It can be used to be associated with data or properties of the volume near it, e.g. whether its location is an obstacle. There has been quite a few research around online 3D reconstruction with the depth cameras via voxels. Here are some of them.* http://delivery.acm.org/10.1145/2050000/2047270/p559-izadi.pdf?ip=128.208.7.188 id=2047270 acc=ACTIVE%20SERVICE key=B63ACEF81C6334F5%2EF43F328D6C8418D0%2E4D4702B0C3E38B35%2E4D4702B0C3E38B35 CFID=830915711 CFTOKEN=23054788 __acm__=1472349664_9fd28ae246d72a507f6a93c5ac84a516KinectFusion: Real-time 3D Reconstruction and Interaction Using a Moving Depth Camera * https://people.mpi-inf.mpg.de/ mzollhoef/Papers/SGASIA2013_VH/paper.pdfReal-time 3D Reconstruction at Scale using Voxel Hashing is a ROS package which provides an implementation of efficient 3D voxel grid data structure that stores voxels with three states: marked, free, unknown. The voxel grid occupies the volume within the costmap region. During each update of the voxel layer's boundary, the voxel layer will mark or remove some of the voxels in the voxel grid based on observations from sensors. It also performs ray tracing, which is discussed next. Note that the voxel grid is not recreated when updating, but only updated unless the size of local costmap is changed.Why ray tracing in obstacle layer and voxel layer? Ray tracing is best known for rendering realistic 3D graphics, so it might be confusing why it is used in dealing with obstacles. One big reason is that obstacles of different type can be detected by robot's sensors. Take a look at figure 17. In theory, we are also able to know if an obstacle is rigid or soft (e.g. grass)[ mentioned in Using Robots in Hazardous Environments by Boudoin, Habib, pp.370].A good blog on voxel ray tracing versus polygong ray tracing: <http://raytracey.blogspot.com/2008/08/voxel-ray-tracing-vs-polygon-ray.html>With the above understanding, let us look into the parameters for the obstacle layer[Some explanations are directly copied from costmap2d ROS Wiki]. These parameters are global filtering parameters that apply to all sensors.* : The maximum height of any obstacle to be inserted into the costmap in meters. This parameter should be set to be slightly higher than the height of your robot. For voxel layer, this is basically the height of the voxel grid. * : The default maximum distance from the robot at which an obstacle will be inserted into the cost map in meters. This can be over-ridden on a per-sensor basis.* : The default range in meters at which to raytrace out obstacles from the map using sensor data. This can be over-ridden on a per-sensor basis. These parameters are only used for the voxel layer (VoxelCostmapPlugin).* : The z origin of the map in meters.* : The z resolution of the map in meters/cell.* : The number of voxels to in each vertical column, the height of the grid is z_resolution * z_voxels.* : The number of unknown cells allowed in a column considered to be "known"* : The maximum number of marked cells allowed in a column considered to be "free".Experiment observations Experiments further clarify the effects of the voxel layer's parameters.We use ASUS Xtion Pro as our depth sensor. We found that position of Xtion matters in that it determines the range of "blind field", which is the region that the depth sensor cannot see anything. In addition, voxels representing obstacles only update (marked or cleared) when obstacles appear within Xtion range. Otherwise, some voxel information will remain, and their influence on costmap inflation remains. Besides,controls how dense the voxels is on the z-axis. If it is higher, the voxel layers are denser. If the value is too low (e.g. 0.01), all the voxels will be put together and thus you won't get useful costmap information. If you set z_resolution to a higher value, your intention should be to obtain obstacles better, therefore you need to increaseparameter which controls how many voxels in each vertical column. It is also useless if you have too many voxels in a column but not enough resolution, because each vertical column has a limit in height. Figure 18-20 shows comparison between different voxel layer parameters setting. § AMCLis a ROS package that deals with robot localization. It is the abbreviation of Adaptive Monte Carlo Localization (AMCL), also known as partical filter localization. This localization technique works like this: Each sample stores a position and orientation data representing the robot's pose. Particles are all sampled randomly initially. When the robot moves, particles are resampled based on their current state as well as robot's action using recursive Bayesian estimation.More discussion on AMCL parameter tuning will be provided later. Please refer to <http://wiki.ros.org/amcl> for more information. For the details of the original algorithm Monte Carlo Localization (MCL), read Chapter 8 of Probabilistic Robotics <cit.>. We now summarize several issues that may affect the quality of AMCL localization[Added on April 8th, 2019. This investigation was done in May, 2017, yet not reported in this guide at the time.].We hope this information makes this guide more complete, and you find it useful. Through experiments, we observed three issues that affect the localization with AMCL. As described in <cit.>, MCL maintains two probabilistic models, a motion model and a measurement model. In ROS , the motion model corresponds to a model of the odometry, while the measurement model correspond to a model of laser scans. With this general understanding, we describe three issues separately as follows.§.§ Header inmessages Messages that are published totopic are of type [See: <http://docs.ros.org/melodic/api/sensor_msgs/html/msg/LaserScan.html>]. This message contains a header with fields dependent on the specific laser scanner that you are using. These fields include (copied from the message documentation)*(float32) start angle of the scan [rad]*(float32) end angle of the scan [rad]*(float32)start angle of the scan [rad]*(float32) time between measurements [seconds] - if your scanner is moving, this will be used in interpolating position of 3d points*(float32) time between scans [seconds]*(float32) minimum range value [m]*(float32) maximum range value [m] We observed in our experiments that if these values are not set correctly for the laser scanner product on board, the quality of localization will be affected (see Figure <ref> and <ref>. We have used two laser scanners products, the SICK LMS 200 and the SICK LMS 291. We provide their parameters below[For LMS 200, thanks to this Github issue (<https://github.com/smichaud/lidar-snowfall/issues/1>)].SICK LMS 200: language=json"range_min": 0.0, "range_max": 81.0, "angle_min": -1.57079637051, "angle_max": 1.57079637051, "angle_increment": 0.0174532923847, "time_increment": 3.70370362361e-05, "scan_time": 0.0133333336562SICK LMS 291: language=json"range_min": 0.0, "range_max": 81.0, "angle_min": -1.57079637051, "angle_max": 1.57079637051, "angle_increment": 0.00872664619235, "time_increment": 7.40740724722e-05, "scan_time": 0.0133333336562§.§ Parameters for measurement and motion modelsThere are parameters listed in thepackage about tuning the laser scanner model (measurement)and odometry model (motion). Refer to the package page for the complete list and their definitions. A detailed discussion requires great understanding of the MCL algorithm in <cit.>, which we omit here. We provide an example of fine tuning these parameters and describe their results qualitatively. The actual parameters you use should depend on your laser scanner and robot.For laser scanner model, the default parameters are:language=json "laser_z_hit": 0.5, "laser_sigma_hit": 0.2, "laser_z_rand" :0.5, "laser_likelihood_max_dist": 2.0To improve the localization of our robot, we increasedandto incorporate higher measurement noise. The resulting parameters are:language=json "laser_z_hit": 0.9, "laser_sigma_hit": 0.1, "laser_z_rand" :0.5, "laser_likelihood_max_dist": 4.0 The behavior is illustrated in Figure <ref> and <ref>. It is clear that in our case, adding noise into the measurement model helped with localization. For the odometry model, we found that our odometry was quite reliable in terms of stability. Therefore, we tuned the parameters so that the algorithm assumes there is low noise in odometry: language=json"kld_err": 0.01, "kld_z": 0.99, "odom_alpha1": 0.005, "odom_alpha2": 0.005, "odom_alpha3": 0.005, "odom_alpha4": 0.005 To verify that the above paremeters for motion model work, we also tried a set of parameters that suggest a noisy odometry model: language=json "kld_err": 0.10, "kld_z": 0.5', "odom_alpha1": 0.008,"odom_alpha2": 0.040,"odom_alpha3": 0.004,"odom_alpha4": 0.025We observed that when the odometry model is less noisy, the particles are more condensed. Otherwise, the particles are more spread-out. §.§ Translation of the laser scanner There is atransform fromtoorthat indicates the pose of the laser scanner with respect to the robot base. If this transform is not correct, it is very likely that the localization behaves strangely. In this situation, we have observed constant shifting of laser readings from the walls of the environment, and sudden drastic change in the localization. It is straightforward enough to make sure the transform is correct; This is usually handled inandspecification of your robot. However, if you are using afile, you may have to publish the transform youself.§ RECOVERY BEHAVIORSAn annoying thing about robot navigation is that the robot may get stuck. Fortunately, the navigation stack has recovery behaviors built-in. Even so, sometimes the robot will exhaust all available recovery behaviors and stay still. Therefore, we may need to figure out a more robust solution. Types of recovery behaviors ROS navigation has two recovery behaviors. They areand . Clear costmap recovery is basically reverting the local costmap to have the same state as the global costmap. Rotate recovery is to recover by rotating 360 degrees in place.Unstuck the robot Sometimes rotate recovery will fail to execute due to rotation failure. At this point, the robot may just give up because it has tried all of its recovery behaviors - clear costmap and rotation. In most experiments we observed that when the robot gives up, there are actually many ways to unstuck the robot. To avoid giving up, we used SMACH to continuously try different recovery behaviors, with additional ones such as setting a temporary goal that is very close to the robot, and returning to some previously visited pose (i.e. backing off). These methods turn out to improve the robot's durability substantially, and unstuck it from previously hopeless tight spaces from our experiment observations[Here is a video demo of my work on mobile robot navigation: <https://youtu.be/1-7GNtR6gVk>].Parameters The parameters for ROS's recovery behavior can be left as default in general. For clear costmap recovery, if you have a relatively high , which means the trajectory is long, you may want to consider increasingparameter, so that bigger area on local costmap is removed, and there is a better chance for the local planner to find a path.§ DYNAMIC RECONFIGURE One of the most flexible aspect about ROS navigation is dynamic reconfiguration, since different parameter setup might be more helpful for certain situations, e.g. when robot is close to the goal. Yet, it is not necessary to do a lot of dynamic reconfiguration.Example One situation that we observed in our experiments is that the robot tends to fall off the global path, even when it is not necessary or bad to do so. Therefore we increased . Since a highwill make the robot stick to the global path, which does not actually lead to the final goal due to tolerance, we need a way to let the robot reach the goal with no hesitation. We chose to dynamically decrease theso thatis emphasized when the robot is close to the goal. After all, doing more experiments is the ultimate way to find problems and figure out solutions.§ PROBLEMS * Getting stuckThis is a problem that we face a lot when using ROS navigation. In both simulation and reality, the robot gets stuck and gives up the goal.* Different speed in different directionsWe observed some weird behavior of the navigation stack. When the goal is set in the -x direction with respect to TF origin, dwa local planner plans less stably (the local planned trajectory jumps around) and the robot moves really slowly. But when the goal is set in the +x direction, dwa local planner is much more stable, and the robot can move faster.I reported this issue on Github here: <https://github.com/ros-planning/navigation/issues/503>. Nobody attempted to resolve it yet. * Reality VS. simulationThere is a difference between reality and simulation. In reality, there are more obstacles with various shapes. For exmaple, in the lab there is a vertical stick that is used to hold to door open. Because it is too thin, the robot sometimes fails to detect it and hit on it. There are also more complicated human activity in reality. * InconsistencyRobots using ROS navigation stack can exhibit inconsistent behaviors, for example when entering a door, the local costmap is generated again and again with slight difference each time, and this may affect path planning, especially when resolution is low. Also, there is no memory for the robot. It does not remember how it entered the room from the door the last time. So it needs to start out fresh again every time it tries to enter a door. Thus, if it enters the door in a different angle than before, it may just get stuck and give up. § THANKS Hope this guide is helpful. Please feel free to add more information from your own experimental observations. * apalike | http://arxiv.org/abs/1706.09068v2 | {
"authors": [
"Kaiyu Zheng"
],
"categories": [
"cs.RO"
],
"primary_category": "cs.RO",
"published": "20170627224403",
"title": "ROS Navigation Tuning Guide"
} |
add1]Tamir Bendory [email protected] add2]Dan Edidin [email protected] add3]Yonina C. Eldar [email protected][add1]The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA [add2] Department of Mathematics, University of Missouri, Columbia, Missouri, USA [add3]The Andrew and Erna Viterbi Faculty of Electrical Engineering, Technion – Israel Institute of Technology, Haifa, Israel[D. Edidin acknowledges support from Simons Collaboration Grant 315460. Y.C. Eldar acknowledges support from the European Union's Horizon 2020 research and innovation program under grant agreement No. 646804–ERCCOG–BNYQ, and from the Israel Science Foundation under Grant no. 335/14.] phase retrieval, phaseless quartic system of equations, ultra-short laser pulse characterization, FROGPhase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging.Unfortunately,phase retrieval isill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the ill-posedness, it is common to use a technique called Frequency-Resolved OpticalGating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. The FROG trace results in a system of phaselessquartic Fourier measurements. In this paper, we prove that it suffices to consider only three translations of the signal to determine almost all bandlimited signals, up to trivial ambiguities.In practice, one usually also has access to the signal's Fourier magnitude. We show that in this caseonly two translations suffice.Our results significantly improve upon earlier work. On Signal Reconstruction from FROG Measurements [ December 30, 2023 ================================================§ INTRODUCTIONPhase retrieval is the problem ofestimating a signal from its Fourier magnitude. This problem plays a key role in many scientific and engineering applications, among them X-ray crystallography, speech recognition, blind channel estimation, alignment tasks and astronomy <cit.>. Optical applications are of particular interest since optical devices, such as a charge-coupled device (CCD) and the human eye, cannot detectphase information of the light wave <cit.>.Almost all one-dimensional signals cannot be determined uniquely from their Fourier magnitude.This immanent ill-posednessmakes this problem substantially more challenging than itsmulti-dimensional counterpart, which is well-posed for almost all signals <cit.>. Two exceptions for one-dimensional signals that are determined uniquely from their Fourier magnitude are minimum phase signals and sparse signals with non-periodic support <cit.>; see also <cit.>. One popular way to overcome the non-uniqueness is by collecting additional information on the sought signal beyond its Fourier magnitude. For instance, this can bedone by taking multiple measurements, each one with a different known mask <cit.>. An important special case employsshifted versions of a single mask. The acquired data is simplythe short-time Fourier transform (STFT) magnitude of the underlying signal. It has been shown thatin many setups, this information is sufficient forefficient and stable recovery <cit.>.For a recent survey of phase retrieval from a signal processing point–of–view, see <cit.>.In this work, we consider an ultra-short laser pulse characterization method,called Frequency-Resolved Optical Gating (FROG).FROG is a simple, commonly-used technique for full characterization of ultra-short laser pulses which enjoys good experimental performance<cit.>. In order to characterize the signal,the FROG method measures the Fourier magnitude of the product of the signal with a translated version of itself, for several different translations. The product of the signal with itself is usually performed using a second harmonic generation(SHG) crystal <cit.>. The acquired data, referred to as FROG trace, is a quartic function of the underlying signal and can be thought of asphaseless quarticFourier measurements.We refer to the problem of recovering a signal from its FROG trace as thequartic phase retrieval problem. Illustration of the FROG setup is given in Figure <ref>.In this paper we provide sufficient conditions on the number of samples required to determine a bandlimited signal uniquely, up to trivial ambiguities, from its FROG trace. Particularly, we show that it is sufficient to consider only three translations of the signal to determinealmost all bandlimited signals. If one also measures the power spectrum of the signal, thentwo translations suffice. The outline of this paper is as follows. In Section <ref> we formulate the FROG problem, discuss its ambiguities and present our main result. Proof of the main result is given in Section <ref> and Section <ref> provides additional proofs for intermediate results. Section <ref> concludes the paper and presents open questions.Throughout the paper we use the following notation. We denote the Fourier transform of a signal z∈ by ẑ_k = ∑_n=0^N-1z_ne^-2πι kn/N, where ι:=√(-1). We further use z, {z} and {z} for its conjugate, real part and imaginary part, respectively. We reserve x∈ to be the underlying signal.In the sequel, all signals are assumed to be periodic with period N and all indices should be considered as modulo N, i.e., z_n = z_n+Nℓ for any integer ℓ∈ℤ.§ MATHEMATICAL FORMULATION AND MAIN RESULTThe goal of this paper is to derive the minimal number of measurements required to determine a signal from its FROG trace. To this end, wefirst formulate the FROG problem andidentify its symmetries, usually called trivial ambiguities in the phase retrieval literature. Then,we introduce and discuss themain results of the paper.§.§ The FROG trace Let us define the signal y_n,m = x_nx_n+mL,whereL is a fixed positive integer. The FROG trace is equal to the one-dimensional Fourier magnitude of y_n,m for each fixed m, i.e., |ŷ_k,m|^2= |∑_n=0^N-1x_nx_n+mLe^-2πι nk/N |^2, k=0, …,N-1,m = 0,…, ⌈ N/L⌉ -1.To ease notation, we assume hereafter that L divides N. Our analysis of the FROG trace holds for bandlimited signals. Formally, we definea bandlimited signal as follows: We say that x∈ℂ^N is a B-bandlimited signal if its Fourier transform ∈ℂ^N containsN-B consecutive zeros. That is, there exits isuch that _i=… = _i+N-B-1=0, where all indices are taken modulo N. The FROG trace is an intensity map ↦ℝ^N×N/L that has three kinds of symmetries. These symmetries form the group of operations acting on the signal for which the intensity map is invariant. The FROG trace is invariant to global rotation, global translation and reflection <cit.>. The first symmetry is continuous, while the latter twogenerally are discrete. These symmetries are similar to equivalent results in phase retrieval, see for instance <cit.>. For bandlimited signals, the global translation symmetry is also continuous: Let x∈ be the underlying signal and let ∈ be its Fourier transform. Let |ŷ_k,m|^2 be the FROG trace of x as defined in (<ref>)for some fixed L. Then, the following signals have the same FROG traceas x: * the rotated signal xe^ιψ for some ψ∈ℝ; * the translated signal x^ℓ obeying x^ℓ_n = x_n-ℓ for some ℓ∈ℤ (equivalently, a signalwith Fourier transform ^ℓ obeying ^ℓ_k= _k e^-2πℓ k/Nfor some ℓ∈ℤ); * the reflected signal x̃ obeying x̃_n = x_-n. For B-bandlimited signals as in Definition <ref> with B≤N/2,the translation ambiguity is continuous. Namely, any signal with a Fourier transform such that ^ψ_k= _k e^ψ k for some ψ∈ℝ has the same FROG trace as x. The result for general signals was derived in <cit.>. The result on the continuity of the translation symmetry for bandlimited signals is a direct corollary of Proposition <ref>, given in Section <ref>.Figure <ref> shows the5-bandlimitedsignal x∈ℝ^11 with Fourier transform given by = (1,,-,0 , 0,0,0,0,0, , - )^T. The second signal is x shifted by three entries.A third signal is a“translated” version of the underlying signal by 1.5 entries. Namely, the kth entry of its Fourier transform is _k e^-2π(1.5) k/N. Clearly, the third signal is not a translated version ofx. Nonetheless, since x is bandlimited, all three signals have the same FROG trace. If x was not a bandlimited signal then the FROG trace of the latter signal would not in general be equal to those of the first two.The symmetries of the FROG trace form a group. Namely, the FROG intensity map →ℝ^N× N/L is invariant under the action of the group G = S^1×μ_N ⋉μ_2, where ⋉ denotes a semi-direct product. The μ_2 corresponds toreflection symmetry andμ_N corresponds totranslation ambiguity, whichrotates _k bye^-2πℓ k /Nfor some integer ℓ∈ℤ. Observe that we usea semi-direct product for the last symmetry sinceμ_2andμ_N do not commute; if one reflects the signal and then multiplies it by a power of e^2π/N, it isnot the sameas multiplying by a power of e^2π/N and thenreflecting.In fact, the semi-direct product of μ_N and μ_2 is the dihedral group D_2N of symmetries of the regular N-gon.If we consider bandlimited signals, then the FROG trace is invariant under the action of the group G = S^1× S^1 ⋉μ_2. That is, the translation ambiguity is continuous. §.§ FROG recovery We are now ready to present the main result of this paper. We assume throughout that the signal is bandlimited, which istypically the case in standard ultra-short pulse characterization experiments <cit.>. We prove that almost any B-bandlimited signal is determined by its FROG trace, up to trivial ambiguities, as long as L≤ N/4. Particularly, we show that we need to consideronly three translations and hence3B measurements are enough to determine the underlying signal. For instance, if L= N/4 then the measurements corresponding to m=0,1,2 determine the signal. If in addition we have access to the signal's power spectrum, then it suffices to choose L≤ N/3. In this case, one may consider only two translations. As an example, if L= N/3, then one can choose m=0,1 (see Remark <ref>). The power spectrum of the sought pulseis often available, or can be measured by a spectrometer, which is integrated into a typical FROG device. Let x be a B-bandlimited signal as defined in Definition <ref> for some B≤N/2. IfN/L≥ 4, then generic signals are determined uniquely from their FROG trace as in (<ref>), modulo the trivial ambiguities (symmetries) of Proposition <ref>, from3Bmeasurements. If in addition we have access to the signal's power spectrum and N/L≥ 3, then 2B measurements are sufficient. By the notion generic, we mean that the set of signals which cannot be uniquely determined, up to trivial ambiguities, is contained in the vanishing locus of a nonzero polynomial. This implies that we can reconstruct almost all signals.This result significantly improves upon earlier workon the uniqueness of the FROG method. In <cit.> it was shown that a continuous signal is determined by its continuous FROG trace and its power spectrum. The uniqueness of the discrete case, as the problem appears in practice, was first considered in <cit.>. It was proven that a discrete bandlimited signal is determined by all N^2 FROG measurements (i.e., L=1) and the signal's power spectrum. Our result requires only 2B FROG measurements if the signal's power spectrum is available, where B is the bandlimit. Furthermore, this is the first result showing that the FROG trace is sufficient to determine the signal even without the power spectrum information.It is interesting to view our results in the broader perspective of nonlinear phaseless systems of equations. In <cit.>, it was shown that4N-4 quadratic equations arising from random frame measurements are sufficient to uniquely determine all signals.Another related setup is the phaseless STFT measurements. This case resembles the FROG setup, where a known reference window replaces the unknown shifted signal. Several works derived uniqueness results for this case under different conditions <cit.>. In <cit.> it was shown that, roughly speaking, it is sufficient to set L≈ N/2(namely, 2N measurements) to determine almost all non-vanishing signals. Comparing to Theorem <ref>, we conclude that, maybe surprisingly, the FROG case is not significantly harder than the phaseless STFT setup. Before moving forward to the proof of the main result, we mention thatseveral algorithms exist to estimate a signal from its FROG trace <cit.>.One popular iterative algorithm is the principal components generalized projections (PCGP) method <cit.>. In each iteration, PCGP performs principal components analysis (PCA) on a data matrix constructed by theprevious estimation. Another approach may be to minimize the least-squares loss function:min_z∈1/2∑_k=0^N-1∑_m=0^N/L-1( |ŷ_k,m|^2 - |∑_n=0^N-1z_nz_n+mLe^-2πι nk/N |^2 )^2.The loss function (<ref>) is a smooth function – a polynomial of degree eight in z – and therefore can be minimized by standard gradient techniques. However, as the function is nonconvex, it is likely that the algorithm willconverge to a local minimum rather than to the global minimum. Figure <ref> examinesthe numerical properties of minimizing the loss function (<ref>) by a trust-region algorithm using the optimization toolbox Manopt <cit.>. The underlying signal x∈ℝ^24 was drawn from a normal i.i.d. distribution with mean zero and variance one. The algorithm wasinitialized from the point x_0 = x + σζ, where σ is a fixed constant andζ takes the values {-1,1} with equal probability. Clearly, for σ=0, x_0=x and therefore any method will succeed. We examined the empirical success for varying values of σ and L. As can be seen, even with L=1, the iterations do not alwaysconverge from an arbitrary initialization.This experiment underscores the challenge in recovering the signal, even in situations where uniqueness is guaranteed. That being said, the results also suggest that for low values of L, namely, large redundancy in the measurements, the algorithm often converges to the global minimum even when it is initialized fairly far away. In other words, thebasin of attraction of (<ref>) is not too small.Theoretical analysis of the basin of attraction in the relatedproblem ofrandom systems of quadratic equations was performed in <cit.>. Recently, nonconvex methods for Fourier phase retrieval, accompanied with theoretical analysis, were proposed in <cit.>. However, in the FROG setup the problem is quartic rather than quadratic. § PROOF OF MAIN RESULT§.§ PreliminariesWe begin the proof by reformulating the measurement model to a more convenient structure. Applying the inverse Fourier transform wewritex_n=1/N∑_k=0^N=1_ke^2π kn/N.Then, according to (<ref>), we haveŷ_k,m =∑_n=0^N-1x_nx_n+mLe^-2π kn/N= 1/N^2∑_n=0^N-1( ∑_ℓ_1 =0^N-1_ℓ_1e^2πℓ_1n/N) ( ∑_ℓ_2 =0^N-1_ℓ_2e^2πℓ_2n/N e^2πℓ_2mL/N)e^-2π k n/N = 1/N^2∑_ℓ_1,ℓ_2 =0^N-1_ℓ_1_ℓ_2e^2πℓ_2mL/N∑_n=0^N-1 e^-2π(k-ℓ_1-ℓ_2)n/N.Since the later sum is equal to N ifk = ℓ_1 + ℓ_2 and zero otherwise, weget ŷ_k,m = 1/N∑_ℓ=0^N-1_ℓ_k-ℓe^2πℓ mL/N = 1/N∑_ℓ=0^N-1_ℓ_k-ℓω^ℓ m,where ω:=e^2π/r,r:=N/L,and we assume that N/L is an integer.Equation (<ref>) implies that, for each fixed k, ŷ_k,m provides r = N/L samples from the (inverse) Fourier transform of _ℓ_k-ℓ.Note that ŷ_k,-m = ∑_ℓ=0^N-1_ℓ_k-ℓω^ℓ m. Because of the reflection ambiguity in Proposition <ref>, it implies that the FROG trace is invariant to sign flip of m. For instance, for r=3, the equations for m=1 and m=2 will be the same since m=2 is equivalent to m=-1. Theproof is based on recursion. We begin by showing explicitly how the first entries ofare determined from the FROG trace. Then, we will show that the knowledge of the first k entries ofand the FROG trace is enough to determine the (k+1)th entry uniquely. Each recursion step is based on the results summarized in the following lemma. The lemma identifiesthe number of solutions of asystem with threephaseless equations. Consider the system of equations | z+v_1| = n_1,| z+v_2| = n_2,| z+v_3| = n_3, for nonnegative n_1,n_2,n_3∈ℝ. *Let v_1,v_2,v_3∈ℂ be distinct and suppose that{v_1-v_2/v_1-v_3}≠ 0.Ifthe system (<ref>) has a solution, then it is unique. Moreover, if n_1,n_2,n_3 are fixed for generic v_1,v_2,v_3∈ℂ then the system will have no solution. *Let v_1,v_2,v_3∈ℝ. If z=a+ b is a solution, then z=a- b is a solution as well. Hence, if the system has a solution, then it has two solutions. Moreover, if n_1,n_2,n_3 are fixed for generic v_1,v_2,v_3∈ℝ then the system will have no solution. See Section <ref>. The notion of generic signals refers here to a set of signals that are not contained in the zero set of some nonzero polynomial in the real and complex parts of each component. Consequently, since the zero set of a polynomial has strictly smaller dimension than the polynomial, this means that the set of signals failing to satisfy the conclusion of the theorem will necessarily have measure zero. Note that Lemma <ref> can be extended to systems of s≥ 3 equations, i.e.,| z+v_1| = n_1,…,| z+v_s|=n_s.If one of the ratios v_1-v_p/v_1-v_q for p,q=2,…,s, p≠ q,is not real then there is at most one solution to the system.§.§ Proof of Theorem <ref> Equipped with Lemma <ref>,wemove forward to the proof of Theorem <ref>.To ease notation, we assume B=N/2,N is even, that _k≠ 0 for k=0,…,N/2-1, and that _k =0 for k=N/2,…,N-1. If the signal's nonzero Fourier coefficients are not in the interval 0…,N/2-1, then we can cyclically reindex the signal without affecting the proof. If N is odd, then one should replace N/2 by ⌊ N/2⌋ everywhere in the sequel. Clearly, the proof carries through for any B≤ N/2. Considering (<ref>), our bandlimit assumption on the signal forms a “pyramid” structure. Here, each row represents fixed k and varying ℓ of _ℓ_k-ℓ for k,ℓ = 0,… N/2-1: [_0^2, 0, … , 0; _0_1, _1 _0, 0, … , 0; _0 _2, _1^2, _2 _0, … , 0 …; ⋮; _N/2-1_0 , _N/2-2_1 , … , _N/2-1_0,0, … , 0; 0, _1 _N/2-1, _2 _N/2-2, … , _N/2-1_1,0, … ,0; ⋮;0, 0, …_N/2-1_N/2-1, …, 0, … ,0. ]Then, _k,m as in (<ref>) is a subsampleof the Fourier transform of each one of the pyramid's rows.From the first row of (<ref>), we see that|_0,0| = 1/N|_0^2|.Because of the continuous rotation ambiguity, we set _0 to be real and, without loss of generality, normalize it so thatN|_0,0| = _0=1. From the second row of (<ref>), we conclude that|_1,0| = 1/N|_0_1+_1_0| = 2/N|_1|.Therefore, we can determine |_1|. Because of the continuous translation ambiguity for bandlimited signals (see Proposition <ref>), we can set arbitrarily _1 = |_1|. Note that this is not true for general signals, where the translation ambiguity is discrete. Our next step is to determine_2 by solving the system for k=2. We denote the unknown variable by z. In this case, we obtain the system of equations for m=0,…,r-1:|_2,m| = 1/N|(1+ω^2m)z + ω^m_1^2 |,where ω is given in (<ref>). Note that for m = (2ℓ+1)r/4 for some integer ℓ∈ℤ, we get ω^2m=-1 so that the system degenerates. If r = N/L ≥ 3 then we can eliminate these equations andthe system|_2,m|/| 1+ω^2m| = 1/N| z + ω^m/1+ω^2m_1^2 |,still has at least two distinct equations. It is easy to see that since |ω|=1, ω^m/1+ω^2m=ω^-m/1+ω^-2m so that this term isself-conjugate and hence real. Since z = _2 is a solution, by the second part of Lemma <ref>, we conclude that the system has two conjugate solutions zand z, corresponding to the reflection symmetry of Proposition <ref>. Hence, we fix _2 to be one of these two conjugate solutions. Fixing _0,_1,_2 up to symmetries, we move forward to determine _3.For k=3, we getthe system of equations form=0,…,r-1,|_3,m| = 1/N| z+ ω^m_1_2 + ω^2m_2_1 +ω^3mz |.As in the previous case,for m = (2ℓ+1)r/6 for some integer ℓ∈ℤwe have ω^3m = -1.In the rest of the cases, we reformulate the equations as|_3,m|/| 1+ω^3m| = 1/N| z+ (ω^m+ω^2m)/1+ω^3m_1_2 |.Again, since |ω|=1, ω^m+ω^2m/1+ω^3m is self conjugate and hence real. Let us denote _2 = |_2| e^θ and divide by e^θtoobtain|_3,m|/| 1+ω^3m| = 1/N| ze^-θ+ (ω^m+ω^2m)/1+ω^3m_1|_2||. Since we set _1 to be real, this is a system of the form of the second part of Lemma <ref>, having two conjugate solutions. Denote these solutionsby z_1,z_2 and recall that the candidate solutions for (<ref>) are z_1e^θ and z_2e^θ. Since_3 is a solution to (<ref>), z_1 = _3e^-θ is one solution. The second solution is given by z_2 = z_1=_3e^θ. Therefore, we conclude that _3e^2θ is a second potential solution to (<ref>). So, currently we have two candidates for _3. Next, we will determine _4 uniquely and show that_3e^2θis inconsistent with the data. This will determine _3 uniquely.For _4 and by eliminating the case of m = r(2ℓ+1)/8 for an integer ℓ∈ℤ (namely, ω^4m = -1), we get the system for m = 0,…,r-1,|_4,m|/| 1+ω^4m| = 1/N| z+ (ω^m+ω^3m)_1_3/1+ω^4m +ω^2m_2^2/1+ω^4m|.To invoke Lemma <ref>, we need three equations. In general, it is most convenient to choose m=0,1,2. If one of these values satisfy ω^4m = -1, then we may always choose another value. Note that for r=3,4, which are of particular interest, ω^4m≠ -1.The following lemma paves the way to determining _4 uniquely: Let v_m = (ω^m+ω^3m)_1_3/1+ω^4m +ω^2m_2^2/1+ω^4m. If r = N/L≥ 4, then for generic _1,_2,_3 the ratio v_0-v_q/v_0-v_p is not real for some distinct p,q ∈{1,…,r-1} with p +q ≠ r. See Section <ref>.Thus, by Lemma <ref> we conclude that if a solution for (<ref>) exists, then it is unique. When r = 3, the system above provides only two distinct equations since the FROG trace ofm=1 and m=2 is the same; see Remark <ref>. However, if |_4| is known, then we get a third equation | z| = |_4| anda similar application of Lemma <ref> shows that _4 is uniquely determined. Recall that currently we have two potential candidates for _3. However, we show in Section <ref> that if we replace _3 by _3e^2θ where θ = (_2) then for generic signals the system of equations (<ref>) has no solution. Therefore, we conclude that we can fix _0,_1,_2,_3,_4 up to trivial ambiguities. The final step of the proof is to show that given _0,_1,…,_k for some k≥ 4, we can determine _k+1 up to symmetries. For an even k=2s we get the system of equations for m=0,…,r-1,|_k+1,m/1+ω^m(k+1)|= 1/N|z + ω^m/1+ω^m(k+1)_1_k+…+ω^ms/1+ω^m(k+1)_s_s+1|,where again we omit the case ω^m(k+1)=-1. To invoke Lemma <ref>, we need three equations. In most cases, one can simply choose m=0,1,2. If one of these values violate the condition ω^m(k+1)≠ -1, then we replace it with larger values of m. For k=2s+1 odd, we obtain the system|_k+1,m/1+ω^m(k+1)|= 1/N|z + ω^m/1+ω^m(k+1)_1_k+…+ω^m(s+1)/1+ω^m(k+1)_s+1^2 |.Let us assume that k=2s is even and denote v_m = ω^m/1+ω^m(k+1)_1_k+…+ω^ms/1+ω^m(k+1)_s_s+1. If r=N/L≥ 4, then the same argument used in the proof of Lemma <ref>shows that for generic values of _0, _1, … , _k, the ratio v_0 -v_pv_0 -v_q will not be real for some distinct values of p,q with p + q ≠ r. Therefore, the system has a unique solution by Lemma <ref>. A similar statement holds for k=2s+1 odd.When r=3, the system provides only two distinct equations; see Remark <ref>. If in addition we assume that knowledge of ||, then we have an additional equation | z| = |_k+1|, ensuring a unique solution. §.§ Example: Determining _4 given_0, _1, _2, _3 To illustrate the method, we describe in more detail the terms used to determine _4 = a_4+ b_4 from _0=1, _1, _2, _3. We are given the following information in a “pyramid form” from which we must determine the unknowns a_4, b_4: [ | (a_4 + b_4 ) + (a_1 + b_1)(a_3+b_3) + (a_2 + b_2 )^2 + (a_3 + b_3) (a_1 + b_1 ) + (a_4 +b_4)|; | (a_4 + b_4 ) + ω (a_1 + b_1)(a_3+b_3) + ω^2(a_2 + b_2 )^2 + ω^3 (a_3 + b_3) (a_1 + b_1 ) + ω^4(a_4 +b_4)|; …; | (a_4 + b_4 ) + ω^r-1 (a_1 + b_1)(a_3+b_3) + ω^2(r-1)(a_2 + b_2 )^2 + ω^3(r-1) (a_3 + b_3) (a_1 + b_1 ) + ω^4r-4(a_4 +b_4)|. ]As we have done throughout the proof, we rearrange the terms as[| (a_4 + b_4 ) + (a_1 + b_1)(a_3 + b_3) +1/2(a_2+b_2)^2|; | (a_4 + b_4) + (ω + ω^3)1 + ω^4 (a_1 + b_1)(a_3 + b_3) + ω^21 + ω^4 (a_2 + b_2)^2|; …; | (a_4 + b_4) + (ω^r-1 + ω^3r-3))1 + ω^4r-4 (a_1 + b_1)(a_3 + b_3) + ω^2r-21 + ω^4r-4 (a_2 + b_2)^2|, ]for any m satisfying ω^4m≠ -1. Suppose that r = N/L ≥ 4 and ω^4,ω^8≠ -1 so we can choose the terms associated with m=0,1,2. If the ratio(_1_3 + 1/2_2^2- (ω + ω^3)1 + ω^4_1 _3 - ω^21 + ω^4_2^2)/( _1_3 + 1/2_2^2- (ω^2 + ω^6)1 + ω^8_1 _3 - ω^41 + ω^8_2^2 )is not real,then _4 = a_4 + b_4 is uniquely determined by the 3 real numbers[ | (a_4 + b_4 ) + (a_1 + b_1)(a_3 + b_3) +1/2(a_2+b_2)^2|;| (a_4 + b_4) + (ω + ω^3)1 + ω^4 (a_1 + b_1)(a_3 + b_3) + ω^21 + ω^4 (a_2 + b_2)^2|; | (a_4 + b_4) + (ω^2 + ω^6)1 + ω^8 (a_1 + b_1)(a_3 + b_3) + ω^41 + ω^8 (a_2 + b_2)^2|. ]If ω^4 = -1 then ω = e^2π/8 and one can determine a_4, b_4 from the terms corresponding to m=0,2,3:[| (a_4 + b_4 ) + (a_1 + b_1)(a_3 + b_3) +1/2(a_2+b_2)^2|; | (a_4 + b_4) + (ω^2 + ω^6)1 + ω^8 (a_1 + b_1)(a_3 + b_3) + ω^41 + ω^8 (a_2 + b_2)^2|; | (a_4 + b_4) + (ω^4 + ω^12)1 + ω^16 (a_1 + b_1)(a_3 + b_3) + ω^81 + ω^16 (a_2 + b_2)^2|. ]By directly substituting ω, these terms reduce to:[ | (a_4 + b_4 ) + (a_1 + b_1)(a_3 + b_3) +1/2(a_2+b_2)^2|; | (a_4 + b_4)-1/2 (a_2 + b_2)^2|; | (a_4 + b_4) - (a_1 + b_1)(a_3 + b_3) + 1/2 (a_2 + b_2)^2|. ]Likewise if ω^8 = -1 then ω = e^2π/16 and we determine a_4 + b_4 from the three real numbers corresponding to m=0,1,4:[| (a_4 + b_4 ) + (a_1 + b_1)(a_3 + b_3) +1/2(a_2+b_2)^2|; | (a_4 + b_4) + (ω + ω^3)1 + ω^4 (a_1 + b_1)(a_3 + b_3) + ω^21 + ω^4 (a_2 + b_2)^2|; | (a_4 + b_4) + (ω^4 + ω^12)1 + ω^16 (a_1 + b_1)(a_3 + b_3) + ω^81 + ω^16 (a_2 + b_2)^2|. ] When r = 3, ω^3 =1 and we only obtain two distinct numbers for m=0,1:[ | (a_4 + b_4 ) + (a_1 + b_1)(a_3 + b_3) +1/2(a_2+b_2)^2|; | (a_4 + b_4) +(a_1 + b_1)(a_3 + b_3) + ω^21 + ω (a_2 + b_2)^2|. ]In this case, there are two possible solutions. However, if we also assume that we know || then we have a third piece of information to uniquely determine _4. § PROOFS OF SUPPORTING RESULTS§.§ On thetranslation symmetry for bandlimited signalsThe following proposition shows that if the signal is bandlimited, then the translation symmetry is continuous. Suppose that xis a B-bandlimited signal with B≤ N/2. Assume without loss of generality that_B=…=_N-1=0. Then, for any μ=e^ψ for some ψ∈[0,2π), any signal with Fourier transform [x̂_0,μx̂_1, μ^2x̂_2,…,μ^B-1x̂_B-1,0,…,0], has the same FROG trace (<ref>) as x. Under the bandlimit assumption, we can substitute p=ℓ and q=k-ℓ and write (<ref>) as ŷ_k,m = 1/N∑_p+q=k 0≤ p,q≤N/2-1_p_qe^2π p mL/N. Now, ifx̂_p is replaced by μ^px̂_p and x̂_q is replaced byμ^qx̂_q then ŷ_k,m is replaced by μ^kŷ_k,m. Hence, the absolute value of ŷ_k,m remains unchanged. Without the bandlimit assumption,q=(k-ℓ) N and thus |ŷ_k,m| is changed unless μ is the Nth root of unity. §.§ Proof of Lemma <ref> The system of equations (<ref>) can be written explicitly as | z|^2 + | v_1|^2+2{ zv_1} = n_1^2, | z|^2 + | v_2|^2+2{ zv_2} = n_2^2,| z|^2 + | v_3|^2+2{ zv_3} = n_3^2. Subtracting thethe second and the third equations from the first, we get { z( v_1 - v_2 ) } = 1/2( n_1^2-n_2^2 + | v_2|^2 - | v_1|^2),{ z( v_1 - v_3 ) } = 1/2( n_1^2-n_3^2 + | v_3|^2 - | v_1|^2). Let z = a +b, v_1- v_2 = c+ d and v_1- v_3 = e+ f. Then, we obtain a system of two linear equations for two variables: ac - bd= 1/2( n_1^2-n_2^2 + | v_2|^2 - | v_1|^2), ae - bf= 1/2( n_1^2-n_3^2 + | v_3|^2 - | v_1|^2). This system has a unique solution provided that the vectors (c,-d) and (e,-f) are not proportional. This is equivalent to the assumption {v_1-v_2/v_1-v_3}≠ 0.We now show that for generic v_1,v_2,v_3 the system (<ref>) has no solutions, so the unique solution to the linear system (<ref>) will not be a solution to (<ref>).If we express v_k = a_k+ b_k, we consider the variety ℐ⊂(ℝ^2)^4 of tuples ((a,b),(a_1,b_1),(a_2,b_2),(a_3,b_3)),such that z=a + b is a solution to (<ref>). Each of theequations in (<ref>) involves a different set of variables, so it imposes an independent condition on the tuples (<ref>). It follows that dimℐ≤ 8-3=5. In particular, ℐ has strictly smaller dimension than the ℝ^6 parametrizing all triples ((a_1,b_1),(a_2,b_2),(a_3,b_3)). Therefore, the system (<ref>) has no solution for generic v_k. Intuitively, this can be seen by noting that the set of z satisfying the equation | z+v_k| = n_k is a circle of radius n_k centered at -a_k- b_k in the complex plane. For generic choices of centers, three circles of fixed radii n_1,n_2,n_3 will have no intersection. For the second part of the lemma, suppose that v_1,v_2,v_3∈ℝ. In this case, (<ref>) is simplified to (a + v_1)^2 + b^2= n_1^2, (a + v_2)^2 + b^2= n_2^2,(a + v_3)^2 + b^2= n_3^2. These equations are invariant under the transformation (a,b)↦(a,-b) so if z=a+b is a solution then z=a - b is a solution as well. Subtracting any pair of equations gives a linear equation in a, so there is at most one value of a solving the system. Hence, if the system has a solution, then it has two conjugate solutions. §.§ Proof of Lemma <ref> Let α_m,1 = ω^3m + ω^m1 + ω^4m and α_m,2 = ω^2m1 + ω^4m.As noted above, α_m,1 and α_m,2 are real if they are well defined. Additionally, note that α_m,i = α_r-m,i for i=1,2. If r = N/L ≥ 4 then we can find p,q with p + q ≠ r so that α_p,1, α_p,2, α_q,1, α_q,2 are well defined. For instance, if ω^4 ≠ -1, ω^8 ≠ -1we take p=1, q=2. Then we can write v_0 -v_p/v_0 -v_q = ( α_0,1 - α_p,1)_1 _3 + (α_0,2 - α_p,2) _2^2/( α_0,1 - α_q,1)_1 _3 + (α_0,2 - α_q,2) _2^2. Moreover, theα_m,i are fixed, so this ratio is well defined as long as the _1, _2,_3 are not solutions to the nonzero quadratic polynomial ( α_0,1 - α_q,1)_1 _3 + (α_0,2 - α_q,2) _2^2. Now we can multiply the numerator and denominator of (<ref>) by (v_0 - v_q) to obtain 1|v_0 - v_q|^2(β_p,1β_q,1|_1_3|^2 + β_p,2β_q,2|_2|^2 + β_p,1β_q,2_1_3_2^2 + β_p,2β_q,1_1_3_2^2), where β_s,i = α_0,i - α_s,i for s = p,q are fixed nonzero real numbers. This expression is real only if ((α_0,1- α_p,1)(α_0,2 - α_q,2)_1_3_2^2 + (α_0,2- α_p,2)(α_0,1 - α_q,1)_1_3_2^2)=0. This condition is a quartic polynomial in the real and complex components of _1, _2, _3. Hence for generic signals the left hand side will not be equal to zero. §.§ Determining_3 uniquely The following lemma shows that given _1,_2 and _4, _3 is determined uniquely from the FROG trace up to symmetries. For a generic signal, let _1∈ℝ, _2,_3,_4∈ℂ with r≥ 4. Consider the following system of equations m = 0,…,r-1: | z + (ω^m+ω^3m)/1+ω^4m_1_3^' + ω^2m/1+ω^4m_2^2| = |_4 + (ω^m+ω^3m)/1+ω^4m_1_3 + ω^2m/1+ω^4m_2^2|.If _3^'=e^2θ_3 and θ = (_2), then the system has no solutions. Moreover, if r=3, then the system of equations:| z |=|_4| | z + _1_3^' + _2^2/2 |=|_4+_1_3+_2^2/2| | z + ω+1/1+ω_1_3^' + ω^2/1+ω_2^2 |=|_4 + ω+1/1+ω_1_3 + ω^2/1+ω_2^2 |has no solutions. The proof is similar to the proof of the first part of Lemma <ref>. Since _1 is generic, we can assume it is nonzero. Let us denote _k = a_k +b_k for k=2,3,4. If the system of equations has a solution, then a_2,b_2,a_3,b_3,a_4,b_4 satisfy a non-trivial polynomial equation. To see this, we argue the following: By assuming _1≠ 0, we can divide the equationsby _1 and then reduce to the case _1 = 1. The proof of Lemma <ref> shows that if a solution z= a+ b exists, then by considering the differences of equations for three distinct values of m, a and b can be expressed as rational functions in the real and complex parts of _2,_3,_3^'.Since _3^' = (_2^2/|_2|)^2_3, (_3^') and (_3^') are also rational functions of a_2,b_2,a_3,b_3. Hence, if z = a+ b is a solution to the system (<ref>), then a = f(a_2,b_2,a_3,b_3,a_4,b_4) and b = g(a_2,b_2,a_3,b_3,a_4,b_4), where f and g are rational functions. In order for z=a+ b to be a feasible solution, it must also satisfy the quadratic equation for m=0 | z + _3^'_1 + _2^2/2 |^2 = |_4 + _3_1 + _2^2/2 |^2. Expanding both sides in terms of (a_2,b_2,a_3,b_3,a_4,b_4), we see that the real numbers (a_2,b_2,a_3,b_3,a_4,b_4) must satisfy an explicit polynomial equation F(a_2,b_2,a_3,b_3,a_4,b_4)=0. As long as F≠ 0, the generic real numbers will not satisfy it. We are left with showing that indeed F≠ 0.For example if ω^4, ω^8 ≠ -1then we can use the equations associated with m=0,1,2:[ |x + _1 _3' + _2^2/2|^2 = |_4 + _1 _3 + _2^2/2|^2; |x + ω + ω^31 +ω^4_1 _3'+ ω^21 + ω^4_2^2|^2 =|_4 + ω + ω^31 +ω^4_1 _3+ω^21 + ω^4_2^2|^2; |x + ω^2 + ω^61 +ω^8_1 _3'+ ω^41 + ω^8_2^2|^2 = |_4 + ω^2 + ω^61 +ω^8_1 _3+ω^21 + ω^4_2^2|^2, ]to show that a_2, b_2, a_3, b_3, a_4, b_4 satisfy a nonzero polynomial F(a_2, b_2, a_3,b_3, a_4, b_4). This is true sinceF(0,1, 1,1,a_4,b_4)(corresponding to _1=1, _2 = , _3 = 1+) has coefficientof a_4^2 given by 4096 (8 t)^2 sin(t/2)^8 sin(t)^2 sin(2 t)^2 (sin(t) + sin(2 t))^2,where ω = cos t + sin t andis the inverse of the sine function. This coefficient is nonzero unless t=0 or t = 2π/3.A similar analysis can be performed in the other cases. For instance, instead of picking m=0,1,2, if ω^4 = -1 one may take m=0,2,4 and if ω^8 =-1 then we choose m=0,1,3. Hence, we conclude that indeed F≠ 0.If r=3, then (<ref>) provides only two distinct equations (see Remark <ref>). We then use the third constraint | z | =|_4 | to derive the same result. § CONCLUSION AND PERSPECTIVEFROG is an important tool forultra-short laser pulse characterization. The problem involves a system of phaseless quartic equations thatdiffers significantly from quadratic systems, appearing in standard phase retrieval problems. In this work, weanalyzed the uniqueness of the FROG method. We have shown that it is sufficient to take only3B FROG measurements in order todetermine a generic B-bandlimited signal uniquely, up to unavoidable symmetries. If the power spectrum of the sought signal is also available, then 2B FROG measurements are enough. Necessary conditions for recovery is an open problem. In addition, this paper did not study computational aspects.An important step in this direction is to analyze the properties of current algorithms used by practitioners, like the PCGP. A natural extension of the FROG modelis called blind FROG or blind phaseless STFT. In this problem, the acquired data is the Fourier magnitude ofy_n,m=x_n^1x_n+mL^2 fortwo signals x^1,x^2∈ <cit.>. The goal is then to estimate both signals simultaneously from their phaseless measurements. Initial results on this model were derived in <cit.>, yet we conjecture that they can be further improved by tighter analysis.§ REFERENCES elsarticle-num | http://arxiv.org/abs/1706.08494v2 | {
"authors": [
"Tamir Bendory",
"Dan Edidin",
"Yonina C. Eldar"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170626173525",
"title": "On Signal Reconstruction from FROG Measurements"
} |
Quantum non-Markovian piecewise dynamics from collision models Jordan M. Horowitz December 30, 2023 ============================================================== Recently, a large class of quantum non-Markovian piecewise dynamics for an open quantum system obeying closed evolution equations has been introduced [B. Vacchini, Phys. Rev. Lett. 117, 230401 (2016)]. These dynamics have been defined in terms of a waiting-time distribution between quantum jumps, along with quantum maps describing the effect of jumps and the system's evolution between them. Here, we present a quantum collision model with memory, whose reduced dynamics in the continuous-time limit reproduces the above class of non-Markovian piecewise dynamics, thus providing an explicit microscopic realization. § INTRODUCTIONPrompted by the growing impact of quantum technologies, the study of non-Markovian (NM) quantum dynamics is currently a topical field <cit.>. Besides the goal of defining, witnessing and even quantifying on a rigorous basis the degree of quantum “non-Markovianity" of an open system dynamics, efforts are under way to advance the longstanding quest for the non-Markovian counterpart of the celebrated Gorini-Kossakowski-Lindblad-Sudarshan master equation (ME) <cit.>. As a pivotal requisite, which may be easily violated <cit.>, a well-defined NM ME must entail a completely positive and trace-preserving (CPT) dynamics for an arbitrary initial state and for suitably large classes of operators and parameters appearing in its expression. While the set of known NM dynamics described by well-behaved MEs (in the above sense) is still relatively small, remarkable progress was made in the last few years.A relatively new approach to quantum NM dynamics is based on quantum collision models (CMs) <cit.>. A CM is a simple microscopic framework for describing the open dynamics of a system S in contact with a bath, where the latter is assumed to consist of a large number of elementary subsystems, the “ancillas". The open dynamics of S resultsfrom its successive pairwise collisions with the bath ancillas, each collision being typically described by a bipartite unitary on S and the involved ancilla. In the continuous-time limit, a CM leads to a Lindblad ME with no need to resort to the Born-Markov approximation <cit.>. Such appealing property prompted NM generalizations of the simplest memoryless CM, whose continuous-time-limit dynamics is ensured by construction to be CPT. A significant instance is the CM in <cit.>, recently extended in <cit.>, which produced a new NM memory-kernel ME. The peculiar structure of this memory-kernel ME and corresponding dynamical map inspired further investigations <cit.> from different viewpoints, which allowed to further enlarge the class of known NM dynamics governed by well-defined MEs. One of these viewpoints builds on the well-known quantum-jumps picture of the Lindblad ME <cit.> to devise a far larger, NM class of piecewise dynamics characterized by a waiting time distribution, a CPT map describing the effect of jumps and a collection of CPT maps accounting for the evolution between jumps <cit.>. This class of piecewise dynamics obeys a memory-kernel ME <cit.>. Given that this general ME encompasses the reduced ME of the CM in <cit.> only as a special case, it is natural to wonder whether a generalized CM can be constructed giving rise to the piecewise-dynamics ME with no restrictions. In this work, we prove that such a CM indeed exists and show that it can be defined as a non-trivial generalization of <cit.> where collisions occur in the form of probabilistic SWAP operations. Among its major distinctive features are the doubling of each ancilla into a pair of subancillas, which allows to introduce the jump map that was fully absent in <cit.>, and the introduction of time-step-dependent swap probabilities, which allows to reproduce waiting time distributions of arbitrary shape unlike <cit.> that was restricted to exponential ones. This extension is of particular importance to comply with possible experimental implementations as well as encompass all the different features of the interaction dynamics that might give rise to non-Markovianity.This paper is organized as follows. In Section <ref>, we review the class of NM quantum dynamics introduced in <cit.>. As anticipated, the main purpose of this work is demonstrating the existence of a quantum CM with memory, which in the continuous-time limit reproduces the above class of NM dynamics. Since this CM is an extension of the one in <cit.>, the latter is reviewed in Section <ref> and a brief introduction to quantum CMs is provided. These introductory sections, in particular, allow us to introduce most of the notation and formalism that we use later in Section <ref>, where the main results of this work are presented. Owing to its central importance, Section <ref> is structured in a number of subsections so as to better highlight the different essential aspects of the proposed CM: the initial state, the way system-ancilla collisions are modelled, the discrete dynamics, its continuous-time limit and, at last, the reduced dynamics of the open system. Our conclusions along with some comments and outlook are given in Section <ref> Some technical proofs are presented in Appendix A.§ REVIEW OF NON-MARKOVIAN PIECEWISE QUANTUM DYNAMICS The prototypical Markovian dynamics of an open quantum system S is described by the Gorini-Kossakowski-Lindblad-Sudarshan ME <cit.>, which readsρ̇=-i[Ĥ,ρ]+∑_kγ_k (L̂_kρL̂^†_k-1/2{L̂^†_kL̂_k,ρ}) ,where ρ(t) is the S density operator, {...,...} stands for the anticommutator, Ĥ is a Hermitian operator, {γ_k} are positive rates, and where {L̂_k} are jump operators. By introducing the maps_t[ρ]=e^R̂tρe^R̂^† t ,[ρ]=∑_kγ_k L̂_kρL̂^†_k ,where we defined the non-Hermitian operator R̂=-i Ĥ-1/2∑_kγ_kL̂^†_kL̂_k, the solution of the Lindblad ME (<ref>) can be written as the Dyson series <cit.>ρ_t = ℛ_t [ ρ_0]+ ∑_j=1^∞∫_0^tdt_j…∫_0^t_2dt_1…ℛ_t-t_j𝒥…𝒥ℛ_t_2 -t_1𝒥ℛ_t_1 [ ρ_0 ]with 0≤ t_1≤ t_2≤...≤ t. (<ref>) shows that the time evolution of S can be viewed as an underlying dynamics described by the evolution map ℛ_t interrupted by jumps each transforming the system state according to the jump map 𝒥. Index j in (<ref>) indeed represents the number of jumps occurred up to time t at instants {t_1,t_2,...,t_j} such that 0≤ t_1≤ t_2≤...≤ t_j≤ t. Note that the maps (<ref>) are not trace-preserving.Both the Lindblad ME (<ref>) and the representation (<ref>) for its exact solution have been taken as a starting point for possible generalizations leading to well-defined dynamics to be described by means of memory kernel MEs, which can describe memory effects in the time evolution. Starting from the seminal work in <cit.>, different approaches have been devised along this line <cit.>. One of us recently extended these results investigating a NM generalization of (<ref>) <cit.>, which in its most general form can be expressed as <cit.>ρ_t =g ( t ) ℰ̅_t [ ρ_0] + ∑_j=1^∞∫_0^tdt_j…∫_0^t_2dt_1f ( t-t_j ) … f ( t_2-t_1 ) g ( t_1 ) ℰ_t-t_j𝒵…𝒵ℰ_t_2 -t_1𝒵ℰ̅_t_1 [ ρ_0].Compared to (<ref>), the jump map J [see (<ref>)] is turned into the CPT map Z, while ℛ_t is replaced by the CPT evolution map ℰ̅_t before any jump has occurred and by the CPT evolution map ℰ_t after the first jump (if any) has taken place. Maps Z, ℰ̅_t and ℰ_t are fully unspecified, but for the requirement of being CPT. Importantly, while in (<ref>) the statistical weight of each possible trajectory is determined by the non-trace-preserving maps ℛ_t, 𝒥 and the initial state <cit.>,in (<ref>) these statistical weights are assigned independently of the maps ℰ̅_t, ℰ_t, Z and the initial state. Indeed, the functions f(t) and g(t), appearing in (<ref>), stand for an arbitrarily chosen waiting time distribution, namely the probabilitydensity for the distribution in time of the jumps, and its associated survival probability g(t)=1-∫_0^tdt' f(t'), that is the probability that no jump has taken place up to time t.The waiting time distribution and the associated survival probability can always be expressed in the formg ( t ) =exp[ - ∫^t_0 s ϕ ( s ) ] , f ( t )=ϕ ( t ) exp[ - ∫^t_0 s ϕ ( s ) ] ,where the positive functionϕ (t) =f ( t )/g ( t )is known as hazard rate function or simply hazard function <cit.>.The meaning of this is that ϕ ( t ) dt provides the probability for a jump to take place in the time interval ( t,t+ dt ], given that no jump has taken place up to time t. Accordingly, the time-dependent coefficient f ( t_-t_j ) … f (t_2-t_1 ) g ( t_1 ) in (<ref>) gives the probability density that j jumps take place at times {t_1,t_2,...,t_j}, while the pre-factor of the first term on the rhs is the probability that no jumps occurred up to time t (this pre-factor indeed multiplies the jump-free evolution map ℰ̅_t). The jumps are thus distributed in time according to a renewal process, which in particular entails that after each jump the process starts anew. Note that, in fact by construction, the dynamical map defined by (<ref>) is ensured to be CPT. Importantly, it can be shown <cit.> that it obeys the memory-kernel MEρ̇ =∫_0^t dt' 𝒲(t-t')[ρ(t')]+ℐ(t)[ρ_0 ] ,where𝒲(t)= d/ dt[f(t)ℰ_t]𝒵+δ(t)f(0)ℰ_0𝒵 ,ℐ(t)= d/ dt[g(t)ℰ̅_t] .The corresponding open dynamics, at variance with the Lindbladian case [see (<ref>) and (<ref>)], is in general NM <cit.>. § COLLISION MODELS WITH MEMORY A quantum CM <cit.> is a simple microscopic model for describing the open dynamics of a system S in contact with a bath B. In its prototypical version, a CM assumes that B comprises a huge number of elementary, identical and non-interacting ancillas all initialized in the same state η. The S-B interaction process occurs via successive pairwise “collisions" between S and the ancillas, each of these collisions being described by a bipartite unitary operation Û_n.By hypothesis, S can collide with each ancilla only once. After n collisions, the state of S is given by ρ_n=Φ^n [ρ_0], where the CPT map Φ is defined asΦ[ρ]= Tr_n {Û_n(ρ⊗η_n)Û_n^†}.Note that, despite the apparent dependance on n(see the partial trace over the nth ancilla), the map Φ does not depend on n since the bath initial state and system-ancilla interaction Hamiltonian are fully homogeneous. It can be shown <cit.> that in the continuous-time limit the dynamics of such a simple CM is described by a Lindblad ME of the form (<ref>), a result which can be expected based on the discrete semigroup property enjoyed by the collision map, Φ^n+m=Φ^nΦ^m. The open dynamics of S corresponding to such a paradigmatic CM is thereby fully Markovian. There are several ways to endow the basic CM just described with memory so as to give rise to a NM dynamics <cit.>. The one of concern to us, given the goals of the present paper, is the CM with memory of <cit.>, which can be regarded as a generalization of a model first put forward in <cit.>. The general structure of the CM in <cit.> is in many respects analogous to the basic memoryless CM described in the previous paragraph except that the system undergoing collisions with the bath ancillas is now bipartite, comprising the very open system under study S plus an auxiliary system M, the “memory", whose Hilbert space dimension is the same as each ancilla's one. A sketch of the CM is given in 1(a). Systems S and M interact all the time according to the pairwise unitary evolution map𝒰_τ[σ]= e^-i Ĥ_SMτσ e^i Ĥ_SMτ ,where Ĥ_SM is the joint S-M Hamiltonian. Here and throughout this paper, σ stands for a joint state of the S-M system and all the bath ancillas.By hypothesis, only M is in direct contact with the bath [see 1(a)]. This interaction takes place through successive collisions, each being described by the pairwise non-unitary quantum map𝒮_n[σ]=p σ+ (1-p)Ŝ_MnσŜ_Mn ,where Ŝ_Mn is the swap unitary operator exchanging the states of M and the nth ancilla. The CPT map (<ref>), which depends parametrically on the probability p, can be interpreted as a probabilistic partial SWAP gate: the memory and ancilla states are either swapped or left unchanged with probability p.The initial state of the overall system (S, M and the bath ancillas) is assumed to beσ_0=(ρ_0⊗η̅_M)⊗η_1⊗η_2⊗···where in particular ρ_0 (η̅_M) is the initial state of S (M). Throughout this paper, tensor product symbols will be omitted whenever possible to avoid using too cumbersome notation.By calling σ_n the overall state at step n, the dynamics proceeds according to σ_n=𝒮_n 𝒰_τ 𝒮_n-1 𝒰_τ…𝒮__2 𝒰_τ 𝒮__1 𝒰_τ[σ_0] ,namely an S-M unitary dynamics goes on all the time, being interrupted at each fixed time step τ by a collision between M and a“fresh" ancilla (one still in the initial state η) described by the non-unitary map (<ref>). Equivalently, one can view each 𝒰_τ itself as embodying the effect of a unitary collision that is however internal to the joint S-M system in such a way that the overall CM dynamics results from subsequent M-ancilla collisions interspersed with internal ones that involve S and M only <cit.>.Like for any CM, the dynamics just defined is discrete. One can however define a continuous-time limit by assuming that the duration of each time step τ becomes very small while the step number n gets very large in such a way that nτ→ t, where t is a continuous time variable. The assumption of a very large number of steps demands an additional prescription for the continuous-time limit of the probability p entering (<ref>) since (<ref>) clearly features p's powers {p^k} for all positive integers k≤ n. This task is carried out by first defining a rate Γ that allows to express p asp= e^-Γτ(which is always possible) and assuming next that Γτ≪1 in such a way that p≃ 1. This ensures that p^k, for any k smaller than n and yet large enough so that k τ→ t'<t is finite, be not washed out in the continuous-time limit. Indeed, this yieldsp^k (p^1/τ)^kτ→ e^-Γ t' .By finally noting that, consistently with the hypothesis Γτ≪ 1, 1- p1- e^-Γτ≃Γτ [(<ref>)] and thatsince the CM is well defined for any choice of η, η̅ and 𝒰_tit is possible to describe the reduced evolution of S by the following CPT mapρ_t= e^-Γ tℰ̅_t [ ρ_0] + ∑_j=1^∞Γ^j e^-Γ t∫_0^tdt_j…∫_0^t_2dt_1ℰ_t-t_j…ℰ_t_2 -t_1ℰ̅_t_1 [ρ_0].which is a special case of (<ref>) for ℰ̅_tρ= Tr_M{𝒰_t[ρ η̅_M]}, ℰ_tρ= Tr_M{𝒰_t[ρ η_M]}, 𝒵=𝕀 , f(t)=Γ e^-Γ t, g(t)=e^-Γ t, ϕ(t)=Γ.In fact it can be shown <cit.> thatthe map (<ref>) obeys a memory-kernel ME of the form (<ref>).Yet, the CM in fact lacks the jump map 𝒵 and is, in addition, apparently constrained to a purely exponential waiting time distribution f(t)=Γ e^-Γ t [the corresponding hazard function ϕ(t) being thus constant].In the next section, we show how to construct a CM with memory whose continuous-time limit yields ME (<ref>) in the most general case, including in particular an arbitrary jump map 𝒵 and an arbitrary waiting time distribution f(t). § A GENERALIZED COLLISION MODEL WITH MEMORY The CM to be defined here is a non-trivial generalization of the CM of <cit.> reviewed in the last section. Just like in <cit.>, the system undergoing collisions with the bath ancillas comprises S and a memory M that are subject to a coherent mutual coupling giving rise to the unitary evolution map (<ref>). At variance with <cit.>, however, now each bath ancilla is bipartite as well, consisting of a pair of “subancillas": one subancilla has the same Hilbert space dimension as S, while the other subancilla has the same dimension as M. A sketch of this generalized CM with memory is displayed in 1(b). §.§ Initial state The initial joint state readsσ_0=(ρ_0⊗η̅_M)⊗(ξ_1⊗η_1)⊗(ξ_2⊗η_2) ⊗… ,where ξ (η) is the initial state of the subancilla having the same dimension as S (M). In full analogy with (<ref>), ρ_0 (η̅_M) is the initial state of S (M). §.§ System-ancilla collisions A further distinctive feature of the generalized CM with memory is that the collisions with the ancillas now involve S as well. By definition, the collision between S-M and the nth bipartite ancilla is described by the non-unitary four-partite CPT map𝒮_n [σ] = p_nσ + (1-p_n) V̂_Sn_1Ŝ_Mn_2σŜ_Mn_2^†V̂_Sn_1^† ,where n_1 and n_2 are the two n's subancillas that are isodimensional to S and M, respectively, while V̂_Sn_1 is a unitary operator acting on S and subancilla n_1.Map (<ref>) therefore swaps the states of M and n_2 and, at the same time, applies the unitary V̂_Sn_1 on S and n_1, or leaves unchanged the state of S, M, n_1 and n_2 with probability p_n. Note that, unlike the CM of the previous section [(<ref>)], now we allow the probability p_n to be in general step-dependent. The reason for this will become clear later on.Based on (<ref>) and the ancilla's initial state [(<ref>)], it is convenient to define a bipartite CPT map on S and M as𝒵 [ ρ__SM]=Tr_n_1 n_2{V̂_Sn_1Ŝ_Mn_2( ρ__SM⊗ξ_n_1⊗η_n_2) Ŝ_Mn_2^†V̂_Sn_1^†} =𝒵[Tr_M{ρ__SM}]⊗ η_M ,where 𝒵 is the CPT map on S defined by𝒵 [ ρ] =Tr_n_1{V̂_Sn_1ρ⊗ξ_n_1V̂_Sn_1^†}.The proof of the last step in (<ref>) is given in Appendix A.(<ref>) and (<ref>) entail that the collision with the nth ancilla [see (<ref>)] changes the reduced state of S and M, ρ__SM, according toTr_n_1n_2{𝒮_n (ρ__SM ξ_n_1 η_n_2)}= p_n ρ__SM + (1-p_n)𝒵 [ρ__SM]= p_n ρ__SM + (1-p_n) 𝒵 [ Tr_M{ρ__SM}]⊗η_M.The essential effect of the collision, thereby, is to either leave with probability p_n the S-M state unchanged or,with probability 1-p_n, to apply the CPT map 𝒵 on S by simultaneously resetting the M's state to η.§.§ Discrete dynamics Similarly to the CM in <cit.> (see previous section), the initial state (<ref>) evolves through an underlying S-M unitary dynamics that is interrupted at each fixed time step τ by a collision described by (<ref>) involving a fresh bipartite ancilla that is still in state ξ⊗η. Accordingly, the overall state at the nth step is given by σ_n=𝒮_nU_τ 𝒮_n-1 𝒰_τ …𝒮_2 𝒰_τ 𝒮_1 𝒰_τ [σ_0].Starting from ρ_SM^(0)=ρ_0 ⊗η̅_M [see (<ref>)], at the end of the first step the reduced S-M state is turned intoρ^(1)_SM=[ρ^(0)_SM] . Next, the collision with ancilla 1 described by map 𝒮_1 [see (<ref>)] takes place followed by another application of the S-M unitary. At the end of the second step, the S-M state thus readsρ^(2)_SM =Tr_1_11_2{𝒰_τ𝒮_1[ρ^(1)_SM ξ_1_1η_1_2]}= Tr_1_11_2{(p_1𝒰_τ+q_1𝒰_τ𝒵̃)[ρ^(1)_SM ξ_1_1η_1_2]}= p_1 [ρ^(1)_SM]+ q_1 𝒵̃[ρ^(1)_SM] ,where the trace is taken over the nth ancilla for n=1 and to simplify the notation we set q_n=1-p_n. By replacing in the last identity the state at the end of the first step (<ref>), (<ref>) can be expressed as a function of the initial S-M state only asρ^(2)_SM=p_1𝒰^2_τ_0jumps[ ρ^(0)_SM]+ q_1𝒰_τ𝒵𝒰_τ_1jump[ ρ^(0)_SM] .Since the elapsed time of the process is an integer multiple of the time step τ and given that a jump (if any) occurs at the end of each time step τ, at the second step either 0 or 1 jumps have taken place. The former and latter cases correspond to the terms featuring zero or one 𝒵 in (<ref>) as highlighted by the captions.At the end of the 3rd step, after the application of maps 𝒮_2 and 𝒰_τ, an analogous calculation leads toρ^(3)_SM= p_2 𝒰_τ[ ρ^(2)_SM] +q_2 𝒰_τ𝒵[ ρ^(2)_SM]=p_2 p_1𝒰^3_τ_0jumps[ ρ^(0)_SM]+ ( p_2 q_1𝒰^2_τ𝒵𝒰_τ +q_2 p_1𝒰_τ𝒵𝒰_τ^2 )_1jump[ ρ^(0)_SM]+ q_2 q_1𝒰_τ𝒵𝒰_τ𝒵𝒰_τ_2jumps[ ρ^(0)_SM],showing that, as expected, 0, 1 or 2 jumps are possible in this case corresponding to as many applications of the map 𝒵.In a similar fashion, at the 4th step we getρ^(4)_SM= p_3 𝒰_τ[ ρ^(3)_SM] +q_3 𝒰_τ𝒵[ ρ^(3)_SM]=p_3 p_2 p_1𝒰^4_τ_0jumps[ ρ^(0)_SM] + ( p_3 p_2 q_1𝒰^3_τ𝒵𝒰_τ p_3 q_2 p_1𝒰^2_τ𝒵𝒰_τ^2 q_3 q_2 p_1𝒰_τ𝒵𝒰^3_τ )_1jump[ ρ^(0)_SM] + (p_3 q_2 q_1𝒰^2_τ𝒵𝒰_τ𝒵𝒰_τ q_3 p_2 q_1𝒰_τ𝒵𝒰^2_τ𝒵𝒰_τ q_3 q_2 p_1𝒰_τ𝒵𝒰_τ𝒵𝒰^2_τ)_2jumps[ ρ^(0)_SM] + q_3 q_2 q_1𝒰_τ𝒵𝒰_τ𝒵𝒰_τ𝒵𝒰_τ_3jumps[ ρ^(0)_SM].In order to write down the nth-step state in a compact form, having in mind the structure of (<ref>), we first note that based on (<ref>) any kth power of the map 𝒰_τ is given by 𝒰_τ^k=𝒰_kτ (in the following we will further set 𝒰_k≡𝒰_kτ to simplify the notation).By induction, the nth-step state for arbitrary n≥2 is given byρ^(n)_SM = (∏_ℓ=1^n-1p_ℓ) 𝒰_n[ρ^(0)_SM]+ ∑_j=1^n-1∑_k_j=1^n-1∑ _k_j-1=1^k_j-1…∑ _k_1=1^k_2-1 π(k_j,…,k_1) 𝒰_n-k_j 𝒵̃ 𝒰_k_j-k_j-1𝒵̃… 𝒵̃ 𝒰_k_2-k_1 𝒵̃ 𝒰_k_1[ρ^(0)_SM],where π(k_j,…,k_1) stands for the probability to perform exactly j jumps at specific steps {k_j,…,k_1} and readsπ(k_j,…,k_1)=(∏_ℓ=k_j1^n1p_ℓ)q_k_j(∏_ℓ=k_j11^k_j1p_ℓ)q_k_j1…q_k_2(∏_ℓ=k_11^k_21p_ℓ)q_k_1(∏_ℓ=1^k_1-1p_ℓ). §.§ Continuous-time limit In order to perform the continuous-time limit, in analogy with (<ref>) we introduce the quantitiesp ( t_k -t_k-1 ) = ^- ∫^t_k -t_k-1_0 s ϕ ( s ) ,q ( t_k -t_k-1 ) =1- ^- ∫^t_k -t_k-1_0 s ϕ ( s ) ,corresponding respectively to the probability of no jump or one jump to take place in each small time interval t_k -t_k-1, which in the case of constant hazard function ϕ ( s ) reduces to a Poisson distribution for the jumps. According to the definition of a renewal process, the jump probabilities thereby depend only on the elapsed time. In this representation, the various contributions appearing in (<ref>), in the limit of a large number of short steps such that the time intervals between steps become increasingly small, can be written as( ∏^j_ℓ =k_ +1p_ℓ) q_k=∏^j_ℓ =k_ +1 ^- ∫^t_l -t_l-1_0 s ϕ ( s )( 1- ^- ∫^t_k -t_k-1_0 s ϕ ( s )) ≈ ^- ∫^t_j -t_k_0 s ϕ ( s ) - ^- ∫^t_j -t_k_ -1_0 s ϕ ( s )≈ ϕ ( t_j -t_k-1 ) ^- ∫^t_j -t_k-1_0 s ϕ ( s ) ( t_k -t_k-1 ) .This shows that the function ϕ ( t ) has indeed the role of hazard rate function [(<ref>], which determines the renewal process describing the time distribution of jumps. Hence, we can thus finally identify( ∏^j_ℓ =k_ +1 p_ℓ) q_k≈ f (t_j -t_k-1 ) dt_k-1.The first term in (<ref>) accordingly becomes∏_ℓ =1^k p_l→ e^- ∫_0^t_kϕ (s) ds =g (t_k ) .Therefore, (<ref>) in the continuous-time limit readsρ_S M (t) = g (t)𝒰_t [ ρ_S M (0)] + ∑_j=1^∞∫_0^tdt_j…∫_0^t_2dt_1 f (t-t_j ) … f (t_2 -t_1 ) g (t_1 ) ×𝒰_t-t_j𝒵…𝒵𝒰_t_2 -t_1𝒵𝒰_t_1 [ ρ_SM (0)] . §.§ Reduced dynamics So far, we have focused on the bipartite system S-M, working out its evolution. We now consider the resulting reduced dynamics for the system S, which embodies the degrees of freedom of the open quantum system of interest. We first recall that ρ_SM (0) = ρ_0⊗η̅_M [see (<ref>)], which ensures the existence of the reduced dynamical map of S. When this expression is replaced in (<ref>) upon taking the trace over M we getρ (t) = g (t) Tr_M{𝒰_t [ ρ_0⊗η̅_M ] }+ ∑_j=1^∞∫_0^tdt_j…∫_0^t_2dt_1 f (t-t_j ) … f (t_2 -t_1 ) g (t_1 )×Tr_M{𝒰_t-t_j𝒵…𝒵𝒰_t_2 -t_1𝒵𝒰_t_1 [ ρ_0⊗η̅_M ] }.By next introducing, according to (<ref>), the CPT maps ℰ_t and ℰ̅_t, whose definition is thus identical to the model in <cit.>, and recalling (<ref>) and (<ref>), we getρ (t) = g (t)ℰ̅_t [ ρ_0 ] + ∑_j=1^∞∫_0^tdt_j…∫_0^t_2dt_1 f (t-t_j ) … f (t_2 -t_1 ) g (t_1 )×Tr_M{𝒰_t-t_j𝒵…𝒵𝒰_t_2 -t_1[ 𝒵 [ ℰ̅_t [ ρ_0 ] ] ⊗η_M ] } .The argument of the partial trace can be expressed by iteration according to𝒰_t-t_j𝒵…𝒵𝒰_t_2 -t_1[ 𝒵[ ℰ̅_t [ ρ_0 ] ] ⊗η_M] =𝒰_t-t_j𝒵…[ 𝒵[ ℰ_t_2 -t_1[ 𝒵[ ℰ̅_t [ ρ_0 ] ]]] ⊗η_M] ,which finally leads to the expressionTr_M{𝒰_t-t_j𝒵…𝒵𝒰_t_2 -t_1[ 𝒵[ ℰ̅_t [ ρ_0 ] ] ⊗η_M] } = ℰ_t-t_j𝒵…𝒵ℰ_t_2 -t_1𝒵ℰ̅_t [ ρ_0 ] ,where for the sake of simplicity we have removed the nested square brackets in the last expression.When this result is replaced in (<ref>), we end up with (<ref>). Accordingly, the reduced dynamics of S in the continuous-time limit necessarily obeys ME (<ref>) with no restrictions.We can therefore conclude that the generalized collision model with memory constructed here is indeed able to reproduce altogether the piecewise NM dynamics with jumps considered in <cit.>. § CONCLUSIONS AND OUTLOOKThe Gorini-Kossakowski-Lindblad-Sudarshan ME has been for over 40 years the workhorse of open quantum systems theory. It embodies the basic reference for open dynamics that lack memory effects. Clearly, though, in the case of strong coupling and/or structured reservoirs a memoryless Markovian description fails to faithfully capture the relevant features of the dynamics. Many non-trivial challenges follow, in particular the need formore general evolution equations that ensure a well-defined(i.e., CPT) dynamics and, at the same time, effectively describe memory effects.On top of this, it is highly desirable that these theoretical descriptions beassociated with corresponding environmental modelsthus providing an underlying microscopic interpretation and, possibly, acontrolled implementation of such non-Markovian dynamics.Both the above aspects were the focus of this paper. Starting from a recently proposed family of memory-kernel MEs corresponding to a largeclass of generally non-Markovian time evolutions, we showed that any such ME admits a microscopic CM from which it can be obtained as the equation governingits continuous-time-limit reduced dynamics. Specifically, the considered time evolutions consist of piecewise dynamics in which a continuous, generally non-Markovian, time evolution is interrupted at random times, distributed according to a general waiting time distribution, by a quantum jump described by a general CPT transformation. These dynamics obey a closed memory-kernel ME.In this work, we showed that one such ME can be obtained as the continuous-time limit of a CMwhere memory effects are due to auxiliary degrees of freedom (which we indeed called memory) mediating the action of the environment on the system. As a distinctive feature of the CM, each bath ancilla isbipartite comprising a pair of subancillas. Each collision occurs in the form of a map that with some probability swaps the state of the memory and one subancilla, while a unitary is at the same time appliedon the system under study and the other subancilla.As a further hallmark of the considered CM, the probability for such swap-and-unitary operation can depend at will on the time step.As remarked in the main text, the ancillas' doubling along with the step-dependance of the aforementioned probability are the crucial features marking the difference between the CM in <cit.> and the one addressed here (which can thus be viewed as a non-trivial generalization of the former). They allow to introduce a jump map as well as a waiting time distribution of arbitrary shape.It is interesting to note that the term “collisional model" was at times used in the literature (see <cit.>) to denote a quantum dynamics that is interrupted at random times by “collisions" – that is jumps in fact – just like in the framework addressed in <cit.>. In this respect, our work provides a connection between this definition of CM, based on random-time collisions, and the one used throughout the paper, where instead collisions occur atfixed times.We finally point out that ME (<ref>) was obtained in <cit.> within a general framework based on the quantization of a family of classical stochastic dynamics. Since this quantization involves non-commuting operators, ME (<ref>) arises only as one of two possible cases corresponding to different operator orderings. The question whether or not a class of underlying CMs can be devised even for the ME arising in the other case <cit.> – which is qualitatively different from ME (<ref>) –is under ongoing investigations. § APPENDIX AA.equationWe here provide the proof of the last identity in (<ref>). Let us first recall the starting point, namely the definition of the map 𝒵 given in the first line of (<ref>), omitting the tensor product symbol to simplify the notation𝒵 [ ρ_SM ] =Tr_n_1 n_2{V̂_Sn_1Ŝ_Mn_2ρ_SMξ_n_1η_n_2V̂_Sn_1^†Ŝ_Mn_2^†} ,and consider two orthonormal bases {| μ⟩_M} and {| ν⟩_n_2} in the Hilbert spaces of M and n_2, respectively. In terms of these vectors, the swap operator Ŝ_Mn_2 is expressed asŜ_Mn_2 = ∑_μ , ν | μ⟩⟨ν |_M⊗| ν⟩⟨μ | _n_2 .Using this expression in (<ref>) the rhs explicitly reads∑_μ , ν μ' , ν'Tr_n_1 n_2{V̂_Sn_1 | μ⟩⟨ν |_M⊗ | ν⟩⟨μ | _n_2ρ_SMξ_n_1η_n_2 | ν' ⟩⟨μ' |_M⊗ | μ' ⟩_⟨ν' |_n_2V̂_Sn_1^†}=∑_μ , ν μ' , ν'Tr_n_1 n_2{V̂_Sn_1 | μ⟩_M⊗ | ν⟩_n_2⟨ν | ρ_SM | ν' ⟩_Mξ_n_1⟨μ | η_n_2 | μ' ⟩_n_2_M⟨μ' | _n_2⟨ν' | V̂_Sn_1^†} ,so that taking the partial trace over n_2 we end up with𝒵 [ ρ_ SM ] = ∑_μ, μ' Tr_n_1 { V̂_Sn_1 | μ⟩_MTr_M {ρ_SM }ξ_n_1 ⟨μ| η_n_2 | μ' ⟩_n_2 _M⟨μ' |V̂_Sn_1^† } .By recalling that the state ρ of the reduced system is just the marginal of ρ_SM and by noting that the expression in square brackets swaps η_n_2 and η_M, we finally get𝒵 [ ρ_SM ] = Tr_n_1{V̂_Sn_1ρξ_n_1η_MV̂_Sn_1^†}= Tr_n_1{V̂_Sn_1ρξ_n_1V̂_Sn_1^†}⊗η_M = 𝒵 [ ρ ] ⊗η_M ,which according to the definition (<ref>) of the map 𝒵 concludes the proof.§ ACKNOWLEDGMENTSWe acknowledge support from the EU Project QuPRoCs (Grant Agreement 641277) and the Fulbright Commission.91 plainreview-breuer H.-P. Breuer, Journal of Physics B 45, 154001 (2012).review-plenio A. Rivas, S. F. Huelga, and M. B. Plenio, Reports on Progress in Physics 77, 094001 (2014).review-vacchini H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016).review-vega I. de Vega and D. Alonso, Rev. Mod. Phys. 89, 015001 (2017).Gorini1976a V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).Lindblad1976a G. Lindblad, Comm.Math. Phys., 48, 119 (1976).barnettPRA2001 S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).maniscalcoPRA2006 S. Maniscalco and F. Petruccione, Phys. Rev. A 73, 012111 (2006).rybarJPB2012 T. Rybar, S. N. Filippov, M. Ziman, and V. Buzek, Journal of Physics B 45, 154006 (2012).giovannettiPRL2012 V. Giovannetti and G. M. Palma, Phys. Rev. Lett. 108, 040401 (2012).giovannettiJPB2012 V. Giovannetti and G. M. Palma, Journal of Physics B 45, 154003 (2012).ciccarelloPRA2013 F. Ciccarello, G. M. Palma, and V. Giovannetti, Phys. Rev. A 87, 040103, (2013).santosPRA2014 N. K. Bernardes, A. R. R. Carvalho, C. H. Monken, and M. F. Santos, Phys. Rev. A 90, 032111 (2014).grimsmoPRL2015 A. L. Grimsmo, Phys. Rev. Lett. 115, 060402 (2015).lorenzoPRA2016 S. Lorenzo, F. Ciccarello, and G. M. Palma, Phys. Rev. A 93, 052111 (2016).luomaPRA2016 S. Kretschmer, K. Luoma, and W. T. Strunz, Phys. Rev. A 94, 012106 (2016).lorenzoARX2017 S. Lorenzo, F. Ciccarello, and G. M. Palma, arXiv:1705.03215. (2017).buzekOSID2005 M. Ziman, P. Stelmachovic, and V. Buzek, Open Syst. Inf. Dyn 12, 81 (2005).ciccarelloPS2013 F. Ciccarello and V. Giovannetti, Physica Scripta T153, 014010 (2013).vacchiniPRA2013 B. Vacchini, Phys. Rev. A 87, 030101 (2013).vacchiniIJQI2014 B. Vacchini, International Journal of Quantum Information 12, 1461011 (2014).dariusPRA2016 D. Chruściński and A. Kossakowski, Phys. Rev. A 94, 020103(R) (2016)vacchiniPRL2016 B. Vacchini, Phys. Rev. Lett. 117, 230401 (2016).Chruscinski2017a D. Chruscinski and A. Kossakowski, Phys. Rev. A 95, 042131(2017).petruccione-book H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems.Oxford: Oxford University Press, 2002.Holevo2001 A. S. Holevo, Statistical Structure of Quantum Theory, Lecture Notes in Physics 67.Berlin: Springer, 2001.Barchielli2009 A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Lecture Notes in Physics 782. Berlin: Springer, 2009.Barchielli1994a A. Barchielli, Some stochastic differential equations in quantum optics and measurement theory: the case of counting processes, Stochastic Evolution of Quantum States in Open Systems and in Measurement Processes, (L. Diósi and B. Lukàcs, eds.), Singapore: World Scientific, 1994.Budini2004a A. A. Budini, Phys. Rev. A. 69, 042107 (2004).Budini2005a A. A. Budini, Phys. Rev. E 72, 056106 (2005).Breuer2008a H.-P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008).Kossakowski2009a A. Kossakowski and R. Rebolledo, Open Syst. Inf. Dyn. 16, 259 (2009).Budini2013a A. A. Budini, Phys. Rev. A 88, 032115 (2013).Budini2013b A. A. Budini, Phys. Rev. A 88, 012124 (2013).Ross2007 S. M. Ross, Introduction to probability models. Burlington: Academic Press, 2007.rauPR1963 J. Rau,Phys. Rev. 129, 1880 (1963).scaraniPRL2002 V. Scarani, M. Ziman, P. ŠŠtelmachovi čč, N. Gisin, and V. Bu žžek, Phys. Rev. Lett. 88, 097905 (2002).zimanPRA2002 M. Ziman, P. ŠŠtelmachovi čč, V. Bu žžek, M. Hillery, V. Scarani, and N. Gisin, Phys. Rev. A 65, 042105 (2002). | http://arxiv.org/abs/1706.09025v1 | {
"authors": [
"Salvatore Lorenzo",
"Francesco Ciccarello",
"G. Massimo Palma",
"Bassano Vacchini"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170627194728",
"title": "Quantum non-Markovian piecewise dynamics from collision models"
} |
Animashree AnandkumarState-by-state Minimax Adaptive Estimation for Nonparametric Hidden Markov Models Luc Lehéricy [email protected] de Mathématiques d'OrsayUniv. Paris-Sud, CNRS, Université Paris-Saclay91405 Orsay, France December 30, 2023 =============================================================================================================================================================================== In this paper, we introduce a new estimator for the emission densities of a nonparametric hidden Markov model. It is adaptive and minimax with respect to each state's regularity–as opposed to globally minimax estimators, which adapt to the worst regularity among the emission densities. Our method is based on Goldenshluger and Lepski's methodology. It is computationally efficient and only requires a family of preliminary estimators, without any restriction on the type of estimators considered. We present two such estimators that allow to reach minimax rates up to a logarithmic term: a spectral estimator and a least squares estimator. We show how to calibrate it in practice and assess its performance on simulations and on real data.hidden Markov model; model selection; nonparametric density estimation; oracle inequality; adaptive minimax estimation; spectral method; least squares method.§ INTRODUCTIONFinite state space hidden Markov models, or HMMs in short, are powerful tools for studying discrete time series and have been used in a variety of applications such as economics, signal processing and image analysis, genomics, ecology, speech recognition and ecology among others. The core idea is that the behaviour of the observations depends on a hidden variable that evolves like a Markov chain. Formally, a hidden Markov model is a process (X_j, Y_j)_j ≥ 1 in which (X_j)_j is a Markov chain on , the Y_i's are independent conditionally on (X_j)_j and the conditional distribution of Y_i given (X_j)_j depends only on X_i. The parameters of the HMM are the parameters of the Markov chain, that is its initial distribution and transition matrix, and the parameters of the observations, that is the emission distributions (ν^*_k)_k ∈ where ν^*_k is the distribution of Y_j conditionally to X_j = k. Only the observations (Y_j)_j are available. In this article, we focus on estimating the emission distributions in a nonparametric setting. More specifically, assume that the emission distributions have a density with respect to some known dominating measure μ, and write f^*_k their densities–which we call the emission densities. The goal of this paper is to estimate all f^*_k's with their minimax rate of convergence when the emission densities are not restricted to belong to a set of densities described by finitely many parameters. §.§ Nonparametric state-by-state adaptivity Theoretical results in the nonparametric setting have only been developed recently. <cit.> and <cit.> introduce spectral methods, and the latter is proved to be minimax but not adaptive–which means one needs to know the regularity of the densities beforehand to reach the minimax rate of convergence. <cit.> introduce a least squares estimator which is shown to be minimax adaptive up to a logarithmic term. However, all these papers have a common drawback: they study the emission densities as a whole and can not handle them separately. This comes from their error criterion, which is the supremum of the errors on all densities: what they actually prove is that max_k ∈f̂_k - f^*_k _2 converges with minimax rate when (f̂_k)_k are their density estimators. In general, the regularity of each emission density could be different, leading to different rates of convergence. This means that having just one emission density that is very hard to estimate is enough to deteriorate the rate of convergence of all emission densities.In this paper, we construct an estimator that is adaptive and estimates each emission density with its own minimax rate of convergence. We call this property state-by-state adaptivity. Our method does so by handling each emission density individually in a way that is theoretically justified–reaching minimax and adaptive rates of convergence with respect to the regularity of the emission densities–and computationally efficient thanks to its low computational and sample complexity.Our approach for estimating the densities nonparametrically is model selection. The core idea is to approximate the target density using a family of parametric models that is dense within the nonparametric class of densities. For a square integrable density f^*, we consider its projection f^*_M on a finite-dimensional space _M (the parametric model), where M is a model index. This projection introduces an error, the bias, which is the distance f^* - f^*_M_2 between the target quantity and the model. The larger the model, the smaller the bias. On the other hand, larger models will make the estimation harder, resulting in a larger variance f̂_M - f^*_M _2^2. The key step of model selection is to select a model with a small total error–or alternatively, a good bias-variance tradeoff. In many situations, it is possible to reach the minimax rate of convergence with a good bias-variance tradeoff. Previous estimators of the emission densities of a HMM perform such a tradeoff based on an error that takes the transition matrix and all emission densities into account. Such an error leads to a rate of convergence that corresponds to the slowest minimax rate amongst the different parameters. In contrast, our method performs a bias-variance tradeoff for each emission density using an error term that depends only on the density in question, which makes it possible to reach the minimax rates for each density.§.§ Plug-in procedure The method we propose is based on the method developed in the seminal papers of <cit.> for density estimation, extended by <cit.> to the white noise and regression models. It takes a family of estimators as input and chooses the estimator that performs a good bias-variance tradeoff separately for each hidden state. We recommend the article of <cit.> for an insightful presentation of this method in the case of conditional density estimation. Our method and assumptions are detailed in Section <ref>. Let us give a quick overview of the method. Assume the densities belong to a Hilbert space . Given a family of subsets of finite-dimensional subspaces of(the models) indexed by M and estimators f̂^(M)_k of the emission densities for each hidden state k and each model M, one computes a substitute for the bias of the estimators byA_k(M) = M'sup{f̂^(M')_k - f̂^(M ∧ M')_k _2 - σ(M') }.for some penalty σ. Then, for each state k, one selects the estimator M̂_k from the model M minimizing the quantity A_k(M) + 2 σ(M).The penalty σ can also be interpreted as a variance bound, so that this penalization procedure can be seen as performing a bias-variance tradeoff. The novelty of this method is that it selects a different M̂_k, that is a different model, for each hidden state: this is where the state-by-state adaptivity comes from. Also note that contrary to <cit.>, we do not make any assumption on how the estimators are computed, provided a variance bound holds. The main theoretical result is an oracle inequality on the selected estimators f̂^(M̂_k)_k, see Theorem <ref>. As a consequence, we are able to get a rate of convergence that is different for each state. These rates of convergence will even be adaptive minimax up to a logarithmic factor when the method is applied to our two families of estimators: spectral estimators and least squares estimators. To the best of our knowledge, this is the first state-by-state adaptive algorithm for hidden Markov models.Note that finding the right penalty term σ is essential in order to obtain minimax rates of convergence. This requires a fine theoretical control of the variance of the auxiliary estimators, in the form of assumption [H(ϵ)] (see Section <ref>). To the best of our knowledge, there is no suitable result in the literature. This is the second theoretical contribution of this paper: we control two families of estimators in a way that makes it possible to reach adaptive minimax rate with our state-by-state selection method, up to a logarithmic term.On the practical side, we run this method and several variants on data simulated from a HMM with three hidden states and one irregular density, as illustrated in Section <ref>. The simulations confirm that it converges with a different rate for each emission density, and that the irregular density does not alter the rate of convergence of the other ones, which is exactly what we wanted to achieve.Better still, the added computation time is negligible compared to the computation time of the estimators: even for the spectral estimators of Section <ref> (which can be computed much faster than the least squares estimators and the maximum likelihood estimators using EM), computing the estimators on 200 models for 50,000 observations (the lower bound of our sample sizes) of a 3-states HMM requires a few minutes, compared to a couple of seconds for the state-by-state selection step. The difference becomes even larger for more observations, since the complexity of the state-by-state selection step is independent of the sample size: for instance, computing the spectral estimators on 300 models for 2,200,000 observations requires a bit less than two hours, and a bit more than ten hours for 10,000,000 observations, compared to less than ten seconds for the selection step in both cases. We refer to Section <ref> for a more detailled discussion about the algorithmic complexity of the algorithms.§.§ Families of estimators We use two methods to construct families of estimators and apply the selection algorithm. The motivation and key result of this part of the paper is to control the variances of the estimators by the right penalty σ. This part is crucial if one wants to get adaptive minimax rates, and has not been adressed in previous papers. For both methods, we develop new theoretical results that allow to obtain a penalty σ that leads to adaptive minimax rates of convergence up to a logarithmic term. We present the algorithms and theoretical guarantees in Section <ref>. The first method is a spectral method and is detailed in Section <ref>. Several spectral algorithms were developed, see for instance <cit.> and <cit.> in the parametric setting, and <cit.> and <cit.> in a nonparametric framework. The main advantages of spectral methods are their computational efficiency and the fact that they do not resort to optimization procedure such as the EM and more generally nonconvex optimization algorithm, thus avoiding the well-documented issue of getting stuck into local sub-optimal minima. Our spectral algorithm is based on the one studied in <cit.>. However, their estimator cannot reach the minimax rate of convergence: the variance bound σ(M) deduced from their results is proportional to M^3, while reaching the minimax rate requires σ(M) to be proportional to M. To solve this issue, we introduce a modified version of their algorithm and show that it has the right variance bound, so that it is able to reach the adaptive minimax rate after our state-by-state selection procedure, up to a logarithmic term. Our algorithm also has an improved complexity: it is at most quasi-linear in the number of observations and in the model dimension, instead of cubic in the model dimension for the original algorithm. The second method is a least squares method and is detailed in Section <ref>. Nonparametric least squares methods were first introduced by <cit.> to estimate the emission densities and extended by <cit.> to estimate all parameters at once. They rely on estimating the density of three consecutive observations of the HMM using a least squares criterion. Since the model is identifiable from the distribution of three consecutive observations when the emission distributions are linearly independent, it is possible to recover the parameters from this density. In practice, these methods are more accurate than the spectral methods and are more stable when the models are close to not satisfying the identifiability condition, see for instance <cit.> for the accuracy and <cit.> for the stability. However, since they rely on the minimization of a nonconvex criterion, the computation times of the corresponding algorithms are often longer than the ones from spectral methods. A key step in proving theoretical guarantees for least squares methods is to relate the error on the density of three consecutive observations to the error on the HMM parameters in order to obtain an oracle inequality on the parameters from the oracle inequality on the density of three observations. More precisely, the difficult part is to lower bound the error on the density by the error on the parameters. Let us write g and g' the densities of the first three observations of a HMM with parameters θ and θ' respectively (these parameters actually correspond to the transition matrix and the emission densities of the HMM). Then one would like to getg - g' _2 ≥(θ) d(θ, θ')where d is the natural ^2 distance on the parameters and (θ) is a positive constant which does not depend on θ'. Such inequalities are then used to lower bound the variance of the estimator of the density of three observations g^* by the variance of the parameter estimators: let g be the projection of g^* and g' be the estimator of g^* on the current approximation space (with index M). Denote θ^*_M and θ̂_M the corresponding parameters and assume that the error g - g' _2 is bounded by some constant σ'(M), then the result will be thatd(θ̂_M, θ^*_M) ≤σ'(M)/(θ^*_M). Such a result is crucial to control the variance of the estimators by a penalty term σ, which is the result we need for the state-by-state selection method. In the case where only the emission densities vary, <cit.> proved that such an inequality always holds for HMMs with 2 hidden states using brute-force computations, but it is still unknown whether it is always true for larger number of states. When the number of states is larger than 2, they show that this inequality holds under a generic assumption. <cit.> extended this result to the case where all parameters may vary. However, the constants deduced from both articles are not explicit, and their regularity (when seen as a function of θ) is unknown, which makes it impossible to use in our setting: one needs the constants (θ^*_M) to be lower bounded by the same positive constant, which requires some sort of regularity on the function θ⟼(θ) in the neighborhood of the true parameters.To solve this problem, we develop a finer control of the behaviour of the difference g - g' _2, which is summarized in Theorem <ref>. We show that it is possible to assumeto be lower semicontinuous and positive without any additional assumption. In addition, we give an explicit formula for the constant when θ' and θ are close, which gives an explicit bound for the asymptotical rate of convergence. This result allows us to control the variance of the least squares estimators by a penalty σ which ensures that the state-by-state method reaches the adaptive minimax rate up to a logarithmic term.§.§ Numerical validation and application to real data sets Section <ref> shows how to apply the state-by-state selection method in practice and shows its performance on simulated data and a comparison with a method based on cross validation that does note estimate state by state.Note that the theoretical results give a penalty term σ known only up to a multiplicative constant which is unknown in practice. This problem, the penalty calibration issue, is usual in model selection methods. It can be solved using algorithms such as the dimension jump heuristics, see for instance <cit.>, who introduce this heuristics and prove that it leads to an optimal penalization in the special case of Gaussian model selection framework. This method has been shown to behave well in practice in a variety of domains, see for instance <cit.>. We describe the method and show how to use this heuristics to calibrate the penalties in Section <ref>. We propose and compare several variants of our algorithm. Section <ref> shows some variants in the calibration of the penalties and Section <ref> shows other ways to select the final estimator. We discuss the result of the simulations and the convergence of the selected estimators in Section <ref>. In Section <ref>, we compare our method with a non state-by-state adaptive method based on cross validation. Finally, we discuss the complexities of the auxiliary estimation methods and of our selection procedures in Section <ref>. In Section <ref>, we apply our algorithm to two sets of GPS tracks. The first data set contains trajectories of artisanal fishers from Madagascar, recorded using a regular sampling with 30 seconds time steps. The second data set contains GPS positions of Peruvian seabird, recorded with 1 second time steps. We convert these tracks into the average velocity during each time step and apply our method using spectral estimators as input. The observed behaviour confirms the ability of our method to adapt to the different regularities by selecting different dimensions for each emission density. Section <ref> contains a conclusion and perspectives for this work. Finally, Appendix A contains the details of our spectral algorithm and Appendix <ref> is dedicated to the proofs.§.§ Notations We will use the following notations throughout the paper. * [K] = {1, …, K} is the set of integers between 1 and K. * (K) is the set of permutations of [K]. * ·_F is the Frobenius norm. We implicitely extend the definition of the Frobenius norm to tensors with more than 2 dimensions. * (A) is the linear space spanned by the family A. * σ_1(A) ≥…≥σ_p ∧ n(A) are the singular values of the matrix A ∈^n × p. * ^2(, μ) is the set of real square integrable measurable functions onwith respect to the measure μ. * For = (f_1, …, f_K) ∈^2(, μ)^K, G() is the Gram matrix of , defined by G()_i,j = ⟨ f_i, f_j ⟩ for all i,j ∈ [K].§ THE STATE-BY-STATE SELECTION PROCEDURE In this section, we introduce the framework and our state-by-state selection method.In Section <ref>, we introduce the notations and assumptions. In Section <ref>, we present our selection method and prove that it satisfies an oracle inequality. §.§ Framework and assumptionsLet (X_j)_j ≥ 1 be a Markov chain with finite state spaceof size K. Let ^* be its transition matrix and π^* be its initial distribution. Let (Y_j)_j ≥ 1 be random variables on a measured space (, μ) with μ σ-finite such that conditionally on (X_j)_j ≥ 1 the Y_j's are independent with a distribution depending only on X_j. Let ν^*_k be the distribution of Y_j conditionally to {X_j = k}. Assume that ν^*_k has density f^*_k with respect to μ. We call (ν^*_k)_k ∈ the emission distributions and ^* = (f^*_k)_k ∈ the emission densities. Then (X_j, Y_j)_j ≥ 1 is a hidden Markov model with parameters (π^*, ^*, ^*). The hidden chain (X_j)_j ≥ 1 is assumed to be unobserved, so that the estimators are based only on the observations (Y_j)_j ≥ 1. Let (_M)_M ∈ be a nested family of finite-dimensional subspaces such that their union is dense in ^2(, μ). The spaces (_M)_M ∈ are our models; in the following we abusively call M the model instead of _M. For each index M ∈, we write ^*, (M) = (f^*, (M)_k)_k ∈ the projection of ^* on (_M)^K. It is the best approximation of the true densities within the model M. In order to estimate the emission densities, we do not need to use every models. Typically there is no point in taking models with more dimensions than the sample size, since they will likely be overfitting. Let _n ⊂ be the set of indices which will be used for the estimation from n observations. For each M ∈_n, we assume we are given an estimator ^(M)_n = (f̂^(M)_n,k)_k ∈∈ (_M)^K. We will need to assume that for all models, the variance–that is the distance between ^(M)_n and ^*, (M)–is small with high probability. In the following, we drop the dependency in n and simply writeand ^(M). The following result is what one usually obtains in model selection. It bounds the distance between the estimators ^(M) and the projections ^*, (M) by some penalty function σ. Thus, σ/2 can be seen as a bound of the variance term. [H(ϵ)] With probability 1 - ϵ,∀ M ∈, inf_τ_n,M∈(K)max_k ∈f̂^(M)_k - f^*, (M)_τ_n,M(k)_2 ≤σ(M, ϵ,n)/2 where the upper bound σ : (M, ϵ,n) ∈× [0,1] ×^* ⟼σ(M, ϵ,n) ∈_+ is nondecreasing in M. We show in Sections <ref> and <ref> how to obtain such a result for a spectral method and for a least squares method (using an algorithm from <cit.>). In the following, we omit the parameters ϵ and n in the notations and only write σ(M). What is important for the selection step is that the permutation τ_n,M does not depend on the model M: one needs all estimators (f̂^(M)_k)_M ∈ to correspond to the same emission density, namely f^*_τ_n(k) when τ_n,M = τ_n is the same for all models M. This can be done in the following way: let M_0 ∈ and letτ̂^(M)∈τ∈(K){max_k ∈f̂^(M)_τ(k) - f̂^(M_0)_k _2 }for all M ∈. Then, consider the estimators obtained by swapping the hidden states by these permutations. In other words, for all k ∈, considerf̂^(M)_k,= f̂^(M)_τ̂^(M)(k).Now, assume that the error on the estimators is small enough. More precisely, write B_M,M_0 = max_k ∈ f^*,(M)_k - f^*,(M_0)_k _2 the distance between the projections of ^* on the models M and M_0 and assume that 2 [ σ(M)/2 + σ(M_0)/2 + B_M,M_0] (that is twice the upper bound of the distance between two estimated emission densities corresponding to the same hidden states in models M and M_0) is smaller than m(^*,M_0) := min_k' ≠ k f^*,(M_0)_k - f^*,(M_0)_k'_2, which is the smallest distance between two different densities of the vector ^*,(M_0).Then [H(ϵ)] ensures that with probability as least 1 - ϵ, for all k, there exists a single component of ^(M) that is closer than σ(M)/2 + σ(M_0)/2 of f^*,(M_0)_k, and this component will be f̂^(M)_τ̂^(M)(k) by definition. This is summarized in the following lemma. Assume [H(ϵ)] holds. Then with probability 1 - ϵ,there exists a permutation τ_n ∈(K) such that for all k ∈ and for all M ∈ such thatσ(M) + σ(M_0) + 2 B_M,M_0 < m(^*,M_0),one hasmax_k ∈f̂^(M)_k,- f^*, (M)_τ_n(k)_2 ≤σ(M)/2.Proof in Section <ref> Thus, this property holds asymptotically as soon as inf tends to infinity and sup_M ∈σ(M) tends to zero. §.§ Estimator and oracle inequality Let us now introduce our selection procedure. This method and the following theorem are based on the approach of <cit.>, but do not require any assumption on the structure of the estimators, provided a variance bound such as Equation (<ref>) holds.For each k ∈ and M ∈, letA_k(M) = M' ∈sup{f̂^(M')_k - f̂^(M ∧ M')_k _2 - σ(M') }.A_k(M) serves as a replacement for the bias of the estimator f̂^(M)_k, as can be seen in Equation (<ref>). This comes from the fact that for large M', the quantity f̂^(M')_k - f̂^(M)_k _2 is upper bounded by the variances f̂^(M')_k - f^*,(M')_k _2 and f̂^(M)_k - f^*,(M)_k _2 (which are bounded by σ(M')/2) plus the bias f^*,(M)_k - f^*_k _2. Thus, only the bias term remains after substracting the variance bound σ(M').Then, for all k ∈, select a model through the bias-variance tradeoffM̂_k ∈M ∈{ A_k(M) + 2 σ(M) }and finally takef̂_k = f̂^(M̂_k)_k.The following theorem shows an oracle inequality on this estimator. Let ϵ≥ 0 and assume equation (<ref>) holds for all k ∈ with probability 1 - ϵ. Then with probability 1 - ϵ,∀ k ∈, f̂_k - f^*_τ_n(k)_2 ≤ 4 inf_M ∈{ f^*, (M)_τ_n(k) - f^*_τ_n(k)_2 + σ(M, ϵ) }.We restrict ourselves to the event of probability at least 1-ϵ where equation (<ref>) holds for all k ∈.The first step consists in decomposing the total error: for all M ∈ and k ∈,f̂^(M̂_k)_k - f^*_τ_n(k)_2 ≤ f̂^(M̂_k)_k - f̂^(M̂_k ∧ M)_k _2 + f̂^(M̂_k ∧ M)_k - f̂^(M)_k _2 + f̂^(M)_k - f^*, (M)_τ_n(k)_2 +f^*, (M)_τ_n(k) - f^*_τ_n(k)_2.From now on, we will omit the subscripts k and τ_n(k). Using equation (<ref>) and the definition of A(M) and M̂, one getsf̂^(M̂) - f^* _2 ≤(A(M) + σ(M̂)) + (A(M̂) + σ(M)) + σ(M) +f^*, (M) - f^* _2≤2 A(M) + 4 σ(M) +f^*, (M) - f^* _2.Then, notice that A(M) can be bounded byA(M) ≤ sup_M'{f̂^(M') - f^*, (M')_2 + f̂^(M ∧ M') - f^*, (M ∧ M')_2 - σ(M') }+ sup_M' f^*, (M') - f^*, (M ∧ M')_2.Since σ is nondecreasing, σ(M ∧ M') ≤σ(M'), so that the first term is upper bounded by zero thanks to equation (<ref>). The second term can be controlled since the orthogonal projection is a contraction. This leads toA(M) ≤ f^* - f^*, (M)_2,which is enough to conclude.The oracle inequality also holds when takingA_k(M) = M' ≥ Msup{f̂^(M')_k - f̂^(M)_k _2 - σ(M') }_+. Note that the selected M̂_k implicitely depends on the probability of error ϵ through the penalty σ.In the asymptotic setting, we take ϵ as a function of n, so that M̂_k is a function of n only. This will be used to get rid of ϵ when proving that the estimators reach the minimax rates of convergence. § PLUG-IN ESTIMATORS AND THEORETICAL GUARANTEESIn this section, we introduce two methods to construct families of estimators of the emission densities. We show that they satisfy assumption [H(ϵ)] for a given variance bound σ.In Section <ref>, we introduce the assumptions we will need for both methods. Section <ref> is dedicated to the spectral estimator and Section <ref> to the least squares estimator.§.§ Framework and assumptionsRecall that we approximate ^2(, μ) by a nested family of finite-dimensional subspaces (_M)_M ∈ such that their union is dense in ^2(, μ) and write f^*,(M)_k the orthogonal projection of f^*_k on _M for all k ∈ and M ∈. We assume that ⊂ and that the space _M has dimension M. A typical way to construct such spaces is to take _M spanned by the first M vectors of an orthonormal basis. Both methods will construct an estimator of the emission densities for each model of this family. These estimators will then be plugged in the state-by-state selection method of Section <ref>, which will select one model for each state of the HMM.We will need the following assumptions.[HX] (X_j)_j ≥ 1 is a stationary ergodic Markov chain with parameters (π^*, ^*); [Hid] ^* is invertible and the family ^* is linearly independent.The ergodicity assumption in [HX] is standard in order to obtain convergence results. In this case, the initial distribution is forgotten exponentially fast, so that the HMM will essentially behave like a stationary process after a short period of time. For the sake of simplicity, we assume the Markov chain to be stationary.[Hid] appears in identifiability results, see for instance <cit.> and Theorem <ref>. It is sufficient to ensure identifiability of the HMM from the law of three consecutive observations. Note that it is in general not possible to recover the law of a HMM from two observations (see for instance Appendix G of <cit.>), so that three is actually the minimum to obtain general identifiability. §.§ The spectral method Algorithm <ref> is a variant of the spectral algorithm introduced in <cit.>. Unlike the original one, it is able to reach the minimax rate of convergence thanks to two improvements. The first one consists in decomposing the joint density on different models, hence the use of two dimensions m and M. The second one consists in trying several randomized joint diagonalizations instead of just one, and selecting the best one, hence the parameter r. These additional parameters do not actually add much to the complexity of the algorithm: in theory, the choice m, r ≈log(n) is fine (see Corollary <ref>), and in practice, any large enough constant works, see Section <ref> for more details.For all M ∈, let (φ_1^M, …, φ_M^M) be an orthonormal basis of _M. Letη_3(m,M)^2 := sup_y, y' ∈^3∑_a,c = 1^m ∑_b=1^M ( φ_a^m(y_1) φ_b^M(y_2) φ_c^m(y_3) - φ_a^m(y'_1) φ_b^M(y'_2) φ_c^m(y'_3) )^2. The following theorem follows the proof of Theorem 3.1 of <cit.>, with modifications that allow to control the error of the spectral estimators in expectation and are essential to obtain the right rates of convergence in Corollary <ref>. Assume [HX] and [Hid] hold. Then there exists a constant M_0 depending on ^* and constants C_σ and n_1 depending on ^* and ^* such that for all ϵ∈ (0,1), for all m,M ∈ such that M ≥ m ≥ M_0 and for all n ≥ n_1 η_3^2(m,M) (-log ϵ)^2, with probability greater than 1 - 6 ϵ,inf_τ∈(K)max_k ∈f̂^(M, ⌈ t ⌉)_k - f^*, (M)_τ(k)_2^2 ≤ C_ση_3^2(m,M) (-log ϵ)^2/nProof in Section <ref>.Note that the constants n_1 and C_σ depend on ^* and ^*. This dependency will not affect the rates of convergence of the estimators (with respect to the sample size n), but it can change the constants of the bounds and the minimum sample size needed to reach the asymptotic regime. Let us now apply the state-by-state selection method to these estimators. The following corollary shows that it is possible to reach the minimax rate of convergence up to a logarithmic term separately for each state under standard assumptions. Note that we need to bound the resulting estimators by some power of n, but this assumption is not very restrictive since α can be arbitrarily large.Assume [HX] and [Hid] hold. Also assume that η_3^2(m, M) ≤ C_η m^2 M for a constant C_η > 0 and that for all k ∈, there exists s_k such that f^*,(M)_k - f^*_k_2 = O(M^-s_k). Then there exists a constant C_σ depending on ^* and ^* such that the following holds.Let α > 0 and C ≥ 2 (1+2α) √(C_η C_σ). Let ^sbs be the estimators selected from the family (^(M, ⌈(1+2α)log(n)⌉))_M ≤ M_max(n) with M_max(n) = n / log(n)^5, m_M = log(n) and σ(M) = C √(M log(n)^4/n) for all M. Then there exists a sequence of random permutations (τ_n)_n ≥ 1 such that∀ k ∈, [ (-n^α) ∨ (f̂^sbs_τ_n(k)∧ n^α) - f^*_k _2^2 ] = O( ( n/log(n)^4)^-2s_k/2s_k+1).The novelty of this result is that each emission density is estimated with its own rate of convergence: the rate -s_k/2s_k+1 is different for each emission density, even though the original spectral estimators did not handle them separately. This is due to our state-by-state selection method. Moreover, it is able to reach the minimax rate for each density in an adaptive way. For instance, in the case of a β-Hölder density on = [0,1]^D (equipped with a trigonometric basis), one can easily check the control of η_3, and the control f^*,(M)_k - f^*_k_2 = O(M^- β / D) follows from standard approximation results, see for instance <cit.>. Thus, our estimators converge with the rate (n/log(n)^4)^-2 β / (2 β + D) to this density: this is the minimax rate up to a logarithmic factor. By aligning the estimators like in Section <ref>, one can replace the sequence of permutations in Corollary <ref> by a single permutation, in other words there exists a random permutation τ which does not depend on n such that∀ k ∈, [ (-n^α) ∨ (f̂^sbs_τ(k)∧ n^α) - f^*_k _2^2 ] = O( ( n/log(n)^4)^-2s_k/2s_k+1).This means that the sequence (f̂^sbs_k)_n ≥ 1 is an adaptive rate-minimax estimator of f^*_k–or more precisely of one of the emission densities (f^*_k')_k' ∈, but since the distribution of the HMM is invariant under relabelling of the hidden states, one can assume the limit to be f^*_k without loss of generality–up to a logarithmic term.At this point, it is important to note that the choice of the constant C ≥ 2(1+2 α) √(C_η C_σ) depends on the hidden parameters of the HMM and as such is unknown. This penalty calibration problem is very common in the model selection framework and can be solved in practice using methods such as the slope heuristics or the dimension jump method which have been proved to be theoretically valid in several cases, see for instance <cit.> and references therein. We use the dimension jump method and explain its principle and implementation in Section <ref>. Using Theorem <ref>, one gets that for all n and for all M ∈ such that n ≥ n_1 η_3^2(m_M,M) (1+2α)^2 log(n)^2, with probability 1 - 6n^-1-2α,inf_τ∈(K)max_k ∈f̂^(M, ⌈ t ⌉)_k - f^*, (M)_τ(k)_2^2 ≤C_ση_3^2(m_M,M) (1+2α)^2 log(n)^2/n ≤(1+α)^2 C_σ C_η M log(n)^4/n ≤ σ(M)^2/4where σ(M) = C √(Mlog(n)^4/n) with C such that C^2 ≥ 4 (1+2α)^2 C_σ C_η.The condition on M becomesn ≥ n_1 log(n)^4 M (1+2α)^2and is asymptotically true for all M ≤ M_max(n) as soon as M_max(n) = o(n / log(n)^4).Thus, [H(6n^-(1+2α))] is true for the family (^(M, ⌈(1+2α)log(n)⌉))_M ≤ M_max(n). Note that the assumption M_max(n) = o(n / log(n)^4) also implies that there exists M_1 such that for n large enough, Lemma <ref> holds for all M ≥ M_1, so that Theorem <ref> implies that for n large enough, there exists a permutation τ_n such that with probability 1 - 6n^-(1+2α), for all k ∈,f̂^sbs_τ_n(k) - f^*_k _2 ≤ 4 inf_M_1 ≤ M ≤ M_max{ f^*,(M)_k - f^*_k _2 + σ(M) } = O ( inf_M_1 ≤ M ≤ M_max{ M^-s_k + √(M log(n)^4/n)}) = O ( ( n/log(n)^4)^-s_k/(1+2s_k)),where the tradeoff is reached for M = (n/log(n)^4)^1/(1+2s_k), which is in [M_1, M_max(n)] for n large enough.Finally, write A the event of probability smaller than 6n^-(1+2α) where [H(6n^-(1+α))] doesn't hold, then for n large enough and for all k ∈,[ (-n^α) ∨ (f̂^sbs_τ_n(k)∧ n^α) - f^*_k _2^2 ] ≤[ _A f̂^sbs_τ_n(k) - f^*_k _2^2 ] + [ _A^c ( n^2α +f^*_k _2^2 ) ] = O ( ( n/log(n)^4)^-2s_k/(1+2s_k)) + O ( n^2α +f^*_k _2^2/n^1+2α) = O ( ( n/log(n)^4)^-2s_k/(1+2s_k)). §.§ The penalized least squares method Letbe a subset of ^2(, μ). We will need the following assumption onin order to control the deviations of the estimators: [HF] ^* ∈^K^*,is closed under projection on _M for all M ∈ and∀ f ∈,f _∞≤f _2 ≤withandlarger than 1.A simple way to construct such a setwhen μ is a finite measure is to take the sets (_M)_M spanned by the first M vectors of an orthonormal basis (φ_i)_i ≥ 0 whose first vector φ_0 is proportional to 1. Then any setof densities such that ∫ f dμ = 1, ∑_i ⟨ f, φ_i ⟩^2 ≤ and ∑_i | ⟨ f, φ_i ⟩ | φ_i _∞≤ for given constantsandand for all f ∈ satisfies [HF]. When ∈^K × K, π∈^K and ∈ (^2(, μ))^K, letg^π, , (y_1, y_2, y_3) = ∑_k_1, k_2, k_3 = 1^K π(k_1) (k_1, k_2) (k_2, k_3) f_k_1(y_1) f_k_2(y_2) f_k_3(y_3).When π is a probability distribution,a transition matrix anda K-uple of probability densities, then g^π, , is the density of the first three observations of a HMM with parameters (π, , ). The motivation behind estimating g^π, , is that it allows to recover the true parameters under the identifiability assumption [Hid], as shown in the following theorem. Letbe the set of transition matrices onand Δ the set of probability distributions on . For a permutation τ∈(K), write _τ its matrix (that is the matrix defined by _τ(i,j) = _{j = τ(i)}). Finally, define the distance on the HMM parameters((π_1, _1, _1), (π_2, _2, _2))^2 = inf_τ∈(K){π_1 - _τπ_2 _2^2 + _1 - _τ_2 _τ^⊤_F^2 + ∑_k ∈ f_1,k - f_2,τ(k)_2^2 }.This distance is invariant under permutation of the hidden states. This corresponds to the fact that a HMM is only identifiable up to relabelling of its hidden states.Let (π^*, ^*, ^*) ∈Δ×× (^2(, μ))^K such that π^*_x > 0 for all x ∈ and [Hid] holds. Then for all (π, , ) ∈Δ×× (^2(, μ))^K,(g^π, ,= g^π^*, ^*, ^*)⇒ ((π, , ),(π^*, ^*, ^*)) = 0.The spectral algorithm of <cit.> applied on the finite dimensional space spanned by the components ofand ^* allows to recover all the parameters even when the emission densities are not probability densities and when the Markov chain is not stationary.Define the empirical contrastγ_n(t) =t _2^2 - 2/n∑_j=1^n t(Z_j)where Z_j := (Y_j, Y_j+1, Y_j+2) and (Y_j)_1 ≤ j ≤ n+2 are the observations. It is a biased estimator of the ^2 loss: for all t ∈ (^2(, μ))^3,[γ_n(t)] =t - g^* _2^2 -g^* _2^2where g^* = g^π^*, ^*, ^*. Since the bias does not depend on the function t, one can hope that the minimizers of γ_n are close to minimizers of t - g^*_2. We will show that this is indeed the case.The least squares estimators of all HMM parameters are defined for each model _M by(π̂^(M), ^(M), ^(M)) ∈π∈Δ,∈,∈ (_M ∩)^K γ_n(g^π, , ).The procedure is summarized in Algorithm <ref>. Note that with the notations of the algorithm,γ_n(g^π, , ^⊤Φ) = _(π, , ) - _M _F^2 - _M _F^2.Then, the proof of the oracle inequality of <cit.> allows to get the following result. Assume [HF], [HX] and [Hid] hold.Then there exists constants C and n_0 depending on ,and ^* such that for all n ≥ n_0, for all t > 0, with probability greater than 1 - e^-t, one has for all M ∈ such that M ≤ n:ĝ^π̂^(M), ^(M), ^(M) - g^π^*, ^*, ^*, (M)_2^2 ≤ C ( t/n + M log(n)/n).In order to deduce a control of the error on the parameters–and in particular on the emission densities–from the previous result, we will need to assume that the quadratic form derived from the second-order expansion of (π, , ) ∈Δ××^K ⟼ g^π, ,- g^* _2^2 around (π^*, ^*, ^*) is nondegenerate. It is still unknown whether this nondegeneracy property is true for all parameters (π^*, ^*, ^*) such that [Hid] and [HX] hold. <cit.> prove it for K = 2 hidden states when only the emission densities are allowed to vary by using brute-force computations. To do so, they introduce an (explicit) polynomial in the coefficients of π^*, ^* and of the Gram matrix of ^* and prove that its value is nonzero if and only if the quadratic form is nondegenerate for the corresponding parameters. The difficult part of the proof is to show that this polynomial is always nonzero.For the expression of this polynomial–which we will write H–in our setting, we refer to Section <ref>. Note that <cit.> proves that this polynomial H is non identically zero: it is shown that there exists parameters (π, , ) satisfying [HX] and [Hid] such that H(π, , ) ≠ 0, which means that the following assumption is generically satisfied: [Hdet] H(π^*, ^*, ^*) ≠ 0. The following result allows to lower bound the ^2 error on the density of three consecutive observations by the error on the parameters of the HMM using this condition. It is an improvement of Theorem 6 of <cit.> and Theorem 9 of <cit.>. The main difference is that the constant c^*(π^*, ^*, ^*, ) does not depend on thearound which the parameters are taken . This is crucial to obtain Corollary <ref>, from which we will deduce [H0]. Note that we do not needto be in a compact neighborhood of ^*. Another improvement is that the constant in the minoration only depends on the true parameters and on the set . * Assume that [HF] holds and that for all f ∈, ∫ f dμ = 1.Then there exist a lower semicontinuous function (π^*, ^*, ^*) ⟼ c^*(π^*, ^*, ^*, ) that is positive when [Hid] and [Hdet] hold and a neighborhoodof ^* in ^K depending only on π^*, ^*, ^* andsuch that for all ∈ and for all π∈Δ, ∈ and ∈^K,g^π, ,- g^π^*, ^*, _2^2 ≥ c^*(π^*, ^*, ^*, ) ((π, , ),(π^*, ^*, ))^2.* There exists a continuous function ϵ : (π^*, ^*, ^*) ↦ϵ(π^*, ^*, ^*) that is positive when [Hid] and [Hdet] hold and such that for all π∈Δ, ∈ and ∈ (^2(, μ))^K a K-uple of probability densities such that ((π, , ),(π^*, ^*, ^*)) ≤ϵ(π^*, ^*, ^*), one hasg^π, ,- g^π^*, ^*, ^*_2^2 ≥ c_0(π^*, ^*, ^*) ((π, , ),(π^*, ^*, ^*))^2.wherec_0(π^*, ^*, ^*) = (inf_k ∈π^*(k) ) σ_K(^*)^4 σ_K(G(^*))^2/4 ∧H(π^*, ^*, ^*)/2 (1 ∧ KG(^*) _∞) (3 K^3 (1 ∨ G(^*) _∞^4))^K^2-K/2.Proof in Section <ref>.Assume [HX], [HF], [Hid] and [Hdet] hold. Also assume that for all f ∈, ∫ f dμ = 1.Then there exists a constant n_0 depending on ,and ^* and constants M_0 and C' depending on , ^* and ^* such that for all n ≥ n_0 and t > 0, with probability greater than 1 - e^-t, one has for all M ∈ such that M_0 ≤ M ≤ n:inf_τ∈(K)max_k ∈f̂^(M)_k - f^*, (M)_τ(k)_2^2 ≤ C' ( M log(n)/n + t/n). Using the second point of Theorem <ref>, one can alternatively take n_0 and M_0 depending on , ^* and ^*, and C' depending on , , ^* and ^* only. For instance, one can take C' =C / c_0(π^*, ^*, ^*) with the notations of Theorems <ref> and <ref>.In particular, this means that the asymptotic variance bound of the least squares estimators (and therefore the rate of convergence of the estimators selected by our state-by-state selection method) does not depend on the set , but only on the HMM parameters and on the boundsandon the square and supremum norms of the emission densities. Note that this universality result is essentially an asymptotic one since it requires n_0 to depend onin a non-explicit way. Letbe the neighborhood given by Theorem <ref>, then there exists M_0 such that for all M ≥ M_0, ^*, (M)∈. Then Theorem <ref> and Theorem <ref> applied to π = π̂^(M), = ^(M), = ^(M) and = ^*, (M) for all M allow to conclude.We may now state the following result which shows that the state-by-state selection method applied to these estimators reaches the minimax rate of convergence (up to a logarithmic factor) in an adaptive manner under generic assumptions. Its proof is the same as the one of Corollary <ref>.Assume [HX], [HF], [Hid] and [Hdet] hold. Also assume that for all f ∈, ∫ f dμ = 1 and that for all k, there exists s_k such that f^*,(M)_k - f^*_k_2 = O(M^-s_k). Then there exists a constant C_σ depending on , , ^* and ^* such that the following holds.Let C ≥ C_σ and let ^sbs be the estimators selected from the family (^(M))_M ≤ n with σ(M) = C √(M log(n)/n) for all M, aligned like in Remark <ref>. Then there exists a random permutation τ which does not depend on n such that∀ k ∈, [ f̂^sbs_τ(k) - f^*_k _2 ] = O( ( n/log(n))^-s_k/2s_k+1).§ NUMERICAL EXPERIMENTSThis section is dedicated to the discussion of the practical implementation of our method. We run the spectral estimators on simulated data for different number of observations and study the rate of convergence of the selected estimators for several variants of our method. Finally, we discuss the algorithmic complexity of the different estimators and selection methods.In Section <ref>, we introduce the parameters with which we generate the observations. In Section <ref>, we discuss how to calibrate the constant of the penalty in practice. In Section <ref>, we introduce two other ways to select the final estimators, the POS and MAX variants. Section <ref> contains the results of the simulations for each variant and calibration method. In Section <ref>, we present a cross validation procedure and compare its results with the one obtained using our method. Finally, we discuss the algorithmic complexity of the different algorithms and estimators in Section <ref>. §.§ Setting and parametersWe take = [0,1] equipped with the Lebesgue measure. We choose the approximation spaces spanned by a trigonometric basis: _M := (φ_1, …, φ_M) withφ_1(x)= 1φ_2m(x)= √(2)cos(2π m x)φ_2m+1(x)= √(2)sin(2π m x)for all x ∈ [0,1] and m ∈^*. We will consider a hidden Markov model with K=3 hidden states and the following parameters:* Transition matrix^*= ([0.70.10.2; 0.080.8 0.12; 0.15 0.150.7 ]) ; * Emission densities (see Figure <ref>) * Uniform distribution on [0;1];* Symmetrized Beta distribution, that is a mixture with the same weight of 2/3 X and 1 - 1/3 X' with X,X' i.i.d. following a Beta distribution with parameters (3,1.6);* Beta distribution with parameters (3,7).We generate n observations and run the spectral algorithm in order to obtain estimators for the models _M with ≤ M ≤, m = 20 and r = ⌈ 2log(n) + 2log(M) ⌉, where = 3 and = 300. Finally, we use the state-by-state selection method to choose the final estimator for each emission density. The main reason for using spectral estimators instead of maximum likelihood estimation or least squares estimation is its computational speed: it is much faster for large n than the least squares algorithm or the EM algorithm, which makes studying asymptotic behaviours possible. We made 300 simulations, 20 per value of n, with n taking values in { 5 × 10^4, 7 × 10^4, 1 × 10^5, 1.5 × 10^5, 2.2 × 10^5, 3.5 × 10^5, 5 × 10^5, 7 × 10^5, 1 × 10^6, 1.5 × 10^6, 2.2 × 10^6, 3.5 × 10^6, 5 × 10^6, 7 × 10^6, 1 × 10^7 }. §.§ Penalty calibration It is important to note that when considering spectral and least squares methods, the penalty σ in the state-by-state selection procedure depends on the hidden parameters of the HMM and as such is unknown in practice. This penalty calibration problem is well known and several procedures exist that allow to solve it, for instance the slope heuristics and the dimension jump method (see for instance <cit.> and references therein). In the following, we will use the dimension jump method to calibrate the penalty in the state-by-state selection procedure. Consider a penalty shape _shape and define M̂_k(ρ) the model selected for the hidden state k by the state-by-state selection estimator using the penalty ρ _shape:M̂_k(ρ) ∈M ∈{ A_k(M) + 2 ρ _shape(M) }.whereA_k(M) = M' ∈sup{f̂^(M')_k - f̂^(M ∧ M')_k _2 - ρ _shape(M') }. The dimension jump method relies on the heuristics that there exists a constant C such that C_shape is a minimal penalty. This means that for all ρ < C, the selected models M̂_k(ρ) will be very large, while for ρ > C, the models will remain small. This translates into a sharp jump located around a value ρ_jump,k = C in the plot of ρ⟼M̂_k(ρ). The final step consists in taking twice this value to calibrate the constant of the penalty, thus selecting the model M̂(2 ρ_jump,k). In practice, we take ρ_jump,k as the position of the largest jump of the function ρ⟼M̂_k(ρ). Figure <ref> shows the resulting dimension jumps for n = 220,000 observations. Each curve corresponds to one of the M̂_k(ρ) and has a clear dimension jump, which confirms the relevance of the heuristics. Several methods may be used to calibrate the constant of the penalty: eachjump. Calibrate the constant independently for each state. This method has the advantage of being easy to calibrate since there is usually a single sharp jump in each state's complexity. However, our theoretical results do not suggest that the penalty constant is different for each state; jumpmax. Calibrate the constant for all states together using only the latest jump. This consists in taking the maximum of the ρ_jump,k to select the final models. Since the penalty is known up to a multiplicative constant and taking a constant larger than needed does not affect the rates of convergence–contrary to smaller constants–this is the “safe” option; jumpmean. Calibrate the constant for all states together using the mean of the positions of the different jumps.We try and compare these calibration methods in Section <ref>. §.§ Alternative selection procedures §.§.§ Variant POS. As mentionned in Section <ref>, it is also possible to select the estimators using the criterionA_k(M) = M' ≥ Msup{f̂^(M')_k - f̂^(M)_k _2 - σ(M') }_+followed byM̂_k ∈M ∈{ A_k(M) + 2 σ(M) }.This positivity condition was in the original Goldenshluger-Lepski method. The theoretical guarantees remain the same as the previous method and both behave almost identically in practice, as shown in Section <ref>.§.§.§ Variant MAX. In the context of kernel density estimation, <cit.> show that the Goldenshluger-Lepski method still works when the biais estimate A_k(M) of the model M is replaced by the distance between the estimator of the model M and the estimator with the smallest bandwidth (the analog of the largest model in our setting). They also prove an oracle inequality for this method after adding a corrective term to the penalty.The following variant is based on the same idea. It consists in selecting the modelM̂_k ∈M ∈{f̂^(M_max)_k - f̂^(M)_k _2 + σ(M) }for each k ∈ and takesf̂_k = f̂^(M̂_k)_k,where σ is the same penalty as the one in the usual state-by-state selection method.An advantage of this algorithm is its lower complexity, since it requires O(M_max) computations of ^2 norms instead of O(M_max^2). We do not study this method theoretically in our setting. However, the simulations (and in particular Figure <ref>) show that it behaves similarly to the standard state-by-state selection method in the asymptotic regime and even has a smaller error for small number of observations. In addition, the dimension jumps are much sharper for this method than for the usual state-by-state selection method (see Figure <ref>), which makes the calibration heuristics easier to use. §.§ ResultsFigure <ref> shows the evolution of the error f̂_k - f^*_k _2 for each state k with respect to the number of observations n, for all penalty calibration methods and all variants of the model selection procedure. Figure <ref> compares the evolution of the median error forthe different calibration methods and for the different selection variants, and Figure <ref> compares two estimators with the oracle estimators.When the number of observations n is large enough, the logarithm of the error decreases linearly with respect to log(n). This corresponds to the asymptotic convergence regime: the error is expected to decrease as a power of the number of observations n when n tends to infinity. The corresponding slopes are listed in Table <ref>.For each state, the confidence intervals of the rates of all estimators–including the oracle estimators–have a common intersection (except for the symmetrized Beta distribution in the jumpmax MAX variant, whose estimators seem to converge faster than the others). This tends to confirm that the calibration and selection variants are asymptotically equivalent. This phenomenon is also visible in Figures <ref> and <ref>: in the asymptotic regime, the errors decrease in a similar way for all methods.Furthermore, the rates of convergence are clearly distinct. The uniform distribution is estimated with a rate of convergence of approximately n^-1/2, which is also the best possible rate (it corresponds to a parametric estimation rate). In comparison, the rate of convergence for the symmetrized Beta distribution is much slower (around n^-0.36). This shows that the algorithm effectively adapts to the regularity of each state and that one irregular emission density does not deteriorate the rates of convergence of the other densities. Note that the above rates are in accordance with the minimax rates as far as the Hölder regularity is concerned. The minimax Hölder rate for the symmetrized Beta (which is 0.6-Hölder) is n^-3/11, or approximately n^-0.27, which means our estimator converges faster than the minimax rate would suggest. The minimax Hölder rate for the Beta distribution (which is 3-Hölder) is n^-3/7, or approximately n^-0.43, which is around the observed value. §.§ Comparison with cross validation In this section, we use a cross validation procedure based on our spectral estimators to check whether our method actually improves estimation accuracy.œ When estimating a density by taking an estimator within some class (the model), two sources of error appear: the bias, that is the (deterministic) distance between the true density and the model, and the variance, that is the (random) error of the estimation within the model. Small models will have a large bias but a small variance, while large models will have a small bias and a large variance. The core issue of model selection is to select a model that minimizes the total error, that is large enough to accurately describe the true densities and small enough to prevent overfitting: in other words, perform a bias-variance tradeoff. Cross validation seeks to achieve such a tradeoff by computing an estimate of the total error. This is done by splitting the sample into two sets, the training sample being used for the calibration of the estimator and the validation sample for measuring the error. Taking the mean of these errors for different splits between training and validation samples provides an estimator of the total error. This method has become popular for its simplicity of use. We refer to the survey of <cit.> for an overview on this method and its guarantees.§.§.§ RiskWe use the least squares criterion of Algorithm <ref> to quantify the error of the estimators. Since the guarantees on spectral estimators rely on the ^2 norm, a least squares criterion is more natural than the likelihood. In addition, the spectral estimators might take negative values depending on the orthonormal basis, which is not a problem as far as ^2 error is concerned but can be an issue for the likelihood. Let us first recall this criterion. Given an orthonormal basis (φ_i)_i ∈ of ^2(, μ), define the coordinate tensor of the empirical distribution of the triplet (Y_1, Y_2, Y_3) on this basis by(a,b,c):= 1/n∑_s=1^nφ_a(Y_s)φ_b(Y_s+1)φ_c(Y_s+2)for all a, b, c ∈. Given a transition matrixof size K, a stationary distribution π ofand a vector of densities = (f_1, …, f_K), define the coordinate matrixofby (b,k) = ⟨φ_b, f_k ⟩. Let _(π, , ) be the coordinate tensor of the distribution of (Y_1, Y_2, Y_3) under the parameters (π, , ), that is_(π, , )(·, b, ·) = [π] [(b,·)] ^⊤ for allb ∈. The empirical least squares criterion is _(π, , ) - _F^2. It corresponds to the ^2 error between the empirical distribution of three consecutive observations and the theoretical distribution under the parameters (π, , ). §.§.§ ImplementationWe use 10-fold cross validation, that is we split the sequence into 10 segments of same size I_1, …, I_10. In order to avoid interferences between samples, we prune the ends of each segment, so that the observations in each segment can be considered independent. In practice, we take a gap of 30 observations between two segments. We ran 150 simulations, 10 per value of n, with the same parameters as in Section <ref>. Each simulation is as follows.For each segment I_j, we run the spectral algorithm on all models _M for ≤ M ≤ using only the observations from the other segments. The transition matrix is estimated using an additional step of the spectral method which is adapted from Steps 8 and 9 of Algorithm 1 of <cit.>. Then, we compute the least squares criterion for the estimated parameters using the segment I_j as observed sample. Finally, for each M, we average this error on all segments I_j, which gives the least squares cross validation error E_VC(M).This cross validation criterion is used to select one model M̂_VC∈_M E_VC(M), from which we construct the final estimators of the emission densities f̂_k = f̂_k^(M̂_VC) for all k. Note that the selected model is the same for all emission densities.§.§.§ ResultsFigure <ref> compares the selected model dimensions for each n using our state-by-state selection method and using the cross validation method. When the number of observations n becomes larger than 10^6, the cross validation tends to always pick the largest model, which means that it does not prevent overfitting as well as our method. The ^2 errors on the emission densities are shown in Figure <ref>. It appears that the cross validation has a lower error for small n (n ≤ 350,000) than our method. However, for larger values of n, the errors becomes larger than the ones of our method (see Figure <ref>) by up to one order of magnitude, and only start decreasing once the selected model is set to the maximum dimension. Finally, the estimated rates of convergence are shown in Table <ref>. Our state-by-state method outperforms the cross validation method for all emission densities. The cross validation estimators only reach the minimax rate of convergence for the less regular density: the symmetrized Beta, and even then they converge slower than the state-by-state estimator. All other emission densities are estimated slower than their minimax rate. §.§ Algorithmic complexityIn the following, we treat K as a constant as far as the algorithmic complexity is concerned. The different complexities are summarized in Table <ref>. §.§.§ Spectral algorithm (see Section <ref>) We consider the algorithmic complexity of estimating the emission densities for all models M such that M_min≤ M ≤ M_max with n observations and auxiliary parameters r and m depending on n and M (upper bounded by m_max and r_max).Step 1 can be computed for all models with O(n M_max m_max^2) operations. It is the only step whose complexity depends on n. Steps 2 and 3 require O(m^3 M) operations for each model and Steps 4 to 7 require O(M r) operations for each model, for a total of O(n M_max m_max^2 + M_max^2 m_max^3 + M_max^2 r_max) operations.In practice, one takes m ∝log(n), r ∝log(n) + log(M) and M_max≤ n, so that the total complexity of the spectral algorithm is O(n log(n)^2 M_max). In comparison, the complexity of the spectral algorithm of <cit.> is O(n M_max^3) because of Step 1. This becomes much larger than our complexity when M_max grows as a power of n (which is necessary in order to reach minimax rates).§.§.§ Least squares algorithm (see Section <ref>) We consider the algorithmic complexity of estimating the emission densities for all models M such that M_min≤ M ≤ M_max with n observations.Step 1 is similar to the one of the spectral algorithm, but with O(n M_max^3) operations. The complexity of Step 2 is more difficult to evaluate. Since the criterion is nonconvex, finding the minimizer requires to run an approximate minimization algorithm whose complexity C_n will depend on the desired precision–which will in turn depend on the number of observations n–and on the initial points. As discussed in <cit.>, this is usually the longest step when computing least squares estimators. Thus, the total complexity of the least squares algorithm is O(n M_max^3 + C_n).Note that despite the worse sample complexity, the least squares algorithm is tractable and can greatly improve the estimation for small sample size. As shown in Section <ref>, the spectral algorithm is unstable for small samples, which makes the state-by-state selection procedure return abnormal results. This can be explained by the matrix inversions of the spectral method, which sometimes lead to nearly singular matrices when the noise is too large. On the other hand, the least squares method does not involve any matrix inversion, and often gives better results than the spectral estimators, as shown in <cit.>, thus making it a relevant choice for small to medium data sets. §.§.§ Selection method and POS variant (see Sections <ref> and <ref>) We consider the algorithmic complexity of selecting estimators from a family (^(M))_M_min≤ M ≤ M_max of estimators. The selection algorithms can be decomposed in two parts. * Compute the distances f̂_k^(M) - f̂_k^(M')_2 for all M, M' and k. This has complexity O(M_max^3): it requires to compute the ^2 distance of at most M_max^2 couples of functions in a Hilbert space of dimension M_max. * Compute ρ̂_k defined as the abscissa of the largest jump of the function ρ⟼M̂_k(ρ) for all k, where M̂_k is defined as in Section <ref>. Note that computing M̂_k(ρ) requires O(M_max^2) operations. An approximate value of ρ̂_k can be computed in O(log(ρ̂_k) M_max^2) operations, which is usually O(M_max^2).Once the ρ̂_k are known, it is possible to calibrate the penalty in constant time for the three calibrations methods (eachjump, jumpmax and jumpmean) and to select the final models in O(M_max^2) operations.Thus, the total complexity of the selection algorithm and of its POS variant is O(M_max^3).§.§.§ Selection method, MAX variant (see Section <ref>) In the MAX variant, the first step of the standard selection procedure is replaced by computing the distances f̂^(M_max)_k - f̂^(M)_k _2 for all M. This has complexity O(M_max^2). The complexity of the other steps remains unchanged.Thus, the total complexity of the MAX variant of the selection algorithm is O(M_max^2). § APPLICATION TO REAL DATA In this section, we present the results of our method on two sets of trajectories. Trajectories are a typical example of dependent data that shows several behaviours depending on the activity of the entity being tracked, which makes hidden Markov models a popular modelling choice. For instance, the movement of a fisher is not the same depending on whether he's travelling to the next fishing zone or actually fishing. The first data set follows artisanal fishers in Madagascar. The second one contains seabird movements. Studying the movements of fishers and seabirds has many applications, for instance understanding the fishing habits of the tracked entity, controlling the fishing pressure on local ecosystems and monitoring the dynamics of coastal ecosystems, see for instance <cit.> and references therein. §.§ Artisanal fisheryWe use GPS tracks of artisanal fishers with a regular sampling period of 30 seconds. These tracks were produced by Faustinato Behivoke (Institut Halieutiques et des Sciences Marines, Université de Toliara, Madagascar) and Marc Léopold (IRD), who recorded artisanal fishers from Ankilibe, in Madagascar. Their fishing method is a seine netting. From this data, we compute the velocity of the fisher during each time step. In order to estimate densities on [0,1], we divide this velocity by an upper bound of the maximum observed velocity. We consider the observation space = [0,1] endowed with the dominating measure δ_0 + Leb, where δ_0 is the dirac measure in zero and Leb is the Lebesgue measure on [0,1]. As a proof of concept, we use the orthonormal basis consisting of the trigonometric basis on [0,1] and the indicator function of {0}, that is the family (φ_m)_m ∈ defined on [0,1] byifx = 0, φ_0(x) = 1φ_m(x) = 0 for allm ∈^* ifx ≠ 0, φ_0(x) = 0φ_1(x) = 1φ_2m(x) = √(2)cos(2π mx) for allm ∈^*φ_2m+1(x) = √(2)sin(2π mx) for allm ∈^* The number of hidden states is chosen using the spectral thresholding method of <cit.>. This methods consists is based on the fact that the rank of the spectral tensor _m,m (with the notations of Algorithm <ref> in Appendix <ref>) is the number of hidden states. This is visible in the spectrum of _m,m by an elbow, as shown in Figure <ref>. Based on these spectra, we use two hidden states.The results using = 1000 are shown in Figures <ref> and <ref>. We took the normalizing velocity large enough that all observed normalized velocities belong to [0,0.8], hence the plot betweeen 0 and 0.8 for the densities. In both cases, the selected model complexities differ greatly depending on the state. This comes from the fact that in both cases, one of the density is spiked, thus requiring more vectors of the orthonormal basis to be approximated. This illustrates that our method is able to estimate the smoother densities with fewer vectors of the basis, thus preventing overfitting. As a side note, we needed considerably less observations than in the simulations: around 10,000, compared to 500,000 in the simulations. This can be explained by the fact that each state is very stable, with an estimated probability of leaving the states below 0.02–compared to 0.3 in the simulations. This is encouraging, as hidden states in real data are expected to be rather stable, especially when the sampling frequency is high, as long as the conditional independance of the observations can be assumed to hold. §.§ Seabird foragingIn this Section, we consider the seabird data from <cit.> and we focus on the tracks named cormorant d in this paper. We apply the same transformation as in the previous section to obtain normalized velocities in [0,0.8] (after removal of anomalous velocities exceeding 150 m/s) and run the spectral algorithm with the trigonometric basis on [0,1] plus the indicator of {0}. The spectral thresholding gives a number of hidden states equal to two; we set it to three to account for more complex behaviours of the seabirds. The results are shown in Figure <ref>.Note that the use of the trigonometric basis allows the estimated densities to take negative values. This is not a problem as far as minimax rates of convergence (in ^2 norm) are concerned, however this can become an issue if one wants to use these densities in a forward-backward algorithm in order to get an estimator of the hidden states. One way to circumvent this problem is to use simplex projection to compute an approximation of the projection of these estimated density on the simplex of all probability densities. Note that since this is an ^2 projection on a convex set which contains the true densities, the projected densities have an even smaller error, thus keeping the minimax rate of convergence of the original estimators. The resulting densities are shown in Figure <ref>The number of observations in this setting is even smaller than for the fishery's data set: our algorithm was able to recover three emission densities from less than 3,000 observations, despite the states being less stable than in the fishery data set: the diagonal terms of the estimated transition matrix using the EM algorithm are (0.83, 0.93, 0.98). In addition, the result of our method is consistent with other estimation methods, as shown in Figure <ref>: estimating the parameters with the EM algorithm using piecewise constant densities leads to a very similar result. § CONCLUSION AND PERSPECTIVES We propose a state-by-state selection method to infer the emission densities of a HMM. Using a family of estimators, our method selects one estimator for each hidden state in a way that is adaptive with respect to this state's regularity. This method does not depend on the type of preliminary estimator, as long as a suitable variance bound is available. As such, it may be seen as a plug-in that takes a family of estimators and the corresponding variance bound and outputs the selected estimator. Note that its complexity does not depend on the number of observations used to compute the estimators, which makes it applicable to arbitrarily large data sets. To apply this method, we present two families of estimators: a least squares estimator and a spectral estimator. For both, we prove a bound on their variance and show that this bound allows to recover the minimax rate of convergence separately on each hidden state, up to a logarithmic factor. The variance bounds are similar to a BIC penalty, with an additional logarithmic factor for the spectral estimators. We carry out a numerical study of the method and some variants on simulated data. We use the spectral estimators, which are both fast and don't suffer from initialization issues, unlike the least squares and maximum likelihood estimators. The simulations show that our selection method is very fast compared to the computation of the estimators and that indeed, the final estimators reach the minimax rate of convergence on each state. Then, we compare our method with a cross validation estimator based on a least square risk. This estimator only reaches the minimax rate corresponding to the worst regularity among the emission densities and fails to select models with small dimensions. It is still noteworthy that the cross validation returns relevant results for small sample sizes, whereas our method requires the sample size to be large enough to work properly. An interesting problem would be to investigate whether cross validation or other methods can be combined with our state-by-state selection method to give an algorithm that is both fast, stable for small sample sizes and optimal in the asymptotic setting. Finally, we apply our algorithm to real trajectory data sets. On this data, our method proves that it is able to match the regularity of the underlying emission densities. In addition, it is able to produce sensible results with far fewer observations than in our simulation study.Our state-by-state selection method can be easily applied to multiview mixture models (also named mixture models with repeated measurement, see for instance <cit.> and <cit.>). Let us first describe the model. A multiview mixture model consists of two random variables, a hidden state U and an observation vector := (Y_i)_i ∈ [m] such that conditionally to U, the components Y_i ofare independent with a distribution depending only on U and i. Let us assume that U takes its values in a finite setof size K and that the Y_i have some density f^*_u,i conditionally to U = u with respect to a dominating measure. A question of interest is to estimate the densities f^*_u,i from a sequence of observed (_n)_n ≥ 1.Our state-by-state selection method can be applied directly to such a model as long as estimators with a proper variance bound are available (see assumption [H(ϵ)] in Section <ref>). Indeed, we never use the dependency structure of the model. Regarding the development of preliminary estimators, multiview mixture models appear closely related to hidden Markov models: <cit.> and <cit.> develop spectral methods that work for both multiview mixtures and HMMs at the same time using the same theoretical arguments. Thus, it seems clear that variance bounds such as the ones we developed can also be written for multiview mixture models. I am grateful to Elisabeth Gassiat and Claire Lacour for their precious advice. I thank Augustin Touron for providing me with a R implementation of the spectral algorithm. I would also like to thank Marie-Pierre Etienne and of course Faustinato Behivoke (Institut Halieutiques et des Sciences Marines, Université de Toliara, Madagascar), Marc Léopold (IRD) and Sophie Bertrand (IRD) for letting me work on their data sets.§ SPECTRAL ALGORITHM, FULL VERSION§ PROOFS §.§ Proof of Lemma <ref> Let τ_n,M be the permutation that minimizes τ⟼max_k ∈f̂^(M)_k - f^*,(M)_τ(k)_2. [H(ϵ)] means that with probability 1-ϵ, one has max_k ∈f̂^(M)_k - f^*,(M)_τ(k)_2 ≤σ(M)/2. Let M ∈. Let us show that f̂^(M)_τ_n,M^-1(k') - f̂^(M_0)_τ_n,M_0^-1(k)_2 > f̂^(M)_τ_n,M^-1(k) - f̂^(M_0)_τ_n,M_0^-1(k)_2 for all k, k' ∈ such that k' ≠ k. If this holds, then the definition of τ̂^(M) implies that τ̂^(M) = τ_n,M^-1∘τ_n,M_0. Thus, one has max_k ∈f̂^(M)_k, new - f^*,(M)_τ_n,M_0(k)_2 ≤σ(M)/2, which is exactly Equation (<ref>) with τ_n = τ_n,M_0. Applying the triangular inequality leads tof̂^(M)_τ_n,M^-1(k) - f̂^(M_0)_τ_n,M_0^-1(k)_2 ≤ f̂^(M)_τ_n,M^-1(k) - f^*,(M)_k_2 +f^*,(M)_k - f^*,(M_0)_k_2 +f^*,(M_0)_k - f̂^(M_0)_τ_n,M_0^-1(k)_2≤ σ(M)/2 + B_M,M_0 + σ(M_0)/2andf̂^(M)_τ_n,M^-1(k') - f̂^(M_0)_τ_n,M_0^-1(k)_2 ≥f^*,(M_0)_k' - f^*,(M_0)_k_2 - f̂^(M)_τ_n,M^-1(k') - f^*,(M)_k'_2 -f^*,(M)_k' - f^*,(M_0)_k'_2 -f^*,(M_0)_k - f̂^(M_0)_τ_n,M_0^-1(k)_2≥m(^*, M_0) - σ(M)/2 - B_M,M_0 - σ(M_0)/2.Thus, the result holds as soon as m(^*, M_0) - σ(M)/2 - B_M,M_0 - σ(M_0)/2 > σ(M)/2 + B_M,M_0 + σ(M_0)/2, which is the condition of Lemma <ref>. §.§ Proof of Theorem <ref> The structure of the proof is the same as the one of Theorem 3.1 of <cit.>.The first difference lies in the fact that we consider different models for each component of the tensors _m,M and _m,M,m in Step 1. As a consequence, we use the left and right singular vectors of _m,M instead of just the right singular vectors of _m,m. A careful reading shows that their proof can be adapted straightforwardly to this situation.The second difference consists in generating several independant random unitary matrices in Step 4 and keeping the one that separates the eigenvalues of all _i(k) best. This allows to replace Lemma F.6 of <cit.> by the following one, based on the independence of the unitary matrices:For all x > 0 and r ∈^*,[ ∀ k, k_1 ≠ k_2, |Λ̂_i_0(k,k_1) - Λ̂_i_0(k,k_2)| ≥2 e^-x/r (1 - ϵ__m,M^2)^1/2/√(e) K^5/2(K-1)γ(_M) ] ≥ 1 - e^-xand[ Λ̂_i_0_∞≥1 + √(2)√(x + log(K^2 r))/√(K)_M _2, ∞] ≤ e^-x,The notations ϵ__m,M (or ϵ__M in the original proof), γ(_M) et _M _2, ∞ are introduced in <cit.>.Using this lemma, their proof leads to our result by taking r = x = t. §.§ Definition of the polynomial H§.§.§ Definition We parametrize the application(π, , ) ∈Δ××(^*)^K ⟼g^π, ,- g^π^*, ^*, ^*_2^2in the following way. For p ∈^K-1, q ∈^K × (K-1) and A ∈^K × (K-1), define the extensions * p̅∈^K defined by p̅(k) = p(k) for all k ∈ [K-1] and p̅(K) = - ∑_k ∈ [K-1] p(k);* q̅∈^K × K by q̅(k,K) = - ∑_k' ∈ [K-1] q(k,k');* A̅∈^K × K by A̅(k,K) = - ∑_k' ∈ [K-1] A(k,k').p̅ corresponds to π - π^*, q̅ to - ^* and A to the components of - ^* on ^* (which is a basis as soon as [Hid] holds). The condition on the last component of p̅ and of each line of q̅ and A̅ follows from the fact that p̅ corresponds to the difference of two probability vectors, q̅ corresponds to the difference of two transition matrices and A̅ correspond to the difference of two vectors of probability densities on a basis of probability densities. Then, consider the quadratic form derived from the Taylor expansion of(p,q,A) ∈^K-1×^K × (K-1)×^(K-1) × K⟼g^π^* + p̅, ^* + q̅,+ A̅^* - g^π^*, ^*, ^*_2^2.Let M be the matrix associated to this quadratic form. We define H as the determinant of M. Direct computations show that H is a polynomial in the coefficients of π^*, ^* and G(^*).§.§.§ Link between H and the quadratic form from Equation (<ref>)The goal of this section is to show how H can be used to lower bound the quadratic form from Equation (<ref>) by a positive constant times the distance between (π, , ) and (π^*, ^*, ^*). We will not need the assumptions [Hid], [HF] or [Hdet] unless specified otherwise.Let us start by the relation between the norms of (p,q,A) and (p̅, q̅, A̅).For all (p,q,A) ∈^K-1×^K × (K-1)×^(K-1) × K,p _2^2 ≤p̅_2^2 ≤ Kp _2^2, q _F^2 ≤q̅_F^2 ≤ Kq _F^2 ,A _F^2 ≤A̅_F^2 ≤ KA _F^2.p _2^2 ≤p̅_2^2 is immediate. Then,p̅_2^2 =p _2^2 + (∑_k ∈ [K-1] p(k) )^2≤p _2^2 + (K-1) ∑_k ∈ [K-1] p(k)^2 =Kp _2^2.The proof is the same for q and A.The next lemma will be used to link the norms of A and A.For all A̅∈^K × K and ^* ∈ (^2(, μ))^K,σ_K(G(^*)) A̅_F^2 ≤∑_k ∈ (A̅^*)_k _2^2 ≤ KG(^*) _∞A̅_F^2For the first inequality, we use that for all k ∈,(A̅^*)_k _2^2 = A̅(k,·) G(^*) A̅(k,·)^⊤ ≥ σ_K(G(^*)) A̅(k,·) _2^2and the inequality follows by summing over k.For the second inequality,∑_k ∈ (A̅^*)_k _2^2 = ∑_k ∈ [K]∫ (A̅^*)_k(x)^2 μ(dx) = ∑_k ∈ [K]∫(∑_j ∈ [K]A̅(k,j) f^*_j(x) )^2 μ(dx)≤ ∑_k ∈ [K]∫ K ∑_j ∈ [K]A̅(k,j)^2 (f^*_j)^2(x) μ(dx)≤K (∑_k,j ∈ [K]A̅(k,j)^2) sup_j ∈∫ (f^*_j)^2(x) μ(dx) =K A̅_F^2G(^*) _∞. Finally, we will use the following result of <cit.> (Section B.2) in order to upper bound the spectrum of the matrix M. For all π_1, π_2 ∈Δ, for all _1, _2 ∈ and for all _1, _2 ∈ (^2(, μ))^K,g^π_1, _1, _1 - g^π_2, _2,_2_2 ≤√(3 K ( G(_1) _∞^3 ∨ G(_2) _∞^3)) ((π_1, _1,_1), (π_2, _2,_2)) Together, these results imply that for all (p,q,A),g^π^* + p̅, ^* + q̅, ^* + A̅^* -g^π^*, ^*, ^*_2^2≤3 K ( G(^* + A̅^*) _∞^3 ∨ G(^*) _∞^3) (p̅_2^2 + q̅_F^2 + ∑_k ∈ (A̅)_k _F^2) ≤3 KG(^*) _∞^3 (1 + K^2A _F^2)^3 (Kp _2^2 + Kq _F^2 + K^2G(^*) _∞ A _F^2)so that σ_1(M) ≤√(3 K^3) (1 ∨ G() _∞^2). Since H = ∏_i=1^(K-1)(2K+1)σ_i(M), one hasσ_(K-1)(2K+1)(M) ≥H/(3 K^3 (1 ∨ G() _∞^4))^K^2-K/2.Now, assume that [Hid] holds, so that σ_K(G(^*)) > 0, theng^π^* + p̅, ^* + q̅, ^* + A̅^* - g^π^*, ^*, ^*_2^2 ≥σ_(K-1)(2K+1)(M) ( p _2^2 +q _F^2 +A _F^2)+ o( p _2^2 +q _F^2 +A _F^2) ≥σ_(K-1)(2K+1)(M)/1 ∧ KG(^*) _∞(p̅_2^2 + q̅_F^2 +∑_k ∈ (A̅^*)_k _F^2 )+ o ( 1/1 ∧σ_K(G(^*))( p̅_2^2 + q̅_F^2 + ∑_k ∈ (A̅^*)_k _F^2 ) )and finallyg^π^* + p̅, ^* + q̅, ^* + A̅^* - g^π^*, ^*, ^*_2^2≥ c_2(π^*, ^*, ^*) (p̅_2^2 + q̅_F^2 +∑_k ∈ (A̅^*)_k _F^2 ) + o ( p̅_2^2 + q̅_F^2 + ∑_k ∈ (A̅^*)_k _F^2 )wherec_2(π^*, ^*, ^*) = H/(1 ∧ KG(^*) _∞) (3 K^3 (1 ∨ G(^*) _∞^4))^K^2-K/2is positive as soon as [Hid] and [Hdet] hold. §.§ Proof of Theorem <ref> LetN_(p, q, ) = g^π^* + p, ^* + q,+- g^π^*, ^*, _2^2and(p,q,) _^2 = ((π^* + p, ^* + q,+ ),(π^*, ^*, ))^2.We want to show that there exists a constant c^* > 0 such that there exists a neighborhoodof ^* such that if one writesc_ := inf_p ∈ (Δ - Δ), q ∈ ( - ),∈ ( - )^KN_(p, q, )/ (p,q,) _^2then inf_∈ c_≥ c^*.The proof follows the structure of the proof of Theorem 6 of <cit.>. It consists of three steps: the first one controls the component ofthat is orthogonal to . This makes it possible to restrictto the finite-dimensional space spanned byin the two other parts. The second step controls the case whenis small, so that the behaviour of N_ is given by its quadratic form, and the last step controls the case whereis far from zero. §.§.§ The orthogonal part.Letbe the orthogonal projection ofon (). ThenN_(p, q, ) = N_(p, q, ) + M_(p, q, ,- )whereM_(p, q, , ) = ∑_i_1, j_1, k_1∑_i_2, j_2, k_2 (π^*+p)(i_1) (^*+q)(i_1, j_1) (^*+q)(j_1, k_1) (π^*+p)(i_2) (^*+q)(i_2, j_2) (^*+q)(j_2, k_2)( ⟨ a_i_1, a_i_2⟩⟨ (f+u)_j_1, (f+u)_j_2⟩⟨ (f+u)_k_1, (f+u)_k_2⟩ + ⟨ (f+u)_i_1, (f+u)_i_2⟩⟨ a_j_1, a_j_2⟩⟨ (f+u)_k_1, (f+u)_k_2⟩ + ⟨ (f+u)_i_1, (f+u)_i_2⟩⟨ (f+u)_j_1, (f+u)_j_2⟩⟨ a_k_1, a_k_2⟩ + ⟨ a_i_1, a_i_2⟩⟨ a_j_1, a_j_2⟩⟨ (f+u)_k_1, (f+u)_k_2⟩ + ⟨ a_i_1, a_i_2⟩⟨ (f+u)_j_1, (f+u)_j_2⟩⟨ a_k_1, a_k_2⟩ + ⟨ (f+u)_i_1, (f+u)_i_2⟩⟨ a_j_1, a_j_2⟩⟨ a_k_1, a_k_2⟩ + ⟨ a_i_1, a_i_2⟩⟨ a_j_1, a_j_2⟩⟨ a_k_1, a_k_2⟩).Let us write Π' the matrix whose diagonal terms are the elements of π^* + p and ' the matrix ^*+q, then M_ can be written asM_(p, q, , ) = ∑_i,j( ((Π' ')^⊤ G() Π' ')_i,j G(+)_i,j ('^⊤ G(+) ')_i,j + ((Π' ')^⊤ G(+) Π' ')_i,j G()_i,j('^⊤ G(+) ')_i,j + ((Π' ')^⊤ G(+) Π' ')_i,j G(+)_i,j ('^⊤ G() ')_i,j + ((Π' ')^⊤ G() Π' ')_i,j G()_i,j ('^⊤ G(+) ')_i,j + ((Π' ')^⊤ G() Π' ')_i,j G(+)_i,j ('^⊤ G() ')_i,j + ((Π' ')^⊤ G(+) Π' ')_i,j G()_i,j ('^⊤ G() ')_i,j + ((Π' ')^⊤ G() Π' ')_i,j G()_i,j ('^⊤ G() ')_i,j).By the Schur product theorem, these terms are nonnegative since they correspond to Hadamard products of three Gram matrices which are nonnegative. Thus, one can lower bound M_(p, q, , ) by the second term of the sum, which leads toM_(p, q, , ) ≥∑_i,j=1^K ((Π' ')^⊤ G(+) Π' ')_i,j ('^⊤ G(+) ')_i,j⟨ a_i, a_j ⟩Assume [Hid] holds for the parameters (π^*+p, ^*+q, +), then the matrices (Π' ')^⊤ G(+) Π' ' and '^⊤ G(+) ' are positive symmetric with respective lowest eigenvalue lower bounded by (inf_k(π^*_k+p_k) σ_K(^* + q))^2 σ_K(G(+)) and σ_K(^* + q)^2 σ_K(G(+)). Therefore, their Hadamard product is positive, and one has (((Π' ')^⊤ G(+) Π' ')_i,j ('^⊤ G(+) ')_i,j)_i,j = (D )^⊤ (D )withan orthogonal matrix and D a diagonal matrix with positive diagonal coefficients. Moreover, the Schur product theorem implies that σ_K(D)^2 ≥ (inf_k(π^*_k+p_k))^2 σ_K(^* + q)^4 σ_K(G(+))^2. ThenM_(p, q, , ) ≥ ∑_i,j=1^K ((D )^⊤ (D ))_i,j⟨ a_i, a_j ⟩ = ∑_j=1^KD _2^2≥ σ_K(D)^2 _2^2≥(inf_k(π^*_k+p_k))^2 σ_K(^* + q)^4 σ_K(G(+))^2 _2^2.Finally, let c_1(π^* + p, ^*+q, +) = (inf_k(π^*_k+p_k))^2 σ_K(^* + q)^4 σ_K(G(+))^2. The application (p, π^*, q, ^*, , ) ↦ c_1(π^* + p, ^*+q, +) is continuous and nonnegative, it is positive when [Hid] holds for the parameters (π^* + p, ^*+q, +), and one hasM_(p, q, , ) ≥ c_1(π^* + p, ^*+q, +) _2^2. We will now control the term N_(p,q,). Two cases appear: when (π^* + p, ^* + q , + ) is close to (π^*, ^*, ^*) in some sense and when it is not. The first case will be solved using the nondegeneracy of the quadratic form ensured by [Hdet]. The second case will be solved using the identifiability of the HMM. §.§.§ In the neighborhood of f*. The Taylor expansion of(p,q,) ∈ (Δ - Δ) × ( - ) × (( - ) ∩())^K ↦ N_(p, q, )around (0,0,0) leads to a nonnegative quadratic form and no linear part. [Hdet], [Hid] and equation (<ref>) ensure that this form is positive for = ^*. Let c_2(^*, π^*, ) be as defined in Section <ref>, then ↦ c_2(^*, π^*, ) is continuous and it is positive in the neighborhood of ^*. Moreover, there exists a positive constant η depending on G() _∞ such that for all (p,q,) such that (p, q, ) _≤ 1, one hasN_(p, q, ) ≥ c_2(^*, π^*, )(p, q, ) _^2 - η (p, q, ) _^3.For instance, η = 4000 K^6G() _∞^3 works: the terms of order 2 or more in the Taylor expansion of N_ are the scalar product of sums of terms of the form ∑_i,j,k ∈π^*(i) ^*(i,j) ^*(j,k) f_i ⊗ f_j ⊗ f_k where zero to three of the f may be replaced by u, zero to two of the ^* by q and π^* may be replaced by p and at least one of them is replaced. There are 63 possibilities, which leads to a sum of (63 K^3)^2 terms, each of which can be bounded by G() _∞^3 (max{ p(i), q(i,j), u_i_2 | i,j ∈})^r where r is the number of replaced terms. By taking the right permutation of states, the max can be bounded by (p, q, ) _, hence the result. Then, using (p,q,) _^2 =(p,q,) _^2 + _2^2 leads toN_(p, q, )/ (p, q, ) _^2≥c_1(^*+q, π^*+p, +) _2^2/ (p,q,) _^2 + _2^2+ c_2(^*, π^*, )(p, q, ) _^2/ (p,q,) _^2 + _2^2 - η (p, q, ) _^2/( (p,q,) _^2 + _2^2)^1/2 ≥c_1(^*+q, π^*+p, +) _2^2/ (p,q,) _^2 + _2^2+ c_2(^*, π^*, )(p, q, ) _^2/ (p,q,) _^2 + _2^2 - η√( (p, q, ) _^2)Let c_0 = min(c_1/2, c_2)/2, then c_0 is continuous and there exists a continuous function (π^*, ^*, ) ↦ϵ(π^*, ^*, ) which is positive as soon as [Hid] and [Hdet] hold for (π^*, ^*, ) and such that(p, q, ) _≤ϵ(π^*, ^*, ) ⇒ N_(p, q, )/ (p, q, ) _^2≥ c_0(^*, π^*, ).Thus, there exists positive constants ϵ_0 and c_near depending on ^*, π^* and ^* such that ∀ (p,q,,) ∈ (Δ-Δ) × (-) × ( - )^K ×^K s.t.(p, q, ) _≤ϵ_0and ∑_k ∈ f_k - f^*_k _2^2 ≤ϵ_0^2, N_(p, q, )/ (p, q, ) _^2≥ c_near.§.§.§ Far from f*.The application(p,q,,) ∈ (Δ-Δ) × (-) × ( - )^K ×^K ⟼ N_(p,q,)restricted to the set of (p,q,,) such that ∈()^K is uniformly continuous for the norm ·_tot defined by(p,q,,) _tot^2 :=p _2^2 +q _F^2 + ∑_k ∈(u_k _2^2 +f_k _2^2 ). Thus, by compactness of (Δ-Δ) × (-) × (( - ) ∩())^K, the applicationc_far: ⟼inf_(p,q,) ∈ (Δ-Δ) × (-) × (( - ) ∩())^Ks.t.(p, q, ) _ > ϵ_0 N_(p, q, )is continuous. Let us now prove that c_far(^*) > 0.Let (p_n, q_n, _n)_n ∈ ((Δ-Δ) × (-) × (( - ) ∩(^*))^K)^ be a sequence such that (p_n, q_n, _n) _^* > ϵ_0 for all n andc_far(^*) = lim_n N_^*(p_n, q_n, _n). By compactness, this sequences converges towards a limit (p,q,) up to taking a subsequence. Necessarily (p, q, ) _^*≥ϵ_0. Since [Hid] holds, Theorem <ref> shows that N_^*(p, q, ) > 0, which implies c_far(^*) > 0 by continuity of N_^*. Note that c_far(^*) may depend onin addition to the parameters π^*, ^* and ^*.Thus, by continuity, there exists ϵ_1 > 0 such that for all ∈^K such that ∑_k ∈ f_k - f^*_k _2^2 ≤ϵ_1^2, c_far() ≥ c_far(^*) / 2.Finally, [HF] implies that there exists a constantdepending only onsuch that (p,q,) _^2 ≤ (p,q,) _^2 ≤ for all (p,q,) ∈ (Δ-Δ) × (-) × ( - )^K. Therefore,∀ (p,q,,) ∈ (Δ-Δ) × (-) × ( - )^K ×^K s.t.(p, q, ) _≥ϵ_0and ∑_k ∈ f_k - f^*_k _2^2 ≤ϵ_1^2, N_(p, q, )/ (p,q,) _^2 ≥N_(p, q, )/≥c_far(^*)/2 .The theorem follows by taking c^*(π^*, ^*, ^*, ) = min(c_far(^*)/2 , c_near) and the neighborhood containing all ∈^K such that ∑_k ∈ f_k - f^*_k _2^2 ≤min(ϵ_0, ϵ_1)^2. Moreover, (π, , ) ⟼ c^*(π, , , ) is lower bounded by this value in a neighborhood of (π^*, ^*, ^*), so that it can be assumed to be lower semicontinuous. Note that the dependency of c^* onappears during this last step and is made non explicit because of the compactness assumption.§.§.§ Proof of Lemma <ref>| N_(p,q,) -N_'(p',q',') | = |g^π^*+p, ^*+q, + - g^π^*, ^*, _2^2 -g^π^*+p', ^*+q', '+' - g^π^*, ^*, '_2^2 |≤2g^π^*+p, ^*+q, + - g^π^*+p', ^*+q', '+'_2^2 + 2g^π^*, ^*,- g^π^*, ^*, '_2^2 + 2 | ⟨ g^π^*+p', ^*+q', '+' - g^π^*, ^*, ' , g^π^*+p, ^*+q, + - g^π^*+p', ^*+q', '+'⟩| + 2 | ⟨ g^π^*+p', ^*+q', '+' - g^π^*, ^*, ' , g^π^*, ^*,- g^π^*, ^*, '⟩|Then, using the fact that g^π, ,- g^π', ', '_2 ≤√(3K)^3(π - π',- ',- ', 0) _tot (see Lemma <ref>), that g^π, , _2 ≤^3 (see for instance Lemma 29 of <cit.>) and the Cauchy-Schwarz inequality,| N_(p,q,) - N_'(p',q',') | ≤6K ^6(p-p', q-q', +-'-', 0) _tot^2 + 6K ^6 (0, 0, 0, -') _tot^2 + 4√(3K)^6(p-p', q-q', +-'-', 0) _tot+ 4√(3K)^6(0, 0, 0, -') _tot^2≤24 K ^6 ((p-p', q-q', -', -') _tot^2 +(p-p', q-q', -', -') _tot),which proves the uniform continuity of the application. | http://arxiv.org/abs/1706.08277v3 | {
"authors": [
"Luc Lehéricy"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170626084237",
"title": "State-by-state Minimax Adaptive Estimation for Nonparametric Hidden Markov Models"
} |
Physics of Fluids Group and Max Planck Center Twente, J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Physics of Fluids Group and Max Planck Center Twente, J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. [email protected] Center for Combustion Energy and Department of Thermal Engineering,Tsinghua University, 100084 Beijing, China. Physics of Fluids Group and Max Planck Center Twente, J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. [email protected] Physics of Fluids Group and Max Planck Center Twente, J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany. In this Letter, we study the motion and wake-patterns of freely rising and falling cylinders in quiescent fluid.We show that the amplitude of oscillation and the overall system-dynamics are intricately linked to two parameters: the particle's mass-density relative to the fluid m^* ≡ρ_p/ρ_fand its relative moment-of-inertia I^* ≡I_p/I_f. This supersedes the current understanding that a critical mass density (m^*≈ 0.54) alone triggers the sudden onset of vigorous vibrations.Using over 144 combinations of m^* and I^*, we comprehensively map out the parameter space covering very heavy (m^* > 10) to very buoyant (m^* < 0.1) particles. The entire data collapses into two scaling regimes demarcated by a transitional Strouhal number, St_t ≈ 0.17. St_t separates a mass-dominated regime from a regime dominated by the particle's moment of inertia. A shift from one regime to the other also marks a gradual transition in the wake-shedding pattern: from the classical 2S (2-Single) vortex mode to a 2P (2-Pairs) vortex mode. Thus, auto-rotation can have a significant influence on the trajectories and wakes of freely rising isotropic bodies.Mass and moment of inertia govern the transition in the dynamics and wakes of freely rising and falling cylinders Detlef Lohse December 30, 2023 ================================================================================================================= Path-instabilities are a common observation in the dynamics of buoyant and heavy particles. Common examples are the fluttering of falling leaves and disks, and the cork-screw and spiral trajectories of air-bubbles rising in water <cit.>.The oscillatory dynamics of such particles can vary a lot depending on the particle's size, shape and its inertia, and the surrounding flow properties. This can be important in a variety of fields ranging from sediment transport and fluidization to multiphase particle- and bubble-column reactors <cit.>. Examples of organisms that exploit path-instabilities are plants and aquatic animals. These often make use of passive appendages attached to their bodies (plumed seeds, barbs, tails, and protrusions) to generate locomotion <cit.>. Recently, Lācis <cit.> demonstrated that the interaction between the wake of a falling bluff body and a protrusion clamped to its rear end can generate a sidewards drift by means of a symmetry-breaking instability similar to that of an inverted pendulum. Such kinds of passive interactions are advantageous to locomotion, since no energy needs to be spent by the animal.Instead,the energy can be extracted through fluid-structure interaction.The simplest case of a rising or falling body in a fluid is a sphere or a cylinder released in quiescent fluid. This problem has traditionally been studied and characterized using two non-dimensional parameters: the solid/fluid density-ratio (m^* ≡ρ_p/ρ_f) and the generalized particle Galileo number, Ga <cit.>. Among these, Ga governs the onset of various kinds of wake-instabilities behind the particle, and m^* governs the motion of the particle in response to these flow instabilities and vortex-induced forces. A number of investigators have studied the influence of these parameters, and various kinds of paths and wakes have been observed for rising and falling bodies <cit.>.However, despite a fair level of understanding of the mechanisms affecting path- and wake-instabilities, it remains unclear as to what factors precisely trigger vigorous path oscillations for a rising or falling object, such as a sphere or a cylinder. The motion of a freely rising/falling particlein a fluid is a complex fluid-structure-interaction (FSI) problem. Wake-induced forces cause the particle to move, which in turn changes the flow field around it. This holds many similarities to the popular subject of vortex-induced-vibrations (VIV) <cit.>. In this area, investigators have explored the dynamical response of elastically mounted and tethered bodies in uniform flows. A spring-mass-damper model was often used to make predictions of the oscillatory response of such systems under various conditions <cit.>. For example, a critical mass-ratio m^*_crit was predicted for the sudden appearance of path-oscillations for elastically mounted spheres and cylinders undergoing VIV <cit.>. For the freely rising body as well, similar predictions were made (m^*_crit = 0.54 for cylinder, and m^*_crit = 0.61 for sphere) by modeling it as a spring-mass-damper system with zero spring-stiffness and zero damping <cit.>. While a dependence on the mass-density ratio is undisputable, others observed wide scatter in their data to the point that there is no consensus with regard to whether a unique critical mass-ratio exists or not <cit.>. Moreover, from a dynamical point of view, the existence of a m^*_crit lies in contradiction with a fundamental concept: i.e, the motion of a light-particle in a fluid should be fluid added-mass dominated (since m_a > m_p) <cit.>. Therefore, it remains surprising that a marginal reduction in mass-density alone would trigger the sudden appearance of large-amplitude oscillations, since the effective mass of the system (actual + added mass) is almost unchanged <cit.>. We will provide a plausible explanation for this anomaly. In this Letter, we study the two-dimensional motion of circular cylinders rising or falling through a quiescent fluid. The body geometry (circular cylinder), the state of the surrounding fluid (quiescent), and the imposition of two-dimensionality of the flow make the problem simplified (see Leontini <cit.> for a discussion on the possible three-dimensional effects). Nevertheless, this model problem can provide important clues about the underlying dynamics of buoyancy-driven bodies in general <cit.>. We performed direct numerical simulations (DNS) using the immersed boundary method. The solver uses the discrete stream-function formulation for the incompressible Navier-Stokes equations <cit.> with a virtual force implementation <cit.>, which enables us to deal with both light and heavy particles. A rectangular computational domain of the size 100D× 16D is used, with a grid width of 0.01L near the cylinder, where D is the cylinder diameter. The fluid motion governed by the incompressible Navier-Stokes equations may be written in the dimensionless form as:∂ u/∂t+ u . ∇u= ∇p +1/Re∇^2 u +f,where u is the velocity vector, p is the pressure, Re is the Reynolds number, and f is the Eulerian body-force that is used to mimic the effects of the immersed body on the flow <cit.>. The direct numerical simulations provide an exact description of the flow-body interaction. However, for modeling purposes, the particle motion may be written in terms of the forces and moments exerted on the body. The Kelvin-Kirchhoff equations expressing linear and angular momentum conservation for the cylinder motion in a fluid may be extended to an incompressible flow containing vorticity <cit.>. The equations read:(m_p + m_a) [d U/dt +Ω× U] =F_v + (m_p - ρ_f 𝒱_p) g ; I dΩ/dt =Γ_v;where m_p is the particle mass, m_a is the added mass, Iis the particle moment of inertia, U is the particle velocity vector, Ω is the angular velocity, ρ_f is the fluid density, 𝒱_p is the particle volume, and g is the acceleration due to gravity. F_v and Γ_v represent the vortex forces and moments on the cylinder in a viscous fluid.Eqs. (<ref>) & (<ref>) point to two parameter-dependences, namely the particle's mass and its moment-of-inertia. In addition, the rotation rate of the particle Ω is linked to I and couples with eq. (<ref>) through a force term. However, the dependence on I was completely neglected for isotropic bodies despite the widespread variation in I in almost all existing experimental studies <cit.>. We therefore map out the [m^*-I^*] parameter space for rising and falling cylinders. We begin with the case of a very buoyant cylinder (m^* = 0.1) rising in a quiescent fluid. Fig. <ref>(a) shows the trajectory of the particle when its rotation is constrained (Ω =0). The amplitude of oscillation A/D ≈ 0.65. Next, we let the same particle auto-rotate while it rises through the fluid. Fig. <ref>(b) shows the trajectory for I^* = 0.1. We observe an 85 % enhancement in the oscillation amplitude due only to relaxing its contraint on rotation. Fig. <ref>(c) & (d) show snapshots of the vorticity field in the vicinity of the particle for the two cases. These observations indicate a clear link between the auto-rotation of the particle and the resulting translational and wake dynamics.Next, we explore systematically the dynamics of heavy and buoyant cylinders, with m^* and I^* both varied in the range [0.1; 10], i.e. covering very heavy (m^* = 10) to very light (m^* = 0.1) particles. It may be noted that in an actual experiment m^* and I^* are linked by the relation I^* = m^* (R_pG^2 /R_fG^2), which imposes an upper bound I^* → 2 m^*. Nevertheless, we vary the I^*independently over a wide range [0.1; 10] since it serves as an independent control parameter to modulate the particle's rotational freedom. The resulting translational and rotational amplitudes of vibration are shown in the contour maps in Fig. <ref>(a) & (b), respectively. The translational and rotational amplitudes depend on both m^* and I^*.The change in wake-pattern as we move from upper-right to lower-left part of the contour map in Fig. <ref>(a) & (b) is shown through the sequence (A)-(D) in Fig. <ref>(c). For large m^*, we always observe a 2S (2-Single) vortex mode (A), which is identical to that of a cylinder fixed in a uniform flow. However, as we lower the m^* and the I^*, we first observe a Mixed mode ((B) & (C)), which gradually transitions to a clear 2P (2-Pairs) mode of wake vortices (D). From the figure, we see that the 2S-2P transition can occur for m^* in the range [0.2-1], depending on the I^*. While a similar 2S-2P transition was also observed in the vortex-induced-vibrations literature <cit.>, the transition itself was thought be sudden at m^* = 0.54. Thus, unlike an elastically mounted or tethered particle, the extra rotational freedom present for a freely rising cylinder could be facilitating this gradual transition from 2S to 2P vortex mode. The same can be said about the oscillation amplitude, which grows gradually when m^* and I^* are reduced. Fig. <ref> demonstrated the existence of regions where the mass did not play a serious role, along with others where the amplitude depended mainly on the mass. To model these, we adopt a simplified approach, where we assume that the vortex force on the cylinder (from eq. (<ref>)) remains similar in intensity across the cases investigated. We describe this vortex force as F_v ≈ F_v0sinω t <cit.>. Next, we assume that the system dynamics is governed by the vortex shedding frequency ω. For the freely rising case, the shedding frequency ω can get modified as we change m^* and/or I^*. In the mass dominated regime, ω should primarily be a function of m_p. Thus, the transverse acceleration of the particle is given as a_p ∼F_v0/m_psinω t <cit.>,which leads to a relation for the transverse motion: x_p ∼F_v0/m_p ω^2sinω t. Non-dimensionalizing with the particle diameter D and the mean rise velocity U, we obtain the dimensionless amplitude, A/D∝ 1/(m^*St^2), where m^* is the dimensionless mass and St≡ωD/2 π U is the Strouhal number or equivalently the dimensionless vortex shedding frequency <cit.>. In Fig. <ref>(a), we plot A/D versus 1/(m^* St^2) for the full range of parameters in the present study. The solid blue curve shows our prediction, where F_v is estimated using the lift coefficient for flow past a fixed cylinder, i.e C_L ≈ 0.84. The prediction matches well with the simulation results up to an oscillation amplitude A/D ≈ 0.5. The inset to Fig. <ref>(a) shows the compensated plot, where the plateau demonstrates the robustness of the scaling. The prediction for the mass dominated regime is valid up to A/D ≈ 0.5.Beyond this, we observe branching of the oscillation amplitude for different values of the particle moment of inertia (right half of Fig. <ref>(a)). A/D almost doubles in this regime (0.6 to 1.2), solely due to a change in I^*. Therefore, we call this the moment of inertia dominated regime. Here the fluid added mass outweighs the particle mass. Therefore, the transverse oscillation amplitude in this regime may be expressed without anm^* dependence, as A/D ∝ 1/St^2. A/D versus 1/St^2 for the full range of parameters is shown in Fig. <ref>(b). A/D scales linearly with 1/St^2,as demonstrated by the red dashed line. When compensated for this, we observe a plateau for 1/St^2 > 32 (or equivalently, St < 0.17) in the inset. The scaling is valid for A/D ≥ 0.6, i.e. when the oscillation amplitudes are large. The above analysis suggests that the growth in oscillation amplitude is linked to the reduction in the dimensionless shedding frequency or St. Thus, a St map (as shown in Fig. <ref>(c)) can summarise the dynamics of the entire family of heavy and light cylinders rising/falling through a quiescent fluid. The clean 2S wake mode is found in the mass-dominated regime (right-side), and the clean 2P mode is restricted to the moment of inertia dominated regime (lower-left). The transition from 2S mode to 2P mode is not sudden, as was found for elastically mounted particles <cit.>, but gradual and marked by the appearance of a Mixed wake mode in the 0.5 < A/D < 0.7 range. The main conclusions of the present study are valid even at a lower Ga (=220). At this Ga (or corresponding Re), the flow can be considered predominantly two-dimensional, as suggested by Leontini <cit.>. Here again, we observe two distinct scaling regimes: a mass-dominated regime for 0 < A/D < 0.3, and a moment of inertia dominated regime for 0.4 < A/D < 0.75, with their respective 1/(m^* St^2) and 1/St^2 scalings. At the same time, the maximum oscillation amplitude is lower (A/D ≈ 0.75), and the wake transition is less distinct, owing to the greater viscous effects. In future work, we will provide a systematic account of the Ga (or Re) effects on the dynamics and wakes of rising/falling cylinders. The simplified model adopted here has enabled us to understand the motions and wakes of heavy and light cylinders rising/falling through still fluid. From the assumption that the vortex forces remain similar in intensity, but change only in the duration of their action, we can explain the growth in the oscillation amplitude. Since the Strouhal number represents the dimensionless frequency of vortex shedding, the fact that it is reduced means that the vortex shedding is retarded <cit.>. While the precise mechanism by which rotation induces this is not clear, one can qualitatively explain the behavior.For this, we compare the rotational response of a high moment of inertia cylinder with that of a low moment of inertia cylinder (Fig. <ref>(c) & (d)). The high moment of inertia cylinder resists rotation, while the low moment of inertia cylinder yields to the fluid torques and rotates. On the side where vorticity is shed (see Fig. <ref>(c) & (d)), the particle's rotation is along the mean flow direction. Therefore, the relative speed at the cylinder surface U_rel= (U_rise - U_rot) is reduced for the low moment of inertia cylinder, which could cause the developing shear-layer to remain attached for longer. Thus the vortex shedding time τ_v ∝ 1/U_rel for the low moment of inertia cylinder, as compared to τ_v ∝ 1/U_rise for the non-rotating cylinder. This corresponds to a reduction in the dimensionless frequency for the low moment of inertia cylinder. In Fig. <ref>, we find evidence that the dimensionless vortex shedding timescale τ_v^* ≡τ_vU_g/D increases almost linearly with the dimensionless inverse relative speed 1/U_rel^* ≡U_g/U_rel, where U_g = √(g D |1-ρ_p/ρ_f|) is a gravitational velocity scale <cit.>. In summary, the present study has demonstrated that the path oscillations of a buoyant particle can be linked to its rotational freedom, as speculated by Ryskin & Leal in 1984 <cit.>. Our findings remain to be experimentally confirmed using cylinder-rising experiments, such as those conducted in the past <cit.>, but with the rotational inerta as an additional control parameter.The insights gained here could be extended to isotropic bodies in 3D, such as buoyant spherical particles <cit.>. In an ongoing work, we have noticedthe validity of this for buoyant spheres rising in still fluid, as well as for buoyant spheres rising through a turbulent flow. These will be the focus of a future work. Varghese Mathai andXiaojue Zhu contributed equally to this work, and their names in the author-list are interchangeable. We thank Guowei He and Xing Zhang for the initial development of the code used here. We gratefully acknowledge J. Magnaudet, L. van Wijngaarden, S. Wildeman, and A. Prosperetti for useful discussions. This work was financially supported by the STW foundation of the Netherlands, and the Foundation for Fundamental Research on Matter (FOM), which is a part of the Netherlands Organisation for Scientific Research (NWO). CS acknowledges the financial support from Natural Science Foundation of China under Grant No. 11672156. We thank the Dutch Supercomputing Consortium SurfSARA, the Italian supercomputer FERMI-CINECA through the PRACE Project No. 2015133124 and the ARCHER UK National Supercomputing Service through the DECI Project 13DECI0246 for the allocation of computing time. 38 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL [Auguste et al.(2013)Auguste, Magnaudet, and Fabre]auguste2013falling authorF. Auguste, authorJ. Magnaudet, and authorD. Fabre, journalJ. Fluid Mech. volume719, pages388 (year2013). [Kelley and Wu(1997)]kelley1997path authorE. Kelley and authorM. Wu, journalPhys. Rev. Lett. volume79, pages1265 (year1997). [Ern et al.(2012)Ern, Risso, Fabre, and Magnaudet]ern2012wake authorP. Ern, authorF. Risso, authorD. Fabre, and authorJ. Magnaudet, journalAnnu. Rev. Fluid Mech. volume44, pages97 (year2012). [Richardson and Zaki(1954)]richardson1954sedimentation authorJ. Richardson and authorW. Zaki, journalChem. Engg. Sci. volume3, pages65 (year1954). [Stringham et al.(1969)Stringham, Simons, and Guy]stringham1969behavior authorG. Stringham, authorD. B. Simons, and authorH. P. Guy, journalUS Government Printing Office(year1969). [Hartman and Yates(1993)]hartman1993free authorM. Hartman and authorJ. G. Yates, journalCollection of Czechoslovak chem. comm. volume58, pages961 (year1993). [Lācis et al.(2014)Lācis, Brosse, Ingremeau, Mazzino, Lundell, Kellay, and Bagheri]lacis2014passive authorU. Lācis, authorN. Brosse, authorF. Ingremeau, authorA. Mazzino, authorF. Lundell, authorH. Kellay, and authorS. Bagheri, journalNature Comm. volume5 (year2014). [Bagheri et al.(2012)Bagheri, Mazzino, and Bottaro]bagheri2012spontaneous authorS. Bagheri, authorA. Mazzino, and authorA. Bottaro, journalPhys. Rev. Lett. volume109, pages154502 (year2012). [Dickinson et al.(2000)Dickinson, Farley, Full, Koehl, Kram, and Lehman]dickinson2000animals authorM. H. Dickinson, authorC. T. Farley, authorR. J. Full, authorM. Koehl, authorR. Kram, and authorS. Lehman, journalScience volume288, pages100 (year2000). [Jenny et al.(2004)Jenny, Dušek, and Bouchet]jenny2004instabilities authorM. Jenny, authorJ. Dušek, and authorG. Bouchet, journalJ. Fluid Mech. volume508, pages201 (year2004). [Veldhuis et al.(2004)Veldhuis, Biesheuvel, van Wijngaarden, and Lohse]veldhuis2004motion authorC. Veldhuis, authorA. Biesheuvel, authorL. van Wijngaarden, and authorD. Lohse, journalNonlinearity volume18, pagesC1 (year2004). [Horowitz and Williamson(2006)]horowitz2006dynamics authorM. Horowitz and authorC. Williamson, journalJ. Fluids Struct. volume22, pages837 (year2006). [Horowitz and Williamson(2008)]horowitz2008critical authorM. Horowitz and authorC. Williamson, journalPhys. Fluids volume20, pages101701 (year2008). [Horowitz and Williamson(2010a)]horowitz2010effect authorM. Horowitz and authorC. Williamson, journalJ. Fluid Mech. volume651, pages251 (year2010a). [Govardhan and Williamson(2005)]govardhan2005vortex authorR. Govardhan and authorC. Williamson, journalJ. Fluid Mech. volume531, pages11 (year2005). [Williamson and Govardhan(2004)]williamson2004vortex authorC. Williamson and authorR. Govardhan, journalAnnu. Rev. Fluid Mech. volume36, pages413 (year2004). [Karamanev et al.(1996)Karamanev, Chavarie, and Mayer]karamanev1996dynamics authorD. Karamanev, authorC. Chavarie, and authorR. Mayer, journalAIChE J. volume42, pages1789 (year1996). [Spelt and Biesheuvel(1997)]spelt1997motion authorP. Spelt and authorA. Biesheuvel, journalJ. Fluid Mech. volume336, pages221 (year1997). [Maxey and Riley(1983)]maxey1983equation authorM. R. Maxey and authorJ. J. Riley, journalPhys. Fluids volume26, pages883 (year1983). [Mathai et al.(2015)Mathai, Prakash, Brons, Sun, and Lohse]mathai2015wake authorV. Mathai, authorV. N. Prakash, authorJ. Brons, authorC. Sun, and authorD. Lohse, journalPhys. Rev. Lett. volume115, pages124501 (year2015). [Ryskin and Leal(1984)]ryskin1984numerical authorG. Ryskin and authorL. G. Leal, journalJ. Fluid Mech. volume148, pages19 (year1984). [Leontini et al.(2007)Leontini, Thompson, and Hourigan]leontini2007three authorJ. S. Leontini, authorM. Thompson, and authorK. Hourigan, journalJ. Fluid Mech. volume577, pages79 (year2007). [Namkoong et al.(2008)Namkoong, Yoo, and Choi]namkoong2008numerical authorK. Namkoong, authorJ. Y. Yoo, and authorH. G. Choi, journalJ. Fluid Mech. volume604, pages33 (year2008). [Zhu et al.(2014a)Zhu, He, and Zhang]zhu2014numerical authorX. Zhu, authorG. He, and authorX. Zhang, journalComp. Fluids volume97, pages1 (year2014a). [Schwarz et al.(2015)Schwarz, Kempe, and Fröhlich]schwarz2015temporal authorS. Schwarz, authorT. Kempe, and authorJ. Fröhlich, journalJ. Comp. Phys. volume281, pages591 (year2015). [supplemental movies available at URL.(2017)]supplemental authorsupplemental movies available at URL. (year2017). [Zhu et al.(2014b)Zhu, He, and Zhang]zhu2014flow authorX. Zhu, authorG. He, and authorX. Zhang, journalPhys. Rev. Lett. volume113, pages238105 (year2014b). [Zhu et al.(2014c)Zhu, He, and Zhang]zhu2014improved authorX. Zhu, authorG. He, and authorX. Zhang, journalJ. Fluids Engg. volume136, pages040903 (year2014c). [Zhu et al.(2014d)Zhu, He, and Zhang]zhu2014flexibility authorX. Zhu, authorG. He, and authorX. Zhang, journalJ. Fluid Mech. volume751, pages164 (year2014d). [Miloh and Landweber(1981)]miloh1981generalization authorT. Miloh and authorL. Landweber, journalPhys. Fluids volume24, pages6 (year1981). [Mougin and Magnaudet(2001)]mougin2001path authorG. Mougin and authorJ. Magnaudet, journalPhys. Rev. Lett. volume88, pages014502 (year2001). [Horowitz and Williamson(2010b)]horowitz2010vortex authorM. Horowitz and authorC. H. Williamson, journalJ. Fluid Mech. volume662, pages352 (year2010b). [Mathai et al.(2016)Mathai, Neut, van der Poel, and Sun]mathai2016translational authorV. Mathai, authorM. W. Neut, authorE. P. van der Poel, and authorC. Sun, journalExp. Fluids volume57, pages1 (year2016). [Shafrir(1965)]shafrir1965horizontal authorU. Shafrir, typeTech. Rep., institutionDTIC Doc. (year1965). [Govardhan and Williamson(2002)]govardhan2002resonance authorR. Govardhan and authorC. Williamson, journalJ. Fluid Mech. volume473, pages147 (year2002). [Bearman(1984)]bearman1984vortex authorP. W. Bearman, journalAnnu. Rev. Fluid Mech. volume16, pages195 (year1984). [Govardhan and Williamson(2004)]govardhan2004critical authorR. Govardhan and authorC. Williamson, journalEur. J. Mech. B/Fluids volume23, pages17 (year2004). [Wu and Gharib(2002)]wu2002experimental authorM. Wu and authorM. Gharib, journalPhys. Fluids volume14, pagesL49 (year2002). | http://arxiv.org/abs/1706.08624v1 | {
"authors": [
"Varghese Mathai",
"Xiaojue Zhu",
"Chao Sun",
"Detlef Lohse"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170626231028",
"title": "Mass and moment of inertia govern the transition in the dynamics and wakes of freely rising and falling cylinders"
} |
corresponding author: [email protected] Group of Complex Systems and Statistical Physics, Physics Faculty, University of Havana, 10400 Havana, Cuba.Using a specially designed experimental set up, we have studied the so-called continuous to intermittent flow transition in sand piles confined in a Hele-Shaw cell where the deposition height of the sand can be controlled. Through systematic measurements varying the height and the input flow, we have established how the size of the pile at which the transition takes place depends on the two parameters studied. The results obtained allows to explain, at least semi-quantitatively, the observations commonly reported in the literature, carried out in experiments where the deposition height is not controlled.PACS numbers Hele-Shaw flows, 47.15.gp; Avalanches (granular systems), 45.70.Ht; Avalanches, phase transitions in, 64.60.av; Granular systems, classical mechanics of, 45.70.-n. Intermittent and continuous flows in granular piles: effects of controlling the feeding height E. Altshuler December 30, 2023 ==============================================================================================Granular media are relevant to many human endeavors: for example, they play a central role in the construction, food and pharmaceutical industries, and also as an important component of the natural environment <cit.>. During the last years granular matter has been increasingly studied from the fundamental point of view by physicists. It has been used, for example, to establish analogies that allow to understand certain phenomena in other areas of physics and engineering, ranging from superconducting avalanches to urban traffic <cit.>.Granular piles –that we will generically call sandpiles– have been used as a model for segreggation phenomena in geophysical scenarios, and also to illustrate the idea of self-organized criticality <cit.>. A particularly attractive configuration of sandpiles is the Hele-Shaw cell: a pile of grain is grown confined between two vertical plates resting on an horizontal surface and separated by a distance w, where grains are poured from above, near a third vertical wall also of width w (see Fig. 1) <cit.>. The height from which the grains are dropped to feed the pile has been rarely controlled, and only by hand <cit.>. Just recently an automatic system has been designed to fully control the dropping height <cit.>.A well known feature of surface flows in granular piles is the existence of continuous flows (where the grains flows uniformly down the slope within a certain depth from the free surface of the pile) and intermittent flows (where an avalanche suddely rolls down the surface of the pile, and accumulates at its lower edge, allowing a front to grow uphill until a new avalanche starts) <cit.>]. The transition between the two regimes as the pile grows is, however, poorly understood.In this communication, we have used the system described in reference <cit.> to study the transition from the continuous to the intermittent regime of granular flows on the surface of a quasi-2D pile as a function of the height, h, from which the granular matter is fed into the system (see Fig. 1). We put special emphasis in unveiling the relation that exists between “conventional“ experiments –i.e., those where the dropping height is not controlled– and those where it is kept constant in time.In every experiment the behavior of the deposition height, h, and the area of the pile were measured as a function of time (the latter allows to compute the input flux, and make sure it is constant along the whole experiment). To obtain the spatial-temporal coordinates of the transition, the evolution in time of a horizontal line of pixels located at a height equal to the half the width of the input flow, from the bottom of the Hele-Shaw cell, was analyzed (see Fig. 2).Two types of experiments were performed. In the first group, we kept the feeding height constant in time, as well as the input flux. In the second, the input flux was kept constant, but the dropping height decreased as the pile height increased.In all experiments, the horizontal size of the pile where the transition from continuous to intermittent flow occurred, Xc , was determined, as shown in Fig. 3.Using the obtained temporal coordinates of the transition it is possible, knowing the critical angle for the sand, the input flow and the height of the container at the beginning, to construct a diagram to predict, at least semi-quantitatively, the instant at which the intermittent regime begins in experiments with non-controlled deposition height. It is also possible to find for any experiment with fixed deposition height an equivalent non-controlled experiment in which transition occurs at the same moment, and vice versa, as shown in Fig. 4.In order to check our predictions, experiments using the same experimental set up were performed, but now without keeping constant the deposition height. The predictions were made applying the mass conservation principle. It was obtained that the deposition height for a non-controlled experiment varies as h_0 - √(2tFtanθ_c), where h_0 is the initial deposition height, F the input flow and θ_c the critical angle as mentioned above. Figure 4 shows, for one input flow, the temporal coordinates tc of the transition for experiments with different fixed deposition heights, as well as the dependence of the deposition height with time for an experiment where it was non-controlled. The black dots follow a lineal dependence with H that delimitates, from left to right, the end of the continuous phase and the beginning of the intermittent one.In summary, we have performed the first systematic study of the transition from the continuous to the intermittent regimes of granular flows on a sand heap, including both conventional experiments, as well as those where the deposition height is controlled to be constant in time. We have demonstrated the relation between the two situations, in such a way that we are able to predict at what size of the pile the transition will take place for a non-controlled deposition height, based on the data taken from experiments with controlled deposition heights. This constitutes a first and necessary step to fully understand the physical nature of the transition.99andreoti B. Andreotti, Y. Forterre and O. Pouliquen, Granular media (Cambridge University Press, Cambridge, United Kingdom, 2013. tejchman J. Tejchman, Confined Granular flow in Silos: Experimental and Numerical Investigations (Springer International Publishing Switzerland, Switzerland, 2013). antony S. J. Antony, W. Hoyle and Y. Ding (Editors), Granular Materials: fundamentals and applications (The Royal Society of Chemistry, Cambridge, United Kingdom, 2004). alt2004 E. Altshuler and T. H. Johansen, Rev. Mod. Phys. 76, 471 (2004). alt2001 E. Altshuler, O. Ramos, C. Martínez, L. E. Flores, and C. Noda, Phys. Rev. Lett. 86, 5490 (2001). aranson I. Aranson and L. S. Tsimring, Rev. Mod. Phys. 78, 641 (2006). alt2003 E. Altshuler, O. Ramos, E. Martínez, A. J. Batista-Leyva, A. Rivera, and K. E. Bassler, Phys. Rev. Lett. 91, 014501 (2003). etien2007 E. Martínez, C. Pérez-Penichet, O. Sotolongo-Costa, O. Ramos, K. J. Måløy, S. Douady, and E. Altshuler, Phys. Rev. E 75, 031303 (2007). alt2008 E. Altshuler, R. Toussaint, E. Martínez, O. Sotolongo-Costa, J. Schmittbuhl, and K. J. Måløy, Phys. Rev. E 77, 031305 (2008). grasselli2000 Y. Grasselli, H. J. Herrmann, G. Oron and S. Zapperi, Granular Matter 2, 97 (2000). grasselli97 Y. Grasselli and H. Herrmann, Physica A 246, 301 (1997). leo L. Domínguez-Rubio, E. Martínez and E. Altshuler, Rev. Cub. Física 32, 111 (2015). dip B. Jähne, Digital Image Processing, 6th edition (Springer-Verlag Berlin Heidelberg, Germany 2005. | http://arxiv.org/abs/1706.08622v2 | {
"authors": [
"L. Alonso-Llanes",
"L. Domínguez-Rubio",
"E. Martínez",
"E. Altshuler"
],
"categories": [
"cond-mat.soft"
],
"primary_category": "cond-mat.soft",
"published": "20170626225953",
"title": "Intermittent and continuos flows in granular piles: effects of controlling the feeding height"
} |
Institute for Parallel and Distributed SystemsUniversity of Stuttgart, [email protected] Today, massive amounts of streaming data from smart devices need to be analyzed automatically to realize the Internet of Things. The Complex Event Processing (CEP) paradigm promises low-latency pattern detection on event streams. However, CEP systems need to be extended with Machine Learning (ML) capabilities such as online training and inference in order to be able to detect fuzzy patterns (e.g. outliers) and to improve pattern recognition accuracy during runtime using incremental model training. In this paper, we propose a distributed CEP system denoted as StreamLearner for ML-enabled complex event detection. The proposed programming model and data-parallel system architecture enable a wide range of real-world applications and allow for dynamically scaling up and out system resources for low-latency, high-throughput event processing. We show that the DEBS Grand Challenge 2017 case study (i.e., anomaly detection in smart factories) integrates seamlessly into the StreamLearner API. Our experiments verify scalability and high event throughput of StreamLearner. <ccs2012> <concept> <concept_id>10010147.10010169.10010170.10010173</concept_id> <concept_desc>Computing methodologies Vector / streaming algorithms</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010919.10010177</concept_id> <concept_desc>Computing methodologies Distributed programming languages</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257</concept_id> <concept_desc>Computing methodologies Machine learning</concept_desc> <concept_significance>300</concept_significance> </concept> <concept> <concept_id>10003752.10003753.10003760</concept_id> <concept_desc>Theory of computation Streaming models</concept_desc> <concept_significance>300</concept_significance> </concept> <concept> <concept_id>10011007.10011006.10011050.10011051</concept_id> <concept_desc>Software and its engineering API languages</concept_desc> <concept_significance>300</concept_significance> </concept> </ccs2012> [500]Computing methodologies Vector / streaming algorithms [500]Computing methodologies Distributed programming languages [300]Computing methodologies Machine learning [300]Theory of computation Streaming models [300]Software and its engineering API languagesGrand Challenge: StreamLearner – Distributed Incremental Machine Learning on Event Streams Christian Mayer, Ruben Mayer, and Majd Abdo December 30, 2023 ============================================================================================ [overlay] [text width=20cm] at ([yshift=-5.0cm]current page.south) (c) Owner 2017. This is the authors' version of the work. It is posted here for your personal use. Not for redistribution. The definitive version is published in Proceedings of ACM International Conference on Distributed and Event-Based Systems 2017 (DEBS '17), http://dx.doi.org/10.1145/3093742.3095103.;§ INTRODUCTION AND BACKGROUNDIn recent years, the surge of Big Streaming Data being available from sensors <cit.>, social networks <cit.>, and smart cities <cit.>, has led to a shift of paradigms in data analytics throughout all disciplines. Instead of batch-oriented processing <cit.>, stream-oriented data analytics <cit.> is becoming the gold standard. This has led to the development of scalable stream processing systems that implement the relational query model of relational data base management systems (RDBMS) as continuous queries on event streams <cit.>, and Complex Event Processing systems that implement pattern matching on event streams <cit.>.Query-driven stream processing, however, demands a domain expert to specify the analytics logic in a deterministic query language with a query that exactly defines which input events are transformed into which output events by an operator. However, an explicit specification is not always possible, as the domain expert might rather be interested in a more abstract query such as “Report me all anomalies that molding machine 42 experiences on the shopfloor.” In this example, it is infeasible to explicitly specify all event patterns that can be seen as an anomaly.There have been different proposals how to deal with this issue. EP-SPARQL employs background ontologies to empower (complex) event processing systems with stream reasoning <cit.> – while focusing on the SPARQL query language. On the other hand, several general-purpose systems for stream processing exist such as Apache Kafka <cit.>, Apache Flink <cit.>, Apache Storm <cit.>, Apache Spark Streaming <cit.>. Although these systems are powerful and generic, they are not tailored towards parallel and scalable incremental model training and inference on event streams.At the same time, an increasing body of research addresses incremental (or online) updates of Machine Learning (ML) models: there are incremental algorithms for all kinds of ML techniques such as support vector machines <cit.>, neural networks <cit.>, or Bayesian models <cit.>. Clearly, a stream processing framework supporting intuitive integration of these algorithms would be highly beneficial – saving the costs of hiring expensive ML experts to migrate these algorithms to the stream processing systems.In this paper, we ask the question: how can we combine event-based stream processing (e.g., for pattern recognition) with powerful Machine Learning functionality (e.g., to perform anomaly detection) in a way that is compatible with existing incremental ML algorithms? We propose the distributed event processing system StreamLearner that decouples expertise of Machine Learning from Distributed CEP using a general-purpose modular API. In particular, we provide the following contributions. * An architectural design and programming interface for data-parallel CEP that allows for easy integration of existing incremental ML algorithms (cf. Section <ref>). * An algorithmic solution to the problems of incremental K-Means clustering and Markov model training in the context of anomaly detection in smart factories (cf. Section <ref>). * An evaluation showing scalability of the StreamLearner architecture and throughput of up to 500 events per second using our algorithms for incremental ML model updates (cf. Section <ref>). § CHALLENGES AND GOALSMachine Learning algorithms train a model using a given set of training data, e.g., building clusters, and then apply the trained model to solve problems, e.g., classifying unknown events. In the course of streaming data becoming available from sensors, models need to be dynamically adapted. That means, that new data is taken into account in the learned model, while old data “fades out” and leaves the model as it becomes irrelevant. This can be modeled by a sliding window over the incoming event streams: Events within the window are relevant for the model training, whereas events that fall out of the window become irrelevant and should not be reflected in the model any longer. Machine Learning on sliding windows is also known as non-stationary Machine Learning, i.e., the problem of keeping a model updated as the underlying streaming data generation “process” underlies a changing probability distribution. To adapt the ML model online, there are different possibilities. For instance, incremental algorithms change the model in a step-by-step fashion. The challenge in doing so is to support incremental processing – i.e., streaming learning. The model should not be re-built from scratch for every new window, but rather incrementally be updated with new data while old data is removed. Another challenge in ML in streaming data is that data from different streams might lead to independent models. For instance, data captured in one production machine might not be suitable to train the model of another production machine. The challenge is to determine which independent models shall be built based on which data from which incoming event streams. Further, the question is how to route the corresponding events to the appropriate model. When these questions are solved, the identified machine learning models can be built in parallel – enabling scalable, low-latency, and high-throughput stream processing. § STREAMLEARNERIn this section, we first give an overview about the StreamLearner architecture, followed by a description of the easy-to-use API for incremental machine learning and situation inference models.§.§ System Overview The architecture of StreamLearner is given in Figure <ref>. In order to parallelize ML-based computation, we have extended the split-process-merge architecture of traditional event-based systems <cit.>. The splitter receives events via the event input stream and forwards them to independent processing units, denoted as tube-ops, according to its splitting logic. Each tube-op atomically performs ML-based incremental stream processing by reading an event from the in-queue, processing the event, and forwarding the output event to the merger. The merger decides about the final events on the event output stream (e.g. sorts the events from the different tube-ops by timestamp to provide a consistent ordering of the event output stream). Due to the independent processing of events, the architecture supports both, scale-up operations by spawning more threads per machine and scale-out operations by adding more machines. Each tube-op processes an event in three phases: shaping, training, and inference. In the shaping phase, it performs stateless preprocessing operations ω_1 and ω_2 (denoted as shaper) to transform the input event into appropriate formats. In the training phase, the stateful trainer module incrementally updates the model parameters of model M (e.g. a neural network in Figure <ref>) according to the user-specified model update function. In the inference phase, the updated model and the preprocessed event serve as an input for the stateful predictor performing a user-defined inference operation and transforming the updated model and the input event to an output event with the model-driven prediction.Note that the StreamLearner API does not restrict application programmers to perform training and inference on different event data. Hence, application programmers are free to use either disjoint subsets, or intersecting subsetsof events in the stream for training and inference. Although it is common practice in ML to separate data that is used for training and inference, we still provide this flexibility, as in real-world streams we might use some events for both, incorporating changing patterns into the ML model and initiating an inference event using the predictor. However, the application programmer can also separate training and inference data by defining the operators in the tube-op accordingly (e.g. generating a dummy event as input for the predictor to indicate that no inference step should be performed). Furthermore, the application programmer can also specify whether the training should happen before inference or vice versa. §.§ Programming Model The application programmer specifies the following functions in order to use the StreamLearner framework in a distributed environment.§.§.§ SplitterGiven an event e_i, the application programmer defines a stateful splitting function split(e_i) that returns a tuple (mid, tid, e_i) defining the tube-op tid on machine mid that receives event e_i.§.§.§ ShapingThe stateless shaper operations ω_1(e_i) and ω_2(e_i) return modified events e_i^1 and e_i^2 that serve as input for the trainer and the predictor module. The default shaper performs the identity operation.§.§.§ TrainerThe stateful trainer operation trainer(e_i^1) returns a reference to the updated model object M'. The application programmer can use any type of machine learning model as long as the model can be used for inference by the predictor. If the model M remains unchanged after processing event e_i^1, the trainer must return a reference to the unchanged model M in order to trigger the predictor for each event. StreamLearner performs a delaying strategy when the application programmer prefers inference before learning. In this case, the tube-op first executes the predictor on the old model M and executes the trainer afterwards to update the model.§.§.§ PredictorThe stateful predictor receives a reference to model M' and input (event) e_i^2 and returns the predicted event e_i^3=predictor(M',e_i^2).§.§.§ MergerThe stateful merger receives predicted output events from the tube-ops and returns a sequence of events that is put to the event output stream, i.e., merger(e_i^3)=f(e_0^3,...,e_j^3,...,e_i^3) for j<i and any function f. Any aggregator function, event ordering scheme, or filtering method can be implemented by the merger. § CASE STUDY: ANOMALY DETECTION IN SMART FACTORIESIn this section, we exemplify usage of our StreamLearner API based on a realistic use case for data analytics posed by the DEBS Grand Challenge 2017[http://www.debs2017.org/call-for-grand-challenge-solutions/] <cit.>. §.§ Problem DescriptionIn smart factories, detecting malfunctioning of production machines is crucial to enable automatic failure correction and timely reactions to bottlenecks in the production line.The goal of this case study is to detect anomalies, i.e., abnormal sequences of sensor events quantifying the state of the production machines. In particular, the input event stream consists of events transporting measurements from a set of production machines P to an anomaly detection operator. The events are created by the set of sensors S that monitor the production machines. We include the time stamps of each measured sensor event by defining a set of discrete time steps DT. Each event e_i=(p_i, d_i, s_i, t_i) consists of a production machine id p_i ∈ P that was monitored, a numerical data value d_i ∈ℝ quantifying the state of the production machine (e.g. temperature, pressure, failure rate), a sensor with id s_i ∈ S that has generated the event, and a time stamp t_i ∈ DT storing the event creation time.The anomaly detection operator has to pass three stages for each event-generating sensor (cf. Figure <ref>).First, it collects all events e_i that were generated within the last W time units (denoted as event window) and clusters the events e_i using the K-means algorithm on the numerical data values d_i for at least M iterations. The standard K-means algorithm iteratively assigns each event in the window to its closest cluster center (with respect to euclidean distance) and recalculates each cluster center as the centroid of all assigned events' numerical data values (in the following we do not differentiate between events and their data values). In the figure, there are five events e_5, e_6, e_7, e_8, e_9 in the event window that are clustered into three clusters C_1,C_2,C_3. With this method, we can characterize each event according to its state, i.e., the cluster it is assigned to.Second, the operator trains a first-order Markov model in order to differentiate normal from abnormal event sequences. A Markov model is a state diagram, where a probability value is associated to each state transition. The probability of a state transition depends only on the current state and not on previous state transitions (independence assumption).These probabilities are maintained in a transition matrix T using the following method: (i) The Markov model consists of K states, one state for each cluster. Each event is assumed to be in the state of the cluster it is assigned to. (ii) The events are ordered with respect to their time stamp – from oldest to youngest. Subsequent events are viewed as state transitions. In Figure <ref>, the events can be sorted as [e_5, e_6, e_7, e_8, e_9]. The respective state transitions are C_2 → C_3 → C_2 → C_2 → C_1. (iii) The transition matrix contains the probabilities of state transitions between any two states, i.e., cluster centers. The probability of two subsequent events being in cluster C_i and transition into cluster C_j for all i,j ∈{1,...,K} is the relative number of these observations. For example the probability of transition from state C_2 to state C_1 is the number of events in state C_2 that transition to state C_1 divided by the total number of transitions from state C_2, i.e., P(C_1|C_2)=#C_2 → C_1/#C_2 →⋆=1/3.Third, an anomaly is defined using the probability of a sequence of observed transitions with length N. In particular, if a series of unlikely state transitions is observed, i.e., the total sequence probability is below the threshold Θ, an event is generated that indicates whether an anomaly has been found. The probability of the sequence can be calculated by breaking the sequence into single state transitions, i.e., in Figure <ref>, P(C_2 → C_3 → C_2 → C_2 → C_1)=P(C_2 → C_3) P(C_3 → C_2) P(C_2 → C_2) P(C_2 → C_1). Using the independence assumption of Markov models, we can assign a probability value to each sequence of state transition and hence quantify the likelihood.§.§ Formulating the Problem in the StreamLearner APIThe scenario fits nicely into the StreamLearner API: for each sensor, an independent ML model is subject to incremental training and inference steps. Therefore, each thread in the StreamLearner API is responsible for all observations of a single sensor enabling StreamLearner to monitor multiple sensors in parallel.§.§.§ SplitterThe splitter receives an event e_i=(p_i, d_i, s, t_i) and assigns the event exclusively to the thread that is responsible for sensor s (or initiates creation of this responsible thread if it does not exist yet). It uses a simple hash map assigning sensor ids to thread ids to provide thread resolution with constant time complexity during processing of the input event stream. With this method, we break the input stream into multiple independent sensor event streams (one stream per sensor). §.§.§ ShapersShapers ω_1 and ω_2 are simply identity operators that pass the event without changes to the respective training or prediction modules.§.§.§ TrainerThe trainer maintains and updates the model in an incremental fashion. The model is defined via the transition matrix T that is calculated using K-means clustering and the respective state transition sequence. Incremental K-Means The goal is to iteratively assign each event to the closest cluster center and recalculate the cluster center as the centroid of all assigned events. The standard approach is to perform M iterations of the K-means clustering algorithm for all events in the event window when triggered by the arrival of a new event. However, this method results in suboptimal runtime due to unnecessary computations that arise in practical settings: * A single new event in the event window will rarely have a global impact to the clustering. In particular, most assignments of events to clusters remain unchanged after adding a new event to the event window. Therefore, the brute-force method of full reclustering can result in huge computational redundancies. * Performing M iterations is unnecessary, if the clustering has already converged in an earlier iteration M'<M. Clearly, we should terminate the algorithm as fast as possible. * The one-dimensional K-means problem is fundamentally easier than the standard NP-hard K-means problem: an optimal solution can be calculated in polynomial time 𝒪(n^2K) for fixed number of clusters K and number of events in the window n <cit.>. Therefore, using a general-purpose K-means algorithm that supports arbitrary dimensionality can result in unnecessary overhead (the trade-off between generality, performance, and optimality). This is illustrated in Figure <ref>. There are four clusters C_1,...,C_4 and events e_5,...,e_9 in the event window. A new event e_10 is arriving. Instead of recomputation of the whole clustering in each iteration, i.e., calculating the distance between each event and cluster center, we touch only events that are potentially affected by a change of the cluster centers. For example, event e_10 is assigned to cluster C_4 which leads to a new cluster center C_4'. However, the next closest event e_6 (left side) keeps the same cluster center C_3. Our basic reasoning is that each event on the left side of the unchanged event e_6 keeps its cluster center as there can be no disturbance in the form of changed cluster centers left-hand of e_6 (only a cascading cluster center shift is possible as C_4 ≥ C_3 ≥ C_2 ≥ C_1 in any phase of the algorithm). A similar argumentation can be made for the right side and also for the removal of events from the window.This idea heavily utilizes the possibility of sorting cluster centers and events in the one-dimensional space. It reduces average runtime of a single iteration of K-means as in many cases only a small subset of events has to be accessed. Combined with the optimization of skipping further computation after convergence in iteration M'<M, incremental updates of the clustering can be much more efficient than naive reclustering. The incremental one-dimensional clustering method is in the same complexity class as naive reclustering as in the worst case, we have to reassign all events to new clusters (the sorting of events takes only logarithmic runtime complexity in the event window size per insertion of a new event – hence the complexity is dominated by the K-means computation).Markov Model The Markov model is defined by the state transition matrix T. Cell (i,j) in the transition matrix T is the probability of two subsequent events to transition from cluster C_i (the first event) to cluster C_j (the second event). Semantically, we count the number of state transitions in the event window to determine the relative frequency such that the row values in T sum to one. Instead of complete recomputation of the whole matrix, we only recalculate the rows and columns of clusters that were subject to any change in the K-means incremental clustering method. This ensures that all state transitions are reflected in the model while saving computational overhead. A reference to the new model T is handed to the predictor method that performs inference on the updated model as presented in the following.§.§.§ PredictorThe predictor module applies the inference step on the changed model for each incoming event. In this scenario, inference is done via the Markov model (i.e., the transition matrix T) to determine whether an anomaly was detected or not. We use the transition matrix to assign a probability value to a sequence of events with associated states (i.e., cluster centers). The brute-force method would calculate the product of state transition probabilities for each sequence of length N and compare it with the probability threshold Θ. However, this leads to many redundant computations for subsequent events.We present an improved incremental method in Figure <ref>. The event window consists of events e_1,...,e_8 sorted by time stamps. Each event is assigned to a cluster C_1 or C_2 resulting in a series of state transitions. We use the transition matrix of the Markov model to determine the probability of each state transition.We calculate the probability of the state transition sequence as the product of all state transitions (the state independence property of Markov models). For instance the probability Π of the first three state transitions is Π=P(C_1|C_1)*P(C_2|C_1)*P(C_2|C_2)=1/3*2/3*3/4=1/4 which is larger than the threshold Θ=0.1. Now we can easily calculate the probability of the next state transition sequence of length N by dividing by the first transition probability of the sequence (i.e., P(C_1|C_1)=1/3) and multiplying with the probability of the new state transition (i.e., P(C_2|C_2)=3/4). Hence, the total probability Π' of the next state transition sequence is Π'=Π/1/3*3/4=9/16>Θ. This method reduces the number of multiplications to N+2(W-N) rather than N(W-N). Finally, the predictor issues an anomaly detection event to the merger (Yes/No).§.§.§ MergerThe merger sorts all anomalies events w.r.t. time stamp to ensure a consistent output event stream using the same procedure as in GraphCEP <cit.>. This method ensures a monotonic increase of event time stamps in the output event stream. § EVALUATIONSIn this section, we present our experiments with StreamLearner on the DEBS Grand Challenge 2017 data set with 50,000 sensor data events. Experimental Setup:We used the following two computing environments. (i) A notebook with 4 × 3.5 GHz (8 threads, Intel Core i7-4710MQ), 8 GB RAM (L1 Cache 256 KB, L2 Cache 1024 KB, L3 Cache 6144 KB), and 64 Bit support.(ii) An in-house shared memory infrastructure with 32 × 2.3 GHz (Quad-Core AMD Opteron(tm) Processor 8356), and 280 GB RAM (L1d cache 64 KB, L1i cache 64 KB, L2 cache 512 KB, L3 cache 2048 KB), and 64 Bit support. Adapting the window size W:In Figure <ref>, we show the absolute throughput of StreamLearner on the y-axis and different window sizes W on the x-axis using the notebook for a different number of threads. Clearly, larger window size leads to lower throughput as computational overhead grows. We normalized this data in Figure <ref> to the interval [0,100] to compare the relative throughput improvements for the different number of threads. Clearly, the benefit of multi-threading arises only for larger window sizes due to the constant distribution overhead that can not be compensated by increased parallelism because each thread has only little computational tasks between points of synchronization (on the splitter and on the merger). Overall scalability is measured in Figure <ref>. It can be seen that StreamLearner scales best for data-parallel problems with relatively little synchronization overhead in comparison to the computational task. For small window sizes (e.g. W=10), throughput does not increase with increasing number of workers. However, for moderate to large window sizes, scaling the number of worker threads has an increasing impact on the relative throughput: scaling from one to nine threads increases throughput by 2.5 ×. In Figure <ref>, we repeated the experiment on the shared-memory infrastructure. The first observation is that the single threaded experiments are four times slower compared to the notebook infrastructure due to the older hardware. Nevertheless, in Figure <ref>, we can see clearly that the relative throughput decreases when using a low rather than a high number of threads (e.g. for larger window sizes W>100). In Figure <ref>, we measure scalability improvements of up to 60%. Nevertheless, it can be also seen that it is not always optimal to use a high number of threads – even if the problem is highly parallelizable. Adapting the number of clusters K:In Figure <ref>, we plot the absolute throughput for a varying number of clusters and different threads. We fixed the window size to W=100. Not surprisingly, an increasing number of clusters leads to reduced throughput due to the increased computational complexity of the clustering problem. Evidently, increasing the number of threads increases the throughput up to a certain point. This is consistent with the findings above.§ CONCLUSIONStreamLearner is a distributed CEP system and API tailored to scalable event detection using Machine Learning on streaming data. Although our API is general-purpose, StreamLearner is especially well-suited to data-parallel problems – with multiple event sources causing diverse patterns in the event streams. For these scenarios, StreamLearner can enrich standard CEP systems with powerful Machine Learning functionality while scaling exceptionally well due to the pipelined incremental training and inference steps on independent models. ACM-Reference-Format | http://arxiv.org/abs/1706.08420v1 | {
"authors": [
"Christian Mayer",
"Ruben Mayer",
"Majd Abdo"
],
"categories": [
"cs.DC"
],
"primary_category": "cs.DC",
"published": "20170626145418",
"title": "StreamLearner: Distributed Incremental Machine Learning on Event Streams: Grand Challenge"
} |
#1 10,1,2,3,4]Benjamin R. FitzpatrickThe work has been supported by the Cooperative Research Centre for Spatial Information (CRCSI), whose activities are funded by the Australian Commonwealth's Cooperative Research Centres Programme. BRF gratefully acknowledges receipt of an Australian Post Graduate Award from the CRCSI. 1,2,3,4]Kerrie Mengersen [1]Mathematical Sciences School, Queensland University of Technology (QUT), Brisbane, QLD 4001, Australia [2]Cooperative Research Centre for Spatial Information (CRCSI), Carlton, VIC 3053, Australia [3]Institute for Future Environments, Queensland University of Technology (QUT), Brisbane, QLD 4001, Australia [4]ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology (QUT), Brisbane, QLD 4001, Australia A network flow approach to visualising the roles of covariates in random forests [ December 30, 2023 ================================================================================ 1 Title We propose novel applications of parallel coordinates plots and Sankey diagrams to represent the hierarchies of interacting covariate effects in random forests. Each visualisation summarises the frequencies of all of the paths through all of the trees in a random forest. Visualisations of the roles of covariates in random forests include: ranked bar or dot charts depicting scalar metrics of the contributions of individual covariates to the predictive accuracy of the random forest; line graphs depicting various summaries of the effect of varying a particular covariate on the predictions from the random forest; heatmaps of metrics of the strengths of interactions between all pairs of covariates; and parallel coordinates plots for each response class depicting the distributions of the values of all covariates among the observations most representative of those predicted to belong that class. Together these visualisations facilitate substantial insights into the roles of covariates in a random forest but do not communicate the frequencies of the hierarchies of covariates effects across the random forest or the orders in which covariates occur in these hierarchies. Our visualisations address these gaps. We demonstrate our visualisations using a random forest fitted to publicly available data and provide a software implementation in the form of an R package. Keywords: Sankey diagram; parallel coordinates plot; ensemble trees; explanatory variable; feature 1.45§ INTRODUCTION Random forests <cit.> have achieved substantial popularity in the data mining and machine learning communities as a modeling method with good performance that is relatively straightforward to tune. Random forests consist of many decision trees each fitted to a separate bootstrapped sample of the training data. These trees are grown by dividing each current terminal node on the basis of one of a random sample of the covariates. An independent random sample of the covariates is drawn for each node to be divided. The division of the observations in a terminal node into two daughter nodes is chosen as that which most reduces the value of some loss function such as the sum of the squared errors associated with the predictions. Each tree is grown in this manner until a stopping criterion is satisfied at which point no further divisions of the terminal nodes are attempted. Stopping criteria include a minimum number of observations in any terminal node, a maximum number of nodes in a tree or an entropy threshold. Trees are grown for typically hundreds to thousands of bootstrap samples of the data resulting in an ensemble of decision trees collectively termed a random forest. The prediction of an observation with a random forest is calculated as the average (or majority vote) of the predictions of this observation with each tree in the random forest. Thus while the process of fitting a random forest is relatively comprehensible, the resulting large ensemble of decision trees makes gaining a complete understanding of the roles of covariates in this model challenging. Understanding the roles of covariates in a random forest is worthwhile even if producing an accurate and generalisable model is the only objective of the modelling. This is because such understanding enables assessment of the model in terms of the current understanding of the system being modelled and a model that better reflects the nature of the system being modelled should generalise more effectively. However, modelling objectives often include both insight into the system being modelled and accurate predictions. Furthermore, models that are not interpretable in terms of the current understanding of the system being modelled may be distrusted regardless of their predictive performance. Visualising a model can assist with such interpretation along with gaining and communicating such insights. <cit.> explain the importance of both `visualisations of the data in the model space' and `visualisations of the model in the data space'. The canonical example of visualisations of the data in the model space are residuals versus fitted values plots. While important for model diagnostics, these plots do not facilitate insights into the roles of covariates in a model. Visualisations of a model in the data space can facilitate insight into the roles of covariates through depiction of the responses of the model to features of the covariates.The high dimensions of the data to which random forest are typically applied has lead the various visualisations of random forests in the associated data spaces to include graphs of the predictions obtained across subspaces or projections of the data space. Such plots typically depict the marginal effects of one or two covariates per plot and do not permit a holistic understanding of the roles of all covariates in the random forest. Another key recommendation by <cit.> is to visualise collections of models rather than just exemplar models with high predictive performance. In the context of visualising random forests this can be interpreted as a recommendation to visualise all trees constituent to a particular random forest. The recommendation to visualise collections of models could also be interpreted as recommending the comparison of visualisations produced for multiple random forests each fitted to the same data using different values for the tuning parameters. The tuning parameters for a random forest include: the size of the random sample of covariates from which a covariate is selected to define a node, those controlling the particulars of the bootstrap sampling and the parameter controlling the stopping criterion. In this paper we propose novel visualisations of the roles of covariates in random forests. We have designed these visualisations to be holistic representations of the structures of random forests that augment existing visualisations of the roles of covariates in random forests. The visualisations we propose are novel applications of parallel coordinates plots and Sankey diagrams that represent all possible paths through the decision trees that constitute a random forest. Our visualisations communicate the identities and orders of the sequences of covariates that define paths through the decision trees of a random forest. Furthermore we have designed these visualisations to foreground the paths most frequently selected across all the decision trees that constitute a random forest. In this way we seek to communicate all of the sequences of interacting covariates that are important to the predictive mechanism of a random forest in a single plot. In Section <ref> we review existing visualisations that facilitate insight into the roles of covariates in a random forest and introduce the visualisations we propose in this context. The construction and interpretation of the visualisations we propose are best explained in terms of an example so we introduce a publicly available data set and describe how we fitted a random forest to these data in Section <ref>. In Section <ref> we explain how we apply parallel coordinates plots to represent the paths through the decision trees of our example random forest. In Section <ref> we explain the how we represent the same information with Sankey diagrams. We open Section <ref> by discussing how our visualisations complement the existing collection of visualisations for investigating the roles of covariates in random forests. We then outline the new insights our visualisations enable that these other visualisations do not. We conclude by proposing elaborations upon our visualisations that could be attempted in future work. A software implementation of our visualisation methods is provided via GitHub in the form of an R <cit.> package. Details regarding how to access this package are provided in the Supplementary Materials§ VISUALISING THE ROLES OF COVARIATES The simplest visualisations of the roles of covariates in a random forest are dot or bar charts that rank covariates by some metric of importance. One such metric relates to the accumulation of the improvement in the splitting criterion across all nodes in the random forest at which the split was defined by the covariate in question <cit.>. Another such metric is calculated from the change in the accuracy with which out of bag samples are predicted when the values of the covariate in question are permuted among them <cit.>. Covariate importance may also be quantified on the basis of the variation in the predictions obtained from a sensitivity analysis conducted with respect to that covariate <cit.>. The feature contribution method may be applied to random forests fitted for classification to quantify the importance of the covariates for predicting observations of each class in turn <cit.>. Summary statistics (e.g. medians) of the contribution of each covariate to prediction of the observations in each class may be visualised as grouped bar charts. While scalar metrics of the importance of individual covariates are a useful starting point, they give little indication of the nature of the roles these covariates play in the random forest beyond the importance of that role. Further insights into the nature of the roles of individual covariates in a random forest may be sought from a family of visualisations which summarise or project the predicted surface for two or three dimensional plotting. Two examples of these sorts of plots are partial dependence (PD) plots <cit.> and variable effect characteristic (VEC) curves <cit.>. The two dimensional versions of both types of plots depict a summary of the effect of a covariate of interest upon the predicted values as a single line. Partial dependence (PD) plots may be applied to a variety of ensemble learning methods including random forests. The most frequently encountered examples are two dimensional plots with the covariate of interest mapped to the horizontal axis and the vertical axis representing either the predicted value in the case of regression or a function of the predicted probability for a response class in the case of classification. The curve in a PD plot is the result of averaging a collection of curves. Each of these curves is a projection of the predicted surface onto a plane defined by the axes for predicted values and the covariate of interest. Each of these projections is produced for the vector of covariate values (excluding the value of the covariate of interest) associated with one of the observations in the training set. Three dimensional analogues are also possible with two covariate axes and one axis for the predictions. Variable effect characteristic curves may be produced as part of a sensitivity analysis for a model <cit.>. A sensitivity analysis involves examining the variation in predicted values obtained from a model supplied with a range of covariate values and thus is not specific to any particular model structure <cit.>. The simplest form of sensitivity analysis examines the sensitivity of the model to a single covariate. Vectors of covariate observations are constructed by combining each of a sequence of values of the covariate of interest with a set of values for the other covariates which are central to their distributions (e.g. the respective means or medians of these covariates over the training data). These vectors are then used to predict the response with the model. The resulting predictions are plotted against the associated values of the covariate of interest in what <cit.> term a variable effect characteristic (VEC) curve. In the case of models fitted for classification tasks one such plot can be produced for the predicted probability of each response class. The motivation of a VEC curve is similar to that of a PD plot. A VEC curve connects a single sequence of predictions from the model where the only variation in the covariate vectors used to calculate these predictions is in the value of the covariate of interest. In contrast, the curve in the PD plot is the average of many such curves produced at each of the set of covariate values associated with an observation in the training data. While depicting overall summaries of the effect of the covariate of interest on the predicted values, any interaction between this covariate and another will not be visible from two dimensional PD plots and VEC curves. All the variation among the parallel projections at different values of the other covariates is compressed into the single average curve displayed in the PD plot and in a VEC curve we see only a single such projection. While the three dimensional analogues of these plots address this concern, they can only display evidence of two way interactions and random forests are ensembles of decision trees that describe interactions between numerous covariates. Sensitivity analyses may be conducted to investigate the sensitivity of a model to any number of covariates simultaneously <cit.>. First, a regular lattice is created which contains all combinations of sequences of values of each of the covariates under investigation. Complete vectors of covariate values are then created by combining the covariate values associated with each point in lattice with a vector containing the central values of all the covariates not under investigation. The model is then used to predict the response from each of these covariate vectors. Two dimensional visualisations can then be constructed for each covariate of interest in turn. In such visualisations the covariate of interest is mapped to the horizontal axis and the values predicted from the model mapped to the vertical axis. The distributions of predicted values obtained from the sensitivity analysis at each value of the covariate of interest are represented as boxplots or intervals on these plots. Three dimensional analogues of these plots can be constructed as heatmaps or contour plots with one covariate of interest on either axis and a summary statistic of the dispersion of predicted values from the sensitivity analysis mapped to the third dimension or colour in the plot. Like PD plots and VEC curves, individual conditional expectation (ICE) plots <cit.> display the predicted values from a model on the vertical axis and the value of a covariate of interest on the horizontal axis. However, rather than displaying an average or interval of predicted values, an ICE plot displays separate curves for the projection of the predicted surface onto these axes at each of the covariate values available in the training data (or a subset thereof). In this way ICE plots are somewhat akin to a data based sensitivity analysis where all covariates are varied. Vertically centred ICE plots and a version that plots the partial derivative of the predicted surface with respect to the covariate of interest on the vertical axis are also available. The curves (or sequences of points) in these plots may be coloured proportionally to the value of a second covariate of interest to search for evidence of two way interactions among the covariates. Among the plots produced by the Random Forest Tool (RAFT) <cit.> is a heatmap depicting a metric of the strengths of interactions between all possible pairs of covariates. These heatmaps use a metric of the strength of an interaction between covariates x_i and x_j defined on the basis of how much more or less likely a decision tree is to contain a binary partition defined using the covariate x_i if above it in the tree is a binary partition defined using the covariate x_j. While such heatmaps of sensitivity metrics for pairs of covariates allow for rapid inspection of many pairs of covariates for evidence of interactions such assessments rely on the quality of the scalar metric of pairwise sensitivity employed. In such situations it would be advisable to produce multiple heatmaps each making use of a different sensitivity metric. Insight into the nature of any interactions detected from these heatmaps would require pairwise plots such as coloured or three dimensional ICE, VEC or PD plots to be produced and inspected for each pair of interacting covariates identified from these heatmaps. The combinatorially large number of potential interactions between even relatively moderate numbers of covariates renders assessment of the nature of each potential interaction by inspection of pairwise plots a time consuming option. Furthermore, such an approach can only detect pairwise interactions. Random forests are ensembles of many decision trees, each of which has the potential to model interactions between many covariates. Consequently, a thorough understanding of the roles of covariates in a random forest will require examination of the potential for interactions of much high order than pairwise interactions. Thus, visualisations of the entire model in the data space that highlight important interactions would seem to be promising tools for identifying important interactions. The high dimensionality of the data to which random forests are often fitted make visual representation of the entire data space challenging. Parallel coordinates plots of the entire data space and multiple dimensional scaling (MDS) plots summarising the entire data space have been used for this task. Linked collections of distinct visualisations organised into interactive dashboards are a method multiple authors have adopted to summarise and convey the large volumes of information involved in visualising a random forest in the data space.The original example of an interactive dashboard for visualising random forests is the Java based random forest tool (RAFT) <cit.>. The RAFT is only available for random forests fitted for classification. <cit.> developed another such interactive dashboard using R and the R package iPlots eXtreme <cit.> for the computational performance improvements it permitted over the RAFT. <cit.> have created a system of linked interactive graphics for diagnosis and exploration of an ensemble classifier which includes some comparisons to random forests fitted to the same data. This dashboard was produced with R, ggplot2 <cit.>, plotly <cit.> and shiny <cit.> and includes visualisations of the models in the data space. The visualisations of the RAFT, <cit.> and <cit.> are useful aids for the discovery of clusters of observations among the training data and understanding covariate importance within these clusters. Both <cit.> and <cit.> use MDS plots to represent the observations in a subspace of the data space. These MDS plots use principal components calculated from the proximity matrix of the random forest being visualised. The ith row of the jth column of the proximity matrix is the proportion of trees in the random forest in which the pair of observations indexed by i and j were assigned to the same terminal node (and thus predicted to have the same response value). In both dashboards observations selected from the MDS plot are then highlighted in the other plots available in these dashboards. In this manner the random forest and the roles of covariates in the random forest can be explored interactively as the user highlights observations and groups of observations of interest. The dashboards of <cit.> and <cit.> both include parallel coordinates plots linked to the MDS plots. These parallel coordinates plots assign each covariate a separate parallel vertical axis. Separate parallel coordinates plots alternatively map to these axes the values of the covariates or the importance of these covariates to the prediction of individual observations. Each observation is represented by a line that connects these parallel vertical axes. In one of these types of parallel coordinates plots the vertical coordinate each line passes through at each axis indicates the value of the covariate represented by that axis associated with the observation represented by the line. In the other type of parallel coordinates plot the vertical coordinate each line passes through at each axis indicates the importance of the associated covariate to the prediction of the observation represented by the line. The points in the MDS plots and lines in the parallel coordinates plots may be coloured by one of a variety of attributes of the observations they represent. These attributes include: the response class of the observation, whether the observation falls above or below a threshold in one covariate and whether or not the observation was correctly classified. Points or lines selected on one plot are highlighted in all plots in the panel. The interactive nature of these visualisations, whereby users can select subsets of observations from the MDS plots and have parallel coordinate plots of just these observation produced, allows the user to examine covariate effects within different subsets of the data. Such subsets could be all the observations predicted to have a particular response class or all the observations correctly predicted to have a particular response class. These parallel coordinates plots may be inspected for common patterns among lines representing observations of particular subsets of the data and contrasts between observations from different subsets of the data. The RAFT includes a third use of parallel coordinates plots that facilitate insight into the roles of covariates in predicting individual response classes in random forests fitted for classification. <cit.> refer to these plots as `prototypes'. For a particular predicted response class, the observation is identified that has the most observations predicted to have this this same class among its nearest neighbours as defined by the proximity matrix for the random forest. The distributions of the values of the covariates among these neighbouring observations are then summarised by intervals drawn on a parallel coordinates plot with axes for each covariate. On the parallel vertical axis for each covariate the 25th percentile of the values of that covariate from the neighbourhood of observations is depicted as the lower bound of the interval, the median of these observations is depicted as the midline and the 75th percentile as the upper bound. Three groups of line segments are drawn between the parallel vertical axes to form the intervals. The first group of line segments connect the 25th percentiles on each axis, a second group connect the medians on each axis and a third group connect the 75th percentiles on each axis. In this way the distributions of covariate values most closely associated with the prediction of the focal response class are represented for all covariates on a single parallel coordinate plot. This plot is termed the `prototype' plot for that response class. This procedure is repeated for successive response classes to create separate prototype plots. Only the observations that have not yet been used to defined a prototype are considered to define the current prototype. The distributions of covariate values associated with groups of observations identified from the proximity matrix of a random forest have also been represented with coxcomb plots. <cit.> apply self organising maps (SOM) to the proximity matrix of a random forest to identify these groups. They then depict the characteristic covariate values associated with each of these groups with a coxcomb plot in the array of coxcomb plots they produce. This contrasts with the method of grouping observations into neighbourhoods to produce the prototypes plots of <cit.> though similar parallel coordinates plots of the distributions of covariate values could be produced for each of the groups of observations identified by the SOM based technique. The visualisations reviewed above collectively facilitate deep insights into the roles of covariates in a random forest. However, none of these visualisations communicate the identities and orders of covariates in the many hierarchies of interacting covariate effects that together constitute a random forest. These hierarchies of covariate effects are defined by the paths from root to leaf nodes through the decisions trees of the random forest. To further aid exploration of the roles of covariates in a random forest we propose novel applications of parallel coordinates plots and Sankey diagrams to represent all possible paths through the decision trees of a random forest. We have designed these visualisations to communicate the identities and orders of the covariates that define these paths and to foreground the paths most frequently selected across the decision trees of a random forest. In this way we present all of the sequences of interacting covariate effects that constitute a random forest in a single plot. This plot also foregrounds the sequences of covariate effects that occur most frequently throughout the random forest.§ AN EXAMPLE RANDOM FOREST We demonstrate our visualisations using a random forest fitted to some publicly available data for a classification problem. The classification problem is that of ground cover classification from remotely sensed imagery. These data consist of 6435 ground truthed pixels of satellite imagery. The response variable is a categorical variable which discretises the type of ground cover present within these pixels into six classes. Each observation of the ground cover class of a pixel is accompanied by the reflectance values of all pixels in a three pixel by three pixel rectangular grid centred on the pixel for which the ground cover class was observed. The reflectance data for each pixel are available in four regions of the electromagnetic spectrum. Thus there are 36 covariates. These data are available from the University of California, Irvine Machine Learning Repository <cit.> as the Statlog Landsat Satellite Data Set and are included in the R package `mlbench' <cit.>. For our example we have used the the `randomForest' package <cit.> which is the reference implementation of random forests in the R language and environment for statistical computing <cit.>. We used the `caret' package <cit.> to construct 100 cross validation folds each with proportions of observations of the ground cover classes that that were similar to those present in the full data set. We then performed 100 fold cross validation with the `caret' package to choose the size of the random sample of covariates that were available to define each binary partition during the model fitting process (the tuning parameter `mtry' in the `randomForest' package). This procedure selected the value of eight for `mtry' to which we set this tuning parameter when we fitted a 500 tree random forest to the full data set. The result was a random forest that correctly classified 75.7% of the observations and had the covariate importance scores plotted in Supplementary Figure 1.§ PARALLEL COORDINATES PLOTS OF PATHS The hierarchies of interacting covariate effects that collectively constitute a random forest define the paths from root nodes to leaf nodes in the decision trees of that random forest. We introduce our parallel coordinates plots of the paths through a random forest by first explaining how such plots can be created for all the paths through a single decision tree. We then explain how this technique may be applied to visualise all the paths through a random forest composed of many such decision trees. Figure <ref> is a circular dendrogram representing the first tree of the 500 tree random forest we fitted to the ground cover classification data. The left plot in Figure <ref> depicts the full dendrogram with a square box outlining the section that has been magnified and reproduced in the right plot in this Figure. The nodes are represented by labels which specify the covariate that defined that node. Edges are represented by arrows exiting each node and terminal nodes have each been labeled `Terminus'. To produce this plot we extracted the first tree from the random forest and converted it into a directed network. We then plotted this network with the `igraph' package<cit.>. In this Figure the root node is located at the centre of the dendrogram and is labeled with the covariate which defined this node: `x.17'. The leaf nodes, each labeled `Terminus', may be seen at the ends of the sequences of arrows emanating from the root node. We also represent all the paths through the first tree of the random forest fitted to the ground cover classification data as a parallel coordinates plot in Figure <ref>.Once the tree was stored as a directed network the `igraph' package was used to determine the paths from the root node to each of the terminal `leaf' nodes. Each of these paths is visible in the dendrogram in Figure <ref>. We then aggregated and transformed the data describing these paths into a format we could plot as a parallel coordinates plot using the `pairs' geometry from the R package `ggplot2' <cit.>. This parallel coordinates plot is presented in Figure <ref>. The data aggregations and transformations necessary to produce each of the visualisations we introduce in this paper are performed with the aid of the R packages `dplyr' <cit.>, `tidyr' <cit.>, `purrr' <cit.>, `magrittr' <cit.> and `forcats' <cit.>. In Figure <ref> the paths from the root node to each of the terminal nodes are represented by sequences of line segments. The orders of nodes along these paths are represented by the horizontal order of the parallel vertical axes from left to right. The root node is represented by the left most vertical axis which is labeled `Node 1'. Here we will refer to the number of nodes along a path from the root node to a particular node as the rank of that particular node. The covariates defining nodes are represented by the gradations on these parallel vertical axes. Terminal nodes of each rank are represented by the first gradation on each parallel vertical axis. The tree represented by the parallel coordinates plot in Figure <ref> has a root node defined by the covariate `x.17'. Subsequently, the representations of each path through this tree originates on the vertical axis labeled Node 1 at the gradation representing the covariate `x.17'. Two edges exit the root node, one of these edges connects to the node defined by the covariate `x.24' and the other connects to the node defined by covariate `x.35'. These two edges are represented by line segments on the parallel coordinates plot between the representation of the root node at the position of the covariate `x.17' on the axis labeled Node 1 and the nodes represented by the positions of the covariates `x.24' and `x.35' on the axis labeled Node 2. The darkness of a line segment in Figure <ref> represents the number of paths through the decision tree that passed through a pair of nodes with ranks represented by the pair vertical axes connected by that line segment and defined by the covariates represented by the vertical coordinates on these axes which the line segment connects. For example, the darkness of the left most pair of line segments in Figure <ref> depicts how a larger proportion of the paths through the decision tree represented in this Figure commenced at the root node and passed through a second node defined by the covariate `x.24' than started at the root node and passed though a second node defined by the covariate `x.35'. This feature of the decision tree may be corroborated by inspection of Figure <ref>. Figure <ref> is a parallel coordinates plot arranged identically to the parallel coordinates plot in Figure <ref>. In Figure <ref> we have represented the first five nodes along all of the paths through all of the 500 decision trees that constitute our example random forest. From this plot it is apparent that the covariates most frequently selected to define the root nodes of these 500 decision trees are `x.17', `x.13', `x.21' and `x.29'. Our software implementation of these parallel coordinates plots gives the user the option of producing a coloured version of these plots with a colour scale from the R package `viridis' <cit.> replacing the grey scale used in Figure <ref>. § SANKEY DIAGRAMS OF PATHS Sankey diagrams have been used to represent the flow of some quantity of interest between nodes in a network. The nodes are represented as rectangular blocks and the flows between nodes are represented as curved links between the blocks. The magnitude of a flow between nodes is represented by the width of the link representing this flow. Conversion of the data used to generate Figure <ref> into a single directed network facilitated the representation of this information as a Sankey diagram with the R package `networkD3' <cit.>.Figure <ref> displays a Sankey diagram of the first five nodes along all paths through all trees in our example random forest.The organisation of this Sankey diagram is similar to that of the parallel coordinates plots in Figures <ref> and <ref>. In Figure <ref> the rectangular blocks represent groups of nodes in the decision trees of the random forest that were the same number of nodes along a path from a root node to a leaf node and were defined by the same covariate. The blocks are organised into columns. The sequence of columns from left to right represents the order of nodes along paths through the decision trees from root nodes to leaf nodes. The left most column in this Sankey diagram contains blocks that represent the various root nodes of the decision trees. The next column to the right contains blocks that represent the second nodes along the paths from root nodes to terminal nodes and so forth. If we count the number of nodes along a path from the root node to a particular node this can be thought of as the rank of that node along the path. In our Sankey diagrams each block is labeled with the rank of the nodes it represents and covariate that defines these nodes. Thus a block labeled `Node.1_x.17' represents all the root nodes defined by the covariate `x.17'. The height of each block reflects the proportion of all the paths through all the trees that had a node at the position along a path represented by the horizontal position of the block where this node was defined by the covariate with which the block is labeled. The width of a link between a pair of blocks represents the proportion of the paths through the trees in which an edge connected nodes with the characteristics encoded in the horizontal positions and labels of these blocks. The `networkD3' package produces interactive visualisations that are displayed in a web browser and may also be written out as a HTML file.Examples of interactive Sankey diagrams produced with the R package that accompanies this paper have been included on the project website. A link to this website is provided on the GitHub page that provides the R package (see Supplementary Materials). These examples include a Sankey diagram of the paths through the trees of a random forest fitted to Anderson's Iris data <cit.>.These data are available in the base installation of R. The Iris data contain a sufficiently small number of covariates that it was feasible to colour blocks and links by the identities of the associated covariates and still be able to visually distinguish these colours. A static version of this Sankey diagram is included as Supplementary Figure 2. Code to produce each of these Sankey diagrams is included among the examples in the R package.§ DISCUSSION We have proposed visualisations that facilitate insights into the roles of covariates in a random forest additional to those available from the collection of visualisations that currently exists for this purpose. Our novel applications of Sankey diagrams and parallel coordinates plots take a network flow approach to representing all possible paths through a random forest in manners that foreground the paths most frequently selected. In this way, our visualisations emphasize the sequences of covariate interactions that are most important to the predictive mechanism of a random forest. This visual communication of the identities and orders of covariates in these sequences along with the frequencies of these sequences has not been achieved in other visualisations of random forests to date. Our visualisations can help a practitioner identify interesting pairs of covariates to examine with pairwise plots to assess interactions between covariates. This will be particularly useful in situations where the number of covariates is large enough that inspection of a separate pairwise plot for every possible pairwise interaction would involve a substantial investment of time. In this manner our plots complement the simple heatmaps of a single scalar metric of interaction strength for each potential pair of covariates allowing for more detailed inspection of the potential for important interactions across the whole forest. Our plots also make apparent important interactions of substantially higher order than pairwise interactions which cannot be discovered from heatmaps of metrics for pairwise interaction strength. Our visualisations complement parallel coordinates plots of covariate `prototypes' for each response class in classification problems. Our plots depict the paths through a random forest in terms of the ordered hierarchies of covariate effects that define these paths. In contrast, prototype plots represent the distributions of covariates from neighbourhoods of observations defined by the proximity matrix of the random forest. Our visualisations display the orders of covariates in the hierarchies of interacting covariate effects involved in the predictive mechanism of a random forest while prototype plots do not display these orders. In contrast, prototype plots display summaries of the values of covariates associated with prediction of particular response classes while our plots do not display these values. The Sankey diagrams and parallel coordinates plots we propose in this paper also provide visual representations of the structure of the random forest as a whole. The random forest fitted in our case study may be seen to have a few covariates that frequently define root nodes, several covariates that occasionally define root nodes and other covariates that rarely or never define root nodes. This asymmetry in the proportions of nodes defined by each covariate at the roots of the trees may be seen to decrease steadily among groups of nodes increasing further from the root nodes. By the fifth node along paths from the root nodes most of the covariates are defining nodes in approximately the same proportion of paths. These visualisations also depict the proportion of nodes of each distance from the root node that are terminal nodes. This provides another overall visual summary of the structure of the random forest as a whole. Our visualisations of the structures of entire random forests would facilitate comparisons of groups of random forests. Comparisons of groups of models are one of the recommendations <cit.>. Groups of random forests it would be pertinent to compare would be those obtained from fitting random forests to the same data set using different values for the tuning parameters. In our case study comparisons of our visualisations to the scalar rankings of covariate importance plotted in Supplementary Figure 1 provides informative contrasts. For example, the covariate that makes the greatest contribution to the predictive performance of the random forest is `x.18'. The covariate `x.18' very rarely defines root nodes but is the most common covariate to define the third node along paths from root to leaf nodes. Thus the contribution of `x.18' to the predictive performance of the random forest can be seen to be in no small part due to the interaction of this covariate with numerous other pairs of covariates that define the second and root nodes of the trees that constitute the random forest. Of the two styles of visualisations we have proposed in this paper, we find the Sankey diagrams more informative. The Sankey diagrams convey the relative frequencies with which nodes of a particular distance from the root node were defined by different covariates through the relative heights of the rectangular blocks representing these groups of nodes. This information is not readily available from the parallel coordinates plots and may only be very approximately inferred from the number and darkness of line segments exiting the representation of groups of nodes of a particular rank that were defined by a particular covariate. In the interactive Sankey diagrams a user may hover the mouse over a rectangular block that represents a group of nodes and have all the links that represent edges connecting to those nodes highlighted. This ability to foreground a particular feature of the Sankey diagram is very useful given the volume of information that is being represented in these plots. Furthermore, highlighting a link results in a text box being displayed. The text box identifies the covariates which define the nodes connected by the edges represented by this link. The text box also specifies the frequency of edges between such nodes across the random forest. Both the readily available interactivity and the additional information available visually from the Sankey diagramslead us to favour these over the parallel coordinates plots as a method for visualising the roles of covariates in a random forest. While our visualisations have been developed for random forests they could be applied to other techniques that utilise one or more decision trees. All paths through a CART <cit.> could be depicted with our techniques. Our techniques could also be extended to visualise the paths through the decision trees of other methods based on ensembles of decision trees such as Gradient Boosted Machines <cit.>. One relatively straightforward elaboration upon our visualisations of random forests for classification would be to produce separate visualisations for each of the predicted response classes.This would involve the production of a panel of visualisations. Each visualisation in the panel would represent all paths through the random forest that lead to prediction of a particular response class. Such panels could accompany an overall visualisation of all the paths and would be of particular interest in situations where there was substantial inequality among the numbers of observations of different response classes. These response class specific visualisations could also suggest revealing horizontal orders for the vertical axes in the associated prototype plots. Another extension of our visualisations could involve giving users the option to apply some threshold so that only the most common nodes at each position along the paths were displayed in the plot. A user could then experiment with values for this threshold to produce visual summaries of the random forest of varying granularity. This would be particularly useful for visualising random forests constructed from large sets of covariates. Further interactivity could be added whereby clicking a particular block or link produced additional visualisations of the characteristics of the random forest represented by that block or link. For example clicking a particular block could produce a visualisation of the empirical probability density of the threshold values of the covariate that defines all the nodes represented by this block. Another example could involve clicking a link between two nodes resulting in the production of visualisations pertinent to investigation of the nature of the interaction between the covariates defining these two nodes. For instance, an ICE plot could be produced for one of these covariates coloured by the second covariate. The observations that were passed down this link could be drawn as points on this ICE plot. Development of such interactive `dashboard' style visualisations could be attempted with a combination of technologies such as R, `plotly' and `Shiny' but would be a substantial project.SUPPLEMENTARY MATERIAL R-package: An R package implementing our methods for visualising the roles of covariates in random forests is provided at <https://github.com/brfitzpatrick/forestviews>. Supplementary Figure 1 A dot plot of the covariate importance scores obtained from our example random forest fitted to the ground cover classification data. Supplementary Figure 2 A Sankey diagram of the first five nodes of all paths through a random forest fitted to Anderson's Iris data. Colour depicts the identities of the covariates associated with the different elements of this diagram.30 urlstyle[Allaire et al.(2017)Allaire, Gandrud, Russell, and Yetman]networkD32017 J. Allaire, C. Gandrud, K. Russell, and C. Yetman. networkD3: D3 JavaScript Network Graphs from R, 2017. URL <https://CRAN.R-project.org/package=networkD3>. R package version 0.4.[Anderson(1935)]Anderson1935 E. Anderson. The irises of the Gaspe Peninsula. Bulletin of the American Iris Society, 59:0 2–5, 1935.[Bache and Wickham(2014)]magrittr2016 S. M. Bache and H. Wickham. magrittr: A Forward-Pipe Operator for R, 2014. URL <https://CRAN.R-project.org/package=magrittr>. R package version 1.5.[Breiman(2001)]Breiman2001 L. Breiman. Random Forests. Machine Learning, 450 (1):0 5–32, 2001.[Breiman and Cutler(2004)]Breiman2004 L. Breiman and A. Cutler. Manual–Setting Up, Using, And Understanding Random Forests V4.0, 2004. URL <https://www.stat.berkeley.edu/ breiman/Using_random_forests_v4.0.pdf>.[Breiman et al.(1984)Breiman, Friedman, Stone, and Olshen]Breiman1984 L. Breiman, J. Friedman, C. Stone, and R. Olshen. Classification and Regression Trees. The Wadsworth and Brooks-Cole statistics-probability series. Taylor & Francis, 1984.[Chang et al.(2017)Chang, Cheng, Allaire, Xie, and McPherson]shiny2017 W. Chang, J. Cheng, J. Allaire, Y. Xie, and J. McPherson. shiny: Web Application Framework for R, 2017. URL <https://CRAN.R-project.org/package=shiny>. R package version 1.0.3.[Cortez and Embrechts(2013)]Cortez2013 P. Cortez and M. J. Embrechts. Using sensitivity analysis and visualization techniques to open black box data mining models. Information Sciences, 225:0 1–17, 2013.[Csardi and Nepusz(2006)]igraph G. Csardi and T. Nepusz. The igraph software package for complex network research. InterJournal, Complex Systems:0 1695, 2006. URL <http://igraph.org>.[da Silva et al.(2017)da Silva, Cook, and Lee]daSilva2017 N. da Silva, D. Cook, and E.-K. Lee. Interactive Graphics for Visually Diagnosing Forest Classifiers in R. 2017. URL <http://arxiv.org/abs/1704.02502>.[Friedman(2001)]Friedman2001 J. H. Friedman. Greedy Function Approximation: A Gradient Boosting Machine. The Annals of Statistics, 290 (5):0 1189–1232, 2001.[Garnier(2017)]viridis2016 S. Garnier. viridis: Default Color Maps from `matplotlib', 2017. URL <https://CRAN.R-project.org/package=viridis>. R package version 0.4.0.[Goldstein et al.(2015)Goldstein, Kapelner, Bleich, and Pitkin]Goldstein2015 A. Goldstein, A. Kapelner, J. Bleich, and E. Pitkin. Peeking Inside the Black Box: Visualizing Statistical Learning with Plots of Individual Conditional Expectation. Journal of Computational and Graphical Statistics, 240 (1):0 44–65, 2015.[Hastie et al.(2009)Hastie, Tibshirani, and Friedman]Hastie2009.15.3.2 T. Hastie, R. Tibshirani, and J. Friedman. Random Forests. In The Elements of Statistical Learning Data Mining, Inference, and Prediction, chapter 15, pages 587–604. Springer, 2nd edition, 2009.[Kuhn et al.(2017)Kuhn, Wing, Weston, Williams, Keefer, Engelhardt, Cooper, Mayer, Kenkel, the R Core Team, Benesty, Lescarbeau, Ziem, Scrucca, Tang, and Candan.]caret M. Kuhn, J. Wing, S. Weston, A. Williams, C. Keefer, A. Engelhardt, T. Cooper, Z. Mayer, B. Kenkel, the R Core Team, M. Benesty, R. Lescarbeau, A. Ziem, L. Scrucca, Y. Tang, and C. Candan. caret: Classification and Regression Training, 2017. URL <https://CRAN.R-project.org/package=caret>. R package version 6.0-76.[Leisch and Dimitriadou(2010)]mlbench2010 F. Leisch and E. Dimitriadou. mlbench: Machine Learning Benchmark Problems, 2010. URL <https://CRAN.R-project.org/package=mlbench>. R package version 2.1-1.[Liaw and Wiener(2002)]randomForest2002 A. Liaw and M. Wiener. Classification and regression by randomforest. R News, 20 (3):0 18–22, 2002. URL <http://CRAN.R-project.org/doc/Rnews/>.[Newman et al.(1998)Newman, Hettich, Blake, and Merz]Newman1998 D. Newman, S. Hettich, C. Blake, and C. Merz. UCI Repository of machine learning databases, 1998. URL <http://www.ics.uci.edu/ mlearn/MLRepository.html>.[Palczewska et al.(2014)Palczewska, Palczewski, Robinson, and Neagu]Palczewska2014 A. Palczewska, J. Palczewski, R. M. Robinson, and D. Neagu. Interpreting Random Forest Classification Models Using a Feature Contribution Method. In T. Bouabana-Tebibel and S. H. Rubin, editors, Integration of Reusable Systems, volume 263, pages 193–218. Springer International Publishing, 2014.[Płoński and Zaremba(2014)]Plonski2014 P. Płoński and K. Zaremba. Visualizing Random Forest with Self-Organising Map. In L. Rutkowski, M. Korytkowski, R. Scherer, R. Tadeusiewicz, L. A. Zadeh, and J. M. Zurada, editors, Artificial Intelligence and Soft Computing: 13th International Conference, ICAISC 2014, Zakopane, Poland, June 1-5, 2014, Proceedings, Part II, pages 63–71. Springer International Publishing, 2014.[Quach(2012)]Quach2012 A. T. Quach. Interactive Random Forests Plots. PhD thesis, Utah State University, 2012.[R Core Team(2017)]R2017 R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2017. URL <https://www.R-project.org/>.[Sievert et al.(2016)Sievert, Parmer, Hocking, Chamberlain, Ram, Corvellec, and Despouy]plotly2017 C. Sievert, C. Parmer, T. Hocking, S. Chamberlain, K. Ram, M. Corvellec, and P. Despouy. plotly: Create Interactive Web Graphics via `plotly.js', 2016. URL <https://CRAN.R-project.org/package=plotly>. R package version 4.5.6.[Urbanek(2011)]Urbanek2011 S. Urbanek. iPlots eXtreme: next-generation interactive graphics design and implementation of modern interactive graphics. Computational Statistics, 260 (3):0 381–393, 2011.[Wickham(2009)]Wickham2009 H. Wickham. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2009. ISBN 978-0-387-98140-6. URL <http://ggplot2.org>.[Wickham(2016)]purrr2016 H. Wickham. purrr: Functional Programming Tools, 2016. URL <https://CRAN.R-project.org/package=purrr>. R package version 0.2.2.[Wickham(2017a)]forcats2017 H. Wickham. forcats: Tools for Working with Categorical Variables (Factors), 2017a. URL <https://CRAN.R-project.org/package=forcats>. R package version 0.2.0.[Wickham(2017b)]tidyr2017 H. Wickham. tidyr: Easily Tidy Data with `spread()' and `gather()' Functions, 2017b. URL <https://CRAN.R-project.org/package=tidyr>. R package version 0.6.2.[Wickham and Francois(2016)]dplyr2016 H. Wickham and R. Francois. dplyr: A Grammar of Data Manipulation, 2016. URL <https://CRAN.R-project.org/package=dplyr>. R package version 0.5.0.[Wickham et al.(2015)Wickham, Cook, and Hofmann]Wickham2015 H. Wickham, D. Cook, and H. Hofmann. Visualizing Statistical Models: Removing the Blindfold. Statistical Analysis and Data Mining: The ASA Data Science Journal, 8:0 203–225, 2015. | http://arxiv.org/abs/1706.08702v1 | {
"authors": [
"Benjamin R. Fitzpatrick",
"Kerrie Mengersen"
],
"categories": [
"stat.OT"
],
"primary_category": "stat.OT",
"published": "20170627075822",
"title": "A network flow approach to visualising the roles of covariates in random forests"
} |
MLMC Method for Statistical Model Checking of Hybrid SystemsS. Esmaeil Zadeh Soudjani R. Majumdar T. NagapetyanMax Planck Institute for Software Systems, Kaiserslautern, Germany{Sadegh,Rupak}@mpi-sws.org Department of Statistics, University of Oxford, United [email protected] Lecture Notes in Computer Science Authors' Instructions Multilevel Monte Carlo Method for StatisticalModel Checking of Hybrid Systems Sadegh Esmaeil Zadeh Soudjani1 Rupak Majumdar1 Tigran Nagapetyan2 December 30, 2023 ===============================================================================We study statistical model checking of continuous-time stochastic hybrid systems. The challenge in applying statistical model checking to these systems is that one cannot simulate such systems exactly. We employ the multilevel Monte Carlo method (MLMC) and work ona sequence of discrete-time stochastic processes whose executions approximate and convergeweakly to that of the original continuous-time stochastic hybrid system withrespect to satisfaction of the property of interest.With focus on bounded-horizon reachability, we recast the model checking problem as the computation of the distribution of the exit time,which is in turn formulated as the expectation of an indicator function.This latter computation involves estimating discontinuous functionals, which reduces the bound onthe convergence rate of the Monte Carlo algorithm.We propose a smoothing step with tunable precision and formally quantify the error of theMLMC approach in the mean-square sense, which is composed of smoothing error, bias, and variance. We formulate a general adaptive algorithm which balances these error terms. Finally, we describe an application of our technique to verify a model of thermostatically controlled loads.§ INTRODUCTION Continuous-time stochastic hybrid systems () are a natural model for cyber-physical systems operating under uncertainty <cit.>.A has a hybrid state space consisting of discrete modes and, for each mode, a set of continuous states (called the invariant).In each mode, the continuous state evolves according to a stochastic differential equation (SDE) in continuous time. Transition from one discrete mode to another may be activated in two ways. The continuous state may hit the boundary of the invariant and make a forced transition according to a discrete stochastic transition kernel. Alternatively, the process may spontaneously change its discrete mode according to a continuous-time Markov chainwhose rates depend on the hybrid state.We consider quantitative analysis of temporal properties of <cit.>. The fundamental analysis problem, called probabilistic reachability, consistsin computing the probability that the state of a exits a given safe set within a given bounded time horizon. Since analytic solutions are not available, there are two common approaches. The first approach is numerical model checking that relies on the exact or approximate computation of the measureof the executions satisfying the temporal property. The second approach, called statistical model checking, relies on finitely many sample executions of the system,and employs hypothesis testing to provide confidence intervals for the estimate of the probability.Statistical model checking has proven to be computationally more efficient than numerical modelchecking as it only requires the system to be executable.Thus, it can be applied to larger classes of systems and of specifications <cit.>. The main underlying assumption in all statistical model checking techniques is the abilityto sample from the space of executions of the system. Unfortunately, we cannot compute exact simulations for the general class of due to the process evolution being continuous in both time and space. In this paper, we describe a statistical model checking approach to usingthe multilevel Monte Carlo (MLMC) method <cit.>,which does not require exact executions of the system.Our procedure works as follows. First, we formulate the quantitative analysis problem as computing the distribution of the first exit time of the system from the given safe set. Then, we build a sequence of approximate models whose executions converge weakly (or in expectation)to the execution of the concrete system. Although these approximate models can be used separately in the classical setting of statistical model checking in order to compute estimates of the exit time, the MLMC method can take advantage of coupling between approximate executions with different time resolutions to provide better convergence rates.An important challenge in applying the MLMC technique to the quantitative analysis of is that a discontinuous function is applied to the first exit time.While MLMC can be applied to discontinuous functions, the convergence rates we can guarantee are poor. We propose a smoothing step that replaces the discontinuous function with a continuous approximation and show that the replacement decreases the overall computation cost. Finally, we analyze the asymptotic computational cost of the MLMC approach for a given error bound. We propose an adaptive algorithm which balances errors due to bias, variance, and smoothing, and which tunes the hyperparameters of the algorithm on the fly.We illustrate our technique on an example model of thermostatically controlled loads.Related work. Formal definitions of various classes of continuous-time probabilistic hybrid models arepresented in <cit.>, together with a comparison. Over such models, <cit.> has formalized the notion of probabilistic reachability,<cit.> has proposed a computational technique based on convex optimization, <cit.> has provided discretization techniques with formal error bounds, and <cit.> has developed an approach based on satisfiability modulo theory.An alternative approach towards formal, finite approximations of continuous-time stochastic models is discussedin <cit.> and extended in <cit.> to switching diffusions. These approaches generally suffer from curse of dimensionality and are not applicable to large dimensional models.For discrete-time stochastic hybrid models probabilistic reachability (and safety) has been fully characterized in <cit.> andcomputed via software tools <cit.> that use finite abstractions. The methods can be extended to more general probabilistic temporal logics <cit.>. These techniques assume discrete-time dynamics and cannot be extended to . An overview of statistical model checking techniques can be found in <cit.>. The paper <cit.> employs statistical model checking for verifying unbounded temporal properties. The paper <cit.> has discussed the use of importance sampling to address the issue of rare events in statistical verification of cyber-physical systems. A distributed implementation of statistical model checking is proposed in <cit.> and a set-oriented method for statistical verification of dynamical systems is presented in <cit.>.Employing multigrid ideas to reduce the computational complexity (in terms of expected number of arithmetic operations) ofestimating an expected value using Monte Carlo path simulations is initially proposed in <cit.>in the context of stochastic differential equations. MLMC has a better asymptotic complexity and by its nature allows to build consecutive approximations,which can balance the bias and variance.The general paradigm with adequate modifications has shown significant gains in modeling jump-diffusion SDEs <cit.>and in fault tolerance applications <cit.>. A more detailed overview of applications of MLMC can be found in <cit.>. The MLMC for estimating distribution functions is described in the recent paper <cit.>, which is adapted to our setting.The article is structured as follows. In Section <ref>, we define the model and the probabilistic reachabilityproblem. In Sections <ref> and <ref>, we discuss the standard Monte Carlo technique and the MLMC method, respectively, and compare their convergence rates.We then discuss two technical modifications: applying a smoothing operator to the discontinuousfunction of exit time (Section <ref>) and anadaptive MLMC algorithm for estimating the hyperparameters (Section <ref>).In Section <ref>, we provide simulation results for an example. § MODEL DEFINITIONWe study statistical model checking for the rich class of continuous-time stochastic hybrid systems (). §.§ Continuous-Time Stochastic Hybrid Systems A continuous-time stochastic hybrid system is a tuple ℋ = (Q,𝒳, b, σ,x_0,r) where the components are defined as follows. States Q is a countable set of discrete states (modes) and 𝒳: Q→𝒫(ℝ^n) maps each mode q∈ Q to an open set 𝒳(q)⊆ℝ^n, called the invariant for the mode q. A state (q,z) with q∈ Q and z∈𝒳(q) is called a hybrid state. The hybrid state space X is defined as X = (q, z)| q∈ Q, z∈𝒳(q). We write ∂ Z for the boundary of a set Z and define ∂ X := (q,z)| q∈ Q, z∈∂𝒳(q). Evolution b: X→ℝ^n is a vector field and σ: X→ℝ^n× m is a matrix-valued function, with n,m∈ℕ_0, where X is the hybrid space defined in (<ref>). For each q∈ Q, define the following SDE: dz(t) = b(q,z(t))dt+σ(q,z(t))dW_t, where (W_t,t≥ 0) is an m-dimensional standard Wiener process in a complete probability space. We assume functions b(q,·):𝒳(q)→ℝ^n and σ(q,·):𝒳(q)→ℝ^n× m are bounded and Lipschitz continuous for all q∈ Q. The assumption ensures the existence and uniqueness of the solution of the SDEs in (<ref>). Initial State x_0∈ X is the initial state of the system; Transition Kernel r:∂ X× Q→[0,1] is a discrete stochastic kernel which governs the switching between the SDEs defined in (<ref>). That is, for all q∈ Q, we assume r(·,q) is measurable and, for all x∈∂ X, the function r(x,·) is a discrete probability measure. Intuitively, an execution of a starts in the initial state x_0, and evolves according to the solution of the diffusion process (<ref>) for the current mode until it hits the boundary of the invariant of the current mode for the first time. At this point, a new mode q' is chosen according to the transition kernel r and the execution proceeds according to the solution of the diffusion process for q', and so on.We need the following definitions. Let z^q(t),q∈ Q be the solution of diffusion process (<ref>) starting from z^q(0)∈𝒳(q). Define t^∗(q) as the first exit time of z^q(t) from the set 𝒳(q),t^∗(q) := inf{t∈ℝ_>0∪{∞},such thatz^q(t)∈∂𝒳(q)}.A stochastic hybrid process, describing the evolution of a , is obtained by theconcatenation of diffusion processes {z^q(t),q∈ Q} together with a jumping mechanism given bya family of first exit times t^∗(q); we make this formal in Definition <ref>. A stochastic process x(t) = (q(t),z(t)) is called an execution of ℋ if there exists a sequence of stopping times T_0 = 0<T_1<T_2<… such that for all k∈ℕ_0: * x(0) = (q_0,z_0)∈ X is the initial state of ℋ; * for t∈[T_k,T_k+1), q(t) = q(T_k) is constant and z(t) is the solution of SDE dz(t) = b(q(T_k),z(t))dt+σ(q(T_k),z(t))dW_t, where W_t is the m-dimensional standard Wiener process; * T_k+1 = T_k+t^∗(q(T_k)) where t^∗(q(T_k)) is the first exit time from the mode q(T_k) as defined in (<ref>); * The probability distribution of q(T_k+1) is governed by the discrete kernel r((q(T_k),z(T_k+1^-)),·) and z(T_k+1) = z(T_k+1^-), where z(T_k+1^-) := lim_t↑ T_k+1z(t). For simplicity of exposition, we have put the following restrictions on the model ℋ in Definition <ref>. First, the model includes only forced jumps activated by reaching the boundaries of the invariant sets ∂𝒳(q), q∈ Q and does not capture spontaneous jumps activated by Poisson processes. Second, the continuous state z(t) remains continuous at the switching times as declared in Definition <ref>. The approach of this paper is still applicable for models without these restrictions by modifying the time discretization scheme presented in Section <ref>.§.§ Example: Thermostatically Controlled Loads Household appliances such as water boilers/heaters, air conditioners, and electric heaters –-all referred to as thermostatically controlled loads (TCLs)-– can store energy due to their thermal mass.TCLs have been extensively studied <cit.> for their role in energy management systems. TCLs generally operate within a dead-band around a temperature set-point and are naturally modeled using . The temperature evolution in a cooling TCL can be characterized by the following SDE: dθ(t) = 1/CR(θ_a - q(t) R P_rate-θ(t))dt+σ(q(t)) dW_t,where θ_a is the ambient temperature, P_rate is the energy transfer rate of the TCL, and R and C are the thermal resistance and capacitance, respectively.The noise term W_t in (<ref>) is a standard Wiener process. The model of the TCL has two discrete modes. When q(t) = 0, we say the TCL is in the OFF mode at time t, and when q(t) = 1, we say it is in the ON mode.The temperature of the cooling TCL is regulated by a controlsignal q(t^+) = f(q(t),θ(t)) based on discrete switching as f(q,θ)= {[ 0, θ≤θ_s - δ_d/2 =: θ_-; 1, θ≥θ_s + δ_d/2 =: θ_+; q,else, ].where θ_s denotes a temperature set-point and δ_d a dead-band. Together, θ_s and δ_d characterize an operating temperature range.The model can be described by the ℋ_TCL = (Q,𝒳, b, σ,x_0,r), where * Q = {0,1} with the invariants 𝒳(0) = (-∞,θ_+) and 𝒳(1) = (θ_-,+∞) * state space of the model X = {0}× (-∞,θ_+)∪{1}× (θ_-,+∞) * b(q,θ) = 1/CR(θ_a - q R P_rate-θ) for all (q,θ)∈ X * σ(0,θ) = σ(0),σ(1,θ) = σ(1) for all (q,θ)∈ X * r(q^+| q,θ) is the Kronecker delta with q^+= f(q,θ). §.§ Problem DefinitionFor a given random variable defined on the executions of a , we study the problem ofestimating its distribution function. Let Y be a real-valued random variable defined on the executions of ℋ. Estimate F_Y(s) := ℙ(Y≤ s), the distribution of Y for a given s∈ℝ. Consider a ℋ with state space X, a safe set A⊂ X, assumed to be measurable, and a time interval [0,s]⊂ℝ_≥ 0. The safety problem asks to compute the probability that theexecutions of ℋ will stay in A during time interval [0,s]. The safety problem is dual to the reachability problem and has a fundamental role in model checking for . By taking Y in Problem <ref> to be the first exit time of the system from A, we reduce the safety problem to Problem <ref>. [Probabilistic Safety] Compute the probability that an execution of the ℋ, with initial condition x_0∈ X, remains within a measurable set A during the bounded time horizon [0,s]: ℙ(ℋ is safe over [0,s]) = ℙ(Y> s) = 1-F_Y(s) where Y:= min{t∈ℝ_≥ 0∪{∞} |x(t)∉ A, x(0) = x_0} and F_Y(s) = ℙ(Y≤ s). The random variable Y defined in Problem <ref> is in fact the first exit time of the system ℋ from the safe set A and its distribution can be represented as the expectation of an indicator functional: F_Y(s)=(1_(-∞,s] (Y)).[Specification of interest for TCL] Although the switching mechanism (<ref>) is designed to keep the temperature inside the interval [θ_-,θ_+], there is still a chance that the temperature goes out of this interval due to the Wiener process W_t. Define a random variable Y=max{θ_t,t∈[0,s]}. We aim to estimate the probability ℙ(Y≤θ_++0.1·δ_d). Analytic solution of Problems <ref>-<ref> is infeasible for the class of .Numerical computation of the solution has been investigated for restrictive subclasses of <cit.>. In this work, we propose an approximate computation technique with a confidence bound.Our technique based on MLMC substantially improves the computational complexity of the standard Monte Carlo method. We first discuss standard Monte Carlo (SMC) method in Section <ref> and then present the MLMC method in Section <ref>. § STANDARD MONTE CARLO METHODIn order to compute the quantities of interest in Problems <ref>-<ref> we need to estimateP = g(Y),where Y is a function of the execution of ℋ, g:ℝ→ℝ is the indicator function over the interval (-∞,s] and P := g(Y) is a one-dimensional random variable. The exact executions of ℋ and thus exact samples of Y are not available is general but it is possible to construct approximate executions and approximate samples that converge to the exact ones in a suitable sense.Alg. <ref> presents a state update routine based on the Euler-Maruyama method that can be used toconstruct approximate executions.Given the model ℋ and the current approximate state (q_k,z_k), this algorithm computesthe approximate state (q_k+1,z_k+1) for the next time step of size Δ. Equation (<ref>) in step <ref> of the algorithm is the Euler-Maruyama approximation of the SDE (<ref>). If z_ is still inside the invariant of the current mode 𝒳(q_k), then the mode remains unchanged and z_ will be the next state (steps <ref>-<ref>). Otherwise, in steps <ref>-<ref> z_ is projected onto the boundary ∂𝒳(q_k) of the invariant and the mode is updated according to the discrete kernel r(q_k,z_k+1).Alg. <ref> generates approximate executions of ℋ and approximate samples of Yusing Alg. <ref>.The algorithm requires the model ℋ, the definition of Y as a function of the execution of of ℋ,and the time interval [0,s]. The output of the algorithm θ^ℓ is an approximate sample of random variable Y.In steps <ref>-<ref>the number of time steps n is selected and the discretization time step Δ is computed. In order to highlight the dependency of the algorithm to the parameter n, we have opted to use ℓ in the representation n = κ 2^ℓ as the superscript of the variables. We call ℓ the level of approximation which is nicely connected to the MLMC terminology discussed in Section <ref>.Alg. <ref> initializes the approximate execution in step <ref> as x^ℓ_0 : = (q_0^ℓ,z_0^ℓ) according to x_0 the initial state of ℋ. Then the algorithm iteratively computes the next approximate state (q_k+1^ℓ,z_k+1^ℓ) by sampling from the m-dimensional standard normal distribution in step <ref> and applying Alg. <ref> to (ℋ, q_k^ℓ,z_k^ℓ,Δ,W_k^ℓ) in step <ref>. Finally, step <ref> constructs the continuous-time approximate execution (q^ℓ(·),z^ℓ(·)) as the piecewise constant version of the discrete execution (q_k^ℓ,z_k^ℓ), which enables the computation of θ^ℓ by applying the definition of Y to (q^ℓ(·),z^ℓ(·)) (step <ref>).Alg. <ref> is parameterized by ℓ.Due to the nature of the Euler-Maruyama method in (<ref>), we expect that the approximate samples θ^ℓconverge to Y as ℓ→∞ in a suitable way. In fact, it is an unbiased estimator in the limit: lim_ℓ→∞ g() =g(Y). The idea behind standard Monte Carlo (SMC) method is to use the empirical mean of g() as an approximation of g(Y). The SMC estimator has the formP̂ = 1N∑_i=1^N g(_i),which is based on N replications of .The replications {θ_i^ℓ,i=1,…,N} can be generated by running Alg. <ref> (with a fixed ℓ) N times, or running any other algorithm that generates such samples (cf. Alg. <ref> in Section <ref>).The SMC method is summarized in Alg. <ref>, which approximates g(Y) based on a general sampling algorithm _ℓ. Note that Alg. <ref> can be used for estimating g(Y) not only with g(·) being the indicator function but also any other functional that can be deterministically evaluated using the executions over the time interval [0,s]. Owing to the randomized nature of algorithm _ℓ embedded in Alg. <ref>,we quantify the quality of its outcome using mean squared error:[ We slightly abuse the notation and indicate by MSE(_ℓ) the mean square error of Alg. <ref> with the embedded sampling algorithm _ℓ.]𝑀𝑆𝐸(_ℓ) ≡[(P̂ -P)^2] =[(P̂ - P̂)^2] + [P̂ -P]^2. The mean square error MSE(_ℓ) is decomposed into two parts:Monte Carlo variance and squared bias error. The latter is a systematic error arising from the fact that we might not sample our random variable exactly, but rather use a suitable approximation,while the former error comes from the randomized nature of the Monte Carlo algorithm. The Monte Carlo variance (first term in (<ref>)) is proportional to N^-1 asP̂ = (1/N∑_i=1^N g(_i))=1/N^2(∑_i=1^N g(_i))=1/N(g()). The cost of Alg. <ref> is typically taken to be the expected runtime in order to achieve a prescribed accuracy 𝑀𝑆𝐸(_ℓ)≤ε. A more convenient approach for theoretical comparison between different methods is to consider the cost associated to sampling algorithm _ℓ, C_ℓ(_ℓ) := [#operations and random number generations to calculateg(θ^ℓ)],which facilitates the definition of convergence rate of the algorithm. We say that Alg. <ref> based on sampling algorithm _ℓ converges with rate γ >0 if lim_ℓ→∞√(MSE (_ℓ)) = 0 and if there exist constants c > 0, η≥0 such that C_ℓ(_ℓ) ≤ c ·(√(MSE (_ℓ)))^-γ·( - log√(MSE (_ℓ)))^η. The definition of convergence rate in (<ref>) indicates that for a desired accuracy MSE (_ℓ)≤ε smaller convergence rate γ implies lower computational cost C_ℓ(_ℓ). The following theorem presents the convergence rate of the SMC method presented in Alg. <ref>. Let θ^ℓ denote the numerical approximation of the random variable Y according to an algorithm _ℓ. Assume there exist positive constants α, ζ, c_1, c_2 such that for all ℓ∈_0 | 𝔼[g(θ^ℓ) - g(Y)] | ≤ c_1 2^-α·ℓ, 𝔼[C_ℓ] ≤ c_22^ζ·ℓ, andg(θ^ℓ) < ∞. Then the standard Monte Carlo method of Alg. <ref> based on sampling algorithm _ℓ converges with rate γ=2+ζα. Recall the role of ℓ in step <ref> of Alg. <ref>. Increasing ℓ results in an exponential increase in the number of time steps thus also in the number of samples. Therefore we have assumed in (<ref>) an exponential bound on the increased cost and an exponential bound in the decreased bias as a function of ℓ.Application to the TCL Case Study. We construct the approximate discrete-time executions as θ^ℓ_k+1 = 1/CR(θ_a - q^ℓ_k R P_rate-θ^ℓ_k)Δ+σ(q^ℓ_k)·√(Δ)· W^ℓ_k,where W^ℓ_k is the sample from the standard normal distribution, Δ=s/n, n = κ2^ℓ, and the discrete mode at any level ℓ is defined as q^ℓ_k+1 := f(q^ℓ_k,θ_k^ℓ) with f(·) defined in (<ref>). This discrete-time updating is slightly different from thefunction of Alg. <ref>, which can be interpreted as follows. Instead of continuous updating of mode, the control signal acts as a digital controller and updates the mode only at the discrete time steps. It is clear, that the cost of simulating one execution of (<ref>) is proportional to the number of the discretization steps, thus setting the parameter ζ=1 in Theorem <ref>.The values of constants α, ζ, c_1, c_2 in Theorem <ref> depend on the regularity of the functional g, sampling algorithm _ℓ and other parameters. In the next section we propose to use MLMC method that improves the convergence rate and substantially reduces the computational complexity of the estimation. We discuss a smoothing in Section <ref> that replaces the indicator function g(·) with a smoothed function and discuss its effect on the algorithm's error. § MULTILEVEL MONTE CARLO METHODThe multilevel Monte Carlo method (MLMC) relies on the simple observation of telescoping sum for expectation:g(θ^L) =g(θ^0) + ∑_l=1^L [ g() - g(θ^ℓ-1)].where θ^0 and θ^L correspond respectively to the coarsest and finest levels of numerical approximation. While any of the approximations {θ^0,θ^1,…,θ^L} can be used individually in Alg. <ref> to approximate Y, instead, the MLMC method independently estimates each of the expectations on the right-hand side of (<ref>) such that the overall varianceis minimized for a given computational cost. The estimator P̂ of g(θ^L) can be seen as a sum of independent estimatorsP̂=∑_ℓ=0^L P^ℓ,where P^0 is an estimator for g(θ^0) based on N_0 samples, and P^ℓ are estimates for [ g() - g(θ^ℓ-1)] based on N_ℓ samples. As we saw in the MSC method of Section <ref>, the simplest forms for P^0 and P^ℓ are the empirical means over all samples:P^0 = 1N_0∑_i=1^N_0 g(θ^0_i), P^ℓ = 1N_ℓ∑_i=1^N_ℓ[g(_i) - g(θ^ℓ-1_i)], ℓ=1,…,L.Using the assumption of having independent estimators {P^0,P^1,P^2,…,P^L} and employing the telescoping sum (<ref>) we can compute respectively the variance of P̂ and bias asP̂ = [∑_ℓ=0^L P^ℓ] = ∑_ℓ=0^LP^ℓ, P - P̂ =P - [∑_ℓ=0^L P^ℓ] =P -g(θ^L).The computation of P^ℓ in (<ref>) requires the samples θ^ℓ_i,θ^ℓ-1_i to be generated from a common probability space. We utilize the fact that sum of normal random variables is still normally distributed.Alg. <ref> presents generation of approximate coupled samples θ^ℓ_i,θ^ℓ-1_ifor the random variable Y defined on the execution of a ℋ.As can be seen in steps <ref>-<ref> and <ref>, the approximate execution for the finerlevel ℓ is constructed exactly the same way as in Alg. <ref> with n_f = κ 2^ℓ time steps. The construction of approximate execution for the coarserlevel (ℓ-1) with n_c = κ 2^ℓ-1 is also similar except that the noise termin step <ref> is obtained by taking the weighted sum of noise terms from the finerlevel (W_2k^ℓ+W_2k+1^ℓ)/√(2).This choice preserves the properties of each approximation level while coupling the executions of levels ℓ-1,ℓ thus also coupling approximate samples θ^ℓ-1,θ^ℓ.Now we are ready to present the MLMC method in Alg. <ref>. The method is parameterized by the number of levels L, number of samples for each level N_ℓ, ℓ= 0,1,…, L (which are gathered in 𝔖), and the initial number of time steps κ. Steps <ref>-<ref> performs the SMC method of Alg. <ref> with embedded sampling algorithm <ref> in order to estimate g(θ^0) with N_0 samples at the initial level ℓ=0. Then the algorithm iteratively estimate [g(θ^l)-g(θ^l-1)] in steps <ref>-<ref> using Alg. <ref> with number of samples N=N_l and with the embedded coupled sampling algorithm <ref>. The sum estimated quantity is reported in step <ref> as the estimation of g(Y). The next theorem gives the convergence rate of MLMC method presented in Alg. <ref>. Let θ^ℓ denote the level ℓ numerical approximation of the random variable Y. Assume the independent estimators P_ℓ used in Alg. <ref> satisfy | [g(θ^ℓ) - g(Y)] | ≤ c_12^-α ℓ and 𝔼[C_ℓ] ≤ c_22^ζ ℓ𝔼[P^ℓ] = {[𝔼[g(θ^0)], ℓ=0; 𝔼[g(θ^ℓ) - g(θ^ℓ-1)], ℓ>0 ]. and [P^ℓ] ≤ c_3N_ℓ^-12^-β ℓ for positive constants α, β, ζ, c_1, c_2, c_3 with α≥1/2 min(β,ζ). Then the MLMC method in Alg. <ref> converges with rate 2+max(ζ-β,0)α. Assumptions in (<ref>) are exactly the same as the ones used in Theorem <ref>. Assumptions in (<ref>) put restriction on the statistical properties of the estimators P^ℓ: they first enables us to use the telescoping property (<ref>) and the second ensures the exponentially decaying variance as a function of level ℓ. In compare with the convergence rate of SMC method in Theorem <ref>, the improvement is due to the non-zero factor β which is the decaying rate of the variance of estimators.Application to the TCL Case Study. We construct the approximate discrete-time executions for the finer lever ℓ as θ^ℓ,f_k+1 = 1/CR(θ_a - q^ℓ,f_k R P_rate-θ^ℓ,f_k)Δ_f+σ(q^ℓ,f_k)·√(Δ_f)· W^ℓ_k,q^ℓ,f_k+1 := f(q^ℓ,f_k,θ_k^ℓ,f), for allk=0,1,…,n_f,where W^ℓ_k is the sample from the standard normal distribution, Δ_f=s/n_f, n_f = κ 2^ℓ, and with f(·) defined in (<ref>).The coupling, which means that we get the dynamics for θ^ℓ,c based on the increments for θ^ℓ,f, is done in a following way: θ^ℓ,c_k+1 = θ_k^ℓ,c+1/CR(θ_a- q^ℓ,c_k R P_rate-θ^ℓ,c_k)Δ_c + σ(q_k^ℓ,c)·√(Δ_c)·1/√(2)·(W^ℓ_2k-1+W^ℓ_2k),q^ℓ,c_k+1 := f(q^ℓ,c_k,θ_k^ℓ,c), for allk=0,1,…,n_c,where Δ_c=s/n_c with n_c = κ 2^ℓ-1. The fact that we have used the same Brownian increments W^ℓ_2k-1,W^ℓ_2k from the finer level (<ref>) in the courser level (<ref>) lays the foundation of having nonzero value ofβ in Theorem <ref>. The cost of simulating one approximate execution in (<ref>)-(<ref>) is proportional to the number of discretization steps, thus setting the parameter ζ=1 in Theorems <ref>-<ref>.Now that we have set up the MLMC method and the coupling technique that improves the convergence rate of the estimation,we focus on the following important problems associated with the approach: * Discontinuity of functional g(Y)=1_(-∞,s](Y), leads to smaller values of α and β in Theorem <ref>. This results in larger convergence rate γ thus larger computational cost for a given accuracy ε. * The optimal choice of parameters N_ℓ, L and the unknown constants in Theorem <ref>.The first issue, discussed in Section <ref>, is resolved through smoothing,which replaces the discontinuous function g with a smoothed function g^δ with Lipschitz constant proportional to δ^-1. The second issue, discussed in Section <ref>, is resolved through an adaptive algorithm.This adaptive algorithm follows <cit.>, and combines the smoothing of discontinuous functionals and the MLMC method. Note that we require an updated set of assumptions and include the search for parameter δ into the adaptive algorithm.§ MLMC WITH SMOOTHED INDICATOR FUNCTIONThe smoothing is based on the function g^δ:→, which are the rescaled translates of a function g^0 : →of the formg^0(x)= 0, x>1 1/2+1/8(5x^3 - 9x),-1≤ x≤11,x<-1, and g^δ (x) = g^0((x-s)/δ), x ∈.Since we add a smoothing step, we need to update the MLMC estimator (<ref>),derive new a MSE decomposition (instead of (<ref>))which incorporates the error due to the smoothing,and update Assumptions (<ref>)-(<ref>) in Theorem <ref>.Note that function (<ref>) is not the only possible choice for a smoothing function (see <cit.>),but in our experience this is the easiest to implement and numerically stable, while still providing significant gainsin computational cost.Recall that the MLMC method is based on a sequence (θ^ℓ)_ℓ∈_0 of random variables, defined on a common probability space together with Y. The new MLMC method that includes smoothing is defined by= 1/N_0·∑_i=1^N_0g^δ (θ^0_i) + ∑_ℓ=1^L1/N_ℓ·∑_i=1^N_ℓ( g^δ (θ^ℓ,f_i) - g^δ (θ^ℓ,c_i) ),with an independent family of ^2-valued random variables (θ^ℓ,f_i,θ^ℓ,c_i) for i=1,…,N_ℓ and ℓ=0,1,…,Lsuch that equality in distribution holds for (θ^ℓ,f_i,θ^ℓ,c_i) and (θ^ℓ,θ^ℓ-1), where we used the notation (θ^0,f_i,θ^0,c_i) = (θ^0_i,0) for the initial level ℓ=0. Note that (<ref>) is the same as the MLMC estimator (<ref>) except using the smoothing function g^δ(·) instead of the indicator function g(·). The next theorem gives the mean square error decomposition for (<ref>). For δ > 0, the error ofin (<ref>) with smoothing function (<ref>) can be decomposed as MSE():= - g(Y)^2 ≤δ^4 + | (g^δ (Y)) - (g^δ (θ^L))|^2+() =: e_1^2+e_2^2+e_3.The error terms in (<ref>) are related to smoothing, bias, and variance, respectively. Note that as δ goes to zero, the Lipschitz constant for g^δ(x) goes to infinity, which has to be taken into account.Hence the assumptions in Theorem <ref> have to be updated.The theoretical analysis and updated assumptions are presented in <cit.>.§ ADAPTIVE MLMC ALGORITHM In this section we present an adaptive algorithm to find the optimal parameters for the MLMC method. For a given >0 we wish to select the parameters of the MLMC algorithm such that its error is at mostand its cost is as small as possible. Our approach to the selection of the replication numbers and of the maximal level follows <cit.>.The adaptive algorithm assumes no prior knowledge on the smoothing parameter δ, along with bias and variance dependencies on it. The smoothing parameter δ is chosen from the discrete set of values δ_m = 1/ 2^m, where m ∈.With a slight abuse of notation we put g^m = g^δ_m.In order to achieveMSE() ≤ we have to assign certain proportions ofto the three sources ofthe error introduced in (<ref>).Specifically we wish to choose the parameters of our algorithm such thate_1 ≤ a_1_*,e_2 ≤ a_2 ·_*,e_3 ≤ a_3^2 ·_*^2, where _* := /a_1+a_2+a_3.The MLMC algorithm is parameterized by thevalue m for smoothing δ_m = 1/2^m, the values of the maximal level L, and the replication numbers 𝔖 = (N_0,…,N_L). We always select L ≥ 2 and N_ℓ≥ 100 for ℓ=0,…,L.By the latter, we ensure a reasonable accuracy in certain estimates to be introduced below. We use y_i,0 to denote actual samples of the random variable θ^0 and (y_i,ℓ,y_i,ℓ-1) to denote the actual samples of the random vector (θ^ℓ,θ^ℓ-1) for ℓ=1,…,L as opposed to θ_i^ℓ,f,θ_i^ℓ,c which were used previously for their respective random variables.Assumptions. Theorem <ref> relies on the assumption of exponential upper bounds in (<ref>)-(<ref>), which in general might be difficult to verify. Instead in this section we study asymptotic upper bounds. For this purpose we use the following notation. For sequences of real numbers u_ℓ and positive real numbers w_ℓ we write u_ℓ≈ w_ℓ if lim_ℓ→∞ u_ℓ/ w_ℓ = 1, and write u_ℓ w_ℓ if lim sup_ℓ→∞ u_ℓ/ w_ℓ≤ 1. We also replace assumptions (<ref>)-(<ref>) with the requirement that for every m there exists c,α > 0 such that| (g^m (θ^ℓ)) - (g^m (θ^ℓ-1)) | ≈ c · 2^-ℓ·α and lim_ℓ→∞ g^m (θ^ℓ) = g^m (Y).This yields the following asymptotic upper bound for the bias at level ℓ| (g^m (Y)) - (g^m (θ^ℓ)) |(2^α-1)^-1·| (g^m (θ^ℓ)) - (g^m (θ^ℓ-1)) |.We put C_r = 2^r+1 with r=3, the degree of polynomial in (<ref>), and suppose that there exists c >0 such that | (g^m (Y)) - (g^m-1 (Y)) | ≈ c ·δ_m^4. This yields the asymptotic upper bound for the smoothing error with parameter δ_m, | g(Y) - (g^m (Y)) |(C_r-1)^-1·| (g^m (Y)) - (g^m-1 (Y))|. Our adaptive MLMC algorithm is based on the intuition that the asymptotic bounds (<ref>) and (<ref>) can be replaced by their corresponding inequalities (≤ instead of ), and estimators for means and variances can be assumed to be nearly exact.Variance Estimation and Selection of the Replication Numbers. To estimate the expectationsand variances we employ the empirical mean and varianceb̂_0 = 1/N_0·∑_i=1^N_0g^m(y_i,0), and b̂_ℓ = 1/N_ℓ·∑_i=1^N_ℓ(g^m(y_i,ℓ) - g^m(y_i,ℓ-1)),v̂_0 = 1/N_0·∑_i=1^N_0|g^m(y_i,0) - b̂_0|^2 and v̂_ℓ = 1/N_ℓ·∑_i=1^N_ℓ|g^m(y_i,ℓ) - g^m(y_i,ℓ-1) - b̂_ℓ|^2.We get that v̂(𝔖) = ∑_ℓ=0^L 1/N_ℓ·v̂_ℓ serves as an empirical upper bound for the variance of the MLMC algorithm with any choice of replication numbers 𝔖 = (N_0,N_1,…,N_L). If, for the present choice of replication numbers,this bound is too large compared to the upper bound for () in (<ref>), i.e., if the variance constraintv̂(𝔖)≤a_3^2 ·_*^2is violated, we determine new values of N'_0,…,N'_L by minimizingc(N_0,…,N_L) subject to the constraint v̂(𝔖) ≤ a_3^2 ·_*^2, which leads toN'_ℓ = v̂_ℓ^1/2(2^ℓ + 1)^1/2·∑_ℓ=0^L( v̂_ℓ· (2^ℓ + 1))^1/2·_*^-2a_3^2,ℓ = 0,1,…, L,and extra samples of θ^0 and (θ^ℓ,θ^ℓ-1) have to be generated accordingly. Bias Estimation and Selection of the Maximal Level. For estimating | (g^m (θ^ℓ)) - (g^m (θ^ℓ-1)) | we can use the values of |b̂_ℓ| already available from (<ref>) for the levels ℓ=1,…,L. We estimate α and c in (<ref>) by a least-squares fit, i.e., we take α̂ and ĉ to minimize (α,c) ↦∑_ℓ∈L( log |b̂_ℓ| +ℓ·α log 2+ log c )^2. While the value of ĉ is irrelevant,an upper bound for | (g^m (θ^L)) - (g^m (θ^L-1))| is given by |b̂_L|, or, more generally, by 2^(ℓ-L) ·α̂· |b̂_ℓ| with ℓ≤ L. This geometric upper bound can be used to set the stopping criterion of increasing the maximal level. Let us defineB̂_2 = max( |b̂_2|, |b̂_1|/2^α̂),for L=2 B̂_L = max( |b̂_L|, |b̂_L-1|/2^α̂, |b̂_L-2|/2^2 α̂) for L ≥ 3. The present value of L is accepted as the maximal level, ifthe bias constraintB̂_L≤a_2 · (2^α̂-1) ·_*is satisfied. Otherwise, L is increased by one, and newsamples will be generated. Selection of the Smoothing Parameter. We wish to determine the smallest value of m, i.e., the largest value of δ_m, such that | (g^m (Y)) - (g^m-1 (Y)) | ≤a_1 · (C_r-1) ·_*is satisfied, which corresponds tothe upper bound for e_1 in (<ref>) together with (<ref>). Initially we try m=2.Actually, Y is approximated by θ^L, so the present value is accepted ifŝ :=| 1/N_L·∑_i=1^N_L(g^m (y_i,L) - g^m-1 (y_i,L))|≤ a_1 · (C_r-1) ·_*. The Adaptive Algorithm. We combine the above results and sum them up in Alg. <ref>, where the desired accuracy ε is the input. § SIMULATION RESULTSRecall Problem <ref> where the goal is to estimate the probability ℙ(Y≤θ_+ + 0.1·δ_d). The random variable Y is defined as Y=max{θ_t, t∈[0,s]}. We set the parameters of the TCL model (<ref>)-(<ref>) according to Table <ref> and select the time horizon s=1 hour.We implement the MLMC Alg. <ref> for target accuracies ε=2^-k, where k∈{3,…,8}. We set the parameters a_1=4, a_2=a_3=2 in (<ref>). With this choice we put less pressure on the smoothing error because the influence of the smoothing parameter δ on the variance and thus on the overal cost is severe. Due to the smoothing step we have to sample executions for the time duration of at least (s+δ) in order to evaluate the functional g(Y). With the selected values of s and ε, sampling executions for 1.5 hours is sufficient.The result of the experiments is presented in Figure <ref>. The left and center plots show the impact of the smoothing coefficient on the variance and mean decays respectively based on 10^6 runs of the algorithm. The data points of the plots with ℓ=1 and with the indicator function are related to the SMC method. These plots indicate that the adaptive MLMC method is beneficial over SMC method due to the strong variance and mean decay with respect to level ℓ as well as the use of smoothing function instead of the indicator function.The computational gain of the MLMC over SMC is presented on the right plot based on 100 runs. The plot compares the expected cost of the SMC method with the estimated cost of the adaptive MLMC method. The cost of SMC method is given by ε^-2-1/α̅ (see Theorem <ref>), which bounds the cost of generating executions and evaluating functionals. We estimate the parameter α̅ through the precalculation and do not take into account the cost of estimating α̅. In this way we assume the parameter α̅ is known in advance and make the comparison more in favor of the SMC method. The plot indicates larger computational gains for higher target accuracies (smaller ε). Note that the curve in the right plot is not monotone because there is an additional cost of updating the smoothing coefficient, hence re-evaluating the functionals with the new value of δ. This additional cost has not been compensated by the MLMC gains as much in compare with the neighboring accuracies. § CONCLUSIONSIn this paper we studied the problem of statistical model checking of continuous-time hybrid systems that do not admit exact simulations. We employed multilevel Monte Carlo method and presented a smoothing step with tunable precision that replaces the desired discontinuous functional with a continuous approximation thus decreasing the overall computational effort of the approach. An adaptive algorithm was designed which balances the errors due to the bias, variance, and smoothing. The approach was demonstrated on the model of thermostatically controlled loads.plain 10 ABKCCL17 A. Abate, L. Bortolussi, M. Kwiatkowska, L. Cardelli, M. Ceska, and L. Laurenti. Reachability computation for switching diffusions: finite abstractions with certifiable and tuneable precision. In Hybrid Systems: Computation and Control, HSCC '17, New York, NY, USA, 2017. ACM. APLS08 A. Abate, M. Prandini, J. Lygeros, and S. Sastry. Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems. Automatica, 44(11):2724–2734, Nov 2008. BHHK03 C. Baier, B. Haverkort, H. Hermanns, and J.-P. Katoen. Model-checking algorithms for continuous-time Markov chains. Trans. on Software Engineering, 29(6):524–541, June 2003. BK08 C. Baier and J.-P. Katoen. Principles of Model Checking. MIT Press, 2008. Bujorianu2002 M.L. Bujorianu and J. Lygeros. Reachability questions in piecewise deterministic Markov processes. In O. Maler and A. Pnueli, editors, Hybrid Systems: Computation and Control, volume 2623 of Lecture Notes in Computer Science, pages 126–140. Springer Verlag, 2003. BujLyg04 M.L. Bujorianu and J. Lygeros. General stochastic hybrid systems: Modelling and optimal control. In in Proc. 43rd IEEE Conf. Decision Control, pages 1872–1877, 2004. Bulychev2012 P. Bulychev, A. David, K.G. Larsen, A. Legay, M. Mikučionis, and D.B. Poulsen. Checking and Distributing Statistical Model Checking, pages 449–463. Springer, Berlin, Heidelberg, 2012. CL06 C.G. Cassandras and J. Lygeros (Eds.). Stochastic Hybrid Systems. Number 24 in Control Engineering. CRC Press, Boca Raton, 2006. Clarke2011 E.M. Clarke and P. Zuliani. Statistical Model Checking for Cyber-Physical Systems, pages 1–12. Springer, Berlin, Heidelberg, 2011. DHKP17 Przemysław Daca, Thomas A. Henzinger, Jan Křetínský, and Tatjana Petrov. Faster statistical model checking for unbounded temporal properties. ACM Trans. Comput. Logic, 18(2):12:1–12:25, May 2017. SSoudjani S. Esmaeil Zadeh Soudjani. Formal Abstractions for Automated Verification and Synthesis of Stochastic Systems. PhD thesis, Technische Universiteit Delft, The Netherlands, Nov 2014. SA13 S. Esmaeil Zadeh Soudjani and A. Abate. Adaptive and sequential gridding procedures for the abstraction and verification of stochastic processes. SIAM Journal on Applied Dynamical Systems, 12(2):921–956, 2013. SA14TCL S. Esmaeil Zadeh Soudjani and A. Abate. Aggregation and control of populations of thermostatically controlled loads by formal abstractions. Control Systems Technology, IEEE Transactions on, 23(3):975–990, May 2015. SADensity14 S. Esmaeil Zadeh Soudjani, R. Majumdar, and A. Abate. Safety verification of continuous-space pure jump Markov processes. In Tools and Algorithms for the Construction and Analysis of Systems (TACAS), pages 147–163. Springer Berlin Heidelberg, 2016. FHT08 M. Fränzle, H. Hermanns, and T. Teige. Stochastic satisfiability modulo theory: A novel technique for the analysis of probabilistic hybrid systems. In Hybrid Systems: Computation and Control, volume 4981 of Lecture Notes in Computer Science, pages 172–186. Springer, 2008. SGEFA14 S. Esmaeil Zadeh Soudjani S. Gerwinn, C. Ellen, M. Fränzle, and A. Abate. Formal synthesis and validation of inhomogeneous thermostatically controlled loads. In Quantitative Evaluation of SysTems, volume 8657 of Lecture Notes in Computer Science, pages 57–73. Springer, 2014. Tigran17 M. B. Giles, T. Nagapetyan, and K. Ritter. Adaptive Multilevel Monte Carlo Approximation of Distribution Functions. ArXiv e-prints, June 2017. giles08 M.B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56(3):607–617, 2008. giles15 M.B. Giles. Multilevel Monte Carlo methods. Acta Numer., 24:259–328, 2015. giles2014multi M.B. Giles, T. Nagapetyan, and K. Ritter. Multilevel Monte Carlo approximation of distribution functions and densities. SIAM/ASA J. Uncertain. Quantif., 3(1):267–295, 2015. heinrich98 S. Heinrich. Monte Carlo complexity of global solution of integral equations. J. Complexity, 14(2):151–175, 1998. MESG13 M. Kamgarpour, C. Ellen, S. Esmaeil Zadeh Soudjani, S. Gerwinn, J.L. Mathieu, N. Mullner, A. Abate, D.S. Callaway, M. Fränzle, and J. Lygeros. Modeling options for demand side participation of thermostatically controlled loads. In International Conference on Bulk Power System Dynamics and Control (IREP), pages 1–15, August 2013. K16 Jan Křetínský. Survey of Statistical Verification of Linear Unbounded Properties: Model Checking and Distances, pages 27–45. Springer International Publishing, Cham, 2016. Larsen14 Kim G. Larsen and Axel Legay. Statistical Model Checking Past, Present, and Future, pages 135–142. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014. SMC10 A. Legay, B. Delahaye, and S. Bensalem. Statistical Model Checking: An Overview, pages 122–135. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. MKC12 J.L. Mathieu, S. Koch, and D.S. Callaway. State estimation and control of electric loads to manage real-time energy imbalance. IEEE Trans. on Power Systems, 28(1):430–440, 2013. pa15 S. Pauli and P. Arbenz. Determining optimal multilevel Monte Carlo parameters with application to fault tolerance. Computers & Mathematics with Applications, 70(11):2638–2651, 2015. PBLB03 G. Pola, M. L. Bujorianu, J. Lygeros, and M. D. Di Benedetto. Stochastic hybrid models: An overview. In In Proceedings IFAC Conference on Analysis and Design of Hybrid Systems, 2003. PJP07 S. Prajna, A. Jadbabaie, and G.J. Pappas. A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Transactions on Automatic Control, 52(8):1415–1428, 2007. TMKA13 I. Tkachev, A. Mereacre, J.-P. Katoen, and A. Abate. Quantitative automata-based controller synthesis for non-autonomous stochastic hybrid systems. In Hybrid Systems: Computation and Control, pages 293–302, New York, NY, USA, 2013. ACM. WRWVD15 Y. Wang, N. Roohi, M. West, M. Viswanathan, and G.E. Dullerud. Statistical verification of dynamical systems using set oriented methods. In Hybrid Systems: Computation and Control, pages 169–178, New York, NY, USA, 2015. ACM. WSBP16 R. Wisniewski, C. Sloth, M. Bujorianu, and N. Piterman. Safety verification of piecewise-deterministic Markov processes. In Hybrid Systems: Computation and Control, pages 257–266, New York, USA, 2016. ACM. xg12 Y. Xia and M.B. Giles. Multilevel path simulation for jump-diffusion SDEs. In Monte Carlo and Quasi-Monte Carlo Methods 2010, pages 695–708. Springer, 2012. ZA14 M. Zamani and A. Abate. Approximately bisimilar symbolic models for randomly switched stochastic systems. Systems & Control Letters, 69:38 – 46, 2014. ZMMAL14 M. Zamani, P. Mohajerin Esfahani, R. Majumdar, A. Abate, and J. Lygeros. Symbolic control of stochastic systems via approximately bisimilar finite abstractions. IEEE Transactions on Automatic Control, 59(12):3135–3150, Dec 2014. | http://arxiv.org/abs/1706.08270v1 | {
"authors": [
"Sadegh Esmaeil Zadeh Soudjani",
"Rupak Majumdar",
"Tigran Nagapetyan"
],
"categories": [
"cs.SY",
"cs.LO",
"math.PR",
"93E03, 68W25,"
],
"primary_category": "cs.SY",
"published": "20170626081109",
"title": "Multilevel Monte Carlo Method for Statistical Model Checking of Hybrid Systems"
} |
Let C be an algebraic space curve defined parametrically by (t)∈ K(t)^n, n≥ 2. In this paper, we introduce a polynomial,the T–function, T(s), which is defined by means of a univariate resultant constructed from (t). We show that T(s)=∏_i=1^n H_P_i(s)^m_i-1, where H_P_i(s), i=1,…,n are polynomials (called the fibre functions) whose roots are the fibre of the ordinary singularitiesP_i∈ C of multiplicity m_i, i=1,…,n. Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.Rational curve parametrization; Singularities of an algebraic curve; Multiplicity of a point; Tangents; Resultant; T–function; Fibre function§ INTRODUCTION Parametrizations of rational curves play an important role in many practical applications in computer aided geometric design where objects are often given and manipulated parametrically(see e.g. <cit.>, <cit.>, <cit.>). In the last years, important advances have been made concerning the information one may obtain from a given rational parametrization defining an algebraic variety. For instance, a complete analysis of the asymptotic behavior of a given curve has been carried out in <cit.>; efficient algorithms for computing the implicit equations that define the curve are provided in<cit.> and <cit.> and the study and computation of the fibre of a point via the parametrizationcan be found in<cit.>. In addition, some aspects concerning the singularities of the curve and their multiplicities are studied in <cit.>, <cit.>, <cit.>, <cit.> and <cit.>. Similar problems, for the case of a given rational parametric surface, are being analyzed. For instance, the computation of the singularities and their multiplicities from the input parametrization is presented in <cit.>, a univariate resultant-based implicitization algorithm for surfaces is provided in <cit.>, and the computation of the fibre of rational surface parametrizations is developed in <cit.>. In this paper, we show how to relate the fibreand the singularities of a given curve defined parametrically, by means of a univariate resultant which is constructed directly from the parametrization. For this purpose, we consider (t)∈ℙ^n(𝕂(t)) a rational projective parametrization of an algebraic curve C over an algebraically closed field of characteristic zero, K.Associated with 𝒫(t), we consider the induced rational map ψ_𝒫:𝕂⟶𝒞⊂ℙ^n(𝕂); t⟼𝒫(t). We denote by (ψ_𝒫) the degree of the rational map ψ_𝒫. The birationality ofψ_, i.e. the properness of (t), is characterizedby (ψ_)=1 (see <cit.> and <cit.>). Intuitively speaking, (t) proper means that (t) traces the curve once, except for at most a finite number of points. We will see that, in fact,these points are the singularities of C.We recall that the degree of a rational map can be seen as the cardinality of the fibre of a generic element (see <cit.>). Weuse this characterization in our reasoning and thus, we denote by ℱ_𝒫(P) the fibre of a point P∈𝒞 via the parametrization 𝒫(t); that is ℱ_𝒫(P)=𝒫^-1(P)={ t∈𝕂 | 𝒫(t)=P }. In order to make the paper more reader–friendly, we first consider the case of a given plane curve C defined parametrically by(t)∈ℙ^2(𝕂(t)) (see Sections<ref> and <ref>)to, afterwards, generalize the results obtained to rational space curves in any dimension (see Section <ref>). We also assume that C has only ordinary singularities (otherwise, one may apply quadratic transformations for birationally transforming the curve into a curve with only ordinary singularities). Non–ordinary singularities have to be treated specially since a non–ordinary singularity might have other singularities in its “neighborhood”. This specific case will be addressed in a future work and in fact, we will show that similar results to those presented in this papercan be stated for curves with non–ordinary singularities. Under these conditions, the main goal of the paper is to prove that a univariate resultant constructed directly from (t), which we will call theT–function, T(t), describes totally the singularities of C. It will be proved that the factorization of T(t) provides the fibre functions of the different singularities ofC as well as their corresponding multiplicities. The fibre function of a point P∈ C via (t) is given by a polynomial H_P(t) which satisfies thatt_0∈ℱ_𝒫(P) if and only if H_P(t_0)=0. In <cit.>, it is proved that if H_P(t)=∏_i=1^n(t-s_i)^k_i then, 𝒞 hasn tangents at P of multiplicities k_1,…,k_n, respectively. In addition, these tangents can be computed using (t) and the roots of each corresponding fibre function. Furthermore, it is shown that _P(𝒞)=(H_P(t)).Taking into account these previous results, in this paper we provethat the T–function can be factorized as T(t)=∏_i=1^n H_P_i(s)^m_i-1, where H_P_i(t) is the fibre functionof the ordinary singularityP_i∈ C and m_i is its multiplicity (for i=1,…,n). Thus, a complete classification of the singularities of a given rational curve, via the factorization of a univariate resultant, is obtained. On finishing this work, we just found a paper by Abhyankar (see <cit.>) that proves the factorization of the T–function for a given polynomial parametrization. In addition, Busé et al., in <cit.>, provide a generalization of Abhyankar's formula for the case of rational parametrizations (not necessarily polynomial). This approach is based on the concept of singular factors introduced in<cit.>, and it involves the construction ofμ–basis. Our approach is totally different, since we generalize Abhyankar's formula by using the methods and techniques presented in <cit.>. This allows us to group the factors of the T–function to easily obtain the fibre functions of the different singularities. In addition, we show how to deal with singularities that are reached by algebraic values of the parameter.As we mentioned above, these results can be stated similarly for the case of rational space curves in any dimension. We remark that the methods developed in this paper generalize some previousresults that partially approach the computation and analysis of singularities for rational parametrized curves(see e.g. <cit.>, <cit.> or <cit.>). Moreover, the ideas presented open several important ways that may be used to obtain significant results concerning rational parametrizations of surfaces.In a future work, this problem will be developed in more detail and some important results are expected to be provided.The structure of the paper is as follows. Sections <ref> and <ref> are devoted to the study of plane curves. In particular, in Section <ref>, we introduce theterminology that will be used throughout this paper as well as some previous results.In Section <ref>, we introduce the T–function and we present the main result of the paper. It claims that the factorization of the T–function provides the fibre functions of the different singularities of the curve. The proof of this result as well as some previous technical lemmas appear in Section <ref>. Section <ref> is devoted to generalize the results in Section <ref> to parametric space curves in any dimension.Throughout the whole paper, we outline all the results obtained with illustrative examples. § ANALYSIS AND COMPUTATION OF THE FIBRE Let 𝒞 be a rational (projective) plane curve defined by the projective parametrization𝒫(t)=(p_1(t):p_2(t):p(t))∈ℙ^2(𝕂(t)),where (p_1,p_2,p)=1, andK is an algebraically closed field of characteristic zero . We assume that 𝒞 is not a line (a line does not have multiple points). Let d_1=(p_1), d_2=(p_2), d_3=(p), and d=max{d_1,d_2,d_3}. Thus, we may writep_1, p_2 and p as{[ p_1(t)=a_0+a_1t+a_2t^2+⋯+a_dt^d; p_2(t)=b_0+b_1t+b_2t^2+⋯+b_dt^d;p(t)=c_0+c_1t+c_2t^2+⋯+c_dt^d. ]. Associated with𝒫(t), we consider the induced rational map ψ_𝒫:𝕂⟶𝒞⊂ℙ^2(𝕂); t⟼𝒫(t). We denote by (ψ_𝒫) the degree of the rational map ψ_𝒫 (for further detailssee e.g. <cit.> pp.143, or <cit.> pp.80). As an important result, we recall thatthe birationality ofψ_, i.e. the properness of (t), is characterizedby (ψ_)=1 (see <cit.> and <cit.>).Also, we recall that the degree of a rational map can be seen as the cardinality of the fibre of a generic element (see Theorem 7, pp. 76 in <cit.>). We will use this characterization in our reasoning. For this purpose, we denote by ℱ_𝒫(P) the fibre of a point P∈𝒞 via the parametrization 𝒫(t); that isℱ_𝒫(P)=𝒫^-1(P)={ t∈𝕂 | 𝒫(t)=P }. In general, it holds that P∈𝒞 if and only if ℱ_𝒫(P)≠∅, although an exception can be found for the limit point of the parametrization.We definethe limit point of the parametrization 𝒫(t) asP_L=lim_t→∞𝒫(t)/t^d=(a_d:b_d:c_d).Note that P_L∈ C since𝒫(t)/t^d=𝒫(t)∈𝒞, for t∈𝕂, and 𝒞 is a closed set. Furthermore, we observe that, given a parametrization (t), there always exists an associated limit point, and it is unique.The limit point is reachable via the parametrization (t), if there existst_0∈ K such that 𝒫(t_0)=P_L. However, the value t_0∈ Kcould not exist, and thenℱ_𝒫(P_L)=∅. Taking into account this statement, if P_L is not an affine point or it is a reachable affine point, we have that (t) is a normal parametrization. Otherwise, we say that (t) is not normal and P_L is the critical point (see Subsection 6.3 in <cit.>). Further properties of the limit point are stated and proved in <cit.>. In Subsection 2.2. in <cit.>, it is stated thatthe degree of a dominant rational map between two varieties of the same dimension is the cardinality of the fiber of a generic element. Therefore, in the case of the mapping ψ_, this implies that almost all points of C (except at most a finite number of points) are generated via (t) by the same number of parameter values, and this number is the degree of ψ_. Thus, intuitively speaking, the degree measures the number of times the parametrization traces the curve when the parameter takes values in K. Taking into account this intuitive notion, the degree of the mapping ψ_ is also called the tracing index of (t). In order to compute the tracing index, the following polynomials are considered,{[ G_1(s,t):=p_1(s)p(t)-p(s)p_1(t); G_2(s,t):=p_2(s)p(t)-p(s)p_2(t); G_3(s,t):=p_1(s)p_2(t)-p_2(s)p_1(t) ].and G(s,t)=(G_1(s,t),G_2(s,t),G_3(s,t)). In the following theorem, we compute thetracing indexof (t) using the polynomial G(s,t) (see Subsection 4.3 in <cit.>). It holds that (ψ_𝒫)=_t(G). We observe that:* The polynomials G_1, G_2 and G_3 satisfy thatG_i(s,t)=-G_i(t,s). Clearly, G(s,t)also has this property. * Taking into account the above statement, it holds that _s(G_i)=_t(G_i) for i=1,2,3, and _s(G)=_t(G).* It holds that _t(G_1)=max{d_1,d_3}. Indeed: if d_1≠ d_3, the statement trivially holds. If d_1=d_3,_t(G_1) may decrease if p_1(s)c_d-p(s)a_d=0. But this would imply that 𝒞 isa line, which is impossible by the assumption. Similarly, it holds that_t(G_2)=max{d_2,d_3}, and _t(G_3)=max{d_1,d_2}.* It holds thatG(s,t)=(G_1(s,t),G_2(s,t)).Indeed: since p(t)G_3(s,t)=p_2(t)G_1(s,t)-p_1(t)G_2(s,t), if h(s,t)∈ K[s,t] divides to G_1(s,t) and G_2(s,t), thenh(s,t) divides to G_3(s,t) or p(t). However, if h(s,t)divides p(t), then h(s,t)=h(t) which would imply that there exists t_0∈𝕂 such that G_1(s,t_0)=G_2(s,t_0)=p(t_0)=0. Hence, p_i(s)/p(s)∈ K, i=1,2, and C would be a line, which is impossible by the assumption. Similarly, it holds thatG(s,t)=(G_1(s,t),G_3(s,t))=(G_2(s,t),G_3(s,t)).Throughout this paper, we assume that𝒫(t) is proper, that is(ψ_𝒫)=1. Otherwise, we can reparametrize the curve using, for instance, the results in <cit.>.Under these conditions, it holds that the degree of C is d (see Theorem 6 in <cit.>). In addition,G(t,s)=t-s (see Theorem <ref>) and the cardinality of the fibre for a generic point of C is 1, although for a particular point it can be different. In order to analyze these special points, in the following, we consider a particular point P=(a,b,c)∈ C. The fibre of P consists of the values t∈𝕂 such that 𝒫(t)= P, that is, those which satisfy the fibre equations, defined as{[ϕ_1(t):=ap(t)-cp_1(t)=0;ϕ_2(t):=bp(t)-cp_2(t)=0; ϕ_3(t):=ap_2(t)-bp_1(t)=0. ].Hence, the fibre ofP is given by the common roots of these equations, which motivates the following definition: Given P∈ℙ^2(𝕂) and the rational parametrization 𝒫(t)∈ℙ^2(𝕂(t)), we definethe fibre function of P at 𝒫(t) asH_P(t):=(ϕ_1,ϕ_2,ϕ_3). Thus,t_0∈ℱ_𝒫(P) if and only if H_P(t_0)=0.Depending on whether P is an affine point or an infinity point, the fibre function can be expressed as follows: * If P is an affine point, thenc≠ 0. Thus, ϕ_3 can be obtained from ϕ_1and ϕ_2 and, therefore, H_P(t)=(ϕ_1(t),ϕ_2(t)).* If P is an infinity point, then c=0. Thus, ϕ_1and ϕ_2 are equivalent to p(t)=0 (note that a≠0 or b≠0) and, therefore,H_P(t)=(p(t),ϕ_3(t)).Note that the functions ϕ_1, ϕ_2 and ϕ_3 depend on Pand 𝒫(t). However, for the sake of simplicity, we do not represent this fact in the notation. In the following, we show how the fibre functionof P is related with the tangents of C at P, and with the multiplicity of P. For this purpose, we first recall thatP is a point of multiplicity ℓ on 𝒞if and only if all the derivatives of F (where F denotes the implicit polynomial defining C) up to and including those of (ℓ-1)–th order, vanish at P butat least one ℓ-th derivative does not vanish at P. We denote it by _P(𝒞). The point Pis called asimplepointon 𝒞if and only if _P(𝒞)=1. If _P(𝒞)=ℓ>1, then we say that P is amultiple or singular point (orsingularity)ofmultiplicity ℓ on 𝒞or an ℓ–fold point. Clearly P∉𝒞 if and only if _P(𝒞)=0.Observe that themultiplicity of 𝒞 at P is given as the order of the Taylor expansion of F at P. The tangentsto 𝒞 at Pare the irreducible factors of the first non–vanishing form in the Taylor expansion of F at P, and themultiplicityof a tangentis themultiplicity of the corresponding factor. If all the ℓ tangents at the ℓ-fold point P are different, then this singularity iscalled ordinary, and non–ordinary otherwise. Thus, we say that the character of P is either ordinary or non-ordinary.In<cit.>, it is shown how to compute the singularities and its corresponding multiplicities from a given parametrization defining a rational plane curve. Furthermore, it is provided a method for computing the tangents and for analyzing the non–ordinary singularities. In particular, the following theorem and corollary are proved.Let 𝒞 be a rational algebraic curve defined by a proper parametrization 𝒫(t), with limit point P_L. Let P≠ P_L be a point of 𝒞 and let H_P(t)=∏_i=1^n(t-s_i)^k_i be its fibre function (under 𝒫(t)). Then, 𝒞 hasn tangents at P of multiplicities k_1,…,k_n, respectively.It cannot be ensured that two different values of t, namely s_i_0 and s_i_1, provide different tangents. Thus, we could have a same tangent (at (s_i_0)=(s_i_1)) of multiplicityk_i_0+k_i_1.Let 𝒞 be a rational algebraic curve defined by a proper parametrization 𝒫(t), with limit point P_L. Let P≠ P_L be a point of 𝒞 and let H_P(t) be its fibre function (under 𝒫(t)). Then, _P(𝒞)=(H_P(t)). Let C be the rational plane curve defined by the projective parametrization𝒫(t)=(-t^3-5t^2-7t-3:t^4+7t^3+17t^2+17t+6:t^4+1)∈ℙ^2(ℂ(t)).Let us compute H_P(t), where P=(0:0:1). Since P is an affine point, we can obtainH_P(t) from ϕ_1 and ϕ_2 (see Remark <ref>). Since ϕ_1(t)=-2 t^3-10 t^2-14 t-6 and ϕ_2(t)=2 t^4+14 t^3+34 t^2+34 t+12, we get thatH_P(t)=(ϕ_1,ϕ_2)=2(t+3)(t+1)^2.Therefore, (-1)=(-3)=P, and applying Theorem <ref>, we deduce that C has at P two different tangents, one of multiplicity 1 and the other one of multiplicity 2. The parametrizations defining these tangents are given asτ_1(t)=𝒫(-3)+𝒫'(-3)t andτ_2(t)=𝒫(-1)+𝒫”(-1)/2t^2,respectively (see <cit.>). Note that these tangents are the linesy=x and y=-x (see Figure <ref>). Finally, we conclude thatP is a non–ordinary point of multiplicity 3 (see Corollary <ref>). § RESULTANTS AND SINGULARITIESIn Section <ref>, we show that, given a rational proper parametrization, 𝒫(t), the multiplicity of a given point, P≠ P_L,is the cardinality of the fibre of𝒫(t)at P (see Corollary <ref>). That is,the multiplicity of P=𝒫(s_0), s_0∈𝕂is given by the cardinality of the setℱ_𝒫(𝒫(s_0))={t∈𝕂:𝒫(t)=𝒫(s_0)}(note that we are assumingthat P≠ P_L). Observe that s_0∈ℱ_𝒫(𝒫(s_0)) and hence, the cardinality of ℱ_𝒫(𝒫(s_0)) is greater than or equal to 1. Thus, 𝒫(s_0) is a singular point if and only ifthe cardinality of ℱ_𝒫(𝒫(s_0)) is greater than 1.Taking into account the above statement, in this section, we show how the different factors of a univariate resultant computed from the polynomials G_i(s,t), i=1,2,3, are exactly the fibre functions of the singularitiesof C. Thus, in particular, the singularitiesof C and its corresponding multiplicities are determined. The idea for the construction of the resultant is that a point 𝒫(s_0)∈ C, s_0∈𝕂, is a singularity if and only if (H_𝒫(s_0)(t))>1 (i.e. the fibre equations of 𝒫(s_0) have more than one common solution).For this purpose, we first assume that 𝒫(s_0) is an affine point. Thus, Remark <ref> implies that the fibre equations are given by{[p_1(t)p(s_0)-p_1(s_0)p(t)=0; p_2(t)p(s_0)-p_2(s_0)p(t)=0. ].Note that this is equivalent to G_1(s_0,t)=G_2(s_0,t)=0, whereG_1(s,t) and G_2(s,t) are the polynomials introduced in (<ref>).Then, 𝒫(s_0) is a singular point if and only if G_1(s_0,t) and G_2(s_0,t) have more than one common root or, equivalently, if and only if the polynomialsG_1(s_0,t)/(t-s_0) and G_2(s_0,t)/(t-s_0) have a common root (we note that s_0 is already a root of G_1(s_0,t) and G_2(s_0,t)). This impliesthat_t(G_1(s_0,t)/t-s_0,G_2(s_0,t)/t-s_0)=0.Hence, given the polynomialR(s)=_t(G_1(s,t)/t-s,G_2(s,t)/t-s),if the point 𝒫(s_0) is singular, then R(s_0)=0. In fact, in <cit.>, it is proved that this resultant provides the product of the fibre functions of the singularities of the curve, in the case that P(t) is a polynomial parametrization. Ageneralization for the case of a given rational parametrization(not necessarily polynomial) is presented in <cit.>.Thus,R(s) can be used to compute the singularities of the curve, but some problems could appear. First, the values s∈𝕂 that provide singular points are roots of the polynomial R but the reciprocal is not true; i.e. a root of R may not provide a singular point. In addition, we are assuming that the singularity is an affine point, but also singularities at infinity have to be detected. The T–function, that we introduce below, improves the properties of R(s) and characterizes the singular points of C (affine and at infinity). In order to introduce it, we need to consider δ_i:=_t(G_i),λ_ij:=min{δ_i,δ_j}, G_i^*(s,t):=G_i(s,t)/t-s∈ K[s,t]andR_ij(s):=_t(G_i^*,G_j^*)∈ K[s]i,j=1,2,3,i<j.We define the T–function of the parametrization𝒫(t) asT(s)=R_12(s)/p(s)^λ_12-1.In the following we show that this function provides essential information concerning the singularities of the given curve C (see Theorem <ref>). To start with, the following proposition claims that the T–function can be defined similarly fromR_13(s) or R_23(s). In addition, inCorollary <ref>, we prove that T(s) is a polynomial.It holds thatT(s)=R_12(s)/p(s)^λ_12-1=R_13(s)/p_1(s)^λ_13-1=R_23(s)/p_2(s)^λ_23-1.Proof: We distinguish two steps to prove the proposition:Step 1First, we show thatR_12(s)/p(s)^λ_12-1=R_13(s)/p_1(s)^λ_13-1.For this purpose, we see the polynomial G_1(s,t)∈ K[s,t] as a polynomial in the variable t that is, G_1(s,t)∈ ( K[s])[t]. Since _t(G_1)=_s(G_1)=δ_1(see Remark <ref>), then G_1 has δ_1 roots (in the variable t), and one of them is t=s. Thus, we may writeG_1(s,t)=_t(G_1)(t-s)(t-α_1(s))⋯ (t-α_δ_1-1(s))andG^*_1(s,t)=_t(G_1)(t-α_1(s))⋯ (t-α_δ_1-1(s)),where _t(·) denotes the leader coefficient with respect to the variable t of a polynomial (·). Now, taking into account the properties of the resultants (see e.g. <cit.>, <cit.>, <cit.>), we get thatR_12(s):=_t(G_1^*,G_2^*)=_t(G^*_1)^δ_2-1∏_i=1^δ_1-1G^*_2(s,α_i(s)).Note that G_2(s,t)=p_2(s)p(t)-p(s)p_2(t), and thus G_2(s,α_i(s))=p_2(s)p(α_i(s))-p(s)p_2(α_i(s)). Furthermore, since G_1(s,α_i(s))=p_1(s)p(α_i(s))-p(s)p_1(α_i(s))=0, we get thatp(α_i(s))=p_1(α_i(s))/p_1(s)p(s).Therefore,G_2(s,α_i(s))=p_2(s)p_1(α_i(s))/p_1(s)p(s)-p(s)p_2(α_i(s))=(p_1(α_i(s))p_2(s)-p_2(α_i(s))p_1(s))p(s)/p_1(s)=G_3(s,α_i(s))p(s)/p_1(s)which implies thatG^*_2(s,α_i(s))=G^*_3(s,α_i(s))p(s)/p_1(s).Now we substitute in (<ref>) and we getR_12(s)=(_t(G^*_1)^δ_2-1∏_i=1^δ_1-1G^*_3(s,α_i(s)))(p(s)/p_1(s))^δ_1-1,which can be expressed asR_12(s)=R_13(s)_t(G^*_1)^δ_2-δ_3(p(s)/p_1(s))^δ_1-1.Hence, we only have to prove that_t(G^*_1)^δ_2-δ_3(p(s)/p_1(s))^δ_1-1=p(s)^λ_12-1/p_1(s)^λ_13-1,and, for this purpose, we consider different cases depending on d_1, d_2 and d_3 (that is, on the degrees of p_1, p_2 and p). Weremind thatδ_1=max{d_1,d_3}, δ_2=max{d_2,d_3} and δ_3=max{d_1,d_2} (see Remark <ref>).* Case 1: Let d_1<d_3. Then, δ_1=d_3≤δ_2, and λ_12=δ_1. In addition, it holds that _t(G^*_1)=_t(G_1)=p_1(s) since G_1(s,t)=p_1(s)p(t)-p(s)p_1(t) and d_1<d_3. Thus,_t(G^*_1)^δ_2-δ_3(p(s)/p_1(s))^δ_1-1=p(s)^δ_1-1/p_1(s)^δ_1-δ_2+δ_3-1 =p(s)^λ_12-1/p_1(s)^δ_1-δ_2+δ_3-1.In order to check that (<ref>) holds, we only have to prove that δ_1-δ_2+δ_3=λ_13. Let us see that this equality holds in the following situations: a) δ_1<δ_3: then,d_2>d_1,d_3 which implies that δ_2=δ_3=d_2 and λ_13=δ_1.b) δ_1>δ_3: then,d_3>d_1,d_2 which implies that δ_1=δ_2=d_3 and λ_13=δ_3.c) δ_1=δ_3: then,d_3=max{d_1,d_2} which implies that d_1<d_2=d_3 and δ_1=δ_2=δ_3. * Case 2: Let d_1>d_3. Then,δ_1=d_1≤δ_3, which implies that λ_13=δ_1. In addition, _t(G^*_1)=p(s), and then_t(G^*_1)^δ_2-δ_3(p(s)/p_1(s))^δ_1-1=p(s)^δ_1+δ_2-δ_3-1/p_1(s)^δ_1-1 =p(s)^δ_1+δ_2-δ_3-1/p_1(s)^λ_13-1.Thus, we only have to prove that δ_1+δ_2-δ_3=λ_12. For this purpose, we reason similarly as in Case 1 by considering the following situations: δ_1<δ_2, δ_1>δ_2 and δ_1=δ_2.* Case 3: Let d_1=d_3<d_2.Then, δ_2=δ_3=d_2, and thus _t(G^*_1)^δ_2-δ_3=1. In addition, δ_1≤δ_2=δ_3, which implies that λ_12=λ_13=δ_1.*Case 4: Let d_2<d_1=d_3. In this case, we have that δ_1=δ_2=δ_3 and then, (<ref>) trivially holds.* Case 5: Let d_1=d_2=d_3. This case is similar to Case 4.Step 2Let us prove thatR_12(s)/p(s)^λ_12-1=R_23(s)/p_2(s)^λ_23-1.For this purpose, we observe that, up to constants in K∖{0}, it holds that R_12(s)=R_21(s). Thus, we may writeR_12(s)=R_21(s)=_t(G^*_2)^δ_1-1∏_i=1^δ_2-1G^*_1(s,β_i(s))whereG^*_2(s,t)=_t(G_2)(t-β_1(s))⋯ (t-β_δ_2-1(s)).Now, we observe that these equalities are equivalent to (<ref>) and (<ref>), respectively. Thus, reasoning similarly as above, we obtain thatR_12(s)=R_23(s)_t(G^*_2)^δ_1-δ_3(p(s)/p_2(s))^δ_2-1and that_t(G^*_2)^δ_1-δ_3(p(s)/p_2(s))^δ_2-1=p(s)^λ_12-1/p_2(s)^λ_23-1.It holds that T(s)∈ K[s]. Proof: Let us assume that T(s) is not a polynomial. Then, we simplify the rational function and we writeR_12(s)/p(s)^λ_12-1=M_12(s)/p̅(s),whereM_12(s)∈ K[s], p(s)∈ K[s]∖ K and(M_12,p̅)=1. Note thatp divides p^λ_12-1. Similarly, from Proposition <ref>, we have that{[ R_13(s)/p_1(s)^λ_13-1=M_13(s)/p̅_1(s)where p_1dividesp_1^λ_13-1,and (M_13,p̅_1)=1; R_23(s)/p_2(s)^λ_23-1=M_23(s)/p̅_2(s) where p_2dividesp_2^λ_23-1,and (M_23,p̅_2)=1. ].Furthermore, we have that (see Proposition <ref>)M_12(s)/p̅(s)=M_13(s)/p̅_1(s)=M_23(s)/p̅_2(s)which implies that M_12(s)p̅_1(s)=M_13p̅(s)M_23(s)p̅_1(s)=M_13p̅_2(s).Taking into account that (M_12,p̅)=(M_13,p̅_1)=(M_23,p̅_2)=1, and the above equalities, we get that p_1=p_2=p. Then, we deduce that p divides p, p_1 and p_2, which is impossible since(p_1,p_2,p)=1. In the following theorem, we show how the ordinary singularities of C can be determined from the T–function. In fact, T(s) describes totally the singularities of the curve, since its factorization provides the fibre functions of each singularity as well as its corresponding multiplicity.From the fibre function, H_P(t), of a point P, one obtains the multiplicity of P, its fibre,and the tangent lines at P (see Section <ref>).An alternative approach for computing this factorization, based on the construction of μ–basis, can be found in <cit.>. In Theorem <ref>, we assume that Chas only ordinary singularities. Otherwise, for applying this theorem, we should apply quadratic transformations (blow-ups) for birationally transforming C into a curve with only ordinary singularities (see Chapter 2 in <cit.>). For such a curve the following theorem holds.(Main theorem) Let 𝒞 be a rational algebraic curve defined by a parametrization 𝒫(t), with limit point P_L. Let P_1,…,P_n be the singular points of C, with multiplicities m_1,…,m_n respectively. Let us assume that they are ordinary singularities and that P_i≠ P_L for i=1,…,n. Then, it holds thatT(s)=∏_i=1^nH_P_i(s)^m_i-1.This theorem will be proved in Section<ref> and a generalization for the case of space curves of any dimension will be presented in Section <ref>. Moreover, in <cit.>, we prove that the theorem holds also if P_L is a singularity. Finally, an analogous result which admits the existence of non–ordinary singularities in the curve will be developed in a future work.Let Cbe a rational plane curve such that all its singularities are ordinary. Let 𝒫(t) be a parametrization of C such that P_L is regular. It holds that (T)=(d-1)(d-2). Proof: From Theorem <ref> and Corollary<ref>, we have that(T)=∑_i=1^nm_i(m_i-1), where P_1,…,P_n are the singular points, andm_1,…,m_n its corresponding multiplicities. Since𝒞 is a rational curve, its genus is 0 and thus ∑_i=1^nm_i(m_i-1)=(d-1)(d-2) (see Chapter 3 in <cit.>).Let 𝒞 be the rational plane curve defined by the projective parametrization𝒫(t)=(t^5-5t^4+5t^3+5t^2-6t:t^5+5t^4+5t^3-5t^2-6t:t^4-13t^2+36)∈ℙ^2(ℂ(t)).We compute the T–function and we get that, up to constants in ℂ,T(s)=(t-2)(t-3)(t+3)(t+2)t^2(t^2+6)(t-1)^2(t+1)^2.Thus, from Theorem <ref>, we deduce that the fibre function of each singularity of𝒞appears in the polynomial T. Let us analyze the different factors of T:* The factors with power 2 correspond to triple points. Indeed: these factors are t, (t-1) and (t+1), and we have that P_1=𝒫(0)=𝒫(1)=𝒫(-1)=(0:0:1). Then, P_1is a triple point whose fibre function isH_P_1(t)=(t-1)(t+1)t. * The factors with power 1 correspond to different double points. In order to determine the associated factors, we should compute the corresponding fibre functions. Forthe factors (t-2) and (t-3), we have that P_2=𝒫(2)=𝒫(3)=(0:1:0) and thusH_P_2(t)=(t-2)(t-3). This implies thatP_2 is a double point atinfinity. * Similarly, if we consider the factors(t+2) and(t+3), we get thatP_3=𝒫(-2)=𝒫(-3)=(1:0:0), and its fibre function isH_P_3(t)=(t+2)(t+3), which implies that P_3 is a double point at infinity. * Finally, the factor (t^2+6)provides the point P_4=𝒫(-I√(6))=𝒫(+I√(6))=(-7/5:7/5:1). Hence, P_4 is an affine double point and its fibre function isH_P_4(t)=(t^2+6).Note that,T(s)=H_P_1(s)^m_1-1 H_P_2(s)^m_2-1 H_P_3(s)^m_3-1 H_P_4(s)^m_4-1.Furthermore, we observe that Corollary <ref> is verified. Indeed, we have that d=5 and _t(T)=12=(d-1)(d-2). In Figure <ref>, we plot the curve𝒞, and a neighborhood of the triple point P_1. Observe that P_4 is an isolated point.In Example <ref>, we have been able to determine the singularities of C and its corresponding multiplicities, by computing the factors of the polynomial T(s). However, ingeneral, one needs to introduce algebraic numbers during the computations. In the following, we present a method that allows us to determine the factors of the polynomial T(s) (and thus, thesingularities of a curve) without directly using algebraic numbers. For this purpose, we introduce the notion offamily of conjugate parametric points (see <cit.>), which generalizes the concept of family of conjugate points (see e.g.Chapter 2 in <cit.>). Theidea is to collect points whose coordinates depend algebraically on all the conjugate roots of the same irreducible polynomial m(t). The computations on such a family of points can be carried out by using the polynomial m(t). Let𝒢={(p_1(α):p_2(α):p(α))|m(α)=0}⊂ℙ^2(𝕂).The set 𝒢 is called a family of conjugate parametric points over 𝕂 if the following conditions are satisfied: * p_1,p_2,p,m∈𝕂[t] and (p_1,p_2,p)=1.* m is irreducible.* (p_1),(p_2),(p)<(m).We denote such a family by 𝒢={𝒫(s)}_m(s)={(p_1(s):p_2(s):p(s))}_m(s). Condition 2 in Definition <ref> can bestated considering that m is only square-free (seeDefinition 12 in<cit.>). However, in order to prove Theorem <ref>, one needsm to be an irreducible polynomial (see Theorem 16 in <cit.>). Hence, using the above definition, in <cit.> it is proved the following theorem. The singularities of thecurve C can be decomposed as a finite union of families of conjugate parametric points over 𝕂 such that all points in the same family have the same multiplicity and character.If some singularities of the given curve are in a family 𝒢={𝒫(s)}_m(s)={(p_1(s):p_2(s):p(s))}_m(s), then the polynomial m(s) will be an irreducible factor of the T–function. In this case, Theorem <ref> allows us to determine the singularities provided by 𝒢 and their corresponding multiplicities. Let m(s) be an irreducible polynomial such that m(s)^k-1, k∈ N, k≥ 1, divides T(s). Then, the roots of m(s) determine the fibre of some singularities of multiplicity k that are defined by afamily of conjugate parametric points. The number of singularities in such a family isn=(m(s))/k. Proof: From Theorem <ref>, we get that the points in 𝒢={𝒫(s)}_m(s) are singularities of multiplicityk. In addition, if 𝒢={P_1,…,P_n}, we have thatm(s)=∏_i=1^nH_P_i(s),where H_P_1,…,H_P_n are the fibre functions of the points P_1, …, P_n, respectively. From Corollary<ref>, we get that (H_P_i)=_P_i(𝒞), i=1,…,n, and since, in this case, _P_i(𝒞)=k, we conclude that (m(s))=nk. Let C be the rational curve defined by 𝒫(t)=(p_1(t):p_2(t):p(t))∈ℙ^2(ℂ(t)), where[ p_1(t)=t^5-13t^4+63t^3-143t^2+152t-60; p_2(t)=t^5-21t^4+157t^3-507t^2+706t-336; p(t)=t^5+7t^4-2t^3+t-1. ]The T–function is given by T(s)=28161216(968585964-2319881360s+2070988203s^2-904722886s^3+208513387s^4-24407436s^5+1145528s^6)(s-1)^2(s-2)^2(s-3)^2. From the polynomial T(s), we deduce that the singularities of C are:* The triple point P_1=𝒫(1)=𝒫(2)=𝒫(3)=(0:0:1). The fibre function of P_1 isH_P_1(s)=(s-1)(s-2)(s-3), and these factors appear with power 2 in the polynomial T(s).* Three double points associated to the irreducible factorm(s)=968585964-2319881360s+2070988203s^2-904722886s^3+208513387s^4-24407436s^5+1145528s^6.Since m(s) appears with power1 in the polynomial T(s), we conclude that it is associated to double points. In addition, using Theorem <ref>, we get that this factor provides three different points(each point has a fibre function of degree 2 and the three fibre functions have power 1; multiplying these fibre functions we obtain the polynomial m). In Figure <ref>, we plot the given curve C, and we can see the singularities obtained (note that each singularity is real and affine).§ THE GENERAL CASE FORRATIONAL SPACE CURVES In this section, we show that Theorem <ref> can be applied for the case that the given curve Cis a rational space curve. In this case, we construct an equivalent polynomial to the T–functionintroduced for plane curves (see Definition <ref>),and we prove that this polynomial, which will be denoted as T_E(s),describes totally the singularities of C, since each factor ofT_E(s)is a power of the fibre function of one singularity of the given curve. This power is, in fact, the multiplicity of the singularity minus 1. We recall that from the fibre function of a point P, one may determine the multiplicity of Pas well as its fibreℱ_𝒫(P)and the tangent lines of C at P (see Section <ref>). The method presented generalizes the results obtained in <cit.>, since a complete classification of the singularities of a given space curve, via the factorization of a univariateresultant, is obtained. In the following, we consider𝒫(t)=(p_1(t):⋯: p_n(t):p(t))∈ℙ^n(𝕂(t)),(p_1,…,p_n,p)=1,a proper parametrization of a given rational space curve C. In addition, we define the associated rational parametrization over K(Z), where Z=(Z_1,…,Z_n-2) and Z_1,…,Z_n-2 are new variables, given by𝒫(t)=(p_1(t):p_2(t):p(t))==(p_1(t):p_2(t)+Z_1p_3(t)+⋯ +Z_n-2p_n(t):p(t))∈ℙ^2((𝕂(Z))(t)).This notation is used for the sake of simplicity, but we note that 𝒫(t) depends on Z. Observe that 𝒫(t) is a proper parametrization of a rational plane curve C defined over the algebraic closure ofK(Z). We can establish a correspondence between the points of 𝒞 and the points of C. More precisely, for each point P=(a_1:a_2: a_3:⋯:a_n: a_n+1)∈ C we have another point P=(a_1:a_2+Z_1a_3+⋯ +Z_n-2a_n: a_n+1) ∈ C. Moreover, this correspondence is bijective for the points satisfying that a_1≠0 or a_n+1≠0. For these points, it holds thatℱ_𝒫(P)=ℱ_𝒫(P), which implies that H_P(s)=H_P(s). Note thatthe polynomial H_Prepresents the fibre function of a point P in the space curve C computed from 𝒫(t); i.e. the roots of H_P are the fibre ofP∈ C (this notion was introduced in Definition <ref> for a given plane curve but it can be easily generalized for space curves). Observe that the above correspondence may also be established between the places of C and C centered at P and P, respectively. That is, for each place φ(t)=(φ_1(t):φ_2(t): φ_3(t):⋯:φ_n(t): φ_n+1(t)) of C centered at P we have the place φ(t)=(φ_1(t):φ_2(t)+Z_1φ_3(t)+⋯+Z_n-2φ_n(t): φ_n+1(t)) of Ccentered at P. Hence, the number of tangents of C at P is the same that the number of tangents of C at P and, as a consequence, _P(𝒞)=_P(𝒞).The correspondence aboveintroduced is not bijective if a_1=a_n+1=0. In this case, we have thatP=(0:1: 0) ∈ C and the corresponding points in C are all the points of the form (0:a_2: a_3:⋯:a_n: 0). Thus,if we have exactly ℓ points P_1,…,P_ℓ∈ C with P_i=(0:a_2,i: a_3,i:⋯:a_n,i: 0), i=1,…,ℓ, thenℱ_𝒫(P)=∪_i=1^ℓℱ_𝒫(P_i). Hence, H_P(s)=∏_i=1^ℓ H_P_i(s) (note that ℱ_𝒫(P_i)∩ℱ_𝒫(P_j)=∅ if i≠ j) and _P(𝒞)=∑_i=1^ℓ_P_i(𝒞).Thus, in order to study the singularities of 𝒞 through those of 𝒞, an additional difficulty arises when 𝒞 contains two or more points of the form (0:a_2: a_3:⋯:a_n: 0). Let us call them bad points. In the following, wemay assume w.l.o.g. that we are not in this case, i.e. 𝒞 does not have two or more bad points. Otherwise, weapply a change of coordinates, and we consider the new parametrization 𝒫^*(t)=(p^*_1(t):p_2(t):⋯: p_n(t):p(t)) of the transformed curve𝒞^*, where p^*_1=∑_i=1^nλ_ip_i,λ_i∈ K. By appropriately choosing λ_1,…,λ_n∈ K, we have that (p^*_1,p)=1 (note that (p_1,…,p_n,p)=1) and thus,𝒞^* does not have bad points.Finally, we also note that additional points, which can not be written in the form (a_1:a_2+Z_1a_3+⋯ +Z_n-2a_n: a_n+1),a_i∈ K, i=1,…,n+1, may appear in the curveC. Such points are obtained as 𝒫(t) for t∈𝕂(Z)∖𝕂 and they do not have a correspondence with any point of C. Under these conditions, let G_1, G_2 and G_3 be the equivalent polynomials to G_1, G_2 and G_3 (defined in (<ref>)), constructed from the parametrization 𝒫(t). In addition, let δ_i:=_t(G_i) and λ_ij:=min{δ_i,δ_j}, i,j=1,2,3, i<j,G_i^*(s,t):=G_i(s,t)/t-s∈ ( K[Z])[s,t], i=1,2,3,andR_ij(s):=_t(G_i^*,G_j^*)∈ ( K[Z])[s],i,j=1,2,3, i<j. Then, the T–function of the parametrization𝒫(t) is given byT(s)=R_12(s)/p(s)^λ_12-1.It holds that T(s)∈ ( K[Z])[s](see Corollary <ref>), and by Proposition <ref> we get thatT(s)=R_12(s)/p(s)^λ_12-1=R_13(s)/p_1(s)^λ_13-1=R_23(s)/p_2(s)^λ_23-1.The following theorem is obtained as a consequence of Theorem <ref> (see Section <ref>), and it shows how the function T(s) can be used to define an equivalent polynomial to the T–functionintroduced for plane curves (see Definition <ref>). This polynomial, will be denoted as T_E(s). Similarly as in the case ofplane curves, we assume that the space curve, C,has only ordinary singularities.The case of space curves with non–ordinary singularities will be analyzed in a future work and an equivalent theorem to Theorem <ref> will be obtained for this case.Finally we remind, that if𝒞has two or more bad points, we consider the transformed curve 𝒞^* introduced above. Note that H_P(s)=H_P^*(s), where P∈ C is moved tothe point P^*∈𝒞^* when the change of coordinates is applied. Undoing this change of coordinates, one recovers the initial singularities P∈ C. Let 𝒞 be a rational algebraic space curve defined by a parametrization 𝒫(t), with limit point P_L. Let P_1,…,P_n be the singular points of C, with multiplicities m_1,…,m_n respectively. Let us assume that they are ordinary singularities and that P_i≠ P_L, for i=1,…,n. Then, it holds thatT_E(s)=∏_i=1^nH_P_i(s)^m_i-1,where T_E(s)=_Z(T(s))∈ K[s], and _Z(T) represents the content of the polynomial T w.r.t Z.Proof: From the above statements, we observe that there exists a bijective correspondence between the points P=(a_1:a_2+Z_1a_3+⋯ +Z_n-2a_n: a_n+1),a_i∈ K, i=1,…,n+1,of C and the pointsP=(a_1:a_2: a_3:⋯:a_n: a_n+1)of C. Consequently, we have that _P(𝒞)=_P(𝒞), which implies that P is a singularity of C of multiplicity m if and only if P is a singularity of C of multiplicity m. Hence, usingTheorem <ref>, we deduce thatT(s)=∏_i=1^nH_P_i(s)^m_i-1L(s,Z).We observe that the factor L(s,Z)∈ K[s,Z]∖ K[s] is a product ofthe fibre functions corresponding to the singularities ofC that can not be written as (a_1:a_2+Z_1a_3+⋯ +Z_n-2a_n: a_n+1),a_i∈ K, i=1,…,n+1 (these singularities do not have an equivalent singularity in C, and its fibre function necessarily is a polynomial in K[s,Z]∖ K[s]). Then, weconclude thatT_E(s)=_Z(T(s))=∏_i=1^nH_P_i(s)^m_i-1.Let 𝒞 be the rational space curve defined by the projective parametrization 𝒫(t)=(p_1(t):p_2(t):p_3(t):p(t))∈ℙ^3(ℂ(t)), where[ p_1(t)=t^5-5t^4+5t^3+5t^2-6t; p_2(t)=t^5+5t^4+5t^3-5t^2-6t; p_3(t)=t^5+7t^4+17t^3+17t^2+6t; p(t)=t^4-13t^2+36. ]We consider the plane curve𝒫(t)=(p_1(t):p_2(t)+Zp_3(t):p(t))∈ℙ^2((ℂ(Z))(t))and compute the corresponding T–function:T(s)=298598400(s-2)(s-3)(s+3)(s+2)s(s+1)L(s,Z),where L(s,Z)=(3Z+1)(2Z+1)(25s^6+25Z^2s^6+50Zs^6+60Z^2s^5+35Zs^5-25s^5+125s^4+322Z^2s^4+375Zs^4+360Z^2s^3-65Zs^3-125s^3-150s^2-185Zs^2+229Z^2s^2+150Zs+300Z^2s+150s-360Z+432Z^2). Removing L(s,Z) (which depends on Z), we getT_E(s)=298598400(s-2)(s-3)(s+3)(s+2)s(s+1).Now, reasoning as in Example <ref>, we deduce that C has three singularities:* The infinity point P_1=(0:1:3:0), with fibre function H_P_1(t)=(t-2)(t-3) (let us remark that P_1 is a bad point; however, it does not represent a problem in this case since there are no more bad points in 𝒞).* The infinity point P_2=(1:0:0:0), with fibre function H_P_2(t)=(t+2)(t+3).* The affine point P_3=(0:0:0:1), with fibre function H_P_3(t)=t(t+1).Note that P_1, P_2 and P_3 are double points of C (see Figure <ref>).§ PROOF OF THE MAIN THEOREM This section is devoted to show the main result of this paper, Theorem <ref> in Section <ref>. For this purpose, we first prove some previous results. In particular, the following lemma is obtained using the main properties of the resultants (see e.g. <cit.>, <cit.>, <cit.>).Let A(s,t), B(s,t), C(s,t)∈ K[s,t], and K(s)∈ K[s]. The following properties hold:* _t(A,K)=K^_t(A).* _t(A,B· C)=_t(A,B)·_t(A,C).* If B dividesA, it holds that _t(A/B,C)=_t(A,C)/_t(B,C).* _t(A,B+CA)=(A)^k_t(A,B), where k=_t(B+CA)-_tB. Proof: First, we remind that if A(t), B(t)∈ K[t], it holds that_t(A,B)=(A)^(B)∏_A(α_i)=0B(α_i).Now, letA(s,t)∈ ( K[s])[t]. The leader coefficient of A(s,t) w.r.t the variable t is _t(A)∈ K[s] and, for each s∈𝕂,the polynomial A(s,t) has _t(A) roots, α_1(s),…,α__t(A)(s), in the algebraic closure of K(s), such that A(s,α_i(s))=0, for i=1,…,_t(A). Then,_t(A,B)=_t(A)^_t(B)∏_A(s,α_i(s))=0B(s,α_i(s)).Now, we provethe four statements of the lemma. * The first statement follows using (<ref>) for the case that B(s,t)∈ K[s]. Then _t(B)=0 and B(s,α_i(s))=B(s) for each i=1,…,_t(A).* In order to prove statement 2, we use (<ref>), and we get that_t(A,B· C)=_t(A)^_t(B· C)∏_A(s,α_i(s))=0B(s,α_i(s))· C(s,α_i(s)).Since _t(B· C)=_t(B)+_t(C), we have that_t(A,B· C)=(_t(A)^_t(B)∏_A(s,α_i(s))=0B(s,α_i(s)))(_t(A)^_t(C)∏_A(s,α_i(s))=0C(s,α_i(s)))=_t(A,B)·_t(A,C). * Reasoning similarly as in statement 2, we get that_t(A/B,C)=_t(A/B)^_t(C)∏_A(s,α_i(s))=0, B(s,α_i(s))≠ 0C(s,α_i(s)).Since B dividesA, we obtain that_t(A/B)^_t(C)∏_A(s,α_i(s))=0, B(s,α_i(s))≠ 0C(s,α_i(s))=_t(A)^_t(C)/_t(B)^_t(C)∏_A(s,α_i(s))=0C(s,α_i(s))/∏_B(s,α_i(s))=0C(s,α_i(s))=_t(A,C)/_t(B,C). * We reason similarly as above, andwe get that_t(A,B+CA)=_t(A)^_t(B+CA)·∏_A(s,α_i(s))=0(B(s,α_i(s))+C(s,α_i(s))A(s,α_i(s)))= =(A)^_t(B+CA)-_tB((A)^_t(B)∏_A(s,α_i(s))=0B(s,α_i(s)))==(A)^k_t(A,B), k=_t(B+CA)-_tB. The following lemma provides a first approach to the proof of the main result presented in this paper (Theorem <ref> in Section <ref>).In particular, it is shown that each factor H_P(s)^m-1, where P is an ordinary singular point of multiplicity m, divides the T–function. Let 𝒞 be a rational algebraic curve defined by a parametrization 𝒫(t), with limit point P_L. Let P≠ P_Lbe an ordinary singular point of multiplicity m. It holds thatT(s)=H_P(s)^m-1T^*(s),where T^*(s)∈ K[s] and (H_P(s),T^*(s))=1.Proof: In order to prove this lemma, we distinguish three different steps. InStep 1, we prove that the lemma holds if P=(0:0:1). In Step 2, we show thatthe lemma holds for any affine singularity. Finally, in Step 3, we prove the lemma for P being a singular point at infinity.Step 1 Let us assume that the given singularity is the point P=(0:0:1). Note that H_P(t)=(ϕ_1,ϕ_2)=(p_1,p_2) since,in this case, a=b=0 (see (<ref>)). Thus, we may write{[p_1(t)=H_P(t)p_1(t); p_2(t)=H_P(t)p_2(t), ].where p_1(t) and p_2(t) are polynomials satisfying that (p_1,p_2)=1. In addition, it holds that(H_P(t),p(t))=1, since (p_1,p_2,p)=1. Hence, from (<ref>), we may write G_3(s,t)=H_P(s)H_P(t)(p_1(s)p_2(t)-p_2(s)p_1(t)) that is,G_3(s,t)=H_P(s)H_P(t)G_3(s,t),where G_3(s,t)=p_1(s)p_2(t)-p_2(s)p_1(t).Observe that since P≠ P_L, Corollary <ref> holds and then(H_P(t))=m≥ 2. Hence,there exist at least two values s_0, s_1∈ Ksuch that H_P(s_0)=H_P(s_1)=0and then, since these roots belong to the fibre ofP, we deduce that((p_1/p)(s_0),(p_2/p)(s_0))=((p_1/p)(s_1),(p_2/p)(s_1))=(0,0).Since P is an ordinary singularity, we have that there can not exist K_1, K_2∈ K such thatK_1((p_1/p)'(s_0),(p_2/p)'(s_0))=K_2((p_1/p)'(s_1),(p_2/p)'(s_1)). We also note that, for i=1,2 and j=0,1, it holds that(p_i/p)'(s_j)=p'_i(s_j)p(s_j)-p_i(s_j)p'(s_j)/p(s_j)^2=p'_i(s_j)/p(s_j),(note that p_i(s_j)=0).In addition, sincep_i(t)=H_P(t)p_i(t), we get thatp'_i(s_j)/p(s_j)=H'_P(s_j)p_i(s_j)+H_P(s_j)p'_i(s_j)/p(s_j)=H'_P(s_j)p_i(s_j)/p(s_j),(note that H_P(s_j)=0). By substituting in(<ref>), we obtain thatK_1(H'_P(s_0)p_1(s_0)/p(s_0),H'_P(s_0)p_2(s_0)/p(s_0))≠ K_2(H'_P(s_1)p_1(s_1)/p(s_1),H'_P(s_1)p_2(s_1)/p(s_1)).Hence, we remark that: * H'_P(s_i)≠ 0, i=0,1. That is,s_0 and s_1 are simple roots ofH_P.* None of the following equalities may be verified:{[ p_1(s_0)=p_2(s_0)=0; p_1(s_1)=p_2(s_1)=0; p_1(s_0)=p_1(s_1)=0; p_2(s_0)=p_2(s_1)=0 ]. * If p_1(s_1)≠ 0 and p_2(s_1)≠ 0, thenp_1(s_0)/p_1(s_1)≠p_2(s_0)/p_2(s_1).Taking into account these remarks, we prove that T(s)=H_P(s)^m-1T^*(s). For this purpose, we first write the T–function asT(s)=R_13(s)/p_1(s)^λ_13-1(see Proposition <ref>). From (<ref>), we get thatR_13(s)=_t(G^*_1(s,t),H_P(s)H_P(t)G_3^*(s,t)),where G_3^*(s,t)=G_3(s,t)/(t-s) (note that G_3^*(s,t)∈ K[s,t] since (t-s) divides G_3(s,t)). By applying statement 2 of Lemma <ref>, we have thatR_13(s)= _t(G^*_1(s,t),H_P(s))_t(G^*_1(s,t),H_P(t))_t(G^*_1(s,t),G_3^*(s,t)).Let us analyse the first two factors: * From statement 1 of Lemma <ref>, we have that_t(G^*_1(s,t),H_P(s))=H_P(s)^_t(G^*_1)=H_P(s)^δ_1-1. * On the other side,_t(G^*_1(s,t),H_P(t))=_t(p_1(s)p(t)-p(s)p_1(t)/t-s,H_P(t))and, by applying statement 3 of Lemma<ref>, we get that_t(p_1(s)p(t)-p(s)p_1(t),H_P(t))/_t(t-s,H_P(t)).Note that _t(t-s,H_P(t))=H_P(s) and, sincep_1(t)=H_P(t)p_1(t), the above expression can be written as_t(p_1(s)p(t)-p(s)H_P(t)p_1(t),H_P(t))/H_P(s).Now, from statements 1 and 4 of Lemma<ref>, we get that_t(G^*_1(s,t),H_P(t))=p_1(s)^m _t(p(t),H_P(t))(H_P(t))^k/H_P(s),where k∈𝕂. Note that,(H_P(t))^k∈ K∖{0} and _t(p(t),H_P(t))∈ K∖{0} (since (p,H_P)=1). Furthermore, we have that p_1(s)=H_P(s)p_1(s). Thus, up to constants inK∖{0}, we deduce that_t(G^*_1(s,t),H_P(t))=H_P(s)^m-1p_1(s)^m. Substituting in (<ref>), we obtain thatR_13(s)=H_P(s)^δ_1-1H_P(s)^m-1p_1(s)^m _t(G^*_1(s,t),G_3^*(s,t)),and thusT(s)=R_13(s)/p_1(s)^λ_13-1=R_13(s)/H_P(s)^λ_13-1p_1(s)^λ_13-1=H_P(s)^m-1T^*(s),whereT^*(s)=_t(G^*_1(s,t),G_3^*(s,t))H_P(s)^δ_1-λ_13/p_1(s)^λ_13-1-m.Note that λ_13=δ_1 since δ_1≤δ_3. Otherwise, if δ_1>δ_3, we would have thatmax{d_1,d_3}>max{d_1,d_2} and then, d_3>d_1,d_2. However, this would imply thatP=P_L (seeDefinition <ref>), which contradicts the assumption. Therefore,T^*(s)=_t(G^*_1(s,t),G_3^*(s,t))/p_1(s)^δ_1-1-m.Now, weprove that T^*(s) ∈ K[s]. We can assume that δ_1-1-m≥ 0, since δ_1=max{d_1,d_3}≥ d_1 and d_1=(p_1)=(H_P·p_1)=m+(p_1). Hence,δ_1≥ m+1 except for the case that (p_1)=0, but in this situation we would have thatT^*(s)= _t(G^*_1(s,t),G_3^*(s,t))∈ K[s].So, let δ_1-1-m≥ 0. Now, we reason similarly as in the proof of Corollary<ref>. Indeed: let us assume thatT^*(s) is not a polynomial. Then, by taking the simplified rational function, we may writeT^*(s)=M_13(s)/p_1(s)where M_13(s)∈ K[s],p_1(s)∈ K[s]∖ Kand (M_13(s),p_1(s))=1. Note that p_1(s) divides p_1(s)^δ_1-1-m. We observe that(<ref>) is obtained from T(s)=R_13(s)/p_1(s)^λ_13-1. However, taking into account Proposition <ref>, we could have considered the expressionT(s)=R_23(s)/p_2(s)^λ_23-1concluding thatT(s)=H_P(s)^m-1T^*(s), whereT^*(s)=_t(G^*_2(s,t),G_3^*(s,t))/p_2(s)^δ_2-1-m.Reasoning similarly as above, we get that there would exist M_23(s)∈ K[s], p_2(s)∈ K[s]∖ Kwith(M_23(s),p_2(s))=1, such thatT^*(s)=M_23(s)/p_2(s).In addition,p_2(s) would divide p_2(s)^δ_2-1-m. Thus, we have thatM_13(s)/p_1(s)=M_23(s)/p_2(s)where(M_13(s),p_1(s))=(M_23(s),p_2(s))=1, which implies thatp_1(s)=p_2(s). Hence (p_1,p_2)≠ 1, which contradicts the definition of these functions.Thus, we have proved that T^* is a polynomial. Finally, we show that(H_P(s),T^*(s))=1. Indeed: if (H_P(s), T^*(s))≠1, there existss_0∈ K such that H_P(s_0)=0 and T^*(s_0)=0, which implies that_t(G^*_1(s,t),G_3^*(s,t))(s_0)=0.Then, by the properties of the resultants, one of the following statements hold: * There existss_1∈ K such that G^*_1(s_0,s_1)=G_3^*(s_0,s_1)=0. This would imply that G^*_3(s_0,s_1)=0, and, then, s_0 and s_1 are elements of the fibre ofP. On the other side,G_3(s_0,s_1)=p_1(s_0)p_2(s_1)-p_2(s_0)p_1(s_1)=0and thus,p_1(s_0)/p_1(s_1)=p_2(s_0)/p_2(s_1).This implies thatP is a non–ordinary singular point (see (<ref>)), which contradicts the assumptions.* It holds that (_t(G^*_1), _t(G_3^*))(s_0)=0. Then, in particular,_t(G^*_1)(s_0)=_t(G_1)(s_0)=p_1(s_0)c_d-p(s_0)a_d=0⇒ a_d=0.Now we reason similarly with the equalityT(s)=R_23(s)/p_2(s)^λ_23-1 and we get(<ref>). From this expression, and reasoning similarly as above, we obtain that(_t(G^*_2), _t(G_3^*))(s_0)=0, which implies that b_d=0. However, if a_d=b_d=0 we deduce that P=P_L, which contradicts the assumptions. Therefore, we conclude that(H_P(s),T^*(s))=1.Step 2Let P=(a:b:1) be a singularity of multiplicitym. In this case, we consider the translation of the curveC defined by the parametrizationP(t)=(p_1(t)-ap(t):p_2(t)-bp(t):p(t)).We have that the pointP=(a:b:1) moves to the point P=(0:0:1), and then H_P(t)=H_P(t) (note that the polynomialH_P(t) is computed from P(t), and the polynomial H_P(t) is computed from P(t)).On the other side, if we compute the polynomial equivalent toG_1(s,t) with the new parametrization P(t), we get thatG_1(s,t)=p_1(s)p(t)-p(s)p_1(t)= =(p_1(s)-ap(s))p(t)-p(s)(p_1(t)-ap(t))=p_1(s)p(t)-p(s)p_1(t)=G_1(s,t).Similarly, one obtains thatG_2(s,t)=G_2(s,t). Thus,R_12(s)=_t(G_1(s,t)/t-s,G_2(s,t)/t-s)=_t(G_1(s,t)/t-s,G_2(s,t)/t-s)=R_12(s),and thenT(s)=R_12(s)/p(s)^λ_12-1=R_12(s)/p(s)^λ_12-1=T(s).Thus, it holds thatT(s)=H_P(s)^m-1T^*(s) and (H_P,T^*)=1, since from Step1, these equalities hold forH_P(s) and T(s). Step 3Let us prove that the lemma holds for a singularity at infinity. For this purpose, we assume that P=(1:0:0). Note that we can reason similarly as in Step 1 taking into account that for this case, H_P(t)=(p(t),ϕ_3(t))=(p(t),p_2(t)) (see Remark <ref>). Hence,G_2(s,t)=H_P(s)H_P(t)(p_2(s)p(t)-p(s)p_2(t)).andR_12(s)=_t(G^*_1,G^*_2)=_t(G^*_1(s,t),H_P(s)H_P(t)G_2^*(s,t))whereG_2^*(s,t)=p_2(s)p(t)-p(s)p_2(t)/t-s.Thus, using the expression T(s)=R_12(s)/p(s)^λ_12-1, we deduce that the lemma also holds if the singularity is the point (1:0:0). A similar reasoning with the expression T(s)=R_23(s)/p_2(s)^λ_23-1 shows that the lemma holds for the point (1:0:0).Finally, let us assume that P=(a:b:0). Then, we reason similarly as in Step2 and we apply a translation such that the point P is moved to the point P=(1:0:0). This translation can be defined parametrically by𝒫(t)=(p_1(t)):p_2(t)-(b/a)p_1(t):p(t)).We assume thata≠ 0; otherwise, it should be b≠ 0 and we would use a translation that would move P to (0:1:0).Under these conditions, we have that H_P(t)=H_P(t). In addition, if we compute the equivalent polynomials to G_1(s,t) and G_3(s,t) with the new parametrization 𝒫(t), we get thatG_1(s,t)=G_1(s,t) and G_3(s,t)=G_3(s,t). Thus, from Proposition<ref>,T(s)=R_13(s)/p_1(s)^λ_13-1=R_13(s)/p_1(s)^λ_13-1=T(s).Therefore T(s)=H_P(s)^m-1T^*(s) and (H_P,T^*)=1, since both equalities hold for H_P(s) and T(s).Proof of the Main Formula (Theorem <ref> in Section <ref>)Taking into account Lemma <ref>, we have that for each singular pointP_i, it holds that T(s)=H_P_i(s)^m_i-1T_i^*(s), where T_i^*(s)∈ K[s] and (H_P_i,T_i^*)=1. In addition, (H_P_i,H_P_j)=1 for i≠ j (otherwise, there would exist s_1∈ K such that P(s_1)=P_i=P_j). Then, we get thatT(s)=∏_i=1^nH_P_i(s)^m_i-1V(s),where V(s)∈ K[s] and (H_P_i,V)=1 for i=1,…,n.Note that ifV(s_0)=0, then T(s_0)=0 and thus,R_12(s_0)=R_13(s_0)=R_23(s_0)=0. From R_13(s_0)=_t(G^*_1(s,t),G^*_3(s,t))(s_0)=0 and using the properties of the resultant, we deduce that one of the following two statements hold: * There exists s_1∈ K such thatG^*_1(s_0,s_1)=G^*_3(s_0,s_1)=0. Thus,H_P(s_1)=0, where P=𝒫(s_0), which is impossible since (V,H_P)=1.* It holds that(_t(G^*_1), _t(G_3^*))(s_0)=0. However, this is also a contraction since we would have that_t(G^*_1)(s_0)=_t(G_1)(s_0)=p_1(s_0)c_d-p(s_0)a_d=0⇒p_1(s_0)/p(s_0)=a_d/c_dand_t(G^*_3)(s_0)=_t(G_3)(s_0)=p_1(s_0)b_d-p_2(s_0)a_d=0⇒p_1(s_0)/p_2(s_0)=a_d/b_d.From both equalities, we deduce thatp_2(s_0)/p(s_0)=b_d/c_d,and then𝒫(s_0)=P_L. This would imply that P_L can be reached by the parametrization P(t) but this implies that it is a singularity (see Proposition 3.4 in <cit.>), which contradicts the assumption of the theorem.Thus, we have thatV∈ K and, hence, we conclude that, up to constants in K∖{0},T(s)=∏_i=1^nH_P_i(s)^m_i-1. 00Abhy Abhyankar S.S., (1990). Algebraic Geometry for Scientists and Engineers. Math. Surveys Monogr., vol. 35, American Mathematical Society, Providence, RI.Blas2 Blasco, A., Pérez-Díaz, S. (2015). Asymptotes of Space Curves. Journal of Computational and Applied Mathematics. Vol. 278, pp. 231–247.MyB-2017(b) Blasco, A., Pérez-Díaz, S. (2017). The Limit Point and the T–Function. See arXiv:1706.09291v2 Buse2010 Busé, L., Thang, L.B. (2010) Matrix-based implicit representations of rational algebraic curves and applications. Computer Aided Geometric Design, 27 vol. 9, 681–699.Buse2012 Busé, L., D'Andrea, C. (2012) Singular factors of rational plane curves. Journal of Algebra, 357, 322–346.Chen2008 Chen, F., Wang, W., Liu, Y. (2008) Computing singular points of plane rational curves. J. Symbolic Comput. 43 (2) 92-–117.Cox1998Cox, D.A.,Little, J.,O'Shea, D.(1998). Using algebraic geometry. Graduate texts in mathematics. Vol.185. Springer–Verlag. Harris:algebraic Harris, J. (1995). Algebraic Geometry. A first Course. Springer-Verlag.Hoffmann Hoffmann,C.(1989). Geometric and Solid Modeling: An Introduction. Morgan Kaufmann Publishers, Inc. California. HSW Hoffmann, C.M., Sendra, J.R., Winkler, F. (1997). Parametric Algebraic Curves and Applications. J. Symbolic Computation. Vol.23.HL97 Hoschek, J., Lasser, D.(1993). Fundamentals of Computer Aided Geometric Design.A.K. Peters Wellesley MA., Ltd. Perez-Repara1 Pérez-Díaz, S. (2006). On the problem of proper reparametrization for rational curves and surfaces. Computer Aided Geometric Design. Vol. 23(4). pp.307–323.JSC-Perez Pérez-Díaz, S. (2007). Computation of the Singularities of Parametric Plane Curves. Journal of Symbolic Computation. Vol. 42/8. pp. 835–857. Fibra-Super Pérez-Díaz, S., Sendra, J.R. (2004). Computation of the Degree of Rational Surface Parametrizations. Journal of Pure and Applied Algebra. Vol. 193/1–3. pp. 99–121.Impli-Super Pérez-Díaz, S., Sendra, J.R. (2008). A univariate resultant-based implicitization algorithm for surfaces. Journal of Symbolic Computation. Vol. 43/2. pp. 118–139.MACOM Pérez-Díaz, S., Sendra, J.R., Villarino, C. (2015). Computing the Singularities of Rational Surfaces. Mathematics of Computation. Vol. 84, N. 294, pp. 1991–-2021.Rubio Rubio, R., Serradilla, J.M., Vélez, M.P. (2009). Detecting real singularities of a space curve from a real rational parametrization. Journal of Symbolic Computation. Vol. 44, Issue 5, pp. 490–498.SWPSendra, J.R., Winkler, F., Pérez–Díaz, S.(2007). Rational Algebraic Curves: A Computer Algebra Approach. Series: Algorithms and Computation in Mathematics.Vol. 22. Springer Verlag. shafa ShafarevichI.R.,(1994). Basic algebraic geometry Schemes; 1 Varieties in projective space. Berlin New York: Springer-Verlag. Vol. 1.Vander van der Waerden, B.L.(1970). Algebra I and II. Springer-Verlag, New York.Walker Walker, R.J.(1950).Algebraic Curves. Princeton University Press. | http://arxiv.org/abs/1706.08430v4 | {
"authors": [
"Angel Blasco",
"Sonia Pérez-Díaz"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170626151206",
"title": "Resultants and Singularities of Parametric Curves"
} |
Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg City, Luxembourg Institute of Physics, Martin Luther University Halle-Wittenberg, 06099 Halle, Germany Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, [email protected] H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom Institut für Physik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 7, 55128 Mainz, GermanyInstitute of Physics, Martin Luther University Halle-Wittenberg, 06099 Halle, GermanyWPI-AIMR and IMR and CSRN, Tohoku University, Sendai, Miyagi 980-8577, Japan Zernike Institue for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The NetherlandsInstitute of Physics, Martin Luther University Halle-Wittenberg, 06099 Halle, Germany Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, GermanyWe present a theory of the Seebeck effect in nanomagnets with dimensions smaller than the spin diffusion length, showing that the spin accumulation generated by a temperature gradient strongly affects the thermopower. We also identify a correction arising from the transverse temperature gradient induced by the anomalous Ettingshausen effect and an induced spin-heat accumulation gradient. The relevance of these effects for nanoscale magnets is illustrated by ab initio calculations on dilute magnetic alloys. 71.15.Rf, 72.15.Jf, 72.25.Ba, 85.75.-dSeebeck Effect in Nanomagnets Ingrid Mertig December 30, 2023 ============================= § INTRODUCTION Spin caloritronics <cit.> addresses the coupling between the spin and heat transport in small structures and devices. The effects addressed so far can be categorized into several groups <cit.>. The first group covers phenomena whose origin is not connected to spin-orbit coupling (SOC). Nonrelativistic spin caloritronics in magnetic conductors addresses thermoelectric effects in which motion of electrons in a thermal gradient drives spin transport, such as the spin-dependent Seebeck <cit.> and the reciprocal Peltier <cit.> effect. Another group of phenomena is caused by SOC and belongs to relativistic spin caloritronics <cit.> including the anomalous <cit.> and spin <cit.> Nernst effects.The Seebeck effect <cit.> or thermopower stands for the generation of an electromotive force or gradient of the electrochemical potential μ by temperature gradients ∇ T. The Seebeck coefficient S parameterized the proportionality when the charge current j vanishes: (∇μ /e)__j=0=S∇ T .In the two-current model for spin-polarized systems, the thermopower of a magnetic metal reads S=σ ^+S^++σ ^-S^-/σ ^++σ ^- ,where σ ^± and S^± are the spin-resolved longitudinal conductivities and Seebeck coefficients, respectively.Here, we study the Seebeck effect in nanoscale magnets on scales equal or less than their spin diffusion length <cit.> as in Figure <ref>. Thermal baths on both sides of the sample drive a heat current in the x direction. Since no charge current flows, a thermovoltage builds up at the sample edges that can be observed non-invasively by tunnel junctions or scanning probes. Note that metallic contacts can detect the thermovoltage at zero-current bias conditions, but this requires additional modelling of the interfaces. We show in the following that in the presence of a thermally generated spin accumulation the thermopower differs from Eq. ( <ref>). We then focus on dilute ternary alloys of a Cu host with magnetic Mn and nonmagnetic Ir impurities. By varying the alloy concentrations we may tune to the unpolarized case S^+=S^-, as well as to spin-dependent S^+ and S^- parameters with equal or opposite signs. The single-electron thermoelectric effects considered here can be distinguished from collective magnon drag effects <cit.> by their temperature dependence.§ THEORY In the two-current model of spin transport in a single-domain magnet <cit.>, extended to include heat transport, the charge (j) and heat (q) current densities read j^±=σ̂^±(∇μ ^±/e)-σ̂^±Ŝ^±∇T^± , q^±=σ̂^±Ŝ^±T(∇μ ^±/e)-κ̂^±∇T^± ,where σ̂^±, Ŝ^±, and κ̂^± are the spin-resolved electric conductivity, Seebeck coefficient, and heat conductivity, respectively. All transport coefficients are tensors that reflect crystalline symmetry and SOC. The “four-current model” Eqs. (<ref>) and (<ref>) can be rewritten as [ j; j^s; q; q^s ] = [σ̂σ̂^sσ̂ŜT σ̂Ŝ ^sT;σ̂^sσ̂σ̂Ŝ^sTσ̂ŜT;σ̂ŜTσ̂Ŝ^sT κ̂T κ̂^sT;σ̂Ŝ^sTσ̂ŜT κ̂^sT κ̂T ][∇μ /e; ∇μ ^s/2e;-∇T/T; -∇T^s/2T ]in terms of the charge j=j^++j^-, spin j^s= j^+-j^-, heat q=q^++q^-, and spin-heat q^s=q^+-q^- current densities. Here, we introduced the conductivity tensors for charge σ̂=σ̂^++σ̂ ^-, spin σ̂^s=σ̂^+-σ̂^-, heat κ̂=κ̂^++κ̂^-, and spin heat κ̂^s= κ̂^+-κ̂^-. The driving forces are ∇μ =1/2(∇μ ^++ ∇μ ^-) , ∇T=1/2( ∇T^++∇T^-)and the gradients of the spin μ ^s=μ ^+-μ ^- <cit.> and spin-heat T^s=T^+-T^- accumulations <cit.>∇μ ^s=1/2(∇μ ^+- ∇μ ^-), ∇T^s=1/2( ∇T^+-∇T^-) .Finally, the tensors Ŝ=σ̂^-1(σ̂^+Ŝ^++σ̂^-Ŝ ^-)and Ŝ^s=σ̂^-1(σ̂^+Ŝ^+-σ̂^-Ŝ^-)in Eq. (<ref>) describe the charge and spin-dependent Seebeck coefficients, respectively. In cubic systems the diagonal component S_ii, where i is the Cartesian component of the applied temperature gradient, reduces to the scalar thermopower Eq. (<ref>).§ RESULTS In the following we apply Eq. (<ref>) to the Seebeck effect in nanoscale magnets assuming their size to be smaller than the spin diffusion length. In this case the spin-flip scattering may be disregarded <cit.>. We focus first on longitudinal transport and disregard ∇T^s. However, we also discuss transverse (Hall) effects as well as the spin temperature gradient below. We adopt open-circuit conditions for charge and spin transport under a temperature gradient. Charge currents and, since we disregard spin-relaxation, spin currents vanish everywhere in the sample: 0 = σ̂ (∇μ/e) + σ̂^s (∇μ^s/2e) - σ̂Ŝ∇T , 0 = σ̂^s (∇μ/e) + σ̂ (∇μ^s/2e) - σ̂Ŝ^s ∇T ,q = T σ̂ [Ŝ (∇μ/e) + Ŝ^s ( ∇μ^s/2e)] - κ̂∇T . The thermopower now differs from the conventional expression given by Eq. ( <ref>). Let us introduce the tensor Σ̂ as . ∇μ/e| __j=0=Σ̂∇T.From Eqs. (<ref>) and (<ref>), we find Σ̂=( σ̂-σ̂^sσ̂^-1σ̂^s) ^-1( σ̂Ŝ-σ̂^sŜ ^s).When the spin accumulation in Eq. (<ref>) vanishes we recover Σ̂→Ŝ. Equation (<ref>) involves only directly measurable material parameters <cit.>, but the physics is clearer in the compact expression Σ̂=(Ŝ^++Ŝ^-)/2 .The spin polarization of the Seebeck coefficient . ∇μ ^s/2e| __j=0=Σ̂^s∇T ,reads Σ̂^s=( σ̂-σ̂^sσ̂^-1σ̂^s) ^-1(σ̂Ŝ^s-σ̂^sŜ) ,or Σ̂^s=(Ŝ^+-Ŝ^-)/2 .The diagonal elements of Σ̂ govern the thermovoltage in the direction of the temperature gradient. The off-diagonal elements of Σ̂ represent transverse thermoelectric phenomena such as the anomalous <cit.> and planar <cit.> Nernst effects. The diagonal and off-diagonal elements of Σ̂^s describe the spin-dependent Seebeck effect <cit.>, as well as (also in non-magnetic systems) the spin and planar-spin Nernst effects <cit.>, respectively. We do not address here anomalous and Hall transport in the purely charge and heat sectors of Eq. (<ref>). §.§ Longitudinal spin accumulation A temperature gradient in x direction ∇T∥e_x induces the voltage in the same direction: (∇ _xμ /e)__j=0=Σ _xx∇ _xT .In order to assess the importance of the difference between Eqs. (<ref>) and (<ref>) and the conventional thermopower Eq. (<ref>) we carried out first-principles transport calculations for the ternary alloys Cu_1-v(Mn_1-wIr_w)_v, where w∈ 0,1] and the total impurity concentration is fixed to v=1 at.% <cit.>. We calculate the transport properties from the solutions of the linearized Boltzmann equation with collision terms calculated for isolated impurities <cit.>. We disregard spin-flip scattering <cit.>, which limits the size of the systems for which our results hold (see below). We calculate the electronic structure of the Cu host by the relativistic Korringa-Kohn-Rostoker method <cit.>. Figure <ref> summarizes the calculated room-temperature (charge) thermopower Eqs. (<ref>) or (<ref>) and (<ref>) and their spin-resolved counterparts, Eqs. (<ref>) and (<ref>). Table <ref> contains additional information for the binary alloys Cu(Mn) and Cu(Ir) with w=0 or w=1 in Fig. <ref>, respectively. Here we implicitly assume an applied magnetic field that orders all localized moments.We observe large differences (even sign changes) between S_xx^+ and S_xx^- that causes significant differences between Σ _xx=(S_xx^++S_xx^-)/2 and the macroscopic S_xx. The complicated behavior of the latter is caused by the weighting of S^+ andS^- by the corresponding conductivities, see Eq. (<ref>). Even though a spin-accumulation gradient suppresses the Seebeck effect, an opposite sign of S_xx^+ and S_xx^- can enhance Σ _xx^s beyond the microscopic as well as macroscopic thermopower. Indeed, Hu et al. <cit.> observed a spin-dependent Seebeck effect that is larger than the charge Seebeck effect in CoFeAl. Our calculations illustrate that the spin-dependent Seebeck effect can be engineered and maximized by doping a host material with impurities. §.§ Hall transport In the presence of spin-orbit interactions the applied temperature gradient ∇T_ext induces anomalous Hall currents. When the electron-phonon coupling is weak, the spin-orbit interaction can, for example, induces transverse temperature gradients. In a cubic magnet the charge and spin conductivity tensors are antisymmetric. With magnetization and spin quantization axis along z: σ̂^(s)=([σ _xx^(s) -σ _yx^(s)0;σ _yx^(s)σ _xx^(s)0;00σ _zz^(s) ]),and analogous expressions hold for Ŝ and Ŝ^s. A charge current in the x direction generates a transverse heat current that heats and cools opposite edges, respectively. A transverse temperature gradient ∇T_ind∥e_y is signature of this anomalous Ettingshausen effect <cit.> gradient. From Eqs. (<ref>), (<ref>), and (<ref>) q=[ Tσ̂(ŜΣ̂+Ŝ^sΣ̂^s)- κ̂] ∇T ,where ∇T=e_x∇ _xT_ext+e _y∇ _yT_ind. Assuming weak electron-phonon scattering, the heat cannot escape the electron systems and q_y=0. Equation (<ref>) then leads to∇ _yT_ind=-A_yx/A_yy∇ _xT_ext ,where A_yx and A_yy are components of the tensor Â=Tσ̂(ŜΣ̂+Ŝ^sΣ̂^s)-κ̂ .Consequently, Eq. (<ref>) leads to a correction to the thermopower (∇ _xμ /e)__j=0=[ Σ _xx-Σ _xyA_yx/ A_yy] ∇ _xT_ext .However, this effect should be small <cit.> for all but the heaviest elements but may become observable when Σ _xx vanishes, which according to Fig. <ref> should occur at around w=0.125. §.§ Spin temperature gradient At low temperatures, the spin temperature gradient ∇ T^s may persist over length scales smaller but of the same order as the spin accumulation <cit.>. From Eqs. (<ref>), (<ref> ), and (<ref>) it follows (∇μ /e)__j=0=Σ̂∇T+Σ̂^s∇T^s/2 , (∇μ^s/2e)__j=0=Σ̂^s∇T+Σ̂∇T^s/2 .Starting with Eq. (<ref>) and employing Eqs. (<ref>) and (<ref>) for the heat and spin-heat current densities we obtain q=Â∇T+B̂∇T^s/2 andq^s=B̂∇T+Â∇T^s/2 ,where B̂=Tσ̂(ŜΣ̂^s+Ŝ^sΣ̂)-κ̂^s ,and  is defined by Eq. (<ref>). With ∇ T^s=e_y∇ _yT_in^s and ∇T= e_x∇ _xT_ex+e_y∇ _yT_in we find (∇ _xμ /e)__j=0 = Σ _xx∇ _xT_ex+Σ _xy∇ _yT_in+Σ _xy^s∇ _yT_in^s/2 = [ Σ _xx-Σ _xyA_yyA_yx-B_yyB_yx/ A_yyA_yy-B_yyB_yy. . -Σ _xy^sA_yyB_yx-B_yyA_yx/ A_yyA_yy-B_yyB_yy] ∇ _xT_exassuming again q_y=0 and q_y^s=0. Similar to Eq. (<ref>), the Hall corrections in Eq. (<ref>) should be significant only when Σ_xx vanishes for w=0.125. However, experimentally it might be difficult to separate the thermopowers Eq. (<ref>) and Eq. (<ref>). §.§ Spin diffusion length and mean free path Our first-principles calculation are carried out for bulk dilute alloys based on Cu and in the single site approximation of spin-conserving impurity scattering. The Hall effects are therefore purely extrinsic. This is an approximation that holds on length scales smaller than various spin diffusion lengths l_sf. On the other hand, the Boltzmann equation approach is valid when the sample is larger than the elastic scattering mean free path l, so our results should be directly applicable for sample lengths L that fulfill l <L≤ l_sf. According to Refs. Gradhand10_2 and Gradhand2012, for the ternary alloy Cu(Mn_0.5Ir_0.5) with impurity concentration of 1 at.% the present results hold on length scales 26 nm <L≤ 60 nm and 100 nm <L≤ 400 nm for Cu(Mn). On the other hand, for nonmagnetic Cu(Ir) the applicability is limited to a smaller interval 10 nm <L≤ 16 nm. We believe that while the results outside these strict limits may not be quantitatively reliable, they still give useful insights into trends. § SUMMARY AND OUTLOOK In summary, we derived expressions for the thermopower valid for ordered magnetic alloys for sample sizes that do not exceed the spin diffusion lengths (that have to be calculated separately). We focus on dilute alloys of Cu with Mn and Ir impurities. For 1% ternary alloys Cu(Mn_1-wIr_w ) with w<0.5 the spin diffusion length is l_sf>60 nm. In this regime the spin and charge accumulations induced by an applied temperature gradient strongly affect each other. By ab initio calculations of the transport properties ofCu(Mn_1-wIr_w) alloys, we predict thermopowers that drastically differ from the bulk value even changing sign. Relativistic Hall effects generate spin accumulations normal to the applied temperature gradient that become significant when the longitudinal thermopower Σ _xx vanishes, for example for Cu(Mn_1-wIr_w) alloys at w≈ 0.125. After having established the principle existence of the various corrections to the conventional transport description it would be natural to move forward to describe extended thin films. A first-principles version of the Boltzmann equation including all electronic spin non-conserving scatterings in extended films is possible, but very expensive for large l_sf. It would still be incomplete, since the relaxation of heat to the lattice by electron-phonon interactions and spin-heat by electron-electron scattering <cit.> are not included. We therefore propose to proceed pragmatically: The regime l<l_sf<L is accessible to spin-heat diffusion equations that can be parameterized by first-principles material-dependent parameters as presented here and relaxation lengths that may be determined otherwise, such as by fitting to experimental results.§ ACKNOWLEDGEMENTS This work was partially supported by the Deutsche Forschungsgemeinschaft via SFB 762 and the priority program SPP 1538 as well as JSPS Grants-in-Aid for Scientific Research (KAKENHI Grant No. 19H00645). M.G. acknowledges financial support from the Leverhulme Trust via an Early Career Research Fellowship (ECF-2013-538) and a visiting professorship at the Centre for Dynamics and Topology of the Johannes-Gutenberg-University Mainz.99 Bauer2010 G. E. W. Bauer, A. H. MacDonald, and S. Maekawa, Spin Caloritronics, Solid State Commun. 150, 459 (2010).Bauer2012 G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spin caloritronics, Nat. Mater. 11, 391 (2012).Boona2014 S. R. Boona, R. C. Myers, and J. P. Heremans, Spin caloritronics, Energy Environ. Sci. 7, 885 (2014).Slachter2010 A. Slachter, F. L. Bakker, J. P. Adam, and B. J. van Wees, Thermally driven spin injection from a ferromagnet into a non-magnetic metal, Nature Phys. 6, 879 (2010).Flipse2012 J. Flipse, F. L. Bakker, A. Slachter, F. K. Dejene, and B. J. van Wees, Direct observation of the spin-dependent Peltier effect, Nature Nanotech. 7, 166 (2012).Goennenwein2012 S. Gönnenwein and G. E. W. Bauer, Electron spins blow hot and cold, Nature Nanotech. 7, 145 (2012).Miyasato2007 T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda, Y. Onose, N. Nagaosa, and Y. Tokura, Crossover Behavior of the Anomalous Hall Effect and Anomalous Nernst Effect in Itinerant Ferromagnets, Phys. Rev. Lett. 99, 086602 (2007).Cheng2008 S.-G. Cheng, Y. Xing, Q.-F. Sun, and X. C. Xie, Spin Nernst effect and Nernst effect in two-dimensional electron systems, Phys. Rev. B 78, 045302 (2008).Liu2010 X. Liu and X. Xie, Spin Nernst effect in the absence of a magnetic field, Solid State Commun. 150, 471 (2010).Ma2010 Z. Ma, Spin Hall effect generated by a temperature gradient and heat current in a two-dimensional electron gas, Solid State Commun.150, 510 (2010).Tauber2012 K. Tauber, M. Gradhand, D. V. Fedorov, and I. Mertig, Extrinsic Spin Nernst Effect from First Principles, Phys. Rev. Lett.109, 026601 (2012).Bose2019 A. Bose and A. A. Tulapurkar, Recent advances in the spin Nernst effect, J. Mag. Mag. Mat. 491, 165526 (2019), and references therein.Seebeck1822 T. J. Seebeck, Magnetische Polarisation der Metalle und Erze durch Temperatur-Differenz, Abh. Deutsch. Akad. Wiss. Berlin, 265-373 (1822).Comment_1 Systems with dimensions larger than the spin-flip length require the solution of a spin-heat diffusion problem <cit.>.Watzmann_2016 J. Watzmann, R. A. Duine, Y. Tserkovnyak, S. R. Boona, H. Jin, A. Prakesh, Y. Zheng, and J. P. Heremans, Magnon-drag thermopower and Nernst coefficient in Fe, Co, and Ni, Phys. Rev. B.94, 144407 (2016)Mott36 N. F. Mott, The Electrical Conductivity of Transition Metals, Proc. R. Soc. London A 153, 699 (1936).Fert68 A. Fert and I. A. Campbell, Two-Current Conduction in Nickel, Phys. Rev. Lett. 21, 1190 (1968).Son87 P. C. van Son, H. van Kempen, and P. Wyder, Boundary Resistance of the Ferromagnetic-Nonferromagnetic Metal Interface, Phys. Rev. Lett. 58, 2271 (1987).Johnson87 M. Johnson and R. H. Silsbee, Thermodynamic analysis of interfacial transport and of the thermomagnetoelectric system, Phys. Rev. B35, 4959 (1987).Johnson88 M. Johnson and R. H. Silsbee, Ferromagnet-Nonferromagnet Interface Resistance, Phys. Rev. Lett. 60, 377 (1988).Valet93 T. Valet and A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B 48, 7099 (1993).Hatami2007 M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly, Thermal spin-transfer torque in magnetoelectronic devices, Phys. Rev. Lett.99, 066603 (2007).Heikkila2010_PRB T. T. Heikkilä, M. Hatami, and G. E. W. Bauer, Spin heat accumulation and its relaxation in spin valves, Phys. Rev. B81, 100408 (2010).Heikkila2010_SSC T. T. Heikkilä, M. Hatami, and G. E. W. Bauer, Electron-electron interaction induced spin thermalization in quasi-low-dimensional spin valves, Solid State Commun. 150, 475 (2010).Dejene2013 F.K. Dejene, J. Flipse, G. E. W. Bauer, and B. J. van Wees, Spin heat accumulation and spin-dependent temperatures in nanopillar spin valves, Nat. Phys. 9, 636 (2013).Marun2014 I. J. Vera-Marun, B. J. van Wees, and R. Jansen, Spin Heat Accumulation Induced by Tunneling from a Ferromagnet, Phys. Rev. Lett.112, 056602 (2014).Wong2015 C. H. Wong, H. T. C. Stoof, and R. A. Duine, Spin-heat relaxation and thermospin diffusion in atomic Bose and Fermi gases, Phys. Rev. A 91, 043602 (2015).Hatami2010 M. Hatami, G. E. W. Bauer, S. Takahashi, and S. Maekawa, Thermoelectric spin diffusion in a ferromagnetic metal, Solid State Commun. 150, 480 (2010).Comment_2 Ŝ and Ŝ^s defined by Eqs. (<ref>) and ( <ref>) can be measured in samples much larger than the spin diffusion length.Pu2006 Y. Pu, E. Johnston-Halperin, D. D. Awschalom, and J. Shi, Anisotropic Thermopower and Planar Nernst Effect in Ga_1-xMn_xAs Ferromagnetic Semiconductors, Phys. Rev. Lett. 97, 036601 (2006).Tauber2013 K. Tauber, D. V. Fedorov, M. Gradhand, and I. Mertig, Spin Hall and spin Nernst effect in dilute ternary alloys, Phys. Rev. B87, 161114(R) (2013).Mertig99 I. Mertig, Transport properties of dilute alloys, Rep. Prog. Phys. 62, 237 (1999).Gradhand10 M. Gradhand, D. V. Fedorov, P. Zahn, and I. Mertig, Extrinsic Spin Hall Effect from First Principles, Phys. Rev. Lett.104, 186403 (2010).Gradhand09 M. Gradhand, M. Czerner, D. V. Fedorov, P. Zahn, B. Y. Yavorsky, L. Szunyogh, and I. Mertig, Spin polarization on Fermi surfaces of metals by the KKR method, Phys. Rev. B 80, 224413 (2009).Hu2014 S. Hu, H. Itoh, and T. Kimura, Efficient thermal spin injection using CoFeAl nanowire, NPG Asia Materials 6, e127 (2014).Hu2013 S. Hu and T. Kimura, Anomalous Nernst-Ettingshausen effect in nonlocal spin valve measurement under high-bias current injection, Phys. Rev. B 87, 014424 (2013).Gradhand10_2 M. Gradhand, D. V. Fedorov, P. Zahn, and I. Mertig, Spin Hall angle versus spin diffusion length: Tailored by impurities, Phys. Rev. B 81, 245109 (2010).Lowitzer2011 S. Lowitzer, M. Gradhand, D. Ködderitzsch, D. V. Fedorov, I. Mertig, and H. Ebert, Extrinsic and Intrinsic Contributions to the Spin Hall Effect of Alloys, Phys. Rev. Lett. 106 , 056601 (2011).Gradhand2011 M. Gradhand, D. V. Fedorov, P. Zahn, and I. Mertig, Skew scattering mechanism by an ab initio approach: Extrinsic spin Hall effect in noble metals, Solid State Phenom. 168-169, 27 (2011).Gradhand2012 M. Gradhand, D. V. Fedorov, P. Zahn, I. Mertig, Y. Otani, Y. Niimi, L. Vila, and A. Fert, Perfect alloys for spin Hall current-induced magnetization switching, SPIN 2, 1250010 (2012). | http://arxiv.org/abs/1706.08753v2 | {
"authors": [
"Dmitry V. Fedorov",
"Martin Gradhand",
"Katarina Tauber",
"Gerrit E. W. Bauer",
"Ingrid Mertig"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.other"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170627094041",
"title": "Seebeck Effect in Nanomagnets"
} |
^1Bocconi Institute for Data Science and Analytics, Bocconi University, Milano, Italy ^2Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Italy ^3International Centre for Theoretical Physics, Trieste, ItalyQuantum annealers aim at solving non-convex optimization problems by exploiting cooperative tunneling effects to escape local minima. The underlying idea consists in designing a classical energy function whose ground states are the sought optimal solutions of the original optimization problem and add a controllable quantum transverse field to generate tunneling processes. A key challenge is to identify classes of non-convex optimization problems for which quantum annealing remains efficient while thermal annealing fails. We show that this happens for a wide class of problems which are central to machine learning. Their energy landscapes is dominated by local minima that cause exponential slow down of classical thermal annealers while simulated quantum annealing converges efficiently to rare dense regions of optimal solutions.Efficiency of quantum versus classical annealing in non-convex learning problems Carlo Baldassi^1,2 and Riccardo Zecchina^1,3 December 30, 2023 ================================================================================§ INTRODUCTION o0.5< g r a p h i c s > Topology of the Suzuki-Trotter vs Robust Ensemble representations. a: the classical objective function we wish to optimize which depends on N discrete variables {σ_j} (N=5 in the picture). b: Suzuki-Trotter interaction topology: y replicas of the classical system (y=7 in the picture) are coupled by periodic 1 dimensional chains, one for each classical spin. c: Robust Ensemble interaction topology: y replicas are coupled through a centroid configuration. In the limit of large N and large y (quantum limit) and for strong interaction couplings all replicas are forced to be close, and the behavior of the two effective models is expected to be similar. Quantum tunneling and quantum correlations govern the behavior of very complex collective phenomena in quantum physics at low temperature. Since the discovery of the factoring quantum algorithms in the 90s <cit.>, a lot of efforts have been devoted to the understanding of how quantum fluctuations could be exploited to find low-energy configurations of energy functions which encode the solutions of non-convex optimization problems in their ground states. This has led to the notion of controlled quantum adiabatic evolution, where atime dependent many-body quantum system is evolved towards its ground states so as to escape local minima through multiple tunneling events <cit.>. When finite temperature effects have to be taken into account, the computational process is called Quantum Annealing (QA). Classical Simulated Annealing (SA) uses thermal fluctuations for the same computational purpose, and Markov Chains based on this principle are among the most widespread optimization techniques across science <cit.>. Quantum fluctuations are qualitatively different from thermal fluctuations and in principle quantum annealing algorithms could lead to extremely powerful alternative computational devices.In the quantum annealing approach, a time dependent quantum transverse field is added to the classical energy function leading to an interpolating Hamiltonian that may take advantage of correlated fluctuations mediated by tunneling. Starting with a high transverse field, the quantum model system can be initialized in its ground state, e.g. all spins aligned in the direction of the field. The adiabatic theorem then ensures that by slowly reducing the transverse field the system remains in the ground state of the interpolating Hamiltonian. At the end of the process the transverse field vanishes and the systems ends up in the sought ground state of the classical energy function. The original optimization problem would then be solved if the overall process could take place in a time bounded by some low degree polynomial in the size of the problem. Unfortunately, the adiabatic process can become extremely slow. The adiabatic theorem requires the rate of change of the Hamiltonian to be smaller than the square of the gap between the ground state and the first excited state <cit.>. For small gaps the process can thus become inefficient. Exponentially small gaps are not only possible in worst case scenarios but have also been found to exist in typical random systems where comparative studies between quantum and classical annealing have so far failed in displaying quantum exponential speed up, e.g. at first order phase transition in quantum spin glasses <cit.> or 2D spin glass systems <cit.>. More positive results have been found for ad hoc energy functions in which global minima are planted in such a way that tunneling cascades can become more efficient than thermal fluctuations <cit.>. As far as the physical implementations of quantum annealers is concerned, studies have been focused on discriminating the presence of quantum effects rather than on their computational effectiveness <cit.>.Consequently, a key open question is to identify classes of relevant optimization problems for which quantum annealing can be shown to be exponentially faster than its classical thermal counterpart.Here we give an answer to this question by providing analytic and simulation evidence of exponential speed up of quantum versus classical simulated annealing for a representative class of random non-convex optimization problems of basic interest in machine learning. The simplest example of this class is the problem of training binary neural networks (described in detail below): very schematically, the variables of the problem are the (binary) connection weights, while the energy measures the training error over a given dataset.These problems have been very recently found to possess a rather distinctive geometrical structure of ground states <cit.>: the free energy landscape has been shown to be characterized by the existence of an exponentially large number metastable states and isolated ground states, and a few regions where the ground states are dense. These dense regions, which had previously escaped the equilibrium statistical physics analysis <cit.>, are exponentially rare, but still possess a very high local internal entropy: they are composed of ground states that are surrounded, at extensive but relatively small distances, by exponentially many other ground states. Under these circumstances, classical SA (as any Markov Chain satisfying detailed balance) gets trapped in the metastable states, suffering ergodicity breaking and exponential slowing down toward the low energy configurations. These problems have been considered to be intractable for decades and display deep similarities with disordered spin glass models which are known to never reach equilibrium.The large deviation analysis that has unveiled the existence of the rare dense regions has led to several novel algorithms, including a Monte Carlo scheme defined over an appropriate objective function <cit.> that bears close similarities with a Quantum Monte Carlo (QMC) technique based on the Suzuki-Trotter transformation <cit.>. Motivated by this analytical mapping and by the geometrical structure of the dense and degenerate ground states which is expected to favor zero temperature kinetic processes <cit.>, we have conducted a full analytical and numerical statistical physics study of the quantum annealing problem, reaching the conclusion that in the quantum limit the QMC process, i.e. Simulated Quantum Annealing (SQA), can equilibrate efficiently while the classical SA gets stuck in high energy metastable states. These results generalize to multi layered networks.While it is known that other quasi-optimal classical algorithms for the same problems exist <cit.>, here we focus on the physical speed up that a quantum annealing approach couldprovide in finding rare regions of ground states. We provide physical arguments and numerical results supporting the conjecture that the real time quantum annealing dynamics behaves similarly to SQA.As far as machine learning is concerned, dense regions of low energy configurations (i.e. quasi-flat minima over macroscopic length scales) are of fundamental interest, as they are particularly well-suited for making predictions given the learned data: on the one hand, these regions are by definition robust with respect to fluctuations in a sizable fraction of the weight configurations and as such are less prone to fit the noise. On the other hand, an optimal Bayesian estimate, resulting from a weighted consensus vote on all configurations, would receive a major contribution from one of such regions, compared to a narrow minimum; the centroid of the region (computed according to any reasonable metric which correlates the distance between configurations with the network outcomes) would act as a representative of the region as a whole <cit.>. In this respect, it is worth mentioning that in deep learning <cit.> all the learning algorithms which lead to good prediction performance always include effects of a systematically injected noise in the learning phase, a fact that makes the equilibrium Gibbs measure not the stationary measure of the learning protocols and drive the systems towards wide minima. We expect that these results can be generalized to many other classes of non convex optimization problems where local entropy plays a role, ranging from robust optimization to physical disordered systems.Quantum gate based algorithms for machine learning exist, however the possibility of a physical implementation remains a critical issue <cit.>.§ ENERGY FUNCTIONS As a working example, we first consider the problem of learning random patterns in single layer neural network with binary weights, the so called binary perceptron problem <cit.>. This network maps vectors of N inputs ξ∈{ -1,+1} ^N to binary outputs τ=±1 through the non linear function τ=sgn(σ·ξ), where σ∈{ -1,+1} ^N is the vector of synaptic weights. Given α N input patterns {ξ^μ} _μ=1^α N with μ=1,...,α N and their corresponding desired outputs {τ^μ} _μ=1^α N, the learning problem consists in finding σ such that all input patterns are simultaneously classified correctly, i.e. sgn(σ·ξ^μ)=τ^μ for all μ. Both the components of the input vectors ξ_i^μ and the outputs τ^μ are independent identically distributed unbiased random variables (P(x)=1/2δ(x-1)+1/2δ(x+1)). In the binary framework, the procedure for writing a spin Hamiltonian whose ground states are the sought optimal solutions of the original optimization problem is well known <cit.>. The energy E of the binary perceptron is proportional to the number of classification errors and can be written asE({σ_j})=∑_μ=1^α NΔ_μ^nΘ(-Δ_μ),Δ_μ≐τ^μ/√(N)∑_j=1^Nξ_j^μσ_jwhere Θ(x) is the Heaviside step function: Θ(x)=1 if x>0, Θ(x)=0 otherwise. When the argument of the Θ function is positive, the perceptron is implementing the wrong input-output mapping. The exponent n∈{ 0,1} defines two different forms of the energy functions which have the same zero energy ground states and different structures of local minima. The equilibrium analysis of the binary perceptron problem shows that in the large size limit and for α<α_c≃0.83 <cit.>, the energy landscape is dominated by an exponential number of local minima and of zero energy ground states that are typically geometrically isolated <cit.>, i.e. they have extensive mutual Hamming distances. For both choices of n the problem is computationally hard for SA processes <cit.>: in the large N limit, a detailed balanced stochastic search process gets stuck in metastable states at energy levels of order O(N) above the ground states.Following the standard SQA approach, we identify the binary variables σ with one of the components of physical quantum spins, say σ^z, and we introduce the Hamiltonian operator of a model of N quantum spins with the perceptron term of Eq. (<ref>) acting in the longitudinal direction z and a magnetic field Γ acting in the transverse direction x. The interpolating Hamiltonian reads:Ĥ=E({σ̂_j^z})-Γ∑_j=1^Nσ̂_j^xwhere σ̂_j^z and σ̂_̂ĵ^x are the spin operators (Pauli matrices) in the z and x directions. For Γ=0 one recovers the classical optimization problem. The QA procedure consists in initializing the system at large β and Γ, and slowly decreasing Γ to 0. To analyze the low temperature phase diagram of the model we need to study the average of the logarithm of the partition function Z=Tr (e^-βĤ). This can be done using the Suzuki-Trotter transformation which leads to the study of a classical effective Hamiltonian acting on a system of y interacting Trotter replicas of the original classical system coupled in an extra dimension:H_eff({σ_j^a} _j,a)=1/y∑_a=1^yE({σ_j^a} _j)-γ/β∑_a=1^y∑_j=1^Nσ_j^aσ_j^a+1-NK/βwhere the σ_j^a=±1 are Ising spins, a∈{ 1,…,y} is a replica index with periodic boundary conditions σ_j^y+1≡σ_j^1, γ=1/2log(βΓ/y) and K=1/2ylog(1/2sinh(2βΓ/y)).The replicated system needs to be studied in the limit y→∞ to recover the so called path integral continuous quantum limit and to make the connection with the behavior of quantum devices <cit.>. The SQA dynamical process samples configurations from an equilibrium distribution and it is not necessarily equivalent to the real time Schrödinger equation evolution of the system. A particularly dangerous situation occurs if the ground states of the system encounter first order phase transitions which are associated to exponentially small gaps <cit.> at finite N. As discussed below, this appears not to be the case for the class of models we are considering.§ CONNECTION WITH THE LOCAL ENTROPY MEASURE The effective Hamiltonian Eq. (<ref>) can be interpreted as many replicas of the original systems coupled through one dimensional periodic chains, one for each original spin, see Fig. <ref>b. Note that the interaction term γ diverges as the transverse field Γ goes to 0. This geometrical structure is very similar to that of the Robust Ensemble (RE) formalism <cit.>, where a probability measure that gives higher weight to rare dense regions of low energy states is introduced. There, the main idea is to maximize Φ(σ^⋆)=log∑_{σ}e^-β E(σ)-λ∑_j=1^Nσ_jσ_j^⋆, i.e. a “local free entropy” where λ is a Lagrange parameter that controls the extensive size of the region around a reference configuration σ^⋆. One can then build a new Gibbs distribution P(σ^⋆)∝ e^yΦ(σ^⋆), where -Φ has the role of an energy and y of an inverse temperature: in the limit of large y, this distribution concentrates on the maxima of Φ. Upon restricting the values of y to be integer (and large), P(σ^⋆) takes a factorized form yielding a replicated probability measure P_RE(σ^⋆,σ^1,…,σ^y)∝ e^-β H_eff^RE(σ^⋆,{σ_j^a}) where the effective energy is given by H_eff^RE(σ^⋆,{σ_j^a} _j,a)=∑_a=1^yE({σ_j^a} _j)-λ/β∑_a=1^y∑_j=1^Nσ_j^aσ_j^⋆ As in the Suzuki-Trotter formalism, H_eff^RE(σ^⋆,{σ_j^a} _j,a) corresponds to a system with an overall energy given by the sum of y individual “real replica energies” plus a geometric coupling term; in this case however the replicas interact with the “reference” configurations σ^⋆ rather than among themselves, see Fig. <ref>c.The Suzuki-Trotter representation and the RE formalism differ in the topology of the interactions between replicas and in the scaling of the interactions, but for both cases there is a classical limit, Γ→0 and λ→∞ respectively, in which the replicated systems are forced to correlate and eventually coalesce in identical configurations. For non convex problems, these will not in general correspond to configuration dominating the original classical Gibbs measure.For the sake of clarity we should remind that in the classical limit and for α<α_c, our model presents an exponential number of far apart isolated ground states which dominate the Gibbs measure. At the same time, there exist rare clusters of ground states with a density close to its maximum possible value (high local entropy) for small but still macroscopic cluster sizes <cit.>. This fact has several consequences: no further subdivision of the clusters into states is possible, the ground states are typically O(1) spin flip connected <cit.> and a tradeoff between tunneling events and exponential number of destination states within the cluster is possible.§ PHASE DIAGRAM: ANALYTICAL AND NUMERICAL RESULTS Thanks to the mean field nature of the energetic part of the system, Eq. (<ref>), we can resort to the replica method for calculating analytically the phase diagram. As discussed in the Appendix Sec. <ref>, this can be done under the so called static approximation, which consists in using a single parameter q_1 to represent the overlaps along the Trotter dimension, q_1^ab=⟨1/N∑_j=1^Nσ_j^aσ_j^b⟩≈ q_1. Although this approximation crudely neglects the dependency of q_1^ab from |a-b|, the resulting predictions show a remarkable agreement with numerical simulations.In the main panel of Fig. <ref>, we report the analytical predictions for the average classical component of the energy of the quantum model as a function of the transverse field Γ. We compare the results with the outcome of extensive simulations performed with the reduced-rejection-rate Monte Carlo method <cit.>, in which Γ is initialized at 2.5 and gradually brought down to 0 in regular small steps, at constant temperature, and fixing the total simulation time to τ Ny·10^4 (as to keep constant the number of Monte Carlo sweeps when varying N and y). The details are reported in the Appendix Sec. <ref>. The size of the systems, the number of samples and the number of Trotter replicas are scaled up to large values so that both finite size effects and the quantum limit are kept under control. A key pointis to observe that the results do not degrade with the number of Trotter replicas: the average ground state energy approaches a limiting value, close to the theoretical prediction, in the large y quantum limit. The results appear to be rather insensitive to both N and the simulation time scaling parameter τ. This indicates that Monte Carlo appears to be able to equilibrate efficiently, in a constant (or almost constant) number of sweeps, at each Γ. The analytical prediction for the classical energy only appears to display a relatively small systematic offset (due to the static approximation) at intermediate values of Γ, while it is very precise at both large and small Γ; the expectation of the total Hamiltonian on the other hand is in excellent agreement with the simulations (see Appendix Sec. <ref>).In the same plot we display the behavior of classical SA simulated with a standard Metropolis-Hastings scheme, under an annealing protocol in β that would follow the same theoretical curve as SQA if the system were able to equilibrate (see Appendix Sec. <ref>): as expected <cit.>, SA gets trapped at very high energies (increasing with problem size; in the thermodynamic limit it is expected that SA would remain stuck at the initial value 0.5N of the energy for times which scale exponentially with N). Alternative annealing protocols yield analogous results; the exponential scaling with N of SA on binary perceptron models had also been observed experimentally in previous results, e.g. in refs. <cit.>.In the inset of Fig. <ref> we report the analytical prediction for the transverse overlap parameter q_1, which quite remarkably reproduces fairly well the average overlap as measured from simulations.In Fig. <ref> we provide the profiles of the the classical energy minima found for different values of Γ in the case of SQA and different temperatures for SA. These results are computed analytically by the cavity method (see Materials and Methods and SI for details) by evaluating which is the most probable energy found at a normalized Hamming distance d from a given configuration. As it turns out, throughout the annealing process, SQA follows a path corresponding to wide valleys while SA gets stuck in steep metastable states. The quantum fluctuations reproduced by the SQA process drive the system to converge toward wide flat regions, in spite of the fact that they are exponentially rare compared to the narrow minima.The physical interpretation of these results is that quantum fluctuations lower the energy of a cluster proportionally to its size or, in other words, that quantum fluctuations allow the system to lower its kinetic energy by delocalizing, see Refs. <cit.> for related results. Along the process of reduction of the transverse field we do not observe any phase transition which could induce a critical slowing down of the quantum annealing process and we expect SQA and QA to behave similarly <cit.>.This is in agreement with the results of a direct comparison between the real time quantum dynamics and the SQA on small systems (N=21): as reported in the Appendix Sec. <ref>, we have performed extensive numerical studies of properly selected small instances of the binary perceptron problem, comparing the results of SQA and QA and analyzing the results of the QA process and the properties of the Hamiltonian. To reproduce the conditions that are known to exist at large values of N, we have selected instances for which a fast annealing schedule SA gets trapped at some positive fraction of violated constraints, and yet the problems display a sufficiently high number of solutions. We found that the agreement between SQA and QA on each sample is excellent. The measurements on the final configurations reached by QA qualitatively confirm the scenario described above, that QA is attracted towards dense low-energy regions without getting stuck during the annealing process. Finally, the analysis of the gap between the ground state of the system and the first excited state as Γ decreases shows no signs of the kind of phenomena which would typically hamper the performance of QA in other models: there are no vanishingly small gaps at finite Γ (cf. the discussion in the introduction). We benchmarked all these results with “randomized” versions of the same samples, in which we randomly permuted the classical energies associated to each spin configuration, so as to keep the distribution of the classical energy levels while destroying the geometric structure of the states. Indeed, for these randomized samples, we found that the gaps nearly close at finite Γ≃0.4, and that correspondingly the QA process fails to track the ground state of the system, resulting in a much reduced probability of finding a solution to the problem.To reproduce the conditions that are known to exist at large values of N we have selected instances for which a fast annealing schedule SA gets trapped at some positive fraction of violated constraints and yet the problems display a sufficiently high number of solutions. We have then compared the behavior of SQA and the real time quantum dynamics studied by the Lanczos method as discussed in <cit.>. The agreement between SQA and QA is ... almost perfect.As concluding remarks we report that the models with n=0 and n=1 have phase diagrams which are qualitatively very similar (for the sake of simplicity, here we reported the n=0 case only). The former presents at very small positive values of Γ a collapse of the density matrix onto the classical one whereas the latter ends up in the classical state only at Γ=0.For the sake of completeness, we have checked that the performance of SQA in the y→∞ quantum limit extends to more complex architectures which include hidden layers; the details are reported in the Appendix Sec. <ref>.§ CONCLUSIONS We conclude by noticing that, at variance with other studies on spin glass models in which the evidence for QA outperforming classical annealing was limited to finite values of y, thereby just defining a different type of classical SA algorithms, in our case the quantum limit coincides with the optimal behavior of the algorithm itself. We believe that these results could play a role in many optimization problems in which optimality of the cost function needs to also meet robustness conditions (i.e. wide minima). As far as learning problems are concerned, it is worth mentioning that for the best performing artificial neural networks, the so called deep networks <cit.>, there is numerical evidence for the existence of rare flat minima <cit.>, and that all the effective algorithms always include effects of systematic injected noise in the learning phase <cit.>, which implies that the equilibrium Gibbs measure is not the stationary measure of the learning protocols. For the sake of clarity we should remark that our results are aimed to suggest that QA can equilibrate efficiently whereas SA cannot, i.e. our notion of quantum speed up is relative to the same algorithmic scheme that runs on classical hardware. Other classical algorithms for the same class of problems, besides the above-mentioned ones based on the RE and the SQA itself, have been discovered <cit.>; however, all of these algorithms are qualitatively different from QA, which can provide a huge speed up by manipulating single bits in parallel. Thus, the overall solving time in a physical QA implementation (neglecting any other technological considerations) would have, at worst, only a mild dependence on N.Our results provide further evidence that learning can be achieved through different types of correlated fluctuations, among which quantum tunneling could be a relevant example for physical devices.The authors thank G. Santoro, B. Kappen and F. Becca for discussions.§ THEORETICAL ANALYSIS BY THE REPLICA METHOD We present here the analytical calculations performed to derive all the theoretical results mentioned in the main text. For completeness, we report all the relevant formulas and definitions here, even those that were already introduced in the main text.The Hamiltonian operator of a model of N quantum spins with an energy term acting in the longitudinal direction z and a magnetic field Γ acting in the transverse direction x is written as:Ĥ=E({σ̂_j^z} _j)-Γ∑_j=1^Nσ̂_j^xwhere σ̂_j^z and σ̂_̂ĵ^x are the spin operators (Pauli matrices) in the z and x directions. We want to study the partition function:Z=Tr (e^-βĤ). By using the Suzuki-Trotter transformation, we end up with a classical effective Hamiltonian acting on a system of y interacting Trotter replicas, to be studied in the limit y→∞:H_eff({σ_j^a} _j,a)=1/y∑_aE({σ_j^a} _j)-γ/β∑_ajσ_j^aσ_j^a+1-NK/βwhere the σ_j^a=±1 are Ising spins, a∈{ 1,…,y} is a replica index with periodic boundary conditions σ_j^y+1≡σ_j^1, and we have defined:γ=1/2log(βΓ/y),K =1/2ylog(1/2sinh(2βΓ/y)). In the following, we will just use σ^a to denote the configuration of one Trotter replica, {σ_j^a} _j; we will always use the indices a or b for the Trotter replicas and assume that they range in 1,…,y; we will also use j for the site index and assume that it ranges in 1,…,N.The effective partition function for a given y reads:Z_eff=∑_{σ^a}e^-β/y∑_aE(σ^a)+γ∑_ajσ_j^aσ_j^a+1+NK. Here, we first study the binary perceptron case in which the longitudinal energy E is defined in terms of a set of α N patterns {ξ^μ} _μ with μ∈{ 1,…,α N}, where each pattern is a binary vector of length N, ξ_j^μ=±1:E(σ)=∑_μ=1^α NΘ(-1/√(N)∑_jξ_j^μσ_j)where Θ(x) is the Heaviside step function: Θ(x)=1 if x>0, Θ(x)=0 otherwise. The energy thus simply counts the number of classification errors of the perceptron, assuming that the desired output for each pattern in the set is 1 (this choice can always be made without loss of generality for this model when the input patterns are random i.i.d. as described below). A different form for the energy function is treated in sec. <ref>.We consider the case in which the patterns entries are extracted randomly and independently from an unbiased distribution, P(ξ_j^μ)=1/2δ(ξ_j^μ-1)+1/2δ(ξ_j^μ+1), and we want to study the typical properties of this system by averaging over the quenched disorder introduced by the patterns. We use the replica method, which exploits the transformation:⟨log Z⟩ _ξ=lim_n→0⟨ Z^n⟩ _ξ-1/n =lim_n→0⟨∏_c=1^nZ⟩ _ξ-1/nwhere ⟨·⟩ _ξ denotes the average over the disorder. We thus need to replicate the whole system n times, and therefore we have two replica indices for each spin. We will use indices c,d=1,…,n for the “virtual” replicas introduced by the replica method,[Note that the parameter n has a different meaning in main text, cf. sec. <ref>.] to distinguish them from the indices a and b used for the Trotter replicas. The average replicated partition function of eq. (<ref>) is thus written as:⟨ Z_eff^n⟩ _ξ=e^nNK⟨∫∏_cajdμ(σ_j^ca)∏_caje^γσ_j^cσ_j^c(a+1)∏_μ ca(Θ[1/√(N)∑_jξ_j^μσ_j^ca](1-e^-β/y)+e^-β/y)⟩ _ξwhere we changed the sum over all configurations into an (n× y× N-dimensional) integral, using the customary notation dμ(σ)=δ(σ-1)+δ(σ+1) with δ(·) denoting the Dirac-delta distribution. Here and in the following, all integrals will be assumed to range over the whole ℝ unless otherwise specified.We introduce new auxiliary variables λ_μ^ca=1/√(N)∑_jξ_j^μσ_j^ca via additional Dirac-deltas: ⟨ Z_eff^n⟩ _ξ= e^nNK∫∏_cajdμ(σ_j^ca)∏_caje^γσ_j^caσ_j^c(a+1)∫∏_μ cadλ_μ^ca∏_μ ca(Θ[λ_μ^ca](1-e^-β/y)+e^-β/y)× ×⟨∏_μ caδ(λ_μ^ca-1/√(N)∑_jξ_j^μσ_j^ca)⟩ _ξ We then use the integral representation of the delta δ(x)=∫dx̂/2πe^ixx̂, and perform the average over the disorder, to the leading order in N:⟨∏_μ caδ(λ_μ^ca-1/√(N)∑_jξ_j^μσ_j^ca)⟩ _ξ= ∫∏_μ cadλ̂_μ^ca/2π∏_μ cae^iλ̂_μ^caλ_μ^ca∏_μexp(-1/2∑_cdabλ̂_μ^caλ̂_μ^db(1/N∑_jσ_j^caσ_j^db)) Next, we introduce the overlaps q^ca,db=1/N∑_jσ_j^caσ_j^db via Dirac-deltas (note that due to symmetries and the fact that the self-overlaps are always 1 we have ny(ny-1)/2 overlaps overall), expand those deltas introducing conjugate parameters q̂^ca,db (as usual for these parameters in these models, we absorb away a factor i and integrate them along the imaginary axis, without explicitly noting this), and finally factorize over the site and pattern indices:⟨ Z_eff^n⟩ _ξ= e^nNK∫∏_c,a>bdq^ca,cbdq̂^ca,cbN/2π∏_c>d,abdq^ca,dbdq̂^ca,dbN/2π× × e^-N∑_c,a>bq^ca,cbq̂^ca,cb-N∑_c>d,abq^ca,dbq̂^ca,db× G_S^N× G_E^α NG_S ≐ ∫∏_cadμ(σ^ca)e^∑_c,a>bq̂^ca,cbσ^caσ^cb+∑_c>d,abq̂^ca,dbσ^caσ^db+γ∑_caσ^caσ^c(a+1)G_E ≐ ∫∏_cadλ^cadλ̂^ca/2π∏_ca(Θ[λ^ca](1-e^-β/y)+e^-β/y)× × e^-1/2∑_ca(λ̂^ca)^2+i∑_caλ^caλ̂^ca-∑_c,a>bλ̂^caλ̂^cbq^ca,cb-∑_c>d,abλ̂^caλ̂^dbq^ca,db We now introduce the replica-symmetric (RS) ansatz for the overlaps:q^ca,db= q_1 if c=dq_0 if c dand analogous for the conjugate parameters q̂^ca,db.Note that this is the so-called “static approximation” since we neglect the dependency of the overlap from the distance along the Trotter dimension; however, we have kept the interaction term γ∑_caσ^caσ^c(a+1) and inserted it in the G_S term (rather than writing it in terms of the overlap q^ca,c(a+1) and inserting it in the G_E term where it would have been rewritten as γ q_1). This difference, despite its inconsistency, is the standard procedure when performing the static approximation, and is justified a posteriori from the comparison with the numerical simulation results. We obtain:⟨ Z_eff^n⟩ _ξ= e^nNK∫∏_c,a>bdq^ca,cbdq̂^ca,cbN/2π∏_c>d,abdq^ca,dbdq̂^ca,dbN/2π× × e^-Nny(y-1)/2q_1q̂_1-Nn(n-1)/2y^2q_0q̂_0× G_S^N× G_E^α NG_S=∫∏_cadμ(σ^ca)e^q̂_1∑_c,a>bσ^caσ^cb+q̂_0∑_c>d,abσ^caσ^db+γ∑_caσ^caσ^c(a+1)G_E=∫∏_cadλ^cadλ̂^ca/2π∏_ca(Θ[λ^ca](1-e^-β/y)+e^-β/y)× × e^-1/2∑_ca(λ̂^ca)^2+i∑_caλ^caλ̂^ca-q_1∑_c,a>bλ̂^caλ̂^cb-q_0∑_c>d,abλ̂^caλ̂^db The entropic term G_S can be explicitly computed asG_S=∫∏_cadμ(σ^ca)e^1/2q̂_1∑_c((∑_aσ^ca)^2-∑_a(σ^ca)^2)+1/2q̂_0((∑_caσ^ca)^2-∑_c(∑_aσ^ca)^2) × e^γ∑_caσ^caσ^c(a+1) = e^-1/2q̂_1ny∫∏_cadμ(σ^ca)e^1/2(q̂_1-q̂_0)∑_c(∑_aσ^ca)^2+1/2q̂_0(∑_caσ^ca)^2+γ∑_caσ^caσ^c(a+1) =∫ Dz_0 e^-1/2q̂_1ny[∫∏_adμ(σ^a)e^1/2(q̂_1-q̂_0)(∑_aσ^a)^2+z_0√(q̂_0)(∑_aσ^a)+γ∑_aσ^aσ^a+1]^n =∫ Dz_0 e^-1/2q̂_1ny[∫ Dz_1∫∏_adμ(σ^a)e^(z_1√(q̂_1-q̂_0)+z_0√(q̂_0))(∑_aσ^a)+γ∑_aσ^aσ^a+1]^nwhere the notation Dz=dz 1/√(2π)e^-x^2/2 is a shorthand for a Gaussian integral, and we used twice the Hubbard-Stratonovich transformation e^1/2b=∫ Dz e^z√(b). The expression between square brackets in the last line is the partition function of a 1-dimensional Ising model of size y with uniform interactions J=γ and uniform fields h=z_1√(q̂_1-q̂_0)+z_0√(q̂_0) and can be computed by the well-known transfer matrix method. Note however that while usually in the analysis of the 1D Ising spin model it is sufficient to keep the largest eigenvalue of the transfer matrix in the thermodynamic limit y→∞, in this case instead we need to keep both eigenvalues, since the interaction term scales with the size of the system. The result is:G_S=∫ Dz_0 e^-1/2q̂_1ny[∫ Dz_1e^γ y∑_w=±1g(z_0,z_1,w)^y]^ng(z_0,z_1,w)≐ cosh(h(z_0,z_1))+w√(sinh(h(z_0,z_1))^2+e^-4γ)h(z_0,z_1)≐z_1√(q̂_1-q̂_0)+z_0√(q̂_0) In the limit of small n we obtain:𝒢_S ≐ 1/nlog G_S+1/2q̂_1y-γ y =∫ Dz_0 log[∫ Dz_1∑_w=±1(cosh(h(z_0,z_1))+w√(sinh(h(z_0,z_1))^2+e^-4γ))^y] Note that in the limit of large y the term γ y tends to-K up to terms of order y^-1.The energetic term G_E is computed similarly, by first performing two Hubbard-Stratonovich transformations which allow to factorize the indices c and a, and then explicitly performing the inner integrals:G_E=∫∏_cadλ^cadλ̂^ca/2π∏_ca(Θ[λ^ca](1-e^-β/y)+e^-β/y)× × e^-1/2∑_ca(λ̂^ca)^2+i∑_caλ^caλ̂^ca-1/2q_1∑_c((∑_aλ̂^ca)^2-∑_a(λ̂^ca)^2)-1/2q_0((∑_caλ̂^ca)^2-∑_c(∑_aλ̂^ca)^2) =∫ Dz_0[∫ Dz_1[∫dλ dλ̂/2π(Θ[λ](1-e^-β/y)+e^-β/y)e^-1-q_1/2(λ̂)^2+iλ̂(λ-z_1√(q_1-q_0)-z_0√(q_0))]^y]^n =∫ Dz_0[∫ Dz_1[1-(1-e^-β/y)H(z_1√(q_1-q_0)+z_0√(q_0)/√(1-q_1))]^y]^nwhere H(x)=1/2erfc(x/√(2)). In the limit of small n and of large y we finally obtain:𝒢_E≐1/nlog G_E=∫ Dz_0log∫ Dz_1exp(-β H(z_1√(q_1-q_0)+z_0√(q_0)/√(1-q_1))) Using equations (<ref>) and (<ref>), we obtain the expression for the action:ϕ≐1/N⟨log Z_eff⟩ =extr_q_0,q_1,q̂_0,q̂_1{1/2y^2q_0q̂_0-1/2y(y-1)q_1q̂_1-1/2q̂_1y+𝒢_S+α𝒢_E} In order to obtain a finite result in the limit of y→∞, we assume the following scalings for the conjugated order parameters:q̂_0=p̂_0/y^2 q̂_1=p̂_1/y^2 With these, we find the following final expressions:ϕ=extr_q_0,q_1,p̂_0,p̂_1{1/2q_0p̂_0-1/2q_1p̂_1+𝒢_S+α𝒢_E} 𝒢_S=∫ Dz_0 log[∫ Dz_1 2cosh(√(k̂(z_0,z_1)^2+β^2Γ^2))] k̂(z_0,z_1) = z_1√(p̂_1-p̂_0)+z_0√(p̂_0) 𝒢_E=∫ Dz_0log∫ Dz_1exp(-β H(k(z_0,z_1)))k(z_0,z_1) =z_1√(q_1-q_0)+z_0√(q_0)/√(1-q_1) The parameters q_0, q_1, p̂_0 and p̂_1 are found by solving the system of equations obtained by setting the partial derivatives of ϕ with respect to those parameters to 0:p̂_0=αβ/√(1-q1)∫ Dz_0∫ Dz_1e^-β H(k(z_0,z_1))G(k(z_0,z_1))(z_1/√(q_1-q_0)-z_0/√(q_0))/∫ Dz_1e^-β H(k(z_0,z_1)) p̂_1=αβ/√((1-q_1)^3(q_1-q_0))× ×∫ Dz_0∫ Dz_1e^-β H(k(z_0,z_1))G(k(z_0,z_1))(z_0√(q_0(q_1-q_0))+z_1(1-q_0))/∫ Dz_1e^-β H(k(z_0,z_1))q_0=1/√(k̂(z_0,z_1)^2+β^2Γ^2)× ×∫ Dz_0∫ Dz_1sinh(√(k̂(z_0,z_1)^2+β^2Γ^2))k̂(z_0,z_1)(z_1/√(p̂_1-p̂_0)-z_0/√(p̂_0))/∫ Dz_1cosh(√(k̂(z_0,z_1)^2+β^2Γ^2))q_1=1/√(k̂(z_0,z_1)^2+β^2Γ^2)× ×∫ Dz_0∫ Dz_1sinh(√(k̂(z_0,z_1)^2+β^2Γ^2))k̂(z_0,z_1)(z_1/√(p̂_1-p̂_0))/∫ Dz_1cosh(√(k̂(z_0,z_1)^2+β^2Γ^2)) Once these are found, we can use them to compute the action ϕ and the average values of the longitudinal energy and the transverse fields, and finally of the Hamiltonian:⟨Ĥ⟩ _ξ= N(E̅-ΓT̅) E̅=1/N⟨ E({σ̂^z})⟩ _ξ=-∂ϕ/∂β=α∫ Dz_0∫ Dz_1e^-β H(k(z_0,z_1))H(k(z_0,z_1))/∫ Dz_1e^-β H(k(z_0,z_1)) T̅=1/N⟨σ̂_j^x⟩=∂ϕ/∂(βΓ)=∫ Dz_0∫ Dz_1βΓ sinh(√(k̂(z_0,z_1)^2+β^2Γ^2))/√(k̂(z_0,z_1)^2+β^2Γ^2)/∫ Dz_1cosh(√(k̂(z_0,z_1)^2+β^2Γ^2))where the notation ⟨·⟩ _ξ denotes the fact that we performed both the average over the quenched disorder and the thermal average.§.§.§ Small Γ limit It can be verified that in the limit Γ→0 the equations (<ref>)-(<ref>) reduce to the classical case, in the RS description. In this limit, q_1→1 (i.e., the Trotter replicas collapse), which leads to:𝒢_E=∫ Dz_0log((1-e^-β)H(z_0√(q_0/1-q_0))+e^-β). For Γ=0 and q_1=1 we also have the identity:[This follows from ∫ Dz_1cosh(a z_1+b z_0)=e^a^2/2cosh(b z_0).]-1/2p̂_1q_1+𝒢_S=-1/2p̂_0+∫ Dz_0log2cosh(z_0√(p̂_0)). Putting these two expressions back in eq. (<ref>) we recover the classical expression where p̂_0 assumes the role of the usual conjugate parameter q̂ in the RS analysis of ref. <cit.>.In order to study in detail how this classical limit is reached, however, we need to expand the saddle point equations around this limit. To to this, we define ϵ=1-q_1≪1. From equation (<ref>), expanding to the leading order, we obtain the scaling p̂_1=ĉ_1/√(ϵ), withĉ_1=[1/√(1-q_0)∫ Dz_0G(z_0√(q_0/1-q_0))/e^-β+(1-e^-β)H(z_0√(q_0/1-q_0))][∫ Dz_1exp(-β H(z_1))z_1].Then, we use this scaling in equation (<ref>) and we expand it, first using βΓ≪1 and then ϵ≪1. We obtain the approximate expression:ϵ=β^2Γ^2/2-√(ĉ_1ϵ)+√(2)(ĉ_1+√(ϵ))ϵ^1/4F(1/√(2)√(ĉ_1/√(ϵ)))/ĉ_1^3/2where F(x)=√(π)/2e^-x^2erfi(x) is the Dawson's function. For a given β (from which we obtain ĉ_1 via eq. (<ref>)), this equation can be solved numerically to obtain ϵ (and thus q_1 and p̂_1) as a function of Γ. This expression has always the solution ϵ=0, which correspond to the purely classical case. There is a critical Γ below which ϵ=0 is also the only solution; above that, two additional solutions appear at ϵ>0, of which the largest is the physical one. Therefore, the classical limit is not achieved continuously, but rather with a first-order transition (although the step is tiny). §.§ Energy function with stability We can generalize the energy function eq. (<ref>) to take into account, for those patterns that are misclassified, by how much the classification is wrong: E(σ)=∑_μ=1^α NΘ(-1/√(N)∑_jξ_j^μσ_j)(-1/√(N)∑_jξ_j^μσ_j)^r. The previous case is recovered by setting r=0. Here, we study the case r=1. Note that this parameter is called n in the main text: that notation was borrowed from ref. <cit.>, but here we change it in order to avoid confusion with the number of replicas. While the ground states in the SAT phase of the classical model are unaffected, the system can have different properties for finite β.This change only affects the 𝒢_E term. Equation (<ref>) becomes (with the definition of eq. (<ref>)):G_E=∫ Dz_0[∫ Dz_1[e^β/y√(1-q_1)(k(z_0,z_1)+1/2β/y√(1-q_1))H(k(z_0,z_1)+β/y√(1-q_1))+.. .+H(-k(z_0,z_1))]^y]^n. In the limit of large y we have the modified version of eq. (<ref>):𝒢_E=1/nlog G_E=∫ Dz_0log∫ Dz_1exp(-β√(1-q_1)[G(k(z_0,z_1))-k(z_0,z_1)H(k(z_0,z_1))]) The saddle point equations (<ref>) and (<ref>) become:p̂_0= -αβ∫ Dz_0∫ Dz_1exp(-β√(1-q_1)A(z_0,z_1))H(k(z_0,z_1))(z_1/√(q_1-q_0)-z_0/√(q_0))/∫ Dz_1exp(β√(1-q_1)A(z_0,z_1)) p̂_1=αβ^2∫ Dz_0∫ Dz_1exp(-β√(1-q_1)A(z_0,z_1))H(k(z_0,z_1))^2/∫ Dz_1exp(β√(1-q_1)A(z_0,z_1))whereA(z_0,z_1)=G(k(z_0,z_1))-k(z_0,z_1)H(k(z_0,z_1)).§.§.§ Small Γ limit As in the previous case, it can be checked that for Γ→0, we have q_1→1 and eq. (<ref>) becomes the expression for the classical model under the RS ansatz:𝒢_E=∫ Dz_0log(e^β√(1-q_0)(k_0(z_0)+1/2β√(1-q_0))H(k_0(z_0)+β√(1-q_0))+H(-k_0(z_0)))where k_0(z_0)=z_0√(q_0/1-q_0). Also, eq. (<ref>) still holds, and p̂_0 takes the role of the usual parameter q̂ in the classical RS analysis. In this case, however, we no longer have p̂_1→∞; rather, it tends to a finite value:p̂_1=αβ^2∫ Dz_0(1-H(-k_0(z_0))/e^β√(1-q_0)(k_0(z_0)+1/2β√(1-q_0))H(k_0(z_0)+β√(1-q_0))+H(-k_0(z_0))) Therefore, the scaling of ϵ=1-q_1 is different in this case. We find (using the definition of eq. (<ref>)):1-q_1=β^2Γ^2∫ Dz_0e^-p̂_1-p̂_0/2/cosh(z_0√(p̂_0))∫ Dz_11/k̂(z_0,z_1)^2(cosh(k̂(z_0,z_1))-sinh(k̂(z_0,z_1))/k̂(z_0,z_1))[alternative version] 1-q_1=β^2Γ^2∫ Dz_0∫ Dz_1e^-p̂_1-p̂_0/2/cosh(k̂(z_0,z_1)√(p̂_0/p̂_1))1/z_0^2p̂_1(cosh(z_0√(p̂_1))-sinh(z_0√(p̂_1))/z_0√(p̂_1))Therefore, the convergence to the classical case is smooth.§ ESTIMATION OF THE LOCAL ENERGY AND ENTROPY LANDSCAPES WITH THE CAVITY METHOD In order to compute the local landscapes of the energy and the entropy around a reference configuration (Fig. <ref>), we used the Belief Propagation (BP) algorithm, a cavity method message-passing algorithm that has been successfully employed numerous times for the study of disordered systems <cit.>. In the case of single-layer binary perceptrons trained on random unbiased i.i.d. patterns, it is believed that the results of this algorithm are exact in the limit of N→∞, at least up to the critical value α_c≈0.83 <cit.>.For a full explanation of the BP equations for binary perceptrons, we refer the interested reader to the Appendix of ref. <cit.>. Here, we provide only a summary. The BP equations involve two sets of quantities (called “messages”), representing cavity marginal probabilities associated with each edge in a factor graph representation of the (classical) Boltzmann distribution induced by the energy function (<ref>). To each edge in the graph linking the variable node i with the factor node μ, are associated two messages, m_i→μ and m̂_μ→ i. These are determined by solving iteratively the following system of equations:m_i→μ=tanh(∑_ν≠μtanh^-1(m̂_ν→ i)) m̂_μ→ i=ξ_i g(a_μ→ i,b_μ→ i)where:g(a,b) =H(a-1/b)-H(a+1/b)/H(a-1/b)+H(a+1/b)a_μ→ i=∑_j≠ iξ_j^μm_j→μb_μ→ i=√(∑_j≠ i(1-m_j→μ^2))(as for the previous section, we used the definition H(x)=1/2erfc(x/√(2)).)Once a self-consistent solution is found, these quantities can be used to compute, using standard formulas, all thermodynamic quantities of interest, in particular the typical (equilibrium) energy and the entropy of the system. A numerically accurate implementation of these equations is available at ref. <cit.>.It is also possible to compute those same thermodynamic quantities in a neighborhood of some arbitrary reference configuration w={ w_i} _i. This is achieved by adding an external field in the direction of that configuration, which amounts at this simple modification of eq. (<ref>):m_i→μ=tanh(∑_ν≠μtanh^-1(m̂_ν→ i)+λ w_i) By varying the auxiliary parameter λ, we can control the size of the neighborhood under consideration (the larger λ, the narrower the neighborhood); the typical normalized Hamming distance from the reference of the configurations that are considered by this modified measure can be obtained from the fixed-point BP messages for any given λ by this formula:d=1/2(1-1/N∑_im_iw_i)where the m_i are the total magnetizations:m_i=tanh(∑_νtanh^-1(m̂_ν→ i)+λ w_i) In order to produce the energy landscape plots of Figs. <ref>a and <ref>b, we simply ran this algorithm at infinite temperature, varying λ and plotting the energy density shift from the center as a function of d. This gives us an estimate of the most probable energy density shift which would be obtained by moving in a random point at distance d from the reference.The plot in Fig. <ref>c was similarly obtained by setting the temperature to 0 and computing the entropy density instead, which in this context is then simply the natural logarithm of the number of solutions in the given neighborhood, divided by N.§ NUMERICAL SIMULATIONS DETAILS OF THE ANNEALING PROTOCOLS§.§ Quantum annealing protocol In this section we provide the details of the QA results presented in Fig. <ref>. The simulations were performed using the RRR Monte Carlo method <cit.>. We fixed the total number of spin flip attempts at τ Ny·10^4 and followed a linear protocol for the annealing of Γ, starting from Γ_0=2.5 and reaching down Γ_1=0. We actually divided the annealing in 30τ steps, where during each step Γ was kept constant and decreased by ΔΓ=Γ_0-Γ_1/30τ after each step. In the figure, we have shown the results for N=4001 and τ=4; the results for N=1001,2001 and for τ=1,2 were essentially indistinguishable at that level of detail. §.§ Classical simulated annealing protocol The results for SA presented in Fig. <ref> used an annealing protocol in β designed to make a direct comparison to QA: we found analytically a curve β_equiv(Γ) such that the classical equilibrium energy would be equal to the longitudinal component of the quantum system energy, eq. (<ref>). The classical equilibrium energy was computed from the equations in ref. <cit.>. The result is shown in Fig. <ref>. The vertical jump to β=20 is due to the transition mentioned in sec. <ref>; as shown in Fig. <ref>, the SA protocol in the regime we tested gets stuck well before this transition.The SA annealing protocol thus consisted in setting β=β_equiv(Γ) and decreasing linearly Γ from 2.5 to 0, like for the QA case. We fixed the total number of spin flip attempts at τ N·10^4 and used τ=4,8,16; as for the QA case, the annealing process was divided in 30τ steps.Other more standard annealing protocols (e.g. linear or exponential or logarithmic) yielded very similar qualitative results, as expected from the analysis of ref. <cit.>.§ ADDITIONAL NUMERICAL RESULTS ON THE ANNEALING PROCESSES§.§ Additional comparisons between theory and simulations Fig. <ref> compares the result of Monte Carlo simulations with the theoretical predictions for the classical component of the energy, eq. (<ref>), and the transverse overlap, eq. (<ref>). Fig. <ref> compares the same simulation results with the analytical curves at finite y instead. This shows a relatively small systematic offset (due to the static approximation) at intermediate values of Γ, while the agreement is good at both large and small Γ.Fig. <ref> shows the comparison with the y→∞ curve for the expectation of the full quantum Hamiltonian, eq. (<ref>), using the same data. The agreement is remarkable, and a close inspection reveals that the curves from the simulation tend towards the theoretical one as y increases, i.e. in the quantum limit.§.§ Experiments with two-layer networks We performed additional experiments using two-layer fully-connected binary networks, the so-called committee machines. Previous results obtained with the robust-ensemble measure <cit.> showed that this case is quite similar to that of single layer networks. In particular, standard Simulated Annealing suffers from an exponential slow-down as the system size increases even moderately, while algorithms that are able to target the dense states do not suffer from the trapping in meta-stable states. Indeed, we found the latter feature to be true in the quantum annealing scenario.The model in this case is defined by a modified energy function (cf. eq. (<ref>)):E(σ)=∑_μ=1^α NΘ(-∑_k=1^Ksgn∑_j=1^N/Kξ_j^μσ_kj)where now the N spin variables are divided in groups of K hidden units, and consequently the spin variables σ_kj have two indices, k=1,…,K for the hidden unit and j=1,…,N/K for the input. Notice that the input size is reduced K-fold with respect to the previous case. The output of these machines is simply decided by the majority of the outputs of the individual units, and the energy still counts the number of errors. The Suzuki-Trotter transformation proceeds in exactly the same way as for the previous cases.Like for the single-layer case, we tested the case of α=0.4 at β=20, and we used K=5 units. We tested different values of N=1005,2005,4005 with different values of the Trotter replicas y=32,64,128 (only y=32 for N=4005) at a fixed overall running time of yNτ·10^4 spin flip attempts, with τ=4 (cf. Fig. <ref>). The MC algorithm and the annealing protocols were also unchanged. The results are shown in Fig. <ref>: all these tests produce curves which are almost indistinguishable at this level of detail for different N, and that seemingly tend to converge to some limit curve for increasing y (while being almost overlapping at small transverse field Γ), consistently with the single-layer scenario. § REAL-TIME QUANTUM ANNEALING ON SMALL SAMPLES§.§ Numerical methods Computing the evolution of the system under Quantum Annealing amounts at solving the time-dependent Schrdinger equation for the system∂/∂ t|ψ(t)⟩ =-iĤ(t)|ψ(t)⟩where we set ℏ=1 for simplicity. In our case, the time dependence of the Hamiltonian H comes in through the varying transverse magnetic field Γ(t). We assume that Γ varies linearly with time between some starting value Γ_0 and 0, in a total time t_max. Therefore, the final Hamiltonian is reduced to the purely classical case, Ĥ(t_max)=E.In the following, we will always work in the basis of the final Hamiltonian, in which every eigenvector |σ⟩ corresponds to a configuration σ∈{ -1,+1} ^N of the spins in the z direction. Therefore, we represent |ψ(t)⟩ with a complex-valued vector of length 2^N with entries ⟨σ|ψ(t)⟩; similarly, the Ĥ(t) operator is represented by a matrix of size 2^N×2^N, H(σ,σ^')=⟨σ|Ĥ|σ^'⟩. The structure of this matrix is very sparse: the diagonal elements H(σ,σ) correspond to the classical energies E(σ), while the only non-zero diagonal elements are those elements H(σ,σ^') such that σ and σ^' are related by a single spin flip, in which case the value is -Γ.In our simulations, the initial state |ψ(0)⟩ was set to the ground state of the system at Γ→∞, i.e. with all the spins aligned in the x direction; in our basis, this corresponds to a uniform vector, ⟨σ|ψ(0)⟩ =N^-1/2 for all σ. We simulated the evolution of the system by the short iterative Lanczos (SIL) method <cit.>: we compute the evolution at fixed Γ for a short time interval Δ t, then lower Γ by a small fixed amount ΔΓ, and iterate. The total evolution time is thus t_max=Γ_0/ΔΓΔ t. Numerical accuracy can be verified by scaling both these steps by a fixed amount and observing no significant difference in the outcome. The evolution is computed by the Lanczos algorithm with enough iterative steps to ensure sufficient accuracy, as determined by observing that increasing the number of steps does not change the outcomes significantly. In the simulations presented here, we set Γ_0=5, ΔΓ=10^-3 and Δ t=0.2, and we used 10 steps in the Lanczos iterations.At the end of the annealing process, we could retrieve the final probability distribution for each configuration of the spins as p(σ)=|⟨σ|ψ(t_max)⟩|^2. §.§ Sample selection Given the exponential scaling with N of the SIL algorithm, simulations are necessarily restricted to small values of N. We used N=21. At these system sizes, there is a very large sample-to-sample variability. Furthermore, the energy barriers are generally small enough for the classical Simulated Annealing to perform well.In order to obtain small but challenging samples, in which we could also study the structure of the solutions, we proceeded as follows: we extracted at random 450 samples with P=17 patterns each (corresponding to α≃0.81, close to the critical value of 0.83 which is valid for large systems), and selected those which had at least a certain minimum number of solutions (note that in such small systems we can easily enumerate all of the 2^21≃2·10^6 configurations and check their energy). We arbitrarily chose 21 solutions as the threshold. We then ran both Simulated Annealing with a fast schedule (with τ=1) and Simulated Quantum Annealing with τ=1 and a large number of Trotter replicas (y=512), and selected those samples in which SA failed while SQA succeeded. This left us with 20 samples, which we then analyzed in detail and over which we performed the real-time QA simulations.§.§.§ Randomized samples For each of the selected samples, we generated a corresponding randomized version by permuting randomly the values of the energy associated to each configuration. This procedure maintains unaltered the spectrum of the energies (so that for example the classical Boltzmann distribution at thermodynamic equilibrium remains unchanged), but completely destroys the geometric features of the energy landscape. We used these randomized samples as a benchmark against the measurements performed in our analysis. §.§ Analysis §.§.§ QA vs SQA We compared the results of real-time QA with the SQA Monte Carlo results, analyzing each of the 20 selected samples individually. In particular, we compared the values of the average longitudinal energy as a function of Γ for the two algorithms. As shown in Fig. <ref>, the agreement is excellent, and the system always gets very close to zero energy. In the same figure, we show that the same annealing protocol however gives substantially different (and rather worse) results on the randomized samples, reflecting the fact that the geometrical features of the landscape are crucial (we verified on a few cases that the results on the randomized samples could be improved by slowing down the annealing process, but we could not get to the same results as for the original systems even with a 100-fold increase in total time). Note that the sample-to-sample variability in these curves appears to be fairly small due to our sample filtering process; we verified in a preliminary analysis that the agreement is generally excellent also without the filtering conditions, e.g. on instances that have no solutions at all. §.§.§ Other measurements We also performed a number of measurements on the final configuration reached by the QA algorithm (both for the original samples and the randomized ones) and studied the properties of the final probability distribution p(σ). These are the quantities that we computed, reported in table <ref>: * The average value of the energy ⟨ E⟩ =∑_σE(σ)p(σ).* The probability of finding a solution P_SOL=∑_σ:E(σ)=0p(σ).* The probability and the energy of the most probable configuration, p(σ^⋆) and E(σ^⋆), where σ^⋆=max_σp(σ^⋆).* The inverse participation ratio IPR=∑_σp(σ)^2, to assess the concentration of the final distribution. (Qualitatively analogous results are obtained using the Shannon entropy.) This measure however does not take into account the geometric structure of the distribution: for instance, if p(σ) were non-zero on just to configurations σ^1 and σ^2, the IPR would be very high, but it would not be able to discriminate between the cases in which σ^1and σ^2 are close to each other or far apart in Hamming distance.* The mean distance between configurations, defined as d̅=∑_σ,σ^'p(σ)p(σ^')d(σ,σ^'), where d(σ,σ^') is the normalized Hamming distance between configurations. This measure is useful since it reflects the geometric features of the final measure: it can only be low if the mass of the probability is concentrated spatially (in particular, it is zero if and only if p(σ) is a delta function).As can be seen from the table, the results are generally in agreement with the qualitative picture described in the main text, especially when compared to the randomized benchmark: the system is able to reach very low energies ⟨ E⟩, the probability of solving the problem P_SOL is very high, the measure is rather concentrated on a few good configurations and those configurations are close to each other (high IPR, low d̅).For the original samples only, we also looked at the final configuration from the Monte Carlo SQA process, σ_SQA, and computed its ranking according to p(σ), which we denoted as r_SQA. A ranking of 1 implies σ_SQA=σ^⋆. All rankings are very small, the largest ones generally corresponding to samples with the largest number of solutions and the less concentrated distributions. This further attests to the good agreement between the QA and SQA processes. §.§.§ Local entropies In order to assess whether the denser ground states were favored in the final configuration with respect to more isolated solutions we compared the mean local entropy curves weighted according to p(σ) with those averaged over all the solutions. More precisely, we define C(n) as the set of the n configurations with highest probability, and n_w as the number of configurations required to achieve a cumulative probability of t, i.e. the lowest n such that ∑_σ∈ C(n)p(σ)≥ w. We also define K(σ,d) as the number of solutions at normalized Hamming distance from σ lower or equal to d. Then the mean local entropy curve weighted with p is then defined as:ϕ_w(d)=1/N∑_σ∈ C(n_w)p(σ)log K(σ,d)/∑_σ∈ C(n_w)p(σ). Denoting by 𝒮={σ|E(σ)=0} the set of all the solutions, we also compute the flat average of the local entropies over 𝒮:ϕ_SOL(d)=1/N|𝒮|∑_σ∈𝒮log K(σ,d). If p concentrates on denser solutions, we expect that the ϕ_w curves should be generally higher than the ϕ_SOL curves. Indeed, the results confirm this scenario, as shown in Fig. <ref>, where we used w=0.9. (This value ensured that C(n_w)⊆𝒮 for all samples and thus that all the local entropies are finite; apart from this, the results are quite insensitive to the choice of w.) Note that, in the limit of large system sizes, the ϕ_SOL curves would be dominated by isolated solutions and display a gap around zero distances; the fact that this is not visible in Fig. <ref> is purely a finite size effect; the ϕ_w curves on the other hand should be roughly comparable to those shown in Fig. <ref>. §.§.§ Energy gaps As mentioned in the introduction of the main text, it is well known that, according to the adiabatic theorem, the effectiveness of the QA process depends on the relation between the rate of change of the Hamiltonian and the size of the gap between the ground state of the system H_0 and the first excited state H_1: smaller gaps require a slower annealing process. Therefore, we performed a static analysis of the energy spectrum of each of the 20 samples at varying Γ, and computed the gap H_1-H_0, comparing the results with those for the randomized versions of the samples. The results are shown in Fig. <ref>. For the original samples, the gap only vanishes in the limit of Γ→0 (which is expected since the ground state at Γ=0 is degenerate). For the randomized samples, on the other hand, the gap nearly closes at non-zero Γ, displaying the characteristics of an “avoided crossing” (see the figure upper inset), which is the type of phenomenon that is known to hamper the performance of QA algorithms. Indeed, the values of Γ where these avoided crossings occur are precisely those at which the mean value of H found by the QA algorithm deviates from the ground state H_0, thereby getting stuck as shown in Fig. <ref>.49 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Shor(1994)]shor1994algorithms author author P. W. Shor, in @noopbooktitle Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on (organization IEEE, year 1994) pp. pages 124–134NoStop [Ray et al.(1989)Ray, Chakrabarti, and Chakrabarti]ray1989sherrington author author P. Ray, author B. K. Chakrabarti,and author A. Chakrabarti, @noopjournal journal Physical Review B volume 39, pages 11828 (year 1989)NoStop [Finnila et al.(1994)Finnila, Gomez, Sebenik, Stenson, and Doll]finnila1994quantum author author A. Finnila, author M. Gomez, author C. Sebenik, author C. Stenson,and author J. Doll, @noopjournal journal Chemical physics letters volume 219, pages 343 (year 1994)NoStop [Kadowaki and Nishimori(1998)]kadowaki1998quantum author author T. Kadowaki and author H. Nishimori, @noopjournal journal Physical Review E volume 58, pages 5355 (year 1998)NoStop [Farhi et al.(2001)Farhi, Goldstone, Gutmann, Lapan, Lundgren, and Preda]farhi2001quantum author author E. Farhi, author J. Goldstone, author S. Gutmann, author J. Lapan, author A. Lundgren,and author D. Preda, @noopjournal journal Science volume 292, pages 472 (year 2001)NoStop [Das and Chakrabarti(2008)]das2008colloquium author author A. Das and author B. K. Chakrabarti, @noopjournal journal Reviews of Modern Physics volume 80, pages 1061 (year 2008)NoStop [Moore and Mertens(2011)]moore2011nature author author C. Moore and author S. Mertens, @nooptitle The nature of computation (publisher Oxford University Press, year 2011)NoStop [Born and Fock(1928)]born1928beweis author author M. Born and author V. Fock,@noopjournal journal Zeitschrift für Physik A Hadrons and Nuclei volume 51,pages 165 (year 1928)NoStop [Landau(1932)]landau1932theorie author author L. Landau, @noopjournal journal Phys. Z. Sowjetunion volume 2, pages 1 (year 1932)NoStop [Zener(1932)]zener1932non author author C. Zener, in @noopbooktitle Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. volume 137 (organization The Royal Society, year 1932) pp. pages 696–702NoStop [Altshuler et al.(2010)Altshuler, Krovi, and Roland]altshuler2010anderson author author B. Altshuler, author H. Krovi, and author J. Roland,@noopjournal journal Proceedings of the National Academy of Sciences volume 107, pages 12446 (year 2010)NoStop [Bapst et al.(2013)Bapst, Foini, Krzakala, Semerjian,and Zamponi]bapst2013quantum author author V. Bapst, author L. Foini, author F. Krzakala, author G. Semerjian,and author F. Zamponi, @noopjournal journal Physics Reports volume 523, pages 127 (year 2013)NoStop [Santoro et al.(2002)Santoro, Martoňák, Tosatti, andCar]santoro2002theory author author G. E. Santoro, author R. Martoňák, author E. Tosatti,and author R. Car, @noopjournal journal Sciencevolume 295, pages 2427 (year 2002)NoStop [Martoňák et al.(2002)Martoňák, Santoro,and Tosatti]martovnak2002quantum author author R. Martoňák, author G. E. Santoro,and author E. Tosatti, @noopjournal journal Physical Review B volume 66, pages 094203 (year 2002)NoStop [Heim et al.(2015)Heim, Rønnow, Isakov, and Troyer]heim2015quantum author author B. Heim, author T. F. Rønnow, author S. V. Isakov,andauthor M. Troyer, @noopjournal journal Science volume 348, pages 215 (year 2015)NoStop [Rønnow et al.(2014)Rønnow, Wang, Job, Boixo, Isakov, Wecker, Martinis, Lidar, and Troyer]ronnow2014defining author author T. F. Rønnow, author Z. Wang, author J. Job, author S. Boixo, author S. V. Isakov, author D. Wecker, author J. M.Martinis, author D. A.Lidar,and author M. Troyer, @noopjournal journal Science volume 345, pages 420 (year 2014)NoStop [Johnson et al.(2011)Johnson, Amin, Gildert, Lanting, Hamze, Dickson, Harris, Berkley, Johansson, Bunyk et al.]johnson2011quantum author author M. W. Johnson, author M. H. Amin, author S. Gildert, author T. Lanting, author F. Hamze, author N. Dickson, author R. Harris, author A. J.Berkley, author J. Johansson, author P. Bunyk, et al., @noopjournal journal Nature volume 473, pages 194 (year 2011)NoStop [Boixo et al.(2014)Boixo, Rønnow, Isakov, Wang, Wecker, Lidar, Martinis,and Troyer]boixo2014evidence author author S. Boixo, author T. F. Rønnow, author S. V. Isakov, author Z. Wang, author D. Wecker, author D. A. Lidar, author J. M. Martinis,and author M. Troyer, @noopjournal journal Nature Physics volume 10, pages 218 (year 2014)NoStop [Langbein et al.(2004)Langbein, Borri, Woggon, Stavarache, Reuter, and Wieck]langbein2004control author author W. Langbein, author P. Borri, author U. Woggon, author V. Stavarache, author D. Reuter,and author A. Wieck, @noopjournal journal Physical Review B volume 69, pages 161301 (year 2004)NoStop [Baldassi et al.(2015)Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina]baldassi_subdominant_2015 author author C. Baldassi, author A. Ingrosso, author C. Lucibello, author L. Saglietti,and author R. Zecchina, @noopjournal journal Physical Review Lettersvolume 115, pages 128101 (year 2015)NoStop [Baldassi et al.(2016a)Baldassi, Borgs, Chayes, Ingrosso, Lucibello, Saglietti, and Zecchina]baldassi_unreasonable_2016 author author C. Baldassi, author C. Borgs, author J. T. Chayes, author A. Ingrosso, author C. Lucibello, author L. Saglietti,and author R. Zecchina, 10.1073/pnas.1608103113 journal journal Proceedings of the National Academy of Sciences volume 113, pages E7655 (year 2016a)NoStop [Baldassi et al.(2016b)Baldassi, Ingrosso, Lucibello, Saglietti, andZecchina]baldassi_local_2016 author author C. Baldassi, author A. Ingrosso, author C. Lucibello, author L. Saglietti,and author R. Zecchina, 10.1088/1742-5468/2016/02/023301 journal journal Journal of Statistical Mechanics: Theory and Experiment volume 2016, pages P023301 (year 2016b)NoStop [Baldassi et al.(2016c)Baldassi, Gerace, Lucibello, Saglietti, and Zecchina]baldassi2016learning author author C. Baldassi, author F. Gerace, author C. Lucibello, author L. Saglietti,and author R. Zecchina, 10.1103/PhysRevE.93.052313 journal journal Physical Review E volume 93, pages 052313 (year 2016c)NoStop [Krauth and Mézard(1989)]krauth-mezard author author W. Krauth and author M. Mézard, @noopjournal journal J. Phys. France volume 50, pages 3057 (year 1989)NoStop [Sompolinsky et al.(1990)Sompolinsky, Tishby, and Seung]sompolinsky1990learning author author H. Sompolinsky, author N. Tishby,and author H. S. Seung,@noopjournal journal Physical Review Letters volume 65, pages 1683 (year 1990)NoStop [Foini et al.(2010)Foini, Semerjian, and Zamponi]foini2010solvable author author L. Foini, author G. Semerjian, and author F. Zamponi,@noopjournal journal Physical review letters volume 105, pages 167204 (year 2010)NoStop [Biroli and Zamponi(2012)]biroli2012tentative author author G. Biroli and author F. Zamponi, @noopjournal journal Journal of Low Temperature Physics volume 168,pages 101 (year 2012)NoStop [Hubara et al.(2016)Hubara, Courbariaux, Soudry, El-Yaniv, and Bengio]hubara2016quantized author author I. Hubara, author M. Courbariaux, author D. Soudry, author R. El-Yaniv,and author Y. Bengio, @noopjournal journal arXiv preprint arXiv:1609.07061 (year 2016)NoStop [Courbariaux et al.(2015)Courbariaux, Bengio, and David]courbariaux2015binaryconnect author author M. Courbariaux, author Y. Bengio,and author J.-P. David, in@noopbooktitle Advances in Neural Information Processing Systems (year 2015) pp. pages 3105–3113NoStop [MacKay(2003)]mackay2003information author author D. J. MacKay, @nooptitle Information theory, inference and learning algorithms (publisher Cambridge university press, year 2003)NoStop [LeCun et al.(2015)LeCun, Bengio, and Hinton]lecun2015deep author author Y. LeCun, author Y. Bengio, and author G. Hinton,@noopjournal journal Nature volume 521, pages 436 (year 2015)NoStop [Aaronson(2015)]aaronson2015read author author S. Aaronson, @noopjournal journal Nature Physics volume 11, pages 291 (year 2015)NoStop [Barahona(1982)]barahona1982computational author author F. Barahona, @noopjournal journal Journal of Physics A: Mathematical and General volume 15, pages 3241 (year 1982)NoStop [Huang and Kabashima(2014)]huang2014origin author author H. Huang and author Y. Kabashima, @noopjournal journal Physical Review E volume 90, pages 052813 (year 2014)NoStop [Horner(1992)]horner1992dynamics author author H. Horner, @noopjournal journal Zeitschrift für Physik B Condensed Matter volume 86, pages 291 (year 1992)NoStop [Bapst and Semerjian(2012)]bapst2012quantum author author V. Bapst and author G. Semerjian, @noopjournal journal Journal of Statistical Mechanics: Theory and Experiment volume 2012, pages P06007 (year 2012)NoStop [Bapst and Semerjian(2013)]bapst2013thermal author author V. Bapst and author G. Semerjian, in @noopbooktitle Journal of Physics: Conference Series, Vol. volume 473 (organization IOP Publishing, year 2013) p. pages 012011NoStop [Baldassi(2017)]baldassi_method_2017 author author C. Baldassi, 10.1088/1742-5468/aa5335 journal journal Journal of Statistical Mechanics: Theory and Experiment volume 2017, pages 033301 (year 2017)NoStop [Baldassi et al.(2007)Baldassi, Braunstein, Brunel, andZecchina]baldassi-et-all-pnas author author C. Baldassi, author A. Braunstein, author N. Brunel,and author R. Zecchina,@noopjournal journal Proceedings of the National Academy of Sciences volume 104, pages 11079 (year 2007)NoStop [Markland et al.()Markland, Morrone, Berne, Miyazaki, Rabani, and Reichman]markland2011 author author T. E. Markland, author J. A. Morrone, author B. J. Berne, author K. Miyazaki, author E. Rabani,and author D. R. Reichman, @noopjournal journal Nature Physics volume 7NoStop [Schneider et al.(2016)Schneider, Guan, and Bartschat]schneider2016chapter author author B. I. Schneider, author X. Guan, and author K. Bartschat,@noopjournal journal Advances in Quantum Chemistry volume 72, pages 95 (year 2016)NoStop [Keskar et al.(2016)Keskar, Mudigere, Nocedal, Smelyanskiy, and Tang]keskar2016large author author N. S. Keskar, author D. Mudigere, author J. Nocedal, author M. Smelyanskiy,and author P. T. P. Tang, @noopjournal journal arXiv preprint arXiv:1609.04836 (year 2016)NoStop [Bottou et al.(2016)Bottou, Curtis, and Nocedal]bottou2016optimization author author L. Bottou, author F. E. Curtis,and author J. Nocedal,@noopjournal journal arXiv preprint arXiv:1606.04838(year 2016)NoStop [Braunstein and Zecchina(2006)]braunstein-zecchina author author A. Braunstein and author R. Zecchina, @noopjournal journal Phys. Rev. Lett. volume 96, pages 030201 (year 2006)NoStop [Baldassi(2009)]baldassi-2009 author author C. Baldassi, @noopjournal journal J. Stat. Phys. volume 136, pages 902 (year 2009)NoStop [Baldassi and Braunstein(2015)]baldassi2015max author author C. Baldassi and author A. Braunstein, http://stacks.iop.org/1742-5468/2015/i=8/a=P08008 journal journal Journal of Statistical Mechanics: Theory and Experiment volume 2015, pages P08008 (year 2015)NoStop [Mézard and Montanari(2009)]mezard_information_2009 author author M. Mézard and author A. Montanari, @nooptitle Information, Physics, and Computation (publisher Oxford University Press,year 2009)NoStop [Mézard(1989)]mezard_space_1989 author author M. Mézard, 10.1088/0305-4470/22/12/018 journal journal Journal of Physics A: Mathematical and General volume 22, pages 2181 (year 1989)NoStop [FBP()]FBP-code @nooptitle Belief Propagation code,howpublished <https://github.com/carlobaldassi/BinaryCommitteeMachineFBP.jl>NoStop | http://arxiv.org/abs/1706.08470v3 | {
"authors": [
"Carlo Baldassi",
"Riccardo Zecchina"
],
"categories": [
"quant-ph",
"cond-mat.dis-nn",
"cs.LG",
"stat.ML"
],
"primary_category": "quant-ph",
"published": "20170626164349",
"title": "Efficiency of quantum versus classical annealing in non-convex learning problems"
} |
unsrt p K^0 K̅^̅0̅ α α̅ β̱ CP-1.80em/∂ σ Σ Γ γ α δ̣ δ Δ Δλ Λ ŁΛ ϵ þθ η ηV_a ω Ω ρ ζβ̱ Bł α ∗α^* α^*2α^*3 ∂/∂x ∂/∂w '∂/∂x' '∂/∂w' | http://arxiv.org/abs/1706.08686v4 | {
"authors": [
"B. B. Jiao"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170627063920",
"title": "Description and prediction of even-A nuclear masses based on residual proton-neutron interactions"
} |
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany Physik-Department, Technische Universität München, Garching, Germany Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan Spin Quantum Rectification Project, ERATO, Japan Science and Technology Agency, Sendai, Japan WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan Spin Quantum Rectification Project, ERATO, Japan Science and Technology Agency, Sendai, Japan Institute for Materials Research, Tohoku University, Sendai, Japan PRESTO, Japan Science and Technology Agency, Saitama, Japan Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany Physik-Department, Technische Universität München, Garching, Germany Nanosystems Initiative Munich, München, Germany Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany Physik-Department, Technische Universität München, Garching, Germany Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany Physik-Department, Technische Universität München, Garching, Germany Nanosystems Initiative Munich,München, Germany Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany Physik-Department, Technische Universität München, Garching, Germany Nanosystems Initiative Munich, München, Germany Institut für Festköper- und Materialphysik, Technische Universität Dresden, Dresden, Germany Center for Transport and Devices of Emergent Materials, Technische Universität Dresden, 01062 Dresden We present experimental control of the magnetic anisotropy in a gadolinium iron garnet (GdIG) thin film from in-plane to perpendicular anisotropy by simply changing the sample temperature.The magnetic hysteresis loops obtained by SQUID magnetometry measurements unambiguously reveal a change of the magnetically easy axis from out-of-plane to in-plane depending on the sample temperature. Additionally, we confirm these findings by the use of temperature dependent broadband ferromagnetic resonance spectroscopy (FMR). In order to determine the effective magnetization, we utilize the intrinsic advantage of FMR spectroscopy which allows to determine the magnetic anisotropy independent of the paramagnetic substrate, while magnetometry determines the combined magnetic moment from film and substrate. This enables us to quantitatively evaluate the anisotropy and the smooth transition from in-plane to perpendicular magnetic anisotropy. Furthermore, we derive the temperature dependent g-factor and the Gilbert damping of the GdIG thin film.Perpendicular magnetic anisotropy in insulating ferrimagnetic gadolinium iron garnet thin films S. T. B. Goennenwein December 30, 2023 ===============================================================================================Controlling the magnetization direction of magnetic systems without the need to switch an external static magnetic field is a challenge that has seen tremendous progress in the past years. It is of considerable interest for applications as it is a key prerequisite to store information in magnetic media in a fast, reliable and energy efficient way. Two notable approaches to achieve this in thin magnetic films are switching the magnetization by short laser pulses<cit.> and switching the magnetization via spin orbit torques<cit.>.For both methods, materials with an easy magnetic anisotropy axis oriented perpendicular to the film plane are of particular interest. While all-optical switching requires a magnetization component perpendicular to the film plane in order to transfer angular momentum<cit.>, spin orbit torque switching with perpendicularly polarized materials allows fast and reliable operation at low current densities<cit.>. Therefore great efforts have been undertaken to achieve magnetic thin films with perpendicular magnetic anisotropy.<cit.> However, research has mainly been focused on conducting ferromagnets that are subject to eddy current losses and thus often feature large magnetization damping. Magnetic garnets are a class of highly tailorable magnetic insulators that have been under investigation and in use in applications for the past six decades.<cit.>The deposition of garnet thin films using sputtering, pulsed laser deposition or liquid phase epitaxy, and their properties are very well understood. In particular, doping the parent compound (yttrium iron garnet, YIG) with rare earth elements is a powerful means to tune the static and dynamic magnetic properties of these materials.<cit.>Here, we study the magnetic properties of a gadolinium iron garnet thin film sample using broadband ferromagnetic resonance (FMR) and SQUID magnetometry. By changing the temperature, we achieve a transition from the typical in-plane magnetic anisotropy (IPA), dominated by the magnetic shape anisotropy, to a perpendicular magnetic anisotropy (PMA) at about 190. We furthermore report the magnetodynamic properties of GdIG confirming and extending previous results.<cit.>§ MATERIAL AND SAMPLE DETAILS We investigate a 2.6 thick gadolinium iron garnet (Gd_3Fe_5O_3, GdIG) film grown by liquid phase epitaxy (LPE) on a (111)-oriented gadolinium gallium garnet substrate (GGG).The sample is identical to the one used in Ref. and is described there in detail.GdIG is a compensating ferrimagnet composed of two effective magnetic sublattices:The magnetic sublattice of the Gd ions and an effective sublattice of the two strongly antiferromagnetically coupled Fe sublattices. The magnetization of the coupled Fe sublattices shows a weak temperature dependence below room temperature and decreases from approximately 190 at 5 to 140 at 300.<cit.> The Gd sublattice magnetization follows a Brillouin-like function and decreases drastically from approximately 800 at 5 to 120 at 300.<cit.>As the Gd and the net Fe sublattice magnetizations are aligned anti-parallel, the remanent magnetizations cancel each other at the so-called compensation temperature =285K of the material.<cit.> Hence, the remanent net magnetization M of GdIG vanishes at .The typical magnetic anisotropies in thin garnet films are the shape anisotropy and the cubic magnetocrystalline anisotropy, but also growth induced anisotropies and magnetoelastic effects due to epitaxial strain have been reported in literature.<cit.> We find that our experimental data can be understood by taking into account only shape anisotropy and an additional anisotropy field perpendicular to the film plane. A full determination of the anisotropy contributions is in principle possible with FMR.Angle dependent FMR measurements (not shown) indicate an anisotropy of cubic symmetry with the easy axis along the crystal [111] direction in agreement with literature.<cit.> The measurements suggest that the origin of the additional anisotropy field perpendicular to the film plane is the cubic magnetocrystalline anisotropy. However, the low signal amplitude and the large FMR linewidth towardsin combination with a small misalignment of the sample, render a complete, temperature dependent anisotropy analysis impossible. In the following, we therefore focus only on shape anisotropy and the additional out-of-plane anisotropy field.§ SQUID MAGNETOMETRY SQUID magnetometry measures the projection of the magnetic moment of a sample on the applied magnetic field direction.For thin magnetic films, however, the background signal from the comparatively thick substrate can be on the order of or even exceed the magnetic moment m of the thin film and hereby impede the quantitative determination of m. Our 2.6 thick GdIG film is grown on a 500 thick GGG substrate warranting a careful subtraction of the paramagnetic background signal of the substrate. In our experiments, H_0 is applied perpendicular to the film plane and thus, the projection of the net magnetization M⃗ = m⃗/V to the out-of-plane axis is recorded as M_⊥. squid shows M_⊥ of the GdIG film as function of the externally applied magnetic field H_0. In the investigated small region of H_0, the magnetization of the paramagnetic substrate can be approximated by a linear background that has been subtracted from the data. The two magnetic hysteresis loops shown in squid are typical for low temperatures (T≲170) and for temperatures close to . The hysteresis loops unambiguously evidence hard and easy axis behavior, respectively. Towards low temperatures (T=170, squid (a)) the net magnetization M=|M⃗| increases and hence, the anisotropy energy associated with the demagnetization field = -M_⊥ [ We use the demagnetization factors of a infinite thin film: N_x,y,z = (0,0,1). ] dominates and forces the magnetization to stay in-plane.At these low temperatures, the anisotropy field perpendicular to the film plane, , caused by the additional anisotropy contribution has a constant, comparatively small magnitude.We therefore observe a hard axis loop in the out-of-plane direction:Upon increasing H_0 from -150 to [retain-explicit-plus]+150, M⃗ continuously rotates from the out-of-plane (oop) direction to the in-plane (ip) direction and back to the oop direction again. The same continuous rotation happens for the opposite sweep direction of H_0 with very little hysteresis. For temperatures close to(T=250, squid (b)),becomes negligible due to the decreasing M whileincreases as shown below. Hence, the out-of-plane direction becomes the magnetically easy axis and, in turn, an easy-axis hysteresis loop is observed:After applying a large negative H_0 [(1) in squid (a)] M and H_0 are first parallel. Sweeping to a positive H_0, M first stays parallel to the film normal and thus M_⊥ remains constant [(2) in squid (a)] until it suddenly flips to being aligned anti-parallel to the film normal at H_0 > + [(3) in squid (a)]. These loops clearly demonstrate that the nature of the anisotropy changes from IPA to PMA on varying temperature. § BROADBAND FERROMAGNETIC RESONANCEIn order to quantify the transition from in-plane to perpendicular anisotropy found in the SQUID magnetometry data, broadband FMR is performed as a function of temperature with the external magnetic field H_0 applied along the film normal.[ The alignment of the sample is confirmed at low temperatures by performing rotations of the magnetic field direction at fixed magnetic field magnitude while recording the frequency of resonance . As the shape anisotropy dominates at low temperatures,goes through an easy-to-identify minimum when the sample is aligned oop. ] For this, H_0 is swept while the complex microwave transmissionof a coplanar waveguide loaded with the sample is recorded at various fixed frequencies between 10 and 25. We perform fits ofto<cit.>(H_0)|_ω = -i Z χ(H_0) + A + B· H_0with the complex parameters A and B accounting for a linear field-dependent background signal of , the complex FMR amplitude Z, and the Polder susceptibility<cit.>χ(H_0) = (H - )/(H - )^2 - ^2 + i/2(H - ).Here, γ is the gyromagnetic ratio, = ω/(γμ_0), and ω is the microwave frequency and the effective magnetization = - /(γμ_0). From the fit, the resonance field H_res and the full width at half-maximum (FWHM) linewidthis extracted. Exemplary data for(data points) and the fits to s21 (solid lines) at two distinct temperatures are shown in the two insets of fmr (a).We obtain excellent agreement of the fits and the data. The insets furthermore show that the signal amplitude is significantly smaller for T=240 than for 110.This is expected as the signal amplitude is proportional to the net magnetization M of the sample which decreases considerably with increasing temperature (cf. fmr (b)). At the same time, the linewidth drastically increases as discussed in the following section. These two aspects prevent a reliable analysis of the FMR signal in the temperature region 250 < T < 300 (i.e. around the compensation temperature).Therefore we do not report data in this temperature region.Nevertheless, FMR is ideally suited to investigate the magnetic properties of the GdIG film selectively, i.e. independent of the substrate, for temperatures below .As all measurements are performed in the high field limit of FMR, the dispersions shown in fmr (a) are linear and we can use the Kittel equation= γμ_0 (- )to extract γ and . It is customary to describe the magnetic anisotropy usingwhich can be related to an anisotropy fieldalong +ẑ as =-=M_⊥ - for positive H_0. Here,is given by the demagnetization field = -M_⊥ (along -ẑ) and the anisotropy fieldof the additional perpendicular anisotropy (along +ẑ). Evidently,can be determined by linearly extrapolating the data to =0. The FMR dispersion and the fit to oop_dispersion (solid lines) are shown for three selected temperatures in fmr (a). At 110 (blue curve)is positive. Therefore, M > indicating that shape anisotropy dominates, and the film plane is a magnetically easy plane while the oop direction is a magnetically hard axis. At 240 (red curve)is negative and hence, the oop direction is a magnetically easy axis.Figure <ref> (b) shows the extracted (T). At 190,changes sign. Above this temperature (marked in red), the oop axis is magnetically easy (PMA) and below this temperature (marked in blue), the oop axis is magnetically hard (IPA). The knowledge of M_⊥(T) obtained from SQUID measurements allows to separate the additional anisotropy fieldfrom(red dots in fmr (b)). =M_⊥- increases considerably for temperatures close towhile at the same time the contribution of the shape anisotropy, = - M_⊥ trends to zero. For T ⪆180,exceedswhich is indicated by the sign change of .Above this temperature, we thus observe PMA. We use the magnetization M determined using SQUID magnetometry from Ref. normalized to the here recordedat 10 in order to quantify . The maximal value μ_0= 0.18 is obtained at 250 which is the highest measured temperature due to the decreasing signal-to-noise ratio towards .We can furthermore extract the g-factor and damping parameters from FMR. The evolution of the g-factor g = γħ/ with temperature is shown in g_damping (a). We observe a substantial decrease of g towards . This is consistent with reports in literature for bulk GIG and can be explained considering that the g-factors of Gd and Fe ions are slightly different such that the angular momentum compensation temperature is larger than the magnetization compensation temperature.<cit.> The linewidth = γ can be separated into a inhomogeneous contribution _0 = (H_0 = 0) and a damping contribution varying linear with frequency with the slope α: = 2 α· + _0.Close to =285K, the dominant contribution to the linewidth is _0 which increases by more than an order of magnitude from 390 at 10K to 6350 at 250K [g_damping (c)]. This temperature dependence of the linewidth has been described theoretically by Clogston et al.<cit.> in terms of a dipole narrowing of the inhomogeneous broadening and was reported experimentally before<cit.>. As opposed to these single frequency experiments, our broadband experiments allow to separate inhomogeneous and intrinsic damping contributions to the linewidth. We find that in addition to the inhomogeneous broadening of the line, also the Gilbert-like (linearly frequency dependent) contribution to the linewidth changes significantly: Upon approaching[g_damping (b)], the Gilbert damping parameter α increases by an order of magnitude.Note, however, that due to the large linewidth and the small magnetic moment of the film, the determination of α has a relatively large uncertainty. [ For the given signal-to-noise ratio and the large linewidth, α and _0 are correlated to a non-negligible degree with a correlation coefficient of C(intercept, slope) = -0.967. ] A more reliable determination of the temperature evolution of α using a single crystal GdIG sample that gives access to the intrinsic bulk damping parameters remains an important task.§ CONCLUSIONSWe investigate the temperature evolution of the magnetic anisotropy of a GdIG thin film using SQUID magnetometry as well as broadband ferromagnetic resonance spectroscopy. At temperatures far away from the compensation temperature , the SQUID magnetometry reveals hard axis hysteresis loops in the out-of-plane direction due to shape anisotropy dominating the magnetic configuration.In contrast, at temperatures close to the compensation point, we observe easy axis hysteresis loops. Broadband ferromagnetic resonance spectroscopy reveals a sign change of the effective magnetization (the magnetic anisotropy field) which is in line with the magnetometry measurements and allows a quantitative analysis of the anisotropy fields. We explain the qualitative anisotropy modifications as a function of temperature by the fact that the magnetic shape anisotropy contribution is reduced considerably close todue to the reduced net magnetization, while the additional perpendicular anisotropy field increases considerably.We conclude that by changing the temperature the nature of the magnetic anisotropy can be changed from an in-plane magnetic anisotropy to a perpendicular magnetic anisotropy. This perpendicular anisotropy close toin combination with the small magnetization of the material may enable optical switching experiments in insulating ferromagnetic garnet materials.Furthermore, we analyze the temperature dependence of the FMR linewidth and the g-factor of the GdIG thin film where we find values compatible with bulk GdIG<cit.>.The linewidth can be separated into a Gilbert-like and an inhomogeneous contribution. We show that in addition to the previously reported increase of the inhomogeneous broadening, also the Gilbert-like damping increases significantly when approaching § ACKNOWLEDGMENTSWe gratefully acknowledge funding via the priority program Spin Caloric Transport (spinCAT), (Projects GO 944/4 and GR 1132/18), the priority program SPP 1601 (HU 1896/2-1) and the collaborative research center SFB 631 of the Deutsche Forschungsgemeinschaft.§ BIBLIOGRAPHY | http://arxiv.org/abs/1706.08488v1 | {
"authors": [
"H. Maier-Flaig",
"S. Geprägs",
"Z. Qiu",
"E. Saitoh",
"R. Gross",
"M. Weiler",
"H. Huebl",
"S. T. B. Goennenwein"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170626172452",
"title": "Perpendicular magnetic anisotropy in insulating ferrimagnetic gadolinium iron garnet thin films"
} |
L[1]> m#1 C[1]> m#1 R[1]> m#1positioning,arrows decorations.pathmorphing decorations.markings | http://arxiv.org/abs/1706.09039v2 | {
"authors": [
"B. C. Allanach",
"D. Bhatia",
"A. M. Iyer"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170627202807",
"title": "Dissecting Multi-Photon Resonances at the Large Hadron Collider"
} |
Elastic Hadron Scattering in Various Pomeron Models Presented by P. Erland at XXIII Cracow Epiphany Conference P. Erland^1, R. Staszewski^2, M. Trzebiński^2,Corresponding author: [email protected], R. Kycia^1 ^1 Cracow University of Technology, Warszawska St. 24, 31-155 Cracow^2 Institute of Nuclear Physics PAN, Radzikowskiego St. 152, 31-342 CracowDecember 30, 2023 ==================================================================================================================================================================================================================================================================In this work the process of elastic hadron scattering is discussed. In particular, scattering amplitudes for the various Pomeron models are compared. In addition, differential elastic cross section as a function of the scattered proton transverse momentum for unpolarised and polarised protons is presented. Finally, an implementation of the elastic scattering amplitudes into the GenEx Monte Carlo generator is discussed. 13.85.Dz§ INTRODUCTIONElastic scattering is the simplest process that one can imagine: in the final state all particles are identical to the initial state ones. This implies that the exchanged object must be a colour singlet and, in particular, that there is no quantum number transfer. In the case of the proton-proton elastic scattering, pp → pp, see Fig. <ref>, such an exchange can be mediated via a photon (electromagnetic interaction) or a Pomeron/Reggeon (strong force).Elastic scattering is a large fraction of total cross section. However, despite many years of research, there are still open questions concerning its nature.There is a strong connection between the elastic scattering amplitude and the total cross section, which is described by the optical theorem. The dependence of the total cross section (σ_tot) on the forward scattering amplitude(f(θ=0), where θ is the scattering angle) is given by: σ_tot=4π/k Imf(0), where k is the wave vector. This fact is widely used in order to precisely determine the total cross section <cit.>.§ SPIN STRUCTURE OF THE POMERONThe differential cross section for unpolarized pp elastic scattering is described by formula:dσ(pp→ pp)/dt=1/16π s(s-4m_p^2)1/4∑_s_1, …, s_4 | ⟨ 2s_3,2s_4|𝒯|2s_1,2s_2 ⟩|^2,where ⟨ 2s_3,2s_4|𝒯|2s_1,2s_2 ⟩ are the helicity amplitudes with a certain spin orientation of each particle (s_i).Contrary to the photons, the nature of Pomerons is not well known – there are still many open questions. For example: the Pomeron spin structure.In the approach of Donnachie and Landshoff, a Pomeron is viewed as a vector object <cit.>. However, as was discussed in <cit.>, such an approach gives a negative x-s value. It is also possible to define it as a scalar or a rank-2 tensor object <cit.>. As was shown in <cit.>, the STAR data <cit.> prefer the tensor over the scalar Pomeron model.§.§ Calculation of the Elastic Scattering AmplitudesThere are 16 helicity amplitudes describing pp elastic scattering for every combination of spins of incoming and outgoing protons. However, only five of them are independent: ψ_1(s,t)= ⟨ ++|𝒯|++ ⟩,ψ_2(s,t)= ⟨ ++|𝒯|–⟩,ψ_3(s,t)= ⟨ +-|𝒯|+- ⟩,ψ_4(s,t)= ⟨ +-|𝒯|-+ ⟩,ψ_5(s,t)= ⟨ ++|𝒯|+- ⟩.ψ_1 and ψ_3 are the amplitudes describing no spin flip, ψ_5 – single flip, ψ_2 and ψ_4 – double flip. These amplitudes can be calculated using a vertex (Γ) and a propagator (Δ) functions, specific for each Pomeron spin (cf. <cit.>):* scalar Pomeron: * vertex: iΓ^(IP_Spp)(p',p)=-i3β_IPNNM_0 F_1[(p'-p)^2],* propagator: iΔ^(IP_S)(s,t)=s/2m_p^2M_0^2(-isα'_IP)^α_IP(t)-1 , * vector Pomeron: * vertex: iΓ_μ^(IP_Vpp)(p',p)=-i3β_IPNNM_0 F_1[(p'-p)^2]γ_μ,* propagator: iΔ_μν^(IP_V)(s,t)=1/M_0^2g_μν(-isα'_IP)^α_IP(t)-1, * tensor Pomeron: * vertex: iΓ_μν^(IP_Tpp)(p',p)=-i3β_IPNNF_1[(p'-p)^2]{1/2[γ_μ(p'+p)_ν+ γ_ν(p'+p)_μ]-1/4g_μν( p'+ p)},* propagator:iΔ_μν,κλ^(IP_S)(s,t)=1/4(g_μκg_νλ+g_μλg_νκ-1/2g_μνg_κλ)(-isα'_IP)^α_IP(t)-1. In these formulas β_IPNN is a coupling constant describing the Pomeron-nucleon interaction, F_1[(p'-p)^2] is a form factor, γ_ν,γ_μ are gamma matrices, p=γ^μ p_μ is a four momentum in a Feynman slash notation, α'_IP=0.25 GeV^-2 is the Pomeron slope and α_IP(t)=1.0808+α'_IP is the Pomeron trajectory.The Pomeron spin structure is visible in its propagator formula. For the tensor Pomeron it depends on four variables (μ,ν,κ,λ), in contrast to the vector (two variables) and the scalar (no variables) Pomeron models.The above formulas have been implemented as a set of C++ classes for future implementation in the MC generator. Such approach allows the calculations of more complicated processes to be made in the future. The outcome of an exemplary calculation is shown in Fig. <ref>, where the absolute value of the imaginary and real part of ψ_2 amplitude is plotted for all three discussed Pomeron models.As can be seen in these figures, the magnitude of the ψ_2 amplitude (real and imaginary part) is similar in those of the tensor and vector models, whereas the scalar model predictions are much higher. A dip located close to t = -0.3 GeV^2 and t = -4.3 GeV^2 for the real and imaginary part of ψ_2 amplitude is due to a change of the sign of the amplitude. The results generated by C++ code were compared with approximate analytic formulas presented in <cit.>. All results are consistent with each other.Since a single amplitude differs a lot between the models, it is interesting to see a cross section integrated over all spin combinations. Results of such calculations are shown in Fig. <ref>. For proton-proton collision the tensor (dotted line) and vector (dashed line) Pomeron gives exactly the same results. For small momentum transfers also the scalar model predictions are comparable. They starts to differ (up to a factor of 10) with the increasing value of the four momentum transfer.§ IMPLEMENTATION IN THE GENEX MONTE CARLO GENERATORMonte Carlo generators are widely used tools in high energy physics since they provide an essential input helping to understand detector effects. In consequence, they provide a way of comparison between the theory and experimental data. Elastic scattering process is present in many recent HEP MC generators. Based on the formulas described in the previous section, the process of elastic scattering has been added to the GenEx MC generator <cit.>.As an example a distribution of the transverse momentum of the final state proton obtained assuming various Pomeron models is shown in Fig. <ref>. The left plot shows the distribution for the unpolarised protons sum of all amplitudes, whereas the right plot illustrates the polarised (i.e. sum of ψ_1, ψ_2 and ψ_5) amplitudes.For both unpolarised and polarised protons the vector and tensor models are in agreement, whereas the scalar model gives slightly different values for larger transverse momentum values.§ SUMMARY AND OUTLOOKThe helicity amplitudes for various Pomeron models for the elastic scattering processes were analysed. It was shown that the differential cross section for the vector and tensor model in proton-proton collisions were in a good agreement, but scalar model differs in region of larger transverse momentum transfer. This difference is also visible in the corresponding, generated MC sample.An analysis of this simplest possible process gives a good starting point for the future studies of exclusive processes. The plans include further developments of the GenEx generator including the non-resonant and resonant soft exclusive production.§ ACKNOWLEDGEMENTSThis work was supported in part by Polish National Science Centre grant UMO-2015/17/D/ST2/03530.99 ALFA_TOTEM ATLAS Collaboration, Measurement of the total cross section from elastic scattering in pp collisions at √(s)=8 TeV with the ATLAS detector, Phys. Lett. B 761 (2016) 158,ATLAS Collaboration, Measurement of the total cross section from elastic scattering in pp collisions at √(s)=7 TeV with the ATLAS detector, Nucl. Phys. B 889 (2014) 486,TOTEM Collaboration, First measurements of the total proton-proton cross section at the LHC energy of √(s) = 7 TeV, EPL 96 (2011) 31002,TOTEM Collaboration, A luminosity-independent measurement of the proton-proton total cross-section at √(s) = 8 TeV, Phys. Rev. Lett. 111 (2013) 012001.jeden A. Donnachie and P. V. Landshoff, pp and pp Elastic Scattering, Nucl. Phys. B 231 (1984) 189.dwa P. Lebiedowicz, O. Nachtmann and A. Szczurek, Exclusive central diffractive production of scalar and pseudoscalar mesons; tensorial vs. vectorial pomeron, Annals Phys. 344 (2014) 301.trzy C. Ewerz, P. Lebiedowicz, O. Nachtmann, A. Szczurek, Helicity in Proton-Proton Elastic Scattering and the Spin Structure of the Pomeron, arXiv:1606.08067.cztery[STAR Collaboration] L. Adamczyk et al., Single Spin Asymmetry A_N in Polarized Proton-Proton Elastic Scattering at √(s) = 200 GeV, Phys. Lett. B 719 (2013) 62.szesc R. A. Kycia, J. Chwastowski, R. Staszewski, J. Turnau, GenEx: A simple generator structure for exclusive processes in high energy collisions, arXiv:1411.6035. | http://arxiv.org/abs/1706.08307v1 | {
"authors": [
"P. Erland",
"R. Staszewski",
"M. Trzebinski",
"R. Kycia"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170626101710",
"title": "Elastic Hadron Scattering in Various Pomeron Models"
} |
label1]Davis D. M. Welakuh [email protected]]Alain M. Dikandécor1 [cor1]Corresponding author [email protected][label1]Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS), Department of Physics, Faculty of Science, University of Buea P.O Box 63 Buea, Cameroon The storage and subsequent retrieval of coherent pulse trains in the quantum memory (i.e. cavity-dark state) of three-level Λ atoms, are considered for an optical medium in which adiabatic photon transfer occurs under the condition of quantum impedance matching. The underlying mechanism is based on intracavity Electromagnetically-Induced Transparency, by which properties of a cavity filled with three-level Λ-type atoms are manipulated by an external control field. Under the impedance matching condition, we derive analytic expressions that suggest a complete transfer of an input field into the cavity-dark state by varying the mixing angle in a specific way, and its subsequent retrieval at a desired time. We illustrate the scheme by demonstrating the complete transfer and retrieval of a Gaussian, a single hyperbolic-secant and a periodic train of time-entangled hyperbolic-secant input photon pulses in the atom-cavity system. For the time-entangled hyperbolic-secant input field, a total controllability of the periodic evolution of the dark state population is made possible by changing the Rabi frequency of the classical driving field, thus allowing to alternately store and retrieve high-intensity photons from the optically dense Electromagnetically-Induced transparent medium. Such multiplexed photon states, which are expected to allow sharing quantum information among many users, are currently of very high demand for applications in long-distance and multiplexed quantum communication.3-level Λ atoms adiabatic photon transfer soliton trains impedance matching condition § INTRODUCTION Progress in the fabrication of materials with high intensity-dependent refractive index <cit.> has motivated a great deal of theoretical interest in nonlinear wave propagation in non-local media<cit.>. In non-local optical systems such as thermal media <cit.>, the nonlinear optical response depends not only on the local intensity at a given point but also on the surrounding intensity profile. While many of these settings require high power laser lights, the phenomenon of electromagnetically-induced transparency (EIT) in multi-level atom systems <cit.> provides a cost-less and highly reliable mechanism to suppress photon loss, and to simultaneously increase light-matter interaction. Combined with sufficiently large nonlinearities, the phenomenon of EIT holds great potential for few-photon nonlinear optics and offers the possibility for many applications in communication and quantum information processing <cit.>.EIT concretely is a quantum interference effect that permits the propagation of light through an otherwise opaque atomic medium. Interesting examples of quantum interference effects have been provided by recent reports on the observation of extremely slow group velocities <cit.>, as well as localization and containment of light pulse within an atomic cloud <cit.>. In addition to a large suppression of optical absorption, EIT interference effect can be used to greatly enhance the efficiency of nonlinear optical processes. An addition to the noticeable effect of EIT is the significant change of dispersion property of optical media and the large reduction of the group velocity of optical wave packets <cit.>. Slow light under conditions of EIThas been observed in resonant multiple-level atoms, semiconductor quantum wells and quantum dots <cit.>. As a result of the enhancement of nonlinear optical processes, many remarkable phenomena such as giant Kerr nonlinearity <cit.>, four-wave mixing, etc, have also been demonstrated <cit.>.The most intriguing manifestation of nonlinearity in excited media is the existence of a specific kind of waves called solitons <cit.>. These unique kind of wave packets occur because of a subtle balance of dispersion by nonlinearity, which results in their undistorted profiles during propagation over long distances. Optical solitons have been extensively investigated since they offer potential interesting applications in optical information processing and transmission, owing to their robustness during propagation. A large class of optical solitons can be produced with intense electromagnetic fields and via far-off resonant excitation schemes <cit.>. Optical solitons produced in such schemes travel with a velocity very close to the speed of light in vacuum.In recent years the idea that slow-light solitons could emerge from three-level EIT systems has been proposed <cit.>. With regards to quantum interference effects in EIT systems, considerable attention has been paid to the weak light solitons in atomic systems <cit.>. For practical applications of optical memory it is desirable to obtain a probe pulse that is robust during its storage and retrieval. Previous analysis on storage of optical solitons has shown that a weak optical soliton pulse can be stored and retrieved in three-level atomic systems via a single EIT <cit.>, as well as in double EIT <cit.>. Investigations of the transfer of quantum correlations from traveling-wave light fields to collective atomic states have been carried out <cit.>. The transfer of single-photon quantum states to and from an optically dense coherently driven medium confined within a resonator, has been suggested <cit.>. In this last work it was demonstrated that well localized single-photon field, represented by the hyperbolic-secant pulse soliton, could be stored and retrieved by an adiabatic rotation of the cavity-dark state. The adiabatic transfer of the quantum state of photons to collective atomic excitations was brought into effect by intracavity EIT <cit.>, we note here that sources of single-photon wave packets have been proposed in ref. <cit.>.In this work we shall be interested in the transfer, storage and retrieval of weak and high-intensity input photon wave packets in the quantum memory of a three-level Λ-atom system, under the condition of dynamical quantum impedance matching. We establish that the explicit dependance of analytic expressions of key parameters governing the transfer process, on the shape of the input wave packet, does not have negative effect on the dark state population but rather provides a well-defined normalization conditions for both the input and output wave packets. We suggest a new possible transfer scheme which involves a train of pulse solitons of finite period <cit.> as the input field, instead of a single-pulse soliton <cit.>. That is, rather than propagating a single hyperbolic-secant pulse into the collective atom-cavity system as previously suggested <cit.>, we shall consider loading a continuous train of time-entangled high-intensity hyperbolic-secant pulses at some finite well-defined time period. By changing the Rabi frequency of the classical driving field the time-multiplexed input fieldwill be transferred into the cavity dark-state, and the population of the dark state is expected to change periodically with a kink profile over each period. The storage of multi-pulse photon solitons in three-level atoms has also been recently considered, using far-off-resonant Raman control scheme <cit.>. In our work we shall utilize the technique of adiabatic transfer <cit.> to map photonic states into collective atomic states. Worth mentioning, for this last scheme the non-destructive and reversible mapping of the quantum information contained in the photon pulses into collective atomic states is achieved using the technique of intracavity EIT <cit.>. Given that in the EIT technique properties of a cavity filled with three-level Λ-type atoms can be controlled by an external field, this enables the storage and retrieval of the periodic train of pulses by periodically switching off and on adiabatically the control field. Our results are particularly relevant for applications in quantum information processing involving relatively weak nonlinearity, where the phenomenon of modulational instability favors periodic optical solitons <cit.> even from a continuous-wave input field.In section <ref> we introduce the model and construct the families of dark states with one cavity photon from quantum states of the Λ-type atom system, coupled to an input field. We illustrate the consistency and reliability of the transfer scheme by considering two different low-intensity input fields, namely a Gaussian wave packet and plane waves. In section <ref> we propose a new dynamical quantum transfer scheme involving a periodic train of high-intensity input pulses, loaded in the three-level Λ atom system at a finite and controllable time period. We formulate the probability-amplitude equations taking into account decays arising due to spontaneous emission, and investigate the time evolutions of characteristic quantities governing the storage and retrieval of the input soliton train. These include the mixing angle, the Rabi frequency and the dark state population. We discuss the connection of these characteristic parameters with results <cit.> for an hyperbolic-secant input pulse. Section <ref> will be devoted to concluding remarks. § THE Λ-TYPE THREE-LEVEL ATOM SYSTEM AND DARK STATEThe mechanism of adiabatic transfer and storage of photon states into a cavity-dark state and vice versa, has been described e.g. in refs. <cit.>. Here we are interested in a distinct photon transfer mechanism which involves time-entangled high-intensity input photon solitons, our main objective being to extend the single-pulse soliton transfer scheme <cit.> to time-multiplexed periodic input photon pulses. We shall exploit the quantum impedance matching condition in order to obtain simplified relations for the transfer, storage and retrieval of any normalizad input field loaded into the atom-cavity system. Consider an optically dense ensemble of N identical three-level atoms confined within an optical cavity. We assume an effective one-dimensional model consisting of a Fabry-Perot cavity with two mirrors, one partially transmitting the input field while the other mirror is totally reflecting, see fig.<ref>. The input-output field is introduced as a continuum field modeled by a set of oscillator modes, denoted by the annihilation operator b̂_k ("k" free-field photon modes), coupled to the cavity mode with a coupling constant κ. The interaction of the cavity field â and the continuum of free-field modes b̂_k is described by the first term of Eq.(<ref>).Of the two dipole-allowed transitions one is coupled by a cavity mode with a coupling constant g, while the other optical transition is driven by a classical field in a coherent state with Rabi frequency Ω(t) (fig.<ref>). The system is initially in state |b⟩, and the fields cause transition between states. The dynamics of the coupled system is described by the Hamiltonian:H = ħκ∑_kâ^†b̂_k + ħ g∑_i = 1^Nâσ_ab^i + ħΩ(t)e^-iν t∑_i = 1^Nσ_ac^i +h.c.,where σ_μν^i=|μ⟩_ii⟨ν| is the flip operator of the i^th atom between states |μ⟩ and |ν⟩ with μ,ν=a,b,c. b̂_k is the annihilation operator of a continuum of free-space modes of the single-photon field, coupled to the selected cavity mode by the creation operator â^†, while κ describes the coupling of the selected modes. We define collective atomic operators <cit.> by the sum of flip operators;σ_ab=1/√(N)∑_i=1^Nσ_ab^i, σ_ac=∑_i=1^Nσ_ac^i,with N the number of atoms. Note that these operators couple only symmetric Dicke-like states <cit.>. With the collective operators defined as above the interaction Hamiltonian describing the transitions between collective states (see fig.<ref>) is given as:H_int = ħ g√(N)âσ_ab+ħΩ(t)e^-iν tσ_ac + h.c.,where the interaction strength of the cavity mode is enhanced by a factor of √(N) which uplifts the stringent requirements of strong-coupling regime of cavity QED. Thus, single photons couple to collective excitations associated with EIT.Of special interest in the atom-cavity system is the dark state, which forms under the condition of two-photon resonance, when the energy difference between the metastable states equals the energy difference per photon of the two fields, i.e. when ω_cb=ν-ν_c, with ν and ν_c the frequencies of the classical drive field and the cavity mode respectively. The resulting families of dark eigenstates with corresponding zero eigenvalues that contains one cavity photon is expressed as:|D,1⟩ = Ω|b,1⟩ -g√(N)|c,0⟩/√(Ω^2+g^2N)= cosθ(t)|b,1⟩ - sinθ(t)|c,0⟩,where cosθ(t)=Ω/√(Ω ^2+ g^2 N) and sinθ(t)=g√(N)/√(Ω ^2+ g^2 N), with θ(t) = arctan(g√(N)/Ω) the mixing angle. It is worth noting that in the limit g√(N)≫Ω(t), the dark state |D,1⟩ is nearly identical to |c,0⟩ (i.e. |D,1⟩∼|c,0⟩). In this limit, a single-photon excitation is shared among the atoms and the effective cavity-dark state decay is reduced as the dark state |D,1⟩ contains only a very small component of (Ω/g√(N)) of the single-photon state |b,1⟩ that is vulnerable to decay. We now discuss the principle of intracavity EIT relevant for our study. Three important mechanisms of dissipation and decay have to be distinguished. The dark state Eq.(<ref>) is immune against decay out of the excited states as it contains no component of the states, but is sensitive to decay of the coherence between the metastable states. This decay γ_bc sets an upper limit for the lifetime of the dark state. In addition to this mechanism the effect of the finite lifetime of the cavity has to be considered. Thus, a bare cavity decay rate γ leads to the effective decay rate γ_D of the dark state |D,1⟩ given by:γ_D = γcos^2θ(t).This relation shows that varying the mixing angle θ(t) will influence the coupling of the cavity dark state to its environment. This is achieved in principle by changing the Rabi-frequency of the classical driving field Ω(t). This property of intracavity EIT will be used to effectively load the cavity system with an excitation resulting from incoming photon wave packets, and to subsequently release this energy after some storage time.We now present a scheme of transfering a single-photon state of the input field into a single-photon cavity dark state (i.e. a single excitation of atom-cavity systems). We consider an input field in a general single-photon state i.e.:|Ψ_in(t)⟩ = ∑_kξ_k^in(t)b̂_k^†|0⟩, where ξ_k^in(t)=ξ_k^in(t_0)e^-iω_k(t-t_0), |0⟩ is the vacuum state of the continuum of modes b̂_k and b̂_k^†|0⟩ =|1_k⟩ is a bosonic Fock state which represents |0,...,1_k,...,0⟩, and ∑_k|ξ_k^in|^2=1. Hereafter we characterize these fields by an envelope wave function Φ_in(z,t) defined by:Φ_in(z,t) = ∑_k⟨ 0_k|b̂e^ikz|Ψ_in(t)⟩ = L/2π c∫ dω_kξ^in(ω_k,t)e^ikz.The normalization condition (L/2π c)∫ dω_k|ξ^in(ω_k,t)|^2=1 of the Fourier coefficients implies the normalization of the input wave function according to:∫dz/L|Φ_in(z,t)|^2= 1.Clearly, Φ_in(z,t) describes a single photon propagating along the z axis.The general state of the combined system of cavity mode and atoms, when the system interacts with the single-photon wave-packet, can be expressed in the compact form:|Ψ(t)⟩ = β(t)|b,1,0_k⟩ + α(t)|a,0,0_k⟩ + ϵ(t)|c,0,0_k⟩ + ∑_kξ_k(t)|b,0,1_k⟩ .where |b,1,0_k⟩ denotes the state corresponding to the atomic system in the collective state|b⟩, the cavity mode in single-photon state |1⟩ and there are no photons in the outside modes |0_k⟩. We stipulate two-photon resonance condition that requires the bare frequency of the cavity mode to coincide with the a-b transition of the atoms, and the carrier frequency of the input wave packet i.e ν_c=ω_ab≡ω_a-ω_b=ω_0 as well as the control field to be in resonance with the a-c transition, i.e ν=ω_ac. The resulting equations of motion for the evolution of slowly-varying amplitudes in the rotating frame are:α̇(t)=-γ_a/2α(t) - ig√(N)β(t) - iΩϵ(t),β̇(t)=-ig√(N)α(t) - iκ∑_kξ_k(t),ϵ̇(t)=-γ_c/2ϵ(t) - iΩα(t),ξ̇_k(t)=-iΔ_kξ_k(t) - iκβ(t),where Δ_k=ω_k-ω_0 is the detuning of the free-field modes from the cavity resonance and ω_0=ω_ab. Here we have included a loss term γ_a/2 and γ_c/2 from state |a⟩ and |c⟩ respectively, representing spontaneous emission. In general, considering the motion of resonant atomic systems, the density matrix equations can be adopted. Nevertheless, for EIT-like coherent atomic systems the density matrix equations can be replaced by the probability amplitude equations without any difference <cit.>. Also Eq.(<ref>) is of no interest for it is not a constituent of the dark state, on the other hand Eqs.(<ref>)-(<ref>) can be rewritten in terms of the dark and orthogonal bright states <cit.>:|D(t)⟩ =-icosθ(t)|b,1,0_k⟩ + isinθ(t)|c,0,0_k⟩,|B(t)⟩ = sinθ(t)|b,1,0_k⟩ + cosθ(t)|c,0,0_k⟩,which yields the following time-evolution equations for the dark and bright-state populations:Ḋ(t)=-iθ̇(t)B(t) + κcosθ(t)∑_kξ_k(t),ξ̇_k(t) =-iΔ_kξ_k(t) - iκsinθ(t)B(t) κcosθ(t)D(t).Following the adiabatic elimination of the bright state and non-adiabatic corrections <cit.>, the remaining amplitudes of dark states and free-field components are given by:Ḋ(t)= κcosθ(t)∑_kξ_k(t), ξ̇_k(t)=-iΔ_kξ_k(t) - κcosθ(t)D(t).By formally integrating Eq.(<ref>) in the continuum limit, substituting the result into Eq.(<ref>) and invoking the standard Markov approximation assuming that no photon arrives the cavity before some reference time t_0, the dark state and output field are found as:D(t) = √(γc/L)∫_t_0^tdτ cosθ(τ)Φ_in(0,τ) e^-[γ/2∫_τ^tcos^2θ(τ')dτ'], Φ_out(0,t)= Φ_in(0,t) - G(t), G(t) = γcosθ(t)∫_t_0^tdτcosθ(τ)Φ_in(0,τ) e^-[γ/2∫_τ^tcos^2θ(τ')dτ'],where γ=κ^2L/c is the empty cavity decay rate. In order to have complete transfer of free-field photons into the dark state, we require an optimization of cosθ(t) such that from Eq.(<ref>), D(t) ∼∫_t_0^tΦ_in(0,τ)dτ. It will be shown later that a result of optimization of the time-dependence of cosθ(t) yields the condition for the normalization of the input field Φ_in(t), which ensures that the dark-state population tends to unity for each input field. To capture and subsequently release a single-photon state of the light field in this way, we start by accumulating the field in a cavity mode and then adiabatically switching off the driving field in such a way that an initial free-space wave packet can be stored in a long-lived atom-like dark state. By adiabatically switching on the Rabi-frequency of the classical driving field, we can release the stored wave packet. The optimization of cosθ(t) in Eq.(<ref>) is acheived under conditions of quantum impedance matching. Taking advantage of the destructive interference of the directly reflected and circulating field components within the cavity, we shall require Φ_out=Φ̇_out=0 which leads to:-d/dtlncosθ(t) + d/dtlnΦ_in(t) = γ/2cos^2θ(t).The above equation is referred to as quantum or dynamical impedance matching. The first term on the left hand side of Eq.(<ref>) describes internal losses due to coherent Raman adiabatic passage, while the second term appears due to time dependence of the input field Φ_in. The right hand side of Eq.(<ref>) can be interpreted as an effective cavity decay rate which is reduced due to intracavity EIT. Solving Eq.(<ref>) for cosθ(t) leads to:cosθ(t) = 1/√(γ)Φ_in(t)/√(∫_t_0^tΦ_in^2(t^')dt^').Quite remarkably, impedance matching presents a viable technique for complete transfer of single-photon state of the free-field into the cavity dark-state by optimizing cosθ(t) as in Eq.(<ref>). The corresponding optimization of cosθ(t) is in principle achieved by changing the Rabi frequency of the classical driving field, which in essence is equivalent to varying the mixing angle θ(t). The Rabi frequency of the classical driving field that optimizes the time-dependence of cosθ(t) derived from Eq.(<ref>) is given by:Ω(t) = g√(N)Φ_in(t)/√(γ∫_t_0^tΦ_in^2(t^')dt^' - Φ_in^2(t)).With this choice of the driving field, the optimization of the time-dependence of cosθ(t) in Eq.(<ref>) yields the dark state: |D(t)|^2 = κ^2/γ∫_t_0^tΦ_in^2(t')dt',where, as already indicated, κ is the coupling of the incoming wave-packet of the free-field into the cavity dark state, γ is the bare cavity decay which are related as √(c/L)=κ/√(γ). An interesting result of the impedance matching condition is the role it plays in optimizing the time dependence such that the dark-state amplitude tends to unity for each incoming wave-packet(s). Formula (<ref>) in particular shows that the explicit dependence of cosθ(t) on the shape of the input pulse Φ_in during the transfer process, ensures complete storage (i.e. a total transferability) of the input photon field in the cavity-dark state.For the retrieval process, an adiabatic rotation of the mixing angle releases the stored photons into free-field photons at some later time t_1. It is relevant to point out that the resulting wave-packet will not necessarily have the same pulse form as the original one. However, the output wave-packet is generated in a well defined form and should correspond, in the ideal limit, to a single-photon Fock state. Therefore, for a time t_1 large enough such that the input wave-packet Φ_in(t) is "completely" stored (i.e. Φ_in(t)=0 for t>t_1), and for cosθ(t_1)=0, we find from the input-output relation:Φ_out(0,t) = -√(γ L/c)D(t_1)cosθ(t)e^-γ/2∫_t_1^tcos^2θ(τ')dτ',where D(t_1) is the dark state population at the retrieval time t_1. By adiabatically switching on the Rabi frequency of the driving field, we obtain the generalized form for the output field Φ_out(t) for any input field i.e.:Φ_out(t) = - Φ_in(t)/∫_t_0^tΦ_in(t')dt'∫_t_0^t_1Φ_in^2(t')dt'.Eq.(26) shows a correspondance between the input and output wave-packet due to time reversal of cosθ(t). The retrieval of the input wave-packet occurs at the time t_1 and Eq.(26) can be written in a closed form as:Φ_out(t) = - Φ_in(t)|D(t_1)|^2/|D(t)|^2. To check the consistency of the proposed total transfer scheme, and to point out the implications of optimizing the time-dependence of cosθ(t) by switching off and on the Rabi frequency of the driving field in an adiabatic fashion, we consider the storage and retrieval of a Gaussian single-photon input pulse sketched in Fig.<ref>, whose normalized intensity profile is given by: Φ_1(t)= Φ_in^(1)(z=0,t)= √(L/cT)(2/π)^1/4exp[-t^2/T^2],where T is the characteristic time. The storage of the input Gaussian wave packet is accomplished by changing the Rabi frequency of the driving field according to Eq.<ref>, resulting in the following optimized time-dependent cosθ(t):cosθ(t) = √(2/γ T)(2/π)^1/4exp[-t^2/T^2]/√(1 + erf[√(2)t/T]),where erf() is the Gauss error function. A consequence of changing the Rabi frequency of the driving field by varying the mixing angle θ(t), is the time evolution of the dark-state population i.e.: |D(t)|^2 = 1/2(1 + erf[√(2)t/T]),which approaches unity as t→∞ (see fig.<ref>).The retrieval of the stored photon into free-field photons occurs at some later time t_1. Thus, by simply reversing the adiabatic rotation of the mixing angle and using Eq.<ref> we obtain the output field:Φ_out^(1)(t) = - √(L/cT)(2/π)^1/41 + erf[√(2)t_1/T]/1 + erf[√(2)t/T]exp[-t^2/T^2].Profile of the output field Φ_out^(1)(t) is sketched in fig.<ref>, assuming a retrieval time t≈ 15 T.§ STORAGE AND RETRIVED OF TIME-ENTANGLED INPUT PULSES TRAIN In the previous section we developed a complete formalism for the storage and subsequent retrieval of a an input photon field in the quantum states k of a three-level Λ atoms system. We obtained the analytic expressions of the dark-state population, the optimized mixing angle and the output field as explicit functnput field. We illustrated the consistency of the proposed scheme by considering a Gaussian wave packet and found that for this specific input field, the dark-state population was kink shaped in time with an asymptotic value of one as t→∞. In this section we use the above analytical results to probe the possibility to store and retrieve a train of single-pulse photon solitons, periodically loaded in the three-level cold atom system at a finite and constant time period. To this last point, while the Gaussian field considered in the previous section are pulse shaped they are nevertheless lower-intensity fields, unlike hyperbolic-secant (i.e. a "sech") pulses which are waves of permanent profile by virtue of their soliton features. In ref. <cit.>, Fleischhauer have addressed the issue of storing and retrieving single "sech" pulse described by the followiwng normalized hyperbolic secant function:Φ_2(t)= Φ_in^(2)(z=0,t)=√(L/cT) sech[2t/T].With this input field, the dark-state population evolves in time according to the formula:D(t) =√(1 + tanh[2t/T]/2),the characteristic feature of which is its smooth-out (i.e. kink) profile extending from 0 to 1 . The corresponding Rabi frequency follows from formula (<ref>);Ω(t) = g√(N)sech(2t/T)/√(γ T/2[1 + tanh(2t/T)] - sech^2(2t/T)).Actually the dark-state population given in formula (<ref>), is associated with the following time-dependent optimized cosθ(t);cosθ(t) = √(2/γ T)sech(2t/T)/√(1 + tanh(2t/T)).Let us think of a transfer scheme in which not just one single "sech" photon pulse, but a packet of identical "sech" pulses of the form (<ref>) is loaded in the cold-atom system, one at a time over a well-defined finite time interval say τ.When the loading period τ is short enough the input pulse train can evolve into a time-entangled pulse multiplex with the profile of a periodic lattice of pulse solitons, so-called soliton crystal <cit.>. Analytically we traduce this in terms of a periodic input field for which the normalized hyperbolic secant pulse Eq.(<ref>) is the fundamental component i.e.:Φ_in(t)= √(L/cT) ∑_ℓ=0^M sech[2/T(t-ℓ τ)],corresponding precisely to a train of M hyperbolic-secant pulses periodically loaded into the atom-cavity system at a finite time interval τ. To find the corresponding Rabi frequency and dark-state population, let us assume the soliton train contains an infinite number of pulses and that the temporal separation τ between neighbour pulses is sufficiently short compared to their propagation time. With this assumption, the sum formula Eq.(<ref>) becomes exact <cit.> leading to:Φ_3(t)=Φ_in^(3)(z=0,t)=2K(m')/π√(L/cT) dn[4K(m')/π t/T],where dn is a Jacobi elliptic function <cit.> and its modulus m is uniquely determined by the transcendental equation <cit.>:τ= π K(m)/2 K(m') T,supplemented with the constraints 0≤ m ≤ 1, m'=1-m. K(m) is the complete elliptic integral of the first kind <cit.>. Fig.<ref> represents the temporal profile of the soliton-crystal photon input field Eq.(<ref>), for m=0.99 (full line) and m=1 (dotted line).When m=0 the dn() function tends to sin() whereas for m=1, the temporal separation τ between two adjacent pulses in the train tends to infinity. In this limit Eq.(<ref>) reduces exactly to Eq.(<ref>).Replacing formula (<ref>) in Eqs.(<ref>) and (<ref>) we find:cosθ(t) = √(χ/γ T)dn[χt/T]/√(E[am(χt/T),m] -E[am(χt_0/T),m]),where χ = 4K(m')/π andam is the Jacobi amplitude function, and:Ω(t) = g√(N)dn[χt/T]/√(γ T/χ[E(am(χt/T),m) -E(am(χt_0/T),m)] - dn^2[χ,t/T]),corresponding respectively to the temporal evolutions of the cosine of the mixing angle θ(t) and the Rabi frequency. As these two quantities depend on the arbitrary parameter t_0, to gain the physics in their analytical expressions we need to fix this parameter. Relying on their asymptotic forms in the single-pulse regime i.e. when m=1, we choose t_0=-τ given that when m=1 the initial loading time t_0→ -∞ consistently with the single-pulse case <cit.>. With this choice Eqs.(<ref>) and (<ref>) become respectively: cosθ(t) = √(χ/γ T)dn[χt/T]/√(E(m) + E[am(χt/T),m]), Ω(t) = g√(N)dn[χt/T]/√(γ T/χ[E(m) + E(am[t/T],m)] - dn^2[χt/T]).In fig. <ref>, we plotted the time evolution of cosθ(t) given by Eq.(<ref>) for m=0.99 (full line) and m=1 (dotted lines). Note that the dotted curve is the temporal profile of cosθ(t) for the single-pulse input field. cosθ(t)→ 0 as t→∞, in agreement with the impedance matching condition.As for the dark-state population, whose time evolution is readily expected to provide the most enlightening insight onto the proposed storage-retrieval process involving the input pulse train, this quantity is obtained by replacing Eq.(<ref>) in Eq.(<ref>) and integrating we find:|D(t)|^2 = χ/4[E(m) + E(am[χt/T],m)].The last quantity is plotted in fig. <ref> versus time, for m=0.99 (full line) and m=1 (dotted line). For m≠ 1, the time evolution of the dark-state population is a periodic train of kinks. The period of the kink lattice is actually the time τ separating two consecutive complete loadings of two identical single-pulse photons. As the figure indicates, over this time scale the dark-state population is fully kink shaped. As m→ 1 the separation between kinks τ→∞, the dark-state population in this limit changes in time as a single kink.The retrieval occurs at a later time t_1 when the input pulse train had been trapped in the dark state, such that no field is observed within the cavity. In this retrieval process, the action of the mixing angle is reversed by adiabatically switching on the classical driving field according to Eq.<ref> at some desired time. From Eq.<ref> we find:Φ_out^(3)(t) =-2K(m')/π√(L/cT) dn[χt/T] E(m) + E[am(χt_1/T) ,m]/ E(m) + E[am(χt/T) ,m]. The output profile is plotted in fig. <ref> for m=0.99 (full line) and m=1 (dotted lines). The plot shows the retrieval at a time t≈ 15T at which each stored pulse is released into free-field photons. Interestingly, the retrived pulse train is identical in shape to the input pulse train. § CONCLUSION Multi-soliton structures have attracted a great deal of attention in recent years, because of their enormous potential in information processing using multiplexed single-mode or multi-mode optical pulses. One of their many virtues is the possibility to simultaneously transmit either identical signals or several distinct signals, to different users. In this work, we investigated the storage and subsequent retrieval of a train of identical high-intensity photons in collective atomic states, by means of the adiabatic transfer mechanism. This process is based on the intracavity EIT, by which properties of a cavity filled with three-level Λ-type atoms can be manipulated by an external classical driving field. Since the mapping of the quantum information contained in the photon field into collective atomic states is achieved using the technique of intracavity EIT, it is non-destructive and reversible. By varying the mixing angle in a specific way, one can store and retrieve wave packets into the dark state of collective atomic system by switching adiabatically off and on the Rabi frequency of the classical driving field. The mechanism of adiabatic photon transfer under the quantum impedance matching condition (i.e. a one-to-one mapping of the input field into the dark state) has been discussedin ref. <cit.>. In this previous work the authors applied the mechanism to the transfer of a "sech"-type pulse. We derived analytic relations for the optimization of cosθ(t) and the corresponding Rabi frequency Ω(t), and obtained a general relation between the dark-state population and any form of input wave packet Φ_in. These relations are valid for any given wave packet, but provided that the arrival time t_0 is well defined. § ACKNOWLEDGMENTSA. M. Dikandé is Alexander von Humboldt Stiftung Alumni.§ REFERENCES00 a1M. Peccianti et al., Opt. Lett. 27 (2002) 1460. a2 C. Conti, M. Peccianti and G. Assanto, Phys. Rev. Lett. 92 (2004) 113902. a3 C. Rotschild et al., Nature Phys. 2 (2006) 769. a4 C. Conti, M. Peccianti and G. Assanto, Phys. Rev. Lett. 91 (2003) 073901. a5 A. M. Dikandé, Europhysics Letters 94 (2011) 44004. a6 M. Fleischhauer et al., Rev. Mod. Phys. 77 (2005) 633. a7 L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi, Nature 397 (1999) 594. a8 C. Liu, Z. Dutton, C. H. Behroozi and L. V. Hau, Nature 409(6819) (2001) 490. a9 M. Fleischhauer, A. Imamoglu and J. P. Marangos, Rev. Mod. Phys. 77 (2005) 633. a10 M. M. Kash et al., Phys. Rev. Lett. 82 (1999) 5229. a11 P. C. Ku, C. J. Chuang-Hasnain and S. L. Chuang, J. Phys. D 40 (2007) R93. a12 H. Schmidt and A. Imamoglu, Opt. Lett. 21 (1996) 1936. a13 S. E. Harris and L. V. Hau, Phys. Rev. Lett. 82 (1999) 4611. a14 H. Wang, D. Goorskey and M. Xiao, Phys. Rev. Lett. 87 (2001) 073601. a15 E. Ignesti, R. Buffa, L. Fini, E. Sali, M. V. Tognetti and S. Cavalieri, Optics Commun. 285 (2012) 1185. a16 S. E. Harris, J. E. Field and A. Imamoglu, Phys. Rev. Lett. 64 (1990) 1107. a17 M. D. Lukin, P. R. Hemmer, M. Loffler and M. O. Scully, Phys. Rev. Lett. 81 (1998) 2675. a18 H. Li and G. Huang, Phys. Rev. A 76 (2007) 043809. a19 H. A. Haus and W. S. Wong, Rev. Mod. Phys. 68 (1996) 423. a20 Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298 (1998) 81. a21 G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 2001), 3rd ed. a22 G. Huang, L. Deng and M. G. Payne, Phys. Rev. E. 72 (2005) 016617. a23 J. Dilley, P. Nisbet-Jones, B. W. Shore and A. Kuhn, Phys. Rev. A 85 (2012) 023834. a24 Y. Wu and L. Deng, Opt. Lett. 29 (2004) 2064. a25 L. G. Si, W. X. Yang, X.-Y. Lü, X. Hao, and X. Yang, Phys. Rev. A 82 (2010) 013836. a26 Y. Chen, Z. Bai and G. Huang, Phys. Rev. A 91 (2014) 023835. a27 M. V. Gromovyi, V. I. Romanenko, S. Mieth, T. Halfmann and L. P. Yatsenko, Optics Commun.284 (2011) 5710. a28 G. Vemuri and K. V. Vasavada, Optics Commun. 129 (1996) 379. a29 V. V. Kozlov and J. H. Eberly, Optics Commun. 179 (2000) 85. a30 G. Buica and T. Nakajima, Optics Commun. 332 (2014) 59. a31 Y. Chen, Z. Chen and G. Huang, Phys. Rev. A 89 (2015) 023820. a32 M. D. Lukin, S. F. Yelin and M. Fleischhauer, Phys. Rev. Lett. 84 (2000) 4232. a33 M. Fleischhauer, S. F. Yelin and M. D. Lukin, Optics. comm. 179 (2000) 395. a34 M. D. Lukin, M. Fleischhauer, M. O. Scully and V. L. Velichansky, Opt. Lett. 23 (1998) 295. a35 C. K. Law and H. J. Kimble, J. Mod. Opt. 44 (1997) 2067. a36 A. M. Dikandé, Phys. Rev. A 81 (2010) 013821. a37 A. M. Dikandé, J. Opt. 13 (2011) 035203. a38 D. Jr. Fandio Jubgang, A. M. Dikandé and A. Sunda-Meya, Phys. Rev. A 92 (2015) 053850. a39 D. Jr. Fandio Jubgang and A. M. Dikandé, J. Opt. Soc. Am. B 34 (2017) 2721. a40 S. Petrosyan and Y. Malakyan, Phys. Rev. A 88 (2013) 063817. a41 R. Dicke, Phys. Rev. 93 (1994) 99. a42 C. Ottaviani, S. Rebi´c, D. Vitali and P. Tombesi, Eur. Phys. J. D 40 (2006) 281. a43 M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions" (Dover Publications), 1965. a44B. A. Malomed, Phys. Rev. B 38 (1988) 9242. a45 C. Cohen-Tannoudji and S. Reynaud, J. Phys. B 10 (1977) 2311. | http://arxiv.org/abs/1706.09023v1 | {
"authors": [
"Davis D. M. Welakuh",
"Alain M. Dikande"
],
"categories": [
"physics.optics",
"nlin.PS"
],
"primary_category": "physics.optics",
"published": "20170627192721",
"title": "Storage and retrieval of time-entangled soliton trains in a three-level atom system coupled to an optical cavity"
} |
Hybrid Photonic Loss Resilient Entanglement Swapping Ryan C. Parker^1, Jaewoo Joo^2,3, Mohsen Razavi^2, Timothy P. Spiller^1=========================================================================== ^1York Centre for Quantum Technologies, Department of Physics, University of York, York, YO10 5DD, U.K.^2School of Electronic and Electrical Engineering, University of Leeds, Leeds, LS2 9JT, U.K.^3School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea We propose a scheme of loss resilient entanglement swapping between two distant parties in lossy optical fibre. In this scheme, Alice and Bob each begin with a pair of entangled non-classical states; these "hybrid states" of light are entangled discrete variable (Fock state) and continuous variable (coherent state) pairs. The continuous variable halves of each of these pairs are sent through lossy optical fibre to a middle location, where these states are then mixed (using a 50:50 beam-splitter) and measured. The detection scheme we use is to measure one of these modes via vacuum detection, and to measure the other mode using homodyne detection. In this work we show that the Bell state |Φ^+⟩=(|00⟩+|11⟩)/√(2) can theoretically be produced following this scheme with high fidelity and entanglement, even when allowing for a small amount of loss. It can be shown that there is an optimal amplitude value (α) of the coherent state when allowing for such loss. We also investigate the realistic circumstance when the loss is not balanced in the propagating modes. We demonstrate that a small amount of loss mismatch does not destroy the overall entanglement, thus demonstrating the physical practicality of this protocol. 2 § INTRODUCTION Distributing entanglement over long distances is a key enabler for quantum communications to be realised on a worldwide scale. Entanglement is an invaluable resource in quantum key distribution<cit.>, quantum secret sharing <cit.> and quantum teleportation <cit.>. Entanglement swapping is performed by two distant parties (Alice and Bob), that each possess a pair of entangled states (modes AB and CD respectively). If they each send one of their systems (B and D) to a central location, a suitable joint measurement entangles the remaining systems (A and C) that Alice and Bob still possess, thus the name entanglement swapping <cit.>. Entanglement swapping (ES) in this way is analagous to a quantum teleportation scheme, where modes B and D are teleported to modes A and C respectively as a result of the joint measurement of modes B and D <cit.>. Currently, ES protocols suffer from sending quantum signals through an optical fibre which introduces decoherence and photon loss <cit.>. Mitigating against this issue takes ES protocols closer to practical implementation, with increased potential for application in quantum repeater <cit.> and quantum relay <cit.> schemes. Furthermore, ES is a perfectly viable method of potentially realising truly long distance quantum communications <cit.> and has recently been demonstrated at a distance of 100 km using optical fibre and time-bin entangled photon-pairs <cit.>, and also at telecom wavelengths with high efficiency <cit.>. ES was initially proposed using discrete variable (DV) states <cit.>, and was shown experimentally using polarised photons <cit.> and vacuum-one-photon quantum states <cit.>. However, as a result of detector inefficiencies lowering success probability (a Bell-State measurement is bounded by 1/2 when using only linear optical elements <cit.>), ES events occur rarely when using only DVs. Research then began on the use of continuous variables (CVs) for ES <cit.>, and was first performed experimentally in 2004 <cit.>. Photonic coherent states work well for ES based on CV states, as coherent states are typically more resilient to photon losses <cit.>. In this paper we investigate the use of entangled hybrid states for application in an ES protocol. These hybrid states of light are entangled discrete and continuous variable quantum states. Hybrid states of light are particularly effective for ES schemes, and have been used in experimental proofs using squeezed states as the CV part<cit.> and also coherent states <cit.>. The DV part uses as basis states the vacuum and single photon Fock (number) states, and the CV part uses the basis states of nearly orthogonal coherent states. This paper is organised as follows. In Section <ref> we introduce the ES protocol used in this work, as well as the detection methods used. In Section <ref> we introduce unequal lossy modes, and parametrise a value for this loss mismatch. In Section <ref> we show that the subsequent entanglement shared by Alice and Bob is not severely damaged when allowing for unequal lossy modes, and show that high levels of fidelity and entanglement can be reached. Our conclusions are given in Section <ref>. § ENTANGLEMENT SWAPPING WITH LOSS §.§ Building Block Entangled States We here use a specific bipartite entangled state (which we refer to as a hybrid entangled state), which has a DV qubit in a spatial mode and a CV qubit in the other mode, as follows: |ψ_HE⟩_AB=1/√(2)(|0⟩_A|α⟩_B+|1⟩_A|-α⟩_B), where the subscript A and B can be replaced with C and D respectively to describe the other initial hybrid entangled state |ψ_HE⟩_CD. The mode B is assumed to be a photonic coherent state going through a photon-lossy channel, while the stationary mode A can be represented by various physical systems. For example, a hybrid photonic state has been recently demonstrated using a vacuum and a single-photon state for mode A in <cit.> as well as using polarisation photons in <cit.>. Instead of the vacuum and single-photon states, for solid-state stationary qubits, atomic ensembles and ions can be excellent candidates to create the state |ψ_HE⟩. For example, a non-maximally entangled state can be created in the hybrid fashion |ϕ_HE⟩_Ap≈√(1-p_c)|G⟩_A|0⟩_p+ √(p_c)|W⟩_A|1⟩_p, where p_c is the success probability of having a single photon in spatial mode p, and |G⟩ and |W⟩ are the hyperfine states of an atomic ensemble (or an ion) <cit.>. Then, we build the optical set-up so that the spatial mode B is matched with one of two directions of pair-wise parametric down-conversion photons from a nonlinear crystal, with efficiency η, while |α⟩_B is injected along the other direction of the pair of photons. |Ψ^tot⟩_ABp ≈√(1-η)√(1-p_c)|G⟩_A|0⟩_p|α⟩_B+√(η)√(1-p_c)|G⟩_A|1⟩_p a^+_B |α⟩_B+ √(1-η)√(p_c)|W⟩_A|1⟩_p|α⟩_B+ √(η)√(p_c)|W⟩_A|2⟩_p a^+_B |α⟩_B If we detect a single photon in mode p and p_c=η, the final state is approximately equal to (|G⟩_A|α⟩_B+|W⟩_A|-α⟩_B)/√(2) <cit.>. §.§ Lossy Modes We use a vacuum state in mode ε_Band ε_D (|0⟩_ε_B and |0⟩_ε_D respectively) as is standard for modelling loss using a beam-splitter, where the second input state is the propagating coherent state in mode B or D which is mixed with the vacuum state to imitate loss. The two lossy modes are mixed at a 50:50 beam-splitter (BS^1/2) and are then measured using a vacuum projection in mode B and a homodyne measurement in mode D. The full ES protocol, including loss, is shown in Fig. <ref>. Through this ES protocol, Alice and Bob can share an entangled pair of qubits that could then be used for quantum communications. In this work we show that this ES scheme is tolerant to low levels of loss in the propagating coherent states, resulting in Alice and Bob ultimately sharing a pair of highly entangled qubits of impressive fidelity when compared to the maximally entangled |Φ^+⟩ Bell state. In this ES scheme we have a beam-splitter (BS) of transmission T described by BS^T_i,j, where i and j are the modes that are mixed at the BS. Let us therefore assume that we have a loss rate of 1-T in a channel, modelled by mixing modes B and D with vacuum states in modes ε_B and ε_D respectively at separate BSs. Each hybrid entangled state is then given by |ψ_loss⟩_ABε_B=BS^T_B,ε_B|ψ_HE⟩_AB|0⟩_ε_B =1/√(2)(|0⟩_A|α√(T)⟩_B|α√(1-T)⟩_ε_B+ |1⟩_A|-α√(T)⟩_B|-α√(1-T)⟩_ε_B), where the hybrid entangled quantum state is given in Eq. <ref>. Note that Eq. <ref> is identical for modelling loss in mode D, using a vacuum state in mode ε_D. After accounting for loss as described above, we then mix the two propagating lossy modes at a 50:50 BS. Mixing two coherent states with a (generalised) BS of transmission t is given by BS^t_B,D |α⟩_B|β⟩_D= |α√(t)-β√(1-t)⟩_B|α√(1-t)+β√(t)⟩_D, where, α and β are complex numbers. In this protocol we mix coherent states of the same amplitude using a 50:50 BS, therefore t=1/2. §.§ Detection Methods For successful ES, we measure mode D via (perfect) balanced homodyne detection, and mode B by a vacuum measurement. It was found that if two homodyne measurements are performed on modes B and D, then the resultant quantum state is a superposition of all possible 2 qubit strings, which is a product state and is therefore undesirable as an outcome for this protocol. A generalised scheme of balanced homodyne detection consists of one 50:50 BS, a strong coherent field |β e^iθ⟩ of amplitude β (where β is real) and two photodetectors; the probe mode (mode D) is combined at a BS with the strong coherent field (local oscillator) of equal frequency, and photodetection is then used to measure the outputs <cit.>. If we perform homodyne detection on an input signal in mode B_1 and the coherent field is injected in mode B_2, then the operator BS^1/2_B_1,B_2 mixes the input state and the coherent field, as shown in Fig. <ref>. The intensity difference (photon number difference) between the two photodetectors (D_B_1 and D_B_2) can be calculated using the two mode operator Î_B_1-B_2=b̂^†_1b̂_2+b̂^†_2b̂_1, with creation and annihilation operator denoted by b̂^†_i and b̂_i respectively, in mode B_i. It therefore follows that, Î_B_1-B_2=2β⟨x̂_θ⟩, where, x̂_θ=1/2(b̂_1e^-iθ+b̂^†_1e^iθ) <cit.>, β is the amplitude of the strong coherent field injected in mode B_2, and the phase of the quadrature x̂_θ is given by the phase of this local oscillator. The probability amplitude of a homodyne measurement on an arbitrary coherent state |α e^iφ⟩ can be described by projecting with an x̂_θ eigenstate, where x̂_θ|x_θ⟩=x_θ|x_θ⟩, for real α <cit.>: ⟨x_θ|α e^iφ|=⟩1/π^1/4exp[-1/2(x_θ)^2+√(2)e^i(φ-θ)α x_θ -1/2e^2i(φ-θ)α^2-1/2α^2], where the subscript on x_θ is indicative of the angle in which the homodyne measurement is performed. In this protocol specifically we will theoretically measure mode D using homodyne detection in the θ=π/2 plane; if we measure in this plane then we are not able to distinguish between the two remaining states (|00⟩_AC and |11⟩_AC), thus leaving them entangled, whereas if one were to measure in the θ=0 plane then these states are distinguishable, which would destroy any entanglement. §.§ Entanglement Swapping with Equal Lossy Modes Measuring a vacuum in mode B and performing homodyne detection in mode D results in the following state, which shows the entangled pair of qubits shared by Alice and Bob after carrying out this protocol in its entirety (prior to tracing out the lossy modes): |ψ_loss⟩_ACε_Bε_D=Ne^(T-1)|α|^2∑_n,m=0^∞(α√(1-T))^n+m/√(n)!√(m)!|n⟩_ε_B|m⟩_ε_D.(e^-2iα x_π/2√(T)|00⟩_AC+(-1)^n+me^2iα x_π/2√(T)|11⟩_AC+e^-T|α|^2((-1)^m|01⟩_AC+(-1)^n|10⟩_AC)), where, N is a normalisation coefficient, and the lossy modes (ε_B and ε_D) are summed over |n⟩_ε_B and |m⟩_ε_D respectively (using the Fock (number) state basis representation of a coherent state, |α⟩=e^-|α|^2/2∑_n=0^∞α^n/√(n)!|n⟩ <cit.>). If one sets the amplitude of the coherent state as T|α|^2>>1 in Eq. <ref>, then the resultant state contains only the diagonal |00⟩_AC and |11⟩_AC terms, as the off-diagonal |01⟩_AC and |10⟩_AC terms are rapidly exponentially dampened by the exponent e^-T|α|^2. After tracing out the lossy modes, and taking these limits of T|α|^2>>1, the resultant density matrix from this quantum state is ρ_AC≈ 1/2e^2(T-1)|α|^2∑_n,m=0^∞((T-1)α^2)^n+m/n!m!.[|00⟩_AC⟨00|+|11⟩_AC⟨11|+(-1)^n+me^4iα x_π/2√(T)|11⟩_AC⟨00|+(-1)^n+me^-4iα x_π/2√(T)|00⟩_AC⟨11|]. Note that the phase factors in Eq. <ref> are known phase factors, set by the measurement outcome x_π/2. These can either be corrected through local operations feeding forward the measurement result, or simply carried through the protocol and dealt with in subsequent post-processing. It will be shown in Section <ref> that the entanglement negativity, fidelity and linear entropy of ρ_AC, with respect to the maximally entangled Bell State |Φ^+⟩=1/√(2)(|00⟩+|11⟩), is optimal for a specific value of the amplitude (α) of the coherent states that propagate through the lossy modes. § ENTANGLEMENT SWAPPING WITH UNEQUAL LOSSY MODES It is important to consider the case of unequal lossy modes in this protocol; in reality the beam-splitters used to mimic lossy optical fibres will not be absolutely equal, the resultant states that are emitted will have different transmission (T) values. However, we show here that the entanglement shared between Alice and Bob after performing ES is not significantly damaged if we consider unequal loss. Firstly, we denote this loss mismatch variable as δ, and we parametrise the transmission in each lossy mode as T_B→T and T_D→T-δ where, like T, δ can only take a value between 0 and 1. In general δ will be a small, positive mismatch to avoid T_D exceeding unity. Performing an analogous derivation to that used to reach Eq. <ref>, and applying the above parametrisation gives |ψ^loss⟩_ACε_Bε_D=Ne^(T-δ/2-1)|α|^2. ∑_n,m=0^∞(α√(1-T))^n(α√(1-T+δ))^m/√(n)!√(m)! |n⟩_ε_B|m⟩_ε_D. (e^-|α𝒯_-|^2/4e^-𝒯_+iα x_π/2 |00⟩_AC +(-1)^me^-|α𝒯_+|^2/4e^-𝒯_-iα x_π/2 |01⟩_AC +(-1)^ne^-|α𝒯_+|^2/4e^𝒯_-iα x_π/2 |10⟩_AC +(-1)^n+me^-|α𝒯_-|^2/4e^𝒯_+iα x_π/2 |11⟩_AC), where, T_±=(√(T)±√(T-δ)).As an example here we consider the case where a system is set up for matched loss (1-T) but there is a small, unknown mismatch. This can be calculated by taking an average over a distribution of δ. To find the averaged density matrix (ρ_AC) of the state |ψ^loss⟩_ACε_Bε_D, for some width in the distribution of the loss mismatch δ, which we label as Δ, we must integrate the density matrix ρ_AC(δ,T,α) over all positive values of δ (where ρ_AC(δ,T,α)=|ψ^loss⟩_ACε_Bε_D⟨ψ^loss|). The distribution of the loss mismatch is a one-sided (positive) Gaussian curve, and so the integral is of the form ρ_AC≡∫_0^∞f(δ,Δ)ρ_AC(δ,T,α)dδ, where, f(δ,Δ)=√(2/πΔ^2)e^-δ^2/2Δ^2 and √(2/πΔ^2) is the normalisation of the function. We will show in the next section that this averaged density matrix provides a high level of entanglement for an optimum α value when considering low levels of loss, and unequal loss in modes ε_B and ε_D. We note that equation (10) could be used directly to model a known mismatch between losses (for example due to unequal lengths of fibre), by choosing a specific value of δ. The results of such calculations show very similar impact on the entanglement to those we give for averaging with a width Δ, so we do not present these. § RESULTS AND DISCUSSION The fidelity (F) of the final density matrix (Eq. <ref>) can be determined using F(|σ⟩,ρ)=⟨σ|ρ|σ⟩, where, |σ⟩=|Φ^+⟩ is the maximally entangled (pure) Bell State, and ρ=ρ_AC is the final averaged density matrix <cit.>. Calculating the closeness (fidelity) of ρ_AC to |Φ^+⟩ confirms that for an optimum amplitude of the coherent state (α≈1.5), T=0.99 in mode B and T=0.98 in mode D, the final state shared by Alice and Bob is of impressive fidelity: F=0.93, where a fidelity of F=1 indicates that the states in comparison are indistinguishable. Intrinsically, the fidelity is unity for the no loss case, but what is promising here is that even for the case with non-negligible loss where T=0.95 in mode B and T=0.94 in D the fidelity reaches a maximum of 0.81 for α=1.3. To evaluate the level of entanglement shared between Alice and Bob after performing entanglement swapping, we apply an entanglement measure called negativity <cit.> using the following: E(ρ_AC)=-2∑_i^λ_i^-, where E denotes the entanglement value of ρ_AC (which can take a value between 0, for no entanglement, and 1, for maximal entanglement), and λ_i^- represents the negative eigenvalues of the partial transpose of the final density matrix, ρ_AC. We also calculate the linear entropy of ρ_AC using S_L(ρ_AC)=1-Tr[ρ_AC^2], where S_L is the linear entropy of the system, and can take any value between 0 (for a pure state) to S_L^max.=1-1/d, where d is the dimension of the system <cit.>. Therefore, in this case the maximum linear entropy will be 0.5, corresponding to a maximally mixed state. The following plots show entanglement and linear entropy as a function of the amplitude (α) of the coherent states used, with fixed transmission (T) values. 2Fig. <ref> shows that for no loss in the system (T=1,Δ=0), the entanglement reaches unity when α>1.7. For finite loss, when T<1, the optimum value of entanglement is approached and is clearly given by a sharp peak as a function of α (see Figs. <ref> and <ref>). Although this shifts to slightly lower values of α when considering higher levels of loss, there is always a clear peak in the plot at a specific amplitude. This is as a result of the analytical expression defining the shared state between Alice and Bob (Eq. <ref>), where the off-diagonal states (with the exception of |00⟩_AC⟨11| and |11⟩_AC⟨00|) are dampened when α>>0. This therefore reduces the entanglement, and also explains why the plots tail off at higher amplitudes for finite T.This is a key point of this paper: to have an optimum α value means that for a practical demonstration of this protocol an experimentalist would know the level of loss that can be tolerated, given the amplitude of the coherent state they have prepared. Furthermore, this optimum value itself is desirable - an amplitude of 2 is not large, but importantly it also is not too close to a vacuum state as to be indistinguishable. Equally, were the amplitudes of the coherent states to be closer to 0 then there is the possibility that these states will overlap at the vacuum, therefore making the superposition of |α⟩_D and |-α⟩_D indistinguishable in a homodyne measurement. Again, this further proves the possibility of performing this protocol experimentally, as a coherent state of this kind of amplitude can be prepared experimentally.When T=0.95, Fig. <ref> shows that even when considering high levels of loss for unequal lossy modes (Δ=0.01) the entanglement value is 0.63 for α≈1.3. Although thestate shared by Alice and Bob is not highly entangled in this case it is nonetheless still useful as a proof-of-principle experiment of this particular entanglement swapping protocol. What is promising in this protocol is that in Fig. <ref>, for a transmission of T=0.99 in one mode and T=0.98 in the other (Δ=0.01) the maximum entanglement value is 0.87, for α≈1.5; these levels of loss are likely to be the most realistic case for a practical implementation of this protocol, and although the entanglement is slightly lessened as a result of this loss, there do still exist methods of increasing entanglement, such as entanglement purification schemes <cit.>. The linear entropy plots compliment the plots of entanglement as a function of α perfectly: it is clear from comparing linear entropy and entanglement plots of the same transmission value that as entanglement increases as function of α, the linear entropy decreases for the same amplitude. What is also worth noting is that in all linear entropy plots, the case where we have significant differences in the lossy modes (Δ=0.1) gives the plots that show the highest level of entropy in the system. This of course arises from the unequal lossy modes causing the overall quantum state shared by Alice and Bob to be more mixed, which in turn is confirmed by the entanglement plots showing lower levels of entanglement for Δ=0.1. Another important quantity to evaluate is the success probability of the protocol. Here we focus on the success probability of the vacuum projection (in mode B) in this ES scheme. Clearly what is of interest is the success probability where the entanglement peaks as a function of the coherent state amplitude α. Calculation of this success probability shows that it is unity for the case of very small α, but drops rapidly and plateaus at 1/2 at the same value of α where the entanglement plots peak (α≈1.5). What is promising here is that the success probability does not decrease as T drops from 1 to 0.95. Furthermore, the loss mismatch does not reduce the success probability in the regime of small α, and only drops to less than 1/2 when α>3, for a significant mismatch in loss (Δ=0.1). Note that as we are assuming a perfect homodyne detection scheme the success probability will inherently be unity in this case. Investigating imperfect homodyne detection will be interesting as future work. § CONCLUSIONS Crucial to this scheme is that the measurements outlined in Section <ref> must be performed specifically as stated (that is, a vacuum projection in mode B and a homodyne detection in D). In doing so, one can theoretically achieve high levels of entanglement for low levels of photon loss. There are three key points to this paper which are worth summarising once more: * Having unequal loss does not significantly impact the entanglement and fidelity values, and the protocol is actually fairly resilient to this * We can reach optimum entanglement, fidelity and linear entropy for a specific value of the amplitude (α) of the propagating coherent states * The most realistic (practical) case is a transmission of T=0.99, and a loss mismatch of Δ=0.01, resulting in an impressive entanglement value of 0.87 for α≈1.5 This work highlights the usefulness of entangled optical hybrid states of light, and shows that the continuous variable part of this hybrid state is particularly resilient to low levels of photon losses. Furthermore, if applied with a suitable entanglement purification scheme <cit.>, this protocol has the potential to be implemented as part of a full quantum repeater protocol. Under the assumption of small losses in a channel, the ES protocol could also be used for entangling two distant superconducting qubits. These can be entangled because the state |ψ_HE⟩ can easily be created between a superconducting qubit and a coherent state inside a superconducting circuit <cit.>. Further work includes investigating cat states (coherent state superpositions) as the propagating continuous variable in the hybrid state, and also investigating the impact of imperfect homodyne detection to this entanglement swapping protocol. § ACKNOWLEDGEMENTS We acknowledge support from EPSRC (EP/M013472/1). J.J. acknowledges support from the KIST Institutional Program (Project No. 2E26680-16-P025). 20 Bennett1984 C. H. Bennett and G. Brassard: In Proc. 1984 IEEE Int. Conf. on Comp., Sys. and Sig. Proc., 175-179 (1984) Ekert1991 A. K. Ekert: Phys. Rev. Lett., 67, 661 (1991) Braunstein2012 S. L. Braunstein and S. Pirandola: Phys. Rev. Lett., 108, 130502 (2015) Gottesman2000 D. Gottesman: Phys. Rev. A, 61, 042311 (2000) Tittel2001 W. Tittel, H. Zbinden and N. Gisin: Phys. Rev. A, 63, 042301 (2001) Bennett1993 C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. K. Wootters: Phys. Rev. Lett., 70, 1895 (1993) Braunstein1998 S. L. Braunstein and H. J. Kimble: Phys. Rev. Lett., 80, 869-872 (1998) Zukowski1993 M. Zukowski, A. Zeilinger, M. A. Horne and A. K. Ekert: Phys. Rev. Lett, 71, 4287-4290 (1993) Bouwmeester2000 D. Bouwmeester, A. K. Ekert and A. Zeilinger: The Physics of Quantum Information, Springer-Verlag Berlin Heidelberg (2000) Gisin2002 N. Gisin, G. Ribordy, W. Tittel and H. Zbinden: Rev. Mod. Phys, 74, 145-195 (2002) Jin2015 R.-B. Jin, M. Takeoka, U. Takagi, R. Shimizu and M. Sasaki: Sci. Rep., 5 (2015) Boone2015 K. Boone, J.-P. Bourgoin, E. Meyer-Scott, K. Heshami, T. Jennewein and C. Simon: Phys. Rev. A, 91, 052325 (2015) Jones2016 C. Jones, D. Kim, M. T. Rakher, P. G. Kwiat and T. D. Ladd: New. Jour. of Phys., 18, 083015 (2016) Zwerger2016 M. Zwerger, B. P. Lanyon, T. E. Northup, C. A. Muschik, W. Dür and N. Sangouard: Quant. Sci. Technol., 2, 044001 (2016) Riedmatten2004 H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, D. Collins and N. Gisin: Phys. Rev. Lett., 92, 047904 (2004) Collins2005 D. Collins, N. Gisin and H. de Riedmatten: Jour. of Mod. Opt., 52, 735-753 (2005) Ngah2014 L. A. Ngah, V. D'Auria, L. Labonte, O. Alibart and S. Tanzilli: In IEEE 16th Int. Conf. Trans. Opt. Net., 1-3 (2014) Khalique2014 A. Khalique and B. C. Sanders: Phys. Rev. A, 90, 032304 (2014) Aspelmeyer2003 M. Aspelmeer, T. Jennewein, M. Pfennigbauer, W. R. Leeb and A. Zeilinger: IEEE Jour. of Sel. Top. in Quant. Elec., 9, 1541-1551 (2003) Riedmatten2005 H. de Riedmatten, I. Marcikic, J. A. W. Van Houwelingen, W. Tittel, H. Zbinden and N. Gisin: Phys. Rev. A, 71, 050302 (2005) Sun2017 Q.-C. Sun et. al.: arXiv preprint arXiv:1704.03960 (2017) Pan1998 J.-W. Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger: Phys. Rev. Lett., 80, 3891-3894 (1998) Pan2001 J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs and A. Zeilinger: Phys. Rev. Lett., 86, 4435-4438 (2001) Sciarrino2002 F. Sciarrino, E. Lombardi, G. Milani and F. de Martini: Phys. Rev. A, 66, 024309 (2002) Calsamiglia2001 J. Calsamiglia and N. Lütkenhaus: App. Phys. B: Las. and Opt., 72, 67-71 (2001) Ralph1999 R. E. S. Polkinghorne and T. C. Ralph: Phys. Rev. Lett., 83, 2095 (1999) Abdi2004 M. Abdi, S. Pirandola, P. Tombesi and D. Vitali: Phys. Rev. A, 89, 022331 (2014) Ban2004 M. Ban: Jour. of Phys. A: Math. and Gen., 37, L385 (2004) Jia2004 X. Jia, X. Su, Q. Pan, J. Gao, C. Xie and K. Peng: Phys. Rev. Lett., 93, 250503 (2004) Joo2016 Y. Lim, J. Joo, T. P. Spiller and H. Jeong: Phys. Rev. A, 94, 062337 (2016) Takeda2015 S. Takeda, M. Fuwa, P. van Loock and A. Furusawa: Phys. Rev. Lett., 114, 100501 (2015) Brask2010 J. B. Brask, I. Rigas, E. S. Polzik, U. L. Andersen and A. S. Sørensen: Phys. Rev. Lett., 105, 160501 (2010) Jeong14 H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph and M. Bellini: Nat. Phot., 8, 564-569 (2014) Laurat14 O. Morin, K. Huang, J. Liu, H. le Jeannic, C. Fabre and J. Laurat: Nat. Phot., 8, 570-574 (2014) DLCZ L.-M. Duan, M. D. Lukin, J. I. Cirac and P, Zoller: Nat., 414, 413-418 (2001) Moonjoo B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt and T. E. Northup: Phys. Rev. Lett., 114, 023602 (2015) Moonjoo1 A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter and T. E. Northup: Nat., 485, 482-485 (2012) Scully1997 M. Scully and M. Zubairy: Quantum Optics, Cambridge University Press (1997) Gerry2005 C. C. Gerry and P. L. Knight: Introductory Quantum Optics, Cambridge University Press (2005) Gardiner2004 C. Gardiner and P. Zoller: Quantum Noise, Springer-Verlag Berlin Heidelberg (2004) Jozsa1994 R. Jozsa: Jour. of Mod. Opt, 41, 2315-2323 (1994) Vidal2002 G. Vidal and R. F. Werner: Phys. Rev. A, 65, 032314 (2002) Manfredi2000 G. Manfredi and M. R. Feix: Phys. Rev. E, 62,4665 (2000) Bennett1996 C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters: Phys. Rev. A, 54, 3824 (1996) Christoph2001 J.-W. Pan, C. Simon, C̆aslav Brukner and A. Zeilinger: Nat., 410, 1067-1070 (2001) Sheng2014 Y.-B. Sheng and L. Zhou: Las. Phys. Lett., 11, 085203 (2014) Sheng2013 Y.-B. Sheng, L. Zhou and G.-L. Long: Phys. Rev. A, 88, 022302 (2013) Xiang2017 Z.-L. Xiang, M. Zhang, L. Jiang and P. Rabl: Phys. Rev. X, 7, 011035 (2017) | http://arxiv.org/abs/1706.08492v2 | {
"authors": [
"Ryan C. Parker",
"Jaewoo Joo",
"Mohsen Razavi",
"Timothy P. Spiller"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170626173157",
"title": "Hybrid Photonic Loss Resilient Entanglement Swapping"
} |
Large-scale Datasets: Faces with Partial Occlusions and Pose Variations in the Wild Tarik Alafif1, Zeyad Hailat2, Melih Aslan3 and Xuewen Chen4 Computer Science Department, Wayne State UniversityDetroit, MI, USA 48120Email: [email protected], [email protected], [email protected], [email protected] 25, 2017 ================================================================================================================================================================================================================================================= Face detection methods have relied on face datasets for training. However, existing face datasets tend to be in small scales for face learning in both constrained and unconstrained environments. In this paper, we first introduce our large-scale image datasets, Large-scale Labeled Face (LSLF) and noisy Large-scale Labeled Non-face (LSLNF). Our LSLF dataset consists of a large number of unconstrained multi-view and partially occluded faces. The faces have many variations in color and grayscale, image quality, image resolution, image illumination, image background, image illusion, human face, cartoon face, facial expression, light and severe partial facial occlusion, make up, gender, age, and race. Many of these faces are partially occluded with accessories such as tattoos, hats, glasses, sunglasses, hands, hair, beards, scarves, microphones, or other objects or persons. The LSLF dataset is currently the largest labeled face image dataset in the literature in terms of the number of labeled images and the number of individuals compared to other existing labeled face image datasets. Second, we introduce our CrowedFaces and CrowedNonFaces image datasets. The crowedFaces and CrowedNonFaces datasets include faces and non-faces images from crowed scenes. These datasets essentially aim for researchers to provide a large number of training examples with many variations for large scale face learning and face recognition tasks.§ INTRODUCTIONFace detection methods have relied on face datasets for training. However, existing face image datasets tend to be in small scales for face learning in both constrained and unconstrained environments. In order to obtain magnificent performance in face detection systems, it require a large number of unconstrained face and non-face training examples with many variations from real life scenes for training. Unconstrained face training examples should have variations in color and grayscale, quality, resolution, illumination, background, illusion, human face, cartoon face, facial expression, light and severe partial facial occlusion, make up, gender, age, and race. Hence, we introduce our large-scale image datasets for faces and non-faces extracted from the wild in the unconstrained environments. The proposed datasets are used for training LSDL face detector <cit.>. The datasets are publicly available at <http://discovery.cs.wayne.edu/lab_website/lsdl/> for research and non-commercial use only. The remainder of this paper is organized as follows. In Section II, we review the existing the labeled face images datasets. In Section III, we first introduce the collection mechanism for obtaining our large-scale image datasets, Large-scale Labeled Face (LSLF) dataset and Large-scale Labeled Non-face (LSLNF) dataset. The LSLF dataset is compared with the existing labeled face image datasets. Then, we introduce our CrowedFaces and CrowedNonFaces image datasets. Finally, we conclude our work in Section IV.§ RELATED WORKExisting labeled face datasets have been available for different goals, but they tend to be in small-scales for face learning in both constrained and unconstrained environments. These datasets are briefly overviewed as follows: Labeled Faces in the Wild (LFW) Dataset <cit.>. The LFW dataset contains of 13,749 labeld face images for 5,749 individuals. The images were automatically labeled. This dataset was collected from web news articles. The face images in LFW dataset have variations in color and grayscale, near frontal pose, lighting, resolution, quality, age, gender, unbalanced race, accessory, partial occlusion, make up, and background. The dataset size is 179 MB. It is publicly available for downloading.WebV-Cele Dataset <cit.>. The WebV-Cele dataset contains of 649,001 face images for 2,427 individuals. Only 42,118 face images are manually labeled. The face images were collected from YouTube videos. The images have variations in color and grayscale, quality, resolution, pose, illumination, background, human face, facial expression, partial occlusion, make up, gender, and race. The dataset is available upon request.CAS-PEAL Dataset <cit.>. The CAS-PEAL dataset contains of 99,594 manually labeled face images for 1,040 individuals. The face images were collected from a studio in constrained environment. The images have variations in color and grayscale , expression, lighting, pose, and accessory. This dataset only contains Chinese faces. The dataset size is nearly 26.6 GB. The dataset is available upon request.Face Recognition Grand Challenge (FRGC) Dataset <cit.>. The FRGC dataset contains of nearly 50,000 manually labeled face images for 466 individuals. The face images were collected from a studio in constrained environment. The images have variations in color, lighting, expression, background, unbalanced race, 3D scan, and image sequence. The dataset size is 3.1 MB. The dataset is available upon request.Multi-PIE Dataset <cit.>. The Multi-PIE dataset contains of 755,370 manually labeled face images for 337 individuals. The face images were collected from a studio in constrained environment. The images have variations in color, resolution, pose, illumination, and expression. The dataset size is 308 MB. The dataset is commercial.FERET Dataset <cit.>. The FERET dataset contains of 14,126 manually labeled face images for 1,199 individuals. The face images were collected from a studio in constrained environment. The images have variations in color, pose, and illumination. The dataset is available upon request.Extended Yale B Dataset <cit.>. The Extended Yale B dataset contains of 16,128 face images for 28 individuals. The face images were collected from a studio in constrained environment. The images have variations in grayscale, pose, and illumination. The dataset is available upon request.§ OUR DATASETSIn this section, we first describe the methodology for obtaining our LSLF dataset and noisy LSLNF dataset from the wild. Second, we introduce our CrowedFaces and CrowedNonFaces datasets. The datasets are briefly explained as follows: §.§ LSLF and LSLNF DatasetsWe first used Wikipedia and other online resources to collect 11,690 popular names for individuals from many countries all over the world. The individual names include celebrities in many categories such as politics, sports, journalism, movies, arts, and educations. Most individual names consist in our datasets are associated with the individual’s first name then last name or vice versa while the rest of the names are famous nick names or popular single names. A sample from individual names is shown in Figure <ref>.We employ a systematic procedure consisting of five phases to obtain noisy LSLF and LSLNF image datasets as shown in Figure <ref>. The five phases consist of automatic YouTube video links crawler, automatic YouTube video downloader, automatic framing extraction, Viola and Jones (VJ) face detection <cit.>, and automatic face and non-face screening. The phases are briefly explained as follows:§.§.§ Automatic YouTube video links crawlerIn this phase, we developed a YouTube video links crawler to read the collected individual names and automatically retrieve and collect YouTube video links for each individual. The individuals’ YouTube video links still can be retrieved even If the ordering of the names is different. The links are associated and written for each labeled individual name in a text file. A sample from YouTube video links is shown in Figure <ref>.§.§.§ Automatic YouTube video downloaderAfter retrieving YouTube video links using our automatic video links crawler, we developed a YouTube video downloader to automatically download the YouTube video links belonging to each individual. The labeled videos were stored to corresponding individual's labeled name. The total number of downloaded labeled videos is 129,435 videos with a total size of 2.96 TB.All the labeled videos were stored in mp4 format for 11,690 individuals where individuals have a minimum of 1 video and a maximum of 68 videos. Our labeled YouTube video dataset has a larger number of labeled videos per person and a larger number of people among other existing labeled video datasets such as YouTube Faces <cit.> and WebV-Cele <cit.>. A sample from YouTube videos is shown in Figure <ref>.§.§.§ Automatic frame extractionAfter downloading the labeled YouTube videos, we randomly selected and extracted a number of frames per video. Following this procedure, we obtained 5,033,177 frames with a total size of 178 GB. All the frames were stored in JPEG format and labeled for 11,690 individuals where individuals have a minimum of 34 frames and a maximum of 2,716 frames. The average number of frames per individual is 430. A sample from the extracted frames is shown in Figure <ref>. §.§.§ VJ face detectionAfter extracting and storing the labeled frames, we automatically applied the VJ face detector to all labeled frames to detect faces. By applying VJ face detection, many face and non-face examples were detected due to a drawback of the VJ face detector that results in many false positives. Each detected example was automatically labeled and associated with the same name as used for the individuals. The number of labeled images is 9,750,456 with a total size of 14.4 GB. All the images were stored in JPEG format for 11,690 individuals where individuals have a minimum of 3 images and a maximum of 6,697 images. The average number of detected images per individual is 834. A sample of images from the results of VJ face detection is shown in Figure <ref>. C>X§.§.§ Automatic face and non-face image screening In this phase, we automatically separated out face and non-face examples as much as possible from the results of the VJ face detection using several automatic screenings. For automatic image screenings, several trained classification models for facial parts were applied to the results of the VJ face detection phase.After completing this screening phase, we obtained the LSLF and LSLNF datasets with slight noise. Here, the noise referred to is the false positives resulted after classification. To be specific, LSLF contains about 1.7% non-faces while LSLNF contains about 10% faces. After completing the screening phase, individuals with 0 images are automatically removed from both datasets. Therefore, our LSLF dataset consists of 1,217,185 labeled face images for 11,478 individuals with about 1.7% noise. These images are stored in JPEG format with a total size of 3.42 GB. Individuals have a minimum of 1 face image and a maximum of 1,177 face images. The average number of face images per individual is 106. On the other hand, our noisy LSLNF dataset consists of 3,468,430 labeled none-face images for 11,682 individuals with about 10% noise. These images are stored in JPEG format with a total size of 3.16 GB. Individuals have a minimum of 1 non-face image and a maximum of 2,282 non-face images. The average number of non-face images per individual is 296. After obtaining the noisy LSLF dataset, we manually removed the noise from it. Therefore, our LSLF dataset consists of 1,195,976 labeled face images for 11,459 individuals. These images are stored in JPEG format with a total size of 5.36 GB. Individuals have a minimum of 1 face image and a maximum of 1,157 face images. The average number of face images per individual is 104. Each image is automatically named as (PersonName_VideoNumber_FrameNumber_ImageNuumber) and stored in the related individual folder. Image samples from the LSLF and noisy LSLNF datasets are shown in Figure <ref>.Our LSLF dataset consists of multi-view faces. Many of these faces have frontal and near frontal poses. These faces have large image variations in color and grayscale, image quality, image resolution, image illumination, image background, image illusion, human face, cartoon face, facial expression, light and severe partial facial occlusion, make up, gender, age, and race. Also, our LSLF dataset has a broad distribution of races from different parts of the world. Many of our face images are partially occluded with accessories such as tattoos, hats, glasses, sunglasses, hands, hair, beards, scarves, microphones, or other objects or persons. These factors essentially make our dataset great for large scale face learning and face recognition tasks. To the best of our knowledge, our LSLF dataset is the largest labeled face image dataset in the literature in terms of the number of labeled images and the number of individuals compared to other existing face image datasets <cit.>. A brief comparison is made in Table <ref>.§.§ CrowdFaces And CrowdNonFaces DatasetsWe introduce our two other datasets extracted from the wild, CrowdFaces and CrowdNonFaces datasets. The objective of these datasets is to obtain multi-view blurred and non-blurred faces, multi-view partially occluded faces, and non-faces to be used for the training. For obtaining these datasets, we manually selected and downloaded thirty YouTube videos that include crowd scenes in streets, sport games, religious gatherings, parties, street fights, and courts. Partially occluded faces are occluded by overlapping objects such as faces, hands, bodies, hats, masks, hair, etc. An overlapping sliding window is applied to these scenes at all locations with different scales to extract non-faces, multi-view blurred and non-blurred faces, and multi-view partially occluded faces. The sliding window starts at size 16 x 16 and increases by 1.5 scaling factor until the window size is no larger than the frame. We manually curated 10,049 faces and 31,662 non-faces at different scales from these sub-windows. Our partially occluded face images include light and severe partially occluded faces. A sample of these images is shown in Figure <ref>.§ CONCLUSIONIn this paper, we first introduce our large-scale image datasets, LSLF and noisy LSLNF. Our LSLF dataset consists of a large number of unconstrained multi-view and partially occluded faces. The faces have many variations in color and grayscale, quality, resolution, illumination, background, illusion, human face, cartoon face, facial expression, light and severe partial facial occlusion, make up, gender, age, and race. Many of the face images are partially occluded with accessories such as tattoos, hats, glasses, sunglasses, hands, hair, beards, scarves, microphones, or other objects or persons. The LSLF dataset is currently the largest labeled face image dataset in the literature in terms of the number of labeled images and the number of individuals compared to other existing face image datasets. Second, we introduced our CrowedFaces and CrowedNonFaces image datasets. The CrowedFaces and CrowedNonFaces datasets include faces and non-faces images from crowed scenes. These datasets essentially aim for researchers to provide a large number of training examples with many variations for large scale face learning and face recognition tasks. ieeetran | http://arxiv.org/abs/1706.08690v1 | {
"authors": [
"Tarik Alafif",
"Zeyad Hailat",
"Melih Aslan",
"Xuewen Chen"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170627070451",
"title": "Large-scale Datasets: Faces with Partial Occlusions and Pose Variations in the Wild"
} |
firstpage–lastpage [NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ======================The SDSS-III APOGEE DR12 is a unique resource to search for stars beyond the tidal radii of star clusters.We have examined the APOGEE DR12 database for new candidates of the young star cluster Palomar 1, a system with previously reported tidal tails (). The APOGEE ASPCAP database includes spectra and stellar parameters for two known members of Pal 1 (Stars I and II), however these do not agree with the stellar parameters determined from optical spectra by <cit.>.We find that the APOGEE analysis of these two stars is strongly affected by the known persistence problem (; ). By re-examining the individual visits, and removing the blue (and sometimes green) APOGEE detector spectra affected by persistence, then we find excellent agreement in a re-analysis of the combined spectra. These methods are applied to another five stars in the APOGEE field with similar radial velocities and metallicities as those of Pal 1. Only one of these new candidates, Star F, may be a member located in the tidal tail based on its heliocentric radial velocity, metallicity, and chemistry. The other four candidates are not well aligned with the tidal tails, and comparison to the Besançon model() suggests that they are more likely to be non-members, i.e.part of the Galactic halo.This APOGEE field could be re-examined for other new candidates if the persistence problem can be removed from the APOGEE spectral database.stars: chemical abundances – techniques: spectroscopy – globular clusters: individual (Palomar 1) – globular clusters: tidal tail § INTRODUCTION Palomar 1 (Pal1) is an unusual globular cluster. It is young (4-6 Gyr; ) and it has a high metallicity ([Fe/H] = -0.6 ± 0.1; ; ); however, itis located 3.6 kpc above the Galactic plane, and 17.2 kpcfrom the Galactic Centre (, 2010 edition). <cit.> examined SDSS and HST photometric fields around Pal1,and detected a dispersed tidal tail extending up to 1^o (∼ 0.4 kpc, or ∼ 80 half-light radii) from eitherside of thecluster centre, with roughly as many stars in the tails as in the central cluster region. Examination of the chemical abundances of the stars in Pal1 can be used to studythe origin of this system. If Pal1 is a globular cluster that has been shredded, then its stars should show a Na-O anti-correlation (). However, if Pal1 is a captured stellar group from a dwarf galaxy, then it can be expected to show lower ratios of the α-elements (amongst other chemical signatures,e.g., see ; ; ).<cit.> determined the elemental abundances of five stars in Pal1fromhigh-resolution HDS Subaru spectroscopy.There was no evidence for aNa-O anti-correlation in the sample, and the [α/Fe] ratios were slightlylower than Galactic field stars at the same metallicity but only with 1σsignificance.These signatures do not favour either scenario for the originof Pal1; however, <cit.> also found high values of [Ba/Y] and [Eu/α]that indicate unique contributions of r-process elements in this system, which seem to differ from most Galactic stars. The physical properties of Pal1 more closely resemble those of young clusters associated with the Sgr stream (i.e. Pal12 and Ter7; ), or the intermediate-age clusters in the LMC (; ; ).Like Pal1, those clusters also have young ages determined from isochrone fitting (; ; ) and show lower [α/Fe] ratios for their metallicities (; ; ).Furthermore, neither Pal1, nor the other young halo clusters, show the sodium-oxygen anti-correlation that <cit.> have shown is typical of globular clusters in the Milky Way. Another interesting sparse and young cluster in the halo is Rup106. Like Pal1, Rup106 also has low [α/Fe] for its metallicity and no Na-O anti-correlation (). Rup106 is not associated with any stellar streams,unlike the Sgr clusters. However, Rup106 also shows low [La/Fe] and [Na/Fe], so does not appear to be directly linked to Pal1.Pal1 may also be linked to the Canis Major over-density based on its chemistry, e.g., high [Ba/Fe] and [La/Fe] (; ; ).If Pal1 is a tidally disrupted globular cluster, this makes it an excellentprobe of the shape of the Milky Way halo.Palomar 5 (Pal5), another low-mass, low-velocity dispersion globular cluster with more spectacular tidal tails, has been used to model the Galactic potential by <cit.>, <cit.> (2016), <cit.>, and <cit.>. Pal5 also shows gaps in the tidal tails that have been examined for constraintsonmini-halo substructure (; ). The tidal tails around Pal1 are much shorter.Characterizing this system further by identifying member stars in the tidal tails,or in a more extended envelope, could be used to better study the shape of the Milky Way halo and the origin and evolution of this cluster.In this paper, we examine the SDSS-APOGEE DR 12 database, which targeted Pal1 as partof its globular cluster ancillary data project. Our search for new members of Pal1required a critical and substantial re-examination of the individual visit spectraand data analysis techniques. In this paper, we present our target selection methods,and cleaning of the combined spectra to remove the persistence problem, and re-analysis of the stellar parameters using the FERRE pipeline. We compare the results with those from <cit.> and <cit.>, as well as with the Besançon model ().§ APOGEE DATA The Apache Point Observatory Galactic Evolution Experiment (APOGEE) is a high-resolution,high signal-to-noise infrared (IR) spectroscopic survey of over 100,000 red giant stars acrossthe full range of the Galactic bulge, bar, disk, and halo (). The survey was carried out at the 2.5-m Sloan Foundation Telescopein New Mexico, covering the wavelength range from 1.5 to 1.7 microns in the H band,with spectral resolution R = 22,500 ().The APOGEE Stellar Parameters and ChemicalAbundances Pipeline (ASPCAP) DR12 () is a data analysis pipeline that produces stellar parametersand abundances for 15 different elements(C, N, O, Na, Mg, Al, Si, S, K, Ca, Ti, V, Mn, Fe and Ni).APOGEE uses the same field size and target positioner as the Sloan Extension forGalactic Understanding and Exploration (SEGUE) of the Sloan Digial Sky Survey (SDSS). It uses a series of 7 squared degree tiles to sample the sky with 2" fibres that observe 300targets simultaneously.One of these tiles was centred on Pal1(RA =53.33^o & Dec =79.58^o, , 2010 edition) with fibers allocated to a variety of targets based on the colours of cool stars (see target selection for the APOGEE program by ). Foreground dwarfs are removedfrom our analysis, as well as objects that are unlikely to be associated with Pal1 based on their metallicity and radial velocity. These include objects with radial velocities outside of -75 ± 15 kms^-1 andmetallicities outside of -1.0 <[Fe/H] <-0.2(i.e., 4σ and 2σ ofthe values for confirmed Pal1 members respectively, e.g., Rosenberg 1998, to account for errors in the APOGEE metallicities and potential kinematic effects along the tidal tails). These targets are shown in Fig. <ref>, where 9% of the starsin this field may be associated with Pal1.Two of these are Stars I and II examined from optical spectra by <cit.>. To further select Pal1 members, we examine acolour-magnitude diagram (CMD) of stars in the central portion of Pal1 from HST ACS photometry (); see Fig. <ref>.Isochrones are generated from the Dartmouth Stellar Evolution Database () are included with agesof 4, 5 and 6 Gyrs, with the distance, reddening, and metallicity from <cit.>, and adopting [α/Fe]=0. However, the APOGEE target selection provides Gunn ugriz and JHK magnitudes of the targets (), requiring conversion to Johnson VI.We have adopted the calibration from Table 4 of <cit.> for Population I stars. [The uncertainties are determined in quadraturegiven the uncertainties for each color index listed in the APOGEE DR12 database and formulae by <cit.>.] The right panel in Fig. <ref> shows the CMD of 47 Tucand an isochrone generated from the Dartmouth Stellar Evolution Database () with an age of 12.2 Gyr.The distance and metallicity are from , with [α/Fe]=0.4 and E(B-V)=0 [The reddening for 47 Tuc of E(B-V)=0.055 from <cit.> does not fit the turn-off well. When no reddening is applied, the fit is better (a lower reddening was similarly found by , E(B-V)=0.03).]. Comparing the CMD of Pal1 to that of 47 Tuc in Fig. <ref> clearly shows that Pal1 is younger and more sparsely populated than a typical globular cluster.The V and I magnitudes from this transformation for Stars I and II are in good agreement with those from the <cit.>; see Table<ref>.An additional five stars (Stars D, E, F, G and H) with radial velocities and metallicities consistent with Pal1 were selected from near the isochrones. We examine the stellar properties of these additional five stars below.§ STARS I AND II The stellar parameters for Stars I and II are shown in Table <ref>, from the opticalanalysis by <cit.>, and the IR analysis of the APOGEE spectrathrough the ASPCAP pipeline. These two sets of results are in very poor agreement, with differences of ΔT_ eff∼ 1000 K and Δlog g ∼1.0,resulting in differences in Δ[Fe/H] ∼ -0.4. In order to understand these differences, the individualvisit spectra for these two stars are examined.There are 24 visits for Star I and 21 visitsfor Star II, with SNR > 6. We find a clear persistence problem in many of the spectra, in additional to some other effects such as poor flat fielding or telluric division problems, poor night sky line removal, and several cosmic ray hits. §.§ Removing Persistence Individual visits for Stars I and II were extracted from the APOGEE database. The alignment of each spectrum was compared to Arcturus, in order to check the radial velocity corrections. Each visit was then broken into the three wavelength regions corresponding to the blue, green, and red detectors. Some of APOGEE's detectors suffered from persistence, which is the contamination of a spectrum by remnants of the previous exposure. The persistence problem is worse on the blue chip (1.514-1.581 μ), see Fig. <ref>. We remove the portion of the spectrum coming from the blue chip detector for any visit that shows persistence. Occasionally it was also necessary to remove the green chip spectrum - we suspect that the green chip itself does not have the persistence problem, but that the data reduction processing of the visit induces a flat fielding problem when persistence is bad on the blue chip. After this process, the remaining spectra from each visit are co-added, i.e., only the non-persistent spectra from the blue, green, and red regions are kept for our analysis.The non-persistent regions of each visit were combined to create the full wavelength range visits, and the cleaned visits were median-combined using IRAF. The final combined spectra for Stars I and II tend to have fewer green spectra than red, and fewer blue than either. This results in a lower SNR for the green than red spectrum, and lowest SNR for the blue spectrum. These spectra were then normalized with a Legendre polynomial (order=8), followed by a k-sigma clipping routine (see ), and sky lines are removed.These steps are illustrated in Fig. <ref>. Since these stars are moderately metal-poor, we found this normalization method to be sufficient for our purposes, but we caution that this is not the same as that used by the ASPCAP pipeline. Stars G and H also have significant persistence on their spectra. We have cleaned them similar to Stars I and II. Stars D, E and F did not have significant persistence problems. These gave us an opportunity to use and test ASPCAP on the original spectra in the APOGEE database. In Fig. <ref>, a portion of the cleaned and combined spectra of our Pal1 members to thatof Arcturus are compared. APOGEE spectra have R=22,500 whereas the Arcturus spectrumfrom ()was convolved with a Gaussian profileto match the lower resolution and has R=24,000.Star G shows broader lines than Arcturus and the other spectra in our sample, which suggests that it is a dwarf star[ The newest APOGEE DR13 grids for dwarfs include rotation models and therefore log g of Star G is removed in the new data release, which supports our claim that Star G is a dwarf star.].In Fig. <ref>, the CN, OH, Mg I, Al I, Si I, and Fe I features in our candidate spectra are highlighted and compared to the Arcturus spectrum. Stars I and II exhibit weaker spectral lines for these species than Arcturus, which can be attributed to their higher surface temperatures. The aforementioned line broadening observed in Star G is present in thesespectral ranges as well. § NEW STELLAR ANALYSES We have carried out a new analysis for all of the stars that may be members of Pal1 based onthe DR12 data. This includes those stars that have a persistence problem, but also those that do not so that we treat the data for all of these objects in a similar way.New stellar parameters are determined,initially from optical and IR photometry using both the <cit.> and <cit.>,colour-temperature relationships.Temperatures and bolometric corrections are determined from theunweighted average of four colours: (B-V), (V-I), (V-K), and (J-K), adopting the metallicity and cluster distance for Pal1 from <cit.>. Reddening estimates are from <cit.>. Surface gravities are determined photometrically as in <cit.>,after adopting a cluster turn-off mass of MA=1.14M_⊙ () corresponding to its young age, such that: logg=4.44+log(MA)+4log(T_ eff/5790) +0.4(M_ bol-4.75) The T_ eff values determined from the two different color-temperaturecalibrations were in excellent agreement for all of the candidates, with the exception of Star F.For this one star, the temperatures differed by ΔT_ eff∼ 1200 K (see Table <ref>). The temperature from <cit.> is much higher, and inconsistent with the position of this star on the colour-magnitude diagram in Fig. <ref>; however, the position of Star F inFig. <ref> depends on a correct V magnitude, which has been flagged in the SDSS database. Without further information on the V magnitude of Star F, we consider both temperatures in the discussion below. The difference between the logg values for two different distance moduli from(1996, 2010 edition) and <cit.> is Δ logg ∼ 0.4, which causes only small to negligible differences in our abundance results. The APOGEE ASPCAP data analysis pipeline uses the least squares template fitting routine, FERRE (), which matches observed spectra to (renormalized) synthetic spectrafrom model atmospheres that have been run through the 1D, LTE, spectrum synthesis code ASSET(; ).FERRE simultaneously determines the stellar parameters, metallicities, and element abundance ratios for a given spectrum. We too have used FERRE[FERRE at Github: https://github.com/jobovy/apogee.] for metallicities and chemical abundances, once where FERRE determines the stellar parameters and a second time where we adopt our photometrically determinedstellar parameters (see Tables <ref>-<ref>). To match the observed spectra to the synthetic spectra, it was necessary to resample the observations to be on the same wavelength scale.This caused the observations to have a slightly lower resolution than the original visits, and the combined spectra hada slightly larger spectral range.This resulted in observations of a few additional absorptionlines (K, Mn) that that are not in the APOGEE DR12 database. For Stars I and II,Table <ref> shows that the photometric stellar parameters yieldchemical abundances and metallicities in excellent agreement with the optical analyses.This implies that persistence is a significant problem in the analysis of thesetwo stars in the DR12 data release (also see discussion of the DR13 data in Section 6.4). This further implies that the analysis of some stars in the APOGEE database can still be improved using the APOGEE spectra themselves.§ STELLAR ABUNDANCES The stellar parameters and chemical abundances for 10 elements have been redetermined in this paper for in a set of Pal1 members and candidates from persistence-cleanedAPOGEE spectra.The results are shown in Tables <ref>-<ref>, including the elements C, O,Mg, Al, Si, S, K, Ca, Mn, and Fe (see Table <ref> for log abundances of all detected lines).The abundance uncertainties are calculated in two ways. When fewer than four lines are available, the error is taken as the standard deviation in [Fe/H]. When there are more than four lines, the measurement error is taken as the standard deviationdivided by root of number of lines. For cases where either of these methods results in an error <0.1 dex, an error of 0.1 dex is adopted since the best synthetic fits have been determined by eye.A few elements require special notes:* Titanium: <cit.> show that the APOGEE (DR12) abundances do not reproduce the [Ti/Fe] trends seen for stars in the solar neighbourhood by <cit.>.This difference is not currently understood, and therefore the ASPCAP titanium lines are to be treated with caution. <cit.> suggested that the Ti line at 15837.8Å, which is not included in the set adopted by ASPCAP, can be considered reliable. We did not use this line in our FERRE estimates. * [α/Fe]:We estimate a mean [α/Fe] ratio by averaging the results for Mg, S, Si and Ca (not O due to the very noisy oxygen lines, and not Ti as discussed above). Overall, the chemical abundances of Stars I and IIare in a good agreement with the optical analysis by <cit.>.Three candidate stars (Stars D, E, and G) have stellarparameters typical of red giants and metallicities of [Fe/H]=-0.6, when determined from the photometric parameters. These values are similar to the members in the core of Pal1. On the other hand, the chemistry of Star H is sufficiently different that it is a likely non-member. Star F warrants special attention due to its position in the tidal tails of Pal1. Two temperatures have been determined from the color-temperature calibrations for this star, based on its photometric uncertainties (see Table <ref>).When the cooler temperature is examined,then its metallicity is significantly different from that of Pal1 such that it would be a non-member. However, if thehotter temperature is adopted, its stellar parameters are typical of a red giant,with a metallicity and chemical abundances that are similar to those of the members of Pal1. Furthermore, with the hotter temperature, then Star F has a low [α/Fe] that isconsistent with the other members of Pal1. Its high [Al/Fe], with slighly low [Mg/Fe], is unusual for a star in Pal1, unless Star F is, or has been contaminated by, an AGB star (e.g., ). § DISCUSSION Using the APOGEE database, we have re-examined the spectra for two known members of Pal1and five new candidate members that are well away from the central region of this cluster. For each member and candidate star, all visits were examined and the blue chips of the spectra with persistence removed, then recombined the clean visits (see section 3.1 for more details). A new stellar analysis has been conducted using FERRE.The results for the cleaned spectra of Stars I and II are in excellent agreement with the optical analysis by <cit.>, whereas the DR12 analyses based on the original spectra are not (see Table <ref>). The chemical abundance and stellar parameters of the candidates are shown in Table <ref>-<ref>.The estimated [α/Fe] ratios for Stars I and II arein good agreement with the optical results of <cit.>. The Na I lines are too weak or noisy in most of the spectra for reliable determinations of [Na/Fe], therefore we do not investigate the Na-O anticorrelation.§.§ Tidal Tails of Pal1 The position of the Pal1 candidates with respect to the tidal tails mapped out by <cit.> based on SDSS photometry are shown in Fig. <ref>. Their contour map is constructed from a probability-weighted star count map of Pal1 candidates from the CMD in the MSTO/MS region. The number of candidates per squarearcmin can be determined as 0.856 at the centre of Pal1 and 0.050 above the background region. The positions of Stars I, II, and D-H are shown relative to new isophots determined by M. Irwin from the same SDSS data in Fig. <ref>. A difference in the adopted bin sizes and isophot levels can suppress the apparent tidal features. Only one of these candidates, Star F, is coincident with the tidal tail found by <cit.>. Given the distance to Pal1 as 14.2 kpc from the Sun (), and the angular separationsof each star from its core, then the minimum distances of each star from the coreof Pal1 range from 220.8 pc (Star F) and 236.9 pc (Star H),to 294.6 pc (Star G), 319.8 pc (Star D), and 326.7 pc (Star E). For these stars to have reached these distances over the lifetime of this cluster (< 6 Gyr) would have required ejection velocities ≤1 kms^-1.These velocities are not particularly large, therefore it is possible that if stars escape from Pal1 then they could be lurking at these angular separations.§.§ Membership Probability AnalysisWe examined the Besançon model () of the stellar populations in the Galaxy to evaluate membership probability of the new Pal1 candidates. The number of stars in the smooth Galactic halo in the direction of Pal1 are estimated based on similar limits in magnitude, colour, radial velocity, and metallicity (Fig. <ref>). To extract this simulated dataset, we runthe model with the following selection criteria:* an H-band range of 7 to 13.8, comparable to the APOGEE target list. * a distance interval from 0 to 50 kpc, to include most of both foreground and background stars. * a 7 sq. deg. field of view, centred on Pal1 to match the SDSS field.* The APOGEE database flags all non-giant stars as dwarfs. To directly compare the Besançon results with APOGEE, MS, WDs and T Tauri stars were removed from our Besançon model and only giant stars were taken into account. These selection criteria result in 1124 total stars in the Besançon model. 129 (12%)stars have radial velocities (-75 ± 15 kms^-1) and metallicities ([Fe/H] = -0.6 ± 0.4) similar to our parameters for theAPOGEE search in the Pal1 field. These should be treated as field contaminants from the smooth halo distribution. In comparison,the SDSS/APOGEE Pal1 field contains 377 giants, of which 33 (9%) have radial velocities and metallicities similar to Pal1. Therefore, the Besançon model predicts a larger fraction of field contaminants (12%) than observedin the APOGEE Pal1 field (9%). This strongly implies that the APOGEE field is representative of the smooth halo, with noevidence for additional stars due to the Pal1 globular cluster.It should be noted that APOGEE's Pal1 field is subject to observational placements, particularly in the fibre limitations. These include(1) crowding in the centre of Pal1 where the bonafide members are located, (2) that not all red giants canbe observed simultaneously, and (3) that only 30% of the total number of good targets were observed.A Monte Carlo approach was also used to randomly examine the potential for extracting Pal1 members from APOGEE Pal1 field. This was done by selecting 30% of stars from Besançon model to account for the APOGEE selections. For each sampling run, the fraction of field stars with our search criteria for Pal1 radial velocities and metallicities (-75 ± 15 kms^-1 and [Fe/H] = -0.6 ± 0.4 dex, respectively) was calculated. 10000 runs were performed and the histogram of the distribution of corrected field contaminants is shown in Fig. <ref>. This histogram shows a well defined Gaussian distribution with a mean fraction of Pal1 contaminants of 0.12 ± 0.02. Considering that number of stars in the RV and [Fe/H] search criteria in the APOGEE Pal1 field yielded 33 out of 377 stars (or 9%), we find that this is consistent with the predicted estimate from our Monte Carlo sampling of the Besançon smooth halo.§.§ Binarity? The velocity variation of the candidates are examined to find any evidencefor a binarity, which could affect the stellar parameters analysis.The radial velocity variations for the two Pal1 members and for all ofour candidate stars are shown in Fig. <ref>. Note that the y-axis represent the RV scatter of the candidates and is the standard deviation of all visits for each target. <cit.> has analysed a Plate-to-Plate RV variation analysis for the APOGEE stars and found that the RVs in the APOGEE database are very stable as the rms scatter is σ = 0.044 kms^-1. They suggest that stars with RV scatter of greater than 1 kms^-1have uncertainty much larger than the typical uncertainties and are possibly in a binary system.Only Stars D and G show a scatter in their radial velocities well above the 2σ limit suggested by <cit.> for detecting binary systems. However, the RV scatter of Star II suggests that it too may be in a binary system. If so, the binary nature of this star does not seem to have affected either its optical analysis, nor our analysis of the corrected IR spectra, since the stellar parameters and chemical abundance ratios are in good agreementwith other stars in Pal1. It should be noted that the binarity of Star D, E, F and H cannot be conclusively established as the sample sizes of these stars are small.§.§ DR13 In the SDSS DR13 release, an attempt to unweight spectra with the persistence problem was established to improve the combined spectral analyses ().When we examine the DR13 database, two more objects could be added to our analysis; however, the results for Stars I and II are still significantly different from the optical results (see Table <ref>).We did not pursue the DR13 data release further. The persistence problem is indeed well named. § SUMMARY Two members of the unusual star cluster Pal1 have been observed in the APOGEE survey. Examination of their ASPCAP database results are in very poor agreement with previously determined optical analysis. We trace this problem to the known persistence problem that affects up to 30% of the spectra in the APOGEE database.By removing those spectra with persistence (and other reduction problems), we have re-analysed the cleaned spectra. Our new analyses for the APOGEE spectra of Stars I and II are in excellent agreement withthe optical analysis by <cit.>.One star, Star F, may be a member of Pal1, based on its heliocentric radial velocity, metallicity and chemical abundances, and location in the tidal tails.However, the temperature of this star is highly uncertain, and it may be (or be contaminated by)an AGB star.All other candidate members found in the APOGEE DR12 database appear tobe part of the smooth Galactic background.§ ACKNOWLEDGMENTSWe would like to thank the anonymous referee for the constructive comments and thorough review of this manuscript. F.J. and K.A.V. acknowledge funding for this work through the NSERC Discovery Grants program. C.M.S. acknowledges funding from the Kenilworth Fund of the New York Community Trust. mnras§ SPECTRAL LINES AND ABUNDANCES | http://arxiv.org/abs/1706.09074v1 | {
"authors": [
"Farbod Jahandar",
"Kim A. Venn",
"Matthew D. Shetrone",
"Mike Irwin",
"Jo Bovy",
"Charli M. Sakari",
"Collin L. Kielty",
"Ruth A. R. Digby",
"Peter M. Frinchaboy"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170627233601",
"title": "The Peculiar Globular Cluster Palomar 1 and Persistence in the SDSS-APOGEE Database"
} |
[1]plain theoremTheorem[section] lem[theorem]Lemma prop[theorem]Proposition cor[theorem]Corollary defn[theorem]Definition definition remarkexampleExample[section] remarkRemark[section] 0 1 ℳ [ ] [x y] 1vec[x y 1]:= Tr #1#1 #1#1 #1#1#1#1 #1 #1#1#1#1 #1#1 #1#1 #1#1 #1#1 #1#1 Semidefinite Programming and Nash Equilibria in Bimatrix Games Amir Ali Ahmadi and Jeffrey Zhang The authors are partially supported by the DARPA Young Faculty Award, the Young Investigator Award of the AFOSR, the CAREER Award of the NSF, the Google Faculty Award, and the Sloan Fellowship.======================================================================================================================================================================================================================================= We explore the power of semidefinite programming (SDP) for finding additive ϵ-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium (NE) problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact NE can be recovered. We show that for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most two and ϵ close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a 5/11-NE can be recovered for any game, or a 1/3-NE for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any NE, and testing whether there exists a NE where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.Keywords: Nash equilibria, semidefinite programming, correlated equilibria.§ INTRODUCTION A bimatrix game is a game between two players (referred to in this paper as players A and B) defined by a pair of m × n payoff matrices A and B. Let _m and _n denote the m-dimensional and n-dimensional simplices_m = {x ∈ℝ^m |x_i ≥ 0, ∀ i, ∑_i = 1^m x_i = 1}, _n = {y ∈ℝ^n |y_i ≥ 0, ∀ i, ∑_i = 1^n y_i = 1}.These form the strategy spaces of player A and player B respectively.For a strategy pair (x,y)∈_m ×_n, the payoff received by player A (resp. player B) is x^TAy (resp. x^TBy). In particular, if the players pick vertices i and j of their respective simplices (also called pure strategies), their payoffs will be A_i,j and B_i,j. One of the prevailing solution concepts for bimatrix games is the notion of Nash equilibrium. At such an equilibrium, the players are playing mutual best responses, i.e., a payoff maximizing strategy against the opposing player's strategy. In our notation, a Nash equilibrium for the game (A,B) is a pair of strategies (x^*, y^*)∈_m ×_n such thatx^*TAy^* ≥ x^TAy^*, ∀ x∈_m,andx^*TBy^* ≥ x^*TBy, ∀ y∈_n.[In this paper we assume that all entries of A and B are between 0 and 1, and argue at the beginning of Section <ref> why this is without loss of generality for the purpose of computing Nash equilibria.]Nash <cit.> proved that for any bimatrix game, such pairs of strategies exist (in fact his result more generally applies to games with a finite number of players and a finite number of pure strategies). While existence of these equilibria is guaranteed, finding them is believed to be a computationally intractable problem. More precisely, a result of Daskalakis, Goldberg, and Papadimitriou <cit.> implies that computing Nash equilibria is PPAD-complete (see <cit.> for a definition) even when the number of players is 3. This result was later improved by Chen and Deng <cit.> who showed the same hardness result for bimatrix games.These results motivate the notion of an approximate Nash equilibrium, a solution concept in which players receive payoffs “close” to their best response payoffs. More precisely, a pair of strategies (x^*, y^*)∈_m ×_n is an (additive) ϵ-Nash equilibrium for the game (A,B) ifx^*TAy^* ≥ x^TAy^* - ϵ, ∀ x∈_m,andx^*TBy^* ≥ x^*TBy- ϵ, ∀ y∈_n. [There are also other important notions of approximate Nash equilibria, such as ϵ-well-supported Nash equilibria <cit.> and relative approximate Nash equilibria <cit.> which are not considered in this paper.] Note that when ϵ=0, (x^*,y^*) form an exact Nash equilibrium, and hence it is of interest to find ϵ-Nash equilibria with ϵ small. Unfortunately, approximation of Nash equilibria has also proved to be computationally difficult. Cheng, Deng, and Teng have shown in <cit.> that, unless PPAD ⊆ P, there cannot be a fully polynomial-time approximation scheme for computing Nash equilibria in bimatrix games. There have, however, been a series of constant factor approximation algorithms for this problem <cit.>, with the current best producing a .3393 approximation via an algorithm by Tsaknakis and Spirakis <cit.>.We remark that there are exponential-time algorithms for computing Nash equilibria, such as the Lemke-Howson algorithm <cit.>. There are also certain subclasses of the problem which can be solved in polynomial time, the most notable example being the case of zero-sum games (i.e. when B=-A). This problem was shown to be solvable via linear programming by Dantzig <cit.>, and later shown to be polynomially equivalent to linear programming by Adler <cit.>. Aside from computation of Nash equilibria, there are a number of related decision questions which are of economic interest but unfortunately NP-hard. Examples include deciding whether a player's payoff exceeds a certain threshold in some Nash equilibrium, deciding whether a game has a unique Nash equilibrium, or testing whether there exists a Nash equilibrium where a particular set of strategies is not played <cit.>.Our focus in this paper is on understanding the power of semidefinite programming[The unfamiliar reader is referred to <cit.> for the theory of SDPs and a description of polynomial-time algorithms for them based on interior point methods.] (SDP) for finding approximate Nash equilibria in bimatrix games or providing certificates for related decision questions. The goal is not to develop a competitive solver, but rather to analyze the algorithmic power of SDP when applied to basic problems around computation of Nash equilibria.Semidefinite programming relaxations have been analyzed in depth in areas such as combinatorial optimization <cit.>, <cit.> and systems theory <cit.>, but not to such an extent in game theory. To our knowledge, the appearance of SDP in the game theory literature includes the work of Stein for exchangeable equilibria in symmetric games <cit.>, of Parrilo on zero-sum polynomial games <cit.>, of Parrilo and Shah for zero-sum stochastic games <cit.>, and of Laraki and Lasserre for semialgebraic min-max problems in static and dynamic games <cit.>. §.§ Organization and Contributions of the PaperIn Section <ref>, we formulate the problem of finding a Nash equilibrium in a bimatrix game as a nonconvex quadratically constrained quadratic program and pose a natural SDP relaxation for it. In Section <ref>, we show that our SDP is exact when the game is strictly competitive (see Definition <ref>). In Section <ref>, we design two continuous but nonconvex objective functions for our SDP whose global minima coincide with rank-1 solutions. We provide a heuristic based on iterative linearization for minimizing both objective functions. We show empirically that these approaches produce ϵ very close to zero (on average in the order of 10^-3). In Section <ref>, we establish a number of bounds on the quality of the approximate Nash equilibria that can be read off of feasible solutions to our SDP. In Theorems <ref>, <ref>, and <ref>, we show that when the SDP returns solutions which are “close” to rank-1, the resulting strategies have have small ϵ. We then present an improved analysis in the rank-2 case which shows how one can recover a 5/11-Nash equilibrium from the SDP solution (Theorem <ref>). We further prove that for symmetric games (i.e., when B = A^T), a 1/3-Nash equilibrium can be recovered in the rank-2 case (Theorem <ref>). We do not currently know of a polynomial-time algorithm for finding rank-2 solutions to our SDP. If such an algorithm were found, it would, together with our analysis, improve the best known approximation bound for symmetric games. In Section <ref>, we show how our SDP formulation can be used to provide certificates for certain (NP-hard) questions of economic interest about Nash equilibria in symmetric games. These are the problems of testing whether the maximum welfare achievable under any symmetric Nash equilibrium exceeds some threshold, and whether a set of strategies is played in every symmetric Nash equilibrium. In Section <ref>, we show that the SDP analyzed in this paper dominates the first level of the Lasserre hierarchy (Proposition <ref>). Some directions for future research are discussed in Section <ref>. The four appendices of the paper add some numerical and technical details. § THE FORMULATION OF OUR SDP RELAXATION In this section we present an SDP relaxation for the problem of finding Nash equilibria in bimatrix games. This is done after a straightforward reformulation of the problem as a nonconvex quadratically constrained quadratic program. Throughout the paper the following notation is used. * A_i, refers to the i-th row of a matrix A. * A_,j refers to the j-th column of a matrix A. * e_i refers to the elementary vector (0,…, 0, 1, 0, …, 0)^T with the 1 being in position i. * _k refers to the k-dimensional simplex. * 1_m refers to the m-dimensional vector of one's. * 0_m refers to the m-dimensional vector of zero's. * J_m,n refers to the m × n matrix of one's. * A ≽ 0 denotes that the matrix A is positive semidefinite (psd), i.e., has nonnegative eigenvalues. * A ≥ 0 denotes that the matrix A is nonnegative, i.e., has nonnegative entries. * A ≽ B denotes that A - B ≽ 0. * 𝕊^k × k denotes the set of symmetric k × k matrices. * (A) denotes the trace of a matrix A, i.e., the sum of its diagonal elements. * A ⊗ B denotes the Kronecker product of matrices A and B. * vec(M) denotes the vectorized version of a matrix M. * For a vector v, diag(v) denotes the diagonal matrix with v on its diagonal. For a square matrix M, diag(M) denotes the vector containing its diagonal entries. We also assume that all entries of the payoff matrices A and B are between 0 and 1. This can be done without loss of generality because Nash equilibria are invariant under certain affine transformations in the payoffs. In particular, the games (A,B) and (cA+dJ_m× n, eB+fJ_m × n) have the same Nash equilibria for any scalars c,d,e, and f, with c and e positive. This is becausex^*TAy≥ x^TAy ⇔ c(x^*TAy^*) + d≥ c(x^TAy^*)+d ⇔ c(x^*TAy^*) + d(x^*TJ_m× ny^*)≥ c(x^TAy^*)+d(x^TJ_m× ny^*) ⇔ x^*T(cA+dJ_m× n)y^*≥ x^T(cA+dJ_m× n)yIdentical reasoning applies for player B. §.§ Nash Equilibria as Solutions to Quadratic ProgramsRecall the definition of a Nash equilibrium from Section <ref>. An equivalent characterizaiton is that a strategy pair (x^*, y^*)∈_m ×_n is a Nash equilibrium for the game (A,B) if and only ifx^*TAy^* ≥ e_i^TAy^*, ∀ i ∈{1,…, m},x^*TBy^* ≥ x^*TBe_i, ∀ i ∈{1,…, n}.The equivalence can be seen by noting that because the payoff from playing any mixed strategy is a convex combination of payoffs from playing pure strategies, there is always a pure strategy best response to the other player's strategy.We now treat the Nash problem as the following quadratic programming (QP) feasibility problem: x ∈ℝ^m, y ∈ℝ^nmin 0 subject to x^TAy ≥ e_i^TAy, ∀ i∈{1,…, m}, x^TBy ≥ x^TBe_j, ∀ j∈{1,…, n}, x_i ≥ 0, ∀ i∈{1,…, m}, y_i ≥ 0, ∀ j∈{1,…, n},∑_i=1^m x_i = 1,∑_i=1^n y_i = 1. Similarly, a pair of strategies x^* ∈_m and y^* ∈_n form an ϵ-Nash equilibrium for the game (A,B) if and only ifx^*TAy^* ≥ e_i^TAy^*-ϵ, ∀ i ∈{1,…, m}, x^*TBy^* ≥ x^*TBe_i-ϵ, ∀ i ∈{1,…, n}. Observe that any pair of simplex vectors (x, y) is an ϵ-Nash equilibrium for the game (A,B) for any ϵ that satisfiesϵ≥max{imax e_i^TAy - x^TAy, imax x^TBe_i - x^TBy}. We use the following notation throughout the paper: * ϵ_A(x,y) := imax e_i^TAy - x^TAy, * ϵ_B(x,y) := imax x^TBe_i - x^TBy, * ϵ(x,y) := max{ϵ_A(x,y), ϵ_B(x,y)},and the function parameters are later omitted if they are clear from the context.§.§ SDP Relaxation The QP formulation in (<ref>) lends itself to a natural SDP relaxation. We define a matrixℳ:= [ X PZ Y],and an augmented matrixℳ' := [ X P xZ Y yx y 1],with X ∈ S^m× m, Z ∈ℝ^n× m, Y ∈ S^n× n, x ∈ℝ^m, y ∈ℝ^n and P = Z^T. The SDP relaxation can then be expressed as SDP1 ℳ' ∈𝕊^m+n+1, m+n+1min 0 subject to(AZ) ≥ e_i^TAy, ∀ i∈{1,…, m}, (BZ) ≥ x^TBe_j, ∀ j∈{1,…, n},∑_i=1^m x_i = 1, ∑_i=1^n y_i = 1, ' ≥ 0, '_m+n+1,m+n+1=1,ℳ' ≽ 0.We refer to the constraints (<ref>) and (<ref>) as the relaxed Nash constraints and the constraints (<ref>) and (<ref>) as the unity constraints. This SDP is motivated by the following observation.Let ' be any rank-1 feasible solution to <ref>. Then the vectors x and y from its last column constitute a Nash equilibrium for the game (A,B).We know that x and y are in the simplex from the constraints (<ref>), (<ref>), and (<ref>).If the matrix ℳ' is rank-1, then it takes the formxx^T xy^T x yx^T yy^T yx^T y^T 1=1vec 1vec^T.Then, from the relaxed Nash constraints we have that e_i^TAy ≤(AZ) = (Ayx^T) = (x^TAy) = x^TAy, x^TAe_i ≤(BZ) = (Byx^T) = (x^TBy) = x^TBy.The claim now follows from the characterization given in (<ref>). Because a Nash equilibrium always exists, there will always be a matrix of the form (<ref>) which is feasible to <ref>. Thus we can disregard any concerns about <ref> being feasible, even when we add valid inequalities to it in Section <ref>. It is intuitive to note that the submatrix P=Z^T of the matrix ' corresponds to a probability distribution over the strategies, and that seeking a rank-1 solution to our SDP can be interpreted as making P a product distribution. The following theorem shows that <ref> is a weak relaxation and stresses the necessity of additional valid constraints. Consider a bimatrix game with payoff matrices bounded in [0,1]. Then for any two vectors x ∈_m and y ∈_n, there exists a feasible solution ' to <ref> with 1vec as its last column. Consider any x,y,γ > 0, and the matrix1vec1vec^T + γ J_m+n,m+n0_m+n0_m+n^T 0.This matrix is the sum of two nonnegative psd matrices and is hence nonnegative and psd. By assumption x and y are in the simplex, and so constraints (<ref>)-(<ref>) of <ref> are satisfied. To check that constraints (<ref>) and (<ref>) hold, note that since A and B are nonnegative, as long as the matrices A and B are not the zero matrices, the quantities (AZ) and (BZ) will become arbitrarily large as γ increases. Since e_i^TAy and x^TBe_i are bounded by 1 by assumption, we will have that constraints (<ref>) and (<ref>) hold for γ large enough. In the case where A or B is the zero matrix, the Nash constraints are trivially satisfied for the respective player.§.§ Valid Inequalities In this subsection, we introduce a number of valid inequalities to improve upon the SDP relaxation in <ref>. These inequalities are justified by being valid if the matrix returned by the SDP is rank-1. The terminology we introduce here to refer to these constraints is used throughout the paper. Constraints (<ref>) and (<ref>) will be referred to as the row inequalities, and (<ref>) and (<ref>) will be referred to as the correlated equilibrium inequalities. Any rank-1 solution ' to <ref> must satisfy the following: ∑_j=1^m X_i,j = ∑_j=1^nP_i,j = x_i, ∀ i ∈{1,…, m}, ∑_j=1^n Y_i,j = ∑_j=1^m Z_i,j = y_i, ∀ i ∈{1,…, n}. ∑_j=1^n A_i,jP_i,j≥∑_j=1^n A_k,jP_i,j, ∀ i,k ∈{1,…, m}, ∑_j=1^m B_j,iP_j,i≥∑_j=1^m B_j,kP_j,i, ∀ i,k ∈{1,…, n}. Recall from (<ref>) that if ' is rank-1, it is of the form[xx^T xy^T x yx^T yy^T yx^T y^T 1] =1vec1vec^T.To show (<ref>), observe that ∑_j=1^m X_i,j = ∑_j=1^m x_ix_j = x_i, ∀ i ∈{1,…, m}. An identical argument works for the remaining matrices P,Z, and Y. To show (<ref>) and (<ref>), observe that a pair (x, y) is a Nash equilibrium if and only if ∀ i, x_i > 0 ⇒ e_i^TAy = x^TAy = imax e_i^TAy, ∀ i, y_i > 0 ⇒ x^TBe_i = x^TBy = imax x^TBe_i. This is because the Nash conditions require that x^TAy, a convex combination of e_i^TAy, be at least e_i^TAy for all i. Indeed, if x_i > 0 but e_i^TAy < x^TAy, the convex combination must be less than imax x^TAy. For each i such that x_i = 0 or y_i = 0, inequalities (<ref>) and (<ref>) reduce to 0 ≥ 0, so we only need to consider strategies played with positive probability. Observe that if ' is rank-1, then ∑_j =1^n A_i,jP_i,j = x_i∑_j =1^n A_i,jy_j = x_ie_i^TAy ≥ x_ie_k^TAy = ∑_j =1^nA_k,jP_i,j, ∀ i,k ∑_j =1^m B_j,iP_j,i = y_i∑_j =1^mB_j,ix_j = y_i x^TBe_i ≥ y_ix^TBe_k = ∑_j =1^mB_j,iP_j,k, ∀ i,k.There are two ways to interpret the inequalities in (<ref>) and (<ref>): the first is as a relaxation of the constraint x_i(e_i^TAy - e_j^TAy) ≥ 0, ∀ i,j, which must hold since any strategy played with positive probability must give the best response payoff. The other interpretation is to have the distribution over outcomes defined by P be a correlated equilibrium <cit.>. This can be imposed by a set of linear constraints on the entries of P as explained next.Suppose the players have access to a public randomization device which prescribes a pure strategy to each of them (unknown to the other player). The distribution over the assignments can be given by a matrix P, where P_i,j is the probability that strategy i is assigned to player A and strategy j is assigned to player B. This distribution is a correlated equilibrium if both players have no incentive to deviate from the strategy prescribed, that is, if the prescribed pure strategies a and b satisfy∑_j=1^n A_i,jProb(b=j|a=i) ≥∑_j=1^n A_k,jProb(b=j|a=i), ∑_i=1^m B_i,jProb(a=i|b=j) ≥∑_i=1^m B_i,kProb(a=i|b=j). If we interpret the P submatrix in our SDP as the distribution over the assignments by the public device, then because of our row constraints, Prob(b=j|a=i)=P_i,j/x_i whenever x_i0 (otherwise the above inequalities are trivial). Similarly, P(a=i|b=j)=P_i,j/y_j for nonzero y_j. Observe now that the above two inequalities imply (<ref>) and (<ref>). Finally, note that every Nash equilibrium generates a correlated equilibrium, since if P is a product distribution given by xy^T, then Prob(b=j|a=i)=y_j and P(a=i|b=j)=x_i. §.§.§ Implied Inequalities In addition to those explicitly mentioned in the previous section, there are other natural valid inequalities which are omitted because they are implied by the ones we have already proposed. We give two examples of such inequalities in the next proposition. We refer to the constraints in (<ref>) below as the distribution constraints. The constraints in (<ref>) are the familiar McCormick inequalities <cit.> for box-constrained quadratic programming. Let zx y. Any rank-1 solution ' to <ref> must satisfy the following: ∑_i = 1^m ∑_j =1^m X_i,j = ∑_i =1^n ∑_j =1^m Z_i,j = ∑_i =1^n ∑_j =1^n Y_i,j = 1. _i,j ≤ z_i, ∀ i,j ∈{1,…,m+n}, _i,j + 1≥ z_i+z_j, ∀ i,j ∈{1,…,m+n}. The distribution constraints follow immediately from the row constraints (<ref>) and (<ref>), along with the unity constraints (<ref>) and (<ref>). The first McCormick inequality is immediate as a consequence of (<ref>) and (<ref>), as all entries ofare nonnegative. To see why the second inequality holds, consider whichever submatrix X, Y, P, or Z that contains _i,j. Suppose that this submatrix is, e.g., P. Then, since P is nonnegative, 0 ≤∑_k=1, ki^m∑_l = 1, lj^n P_k,l(<ref>)=∑_k=1, ki^m (x_k - P_k,j)(<ref>)= (1-x_i) - (y_j - P_i,j) = P_i,j+1-x_i-y_j. The same argument holds for the other submatrices, and this concludes the proof. §.§ Simplifying our SDP We observe that the row constraints (<ref>) and (<ref>) along with the correlated equilibrium constraints (<ref>) and (<ref>) imply the relaxed Nash constraints (<ref>) and (<ref>). Indeed, if we fix an index k ∈ {1, …, m}, then(AZ) = ∑_i=1^m ∑_j=1^n A_i,j P_i,j(<ref>)≥∑_i=1^m ∑_j=1^n A_k,j P_i,j≥∑_j=1^n A_k,j (∑_i=1^m P_i,j) (<ref>), P = Z^T≥∑_j=1^n A_k, j y_j = e_k^TAy.The proof for player B proceeds identically. Then, after collecting the valid inequalities and removing the relaxed Nash constraints, we arrive at an SDP given by SDP1' ' ∈ S^(m+n+1)×(m+n+1)min 0 subject to (<ref>)-(<ref>), (<ref>)-(<ref>).We make the observation that the last row and column of ' can be removed from this SDP, that is, there is a one-to-one correspondence between solutions to <ref> and those to the following SDP (where ℳ:= [ X PZ Y], with P = Z^T): SDP2 ∈ S^(m+n)×(m+n)min 0 subject to≽ 0,≥ 0,∑_i=1^n ∑_j=1^n P_i,j = 1,∑_j=1^m X_i,j = ∑_j=1^nP_i,j, ∀ i ∈{1,…, m},∑_j=1^n Y_i,j = ∑_j=1^m Z_i,j, ∀ i ∈{1,…, n},∑_j=1^n A_i,jP_i,j≥∑_j=1^n A_k,jP_i,j, ∀ i,k ∈{1,…, m},∑_j=1^m B_j,iP_j,i≥∑_j=1^m B_j,kP_j,i, ∀ i,k ∈{1,…, n}.Indeed, it is readily verified that the submatrixfrom any feasible solution ' to <ref> is feasible to <ref>. Conversely, letbe any feasible matrix to <ref>. Consider an eigendecomposition = ∑_i=1^k λ_i v_i v_i^T and let x y1_m+n/2. Then the matrix' x yx^T y^T 1= ∑_i=1^k λ_iv_i1_m+n^Tv_i/2v_i1_m+n^Tv_i/2 ^Tis easily seen to be feasible to <ref>.Given any feasible solutionto <ref>, observe that the submatrix P is a correlated equilibrium. We take our candidate approximate Nash equilibrium to be the pair x = P1_n and y = P^T1_m. If the correlated equilibrium P is rank-1, then the pair (x,y) so defined constitutes an exact Nash equilibrium. In Section <ref>, we will add certain objective functions to <ref> with the interpretation of searching for low-rank correlated equilibria.§ EXACTNESS FOR STRICTLY COMPETITIVE GAMES In this section, we show that <ref> recovers a Nash equilibrium for any zero-sum game, and that <ref> recovers a Nash equilibrium for any strictly competitive game (see Definition <ref> below). Both these notions represent games where the two players are in direct competition, but strictly competitive games are more general, and for example, allow both players to have nonnegative payoff matrices. These classes of games are solvable in polynomial time via linear programming. Nonetheless, it is reassuring to know that our SDPs recover these important special cases. A zero-sum game is a game in which the payoff matrices satisfy A = -B. For a zero-sum game, the vectors x and y from the last column of any feasible solution ' to <ref> constitute a Nash equilibrium. Recall that the relaxed Nash constraints (<ref>) and (<ref>) read (AZ) ≥ e_i^TAy, ∀ i ∈{1,…, m}, (BZ) ≥ x^TBe_j, ∀ j ∈{1,…, n}. Since B=-A, the latter statement is equivalent to (AZ) ≤ x^TAe_j, ∀ j ∈{1,…, n}. In conjunction these imply e_i^TAy ≤(AZ) ≤ x^TAe_j, ∀ i ∈{1,…, m}, j ∈{1,…, n}. We claim that any pair x ∈_m and y ∈_n which satisfies the above condition is a Nash equilibrium. To see that x^TAy ≥ e_i^TAy, ∀ i ∈{1,…, m}, observe that x^TAy is a convex combination of x^TAe_j, which are at least e_i^TAy by (<ref>). To see that x^TBy ≥ x^TBe_j ⇔ x^TAy ≤ x^TAe_j, ∀ j ∈{1,…, n}, observe that x^TAy is a convex combination of e_i^TAy, which are at most x^TAe_j by (<ref>). A game (A, B) is strictly competitive if for all x, x' ∈_m, y, y' ∈_n, x^TAy - x'^TAy' and x'^TBy' - x^TBy have the same sign. The interpretation of this definition is that if one player benefits from changing from one outcome to another, the other player must suffer. Adler, Daskalakis, and Papadimitriou show in <cit.> that the following much simpler characterization is equivalent. A game is strictly competitive if and only if there exist scalars c,d,e, and f, with c > 0, e > 0, such that cA+dJ_m × n = -eB + fJ_m × n. One can easily show that there exist strictly competitive games for which not all feasible solutions to <ref> have Nash equilibria as their last columns (see Theorem <ref>). However, we show that this is the case for <ref>. For a strictly competitive game, the vectors xP1_n and yP^T1_m from any feasible solutionto <ref> constitute a Nash equilibrium. To prove Theorem <ref> we need the following lemma, which shows that feasibility of a matrixin <ref> is invariant under certain transformations of A and B. Let c,d,e, and f be any set of scalars with c > 0 and e > 0. If a matrixis feasible to <ref> with input payoff matrices A and B, then it is also feasible to <ref> with input matrices cA+dJ_m × n and eB + fJ_m × n. It suffices to check that constraints (<ref>) and (<ref>) of <ref> still hold, as only the correlated equilibrium constraints use the matrices A and B. We only show that constraint (<ref>) still holds because the argument for constraint (<ref>) is identical.Note from the definition of x that for each i ∈{1,…, m}, x_i = ∑_j=1^n (J_m× n)_i,jP_i,j. To check that the correlated equilibrium constraints hold, observe that for scalars c>0,d, and for all i,k ∈{1,…, m},∑_j=1^n A_i,jP_i,j ≥∑_j=1^n A_k,jP_i,j ⇔ c∑_j=1^n A_i,jP_i,j + d ∑_j=1^n P_i,j ≥ c∑_j=1^n A_k,jP_i,j + d ∑_j=1^n P_i,j ⇔ c∑_j=1^n A_i,jP_i,j + d∑_j=1^n (J_m × n)_i,jP_i,j ≥ c∑_j=1^n A_k,jP_i,j + d∑_j=1^n (J_m × n)_k,jP_i,j ⇔∑_j=1^n (cA_i,j+dJ_m × n)_k,jP_i,j ≥∑_j=1^n (cA_i,j+dJ_m × n)_k,jP_i,j. Let A and B be the payoff matrices of the given strictly competitive game and letbe a feasible solution to <ref>. Since the game is strictly competitive, we know from Theorem <ref> that cA + dJ_m× n = -eB + fJ_m× n for some scalars c>0,e>0,d, f. Consider a new game with input matrices à = cA+dJ_m× n and B̃ = eB - fJ_m× n. By Lemma <ref>,is still feasible to <ref> with input matrices à and B̃. By the arguments in Section <ref>, the matrix ' x yx^T y^T 1 is feasible to <ref>, and hence also to <ref>. Now notice that since à = -B̃, Theorem <ref> implies that the vectors x and y in the last column form a Nash equilibrium to the game (Ã, B̃). Finally recall from the arguments at the beginning of Section <ref> that Nash equilibria are invariant to scaling and shifting of the payoff matrices, and hence (x, y) is a Nash equilibrium to the game (A, B). § ALGORITHMS FOR LOWERING RANK In this section, we present heuristics which aim to find low-rank solutions to <ref> and present some empirical results. Recall that our <ref> in Section <ref> did not have an objective function. Hence, we can encourage low-rank solutions by choosing certain objective functions, in particular the trace of the matrix , which is a general heuristic for minimizing the rank of symmetric matrices <cit.>.This simple objective function is already guaranteed to produce a rank-1 solution in the case of strictly competitive games (see Proposition <ref> below).For general games, however, one can design better objective functions in an iterative fashion (see Section <ref>).Notational Remark: For the remainder of this section, we will use the shorthand xP1_n and yP^T1_m, where P is the upper right submatrix of a feasible solutionto <ref>. For a strictly competitive game, any optimal solution to <ref> with () as the objective function must be rank-1. Let:=X PP^T Ybe a feasible solution to <ref>. In the case of strictly competitive games, from Theorem <ref> we know that that (x, y) is a Nash equilibrium. Then because the matrixis psd, from (<ref>) and an application of the Schur complement (see, e.g. <cit.>) to x yx^T y^T 1, we have that ≽^T. Hence, =xx^T xy^T yx^T yy^T+ 𝒫 for some psd matrix 𝒫 and the Nash equilibrium (x, y). Given this expression, the objective function () is then x^Tx + y^Ty + (𝒫). As (x,y) is a Nash equilibrium, the choice of 𝒫 = 0 results in a feasible solution. Since the zero matrix has the minimum possible trace among all psd matrices, the solution will be the rank-1 matrix ^T. If the row constraints and the nonnegativity constraints on X and Y are removed from <ref>, then this SDP with () as the objective function can be interpreted as searching for a minimum-rank correlated equilibrium P via the nuclear norm relaxation; see <cit.>.§.§ Linearization AlgorithmsThe algorithms we present in this section for minimzing the rank of the matrixin <ref> are based on iterative linearization of certain nonconvex objective functions. Motivated by the next proposition, we design two continuous (nonconvex) objective functions that, if minimized exactly, would guarantee rank-1 solutions. We will then linearize these functions iteratively. Let the matrices X and Y and vectors xP1_n and yP^T1_m be taken from a feasible solution to <ref>. Then the matrixis rank-1 if and only if X_i,i = x_i^2 and Y_i,i = y_i^2 for all i.Note that ifis rank-1, then it can be written as zz^T for some z ∈ℝ^m+n. The i-th diagonal entry in the X submatrix will then be equal toz_i^2 (<ref>)=1/4z_i^2(1_m+n^Tzz^T1_m+n) = (1/2_i,1_m+n)^2 (<ref>)= (P_i,1_n)^2 = x_i^2,where the second equality holds because _i,—the i-th row of —is z_iz^T. An analogous statement holds for the diagonal entries of Y, and hence the condition is necessary. To show sufficiency, let z. Sinceis psd, we have that _i,j≤√(_i,i_j,j), which implies _i,j≤ z_iz_j by the assumption of the proposition. Recall from the distribution constraint (<ref>) that ∑_i=1^m+n∑_j=1^m+n_i,j=4. Further, the same constraint along with the definitions of x and y imply that ∑_i=1^m+n z_i = 2, which means that ∑_i=1^m+n∑_j=1^m+n z_iz_j = 4. Hence in order to have the equality 4 = ∑_i=1^m+n∑_j=1^m+n_i,j≤∑_i=1^m+n∑_j=1^m+n z_iz_j = 4, we must have _i,j = z_iz_j for each i and j. Consequentlyis rank-1.We focus now on two nonconvex objectives that as a consequence of the above proposition would return rank-1 solutions:All optimal solutions to <ref> with the objective function ∑_i=1^m+n√(_i,i) or () - x^Tx - y^Ty are rank-1.We show that each of these objectives has a specific lower bound which is achieved if and only if the matrix is rank-1. Observe that since ≽^T, we have √(X_i,i)≥ x_i and √(Y_i,i)≥ y_i, and hence∑_i=1^m+n√(_i,i)≥∑_i=1^m x_i + ∑_i=1^n y_i= 2. Further note that() - ^T≥^T - ^T = 0. We can see that the lower bounds are achieved if and only if X_i,i = x_i^2 and Y_i,i = y_i^2 for all i, which by Proposition <ref> happens if and only ifis rank-1.We refer to our two objective functions in Proposition <ref> as the “square root objective” and the “diagonal gap objective” respectively. While these are both nonconvex, we will attempt to iteratively minimize them by linearizing them through a first order Taylor expansion. For example, at iteration k of the algorithm,∑_i=1^m+n√(_i,i^(k))≃∑_i=1^m+n√(_i,i^(k-1)) + 1/2√(_i,i^(k-1))(_i,i^(k) - _i,i^(k-1)).Note that for the purposes of minimization, this reduces to minimizing ∑_i=1^m+n1/√(_i,i^(k-1))_i,i^(k).In similar fashion, for the second objective function, at iteration k we can make the approximation()-^(k)T^(k)≃()-^(k-1)T^(k-1)T - 2^(k-1)T(^(k)-^(k-1)).Once again, for the purposes of minimization this reduces to minimizing ()-2^(k-1)T^(k). This approach then leads to the following two algorithms.[An algorithm similar to Algorithm <ref> is used in <cit.>.] Note that the first iteration of both algorithms uses the nuclear norm (i.e. trace) ofas the objective. The square root algorithm has the following property. Let ^(1), ^(2), … be the sequence of optimal matrices obtained from the square root algorithm. Then the sequence {∑_i=1^m+n√(_i,i^(k))} is nonincreasing and is lower bounded by two. If it reaches two at some iteration t, then the matrix ^(t) is rank-1. Observe that for any k>1, ∑_i=1^m+n√(_i,i^(k))≤1/2∑_i=1^m+n (_i,i^(k)/√(_i,i^(k-1))+√(_i,i^(k-1))) ≤1/2∑_i=1^m+n (_i,i^(k-1)/√(_i,i^(k-1))+√(_i,i^(k-1))) = ∑_i=1^m+n√(_i,i^(k-1)), where the first inequality follows from the arithmetic-mean-geometric-mean inequality, and the second follows from that _i,i^(k) is chosen to minimize ∑_i=1^m+n_i,i^(k)/√(_i,i^(k-1)) and hence achieves a no larger value than the feasible solution ^(k-1). This shows that the sequence is nonincreasing. The proof of Proposition <ref> already shows that the sequence is lower bounded by two, and Proposition <ref> itself shows that reaching two is sufficient to have the matrix be rank-1.The diagonal gap algorithm has the following property. Let ^(1), ^(2), … be the sequence of optimal matrices obtained from the diagonal gap algorithm. Then the sequence{(^(k))- ^(k)T^(k)} is nonincreasing and is lower bounded by zero. If it reaches zero at some iteration t, then the matrix ^(t) is rank-1. Observe that (^(k)) - ^(k)T^(k) ≤(^(k))-^(k)T^(k) + (^(k)- ^(k-1))^T(^(k)- ^(k-1))=(^(k))-2^(k)T^(k-1)+^(k-1)T^(k-1)≤(^(k-1))-2^(k-1)T^(k-1)+^(k-1)T^(k-1)=(^(k-1))-^(k-1)T^(k-1), where the second inequality follows from that ^(k) is chosen to minimize(^(k-1))-2^(k-1)T^(k-1)and hence achieves a no larger value than the feasible solution ^(k-1). This shows that the sequence is nonincreasing. The proof of Proposition <ref> already shows that the sequence is lower bounded by zero, and Proposition <ref> itself shows that reaching zero is sufficient to have the matrix be rank-1. We also invite the reader to also see Theorem <ref> in the next section which relates the objective value of the diagonal gap minimization algorithm and the quality of approximate Nash equilibria that the algorithm produces.§.§ Numerical ExperimentsWe tested Algorithms <ref> and <ref> on games coming from 100 randomly generated payoff matrices with entries bounded in [0,1] of varying sizes. Below is a table of statistics for 20 × 20 matrices; the data for the rest of the sizes can be found in Appendix <ref>.[The code that produced these results is publicly available at <aaa.princeton.edu/software>. The functioncomputes an approximate Nash equilibrium using one of our two algorithms as specified by the user.] We can see that our algorithms return approximate Nash equilibria with fairly low ϵ (recall the definition from Section <ref>). We ran 20 iterations of each algorithm on each game. Using the SDP solver of MOSEK <cit.>, each iteration takes on average under 4 seconds to solve on a standard personal machine with a 3.4 GHz processor and 16 GB of memory. The histograms below show the effect of increasing the number of iterations on lowering ϵ on 20 × 20 games. For both algorithms, there was a clear improvement of the ϵ by increasing the number of iterations.§ BOUNDS ON Ε FOR GENERAL GAMESSince the problem of computing a Nash equilibrium to an arbitrary bimatrix game is PPAD-complete, it is unlikely that one can find rank-1 solutions to this SDP in polynomial time. In Section <ref>, we designed objective functions (such as variations of the nuclear norm) that empirically do very well in finding low-rank solutions to <ref>. Nevertheless, it is of interest to know if the solution returned by <ref> is not rank-1, whether one can recover an ϵ-Nash equilibrium from it and have a guarantee on ϵ. Our goal in this section is to study this question.Notational Remark: Recall our notation for the matrix ℳ:= [ X PZ Y].Throughout this section, any matrices X, Z, P=Z^T and Y are assumed to be taken from a feasible solution to <ref>. Furthermore, x and y will be P1_n and P^T1_m respectively.The ultimate results of this section are the theorems in Sections <ref> and <ref>. To work towards them, we need a number of preliminary lemmas which we present in Section <ref>. §.§ Lemmas Towards Bounds on ϵ We first observe the following connection between the approximate payoffs (AZ) and (BZ), and ϵ(x,y), as defined in Section <ref>.Consider any feasible solution to <ref>. Then ϵ(x,y) ≤max{(AZ)-x^TAy, (BZ) - x^TBy}. Recall from the argument at the beginning of Section <ref> that constraints (<ref>) and (<ref>) imply (AZ) ≥ e_i^TAy and (BZ) ≥ x^TBe_i for all i. Hence, we have ϵ_A ≤(AZ)-x^TAy and ϵ_B ≤(BZ) - x^TBy.We thus are interested in the difference of the two matrices P=Z^T and xy^T. These two matrices can be interpreted as two different probability distributions over the strategy outcomes. The matrix P is the probability distribution from the SDP which generates the approximate payoffs (AZ) and (BZ), while xy^T is the product distribution that would have resulted if the matrix had been rank-1. We will see that the difference of these distributions is key in studying the ϵ which results from <ref>. Hence, we first take steps to represent this difference.Consider any feasible matrixto <ref> with an eigendecomposition = ∑_i=1^k λ_i v_iv_i^T =:∑_i = 1^k λ_ia_i b_i a_i b_i^T, so that the eigenvectors v_i∈ℝ^m+n are partitioned into vectors a_i ∈ℝ^m and b_i∈ℝ^n. Then for all i,∑_j=1^m (a_i)_j = ∑_j=1^n (b_i)_j. We know from (<ref>), (<ref>), and (<ref>) that ∑_i=1^k λ_i 1_m^T a_ia_i^T 1_m (<ref>),(<ref>)= 1,∑_i=1^k λ_i 1_m^T a_ib_i^T 1_n (<ref>)= 1, ∑_i=1^k λ_i 1_n^T b_ia_i^T 1_m (<ref>)= 1, ∑_i=1^k λ_i 1_n^T b_ib_i^T 1_n (<ref>),(<ref>)= 1. Then by subtracting terms we have (<ref>)-(<ref>)=∑_i=1^k λ_i 1_m^Ta_i (a_i^T1_m - b_i^T1_n) = 0, (<ref>)-(<ref>)=∑_i=1^k λ_i 1_n^Tb_i (a_i^T1_m - b_i^T1_n) = 0. By subtracting again these imply (<ref>)-(<ref>)=∑_i=1^k λ_i (1_m^T a_i - 1_n^Tb_i)^2 = 0. As all λ_i are nonnegative due to positive semidefiniteness of , the only way for this equality to hold is to have 1_m^Ta_i = 1_n^Tb_i, ∀ i. This is equivalent to the statement of the claim. From Lemma <ref>, we can let s_i ∑_j=1^m (a_i)_j = ∑_j=1^n (b_i)_j, and furthermore we assume without loss of generality that each s_i is nonnegative. Note that from the definition of x we havex_i = ∑_j = 1^m P_ij = ∑_l = 1^k ∑_j = 1^m λ_l (a_l)_i (b_l)_j = ∑_j = 1^k λ_j s_j (a_l)_i.Hence,x = ∑_i =1^k λ_i s_i a_i.Similarly,y = ∑_i=1^k λ_i s_i b_i.Finally note from the distribution constraint (<ref>) that this implies∑_i=1^k λ_is_i^2 = 1. Let= ∑_i=1^k λ_ia_i b_ia_i b_i ^T,be a feasible solution to <ref>, such that the eigenvectors ofare partitioned into a_i and b_i with ∑_j=1^m (a_i)_j = ∑_j=1^n (b_i)_j=s_i, ∀ i. ThenP-xy^T= ∑_i = 1^k ∑_j>i^k λ_iλ_j (s_ja_i - s_ia_j)(s_jb_i - s_ib_j)^T. Using equations (<ref>) and (<ref>) we can write P - xy^T= ∑_i=1^k λ_i a_ib_i^T - (∑_i =1^k λ_i s_i a_i)(∑_j=1^k λ_j s_j b_j)^T= ∑_i=1^k λ_ia_i(b_i - s_i∑_j=1^k λ_js_jb_j)^T(<ref>)=∑_i=1^k λ_ia_i(∑_j=1^k λ_js_j^2b_i - s_i∑_j=1^k λ_js_jb_j)^T=∑_i=1^k∑_j=1^k λ_iλ_j a_is_j(s_jb_i-s_ib_j)^T=∑_i = 1^k ∑_j > i^k λ_iλ_j (s_ja_i - s_ia_j)(s_jb_i - s_ib_j)^T, where the last line follows from observing that terms where i and j are switched can be combined. We can relate ϵ and P-xy^T with the following lemma. Let the matrix P and the vectors xP1_n and yP^T1_m come from any feasible solution to <ref>. Thenϵ≤P-xy^T_1/2,where ·_1 here denotes the entrywise L-1 norm, i.e., the sum of the absolute values of the entries of the matrix. Let D := P - xy^T. From Lemma <ref>,ϵ_A ≤(AZ) - x^TAy = (A(Z-yx^T)).If we then hold D fixed and restrict that A has entries bounded in [0,1], the quantity (AD^T) is maximized whenA_i,j =1 D_i,j≥ 0 0 D_i,j < 0.The resulting quantity (AD^T) will then be the sum of all nonnegative elements of D. Since the sum of all elements in D is zero, this quantity will be equal to 1/2D_1. The proof for ϵ_B is identical, and the result follows from that ϵ is the maximum of ϵ_A and ϵ_B.§.§ Bounds on ϵ We provide a number of bounds on ϵ(x,y) for xP1_n and yP^T1_m coming from any feasible solution to <ref>. Our first two theorems roughly state that solutions which are “close” to rank-1 provide small ϵ. Consider any feasible solutionto <ref>. Supposeis rank-k and its eigenvalues are λ_1 ≥λ_2 ≥ ... ≥λ_k > 0. Then x and y constitute an ϵ-NE to the game (A,B) with ϵ≤m+n/2∑_i=2^k λ_i. By the Perron Frobenius theorem (see e.g. <cit.>), the eigenvector corresponding to λ_1 can be assumed to be nonnegative, and hences_1 = a_1_1 = b_1_1. We further note that for all i, since a_i b_i is a vector of length m+n with 2-norm equal to 1, we must have a_i b_i_1 ≤√(m+n). Since s_i is the sum of the elements of a_i and b_i, we know that s_i ≤min{a_i_1, b_i_1}≤√(m+n)/2. This then gives us s_i^2 ≤a_i_1b_i_1 ≤m+n/4, with the first inequality following from (<ref>) and the second from (<ref>). Finally note that a consequence of the nonnegativity of ·_1 and (<ref>) is that for all i, j, a_i_1b_j_1 + b_i_1a_j_1 ≤(a_i_1+b_i_1)(a_j_1+b_j_1)= a_i b_i_1 a_j b_j_1 (<ref>)≤ m+n. Now we let D := P - xy^T and upper bound 1/2D_1 using Lemma <ref>. 1/2D_1= 1/2∑_i = 1^k ∑_j>i^k λ_iλ_j (s_ja_i - s_ia_j)(s_jb_i - s_ib_j)^T_1≤1/2∑_i = 1^k ∑_j>i^k λ_iλ_j (s_ja_i - s_ia_j)(s_jb_i - s_ib_j)^T_1≤1/2∑_i = 1^k ∑_j > i^k λ_iλ_j s_ja_i - s_ia_j_1s_jb_i - s_ib_j_1≤1/2∑_i = 1^k ∑_j > i^k λ_iλ_j (s_ja_i_1 + s_ia_j_1)(s_jb_i_1 + s_ib_j_1) (<ref>),(<ref>)≤1/2∑_j=2^kλ_1s_1^2λ_j(s_j+a_j_1)(s_j+b_j_1)+1/2∑_i = 2^k ∑_j > i^k λ_iλ_j (s_j^2m+n/4 + s_i^2m+n/4 + s_is_ja_i_1b_j_1+ s_is_ja_j_1b_i_1)(<ref>),(<ref>),(<ref>)≤m+n/2λ_1s_1^2 ∑_i=2^k λ_i+ 1/2∑_i=2^k ∑_j > i^k λ_iλ_jm+n/4(s_i^2 + s_j^2) + λ_iλ_js_is_j (m+n)AMGM≤m+n/2λ_1s_1^2 ∑_i=2^k λ_i + m+n/2∑_i=2^k ∑_j > i^k λ_iλ_j (s_i^2 + s_j^2/4 + s_i^2+s_j^2/2)= m+n/2λ_1s_1^2 ∑_i=2^k λ_i + 3(m+n)/8∑_i=2^k ∑_j > i^k λ_iλ_j (s_i^2+s_j^2)= m+n/2λ_1s_1^2 ∑_i=2^k λ_i + 3(m+n)/8(∑_i=2^kλ_is_i^2 ∑_j > i^k λ_j + ∑_i=2^k λ_i ∑_j > i^k λ_js_j^2)= m+n/2λ_1s_1^2 ∑_i=2^k λ_i + 3(m+n)/8(∑_j = 2^k λ_j ∑_2≤ i<j^kλ_is_i^2 + ∑_i=2^k λ_i ∑_j > i^k λ_js_j^2)≤m+n/2λ_1s_1^2 ∑_i=2^k λ_i + 3(m+n)/8(∑_j =2^k λ_js_j^2)∑_i=2^k λ_i(<ref>)=m+n/2λ_1s_1^2 ∑_i=2^k λ_i + 3(m+n)/8(1-λ_1s_1^2)∑_i=2^k λ_i= m+n/8(3+λ_1s_1^2)∑_i=2^k λ_i(<ref>)≤m+n/2∑_i=2^k λ_i. AMGM is used to denote the arithmetic-mean-geometric-mean inequality. The following theorem quantifies how making the objective of the diagonal gap algorithm from Section <ref> small makes ϵ small. The proof is similar to the proof of Theorem <ref>. Letbe a feasible solution to <ref>. Then, x and y constitute an ϵ-NE to the game (A,B) with ϵ≤3(m+n)/8(()-x^Tx-y^Ty). Letbe rank-k with eigenvalues λ_1 ≥λ_2 ≥…≥λ_k > 0 and eigenvectors v_1, …, v_k partitioned as in Lemma <ref> so that v_i =a_ib_i with ∑_j=1^m (a_i)_j = ∑_j=1^n (b_i)_j for i = 1, …, k. Let s_i ∑_j=1^m (a_i)_j. Then we have () = ∑_i=1^k λ_i, and x^Tx + y^Ty (<ref>),(<ref>)= (∑_i=1^k λ_i s_i v_i)^T(∑_i=1^k λ_i s_i v_i) = ∑_i=1^k λ_i^2s_i^2. We now get the following chain of inequalities (the first one follows from Lemma <ref> and inequality (<ref>)): ϵ ≤1/2∑_i = 1^k ∑_j > i^k λ_iλ_j (s_ja_i_1 + s_ia_j_1)(s_jb_i_1 + s_ib_j_1)(<ref>),(<ref>)≤1/2∑_i = 1^k ∑_j > i^k λ_iλ_j (s_j^2m+n/4 + s_i^2m+n/4 + s_is_ja_i_1b_j_1+ s_is_ja_j_1b_i_1)(<ref>)≤1/2∑_i=1^k ∑_j > i^k λ_iλ_jm+n/4(s_i^2 + s_j^2) + λ_iλ_js_is_j (m+n)AMGM≤m+n/2∑_i=1^k ∑_j > i^k λ_iλ_j (s_i^2 + s_j^2/4 + s_i^2+s_j^2/2)= 3(m+n)/8∑_i=1^k ∑_j > i^k λ_iλ_j (s_i^2+s_j^2)= 3(m+n)/8(∑_i=1^kλ_is_i^2 ∑_j > i^k λ_j + ∑_i=1^k λ_i ∑_j > i^k λ_js_j^2)= 3(m+n)/8(∑_j = 1^k λ_j ∑_1≤ i<j^kλ_is_i^2 + ∑_i=1^k λ_i ∑_j > i^k λ_js_j^2)= 3(m+n)/8(∑_i = 1^k λ_i ∑_jiλ_js_j^2)(<ref>)=3(m+n)/8(∑_i = 1^k λ_i (1-λ_is_i^2))= 3(m+n)/8(∑_i=1^k λ_i - ∑_i=1^k λ_i^2s_i^2) (<ref>)=3(m+n)/8(()-x^Tx-y^Ty). We now give a bound on ϵ which is dependent on the nonnegative rank of the matrix returned by <ref>. Our analysis will also be useful for the next subsection. To begin, we first recall the definition of the nonnegative rank. The nonnegative rank of a (nonnegative) m × n matrix M is the smallest k for which there exist a nonnegative m × k matrix U and a nonnegative n × k matrix V such that M = UV^T. Such a decomposition is called a nonnegative matrix factorization of M. Consider the matrix P from any feasible solution to <ref>. Suppose its nonnegative rank is k. Then xP1_n and yP^T1_m constitute an ϵ-NE to the game (A,B) with ϵ≤ 1-1/k.Since P has nonnegative rank k and its entries sum up to 1, we can write P = ∑_i=1^k σ_ia_ib_i^T, where a_i ∈_m, b_i ∈_n, and ∑_i=1^k σ_i = 1. From Lemma <ref> and inequality (<ref>) (keeping in mind that s_i = 1, ∀ i) we haveϵ ≤1/2∑_i=1^k ∑_j > i^k σ_iσ_j (a_i_1 + a_j_1)(b_i_1 + b_j_1)≤ 2 ∑_i=1^k ∑_j > i^k σ_iσ_j= 2(1/2(∑_i=1^k σ_i ∑_j=1^k σ_j - ∑_i=1^k σ_i^2))= 1 - ∑_i=1^k σ_i^2≤ 1 - 1/k,where the last line follows fromthe fact that v_2^2 ≥1/k for any vector v ∈_k.§.§ Bounds on ϵ in the Rank-2 Case We now provide a number of bounds on ϵ(x,y) with xP1_n and yP^T1_m which hold for rank-2 feasible solutionsto <ref> (note that P will have rank at most 2 in this case). This is motivated by our ability to show stronger (constant) bounds in this case, and the fact that we often recover rank-2 (or rank-1) solutions with our algorithms in Section <ref>. Furthermore, our analysis will use the special property that a rank-2 nonnegative matrix will have nonnegative rank also equal to two, and that a nonnegative factorization of it can be computed in polynomial time (see, e.g., Section 4 of <cit.>). We begin with the following observation, which follows from Theorem <ref> when k = 2. If the matrix P from a feasible solution to <ref> is rank-2, then x and y constitute a 1/2-NE.We now show how this pair of strategies can be refined. If the matrix P from a feasible solution to <ref> is rank-2, then either x and y constitute a 5/11-NE, or a 5/11-NE can be recovered from P in polynomial time. We consider 3 cases, depending on whether ϵ_A(x,y) and ϵ_B(x,y) are greater than or less than .4. If ϵ_A ≤ .4, ϵ_B ≤ .4, then (x, y) is already a .4-Nash equilibrium. Now consider the case when ϵ_A ≥ .4, ϵ_B ≥ .4. Since ϵ_A ≤(A(P-xy^T)^T) and ϵ_B ≤(B(P-xy^T)^T) as seen in the proof of Lemma <ref>, we have, reusing the notation in the proof of Theorem <ref>, σ_1σ_2(a_1-a_2)^T A (b_1-b_2) ≥ .4, σ_1σ_2(a_1-a_2)^T B (b_1-b_2) ≥ .4. Since A, a_1, a_2, b_1, and b_2 are all nonnegative and σ_1σ_2 ≤1/4, a_1^TAb_1 + a_2^TAb_2 ≥ (a_1-a_2)^T A (b_1-b_2) ≥ 1.6,and the same inequalities hold for for player B. In particular, since A and B have entries bounded in [0,1] and a_1,a_2,b_1, and b_2 are simplex vectors, all the quantities a_1^TAb_1, a_2^TAb_2, a_1^TBb_1, and a_2^TBb_2 are at most 1, and consequently at least .6. Hence (a_1,a_2) and (a_2,b_2) are both .4-Nash equilibria. Now suppose that (x,y) is a .4-NE for one player (without loss of generality player A) but not for the other (without loss of generality player B). Then ϵ_A ≤ .4, and ϵ_B ≥ .4. Let y^* be a best response for player B to x, and let p = 1/1+ϵ_B - ϵ_A. Consider the strategy profile (x̃,ỹ)(x, py + (1-p)y^*). This can be interpreted as the outcome (x,y) occurring with probability p, and the outcome (x,y^*) happening with probability 1-p. In the first case, player A will have ϵ_A(x,y) = ϵ_A and player B will have ϵ_B(x,y) = ϵ_B. In the second outcome, player A will have ϵ_A(x,y^*) at most 1, while player B will have ϵ_B(x,y^*) = 0. Then under this strategy profile, both players have the same upper bound for ϵ, which equals ϵ_B p = ϵ_B/1 + ϵ_B - ϵ_A. To find the worst case for this value, let ϵ_B = .5 (note from Theorem <ref> that ϵ_B ≤1/2) and ϵ_A = .4, and this will return ϵ = 5/11. We now show a stronger result in the case of symmetric games. A symmetric game is a game in which the payoff matrices A and B satisfy B=A^T.A Nash equilibrium strategy (x,y) is said to be symmetric if x=y. Every symmetric bimatrix game has a symmetric Nash equilibrium. For the proof of Theorem <ref> below we modify <ref> so that we are seeking a symmetric solution. We also need a more specialized notion of the nonnegative rank. A matrix M is completely positive (CP) if it admits a decomposition M=UU^T for some nonnegative matrix U.The CP-rank of an n × n CP matrix M is the smallest k for which there exists a nonnegative n × k matrix U such that M = UU^T. A rank-2, nonnegative, and positive semidefinite matrix is CP and has CP-rank 2. It is also known (see e.g., Section 4 in <cit.>) that the CP factorization of a rank-2 CP matrix can be found to arbitrary accuracy in polynomial time. Suppose the constraint P ≽ 0 is added to <ref>. Then if in a feasible solution to this new SDP the matrix P is rank-2, either x and y constitute a symmetric 1/3-NE, or a symmetric 1/3-NE can be recovered from P in polynomial time. If (x,y) is already a symmetric 1/3-NE, then the claim is established. Now suppose that (x,y) does not constitute a 1/3-Nash equilibrium. Similarly as in the proof of Theorem <ref>, we can decompose P into ∑_i=1^2 σ_i a_ia_i^T, where ∑_i=1^2 σ_i = 1 and each a_i is a vector on the unit simplex. Then we have σ_1σ_2(a_1-a_2)^T A (a_1-a_2) ≥1/3. Since A, a_1, and a_2 are all nonnegative, and σ_1σ_2 ≤1/4, we get a_1^TAa_1+ a_2^TAa_2 ≥ (a_1-a_2)^T A (a_1-a_2) ≥4/3. In particular, at least one of a_1^TAa_1 and a_2^TAa_2 is at least 2/3. Since the maximum possible payoff is 1, at least one of (a_1,a_1) and (a_2,a_2) is a (symmetric) 1/3-Nash equilibrium. For symmetric games, instead of the construction stated in Theorem <ref>, one can simply optimize over a smaller m × m matrix (note m=n). This is the relaxed version of exchangeable equilibria <cit.>, with the completely positive constraint relaxed to a psd constraint. The statements of Corollary <ref>, and Theorem <ref>, and Theorem <ref> hold for any rank-2 correlated equilibrium. Indeed, given any rank-2 (equivalently, nonnegative-rank-2) correlated equilibrium P, one can complete it to a (rank-2) feasible solution to <ref> as follows. Let P = ∑_i=1^2 σ_ia_ib_i^T, where a_i ∈_m, b_i ∈_n, and σ_1 + σ_2 = 1. It is easy to check that∑_i=1^2 σ_ia_i b_ia_i b_i ^Tis feasible to <ref>. § BOUNDING PAYOFFS AND STRATEGY EXCLUSION IN SYMMETRIC GAMESIn addition to finding ϵ-additive Nash equilibria, our SDP approach can be used to answer certain questions of economic interest about Nash equilibria without actually computing them. For instance, economists often would like to know the maximum welfare (sum of the two players' payoffs) achievable under any Nash equilibrium, or whether there exists a Nash equilibrium in which a given subset of strategies (corresponding, e.g., to undesirable behavior) is not played. Both these questions are NP-hard for bimatrix games <cit.>, even when the game is symmetric and only symmetric equilibria are considered <cit.>. In this section, we consider these two problems in the symmetric setting and compare the performance of our SDP approach to an LP approach which searches over symmetric correlated equilibria. For general equilibria, it turns out that for these two specific questions, our SDP approach is equivalent to an LP that searches over correlated equilibria. §.§ Bounding PayoffsWhen designing policies that are subject to game theoretic behavior by agents, economists would often like to find one with a good socially optimal outcome, which usually corresponds to an equilibrium giving the maximum welfare. Hence, given a game, it is of interest to know the highest achievable welfare under any Nash equilibrium. For symmetric games, symmetric equilibria are of particular interest as they reflect the notion that identical agents should behave similarly given identical options.Note that the maximum welfare of a symmetric game under any symmetric Nash equilibrium is equal to the optimal value of the following quadratic program:x ∈_mmax 2x^TAxsubject to x^TAx ≥ e_i^TAx, ∀ i∈{1,…, m}.One can find an upper bound on this number by solving an LP which searches over symmetric correlated equilibria: LP1 P ∈𝕊^m, mmax(AP^T) subject to∑_i=1^m ∑_j=1^m P_i,j = 1 ∑_j=1^m A_i,jP_i,j≥∑_j=1^m A_k,jP_i,j, ∀ i,k ∈{1,…, m}, P ≥ 0. A potentially better upper bound on the maximum welfare can be obtained from a version of <ref> adapted to this specific problem:SDP3 P ∈𝕊^m, mmax(AP^T) subject to (<ref>), (<ref>), (<ref>)P ≽ 0. To test the quality of these upper bounds, we tested this LP and SDP on a random sample of one hundred 5× 5 and 10 × 10 games[The matrix A in each game was randomly generated with diagonal entries uniform and independent in [0,.5] and off-diagonal entries uniform and independent in [0,1].]. The resulting upper bounds are in Figure <ref>, which shows that the bound returned by <ref> was exact in a large number of the experiments.[The computation of the exact maximum payoffs was done with thesoftware <cit.>, which computes all extreme Nash equilibria. For a definition of extreme Nash equilibria and for understanding why it is sufficient for us to compare against extreme Nash equilibria (both in Section <ref> and in Section <ref>), see Appendix <ref>. The computation of the SDP upper bound has been implemented in the file nashbound.m, which is publicly available at <aaa.princeton.edu/software>. This file more generally computes an SDP-based lower bound on the minimum of an input quadratic function over the set of Nash equilibria of a bimatrix game. The file also takes as an argument whether one wishes to only consider symmetric equilibria when the game is symmetric.] §.§ Strategy ExclusionThe strategy exclusion problem asks, given a subset of strategies 𝒮=(𝒮_x,𝒮_y), with 𝒮_x ⊆{1,…,m} and 𝒮_y⊆{1,…,n}, is there a Nash equilibrium in which no strategy in 𝒮 is played with positive probability. We will call a set 𝒮 “persistent” if the answer to this question is negative, i.e. at least one strategy in 𝒮 is played with positive probability in every Nash equilibrium. One application of the strategy exclusion problem is to understand whether certain strategies can be discouraged in the design of a game, such as reckless behavior in a game of chicken or defecting in a game of prisoner's dilemma. In these particular examples these strategy sets are persistent and cannot be discouraged.As in the previous subsection, we consider the strategy exclusion problem for symmetric strategies in symmetric games (such as the aforementioned games of chicken and prisoner's dilemma). A quadratic program which addresses this problem is as follows: x∈_mmin∑_i ∈𝒮_x x_isubject to x^TAx ≥ e_i^TAx, ∀ i∈{1,…, m}.Observe that by design, 𝒮 is persistent if and only if this quadratic program has a positive optimal value. As in the previous subsection, an LP relaxation of this problem which searches over symmetric correlated equilibria is given by LP2 P ∈𝕊^m, mmin∑_i ∈𝒮_x∑_j=1^m P_ij subject to (<ref>), (<ref>), (<ref>).The SDP relaxation that we propose for the strategy exclusion problem is the following: SDP4 P ∈𝕊^m, mmin∑_i ∈𝒮_x∑_j=1^m P_ij subject to (<ref>), (<ref>), (<ref>)P ≽ 0.Our approach would be to declare that the strategy set 𝒮_x is persistent if and only if <ref> has a positive optimal value.Note that since the optimal value of <ref> is a lower bound for that of (<ref>), <ref> carries over the property that if a set 𝒮 is not persistent, then the SDP for sure returns zero. Thus, when using <ref> on a set which is not persistent, our algorithm will always be correct. However, this is not necessarily the case for a persistent set. While we can be certain that a set is persistent if <ref> returns a positive optimal value (again, because the optimal value of <ref> is a lower bound for that of (<ref>)), there is still the possibility that for a persistent set <ref> will have optimal value zero. The same arguments hold for the optimal value of <ref>.To test the performance of <ref> and <ref>, we generated 100 random games of size 5× 5 and 10× 10 and computed all their symmetric extreme Nash equilibria[The exact computation of the exact Nash equilibria was done again with thesoftware <cit.>, which computes extreme Nash equilibria. To understand why this suffices for our purposes see Appendix <ref>.]. We then, for every strategy set 𝒮 of cardinality one and two, checked whether that set of strategies was persistent, first by checking among the extreme Nash equilibria, then through <ref> and <ref>. The results are presented in Tables <ref> and <ref>. As can be seen, <ref> was quite effective for the strategy exclusion problem. § CONNECTION TO THE SUM OF SQUARES/LASSERRE HIERARCHY In this section, we clarify the connection of the SDPs we have proposed in this paper to those arising in the sum of squares/Lasserre hierarchy. We start by briefly reviewing this hierarchy. §.§ Sum of Squares/Lasserre HierarchyThe sum of squares/Lasserre hierarchy[The unfamiliar reader is referred to <cit.> for an introduction to this hierarchy and the related theory of moment relaxations.] gives a recipe for constructing a sequence of SDPs whose optimal values converge to the optimal value of a given polynomial optimization problem. Recall that a polynomial optimization problem (pop) is a problem of minimizing a polynomial over a basic semialgebraic set, i.e., a problem of the formx ∈ℝ^nmin f(x) subject to g_i(x) ≥ 0, ∀ i ∈{1,…,m},where f,g_i are polynomial functions.In this section, when we refer to the k-th level of the Lasserre hierarchy, we mean the optimization problem γ_sos^kγ,σ_imaxγsubject to f(x)-γ = σ_0(x) + ∑_i=1^m σ_i(x)g_i(x),σ_iis sos, ∀ i ∈{0,…,m},σ_0, g_iσ_ihave degree at most2k, ∀ i ∈{1,…,m}.Here, the notation “sos” stands for sum of squares. We say that a polynomial p is a sum of squares if there exist polynomials q_1,…,q_r such that p= ∑_i=1^r q_i^2. There are two primary properties of the Lasserre hierarchy which are of interest. The first is that any fixed level of this hierarchy gives an SDP of size polynomial in n. The second is that, if the set {x∈ℝ^n|g_i(x)≥ 0} is Archimedean (see, e.g. <cit.> for definition), then k →∞limγ_sos^k = p^*, where p^* is the optimal value of the pop in (<ref>). The latter statement is a consequence of Putinar's positivstellensatz <cit.>, <cit.>.§.§ The Lasserre Hierarchy and <ref> One can show, e.g. via the arguments in <cit.>, that the feasible sets of the SDPs dual to the SDPs underlying the hierarchy we summarized above produce an arbitrarily tight outer approximation to the convex hull of the set of Nash equilibria of any game. The downside of this approach, however, is that the higher levels of the hierarchy can get expensive very quickly. This is why the approach we took in this paper was instead to improve the first level of the hierarchy. The next proposition formalizes this connection. Consider the problem of minimizing any quadratic objective function over the set of Nash equilibria of a bimatrix game. Then, <ref> (and hence <ref>) gives a lower bound on this problem which is no worse than that produced by the first level of the Lasserre hierarchy. To prove this proposition we show that the first level of the Lasserre hierarchy is dual to a weakened version of <ref>. Explicit parametrization of first level of the Lasserre hierarchy. Consider the formulation of the Lasserre hierarchy in (<ref>) with k=1. Suppose we are minimizing a quadratic functionf(x,y)=1vec^T 𝒞1vecover the set of Nash equilibria as described by the linear and quadratic constraints in (<ref>). If we apply the first level of the Lasserre hierarchy to this particular pop, we get Q, α, χ, β, ψ, ηmaxγ subject to1vec^T 𝒞1vec-γ = 1vec^T Q 1vec+∑_i=1^m α_i(x^TAy - e_i^TAy)+ ∑_i=1^n β_i(x^TBy - x^TBe_i)+ ∑_i=1^m χ_i x_i + ∑_i=1^n ψ_i y_i+η_1 (∑_i=1^m x_i - 1)+ η_2 (∑_i=1^n y_i -1),Q≽ 0, α, χ, β, ψ ≥ 0, where Q ∈𝕊^m+n+1 × m+n+1,α, χ∈ℝ^m, β, ψ∈ℝ^n, η∈ℝ^2. By matching coefficients of the two quadratic functions on the left and right hand sides of (<ref>), this SDP can be written as γ,α,β,χ,ψ,ηmaxγsubject toℋ≽ 0,α, β, χ, ψ≥ 0, where ℋ:= 1/2 0 (-∑_i=1^m α_i )A + (-∑_i=1^m β_i)B∑_i=1^n β_i B_,i-χ - η_11_m (-∑_i=1^m α_i)A + (-∑_i=1^n β_i)B 0∑_i=1^m α_i A_i,^T-ψ - η_21_n ∑_i=1^n β_iB_,i^T-χ^T-η_11_m^T∑_i=1^m α_i A_i,-ψ^T - η_21_n^T2η_1+2η_2-2γ+𝒞. Dual of a weakened version of SDP1. With this formulation in mind, let us consider a weakened version of <ref> with only the relaxed Nash constraints, unity constraints, and nonnegativity constraints on x and y in the last column (i.e., the nonegativity constraint is not applied to the entire matrix). Let the objective be (C'). To write this new SDP in standard form, let 𝒜_i := 1/2 0 A 0A^T 0 -A_i,^T0 -A_i,0, ℬ_i := 1/2 0 B -B_,iB^T 0 0 -B_,i^T 0 0 , 𝒮_1 := 1/2 0 0 1_m0 0 0 1_m^T 0 -2, 𝒮_2 := 1/2 0 0 00 0 1_n0 1_n^T -2 . Let 𝒩_i be the matrix with all zeros except a 1/2 at entry (i,m+n+1) and (m+n+1,i) (or a 1 if i=m+n+1). Then this SDP can be written as SDP0 'min(𝒞') subject toℳ' ≽ 0,(𝒩_i') ≥ 0, ∀ i ∈{1,…,m+n}, (𝒜_i') ≥ 0, ∀ i ∈{1,…,m}, (ℬ_i') ≥ 0, ∀ i ∈{1,…,n}, (𝒮_1') = 0, (𝒮_2') = 0, (𝒩_m+n+1) = 1 . We now create dual variables for each constraint; we choose α_i and β_i for the relaxed Nash constraints (<ref>) and (<ref>), η_1 and η_2 for the unity constraints (<ref>) and (<ref>), χ for the nonnegativity of x (<ref>), ψ for the nonnegativity of y (<ref>), and γ for the final constraint on the corner (<ref>). These variables are chosen to coincide with those used in the parametrization of the first level of the Lasserre hierarchy, as can be seen more clearly below. We then write the dual of the above SDP as α, β, λ, γmaxγsubject to∑_i = 1^m α_i 𝒜_i + ∑_i = 1^n β_i ℬ_i + ∑_i = 1^2 η_i𝒮_i+∑_i=1^m 𝒩_i+nχ_i+∑_i=1^n 𝒩_iψ_i + γ𝒩_m+n+1≼𝒞,α, β, χ, ψ≥ 0. which can be rewritten as α, β, χ, ψ, γmaxγsubject to𝒢≽ 0,α, β, χ,ψ≥ 0, where 𝒢1/2 0 (-∑_i=1^m α_i )A + (-∑_i=1^m β_i)B∑_i=1^n β_i B_,i-χ - η_11_m (-∑_i=1^m α_i)A + (-∑_i=1^n β_i)B 0∑_i=1^m α_i A_i,^T-ψ - η_21_n ∑_i=1^n β_iB_,i^T-χ^T-η_11_m^T∑_i=1^m α_i A_i,-ψ^T - η_21_n^T2η_1+2η_2-2γ+𝒞. We can now see that the matrix 𝒢 coincides with the matrix ℋ in the SDP (<ref>). Then we have(<ref>)^opt=(<ref>)^opt = (<ref>)^opt≤ <ref>^opt≤ <ref>^opt, where the first inequality follows from weak duality, and the second follows from that the constraints of <ref> are a subset of the constraints of <ref>. The Lasserre hierarchy can be viewed in each step as a pair of primal-dual SDPs: the sum of squares formulation which we have just presented, and a moment formulation which is dual to the sos formulation <cit.>. All our SDPs in this paper can be viewed more directly as an improvement upon the moment formulation. One can see, either by inspection or as an implication of the proof of Theorem <ref>, that in the case where the objective function corresponds to maximizing player A's and/or B's payoffs[This would be the case, for example, in the maximum social welfare problem of Section <ref>, where the matrix of the quadratic form in the objective function is given by𝒞= 0 -A-B 0 -A-B 0 0 0 0 0. ], SDPs (<ref>) and (<ref>) are infeasible. This means that for such problems the first level of the Lasserre hierarchy gives an upper bound of +∞ on the maximum payoff. On the other hand, the additional valid inequalities in <ref> guarantee that the resulting bound is always finite. § FUTURE WORKOur work leaves many avenues of further research. Are there other interesting subclasses of games (besides strictly competitive games) for which our SDP is guaranteed to recover an exact Nash equilibrium? Can the guarantees on ϵ in Section <ref> be improved in the rank-2 case (or the general case) by improving our analysis? Is there a polynomial time algorithm that is guaranteed to find a rank-2 solution to <ref>? Such an algorithm, together with our analysis, would improve the best known approximation bound for symmetric games (see Theorem <ref>). Can this bound be extended to general games? We show in Appendix <ref> that some natural approaches based on symmetrization of games do not immediately lead to a positive answer to this question. Can SDPs in a higher level of the Lasserre hierarchy be used to achieve better ϵ guarantees? What are systematic ways of adding valid inequalities to these higher-order SDPs by exploiting the structure of the Nash equilibrium problem? For example, since any strategy played with positive probability must give the same payoff, one can add a relaxed version of the cubic constraintsx_ix_j(e_i^TAy - e_j^TAy)=0, ∀ i,j ∈{1,…,m}to the SDP underlying the second level of the Lasserre hierarchy. What are other valid inequalities for the second level? Finally, our algorithms were specifically designed for two-player one-shot games. This leaves open the design and analysis of semidefinite relaxations for repeated games or games with more than two players. § ACKNOWLEDGMENTS.We would like to thank Ilan Adler, Costis Daskalakis, Georgina Hall, Ramon van Handel, and Robert Vanderbei for insightful exchanges. We are also extremely grateful to an anonymous referee for various insightful questions and comments which have led to a significantly improved version of this manuscript. abbrv § STATISTICS ON Ε FROM ALGORITHMS IN SECTION <REF>Below are statistics for the ϵ recovered in 100 random games of varying sizes using the algorithms of Section <ref>. § COMPARISON WITH AN SDP APPROACH FROM <CIT.>In this section, at the request of a referee, we compare the first level of the SDP hierarchy given in <cit.> to <ref> using (M) as the objective function on 100 randomly generated games for each size given in the tables below. The first level of the hierarchy in <cit.> optimizes over a matrix which is slightly bigger than the one in <ref>, though it has a number of constraints linear in the size of the game considered, as opposed to the quadratic number in <ref>. We remark that the approach in <cit.> is applicable more generally to many other problems, including several in game theory.The scalar ϵ reported in Table <ref> is computed using the strategies (x,y) extracted from the first row of the optimal matrix M_1 as described in Section 4.1 of <cit.>. The scalar ϵ reported in Table <ref> is computed using x = P1_n and y = P^T1_m from the optimal solution to <ref> with () as the objective function.We also ran the second level of the hierarchy in <cit.> on the same 100 5 × 5 games. The maximum ϵ observed was .3362, while the mean was .1880 and the median was .1800. The size of the variable matrix that needs to be positive semidefinite for this level is 78 × 78.§ LEMMAS FOR EXTREME NASH EQUILIBRIAThe results reported in Section <ref> were found using the <cit.> software which computes extreme Nash equilibria (see definition below). In particular the true maximum welfare and the persistent strategy sets were found in relation to extreme symmetric Nash equilibria only. We show in this appendix why this is sufficient for the claims we made about all symmetric Nash equilibria. We prove a more general statement below about general games and general Nash equilibria since this could be of potential independent interest. The proof for symmetric games is identical once the strategies considered are restricted to be symmetric. An extreme Nash equilibrium is a Nash equilibrium which cannot be expressed as a convex combination of other Nash equilibria. All Nash equilibria are convex combinations of extreme Nash equilibria. It suffices to show that any extreme point of the convex hull of the set of Nash equilibria must be an extreme Nash equilibrium, as any point in a compact convex set can be written as a convex combination of its extreme points. Note that this convex hull contains three types of points: extreme Nash equilibria, Nash equilibria which are not extreme, and convex combinations of Nash equilibria which are not Nash equilibria. The claim then follows because any extreme point of the convex hull cannot be of the second or third type, as they can be written as convex combinations of other points in the hull. The next lemma shows that checking extreme Nash equilibria are sufficient for the maximum welfare problem. For any bimatrix game, there exists an extreme Nash equilibrium giving the maximum welfare among all Nash equilibria. Consider any Nash equilibrium (x̃, ỹ), and let it be written as x̃ ỹ = ∑_i=1^r λ_ix^i y^i for some set of extreme Nash equilibria x^1 y^1 , …,x^r y^r and λ∈_r. Observe that for any i,j,x^iTAy^j≤ x^jTAy^j, x^iTBy^j≤ x^iTBy^i,from the definition of a Nash equilibrium. Now note that x̃^T(A+B)ỹ= (∑_i=1^r λ_i x^i)^T(A+B)(∑_i=1^r λ_i y^i)=∑_i=1^r ∑_j=1^r λ_iλ_j x^iT(A+B)y^j= ∑_i=1^r ∑_j=1^r λ_iλ_j x^iTAy^j + ∑_i=1^r ∑_j=1^r λ_iλ_j x^iTBy^j(<ref>)≤∑_i=1^r ∑_j=1^r λ_iλ_j x^jTAy^j + ∑_i=1^r ∑_j=1^r λ_iλ_j x^iTBy^i= ∑_i=1^r λ_i x^iTAy^i + ∑_i=1^r λ_i x^iTBy^i= ∑_i=1^r λ_i x^iT(A+B)y^i. In particular, since each (x^i, y^i) is an extreme Nash equilibrium, this tells us for any Nash equilibrium (x̃, ỹ) there must be an extreme Nash equilibrium which has at least as much welfare. Similarly for the results for persistent sets in Section <ref>, there is no loss in restricting attention to extreme Nash equilibria. For a given strategy set 𝒮, if every extreme Nash equilibrium plays at least one strategy in 𝒮 with positive probability, then every Nash equilibrium plays at least one strategy in 𝒮 with positive probability. Let 𝒮 be a persistent set of strategies. Since all Nash equilibria are composed of nonnegative entries, and every extreme Nash equilibrium has positive probability on some entry in 𝒮, any convex combination of extreme Nash equilibria must have positive probability on some entry in 𝒮. § A NOTE ON REDUCTIONS FROM GENERAL GAMES TO SYMMETRIC GAMES An anonymous referee asked us if our guarantees for symmetric games transfer over to general games by symmetrization. Indeed, there are reductions in the literature that take a general game, construct a symmetric game from it, and relate the Nash equilibria of the original game to symmetric Nash equilibria of its symmetrized version. In this Appendix, we review two well-known reductions of this type <cit.> and show that the quality of approximate Nash equilibria can differ greatly between the two games. We hope that our examples can be of independent interest.§.§ The Reduction of Griesmer, Hoffman, and Robinson <cit.> Consider a game (A,B) with A, B > 0 and a Nash equilibrium (x^*, y^*) of it with payoffs p_A x^*TAy^* and p_B x^*TBy^*. Then the symmetric game (S_AB, S_AB^T) with S_AB 0 AB^T 0admits a symmetric Nash equilibrium in which both players play p_A/p_A+p_Bx^*p_B/p_A+p_By^*. In the reverse direction, any symmetric equilibrium (xy,xy) of (S_AB, S_AB^T) yields a Nash equilibrium (x/1_m^Tx, y/1_n^Ty) to the original game (A,B).To demonstrate that high-quality approximate Nash equilibria in the symmetrized game can map to low-quality approximate Nash equilibria in the original game, consider the game given by (A,B) = ( ϵ0 1 1, ϵ^2 0 0 1 ) for some ϵ > 0. The symmetric strategy( 1/1 + ϵ0ϵ/1+ϵ0 , 1/1 + ϵ0ϵ/1+ϵ0 )is an ϵ1-ϵ/1+ϵ-NE for (S_AB, S_AB^T), but the strategy pair ( 1 0 ,1 0 ) is a (1-ϵ)-NE for (A,B). §.§ The Reduction of Jurg, Jansen, Potters, and Tijs <cit.>Consider a game (A,B) with A > 0, B < 0 and a Nash equilibrium (x^*, y^*) of it with payoffs p_A x^*TAy^* and p_B x^*TBy^*. Then the symmetric game (S_AB, S_AB^T) with S_AB 0_m × mA -1_mB^T 0_n × n1_n1_m^T -1_n^T 0admits a symmetric Nash equilibrium in which both players playx^*/2 - p_B y^*/2+p_A1 - 1/2-p_B - 1/2+p_A.In the reverse direction, any symmetric equilibrium (xy z ,xy z) of (S_AB, S_AB^T) yields a Nash equilibrium (x/1_m^Tx, y/1_n^Ty) to the original game (A,B). This reduction has some advantages over the previous one (see <cit.>).To demonstrate that high-quality approximate Nash equilibria in the new symmetrized game can again map to low-quality approximate Nash equilibria in the original game, consider the game given by (A,B) = (0 0 0 1,-1 -1 0 0 ). Let ϵ∈ (0,1/2). The symmetric strategy( ϵ0 1 - ϵ0 0, ϵ0 1 - ϵ0 0)is an ϵ/2(1-ϵ)-NE[Note that approximation factor is halved since the range of the entries of the payoff matrix in the symmetrized game is [-1,1].] for (S_AB, S_AB^T), but the strategy pair ( 1 0 ,1 0 ) is a 1-NE for (A,B). | http://arxiv.org/abs/1706.08550v3 | {
"authors": [
"Amir Ali Ahmadi",
"Jeffrey Zhang"
],
"categories": [
"math.OC",
"cs.DS",
"cs.GT",
"90C90 (Primary) 90C22, 91A5, 91A10 (Secondary)",
"G.1.6"
],
"primary_category": "math.OC",
"published": "20170626181858",
"title": "Semidefinite Programming and Nash Equilibria in Bimatrix Games"
} |
Física Teórica y Materia Condensada, UAM-Azcapotzalco, C.P. 02200, Ciudad de México, México The reason why the effective-mass approximation works so well with nanoscopic structures has been an enigma and a challenge for theorists. To explain this issue, we re-derive the effective-mass approximation using, instead of the wave functions for infinite-periodic-systems and the ensuing continuous bands, the eigenfunctions and eigenvalues obtained in the theory of finite periodic systems, where the finiteness of the number of primitive cells in the nanoscopic layers, is a prerequisite and an essential condition. This derivation justifies and shows why this approximation works so well for nano-structures. We show also with explicit optical-response calculations that the rapidly varying eigenfunctions Φ_ϵ_0,η_0(z) of the one-band wave functions Ψ^ϵ_0,η_0_μ,ν(z)= Ψ^ϵ_0_μ,ν(z) Φ_ϵ_0,η_0(z), can be safely dropped out for the calculation of inter-band transition matrix elements. Why the effective-mass approximation works so well for nano-structures Pedro Pereyra December 30, 2023 ======================================================================§ INTRODUCTIONThe effective-mass approximation (EMA) is, without a doubt, the most recurrent and widely used approximation in theoretical calculations involving semiconductor structures. The formal justification of why this approximation, where the wave packets are constructed in terms of infinite periodic system wave functions,<cit.> works so well for finite micro and nano-structures, has been an enigma and a challenge for theorists. Despite the various guises of the EMA, the correct explanation has remained elusive.M. G. Burt in a number of papers<cit.> analysed critically the drawbacks of the “conventional" EMA,and tried to overcome these attempts providing a “new" envelope-function method, using again wave functions of infinite periodic systems. Now that the theory of finite periodic systems (TFPS) has evolved and has shown the ability to obtain the true, bona fide, energy eigenvalues and eigenfunctions of finite periodic structures with a finite number of unit cells,<cit.> it is worth reviewing and re-deriving the EMA within the TFPSto understand why it works so well. The purpose of this letter is to re-derive the effective mass approximation taking into account the system and layers finiteness as the fundamental requisite.Superlattices and layered structures are characterized by the simultaneous presence of two length scales: the crystalline unit cells in the semiconductor layers of atomic size and the layers widths. While the primitive cells lengths are of the order of 0.5nm, depending on the the semiconductor, the layers widths are of the order of 5nm, depending on the number of atomic cells per layer. This important difference in size is behind the factorization of the heterostructure wave function (HWF) in terms of rapid and slowly varying functions. The finiteness of the number n_X of primitive cells, in the direction of growth, of layer X (=A,B,...), and the finiteness of the number of layers in the heterostructure or number of superlattice (SL) unit cells n_S, is not only an obvious characteristic, but also an essential requisite in the TFPS.§ FINITENESS OF PERIODIC LAYERS. AN OUTLINE OF THE TFPS Soon after the semiconductor SLs were introduced,<cit.> and the subbands (or minibands) structures of direct and indirect band gap semiconductors were experimentally and theoretically confirmed,<cit.> Leo Esaki noticed that whereas in reality SLs contain a finite number of layers, with a finite number of atomic cells each, the standardtheoretical approaches tacitly assume that SLsare infinite-periodic structures with alternating layers containing also aninfinite number of atomic cells.<cit.> In fact, the HWF and SL wave functions are generally<cit.> written as ψ ( r)=∑_l u_n_l( r)f_l( r),with u_n_ l( r) the periodic part of the host-semiconductor Bloch's function at band n_l, and f_l( r)∝exp[ik_⊥· r_⊥]χ_l(z) the envelope wave function, with k_⊥=k_x+ k_y the perpendicular wave number assumed, generally, a constant of motion.<cit.> At the end, it is common to assume wave functions ψ( r) set up from wave functions u_n_0 of only one band, evaluated at the center of the Brillouin zone or at the subband edge k = 0. For SLs the envelope function is, again, written in terms of Bloch-type functions χ_μ(z) =exp(iqz)u_μ(z), characterized by a subband index μ and a continuous wave number q that is then artificially discretized, via the cyclic boundary condition.On the other side, the theory of finite periodic systems has grown, and has been generalized to include periodic structures with arbitrary potential profiles, arbitrary but finite number n of unit cells and arbitrary but finite number N of propagating modes for open, bounded and quasi-bounded periodic structures.<cit.> The TFPS is based on the transfer matrix properties and the rigorous fulfillment of continuity conditions, that make possible to express the n-cells transfer matrix M_n as M^n, where M, for time reversal invariant systems, is the single-cell transfer matrix of dimention 2N×2NM(z_i+1,z_i)=( [α ββ^*α^* ]).The accurate calculation of this matrix is crucial in this approach. The complex matrix functions α and β depend strongly on the atomic or heterostructure potential profiles. The relationM_n=M^n=( [ α_n β_n β_n^* α_n^* ]),that was the source of errors in numerical calculations,<cit.> has been rigorously transformed, after defining the matrix function p_n-1=β^-1β_n, into the matrix-recurrence relation<cit.>p_n-(β^-1αβ+ α^*)p_n-1+p_n-2=0,with analytic solutions. In the single mode approximation, of interest here, this relation becomes the recurrence relation of Chebyshev polynomials of the second kind U_n, evaluated at the real part of α=α_R+iα_I. The n-cell transfer matrix elements, α_n and β_n, can straightforwardly be determined, through the simple relationsα_n=U_n-α^*U_n-1, andβ_n=β U_n-1.The eigenvalues of any quasi-bounded (qb) periodic system defined between z_L and z_R, see figure 1, with z_0-z_L=z_R-z_n=d/2,can be obtained by solving the equation<cit.>Re(α_ne^ikd)-k^2-q_w^2/2q_wk Im(α_ne^ikd)-k^2+q_w^2/2q_wkβ_nI=0.q_w and k are the wave numbers at the left (right) and right (left) of the discontinuity point z_L (z_R) and β_nI the imaginary part of β_n. The eigenfunctions of the quasi-bounded superlattice are given by<cit.>Ψ_μ,ν^qb(z)=a_o e^i k d/2/2k[((α_p+γ _p)α _j+(β_p+δ_p)β_j^∗)(k-iq_w ). + .((α _p+γ_p)β_j+(β_p+δ _p)α _j^∗) e^-i k d(k+iq_w)]_E=E_μ,ν,with a_o a normalization constant and z any point in the j+1 cell. α_j, β_j,... are matrix elements of the transfer matrix M_j(z_j,z_0) that connects the state vectorsat points separated by exactly j unit cells. α_p, β_p ... , where p stands for part of a unit cell, are the matrix elements of the transfer matrix M_p(z,z_j) that connects the state vectors at z_j and z, for z_j≤ z ≤ z_j+1.Our purpose here is to derive the effective mass approximation for the Schrödinger equation of a layered semiconductor heterostructure A/B/C..., using the eigenvalues andeigenfunctions obtained in the TFPS. We will assume, without loss of generality, that our system is a binary structure A/B/A...B/A, where the periodic semiconductor layers A=(a_A)^n_A and B=(b_B)^n_B contain n_A and n_B unit cells a_A and b_B, respectively, in the growing direction z. We will show that the effective-mass approximation (EMA) can be derived when the heterostructure wave function ψ(z) is written as the product Φ_ϵ_0,κ_0(z) Ψ^ϵ_0_μ,ν(z), where Ψ^ϵ_0_μ,ν(z) is the envelope function and Φ_ϵ_0,κ_0(z) is the fast-varying function obtained in the TFPS, evaluated at the band-edges defined by the energy band index ϵ_0 and the intra-band (or wave number) index κ_0. In the particular case of periodic heterostructures, i.e. of SLs (AB)^n=((a_A)^n_A(b_B)^n_B)^n, the envelope functions are straightforwardly obtained in the EMA and the TFPS. It is worth emphasizing that since the transfer matrices are the matrix representation of the continuity and boundary conditions and the phase evolution of the quantum states, it is clear that the fast-varying and envelope wave functions, obtained in the TFPS, fulfill the continuity and boundary conditions. We will show also, for a specific example, that the optical response calculated with the matrix elements ⟨Ψ^ϵ'_0_μ',ν'Φ^A_ϵ',κ'(z)|H_ int|Ψ^ϵ_0_μ,νΦ^A_ϵ,κ(z)⟩ is practically the same as the optical response obtained with the matrix elements ⟨Ψ^ϵ'_0_μ',ν'|H_ int|Ψ^ϵ_0_μ,ν⟩, were the fast-varying wave functions Φ^A_ϵ,κ(z) are ignored.§ AN ALTERNATIVE DERIVATION OF THE EFFECTIVE-MASS APPROXIMATIONSuppose now that for each layer X (with X equal A or B) we can write the one-particle Schrödinger equation(p^2/2m + V_X( r) ) Φ^X( r)=EΦ^X( r),where the potential V_X( r) is periodic, at least in the growing direction z. To simplify this problem we can follow the confined geometry method in Ref. [Bagwell] and the multichannel transfer matrix method in Refs. [PereyraPRL] and [PereyraJPA]. If we assume that the transverse widths are w_x and w_y and we write the potential V_X( r) as the sum of a confining potential V_X^C(x,y), which is infinite for |x|>w_x/2 and |y|>w_y/2, and the function V_X^L(x,y,z) periodic in z, the orthonormal wave functions χ_j(x,y), which are solutions of(-ħ^2/2m(∂^2/∂ x^2+∂^2/∂ y^2)+V_X^C(x,y))χ_j^X(x,y)=ε_j^Xχ_j^X(x,y),can be used to express the wave function Φ^X( r) asΦ^X( r)=∑_iχ_i^X(x,y)ϕ_i^X( z).If we replace this function in the Schrödinger equation (<ref>), multiply from the left by χ_j^X*(x,y) and integrate upon x and y, we obtain the set of coupled equations-ħ^2/2m∂^2/∂ z^2ϕ_j^X( z)+∑_i=1^N_XV_ij^X(z) ϕ_i^X( z)=(E-ε_j^X)ϕ_j^X( z). Here N_X is the number of propagating modes in layer X, or the number of open channels (defined by the condition E>ε_j^X), andV_ij^X(z)=∫_0^w_x∫_0^w_ydxdyχ_j^X*(x,y) V_X^L(x,y,z)χ_j^X(x,y),are the coupling-channels matrix elements. In this way the 3D multichannel problem is reduced into the 1D multichannel problem. It was shown in Refs. [PereyraPRL] and [PereyraJPA], and mentioned before, that a general solution for the 1D multichannel periodic system can be obtained in terms of the matrix polynomials p_n, when the single-cell transfer matrix M(z_i+1,z_i) is known. In actual semiconductor layers, the number of propagating modes depends on the Fermi energy and the cross section w_xw_y. When the multichannel problem for a specific semiconductor X, with n_X unit cells is solved, one obtains the N_Xn_X energy eigenvalues E^X_ϵ,η (which determine the conduction and valence bands) and the corresponding eigenfuntions ϕ^X_ϵ,η (z). In the widely used 1D one channel approximation, with V_X(z)=V_11^X(z), E^X=E-ε_1^X and ϕ^X(z)=ϕ_1^X( z), equation (<ref>) becomes( p_z^2/2m + V_X(z) ) ϕ^X( z)=E^Xϕ^X(z).In this limit and given the periodic atomic potentials V_A(z) and V_B(z), in the semiconductor layers A=(a_A)^n_A and B=(b_B)^n_B, one can obtain the unit-cell transfer matrices M_a and M_b and determine, applying the TFPS, the band structures E^A_ϵ,η and E^B_ϵ,η, andusing the Eq. (<ref>), the eigenfunctions ϕ^A_ϵ,η (z) and ϕ^B_ϵ,η (z). A very good approximation for the atomic potentials V_A(z) and V_B(z), are the effective potentials in the Hartree-Fock approximation. The quantum numbers ϵ denote the bands, and the quantum numbers η the intra-band energy levels.We will denote the valence and the conduction bands with ϵ=c=1 and ϵ=v=2, respectively.The intra-band energy levels correspond to η=1, 2, ... , n_X+1. In terms of these energies the fundamental energy gap in layer X is given byE^X_g=E^X_1,1-E^X_2,n_X+1≡E^X_ c,1-E^X_ v,n_X+1X=A,B.with E^X_ c,1 the first energy eigenvalue of the conduction band, i.e. the conduction band-edge denoted later as E^X_ϵ_0X, and E^X_ v,n_X+1 the last energy eigenvalue of the valence band, i.e. the upper-edge of the valence band. As is well known, the band edges of layers A and B do not coincide, in general (see figure <ref>), and their difference gives rise to the conduction and valence band split offs, as well as, to piecewise constant superlattice or heterostructure potential. <cit.> We will assume from here on that the semiconductor layers A and B are such that E^A_g < E^B_g. If energies are below the barrier height (E<V_b), see inset in figure <ref>, the eigenfunctions ϕ^A_ϵ,κ(z)=ϕ^A(z,E)|_E=E^A_ϵ,η are propagating functions while ϕ^B(z,E)|_E=E^A_ϵ,η are evanescent.<cit.>For each value of the quantum number η we have the corresponding wave number κ_η. To keep some analogy with conventional notation, we can represent the energy eigenvalues E_ϵ,η as E_ϵ,κ_η or just as E_ϵ,κ, that can be written also as E_ϵ(κ), keeping in mind that κ is discrete.It is clear that if we are able to determine the eigenvalues E^A,B_ϵ,κ and eigenfunctions ϕ^A,B_ϵ,κ, we are close to obtain the full solution for the heterostructure or SL. Having the wave functions ϕ^A,B_ϵ,κ, we must still fulfill the continuity and boundary conditions at the layered structure interfaces. Although this task could, in principle, be accomplished, it is not so simple for these functions (as for the envelope functions) and it is not our purpose here. We will, instead, turn our attention into the derivation of the effective mass approximation based on the existence of the set of rapidly-varying orthogonal functions ϕ^A,B_ϵ,κ.To derive the EMA in the TFPS we need to expand the heterostructure or SL wave functions ψ(z) in terms of the local wave functions ϕ^A_ϵ,κ(z) and ϕ^B_ϵ,κ(z), defined inside the layers A and B respectively. To simplify the discussion let us assume that we have the SL (AB)^nA. Ifζ = zmod [ l_c]-a, with a the width of layer A, l_c=a+b the length of the SL unit-cell, and H(w) is the Heaviside function, we can write a rapidly-varying wave function as (see figure <ref>)Φ^ϵ_B,κ_B_ϵ_A,κ_A(z) =H(-ζ )ϕ^A_ϵ_A,κ_A(zmod [l_c ])+ H(ζ )ϕ^B_ϵ_B,κ_B(zmod [l_c ]). As mentioned before, in the conventional derivations of the effective-mass approximation, the wave functions inside each layer are expanded in terms of the periodic parts of the band-edge Bloch functions, u_l,k_0^A or u_l,k_0^B, which are generally assumed to be equal.<cit.> Setting up the SL wave function ψ(z), the assumptions of only one-band and small k-vectors are also made.<cit.> In the theory of finite periodic systems, the bands and wave functions ϕ^A_ϵ,κ(z) and ϕ^B_ϵ,κ(z) are the energy eigenvalues and the eigenfunctions of the periodic systems (a_A)^n_A, (b_B)^n_B. In figure <ref> we show a simplified calculation in the TFPS of the energy spectrum<cit.> and transmission coefficients for a specific (confined and open) semiconductor A=(a_A)^n_A, with energy gap E_gA≃2.6eV and unit-cell length l_A= 5.15nm.On the left hand side of figure <ref>, we show the valence and the conduction bands (VB and CB) of the periodic sequence (a_A)^n_A bounded by cladding layers C, and, on theright hand side, the transmission coefficients through the same semiconductor but open. At the top of the left hand side column, we plot also the subbands (or minibands) of the SL (AB)^n for E_gA≃2.6eV, E_gB≃2.9eV, l_A∼ l_B 5.15nm and n=10. These graphs show that as the layer widthw_A=l_An_A gets thinner, the energy levels separation, Δ E_c, and theenergy-levels widths, Γ E_μ, increase. On the other hand, it is known that whereas the energy gap E_gA remains constant when the number of unit cells n_A varies, the subbands of the superlattice (AB)^n, for a fixed barrier width w_B, move with the band-edge energy level upwards when n_A decreases, and downwards when n_A, hence w_A, increases.This behavior of the energy spectra, justifies the one-band `ansatz' and strengthens the relevance of the band-edge functions as the number of unit cells n_A gets smaller. In the specific example of figure <ref>, the level widthΔΓ_1is of the order of the subband widths ∼ 10meV), and the energy levels separation for a semiconductor with n_A ∼5 (w_A∼ 25nm) is approximately 600meV, which is much larger than the bands split off in the conduction and valence bands of layers A and B. Thus, in order to define the heterostructure or SL wave function ψ (z) in terms of the envelope and the fast-varying functions, it is justified to consider the band-edge and one-band assumptions. Therefore, we can consider the expansionψ(z)= ∑_κ^A_0,κ^B_0⟨ϵ_0,κ_0 |ψ⟩Φ_ϵ_0,κ_0(z).Here and in the following,the quantum numbers ϵ_0 and κ_0 represent the set ϵ^A_0,ϵ^B_0 and κ^A_0,κ^B_0, respectively. For a simple and compact notation, we will denote the expansion coefficient ⟨ϵ_0,κ_0 |ψ⟩, known also as the envelope function, as φ^ϵ_0_κ_0(z) or φ^ϵ_0(κ_0,z). If we introduce thefunction ψ(z) of Eq. (<ref>) into the SL Schrödinger equation( p_z^2/2m + V_SL(z) ) ψ(z)=Eψ(z),whereV_SL(z)= H(-ζ )V_A(zmod[l_c ])+ H(ζ )V_B(z mod [l_c ]),multiply by Φ_ϵ_0,κ'_0(z) and integrate, we have∑_κ^A_0,κ^B_0[ H(-ζ )E_ϵ^A_0,κ^A_0δ_κ^A_0,κ^A'_0 +H(ζ )E_ϵ^B_0,κ^B_0δ_κ^B_0,κ^B'_0]⟨ϵ_0,κ_0 |ψ⟩ = E ⟨ϵ_0,κ_0 |ψ⟩.SinceE_ϵ^B_0,κ^B_0=E_ϵ^A_0,κ^A_0+ V_κ^A_0,κ^B_0= E_ϵ^A_0,κ^A_0+⟨κ^A_0|V_ϵ P|κ^B_0⟩ ,the sectionally constant periodic potential V_P(z), known as the split off, appears here naturally as a consequence of the difference in the energy band structures oflayers A and B, both in the conduction and valence bands. Therefore, we are left withE^A_ϵ^A_0(κ^A_0)φ^ϵ_0(κ^A_0) +∑_κ^B_0⟨κ^A_0|V_ϵ P|κ^B_0⟩φ^ϵ_0(κ^B_0) = E φ^ϵ_0(κ^A_0).We can now, as usual, multiply by (1/Ω) e^i κ zand sum the Fourier series to obtainE^A_ϵ_0(-i∂/∂ z)Ψ^ϵ_0(z) +V_P (z) Ψ^ϵ_0(z) = E Ψ^ϵ_0(z).If we further approximate E_ϵ_0(-i∂/∂ z) by a quadratic function of -i∂/∂ z, near the band edge, assuming that the k-vector at the edge is small and an effective mass m^*_ϵ_0, defined as usual for each layer, we have[p_z^2/2 m^*_ϵ_0+V_P (z)] Ψ^ϵ_0_μ,ν(z) = (E -E^A_ϵ_0,η_0)_μ,νΨ^ϵ_0_μ,ν(z),with ϵ_0=c and η_0=1 for the conduction band and ϵ_0=v and η_0=n_A+1 for the valence band. If we define the energy eigenvaluesE_μ,ν=(E -E^A_ϵ_0,η_0)_μ,ν,measured from the band edges, we can write the Schrödinger equation in the effective mass approximation[p_z^2/2 m^*_ϵ_0+V_P (z)] Ψ^ϵ_0_μ,ν(z) = E_μ,νΨ^ϵ_0_μ,ν(z),that we were looking for and was used for SLs and heterostures, without a specific proof. As mentioned before, for SLs we can use the TFPS to solve this equation and to determine the eigenvalues E_μ,ν and the eigenfunctions Ψ^ϵ_0_μ,ν(z), known as envelope functions. It is worth noting that this derivation of EMA does not require that the layered structure be periodic. Therefore, the EMA is valid for any layered heterostructure. All the assumptions behind this derivation imply that the wave functions ψ(z) can be written asψ(z)→Ψ^ϵ_0_μ,ν(z)Φ_ϵ_0,η_0(z)with Ψ^ϵ_0_μ,ν(z) the SL eigenfunction (envelope functions) andΦ_ϵ_0,η_0(z) the rapid oscillating wave functions. In figure <ref> we plot the functions Ψ^c_1,1(z) and Φ_c,1(z), in the conduction band, and the functions Ψ^v_2',1'(z) and Φ_v,1(z) of the valence band. these functions can in principle be determined within the TFPS.Dealing with transport properties, one can neglect thefunction Φ_ϵ_0,η_0(z), however, for calculations involving two bands, the whole wave function ψ (z) should, in principle, be considered. We will show now that the fast-varying factor Φ_ϵ_0,η_0(z) can effectively be ignored in optical response calculations. § ON THE REDUNDANCY OF THE FAST-VARYING FUNCTIONS To determine the effect of the rapidly-oscillating factor Φ_ϵ_0,η_0(z) on the optical response, let us consider the blue emitting (In_0.2Ga_0.8N\ In_0.05Ga_0.95N)^10\ In_0.2Ga_0.8N superlattice studied in Refs. [NakamuraPaper] and [PereyraEPL]. We will calculate the optical response=cmr6 0=χ__ΦΨ= ∑_ν,ν'f_eh|⟨ψ^v_ f |H_ int|ψ^c_ i⟩|^2/(ħω-E_1,ν^c+E_2',ν'^v+E_B)^2+Γ^2taking into account the fast-varying functions Φ_ϵ_0,η_0(z), which means ψ^c_ i = ψ^c,1_1,ν(z) = Φ_c,1(z)Ψ^c_1,ν(z) and ψ^v_ f = ψ^v,n_A+1_2',ν'(z) = Φ_v,n_A+1(z)Ψ^v_2',ν'(z). These results are compared in figure (<ref>) with the optical response=cmr12 12=χ__Ψ= ∑_ν,ν'f_eh|⟨Ψ^v_2',ν' |H_ int|Ψ^c_1,ν⟩|^2/(ħω-E_1,ν^c+E_2',ν'^v+E_B)^2+Γ^2calculated in Ref. [PereyraEPL], ignoring the fast-varying functions. As was shown in this reference and can be seen in figure <ref>, this optical response agrees extremely well with the experimental results in panel (c). <cit.> In (<ref>) and (<ref>), ħω is the emitted photon energy,E^c_1,ν the energy levels in the first subband of the CB, E^v_2',ν' the (heavy hole) energy levels in the second subband of the VB, E_B the exciton binding energy, f_eh the occupation probabilities and Γ the level broadening energy. Besides the overall amplification, by a factor of ≃ 2.4, our calculations show that the rapidly-varying functions have no effect on the optical spectrum.According with the mean value theorem for definite integrals, the optical response =cmr6 0=χ__ΦΨ in equation (<ref>) can be written as=cmr6 0=χ__ΦΨ= ∑_ν,ν'ϕ^c,ν_v,ν'(z_0)=cmr12 12=χ__Ψwith ϕ^c,ν_v,ν'(z_0) a number, which in principle depends on the quantum numbers ν and ν'. Specific calculations show that this factor is almost constant (see figure <ref>), and consistent with the differences in the numerical values of the optical responses χ__ΦΨ and χ__Ψ in figure <ref>. § CONCLUSIONS We have derived the effective mass approximation for the Schrödiger equation of layered hetrostructures, based on the energy eigenvalues and rapidly- oscillating eigenfunctions obtained, for each layer, in the theory of finite periodic systems. This derivation that is based on physical quantities of finite structures explains why the EMA works so well when applied to this kind of systems. We have shown also that, in order to calculate interband transition matrix elements, the rapidly-oscillating wave functions Φ_ϵ_0,η_0(z), that should be multiplied by the envelope functions, Ψ^ϵ_0_μ,ν(z), can safely be ignored.§ ACKNOWLEDGEMENT I acknowledge the useful comments of Herbert P. Simanjuntak. Wannier G. H. Wannier Phys. Rev. 52 191 (1937). Luttinger J. M. Luttinger and W. KohnPhys. Rev. 97 869 (1955) Altarelli M. Allarelliand F. Bassani Handbook on Semiconductom vol I Band Theory and Transport Properties vol 1, ed W. Paul (Amsterdam: North-Holland 1982) p 269. AltarelliLesHuches M. Altarelli inHeterojunctions and Semiconductor Superlattices: Proceedings of the Winter School Les Houches Ed. byGuy Allan and Gerald Bastard, France, March 12-21, 1985. PollakF. H. Pollak and M. Cardona J. Phys. Chem. 27 423 (1966) Dingle R. Dingle, W. Wiegmann and C. H. HenryPhys. Rev. Lett. 33, 827 (1974). Slater J. C. Slater, Phys. Rev. 76, 452 (1949). BurtM. G. Burt J. Phys: Condens. Matter 4 6551490(1992), and references therein. AbelesF. Abeles, Ann. Phys. (Paris) 5, 596 (1950);5, 706 (1950). Erdos P. Erdös and R. C. Herndon, Adv. Phys. 31, 65 (1982). Claro1982 M. Pacheco and F. Claro, Phys: Status Solidi B 114, 399 (1982). Ricco B. Ricco and M.Ya. Azbel, Phys. Rev. B 29, 1970 (1984). Vezzetti D. J. Vezzetti and M. Cahay, J. Phys. D 19, L53 �55 (1986). Kolatas T. H. Kolatas and A. R. Lee, Eur. J. Phys. 12, 275 (1991). Griffiths D. J. Griffiths and N. F. Taussing, Am. J. Phys. 60, 883 (1992). PeisakovichYu. G. Peisakhovich, J. Phys. A, 29, 5103 (1996)PereyraPRL P. Pereyra, Phys. Rev. Lett. 80, 2677(1998). PereyraJPA P. Pereyra,J. Phys. A 31, 4521 (1998). PereyraCastilloP. Pereyra and E Castillo, Phys. Rev. B 65, 205120 (2002).Keldysh1962L. V. Keldysh Fiz. Tverd. Tela 4, 2265 (1962), English translation, in Sov. Phys. -Solid State 4, 1658 (1963). EsakiTsu1970 L. Esaki and R. Tsu, IBM J. Res. Develop. 14, 61 (1970); Esaki1972 L. Esaki, L. L. Chang, W. E. Howard, and V. L. Bideout,in Proceedings of the Eleventh International Conference on th'e Physics of Semiconductors, Warsaw, Poland, 1972, edited by The Polish Academy of Sciences (PWN-Polish Scientific Publishers, Warsaw, Poland, 1972), p. 431Dingle1974 R. Dingle, W. Wiegmann and C. H. HenryPhys. Rev. Lett. 33, 827 (1974). Mukherji1975 D. Mukherji, Phys. Rev. B 12, 4338 (1975). Miller1976 R. C. Miller, R. Dingle, A. C. Gossard, R. A. Logan, W. A. Nordland Jr. and W. Wiegmann, J. Appl. Phys. 47, 4509 (1976). Chang1977 L. L. Chang, H. Sakaki, C. -A. Chang and L. Esaki, Phys. Rev. Lett. 45, 1489 (1977). SaiHalaszChang1978G. A. Sai-Halasz, L.L. Chang, J. -M. Welter, C. -A. Chang and L. EsakiSol. Stat. Comm. 27, 935 (1978). Capasso1986 F. Capasso, K. Mohammed and A. Y. Cho, IEEE J. Quantum Elect. QE22, 1853 (1986). LuoFurdyna1990 H. Luo and J. K. Furdyna, Phys. Rev. B 41, 5188 (1990). Rauch1997 C. Rauch, G. Strasser, K. Unterrainer, E. Gornik and B. Brill,Appl. Phys. Lett. 70, 679 (1997). Petrov1997 M. V. Petrov and S. Lyon,Appl. Phys. Lett. 70, 3269 (1997). Heer1998 R. Heer, J. Smoliner, G. Strasser and E. Gornik,Appl. Phys. Lett. 73, 3138 (1998).EsakiLesHuchesL. Esaki, inHeterojunctions and Semiconductor Superlattices: Proceedings of the Winter School Les Houches Ed. byGuy Allan and Gerald Bastard, France, March 12-21, 1985.DresselhaussG. Dresselhaus, J. Phys. Chem. Solids 1, 14 (1956). BreiteneckerM. Breitenecker, R. Sexl and W. Thirring,ZeitSchrifty. Phys. 182, 123 (1964). Sanders G. D. Sanders and Y-Ch. Chang Phys. Rev. B 32, 5517 (1985).Bastard1987NATO G. Bastard, in Proceedings of the NATO Advanced Study Institute on Molecular Beam Epitaxy (MBE) and Heterostructures, Series E: Applied Sciences- No. 87, Ed. by L.L. Chang and K. Ploog, Erice, Italy, March 7-19, 1983. SmithD. L. Smith and C. Mailhiot, Rev. Mod. Phys.62, 173 (1990). BaraffG. A. Baraff and D. Gershoni, Phys. Rev. B 43, 4011 (1991).Pereyra2005 P. Pereyra, Ann. Phys. 320, 1 (2005). LuqueJ. M. Luque-Raigon, J. Halme, H. Miguez and G. Lozano, J. Opt. 15, 125719 (2013). BagwellP. F. Bagwell, Phys. Rev. B 41, 10354 (1990). Enderleinpg252 R. Enderlein and N.J.M. Horing, Fundamentals of semiconductor physics and devices, (World Scientific, Singapure, New Jersey, 1997). Bastardpg67G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, (Les Editions de Physique, Les Ulis Cedex, France 1988). simplified The spectrum and transmission coefficients in this figure were obtained for a simplified and approximate piecewise constant potential. The potential parameters were chosen to fit the accurate results that will be published elsewhere. NakamuraPaper Nakamura S., Senoh M., Nagahama S., Iwasa N., Yamada Ta., Matsushita T., App. Phys. Lett. 38, 3269-3271 (1996). PereyraEPL P. Pereyra, Europhysics Letters 118, 14002 (2017). | http://arxiv.org/abs/1706.08673v2 | {
"authors": [
"Pedro Pereyra"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170627051316",
"title": "Why the effective-mass approximation works so well for nano-structures"
} |
M.B. BARBARO et al.NUCLEAR DEPENDENCE OF THE 2P2H RESPONSE1,2, J.E. Amaro3, J.A. Caballero4, A. De Pace2, T.W. Donnelly5, G.D. Megías4, I. Ruiz Simo3 Dipartimento di Fisica, Università di Torino, Italy1 Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Italy2 Departamento de Fisica Atomica, Molecular y Nuclear and Instituto de Fisica Teorica y Computacional Carlos I, Universidad de Granada, Spain3 Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Sevilla, Spain4 Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, USA5We present the results of a recent study <cit.> of meson-exchange two-body currents in lepton-nucleus inclusive scattering at various kinematics and for different nuclei within the Relativistic Fermi Gas model.We show that the associated nuclear response functions at their peaks scale as A k_F^2, for Fermi momentum k_F going from 200 to300 MeV/c and momentum transfer q from 2k_F to 2 GeV/c. This behavior is different from what is found for the quasielastic response, which scales as A/k_F. This result can be valuable in the analyses of long-baseline neutrino oscillation experiments, which need to implement these nuclear effects in Monte Carlo simulations for different kinematics and nuclear targets.13.15.+g, 25.30.Pt The study of two-particle two-hole (2p-2h) excitations in lepton-nucleus scattering has gathered renewed attention over the last few years owing to its importance for neutrino oscillation experiments. While the main purposes of these experiments are the precise measurement of neutrino properties and the exploration of new physics beyond the Standard Model of particle physics, the data analysis strongly relies on the input from nuclear physics. The analysis of the currently operating (T2K, NOvA) and future (T2HK, DUNE) long-baseline neutrino oscillation experiments requires indeed a very precise knowledge of neutrino-nucleus cross sections over an energy regime going from hundreds to thousands of MeV. The main difficulty in the interpretation of the data arises from the ignorance of the exact incoming neutrino energy, which is widely distributed around its average value according to a certain flux Φ(E_ν). The reconstruction of the neutrino energy from the observed reaction products is obviously strongly dependent upon the nuclear model used in the analysis.Hence the need for an accurate description of the many-body nuclear system and of different reaction mechanisms, from the quasielastic (QE) region up to the deep-inelastic scattering one. This has motivated an intense recent activity on this subject. An updated review of the status and challenges of the field can be found in Ref. <cit.>.2p2h excited states have been extensively explored in the past <cit.>-<cit.>in electron scattering studies: they correspond to the ejection of two nucleons above the Fermi level, with the associated creation of two holes inside the Fermi sea, and give a large contribution to the inclusive (e,e') cross section in the so-called “dip region” lying between the QE and Δ(1232) excitation peaks (see <cit.> for an exhaustive comparison with electron scattering data on ^12C). In neutrino scattering,2p-2h excitations have been shown <cit.>-<cit.> to play a crucial role in explaining the cross sections measured in the MiniBooNE, MINERvA and T2K experiments <cit.>.Whereas most of the existing work on the 2p2h contribution to neutrino-nucleus cross section refers to a carbon target <cit.>,it is becoming more and more important to extend the calculationsto heavier nuclei, in particular argon and oxygen, which also are and will be used as targets in neutrino oscillation experiments.The exact evaluation of the 2p2h cross section involves a 7-dimensional integral for each value of the energy and momentum transfer, and an additional integral over the experimental neutrino flux should be performed before comparing with the data. Although this can be done in principle for different nuclear targets, it is useful to provide an estimate of the density dependence of these contributions, which can be used to extrapolate the results from one nucleus to another. This is the main motivation of the present study.The lepton-nucleus inclusive cross section can be described in terms of response functions, which embody the nuclear dynamics.There aretwo response functions in the case of electron scattering, R^L and R^T, and five in the case of charged-current (anti)neutrino scattering, R^CC, R^CL, R^LL, R^T, and R^T', all of them depending upon the momentum and energy transfer (q,ω). Each response function is related to specific components of the hadronic tensor W^μν(q,ω) and receives contributions from different reaction mechanisms (one-body knockout, two-body knockout, resonance excitation, etc.) depending on the kinematics.Before examining the two-body contribution, it is worth reminding how the one-body cross section depends on the nuclear density. In <cit.> inclusive electron scattering data from various nuclei were analyzed in terms of “superscaling”: it was shown that, for energy loss below the quasielastic peak, the scaling functions, represented versus an appropriate dimensionless scaling variable, are not only independent of the momentum transfer (scaling of first kind), but they also coincide for mass number A≥4 (scaling of second kind).More specifically, the reduced QE cross section was found to scale as A/k_F, k_F being the Fermi momentum.It was also shown that for higher energy transfers superscaling is broken and that its violations reside in the transverse channel rather than in the longitudinal one. Such violations must be ascribed to reaction mechanisms different from one-nucleon knockout. Two-particle-two-hole excitations, which are mainly transverse and occur in the region between the quasielastic and Δ production peaks, are – at least in part – responsible for this violation. The model we use to evaluate the 2p-2h nuclear responses is based on the Relativistic Fermi Gas (RFG), where it is possible to perform an exact relativistic calculation. It should be stressed that in the GeV energy regime we are interested in these effects cannot be ignored. In the RFG model relativity affects not only the kinematics, but also the nuclear current matrix elements, which are different from the non-relativistic ones. The two-body meson exchange currents (MEC) used in this work are represented in Fig. <ref> for the weak case (where the wavy line represents a W-boson) and are deduced from a fully relativistic Lagrangian including nucleons (solid lines), pions (dashed lines) and Δ(thick lines in diagrams f-i) degrees of freedom. In the electromagnetic case the diagrams (d) and (e) are absent and the wavy line represents the exchanged photon. These elementary diagrams give rise to a huge number of many-body diagrams, each of them involving a 7-dimensional integral. In order to speed up the calculation we have recently proposed and tested approximate schemes, capable of reducing the dimensionality of the integralto 1 <cit.> or 3 <cit.>. However here we employ the exact results. Further details of the model can be found in <cit.> for the electromagnetic case and in <cit.> for the extension to the weak sector.Since the behavior with density of the nuclear response is not expected to depend very much on the specific channel or on the nature of the probe, for sake of illustration we focus on the electromagnetic 2p-2h transverse response, which largely dominates over the longitudinal one.Our starting point is therefore the electromagnetic transverse response, R^T_ MEC, which is displayed in Fig. <ref> as a function of the energy transfer ω for momentum transfers q ranging from 50 to 2000 MeV/c and three values of k_F from 200 to 300 MeV/c, the typical range of Fermi momenta to which most nuclei belong <cit.>. In order to appreciate the relevance of the MEC response as compared to QE one, in Fig. <ref> we show both response functions for two values of the momentum transfer and two different k_F. The MEC response is also splitted into itsΔΔ, ππ and πΔ-interference components, showing that the ΔΔ contribution is dominant in all cases. It appears that, while for a relatively light nucleus (k_F=200 MeV/c) the MEC peak drops from ∼30% to ∼10% of the QE one when the momentum transfer q goes from 0.8 to 2 GeV/c, for a heavy nucleus (k_F=300 MeV/c) the MEC peak is almost as high as the QE one at q=0.8 GeV/c and about half of it for q=2 GeV/c. This shows that the k_F-behavior of the 2p2h response is very different from the above mentioned 1/k_F scaling law typical of the QE response (scaling of second kind). The results in Fig. <ref> also show that the 2p2h response drops faster than the quasielastic one as the momentum transfer increases, according to the fact that the MEC also break scaling of first kind. In order to explore the k_F-behavior of R^T_ MEC, we first remove the single-nucleon physics from the problem (which also causes the fast growth of the response as ω approaches the light-cone) and we define the following reduced response (per nucleon) F^T_ MEC(q,ω) ≡R^T_ MEC(q,ω)/ Z G_Mp^2(τ) + N G_Mn^2(τ) ,where τ≡ (q^2-ω^2)/(4m_N^2) and G_Mp and G_Mn are the proton and neutron magnetic form factors. For simplicity here we neglect in the single-nucleon dividingfactor small contributions coming from the motion of the nucleons, where the electric form factor contributes, which depend on the Fermi momentum <cit.>. In the upper panels of Fig. <ref> we show the response R^T_ MEC and the reduced response F^T_ MEC for q=800 MeV/c and the same three values of k_F used above. It clearly appears that the 2p-2h response, unlike the 1-body quasielastic one, increases as the Fermi momentum increases. In the lower panels of Fig. <ref> we display the scaled 2p-2h MEC response, defined asF^T_ MEC(ψ^'_ MEC) ≡F^T_ MEC/η_F^2 ,namely the reduced response divided by η_F^2 ≡ (k_F/m_N)^2, as a function of the MEC scaling variable ψ^'_ MEC(q,ω,k_F) (left panel) and of the quasielatic one ψ^'_ QE(q,ω,k_F) (right panel). The MEC scaling variable is defined in Ref. <cit.>, in analogy with the usual QE scaling variable <cit.>.The results show that the reduced 2p-2h response per nucleon roughly scales as k_F^2 when represented as a function of ψ^'_ MEC (Fig. <ref>c), i.e., the scaled 2p-2h MEC response shown there coalesces at the peak into a universal result. This scaling law is very accurate at the peak of the 2p-2h response, while it is violated to some extent at large negative values of the scaling variable. Fig. <ref>d shows that in this “deep scaling” region it is more appropriate to use the usual scaling variable ψ^'_ QE devised for quasielastic scattering. This latter region was previously investigated in <cit.>, where the specific cases of ^12C and ^197Au were considered and the results were compared withJLab data at electron energy ϵ=4.045 GeV: there it was shown that at very high momentum transfers the 2p-2h MEC contributions are very significant in this deep scaling region, to the extent that they may even provide the dominant effect. Nevertheless, the scaling violations associated to them were shown to be reasonably compatible with the spread found in the data.A closer inspection of the scaling properties of the 2p-2h response, performed in Ref. <cit.>, has also shown that all the contributions (ΔΔ, ππ and πΔ) roughly grow as k_F^2, the quality of scaling being better for the ΔΔ piece than for the other two contributions. Furthermore, at high momentum transfer the total MEC response scales better than the pure Δ piece around the peak, indicating a compensation of scaling violations between the three terms.In Fig. <ref> the scaled 2p-2h MEC response is now plotted versus ψ^'_ MEC for four values of q. Here we seethat the same k_F-dependence is valid for different values of q as long as Pauli blocking is not active, namely q>2 k_F. At lower q and in the deep scaling region this type of scaling is seen to be broken.Finally, focusing on practical cases, in Fig. <ref> we show R^T_ MEC versus ω, together with F^T_ MEC andf^T_MEC≡ F^T_MEC× k_Fversus ψ^'_ QE for three values of q and for the symmetric nuclei ^4He, ^12C, ^16O and ^40Ca. The case of asymmetric nuclei, Z≠ N requires more involved formalism and will be addressed in future work, although preliminary studies indicate that the qualitative behavior with k_F does not change dramatically unless N-Z is very large.The cases of ^12C and ^16O are clearly relevant for ongoing neutrino oscillation studies, whereas the case of ^40Ca is a symmetric nucleus lying close to the important case of ^40Ar. For comparison, ^4He is also displayed and, despite its small mass, is seen to be “typical”. In contrast, the case of ^2H, whose Fermi momentum is unusually small (k_F= 55 MeV/c), was also explored and found to be completely anomalous: the MEC responses (R^T_ MEC) and superscaling results (f^T_ MEC) were both too small to show in the figure. Summarizing, the 2p-2h MEC response function per nucleon roughly grows as k_F^2 for Fermi momenta varying from 200 to 300 MeV/c.This scaling law is excellent around the MEC peak for high values of q, it starts to break down around q = 2 k_F, and gets worse and worse as q decreases.This behavior must be compared with that of the 1-body response, which scales as 1/k_F: hence the relative importance of the 2p-2h contribution grows as k_F^3.This result allows one to get an estimate of the relevance of these contributions for a variety of nuclei, of interest in ongoing and future neutrino scattering experiments, and should facilitate the implementation of 2p-2h effects in event generators. § ACKNOWLEDGEMENTS This work was partially supported by the INFN under Project MANYBODY, by the University of Turin under Project BARM-RIC-LOC-15-02, by the Spanish Ministerio de Economia y Competitividad and ERDF (European Regional Development Fund) under contracts FIS2014-59386-P, FIS2014-53448-C2-1, by the Junta de Andalucia (grants No. FQM-225, FQM160), and part (TWD) by the U.S. Department of Energy under cooperative agreement DE-FC02-94ER40818. IRS acknowledges support from a Juan de la Cierva-incorporacion program from Spanish MINECO. GDM acknowledges support from a Junta de Andalucia program (FQM7632, Proyectos de Excelencia 2011).99 Amaro:2017eahJ. E. Amaro, M. B. Barbaro, J. A. Caballero, A. De Pace, T. W. Donnelly, G. D. Megias and I. Ruiz Simo,Phys. Rev. C95, no. 6, 065502 (2017). WP L. Alvarez-Ruso et al., “NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering,” arXiv:1706.03621 [hep-ph]. Donnelly:1978xaT. W. Donnelly, J. W. Van Orden, T. De Forest, Jr. and W. C. Hermans,Phys. Lett.76B, 393 (1978). Van81 J. W. Van Orden and T. W. Donnelly, Annals Phys. 131 (1981) 451.Alb84 W.M. Alberico, M. Ericson, and A. Molinari,Annals Phys. 154 (1984) 356.Ama93 J. E. Amaro, G. Co', A. M. Lallena, Annals Phys. 221 (1993) 306.Ama94 J. E. Amaro, A. M. Lallena and G. Co, Nucl. Phys. A 578 (1994) 365.Alb91 W.M. Alberico, A. De Pace, A. Drago, and A. Molinari, Riv. Nuov. Cim. vol. 14, (1991) 1.Gil97 A. Gil, J. Nieves, and E. Oset, Nucl. Phys. A 627 (1997) 543.Dek91 M.J. Dekker, P.J. Brussaard, and J.A. Tjon, Phys. Lett. B 266 (1991) 249.Dek92 M.J. Dekker, P.J. Brussaard, and J.A. Tjon, Phys. Lett. B 289 (1992) 255.Dek94 M.J. Dekker, P.J. Brussaard, and J.A. Tjon, Phys. Rev. C 49 (1994) 2650. DePace03 A. De Pace, M. Nardi, W. M. Alberico, T. W. Donnelly and A. Molinari, Nucl. Phys. A 726, 303 (2003).DePace04 A. De Pace, M. Nardi, W. M. Alberico, T. W. Donnelly and A. Molinari, Nucl. Phys. A 741, 249 (2004).Meg16e G. D. Megias, J. E. Amaro, M. B. Barbaro, J. A. Caballero and T. W. Donnelly, Phys. Rev. D94 (2016) 013012.Mar09 M. Martini, M. Ericson, G. Chanfray, J. Marteau, Phys. Rev. C80 (2009) 065501.Ama10bJ.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Donnelly and C.F. Williamson,Phys. Lett. B696 (2011) 151.Mar10 M. Martini, M. Ericson, G. Chanfray, J. Marteau,Phys.Rev. C81 (2010) 045502. Nie11 J. Nieves, I. Ruiz Simo, M.J. Vicente Vacas,Phys.Rev. C83 (2011) 045501.Nie12 J. Nieves, I. Ruiz Simo, M.J. Vicente Vacas,Phys.Lett. B707 (2012) 72.Ama11J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Donnelly,C.F. Williamson, Phys.Lett. B696 (2011) 151.Ama12 J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Donnelly,Phys.Rev.Lett. 108 (2012) 152501.Van16T. Van Cuyck, N. Jachowicz, R. Gonzalez-Jimenez, M. Martini, V. Pandey, J. Ryckebusch and N. Van Dessel, Phys. Rev. C 94, 024611 (2016). Van17 T. Van Cuyck, N. Jachowicz, R. Gonzalez-Jimenez, J. Ryckebusch and N. Van Dessel, arXiv:1702.06402 [nucl-th].Ama10a J. E. Amaro, C. Maieron, M. B. Barbaro, J. A. Caballero and T. W. Donnelly, Phys. Rev. C 82 (2010) 044601.Mini1 A.A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], Phys. Rev. D81 (2010) 092005.Mini2 A.A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], Phys. Rev. D88 (2013) 032001.Minerva1 P.A. Rodrigues et al. [MINERvA Collaboration], Phys. Rev. Lett. 116, 071802 (2016).Minerva2 G.A. Fiorentini et al. (MINERvA Collaboration),Phys. Rev. Lett. 111 (2013) 022502.T2K1 K. Abe et al. (T2K Collaboration),Phys. Rev. D87 (2013) 092003.T2K2 K. Abe et al. (T2K Collaboration),Phys. Rev. D90 (2014) 052010. Ben15O. Benhar, A. Lovato and N. Rocco, Phys. Rev. C92 (2015) 024602.Roc16 N. Rocco, A. Lovato and O. Benhar, Phys. Rev. C94 (2016) 065501.Meg16nuG. D. Megias, J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly and I. Ruiz Simo,Phys. Rev. D 94 (2016) 093004. Ivanov:2015aya M. V. Ivanov, G. D. Megias, R. González-Jiménez, O. Moreno, M. B. Barbaro, J. A. Caballero and T. W. Donnelly,J. Phys. G43 (2016) no.4,045101. Ivanov:2015wpa M. V. Ivanov et al.,Phys. Rev. C91 (2015) no.3,034607.Ivanov:2016vul M. V. Ivanov et al.,J. Phys. Conf. Ser.724 (2016) no.1,012020. Ivanov:2016mzo M. V. Ivanov et al.,AIP Conf. Proc.1722 (2016) 030005. Sim16 I. Ruiz Simo, J. E. Amaro, M. B. Barbaro, A. De Pace, J. A. Caballero and T. W. Donnelly, J. Phys. G44 (2017) 065105.Kat16T. Katori and M. Martini,arXiv:1611.07770 [hep-ph].Don99-1T. W. Donnelly and I. Sick, Phys. Rev. Lett. 82 (1999) 3212.Don99-2T. W. Donnelly and I. Sick, Phys. Rev. C60 (1999) 065502. RuizSimo:2017onbI. Ruiz Simo, J. E. Amaro, M. B. Barbaro, J. A. Caballero, G. D. Megias and T. W. Donnelly,Phys. Lett. B 770, 193 (2017).MCA I. Ruiz Simo, J. E. Amaro, M. B. Barbaro, J. A. Caballero, G. D. Megias and T. W. Donnelly,arXiv:1706.06377 [nucl-th].ChiaraC. Maieron, T. W. Donnelly and I. Sick, Phys. Rev. C 65 (2002) 025502.Amaro:2004bsJ. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, A. Molinari and I. Sick, Phys. Rev. C 71 (2005) 015501.scaling W. M. Alberico, A. Molinari, T. W. Donnelly, E. L. Kronenberg and J. W. Van Orden,Phys. Rev. C 38 (1998) 1801. | http://arxiv.org/abs/1706.08887v1 | {
"authors": [
"M. B. Barbaro",
"J. E. Amaro",
"J. A. Caballero",
"A. De Pace",
"T. W. Donnelly",
"G. D. Megias",
"I. Ruiz Simo"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170627143419",
"title": "Nuclear dependence of the 2p2h electroweak response in the Relativistic Fermi Gas model"
} |
rsfs OMScmsymn | http://arxiv.org/abs/1706.08480v4 | {
"authors": [
"Laura Bernard",
"Luc Blanchet",
"Alejandro Bohé",
"Guillaume Faye",
"Sylvain Marsat"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170626170145",
"title": "Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order"
} |
Saarland University, Fachbereich Mathematik,66041 Saarbrücken, Germany [email protected], [email protected] The second author was partially funded by the ERC Advanced Grant NCDFP, held by Roland Speicher. This work was part of the first author's Master's thesis. This work was also supported by the DFG project Quantenautomorphismen von Graphen [2010]46LXX (Primary); 20B25, 05CXX (Secondary) The study of graph C^*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C^*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C^*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.Quantum symmetries of graph C^*-algebras Moritz Weber December 30, 2023 ========================================§ INTRODUCTION Symmetry constitutes one of the most important properties of a graph. It is captured by its automorphism group(Γ):={σ∈ S_n | σϵ=ϵσ}⊂ S_n,where Γ=(V,E) is a finite graph with n vertices and no multiple edges, ϵ∈ M_n({0,1}) is its adjacency matrix, and S_n is the symmetric group. In modern mathematics, notably in operator algebras, symmetries are no longer described only by groups, but by quantum groups. In 2005, Banica <cit.> gave a definition of a quantum automorphism group of a finite graphwithinWoronowicz's theory of compact matrix quantum groups <cit.>. In our notation,(Γ) is based on the C^*-algebra C( (Γ)):= C(S_n^+) / ⟨ uϵ=ϵ u⟩=C^*(u_ij, i,j=1,…,n |u_ij=u_ij^*=u_ij^2, ∑_l u_il=1=∑_l u_lj, R_Ban hold),where S_n^+ is Wang's quantum symmetric group <cit.> and R_Ban are the relations ∑_k u_ikϵ_kj=∑_k ϵ_iku_kj.Earlier, in 2003, Bichon <cit.> defined a quantum automorphism group (Γ) viaC( (Γ)):=C^*(u_ij, i,j=1,…,n |u_ij=u_ij^*=u_ij^2, ∑_l u_il=1=∑_l u_lj, R_Bic hold),where R_Bic are the relations∑_k u_ikϵ_kj=∑_k ϵ_iku_kj,u_s(e)s(f)u_r(e)r(f)=u_r(e)r(f)u_s(e)s(f) fore,f∈ E,and r:E→ V and s:E→ V are range and source maps respectively. We immediately see that(Γ)⊂(Γ)⊂(Γ)holds, in the sense that there are surjective ^*-homomorphisms:C((Γ))→C((Γ)) →C((Γ))u_ij↦u_ij↦(σ↦σ_ij)Relatively little is known about these two quantum automorphism groups of graphs and we refer to Section <ref> for an overview on all published articles in this area.Graph C^*-algebras in turn are well-established objects in operator algebras. They emerged from Cuntz and Krieger's work <cit.> in the 1980's and they developed to be one of the most important classes of examples of C^*-algebras, see for instance Raeburn's book for an overview <cit.>. Given a finite graph Γ=(V,E) the associated graph C^*-algebra C^*(Γ) is defined as C^*(Γ):=C^*(p_v, v∈ V, s_e, e∈ E | p_v=p_v^*=p_v^2,p_vp_w=0forv≠ w,s_e^*s_e=p_r(e), ∑_e∈ Es(e)=v s_es_e^*=p_v,ifs^-1(v)≠∅). A natural question is then:What is the quantum symmetry group of the graph C^*-algebra and is it one of the above two quantum automorphism groups of the underlying graphs? The answer is: It is given by the onedefined by Banica. Note however, that Bichon's definition has its justification in other contexts such as in <cit.> or in the recent work by Speicher and the second author <cit.>. Moreover, Bichon's work <cit.> inspired us how to formulate our main theorem, see also Remark <ref>.§ MAIN RESULT Intuitively speaking, our main result is that the quantum symmetry of a finite graph without multiple edges coincides with the quantum symmetry of the associated graph C^*-algebra. In other words, the following diagram is commutative: finite graphs [dr]_Γ↦(Γ) [rr]^Γ↦C^*(Γ)graph C^*-algebras [dl]^C^*(Γ)↦QSym(C^*(Γ)) quantum symmetry groupsMore precisely, we have the following result. Let Γ be a finite graph with n vertices V={ 1, ..., n} and m edges E = { e_1, ..., e_m} having no multiple edges. The maps α: C^*(Γ)→ C((Γ))⊗ C^*(Γ) ,p_i↦∑_k = 1^n u_ik⊗ p_k, 1 ≤ i ≤ n,s_e_j ↦∑_l=1^mu_s(e_j)s(e_l)u_r(e_j)r(e_l)⊗ s_e_l, 1 ≤ j ≤ m, andβ: C^*(Γ)→C((Γ)) ⊗ C^*(Γ) ,p_i↦∑_k = 1^n u_ki⊗p_k, 1 ≤ i ≤ n,s_e_j ↦∑_l=1^mu_s(e_l)s(e_j)u_r(e_l)r(e_j)⊗ s_e_l,1 ≤ j ≤ m define a left and a right action of (Γ) on C^*(Γ), respectively. Moreover, whenever G is a compact matrix quantum group acting on C^*(Γ) in the above way, we have G ⊂(Γ). In this sense, the quantum automorphism group (Γ) of Γ is the quantum symmetry group of C^*(Γ), see also Remark <ref>. We also provide some tools for comparing and dealing with the two definitions of quantum automorphism groups of graphs, (Γ) and (Γ), notably depending on the complement Γ^c of Γ, see Section <ref>. Moreover, we provide a list of all (Γ), (Γ) and (Γ) for undirected graphs Γ on four vertices, having no multiple edges and no loops, see Section <ref>.§ PRELIMINARIES§.§ Graphs We fix some notations for graphs used throughout this article. A graph Γ=(V,E) is finite, if the set V of vertices and the set E of edges are finite. We denote by r:E→ V the range map and by s:E→ V the source map. Agraph is undirected if for every e ∈ E there is a f ∈ E with s(f) = r(e) and r(f) = s(e); it is directed otherwise. Elements e ∈ E with s(e) = r(e) are called loops.A graph without multiple edges is a directed graph, where there are no e,f ∈ E, e ≠ f, such that s(e) = s(f) and r(e) =r(f).For a finite graph Γ=(V,E) with V={1,…,n}, its adjacency matrix ϵ∈ M_n(_0) is defined as ϵ_ij:=#{e∈ E |s(e)=i, r(e)=j}. Here _0={0,1,2,…}. Throughout this article we restrict to finite graphs having no multiple edges.IfΓ = (V,E) is a directed graph without multiple edges, we denote by Γ^c = (V,E') the complement of Γ, where E' = (V × V)\ E. Within the category of graphs having no loops, the complement Γ^c is defined using E' = (V × V)\ (E ∪{(i,i);i ∈ V}). §.§ Automorphism groups of graphs For a finite graph Γ=(V,E) without multiple edges, a graph automorphism is a bijective map σ: V → V such that (σ(i), σ(j)) ∈ E if and only if (i,j) ∈ E. In other words, ϵ_σ(i)σ(j)=1 if and only if ϵ_ij=1. The set of all graph automorphisms of Γ forms a group, the automorphism group (Γ). We can view (Γ) as a subgroup of the symmetric group S_n, if Γ has n vertices:Aut(Γ) = {σ∈ S_n | σε = εσ}⊂ S_n§.§ Graph C^*-algebras The theory of Graph C^*-algebras has its roots in Cuntz and Krieger's work<cit.> in 1980. Nowadays, it forms a well-developed and very active part of the theory of C^*-algebras, see <cit.> for an overview or <cit.> for recent developments. For a finite, directed graph Γ=(V,E) without multiple edges, the graph C^*-algebra C^*(Γ) is the universal C^*-algebra generated by mutually orthogonal projections p_v, v ∈ V and partial isometries s_e, e ∈ E such that(i) s_e^* s_e = p_r(e) for all e ∈ E(ii) and p_v = ∑_e∈ E : s(e)=v s_e s_e^* for every v ∈ V with s^-1(v) ≠∅.It follows immediately, thats_e^*s_f = 0 for e ≠ f and ∑_v∈ V p_v = 1hold true in C^*(Γ). §.§ Compact matrix quantum groups Compact matrix quantum groups were defined by Woronowicz <cit.>in 1987. They form a special class of compact quantum groups, see <cit.> for recent books. A compact matrix quantum group G is a pair (C(G),u), where C(G) is a unital (not necessarily commutative) C^*-algebra which is generated by u_ij, 1 ≤ i,j ≤ n, the entries of a matrix u ∈ M_n(C(G)). Moreover, the *-homomorphism Δ: C(G) → C(G) ⊗ C(G), u_ij↦∑_k=1^n u_ik⊗ u_kj must exist, and u and its transpose u^t must be invertible.As an example, consider the quantum symmetric group S_n^+ = (C(S_n^+), u) as defined by Wang <cit.> in 1998. It is the compact matrix quantum group given byC(S_n^+) := C^*(u_ij|u_ij = u_ij^* = u_ij^2,∑_l=1^n u_il = 1 = ∑_l=1^n u_li for all 1 ≤ i,j ≤ n).One can show that the quotient of C(S_n^+) by the relations that all u_ij commute is exactly C(S_n). Moreover, the symmetric group S_n may be viewed as a compact matrix quantum group S_n=(C(S_n),u), where u_ij:S_n→ are the evaluation maps of the matrix entries. This justifies the name “quantum symmetric group”. If G=(C(G),u) and H=(C(H), v) are compact matrix quantum groups with u ∈ M_n(C(G)) and v ∈ M_n(C(H)), we say that G is a compact matrix quantum subgroup of H, if there is a surjective *-homomorphism from C(H) to C(G) mapping generators to generators. In this case we write G ⊆ H. As an example: S_n⊂ S_n^+.The compact matrix quantum groups G and H are equal as compact matrix quantum groups, writing G =H, if we have G ⊆ H and H ⊆ G.§.§ Actions of quantum groups Let G=(C(G), u) be a compact matrix quantum group and let B be a C^*-algebra.A left action of G on B is a unital *-homomorphism α: B → C(G) ⊗ B such that (i) (Δ⊗id) ∘α = (id⊗α) ∘α(ii) and α(B)(C(G) ⊗ 1) is linearly dense in C(G) ⊗ B.A right action is a unital *-homomorphism β: B → C(G) ⊗ B with (i) ((F ∘Δ) ⊗id)) ∘β = (id⊗β) ∘β(ii) and β(B)(C(G) ⊗ 1) is linearly dense in C(G) ⊗ B,where F is the flip map F: C(G) ⊗ C(G) → C(G) ⊗ C(G), a ⊗ b ↦ b ⊗ a.Note that in some articles (for instance in <cit.>), the property (ii) is replaced by(ii') (ϵ⊗id) ∘α = id(iii') and there is a dense *-subalgebra of B, such that α restricts to a right coaction of the Hopf *-algebra on the *-subalgebra.One can show that (ii') and (iii') are equivalent to (ii), see <cit.>.§.§ Quantum symmetry group of n points According to Wang's work <cit.>, we know that S_n^+ (from Example <ref>) is the quantum symmetry group of n points in the sense that(i) S_n^+ acts from left and right onC^*(p_1,…,p_n | p_i=p_i^*=p_i^2, ∑_l p_l=1)by α(p_i):=∑_k=1^n u_ik⊗ p_k and β(p_i):=∑_k=1^nu_ki⊗ p_k(ii) and S_n^+ is maximal with these actions, i.e. any other compact matrix quantum group with actions defined as α and β is a compact matrix quantum subgroup of S_n^+.See also <cit.> for similar questions around quantum symmetries.§ QUANTUM AUTOMORPHISM GROUPS OF GRAPHS Wang's work in the 1990's was the starting point of the investigations of quantum symmetry phenomena for discrete structures (within Woronowicz's framework). Note that n points may be viewed as the totally disconnected graph on n vertices. A decade later, Banica and Bichon extended Wang's approach to a theory of quantum automorphism groups of finite graphs. In the sequel, we restrict to finite graphs having no multiple edges. §.§ Bichon's quantum automorphism group of a graph In 2003, Bichon <cit.> defined a quantum automorphism group as follows. Let Γ = (V, E) be a finite graph with n vertices V = {1, ... , n } and m edges E = { e_1, ... , e_m }. The quantum automorphism group (Γ) is the compact matrix quantum group (C((Γ)), u), where C((Γ)) is the universal C^*-algebra with generators u_ij, 1 ≤ i,j ≤ n and relationsu_ij = u_ij^*,u_iju_ik = δ_jku_ij ,u_jiu_ki = δ_jku_ji, 1 ≤ i,j,k ≤ n,∑_l=1^n u_il = 1 = ∑_l=1^n u_li, 1 ≤ i ≤ n,u_s(e_j)iu_r(e_j)k = u_r(e_j)ku_s(e_j)i = 0, e_j ∈ E, (i,k) ∉ E,u_is(e_j)u_kr(e_j) = u_kr(e_j)u_is(e_j) = 0, e_j ∈ E, (i,k) ∉ E,u_s(e_j)s(e_l)u_r(e_j)r(e_l) = u_r(e_j)r(e_l)u_s(e_j)s(e_l), e_j, e_l ∈ E. In the original definition of Bichon, there is actually another relation which is implied by the others:∑_l=1^m u_s(e_l)s(e_j)u_r(e_l)r(e_j) = 1 = ∑_l=1^m u_s(e_j)s(e_l)u_r(e_j)r(e_l),e_j ∈ EIndeed, Relations (<ref>) are implied by Relations (<ref>), (<ref>) and (<ref>):∑_l=1^m u_s(e_l)s(e_j)u_r(e_l)r(e_j) = ∑_i,k =1^n u_is(e_j) u_kr(e_j) =(∑_i=1^n u_is(e_j))(∑_k=1^n u_kr(e_j))=1 §.§ Banica's quantum automorphism group of a graph Two years later, Banica <cit.> gave the following definition. Let Γ =(V, E) be a finite graph with n vertices and adjacency matrix ε∈ M_n({0,1}).The quantum automorphism group (Γ) is the compact matrix quantum group (C((Γ)),u), where C((Γ)) is the universal C^*-algebra with generators u_ij, 1 ≤ i,j ≤ n and Relations (<ref>), (<ref>) together withu ε = ε u ,which is nothing but ∑_ku_ikϵ_kj=∑_kϵ_iku_kj. §.§ Link between the two definitions It is easy to see (<cit.> or <cit.>) that Banica's definition may be expressed as:C((Γ)) = C^*( u_ij|Relations(<ref>)–(<ref>))We thus haveAut(Γ) ⊆(Γ) ⊆(Γ)in the sense of compact matrix quantum subgroups, see Section <ref>. Equality holds, if C((Γ)) and C((Γ)) are commutative. Moreover, note that (see Example <ref>): C(S_n^+) = C^*(u_ij|Relations(<ref>) and (<ref>))As an example, let Γ be the complete graph (i.e. E=V× V). Then:(Γ)=(Γ)=S_n,(Γ)=S_n^+For its complement Γ^c (i.e. E=∅), we have:(Γ^c)=S_n,(Γ^c)=(Γ^c)=S_n^+ §.§ Review of the literature on quantum automorphism groups of graphsAt the moment there are only few articles regarding quantum automorphism groups of graphs. Some results are the following. In <cit.>, Bichon defined the hyperoctahedral quantum group and showed that this group is the quantum automorphism group of some graph. Banica computed the Poincaré series of (Γ) for homogenous graphs with less than eight vertices in <cit.>. Banica, Bichon and Chenevier considered circulant graphs having p vertices for p prime in <cit.>. They proved (Γ) = Aut(Γ) if the graph Γ does fulfill certain properties. Banica and Bichon investigated(Γ) for vertex-transitive graphs of order less or equal to eleven in <cit.>. They also computed (Γ) for the direct product, the Cartesian product and the lexicographic product of specific graphs. Chassaniol also studied the lexicographic product of graphs in <cit.>. In her PhD thesis <cit.>, Fulton studied undirected trees Γ such that Aut(Γ)= ℤ_2 ×ℤ_2 × ... ×ℤ_2, where we have k kopies of the cyclic group ℤ_2=/2. She proved Aut(Γ) =(Γ)=(Γ) for k=1 and Aut(Γ) ≠(Γ)=(Γ) for k≥ 2. See also <cit.> for links to quantum isometry groups. §.§ Comparing with the complement of the graph As can be seen from Section <ref>, the theory of quantum automorphism groups of graphs is still in its infancy. We now provide some basic results on the link between (Γ) and (Γ^c). Note that while we have(Γ)=(Γ^c)and(Γ)=(Γ^c) for all graphs Γ (using ϵ_Γ^c=A-ϵ_Γ for the adjacency matrices, with A∈ M_n({1}) the matrix filled with units, and uA=A=Au by Relation (<ref>)), we may have (Γ)≠(Γ^c), for instance when Γ is the complete graph, see Example <ref>. If (Γ) ⊂(Γ^c), then (Γ) = Aut(Γ).Relation (<ref>) in C((Γ^c)) implies that u_ik and u_jl commute inC((Γ)) whenever (i,j)∉ E and (k,l)∉ E. Together with Relations (<ref>), (<ref>) and (<ref>) in C((Γ)) this yields commutativity of all generators.If (Γ^c) = (Γ^c), then (Γ) = Aut(Γ). We have (Γ)⊂(Γ)=(Γ^c)=(Γ^c) and apply Lemma <ref>. The next lemma shows that the quantum automorphism groups of a graph without loops does not change if we add those. Let Γ =(V,E) be a finite graph without loops. Consider Γ' = (V, E') with E' = E ∪{ (i,i), i ∈ V }. It holds (i) (Γ) = (Γ'),(ii) (Γ) = (Γ').For (i), we use ϵ_Γ'=1+ϵ_Γ, where 1 is the identity matrix in M_n({0,1}). Thus, u ε_Γ = ε_Γ u is equivalent to u ε_Γ' = ε_Γ' u. For (ii), all we need to check is thatu_is(e_j)u_ir(e_j) = u_ir(e_j)u_is(e_j) is fulfilled in C((Γ)) for all i ∈ V, e_j ∈ E, which is true due to Relation (<ref>).§.§ Quantum automorphism groups on four vertices For a small number of vertices of undirected graphs, a complete classification of (Γ) and (Γ) is possible. For n∈{1,2,3}, we have C(S_n^+)=C(S_n), hence (Γ)=(Γ)=(Γ). For n=4, we now provide a complete table for graphs having no loops. We restrict to undirected graphs in order to keep it simple. We need the following lemma to compute the quantum automorphism groups. Let Γ = (V,E) be a finite graph with V={ 1, ..., n} and let e_j ∈ E. Let q∈ V with s^-1(q)=∅. For the generators of C((Γ)) it holdsu_q s(e_j) = 0 = u_s(e_j)q. By Relations (<ref>) and (<ref>), we get u_q s(e_j) =u_qs(e_j)(∑_i =1^n u_ir(e_j)) = ∑_i=1^nu_qs(e_j) u_ir(e_j)= 0,because (q,i) ∉ E for all i ∈ V. Likewise, we get u_s(e_j)q = 0. In the following, D_4 denotes the dihedral group defined as D_4 := ⟨ x,y|x^2=y^2=(xy)^4 = e ⟩,H_2^+ denotes the hyperoctahedral quantum group defined by Bichon in <cit.> and _2 denotes the cyclic group /2. The quantum group _2 *_2=(C^*(_2*_2),u) is understood as the compact matrix quantum group with matrix [ p 1-p 0 0; 1-p p 0 0; 0 0 q 1-q; 0 0 1-q q ]where C^*(_2*_2) is seen as the universal unital C^*-algebra generated by two projections p and q. Recall that (Γ)=(Γ^c) and (Γ)=(Γ^c), where Γ^c is the complement of Γ within the category of graphs having no loops.Parts of the following table were also computed in <cit.> and <cit.>. (0.7,0.5)(-0.5,-0.5)∙(-0.5,0.5)∙(0.5,-0.5)∙(0.5,0.5)∙(0.7,0.5)(-0.25,0.75)(1,0)1 (0.7,0.5)(-0.25, -0.25)(1,0)1(0.7,0.5)(-0.35, -0.25)(0,1)1(0.7,0.5)(0.7, -0.25)(0,1)1 (0.7,0.5)(0.7, -0.25)(-1,1)1 (0.7,0.5)(-0.25, -0.25)(1,1)1 Let Γ be an undirected graphon four vertices having no loops and no multiple edges. Then:t]c p0.5cm p0.5cm c c c cΓ Γ^cAut(Γ) (Γ^c) (Γ) (Γ) (1)(0,0) (0,0) (0,0) (0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)S_4 S_4 S_4^+S_4^+ (2)(0,0) (0,0)(0,0) (0,0) (0,0)(0,0)(0,0)(0,0)(0,0)(0,0)ℤ_2 ×ℤ_2 ℤ_2 ×ℤ_2 ℤ_2 * ℤ_2 ℤ_2 * ℤ_2 (3)(0,0) (0,0)(0,0)(0,0) (0,0) (0,0)(0,0)(0,0)(0,0)(0,0)ℤ_2 ℤ_2 ℤ_2 ℤ_2(4)(0,0) (0,0)(0,0)(0,0)(0,0) (0,0)(0,0)(0,0)(0,0)(0,0) D_4 D_4 H_2^+ H_2^+(5)(0,0) (0,0)(0,0)(0,0)(0,0) (0,0) (0,0)(0,0)(0,0)(0,0) S_3 S_3 S_3 S_3(6) (0,0) (0,0)(0,0)(0,0)(0,0)(0,0) (0,0)(0,0)(0,0)(0,0)ℤ_2 ℤ_2 ℤ_2 ℤ_2 For every row of the table, we compute (Γ) and we show (Γ)=(Γ). We then obtain (Γ^c) by using Lemma <ref>. We label the points of the graphs as follows:(3,3)(1,1) (0,2)1 (2.5,2)2 (0,0)3 (2.5,0)4 (1) Obvious, see Example <ref>. (2) Let (u_ij)_1 ≤ i,j ≤ 4 be the generators of C((Γ)). Lemma <ref> yields u_31= u_32= u_41= u_42= u_13= u_23 =u_14 =u_24= 0.With Relations (<ref>) we deduceu=[ u_11 1 - u_1100; 1 - u_11 u_1100;00 u_331- u_33;001- u_33 u_33 ].Thus(Γ) = ℤ_2 * ℤ_2.Since u_iju_kl = u_klu_ij holds for (i,k), (j,l) ∈{(1,2), (2,1)} in C((Γ)), we get (Γ)=(Γ). (3) Lemma <ref> yields u_14 = u_24 = u_34 = u_41 = u_42 = u_43 = 0.This implies(Γ) ⊆ S_3^+ = S_3,thus (Γ) is commutative and hence (Γ)=(Γ)=(Γ)=_2.(4)Let Δ and Δ' be the comultiplication maps of (Γ) and H_2^+, respectively. We first show that these two quantum groups coincide as compact quantum groups, i.e. there is a ^*-isomorphism ϕ: C(H_2^+)→ C((Γ))such that Δ'∘ϕ=(ϕ⊗ϕ)∘Δ. Step 1: The map ϕ exists and we have Δ'∘ϕ=(ϕ⊗ϕ)∘Δ.From ε u = u ε we getu = [ u_11 u_12 u_13 u_14; u_12 u_11 u_14 u_13; u_31 u_32 u_33 u_34; u_32 u_31 u_34 u_33 ].Define v_11 := u_11 - u_12, v_12 := u_13 - u_14, v_21 := u_31 - u_32 and v_22 := u_33 - u_34. One can compute thatv_ij, i,j =1,2 fulfill the relations of C(H_2^+) and with the universal property we get a *-homomorphism φ: C(H_2^+) → C((Γ)). Since Δ' ∘φ = (φ⊗φ) ∘Δ also holds, we get that (Γ) is a quantum subgroup of H_2^+. Step 2: The map ϕ is a ^*-isomorphism.Let (v_ij)_i,j=1,2 be the generators of C(H_2^+). Define u_11 := u_22:= v_11^2 + v_11/2,u_12 := u_21:= v_11^2 - v_11/2,u_13 := u_24:= v_12^2 + v_12/2,u_14 := u_23:= v_12^2 - v_12/2,u_31 := u_42:= v_21^2 + v_21/2,u_41 := u_32:= v_21^2 - v_21/2,u_33 := u_44:= v_22^2 + v_22/2,u_34 := u_43:= v_22^2 - v_22/2.One can show that the (u_ij)_1 ≤ i,j ≤ 4 fulfill the relations of C((Γ)). The universal property now gives us a *-homomorphism φ' : C((Γ)) → C(H_2^+) and ϕ' is the inverse of ϕ and vice versa.Step 3: We have (Γ) = (Γ).We have seen in Step 1, thatu_11 = u_22, u_12 = u_21,u_13 = u_24,u_14 = u_23, u_31 = u_42,u_32 = u_41,u_33 = u_44,u_34 = u_43 and therefore we get u_ij u_kl = u_kl^2 = u_kl u_ijfor all (i,k), (j,l) ∈ E. Thus (Γ) = (Γ).(5) We conclude as in (3). (6) Some direct computations using ε u = u ε and Relations (<ref>) showu = [ u_33 1-u_3300; 1-u_33 u_3300;00 u_33 1-u_33;00 1-u_33 u_33 ].Thus (Γ) is commutative. § PROOF OF THE MAIN RESULT We now prove the main result of this article (see Section <ref>) for a finite graph Γ with vertices V={1,…,n} and edges E={e_1,…,e_m} having no multiple edges. We define the quantum symmetry group QSym(C^*(Γ)) of C^*(Γ) to be the maximal compact matrix quantum group G acting on C^*(Γ) by α: C^*(Γ) → C(G)⊗ C^*(Γ) and β: C^*(Γ) → C(G)⊗ C^*(Γ) as defined in the statement of our main theorem. We thus have to show that (Γ) acts on C^*(Γ) via α and β (see Sections <ref> and <ref>) and that it is maximal with these actions (see Section <ref>).§.§ Existence of the maps α and β In order to prove that α: C^*(Γ)→ C((Γ))⊗ C^*(Γ) p_i↦ p_i':=∑_k = 1^n u_ik⊗ p_k, 1 ≤ i ≤ ns_e_j ↦ s_e_j':=∑_l=1^mu_s(e_j)s(e_l)u_r(e_j)r(e_l)⊗ s_e_l, 1 ≤ j ≤ m defines a *-homomorphism, all we have to show is that the relations of C^*(Γ) hold for p_i' and s_e_j'. We may then use the universal property of C^*(Γ). The proof for the existence of β is analogous.§.§.§ The p_i' are mutually orthogonal projections Obviously, p_i' = (p_i')^* holds. Moreover, using p_kp_l=δ_klp_k and Relations (<ref>), we havep_i'p_j'=∑_k,l=1^n u_iku_jl⊗ p_kp_l=∑_k=1^n u_iku_jk⊗ p_k=δ_ijp_i'. §.§.§ The s_e_j' are partial isometries with (s_e_j')^* s_e_j' = p_r(e_j)' Using s_e_l^*s_ei=δ_ilp_r(e_i) (see Section <ref>) and Relations (<ref>), we have(s_e_j')^* s_e_j'=∑_l,i = 1^m u_r(e_j)r(e_l)u_s(e_j)s(e_l)u_s(e_j)s(e_i)u_r(e_j)r(e_i)⊗ s_e_l^*s_e_i=∑_i = 1^m u_r(e_j)r(e_i)u_s(e_j)s(e_i)u_r(e_j)r(e_i)⊗ p_r(e_i).By Relations (<ref>) we have u_r(e_j)j'u_s(e_j)i' u_r(e_j)j' = 0 for (i',j') ∉ E. This yields∑_i = 1^mu_r(e_j)r(e_i)u_s(e_j)s(e_i)u_r(e_j)r(e_i)⊗ p_r(e_i)= ∑_i', j' =1^n u_r(e_j)j'u_s(e_j)i'u_r(e_j)j'⊗ p_j'.Using Relations (<ref>), we obtain ∑_i=1^nu_s(e_j)i'=1 and thus(s_e_j')^* s_e_j'=∑_i', j' =1^n u_r(e_j)j'u_s(e_j)i'u_r(e_j)j'⊗ p_j'=∑_j' = 1^n u_r(e_j)j'⊗ p_j'=p_r(e_j)'. §.§.§ We have ∑_j: s(e_j)= v s_e_j' (s_e_j')^* = p_v' for s^-1(v)≠∅Using Relations (<ref>), we get for v∈ V with s^-1(v)≠∅:∑_j∈{1,…,m} s(e_j)= v s_e_j' (s_e_j')^* =∑_j∈{1,…,m} s(e_j)= v∑_i,l=1^m u_vs(e_l)u_r(e_j)r(e_l)u_r(e_j)r(e_i)u_vs(e_i)⊗ s_e_ls_e_i^*= ∑_l=1^m ∑_i∈{1,…,m} r(e_i)=r(e_l) u_vs(e_l)(∑_j∈{1,…,m} s(e_j)= vu_r(e_j)r(e_l))u_vs(e_i)⊗ s_e_ls_e_i^*Now,∑_j∈{1,…,m} s(e_j)= vu_r(e_j)r(e_l)=∑_q∈ V (v,q)∈ Eu_qr(e_l) and forq∈ V with(v,q) ∉ E we have u_vs(e_l)u_qr(e_l)=0 by Relations (<ref>).Thus, for any l∈{1,…,m}, we have using Relations (<ref>)u_vs(e_l)∑_q∈ V (v,q)∈ Eu_qr(e_l)=u_vs(e_l)∑_q∈ Vu_qr(e_l)=u_vs(e_l)and hence:∑_j∈{1,…,m} s(e_j)= v s_e_j' (s_e_j')^* = ∑_l=1^m ∑_i∈{1,…,m} r(e_i)=r(e_l)u_vs(e_l)u_vs(e_i)⊗ s_e_ls_e_i^*Since Γ has no multiple edges by assumption, r(e_i)=r(e_l) and s(e_i)=s(e_l) implies e_i=e_l. We thus infer using Relations (<ref>):∑_j∈{1,…,m} s(e_j)= v s_e_j' (s_e_j')^* = ∑_l=1^mu_vs(e_l)⊗ s_e_ls_e_l^*Now, for V':={q∈ V |s^-1(q)≠∅}, we have, using the relations in C^*(Γ):∑_l=1^mu_vs(e_l)⊗ s_e_ls_e_l^*=∑_q ∈ V'∑_l∈{1,…,m} s(e_l) =q u_vq⊗ s_e_ls_e_l^*=∑_q∈ V'u_vq⊗ p_q Since we know that u_vq = 0 for q ∉ V' by Lemma <ref>, we finally conclude∑_j∈{1,…,m} s(e_j)= v s_e_j' (s_e_j')^* =∑_q =1^n u_vq⊗ p_q =p_v'.This settles the existence of α.§.§ The map α is a left action and β is a right actionWe only prove this claim for α, the proof for β being analogous.§.§.§ (Δ⊗id) ∘α=(id⊗α)∘α holds and α is unital Using Relations (<ref>), this is straightforward to check.It remains to show that𝒮:=span α(C^*(Γ))(C((Γ)) ⊗ 1)is dense in C((Γ)) ⊗ C^*(Γ), which we will do in the sequel.§.§.§ The elements 1⊗ p_l,1⊗ s_e_l and 1⊗ s_e_l^* are in 𝒮 Using Relations (<ref>) and (<ref>) we infer:𝒮∋∑_i=1^n α(p_i)(u_il⊗ 1) = ∑_i=1^n ∑_j=1^n u_ij u_il⊗ p_j = ∑_i=1^n u_il⊗ p_l = 1 ⊗ p_lMoreover, for e_l ∈ E we get, using Relations (<ref>) and V':={v∈ V |s^-1(v)≠∅}:∑_v ∈ V' ∑_j∈{1,…,m} s(e_j)= vα(s_e_j)(u_r(e_j)r(e_l)u_vs(e_l)⊗ 1)=∑_v ∈ V'∑_j∈{1,…,m} s(e_j)= v(∑_k=1^m u_vs(e_k)u_r(e_j)r(e_k)u_r(e_j)r(e_l)u_vs(e_l)⊗ s_e_k)=∑_v ∈ V'(∑_k∈{1,…,m} r(e_k)=r(e_l) u_vs(e_k)(∑_j∈{1,…,m} s(e_j)= v u_r(e_j)r(e_l))u_vs(e_l)⊗ s_e_k)We proceed similar to Step <ref>. By Relations (<ref>), we know u_qr(e_l)u_vs(e_l) =0 for (v,q) ∉ E. Thus, by Relations (<ref>) and (<ref>) and using that Γ has no multiple edges, we obtain:∑_v ∈ V' ∑_j∈{1,…,m} s(e_j)= vα(s_e_j)(u_r(e_j)r(e_l)u_vs(e_l)⊗ 1)=∑_v ∈ V'(∑_k∈{1,…,m} r(e_k)=r(e_l) u_vs(e_k)(∑_q=1^n u_qr(e_l))u_vs(e_l)⊗ s_e_k)=∑_v ∈ V'(∑_k∈{1,…,m} r(e_k)=r(e_l) u_vs(e_k)u_vs(e_l)⊗ s_e_k)=∑_v ∈ V' u_vs(e_l)⊗ s_e_lFinally, Lemma <ref> yields u_vs(e_l)=0 for v∉ V'. Hence, using Relations (<ref>):𝒮∋∑_v ∈ V'∑_j∈{1,…,m} s(e_j)= vα(s_e_j)(u_r(e_j)r(e_l)u_s(e_j)s(e_l)⊗ 1) =∑_i=1^n u_is(e_l)⊗ s_e_l = 1 ⊗ s_e_l Define V” := {v ∈ V|r^-1(v)≠∅}. Similar to the computations above, we get𝒮∋∑_v ∈ V”∑_j∈{1,…,m} s(e_j)= vα(s_e_j^*)(u_s(e_j)s(e_l)u_r(e_j)r(e_l)⊗ 1) = 1 ⊗ s_e_l^*. §.§.§ If 1⊗ x,1⊗ y∈𝒮, then also 1⊗ xy∈𝒮The remainder of the proof of Step <ref> consists in general facts for actions of compact matrix quantum groups.We may write 1⊗ x∈𝒮 and 1⊗ y∈𝒮 as1 ⊗ x = ∑_i=1^l α(z_i)(w_i ⊗ 1),1 ⊗ y = ∑_j=1^k α(t_j)(v_j ⊗ 1)for some z_i,t_j ∈ C^*(Γ) and w_i, v_j ∈ C((Γ)). Therefore1 ⊗ xy= ∑_i=1^l α(z_i)(w_i ⊗ 1)(1 ⊗ y)= ∑_i=1^l α(z_i)(1 ⊗ y) (w_i ⊗ 1)= ∑_i=1^l ∑_j=1^k α(z_i t_j) (v_j w_i ⊗ 1)∈𝒮 §.§.§ 𝒮 is dense in C((Γ)) ⊗ C^*(Γ)Summarizing, we get that 1 ⊗ w ∈𝒮 for all monomials w in p_i, s_e_j, s_e_j^*, 1 ≤ i ≤ n, 1 ≤ j ≤ m. Since α is unital, we also have:C((Γ)) ⊗ 1 ⊆α(C^*(Γ))(C((Γ))⊗ 1)⊆𝒮We conclude that 𝒮 is dense in C((Γ)) ⊗ C^*(Γ), which settles Step <ref>. §.§ The quantum group (Γ) acts maximally on C^*(Γ) For proving the maximality, let G= (C(G), u) be another compact matrix quantum group acting on C^*(Γ) by α': C^*(Γ) → C(G) ⊗ C^*(Γ) and β': C^*(Γ) →C(G) ⊗ C^*(Γ) in the way (Γ) acts on C^*(Γ) viaα and β. We want to show that there is a *-homomorphism C((Γ)) → C(G) sending generators to generators. Thus, we need to compute that the generators u_ij of C(G) fulfill the relations of C((Γ)). §.§.§ The Relations (<ref>) hold in C(G) The equation ∑_k=1^n u_ik⊗ p_k = α'(p_i) = α'(p_i)^* = ∑_k =1^n u_ik^* ⊗ p_kyields u_ij = u_ij^* after multiplying from the left with 1⊗ p_j. We also have ∑_i=1^n u_ji u_ki⊗ p_i = ∑_i,l=1^n u_ji u_kl⊗ p_i p_l= α'(p_j) α'(p_k)= δ_jkα'(p_j)= ∑_i=1^n δ_jk u_ji⊗ p_ifrom which we infer u_jiu_ki=δ_jku_ji. Using β', we also obtain u_iju_ik=δ_jku_ij.§.§.§ The Relations (<ref>) hold in C(G) From∑_k =1^n 1 ⊗ p_k = 1 ⊗ 1 = α'(1) = ∑_i=1^n α'(p_i) = ∑_k=1^n (∑_i=1^n u_ik) ⊗ p_kwe deduce ∑_i=1^n u_ik=1, and likewise ∑_i=1^nu_ki=1 using β'.§.§.§ The Relations (<ref>) hold in C(G) Using s_e_l^*s_e_t=δ_ltp_r(e_l) (see Section <ref>) and Relations (<ref>) in C(G), we obtain for any j:∑_q=1^n u_r(e_j)q⊗ p_q=α'(p_r(e_j))=α'(s_e_j^*s_e_j)= ∑_l,t=1^m u_r(e_j)r(e_l) u_s(e_j)s(e_l)u_s(e_j)s(e_t)u_r(e_j)r(e_t)⊗ s_e_l^*s_e_t=∑_l=1^m u_r(e_j)r(e_l)u_s(e_j)s(e_l)u_r(e_j)r(e_l)⊗ p_r(e_l)Multiplication with 1⊗ p_k yields:u_r(e_j)k=∑_l∈{1,…,m} r(e_l)=k u_r(e_j)ku_s(e_j)s(e_l)u_r(e_j)kIf r^-1(k)=∅, then u_r(e_j)k=0 and hence u_s(e_j)iu_r(e_j)k=u_r(e_j)ku_s(e_j)i=0 for all i∈ V.Otherwise, if r^-1(k)≠∅, we use Relations (<ref>) and (<ref>) in C(G) and get∑_l∈{1,…,m} r(e_l) =k u_r(e_j)ku_s(e_j)s(e_l)u_r(e_j)k= u_r(e_j)k= u_r(e_j)k^2=∑_i=1^n u_r(e_j)k u_s(e_j)i u_r(e_j)k and therefore∑_i∈ V (i,k)∉ Eu_r(e_j)k u_s(e_j)i u_r(e_j)k =0. Sinceu_r(e_j)k u_s(e_j)i u_r(e_j)k =(u_s(e_j)i u_r(e_j)k)^*u_s(e_j)i u_r(e_j)kholds, the above is a vanishing sum of positive elements – and hence each summand vanishes. This yields u_s(e_j)i u_r(e_j)k= 0 for all (i,k) ∉ E.§.§.§ The Relations (<ref>) hold in C(G) The argument is analogous to the one for proving Relations (<ref>) when replacing α' by β'.The proof of the main theorem is complete.Let Γ be a finite graph with n vertices V={ 1, ..., n} and m edges E = { e_1, ..., e_m}. In <cit.>, Bichon showed that (Γ) is the quantum symmetry group of Γ in his sense, where β_V: C(V) → C((Γ))⊗ C(V) , g_i ↦∑_k=1^nu_ki⊗ g_k,β_E: C(E) → C((Γ))⊗ C(E) , f_j ↦∑_l=1^m u_s(e_l)s(e_j)u_r(e_l)r(e_j)⊗ f_l,define actions of (Γ) on C(V) and C(E), respectively. Those actions inspired us, how an action of a compact matrix quantum group on C^*(Γ) should look like. However, note that edges in the commutative C^*-algebra C(E) of continuous functions on E are represented as projections unlike in the case of C^*(Γ). Therefore, the quantum symmetry group of C^*(Γ) is (Γ) rather than (Γ). On the other hand, if we consider the quotient of C^*(Γ) by the relations s_e=s_e^*, its quantum symmetry group is (Γ). Indeed, selfadjointness of s_e yields∑_l=1^mu_s(e_j)s(e_l)u_r(e_j)r(e_l)⊗ s_e_l = α(s_e_j) = α(s_e_j)^* = ∑_l=1^mu_r(e_j)r(e_l)u_s(e_j)s(e_l)⊗ s_e_l,from which we obtain Relations (<ref>) by multiplication with (1 ⊗ s_e_i^*) from the left.plain | http://arxiv.org/abs/1706.08833v2 | {
"authors": [
"Simon Schmidt",
"Moritz Weber"
],
"categories": [
"math.OA",
"math.FA"
],
"primary_category": "math.OA",
"published": "20170627132701",
"title": "Quantum Symmetries of Graph C*-Algebras"
} |
Departamento de Física Teórica II, Universidad Complutense de Madrid, 28040 Madrid, [email protected], [email protected], [email protected] We introduce a class of generalized Lipkin–Meshkov–Glick (gLMG) models with (m) interactions of Haldane–Shastry type. We have computed the partition function of these models in closed form by exactly evaluating the partition function of the restriction of a spin chain Hamiltonian of Haldane–Shastry type to subspaces with well-defined magnon numbers. As a byproduct of our analysis, we have obtained strong numerical evidence of the Gaussian character of the level density of the latter restricted Hamiltonians, and studied the distribution of the spacings of consecutive unfolded levels. We have also discussed the thermodynamic behavior of a large family of (2) and (3) gLMG models, showing that it is qualitatively similar to that of a two-level system. Keywords: integrable spin chains and vertex models, solvable lattice modelsGeneralized Lipkin–Meshkov–Glick models of Haldane–Shastry type José A. Carrasco, Federico Finkel, Artemio González-López December 30, 2023 ===============================================================§ INTRODUCTION One of the first, and still one of the few, quantum mechanical many-body models that has been solved in the literature is the Lipkin–Meshkov–Glick (LMG) model <cit.>, which describes a system of N fermions with two N-fold degenerate one-particle levels. The original motivation for introducing this model was testing the validity of different approximation schemes from solid state physics or field theory in the context of nuclear physics. Over the years, the LMG model has appeared in connection with a wide range of problems of physical interest, including shape transitions in nuclei <cit.>, trapped ion and optical cavity experiments <cit.>, two-modes Bose–Einstein condensates <cit.>, and quantum information theory <cit.>. In particular, it has been shown that the von Neumann entanglement entropy of its ground state grows logarithmically with the size of the subsystem, as is the case for one-dimensional critical systems <cit.> (although this model is actually not critical <cit.>).As already noted in the original papers, the key to the solvability of the LMG model is the fact that it can be mapped to a system of N spin-1/2 particles with constant long-range interactions of XY type in an external transverse magnetic field. In the isotropic (XX) case the Hamiltonian of this effective model is a polynomial in ^2 and J_z, whereis the the total spin operator, and can thus be exactly solved for arbitrary N. The general (non-isotropic) LMG model can be solved in principle via the Bethe ansatz <cit.>, though in practice this is less efficient than brute-force numerical diagonalization. In the thermodynamic limit, however, the density of states of the latter model in the highest spin sector (J=N/2) has been derived by means of a spin-coherent-state formalism <cit.>.A wide family of models with long-range interactions of (m) type generalizing the isotropic LMG model was recently introduced in Ref. <cit.>. In analogy with the latter model, the non-degenerate ground state of these novel models is given by a Dicke state whose reduced density matrix for a subsystem of L<N spins can be computed in closed form, which in turn yields the entanglement entropy in the thermodynamic limit N→∞ with L/N= finite. Although both the von Neumann and the Rényi entanglement entropies grow logarithmically with the size L of the subsystem, the corresponding prefactor is independent of the Rényi parameter, which implies that none of these models can be critical. Interestingly, for m>3 there is at least one quantum phase whose Tsallis entanglement entropy <cit.> becomes extensive for a suitable value of the Tsallis parameter. However, the full spectrum of these models in general cannot be evaluated in closed form.In this paper we introduce a family of generalized Lipkin–Meshkov–Glick (gLMG) models, with interactions governed by an (m) integrable spin chain of Haldane–Shastry type. The latter chains are the celebrated Haldane–Shastry (HS) (m) spin chain <cit.>, which describes a circular array of equispaced spins with two-body long-range interactions inversely proportional to the square of the (chord) distance, and its rational <cit.> and hyperbolic <cit.> analogues. Although the HS chain was originally introduced as a model whose exact ground state coincides with Gutzwiller's variational wave function for the Hubbard model in the limit of large on-site interaction <cit.>, it soon proved of interest per se in condensed matter and theoretical physics. Indeed, as pointed out by Haldane <cit.>, the spinon excitations of this chain provide one of the simplest examples of a quantum system featuring fractional statistics (see also <cit.>). The HS chain is closely connected to important conformal field theories like the k=1 Wess–Zumino–Novikov–Witten model <cit.>, and has recently been related to infinite matrix product states <cit.>. Integrable extensions of the Haldane–Shastry chain with long-range interactions involving more than two spins also play a key role for describing non-perturbatively the spectrum of planar =4 gauge theory in the context of the AdS-CFT correspondence <cit.>. The interest in spin chains of HS type has been further reinforced by recent developments in quantum simulation, as witnessed by the proposal of an experimental realization of the HS chain using two internal atomic states of atoms trapped in a photonic crystal waveguide <cit.>.One of the key features of spin chains of Haldane–Shastry type is the fact that their partition functions can be exactly computed for any number of spins <cit.> by exploiting their connection with a corresponding spin dynamical model of Calogero–Sutherland type <cit.> by means of a mechanism known as the Polychronakos “freezing trick” <cit.>. This has made it possible to check the validity of several fundamental conjectures on the characterization of quantum chaos vs. integrability <cit.>. In particular, it has been shown that spin chains of HS type do not behave as expected for a `generic' integrable system, in the sense that the distribution of the spacings between consecutive levels is not Poissonnian <cit.>.The gLMG models that we introduce in this paper can also be regarded as a deformation of the (m) spin chains of HS type. More precisely, we add to the HS-type Hamiltonian a term depending on the generators of the standard (m) Cartan subalgebra, which commutes with the former Hamiltonian. In particular, when this extra term is linear in the Cartan generators it can be interpreted as an (m) external magnetic field, and the corresponding models are the ones studied in Ref. <cit.>. Likewise, when the extra term is a suitable quadratic combination of the Cartan generators we recover the models introduced in Ref. <cit.>, which include the isotropic LMG model. We shall see that the Hilbert space of a general gLMG model decomposes as a direct sum of subspaces with fixed magnon numbers, which are separately invariant under the action of both the original HS-type Hamiltonian and the new term. By suitably adapting the freezing trick, we shall be able to compute the partition function of the restriction of the Hamiltonians of the three spin chains of HS type to the latter invariant subspaces. This in turn yields the partition function of the full gLMG Hamiltonian, since the Cartan generators are proportional to the identity on these subspaces. The knowledge of the partition function of the gLMG models of HS type, as well as the restricted partition functions of the corresponding spin chains, enables one to study several statistical properties of the spectrum of the latter models. In particular, we have obtained strong numerical evidence that the level density of the restriction of the HS-type chain Hamiltonians to subspaces with fixed magnon numbers follows a Gaussian distribution in the large N limit, as is known to be the case for the full spectrum of these models <cit.>. We have also studied the distribution of the spacings between consecutive levels of the restrictions of these models to the invariant subspaces, showing that it follows the characteristic law for an approximately equispaced spectrum with normally distributed energy levels <cit.>. Finally, we have numerically computed the thermodynamic functions of gLMG models of HS type whose extra term is quadratic in the Cartan generators, comparing them with the exact results for the original (HS-type) chains in the thermodynamic limit derived in Ref. <cit.>.§ THE MODELS The models we shall study in this paper are deformations of (m) spin chains with Hamiltonians of the formH_0=∑_1≤ i<j≤ N h_ij(1- S_ij) , =+,- , with h_ij∈. In the latter equation S_ij is the operator permuting the (m) spins of the i-th and j-th particles, whose action on the canonical (m) spin basis={|s_1⟩⊗⋯⊗|s_N⟩≡|s_1,…,s_N⟩| s_i=1,…,m ,1≤ i≤ N} ,is given byS_ij|s_1,…,s_i,…,s_j,…,s_N⟩=|s_1,…,s_j,…,s_i,…,s_N⟩ .These operators can be expressed in terms of the local (Hermitian) generators t^a_k (a=1,…,m^2-1) of the fundamental representation of the (m) algebra acting on the k-th site (with the normalization (t^a_kt^b_k)=1/2 _ab) asS_ij=1/m+2∑_a=1^m^2-1t^a_it^a_j≡1/m+2 _i·_j .We can thus write[Here and throughout the paper, all sums and products run from 1 to N unless otherwise specified.]H_0=-∑_i jh_ij _i·_j+E_0 ,with E_0=(1-/m)∑_i<jh_ij. In particular, for m=2 we have _k=1/2 _k, where _k=(_k^1,_k^2,_k^3) are the three Pauli matrices at the k-th site.Let _a denote the a-th magnon number operator defined by_a|s_1,…,s_N⟩=N_a|s_1,…,s_N⟩ , 1≤ a≤ m ,where[We shall denote in what follows by |A| the cardinal of the set A.]N_a=|{k=1,…,N| s_k=a}| .The latter operators are related to the Hermitian generators of the standard Cartan subalgebra of the Lie algebra (m), as we shall now explain. Indeed, let J^a_k denote the operator whose action on the Hilbert space of the k-th particle is given byJ_k^a|s_k⟩=(_a,s_k-_m,s_k)|s_k⟩ , 1≤ a≤ m-1 .The m-1 commuting operators J^a_k generate the standard Cartan subalgebra[This choice of the generators of the standard Cartan subalgebra of (m) is simply a matter of convenience. Note, however, that these generators are not orthogonal with respect to the usual Killing–Cartan scalar product, i.e., (J_k^aJ_k^b)0 for a b.] of (m) at each site k. We then define the global (Hermitian) Cartan generatorsJ^a≡∑_k=1^N J_k^a , 1≤ a≤ m-1 .From Eq. (<ref>) it then follows thatJ^a=_a-_m , 1≤ a≤ m-1 .Summing over a and taking into account that ∑_a=1^m_a=N we obtain∑_a=1^m-1J^a=N-m _m , 1≤ a≤ m-1 .Using the last two equations we can express the magnon number operators in terms of the Cartan subalgebra generators as_a=J^a(1-_am)-+N/m , 1≤ a≤ m ,where≡1/m∑_a=1^m-1J^a . We shall consider in what follows deformations H=H_0+H_1 of (<ref>) in whichH_1=h(_1,…,_m)is an analytic function of the magnon number operators _a. Note, first of all, that the previous expression for H_1 is not ambiguous, since [_a,_b]=0 for 1≤ a,b≤ m. It is also clear that H_1 lies in the enveloping algebra of the (m) Cartan subalgebra on account of Eq. (<ref>). For this reason, we shall say thatH=H_0+H_1 =∑_i<j h_ij(1- S_ij)+h(_1,…,_m)=-∑_i jh_ij _i·_j+h(_1,…,_m)+E_0 .is an (m) generalized Lipkin–Meshkov–Glick (gLMG) model. In particular, when h_ij>0 for all i<j, =+ and h is the quadratic polynomialh(x_1,…,x_m)=∑_a=1^m-1c_a(x_a-x_m-Nh_a)^2 ,with h_a∈ ,c_a>0 ,we obtain the models whose ground state entanglement entropy was computed in closed form in Ref. <cit.>. The latter models include the original ((2), isotropic) LMG model when h_ij=2/N for all i<j and c_1=1/(2N), up to a constant energy.One of the fundamental properties of the Hamiltonian (<ref>) is that it preserves the subspaces of the Hilbert space ≡(^m)^⊗ N with a fixed magnon configuration. Indeed, let us denote by (), where =(N_1,…,N_m) and ||≡ N_1+⋯+N_m=N, the subspace of whose elements are linear combinations of basis states |s_1,…,s_N⟩≡|⟩ with magnon numbers N_a (cf. Eq. (<ref>)). Clearly H_0 leaves () invariant, since each permutation operator S_ij does. On the other hand, _a|s⟩=N_a|s⟩ on () by construction, and thereforeH_1=h()on () .Thus H=H_0+H_1 preserves (), as stated. It is also clear from the above discussion that [H_0,H_1]=0, and that the eigenvalues of H^≡ H|_() can be expressed asE^0_i()+h() , 1≤ i≤() ,where {E^0_i()}_1≤ i≤() is the spectrum of H_0^≡ H_0|_(). Hence the partition function Z^(T) of H^ is given byZ^(T)=q^h()∑_i=1^()q^E_i^0()≡ q^h()Z_0^(T) , q≡^-1/k_BT ,where Z_0^(T) is the partition function of H_0^ . Since=⊕_||=N() ,the partition function of H is given byZ(T)=∑_||=NZ^(T)=∑_||=Nq^h()Z_0^(T) .Thus the partition function of the model (<ref>) is completely determined by the partition functions Z^_0(T) of the restrictions of the spin chain Hamiltonian H_0 to each of the subspaces (). We shall see in the following sections that the latter partition functions can be computed in closed form when H_0 is the Hamiltonian of one of the three spin chains of HS type, namely the Haldane–Shastry <cit.>, Polychronakos–Frahm (PF) <cit.> and Frahm–Inozemtsev (FI) <cit.> chains. The chain sites of these integrable spin chains can be expressed asz_k=kπ/N ,for the HS chain _k ,for the PF chain ^2ξ_k ,for the FI chain ,where _k and ξ_k respectively denote the k-th zero of the Hermite polynomial of degree N and the generalized Laguerre polynomial L_N^-2N+1 with >2(N-1). In all three cases, the interaction strength is a function h_ij=h(z_i-z_j) of the difference z_i-z_j, namelyh(x)= 12sin^-2x ,for the HS chainx^-2 ,for the PF chain 12sinh^-2x ,for the FI chain .Remarkably, the (total) partition function Z_0(T)=∑_||=NZ_0^(T) of all of these models can be computed in closed form by exploiting their close connection with their associated spin Calogero–Sutherland models (see, e.g., <cit.>). In the following sections we shall adapt this technique, known in the literature as Polychronakos's freezing trick <cit.>, to evaluate the restricted partition functions Z_0^(T). § THE FREEZING TRICK In this section we shall outline the computation of the restricted partition function Z_0^ for the Haldane–Shastry spin chain, which is the best known of these models and presents certain technical subtleties stemming from its translation invariance. To this end, we first recall that in this case H_0 is related to the strong interaction limit of the spin Sutherland model=-Δ+a∑_i≠ jsin^-2(x_i-x_j)(a- S_ij) , a>0 ,where ≡∑_i ^2_x_i. Indeed, we can write=+4a() ,where =(x_1,…,x_N),=-Δ+a(a-1)∑_i≠ jsin^-2(x_i-x_j)is the scalar Sutherland model and()=1/2∑_i<jsin^-2(x_i-x_j)(1- S_ij)is obtained from H_0 replacing the chain sites z_i by the dynamical variables x_i. Since and are translation invariant, the total momentum is conserved and can be set to zero by working in the center of mass frame. In the strong interaction limit a→∞ the eigenfunctions of become sharply peaked at the coordinates of the minimum of the scalar potentialU()=∑_i jsin^-2(x_i-x_j)in the configuration space (A_N-1 Weyl chamber)A={∈^N| x_1<⋯<x_N} ,which (up to an overall translation) coincide with the chain sites z_k=kπ/N . Thus, when a≫1 the eigenvalues of are approximately given byE_ij≃ E^sc_i+4aE^0_j , a≫1 ,where E^sc_i and E^0_j respectively denote two arbitrary eigenvalues of and H_0. From the latter equation it immediately follows that the partition function Z_0(T) of the Haldane–Shastry chain is given by the freezing trick formulaZ_0(T)=lim_a→∞(4aT)/(4aT) .This is the basis for the computation of Z_0(T) in Ref. <cit.>. We shall now show that essentially the same procedure can be carried out to compute the restricted partition functions Z_0^(T). Essentially, this is due to the fact that the spin Hamiltonian preserves the subspaces L^2(A)⊗() of its Hilbert space L^2(A)⊗. Thus, Z_0^ can be obtained from the analogue of Eq. (<ref>), namelyZ_0^(T)=lim_a→∞^(4aT)/(4aT) ,where ^ is the partition function of ^=|_L^2(A)⊗().To begin with, note that the Hamiltonian is equivalent to its symmetric/antisymmetric extension to the Hilbert space _±(L^2(^N)⊗), where _+ (resp. _-) is the symmetrizer (resp. antisymmetrizer) with respect to permutations of the particles' coordinates and (m) spin variables. This is basically due to the fact that any point ∈^N not lying on the singular hyperplanes x_i-x_j=0 can be mapped in a unique way to a point in A by a suitable permutation. As we shall see below, it shall be convenient for what follows to identify with its symmetric (resp. antisymmetric) extension when =1 (resp. =-1). With this identification, it can be shown <cit.> that is represented by an upper triangular matrix in the appropriately ordered (non-orthonormal) basis with elements|,⟩=_(^2·∏_i<jsin(x_i-x_j)^a |⟩) ,where |⟩∈ and ≡(p_1,…,p_N)∈^N satisfy the following conditions: * The differences n_i≡ p_i-p_i+1 (1≤ i≤ N-1) are nonnegative integers.* If p_i=p_i+1 then s_i≺ s_i+1. * The total momentum of the state |,⟩ vanishes, i.e., ∑_i p_i=0.In the second condition, the notation s_i≺ s_j stands for s_i<s_j when =-1 and s_i≤ s_j when =1. The first condition is justified in Ref. <cit.>, the second one can be arranged due to the symmetric/antisymmetric nature of the states (<ref>), while the last one simply reflects that we are working in the center of mass frame. As shown in the latter reference, the states |,⟩ should be ordered in such a way that |,⟩ precedes |','⟩ whenever <', where the last notation means thatprecedes ' in the lexicographic order. With this partial order, the action of on the basis (<ref>) is upper triangular. More precisely <cit.>,|,⟩=E()|,⟩+∑_'<; 'c(',') |','⟩ ,with c(',')∈ andE()=∑_i[2p_i+a(N+1-2i)]^2 .Since preserves (), if the vector in Eq. (<ref>) is such that |⟩∈() then |'⟩∈() for all vectors ' appearing in the RHS of the latter equation. In other words, ^ is also upper triangular with respect to the basis (<ref>), where |⟩∈∩() and the quantum numbers (,) satisfy conditions i)–iii) above, ordered as previously explained. Moreover, by Eq. (<ref>) the eigenvalues of ^ are given by Eq. (<ref>). Expanding the latter equation in powers of a we obtainE()=+4a∑_ip_i(N+1-2i)+(1) ,where=a^2∑_i(N+1-2i)^2=a^2/3 N(N^2-1)is the ground state energy of the ferromagnetic model (=1). Thus in the limit a→∞ we havelim_a→∞q^-/4a^(4aT)=∑_,q^∑_i p_i(N+1-2i) ,where the sum is extended to all (,) satisfying conditions i)–iii) above with |⟩∈∩(). Since the exponent is independent of the spin variables , the sum over can be immediately carried out, namelylim_a→∞q^-/4a^(4aT)=∑_d(,,) q^∑_i p_i(N+1-2i) ,where the spin degeneracy factor d(,,) is the number of multiindices satisfying condition ii) above for a given such that |⟩∈∩(). In other words,d(,,)=|S(,)∩()|whereS(,)≡{|⟩∈| p_i=p_i+1⇒ s_i≺ s_i+1 , 1≤ i≤ N-1} .In order to evaluate the sum in Eq. (<ref>), we note that by conditions i) and iii) above we can write the multiindexas=(ρ_1,…,ρ_1^k_1,…, ρ_r,…,ρ_r^k_r) ,withk_1+⋯+k_r=N,k_1ρ_1+⋯+k_rρ_r=0 ,ρ_i>ρ_i+1,ρ_i-ρ_i+1∈ .Thus the multiindexconsists of r blocks of lengths k_1,…,k_r. CallingK_i=∑_j=1^ik_j , 0≤ i≤ r ,we have∑_i p_i(N+1-2i)=∑_i=1^rρ_i∑_j=K_i-1+1^K_i(N+1-2j)=∑_i=1^rρ_ik_i(N-2K_i+k_i).Since d(,,) obviously depends on only through ≡(k_1,…,k_r), we can rewrite Eq. (<ref>) aslim_a→∞q^-/4a^(4aT)=∑_r=1^N∑_∈^r_Nd(,,) ∑_ρ_1>⋯>ρ_r, ρ_i-ρ_i+1∈ k_1ρ_1+⋯+k_rρ_r=0q^∑_i=1^rρ_ik_i(N-2K_i+k_i) ,where _N^r denotes the set of all partitions of the integer N in r parts with order taken into account. The inner sum in Eq. (<ref>) was evaluated in Ref. <cit.>, with the result∑_ρ_1>⋯>ρ_r, ρ_i-ρ_i+1∈ k_1ρ_1+⋯+k_rρ_r=0q^∑_i=1^rρ_ik_i(N-2K_i+k_i)= ∏_i=1^r-1q^(K_i)/1-q^(K_i) ,where(k)=k(N-k) .Substituting Eq. (<ref>) into Eq. (<ref>) we obtainlim_a→∞q^-/4a^(4aT)=∑_r=1^N∑_∈^r_Nd(,,) ∏_i=1^r-1q^(K_i)/1-q^(K_i) ,where is any multiindex of the form (<ref>). The partition function for the scalar Hamiltonian was also evaluated in Ref. <cit.> in the large a limit, namelylim_a→∞q^-/4a(4aT)=∏_i=1^N(1-q^(i))^-1 .Combining Eqs. (<ref>)-(<ref>) with Eq. (<ref>) we finally obtain the following explicit formula for the restricted partition function Z_0^(T):Z_0^(T)=∑_r=1^N∑_∈^r_Nd(,,) ∏_i=1^r-1q^(K_i)·∏_j=1^N-r(1-q^(K'_j)) ,where{K_1',…,K'_N-r}={1,…,N-1}-{K_1,…,K_r-1}and is determined by through Eq. (<ref>). Following a similar procedure for the PF and FI chains we again obtain Eq. (<ref>), but with (k) in Eq. (<ref>) respectively given by k and k(-2N+k+1) (see Refs. <cit.> for more details). In summary, the restricted partition function Z_0^(T) for the three chains of HS type is given by Eq. (<ref>), with dispersion relation(k)= k(N-k) ,for the HS chaink ,for the PF chaink(-2N+k+1) ,for the FI chain .Equations (<ref>)-(<ref>) yield an explicit formula for the partition function of the (m) gLMG model (<ref>) with interactions h_ij=h(z_i-z_j) given by Eqs. (<ref>)-(<ref>), once the degeneracy factor d(,,) is known.§ DEGENERACY FACTOR As we have seen in the previous section, in order to evaluate the partition function Z(T) of an (m) gLMG model of HS type through Eqs. (<ref>)-(<ref>), we only need to determine the degeneracy factor d(,,) defined in Eq. (<ref>). To this end, let us fix in Eq. (<ref>) (with ∈_N^r) and take =(N_1,…,N_m) such that N_a∈_0≡∪{0} and ||=N. The degeneracy factor d(,,) is obviously much easier to compute in the antiferromagnetic case (=-1), since by Pauli's principle the (m) spins in each block of length k_1,…,k_r in which the components of are equal must all be different (in fact, arranged in a strictly increasing sequence according to condition ii) in the previous section). §.§ Anti-ferromagnetic case Let us define the vector =(r_1,…,r_m) byr_i≡|{j=1,…,r| k_j=i}|∈_0, 1≤ i≤ m,so thatr_1+⋯+r_m=r , r_1+2r_2+⋯+mr_m=N .In other words, r_i is the number of blocks of length i in the expression (<ref>) for . Obviously d(,,-)≡ D_m(,), where D_m(,) denotes the number of ways one can distribute N_1 spins |1⟩, N_2 spins |2⟩, … , N_m spins |m⟩ in r_1 blocks of one site, r_2 blocks of two sites, … , r_m blocks of m sites, with all spins different in each block.For 1≤ i,j≤ m, let us denote by N_i,j∈_0 the number of spins |i⟩ in the r_j blocks of j sites, and define _i=(N_i,1,…,N_i,m) such that |_i|=N_i for 1≤ i≤ m. We can find an expression for the degeneracy factor by counting the number of ways one can fill the pattern of blocks so that all the spins in each block are different. To this end, we start with an empty pattern and fill it as follows: * Fill all the r_m blocks of m sites.In the r_m blocks of m sites there must be r_m spins of each type. We are left with N_1-r_m spins |1⟩, N_2-r_m spins |2⟩,…, N_m-r_m spins |m⟩ and a pattern of r_1 blocks of one site, r_2 blocks of two sites, …, r_m-1 blocks of m-1 sites. * Distribute the remaining N_m-r_m spins of type |m⟩ in the r-r_m empty blocks left.As in the previous step, we next fix a vector=(x_1,…,x_m-1)with x_i≡ N_m,i∈_0 and ||=N_m-r_m. Clearly, the number of ways of distributing the N_m-r_m spins |m⟩ in the available r-r_m blocks is given by the product of binomial coefficients ∏_i=1^m-1r_ix_i.* For eachin step ii), we are left with a new pattern ∈_0^m-1 and new spins of types 1,…,m-1 with magnon numbers (N̂_1,…,N̂_m-1)≡.Remarkably, the new patternhas no blocks of m sites and the new vectorhas no spins |m⟩. More precisely, for i=1,…,m-1 there are now _i=r_i-x_i+x_i+1 blocks of i sites, i.e., the previous r_i minus the occupied blocks of i sites plus the occupied blocks of i+1 sites (note that we must take x_m=0, since all the blocks of m sites were filled up in the first step). Thus, the new pattern ≡() and magnon vectorare given by_i =r_i-x_i+x_i+1 , 1≤ i≤ m-2 ;_m-1=r_m-1-x_m-1, _i =N_i-r_m ,1≤ i≤ m-1,and thereforeD_m(,)=∑_||=N_m-r_m1.25em∏_i=1^m-1r_ix_i· D_m-1((),) .Note that the new vectors and satisfy a relation analogous to the last Eq. (<ref>), namely (by Eqs. (<ref>)-(<ref>))||=N-N_m-(m-1)r_m =_1+2 _2+⋯+(m-1) _m-1 . * Iterate the process described above.By Eq. (<ref>), we can express the degeneracy factorD_m(,)≡ D_m(^(m),^(m))as a linear combination of degeneracy factorsD_m-1(,)≡ D_m-1(^(m-1),^(m-1)) .This process can be iterated, by expressing each term D_m-1(^(m-1),^(m-1)) in Eq. (<ref>) in terms of degeneracy factorsD_m-2((^(m-1)), (^(m-1)) )≡ D_m-2(^(m-2),^(m-2)) ,and so on. We thus obtain the recursion relationD_k(^(k),^(k))=∑_||=N_k^(k)-r_k^(k)1.6em ∏_i=1^k-1r_i^(k)x_i· D_k-1(^(k-1)(),^(k-1)),where^(k),^(k)∈_0^k ;, ^(k-1)(),^(k-1)∈_0^k-1 ,withr_i^(k-1)()=r_i^(k)-x_i+x_i+1(x_k≡0) , N_i^(k-1)=N_i^(k)-r_k^(k)and ^(m)≡, ^(m)≡. The above recursion relation, together with the obvious initial condition D_1=1, fully determines D_m(,).In Section <ref> we shall illustrate the above procedure for computing the degeneracy factor d(,,-)≡ D_m(,) with several examples. Once D_m is determined, the restricted partition Z_0^,(-)(T) in the antiferromagnetic case is obtained from Eq. (<ref>), namelyZ_0^,(-)(T)=∑_r=⌈ N/m⌉^N1.4em∑_∈^r_ND_m(,) ∏_i=1^r-1q^(K_i)·∏_j=1^N-r(1-q^(K'_j)) ,where the range of the last sum comes from condition ii) above, since in the antiferromagnetic case the lengths k_i of the blocks in Eq. (<ref>) are all at most equal to m. The partition function of the corresponding gLMG model of HS type (<ref>) can then be computed from Eq. (<ref>), with the resultZ^(-)(T)=∑_||=Nq^h()∑_r=⌈ N/m⌉^N1.4em∑_∈^r_ND_m(,) ∏_i=1^r-1q^(K_i)·∏_j=1^N-r(1-q^(K'_j)) .§.§ Ferromagnetic case A similar procedure could be followed in principle to compute the degeneracy factor d(,,+) in the ferromagnetic case =1. The main difference is that now each value of the (m) spin can be used more than once to fill the blocks of length k_1,…,k_r determined by the multiindex in Eq. (<ref>), which considerably complicates matters.In practice, it is much easier to derive the ferromagnetic partition function Z^(+) from the antiferromagnetic one Z^(-) computed in the previous subsection by means of the identityH_0^(+)+H_0^(-)=∑_i jh_ij≡(N) ,where H_0^(±) denotes the Hamiltonian (<ref>) with =±. The constant (N), which is the maximum energy of H_0^(-), can be easily computed in closed form for each of the interactions (<ref>)-(<ref>) taking into account the identity <cit.>(N)=∑_i=1^N-1(i) ,namely=N/6 (N^2-1) , for the HS chain N/2 (N-1) ,for the PF chain N/6 (N-1)(3-4N+2) ,for the FI chain .From Eq. (<ref>) it immediately follows that the restricted partition functions Z_0^,(±) of H_0^(±) are related byZ_0^,(+)(q)=q^(N)Z_0^,(-)(q^-1) .Using Eqs. (<ref>) and (<ref>) we easily obtainZ_0^,(+)(T) =q^(N)∑_r=⌈ N/m⌉^N1.4em∑_∈^r_N D_m(,) ∏_i=1^r-1q^-(K_i)·∏_j=1^N-r(1-q^-(K'_j))=∑_r=⌈N/m⌉^N(-1)^N-r∑_∈^r_ND_m(,)∏_i=1^N-r(1-q^(K'_i)) ,where D(,) is the antiferromagnetic degeneracy factor computed in the previous subsection. By Eq. (<ref>), the partition function of the ferromagnetic Hamiltonian H^(+)≡ H_0^(+)+H_1 is given byZ^(+)(T)=∑_||=Nq^h()∑_r=⌈N/m⌉^N(-1)^N-r∑_∈^r_ND_m(,) ∏_i=1^N-r(1-q^(K'_i)) . § EXAMPLES§.§ (2) In this case =(r_1,r_2), =(N_1,N_2), and the recursion relation (<ref>) with D_1=1 immediately yieldsD_2(,)=r_1N_2-r_2 .Expressing r_1,r_2 in terms of r and N by means of the relations r=r_1+r_2, N=N_1+N_2=r_1+2 r_2 we finally obtainD_2(,)=2r-Nr-N_1=2r-Nr-N_2 .Thus the restricted partition function of the (2) chains (<ref>) of HS type is given byZ_0^(T)=∑_r=1^N(-1)^1+/2(N-r)∑_∈^r_N2r-Nr-N_1q^1-/2∑_i=1^r-1(K_i)∏_i=1^N-r(1-q^(K'_i)) .By Eq. (<ref>), the partition function of the corresponding (2) gLMG model readsZ(T)=∑_N_1=0^Nq^h(N_1,N-N_1)∑_r=1^N(-1)^1+/2(N-r)∑_∈^r_N2r-Nr-N_1q^1-/2∑_i=1^r-1(K_i)∏_i=1^N-r(1-q^(K'_i)) .§.§ (3)Let =(r_1,r_2,r_3) and =(N_1,N_2,N_3) such that r=r_1+r_2+r_3, N=N_1+N_2+N_3 and r_1+2 r_2+3 r_3=N. We then haveD_3(,)=∑_x_1+x_2=N_3-r_3r_1x_1r_2x_2D_2(,) =∑_x_1+x_2=N_3-r_3r_1x_1r_2x_22 - -_2=∑_x_1+x_2=N_3-r_3r_1x_1r_2x_22r-2x_1-N+N_3r-x_1-N_2,where we have used the identities =r-r_3-x_1, =N-N_3-2 r_3 and _2=N_2-r_3. §.§ (4) Let =(r_1,r_2,r_3,r_4) and =(N_1,N_2,N_3,N_4) such that r=r_1+r_2+r_3+r_4, N=N_1+N_2+N_3+N_4=r_1+2 r_2+3 r_3+4 r_4. Using Eq. (<ref>) with (,)≡(^(3),^(3)) in place of (,) we easily obtainD_4(,) =∑_x_1+x_2+x_3=N_4-r_4r_1x_1r_2x_2r_3x_3D_3(,)=∑_x_1+x_2+x_3=N_4-r_4r_1x_1r_2x_2r_3x_3 .5em ∑_y_1+y_2=N_3-r_4-r_3+x_3r_1-x_1+x_2y_1r_2-x_2+x_3y_22 r-2 x_1-2 y_1-N_1- N_2r-x_1-y_1-N_2. § THE LMG-PF MODEL When H_0 is the Hamiltonian of the PF chain the restricted partition function Z_0^(T), and hence the partition function Z(T) of the corresponding LMG-PF model (<ref>), can be considerably simplified. Indeed, in this case=-Δ+a r^2+∑_i ja(a- S_ij)/(x_i-x_j)^2 =+2aĤ_0() ,where r^2≡∑_ix_i^2,=-Δ+a r^2+∑_i ja(a-1)/(x_i-x_j)^2is the scalar Calogero model andĤ_0()=∑_i<j1- S_ij/(x_i-x_j)^2is obtained from H_0 by the formal substitution z_i↦ x_i. Proceeding as in Section <ref> we obtain the analogue of Eq. (<ref>), namelyZ_0^(T)=lim_a→∞^(2aT)/(2aT) ,where the partition function (2aT) of the scalar Calogero model is given by <cit.>(2aT)=q^/2a∏_i(1-q^i)^-1 ,≡ a^2N(N-1)+aN .In order to compute ^(2aT), we note <cit.> that the Hamiltonian (<ref>) of the spin Calogero model is upper triangular in the basis with elements|,⟩=^-ar^2/2∏_i<j|x_i-x_j|^a_(∏_ix_i^p_i|⟩)partially ordered by the total degree ||, with corresponding eigenvaluesE()=2a(p_1+⋯+p_N)+ .Of course, we must choose the quantum numbers (,) in such a way that the states (<ref>) are actually a basis. The main difference with the HS and FI models is that only in this case E() and the admissible partial order of the basis states (<ref>) do not depend on the ordering of the components of <cit.>. As a consequence, we can choose the quantum numbers (,) in each subspace _[L^2(^N)⊗()] as follows: * We first order the components of the spin quantum number increasingly, so that=(1,…,1^N_1,…,m,…,m^N_m)is now fixed.* In each block of with fixed magnon number |a⟩ we order the corresponding components of the vector also increasingly, so that =(^1,…,^m) with^j≡(ρ^j_1,…,ρ^j_N_j)and ρ^j_i ≺ ρ^j_i+1.We thus haveE()=+2a∑_i=1^Np_i=+2a∑_j=1^m∑_i=1^N_jρ^j_i ,and thereforeq^-/2a^(2aT)=∑_ρ^k_i≺ρ^k_i+1∏_j=1^mq^ρ^j_1+⋯+ρ^j_N_j =∏_j=1^m 1.75em∑_0≤ρ^j_1≺ ⋯ ≺ρ^j_N_jq^ρ^j_1+⋯+ρ^j_N_j .The inner sum in the latter formula can be computed in closed form, with the result∑_0≤ρ^j_1≺⋯≺ρ^j_N_jq^ρ^j_1+⋯+ρ^j_N_j= q^1-/4N_j(N_j-1)∏_i=1^N_j(1-q^i)^-1≡ q^1-/4N_j(N_j-1)(q)_N_j^-1 ,and thus^(2aT)=∏_j=1^mq^1-/4N_j(N_j-1)(q)_N_j^-1=q^1-/4∑_j=1^mN_j(N_j-1)∏_j=1^m(q)_N_j^-1 .From this equation and Eqs. (<ref>)-(<ref>) we obtain the following closed-form expression for the restricted partition function of the PF chain:Z^_0(T)=q^1-/4∑_j=1^mN_j(N_j-1)(q)_N/∏_j=1^m(q)_N_j .Finally, by Eq. (<ref>) the partition function of the LMG-PF model is given byZ(T) =∑_N_1+⋯+N_m=Nq^h()+1-/4∑_j=1^mN_j(N_j-1)(q)_N/∏_j=1^m(q)_N_j≡∑_N_1+⋯+N_m=Nq^h()+1-/4∑_j=1^mN_j(N_j-1)NN_1,…,N_mq .In particular, for h=0 we recover the well-known formula for the partition function of the PF chain in Ref. <cit.>.§ ANALYSIS OF THE SPECTRUM AND THERMODYNAMICS In this section we shall take advantage of the knowledge of the restricted partition function of the gLMG models (<ref>) to study several statistical properties of their spectrum and analyze the behavior of their thermodynamic functions for large N. To begin with, we have examined the level density of the restriction of the Hamiltonian to subspaces with a fixed magnum content. Since H_1 is constant on these subspaces, this is of course equivalent to studying the level density of the corresponding spin chains of HS type. It is well-known in this respect <cit.> that the level density of the complete spectrum of the latter models becomes normally distributed in the N→∞ limit, essentially due to the existence of a description of the spectrum in terms of Haldane's motifs <cit.>. We have computed the spectrum of the HS chain for up to N=26 for (2) and N=24 for (3) in the largest subspace () (with N_i=N/m for all i). Our results clearly indicate that the spectrum of the restriction of H_0 to this subspace is also normally distributed (see Fig. <ref>, left), with parameters μ andgiven by the mean and standard deviation of the restricted spectrum. For the FI and PF chains we have obtained similar results. This fact suggests <cit.> that in all three cases there might be a formula for the energies in each sector of the spectrum with fixed magnon numbers in terms of motifs.Since the continuous part of the cumulative level density in each sector can be well approximated by a Gaussian distribution, the energies of the “unfolded” spectrum <cit.> can be taken asη_i=∫_-∞^E_ig(t) t , g(E)=1/√(2π) ^-(E-μ)^2/2^2 , i=1,…,n .According to a long-standing conjecture due to Berry and Tabor <cit.>, the distribution of the (normalized) spacings between consecutive levels of the unfolded spectrum, defined ass_i=(n-1)η_i+1-η_i/η_n-η_1 , i=1,…,n-1,is expected to be Poissonian for `generic' integrable systems. On the other hand, for a chaotic system the well-known Bohigas–Giannoni–Schmit conjecture posits that this distribution should be given by the Wigner distribution corresponding to the appropriate ensemble of random matrices <cit.>. In Refs. <cit.> it was observed that the distribution p(s) of the spacings between consecutive levels of the whole spectrum of all three chains of HS type follows none of the above distributions, but is typically given by the `square root of a logarithm' lawP(s)=1-2/√(π) s_max√(log(s_max/s)) ,where P(s)=∫_0^sp(s') s' is the cumulative distribution and s_max is the maximum spacing. As shown in Refs. <cit.>, this is due to the fact that the raw spectrum of the latter chains is approximately equispaced and normally distributed. We have computed the distribution of consecutive (normalized) spacings in the subspaces mentioned above for the HS, PF and FI chains for m=2 and 3. In all cases, the cumulative spacings distribution P(s) fits Eq. (<ref>) with remarkable accuracy (see the insets Fig. <ref>, right, for the HS chain). This clearly suggests that the (raw) spectrum of the restriction of the three HS-type chains to subspaces with fixed magnon content is also approximately equispaced. We have also verified that this conclusion is indeed correct for all three chains of HS type. For instance, for the (2) HS chain with N_1=N_2=13 (cf. Fig. <ref>, top right) 93.8% of the spacings between consecutive levels of the raw spectrum is equal to 1, while for the (3) HS chain with N_1=N_2=N_3=8 (cf. Fig. <ref>, bottom right) the predominant spacing is again 1 and occurs 95.7% of the times. We shall next analyze the thermodynamics of a class of LMG models of HS type whose deformation Hamiltonian (<ref>) is given byh(x_1,…,x_m)=1/N ∑_a=1^m(x_a-n_aN)^2 ,where the parameters n_a (1≤ a≤ m) are assumed to lie in the interval (0,1) and n_1+⋯+n_m=1. These parameters thus represent the magnon densities of the ground state in the ferromagnetic case (=1). The motivation for considering a quadratic deformation Hamiltonian is, first of all, that in the original, isotropic LMG model the external term H_1 is precisely of this form. More recently, generalized LMG models with a quadratic external term have proved of interest in the context of quantum information theory, since they are some of the few systems for which the bipartite entanglement entropy of the ground state can be computed in closed form <cit.>. Using the exact formulas (<ref>)-(<ref>) and (<ref>), we have evaluated the partition function of this class of models for a relatively large number of spins, of the order of 100 (resp. 50) for the (2) (resp. (3)) ferromagnetic LMG-PF models. From the resulting expression, we have computed the free energy f, the internal energy u, the entropy s and the specific heat c (per spin, in all cases) via the formulasf(T) =-T/N log Z(T) , u(T)=T^2/N ∂log Z(T)/∂ T ,s(T) =∂/∂ T(T/Nlog Z(T)) , c(T)=2T/N ∂log Z(T)/∂ T+T^2/N∂^2log Z(T)/∂ T^2 ,where we have taken Boltzmann's constant k_B=1. We have first verified that the thermodynamic functions are practically independent of N for N≲ 100 (in the (2) case) and N≲ 50 (in the (3) case). Thus the thermodynamic functions for N=100 (in the (2) case) and N=50 (in the (3) case) can be regarded as a reasonable approximation of their N→∞ counterparts. As an additional check, we have compared the results for the (2) PF chain with no deformation Hamiltonian and N=100 spins with the exact N→∞ formulas derived in Ref. <cit.>, finding them in excellent agreement (cf. Fig. <ref>). In particular, the extensive behavior of the thermodynamic entropy contrasts with the logarithmic growth of the ground-state entanglement entropy of the ferromagnetic “quadratic” gLMG models studied in Ref. <cit.>.In Figs. <ref> and <ref> we present the plots of the free and internal energies, the entropy and the specific heat (per spin) respectively of the (2) and (3) models (<ref>)-(<ref>) in the PF case. It is apparent from these figures that both the (2) and the (3) thermodynamic functions qualitatively behave like those of a two-level system, as for instance the one-dimensional Ising model at zero magnetic field or a paramagnetic spin 1/2 ion <cit.>. In particular, from Figs. <ref> and <ref> we see that the specific heat exhibits the Schottky peak characteristic of the latter systems. Finally, it may seem surprising that the entropy per spin does not appear to vanish at T=0 in some cases, especially when h=0 (see, e.g., Fig. <ref>). Of course, the explanation for this behavior is that the number of spins N is finite (though large), so that s(0)=(log d(m,N))/N, where d(m,N) is the ground state degeneracy. In the ferromagnetic case under consideration, it follows from Eq. (<ref>) that when h=0 the ground states are the symmetric states, so thatd(m,N)=N+m-1m-1≃N^m-1/(m-1)! ,and thus s(0)≃(m-1)(log N)/N is small but nonzero. On the other hand, when h does not vanish identically the H_1 term in Eq. (<ref>) breaks the ground state degeneracy almost completely (the more so in the less symmetric cases, in which the densities n_a are all different), so that s(0) is significantly smaller than its h=0 counterpart.§ CONCLUSIONS We shall finish this paper with a brief summary of its main results. We have introduced a family of generalized (m) Lipkin–Meshkov–Glick models whose interacting term is a spin chain of Haldane–Shastry type, which can be equivalently regarded as the deformation of a spin chain of HS type H_0 by the addition of a term H_1 in the enveloping algebra of the Cartan subalgebra of (m). The Hilbert space of the system is a direct sum of subspaces () with fixed magnon numbers, in which the action of the deformation term is diagonal, so that the model's partition function decomposes as in Eq. (<ref>). By a suitable adaptation of Polychronakos's freezing trick, we have been able to compute in closed form the partition functions of the restrictions of the spin chain Hamiltonian H_0 to the subspaces (). In view of the previous remarks, this immediately yields the partition function of the associated gLMG model. In particular, when H_0 is the Hamiltonian of the Polychronakos–Frahm spin chain we have obtained an alternative, simpler expression for the partition function akin to Polychronakos's formula <cit.> for the case H_1=0. This closed-form expression for the partition function of the restriction of H_0 to the subspaces () has been used in numerical calculations to provide strong evidence that the level density of the latter restriction is Gaussian when the number of spins tends to infinity. In view of the results of Ref. <cit.>, this suggests that there exists a description of the spectrum of H_0|_() in terms of motifs, a fact that deserves further investigation. We have also numerically studied the distribution of the spacings of consecutive unfolded levels of H_0|_(), showing that it follows the same characteristic law previously found for the complete spectrum. As a final application, we have computed the free and internal energies, the entropy and the specific heat per spin of a class of (2) and (3) gLMG models with quadratic H_1. We have checked that these functions are virtually independent of the number of spins N when this number is sufficiently large, which indicates that they yield reasonable approximations to their respective thermodynamic limits. Our analysis shows that the thermodynamic functions of these models are qualitatively similar to those of a two-level system, as already observed in Ref. <cit.> for the (2) chains of HS type. In the latter chains, this similarity is ultimately due to the existence of a description of the spectrum in terms of motifs, which leads to simple closed formulas for the thermodynamic functions in terms of the dispersion relation. This again suggests that a description of this type should also exist for the more general models studied in this paper. § REFERENCES10 LMG65 Lipkin H J, Meshkov N and Glick A J, Validity of many-body approximation methods for a solvable model: (I). Exact solutions and perturbation theory,1965 Nucl. Phys. 62 188MGL65 Meshkov N, Glick A J and Lipkin H J, Validity of many-body approximation methods for a solvable model: (II). Linearization procedures,1965 Nucl. Phys. 62 199GLM65 Glick A J, Lipkin H J and Meshkov N, Validity of many-body approximation methods for a solvable model: (III). Diagram summations,1965 Nucl. Phys. 62 211RS80 Ring P and Schuck P, The Nuclear Many-Body Problem (Berlin: Springer-Verlag), first edition 1980UF03 Unanyan R G and Fleischhauer M, Decoherence-free generation of many-particle entanglement by adiabatic ground-state transitions,2003 Phys. Rev. Lett. 90 133601(4)CLJ09 Chen G, Liang J Q and Jia S, Interaction-induced Lipkin–Meshkov–Glick model in a Bose–Einstein condensate inside an optical cavity,2009 Opt. Express 17 19682UZ92 Ulyanov V V and Zaslavskii O B, New methods in the theory of quantum spin chains,1992 Phys. Rep. 216 179ORTR16 Opanchuk B, Rosales-Zárate L, Teh R Y and Reid M D, Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose–Einstein condensates,2016 Phys. Rev. A 94 062125(14)RCCP17 Romera E, Castaños O, Calixto M and Pérez-Bernal F, Delocalization properties at isolated avoided crossings in Lipkin–Meshkov–Glick type Hamiltonian models,2017 J. Stat. Mech. Theory-E. 2017 P013101(19)PS05 Popkov V and Salerno M, Logarithmic divergence of the block entanglement entropy for the ferromagnetic Heisenberg model,2005 Phys. Rev. A 71 012301(4)LORV05 Latorre J I, Orús R, Rico E and Vidal J, Entanglement entropy in the Lipkin–Meshkov–Glick model,2005 Phys. Rev. A 71 064101(4)BDV06 Barthel T, Dusuel S and Vidal J, Entanglement entropy beyond the free case,2006 Phys. Rev. Lett. 97 220402(4)ODV08 Orus R, Dusuel S and Vidal J, Equivalence of critical scaling laws for many-body entanglement in the Lipkin–Meshkov–Glick model,2008 Phys. Rev. Lett. 101 25701(4)WVVD12 Wilms J, Vidal J, Verstraete F and Dusuel S, Finite-temperature mutual information in a simple phase transition,2012 J. Stat. Mech.-Theory E. 2012 P01023(21)VLRK03 Vidal G, Latorre J I, Rico E and Kitaev A, Entanglement in quantum critical phenomena,2003 Phys. Rev. Lett. 90 227902(4)HLW94 Holzhey C, Larsen F and Wilczek F, Geometric and renormalized entropy in conformal field theory,1994 Nucl. Phys. B 424 443Ko04 Korepin V E, Universality of entropy scaling in one dimensional gapless models,2004 Phys. Rev. Lett. 92 096402(3)RM04 Refael G and Moore J E, Entanglement entropy of random quantum critical points in one dimension,2004 Phys. Rev. Lett. 93 260602(4)CFGRT16-LMG Carrasco J A, Finkel F, González-López A, Rodríguez M A and Tempesta P, Generalized isotropic Lipkin–Meshkov–Glick models: ground state entanglement and quantum entropies,2016 J. Stat. Mech. Theory-E. 2016 033114(33)PD99 Pan F and Draayer J P, Analytical solutions for the LMG model,1999 Phys. Lett. B 451 1MOPN06 Morita H, Ohnishi H, da Providência J and Nishiyama S, Exact solutions for the LMG model Hamiltonian based on the Bethe ansatz, 2006 Nucl. Phys. B 737 337RVM07 Ribeiro P, Vidal J and Mosseri R, Thermodynamical limit of the Lipkin–Meshkov–Glick model,2007 Phys. Rev. Lett. 99 050402(4)RVM08 Ribeiro P, Vidal J and Mosseri R, Exact spectrum of the Lipkin–Meshkov–Glick model in the thermodynamic limit and finite-size corrections,2008 Phys. Rev. E 78 021106(13)Ts88 Tsallis C, Possible generalization of Boltzmann–Gibbs statistics, 1988 J. Stat. Phys. 52 479Ts09 Tsallis C, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World (Berlin: Springer) 2009Ha88 Haldane F D M, Exact Jastrow–Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r^2 exchange,1988 Phys. Rev. Lett. 60 635Sh88 Shastry B S, Exact solution of an S=1/2 Heisenberg antiferromagnetic chain with long-ranged interactions,1988 Phys. Rev. Lett. 60 639HHTBP92 Haldane F D M, Ha Z N C, Talstra J C, Bernard D and Pasquier V, Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory,1992 Phys. Rev. Lett. 69 2021Po93 Polychronakos A P, Lattice integrable systems of Haldane–Shastry type,1993 Phys. Rev. Lett. 70 2329Fr93 Frahm H, Spectrum of a spin chain with inverse-square exchange,1993 J. Phys. A: Math. Gen. 26 L473FI94 Frahm H and Inozemtsev V I, New family of solvable 1D Heisenberg models,1994 J. Phys. A: Math. Gen. 27 L801Gu63 Gutzwiller M C, Effect of correlation on the ferromagnetism of transition metals,1963 Phys. Rev. Lett. 10 159GV87 Gebhard F and Vollhardt D, Correlation functions for Hubbard-type models: the exact results for the Gutzwiller wave function in one dimension,1987 Phys. Rev. Lett. 59 1472Ha91b Haldane F D M, “Fractional statistics” in arbitrary dimensions: a generalization of the Pauli principle,1991 Phys. Rev. Lett. 67 937GS05 Greiter M and Schuricht D, No attraction between spinons in the Haldane–Shastry model,2005 Phys. Rev. B 71 224424(4)Gr09 Greiter M, Statistical phases and momentum spacings for one-dimensional anyons,2009 Phys. Rev. B 79 064409(5)BBS08 Basu-Mallick B, Bondyopadhaya N and Sen D, Low energy properties of the SU(m|n) supersymmetric Haldane–Shastry spin chain,2008 Nucl. Phys. B 795 596CS10 Cirac J I and Sierra G, Infinite matrix product states, conformal field theory, and the Haldane–Shastry model,2010 Phys. Rev. B 81 104431(4)BBL08 Bargheer T, Beisert N and Loebbert F, Boosting nearest-neighbour to long-range integrable spin chains,2008 J. Stat. Mech. Theory-E. 2008 L11001(9)BBL09 Bargheer T, Beisert N and Loebbert F, Long-range deformations for integrable spin chains,2009 J. Phys. A: Math. Theor. 42 285205(58)HGCK16 Hung C L, González-Tudela A, Cirac J I and Kimble H J, Quantum spin dynamics with pairwise-tunable, long-range interactions,2016 Proc. Natl. Acad. Sci. U. S. A. 113 E4946Po94 Polychronakos A P, Exact spectrum of SU(n) spin chain with inverse-square exchange,1994 Nucl. Phys. B 419 553FG05 Finkel F and González-López A, Global properties of the spectrum of the Haldane–Shastry spin chain,2005 Phys. Rev. B 72 174411(6)BFGR10 Barba J C, Finkel F, González-López A and Rodríguez M A, Inozemtsev's hyperbolic spin model and its related spin chain,2010 Nucl. Phys. B 839 499Ca71 Calogero F, Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials,1971 J. Math. Phys. 12 419Su71 Sutherland B, Exact results for a quantum many-body problem in one dimension,1971 Phys. Rev. A 4 2019Su72 Sutherland B, Exact results for a quantum many-body problem in one dimension. II,1972 Phys. Rev. A 5 1372In96 Inozemtsev V I, Exactly solvable model of interacting electrons confined by the Morse potential,1996 Phys. Scr. 53 516BT77 Berry M V and Tabor M, Level clustering in the regular spectrum,1977 Proc. R. Soc. London Ser. A 356 375BGS84 Bohigas O, Giannoni M J and Schmit C, Characterization of chaotic quantum spectra and universality of level fluctuation laws,1984 Phys. Rev. Lett. 52 1BFGR08epl Barba J C, Finkel F, González-López A and Rodríguez M A, The Berry–Tabor conjecture for spin chains of Haldane–Shastry type, 2008 Europhys. Lett. 83 27005(6)EFG12 Enciso A, Finkel F and González-López A, Thermodynamics of spin chains of Haldane–Shastry type and one-dimensional vertex models,2012 Ann. Phys.-New York 327 2627EFG09 Enciso A, Finkel F and González-López A, Spin chains of Haldane–Shastry type and a generalized central limit theorem,2009 Phys. Rev. E 79 060105(4)EFG10 Enciso A, Finkel F and González-López A, Level density of spin chains of Haldane–Shastry type,2010 Phys. Rev. E 82 051117(6)BBH10 Basu-Mallick B, Bondyopadhaya N and Hikami K, One-dimensional vertex models associated with a class of Yangian invariant Haldane–Shastry like spin chains,2010 SIGMA 6 091(13)Ha01 Haake F, Quantum Signatures of Chaos (Berlin: Springer-Verlag), second edition 2001BFGR08 Barba J C, Finkel F, González-López A and Rodríguez M A, Polychronakos–Frahm spin chain of BC_N type and the Berry–Tabor conjecture,2008 Phys. Rev. B 77 214422(10)BFGR09 Barba J C, Finkel F, González-López A and Rodríguez M A, An exactly solvable supersymmetric spin chain of BC_N type,2009 Nucl. Phys. B 806 684Mu10 Mussardo G, Statistical Field Theory: an Introduction to Exactly Solved Models in Statistical Physics (Oxford: Oxford University Press) 2010 | http://arxiv.org/abs/1706.08751v2 | {
"authors": [
"Jose A. Carrasco",
"Federico Finkel",
"Artemio Gonzalez-Lopez"
],
"categories": [
"cond-mat.stat-mech",
"math-ph",
"math.MP",
"nlin.SI"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170627092957",
"title": "Generalized Lipkin-Meshkov-Glick models of Haldane-Shastry type"
} |
[Email: ][email protected] Department of Mathematics, University of North Carolina, Chapel Hill, NC USA Departments of Mathematics and Biomedical Engineering and McAllister Heart Institute, University of North Carolina, Chapel Hill, NC USAIn standard models of cardiac electrophysiology, including the bidomain and monodomain models, local perturbations can propagate at infinite speed. We address this unrealistic propertyby developing a hyperbolic bidomain model that is based on a generalization of Ohm's law with a Cattaneo-type model for the fluxes.Further, we obtain a hyperbolic monodomain model in the case that the intracellular and extracellular conductivity tensors have the same anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is equivalent to a cable model that includes axial inductances, and the relaxation times of the Cattaneo fluxes are strictly related to these inductances.A purely linear analysis shows that the inductances are negligible, but models of cardiac electrophysiology are highly nonlinear, and linear predictions may not capture the fully nonlinear dynamics. In fact, contrary to the linear analysis, we show that for simple nonlinear ionic models, an increase in conduction velocity is obtained for small and moderate values of the relaxation time. A similar behavior is also demonstrated with biophysically detailed ionic models. Using the Fenton-Karma model along with a low-order finite element spatial discretization, we numerically analyze differences between the standard monodomain model and the hyperbolic monodomain model. In a simple benchmark test, we show that the propagation of the action potential is strongly influenced by the alignment of the fibers with respect to the mesh in both the parabolic and hyperbolic models when using relatively coarse spatial discretizations. Accurate predictions of the conduction velocity require computational mesh spacings on the order of a single cardiac cell. We also compare the two formulations in the case of spiral break up and atrial fibrillation in an anatomically detailed model of the left atrium, and we examine the effect of intracellular and extracellular inductances on the virtual electrode phenomenon.Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology Boyce E. Griffith=============================================================================Since its introduction by <cit.>, the cable equation and its higher-dimensional generalizations <cit.> have been common models of electrical impulse propagation in excitable media, including neurons and muscle. The effects of inductances in these systems are considered to be relatively small, and so they are neglected in classical versions of these models. By omitting such terms, the standard equations of cardiac electrophysiology become parabolic, but, as in all parabolic equations, local perturbations can propagate at infinite speed. This unrealistic property has been addressed in models of neurons by <cit.>, and hyperbolic models including inductances have been proposed by <cit.>, and <cit.>. These models are also supported by the fact that neurons, skeletal muscle cells, and cardiomyocytes show the typical resonance effects due to inductances, as demonstrated by the studies of <cit.> and <cit.>. The common conclusion that inductances are negligible, which is based on the linear analyses for neurons by Scott <cit.> and the numerical results of <cit.>, may not be valid in the complex arrangement of cardiac tissue, where the inhomogeneities together with highly nonlinear reactions can lead to reentrant waves and chaotic behavior. We thus derive a hyperbolic model for cardiac electrophysiology, and we compare the solutions with parabolic models in several cases, including simple excitation patterns as well as for spiral waves, atrial fibrillation, and the virtual electrode phenomenon.The heart is a complex organ with a highly heterogeneous structure. Muscle cell are embedded in an extracellular compartment that includes many components, including capillaries, connective tissue, and collagen. The structural arrangement of the tissue is known to influence electrical impulse propagation <cit.>. The standard models of electrophysiology neglect all these complexities, and the resulting equations have the unrealistic feature that local perturbations can propagate infinitely fast. These models are fundamentally based on the cable equation. The works of <cit.> and <cit.> were the first to suggest that the cable model for neurons should contain inductances because of the three-dimensional nature of the axon. Several authors, including <cit.>, and <cit.>, have investigated this hypothesis for nerves. Their conclusions were that inductances in neurons are of the order of few μH and therefore are negligible, following a linear analysis of a one-dimensional nerve discussed by <cit.>. Several years later, further studies by <cit.>, and <cit.> showed that the cell membranes of excitable tissue exhibit self-inductance. In particular, <cit.> showed that the impedance of the embryonic heart cell membrane resonates at a frequency around 1 Hz, thereby enhancing homogeneity of the voltage. To our knowledge, there has been no experimental study that addresses the role of inductances for reentrant waves. In this chaotic scenario, inductances may have an important role in the system dynamics. Excitable tissues have a characteristic speed of transmission that limits the velocity at which signals can propagate. Any signal, including action potential wavefronts, cannot propagate faster than this characteristic speed. On the other hand, it is important to notice that propagation of the action potential is related to the nonlinear dynamics of the system, and not to the wave-like behavior that may be induced by inductances. One way to transform the cable equation into a hyperbolic equation is to simply add an “inertial” term proportional to the persistence time of the diffusive process, as done by <cit.> Unfortunately, this type of model cannot be derived from reasonable physical assumptions. However, as we show here, it is possible to perform a phenomenological derivation of a hyperbolic reaction-diffusion model by using the Cattaneo model for the fluxes. This formulation was originally introduced by <cit.> to eliminate the anomalies found using Fourier's law in the heat equation, and this model has subsequently been used in a wide range of applications, including forest fire models <cit.>, chemical systems <cit.>, thermal combustion <cit.>, and the spread of viral infections <cit.>. Cattaneo-type models for the fluxes have been derived in several ways, ranging from phenomenological and thermodynamical derivations to isotropic and anisotropic random walks with reactions <cit.>. The use of Cattaneo-type fluxes in a monodomain model of cardiac electrophysiology leads to two additional terms in the equations proportional to the characteristic relaxation time of the medium: the second derivative in time of the potential, which is associated with “inertia”, and the time derivative of the ionic currents. Even if the relaxation time is small, the rapid variation of the ionic currents can impart a contribution that may not be negligible. This is particularly relevant near the front of the wave, where fast currents give rise to the upstroke of the action potential. Verification and validation remain major challenges in computational electrophysiology. <cit.> propose that the “wave speed should not be sensitive to choices of numerical solution protocol, such as mesh density, numerical integration scheme, etc.” Although this is a fundamentally desirable criterion, it is also nearly impossible to achieve in practice. What does seem reasonable is to require that that the error in the wave speed be “small”, but how small the error must be taken clearly depends on the case under consideration. For example, for a single heart beat, an error of 5% might represent a reasonable approximation. On the other hand, models of cardiac fibrillation may require a much smaller error. In fact, in this case, a 5% error in the conduction velocity can determine whether reentrant waves form, or when and how they break up. Although this problem has been discussed in detail in prior work <cit.>, the common belief within the field seems to be that spatial discretizations on the order of 200 μm are sufficient to capture the conduction velocities of the propagating fronts. This estimate seems to hold for isotropic propagation at normal coupling strengths, but in many important cases, the most relevant propagation of the electrical signal is transverse to the alignment of the cardiac cells, where the conductivities are typically 8 times smaller than in the longitudinal direction <cit.>. As shown by <cit.>, the mesh sizes needed to resolve transverse propagation are actually closer to 25 μm. Such a small mesh spacing requires the use of large-scale simulations and highly efficient codes. Further, the need to use such high spatial resolutions challenges the fundamental idea of using a continuum model for the description of cardiac electrophysiology, and multiscale models have been proposed <cit.>. This paper shows that the transverse conduction velocities are sensitive to the grid size and to the mesh orientation for both regular and hyperbolic versions of the model, and that the hyperbolic model has similar mesh size requirements as the standard model.§ PHENOMENOLOGICAL DERIVATION OF THE HYPERBOLIC BIDOMAIN EQUATIONS In the 1970's, <cit.> formulated a bidomain model of the propagation of the action potential in cardiac muscle. This tissue-scale model considers the myocardium to be composed of continuous intracellular and extracellular compartments, coupled via a continuous cellular membrane. The bidomain equations can be derived from a model that accounts for the tissue microarchitecture <cit.>, but the bidomain model is a fundamentally homogenized description of excitation propagation that neglects the details of this microarchitecture. Instead, the bidomain equations describe the dynamics of a local average of the voltages in the intracellular and extracellular compartments over a control volume. One of the assumptions required by the homogenization procedure is that the control volume is large compared to the scale of the cellular microarchitecture but small compared to any other important spatial scale of the system, such as the width of the action potential wavefront. Although the validity of this model has been questioned, for example by <cit.>, this approach has been extremely successful, and at present, most simulations of cardiac electrophysiology use such models. For a detailed review of the bidomain model and other models of electrophysiology, we refer to <cit.>. Here, we assume that the homogenization assumptions hold, and we derive the hyperbolic bidomain model phenomenologically, starting from charge conservation in quasistatic conditions. In the hyperbolic bidomain model, as in the standard bidomain model, the extracellular and intracellular compartments are characterized by the anisotropic conductivity tensors, respectively σ_e and σ_i. Introducing a local orthonormal basis, {f_0,s_0,n_0}, we assume the conductivity tensors can be written as σ_j=σ_j^ff_0⊗f_0+σ_j^ss_0⊗s_0+σ_j^nn_0⊗n_0, for j=e, i. Typical values of the conductivity coefficients range from 10^-2 mS/cm in the cross-fiber direction to 1 mS/cm in the fiber direction (for example, <cit.> use σ^f=1.2 mS/cm, σ^s=0.346 mS/cm, and σ^n=0.0435 mS/cm).We model the current density fluxes using a Cattaneo-type equation, so thatτ_e∂J_e∂ t+J_e= -σ_e∇ V_e, τ_i∂J_i∂ t+J_i= -σ_i∇ V_i,where V_i (V_e), J_i (J_e), and τ_i (τ_e) are the intracellular (extracellular) potential, the intracellular (extracellular) flux, and the intracellular (extracellular) relaxation time, respectively. As shown by <cit.>, relations (<ref>) and (<ref>) can be also derived from the simplest model of ionic conduction in a dilute system. The derivation of these evolution laws for the fluxes from the generalized Gibbs equation <cit.> shows that they are consistent with the theory of extended irreversible thermodynamics. Alternatively, equations (<ref>) and (<ref>) can be interpreted as arising from a circuit model that includes inductances, such as the one depicted in Fig. <ref>. The derivation of the cable equation for the circuit model shown in Fig. <ref> is performed in Appendix <ref>. In particular, we show in equation (<ref>) the relationship between inductance and the relaxation time. To derive the higher-dimensional model equations, we begin by taking the divergence of the fluxes (<ref>) and (<ref>), so thatτ_e∂∂ t∇·J_e+∇·J_e= -∇·σ_e∇ V_e, τ_i∂∂ t∇·J_i+∇·J_i= -∇·σ_i∇ V_i.As in similar derivations of the bidomain model, we impose a quasistatic form of charge conservation <cit.>, yielding∇·(J_i+J_e)=0.Additionally, the current leaving each compartment needs to enter the other, so that -∇·J_i=I_t=∇·J_e,where I_t=χ(C_m∂ V∂ t+I_ion) is the usual transmembrane current density, with χ the membrane area per unit volume of tissue, C_m the membrane capacitance, and I_ion the transmembrane ionic current. Using equation (<ref>) in (<ref>) and (<ref>), we obtain the system of equations,τ_e∂ I_t∂ t+I_t= -∇·σ_e∇ V_e,-τ_i∂ I_t∂ t-I_t= -∇·σ_i∇ V_i.Defining the transmembrane potential V=V_i-V_e to eliminate V_i from the equations yieldsτ_e∂ I_t∂ t+I_t= -∇·σ_e∇ V_e, τ_i∂ I_t∂ t+I_t=∇·σ_i∇ V+∇·σ_i∇ V_e.Expanding the transmembrane currents, we finally obtain the hyperbolic bidomain model,τ_eC_m∂^2V∂ t^2+C_m∂ V∂ t+∇·D_e∇ V_e=-I_ion-τ_e∂ I_ion∂ t, τ_iC_m∂^2V∂ t^2+C_m∂ V∂ t-∇·D_i∇ V-∇·D_i∇ V_e=-I_ion-τ_i∂ I_ion∂ t,where we set D_i=σ_i/χ and D_e=σ_e/χ. Alternatively, the model equations can be written asτ_iC_m∂^2V∂ t^2+C_m∂ V∂ t-∇·D_i∇ V-∇·D_i∇ V_e=-I_ion-τ_i∂ I_ion∂ t, (τ_e-τ_i)C_m∂^2V∂ t^2+∇·D_i∇ V+∇·(D_e+D_i)∇ V_e= (τ_i-τ_e)∂ I_ion∂ t.Notice that if τ_i=τ_e=τ, then these equations reduce to a hyperbolic-elliptic system that is similar to the parabolic-elliptic form of the standard bidomain model,τ C_m∂^2V∂ t^2+C_m∂ V∂ t-∇·D_i∇ V-∇·D_i∇ V_e= -I_ion-τ∂ I_ion∂ t, ∇·D_i∇ V+∇·(D_e+D_i)∇ V_e= 0.If we further take τ=0, we retrieve the usual bidomain model in its parabolic-elliptic form,C_m∂ V∂ t-∇·D_i∇ V-∇·D_i∇ V_e=- I_ion, ∇·D_i∇ V+∇·(D_e+D_i)∇ V_e= 0. § REDUCTION TO THE HYPERBOLIC MONODOMAIN The hyperbolic bidomain model can be simplified by assuming the extracellular and intracellular compartments have the same anisotropy ratios, so that D=D_e=λD_i. If we make this assumption, we obtain from (<ref>)-∇·D∇ V_e=1λ+1∇·D∇ V+λλ+1(τ_e-τ_i)C_m∂^2V∂ t^2-λλ+1(τ_i-τ_e)∂ I_ion∂ t,Substituting in (<ref>), we find[τ_i+λλ+1(τ_e-τ_i)]C_m∂^2V∂ t^2+C_m∂ V∂ t-λλ+1∇·D∇ V=-I_ion-[τ_i+λλ+1(τ_e-τ_i)]∂ I_ion∂ t.Defining τ=τ_i+λ(τ_e-τ_i)/(λ+1), we obtain the hyperbolic monodomain model,τ C_m∂^2V∂ t^2+C_m∂ V∂ t-∇·D∇ V=-I_ion-τ∂ I_ion∂ t,where here we have absorbed the term λ/(λ+1) into D. The relaxation time τ of the monodomain model is always positive, and it is zero only if both τ_i and τ_e are zero. In fact, if τ_e=0, then τ=τ_i/(λ+1), and if τ_i=0, then τ=λτ_e/(λ+1). However, if τ_i=τ_e, then τ=τ_i=τ_e. Introducing a new variable Q = ∂ V∂ t, we transform the hyperbolic monodomain equations into the first-order system, ∂ V∂ t= Q, τ C_m∂ Q∂ t+C_mQ-∇·D∇ V = -I_ion-τ∂ I_ion∂ t.Equations (<ref>)–(<ref>) are usually supplemented with insulation boundary conditions, such that ∇ V·N=0 on the boundary of the domain, where the vector N is the normal to the boundary.Following <cit.>, we say that a solution of the hyperbolic monodomain equations has a finite propagation speed if, given compactly supported initial conditions for V at time t=0, V(·,t) is also compactly supported for any t > 0. The compact support is taken with respect to the resting potential V_0. By contrast, a solution has infinite propagation speed if the initial data are compactly supported, but for any t > 0 and any R > 0, the setM_R,t = {x: ||x||_2 ≥ R, V(x, t) > V_0 } has positive measure. For anyrelaxation time τ≠0, (<ref>) ishyperbolic, and it thereby has the property of finite propagation speed. Setting τ = 0, the equations become parabolic and have solutions with infinite propagation speeds. Specifically, in the standard parabolic model, local perturbations in V, even those that do not generate a propagating front, will travel at infinite speed. § IONIC MODELS Transmembrane ionic fluxes through ion channels, pumps, and exchangers are responsible for the cardiac action potential. The action potential is initiated by a fast inward sodium current that depolarizes the cellular membrane. After depolarization phase, slow inward currents (primarily calcium currents) and slow outward currents (primarily potassium currents) approximately balance each other, prolonging the action potential and creating a plateau phase. Ultimately, the slow outward currents bring the transmembrane potential difference back to its resting value of approximately -80 mV. The bidomain and monodomain models must be completed by specifying the form of I_ion, which accounts for these transmembrane currents.The states of the transmembrane ion channels are described by a collection of variables, w, associated with the ionic model, so that I_ion = I_ion(V,w). Typically, the state variable dynamics are determined by a spatially decoupled system of nonlinear ordinary differential equations,∂w∂ t=g(V,w),where the form of g depends on the details of the particular ionic model. We consider the simplified piecewise-linear model of <cit.>, the two-variable model of <cit.>, the three-variable model of <cit.>, and the biophysically detailed models of <cit.> for the ventricles (20 variables) and of <cit.> for the atria (57 variables). In the two-variable model of <cit.>, w={ r}, with∂ r∂ t=(ε+μ_1rμ_2+V)(-r-kV(V-b-1)),and the ionic current is I_ion=kV(V-α)(V-1)+rV. The parameters used in this case are shown in Table <ref>. In the three-variable model of <cit.>, w={ v,w}, with∂ v∂ t=1τ_v^-(V)(1-p)(1-v)-1τ_v^_+pv, ∂ w∂ t=1τ_w^-(1-p)(1-w)-1τ_w^_+pw,where τ_v^-(V)=(1-q)τ_v1^-+qτ_v2^-, p=H(V-V_c), q=H(V-V_v), and H(·) is the Heaviside function. The total ionic current is the sum of three currents, I_ion=-I_fi(V,v)-I_so(V)-I_si(V,w), withI_fi(V,v) = -1τ_dvp(V-V_c)(1-V),I_so(V) =1τ_0V(1-p)+1τ_rp,I_si(V,w) = -12τ_siw(1+tanh(k(V-V_c^si))).The sets of parameters used in this paper for this model are shown in Table <ref>.We refer to the original papers <cit.> for the statements of the equations and parameters of the biophysically detailed ionic models.§ NUMERICAL RESULTS The hyperbolic monodomain and bidomain models are discretized using a low-order finite element scheme described in Appendix <ref> that is implemented in the open-source parallel C++ code BeatIt (available at <http://github.com/rossisimone/beatit>), which is based on the libMesh finite element library<cit.> and relies on linear solvers provided by PETSc <cit.>. All the code used for the following tests is contained in the online repository, and all tests can be replicated directly from those codes. The only exception is the atrial fibrillation test, whichuses a patient-specific mesh that is not contained in the repository. One- and two-dimensional simulations were run using a Linux workstation with two Intel Xeon E5-2650 v3 processors (up to 40 threads) and 32 GB of memory. Three-dimensional simulations were run on the KillDevil Linux cluster at the University of North Carolina at Chapel Hill. We used Matlab<cit.> to visualize the one-dimensional results and Paraview<cit.> for the two- and three-dimensional simulations. §.§ Comparison with an exact solutionWe start by considering a simple piecewise-linear bistable model for the ionic currents, withI_ion(V) = kV - k[V_2H(V-V_1)+V_0(1-H(V-V_1))],where V_0 is the resting potential, V_1 is the threshold potential, V_2 is the depolarization potential, and H(·) is the Heaviside function. This model (<ref>) reduces to the piecewise-linear model of <cit.> after a simple dimensional analysis, after which the nondimensional ionic currents take the formÎ_ion(V̂)=V̂-H(V̂-α).The nondimensional potential V̂ is related to the dimensional potential by V=(V_2-V_0)V̂+V_0.In this model, α=(V_1-V_0)/(V_2-V_0) describes the excitability of the tissue. As we show in Appendix <ref>, using model (<ref>), it is possible to find the analytic solution of a propagating front for the nondimensional hyperbolic monodomain problem. In particular, we find that the front propagation speed in an unbounded domain isc=(1-2α)√(μ+(α-α^2)(μ-1)^2)<√(1μ)=c_s.The speed c_s=√(1/μ)=√(C_m/τ k) represents the maximum speed at which a perturbation can travel in the system. When the relaxation time τ goes to zero, the local perturbations can travel at infinite speed. The parameter μ = τ k / C_m is a nondimensional number representing the ratio between the relaxation time and the characteristic time of the reactions and characterizing the effects of the inductances in the system (see Appendix <ref>).The corresponding dimensional speed isv=√(σ kχ C_m^2)(1-2α)√(μ+(α-α^2)(μ-1)^2)<√(σχτ C_m). As a verification test, we consider the spatial interval Ω=[0,50]. An initial stimulus of amplitude 1 is applied at x∈[24.5,25.5] for the interval t∈[0.03,1.03]. The system of equations is solved using a mesh size h=0.03125 and a time step size Δ t=0.003653. We show the nondimensional solutions V̂ and Q̂ at t=10 in Fig. <ref>(center and right). We register the activation time at a particular spatial location whenever V̂ crosses a threshold of 0.9. The conduction velocities are measured by picking the distance between two points x_1 and x_2 ∈Ω and dividing it by the time interval between the activation times at these two locations. Because (<ref>) is obtained by assuming that Ω is the entire real line, we need to measure conduction velocities far from the boundaries. For this reason, we choose x_1=30 and x_2=32. The comparison between the exact conduction velocities defined by equation (<ref>) and those obtained by the simulations is shown in Fig. <ref>(left).We also examine the effect of the relaxation time on the conduction velocities for three values of the excitability parameter, α∈{ 0.1, 0.2, 0.3}. With this simple ionic model, the larger the relaxation time, the slower the wave speed. The limit speed c_s at which local perturbations can travel is also shown in Fig. <ref>(left). Thus, in this piecewise-linear model, the effect of inductances is to slow down the propagation of the front. By contrast, the following tests will show that with fully nonlinear models, inductances can also enhance propagation. Using the analytic solution derived in Appendix <ref>, we also perform a space-time convergence study. We consider the spatial interval Ω=[-25,25] and set the initial conditions according to (<ref>) and (<ref>), assuming the front at time t=0 is at x=0 and α = 0.1. The system is discretized in time using implicit-explicit (IMEX) Runge-Kutta (RK) time integrators<cit.>. In particular, we use the forward/backward Euler schemes for the first-order time integrator and the Heun/Crank-Nicholson schemes for the second-order time integrators. For more details on the time integrator, see Appendix <ref>. To ensure the validity of the solutions (<ref>) and (<ref>) in the considered bounded domain, we run the simulation from t=0 to t=1, so that the front is still far from the boundaries.The time step size Δ t is taken to be linearly proportional to the mesh size h, with a value of Δ t = 0.05 in the coarsest cases. The time step size was chosen to guarantee a large enough number of time iterations and while satisfying the CFL condition (see Appendix <ref>). Fig. <ref> shows the errors for the potential V and its time derivative Q using the first-order (Fig. <ref> left) and second-order (Fig. <ref> center) time stepping schemes. Because the ionic currents and their derivative in this case are not smooth functions, we do not expect to obtain full second-order convergence. On the other hand, whereas the convergence rates for the parabolic monodomain model are always first order in both V and Q, the hyperbolic monodomain model with the second-order time stepping scheme converges quadratically in V and linearly only in Q.The simulations were run with and without regularization of the Heaviside and Dirac-δ functions. The errors reported in Fig. <ref>correspond to the more accurate simulations. We show in Fig. <ref>(right) that the slow convergence results from the non-smoothness of the ionic currents: we compare the convergence of the piecewise-linear reaction model with the monodomain model with the cubic reaction termÎ_ion(V̂)=V̂(V̂-1)(V̂-α).The exact solution for the cubic reaction in the parabolic monodomain has been determined previously <cit.>. We are not aware of the existence of an analytical solution for the hyperbolic monodomain with a cubic reaction term. We show in Fig. <ref> that the cubic model converges quadratically (using a second-order time stepping scheme) whereas the piecewise-linear model converges linearly. To conclude, at least in these tests, the discretization errors of the parabolic mondomain model are larger than the discretization errors in the hyperbolic monodomain model, indicating a better accuracy for the hyperbolic model. §.§ Effect of the relaxation time on the conduction velocity for different ionic models We now analyze how the inductance terms influence the conduction velocities of cardiac action potentials. We consider four ionic models: the two-variable model of <cit.>, the three-variable model of <cit.>, the ventricular M-cell model of <cit.> (20 variables), and the atrial model of <cit.> (57 variables). For the two-variable Aliev-Panfilov model, we use the parameters of <cit.>, as reported in Table <ref>. We consider two values for the excitability parameter, α=0.1 and α=0.2. For the three-variable Fenton-Karma model, we consider parameters sets 3, 4, 5, and 6 given by <cit.> and reported in Table <ref>. For the biophysically detailed ionic model of Ten tusscher et al. and Grandi et al., our simulations used the C++ implementations of the models provided by the authors, using C_m=1 μF/cm^2 for the membrane capacitance in both models. For all cases, the domain is the interval Ω=[0,5] cm. The simulations use a mesh size h=31.25 μm and a time step size Δ t=0.003653 ms. An initial stimulus is applied at the center of the domain, x∈[2.45,2.55] cm, during the time interval t∈[0.03,1.03] ms. The conduction velocities are measured, as in the previous test case, by dividing the distance between two points x_1 and x_2 by the time interval between the activation times at these two selected locations. The computed conduction velocities of different ionic models are shown in Fig. <ref> for relaxation times in the range [0,1] ms. Contrary to the results for the piecewise-linear ionic model, in this case, small and moderate values of the relaxation time act to enhance propagation.We also compare the computational time required by the parabolic and hyperbolic monodomain models in two and three spatial dimensions. Table <ref> shows the number of iterations taken by the linear solvers (conjugate gradient preconditioned by successive over-relaxation) along with the relative computational times of the hyperbolic monodomain model (with respect to the parabolic model) for the reaction step andthe diffusion step. In the reaction step, we solve the ionic model, and we also evaluate the ionic currents and their time derivatives. For this reason, we expect this step to take longer in the hyperbolic monodomain model. In fact, with simple ionic models, the amount of work required to evaluate the ionic currents and their time derivatives is almost doubled. This is natural because, for such simple models, the time derivative of the ionic currents do not use any quantities already computed in the evaluation of the ionic currents. By contrast, for biophysically detailed models, most of the computation is needed for the evaluation of the ionic currents, and many quantities can be reused for the evaluation of their time derivatives. This is reflected in Table <ref> by a small increase (less than 7%) in the computational time of the hyperbolic model for the Ten Tusscher '06 model.Unexpectedly, the solution of the linear system take fewer iterations in the hyperbolic monodomain model. This is reflected by a speedup of about 20%of the computational time in the diffusion step, where we assembled the right hand side, solve the linear system, and update the variables Q and V.§.§ Conduction velocity anisotropyThis section analyzes whether inductances affect the anisotropy in conduction velocity. We define the anisotropy ratio as the ratio between the conduction velocities in the transverse and longitudinal directions, and to obtain a simple estimate for this ratio, we consider plane-wave solutions propagating in these directions.We use the Fenton-Karma model with parameter set 3 and consider four values of the conductivity: σ_1 = 0.1 mS/mm, σ_2 = 0.05 mS/mm, σ_3 = 0.025 mS/mm, and σ_4 = 0.0125 mS/mm. Evaluating the wave speeds v_1, v_2, v_3, and v_4 corresponding to theconductivities σ_1, σ_2, σ_3, and σ_4, respectively, we show how the anisotropy ratio changes with respect to the relaxation time for several conductivity ratios. In fact, as shown inTable <ref>, σ_1 and σ_4 are the typical longitudinal and transverse conductivities for this model. Considering the interval Ω=[0,5] cm, we use a mesh sizeh=25 μm and a time step size Δ t=0.0025 ms.Fig. <ref>(left) shows how the velocities at different σ compare to each other. To make the comparison more clear, we normalize the results with respect to the conduction velocities of the parabolic monodomain model, i.e., corresponding to τ = 0 ms. The results shown in Fig. <ref>(left) should be understood as the percentage difference of the wave speed with respect to the parabolic case. It is clear that the curves are similar for the conductivities considered. On the other hand, the relaxation time at which the conduction velocity of the hyperbolic monodomain model matches the conduction velocity of the parabolic monodomain model decreases with smaller conductivities. Fig. <ref>(right) shows the relaxation time needed to maintain the same conduction velocity as in the parabolic monodomain model. In particular, for σ_4, we find that the relaxation time needed in the hyperbolic monodomain model to yield the same velocities as in the parabolic monodomain models is about τ = 0.38 ms. The differences in the velocities computed with τ =0.4 ms and τ = 0.38 ms are smaller than 2%. This difference is typically smaller than the numerical error in the simulations. For this reason, in the following tests, our comparison will only use the value τ = 0.4 ms. Fig. <ref>(center) shows how the anisotropy ratio is influenced by the relaxation time.Although the ratio is not constant, the variation over the relaxation times considered never exceeds 5%. For the conductivity ratios 8:1 (σ_1:σ_4), 4:1 (σ_1:σ_3), and 2:1 (σ_1:σ_2), we find that the ratios between the conduction velocities in the transverse and in the longitudinal directions are approximately1/√(8) (v_4/v_1), 1/2 (v_3/v_1), and 1/√(2) (v_2/v_1).This behavior is expected because the conduction velocity depends on the square root of the conductivity. We conclude that the effect of the inductances on the anisotropy ratio is negligible. §.§ Discretization error: a simple two-dimensional benchmark It is important to be aware of the limitations of the numerical method used in the simulations. For example, spiral break up can occur easily if the wavefront is not accurately captured. In fact, numerical error can introduce spatial inhomogeneities that lead both to spiral wave formation and also to spiral break up. This numerical artifact disappears under grid refinement, and on sufficiently fine computational meshes, spiral wave break up results only from physical effects, such as conduction block. To examine the role of spatial discretization on the system dynamics, we consider a square slab of tissue, Ω=[0,12]×[0,12] cm, and we use the Fenton-Karma model with the parameter set 3. Reentry is induced by an S1-S2 protocol.First, an external stimulus of unitary magnitude is applied in the left bottom corner, Ω_stim={x∈Ω:‖x‖ _1≤1} at t=0 ms. A second stimulus with the same amplitude and in the same region is applied after 300 ms. We consider two cases: 1) the fiber field is aligned with some of the mesh edges; and 2) the fiber field is not aligned with any edge in the mesh. Figs. <ref>(top) and <ref>(top) demonstrate that this can be easily achieved by fixing the fiber field and rotating the mesh. Those figures show the fiber field in green on top of the computational grid. The red region at the bottom left of the domain is the stimulus region, Ω_stim. In the first test (Fig. <ref>), the fibers are set to be orthogonal to the propagating front. Using 512 elements per side, which is equivalent to a mesh size h of approximately 234 μm (163 μm in the direction of wave propagation), the solutions obtained on the two meshes differ substantially. Large differences can also be found when we use 1024 elements per side, which is equivalent to a mesh size h of approximately 117 μm (82.5 μm in the direction of wave propagation). It is clear from Fig. <ref> that whenever the fiber field is not aligned with the mesh, the conduction velocity is largely overestimated. The second test is similar to the first one, but the fibers are now rotated by 90^∘; see Fig. <ref>. It is clear that the error on the conduction velocity in the fiber direction is much smaller than the error in the transverse direction. In fact, the solutions for h≈234 μm and h≈117 μm show the greatest differences for transverse propagation.Although similar results have already been reported in prior work <cit.>, it is nonetheless generally believed that a mesh size of the order of 200 μm is usually sufficient for cardiac computational electrophysiology<cit.>.This belief is supported by numerical simulations using a mesh size of 200 μm that show that the error on the wave speed in the fiber direction is less than 5%. On the other hand, the most interesting dynamics of the propagating front often will occur perpendicular to the fiber direction. For example, during the normal electrical activation of the ventricles, the signal spreads from the endocardium to the epicardium traveling across the ventricular wall, perpendicular to the fibers. The reduced conductivity in the transversal direction, usually taken to be about 8 times smaller, requires a finer resolution of the grid. As noted by <cit.>, the required grid resolution to capture the transverse conduction velocity with an error smaller than 5% is about 25 μm. These results strongly indicate that to capture correctly the wave speeds, the mesh discretization needs to be about the size of a single cardiomyocyte, as also discussed by <cit.>.§.§ Effect of the relaxation time on spiral break upOur next tests explore the effect of relaxation time on spiral wave break up. We consider a square slab of tissue, Ω=[0,12]×[0,12] cm. An initial stimulus is applied in the region y<0.5 cm for 1 ms at t=0 ms to generate a wave propagating in the y-axis. A second stimulus is then applied in the region { x<6 cm ∧ y<7 cm} for 1 ms at time t=320 ms to initiate a spiral wave. Spiral break up is easily obtained using the parameter set 3 in Table <ref>, because of the steep action potential duration (APD) restitution curve. In this case, the back of the wave forms scallops, and when the turning spiral tries to invade these regions, it encounters refractory tissue and breaks. Fig. <ref> shows the formation of the spiral wave in the case of isotropy. With τ =0.4 ms and the Fenton-Karma model with parameter set 3, the conduction velocity of the hyperbolic monodomain model is very close to the conduction velocity of the parabolic monodomain model. In fact, the evolution of the spiral waves in the two cases is very similar.Fig. <ref> compares spiral break up obtained using the monodomain model and the hyperbolic monodomain model in the anisotropic case, with fibers aligned with the y-axis. The relaxation time for the hyperbolic monodomain model is chosen to be τ=0.4 ms. As explained in Sec. <ref>, although the conduction velocities in the transverse direction are different in the hyperbolic and parabolic models for τ=0.4 ms, the difference is smaller than 2%. This difference is smaller than the spatial error and, for this reason, we consider the two cases to yield essentially the same anisotropy ratio. It is clearly shown in Fig. <ref> that in the initial phase (up to t=600 ms), the spirals are almost identical. After the first rotation, however, the spiral wave starts to break, and the relaxation time of the hyperbolic monodomain model shows quantitative differences in the form of the break up. Fig. <ref> shows the dynamics of Q=∂_tV. This variable can be used to define the fronts (red) and the tails (light blue) of the waves. §.§ Atrial fibrillation As a more complex application of the model, we use the hyperbolic monodomain model to simulate atrial fibrillation. Although a rigorous study of atrial fibrillation requires the use of a bidomain model, similar to the one proposed in equations (<ref>) and (<ref>), the results given by the monodomain model can represent a reasonable approximation to the dynamics of the full bidomain model. For instance, <cit.> provide a comparison of the dynamics of the bidomain and monodomain models.The anatomical geometry used in these simulations was based on the human heart model constructed by <cit.> using a 4D extended cardiac-torso (XCAT) phantom. From their data, we extracted and reconstructed a geometrical representation of the left atrium using SOLIDWORKS. We then used Trelis to generate a simplex mesh of the left atrium consisting of approximately 3.5 million elements.Our simulations use the Fenton-Karma model with parameter set 3.A major challenge in modeling the atria is the definition of the fiber field. In work spanning the past 100 years, several studies have tried to characterize the fiber architecture of the atria <cit.>, but the structure of the muscle in the atria is so complex (as can be seen from the diagrams by <cit.>) that developing a set of mathematical rules to reproduce a realistic fiber field is challenging. Our strategy for modeling the fiber field is to use the gradient of a harmonic function that satisfies prescribed boundary conditions. We divided the left atrium into several regions, and solved three Poisson problems: one for the left atrial appendage; one for the Bachmann's bundle; and one for the remaining parts of the left atrium.In particular, for the left atrial appendage, we used a method similar to the one proposed by <cit.> In the other regions, similar to work by <cit.> and <cit.>, we used the normalized gradient of the solution of the Poisson problem to reconstruct a fiber field similar to the one shown in the studies by Ho et al. <cit.>. The approximate fiber field of the left atrium reconstructed using this strategy is shown in Fig. <ref>, where the colors represent the magnitude of the x-component of the fibers and are used only to highlight changes in direction. To initiate atrial fibrillation, we use an S1-S2 stimulation protocol that is applied at the junction with the inter-atrial band, as described by <cit.>. The S1 stimulus is applied with a cycle length of 350 ms followed by a short S2 stimulus with a cycle length of 160 ms. Fig. <ref>(right) shows that the initial electrical activation times are similar for the monodomain and hyperbolic monodomain models. Because this set of parameters gives a velocity of about 45 cm/s, roughly 35% slower than the expected conduction velocity <cit.>, it takes longer for the atrium to be fully activated. When the second stimulus is applied, spiral waves form in the hyperbolic monodomain system but not in the parabolic monodomain model; see Fig. <ref> at 420 ms. Spiral waves are generated for τ=0 ms only after the third stimulus is applied. Fig. <ref> also shows the fronts (red) and tails (light blue) of the waves, as indicated by Q.§.§ Virtual electrodes The reduction of the bidomain model to the monodomain model is not valid if the extracellular and intracellular compartments have different anisotropy ratios. As shown in experimental studies<cit.>, when a current is supplied by an extracellular electrode, adjacent areas of depolarization and hyperpolarization are formed in the case of unequal anisotropy ratios. Applying a stimulation current I_e,stim in the extracellular compartment, the region of hyperpolarization near a cathode (I_e,stim<0) is called a virtual anode, while the region of depolarization near an anode (I_e,stim>0) is called a virtal cathode. The existence of virtual cathodes and anodes is important to understand the four mechanisms of cardiac stimulation and to study the mechanisms of defribillation. In this section, we focus only on a unipolar cathodal stimulation to investigate the cathode formation mechanism in the hyperbolic bidomain model. We refer the reader interested in this virtual electrode phenomenon to the more detailed numerical studies of <cit.> and <cit.>.Following <cit.>, we consider the domain Ω = [-2,2] cm×[-0.8,0.8] cm, and consider this to correspond to the epicardial surface. We apply a unipolar cathodal current in the region Ω_stim = [-0.5,0.5] mm×[-0.1,0.1] mm. We use the Fenton-Karma model with parameter set 3, and the corresponding nondimensionalized current stimulus amplitude is -100. We choose σ^f_e = 1.5448, σ^s_e=σ^n_e =1.0438 and σ^f_i = 2.3172, σ^s_i=σ^n_i = 0.2435, with the fibers aligned with the x-axis. Fig. <ref> shows the pattern formed by the transmembrane potential 2 ms after the extracellular stimulus is applied. Red regions are areas of depolarization, and blue regions are areas of hyperpolarization. The depolarization and hyperpolarization regions are affected by the choice of the extracellular and intracellular relaxation times.For this test, we consider the relaxation times τ_i and τ_e to take the values 0, 0.2, and 0.4 ms. Note that if τ_i = τ_e, the hyperbolic bidomain equations (<ref>) and (<ref>) reduce to the hyperbolic-elliptic system of equations (<ref>) and (<ref>).Although this test shows the ability of the hyperbolic bidomain model to reproduce the virtual electrode phenomenon, the influence of the relaxation times in the stimulation remains unclear and necessitates further investigation.§ CONCLUSIONS Local perturbation in the standard bidomain and monodomain models propagate with an infinite speed. To correct this unrealistic feature, we developed a hyperbolic version of the bidomain model, and in the case that intracellular and extracellular conductivity tensors have equal anisotropy ratios, we reduced this model to a hyperbolic monodomain model. Our derivation relies on a Cattaneo-type model for the fluxes, described by equations (<ref>) and (<ref>). The Cattaneo-type fluxes are equivalent to introducing self-inductance effects, as shown by the schematic diagram in Fig. <ref>. As shown in Appendix <ref>, relaxation times introduced in the Cattaneo fluxes represent the ratio between the inductances and the resistances of the extracellular and intracellular compartments. Although the hyperbolic monodomain model reduces to the classical parabolic model in the case that the relaxation times of both the intracellular and extracellular compartments are zero, the models differ if at least one of these relaxation times is nonzero. Although hyperbolic bidomain and monodomain models do not appear tohave been previously discussed in the literature, the work of <cit.> does address the problem of propagation with infinite speed. Their approach uses a nonlinear model for the fluxes based on porous medium assumptions. Because of the nonlinear nature of the fluxes, the porous medium approach to cardiac electrophysiology is difficult to analyze, even for simple linear reactions. <cit.> use the simplified ionic model proposed by <cit.>. It would be interesting to see the extension to biophysically detailed models. Moreover, it is clear that the porous medium approach is not incompatible with the theory developed here. In fact, the two models could be combined using a nonlinear Cattaneo-type model for the fluxes. As discussed by <cit.>, abnormal conduction velocities may contribute to arrhythmogenesis. We have studied how the conduction velocities of the propagating action potential change with respect to the relaxation time of the Cattaneo-type fluxes. In the case of the piecewise-linear model of <cit.>, it is possible to find an analytical expression for the conduction velocity of the hyperbolic monodomain model. (Unfortunately, the wave speed of this bistable model has been reported incorrectly in some prior studies <cit.>.) We have shown that for the hyperbolic monodomain model, signals cannot travel faster than the characteristic propagation speed of the medium. Our numerical results match the theoretical predictions for different values of the excitability parameter. In this simple case, conduction velocity is a monotone decreasing function of relaxation time. Additionally, we have shown that, for this linear case, discretization errors are lower in the hyperbolic model.Using simple nonlinear ionic models <cit.>, however, we found that the relationship between the relaxation time and the conduction velocity is not always monotone. In fact, for small and moderate values of the relaxation time, we found that waves propagate faster than in the standard monodomain model. With these ionic models, a maximum conduction velocity is found at an optimal relaxation time.At small relaxation times, the second derivative terms are relatively small.Consequently, this phenomenon likely results from the time derivative of the ionic currents. For larger values of the relaxation times, inertial effects dominate, and the conduction velocity monotonically decreases to zero. This implies that we can find a value of the relaxation time for which the action potential propagates at the same velocity as in the standard monodomain model. This fact allowed us to directly investigate the influence of inductances in the monodomain model. In the biophysically detailed ionic models for ventricular myocytes <cit.> and atrial myocytes <cit.> considered here, a similar behavior can be found, although only for very small values of the relaxation time. For this reason, we compared the parabolic and hyperbolic monodomain models in two and three spatial dimensions using the Fenton-Karma model <cit.>.Changing the type of equation may have important consequences in terms of the stability of the numerical approximations.We did not encounter any substantial numerical difficulties beyond those already present in the standard parabolic models.On the contrary, we have shown that the linear system is easier to solve and the solutions are more accurate when the hyperbolic system is used. Moreover, we believe that the solution of the parabolic monodomain and bidomain model already have many of the numerical difficulties usually associated with hyperbolic systems: the propagating action potential in the parabolic models already have sharp fronts which represent one of the main difficulties in hyperbolic systems.Additionally, the hyperbolic monodomain and bidomain models are damped wave equations, where the damping term isdominant.In conclusion, the introduction of hyperbolic terms to the monodomain and bidomain models do not appear to add substantial numerical difficulties. We performed several qualitative comparisons between the parabolic and the hyperbolic monodomain models. In particular, we considered the case of spiral break up resulting from conduction block caused by a steep APD restitution curve. Using parameter set 3 of the Fenton-Karma model, the back of the wave easily forms a series of indentations or scallops. When a spiral wave invades this after turning, it encounters refractory tissue that causes conduction block, leading to a break up of the wave. This behavior, already observed for the standard monodomain model and explained by <cit.>, is still present in the hyperbolic monodomain model. We showed spiral break up in a simple two-dimensional test case and also in an anatomically realistic three-dimensional model of the left atrium. We reconstructed on the left atria a fiber field qualitatively in accordance with the data reported by <cit.>, and <cit.> We applied a fast-pacing S1-S2 stimulation protocol to the left atrium to induce spiral waves and spiral break up. Although the initial activation sequences were similar, for the hyperbolic monodomain model, spiral waves appeared already after the second stimulus was applied, whereas for the parabolic monodomain at least three stimuli were needed.We also used a simple two-dimensional benchmark to investigate how the alignment of the fiber direction with respect to the elements of the finite element mesh influences wave propagation. Under mesh refinement, spatial discretization errors become smaller and smaller, but the common assumption that a grid resolution of 200 μm is sufficient to correctly capture the conduction velocity <cit.> seems to be overly optimistic. In our tests, we analyzed the propagation of a simple wave using an anisotropic conductivity tensor. Fixing the fiber field and rotating the mesh, we were able to show that using a mesh size of about 234 μm (163 μm in the direction of wave propagation), the propagation was greatly influenced by the mesh orientation relative to the fiber alignment. Even on a finer mesh with a mesh size of about 117 μm (82.5 μm in the direction of wave propagation), visible differences between the propagation patterns obtained with the different orientations remained. These tests highlight the fact that if the transverse conductivity coefficients are taken to be about 8 times smaller than the longitudinal conductivity, as is typically done in practice <cit.>, the mesh size needs to be much smaller than 200 μm in order to yield resolved dynamics. Our results are in accordance with the convergence study by <cit.> for propagation in the transverse direction, in which the authors showed that the mesh size in the transverse direction needs to be about 25 μm. Because such mesh resolutions require elements only slightly larger than the actual cardiac cells, it is natural to ask whether the use of a continuum model is even appropriate: for a similar computational cost, it could be possible to construct a discrete model of the heart. Although the difficulties in representing correctly the conduction velocities are well documented in the literature <cit.> and have been thoroughly analyzed by <cit.>, the focus generally seems to have been on the conduction in the fiber direction, as is clear from the choice of the three-dimensional benchmark test proposed by <cit.> Other solution strategies for the monodomain and bidomain models using adaptive mesh refinement <cit.> and high-order elements <cit.> have been proposed, but their application so far appears to be relatively limited.Finally, we showed that the hyperbolic bidomain model can capture the virtual electrode phenomenon observed in the standard model.By stimulating the epicardial surface using a unipolar cathodal current, we have shown how the pattern of the transmembrane potential is influenced by the intracellular and extracellular relaxation times. Nonetheless, a more detailed study on how the relaxation times affect the virtual electrodes is needed to fully understand the hyperbolic bidomain model.Comparing the stimulation patterns to experimental data could even serve to calibrate the magnitude of the relaxation times in the hyperbolic bidomain model.Despite significant progress in models of cardiac electrophysiology, the complex effects of nonlinearity and heterogeneity in the heart are far from being fully understood. We have shown that the nonlinearities of the underlying physics can give unexpected results, in contradiction to the linear case. Inductances in tissue propagation could play an important role in cardiac electrical dynamics, especially close to the wavefronts, where the fast currents responsible for the initiation of the action potential can give a small but non-negligible contribution.The main phenomenon we observed in the hyperbolic model is the enhancement of the conduction velocity at small relaxation times because of the presence of the time derivative of the ionic currents. This phenomenon would allow electric signals to propagate at the same speed with lower conductance as compared to the standard models. Future experimental studies are needed to confirm the importance of these effects. § ACKNOWLEDGEMENTS The authors gratefully acknowledge research support from NIH award HL117063 and from NSF award ACI 1450327. The human atrial model used in this work was kindly provided by Prof. W. P. Segars. We also thank Prof. C. S. Henriquez and Prof. J. P. Hummel for extended discussions on atrial modeling. We also gratefully acknowledge support by the libMesh developers in aiding in the development the code used in the simulations reported herein.§ CABLE EQUATION WITH INDUCTANCES We follow the derivation of the cable equation by <cit.>. As shown in Fig. <ref>, we assume there exist inductance effects in both the extra- and intracellular axial currents. We haveV_e(x)-V_e(x+Δ x) = -L_e∂_tI_eΔ x-R_eI_eΔ x,V_i(x)-V_i(x+Δ x) = -L_i∂_tI_iΔ x-R_iI_iΔ x,where I_e and I_i are the extracellular and intracellular axial currents, respectively. Dividing by Δ x and taking the limit as Δ x→0, we find∂_xV_e= -L_e∂_tI_e-R_eI_e, ∂_xV_i= -L_i∂_tI_i-R_iI_i.The minus sign on the right hand sides is a convention that ensures that positive charges flow from left to right. Applying Kirchhoff's current law, we have that I_e(x)-I_e(x+Δ x)+I_tΔ x = 0,I_i(x)-I_e(x+Δ x)-I_tΔ x = 0,which, in the limit Δ x→0, becomes I_t=∂_xI_e=-∂_xI_i.Differentiatingequations (<ref>) and (<ref>) with respect to x, and equation (<ref>) with respect to t, we find that for the extracellular compartment,∂_xxV_e= -L_e∂_xtI_e-R_e∂_xI_e, ∂_xtI_e=∂_tI_t,and for the intracellular compartment,∂_xxV_i= -L_i∂_xtI_i-R_i∂_xI_i, ∂_xtI_e= -∂_tI_t.Substituting the mixed derivative of the extracellular and intracellular currents, we find∂_xxV_e=-L_e∂_tI_t-R_eI_t,∂_xxV_i=L_i∂_tI_t+R_iI_t.Defining V=V_i-V_e, and taking the difference between the intracellular and the extracellular equations, we can write a single equation for V,∂_xxV=(L_i+L_e)∂_tI_t+(R_i+R_e)I_t.Recalling that I_t=χ(C_m∂_tV+I_ion),we finally find the hyperbolic form of the cable equation,τ C_m∂^2V∂ t^2+C_m∂ V∂ t-∂∂ x(D∂ V∂ x)=-I_ion-τ∂ I_ion∂ t,where we define the conductivity as D=1/χ(R_i+R_e) and the relaxation time byτ=L_i+L_eR_i+R_e. § EXACT SOLUTION FOR THE PIECEWISE-LINEAR BISTABLE MODEL Consider the simplified piecewise-linear bistable model, I_ion(V) = kV - k[V_2H(V-V_1)+V_0(1-H(V-V_1))].The nondimensional form of equation (<ref>) is the simplified model proposed by <cit.>, Î_ion(V̂)=V̂-H(V̂-α).Using (<ref>) in the hyperbolic monodomain model (<ref>) and considering only one spatial dimension, we have μ∂^2V̂∂t̂^2+(1+μ)∂V̂∂t̂-∂^2V̂∂x̂^2+V̂=0,V̂<α, μ∂^2V̂∂t̂^2+(1+μ)∂V̂∂t̂-∂^2V̂∂x̂^2+V̂=1,V̂>α, where we have introduced the nondimensional variablest̂=1Tt̂,x̂=1Lx,V̂=V-V_0V_2-V_0.In particular, we defineT=C_mk, L=√(σkχ),μ=τ kC_m,where μ is a nondimensional number that characterizes the effect of the inductances in the system. Specifically, μ isthe ratio between the relaxation time of the system and the characteristic time of the reactions: the larger μ, the more important the effects of the inductances become.Introducing the change of coordinates z=x-ct, such that U(z)=U(x-ct)=V̂(x,t),the system (<ref>) is transformed into(c^2μ-1)U_zz-c(1+μ)U_z+U=0, z>0, (c^2μ-1)U_zz-c(1+μ)U_z+U=1, z<0.Consider first the case z>0. Using γ=c^2μ-1 and β=-c(1+μ), this equation reads γ V”+β V'+V=0,for which the roots of the characteristic polynomial areλ_±=-β±√(β^2-4γ)2γ,so that the solution is U(z)=A_+e^λ_+z+A_-e^λ_-z. It is easy to verify that the case z<0 has solution U(z)=B_+e^λ_+z+B_-e^λ_-z+1. The global solution, therefore, isU(z)= A_+e^λ_+z+A_-e^λ_-z, z>0,B_+e^λ_+z+B_-e^λ_-z+1, z<0.For (<ref>) to represent a propagating front, it is necessary that the solution is real and bounded for any value of z. This implies that either λ_-<0<λ_+ or λ_+<0<λ_-. Requiring β^2-4γ>0, we find the condition γ<β^2/4, which is always satisfied for the model (<ref>) because β^2-4γ=c^2(1-μ)^2+4>0. If c>0 then β < 0. Then λ_->0 and λ_+<0 if c<√(1μ).Imposing U(∞)=0, we find A_-=0, and imposing U(-∞)=1, we find B_+=0, so thatU(z)= A_+e^λ_+z, z>0,B_-e^λ_-z+1, z≤0.To ensure the continuity of the solution at z=0, we fix U(0)=α, so that A_+=α=1+B_- andU(z)=α e^λ_+z, z>0, (α-1)e^λ_-z+1, z≤0.The first derivative of U isU'(z)=αλ_+e^λ_+z, z>0, (α-1)λ_-e^λ_-z, z≤0.Imposing the continuity of U' at z=0, we find αλ_+=(α-1)λ_- and therefore γβ^2=(α^2-α)(2α-1)^2. Using the definitions of γ and β in (<ref>), we finally obtain an expression for the speed c in terms of μ and α,c=(1-2α)√(μ+(α-α^2)(μ-1)^2)≤√(1μ).Notice that the characteristic propagation speed is bounded from above by √(1/μ).§ FINITE ELEMENT DISCRETIZATION The common practice to solve the coupled nonlinear system of cardiac electrophysiology is to split the monodomain (or bidomain) model from the ionic model. Motivated by the work of <cit.>, we will also decouple the two systems.For brevity, we show the discretization only for the monodomain model with a first-order time integrator, because first-order time integrators are still widely used in the community. The same approach can be used to discretize the hyperbolic bidomain model.Given the subinterval [t^n, t^n+1], with t^0=0, we define two separate subproblems, one for the ionic model and one for the monodomain equation connected by the initial conditions. In the first step, we solve the ionic model (<ref>) for w^n+1, and we compute the ionic currents using the updated values of the state variables.The ionic currents computed in this way are then used to solve the monodomain system(<ref>)–(<ref>).As shown in <cit.>, the most efficient numerical algorithm to solve the stiff ODEs of the ionic model strongly depends on the model considered. We use the forward Euler method for the simplified ionic models of Aliev-Panfilov and Fenton-Karma.For the ventricular ionic model of <cit.>, we use the Rush-Larsen method <cit.>. Theatrial ionic model of <cit.> has a stricter restriction on the time step size.Consequently, we use the Rush-Larsen method in combination to the backward Euler method for linear equations.The time step size is defined as Δ t = t^n+1-t^n.To obtain a second-order time discretization scheme, the ionic model can be solved using the explicit trapezoidal method (i.e., Heun's method), which is a second-order accurate strong stability preserving Runge-Kutta method.Once the solution of the ionic model has been found, we solve the monodomain model using a low-order finite element discretization. Because (<ref>) is a simple ordinary differential equation, the computational costs for solving (<ref>) or the system (<ref>)–(<ref>) is comparable. Denoting by (v,w)_Ω=∫_Ωvw the L^2(Ω) inner product and using the boundary conditions (<ref>), the Galerkin approximation of equations (<ref>)–(<ref>) is to find V^h, Q^h ∈𝒮^h such that(∂ V^h∂ t,ψ^h)_Ω=(Q^h,ψ^h)_Ω, (τ C_m∂ Q^h∂ t+C_mQ^h,ϕ^h)_Ω+(D∇ V,∇ϕ^h)_Ω=(I_ion^h,ϕ^h)_Ω+(τ∂ I_ion^h∂ t,ϕ^h)_Ω,for all ψ^h, ϕ^h ∈𝒮^h, where𝒮^h={ v^h∈ C^0(Ω̅):.v^h|_K∈𝒫^1(K), ∀ K∈𝒯^h} .In practice, we look for a continuous solution that is linear in every simplex element K in the triangulation 𝒯^h of Ω. Introducing the basis of nodal shape functions { N_A} _A=1^M, with M=dim(𝒮^h), the solution fields are discretized asV^h=∑_A=1^MN_A𝖵_A, Q^h=∑_A=1^MN_A𝖰_A.Equations (<ref>) and (<ref>) yield the matrix system𝖵̇=𝖰,C_m𝖬(τ𝖰̇+𝖰)+𝖪𝖵=𝖥+τ𝖫.We use the half-lumping scheme proposed by <cit.>, so that𝖥=𝖬𝖨,𝖫=𝖬𝖩and𝖵̇=𝖰,C_m𝖬_L(τ𝖰̇+𝖰)+𝖪𝖵=𝖬(𝖨+τ𝖩),where[𝖬_AB]=(N_A,N_B)_Ω,[𝖪_AB]=(∇ N_A,D∇ N_B)_Ω,𝖨 and 𝖩 are the nodal evaluation of I_ion and ∂ I_ion/∂ t, and 𝖬_L is the lumped mass matrix obtained by row summation of the mass matrix. Using a simple first-order implicit-explicit time integrator, the fully discrete system of equations reads𝖵^n+1=𝖵^n+Δ t𝖰^n+1,C_m(τ+Δ t)𝖬_L𝖰^n+1+Δ t𝖪𝖵^n+1=Δ t𝖬(𝖨^*+τ𝖩^*) +τ C_m𝖬_L𝖰^n,where 𝖨^* and 𝖩^* indicate the values of the ionic currents, and of their derivatives, evaluated using the updated values for w^n+1 obtained by solving the ionic model. Introducing (<ref>) in (<ref>), we arrive at a single system for 𝖰^n+1,[C_m(τ+Δ t)𝖬_L+Δ t^2𝖪]𝖰^n+1= τ C_m𝖬_L𝖰^n-Δ t𝖪𝖵^n+Δ t𝖬(𝖨^*+τ𝖩^*).The time derivative of the ionic currents, 𝖩^*, is approximated nodally using the chain rule as∂ I_ion∂ t(t^n)≈ ∂ I_ion∂ V(V^n,w^n+1)Q^n+∂I_ion∂w(V^n,w^n+1)·g(V^n,w^n+1),where the function g(V, w) is the right hand side of the ionic model system defined in equation (<ref>). After solving (<ref>), we update the voltage equation (<ref>). The scheme used to obtain (<ref>) and (<ref>) is the simplest method – ARS(1,1,1) – of a series of implicit-explicit (IMEX) Runge-Kutta (RK) algorithms that are popular for hyperbolic systems <cit.>. For a second-order time integrator, we use the H-CN(2,2,2) scheme<cit.>, which combines Heun's method for the explicit part and the Crank-Nicholson method for the implicit part.We have not experienced any difference in time step size restriction between the parabolic and hyperbolic models. In fact, we have used an IMEX-RK method where only the ionic currents and their time derivatives are treated explicitly in both parabolic and hyperbolic models. The time step size restriction is dictated only by the ionic model. Nonetheless, the time step size should be chosen to accurately capture the propagation of the action potential. Denoting with v the conduction velocity and with h_mthe smallest element size, we chose our time step size such that the CFL condition Δ t ≤ h_m / v holds.From a purely numerical perspective, in the first-order scheme, the dissipative nature of the backward Euler method can be used to remove numerical oscillations, while the second-order Crank-Nicholson method will keep such oscillations. A dissipative second-order time integrator could be used in place of the Crank-Nicholson method. In any case, the physical dissipation of the hyperbolic monodomain model reduces such artifacts for first- and second-order schemes.One of the main difficulties arising from hyperbolic systems is their tendencies to form discontinuities for whichnumerical approximations typically develop spurious oscillations. Although the standard monodomain and bidomain models are parabolic systems, if thewavefront is not accurately resolved then the upstroke of the action potential can act as a discontinuity. For this reason, the numerical methods used for cardiac electrophysiology should be already suitable for the hyperbolic systems considered here. abbrv agsm | http://arxiv.org/abs/1706.08490v2 | {
"authors": [
"Simone Rossi",
"Boyce E. Griffith"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170626172908",
"title": "Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology"
} |
http://arxiv.org/abs/1706.08958v2 | {
"authors": [
"Fixiang Li",
"Chen Sun",
"Vladimir Y. Chernyak",
"Nikolai A. Sinitsyn"
],
"categories": [
"quant-ph",
"cond-mat.mes-hall",
"math-ph",
"math.MP"
],
"primary_category": "quant-ph",
"published": "20170627174407",
"title": "Multistate Landau-Zener models with all levels crossing at one point"
} |
|
[correspAuth]Corresponding author [email protected][myfootnote]Materials Science Division, Argonne National Laboratory, 9700 Cass Ave, Lemont, IL 60439 [myfootnote1]ECE Department, Purdue University, 465 Northwestern Ave, West Lafayette, IN 47907 [myfootnote2]Mathematics and Computer Science Division, Argonne National Laboratory, 9700 Cass Ave, Lemont, IL 60439[myfootnote3]Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL 60208 Analytical electron microscopy and spectroscopy of biological specimens, polymers,and other beam sensitive materialshas been a challenging area due to irradiation damage. There is a pressing need to develop novel imaging and spectroscopic imaging methods that will minimize such sample damage as well as reduce the data acquisition time. The latter is useful for high-throughput analysis of materials structure and chemistry. In this work, we present a novel machine learning based method for dynamic sparse sampling of EDS data using a scanning electron microscope. Our method, based on the supervised learning approach for dynamic sampling algorithm and neural networks based classification of EDS data, allows a dramatic reduction in the total sampling of up to 90%, while maintaining the fidelity of the reconstructed elemental maps and spectroscopic data.We believe this approach will enable imaging and elemental mapping of materials that would otherwise be inaccessible to these analysis techniques. scanning electron microscopy (SEM) Energy dispersive spectroscopy (EDS) dynamic sampling SLADS Neural Networks dose reduction § INTRODUCTIONAnalytical electron microscopy based on energy dispersive X-ray spectroscopy (EDS) is a very versatile and successful technique for exploring elemental composition in microanalysis from the sub-nanometer scale to the micron scale <cit.>.Modernscanning electron microscopes (SEM) equipped with EDS detectors are routinely used for qualitative, semi-quantitative or quantitative elemental mappingof various materials ranging from inorganic to organic, and including biological specimens. Although EDS allows us to identify the elemental composition at a given location with high accuracy, each spot measurement can take anywhere from 0.1-10 s to acquire. As a result, if one wants to acquire EDS maps on a rectilinear grid with 256 × 256 grid points, the total imaging time could be on the order of tens to hundreds of hours. Furthermore, during the acquisition process, the sample gets exposed to a highly focused electron beam that can result in unwanted radiation damage such as knock-on damage, radiolysis, sample charging or heating. Organic and biological specimens are more prone to such damage due to electrostatic charging. Therefore minimizing the total radiation exposure of the sample is also of critical importance. One approach to solve this problem is to sample the rectilinear grid sparsely. However, it is critical that elemental composition maps reconstructed from these samples are accurate. Hence the selection of the measurement locations is of critical importance. Sparse sampling techniques in the literature fall into two main categories – Static Sampling and Dynamic Sampling (DS). In Static Sampling the measurement locations are predetermined. Such methods include object independent static sampling methods such as Random Sampling strategies <cit.> and Low-discrepancy Sampling strategies <cit.>, and sampling methods based on a model of the object being sampled such as those described in <cit.>. In Dynamic Sampling, previous measurements are used to determine the next measurement or measurements. Hence, DS methods have the potential to find a sparse set of measurements that will allow for a high-fidelity reconstruction of the underlying sample. DS methods in the literature include dynamic compressive sensing methods <cit.> which are meant for unconstrained measurements, application specific DS methods <cit.>, and point-wise DS methods <cit.>. In this paper, we use the dynamic sampling method described in <cit.>, Supervised Learning Approach for Dynamic Sampling (SLADS). SLADS is designed for point-wise measurement schemes, and is both fast and accurate, making it an ideal candidate for EDS mapping. In SLADS, each measurement is assumed to be scalar valued, but each EDS measurement, or spectrum, is a vector, containing the electron counts for different energies. Therefore, in order to apply SLADS for EDS, we need to extend SLADS to vector quantities or convert the EDS spectra into scalar values. In particular, we need to classify every measured spectrum as pure noise or as one of L different phases. To determine whether a spectrum is pure noise, we use a Neural Network Regression (NNR) Model <cit.>. For the classification step we use Convolutional Neural Networks (CNNs). Classification is a classical and popular machine learning problem in computer science for which many well-established models and algorithms are available. Examples include logistic regression and Support Vector Machines (SVM) which have been proven very accurate for binary classification <cit.>. Artificial neural networks, previously known as multilayer perceptron, have recently gained popularity for multi-class classification particularly because of CNNs <cit.> that introduced the concept of deep learning. The CNNs architecture has convolution layers and sub-sampling layers that extract features from input data before they reach fully connected layers, which are identical to traditional neural networks. CNNs-based classification has shown impressive results for natural images, such as those in the ImageNet challenge dataset <cit.>, the handwritten digits (MNIST) dataset <cit.> and the CIFAR-10 dataset <cit.>. CNNs are also becoming popular in scientific and medical research, in areas such as tomography, magnetic resonance imaging, genomics, protein structure prediction etc. <cit.>. It is because of the proven success of CNNs that we chose to use one for EDS classification. In this paper, we first introduce the theory for SLADS and for detection and classification of EDS spectra. Then, we show results from four SLADS experiments performed on EDS data. In particular, we show experiments on a 2-phase sample measured at two different resolutions and experiments on a 4-phase sample measured at two different resolutions. We also evaluate the performance of our classifier.§ THEORETICAL METHODS In this section we introduce the theory behind dynamic sampling as well as how we adapt it for EDS. §.§ SLADS Dynamic SamplingSupervised learning approach for Dynamic Sampling (SLADS) was developed by Godaliyadda et al. <cit.>. The goal of dynamic sampling, in general, is to find the measurement which, when added to the existing dataset, has the greatest effect on the expected reduction in distortion (ERD). It is important to note that in this section we assume, as in the SLADS framework, that every measurement is a scalar quantity. We later elaborate how we generalize SLADS for EDS, where measurements are vectors. First, we define the image of the underlying object we wish to measure as X ∈ℝ^N, and the value of location s as X_s. Now assume we have already measured k pixels from this image. Then we can construct a measurement vector,Y^(k) =[[ s^(1), X_s^(1);⋮; s^(k), X_s^(k) ]].Using Y^(k) we can then reconstruct an image X̂^(k).Second, we define the distortion between the ground-truth X and the reconstruction X̂^(k) as D (X,X̂^(k)). Here D (X,X̂^(k)) can be any metric that accurately quantifies the difference between X and X̂^(k). For example, if we have a labeled image, where each label corresponds to a different phase, then,D (X,X̂^(k)) = ∑_i=1^N I ( X_i,X̂^(k)_i ),where I is an indicator function defined as I ( X_i,X̂^(k)_i) = 0 X_i=X̂^(k)_i 1X_i ≠X̂^(k)_i. Assume we measure pixel location s, where s ∈{Ω∖𝒮}, where Ω is the set containing indices of all pixels, and 𝒮 is the set containing pixel locations of all measured pixels. Then we can define the reduction in distortion (RD) that results from measuring s as,R^(k;s) = D ( X , X̂^(k) ) - D ( X , X̂^(k;s) ).Ideally we would like to take the next measurement at the pixel that maximizes the RD. However, because we do not know X, i.e. the ground-truth, the pixel that maximizes the expected reduction in distortion (ERD) is measured in the SLADS framework instead. The ERD is defined as, R̅^(k;s)= 𝔼[ R^(k;s)| Y^(k)].Hence, in SLADS the goal is to measure the location, s^(k+1)= max_s ∈Ω{R̅^(k;s)}. In SLADS the relationship between the measurements and the ERD for any unmeasured location s is assumed to be given by, 𝔼[ R^(k;s)| Y^(k)]= θ̂ V^(k)_s.Here, V^(k)_s is a t × 1 feature vector extracted for location s and θ̂ is 1 × t vector that is computed in training. The training procedure is detailed in <cit.> and therefore will not be detailed here.§.§ Adapting SLADS for Energy-Dispersive SpectroscopyIn SLADS it is assumed that a measurement is a scalar value. However, in EDS, the measurement spectrum is a p × 1 vector. So to use SLADS for EDS we either need to redefine the distortion metric, or convert the p × 1 vector spectra to a labeled discrete class of scalar values. In this paper, we use the latter approach. In order to make a meaningful conversion, the scalar value should be descriptive of the measured energy spectrum, and ultimately allow us to obtain a complete understanding of the underlying object. The objective in this work is to identify the distribution of different phases in the underlying object. Note that a phase is defined as the set of all locations in the image that have the same EDS spectrum. So if we can classify the measured spectra into one of L classes, where the L classes correspond to the L different phases, and hence L different spectra, then we can readily adapt SLADS for EDS. Figure <ref>(a) shows this adaptation in more details. The method we used for classifying spectra is detailed in the next section. §.§ Classifying Energy Dispersive SpectraAssume the EDS measurement from a location s is given by Z_s ∈ℝ^p, where p>1. Hence we need to convert Z_s to a discrete integer class X_s, where X_s is a label that corresponds to the elemental composition (phase) at location s. Also, let us assume that the sample we are measuring has L different phases, and therefore, X_s ∈{ 0,1,L }, where 0 corresponds to an ill-spectrum. An ill-spectrum can be caused by a sample defect, equipment noise or other undesired phenomena, and therefore is not from one of the L phases that are known. In this paper, to classify Z_s in to one of L+1 classes we use a two step approach as shown in Figure <ref>(b). In the first step, we determine if the measured spectrum is an ill-spectrum. We call this step the detection step. If we determine that Z_s is an ill-spectrum then we let X_s=0. If not we move on to the second step of determining which of the L phases Z_s belongs to and assign that label to X_s. We call this second step the classification step. In the next two sections we will explain the algorithms we used for the Detection and Classification steps. §.§.§ Detection using Neural Network Regression In the detection step, we use a neural network regression (NNR) model to detect the ill-spectrum class <cit.>. The NNR model we use has Q neurons in each hidden layers as well as the output layer. Assume that we have M training spectra for each of the L phases. The goal of training is to find a function f̂(·) that minimizes the Loss function and project training spectra onto a pre-set straight line f, where:Loss = 1/2∑_r ∈{ 1,2,, LM } || f -f̂(Z_s) ||^2Here, LM is the total number of training spectra and Z_s where s ∈{ 1,2,, LM } denotes one training spectrum. It is important to note that since this is a neural network architecture, by saying we find f̂(·), it is understood that we find the weights of the neural networks, that correspond to f̂(·).To determine if a spectrum Z_s is an ill-spectrum or one which belongs to one of L phases, we first compute,g_s = | f - f̂(Z_s) |.where, g_s ∈ℝ^Q. Then we compute the variance metric, σ^2 ( Z_s ) = 1/Q∑_i=1^Q[ g_s,i - μ_s ]^2where, g_s,i is the i^th element of the vector g_s and μ_s = 1/Q∑_i=1^Q g_s,iThen a pre-set threshold is applied to the variance metric to decide whether the Z_s is a ill-spectrum i.e.X̂_s =0,σ^2(Z_s) > T{ 1,2, L }, σ^2(Z_s) ≤ T. §.§.§ Classification using Convolutional Neural NetworksThe next task at hand is to classify the spectrum according to one of the L labels, given that we found a spectrum Z_s for some location s, which is not an ill-spectrum. For this classification problem we use the convolutional neural networks (CNNs) as described in <cit.> and implement using tensorflow <cit.>.The CNNs we use in this paper has two convolution layers, each followed by a max-pooling layer followed by three fully connected layers, shown in Figure <ref>. The first convolution layer has u_1 1 × k kernels sliding across the input spectrum with a stride v to extract u_1 features, each of size 1 × n_1.The max-pooling layer that follows this layer again operates with the same stride and kernel size, resulting in u_1 features, each of size 1 × m_1.The second convolution layer increases the number of features from u_1 to u_2, where, u_2u_1 =0, by the application of u_2 kernels of size 1 × k at the same stride (v) to the output of the first max-pooling layer.The max pooling layer that follows is identical to the previous max-pooling layer.After the convolution and max-pooling layers, all feature values are stacked into a single vector known as a flat layer. The flat layer is the transition into the fully connected layers that follow.These layers have the same architecture as typical neural networks.In the fully connected layers, the number of neurons in each layer is reduced in our implementation.The output of the fully connected layers is a 1 × L vector, X_s^1. Each entry of this vector is then sent through a SoftMax function to create again an 1 × L vector, which we will denote as X_s^2, for a location s.X_s,i^2 = exp( X_s,i^1)/∑_j=1^Lexp( X_s,j^1) .Here, X_s,j^1 corresponds to the j^th component of X_s^1.Now assume we have the same training examples as in the previous section i.e. M spectra from the L phases. When training the CNNs we minimize the Cross-Entropy, defined as,CE = - ∑_r=1^LM X_s^one-hotlog X_s^2where, LM is the total number of training samples, and X_s^one-hot is the “one-hot" representation of X_s. Here, the “one-hot" notation of label m, when L labels are available, is a L × 1 dimensional vector with 1 at location m and zeros everywhere else. § EXPERIMENTAL METHODS AND RESULTSIn this section, we will first describe the experimental methods used in the simulated experiments in Section <ref> and then present the results for the simulated experiments in Section <ref>.§.§ Experimental Methods Here we first present how we generate images for training and simulated objects to perform SLADS on and finally how we train the classifier we described in Section <ref>. For all the experiments we used a Phenom ProX Desktop SEM. The acceleration voltage of the microscope was set to 15 kV. To acquire spectra for the experiments, we used the EDS detector in spot mode with an acquisition time of 10 s.§.§.§ Constructing Segmented Images for TrainingWe first acquired representative images from the object with L different phases using the back scattered detector on the Phenom. Then we segmented this image so that each label would correspond to a different phase. Finally we denoised the image using an appropriate denoising scheme to create a clean image. §.§.§ Constructing a Simulated Object In this paper, we want to dynamically measure an object in the EDS mode. This means that if the beam is moved to a location s, the measurement extracted, Z_s, is a p dimensional vector. So we can think of the problem as sampling an object with dimensions N × N × p, where we can only sample sparsely in the spatial dimension, i.e. the dimension with N × N points. This hypothetical object is what we call here a simulated object.To create the simulated object, we first acquire an SEM image, and segment it just as in the previous section. We then add noise to this image by assigning the label 0 to a randomly selected set of the pixels. Then we experimentally collect M different spectra from each of the L different phases using the EDS detector in spot mode. Finally we raster through the segmented and noise added image and assign a spectrum to each pixel location in the following manner: If the value read at a pixel location s is 0 we assign a pure noise spectrum to that location. If the value read is l ∈{ 1, 2,L }, we randomly pick one of the M spectra we acquired previously for phase l and add Poisson noise to it. Then we assign the noise added spectrum to location s. It is important to note that the noise added to each location is independent of location and is different for each pixel location. So now we have an object of size N × N × p to use in our SLADS experiment.§.§.§ Specification of the Neural Networks to Classify EDS Spectra In order to train and validate the detection and classification neural networks, we again collected M^train spectra for each phase. Then we added Poisson noise to each spectrum and then used half of the spectra to train the NNR and the CNNs, and used the other half to validate, before using it in SLADS.The NNR network we used has 5 fully connected hidden layers, each with 100 neurons. The CNNs classification system we used has kernel size k = 10 and a stride of 2 for all convolution and max-pooling layers. The number of features in the first and second convolution layers are 8 and 16 respectively. The flat layer stacks all features from second max-pooling layer into a 2048 dimensional vector. The number of neurons for the following 3 fully connected layers are 100, 32 and 8.§.§ Results In this section, we will present results from SLADS experiments performed on 4 different simulated objects. To quantify the performance of SLADS, we will use the total distortion (TD) metric. The TD after k measurements are made is defined as, TD_k = 1/|Ω| D ( X,X̂^(k)).We will also evaluate the accuracy of the classification by computing the misclassification rate. §.§.§ Experiment on Simulated 2-Phase Object with Pb-Sn Alloy Here we will present results from sampling two 2-phase simulated objects, one with dimensions 128 × 128 × p, and the other with dimensions 1024 × 1024 × p created using spectra and SEM images acquired from a Pb-Sn eutectic alloy sample <cit.>. It is important to note that the dimension p here corresponds to the dimension of the spectrum, i.e. in the simulated object at each pixel location we have a p dimensional spectrum. One of the phases has Pb and Sn, and the other only Sn. Both these objects, as well as the training data for SLADS, were created using SEM images taken at1024 × 1024 resolution. We created 128 × 128 images by down-sampling the original 1024 × 1024 images. Then we used a simple thresholding scheme to segment all the SEM images. Then we added noise only to the testing images. The images we used for testing and training are shown in Figure <ref>.To train and validate the neural networks we acquired 24 spectra from each phase. To create the simulated testing object we acquired and used 12 (different) spectra for each phase. The noise added to the spectra while creating the simulated object is Poisson noise with λ = 2. The ill-spectrum we generated were also Poisson random vectors, with independent elements and λ = 20.The results after 15% of samples were collected from the 128 × 128 image is shown in Figure <ref>. The TD with 15% of samples was 0.0015. From this figure it is clear that we can achieve near perfect reconstruction with just 15% of samples. The misclassification rate of the detection and classification system was computed to be 0.0002, which tells us that the detection and classification system is also very accurate.The results after 5% of samples were collected from the 1024 × 1024 image is shown in Figure <ref>. The TD with 5% of samples was 0.0013.Here we see that even for the same object, if we sample at a higher spatial resolution we can achieve similar results with just 5% of measurements. In this experiment the misclassification rate of the detection and classification system was computed to be 0. §.§.§ Experiment on Simulated 4-phase ObjectFor this experiment we will again sample two simulated objects, one with dimensions 256 × 256 × p, and the other with dimensions 1024 × 1024 × p created using spectra and SEM images from a micro-powder mixture with 4 phases i.e. CaO, LaO,Si and C.The testing and training images, shown in Figure <ref> were created in exactly same manner as in the previous experiment. The testing object was again created with 12 spectra from each phase. The neural networks were also trained validated just as before once more using 24 spectra from each phase.The results after 20% of samples were collected from the 256 × 256 image is shown in Figure <ref>. The TD with 20% of samples was 0.006. Again we see that we can achieve near perfect reconstruction with 20% of samples. The misclassification rate of the detection and classification system was computed to be 0.005, which again tells us that the detection and classification system is very accurate. However, we do note that this is not as accurate as in the 2-phase case.The results after 5% of samples were collected from the 1024 × 1024 image is shown in Figure <ref>. The TD with 5% of samples was 0.02. In this experiment the misclassification rate of the detection and classification system was computed to be 0.0009. § DISCUSSION It is clear that by using SLADS to determine the sampling locations, we can reduce the overall exposure to anywhere between 5-20%, and still obtain a near perfect reconstruction. These results also show that the SLADS method is better suited for higher pixel resolution mapping which maximizes the resolution capability of the instrument and detector. This method would be useful for investigation of biological and beam-sensitive samples, such as live cell imaging, as well as for high-throughput imaging of large samples, such as fabrication by additive manufacturing and defects metrology in chemical and structural study.§ CONCLUSIONIn conclusion, we have shown that integrating dynamic sampling (SLADS) with EDS classification (CNNs) offers significant advantage in terms of dose reduction and the overall data acquisition time. We have shown that when imaging at lower pixel resolution i.e. 128 × 128 or 256 × 256 we can achieve a high-fidelity reconstructions with approximately 20% samples and when imaging at higher pixel resolution i.e 1024 × 1024 we can achieve a high-fidelity reconstruction with just 5% samples. We have also shown that our classification algorithm performs remarkably well in all the experiments. For future work, we will expand our EDS training database by including analytically simulated EDS data for more commonly used elements. We will use pure simulated EDS data to train the classification system, which enables more automated, high-throughput acquisition and characterization across different microscopic and spectroscopic platforms. § ACKNOWLEDGEMENTThis material is based upon work supported by Laboratory Directed Research and Development (LDRD) funding from Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-06CH11357. 10 url<#>1urlprefixURL href#1#2#2 #1#1goldstein2012scanning J. Goldstein, D. E. Newbury, P. Echlin, D. C. Joy, A. D. Romig Jr, C. E. Lyman, C. Fiori, E. Lifshin, Scanning electron microscopy and X-ray microanalysis: a text for biologists, materials scientists, and geologists, Springer Science & Business Media, 2012.Allen2010 A. J. D'Alfonso, B. Freitag, D. Klenov, L. J. Allen, https://link.aps.org/doi/10.1103/PhysRevB.81.100101Atomic-resolution chemical mapping using energy-dispersive x-ray spectroscopy, Phys. Rev. B 81 (2010) 100101. http://dx.doi.org/10.1103/PhysRevB.81.100101 doi:10.1103/PhysRevB.81.100101. <https://link.aps.org/doi/10.1103/PhysRevB.81.100101>Krivanek2012 T. C. Lovejoy, Q. M. Ramasse, M. Falke, A. Kaeppel, R. Terborg, R. Zan, N. Dellby, O. L. Krivanek, http://dx.doi.org/10.1063/1.3701598Single atom identification by energy dispersive x-ray spectroscopy, Applied Physics Letters 100 (15) (2012) 154101. http://arxiv.org/abs/http://dx.doi.org/10.1063/1.3701598 arXiv:http://dx.doi.org/10.1063/1.3701598, http://dx.doi.org/10.1063/1.3701598 doi:10.1063/1.3701598. <http://dx.doi.org/10.1063/1.3701598>Hyrum13 H. S. Anderson, J. Ilic-Helms, B. Rohrer, J. Wheeler, K. Larson, Sparse imaging for fast electron microscopy, IS&T/SPIE Electronic Imaging (2013) 86570C–86570C.LDSampling R. Ohbuchi, M. Aono, Quasi-monte carlo rendering with adaptive sampling.Mueller2011 K. Mueller, Selection of optimal views for computed tomography reconstruction (Jan. 28 2011).wang2010variable Z. Wang, G. R. Arce, Variable density compressed image sampling, Image Processing, IEEE Transactions on 19 (1) (2010) 264–270.seeger2008compressed M. W. Seeger, H. Nickisch, Compressed sensing and bayesian experimental design, in: Proceedings of the 25th international conference on Machine learning, ACM, 2008, pp. 912–919.carson2012communications W. R. Carson, M. Chen, M. R. D. Rodrigues, R. Calderbank, L. Carin, Communications-inspired projection design with application to compressive sensing, SIAM Journal on Imaging Sciences 5 (4) (2012) 1185–1212.Seeger10 M. Seeger, H. Nickisch, R. Pohmann, B. Schölkopf, Optimization of k-space trajectories for compressed sensing by bayesian experimental design, Magnetic Resonance in Medicine 63 (1) (2010) 116–126.joost2012dynamic K. Joost Batenburg, W. J. Palenstijn, P. Balázs, J. Sijbers, Dynamic angle selection in binary tomography, Computer Vision and Image Understanding.Vanlier2012 J. Vanlier, C. A. Tiemann, P. A. J. Hilbers, N. A. W. van Riel, A bayesian approach to targeted experiment design, Bioinformatics 28 (8) (2012) 1136–1142.merryman2005adaptive T. E. Merryman, J. Kovacevic, An adaptive multirate algorithm for acquisition of fluorescence microscopy data sets, IEEE Trans. on Image Processing 14 (9) (2005) 1246–1253.godaliyaddaMBDS G. M. D. Godaliyadda, G. T. Buzzard, C. A. Bouman, A model-based framework for fast dynamic image sampling, in: proceedings of IEEE International Conference on Acoustics Speech and Signal Processing, 2014, pp. 1822–6.Godaliyadda2 G. D. Godaliyadda, D. Hye Ye, M. A. Uchic, M. A. Groeber, G. T. Buzzard, C. A. Bouman, A supervised learning approach for dynamic sampling, in: IS&T Imaging, International Society for Optics and Photonics, 2016.specht1991general D. F. Specht, A general regression neural network, IEEE transactions on neural networks 2 (6) (1991) 568–576.suykens1999least J. A. Suykens, J. Vandewalle, Least squares support vector machine classifiers, Neural processing letters 9 (3) (1999) 293–300.hinton2006reducing G. E. Hinton, R. R. Salakhutdinov, Reducing the dimensionality of data with neural networks, science 313 (5786) (2006) 504–507.lecun2015deep Y. LeCun, Y. Bengio, G. Hinton, Deep learning, Nature 521 (7553) (2015) 436–444.krizhevsky2012imagenet A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, in: Advances in neural information processing systems, 2012, pp. 1097–1105.hadsell2006dimensionality R. Hadsell, S. Chopra, Y. LeCun, Dimensionality reduction by learning an invariant mapping, in: Computer vision and pattern recognition, 2006 IEEE computer society conference on, Vol. 2, IEEE, 2006, pp. 1735–1742.he2016deep K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 770–778.hua2014computer K.-L. Hua, C.-H. Hsu, S. C. Hidayati, W.-H. Cheng, Y.-J. Chen, Computer-aided classification of lung nodules on computed tomography images via deep learning technique., OncoTargets and therapy 8 (2014) 2015–2022.wang2016perspective G. Wang, A perspective on deep imaging, IEEE Access 4 (2016) 8914–8924.park2015deep Y. Park, M. Kellis, Deep learning for regulatory genomics, Nat Biotechnol 33 (8) (2015) 825–6.zhou2014deep J. Zhou, O. Troyanskaya, Deep supervised and convolutional generative stochastic network for protein secondary structure prediction, in: International Conference on Machine Learning, 2014, pp. 745–753.Godaliyadda3 G. D. Godaliyadda, D. Hye Ye, M. A. Uchic, M. A. Groeber, G. T. Buzzard, C. A. Bouman, A framework for dynamic image sampling based on supervised learning (slads), ARXIV, 2017.Garth N. M. Scarborough, G. M. D. P. Godaliyadda, D. H. Ye, D. J. Kissick, S. Zhang, J. A. Newman, M. J. Sheedlo, A. U. Chowdhury, R. F. Fischetti, C. Das, G. T. Buzzard, C. A. Bouman, G. J. Simpson, Dynamic x-ray diffraction sampling for protein crystal positioning, Journal of Synchrotron Radiation 24 (1) (2017) 188–195. http://dx.doi.org/10.1107/S160057751601612X doi:10.1107/S160057751601612X.abadi2016tensorflow M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al., Tensorflow: Large-scale machine learning on heterogeneous distributed systems, arXiv preprint arXiv:1603.04467.rowenhorst2006three D. Rowenhorst, J. Kuang, K. Thornton, P. Voorhees, Three-dimensional analysis of particle coarsening in high volume fraction solid–liquid mixtures, Acta materialia 54 (8) (2006) 2027–2039. | http://arxiv.org/abs/1707.03848v1 | {
"authors": [
"Yan Zhang",
"G. M. Dilshan Godaliyadda",
"Nicola Ferrier",
"Emine B. Gulsoy",
"Charles A. Bouman",
"Charudatta Phatak"
],
"categories": [
"cs.LG",
"cs.CV"
],
"primary_category": "cs.LG",
"published": "20170627150514",
"title": "Reduced Electron Exposure for Energy-Dispersive Spectroscopy using Dynamic Sampling"
} |
[email protected] of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, PolandWe present an optical method to measure radio-frequency electro-optic phase modulation profiles by employing spectrum-to-time mapping realized by highly chirped optical pulses. We directly characterize temporal phase modulation profiles of up to 12.5 GHz bandwidth, with temporal resolution comparable to high-end electronic oscilloscopes. The presented optical setup is a valuable tool for direct characterization of complex temporal electro-optic phase modulation profiles, which is indispensable for practical realization of deterministic spectral-temporal reshaping of quantum light pulses. Measurement of radio-frequency temporal phase modulation using spectral interferometry Michał Karpiński 21 September 2017 ====================================================================================== § INTRODUCTIONEncoding of quantum information in the spectral-temporal degree of freedom of optical pulses has been recently recognized as a promising platform for high-dimensional quantum information processing and quantum communication, that is naturally compatible with fibre-optic infrastructure and integrated optical setups <cit.>. Realization of general spectral-temporal photonic quantum information processing and measurement requires spectral-temporal shaping of optical pulses that does not involve spectral or temporal filtering. A recipe for such manipulations is well-known within the field of classical ultrashort pulse shaping <cit.>. It relies on successive imprints of spectral and temporal phases onto the optical pulse in a controllable manner. While the former is achieved simply by propagating the pulse through a dispersive medium, the latter requires elaborate active modulation techniques, which may be implemented e. g. by nonlinear-optical <cit.> or electro-optic methods <cit.>. The substantial advantage of electro-optic, and the recently demonstrated electro-optomechanical approach <cit.>, lies in its low-noise and deterministic operation, which are key characteristics for applications in optical quantum technologies <cit.>.In the electro-optic approach a well-defined temporal phase imprint is applied to the optical pulse by driving an electro-optic phase modulator (EOPM) with an appropriate radio-frequency (RF) field, of up to tens of GHz bandwidths. Detailed characterization of the imprinted temporal phase becomes thus a key necessity, especially given recent efforts to diversify from standard single-tone RF modulation techniques <cit.> to more complex temporal phase profiles<cit.>, in particular in guided-wave electronic systems. Here we experimentally show the reconstruction of a fast temporal phase modulation profile performed using the spectral encoding method. The spectral encoding technique relies on mapping a temporal waveform of interest onto the spectrum of an optical pulse. This can be achieved by highly chirping the pulse and subsequently modulating either its spectral amplitude or spectral phase. The former variation is known as the photonic time stretch (PTS) technique and has been extensively used for the analysis of RF signals <cit.> as well as for detailed characterization of THz frequency electric-field pulses <cit.>. Since we are directly interested in the temporal phase profiles we use the latter variation of this method.We use a highly chirped optical pulse to probe the phase imprinted by an RF field applied to the EOPM. The spectral phase of the optical pulse is afterwards recovered using standard spectral interferometry <cit.> and, after appropriate chirp calibration, translated into the temporal profile of the modulating pulse with temporal resolution comparable to or surpassing the fastest electronic oscilloscopes available. The temporal phase modulation profiles are directly probed without the need for electro-optic response calibration of the EOPM. Its calibration can be optionally employed to translate the phase modulation profile into the electronic waveform inducing it, which effectively turns the setup into a high-bandwidth optical oscilloscope.Compared with the hitherto experiments using the spectral encoding approach our work is related to the scheme presented in ref. <cit.>, where free-space THz waveforms were characterized. We extend the applications of THz-pulse-characterization methods to direct measurement of radio-frequency temporal phase modulation profiles in a guided-wave platform. Since our solution can also be employed for the reconstruction of RF signals themselves, it can be alternatively treated as a variation of PTS technique relying on phase rather than intensity modulation of a chirped optical pulse.§ MEASUREMENT SCHEME Our experimental implementation relies on a Mach-Zehnder interferometer with a lithium-niobate-waveguide EOPM placed in one of its arms and a variable delay line τ placed in the other arm. The interferometer is fed with heavily chirped optical pulses and one of its output ports is monitored by a spectrometer, such as an optical spectrum analyser (OSA). Assuming the temporal mismatch between interferometer arms τ, equal transmission of the interferometer arms over the wavelength range used in the experiment, the spectral phase imprint resulting from temporal RF modulation φ(ω) through time-to-frequency mapping induced by the pulse chirp, the spectral intensity of the output field is given by:I(ω)∝|E(ω)+E(ω)e^iωτ+iΔψ(ω)+iφ(ω)|^2= = 2|E(ω)|^2+2|E(ω)|^2cos[ωτ+Δψ(ω)+φ(ω)],where E(ω) is the initial spectral amplitude of the pulse entering the interferometer and Δψ(ω) is the difference in spectral phases imprinted by the interferometer arms. In the Fourier domain, the two terms in the above equation correspond to three distinct peaks, i. e. the baseband peak and two symmetrically located sidebands. By choosing one of the sidebands, performing an inverse Fourier transformone can recover the spectral phase profile ωτ+ Δψ(ω) + φ(ω) <cit.>, which is a standard phase retrieval method used e. g. in spectral interferometry for femtosecond pulse characterization <cit.>. Retrieval of φ(ω) requires calibrating the value of τ and the differential Δψ(ω)term. The sum of both contributions can be easily determined by retrieving a reference spectral phase profile with the RF modulation switched off.The reference spectral phase profile is afterwards subtracted from the actual measurement with the RF signal switched on. Additionally, this procedure avoids the need for precise spectral calibration of the OSA <cit.>. Once the reference phase profile for a given delay τ is known, the RF waveforms can be recovered from single spectral measurements. Provided the pulses entering the interferometer are chirped such that temporal far-field condition is satisfied <cit.>, the spectral phase φ(ω) can be directly translated into the temporal profile of RF modulation using the linear relation ω=ω_0+at, where a is the chirping rate and ω_0 is the central angular frequency of the pulse. Both the parameters can be easily determined experimentally, as we detail further in the text. § EXPERIMENTAL SETUPThe scheme of our experimental setup is presented in Fig. <ref>. We use a polarization-maintaining-fibre Mach-Zehnder (MZ) interferometer with the EOPM placed in one of its arms. Its advantage with respect to the intrinsically stable common-path polarization interferometer <cit.> lies in the fact that optical field in the reference arm is not subjected to any temporal phase modulation. Additionally, in this setting the delay τ can be easily adjusted using a regular air-gap delay line. A pulsed beam from a fibre femtosecond oscillator (Menlo Systems C-Fiber HP, 1550 nm central wavelength, 50 fs pulse duration) is fibre-coupled and divided in two parts, of which one is sent through a free-space variable delay line and coupled into a 12.5-GHz-bandwidth photodiode detector (PD, EoTech-3500FEXT), generating the RF pulse to be characterized, while the other is chirped in a length of single-mode fibre (SMF) and directed into the MZ interferometer. The RF pulse generated by the PD is amplified by a pair of broadband (0-40 GHz) RF amplifiers connected in series and fed into the EOPM (EO-Space PM-DV5-40-PFU-PFU-LV-UL) placed in one of the interferometer arms. Inside the EOPM the RF pulse temporally overlaps with the optical pulse chirped during the propagation through either 250 m or 500 m of SMF (Corning SMF-28). The delay between the RF and optical pulses is tuned using a 1.5 m-long air-gap variable delay line placed before the PD. One of the output ports of the interferometer is monitored by an OSA (Yokogawa AQ6370D) whereas the second output port is sent through a bandpass filter (3 nm full-width-half-maximum bandwidth) and directed to an auxiliary photodiode detector (not presented in the experimental setup scheme). Simultaneous monitoring of interference fringes in the spectral and temporal domains simplifies the initial alignment of the interferometer in a balanced position, which facilities the synchronization between the RF and optical pulses. For the actual measurements the interferometer is used in an unbalanced setting (τ≠ 0), with only the OSA being monitored. The delay τ, which defines the spectral fringe spacing, needs to be large enough to yield a clearly distinguishable sideband in Fourier domain and small enough to avoid detrimental effects resulting from the finite resolution of the spectrometer. Our delay setting resulted in fringe spacing of 1.5 nm. Setting the resolution of the OSA to 0.1 nm allowed us minimize the effects finite spectrometer resolution of data interpolation on the reconstructed spectral phase <cit.>. § RESULTS AND DISCUSSION In Fig. <ref> we present exemplary interferometer output spectra (τ≠0) collected using the OSA with RF modulation switched off and on for the case of a 250-m-long chirping SMF. The phase imprinted by the RF pulse significantly modifies the positions of minima and maxima of spectral interference fringes over bandwidth range of approximately 50 nm. In further experiments we used larger values of τ, resulting in denser spectral fringes, which allowed us to easily separate one of the spectral sidebands after its transformation to the Fourier domain. In our experiment the acquisition time of a single OSA measurement is much longer than the time interval between two subsequent optical pulses. Single-shot acquisition could be achieved by employing a spectrometer with a multipixel detector array and reducing the repetition rate of the fs oscillator to match the spectrometerrefresh rate (e. g. by an external pulse picker) <cit.>. Alternatively, in analogy to the PTS approach, frequency-to-time mapping via the dispersive Fourier transform principle could be used to record the spectrum using a single fast photodiode <cit.>, possibly with an appropriate reduction of repetition rate. To retrieve the chirp value a and verify the repeatability of our measurement scheme we retrieved the phase modulation profiles for a series of delays between the RF waveform and the optical pulse. This has been achieved by adjusting the air-gap variable delay line in a series of 4-mm steps corresponding to 13.34 ps of delay. In Fig. <ref>(a,b) we present the result of this procedure for the 250 m and 500 m lengths of the chirping SMF. The position of modulation maximum has been recorded to recover the SMF group delay dispersion values yielding-11.1 ps^2 for 500 m of SMF, Fig. <ref>(c), and -5.73 ps^2 for 250 m of SMF, Fig. <ref>(d). The final result of the experiment is shown in Fig. <ref>, where we present the RF modulation profiles retrieved for the two values of optical pulse chirp, Fig. <ref>(a). The reconstructed waveforms agree very well with each other. Naturally the temporal interval covered by the chirped optical pulse is longer for a larger GDD value. We verified that the sensitivity of our setup allows measuring pulses generated directly by the PD with a peak voltage of 100 mV. For the purpose of spectral-temporal light shaping the most relevant information is the temporal phase modulation profile induced by the RF pulse. It can however be converted to retrieve the actual electronic waveform by using the EOPM electro-optic response profile. We used the response profile provided the EOPM manufacturer in the 0-30 GHz frequency range. The EOPM response can be independently verified using spectral sideband analysis <cit.>. In the temporal domain the measured phase modulation profile is a convolution of the electronic signal feeding the EOPM with its electro-optic response function. Thus, the retrieval of electronic waveform requires a reciprocal deconvolution operation, which is especially easy to implement in the spectral domain, where it corresponds to division by the EOPM spectral response function. The division performed in the spectral domain is also beneficial for the deconvolution fidelity since this is the domain where the instrument response function is directly measured. The results of the deconvolution procedure, based on the spectral response profile provided by the EOPM manufacturer, are presented in Fig. <ref>(b). The temporal resolution achievable through chirped-pulse spectral-temporal mapping is given by (t_bt_a)^1/2, where t_b and t_a are the duration of the optical pulse before and after the chirp <cit.>. For the optical pulses employed in our setup this value equals approximately 8.7 ps for 500 m and 6.1 ps for 250 m of SMF. Although the expected optical temporal resolution can easily reach single ps, in practice the resolution is limited by the finite bandwidth of EOPMs. Whereas this limitation is irrelevant for the direct characterization of phase-modulation profiles, for probing of electronic waveforms it imposes a trade-off between the sensitivity and temporal resolution of the method, as the EOPM bandwidth could be increased at the cost of reduced sensitivity by using a shortermodulator. This can be mitigated by the development of novel types of electro-optic modulators offering both exceptionally low half-wave voltage and high modulation bandwidth <cit.>.§ CONCLUSIONS AND OUTLOOK We present a robust method for the direct characterization of RF electro-optic temporal phase modulation profiles by means of chirped-pulse sampling method. It extends the applications of THz field characterization methods and spectral encoding techniques to radio-frequency temporal phase profiles in an integrated optical platform, which are routinely employed for electro-optic spectral shaping of classical and quantum light. Alternatively it can be used for the reconstruction of RF electronic signals inducing phase modulation, which presents a variation of the PTS technique relying on phase rather than intensity modulation, involving a phase-sensitive detection scheme. Our work paves the way towards deterministic realization of complex spectral-temporal mode transformations, including general unitary transformations <cit.>, which will lead to important developments in spectrally-encoded photonic quantum information processing and communication <cit.>. Thanks to the reconfigurability of electro-optic phase modulation patterns we also expect it to become a valuable tool for realization of adaptive spectral-temporal shaping strategies, in particular of quantum light <cit.>.§ ACKNOWLEDGEMENTS We thank K. Banaszek, C. Radzewicz and A. O. C. Davis for careful reading of the manuscript. This work has been supported by the National Science Centre of Poland (project no. 2014/15/D/ST2/02385). M. J. was supported by the Foundation for Polish Science. The authors declare that they have no conflict of interest.45Brecht2015 Brecht, B.; Reddy, D. V.; Silberhorn, C.; Raymer, M. G., Phys. Rev. X 2015, 5, 041017.Humphreys2013 Humphreys, P. C.; Metcalf, B. J.; Spring, J. B.; Moore, M.; Jin, X.-M.; Barbieri, M.; Kolthammer, W. S.; Walmsley, I. A., Phys. Rev. Lett. 2013, 111, 150501.Lukens2017 Lukens, J. M.; Lougovski, P., Optica 2017, 4, 8–16.Edition Agrawal, G. P. Fiber-Optic Communication System; Wiley: Hoboken, NJ, 2010.Torres-Company2011 Torres-Company, V.; Lancis, J.; Andrés, P. Prog. Opt. 2011, 56, 1–80.Salem Salem, R.; Foster, M. A.; Gaeta, A. L. Adv. Opt. Photonics 2013, 5, 274–317.Agha2014 Agha I., Ates S., Sapienza L., Srinivasan K., Opt. Lett. 2014, 39, 5677–5680Zhu13 Zhu, Y.; Kim, J.; Gauthier, D. J.; Phys. Rev. A 2013, 87, 043808.Donohue2016 Donohue, J. M.; Mastrovich, M.; Resch, K. J. Phys. Rev. Lett. 2016, 117, 243602. Matsuda2016 Matsuda, N., Science Adv. 2016, 2, e1501223.Jachura2016 Karpiński, M.; Jachura, M.; Wright, L. J.; Smith, B. J. Nat. Photon. 2016, 11, 53–57. Li2013 Li, B.; Li, M.; Lou, S.; Azaña, J. Opt. Express 2013, 21, 16814–16830.Poberezhskiy2005 Poberezhskiy, I. Y.; Bortnik, B.; Chou, J.; Jalali, B.; Fetterman, H. R. IEEE J. Quant. Electron 2005, 41, 1533-1539.Wright2017 Wright, L. J.; Karpiński, M.; Söller, C.; Smith, B. J., Phys. Rev. Lett. 2017, 118, 023601.Fan2016 Fan, L.; Zou, C.-l.; Poot, M.; Cheng, R.; Guo, X.; Han, X.; Tang, H. X. Nat. Photon. 2016, 10, 766-770.Olislager Olislager, L.; Mbodji, I.; Woodhead, E.; Cussey, J.; Furfaro, L.; Emplit, P.; Massar, S.; Huy, K. P.; Merolla, J.-M. New J. Phys. 2012, 14, 043015.Weiner Lukens, J. M.; Odele, O. D.; Leaird, D. E.; Weiner, A. M. Opt. Lett. 2015, 40, 5331-5334.Lansinis2015 Plansinis, B. W.; Donaldson, W. R.; Agrawal, G. P. J. Opt. Soc. Am. B 2015, 32 , 1550-1554.Agha2016 Pedrotti, L. M.; Agha, I. Opt. Express 2016, 24, 16687-16694.Jalali1999Coppinger, F.; Bhushan, A.S.; Jalali, B. IEEE Trans. Microw. Theory Tech. 1999, 47, 1309-1314.Jalali2007 Chou, J.; Boyraz, O.; Solli, D.; Jalali, B. Appl. Phys. Lett. 2007, 91, 161105.Jalali2013 Buckley, B. W.; Madini, A.M.; Jalali, B. Opt. Express 2013, 21, 21618-21627.Jiang2011 Jiang, Z.; Zhang, X.-C. Appl. Phys. Lett. 1998, 72, 1945-1947.THZ2000 Yan, X.; MacLeod, A. M.; Gillespie, W. A.; Knippels, G. M. H.; Oepts, D.; van der Merr, A. F. G.; Seidel, W.; Phys. Rev. Lett. 2000, 85, 3404–3407.THZ2011Matlis, N. H.; Plateau, G. R.; van Tilborg, J.; Leemans, W. P. J. Opt. Soc. Am. B 2011, 28, 23–27.THZ2015 Roussel, E.; Evain, C.; Parquier, M. L.; Szwaj, C.; Bielawski, S.; Manceron, L.; Brubach J.-B.; Tordeaux, M.-A.; Ricaud, J.-P.; Cassinari, L.; Labat, M; Couprie, M.-E.; Roy, P. Sci. Rep. 2015, 5, 10330.Takeda1982 Takeda, M.; Ina, H.; Kobayashi, S. J. Opt. Soc. Am. A 1982, 72, 156–160. Drescher Zhou, M.-L.; Feng, L.; Chun, L.; Fei, D.; Li, Y.-T.; Wang, W.-M.; Sheng, Z.-M.; Chen, L.-M.; Ma, J.-L.; Lu, X.; Dong, Q.-L.; Zhang, J. Chin. Phys. Lett 2012, 29, 015202. Iaconis1999 Iaconis, C.; Walmsley, I. A. IEEE J. Quant. Electron 1999, 35, 501–509.Dorrer1 Dorrer, C. J. Opt. Soc. Am. B 1999, 16, 1160–1168.Dorrer2 Dorrer, C.; Belabas, N.; Likforman, J. P.; Joffre, M. J. Opt. Soc. Am. B 2000, 17, 1795–1802.Kim2016 Kim, K. Y.; Yellampalle, B.; Rodriguez, G.; Averitt, R. D.; Taylor, A. J.; Glownia, J. H. Appl. Phys. Lett. 2006, 88, 041123.kalibracja Shi, Y.; Yan, L.; Willner, A. E. J. Lightwave Technol. 2003, 21, 2358–2367.Sun2016 Sun, F. G.; Jiang, Z.; Zhang, X.-C. Appl. Phys. Lett. 1998, 73, 2233–2235.Enami Enami, Y.; Mathine, D.; DeRose, C. T.; Norwood, R. A.; Luo, J.; Jen, A. K.-Y.; Peyghambarian, N. Appl. Phys. Lett. 2007, 91, 093505. Melikyan Melikyan, A.; Alloatti, L.; Muslija, A.; Hillerkuss, D.; Schindler, P. C.; Li, J.; Palmer, R.; Korn, D.; Muehlbrandt, S.; Van Thourhout, D.; Chen, B.; Dinu, R.; Sommer, M.; Koos, C.; Kohl, M.; Leuthold, J. Nat. Photon. 2014, 8, 229–233. Morizur2010 Morizur, J.-F.; Nicholls,L.; Jian, P.; Armstrong, S.; Treps, N.; Hage, B.; Hsu, M.; Bowen, W.; Janousek, J.; Bachor, H.-A. J. Opt. Soc. Am. A 2010, 27, 2524–2531.Bellini2012 Polycarpou, C.; Cassemiro, K. N.; Venturi, G.; Zavatta, A.; Bellini, M. Phys. Rev. Lett. 2012, 109, 053602. | http://arxiv.org/abs/1706.09881v2 | {
"authors": [
"Michał Jachura",
"Jan Szczepanek",
"Wojciech Wasilewski",
"Michal Karpinski"
],
"categories": [
"physics.ins-det",
"physics.optics"
],
"primary_category": "physics.ins-det",
"published": "20170627180533",
"title": "Measurement of radio-frequency temporal phase modulation using spectral interferometry"
} |
1*]Edmund Barter 1]Thilo Gross [1]University of Bristol, Department of Engineering Mathematics, Bristol, UK [*][email protected] ecology it is widely recognised that many landscapes comprise a network of discrete patches of habitat. The species that inhabit the patches interact with each other through a foodweb, the network of feeding interactions. The meta-foodweb model proposed by Pillai et al. combines the feeding relationships at each patch with the dispersal of species between patches, such that the whole system is represented by a network of networks. Previous work on meta-foodwebs has focussed on landscape networks that do not have an explicit spatial embedding, but in real landscapes the patches are usually distributed in space. Here we compare the dispersal of a meta-foodweb on networks, that do not have a spatial embedding, and random geometric networks, that do have a spatial embedding. We found that local structure and large network distances in spatially embedded networks, lead to meso-scale patterns of patch occupation by both specialist and omnivorous species. In particular, we found that spatial separations make the coexistence of competing species more likely. Our results highlight the effects of spatial embeddings for meta-foodweb models, and the need for new analytical approaches to them.Spatial effects in meta-foodwebs [ December 30, 2023 ================================§ INTRODUCTION Foodwebs, the networks of trophic (feeding) interactions among a community of species, are among the paradigmatic examples of complex networks. Their composition and dynamics have been studied extensively.<cit.> In nature, communities are often not isolated but are embedded in a complex structured environment that consists of distinct patches of habitat.<cit.> Depending on the system under consideration the patches may be lakes, islands, or actual patches of forest left in an agricultural landscape. In typical environments, the communities at many similar patches interact through the dispersal of individuals between neighbouring patches. The aggregations of the foodwebs at patches related by a complex spatial network are called meta-foodwebs.<cit.> These systems, comprising local foodwebs joined to each other by links between patches, can be represented by a network of networks <cit.>(Fig <ref>b).The study of spatial interactions has a long history in Ecology. For instance in explaining the global coexistence of similar competitors<cit.> and the survival of multiple species on the same limiting resource.<cit.> Studies in this area often account for the presence or absence of the species at each patch as a binary variable.<cit.> In these, so-called patch-dynamic, models the state of each patch changes in time due to local colonization and extinction events. Most patch-dynamic models focus on simple cases such as single populations or competitive interactions between similar species. <cit.> However, recently Pillai et al.<cit.> set out a framework that incorporates trophic interactions between species into patch-dynamic models. This meta-foodweb model considers complex foodwebs of many species with predator-prey and competitive interactions.The meta-foodweb model of Pillai et. al. has been used to demonstrate that general spatial heterogeneity can increase stability of complex foodwebs.<cit.> More specifically the number and distribution of links in the network of patches have been shown to have non-trivial effects on the distribution of foodwebs among the patches.<cit.> We have previously shown that species at different levels of a food chain may benefit from different distributions of patch degrees (number of links to other patches).<cit.> Other previous work addressed the prominent question of whether, and under what conditions an omnivore predator can coexist with a specialist, who is a stronger competitor.<cit.> This demonstrated that an omnivore can persist only when the average number of links at each patch, the mean degree, is within a particular range. Previous theoretical works investigated meta-foodwebs where the underlying patch network was assumed to be an random graph or configuration model network.<cit.> In such networks any given pair of patches has a fixed chance of interacting and therefore the networks typically have a small diameter, a measure of the maximum distance between nodes.<cit.> In these networks the shortest path (series of links) between any given pair of nodes is small <cit.> and they are subsequently termed small-worlds. <cit.>By contrast, real world meta-foodwebs are constrained by geography and the individual's ability to travel between patches. Such a spatial embedding constrains the possible networks aspatches are only linked if they are close enough together for individuals to disperse between them. Geometric distances are translated in network distances and in the resulting network the shortest path between two patches can be relatively long.<cit.> In a small world network, a species can quickly disperse from any node to every other node. By contrast, in a large world pronounced geographical barriers, characterized by long network path lengths, may exists that impede rapid colonization of distant parts. While we will provide a more detailed analysis below it is intuitively conceivable that the large world nature of spatially embedded networks of patches creates spatial niches in which a species can survive with relatively little danger from competitors. Moreover, in small worlds the neighbours of any particular node tend to be a representative sample from the network. A node in a small world is thus exposed to colonization from the full range of communities that the system supports. By contrast, in large worlds the neighbours of a node are located in the same region of the network as the focal node, and most colonization will be from communities that are very similar to the one established in the focal node. This reinforcement of communities may further promote species persistence.Here we investigate the effects of spatial nature of a patchy environment (i.e. the large-worldishness) on the dispersal of foodwebs. We build our analysis on a comparison of non-spatial networks<cit.> and explicitly spatial, random geometric patch networks.<cit.> We find two results: First, specialist consumers are less abundant (occupy fewer patches) on the spatial patch networks. Second, when the landscape is also occupied by a competitor, generalist consumers are more abundant on the spatial patch networks than on non-spatial patch networks. We conclude that these results are predominantly due to the larger distances between patches in the spatial networks. § THE MODEL We study a version of the model proposed by Pillai et al..<cit.> The model describes a set of species, each of which either occupies or is absent from each patch in a spatial network at each moment in time. Trophic interactions between the species are represented by a global meta-foodweb (Fig. <ref>a). Following Pillai et al.<cit.> we assume each patch contains only a subset of the species of the global foodweb, and these comprise a local food chain (Fig. <ref>b). The global meta-foodweb comprising all the species only becomes evident when an aggregation of the spatial system is considered.The trophic interactions of the meta-foodweb have implications on the ability of species to occupy each patch. A species must be able to feed at every patch it occupies. Primary producers can occupy an empty patch, but all other species can only occupy patches where their prey is present. Furthermore a species cannot share a patch with another species competing for the same prey. Following Pillai et al.<cit.> we assume that specialist consumers outcompete their generalist counterparts for a particular prey all of the time. Hence, all local foodwebs are linear chains. A patch that contains at least one food source that can be utilized by a given species, and does not contain a superior competitor to that species, can be occupied by it and we say the patch is available to that species.The species that occupy a particular patch change in time due to the colonization of the patch by species from neighbouring patches and the extinction of the local populations at the patch. When established on a patch, species X colonises neighbouring patches that are available to it at the rate c_X. When established at a patch, species X also goes extinct on that patch at the rate e_X. In the following we set e_X=1 (arbitrary units) and further assume that c_X=c for all species in the meta-foodweb. At all times the local foodwebs must satisfy the restrictions imposed on species by trophic interactions. Therefore, when species X is established on a patch where its only prey is species Y, and species Y goes extinct on that patch, species X will also go extinct, we call this process indirect extinction. When a specialist predator, species X, colonises a patch that is occupied by an omnivore species Y, and species X and species Y share a food source,then species Y may no longer exist on that food source. This will lead to the extinction of species Y on that patch, unless it can prey on species X, so has a food source it does not compete for. We call the extinction of a generalist by an arriving specialist driven extinction.To study the effects of space on species coexistence we consider dynamics on two different types ofpatch networks. The first type of network are those from the () ensemble.<cit.> These networks are generated by taking a set of nodes and assigning a link between each pair with a probability p=z/N, where N is the number of nodes and z is the desired mean degree, i.e. the expectation value of the number of neighbours for a randomly chosen node.<cit.>A central property of networks that determines how easy a particular network is to colonize is the degree distribution p_k, the probability distribution that a given node has k links.<cit.> Fornetworks the degree distribution is poisson distributed, so p_k= z^ke^-z/k!. Because links are added randomly between nodes theis a non-spatial network and every node can be reached from every other node in a small number of steps, which scales as log(N).<cit.>The second type of network considered here are from the ensemble of 2-dimensional random geometric () networks .<cit.> These networks are generated by, first, randomly distributing N nodes in a unit square with uniform distribution and then adding links between all pairs of nodes that are within a distance r of each other. The degree distribution ofnetworks is the same as that fornetworks, p_k= z^ke^-z/k! where the mean degree is given by z=π r^2 N.The spatial distances between points innetworks are translated into network distances and therefore the shortest path between some pairs of nodes is large and scales with √(N)/r.<cit.>Although theandnetworks have the same degree distribution, they have different levels of degree correlations. Innetworks a link is equally likely between any pair of nodes, and so the degrees of neighbouring nodes are uncorrelated. Innetworks each node is at the centre of a circle with radius r that contains all its neighbours, we call this its spatial neighbourhood. A node's degree is the number of nodes in its spatial neighbourhood. Two nodes that have a link between them in the network must have overlapping spatial neighbourhoods. As the number of nodes in the intersection of their spatial neighbourhoods is the same, the degree of a node in a random geometric graph is correlated with the degree of its network neighbours.We simulate the dispersal of the species in the meta-foodweb on networks from each ensemble using a Gillespie-type algorithm .<cit.> By choosing ensembles with matching degree distributions we eliminate the effects on dispersal considered previously, <cit.> and focus on the differences due to the difference in path lengths between the ensembles.For each species in the system there are two possible equilibrium states, either the species is extinct at the metapopulation level, i.e. does not occupy any patches, or the metapopulation is a constant size, i.e. it does. After the simulation has reached this equilibrium for all species we record the time for which each patch was occupied by each possible configuration of the species over the rest of the simulation. From these states we calculate the fraction of time each species occupies each patch (the patch occupation), the sum of which over all patches we term the species abundance.§ SINGLE SPECIES To begin we compare the dispersal of a single species on networks fromandensembles. We find that above the persistence threshold species abundance is higher onnetworks thannetworks, see Fig. <ref>. This means that populations of the species exist at more patches innetworks thannetworks. Therefore, the global extinction risk for a single species is comparatively greater onthannetworks.The difference in species abundance between the network ensembles is greatest close to the threshold, while for colonisation rates far above the threshold abundance is similar on both ensembles, see Fig. <ref>.In general the risk of global extinction is higher when abundance is lower. Therefore, the increase in extinction risk to a species on , in comparison tonetworks, is greatest when the risk of global extinction is greatest.Let us now try to explain these findings with respect to the spatial structure ofnetworks. Considering parameters near the threshold, we find that the distribution of patch-wise occupations is broader onthannetworks, see Fig. <ref>a. Some patches innetworks are occupied for a larger fraction of the time than any patches innetworks, while others are occupied for a smaller fraction of the time than any innetworks. As extinction is the same at all patches this means that some patches in thenetworks are, when empty, colonised more quickly by the species than others.To explain the variation in colonisation between patches we must consider which properties of a patch determine how easy it is to colonise. It is well understood that a patch's degree is correlated with its occupation.<cit.> The distribution of patch occupations onnetworks has a similar shape to the degree distribution, reflecting the influence of degree on patch occupation. The broad distribution of patch occupation innetworks reflects the greater prominence non-degree effects in its determination. This indicates that in the large world network the detailed spatial structure, rather than just the overall connectivity, is of greater importance. The probability of an unoccupied patch being colonised is affected by the occupation of its neighbours. An unoccupied patch with frequently occupied neighbours will be colonisedquickly, and so be unoccupied for a relatively short period. On the other hand, an unoccupied patch with infrequently occupied neighbours may not be colonised for a longer period. Therefore the properties of the neighbourhood of a patch can determine how often it is occupied.The observed differences between the distribution of patch occupations inandnetworks can thus be intuitively explained by the different average distances in these networks. In thenetworks the short distances mean the neighbourhoods of all patches are similar. In thenetworks the long distances mean local neighbourhoods can be more varied between nodes.We test the effect of differences between neighbourhoods explicitly by considering the occupation of nodes with only the degree 10 in Fig. <ref>a. We find that the occupation of patches with degree 10 innetworks has a distribution that is narrower than the distribution for all patches. Fornetworks we find that the the occupation of patches with degree 10 has a relatively broad distribution, only a small amount narrower than the distribution of all patches. Therefore, patches of the same degree have more varied occupations innetworks than innetworks. This implies that, in spatial networks, the structure of a patch's neighbourhood has a large influence on its occupation by the species. By observing the occupation of patches on example networks, such as in Fig. <ref>a, we find meso-scale patterns of patch occupation. There are distinct regions of high occupation patches separate from regions of low occupation patches. In the large world network the species is distributed unevenly across the landscape, as regions with different local structures are separated by large spatial distances. Innetworks there are often more low occupation regions than high occupation regions, and as such the species abundance is lower on these than equivalentnetworks. Spatial distances reduce the number of patches the species can easily colonise and so abundance is lower in the large world networks.We now consider the situation at higher values of c, for which the species has similar abundance on both types of network. At these parameter values the distribution of patch occupations is similar on both types of network, see Fig. <ref>b. Furthermore, the distributions are similar when considering only patches of degree 10, see Fig. <ref>b, suggesting that at these parameter values degree is a good indicator of occupation onnetworks as well asnetworks. When overall abundance is high the neighbourhoods of nodes in the large world are more similar and so have a lesser impact on the occupation of individual patches.Observing example networks shows that spatial variations do exist in networks at high overall abundance, even though degree is a good indicator of patch occupation. This is because thenetworks have degree correlations. Innetworks high degree nodes have more high degree neighbours and low degree nodes have more low degree neighbours. Therefore spatial regions with many high degree nodes are more highly occupied that spatial regions with many low degree nodes. In thethe degree correlations translate to spatial correlations so the species abundance varies between different regions.Furthermore, we find that the mean occupation of patches innetworks with relatively low degree is lower than for those innetworks,while the mean occupation of patches innetworks with relatively high degrees is higher than for those innetworks, see Fig. <ref>. The effects of degree on occupation are enhanced onnetworks due to degree correlations. Typical high degree patches innetworks also have high degree neighbours and are colonised more easily than typical high degree patches innetworks. Therefore, as well as the regional variations due to large network distances, in spatial networks the species distribution is uneven due to the structural correlations of nearby nodes.To summarise, both long network distances and degree correlations cause the emergence of meso-scale patterns in the occupation of patches in the random geometric network. At low overall abundance the spatial separations have the dominant effect, and the results is large variations between the occupation of different patches. At high overall abundancespatial patterns are predominantly due to degree correlations, and these result in smaller spatial variations in patch-wise occupation. § FOUR SPECIES FOODWEB We now extend our study to the meta-foodweb shown in Fig. <ref>a, which includes both predator-prey feeding and specialist-omnivore competition interactions.Species A, a primary producer, that can survive on abiotic resources; species B, a secondary consumer, that feeds upon the primary producer; species C, a top predator, that feeds upon the secondary consumer; and species O, an omnivore that preys upon the primary producer and secondary consumer. Species B, C and O can only occupy patches alongside their prey. In addition, species O is out competed by the specialist predators and therefore cannot survive on patches where species C is established. When both species B and species O occupy a patch, species B out-competes species O for feeding on species A, however species O can still survive by feeding on species B instead. For each species, X, there is a range of values of c_X/e_X for which that species is unable to persist in a network. For specialist species the range is characterized by a critical value of μ_X. When c_X/e_X<μ_X the species cannot persist in the network and will become globally extinct, but when c_X/e_X>μ_X the species will persist and be expected to occupy a quasi-steady fraction of the patches. <cit.> For an omnivore species Y which is a competitor to species X there are two critical values. Species Y can only persist in the network when bothc_Y/e_Y>μ_Y, andc_X/e_X<ν_Y, where the limit ν_Y is itself dependent on c_Y/e_Y.When c_X/e_X>ν_Y the omnivore's competitor occupies a large fraction of the patches and the omnivore is unable to persist anywhere in the network, becoming globally extinct. Though we have set c_X/e_X=c for all species, due to indirect extinctions the threshold value, μ_X, is not the same for all species.Using the foodweb, we can establish how the spatial embedding affects species that interact with feeding or competitive interactions. We investigate: a) how the distribution of a predator is affected by the uneven distribution of its prey, and b) how the distribution of an omnivore is affected by the uneven distribution of its superior competitor.We start by studying the food chain of species A, B and C that contains only predator-prey relationships. We find that all these specialist species have similar behaviour to a single species, and are less abundant inthannetworks with the same degree distributions under the same dispersal conditions, see Fig. <ref>. The set of patches occupied by a predator is a subset of the set of patches occupied by its prey. Furthermore the effective extinction rate of the predator is larger than that of its prey. Therefore it is harder for a predator to disperse and the abundance of a predator cannot be greater than the abundance of the prey species it consumes. We find that the abundance of predators is lower onnetworks even at parameter values where the abundance of their prey is similar in both ensembles.This suggests that the spatial effects of the underlying network experienced by the single species are experienced by all species in the chain.For the single species onnetworks, we found that structural properties lead to spatial variations in patch abundance.We find that a predator utilising a species as its food source has an even more uneven distribution than its prey, see Fig. <ref>. The variation in the in habitability of patches in the underlying network is exaggerated by the dispersal of intermediate species in the chain. For species A, its prey occupies all patches equally, but structural properties of the network mean it will find some patches are more hospitable than others. For species B, preying on species A, the structural properties have the same effects, and in addition its prey is unevenly distributed. As species A and B undergo similar dispersal processes the uneven distribution of prey aligns with the structural variations. Therefore, species B experiences greater variation inhabitability, and therefore occupation, across the patches in the network. The local structures in large world networks have a greater impact on the distribution of species higher in the food chain. Now let us focus on the effects on the distribution of the omnivore species O due to coexisting with the meta-population of the species C. The two species coexist only for colonisation rates in the range μ_C<c<ν_O. When c<μ_C, species C cannot persist in the meta community and so the patches available to species O are the same patches available to species B, and the omnivore behaves identically to the specialist species B. Whenc>ν_O the omnivore becomes extinct due to driven extinction from competition with the specialist.When the specialist and omnivore coexist, the omnivores abundance is greater onnetworks than onnetworks, see Fig. <ref>. We previously saw that the large distances in spatial networks hinder the occupation of patches by the specialists, by contrast these distances aid the occupation of patches by the omnivore. One reason the omnivore is more abundant onthannetworks, in the coexistence regime, is the lower overall abundance of species C. More of the patches are available to the omnivore for more of the time in thenetworks. Further, colonisation of a patch that is occupied by species O by species C,causes the local extinction of species O. Therefore when a patch which is occupied by species O and species B neighbours a patch occupied by species C the colonisation rate of species C acts to increase the effective extinction rate of species O. In spatial networks omnivore populations encounter specialist populations less frequently, and competition has a smaller effect on the occupation of patches by the omnivores.We have seen that, on bothandnetworks occupation by specialist species increases with node degree, see Fig. <ref>. However the variation isgreater innetworks, with high degree nodes more likely to be colonised than lower degree nodes. We now investigate whether the greater variation in patch occupation by the specialist onnetworks, affects the distribution of the omnivore.On both types of network, at c=0.8, occupation by the omnivore is largest on nodes with an intermediate degree. Low degree nodes are hard for all species to occupy, while the large amount of time species C occupies high degree nodes means that they are often unavailable to the omnivore. The effect of competitor occupation on omnivore occupation of high degree nodes isgreater onthan onnetworks. Figure <ref>b fornetworks, has a pronounced peak in omnivore occupation at intermediate degrees. The high occupation by the specialist of high degree patches makes them almost always unavailable to the omnivore and the omnivore rarely occupies them. In spatial networks some patches are occupied by the specialist so often that a population of omnivores is almost never established on them.Patches with intermediate degrees are relatively unlikely to be occupied by the specialist onnetworks compared tonetworks. Subsequently, the occupation of these patches by the omnivore is high, almost as high as occupation by the omnivore in the absence of competition, which is equivalent to occupation by species B. The large network distances mean that the omnivore is able to colonise these nodes and establish populations that rarely encounter competition from specialist populations.Bothandnetworks have poisson degree distributions, and many more patches with intermediate degree than with the highest degrees. Subsequently, the overall abundance of the omnivore is greater on thethan thenetworks. The spatial distances in large worlds make more patches more hospitable to the omnivore than they make less hospitable, and therefore decrease the chance of global omnivore extinction.To summarise, the large path lengths innetworks mean that some regions of the graph are hospitable to the omnivore but inhospitable to the specialist. Therefore, the omnivore can colonise these regions without suffering regular competitive extinctions, and consequentially, occupy them a large fraction of the time. In other words, the omnivore can fill in gaps in the specialists dispersal, see Fig. <ref>b.§ CONCLUSIONS AND DISCUSSION We used agent-based simulations to investigate the dispersal of a foodweb, comprising predator-prey and competitive interactions, on spatially embedded patch networks.The abundance of primary producers and specialist consumers, which are only affected by predator-prey interactions, is lower on spatially embedded random geometric patch networks than non-spatial patch networks. Furthermore, in spatially embedded networks there is a greater variation between the fraction of time different patches are occupied by any species. By visualising patch abundance of specialists, we found that these variations correspond meso-scale patters of patch occupation. Regions dense with patches of high occupation are spatially separated from regions of patches with low occupation.A specialist consumer only occupies patches that its prey also occupies. In general its prey is also a specialist consumer. The local factors, such as high degree, that make a patch easy for the prey species to occupy it also make it easy for its predator to occupy. Further, spatial variations in prey occupations reduce the availability of patches for its predator in some regions of spatial networks. Consequently, the difference in the abundance of a predator between and random geometric networks is larger than the difference in the abundance of its prey.We assume that generalist species, such as omnivores, are weaker competitors than specialist species with the same prey¸ such that the generalist is driven to local extinction in any interaction. Subsequently, reduced overall specialist abundance in random geometric networks aids generalist persistence. Further, we identify that the meso-scale structure in spatially embedded networks aids the persistence of generalist species in three ways. Firstly, patches can have relatively high degrees but low specialist abundance due to network separation from the most occupied regions of the network and patches with higher degree are easier for the generalist to colonise. Secondly, patches with low specialist occupation are often grouped together and the generalist can persist in such a region without stochastic extinction. Thirdly, regions with low specialist occupation are separated by long distances from the regions of high specialist occupation and so the generalist populations there are rarely threatened by driven extinction. The combination of these factors leads to greater abundance of generalist species on random geometric networks, compared to networks.Two properties of the spatially embedded networks cause the significant regional variations in patch occupation: long distances between nodes and degree correlations. Previous work has indicated that long distances have a greater effect than degree correlations on dispersal processes,<cit.> and our results also support this conclusion. When regional variations are large, degree is a poor indicator of patch occupation, but when regional variations are small it is a good indicator. This suggests that when regional variations are larger, they are being primarily caused by long network distances and when regional variations are smaller we are seeing the effect only of degree correlations. In other words, long distances have a greater effect than degree correlations on species dispersal over spatial networks.Real patch landscapes are usually inherently spatial and these results suggest that structures resulting from this have many implications for environmental management. For example, our results demonstrate the contrasting results possible from measuring diversity at different spatial scales.<cit.>Further, the ability of a species to inhabit a patch is dependent on the structure of surrounding patches and long range effects. In situations, such as deforestation, where a managed removal of patches attempts to preserve habitat by preserving a single, well populated patch, these attempts may fail if the neighbourhood of that patch is destroyed such that recolonisation becomes more difficult. Alternatively, well intentioned environmental management to maintain a specialist species can destroy the spatial separation that allows omnivores to persist, sheltered from specialist populations. The spatial embedding of agents is a property of many networks found in ecology <cit.> and elsewhere. <cit.> However, dynamic process on networks incorporating spatial structure are relatively poorly understood. Most existing analytical approaches for dispersal processes on networks focus on local structure,<cit.>such as degree correlations. Our results show that local properties in spatial networks often have a smaller impact than long network distances. Though there has been some recent development of methods for considering spatial separation of nodes in networks,<cit.> further advances are required to characterise space in network models, and understand its effects on dynamic processes.One particular challenge for analytical approaches to spatial networks is how to characterise their spatial structure. The most popular model for generating spatial networks, using random geometric graphs, results in networks with several notable properties that make them different from random networks. For instance higher clustering coefficients, larger diameters, larger average path length and positive degree correlations.<cit.> These properties are not independent but are jointly caused by the spatial embedding. For example, nodes near each other are neighbours and the radius containing their neighbourhoods largely intersect, resulting in clustering and also degree correlations. Our results show that both long separations and local properties are simultaneously important in explaining the dynamical behaviour of systems on spatial networks. Hence,methods to better understand these systems should be capable of accounting for the combination of these properties.We finish by suggesting how this may be accomplished. It may be possible to approximate the regions of a spatial graph of homogeneous patches by a graph of heterogeneous meta-patches. The properties of each of these meta-patches would be constructed to reflect the region of the underlying spatial graph that the particular patch represents. A random geometric graph in this model would have meta-patches with properties drawn from a distribution determined by the formation processes of a random geometric graph. For example, each meta-patch could have a local extinction probability, based on the number of patches included and the number of links between them. The graph of meta-patches contains many fewer patches, and could be potentially be analysed using available numerical methods. The sensitivity of the model to meta-patch properties may be analysed to determine the systems sensitivity to the underlying properties of spatial patch networks that inform them. § DATA AVAILABILITYThis study did not involve any underlying data.§ ACKNOWLEDGEMENTSThis work was supported by the EPSRC under grant codes EP/N034384/1 and EP/I013717/1.§ ADDITIONAL INFORMATION There is no competing interest for any of the authors.§ AUTHOR CONTRIBUTIONAll authors contributed to and reviewed the manuscript. E.B. prepared the figures. | http://arxiv.org/abs/1706.08422v2 | {
"authors": [
"Edmund Barter",
"Thilo Gross"
],
"categories": [
"q-bio.PE",
"physics.bio-ph"
],
"primary_category": "q-bio.PE",
"published": "20170626145601",
"title": "Spatial effects in meta-food-webs"
} |
Current address: Division of Chemistry and Biological Chemistry, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, JapanCentre for Micro-Photonics, Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Hawthorn, VIC 3122, Australia [email protected] Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, JapanUsing polarization-resolved transient reflection spectroscopy, we investigate the ultrafast modulation of light interacting with a metasurface consisting of coherently vibrating nanophotonic meta-atoms in the form of U-shaped split-ring resonators, that exhibit co-localized optical and mechanical resonances. With a two-dimensional square-lattice array of these resonators formed of gold on a glass substrate, we monitor the visible-pump-pulse induced gigahertz oscillations in intensity of reflected linearly-polarized infrared probe light pulses, modulated by the resonators effectively acting as miniature tuning forks. A multimodal vibrational response involving the opening and closing motion of the split rings is detected in this way. Numerical simulations of the associated transient deformations and strain fields elucidate the complex nanomechanical dynamics contributing to the ultrafast optical modulation, and point to the role of acousto-plasmonic interactions through the opening and closing motion of the SRR gaps as the dominant effect. Applications include ultrafast acoustooptic modulator design and sensing. Gigahertz optomechanical modulation by split-ring-resonator nanophotonic meta-atom arrays O. B. Wright December 30, 2023 =========================================================================================Electromagnetic metamaterials are artificial media composed of arrays of unit structures much smaller than the optical wavelength, and can exhibit non-intuitive properties such as negative refractive index<cit.>, magnetism at optical frequencies<cit.> and cloaking<cit.>. Such behavior is determined by the electrical or magnetic character of the constitutent meta-atoms. Following on from the first demonstration of negative permittivity and permeability at microwave frequencies<cit.> in a hybrid split-ring resonator (SRR) and rod-array structure, a great deal of research on metamaterialshas been carried out up to optical frequencies, with potential applications in telecommunications and sensing techonologies, for example. Meta-atoms at optical frequencies have been proposed in many forms, such as three-dimensional (3D) fishnets<cit.>, cut-wire pair arrays<cit.> as well as the generic split-ring resonators (SRR)<cit.>, to name only a few. At such frequencies plasmonic effects play an essential role in determining the effective parameters. Another promising avenue for applications involves the use of active metamaterials for wave control<cit.>; at optical or infrared frequencies, for example, various methods of modulating the effective parameters of the metamaterial to this end have been demonstrated, such as photoswitching<cit.>, heating and cooling (i.e thermal control)<cit.>, the use of phase-change materials<cit.>, electrooptic materials<cit.>, or coupling with microelectromechanical systems<cit.>. In particular, in the case of SRR-based metamaterials, active control of their optical or infrared properties has been conducted by many groups using photoswitching or electrooptic control<cit.>, phase-change materials<cit.>, or mechanical deformations<cit.>, for example.A different route to such modulation is the use of phonons, which promise ultrahigh frequency operation and control. Ulbricht et al. <cit.> used GHz acoustic phonons to modulate the transmission of a metalayer consisting of an array of nanoholes in a gold film, and O'Brien et al.<cit.> made use of GHz acoustic phonons in nanoscale gold Swiss-cross arrays with different lengths of horizontal and vertical arms to modulate linearly polarized light. Shelton et al.<cit.> demonstrated the coupling between gold SRR-array infrared resonances and THz optical phonons in thin dielectric layers, thereby modulating the infrared transmission spectrum, and Dong et al.<cit.> reported the modulation of light with GHz acoustic phonons in U-shaped nanowire arrays.However, in spite of the intense interest in the fascinating properties of SRRs, to our knowledge GHz acoustic phonons have not been used to modulate the optical properties of SRR meta-atoms. In this paper we report on the GHz acoustic modulation of a metamaterial consisting of a 2D array of U-shaped nanoscale gold SRRs at near-infrared optical wavelengths using a combination of both experiment, based on a femtosecond polarization-resolved pump-probe technique, and numerical simulations of the transient deformation and strain fields.Our metamaterial consists of an array of sub-micron SRRs, as shown schematically in Figs. 1(a) and (b). The original Pendry-et-al.<cit.> SRRs were proposed to achieve effective magnetic permeability through electromagnetic resonances at GHz electromagnetic frequencies. The fundamental magnetic resonance of an SRR can be approximated as an LC circuit consisting of an inductance (L) and a capacitance (C). We have chosen U-shaped gold nanoscale SRRs of dimensions as shown in the inset of Fig. 1(c), so that their resonant frequency is in the near-infrared. Using the analytical LC model of Linden et al.<cit.>, we calculate this fundamental resonance to be at a wavelength of ∼1.5 μm. The SRR dimensions chosen, with reference to Fig. 1(b), are side l=230 nm, gap d=84 nm, bottom-width w=92 nm and thickness t=60 nm. They are arranged in a square lattice of pitch 326 nm, as shown by the electron micrograph in Fig. 1(b). The structures are patterned using electron-beam lithography and standard lift-off procedures<cit.>. A 0.5 mm thick slab of BK7 glass is used as a substrate, and a 2 nm Cr layer is incorporated to improve adhesion.We first characterize the SRR array by normal-incidence white-light optical transmission spectra for horizontal and vertical polarizations, as shown in Fig. 2 for both experiment (solid red lines) and finite-element method (FEM) electromagnetic simulations (COMSOL Multiphysics, solid green lines) using periodic boundary conditions and literature values of the refractive indicies<cit.>. We also show for reference the simulated reflection spectrum (dotted blue lines) and absorption spectrum (dashed orange lines). Details of the simulations are given in the Supplementary Material. The horizontal optical polarization configuration of Fig. 2(a) is expected from analytical considerations to excite the fundamental magnetic resonance at ∼1.5 μm for our sample, but this resonance is out of the measured wavelength range. However, for this optical polarization we find first and second orders of plasmonic resonance at 808 nm and 572 nm in the simulations, labeled (1) and (2), respectively, in the reflection spectrum, and experimental transmission dips are correspondingly observed at wavelengths closely matchingthose predicted in transmission. These resonances have been reported by other groups for similar U-shaped SRR arrays in the near infrared<cit.>. Figure 2(b) shows the equivalent spectra for vertical polarization. This polarization does not excite a circulating current component in the split ring owing to symmetry. The directions and strengths of current flow at selected locations at a given time for the representative plasmonic resonances, calculated from electromagnetic simulations (COMSOL Multiphysics), are also shown in Fig. 2. (See the Supplementary Material for details and plots.) There is in general very good overall agreement between the experimental and simulated transmission spectra for both probe polarizations.We use an optical pump-probe technique to generate GHz vibrational modes and to detect the modulated optical reflectance, as shown in Fig. 3(a). A mode-locked Ti:sapphire laser with a repetition rate of 80 MHz and optical pulse duration of ∼200 fs provides probe pulses at a wavelength of 800 nm as well as frequency-doubled pump pulses at a wavelength of 400 nm by use of a BBO (beta barium borate) crystal. At both 400 and 800 nm, gold is a good absorber with an optical absorption depth ∼15 nm<cit.>. The pump beam, chopped at 1 MHz by an acoustooptic modulator for lock-in detection, is focused onto the sample surface at normal incidence through a 20× objective lens to a spot diameter ∼6 μm FWHM (full width at half-maximum) with vertical linear polarization, as shown in Fig. 3(a) by the dashed (blue) arrow. Ultrafast electron diffusion in gold<cit.> rapidly transfers energy from the optical absorption depth to the whole depth of the SRR. The resulting stress field coherently excites the vibrational modes of ∼300 SRRs, corresponding to near k=0 acoustic wave vectors of the SRR array, which act as a phononic crystal. A single pump pulse has an energy of ∼50 pJ, which leads to acoustic strains up to ∼10^-4 or displacements of ∼20 pm, corresponding to transient temperature rises of ∼20 K. The probe pulses, of the same pulse energy, are passed through a motorized delay line, and are then focused at normal incidence to a similar spot size as the pump in order to monitor the transient reflectivity changes. The linear polarization of the probe beam is aligned alternatively horizontally and vertically, as shown in Fig. 3(a) (solid red arrows). The probe wavelength of 800 nm is conveniently set off the first plasmonic resonance of 780 nm for horizontal polarization (which produces electric-field localization in the SRR gap) to provide an enhanced sensitivity of the SRR reflectivity to deformation-induced changes in its geometry for this polarization. Variations in intensity ∼10^-4 of the probe beam reflected from the sample are monitored with a photodiode and a lock-in amplifer tuned to the chopping frequency.The measured relative reflectivity variations δ R(t)/R induced by the GHz vibrations in the SRR array are shown in Fig. 3(b) for horizontal (dotted line) and vertical (solid line) probe polarizations. At zero delay time, δR/R shows rapid changes owing to electronic excitation and subsequent heating of the gold. After that, a complex oscillatory damped variation in δ R/R is evident, ∼10 times larger in the horizontal polarization case than in the vertical. We attribute this difference to the much sharper plasmon resonance close to the probe wavelength λ for horizontal polarization compared to vertical (leading to a ∼10 times larger gradient |dR/dλ|—that governs the amplitude of the modulation in δ R/R when the plasmon resonance curve is shifted by strain or deformation—for horizontal polarization compared to vertical polarization at λ=800 nm).The corresponding moduli of the temporal Fourier transforms (|FT|) of δ R/R (after subtracting the thermal background) are shown on a normalized scale vs frequency in Fig. 3(c). In particular a main resonance is evident in both polarizations at 4.8 GHz, and smaller resonances in the horizontal polarization at 3.2, 6.5. 9.5, 13.5 and 20.5 GHz appear. (The amplitude of the vertical probe polarization data is too small to extract other resonances.) Notably absent for both polarizations are Brillouin oscillations, that can arise from probe light scattered from coherent longitudinal strain pulses launched into the substrate. (These oscillations are expected to be very close in frequency to 22 GHz<cit.>.) As we discuss in detail below, in the present experiment it turns out that plasmonic coupling dominates the detection process.In order to better understand these results, transient deformation and strain distributions in the SRR structure are calculated using frequency-domain FEM simulations (COMSOL, Multiphysics) with a mesh size of ∼8 nm, a time step of 1.0 ps, and a total calculation time of 12 ns. We implement periodic boundary conditions (BCs) with a unit cell consisting of a gold SRR (as defined by the experimental geometry) on a BK7 glass substrate section of dimensions 326 nm × 326 nm × 755 nm (ignoring the 2 nm Cr adhesion layer). Low-acoustic-reflection BCs are used over the bottom surface of the substrate. As an approximation to the thermoelastic excitation in gold, the whole SRR is subject to an initial isotropic stress that initiates an expansion. This assumption preserves the required left-right symmetry of the optical excitation in experiment with vertically-polarized normal optical incidence. In the simulations, literature values of longitudinal and shear sound velocities as well as densities of gold and BK7 are used: v_l = 3240 m/s, v_t = 1200 m/s, and ρ = 19300 kg/m^3 and v_l = 6050 m/s, v_t = 3680 m/s, and ρ = 2510 kg/m^3, respectively<cit.>. (The accuracy of the simulations were checked by using different discretizations.)In order to obtain some quantitative parameters that can characterize the SRR vibrational motion, we plot vs frequency in Fig. 4(a) the normalized modulus of the temporal FT of the calculated volumetric strain δ V/V (dashed purple line) and displacement U_x (solid green line), both averaged over half of the top surface of the SRR (i.e. exploiting the mirror symmetry), where x and y are the in-plane horizontal and vertical coordinates, respectively. For reference we also include the |FT| of the experimental reflectivity change δ R/R (dotted red line) in the plot for horizontal probe polarization. Individual tensile strain component vibrational spectra η_xx, η_yy and η_zz, averaged over half of the top surface of the SRR, are shown in Fig. 4(b), and all three displacement component spectra are shown in Fig. 4(c). In each case the relative values are accurately represented.Several resonant frequencies are revealed that are very close to the experimental values. Since on resonance U_x in the chosen frequency range is signficantly larger than U_y and U_z (see Fig. 4(c) noting the different vertical scales), we choose to reproduce its variation in Fig. 4(a) next to the experimental data. Deformations can contribute to the optical modulation in general in two ways: 1) by changing the probe plasmonic resonance frequency through, for example, a change in the SRR gap width; 2) by their role in changing the sample geometry, thus affecting the reflected angular distribution of the probe light, and leading to an intensity modulation owing to the finite optical solid angle defined by the collection optics. Effect 1) is likely to be more important in our SRR sample owing to the proximity of the horizontally-polarized probe wavelength to a plasmonic resonance. Likewise, strains can contribute to the optical modulation through strain-induced variations in the refractive index in the SRR or the substrate, although the photoelastic effect in gold at the 800 nm probe wavelength is known to be small, and has never been seen to give rise to any opto-acoustic interaction in experiments at such optical wavelengths<cit.>. However, since similar strains are coupled to the glass substrate, the light reflected from the substrate around the SRR could well contribute to the optical modulation. For this reason we have included the above-mentioned volumetric strain δ V/V=η_xx+η_yy+η_zz in the plot of Fig. 4(a) for comparison. However, since η_xx and η_yy show similar amplitudes in Fig. 4(b) (and the same being true for the uncovered glass regions of the unit cell, as verified by simulation), one would expect the photoelastic effect in the glass substrate to modulate δ R/R with a similar amplitude for both horizontal and vertical probe polarizations. This can be seen from the following definitions:δϵ_xx=P_11η_xx + P_12η_yy + P_12η_zz, δϵ_yy=P_12η_xx + P_11η_yy + P_12η_zz,where ϵ_ij are the dielectric constants and P_ij the photoelastic constants (P_12/P_11∼2 in glass<cit.>), showing that one expects the change δϵ_xx (probed by horizontal polarization) to be of the same order as δϵ_yy (probed by vertical polarization) when η_xx∼η_yy, a situation that leads to a similar value of δ R/R for the two probe polarizations. Since this is definitely not the case in experiment, one can conclude that it is likely to be the deformations of the SRR, i.e. particularly the component U_x, that dominate in the optical modulation. This conclusion is also consistent with the above-mentioned observed absence of Brillouin oscillations, which depend on the photoelastic effect. The simulated FT spectra of both δ V/V and U_x show easily recognizable resonances at 2.8, 4.5, 6.0, 10.0, 12.8 and 21.3 GHz. We have labeled as peaks (1)-(6) in Fig. 4(a) the ones that are close to maxima in experiment at 3.2, 4.8, 6.5. 9.5, 13.5 and 20.5 GHz, respectively, in the same figure. Whether a particular peak appears in experiment depends on the detailed thermoelastic coupling of the SRR to the substrate as well as on the specfic detection mechanism, so we do not expect the simulated peak heights to be the same as those in experiment; a comprehensive theory of the opto-acoustic nanoscale interaction is beyond the scope of this paper. The predicted peaks are much narrower than in experiment, presumably owing to variations in the SRR geometry (inhomogeneous broading) and to GHz ultrasonic attenuation in gold and in glass. The nature of these six labelled modes can be gleaned from the corresponding simulated deformations and strain distributions shown in Fig. 5. Resonance (1) corresponds to the fundamental tuning-fork-like mode, whereas modes (2)-(6) are of higher order. Their motion can be viewed as animations (see Supplementary Material), showing that all influence the SRR gap geometry. As prescribed by the reflection symmetry of the SRR in the bisector plane parallel to the y axis, all the modes extracted show this required left-right mirror symmetry, and appear to all intents and purposes to be independent modes of isolated SRRs. Concerning the large difference in experimental modulation amplitudes between horizontal and vertical probe polarizations, it is likely, as mentioned above, that plasmonic effects (i.e. perturbation of the enhanced E-fields) associated with the variation in the SRR gap, which are much more important for horizontal probe polarization, are mainly responsible. The tuning-fork-like vibration of the SRR “prongs” for the lower-order vibrational modes produces alternative blue-shifting (gap-opening) and red-shifting (gap-closing) of the plasmonic resonance, thus modulating the transient reflectivity<cit.>. To further investigate the origin of the vibrational modes dominant in the spectra for displacement and tensile strain, in the simulation we separately varied the SRR pitch a the thickness t around the chosen values, as shown in Fig. 6(a) and (b), respectively. The frequencies of mode (1)-(6) were independent of the pitch over the investigated range a = 326 ± 30 nm, confirming our conclusion above that these modes can be treated to a good approximation as individual resonances of a single SRR in spite of being k-near-zero collective modes of the phononic-crystal structure. In contrast to the variation with a, the frequencies of all six modes depend on thickness t over the investigated range t = 60 ± 10 nm, decreasing with increasing t owing to the increased mass loading. Plots of the mode frequencies against 1/t in Fig. 6(b) show that for modes (1) and (6) the variation is approximately ∝ 1/t, although we do not have a simple model to mimick this behaviour. Such variations of GHz mode frequency with nanoscale geometry have previously been investigated in phononic crystals (see, for example, Refs. sakuma2012vibrational and robillard2008collective). In conclusion, we have measured the ultrafast acoustooptic response of SRR nanophotonic meta-atoms for the first time by means of femtosecond polarization-resolved transient reflection spectroscopy in the near-infrared. Ultrafast changes in optical reflectivity arise from GHz acoustic vibrations of a square lattice of suboptical-wavelength size thin gold U-shaped SRRs attached to a glass substrate. Individual vibrational modes of the SRRs in the frequency range ∼3-20 GHz are clearly identified to be photoexcited by the optical pump pulses and to give rise to a modulation in reflection δ R that is much larger for linear probe polarizations aligned across the SRR gap (i.e. horizontally rather than vertically). Simulations of the complex transient deformations and strains in the sample give reasonable overall agreement with the six resonant frequencies observed in experiment. Our analysis suggests that in this structure it is the deformations that tend to open and close the SRR gaps that contribute most to δ R through acousto-plasmonic effects rather than through the photoelastic effect, demonstrating the importance of the optomechanical interaction of the enhanced localized electric field in the gap with the GHz gap variations. In future, it would be interesting to determine how the acoustic modulation affects the angular distribution of optical intensity scattered from the SRR structures, both in reflection and transmission. Moreover, by tailoring the geometry and choice of material for the SRR, for example by extending the structure in 3D<cit.>, it should be possible to enhance the acousto-plasmonic interaction, thereby opening the way for efficient ultrafast acoustic modulation using SRR meta-atoms. Our study also opens new vistas for the design of coherent phonon devices sensitive to variations in the phononic, electronic or thermal environment. We are grateful to Kentaro Fujita for stimulating discussions. We acknowledge Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) and well as support from the Japanese Society for the Promotion of Science (JSPS).47 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Shelby, Smith, and Schultz(2001)]shelby2001 author author R. A. Shelby, author D. R. Smith, and author S. Schultz, 10.1126/science.1058847 journal journal Science volume 292, pages 77 (year 2001)NoStop [Linden et al.(2006)Linden, Enkrich, Dolling, Klein, Zhou, Koschny, Soukoulis, Burger, Schmidt, and Wegener]linden2006photonic author author S. Linden, author C. Enkrich, author G. Dolling, author M. W. Klein, author J. Zhou, author T. Koschny, author C. M.Soukoulis, author S. Burger, author F. Schmidt, and author M. Wegener,@noopjournal journal IEEE J. Sel. Top. Quant. Electron. volume 12, pages 1097 (year 2006)NoStop [Schurig et al.(2006)Schurig, Mock, Justice, Cummer, Pendry, Starr, and Smith]schurig2006metamaterial author author D. Schurig, author J. Mock, author B. Justice, author S. A. Cummer, author J. B. Pendry, author A. Starr,and author D. Smith, @noopjournal journal Science volume 314, pages 977 (year 2006)NoStop [Smith et al.(2000)Smith, Padilla, Vier, Nemat-Nasser,and Schultz]Smith2000 author author D. R. Smith, author W. J. Padilla, author D. C. Vier, author S. C. Nemat-Nasser,andauthor S. Schultz, 10.1103/PhysRevLett.84.4184 journal journal Phys. Rev. Lett. volume 84, pages 4184 (year 2000)NoStop [Valentine et al.(2008)Valentine, Zhang, Zentgraf, Ulin-Avila, Genov, Bartal, andZhang]valentine2008three author author J. Valentine, author S. Zhang, author T. Zentgraf, author E. Ulin-Avila, author D. A. Genov, author G. Bartal,and author X. Zhang, @noopjournal journal Nature volume 455,pages 376 (year 2008)NoStop [Dolling et al.(2005)Dolling, Enkrich, Wegener, Zhou, Soukoulis, and Linden]dolling2005cut author author G. Dolling, author C. Enkrich, author M. Wegener, author J. Zhou, author C. M. Soukoulis,and author S. Linden, @noopjournal journal Opt. Lett. volume 30, pages 3198 (year 2005)NoStop [Enkrich et al.(2005)Enkrich, Wegener, Linden, Burger, Zschiedrich, Schmidt, Zhou, Koschny, and Soukoulis]enkrich2005magnetic author author C. Enkrich, author M. Wegener, author S. Linden, author S. Burger, author L. Zschiedrich, author F. Schmidt, author J. Zhou, author T. Koschny,and author C. Soukoulis, @noopjournal journal Phys. Rev. Lett. volume 95, pages 203901 (year 2005)NoStop [Zheludev and Kivshar(2012)]zheludev2012metamaterials author author N. I. Zheludev and author Y. S. Kivshar, @noopjournal journal Nat. Mater. volume 11, pages 917 (year 2012)NoStop [Dani et al.(2009)Dani, Ku, Upadhya, Prasankumar, Brueck, and Taylor]dani2009subpicosecond author author K. M. Dani, author Z. Ku, author P. C. Upadhya, author R. P. Prasankumar, author S. Brueck,and author A. J. Taylor, @noopjournal journal Nano Lett. volume 9, pages 3565 (year 2009)NoStop [Xiao et al.(2010)Xiao, Drachev, Kildishev, Ni, Chettiar, Yuan, and Shalaev]xiao2010loss author author S. Xiao, author V. P. Drachev, author A. V. Kildishev, author X. Ni, author U. K. Chettiar, author H.-K. Yuan,and author V. M. Shalaev, @noopjournal journal Nature volume 466,pages 735 (year 2010)NoStop [Liu and Padilla(2016)]liu2016thermochromic author author X. Liu and author W. J. Padilla, @noopjournal journal Adv. Mater. volume 28, pages 871 (year 2016)NoStop [Ou et al.(2016)Ou, Plum, Zhang, and Zheludev]ou2016giant author author J.-Y. Ou, author E. Plum, author J. Zhang,and author N. I. Zheludev, @noopjournal journal Adv. Mater. volume 28, pages 729 (year 2016)NoStop [Pang et al.(2017)Pang, Wang, Cheng, Xia, Zhou, Xu, Cui, andQu]pang2017thermally author author Y. Pang, author J. Wang, author Q. Cheng, author S. Xia, author X. Y. Zhou, author Z. Xu, author T. J.Cui,and author S. Qu, @noopjournal journal Appl. Phys. Lett. volume 110, pages 104103 (year 2017)NoStop [Sautter et al.(2015)Sautter, Staude, Decker, Rusak, Neshev, Brener, and Kivshar]sautter2015active author author J. Sautter, author I. Staude, author M. Decker, author E. Rusak, author D. N. Neshev, author I. Brener,and author Y. S. Kivshar, @noopjournal journal ACS Nano volume 9,pages 4308 (year 2015)NoStop [Yin et al.(2015)Yin, Schaäferling, Michel, Tittl, Wuttig, Taubner, and Giessen]yin2015active author author X. Yin, author M. Schäferling, author A.-K. U.Michel, author A. Tittl, author M. Wuttig, author T. Taubner,andauthor H. Giessen, @noopjournal journal Nano Lett. volume 15, pages 4255 (year 2015)NoStop [Yao et al.(2014)Yao, Shankar, Kats, Song, Kong, Loncar, and Capasso]yao2014electrically author author Y. Yao, author R. Shankar, author M. A. Kats, author Y. Song, author J. Kong, author M. Loncar,and author F. Capasso, @noopjournal journal Nano Lett. volume 14, pages 6526 (year 2014)NoStop [Valente et al.(2015)Valente, Ou, Plum, Youngs,and Zheludev]valente2015magneto author author J. Valente, author J.-Y. Ou, author E. Plum, author I. J. Youngs,and author N. I. Zheludev, @noopjournal journal Nat. Commun. volume 6, pages 7021 (year 2015)NoStop [Gutruf et al.(2015)Gutruf, Zou, Withayachumnankul, Bhaskaran, Sriram, and Fumeaux]gutruf2015mechanically author author P. Gutruf, author C. Zou, author W. Withayachumnankul, author M. Bhaskaran, author S. Sriram,and author C. Fumeaux, @noopjournal journal ACS Nano volume 10, pages 133 (year 2015)NoStop [Ee and Agarwal(2016)]ee2016tunable author author H.-S. Ee and author R. Agarwal,@noopjournal journal Nano Lett.volume 16, pages 2818 (year 2016)NoStop [Lee et al.(2012)Lee, Choi, Kim, Lee, Liu, Yin, Choi, Lee, Choi, Choi, Zhang, andMin]lee2012switching author author S. H. Lee, author M. Choi, author T.-T. Kim, author S. Lee, author M. Liu, author X. Yin, author H. K. Choi, author S. S. Lee, author C.-G. Choi, author S.-Y. Choi, author X. Zhang,and author B. Min, @noopjournal journal Nat. Mater. volume 11, pages 936 (year 2012)NoStop [Gu et al.(2012)Gu, Singh, Liu, Zhang, Ma, Zhang, Maier, Tian, Azad, Chen, Taylor, Han, and Zhang]gu2012active author author J. Gu, author R. Singh, author X. Liu, author X. Zhang, author Y. Ma, author S. Zhang, author S. A. Maier, author Z. Tian, author A. K. Azad, author H.-T. Chen, author A. J. Taylor, author J. Han,and author W. Zhang, @noopjournal journal Nat. Commun. volume 3, pages 1151 (year 2012)NoStop [Liu et al.(2015)Liu, Luxmoore, Mikhailov, Savostianova, Valmorra, Faist, andNash]liu2015highly author author P. Q. Liu, author I. J. Luxmoore, author S. A. Mikhailov, author N. A. Savostianova, author F. Valmorra, author J. Faist,and author G. R. Nash, @noopjournal journal Nat. Commun. volume 6, pages 8969 (year 2015)NoStop [Meng, Zhong, and Zhang()]meng2017hybrid author author Q. Meng, author Z. Zhong,andauthor B. Zhang, @noopjournal journal Sci. Rep. volume 7, pages 45708NoStop [Samson et al.(2010)Samson, MacDonald, De Angelis, Gholipour, Knight, Huang, Di Fabrizio, Hewak, and Zheludev]samson2010metamaterial author author Z. Samson, author K. MacDonald, author F. De Angelis, author B. Gholipour, author K. Knight, author C. Huang, author E. Di Fabrizio, author D. Hewak,and author N. Zheludev, @noopjournal journal Appl. Phys. Lett. volume 96,pages 143105 (year 2010)NoStop [Kodama and Coutu Jr(2016)]kodama2016tunable author author C. Kodama and author R. Coutu Jr, @noopjournal journal Appl. Phys. Lett. volume 108, pages 231901 (year 2016)NoStop [Pryce et al.(2010)Pryce, Aydin, Kelaita, Briggs, andAtwater]pryce2010highly author author I. M. Pryce, author K. Aydin, author Y. A. Kelaita, author R. M. Briggs,and author H. A. Atwater, @noopjournal journal Nano Lett. volume 10, pages 4222 (year 2010)NoStop [Ou et al.(2011)Ou, Plum, Jiang, and Zheludev]ou2011reconfigurable author author J.-Y. Ou, author E. Plum, author L. Jiang,and author N. I. Zheludev, @noopjournal journal Nano Lett. volume 11, pages 2142 (year 2011)NoStop [Pitchappa et al.(2016)Pitchappa, Manjappa, Ho, Qian, Singh, Singh, and Lee]pitchappa2016active author author P. Pitchappa, author M. Manjappa, author C. P. Ho, author Y. Qian, author R. Singh, author N. Singh,and author C. Lee, @noopjournal journal Appl. Phys. Lett. volume 108, pages 111102 (year 2016)NoStop [Ulbricht et al.(2017)Ulbricht, Sakuma, Imade, Otsuka, Tomoda, Matsuda, Kim, Park, and Wright]ulbricht2017 author author R. Ulbricht, author H. Sakuma, author Y. Imade, author P. H. Otsuka, author M. Tomoda, author O. Matsuda, author H. Kim, author G. Park,and author O. B. Wright, @noopjournal journal Appl. Phys. Lett. volume 110, pages 091910 (year 2017)NoStop [O'Brien et al.(2014)O'Brien, Lanzillotti-Kimura, Rho, Suchowski, Yin, and Zhang]o2014ultrafast author author K. O'Brien, author N. Lanzillotti-Kimura, author J. Rho, author H. Suchowski, author X. Yin,and author X. Zhang, @noopjournal journal Nat. Commun. volume 5, pages 4042 (year 2014)NoStop [Shelton et al.(2011)Shelton, Brener, Ginn, Sinclair, Peters, Coffey, andBoreman]shelton2011strong author author D. J. Shelton, author I. Brener, author J. C. Ginn, author M. B. Sinclair, author D. W. Peters, author K. R. Coffey,and author G. D. Boreman, @noopjournal journal Nano Lett. volume 11, pages 2104 (year 2011)NoStop [Dong et al.(2016)Dong, Chen, Zhou, Wang, Zhang, and Sun]dong2015 author author B. Dong, author X. Chen, author F. Zhou, author C. Wang, author H. F. Zhang,and author C. Sun, @noopjournal journal Nano Lett. volume 16, pages 7690 (year 2016)NoStop [Pendry et al.(1999)Pendry, Holden, Robbins, and Stewart]pendry1999magnetism author author J. B. Pendry, author A. J. Holden, author D. Robbins,andauthor W. Stewart, @noopjournal journal IEEE Trans. Microw. Theory Techn. volume 47, pages 2075 (year 1999)NoStop [Gervinskas et al.(2013)Gervinskas, Seniutinas, Rosa, andJuodkazis]Gervinskas2013 author author G. Gervinskas, author G. Seniutinas, author L. Rosa, and author S. Juodkazis,@noopjournal journal Adv. Opt. Mater.volume 1, pages 456 (year 2013)NoStop [Hagemann, Gudat, and Kunz(1975)]hagemann1975optical author author H.-J. Hagemann, author W. Gudat, and author C. Kunz, @noopjournal journal J. Opt. Soc. Am. volume 65, pages 742 (year 1975)NoStop [Sch()]Schott @noopnote http://www.schott.com/d/advanced_optics/ac85c64c-60a0-4113-a9df-23ee1be20428/1.1/schott-optical-glass-collection-datasheets-english-17012017.pdfNoStop [Rockstuhl et al.(2006)Rockstuhl, Zentgraf, Guo, Liu, Etrich, Loa, Syassen, Kuhl, Lederer, and Giessen]rockstuhl2006resonances author author C. Rockstuhl, author T. Zentgraf, author H. Guo, author N. Liu, author C. Etrich, author I. Loa, author K. Syassen, author J. Kuhl, author F. Lederer,andauthor H. Giessen, @noopjournal journal Appl. Phys. B volume 84, pages 219 (year 2006)NoStop [Corrigan et al.(2008)Corrigan, Kolb, Sushkov, Drew, Schmadel, and Phaneuf]corrigan2008optical author author T. Corrigan, author P. Kolb, author A. Sushkov, author H. Drew, author D. Schmadel,and author R. Phaneuf, @noopjournal journal Opt. Express volume 16, pages 19850 (year 2008)NoStop [Wright(1994)]wright1994ultrafast author author O. B. Wright, @noopjournal journal Phys. Rev. B volume 49, pages 9985 (year 1994)NoStop [Lide(2004)]CRC editor D. R. Lide, ed.,@nooptitle CRC Handbook of Chemistry and Physics, edition 85th ed. (publisher CRC Press, address Boca Raton, Fla., year 2004)NoStop [Garfinkel, Tiemann, andEngeler(1966)]garfinkel1966piezoreflectivity author author M. Garfinkel, author J. Tiemann,and author W. Engeler,@noopjournal journal Phys. Rev.volume 148, pages 695 (year 1966)NoStop [He et al.(2017)He, Ristow, Grossmann, Brick, Guo, Schubert, Hettich, Gusev, and Dekorsy]he2017acoustic author author C. He, author O. Ristow, author M. Grossmann, author D. Brick, author Y. Guo, author M. Schubert, author M. Hettich, author V. Gusev,and author T. Dekorsy, @noopjournal journal Phys. Rev. B volume 95, pages 184302 (year 2017)NoStop [Dixon(1967)]dixon1967photoelastic author author R. Dixon, @noopjournal journal J. Appl. Phys. volume 38, pages 5149 (year 1967)NoStop [Huang et al.(2017)Huang, Qian, Jain, and El-Sayed]huang2007 author author W. Huang, author W. Qian, author P. K. Jain,andauthor M. A. El-Sayed,@noopjournal journal Nano. Lett.volume 10, pages 3227 (year 2017)NoStop [Sakuma et al.(2012)Sakuma, Tomoda, Otsuka, Matsuda, Wright, Fukui, Tomioka, andVeres]sakuma2012vibrational author author H. Sakuma, author M. Tomoda, author P. H. Otsuka, author O. Matsuda, author O. B. Wright, author T. Fukui, author K. Tomioka,and author I. A. Veres, @noopjournal journal Appl. Phys. Lett. volume 100, pages 131902 (year 2012)NoStop [Robillard et al.(2008)Robillard, Devos, Roch-Jeune, andMante]robillard2008collective author author J.-F. Robillard, author A. Devos, author I. Roch-Jeune,andauthor P. Mante, @noopjournal journal Phys. Rev. B volume 78, pages 064302 (year 2008)NoStop [Liu et al.(2008)Liu, Guo, Fu, Kaiser, Schweizer, and Giessen]liu2008three author author N. Liu, author H. Guo, author L. Fu, author S. Kaiser, author H. Schweizer,and author H. Giessen, @noopjournal journal Nat. Mater. volume 7, pages 31 (year 2008)NoStop§ SUPPLEMENTARY MATERIAL§ OPTICAL 3D FINITE-ELEMENT MODELLINGThe optical reflectance, transmittance and absorption spectra of the sample for normal incidence are simulated with a frequency-domain finite-element method (COMSOL Multiphysics) to solve the Maxwell equations. The unit cell consists of a BK7 glass substrate, a gold split-ring resonator (SRR) and the vacuum above them, as shown in Fig. S1. The SRR dimensions chosen, with reference to Fig. 1(a) of the main text, are side l=230 nm, gap d=84 nm, bottom-width w=92 nm and thickness t=60 nm. We implement periodic boundary conditions (BCs) with an unit cell of a gold SRR set on the interface between the vacuum and the BK7 glass substrate. S-parameter ports are used over two parallel planes, termed the input and output planes, defined to be 400 nm from this interface, and optical scattering BCs, set 100 nm outside these ports, are also used, as shown in Fig. S1.A total of approximately 1,150,000 tetrahedral mesh elements are used for the discretization, of maximum side 63 nm in the vacuum, maximum 42 nm in the BK7 substrate and maximum 15 nm in the gold SRR. (These mesh sizes are six times smaller than the minimum simulated optical wavelength in each medium.) A finer mesh was used near the corners in the gold SRR, where the electric fields are particularly enhanced. The refractive indices of gold and BK7 glass are taken from the literature<cit.>Simulations were performed independently with electric field E polarized along the x axis (horizontal polarization) and with E polarized the y axis (vertical polarization).The simulated reflectance, transmittance, and absorption for the two polarizations are plotted in Fig. 2(a) and (b) of the main text, respectively. Vector E-field and current density J plots at the optical wavelengths corresponding to the three peaks identified in Fig. 2(a) and (b) of the main text are shown in Fig. S2. 1 url<#>1urlprefixURLSCHOTT notehttp://www.schott.com/d/advanced_optics/ac85c64c-60a0-4113-a9df-23ee1be20428/1.1/schott-optical-glass-collection-datasheets-english-17012017.pdf.hagemann1975optical authorHagemann, H.-J., authorGudat, W. & authorKunz, C. journalJ. Opt. Soc. Am. volume65, pages742–744 (year1975). | http://arxiv.org/abs/1706.08909v1 | {
"authors": [
"Y. Imade",
"R. Ulbricht",
"M. Tomoda",
"O. Matsuda",
"G. Seniutinas",
"S. Juodkazis",
"O. B. Wright"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170627153008",
"title": "Gigahertz optomechanical modulation by split-ring-resonator nanophotonic meta-atom arrays"
} |
-0.5cm 0cm -0cm 22.5cm 16.5cmplain theoremTheorem proposition[theorem]Proposition corollary[theorem]Corollary lemma[theorem]Lemma conjecture[theorem]Conjecturedefinition remark[theorem]Remark notation[theorem]Notation definition[theorem]Definition question[theorem]Question problem[theorem]Problem example[theorem]Exampleí ő ű →[ ℝ ℤ ℕ 𝔼 ℂ 11 𝒫 𝒱 ℰ 𝒵 𝒩 ℒε[1]#1Mixing time of an unaligned Gibbs sampler on the square Balázs GerencsérB. Gerencsér is with the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences and the ELTE Eötvös Loránd University, Department of Probability Theory and Statistics, [email protected]. His work is supported by NKFIH (National Research, Development and Innovation Office) grant PD 121107.December 30, 2023 ====================================================================================================================================================================================================================================================================================================================================================== The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square [0,1]^2 to approach a stationary distribution with density proportional to exp(-A^2(u-v)^2) for (u,v)∈ [0,1]^2 with some large parameter A.Diaconis conjectured the mixing time of this process to be O(A^2) which we confirm in this paper. This improves on the currently known O(exp(A^2)) estimate.§ INTRODUCTION A standard use of Markov chains is to sample from a probability distribution that would be otherwise hard to access. This can happen when the distribution is supported on a set implicitly defined by some constraints, e.g., a convex body in a high dimensional space <cit.>, <cit.>, proper colorings of a graph <cit.>, <cit.>, etc. Several frameworks have been designed to achieve this goal including the Metropolis algorithm and the Gibbs sampler and their variants.There is a vast range of applications and studies, we refer the reader to <cit.>, <cit.> for orientation.A central and recurring question is the efficiency of these algorithms in the different settings. We highlight two phenomena that can decrease the performance of such algorithms. First, the incremental change the Markov chain allows is usually quite rigid and given by the structure of the state space. However, the desired stationary distribution does not need to be aligned with the directions where the Markov chain mixes fast. Second, some boundary effects might occur if the Markov chain can get trapped in some remote part of the state space.In this paper we analyze an example of the Gibbs sampling procedure proposed by Diaconis which is surprisingly simple considering it captures both of the two phenomena above. We call the coordinate Gibbs sampler for the diagonal distribution the following process. Fix a large positive constant A and on [0,1]^2 define the distribution π with density proportional to exp(-A^2(u-v)^2) for (u,v)∈ [0,1]^2. At each step randomly choose coordinate u or v and rerandomize it according to the conditional distribution of π.Notice that the distribution of this Markov chain is mostly concentrated near the diagonal of the unit square, while only horizontal and vertical transitions are allowed.Furthermore, near (0,0) and (1,1) we see that both the high density of π and also the boundaries of the square hinder the movement of the chain.The efficiency of the algorithm is quantified by the mixing time of the Markov chain. For any Markov chain X(0),X(1),… on some state space Ω (which is [0,1]^2 in our case) let (X(t)) denote the distribution of the state at time t and η be the stationary distribution assuming it is unique (denoted by π for our case). Using the total variation distance between measures, ρ-σ_ TV := sup_S⊆Ω |ρ(S)-σ(S)| we define the mixing time ast_ mix(X,):=sup_X(0)∈Ωmin{t:(X(t))-η_ TV≤}. Diaconis conjectured that the mixing time of the example proposed is O(A^2), the goal of this paper is to confirm this bound.Let X(t) follow the coordinate Gibbs sampler for the diagonal distribution. For any 0< α <1 there exists β > 0 such that for large enough A,t_ mix(X, α) ≤β A^2. Up until now only O(exp(A^2)) was known which easily follows from a minorization condition of the transition kernel.Observe that the diagonal nature of the distribution plays an important role in the mixing behavior, making the distribution and the randomization steps unaligned. If we took the distribution with density proportional to exp(A^2(u - 1/2)^2) for (u,v)∈ [0,1]^2, then the mixing time would decrease to be O(1). Indeed, this is a product distribution, product of one for u and one (uniform) for v, consequently after a rerandomization is performed along both coordinates, the distribution of the process will exactly match the prescribed one. This will happen with probability arbitrarily close to 1 within a corresponding finite number of steps, not depending on the value of A.The rest of the paper is organized as follows. In Section <ref> a formal definition of the process of interest is provided and further variants are introduced that help the analysis. Section <ref> provides the building blocks for the proof, to understand the short-term behavior of the process based on the initialization. Afterwards, the proof of Theorem <ref> is aggregated in Section <ref>. Finally, a complementing lower bound demonstrating that Theorem <ref> is essentially sharp is given in Section <ref> together with some numerical simulations.§ PRELIMINARIES, ALTERNATIVE PROCESSESWe now formally define the coordinate Gibbs sampler for the diagonal distribution which we denote by X(t), then we introduce variants that will be more convenient to handle.Let φ(x):=exp(-A^2x^2) for some large A>0 and let π be the probability distribution on [0,1]^2 with density ^-1φ(u-v) at (u,v)∈ [0,1]^2 (where =∫_[0,1]^2φ(u-v)). We write π(·,v) for the conditional distribution of the u coordinate when v is fixed (similarly for π(u,·)). Denote by π_u the projection of π, that is, the overall distribution of the u coordinate. When defining the coordinate Gibbs sampler for the diagonal distribution, we separate the decision of the direction of randomization and the randomization itself. For t=1,2,… let r(t) be an i.i.d. sequence of variables of characters U,V taking both with probability 1/2.Given some initial point X(0)∈ [0,1]^2 the random variable X(t) = (X_u(t),X_v(t)) is generated as a Markov chain from X(t-1) by randomizing along the axis given by r(t). Formally,X(t) := (u^+,X_v(t-1)), ifr(t)=U, whereu^+∼π(·,X_v(t-1)),(X_u(t-1), v^+), ifr(t)=V, wherev^+∼π(X_u(t-1),·), ,where u^+, v^+ are conditionally independent of the past at all steps. Note that when multiple U's follow each other in the series r(t) (similarly for V), the values u^+ are repeatedly overwritten and forgotten, with no further mixing happening for the overall distribution. Therefore we define an alternative process where this effect does not occur, but rather the directions of randomization are deterministic.Let X^*(0):=X(0), then the following process is generated:X^*(2s):= (u^+,X^*_v(2s-1)),whereu^+∼π(·,X^*_v(2s-1)),X^*(2s+1):= (X^*_u(2s), v^+), wherev^+∼π(X^*_u(2s),·).It would be convenient for the analysis if it wasn't necessary to distinguish the steps based on the parity of the time index.For that reason, consider the following modification. At every even step take X^*(2s) as before, at every odd step take X^*(2s+1) flipped along the diagonal of the square (exchange the two coordinates). Equivalently, flip the process at every step while generating. As a result, the randomization happens in the same direction at every step.Note that the target distribution π is symmetric along the diagonal therefore no adjustment is needed for the flipping. Formally, the process described is the following: Let Y(0):=X(0), then the random variables Y(t) are generated from Y(t-1) as followsY(t) := (u^+,Y_u(t-1)), whereu^+ ∼π(·,Y_u(t-1)).Observe that the scalar process Y_u(t) is a Markov chain by itself simply because Y(t) depends on Y(t-1) only through Y_u(t-1). § DYNAMICS OF Y_U(T)In this section we prove two properties of the evolution of Y_u(t), which will be the key elements to compute the mixing time bounds. First, we show that the process cannot stay arbitrarily long at the sides of the unit interval, in [0,1/2-δ) or (1/2+δ,1], where some small enough parameter δ>0 will be chosen. Second, we prove that starting from a point in the middle part [1/2-δ,1/2+δ], the distribution of the process quickly approaches the stationary distribution. §.§ Reaching the middleWe work on the case when the Y_u(0) is away from the middle of [0,1].We want to ensure that the process does not stay near the boundaries for a long period. To quantify this, the time to reach the middle is defined as follows: Let ν_m := min{s: Y_u(s) ∈[1/2-δ, 1/2+δ]}. Without the loss of generality we may assume that Y_u(0) is on the left part of [0,1], thanks to the symmetry of π w.r.t. (1/2, 1/2). Therefore we start from Y_u(0)<1/2-δ.For this period before reaching the middle we introduce a slightly simplified process Y', where both coordinates are allowed to take values in [0,∞) in principle. This is not supposed to have a substantially different behavior, but will allow more convenient analytic investigation as fewer boundaries are present.For any v∈ let σ_v be the measure on [0,∞) with density proportional to φ(u-v) conditioned on u∈ [0,∞). Let Y'(0):=X(0), then define the Markov chain Y'(t) as follows:Y'(t) := (u^+,Y'_u(t-1)), whereu^+ ∼σ_Y'_u(t-1). We can generate Y'(t) to be coupled to Y(t) as long as possible. For a fixed v, π(u,v) is proportional to φ(u-v) conditioned on u∈ [0,1]. Therefore, when we need to generate u^+ we draw a random sample from σ_Y'_u(t-1) and use it for both Y(t) and Y'(t) if u^+<1. Otherwise, we use it for Y'(t) but for Y(t) we draw a new independent sample from π(·,Y'_u(t-1)). It is easy to verify this is overall a valid method for generating a random variable of distribution π(·,Y'_u(t-1)).In the latter case, we also signal decoupling by setting a stopping time ν_c^1=t. We show this rarely happens, when governed by a variant of ν_m. Let ν̃_m := min{s: Y_u(s) ≥ 1/2-δ}.For any α_1>0 there is β_1>0 such that P(ν_c^1 < min(ν̃_m, α_1 A^2)) = O(exp(-β_1 A^2)). We want to bound the probability of decoupling at every point in time.When u^+ is drawn, Y'_u(t-1)<1/2-δ is ensured as ν̃_m has not yet occurred. For any v<1/2-δ we have σ_v({u^+>1})≤ 2 P(u > 1, u∼(v,1/(2A^2)))≤ 2exp(-A^2(1/2+δ)^2)/2√(π)A(1/2+δ).Here we use that the conditional probability is at most twice the unconditional one (because of v≥ 0), use the monotonicity in v, then apply a standard tail probability estimate for the Gaussian distribution.These exceptional events may occur at most at α_1 A^2 different times, therefore by using the union bound the overall probability is O(exp(-β_1 A^2)) for any β_1< (1/2+δ)^2. There exists β_2>0 constant such that P(ν_m ≠ν̃_m) = O(exp(-β_2 A^2). By a similar argument as above this bad event {ν_m ≠ν̃_m} happens when Y_u(t-1)<1/2-δ but Y_u(t)>1/2+δ when ν̃_m occurs, then a Gaussian tail probability estimate gives an upper bound of2exp(-A^2(2δ)^2)/2√(π)A(2δ).The lemma holds with β_2=(2δ)^2. Handling Y'(t) is still challenging due to the conditional distributions included in the definition. Therefore we introduce the following process that will be both convenient to handle and to relate to Y'(t). Let Z̃(t) be a random walk with i.i.d.(0,1/(2A^2)) increments, starting from Z̃(0) := X_u(0).Let Z(t) := |Z̃(t)|. Let us denote by ϕ the distribution of the centered Gaussian with variance 1/(2A^2). During the analysis of Z(t) we will also need to use the distribution of the absolute value of a Gaussian distribution with variance 1/(2A^2). We denote it by ϕ_x when the original one is centered at x and it is easy to verify that we can express it for any A⊂ [0,∞) by ϕ_x(A) = ϕ(A-x)+ϕ(-A-x).Z(t) and Y'_u(t) can be coupled such that Z(t)≤ Y'_u(t) for all t≥ 0.At 0 we have Z(0) = Y'_u(0). We construct the coupling iteratively, assuming Z(t-1) ≤ Y'_u(t-1) we perform the next step of the coupling which will satisfy Z(t) ≤ Y'_u(t).We will use the monotone coupling between the two. For two probability distributions ρ,ρ' the monotone coupling is the one assigning x to x' when ρ((-∞,x])=ρ'((-∞,x']). (We now skip currently irrelevant technical details about continuity, etc.). It is easy to verify that x≤ x' is maintained through this coupling exactly if ρ((-∞,y]) ≥ρ'((-∞,y]) for all y. In our case we will need the following:For any v≥v̅≥ 0 and u≥ 0:ϕ_v̅([0,u])≥σ_v([0,u]). Here v̅ corresponds to Z(t-1) and v to Y'_u(t-1) and we compare the distributions for step t. We are going to prove the following two inequalities:ϕ_v̅([0,u])≥ϕ_v([0,u]), ϕ_v([0,u])≥σ_v([0,u]). For the first of the two we compute ∂_v ϕ_v([0,u]):∂_v ϕ_v([0,u]) = ∂_v(ϕ([-v-u,-v+u])) = ∂_v (1/∫_-∞^∞φ∫_-v-u^-v+uφ) = 1/∫_-∞^∞φ(-φ(-v+u)+φ(-v-u))≤ 0.This last inequality holds because |-v+u|≤ |-v-u| and φ(x) is decreasing in |x|. Consequently, when v̅ is increased to v, the measure of [0,u] decreases confirming the first inequality.This intuitively means that when a Gaussian distribution is shifted to the right then even the reflected Gaussian is shifted (if it was centered at a non-negative point).The second inequality to confirm is the following:ϕ_v([0,u]) = ϕ([-v-u,-v+u]) ≥σ_v([0,u]). We rearrange and cancel out as much as possible from the domain of integrations.. ∫_-v-u^-v+uφ/ ∫_-∞^∞φ.≥. ∫_-v^-v+uφ/ ∫_-v^∞φ. ∫_-v-u^-v+uφ·∫_-v^∞φ ≥∫_-v^-v+uφ·∫_-∞^∞φ (∫_-v-u^-vφ + ∫_-v^-v+uφ) ·∫_-v^∞φ ≥∫_-v^-v+uφ·(∫_-∞^-vφ + ∫_-v^∞φ) ∫_-v-u^-vφ·∫_-v^∞φ ≥∫_-v^-v+uφ·∫_-∞^-vφ ∫_-v-u^-vφ·(∫_-v^-v+uφ + ∫_-v+u^∞φ)≥∫_-v^-v+uφ·(∫_-∞^-v-uφ + ∫_-v-u^-vφ) ∫_-v-u^-vφ·∫_-v+u^∞φ ≥∫_-v^-v+uφ·∫_-∞^-v-uφ We substitute the functions to integrate and transform them to compare them on the same domain.∫_-v-u^-ve^-A^2x^2dx ·∫_-v+u^∞ e^-A^2y^2dy≥∫_-v^-v+ue^-A^2x^2dx ·∫_-∞^-v-u e^-A^2y^2dy ∫_0^ue^-A^2(x+v)^2dx ·∫_u^∞ e^-A^2(y-v)^2dy≥∫_0^ue^-A^2(x-v)^2dx ·∫_u^∞ e^-A^2(y+v)^2dy ∫_0^u∫_u^∞ e^-A^2(x^2+y^2+2v^2 + 2v(x-y)) dy dx≥∫_0^u∫_u^∞ e^-A^2(x^2+y^2+2v^2 - 2v(x-y)) dy dx On all the domain of integration we have x≤ y. Therefore the exponent is larger at every point for the left hand side, which confirms the second inequality, completing the proof of the lemma. Lemma <ref> thus ensures that the monotone coupling preserves the ordering, and we can indeed use the recursive coupling scheme while keeping Z(t)≤ Y'_u(t) at every step.For any α_3>0 there exists β_3 > 0 with the following. For large enough A with probability at least 1-α_3 we have ν_m < β_3 A^2. First we look at the hitting time analogous to ν̃_m for Y'_u defined as ν̂_m = min{s: Y'_u(s) ≥ 1/2-δ}. Without aiming for tight estimates ν̂_m≤ t can be ensured by Y'_u(t)≥ 1/2-δ and by Proposition <ref> this holds whenever Z(t)≥ 1/2-δ. The latter is equivalent to Z̃(t)∉ [-1/2+δ,1/2-δ].For some β_3 >0, the distribution of Z̃(β_3A^2) is (X_u(0),β_3/2). Choosing β_3 large enough, the probability of this falling into [-1/2+δ,1/2-δ] can be made below α_3/2 and this event is a superset of ν̂_m>β_3A^2.Now apply Lemma <ref> with α_1 = β_3. Note that ν̃_m ≠ν̂_m can only happen if ν_c^1 <ν̃_m. Also Lemma <ref> ensures that ν_m and ν̃_m almost always coincide. Altogether, we have ν_m = ν̃_m = ν̂_m < β_3 A^2 with an exceptional probability at most O(exp(-β_2 A^2)) + O(exp(-β_1 A^2)) + α_3/2, this stays below α_3 when A is large enough, which completes the proof.§.§ Diffusion from the middleIn the previous subsection we have seen that the process Y_u(t) eventually has to reach the middle of the interval [0,1] as formulated in Proposition <ref>. Now we complement the analysis and consider the case when the process is initialized from the middle, meaning Y_u(0) ∈ [1/2-δ, 1/2+δ].Intuitively, we expect the process to evolve as a random walk with independent Gaussian increments. However, we have to be careful as boundary effects might alter the behavior of Y_u(t) when it moves near the ends of the interval [0,1].In this subsection we provide the techniques to estimate these boundary effects which will allow to conclude that the mixing of a random walk still translates to comparable mixing of Y_u(t).Let W(t) be a random walk with i.i.d.(0,1/(2A^2)) increments, starting from W(0) := Y_u(0).Our goal is to couple W(t) with Y_u(t) which only has a chance as long as W(t) stays within [0,1].Let ν_c^2 := min{s:W(s)∉ [0,1]}.There exist a coupling of the processes Y_u and W such that Y_u(t)=W(t) whenever t < ν_c^2. Assume the coupling holds until t-1, having Y_u(t-1)=W(t-1). Let ζ∼(0,1/(2A^2)) be independent from the past, then define W(t) = W(t-1)+ζ. For Y_u(t), accept Y_u(t-1)+ζ if it is in [0,1] otherwise redraw it according to π(·,Y_u(t-1)).The same values are obtained for the two processes at t except if W(t) is outside [0,1]. This is exactly the event we wanted to indicate with ν_c^2 when we allow the two processes to decouple. For any α_4 > 0 there exists β_4>0 with the following property. For A large enough, if Y_u(0)∈ [1/2-δ, 1/2+δ] there holds P(ν_c^2 < α_4A ^2) < β_4. We also have β_4 0 as we choose α_4 0. We need to control the minimum and the maximum of a random walk where we use the following result of Erds and Kac <cit.>: Let ξ_1,ξ_,… i.i.d. random variables, ξ_k = 0, D^2 ξ_k = 1. Let S_k = ξ_1 + ξ_2 + … + ξ_k. Then for any α≥ 0lim_n∞ P(max(S_1,S_2,…,S_n) < α√(n)) = √(2/π)∫_0^αexp(-x^2/2)dx. Translating to the current situation, now that we use an initial value Y_u(0)∈ [1/2-δ,1/2+δ] as a reference, we want an upper bound on the probability that the partial sums generating W(t) never exceed 1/2-δ (nor they go below -1/2+δ). The increments have variance 1/(2A^2) and the number of steps is α_4A^2. Formally,P(max(0,W(1)-W(0),…, W(α_4A^2)-W(0)) < 1/2-δ) = P(max(0,W(1)-W(0),…, W(α_4A^2)-W(0))√(2)A < 1-2δ/√(2α_4)√(α_4A^2)) √(2/π)∫_0^1-2δ/√(2α_4)exp(-x^2/2)dx.Now ν_c^2<α_4A^2 can only occur if this event fails and the maximum exceeds 1/2-δ, meaning W(t) might exceed 1, or alternatively, the minimum of the process goes below -1/2+δ corresponding to W(t) possibly leaving [0,1] at 0. Consequently, we may fix any small > 0, then for any large enough A we getP(ν_c^2 < α_4 A^2) ≤ 2(1 - √(2/π)∫_0^1-2δ/√(2α_4)exp(-x^2/2)dx ) +=: β_4.Observe that the right hand side of the expression indeed converges to 0 as α_4 0. There exists a constant α_5 > 0 such that for A large enough, if Y_u(0)∈ [1/2-δ, 1/2+δ] we have (Y_u(α_5A^2)) -π_u_ TV < 1/3.We introduce α_5 as a parameter. We will find sufficient conditions that ensure the claim of the proposition to hold, then pick a α_5 that satisfies the conditions found.We first compare two simpler distributions, that of W(α_5A^2) and the uniform μ.By the definition of W(t), the distribution of W(α_5A^2) is (Y_u(0),α_5/2).(W(α_5A^2))-μ_ TV = 1/2∫_-∞^∞|exp(-(x-Y_u(0))^2/α_5)/√(α_5π)-1_[0,1](x)|dxThe integrand has the form |a-b| which we replace by a+b-2min(a,b) (knowing these variables are non-negative). Also, as the probability density functions integrate to 1, we get(W(α_5A^2))-μ_ TV = 1 - ∫_-∞^∞min(exp(-(x-Y_u(0))^2/α_5)/√(α_5π),1_[0,1](x)) dx= 1 - ∫_0^1 min(exp(-(x-Y_u(0))^2/α_5)/√(α_5π),1) dx≤ 1+2δ - ∫_-δ^1+δmin(exp(-(x-1/2)^2/α_5)/√(α_5π),1) dx =: γ.The last inequality follows because the constant term is increased by 2δ, so is the length of the domain of the integration but the integrand is bounded above by 1. This step also involves an implicit change of variable depending on Y_u(0), and it results in a final expression independent of this starting condition. The γ we get is also independent of A, it does depend on δ but has a limit as δ 0.The claim of the lemma is about two other distributions, now we relate them to the ones just compared. Using Lemma <ref> for α_4=α_5 we know that Y_u(t) and W(t) can be coupled well up to t=α_5A^2, which directly implies(Y_u(α_5A^2))-(W(α_5A^2))_ TV≤β_4,where β_4 is the constant given by Lemma <ref>.To compare π_u with μ we show π_u converges to μ in total variation as A∞. For every x∈ [0,1] definep_u(x) = A/√(π)∫_-x^1-xφ(y)dy,this is a function proportional to the density of π_u. By standard Gaussian tail estimates for all x∈ (0,1) we get1 - exp(-A^2x^2)/2√(π)Ax - exp(-A^2(1-x)^2)/2√(π)A(1-x)≤ p_u(x) ≤ 1.Hence for all x∈ (0,1), p_u(x) 1 as A∞. These are uniformly bounded functions, so ∫_0^1p_u 1. The expression to consider for the convergence of the distributions isμ - π_u_ TV = 1/2∫_0^1 |1-p_u(x)/∫_0^1 p_u|dx.Here 1/∫_0^1 p_u is converging to 1 and is therefore bounded after some threshold, so the functions are eventually uniformly bounded and pointwise converging to 0. Thus the integrals also converge, and we getlim_A∞μ - π_u_ TV = 0. We can now combine our bounds of (<ref>), (<ref>) and (<ref>):(Y_u(α_5A^2)) -π_u_ TV ≤(Y_u(α_5A^2)) - (W(α_5A^2))_ TV + (W(α_5A^2))-μ_ TV+μ - π_u_ TV < β_4 + γ + ,where >0 can be as small as wanted by setting A large enough. The proposition holds if we can ensure this sum to be small enough.Note that a strong compromise is present for the choice of the constant α_5. In (<ref>) we want to limit how likely the boundaries of the unit interval are to be reached, at the same time in (<ref>) we want to show that Y_u(s) is already spread out to some extent.Still, a specific choice is possible. For α_5 = 0.10 Lemma <ref> provides β_4≈ 0.051 when using δ==0 and computer calculations for (<ref>). By choosing δ,>0 but small enough, trusting computers but not too much, we can safely say β_4 < 0.06. In (<ref>) using the same choice of α_5 we numerically get γ≈ 0.263 for δ==0. Once again we allow a safety margin to only claim β_4+γ+<1/3. § OVERALL MIXINGWe are now ready to establish mixing time bounds for the process we understand the best, Y_u(t), then we will translate those results to the original process of interest X(t).Let us defined(t) := sup_Y_u(0)∈[0,1](Y_u(t))-π_u_ TV,which measures the distance from the stationary distribution from the worst starting point. We can give good bounds based on the previous sections:There exists a constant β_6>0 such that d(β_6 A^2) < 4/9. Intuitively, from any starting point we can first wait for the process to reach the middle and then let the diffusion happen from there, as these are components we can already control.Let us apply Proposition <ref> with α_3 = 1/9 providing a certain β_3. Once the process is in the middle part [1/2-δ,1/2+δ] we know by Proposition <ref> that in the subsequent α_5A^2 steps sufficient diffusion occurs. Let β_6=β_3+α_5.Formally, fix Y_u(0)∈ [0,1]. We perform our calculations by conditioning on the value of ν_m.(Y_u((β_3+α_5)A^2))-π_u _ TV = ∑_s=0^∞P(ν_m=s) (Y_u((β_3+α_5)A^2) | ν_m=s) - π_u _ TV.Conditioned on ν_m=s, Y_u(s)∈ [1/2-δ,1/2+δ], therefore Proposition <ref> provides (Y_u(s+α_5A^2) | ν_m=s)-π_u_ TV < 1/3. We use this for s≤β_3A^2, then performing β_3A^2-s more steps can only decrease this distance, see <cit.> for a detailed discussion about this. For s>β_3A^2 we use the trivial bound on the total variation distance. We get(Y_u((β_3+α_5)A^2))-π_u _ TV≤∑_s=0^β_3A^2 P(ν_m=s) ·1/3+ P(ν_m>β_3A^2)· 1 ≤1/3 + α_3 = 4/9.A slight variation of d(t) compares the distribution of the process when launched from two different starting points.d̅(t) := sup_Y^1_u(0),Y^2_u(0)∈[0,1](Y^1_u(t))-(Y^2_u(t))_ TV,Standard results provide the inequalities d(t) ≤d̅(t) ≤ 2d(t) and the submultiplicativity d̅(s+t)≤d̅(s)d̅(t), see <cit.>. The results therein are given for finite state Markov chains but are straightforward to translate to the current case of absolutely continuous distributions and transition kernels.For any 0<α_7<1 there exists β_7>0 such thatt_ mix(Y_u, α_7) ≤β_7 A^2.Using Lemma <ref> for any k≥ 1 we getd(k β_6 A^2) ≤d̅(k β_6 A^2) ≤ (d̅(β_6 A^2))^k ≤ (2d(β_6 A^2))^k ≤(8/9)^k.For k = ⌈logα_7 / log (8/9)⌉ this is less than α_7 thus by setting β_7 = β_6 ⌈logα_7 / log (8/9)⌉the process will be close enough to the stationary distribution as required at t=β_7A^2.The mixing time of Y_u and Y are nearly the same, for any 0<α_7<1t_ mix(Y, α_7) = t_ mix(Y_u, α_7) + 1.First, we use that the total variation distance between the marginals is at most the distance between the overall distributions. Consequently, for any t we have (Y_u(t-1))-π_u_ TV≤(Y(t))-π_ TV. This gives t_ mix(Y, α_7) ≥ t_ mix(Y_u, α_7) + 1.For the other direction, assume (Y_u(t))-π_u_ TV≤α_7 for some t. This means there is an optimal coupling with a random variable Ỹ_u^1 having distribution π_u such that P(Y_u(t) ≠Ỹ_u^1) ≤α_7. As Ỹ_u^1 has distribution π_u, it is possible to draw an additional random variable Ỹ_u^2 to get (Ỹ_u^2,Ỹ_u^1) with distribution π.This is the same step when generating Y_u(t+1) from Y_u(t) thus we may keep the above coupling whenever already present. Therefore we have P((Y_u(t+1),Y_u(t)) ≠ (Ỹ_u^2,Ỹ_u^1))≤α_7 which can also be written as (Y(t+1))-π_ TV≤α_7. This implies t_ mix(Y, α_7) ≤ t +1, completing the proof. We are now ready to prove the main theorem of the paper, as stated in the introduction.*thm:XmixTheorem <ref> \beginthm:Xmix Let X(t) be the coordinate Gibbs sampler for the diagonal distribution. For any 0<α<1 there exists β > 0 such that for large enough At_ mix(X, α) ≤β A^2.\endthm:XmixWe use Proposition <ref> with α_7=α/2 and get a constant β_7 such that t_ mix(Y_u,α/2)≤β_7A^2 and by Lemma <ref> also t_ mix(Y, α/2)≤β_7A^2 + 1. At each step the distribution of X^* and Y might differ only by flipping along the diagonal, which does not change the distance from the (symmetric) π thus also leaves the mixing time the same so we get t_ mix(X^*, α_7/2)≤β_7A^2 +1.The definition of X^* was based on the observation that when the same coordinate is rerandomized repeatedly, no additional mixing happens and the values at that coordinate simply get overwritten. Let us now quantify this effect, counting how many times did the direction of randomization change:N(t):=|{s : 1≤ s ≤ t-1, r(s)≠ r(s+1)}|.With this notation we see that (X(t) | N(t)=k,r(1)=V) = (X^*(k+1)) for all t≥ 1.Without the loss of generality we now focus on the case of r(1)=V. Let us express the distribution of X(t) conditioning on the value of N(t).(X(t) | r(1)=V) = ∑_k=0^t-1P(N(t)=k)(X^*(k+1)) = ∑_k=0^t-11/2^t-1t-1k(X^*(k+1)).We substitute t=3β_7 A^2 and evaluate the total variation distance from π.(X(3β_7 A^2) | r(1)=V)-π_ TV≤∑_k=0^t-11/2^t-1t-1k(X^*(k+1)) - π_ TV≤ P(Binom(3β_7 A^2-1,1/2) < β_7A^2) · 1 + 1 ·(X^*(β_7A^2+1)) - π_ TV≤exp(-β_7 A^2) + α/2.The last line holds with some positiveby Hoeffding's inequality for the Binomial distribution and by substituting the upper bound on the total variation distance when we know k is above the mixing time. For large enough A this is below α.By symmetry, the same bound holds for (X(3β_7A^2) | r(1)=U) and by convexity it is also true for the mixture of the two, the unconditional distribution of X(3β_7A^2). This concludes the proof with β=3β_7. Finally, let us comment on the multitude of constants α_i,β_i appearing throughout the proofs, verifying that they can be consistently chosen when needed. First, a small enough δ>0 has to be picked for the proof of Proposition <ref> which also relies on Lemma <ref>. Once it is fixed, observe that in the remaining Sections <ref> and <ref> all the constants only depend on other ones with lower indices, with the last α, β of Theorem <ref> also depending on some previous ones. This excludes the issue of circular dependence.§ FURTHER ESTIMATESIn this section we complement the main result Theorem <ref> by a lower bound showing that the order of A^2 is exact and by demonstrating the evolution of the distribution via numerical simulations.Such a lower bound is plausible once having Lemma <ref> and Lemma <ref>, these roughly say that when starting from the middle Y_u behaves like a random walk for order of A^2 steps and reaches only constant distance in order of A^2 steps. Let us proceed by forming a formal argument.Let X(t) be the coordinate Gibbs sampler for the diagonal distribution. There exists constants α', β' > 0 such that for large enough A t_ mix(X,α') > β'A^2.First of all, to bound the mixing time from below it is sufficient to give a lower bound on the number of steps needed for a single starting point. In this spirit, we set X(0) = (1/2,1/2). With this choice, the arguments in Section <ref> can be applied.Set S = [0,1/4]^2 ∪ [3/4,1]^2. Once we prove π(S)-P(X(β'A^2)∈ S) > α' for a proper choice of α',β' that warrants a large total variation distance at the time β'A^2 and confirms our bound for the mixing time.π(S) ≥ 1/8. If we divide the unit square to 4-by-4 equal size smaller squares, then S is composed of two of these smaller squares, see Figure <ref>. It is enough to show that the selected squares forming S have greater or equal probability than the other squares w.r.t. π, this directly confirms π(S)≥ 2/16 = 1/8.To verify this, we compare the unnormalized density φ on them. We use the simple inequality that for u,v∈ [0,1/4] and any x≥ 0 we haveφ(u-v) ≥φ(u-(1/2-v+x)).Indeed, note that φ(y) is monotone decreasing with |y|. Then for u≥ v, x=0 an easy comparison of the arguments provides the bound, while the other cases follow similarly. Observe that for x=0 this inequality compares φ at some point and its reflection to the line v=1/4. Setting x>0 corresponds to a further shift increasing the v coordinate after the reflection.Using this we see that the points of the small square labeled by (a,a) in Figure <ref> correspond to the points of (a,b) after a reflection so φ is pointwise larger on (a,a) by the above inequality. The same comparison holds against (a,c), (a,d) where an additional shift is necessary besides the reflection. Consequently, π is maximal for the (a,a) square compared to the others in its column.Additionally, note that φ(u-v) is invariant under the shift of (u,v) in the direction (1,1). Therefore π is exactly the same for the four squares on the diagonal. Furthermore, all the other squares are diagonally shifted and/or reflected (w.r.t. the diagonal) copies of the ones considered above, where we have seen that their probability is upper bounded by the probability of the square (a,a). The distribution π is symmetric w.r.t. the diagonal, so we conclude that (a,a) (and therefore also (d,d)) have indeed maximal probability among all squares. For any α'_1>0 there exists β'_1>0 such that for large enough A any t≤β'_1 A^2 satisfiesP(Y(t) ∈ S) < α'_1.We want to rely on the previous observations that Y_u(t) behaves like a random walk for a while with certain Gaussian increments. Using Lemma <ref> we can choose α_4>0 so that the corresponding β_4 goes below α'_1/2. Let us denote this α_4 by β'_2 for convenience.Also, there exists β'_3>0 so thatP((1/2, β'_3/2) ∈ [0,1/4]∪ [3/4,1]) < α'_1/2,and clearly the same probability bound holds if the variance is decreased. Fixing Y_u(0)=W(0)=1/2, the distribution of W(β'_3 A^2) is exactly (1/2, β'_3/2).To join our estimates we formP(Y(t) ∈ S) ≤ P(Y_u(t) ∈ [0,1/4]∪ [3/4,1]) ≤ P(W(t)≠ Y_u(t)) + P(W(t) ∈ [0,1/4]∪ [3/4,1]).For t≤β'_2 A^2 the first term is below α'_1/2 as it is an upper bound for the decoupling of W,Y_u to happen. For t≤β'_3 A^2 the second term is below α'_1/2. Altogether, if t ≤min(β'_2,β'_3) A^2,P(Y(t) ∈ S) < α'_1.Therefore by choosing β'_1 = min(β'_2,β'_3) we complete the proof. Apply Lemma <ref> with α'_1 = 1/16 to get some β'_1. The distribution of X(β'_1A^2) is a mixture of the distributions of Y(t) and their diagonally flipped version for t≤β'_1A^2, where t corresponds to how many times the rerandomization happened in a new direction. The set S is symmetric w.r.t. the diagonal so for P(X(β'_1 A^2)∈ S) we can simply say it is a convex combination of P(Y(t)∈ S), t≤β'_1 A^2 without needing any correction for the diagonal flip. Now by Lemma <ref> each of these probabilities are below α'_1, therefore it follows thatP(X(β'_1 A^2)∈ S) < α'_1 = 1/16. Comparing this with the statement of Lemma <ref> we getπ(S) - P(X(β'_1 A^2)∈ S) > 1/16.Consequently (X(β'_1 A^2)) - π_ TV > 1/16, so t_ mix(X,1/16) > β'_1 A^2. Thus the theorem holds with the choice α'=1/16, β'=β'_1. Finally, we present numerical approximations of the evolution of the distribution over time for different values of A. The unit square is discretized with a resolution of 500× 500 and the distribution is calculated along these points. The starting point is always (0,0) at the lower left corner. The results are presented in Figure <ref> for different A and different t. Both the convergence to the stationary distribution is visible and also how this distribution becomes more concentrated along the diagonal for higher values of A. We also computed the time necessary to get within a total variation distance of 1/4 of the stationary distribution, for A=10,t=71, for A=50, t=1858, for A=250, t=47233 is needed. This is a good proxy for the mixing time, note that only a single (but intuitively bad) starting point was tested and the discretization might have introduced some error. Still, the quadratic growth of t with respect to the increase of A is already apparent. § ACKNOWLEDGMENTSThe author would like to express his thanks to Persi Diaconis and György Michaletzky for their inspiring comments and to the American Institute of Mathematics for the stimulating workshop they hosted and organized. siam ] | http://arxiv.org/abs/1706.08694v2 | {
"authors": [
"Balázs Gerencsér"
],
"categories": [
"math.PR",
"60J10, 37A25, 60J20"
],
"primary_category": "math.PR",
"published": "20170627071927",
"title": "Mixing time of an unaligned Gibbs sampler on the square"
} |
Attractive and repulsive polymer-mediated forces between scale-free surfaces Mehran Kardar December 30, 2023 ============================================================================§ INTODUCTION The phenomenon of holographic duality, whereby a strongly interacting boundary quantum field theory is dual to a bulk quantum gravity theory in the semiclassical limit (and vice versa) has lead to an extraordinarily powerful way to reason about quantum gravity theories. This is due in no small part to the work of Maldacena <cit.> who first found a quantitative argument that there is an equivalence between string theory on AdS_5× S^5 and 𝒩=4 supersymmetric Yang–Mills theory on the four-dimensional boundary. Maldacena's papers generated an explosion of work consolidating and exploring holographic dualities, starting with the first works of Gubser, Klebanov, and Polyakov <cit.>, and Witten <cit.> and continuing unabated to the current day.One striking result of the AdS/CFT correspondence is the duality between bulk geometry and the pattern of quantum entanglement in the boundary theory. This connection manifests itself in the formula of Ryu and Takayanagi <cit.> expressing the entropy of a region in the boundary CFT with the area of a specific bulk minimal surface. This observation has been considerably expanded and developed in recent years commencing with the proposals of Van Raamsdonk <cit.> and Swingle <cit.> and strengthened by the ER=EPR proposal of Susskind and Maldacena <cit.>.These ideas have recently sparked a very fertile line of enquiry in which results from quantum information theory developed to quantify quantum entanglement and Hamiltonian complexity theory are used to understand the quantum dynamics of black holes <cit.>.Quantum information theory is also playing an increasingly important role in the study of holographic duality. For example, it was realised that new techniques would be required to address some apparantly paradoxical features of the bulk/boundary correspondence, in particular, in understanding the boundary dual of local bulk operators. In a prescient and influential work, Almheiri, Harlow, and Dong <cit.> exploited the theory of quantum error correction to resolve these ambiguities. They argue that bulk local operators should manifest themselves as logical operators on subspaces of the boundary CFT's AdS_3 space.The connection between quantum error correction and the AdS/CFT correspondence, as highlighted by Almheiri, Harlow, and Dong, was dramatically illustrated by the construction of a beautiful toy model of holographic duality known as the holographic code <cit.>. This is a discrete model of the kinematical content of the AdS/CFT correspondence built on special tensors known as perfect tensors arising from certain quantum error correcting codes. The holographic code has proved extremely helpful as a testbed for conjectures and also as a sandbox for refinements of the holographic dictionary. In the two years since its inception there have been numerous papers investigating and generalising both holographic codes <cit.>and perfect tensors <cit.>. Noticing that holographic codes are tensor networks with a particular causal structure, it is tempting to hope that there is a more general manifestation of the AdS/CFT correspondence arising from the network structure alone <cit.>.One key question is largely open in the context of realising a full dynamical toy model of the AdS/CFT correspondence via quantum codes and perfect tensors: How can one realise dynamics for holographic codes? It is this question that we address in this paper in the context of the simplest possible holographic code, namely the holographic state coming from a tree tensor network built with a 3-leg perfect tensor. To do this we leverage powerful new results <cit.> of Jones on unitary representations of a discrete analogue of the conformal group known as Thompson's group T <cit.>. We argue that dynamics for the holographic state should be given precisely as a unitary representation of T. It is worth emphasising that this approach is distinct from the p-adic AdS/CFT correspondence <cit.> (see also <cit.>) as the group of symmetries in this case is not isomorphic to T.This paper is largely synthesised from a great deal of hard work done by Penner, Funar, Sergiescu, and Jones <cit.>. We make no claim on the originality of the unitary representations described herein, nor on the connection between Thompson's group T and the Ptolemy group(oid). The main contribution of this work is to notice that when Penner's work on the Ptolemy group(oid) is combined with Jones's unitary representations via a holographic state, we get a dynamical toy model of the AdS/CFT correspondence. We also discuss how one could generalise these results to holographic states built on more general tessellations of the hyperbolic plane.§ THE BUILDING BLOCKS In this section we explain how to form a Hilbert space for states that are, in a suitable sense, rotation-invariant on a discrete approximation of AdS_3. §.§ Perfect tensors The notion of a perfect tensor was introduced in <cit.>. These highly nongeneric objects capture a discrete version of rotation invariance which is extremely useful in building network approximations to continuous manifolds.We employ a tensor-network representation for n-index tensors: Suppose that T_j_1j_2⋯ j_n is a tensor with n indices, each ranging from 0 to d-1. Then T is depicted as a vertex with d legs, where each leg represents one of the indices in counter-clockwise order (figure <ref>). By convention, the first leg, here indexed by j_1, is the one directly following the label “T” in counter-clockwise order. We now come to perfect tensors.An n-index tensor T_j_1j_2… j_n is a perfect tensor if, for any bipartition of its indices into a pair of complementary sets {j_1, j_2, …, j_n}=A∪ A^c such that, without loss of generality, |A|≤ |A^c|, T is proportional to an isometry from the Hilbert space associated with A to the Hilbert space associated with A^c.If the legs have dimension d then, for a given partition {j_1, j_2, …, j_n}= A∪ A^c, T is a linear mapT⊗_j∈ Aℂ^d →⊗_j∈ A^cℂ^d.In particular, if A = ∅ and A^c = {j_1, j_2, …, j_n}, then T is a vector in the Hilbert space (ℂ^d)^⊗ n. From now on we write d for the (constant) dimension of the legs of a perfect tensor.The perfect tensor condition is nontrivial and it is far from obvious whether perfect tensors exist at all (they do). Throughout this paper wefocus on the simplest case of n=3. In this particularly simple setting the definition of a perfect tensor reduces to that of a 3-leg tensor V which is an isometry in all three possible directions, as depicted in figure <ref>.Here are some interesting examples of perfect tensors to keep in mind in the sequel.For the case n=2 any unitary U is perfect. Let n=3 and d =4 and define the map Vℂ^2⊗ℂ^2 →ℂ^2⊗ℂ^2⊗ℂ^2⊗ℂ^2 byV|j⟩|k⟩ = 1/2|j⟩ |Ψ^-⟩ |k⟩,where |Ψ^-⟩≡1/√(2)(|01⟩ - |10⟩) is the singlet state. This example may be generalised to the Fuss-Catalan planar algebra <cit.>. Also for the case n=3 but with d=3 define the map Vℂ^3→ℂ^3⊗ℂ^3 by⟨ jk|V|l⟩ =0 if j=k, k=l, or j=l,1 otherwise.This example comes from the 4-colour theorem <cit.>. Another example for n=4 and d=3 is based on the 3-qutrit code, given by the 4-index tensor T defined byT|x⟩|y⟩ = |2x+y mod 3⟩|x+y mod 3⟩.There are now several constructions and generalisations of perfect tensors. A partial list includes <cit.>.In this paper, we concentrate on the simplest possible case, where the 3-leg perfect tensor V is additionally rotation-invariant, so that the three conditions for perfectness collapse to only the isometry condition. The isometry defined by (<ref>) provides a nontrivial example of such a perfect tensor.§.§ AdS3 and a little hyperbolic geometry The introductory material described here is adapted from <cit.>.§.§.§ The hyperbolic planeDenote the upper half plane model of two-dimensional hyperbolic space by ℍ^2 and the Poincaré disc model of ℍ^2 by 𝔻. These two models are related via the conformal transformationf(z) = z-i/z+i,known as the Cayley transformation (figure <ref>).The asymptotic boundary ∂𝔻 of the Poincaré disc 𝔻 is the circle S^1 at infinity. The metric for the disc model is given byds^2 = 4dx^2 + 4dy^2/(1-x^2-y^2)^2.Geodesics in 𝔻 are circles which meet the boundary at right angles.Isometries of the hyperbolic plane are realised by the group PSL(2,ℝ) of Möbius transformations. These transformations act naturally on the upper half plane model ℍ^2 via fractional linear transformationsz↦az+b/cz+d.Hence, after conjugation with the conformal map (<ref>), they can also act on 𝔻.§.§.§ (2+1)-dimensional anti de Sitter spaceMuch of the discussion in this paper concerns (2+1)-dimensional anti de Sitter space AdS_3. This space may be understood as the quadric surface described byX^2+Y^2-U^2-V^2=-1in flat four-dimensional spacetime with two space and two time dimensions:ds^2 = dX^2+dY^2-dU^2-dV^2.To actually visualise AdS_3 it is convenient to exploit sausage coordinates <cit.>X= 2ρ/1-ρ^2cos(φ), Y= 2ρ/1-ρ^2sin(φ), U= 1+ρ^2/1-ρ^2cos(t), V= 1+ρ^2/1-ρ^2sin(t),where 0 ≤ρ< 1, 0 ≤φ < 2π, and -π≤ t <π. The metric for AdS_3 in terms of sausage coordinates is given byds^2 = -(1+ρ^2/1-ρ^2)^2 dt^2 + 4/(1-ρ^2)^2(dρ^2 + ρ^2dϕ^2).In terms of these coordinates AdS_3 may be visualised as a cylinder whose equal-time slices are copies of the Poincaré disc 𝔻 and whose end caps are identified.The boundary of AdS_3 is timelike and is seen to be the two-dimensional surface of a cylinder. This is topologically S^1× S^1 and hence identified with the conformal compactification S^1,1≅ S^1× S^1 of Minkowski space ℝ^1,1. The boundary is called conformal infinity.Geodesics within AdS_3 are found via intersection of the quadric X^2+Y^2-U^2-V^2=-1 and hyperplanes containing the origin, i.e., surfaces defined viaasin(α)X+acos(α)Y-bsin(β)U-bcos(β)V = 0.The two-dimensional plane containing a geodesic is thus given by2ρ/1+ρ^2sin(φ+α) = b/asin(t+β)which is timelike if |b/a| < 1, lightlike if |b/a|=1, and spacelike if |b/a|>1. Lightlike geodesics in the boundary propagate around the boundary at the speed of light and hence realise spirals or helices.§.§.§ Black holes in 2+1 dimensions Einstein's equations do not admit gravitational wave solutions in 2+1 dimensions. However, there are black hole type solutions in the case of a negative cosmological constant. These were discovered by Bañados, Teitelboim, and Zanelli <cit.> (see <cit.> for further details). The black hole solutions we consider here all arise as quotients of AdS_3 by discrete isometries.The t=0 surface of the BTZ black hole solution arises as the result of a quotient by a discrete hyperbolic transformation of the Poincaré disk: one takes a geodesic γ, and its image γ' under the transformation and identifies them. The BTZ solution arises from the action of the following transformation:X'= X Y'= Usinh(2π√(M)) + Ycosh(2π√(M)) V'= Ucosh(2π√(M)) + Ysinh(2π√(M)) U'= U. The topology of the BTZ solution is different from that of AdS_3: the t=0 slice is now a cylinder (although it is still locally AdS) and has an event horizon (figure <ref>).§.§ Tessellations We build discretised models of 𝔻 via tessellations, that is, we cover 𝔻 with a grid of polygons. Throughout this paper we focus on triangles, but everything we say in the sequel has a natural generalisation to other tessellations via, e.g., pentagons and so on. A tessellation 𝒫 of 𝔻 (or, a subset A⊂𝔻) is a collection of convex polygons in 𝔻 (respectively, A) such that * the interiors of the polygons in 𝒫 are mutually disjoint; * the union of the polygons in 𝒫 is 𝔻 (respectively, A); * the collection 𝒫 is locally finite.[A collection 𝒫 of polygons is locally finite if and only if for each point x∈𝔻 there is an open neighbourhood of x which nontrivially intersects with only finitely many elements of 𝒫.] We say that the tessellation 𝒫 is exact if and only if every side of a polygon is a side of exactly two polygons in 𝒫. A regular tessellation of 𝔻 is then an exact tessellation consisting of congruent regular polygons.We also define a tessellation to be an ideal regular triangulation τ of 𝔻 if it is a countable locally finite collection of geodesics in 𝔻 such that each connected region in 𝔻∖τ is an ideal triangle. (An ideal triangle is a hyperbolic triangle all of whose vertices lie on the boundary of the Poincaré disc model.) The vertices τ^(0) of the triangulation are the asymptotes of the geodesics comprising the edges of the triangulation, regarded as points of the circle at infinity. We denote by τ^(2) the collection of all the complementary triangles in 𝔻∖τ.There are many examples of ideal regular triangulations of 𝔻. The Farey tessellation τ_*, for instance, is generated by the action of 𝑃𝑆𝐿(2,ℤ) on the basic ideal triangle with vertices at 1,-1, and i (figure <ref>).The Farey triangulation has many mathematical advantages. However, to discuss holographic codes and symmetries it is actually more convenient to use the dyadic tessellation (figure <ref>).Its vertices on ∂𝔻 are determined by dyadic subdivision. This means the following: First we realise ∂𝔻 as the unit interval with endpoints identified. Then the coordinates of the points on the boundary have the form a/2^n for a∈ℤ^+ and n∈ℤ^+. Both the Farey and dyadic tessellations yield equivalent results in the sequel – this is a consequence of results of Imbert, see e.g. the volume <cit.> and the papers of Penner, Imbert, and Lochak and Schneps therein. The mapping that identifies the Farey and dyadic tessellations is the Minkowski question mark function ?(x).Just as in Euclidean space, it is often important to distinguish an edge, analogous to the origin, to set a reference. This allows us to distinguish transformations preserving the tessellation. In the context of the tessellations considered here this is achieved by choosing an edge e along with a preferred orientation of e as a distinguished oriented edge. The central objects of study in this paper are then pairs (τ, e) of a tessellation τ together with a specific chosen distinguished oriented edge e. The standard tessellation with distinguished oriented edge (τ_0, e_0) is shown in figure <ref>.The edge e_0 is taken to be the geodesic joining the points with coordinates 0/1 and 1/0 in the upper half plane model.We will also encounter nonregular tessellations with distinguished oriented edge. These are all built by applying a finite sequence of Pachner moves <cit.> to the standard tessellation. What a Pachner move means in the present context is this: we isolate an ideal quadrilateral formed by two ideal triangles in the tessellation, remove the geodesic joining the diametrically opposed vertices, and then add in a new geodesic between the other pair of opposite vertices. An example of such a tessellation is shown in figure <ref>.As long as the diagonal geodesic is not the distinguished oriented edge, such a Pachner move is its own inverse. In the case where the diagonal geodesic is the distinguished oriented edge when we apply the Pachner move, we use the orientation given by rotating the edge by 90^∘. This means that it takes four such Pachner moves to return to the original tessellation (figure <ref>).We call by admissible any tessellation (τ, e) with oriented edge that arises from the standard tessellation via a finite sequence of such Pachner flips.The main reason for introducing a distinguished oriented edge for a tessellation is that it gives us a mechanism to compare two tessellations without requiring the introduction of a cutoff (a concept we discuss in the next subsection). The way this works is as follows. Suppose that (τ, e) is an admissible tessellation with oriented edge; we will build a homeomorphism f S^1→ S^1 of the boundary S^1≅∂𝔻 of the disk which produces this tessellation from the canonical Farey tessellation (τ_*,e_*). (After you see this construction you will be able to compose two such homeomorphisms to build a map between any two tessellations with oriented edge.) This works because all of the triangles in our tessellations are ideal; therefore they are unambiguously specified once we say where the boundary points are located. Any homeomorphism f S^1→ S^1 produces a new (not necessarily admissible) tessellation (f(τ_*), f(e_*)) with oriented edge.The first step in the construction of the homeomorphism f producing (τ, e) from the Farey tessellation is to label all the vertices of the Farey tessellation with rational numbers according to the following recipe. In the upper half plane model the vertex at z = 1 corresponds to x=∞ of ℍ^2, -1 to the origin x=0 of ℍ^2, and √(-1) to x = -1. We use this identification to iteratively label the vertices of the Farey triangulation: take the edge connecting 0/1 and ∞≡ 1/0 in the upper half plane model and label the third vertex underneath corresponding to 1/1 = (0+1)/(1+0) in the upper half plane. Now continue this process: for every pair of previously labelled vertices (p/q, r/s) of a triangle, label the remaining vertex with the mediant (p+r)/(q+s). This process leads to a bijection between the rational numbers ℚ and the vertices τ_*^(0), see figure <ref>.We can now recursively build the map f: start with the endpoints 01 and 10: these are mapped to the endpoints of the distinguished oriented edge e in (τ, e). There is always triangle to the right of the distinguished oriented edge in any tessellation. Let f map the point 11 to the vertex of this triangle which isn't on e. Do the same with the triangle to the left. Now recursively visit all of the triangles to the left and right, all the while mapping the corresponding vertices from the Farey triangulation to the vertices of the new triangles. This procedure creates a bijection between the vertex sets. It turns out that this identification may be extended to a homeomorphism f_(τ,e) called the characteristic mapping of (τ,e) <cit.>. The argument presented here is a powerful manifestation of the bulk boundary correspondence: we've established a bijection between two sets, namely, the bulk𝖳𝖾𝗌𝗌 = {tessellations of 𝔻 with distinguished oriented edge},and a set of objects acting on the boundary, i.e.,Homeo_+ = {orientation-preserving homeomorphisms of ∂𝔻}.This is the content of the following theorem. The characteristic mapping (τ,e) ↦ f_(τ,e) induces a bijection between 𝖳𝖾𝗌𝗌 and Homeo_+.§.§ CutoffsA crucial role throughout this work is played by cutoffs. What are these? We think of imposing a UV cutoff on the system defined on a timeslice 𝔻 by truncating it to a region whose boundary is in the interior of 𝔻 apart from a finite number of isolated points on the boundary. A consequence of this is that we have restricted the system to have finite volume. Such a cutoff need not be rotation invariant, indeed, it is very convenient to allow the cutoff boundary to have angular dependence.We define a cutoff as follows: Take a list of geodesics γ≡ (e_1,e_2,…, e_n), where the geodesics come from a tessellation τ of 𝔻. Every geodesic partitions the disc into two halfspaces. The cutoff A_γ associated with γ is the finite-volume convex region that is given by the intersection of the halfspaces of the geodesics in γ. The requirement of finite volume stems from the fact that we want to exclude regions which include a subset of the boundary with nonzero measure. We further require that the geodesics (together with the endpoints on ∂𝔻) comprising the boundary ∂ A_γ of A_γ form a clockwise oriented cycle. (The orientation dictates which side of the geodesic the halfspace is on.) An example of a cutoff with boundary γ = (e_1,e_2,…, e_7) is shown in figure <ref>.For simplicity we'll also refer to the boundary γ defining a cutoff A_γ as a cutoff.A important feature of the set of all such cutoffs 𝒫 is that it is a directed set. This means that we can define a partial order ≼ on 𝒫 where we say that a cutoff γ is smaller than the cutoff γ', written γ≼γ', if A_γ⊆ A_γ'.Further, given two cutoffs γ and γ' we can always find a third cutoff γ” such that γ≼γ” and γ'≼γ”. It is worth noting that since all of our tessellations agree with the dyadic tessellation τ_0 sufficiently close to the boundary (they only differ by a finite number of edge flips/Pachner moves), we can always find a cutoff γ” in the standard tessellation τ_0 bigger than γ and γ' coming from arbitrary tessellations (figure <ref>). §.§ Holographic statesGiven a perfect tensor V, a tessellation τ, and a cutoff γ coming from τ, we can build a special quantum state |ψ_γ⟩ according to the following recipe: for every triangle inside A_γ we associate one copy of V, with one leg per edge, and for every adjacent pair of triangles we contract the legs of V associated with the common edge, as in figure <ref>. We can associate a Hilbert space ℋ_γ with each cutoff γ = (e_1,e_2,…, e_n) in a natural way: Simply take the tensor product of the leg Hilbert space ℂ^d over each edge of the cutoff γ, i.e.,ℋ_γ = ⊗_j=1^n ℂ^d.If we bipartition this boundary system {j_1, j_2, …, j_n}≡ A∪ A^c with A ≡∅, then we can regard the tensor network associated to τ and γ as a state |ψ_γ⟩∈ℋ_γ, called the holographic state (figure <ref>).In this way we see that a holographic state associated with a tessellation τ and a perfect tensor V is really a family of states |ψ_γ⟩, one per cutoff γ from the tessellation τ.§ THE SEMICONTINUOUS LIMIT A holographic state |ψ_γ⟩ built from a perfect tensor V and a tessellation τ with cutoff γ should, in a sense to be specified presently, be equivalent to a holographic state |ψ_γ'⟩ built from the same tessellation and tensor V but with a larger cutoff γ'≽γ. The question is: How can we compare two states in different Hilbert spaces? The answer is to build an equivalence relation on the set of all boundary Hilbert spaces.The physical intuition behind our equivalence relation is the following. A cutoff γ induces a (generally nonregular) lattice structure on the boundary space. A larger cutoff γ'≽γ induces a finer lattice structure on the boundary, and some of the cells in the original lattice have been subdivided, or fine-grained. In the real-space picture we are working with here, a fine-graining corresponds to a real-space renormalisation group transformation, which is taken to be an isometry.We now have a viable notion of equivalence between two states |ϕ_γ⟩∈ℋ_γ and |ψ_γ'⟩∈ℋ_γ' in two possibly different Hilbert spaces. First suppose that γ≼γ'. In this case there should be a fine-graining isometry T^γ_γ'ℋ_γ→ℋ_γ' which fine-grains any state |ϕ_γ⟩∈ℋ_γ into a new state T^γ_γ'|ϕ_γ⟩ living in the finer Hilbert space ℋ_γ'. Although initially these two states are mathematically different, they are physically equivalent. From the viewpoint of physics, they represent the same state – just living in Hilbert spaces with different cutoffs. In this context we think of the fine-graining operation T^γ_γ' as adding no further information – i.e., correlations – above the cutoff γ. We thus have a method to compare |ϕ_γ⟩ and |ψ_γ'⟩: First fine-grain |ϕ_γ⟩ via T^γ_γ' to a state T^γ_γ'|ϕ_γ⟩∈ℋ_γ', then compare T^γ_γ'|ϕ_γ⟩ with |ψ_γ'⟩ by use of the inner product defined on ℋ_γ'.To obtain a general equivalence relation on the set of all Hilbert spaces ℋ_γ associated with cutoffs γ∈𝒫 we need to exploit one more feature of the space of cutoffs, namely, that it is a directed set. Suppose we want to compare two states |ϕ_γ⟩∈ℋ_γ and |ψ_γ'⟩∈ℋ_γ' but neither γ≼γ' nor γ'≼γ. To do this we realise that there is always a larger cutoff γ” which refines both, i.e., γ”≥γ and γ”≥γ'. We then fine-grain both of the states into a common Hilbert space ℋ_γ” and compare them there. Their overlap is given by⟨ϕ_γ|(T^γ_γ”)^† T^γ'_γ”|ψ_γ'⟩. For all of this to be well defined we need the fine-graining operation T^γ_γ' to satisfy two consistency conditions: * If γ = γ' then T^γ_γ' = 𝕀; * For all γ≼γ'≼γ” we have that T^γ_γ” = T^γ'_γ”T^γ_γ'. It is straightforward to check that when we have a fine-graining operation obeying conditions (1) and (2) we get a well-behaved equivalence relation ∼ on the set ℋ of all states in some Hilbert space ℋ_γ with some cutoff γ. The correct space to represent the set of all states with some cutoff is the disjoint union of the Hilbert spaces ℋ_γ,ℋ≡_γ∈𝒫ℋ_γ.Why not use the simple union? The problem is when we have two incomparable cutoffs γ and γ' with the same number of edges: in this case the Hilbert spaces ℋ_γ and ℋ_γ' are isomorphic, and hence would collapse to the same element in the standard union. However, these two spaces are physically very different; we don't want to forget the cutoff associated to each space. This is achieved by tagging each element of the union with its corresponding cutoff, which is exactly the disjoint union.When we mod out ℋ by the equivalence relation ∼we end up with a bona fide Hilbert spaceℋ = ℋ_γ= (_γ∈𝒫ℋ_γ/∼)^·= ( 0cm3em[c][4em][c]17em the disjoint union of ℋ_γ over all γ∈𝒫 modulo the equivalence relation that |ϕ_γ⟩∼ |ψ_γ'⟩ if there are γ”≽γ and γ”≽γ' such that T^γ_γ”|ϕ_γ⟩=T^γ'_γ”|ψ_γ'⟩. )^·Here (…)^· denotes the completion with respect to the standard norm ·. The Hilbert space ℋ is known as the direct limit of the directed system (ℋ_γ, T^γ_γ') of Hilbert spaces. It is an infinite-dimensional separable Hilbert space.There is a great deal of arbitrariness in choosing the fine-graining isometries. However, in our case we want the holographic states |ψ_γ⟩ to all be equivalent. This is achieved by setting, for γ≼γ', the isometry T^γ_γ' to be the tensor network built from V associated with the region bounded by the curves γ and γ' (figure <ref>). When we take for our directed set of possible cutoffs the set 𝒫_0 of cutoffs coming from the standard tessellation τ_0 and we use for T^γ_γ' the tensor network built from a perfect tensor V, we call the resulting direct limit of Hilbert spacesℋ≡ℋ_γthe semicontinuous limit <cit.>. Elements of the semicontinuous limit are equivalence classes[|ϕ_γ⟩] ≡{(γ', |ψ_γ'⟩)| T^γ'_γ”|ψ_γ'⟩ = T^γ_γ”|ϕ_γ⟩ for some γ”∈𝒫_0} of states coming from boundary systems with a finite cutoff.What is the physical intuition for a resident of the semicontinuous limit? To get an intuition for this we first note that any vector |ϕ_γ⟩∈ℋ_γ has a natural image – it is isometrically embedded – in ℋ as[|ϕ_γ⟩].You should think of the state [|ϕ_γ⟩] ∈ℋ as the UV completion of |ϕ_γ⟩: it is essentially the state |ϕ_γ⟩ which has been infinitely fine-grained via T^γ_γ' as γ' gets closer and closer to the actual boundary ∂𝔻 (figure <ref>). How does one work with the semicontinuous limit ℋ in practice? Many calculations we need to carry out will not change the cutoff. Since every state |ϕ_γ⟩∈ℋ_γ is isometrically embedded in ℋ we can forget about ℋ and pretend we are working just in ℋ_γ. There are, however, occasions where an operation will lead to a change of cutoff. In this case an initial state |ϕ_γ⟩∈ℋ_γ might end up in the space ℋ_γ' of boundary states with a different cutoff γ'. We exploit the inner product defined by (<ref>), i.e.,([|ϕ_γ⟩],[|ψ_γ⟩]) ≡⟨ϕ_γ|(T^γ_γ”)^† T^γ'_γ”|ψ_γ'⟩,and then work with respect to the bounary space ℋ_γ”. Thus we can effectively work in a finite-dimensional Hilbert space throughout, increasing the cutoff via T^γ_γ' when necessary. Physically this is really no different to how one works with digital images: suppose we have an image defined at some resolution (i.e., cutoff) and we want to prepare an image at a different resolution. Here we fine-grain via interpolation – this is the analogue of T^γ_γ' for digital images – and then work on a higher-resolution image. This allows us to cut and paste together images with incommensurate resolutions, i.e., compare them.Thus, from now on, we define the semicontinuous limit ℋ to be the kinematical space for the boundary theory of a holographic state.§.§ States with geometry Let (ℋ_γ, T^γ_γ') be a directed system of Hilbert spaces associated with the holographic state built from aperfect tensor V. Suppose we have a state [|ϕ_γ⟩] ∈ℋ in our boundary kinematical space. The state |ϕ_γ⟩ is allowed to be any possible state in the boundary Hilbert space ℋ_γ, which is a d^|γ|-dimensional Hilbert space. However, it could be that |ϕ_γ⟩ is very special, i.e., it could be that |ϕ_γ⟩ arises as the contraction of the perfect tensor V according to some tessellation of the region A_γ, for example:< g r a p h i c s >Or it could be that |ϕ_γ⟩ arises from a different tessellation such as< g r a p h i c s > If a state [|ϕ_γ⟩] ∈ℋ arises from the contraction of the tensor V according to some tessellation τ of A_γ, then we say that it has geometry (τ, A_γ).There is an equivalent way to specify the geometry of a state [|ϕ⟩] ∈ℋ (if it has one) which makes no reference to cutoffs. In this case we say that [|ϕ⟩] has geometry (τ, e) if it arises as a sequence of states coming from contracting the perfect tensor V according to the tessellation τ. It is not immediate that this approach gives a well-defined state within the semicontinuous limit Hilbert space ℋ. That this is so can be seen as follows. Given an admissible tessellation (τ,e), we know that far enough away from the origin the tessellation τ agrees with the standard tessellation τ_0. This is because an admissible tessellation is generated by a finite number of Pachner flips. Now choose a common cutoff γ from both τ and τ_0 (which is guaranteed to exist for sufficiently large A_γ). Consider the tessellation τ restricted to A_γ: this is an instance of the previous scenario, so define the state |ϕ_γ⟩∈ℋ_γ by contracting the perfect tensor V according to the tessellation τ and A_γ. This recipe is well defined because increasing the cutoff corresponds precisely with the equivalence relation ∼ employed to define ℋ in the first place.We now introduce a distinguished subset 𝒢 of ℋ defined to be the set of all states with some geometry (τ, e), or equivalently, (τ, A_γ). § DYNAMICS AS A UNITARY REPRESENTATION OF SYMMETRIES What does it mean for a quantum system to “have dynamics”? One answer is as follows. The operation “wait for t units of time” is a symmetry in quantum mechanics, in the sense that waiting t units of time should not change the inner products between pairs, i.e., distinguishability, of states. Hence, thanks to Wigner's theorem (see, e.g., <cit.>), this operation must be represented by a (projective) unitary or antiunitary operator U_t on the kinematical Hilbert space ℋ.[We focus only on the unitary case from now on.] Actually, we get something a little stronger by imposing the condition that first waiting t_1 units of time and then waiting t_2 units of time is equivalent to waiting t_1+t_2 units of time. Further demanding that waiting 0 units of time corresponds to the identity yields the observation that a quantum system with kinematical Hilbert space ℋ “has dynamics” if it affords a (projective) unitary representation of the time translation group ℝ, i.e., we have a family of unitary operators U_tℝ→𝒰(ℋ) such thatU_t_1+t_2 = e^iϕ(t_1,t_2)U_t_1U_t_2,t_1,t_2∈ℝ,where U_0 = 𝕀 and ϕ must satisfy some nontrivial conditions.It is straightforward to find many such representations for any quantum system: just choose a random Hermitian operator on ℋ and buildU_t = e^-itH.This construction is rather arbitrary and therefore not satisfying. To get a more interesting answer we need to impose additional constraints on what we want our dynamics to do. These constraints are typically that the system must exhibit more than just the time-translation symmetry. Indeed, we usually demand that relativistic quantum systems exhibit the full group of Poincaré symmetries. Thus, arguing as above, we would say our quantum system is Poincaré invariant if we can find a (projective) unitary representation of the (universal cover of the) Poincaré group ℝ^1,3⋊SL(2,ℂ). This group contains, as a subgroup, our original group ℝ of time-translation symmetries. However, it is important to note that the group of time-translation symmetries does not commute with general Poincaré transformations. Thus it is not sufficient to find a Hamiltonian H that commutes with the generators of boosts etc. This is precisely why building representations of the Poincaré group is much harder than building symmetric models in nonrelativistic quantum mechanics: we have to do everything at once in the relativistic setting.In this paper we are looking for something much stronger than just a relativistic quantum system. Indeed, we want our quantum system to correspond to a (1+1)-dimensional conformally invariant quantum system. A rather strong interpretation of this is that our quantum system should not only give us a unitary representation of the Poincaré group for ℝ^1,1 but also a unitary representation of local conformal transformations. The group (ℝ^1,1) of all local conformal transformations of ℝ^1,1 is given by (see, e.g., <cit.>)(ℝ^1,1) = (_+(ℝ)×_+(ℝ))∪(_-(ℝ)×_-(ℝ))Hence the group of all conformal diffeomorphisms consists of two connected components. We now restrict our attention to the connected component _+(ℝ)×_+(ℝ). This group may be understood in terms of _+(ℝ), which is concrete enough. However, _+(ℝ) is extremely large and we rather study the conformal diffeomorphisms of the conformal compactification of ℝ^1,1. Thus the group of orientation-preserving conformal diffeomorphisms of the conformal compactification S^1,1 is isomorphic to the group(_+(S^1)×_+(S^1))∪(_-(S^1)×_-(S^1)).One then redefines “the” conformal group (ℝ^1,1) of ℝ^1,1 to be the connected component of (S^1,1), namely(ℝ^1,1) ≅_+(S^1)×_+(S^1).Unitary representations of (ℝ^1,1) are all built from representations of _+(S^1), known as the chiral conformal group. So, as is typical, we focus on _+(S^1).Thus our goal, at its most ambitious, is to find a (projective) unitary representation of _+(S^1) on our kinematical semicontinuous limit space ℋ. If we could actually do this then we'd have arguably built a full conformal field theory: to get the Hamiltonian, for example, one just needs to differentiate the representation of the one-parameter group of time translations.That the semicontinuous limit Hilbert space ℋ might be a natural place to look for a unitary representation of _+(S^1) comes from the observation that _+(S^1)⊂Homeo_+ which, in turn, may be identified with the space 𝖳𝖾𝗌𝗌. This leads to the naive idea of representing the action of f∈_+(S^1) on ℋ by an operator π(f) that takes a state |ψ_(τ,e)⟩ with geometry (τ, e) to a state with geometry (f(t),f(e)) and then extending by linearity. This idea almost works, but runs into the problem that when f∈_+(S^1) acts on the boundary S^1 it typically takes an admissible tessellation with distinguished oriented edge (τ, e) to an inadmissible tessellation. There seems no easy way to add in states with the geometry of these inadmissible tessellations without leading to a nonseparable Hilbert space!The approach we take instead is to understand exactly what transformations do preserve the admissability of a tessellation and instead focus on the group these transformations generate. Astonishingly, this group, known as Thompson's group T, while not containing _+(S^1), does contain sequences which can approximate any f∈_+(S^1) arbitrarily well. §.§ Thompson's group T We review in this subsection the definition and some of the basic properties of Richard Thompson's groups F and T. The material presented here is adapted from the canonical reference <cit.> of Cannon, Floyd, and Parry.We start with the definition of a group known as Thompson's group F, which is a subgroup of the group T which we work with in the sequel. We call by Thompson's group F the group of piecewise linear homeomorphisms from [0,1] to itself which are differentiable except at finitely many dyadic[A dyadic rational is a rational number of the form a/2^n, with a and n integers.] rational numbers and such that on the differentiable intervals the derivatives are powers of 2. That F is indeed a group follows from the following observations. Let f∈ F. Since the derivative of f, where it is defined, is always positive, it preserves the orientation of [0,1]. Suppose that 0 = x_0 < x_1 < ⋯ < x_n = 1 are the points where f is not differentiable. Then f(x) = a_1x, x_0≤ x ≤ x_1, a_2x+ b_2, x_1≤ x ≤ x_2, ⋮ a_nx+ b_n, x_n-1≤ x ≤ x_n, where, for all j=1, 2, …, n, a_j is a power of two and b_j is a dyadic rational (and we set b_1 = 0). The inverse f^-1 also has power-of-two derivatives except on dyadic rational points and, since f maps the set of dyadic rationals to itself, we deduce that F is a group under composition. To define T we regard S^1 as the interval [0,1] with the endpoint 1 identified with 0. Thompson's group T is then the collection of piecewise linear homeomorphisms from S^1 to S^1 taking dyadic rational numbers to dyadic rational numbers and which are differentiable except at a possibly finite number of locations which are also dyadic rational, and whose slopes are given by powers of 2. That T is a group follows from an argument identical to that presented for F above. Since every element of F also satisfies the conditions to be an element of T we have that it is a subgroup F≤ T.It turns out <cit.> that Thompson's group F is generated by two elements, A and B, defined byA(x)= 1/2 x, x∈ [0,12), x-1/4, x∈ [12, 34), 2x-1, x∈ [34, 1], B(x)= x, x∈ [0,12), x/2+1/4, x∈ [12, 34), x-1/8, x∈ [34, 78 ), 2x-1, x∈ [78, 1].Thompson's group T is generated by A and B together with a third element C, defined byC(x) = x/2 + 3/4, x∈ [0,12), 2x-1, x∈ [12, 34), x-1/4, x∈ [34, 1],A, B, and C are illustrated in figure <ref>. There are many alternative representations that have been developed to work with elements of F and T. One of the most convenient for us will be via tree diagrams. To describe these we first introduce the tree 𝒯 of standard dyadic intervals. A standard dyadic interval is an interval in [0,1] of the form [a/2^n, a+1/2^n], where a,n∈ℤ^+.We build the tree 𝒯 of standard dyadic intervals by introducing a node for each standard dyadic interval and connecting two nodes if one is included in the other (figure <ref>). Here one can think of edges as denoting subdivisions.A finite ordered rooted binary subtree with root [0,1] of 𝒯 is called a 𝒯-tree. From now on we suppress the labellings of the nodes via standard dyadic intervals as they may be reconstructed from context. Here are two examples of 𝒯-trees:< g r a p h i c s >There is a one-to-one correspondence between 𝒯-trees and certain partitions of the unit interval: Let Γ≡{[x_0,x_1), [x_1,x_2), …, [x_n-1,x_n]}, with 0 = x_0 < x_1 < ⋯ < x_n = 1, be a partition of [0,1]. We define [x_j,x_j+1], j = 0,1, …, n-1, to be the intervals of the partition (note the inclusion of the end points). A partition Γ of [0,1] is called a standard dyadic partition if all the intervals of Γ are standard dyadic intervals.The leaves of a 𝒯-tree describe the intervals of a standard dyadic partition and there is a bijection between standard dyadic partitions and 𝒯-trees.The utility of 𝒯-trees in discussing Thompson's groups comes from the following lemma. Let f∈ F. There is a standard dyadic partition Γ = {0 = x_0 < x_1 < ⋯ < x_n = 1} such that f is linear and differentiable on every interval of Γ and f(Γ) = {f(0) = f(x_0) < f(x_1) < ⋯ < f(x_n) = f(1)} is also a standard dyadic partition.This lemma provides us with the motivation to introduce the notion of a tree diagram. This is a pair (R,S) of 𝒯-trees such that R and S have the same number of leaves. You should think of this pair as a fraction R/S. The tree R is the domain tree or numerator tree and S is the range tree or denominator tree. Here is an example of a tree diagram:(-2pt < g r a p h i c s > ,< g r a p h i c s > ).Because of lemma <ref> we know that there exist standard dyadic partitions Γ and f(Γ) such that f is differentiable and linear on the intervals of Γ and maps them to the intervals of f(Γ). We associate to f the tree diagram (R,S) where we get R from the partition Γ by representing it as a 𝒯-tree, and S is similarly the 𝒯-tree associated to f(Γ).Note that one can associate many different tree diagrams (R,S) to f∈ F: these can be obtained by simultaneously adjoining carets to the leaves of R and S. What this means is that we take the jth leaves of both R and S and simultaneously adjoin the binary tree with two leaves to these leaves. This process is illustrated here, highlighting the adjoined carets in red:(-2pt < g r a p h i c s > ,< g r a p h i c s > )↦(-4pt < g r a p h i c s > ,< g r a p h i c s > ).The caret adjunction process builds a new standard dyadic partition Γ' from Γ which has the same intervals as Γ except that the jth interval has been symmetrically subdivided into two intervals. Call the interval in Γ represented by the jth leaf I and the corresponding interval in f(Γ) by J=f(I). The new intervals in the partition Γ' represented by the leaves of the adjoined carets are called I_1 and I_2. Since the map f is linear and differentiable on the interval I represented by the jth leaf, we deduce that f(I_1) = J_1 and f(I_2) = J_2. Thus the tree diagram (R',S') arising from the caret adjunction is also a tree diagram for f.One can reduce a tree diagram (R,S) by eliminating common carets. If there are no common carets that can be eliminated, the diagram is said to be reduced. It turns out <cit.> that there is exactly one reduced tree diagram for every f∈ F and vice versa.Here are the reduced tree diagrams for A and B:A = (-2pt < g r a p h i c s > ,< g r a p h i c s > ),B= (-4pt < g r a p h i c s > ,< g r a p h i c s > ). This entire discussion can be repeated for the case of Thompson's group T with essentially no modification. The only difference is due to the fact that elements of T can move the origin. To keep track of this, when an element of T does map the origin to a different point we denote the image of the interval in the domain tree in the range tree with a small circle. This is illustrated for C here:C = (-2pt < g r a p h i c s > ,< g r a p h i c s > ). Since we are free to adjoin as many carets as we like, we can understand Thompson's groups F and T as rewriting rules for infinite trees: an element of one of Thompson's groups will take an infinitely large domain tree, cut off the infinite base tree leaving a finite domain tree, replace the reduced domain tree with the range tree, and then add the infinite base by adjunction of carets. This action is in sharp contrast to the way other natural groups act on trees, in particular, the p-adic groups which have featured in a related model of the AdS/CFT correspondence <cit.>. §.§ Approximating diffeomorphisms In this subsection we describe a fundamental result which explains the precise connection between Thompson's group T and the chiral conformal group. Both groups are subsets of the group Homeo(S^1) of homeomorphisms of the circle, and our ambition is that by studying T we can learn about _+(S^1) because the two groups are morally “close” to one another (in the extremely coarse sense that they are subgroups of the same group).The most direct hope for using Thompson's group T to study the chiral conformal group _+(S^1) would have been, e.g., that T is a subgroup of the conformal group T≤_+(S^1). However, this hope was doomed to failure from the outset as the elements of T are not differentiable. The next best thing, therefore, is to try and approximate one group by the other. This is indeed possible and is captured by the following striking proposition <cit.>. Let f∈_+(S^1). Then there exists a sequence g_n∈ T, n∈ℕ, such that lim_n→∞f-g_n_∞ = lim_n→∞sup_x∈ S^1 |f(x)-g_n(x)| = 0.Thanks to this proposition we can be encouraged that studying T will give us some insight into _+(S^1). Suppose we have a general purpose way to build (projective) unitary representations π of T, then, supposing that the representation π is “sufficiently continuous” then we'd have a general purpose procedure to build representations of _+(S^1) according toπ(f) “≡” lim_n→∞π(g_n).The scare quotes here indicate that we currently do not understand how to find such sufficiently continuous representations. If we could do this, however, then we'd have constructed a new procedure to build conformal field theories.The emphasis in this paper is to promote a bug to a feature and study projective unitary representations of T in their own right as a toy model of the AdS/CFT correspondence. The benefit of this is the resulting representations are so explicit that we can calculate everything of interest explicitly. §.§ The action of Thompson's group T on tessellations Thompson's group T acts in a natural way on the boundary S^1 = ∂𝔻 of the Poincaré disc and, in particular, on points with dyadic rational coordinates. According to this action our standard dyadic tessellation with distinguished oriented edge (τ,e) is mapped to another tessellation with vertices on dyadic rationals. It is a remarkable consequence of the work of Imbert, Penner, and Lochak and Schneps <cit.> that every element of T takes any admissible tessellation with distinguished oriented edge to another admissible tessellation with distinguished oriented edge, i.e., via Pachner flip. We don't revisit this proof here. Instead, we present the action of the generators A, B, and C on the standard dyadic tessellation as this is illustrative enough to imagine how an arbitrary Thompson group element acts (figure <ref>). §.§ Unitary representations of T from perfect tensors In this subsection we review a general purpose procedure, due to Jones <cit.>, to build unitary representations of T from perfect tensors. There are a couple of ways to understand this construction, by far the most elegant of which is a categorical argument based on the localisation functor <cit.>. A general mathematical framework for this is described in <cit.>.The Hilbert space that furnishes our representation is none other than the semicontinuous limit space ℋ. Recall that elements of this space are (equivalence classes of) cutoff+state pairs (γ, |ψ_γ⟩), where |ψ_γ⟩∈ℋ_γ, subject to the equivalence relation ∼. Since our cutoffs γ are defined by the dyadic tessellation τ the endpoints of each geodesic in the cutoff lies on dyadic rationals. Thompson's group T acts naturally on geodesics with dyadic rational endpoints by simply moving the endpoints. Hence T acts naturally on cutoffs γ = (e_1, e_2, …, e_n) as we simply define f(γ) to be the cutoff (f(e_1), f(e_2), …, f(e_n)). In this fashion we can allow T to act on elements of ℋ: suppose we have an element of ℋ with representative (γ, |ψ_γ⟩). Then we could build an action by settingπ(f)(γ, |ψ_γ⟩)“=” (f(γ), |ψ_γ⟩).The reason there are scare quotes is due to the fact that since T acts via Pachner flips of the standard dyadic tessellation we are only guaranteed that f(γ) is a cutoff coming from an admissible tessellation. The only subtlety is that sometimes f(γ) will not be a cutoff in 𝒫_0, i.e., one coming from the standard dyadic tessellation τ_0. This is a problem for defining an action of T which maps elements of ℋ to elements of ℋ, because (f(γ), |ψ_γ⟩) is a member of ℋ only when f(γ)∈𝒫_0. However, it is always possible to remedy this problem: just take a finer cutoff γ'≽γ such that f(γ') also comes from the standard dyadic partition τ_0. That this is always possible follows from Lemma <ref>: One must first identify cutoffs with partitions of the circle S^1 via the intervals determined by the endpoints of the geodesics. For example, the cutoff defined by γ = (e_1,e_2,…, e_7),< g r a p h i c s >is naturally associated to the partitionΓ = {[0,14), [14, 38), [38,12),[12,58),[58,34),[34,78), [78, 1)}of S^1.Therefore, the action of T is defined as follows: first take a representative (γ, |ψ_γ⟩) of the element [|ψ_γ⟩]∈ℋ you want to act on, then refine it to an equivalent representative(γ, |ψ_γ⟩) ∼ (γ', T^γ_γ'|ψ_γ⟩)such that f(γ') ∈𝒫_0, i.e. it corresponds to a standard dyadic partition, and then defineπ(f)([|ψ_γ⟩]) ≡π(f)(γ, |ψ_γ⟩) ≡ (f(γ'), T^γ_γ'|ψ_γ⟩).One still needs to check that everything is well defined, i.e., that refinement plays nicely with the action of T. This can be done and is detailed in <cit.>.It is worth working through one example in detail in order to convince yourself that the action T we've defined actually leads to something nontrivial. To do this we focus on the simplest possible example state, namely the state determined by the perfect tensor V itself on an ideal triangle:< g r a p h i c s >A representative for this state is (γ, |ϕ_γ⟩) where γ = (e_1, e_2, e_3) corresponds to the partition {[0,12), [12, 34), [34, 1)} and |ϕ_γ⟩∈ℂ^d⊗ℂ^d⊗ℂ^d (note that ⟨ jkl|ϕ_γ⟩ = 1/√(d)V^j_kl, i.e., |ϕ_γ⟩ is proportional to V in order it be a normalised state). We apply the generator B to this representative. The first difficulty we encounter is that B(γ) is not an admissible region. Therefore, to correctly apply the transformation we first need to refine γ via T^γ_γ' = 𝕀⊗𝕀⊗ V to a new cutoff γ' = (e_1, e_2, f_3, f_4) corresponding to the partition {[0,12), [12, 34), [34, 78), [78, 1)}, giving us the equivalent representative (γ', 𝕀⊗𝕀⊗ V|ϕ_γ⟩) which is illustrated here:< g r a p h i c s > The generator B, when applied to γ', now does give us an admissible region B(γ') corresponding to the new partition {[0,12), [12, 58), [58, 34), [34, 1)}. The action of π(B) is thereforeπ(B)(γ', 𝕀⊗𝕀⊗ V|ϕ_γ⟩) = (B(γ'), 𝕀⊗𝕀⊗ V|ϕ_γ⟩).The result of the transformation is< g r a p h i c s > Although it might look as though B did nothing, the pair (B(γ'), 𝕀⊗𝕀⊗ V|ϕ_γ⟩) is in general the representative for a different state in the semicontinuous limit space. This is because B moved the legs around: the pair (B(γ'), 𝕀⊗𝕀⊗ V|ϕ_γ⟩) has the same state 𝕀⊗𝕀⊗ V|ϕ_γ⟩ but now associated to a different region with boundary B(γ'). That is, the state 𝕀⊗𝕀⊗ V|ϕ_γ⟩ has been moved from the subspace ℋ_γ' into a different subspace of ℋ, namely, ℋ_B(γ').Let's see if the new state π(B)(γ', 𝕀⊗𝕀⊗ V|ϕ_γ⟩) after the action of B is any different to the original state (γ, |ϕ_γ⟩): to do this we need to compare them within the same subspace of ℋ. This is rather simple in this case because γ≼ B(γ'), so all we need to do is refine our original representative into B(γ') via T^γ_B(γ') = 𝕀⊗ V⊗𝕀. On this common subspace ℋ_B(γ') we find that the original state and the final state have representatives(B(γ'), 𝕀⊗ V⊗𝕀|ϕ_γ⟩), and (B(γ'), 𝕀⊗𝕀⊗ V|ϕ_γ⟩),respectively. The inner product between the two states is thus⟨ϕ_γ|(𝕀⊗ V^†⊗𝕀)(𝕀⊗𝕀⊗ V)|ϕ_γ⟩which, depending on V, is not always equal to 1. In particular, for the example (<ref>) we find that the inner product is given by 1/2. Thus we have shown that the action of T on the Hilbert space ℋ can be nontrivial.Now that we have an action of T on ℋ we need to establish that it is unitary. This is relatively easy to do by confirming the invariance of the inner product for all representatives (γ, |ϕ_γ⟩):(π(f)[|ϕ_γ⟩], π(f)[|ϕ_γ⟩]) = ([|ϕ_γ⟩], [|ϕ_γ⟩]).That this is true is a consequence of the fact that V is an isometry, which in turn follows from the perfect tensor condition. We will not deny the reader the pleasure of confirming this result for themselves.One might wonder if the unitary representation of T thus constructed is generated by a Hamiltonian. To this end, note that in the continuum case, time evolution amounts to a (strongly continuous) unitary representation t↦ U_t, t∈ℝ, of the time-translation group ℝ. The Hamiltonian can then be obtained as H=iħd/dtU_t, so that U_t=e^-iHt/ħ. But what is the analogue of time in our discrete setting? Since the modular group 𝑃𝑆𝐿_2(ℤ) is a subgroup of T (generated by the elements A and C of T), the only translations of the form z↦az+b/cz+d with a,b,c,d∈ℤ are those for which a=d=1 and c=0, that is, only translations z↦ z+b by integers b are possible. It is therefore reasonable to say that if there is an analogue of time-translations in the discrete theory, it is given by ℤ. But then the time-derivative d/dtU_t and therefore the notion of Hamiltonian are not well-defined, so we will have to be content with the unitary representation as dynamics. Furthermore, note that the absence of a Hamiltonian is a commonly accepted and expected behaviour of discrete theories such as those of quantum walks or quantum cellular automata, and therefore not at all surprising. §.§ The bulk Hilbert space In this subsection we build a subspace ℋ_bulk⊂ℋ which we later argue corresponds to the Hilbert space for the dual bulk gravitational theory. For the mathematically inclined: what we will do here is build the GNS representation of the von Neumann group algebra built on T.In the previous subsection we described an action of T on the semicontinuous limit space ℋ. To build our bulk Hilbert space we single out the state[|Ω⟩] with the geometry of the standard tessellation. This is a state in the boundary Hilbert space. We build up the bulk space ℋ_bulk by simply adjoining states to ℋ_bulk which result from the action of f, i.e., we setℋ_bulk≡span{π(f)[|Ω⟩] |f∈ T}.This space is the subspace of ℋ generated by all “conformal-like” transformations from T acting on the trivial vacuum state. Because this space is closed under the action of T it also gives us a unitary representationπ(f)|_ℋ_bulkℋ_bulk→ℋ_bulk.The physical analogy to keep in mind here is that ℋ should be thought of as the full CFT Hilbert space, i.e., as corresponding to the CFT vacuum plus all excitations built from the vacuum given by applying primary, secondary, tertiary fields etc. The bulk subspaceℋ_bulk corresponds to the subspace of the full boundary CFT Hilbert space generated by the conformal vacuum plus only those states generated from the conformal vacuum by application of Virasora algebra elements, i.e., as all states built from the vacuum via local conformal transformations.Due to the particular way Thompson's group T acts we can describe an overcomplete basis of ℋ_bulk by labelling kets with binary trees. The first observation is that the inner product between two states of the form π(f)[|Ω⟩] and π(g)[|Ω⟩], f,g∈ T, in ℋ_bulk may be computed by using the group property of T as follows(π(f)[|Ω⟩], π(g)[|Ω⟩]) = ([|Ω⟩], π(f^-1g)[|Ω⟩]) =some function of the reduced tree diagram (R,S) for f^-1g∈ T.Using this observation we notice that we can understand all matrix elements of an operator π(f) in ℋ_bulk in terms of vacuum matrix elements via knowledge of just ([|Ω⟩], π(h)[|Ω⟩]) for all h∈ T. The next step is to define the following special vectors in ℋ_γ: notice that any cutoff γ determines a binary tree if we first build the partition of the unit interval and then associate to the partition to its subtree in the tree of standard dyadic intervals. If γ has n intervals this tree will have n leaves; we write R for this tree. Let S be an arbitrary connected binary tree with n leaves and define the following ket|R,S⟩∈ℋ_bulkfirstly via its inner products with kets built from other pairs of binary trees (R,S') with R the same but S' an arbitrary binary tree with n leaves according to⟨ R,S'|R,S⟩ + reflect S and join its leaves to S'; replace vertices with V and contract.For example⟨-2pt < g r a p h i c s > ,< g r a p h i c s >| -2pt < g r a p h i c s > ,< g r a p h i c s > ⟩ =-10pt < g r a p h i c s > .This definition may be extended to give the inner product between arbitrary kets |R,S⟩ and |R',S'⟩: the only difference is that if R≠ R' we first simultaneously subdivide the pairs of trees (R,S) and separately the pair (R',S') by adding carets to S and R' until they become equal.It is possible to argue that the kets |R,S⟩ defined by pairs of trees uniquely determine elements of ℋ_bulk, i.e., the process of adding carets is consistent with the equivalence relation ∼ on ℋ.§ THE P/T CORRESPONDENCE We have focussed so far on building dynamics for the boundary theory of a holographic state, and what that means. We have argued that the Hilbert space of the boundary (which would be Hilbert space of the CFT part of the AdS/CFT correspondence) should be given by the semicontinuous limit and that the role of the dynamics of the boundary should be taken by Thompson's group T. However, we have neglected all discussion of the bulk, and what the analogous objects are here. In this section we argue that there is an analogy between the group of (large) asymptotic diffeomorphisms of the bulk on the continuous side and something called the Ptolemy groupoid on the semicontinuous side. The object that takes the place of a quantum gravity in bulk then consists of: (1) an isometry from the subspace of “semiclassical” bulk states into the boundary Hilbert space (the semicontinuous limit ℋ) Φℋ_bulk→ℋ_boundary≡ℋ; and (2) a unitary representation of a group of large “discretised” diffeomorphisms, known as the Ptolemy group 𝑃𝑡, on the subspace ℋ_bulk. Further, these discretised bulk diffeomorphisms will correspond precisely with the group T of Thompson “conformal” transformations of the boundary. This is the closest analogue to the situation in the standard AdS/CFT correspondence: since small bulk diffeomorphisms are gauge transformations for the bulk the only transformations which can generate physically different states are those bulk diffeomorphisms with a nontrivial asymptotic action, in which case the bulk diffeomorphisms are dual to their restrictions on the boundary <cit.>. The results in this section build heavily on the work of Penner and coworkers, see especially the volume <cit.> for details.Before we define the Ptolemy group, we first have to agree on what a “discretised diffeomorphism” ought to be. For this paper we think of a tessellation with distinguished oriented edge (τ,e) as defining something like a geometrical structure for 𝔻 and hence, by specifying geodesics in the boundary of AdS_3 via the vertices of the tessellation, for AdS_3 (see <ref> for the description of geodesics in the boundary of AdS). If tessellations with distinguished oriented edge are like geometries then a “discretised diffeomorphism” ought to be a map F which takes one tessellation with distinguished oriented edge (τ, e) and gives us another (τ', e') = F(τ,e). Such a map F is currently only partially defined by this prescription: it only makes sense if you give it precisely the tessellation with distinguished oriented edge (τ, e) and is otherwise undefined. We will remedy this defect in three stages.The first stage is to agree what elementary moves we are going to allow and then build our possible maps F from these moves by composition. This will already give us the structure of a groupoid (which is just a fancy word for a set with only a partially defined group product structure). We've already identified a special move, namely, the Pachner flip (figure <ref>): take an edge γ∈τ with its two neighbouring triangles which together form an ideal quadrilateral. The edge γ is the diagonal of the quadrilateral. The Pachner flip is then the new tessellation τ_γ formed by removing this diagonal and replacing it with the other diagonal γ':τ_γ = (τ∪{γ'}) ∖{γ}.The tessellation τ_γ is said to have resulted from an elementary move along γ∈τ. As long as γ is not the distinguished oriented edge this definition extends to tessellations with distinguished oriented edge by simply defining the oriented edge of τ_γ to be that inherited from (τ,e). If γ is the distinguished oriented edge then we take the Pachner flip to act in counter-clockwise direction. Thus a Pachner flip of the quadrilateral containing the edge e as its diagonal has order 4. We now define the Ptolemy groupoid Pt' to be a collection of sets Mor((τ,e),(τ',e')) of partially defined maps F(τ,e) → (τ',e') between pairs of tessellations with distinguished oriented edge. The set of allowed maps between a given pair of tessellations with distinguished oriented edge is given by all finite sequences of Pachner flips which produce (τ',e') from (τ,e). Note that this is nontrivial as there can be more than one such sequence. Any allowed map in Mor((τ,e),(τ',e')) is invertible because each of the Pachner flip moves are invertible. This is a groupoid because the product operation given by composition is, at this stage, only partially defined.We haven't yet solved the problem of how to independently specify a discrete diffeomorphism that can act on any input tessellation with distinguished edge (τ,e) since we have only agreed on a set of allowed maps between a given pair of tessellations with distinguished oriented edge. We want to find a recipe to select the “same” map from each of these sets. Thus we want to find a “universal way” to specify a Pachner flip which makes no reference to the tessellation it acts on. To do this realise that every edge γ∈τ can be specified in terms of a vertex in the Farey tessellation: in describing Theorem <ref> we built a labelling, called the characteristic map, of the vertices of a tessellation with distinguished oriented edge via rational numbers. Our universal specification then proceeds as follows: take the unique ideal triangle containing an edge γ which is in the component of 𝔻∖γ which does not contain the distinguished oriented edge. One of the ideal vertices of this triangle does not live on γ. It is labelled by a rational number ℚ in the Farey tessellation. That rational number q(γ) uniquely specifies γ in (τ, e). Figure <ref> shows an example of this labelling.Given a rational number q we can now invert the process described above to uniquely specify an edge in any admissible tessellation (τ,e) with distinguished oriented edge. Suppose you have some rational number q and some arbitrary admissible tessellation (τ,e), then first label all the vertices via the characteristic mapping f_(τ,e). Next find the vertex labelled q, then select the edge γ whose label is q(γ). As long as q≠±1 this recipe will uniquely select an edge from (τ,e).We now have a correspondence between ℚ≡ℚ∖{-1,+1} and (τ,e) which identifies an edge γ_q in (τ,e) for every q∈ℚ. Denote by ϕ_q, q∈ℚ, the map that carries out the Pachner flip of the edge γ_q in any admissible tessellation with distinguished oriented edge. There is a natural composition law for such ϕ_q's. The set of all such maps, and their compositions, is denoted M. Now two such maps ϕ, ϕ' ∈ M are equivalent if they act identically on all admissible tessellations with distinguished oriented edge. Since this is an equivalence relation ∼ we now only consider maps ϕ up to this relation ∼. Let [ϕ] denote the equivalence class of ϕ∈ M and let K = {ϕ∈ M | ϕ∼id}. The Ptolemy group is now the group given by the quotient 𝑃𝑡≡ M/K. It may be argued that this is a group, i.e., K is big enough to allow every element to have an inverse.It turns out that the Ptolemy group 𝑃𝑡 is isomorphic to none other than Thompson's group T. This isomorphism is not hard to guess now that we've set up all of the machinery: one has to verify that to every element g∈𝑃𝑡 we can associate a unique element f_g∈ T (and vice versa) in a homomorphic way. The way to do this is via the characteristic mapping: an element g∈𝑃𝑡 corresponds to a sequence of Pachner flips which, in turn, are associated to a homeomorphism f_g of the boundary ∂𝔻≅ S^1.Because the Ptolemy group 𝑃𝑡 acts via the same Pachner flips as T we can directly take it to act on ℋ_bulk; we now have the strongest possible manifestation of the bulk/boundary correspondence. The “discrete conformal group” T of the boundary is precisely the group of “discrete diffeomorphisms” of the bulk. § BLACK HOLES AND THE ER=EPR CORRESPONDENCE One core limitation of the dyadic tessellation is that we cannot represent the geometries for gravitational solutions corresponding to particles. These are conical (see, e.g., <cit.> for a description) and require a more general hyperbolic tessellation. We consider the generalisation of our results to this situation in the next section. However, one setting we can discuss in the context of our 𝑃𝑡/T correspondence is that of black hole solutions.In 2+1 dimensions there is a well-known family of black-hole solutions of Einstein's equations in the presence of a negative cosmological constant due to Bañados, Teitelboim, and Zanelli. These geometries are multiply connected and may be produced from 𝔻 by quotienting out by a discrete subgroup of the group of isometries of 𝔻. We illustrate the simplest example here. This solution corresponds to the thermofield double (TFD) state according to the standard AdS/CFT correspondence. Here we describe the analogue of the TFD for the 𝑃𝑡/T correspondence.A tessellation for the BTZ spacetime can be built from the dyadic tessellation by choosing two opposite geodesics and identifying them according to the procedure outlined in <ref>, see figure <ref>.The two identified geodesics are indicated with arrows. The result of this procedure is a tessellation of the cylinder with two boundaries A and B at spatial infinity. The two boundaries may each be identified with S^1.By associating the perfect tensor V to each triangle in the BTZ tessellation we obtain the tensor network shown in figure <ref>.This network can be thought of in two ways. Firstly, it may be understood as a state |Φ_AB⟩ with the geometry τ_BTZ: here we are thinking of the Hilbert space of the entire system given by ℋ_AB≅ℋ_A⊗ℋ_B, the semicontinuous limit built on the two boundaries A and B at spatial infinity. Note that |Φ_AB⟩ is not a product state with respect to the tensor product over A and B, it is an entangled state. This gives rise to the second interpretation of |Φ_AB⟩, namely, as an entangled state of the two distinct subsystems A and B. This equivalence between entanglement and geometry is the manifestation of the ER=EPR proposal <cit.> for the 𝑃𝑡/T correspondence.§ GENERALISATIONS In this paper, we have detailed the construction of the semicontinuous limit, and dynamics thereof, for the specific case of triangular tessellations. There are now different directions for generalisation, each coming with different challenges.One of the most important underlying features of the triangular tessellations we use is that the constructed tensor networks have a tree-like structure. This tree-like structure is the main reason it is possible to take the semicontinuous limit. Therefore, a first step would be to consider other tessellations whose underlying graph structure is that of a tree. The analogues of the modular group from our example are then given by certain Coxeter groups which are isomorphic to modular groups over extended integer rings. It is not yet known what the analogue of Thompson's group would be in this case, but a starting point might be to find a generalisation of the Ptolemy group, whose definition appears much simpler.In general, for arbitrary regular tilings of the Poincaré disk the symmetry group will be some Fuchsian group. For example, in the case of a hexagonal tiling with four hexagons meeting at a vertex (figure <ref>) it turns out that using a 6-leg perfect tensor will give us a unitary representation of the isometry group of this tessellation, namely, a Fuchsian group G given by a subgroup of the quaternions ℚ defined over the field ℚ[√(2)+√(3)].This is not exactly a “big” group in the sense that Thompson's group T is the group of all piecewise 𝑃𝑆𝐿(2,ℤ) functions, but it is still pretty big. The physical analogy here is that the group G is a group of only global symmetries, whereas Thompson's group T is analogous to the conformal group which is a group of local symmetries. We very nearly do get unitary representations of an analogous group of piecewise G transformations, but this just fails (it turns out that the semicontinuous limit is not continuous enough for these transformations to admit unitary representations).In physical terms what this means is that there is an analogue of the group of global conformal transformations which acts unitarily, but that local conformal transformations do not directly act unitarily on the semicontinuous limit Hilbert space unless we either: (1) find special perfect tensors that satisfy additional constraints; or (2) we augment the semicontinuous limit Hilbert space by adding in states via a process known as completion. The second option is not entirely desirable as it is completely unclear whether the resulting Hilbert space is separable.One benefit to considering more general tessellations such as the {6, 4} tessellation here is that we can represent bulk particle solutions. This is easy to do by forming conical geometries by deleting and gluing. For example, figure <ref> shows a massive particle solution built by cutting out a quarter of the tessellation and gluing the exposed edges.By substituting a 6-leg perfect tensor in place of every polygon in this tessellation and contracting we obtain a semicontinuous Hilbert space etc. We also get a unitary action of a subgroup of the group G.Finally, regarding holographic codes instead of states, there is really no work to do: the semicontinous limit can be constructed in the same way as before since additional bulk legs do not interfere with the procedure. In the limit we then obtain a space of linear maps (instead of a Hilbert space) containing holographic codes; Thompson's group T acts in exactly the same way as before. § CONCLUSIONS AND OUTLOOK In this paper we have commenced the construction of a dynamical toy model of the AdS/CFT correspondence built on holographic codes. Our findings are summarised in table <ref>. Many obvious steps remain incomplete, including, the discussion of fields and their holographic duals, the Ryu-Takayanagi formula, and others. §.§ Fields for Thompson's groupsWe have argued that there is a strong analogy between the conformal group and Thompson's group T. One might hope that this analogy is not accidental and can be extended to build a theory of Thompson-symmetric fields in parallel with conformal field theory. It turns out that one can define analogues of primary field operators for Thompson's group T and that they transform in an analogous way to their conformal counterparts under Thompson group transformations. Additionally, these field operators enjoy fusion rules and give rise to descendant fields as one might hope <cit.>. §.§ Bulk fieldsIt is straightforward to add bulk fields by exploiting pluperfect tensors <cit.>. This should give rise to something like an intertwiner for Thompson group representations. We have not explored this in any detail. §.§ More general tessellations and MERAVery recent work of Evenbly <cit.> has lead to the construction of MERA-like networks with perfect-like properties. These seem a particularly promising place to look for Thompson-like discrete local symmetry groups which are not based on trees.Firstly, we'd like to sincerely thank Vaughan Jones and Yunxiang Ren for many helpful discussions and for extremely valuable guidance, especially with the mathematical theory of subfactors, planar algebras, and the Thompson group. We are also grateful to Cédric Bény, Gemma De las Cuevas, Robert König, Fernando Pastawski, and Mario Szegedy for numerous comments and suggestions. Finally, we thank the anonymous referees of the Quantum Information Processing (QIP) conference 2018 in Delft for their helpful reports.This work was supported by the DFG through SFB 1227 (DQ-mat) and the RTG 1991, the ERC grants QFTCMPS and SIQS, the cluster of excellence EXC201 Quantum Engineering and Space-Time Research, and the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQUS, CE170100009). unsrt | http://arxiv.org/abs/1706.08823v4 | {
"authors": [
"Tobias J. Osborne",
"Deniz E. Stiegemann"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP",
"math.OA"
],
"primary_category": "quant-ph",
"published": "20170627124528",
"title": "Dynamics for holographic codes"
} |
headingsThe NA62 RICH detectorconstruction and performanceAndrea BizzetiAndrea Bizzeti Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia Istituto Nazionale di Fisica Nucleare – Sezione di Firenze, ItalyE-mail: [email protected] NA62 RICH detector Andrea Bizzeti1,2 on behalf of the NA62 RICH Working Group: A. Bizzeti,F. Bucci, R. Ciaranfi, E. Iacopini, G. Latino, M. Lenti, R. Volpe,G. Anzivino, M. Barbanera, E. Imbergamo, R. Lollini, P. Cenci, V. Duk, M. Pepe, M. PicciniJune 23, 2017 ================================================================================================================================================================================================================================================== The RICH detector of the NA62 experiment at CERN SPS is required to suppress μ^+ contamination inK^+ →π^+ νν̅ candidate eventsby a factor at least 100 between 15 and 35 GeV/c momentum, to measure the pion arrival time with ∼ 100 ps resolutionand to produce a trigger for a charged track. It consists of a 17 m long tank filled with Neon gas at atmospheric pressure. Čerenkov light is reflected by a mosaic of 20 spherical mirrorsplaced at the downstream end of the vessel and is collected by 1952 photomultipliers placed at the upstream end. The construction of the detector will be described and the performance reached during first runs will be discussed. § THE NA62 RICH DETECTOR The NA62 experiment<cit.> at CERN SPS North Areahas been designed to study charged kaon decaysand particularly to measure the branching ratio (≈ 10^-10) of the very rare decay K^+ →π^+ νν̅with a 10% precision.The NA62 experimental apparatus is described in detail in <cit.>. The largest background toK^+ →π^+ νν̅ comes from the K^+ →μ^+ ν_μ decay, which is 10 orders of magnitude more abundant. This hugebackground is mainly suppressed using kinematical methods and the very different response of calorimeters to muons and charged pions. Another factor 100 in muon rejection is needed in the momentum range between 15 and 35 GeV/c. A dedicated Čerenkov detector, the RICH, has been designed and built for this purpose. Neon gas at atmospheric pressure is used as radiator, with refraction index n = 1 + 62.8 × 10^-6 at a wavelength λ=300μm,corresponding to a Čerenkov threshold for charged pionsof 12.5 GeV/c. Two full-length prototypeswere built and tested with hadron beams to study the performanceof the proposed layout<cit.>.The RICH radiator container (“vessel”)is a 17 m long vacuum-proof steel tank,composed of 4 cylindrical sections of diameterupto 4 m, closed by two thin aluminium windows to minimize the material budget crossed by particles. A sketch of the RICH detector is shown in Fig.<ref>. Fresh neon at atmospheric pressureare injected in the 200 m^3 vessel volume after it has been completely evacuated. No purification or recirculation system is used. A mosaic of spherical mirrors with 17.0 m focal length,18 of hexagonal shape (350 mm side)and two semi-hexagonalwith a circular opening for the beam pipe, is placed at the downstream end of the vesselto reflect and focus Čěřěňǩǒv̌ light towards the two regions equipped with photomultipliers (PMTs) at the upstream end of the vessel (see Fig.<ref>).Each mirror is supported using a dowel inserted in a 12 mm diameter cylindricalholedrilled in the rear face of the mirrorclose to its barycentre. Dowels are connected to the RICH vessel by means of a vertical support panel, made of an aluminum honeycomb structure to minimize the material budget. Two thin aluminium ribbons, each onepulled by a micrometric piezo-electric motor,keep the mirror in equilibrium and allow to modify its orientation. A third vertical ribbon, without motor, avoids on-axis rotation. The mirror orientationis measured by comparing the positionof the centre of the Čerenkov ring reconstructed by the RICH PMTs with its expected position based on the track direction reconstructed by the spectrometer and can be finely tuned using piezomotors(see Fig.<ref>). Two arrays of 976 Hamamatsu R7400-U03 PMTs are locatedat the upstream end of the vessel,left and right of the beam pipe. The PMTs have an 8 mm diameter active region and are packed in an hexagonal lattice with 18 mm pixel size. Each PMT has a bialkali cathode,sensitive between 185 and 650 nm wavelength with about 20% peak quantum efficiency at 420 nm. Its 8-dynode system provides a gain of 1.5 × 10^6at 900 V supply voltage, with a time jitter of 280 ps FWHM. PMTs are located in air outside the vesseland are separated from neon by a quartz window;an aluminized mylar Winston cone<cit.> is used to reflect incoming light to the active area of each PMT. The front-end electronics consists of 64 custom made boards, each of them equipped with four 8-channels Time-over-Threshold NINO discriminator chips<cit.>. The readout is provided by 4 TEL62 boards,each of them equipped with sixteen 32-channels HPTDC<cit.>;a fifth TEL62 board receives a multiplicity output (logic OR of the 8 channels) from each NINO discriminatorand is used for triggering.The time resolution of Čerenkov rings has been measured by comparing the average times of two subsets of the PMT signals, resulting in σ_t= 70 ps.§ PARTICLE IDENTIFICATION In order to assess the RICH performance, the Čerenkov ring radius (which depends on particle velocity)measured by the RICH is related to the track momentummeasured by the magnetic spectrometer. Figure <ref>(left)shows a clear separation between different particles in the momentum range 15–35 GeV/c. Pion-muon separation is achived by cutting on the particle mass,calculated from the measured values of the particle velocity (from the Čerenkov ring radius)and momentum. The charged pion identification efficiency ε_πand muon mis-identification probability ε_μ are plotted in Fig. <ref>(right)for several values of the mass cut.At ε_π=90% the muon mis-identification probabilityis ε_μ≃ 1%. § CONCLUSIONS The NA62 RICH detector was installed in 2014 and commissioned in autumn 2014 and 2015; it is fully operational since the 2016 run. First performance studies with collected datashow that the RICH fulfilled the expectations,achieving a time resolution of 70 psand a factor ∼ 100 in muon suppression. § ACKNOWLEDGEMENTS The construction of the RICH detector would not have been possible without the enthusiastic work of many technicians from University and INFN of Perugia and Firenze, the staff of CERN laboratory,the collaboration with Vito Carassiti from INFN Ferrara. A special thank to the NA62 collaboration for the full dedication to the construction, commissioning and running of the experiment. 99ref:na62_proposal G. Anelli , Proposal to measure the rare decay K^+→π^+νν̅ at the CERN SPS, CERN-SPSC-2005-013, CERN-SPSC-P-326 (2005).ref:na62_detector E. Cortina Gil (NA62 Collaboration),J. Instr. 12 (2017) P05025,doi:10.1088/1748-0221/12/05/P05025.ref:buras A. J. Buras , JHEP 1511 (2015) 166.ref:e949 A.V. Artamonov (E949 Collaboration),Phys. Rev. D 79 (2009) 092004.ref:rich_proto1 G.Ânzivino ,Nucl. Instr. Meth. Phys. Res. A 538 (2008) 314.ref:rich_proto2 B. Angelucci ,Nucl. Instr. Meth. Phys. Res. A 621 (2010) 205.ref:winston H. Hinterberger and R. Winston,Rev. Sci. Instr.37 (1966) 1094.ref:nino F. Aghinolfi , Nucl. Instr. Meth. Phys. Res. A 621ref:hptdc J. Christiansen,High Performance Time to Digital Converter,CERN/EP-MIC (2004), <https://cds.cern.ch/record/1067476/files/cer-002723234.pdf> | http://arxiv.org/abs/1706.08496v1 | {
"authors": [
"Andrea Bizzeti"
],
"categories": [
"physics.ins-det",
"hep-ex"
],
"primary_category": "physics.ins-det",
"published": "20170626174039",
"title": "The NA62 RICH detector"
} |
APS/[email protected]@nju.edu.cn Collaborative Innovation Center of Advanced Microstructures and Key Laboratory of Modern Acoustics, MOE, Institute of Acoustics, Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of [email protected] of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583, Singapore We use metasurfaces to enable acoustic orbital angular momentum (a-OAM) based multiplexing in real-time, postprocess-free and sensor-scanning-free fashions to improve the bandwidth of acoustic communication, with intrinsic compatibility and expandability to cooperate with other multiplexing technologies. The mechanism relied on encoding information onto twisted beams is numerically and experimentally demonstrated by realizing the real-time picture transfer, which differs from existing static data transfer by encoding data onto OAM states. Our study can boost the capacity of acoustic communication links and offer potential to revolutionize relevant fields.Twisted Acoustics Cheng-Wei Qiu August 25, 2017 ===================Acoustic communication is pivotal in applications such as ocean exploration, where sound is the dominant information carrier due to the prominent loss of light in ocean <cit.>. Unlike optical communication <cit.> with the properties of high frequency and light speed, information transfer based on sound is subject to deficiencies of low frequency and velocity, limiting the development of more advanced acoustic communication <cit.>. Although remarkable progress has been made by introducing wavelength-division multiplexing (WDM), time-division multiplexing (TDM) and multilevel amplitude/phase modulation <cit.>, data-rate of acoustic communication is approaching its current limit, due to that sound, as a scalar wave, bears no polarization or spin, as opposed to its electromagnetic counterparts. It is stringent to exploit plausible multiplexing mechanisms to encode information in a scalar field with multiple states orthogonal and compatible to existing degree of freedoms (DOFs). Orbital angular momentum (OAM), with the infinite dimensionality of its Hilbert space and unbounded orthogonal states, is a promising candidate.Many efforts have been made in optical OAM-based multiplexing <cit.>, such as spiral phase masks <cit.>, Dammann gratings <cit.>, q-plates <cit.> or interferometers <cit.>. The precise control of sound would result in bulky device if we directly translate the optical mechanisms into acoustics. Previous works on a-OAM beams <cit.> primarily exploit their mechanical effects such as for particle trapping and manipulation <cit.>, only one work has attempted to utilize OAM in acoustic communication <cit.>. However, the method in Ref. <cit.> relies on encoding data on OAM states and only employs this single dimension to realize a static data transfer, while in reality the fast and continuous information transmission should work in a real-time and dynamic manner. In addition, active sensors array is required to scan the spatial field and complex algorithms should be conducted for data decoding, which will not only impose extra loads to the existing communication link both in hardware and software, but also limit the transmission speed due to the time consumption in post data-processing. The active decoding method would also restrict the transmission accuracy since the bit error rate highly dependents on the transducers number in the sensor array<cit.>.We theoretically propose and experimentally validate the twisted acoustic beam with OAM for real-time information transfer in a passive, postprocess-free and sensor-scanning-free paradigm with metasurfaces <cit.> to overcome those aforementioned issues. Rather than encoding data onto OAM states, here we use the a-OAM beams as the data carriers, exhibiting the instinct compatibility with pre-existing DOFs. We enter the null of the twisted beams and take full advantage of this trait, which is usually less significant. A subwavelength acoustic de-multiplexing metasurface (a-DMM) with a thickness of 0.5λ and radius of 0.53λ (λ is sound wavelength, see Supplementary Material for details), as a passive and compact de-multiplexing component, is designed to directly and promptly decode data by a single transducer. Comparisons among the a-DMM-based information transfer in this work, the methods in Ref. <cit.> and other milestone references are shown in Fig. <ref>(a). The pressure transmittance is 91.5% for one a-DMM and a nearly 100% data transmission accuracy is achieved. Advantages of free of signal postprocess and field scanning, passive and compactness, high capacity and accuracy in real-time data transmission will be demonstrated both numerically and experimentally in what follows.Schematic of the a-OAM multiplexing and de-multiplexing mechanism is illustrated in Fig. 1(b), where twisted beams with different topological charge m propagate in an overlaid fashion as orthogonal channels. Note that OAM beam has a spiral phase dislocation e^imθ (θ is azimuthal angle) and null core, which is a crucial feature in our scheme. The multiplexing signal comprising of twisted beams with different m can be expressed as p(r,θ,z,t)=∑_mA_m(t)e^i(mθ+k_z z+ϕ_m(t)), which propagates along z direction in waveguide to avoid attenuation due to diffraction. Here k_z is the wavenumber, the time-dependent amplitude A_m(t) and phase ϕ_m(t) can be merged in the multilevel formats, suggesting the handy combination with pre-existing technologies (e.g., WDM and TDM) without introducing extra loads. We use the angular spectrum (AS) to yield the exact input in transmitting end (see Supplementary Material), similar with that for calculating the profile in acoustic hologram <cit.>. The continuous azimuthal-dependent p_in is discretized by dividing the entrance into eight fan-like sections, individually accessing to eight inputs. As a result of the spatial multiplexing, an enhancement of spectral efficiency by a factor of N can be expected, with N being the total number of a-OAM beams.For separating twisted beams with different m and thereby decoding the data in each a-OAM channel, a mechanism based on acoustic resonance is proposed: N layers of identical passive metasurfaces denoted as a-DMMs, capable of converting acoustic resonances to a-OAM, are impressed successively at the receiving terminal, with the output detected by a single transducer at the center after each layer. Each a-DMM has an OAM of -1, leading the order of all the transmitted a-OAM modes to be lowered by 1. Consequently, the order of a-OAM after the n^th (1 ≤ n≤ N) layer is the linear superposition of the a-OAM value of the incident beams m and the total a-OAM provided by these n structures, reading as m-n. Considering the inherent ‘doughnut’-shaped intensity profile of a twisted beam, only by using m layers of a-DMMs to convert the spiral phase pattern of a m^th order beam to a planar shape can we remove the azimuthal phase term and observe a non-zero intensity at the core. An example is illustrated in Fig. 1(c), which is the equiphase surface of the 1^st OAM beam before and after an a-DMM in waveguide. On the contrary, beams with other a-OAM values still remain spiral phase and null core. Therefore, we can precisely detect the information encoded onto the m^th order beam exclusively after the m^th a-DMM by a single transducer where the updated a-OAM order is exactly zero.We present a demonstration via merging the data in multilevel phase (differential binary phase shift keying, DBPSK) and amplitude (quadrature amplitude modulation, QAM) formats. The notation m^th(A_m,ϕ_m) indicates the m^th a-OAM beam with the amplitude A_m and phase ϕ_m hereafter. The input multiplexing signal is the superposition of the 1^st and 2^nd a-OAM beams carrying the objective data. Acoustic pressure p along central axis of the waveguide in region L1 and L2 after de-multiplexed by the a-DMMs are illustrated in Figs. 1(d)-(f), where the multiplexing signals in the resonance frequency f_0=2287Hz of the a-DMMs carry different information, as function of the distance d from output surfaces of the corresponding layers. The pressure transmittance is 91.5% after one a-DMM, guaranteeing a high transmission efficiency. For better comparison, all the values are normalized by the maximum absolute pressure |p| in 1^st(1,0)+2^nd(1,0) case. The objective data carried by the 1^st and 2^nd twisted beams are perfectly restored after de-multiplexed by the two a-DMMs in regions L1 and L2, respectively. For example, in Fig. 1(d) the received signals in L2 have a phase shift of π which is exactly the phase difference of the 2^nd a-OAM beams between the two input signals, and in Fig. 1(f) the amplitude in the 1^st(2,0)+2^nd(1,0) case is two times (half) of that in the 1^st(1,0)+2^nd(2,0) case in L1 (L2). These results verify that the orthogonality between the a-OAM modes effectively avoids mode coupling and the associated crosstalk in the spatially independent channels. Moreover, they substantially prove that the a-OAM states are also essentially orthogonal to the pre-existing dimensions of phase and amplitude. Meanwhile, due to the resonant nature and consequent high frequency selectivity of the metasurface <cit.>, only the a-OAM beams in the resonant frequency f_0 of the particular a-DMM can be converted to plane wave and detected, which ensures the orthogonality with frequency as well. The frequency selectivity helps to simplify the terminal configuration when combined with the WDM technology in which complicated equipment is required to filter <cit.>, increasing the speed and reducing the burden of the post data processing.We demonstrate the a-OAM multiplexing and de-multiplexing via real-time data transfer. As a visualized example, we use the 1^st and 2^nd a-OAM beams as two channels Ch1 and Ch2, and independently encode the pixels of two pictures, the images of letters “A" and “a", into the phase of a-OAM beams in DBPSK format through numerical simulations, where each pixel is encoded as a binary data (see Supplementary Material for simulation detail). For real-time communication, the multiplexing signal is in pulse modulation, with the central frequency f_0 (period T_0 and wavelength 15 cm), pulse period 20T_0 and duty ratio 0.7, where each pulse cycle contains one-bit data. The eight inputs [cf. Fig. 1(b)] for generating the multiplexing signal and simultaneously enabling the two channels are displayed in Fig. 2(b). In receiving terminal, two a-DMMs are cascaded to sort the 1^st and 2^nd a-OAM beams and two microphones (Mics) are placed to detect the signals in region L1 and L2. The received real-time signal in Ch1 by a Mic in L1 as function of time in each pulse period is partly (first 12 cycles) plotted in Fig. 2(c), where the blue plane is the reference surface for phase comparing and extracting the data in each cycle of the pulse-modulated signal. Data in other cycles and Ch2 is obtained in similar way. The decoded dataflows in Ch1 (1^st beam) and Ch2 (2^nd beam) are displayed in Fig. 2(a) in comparison with the objective. Two images are reconstructed with the received dataflows as shown in Figs. 2(d)-(e), which undistortedly reproduce the pictures of letters “A" and “a".Experiments are conducted to verify the real-time communication based on a-OAM. Photographs of an a-DMM made of UV resin and experimental setup are shown in Figs. 3(a) and (b) (see Supplementary Material). Two a-DMMs, with the radius 0.53λ and thickness 0.5λ, are sequentially inserted in cylindrical waveguide, and two Mics are placed centrally in L1 and L2. As a proof-of-concept experiment, we consider the independent transmission of two images each with 4×4 pixels encoded in the 1^st and 2^nd a-OAM beams. The experimentally received data extracted from the real-time transmitted signals (shown in Supplementary Material) in Ch1 and Ch2, comparing with the objective are displayed in Fig. 3(c). Images retrieved from the two dataflows are illustrated in Figs. 3(d) and (e) where perfect reconstructions are observed, demonstrating the experimental viability of the data transfer based on twisted beams.Furthermore, we combine the a-OAM multiplexing with the multi-carrier modulation (MCM) technology to increase the transmission efficiency within the limited available bandwidth, which is crucial to transfer some urgent information and particularly beneficial in the varying fading conditions <cit.>. The high-speed data stream is separated into several parallel flows of a relatively lower speed, encoded onto different a-OAM beams transmitting simultaneously, and then the decoded data are assembled accordingly. Here we demonstrate the MCM data transfer of the image of letters “NJU", by encoding the image pixels alternately into the 1^st and 2^nd beams. The assembled data stream measured experimentally, in comparison with the simulation results and the objective are displayed in Fig. 4(a), where the inset partly shows the enlarged view of the comparison. Figure 4(b) shows the retrieved image with the received data, which is the exact reproduction of the original picture. The combination of MCM and twisted acoustics would facilitate the high-speed data transfer and improve the efficiency of post processing.To conclude, we propose and experimentally demonstrate a simple and cost-effective scheme of twisted acoustics for real-time postprocess-free, sensor-scanning-free and high-capacity communication, where twisted beams of different OAM values serve as the spatially independent channels to carry information, with the compatibility with pre-existing multiplexing technologies. Subwavelength acoustic metasurfaces are designed as the passive and efficient de-multiplexing component for direct and prompt data decoding without sensors array and post-processing. We prove the effectiveness of the scheme via numerical simulations and experiments. It is noteworthy that the proposed scheme is universal since data transfer with this method is in principle not restricted by the number of OAM beams or transmitting distance, which can be extended to contain more a-OAM channels. Further improvements might lead to encode data by higher-order phase shift keying technology to achieve high capacity and spectral-efficiency acoustic communication.In addition, the a-OAM based data transfer bears the security advantage of resisting to eavesdropping. Conventionally, the information would be covertly intercepted with additional receiver due to the atmospheric scattering, which require extra mathematical encryption. Our scheme offers the security enhancement as it is difficult to read the data without positioning the detector directly in the path of the intended receiver <cit.>. In other words, the recovery is non-trivial. With the intrinsic orthogonality, high decoding efficiency and transmitting accuracy, increased integration density and the potential resistance to eavesdropping, twisted acoustics with OAM would take the acoustic communication to new heights, providing potential to improve the capacity and security of information transmission.This work was supported by the National Key R&D Program of China, (Grant No. 2017YFA0303700), National Natural Science Foundation of China (Grants No. 11634006 and No. 81127901) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. | http://arxiv.org/abs/1706.08944v2 | {
"authors": [
"Xue Jiang",
"Bin Liang",
"Jian-chun Cheng",
"Cheng-Wei Qiu"
],
"categories": [
"physics.app-ph",
"cond-mat.mtrl-sci",
"physics.class-ph"
],
"primary_category": "physics.app-ph",
"published": "20170627171740",
"title": "Twisted Acoustics"
} |
[email protected] Physics of Living Systems Group, Department of Physics, Massachusetts Institute of Technology, 400 Technology Square, Cambridge, MA 02139 Current is a characteristic feature of nonequilibrium systems. In stochastic systems, these currents exhibit fluctuations constrained by the rate of dissipation in accordance with the recently discovered thermodynamic uncertainty relation. Here, we derive a conjugate uncertainty relationship for the first passage time to accumulate a fixed net current. More generally, we use the tools of large-deviation theory to simply connect current fluctuations and first-passage-time fluctuations in the limit of long times and large currents. With this connection, previously discovered symmetries and bounds on the large-deviation function for currents are readily transferred to first passage times.05.70.Ln,05.40.-aFundamental Bounds on First Passage Time Fluctuations for Currents Jordan M. Horowitz December 30, 2023 ================================================================== Introduction.—Thermodynamics constrains the fluctuations of nonequilibrium systems, as evidenced by a growing collection of universal predictions connecting dissipation to fluctuations.Examples include the fluctuation theorems <cit.>, nonequilibrium fluctuation-dissipation theorems <cit.>, and, more recently, the thermodynamic uncertainty relation <cit.>. Remarkably, all these results can be viewed through one unifying lens, namely large-deviation theory <cit.>. In fact, over the past two decades this formalism has proven to be an essential tool for characterizing the dynamical fluctuations of nonequilibrium systems <cit.>.Recently, these techniques have revealed a universal inequality between the far-from-equilibrium fluctuations in current—such as the flow of particles, energy or entropy—with the near-equilibrium fluctuations predicted by linear-response theory <cit.>.A useful corollary is the thermodynamic uncertainty relation <cit.>, which offers a fundamental trade-off between typical current fluctuations and dissipation [In the long-time limit, the typical fluctuations exhibit small deviations about the steady-state current.]. Specifically, a nonequilibrium Markov process generating an average time-integrated current <J> during a long observation time T_ obs has a variance Var(J) constrained by the mean entropy-production rate σ (with Boltzmann's constant k_ B=1):Var(J)/⟨ J⟩^2≥2/T_ obsσ.Thus, reducing fluctuations comes with an energetic cost.A significant body of recent work has analyzed such current fluctuations for a fixed observation time <cit.>. In this Letter, we consider the complementary problem, analyzing the fluctuations of first passage times T to reach a large threshold time-integratedcurrent J_ thr (see Fig. <ref>). We show that properties of the first passage time distribution for asymptotically large J_ thr follow simply from knowledge of the current fluctuations. This conjugate relationship between fixed-time and fixed-current trajectory ensembles mirrors the study of inverse or adjoint processes in queuing theory <cit.>, and it extends Garrahan's work on first passage time fluctuations of dynamical activity—a monotonically increasing counting variable <cit.>—to current variables which can grow or shrink. By relating the conjugate problems, we are able to transform inequalities governing current fluctuations into associated inequalities for passage-time fluctuations, as well as offer fresh insight into recent predictions for entropy-production first passage times <cit.>. For instance, we show that the distribution for the time T to first hit a large threshold current J_ thr must satisfy a corresponding uncertainty relation:Var(T)/⟨ T⟩^2≥2/⟨ T ⟩σ. The two faces of the thermodynamic uncertainty relationship can be viewed as two ways to infer a bound on the entropy-production rate—one utilizing the current fluctuations in a fixed-time ensemble and the other utilizing the time fluctuations in a fixed-current ensemble. Though these two sets of fluctuations contain equivalent information, we emphasize that the physical measurements are quite distinct.Setup.— To make the notions concrete, we focus our presentation on nonequilibrium systems that can be modeled as Markov jump processes. Specifically, we have in mind a mesoscopic system with states i=1,…, N, whose time-varying probability density p={p_i}_i=1^N evolves according to the master equation ṗ = 𝕎p, where 𝕎_ij is the probability rate to transition from j→ i, and -𝕎_ii=∑_j≠ i𝕎_ji is the exit rate from i. We assume that 𝕎 is irreducible – so that a unique steady-state exists – and that every transition is reversible, that is 𝕎_ij≠ 0 only when 𝕎_ji≠ 0. Thermodynamics enters by requiring transitions to satisfy local detailed balance. The ratio of rates for each transition can then be identified with a generalized thermodynamic force ℱ_ij=ln(𝕎_ij/𝕎_ji) [The thermodynamic force may alternatively be defined in terms of the steady state density π as ℱ_ij = ln(𝕎_ijπ_j / 𝕎_jiπ_i). These two definitions differ by the change in Shannon entropy ln(π_j / π_i) which averages to zero over a long trajectory.], which quantifies the flow of free energy into the surrounding environment <cit.>.Fluctuating currents represent the net buildup of transitions between the system's mesoscopic states. Indeed, in any given stochastic realization of our system's evolution there will be some random number of net transitions, or current, between every pair of states j→ i, which we label as J_ij. Our interest though is in generalized currents obtained as superpositions of mesoscopic transitions, J ≡∑_i>j d_ij J_ij, where the d_ij indicate how much a particular transition contributes. Such generalized currents often represent a measurable global flow through the system, such as the ATP consumption throughout a biochemical network, or the net flow of heat between multiple thermal reservoirs <cit.>. A particularly important example is the fluctuating environmental entropy production Σ obtained by choosing d_ij = ln𝕎_ij / 𝕎_ji. Its average rate σ=lim_T_ obs→∞⟨Σ⟩/T_ obs measures the time irreversibility of the dynamics.For long observation times T_ obs, the probability of observing a current J satisfies a large-deviation principle P(J|T_ obs)≍ e^-T_ obsI(J/T_ obs) with large-deviation rate function I(j) <cit.>, where the lowercase letter j ≡ J / T_ obs represents an intensive quantity. The large-deviation function I captures not just the typical fluctuations predicted by the central-limit theorem but also the relative likelihood of exponentially rare events. A useful complementary characterization of the fluctuations is through the scaled cumulant generating function (SCGF) ψ(λ)=lim_T_ obs→∞(1/T_ obs)ln⟨ e^-λ J⟩, with the expectation taken over trajectories of length T_ obs. Derivatives of ψ at the origin encode all the long-time current cumulants. The pair I and ψ are intimately related through the Legendre-Fenchel transform, as graphically illustrated in Fig. <ref> <cit.>.Universal symmetries and bounds on I(commensurately ψ) have refined our understanding of the thermodynamics of nonequilibrium systems. In the following, we develop a complementary point of view based on current first passage times. First passage time fluctuations for large current.— We now consider a large (in magnitude) fixed amount of accumulated current J_ thr and seek the time at which that threshold current is first reached. As seen in Fig. <ref>, the mean first passage time scales extensively with the magnitude of J_ thr, suggesting a large-deviation form for the first passage time distribution F(T|J_ thr). We note, however, that J_ thr can be either positive or negative, and introduce two different rate functions, ϕ_+(t) and ϕ_-(t), to handle these cases:F(T|J_ thr)≍ e^-J_ thrϕ_+(T/J_ thr),J_ thr > 0e^J_ thrϕ_-(-T/J_ thr),J_ thr < 0.Correspondingly, there are now two different SCGFs g_±(μ) = lim_J_ thr→±∞ (1 / J_ thr) ln<e^-μ T>, with the expectation computed over trajectories having a fixed time-integrated current J_ thr. Without loss of generality, we assume a choice of {d_ij} such that <J> > 0. In this case, the + subscript corresponds to branches quantifying typical (positive-current) fluctuations and the - subscript corresponds to rare (negative-current) branches. It is useful to also split ψ into two branches, ψ_+ with negative slope and ψ_- with positive slope (see Fig. <ref>). Our central result is that the large deviations in scaled first passage times t≡ T/|J_ thr| are completely determined by the large-deviation functions for current fluctuations:ϕ_±(t)=tI(± 1/t), g_±(μ)=ψ_±^-1(μ).Analogous relations have appeared for counting variables <cit.> and for entropy-production fluctuations <cit.>, but we show these connections are, in fact, more general and extend to all currents. Thus, all known properties of I—most notably, symmetries and bounds—can naturally be translated to ϕ.Here, we offer a heuristic argument for Eq. (<ref>) assuming positive current. A sketch of a proof is included at the end of the Letter, and a more detailed proof is provided in the Supplemental Material (SM). To start, we write 𝒫(γ) to denote the probability distribution for a mesoscopic trajectory γ—that is a sequence of states visited by the system and their jump times. Then the likelihood of a large first passage time T=tJ to a large current J can be conveniently expressed as P(T=tJ)=∫ dγ δ(T-tJ) 𝒫(γ),where the integral is over all trajectories. However, the only trajectories that can contribute to this integral have current J. Furthermore, large current can only be attained after a long time. Taken together these observations suggest we can replace 𝒫 with the large-deviation form for large T [In passing from 𝒫(γ) to e^-T I(J/T) we must recognize that I(J/T) measures the asymptotic probability of a trajectory with net current J in time T, including trajectories which have already hit J at earlier times. Provided |<J>| > 0, the probability that the trajectory is making a first passage dwarfs the probability of repeated passages in the large J limit.]:P(T=tJ)≍∫ dJ δ(T-tJ) e^-TI(J/T)=e^-JtI(1/t),which implies ϕ_+(t) = t I(1/t), and g_+(μ) follows by Legendre-Fenchel transform. Put simply, switching from current to first passage time is a change of variables where we replace current by its inverse.We now turn to the implications of Eq. (<ref>). For any generalized current, itslong-time fluctuations are constrained by the entropy-production rate via Eq. (<ref>). This constraint actually follows from an inequality on the large-deviation rate function,I(j)≤(j-⟨ j⟩)^2/4⟨ j⟩^2σ≡ I_ bnd(j).Translating to first passage time fluctuations, we haveϕ_+(t)≤(t-⟨ t⟩)^2/4tσ≡ϕ_ bnd(t),after noting that the typical behavior ⟨ j⟩=1/⟨ t⟩ does not depend on the choice of ensemble – fixed T_ obs versus fixed J_ thr. Equation (<ref>) follows since the large J_ thr variance is computed in terms of derivatives of the large-deviation function as Var(T) = J_ thr / ϕ_+”(<t>) <cit.>. Thus, dissipation is a fundamental constraint to controlling first passage time fluctuations as well as current fluctuations.Together Eqs. (<ref>) and (<ref>) point to a remarkable property of the stochastic evolution of currents, which is best appreciated by normalizing the large-deviation forms e^-T_ obsI_ bnd(j) and e^-J_ thrϕ_ bnd(t). For currents, we have a Gaussian distributionP_ bnd(j) = √(T_ obsσ/4 π<j>^2)exp[-T_ obs (j - <j>)^2 σ/4 <j>^2],whereas the first passage time distribution is an inverse GaussianF_ bnd(t) = √(J_ thrσ<t>^2 /4 π t^3)exp[-J_ thr(t - <t>)^2 σ/4 t].Remarkably, these are the distributions we would have predicted if we had simply treated the evolution of the current as a one-dimensional diffusion process with constant drift ⟨ j⟩ and diffusion coefficient σ/⟨ j⟩^2 <cit.>. This observation suggests that while the precise dynamics of the currents is generally complex, there is a simple auxiliary diffusion process that constrains it, reminiscent of the universal form observed for the stochastic evolution of the entropy production as a drift-diffusion process <cit.>.First passage time fluctuations for negative current and the fluctuation theorem.— We have focused primarily on first passage times to reach a (typical) positive current. We can also consider the first passage time to the exponentially suppressed negative currents that arise due to trajectories that appear to run backwards in time. The distribution for the time to reach J_ thr < 0 scales according to ϕ_-(t), which can be related to ψ_-(λ) (see Fig. <ref>). This connection is especially interesting when ψ posses a symmetry that relates its two branches ψ_+ and ψ_-, because this naturally translates to a relationship between ϕ_+ and ϕ_-.Generically, ψ_- vanishes at some λ^*. For certain currents it also satisfies ψ_+(λ) = ψ_-(λ^* - λ). As an example, the fluctuation theorem implies such a symmetry with λ^*=1 for the entropy production (itself a generalized current) <cit.>. Symmetry of ψ yields a corresponding symmetry in g_±: g_+(μ) = -g_-(μ) + λ^*. Taking the Legendre-Fenchel transform givesϕ_+(t) = ϕ_-(t) - λ^*,indicating that ϕ_+ and ϕ_- differ by a constant offset when the SCGF symmetry is present. Equation (<ref>) must be interpreted carefully, as it compares large-deviation functions for two different distributions. Typically, large-deviation rate functions are shifted such that their minimum equals zero. In this case, a symmetrical ψ implies that ϕ_+ and ϕ_- are identical, and the large-current first passage time distribution F(T|J_ thr) is the same for both positive and negative J_ thr. While the constant offset in Eq. (<ref>) does not affect the form of F(T|J_ thr), it reflects the fact that the probability of reaching |J_ thr| exceeds that of reaching -|J_ thr| by a factor of e^λ^* |J_ thr|. Using the same methods as those in this Letter, Saito and Dhar reached similar conclusions for the case that the generalized current is the entropy production <cit.>, and Neri et al. have proven a corresponding fluctuation theorem for entropy production stopping times using Martingale theory <cit.>. Our result, Eq. (<ref>), extends more generally to any current satisfying a SCGF symmetry about λ^*, including the example of the next section.Illustrative example.— To demonstrate the bounds in a more explicit context, we solve for the large-deviation behavior of a minimal model for an enzyme-mediated reaction from reactant R to product P. The enzyme can be either in a ground state E or an activated state E^*, and the E↔ E^* transformations proceed via one of three pathways: (1) the enzyme exchanges heat with a thermal bath, (2) the enzyme accepts free energy by converting an activated fuel molecule F^* into a deactivated form F, or (3) the activated enzyme converts R → P. Each of these pathways proceeds forward or backward, as depicted in Fig. <ref>, with six rate constants defining the model. We follow the net transformations of R into P as the accumulated current J, so the first passage time can be interpreted as the time to generate J product molecules.The analytical solution of this model using standard methods is outlined in the SM. Figure <ref> graphically shows the large-deviation function bound, Eq. (<ref>), as well as the uncertainty bound, Eq. (<ref>) (see inset). The analytical calculations are supplemented by trajectory sampling with finite J_ thr, the results of which are plotted with colored markers in Fig. <ref>. Motivated by the t^-3/2 prefactor in Eq. (<ref>), we extract estimates for ϕ_+(t) from the sampled trajectories by first approximating F(T|J_ thr) with a histogram and then computingϕ_+^ est(t) = -1/J_ thr(ln F(tJ_ thr|J_ thr) + 3/2ln t) + C_ off,where C_ off is a constant offset used to set the minimum of ϕ_+^ est to zero. We observe that the large-deviation form (and, consequently, the thermodynamic uncertainty relation) remain valid even for small J_ thr.Conclusion.— In the large-deviation limit, we have shown that current fluctuations with fixed observation time are intimately related to the fluctuations in first passage times to large current. As a result, we have seen how the thermodynamic uncertainty relation and the fluctuation theorem for entropy production naturally lead to a universal symmetry and bounds on first passage time fluctuations. Tighter-than-quadratic bounds on current large-deviation fluctuations <cit.> also readily translate to corresponding first passage time bounds.Practically, we anticipate that it will be useful to convert between fixed-time and fixed-current ensembles since some experiments are more naturally suited to one than the other. For example, imagine we seek a dissipation bound for the enzyme-mediated reaction in Fig. <ref>. Fluctuations in product formation after time T_ obs could be measured spectroscopically, assuming Beer's law and a calibrated mapping from fluorescence intensity to product concentration. But the fixed J_ thr ensemble offers an advantage. By measuring first passage time fluctuations to reach a fixed fluorescence intensity, the mapping between fluorescence and concentration could be avoided altogether. More ambitiously, we expect the fluctuating time ensemble to be a natural way to analyze the role of dissipation in Brownian clocks <cit.>.Sketch of a proof for Eq. (<ref>).—The main result, Eq. (<ref>), consists of two relations: one connects the large-deviation rate function I with ϕ_±, the other connects ψ with g. Here we sketch a proof of g_±(μ) = ψ_±^-1(μ). The relationship between I and ϕ_± follows by applying the Gärtner-Ellis theorem to compute I from ψ and ϕ_± from g_±. More details are presented in the SM.The basic strategy is to express both g and ψ in terms of spectral properties of a tilted rate matrix 𝕎(λ), whose elements are given by 𝕎_ij(λ) = 𝕎_ij e^-λ d_ij. The first half of this connection is well known; the largest eigenvalue of 𝕎(λ) is the SCGF ψ(λ). <cit.>. Expressing g in terms of the tilted rate matrix requires a slightly more involved calculation following the general strategy of <cit.>.Let F_ij(T|J) be the distribution of times T to first accumulate J current with a jump to i, conditioned upon a start in j. We connect F_ij to the transition probability P_ij(J, T) to go from j → i in time T, having accumulated current J via the renewal equation: P(J,T)=∫_0^T dtP(0,T-t)· F(t|J), written in matrix notation. The convolution is simplified by Laplace transform (denoted with a tilde) to convert from T to μ, ultimately yielding e^-J g_±(μ)≍⟨ e^-μ T⟩ = F̃(μ|J) ≍P̃(J, μ). Furthermore, P̃(J, μ) can be expressed in terms of the tilted rate matrix via an inverse Laplace transform of P̃(λ, μ) = 1 / (𝕎(λ) - μ𝕀), where the caret denotes a Laplace transform from J to λ. Using complex analysis to perform the inverse transform, we obtain e^-J g_±(μ)≍ e^λ̅J, where λ̅ = ψ_+^-1(μ) for J>0 and λ̅ = ψ_-^-1(μ) for J<0. Hence, g_± and ψ_± are inverses. We gratefully acknowledge the Gordon and Betty Moore Foundation for supporting TRG and JMH as Physics of Living Systems Fellows through Grant GBMF4513. § SUPPLEMENTAL MATERIAL§ DERIVATIONS OF MAIN RESULTThe main result of the main text, Eq. (4), consists of two relations: one connects the large-deviation rate function I with ϕ_±, the other connects ψ with g. We first prove g_±(μ) = ψ_±^-1(μ). The relationship between I and ϕ_± follows by applying the Gärtner-Ellis theorem to compute I from ψ and ϕ_± from g_±. §.§ Scaled cumulant generating functions g and ψ are inversesThe basic strategy is to express both g and ψ in terms of spectral properties of a tilted rate matrix 𝕎(λ), whose elements are given by 𝕎_ij(λ) = 𝕎_ij e^-λ d_ij. The first half of this connection is well-known; starting with initial density ρ, the generating function for currents is obtained by the averaging over trajectories of length T as <e^-λ J> = 1· e^𝕎(λ) T·ρ, where 1 = {1, , 1} <cit.>. It follows that the largest eigenvalue of 𝕎(λ) is the scaled cumulant generating function ψ(λ)=lim_T→∞(1/T)ln<e^-λ J>.Expressing g in terms of 𝕎(λ) requires a slightly more involved calculation. We follow the general strategy of <cit.>. First, we recall that g is naturally expressed in terms of the Laplace transform of the first passage time distribution F̃(μ|J) ≡∫_0^∞ dT e^-μ T F(T|J) asg_±(μ)=lim_J→±∞(1/J)lnF̃(μ|J).Thus, our goal is to express the large J asymptotics of F̃ in terms of 𝕎(λ).To this end, we introduce F_ij(T|J) as the distribution of times to first reach J current by a transition to i, given a start in j. We connect F_ij to the transition probability P_ij(J, T) to go from j → i in time T, having accumulated current J via the renewal equation:P_ij(J,T)=∫_0^T dt ∑_k P_ik(0,T-t) F_kj(t|J).The convolution is made simpler by performing the Laplace transform (denoted with a tilde) to convert from T to conjugate field μ. After minor rearrangement, the Laplace-transformed renewal equation leads toF̃(μ|J)= 1·𝐅̃(μ|J)·ρ=1·P̃(0,μ)^-1·P̃(J,μ)·ρ,where 𝐅̃ and 𝐏̃ are matrices with ij matrix elements F̃_ij and P̃_ij, respsectively. The only term that contributes for large J is P̃(J,μ), which we analyze by taking an additional (two-sided) Laplace transform (denoted with a caret), this time a transform that converts from J to a conjugate field λ:P̃(λ,μ) =∫_-∞^∞ dJe^-λ J∫_0^∞ dTe^-μ T P(J,T).By first performing the integral over J, we obtainP̃(λ,μ)=∫_0^∞ dTe^-(μ𝕀-𝕎(λ))T=1/𝕎(λ)-μ𝕀.The integral is convergent only in the region ψ_+^-1(μ)<λ<ψ_-^-1(μ). We obtain P̃(J, μ) by using a complex integral to invert the two-sided Laplace transform:P̃(J, μ) = 1/2 π i∫_C dλ P̃(λ, μ) e^λ J= 1/2 π i∫_C dλ e^λ J/𝕎(λ) - μ𝕀.The contour C is chosen to be an infinite semicircle centered at a value of λ chosen to fall inside the region of convergence. So that the contour integral along the semicircular arc vanishes, C must enclose the right half plane for J<0 or the left half plane for J>0. The integral can then be performed using the residue theorem. The asymptotic form for large J is determined by the dominant pole, which comes from the the largest eigenvalue ψ(λ) of 𝕎(λ).Hence, P̃(J, μ) ≍ e^λ̅ J, where λ̅ = ψ_+^-1(μ) for J > 0 and λ̅ = ψ_-^-1(μ) for J < 0. Using Eq. (<ref>), we get the large J asymptotic scaling of the Laplace-transformed first-passage-time distribution, F̃(μ | J) ≍ e^λ̅ J, and from Eq. (<ref>) the SCGF g_±(μ) = λ̅ = ψ_±^-1(μ). We see that ψ and g are indeed inverses. §.§ Large-deviation rate functions are related by ϕ_±(t) = t I(± 1 / t) By the Gärtner-Ellis theorem, ϕ_± and g_± are related by a Legendre-Fenchel transform <cit.>. Hence,ϕ_±(t)= ∓min_μ (g_±(μ) ±μ t)= ∓ g_±(μ̃) - μ̃ t, with g_±'(μ̃) = ∓ t,where μ̃ is the exponential bias that renders t = T / |J_ thr| typical. Similarly, in the fluctuating current ensemble, we define the exponential bias λ̃ that renders j = J / T_ obs typical. The Legendre-Fenchel transform relates I to ψ in terms of this λ̃:I(j)= -min_λ (ψ(λ) + λ j)= - ψ(λ̃) - λ̃ j,with ψ'(λ̃) = -j.To connect Eqs. (<ref>) and (<ref>), we note that the derivatives of g are related to those of ψ since g and ψ are inverses, g_±(ψ_±(λ̃)) = λ̃. Differentiating both sides of this equation and rearranging gives g_±'(ψ_±(λ̃)) = 1 / ψ_±'(λ̃). Note that the condition defining μ̃ in Eq. (<ref>), g_±'(μ̃) = ∓ t, can now be expressed as a condition on ψ: when μ̃ = ψ_±(λ̃), then ψ_±'(λ̃) = ∓ 1/t. Inserting this back into Eq. (<ref>) givesϕ_±(t)= ∓ g_±(ψ_±(λ̃)) - ψ_±(λ̃) t,where ψ_±'(λ̃) = ∓ t^-1= - ψ_±(λ̃) t ∓λ̃,where ψ_±'(λ̃) = ∓ t^-1= t (- ψ_±(λ̃) ∓λ̃ t^-1),where ψ_±'(λ̃) = ∓ t^-1= t I(± 1 / t),with the last line following from Eq. (<ref>).§ TWO-STATE, THREE-PATHWAY MODEL Analytical forms for ψ, I, g_±, and ϕ_± can be found for the two-state, three-pathway model of the main text. We take d_12^ rxn = 1, d_21^ rxn = -1 and d_12^ therm = d_21^ therm = d_12^ fuel = d_21^ fuel = 0. Thus we monitor the rate of net current from reactant to products, which has a steady-state value<j> = (β - α) / S,whereS= k^ therm_12 + k^ therm_21 + k^ fuel_12 + k^ fuel_21 + k^ rxn_12 + k^ rxn_21, α = k^ rxn_21 (k^ therm_12 + k^ fuel_12), β = k^ rxn_12 (k^ therm_21 + k^ fuel_21). The tilted rate matrix for this reactant to product current is𝕎(λ) = [- k_21^ rxn - k_21^ therm - k_21^ fuel k_12^ rxn e^-λ + k_12^ therm + k_12^ fuel;k_21^ rxn e^λ + k_21^ therm + k_21^ fuel -k_12^ rxn - k_12^ therm - k_12^ fuel ].The scaled cumulant generating function (SCGF) for current is found as the maximum eigenvalue of 𝕎(λ):ψ(λ) = -S/2 + 1/2√(S^2 + 4(1 - e^-λ)(α e^-λ - β)).In this case, ψ^-1 = g can be computed analytically. As clear from Fig. 2 of the main text, the inversion requires us to define a “+” and “-” branch of g:g_±(μ) = ln(α + β + S μ + μ^2 ±√((α + β + S μ + μ^2)^2 - 4 αβ)/2 α).Using the Gärtner-Ellis theorem, we compute I and ϕ_± with Legendre-Fenchel transforms,I(j)= -min_λ(ψ(λ) + λ j) ϕ_±(t)= ∓min_μ(g_±(μ) ±μ t).For this two-state model, the minimizations can be carried out analytically with a moderate amount of algebra. For compactness, we define two new functions:γ(j) = 2 + √(4 + j^-2(S^2 - 4(α + β) + 4 αβ j^-2))andδ(j) = √((S^2 - 4(α + β))j^-2 + 4 γ(j)).In terms of γ and δ we find the rate functions: I(j) = j/2(S/j - δ(j) + 2 ln (2 α j^-2) - 2 ln [γ(j) - δ(j)]),j ≥ 0-j/2(-S/j - δ(j) + 2 ln (2 α j^-2) - 2 ln [γ(j) - δ(j)]) + ln (α / β),j < 0 , ϕ_+(t) = 1/2(S t - δ(t^-1) + 2 ln(2 α t^2) - 2 ln[γ(t^-1) - δ(t^-1)]).andϕ_-(t) = ϕ_+(t) + ln(β / α) Observe that this final equation agrees with Eq. (11) of the main text, where λ^* = ln(α / β). As discussed in the main text, the fact that ψ_+(t) and ψ_-(t) have identical t-dependence is a consequence of the symmetry ψ_+(λ) = ψ_-(λ^* - λ). | http://arxiv.org/abs/1706.09027v2 | {
"authors": [
"Todd R. Gingrich",
"Jordan M. Horowitz"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170627195018",
"title": "Fundamental Bounds on First Passage Time Fluctuations for Currents"
} |
Dimensional crossover of Bose-Einstein condensation phenomenain quantum gases confined within slab geometries Francesco Delfino and Ettore Vicari December 30, 2023 ================================================================================================================= Understanding the properties of novel solid-state quantum emitters is pivotal for a variety of applications in field ranging from quantum optics to biology. Recently discovered defects in hexagonal boron nitride are especially interesting, as they offer much desired characteristics such as narrow emission lines and photostability. Here, we study the dependence of the emission on the excitation wavelength. We find that, in order to achieve bright single photon emission with high quantum efficiency, the excitation wavelength has to be matched to the emitter. This is a strong indication that the emitters possess a complex level scheme and cannot be described by a simple two or three level system. Using this excitation dependence of the emission, we thus gain further insight to the internal level scheme and demonstrate how to distinguish different emitters both spatially as well as in terms of their photon correlations.Research on quantum emitters in solid-state materials has gainedmomentum with the discovery of a variety of new emittersin recent yearsand the first successful attempts toengineer their properties. <cit.> Such solid-state emittersare a promising alternativeto trapped atoms and ions in quantum information processing <cit.> as they feature narrowtransition lines and long coherence times while having advantages in many aspects, in particularin terms of scalability and miniaturization. <cit.> As emitters in a solid material can easily be moved around and brought into the vicinity ofother structures, they resemble nearly ideal probes and can be used invarious sensing experiments, e.g., for measuring their electric and magneticenvironment. <cit.> Furthermore, solid-state quantum emitters can be extremely photostable andcan serve as an alternative to organic dyes as biomarkers. <cit.>Recently, quantum emitters hosted in atomically thin, socalled two-dimensional, materials have beendiscovered. <cit.> These materials are promising for a variety of applications, rangingfrom optical switching to biomedical applications. <cit.> One of these emitters, found in hexagonal boron nitride (hBN), has turned out to be a bright, photostable, room temperature single photon source. Furthermore, it possesnarrow emission lines and can be excited using non-linear processes. <cit.> Thanks to all these attractive properties, first attempts to integrate such defects inphotonic structures were successfully carried out. <cit.> Nevertheless, up to date the details of the emitters'level structure remain elusive.So far, single photon emitters in hBN have been studied at differenttemperatures <cit.> and different emission wavelengths <cit.> and mechanisms to alter the emissionwavelength optically has been investigated. <cit.> In addition, first principle calculations using group and density-functional theoryof the energy levels have been carried out <cit.> to get insightinto the atomic structure of the defects.A study of the polarization selection rules of the zero phonon line of the defects revealed a misalignment of emission and absorption dipole – a strong indication of a multi-level system.<cit.> In order to control the creation of these defects and to understand their atomic origin,defect formation has been studied for example at different annealing temperatures, <cit.> and different atmospheres, <cit.>and with etching and ion implantation. <cit.>Despite these efforts, the atomic origin remains unknown and more informationont he properties of the emitters needs to be gathered.Here, we study the level structure of defects in multilayer hBN flakes by photoluminescence excitation (PLE)spectroscopy on single emitters. By varying the excitation wavelength while monitoring emissionintensity and emission spectrum (Figure <ref>a), we gain knowledge on thelevel structure of the emitter and demonstrate the spectral dependence of the quantum efficiency.The optical setup consists of a home-built confocal microscope, fiber coupledto an optical parametric oscillator (OPO, Opium, Radiantis) in another lab, a spectrometer (Andor Shamrock 303i) with a cooled CCD camera (Andor iDus),and two avalanche photodiodes (APDs, Micro Photon Devices) in a Hanbury Brown and Twiss (HBT) configuration.The OPO emission was used to excite the hBN nanoflakes through a 0.85 NA 60X magnification objective(Edmund Optics).The laser powers are measuredin front of the slightly overfilled objective lens. The emitted photoluminescence was collected through the sameobective and separated from the excitation light by means of a 50:50 beamsplitter.The photoluminescence was filtered by a 633 longpass filterand fiber coupled either to the spectrometer or the APDs. The samples was mounted on a x-y-z piezo-translationstage to facilitate confocal photoluminescence scans.The samples used were hBN nanoflakes (in ethanol/water, Graphene Supermarket)drop casted on clean silicon wafers. After evaporation of the solvent,the samples were annealed in a nitrogen environment at 500 for four hours in an oven (Unitemp RTP-150).The annealing increase the number of emitters on the substrates, although brightsingle photon emitters were already found on non-annealed samples.The scanning electron microscope (SEM) images (Figure <ref>b-c) show typical nanoflakesaround 100-500 in size, with thicknesses estimatedaround 10-40.Additional TEM and Raman characterisation of the nanoflakes (available in the Supporting Information) confirms the composition and crystallinity of the nanoflakes. Panels (d-f) of Figure <ref> illustrate the effect that is investigated in more detail in the following.Confocal images show two different, adjacent, quantum emitterslocated by the white and yellow circles. Specifically, the brightest emitter in panel (d),corresponding to 530 excitation laser wavelength, is only weakly visible inthe scans using 600 illumination. Conversely, the strongest emitter in the 600 case is not visible in the 530 scans.However, both these emitters appear in the confocal scans using 550 light.These strikingly different images imply that they have very different excitation spectra and, therefore, level structure. To investigate this effect in more detail we studied the photoluminescence of severalnanoflakes with emission lines ranging from 630 to 740,using 530-620 excitationfrom the OPO. An overview of the photolumiscence spectra of a nanoflake as a function ofexcitation wavelength can be seen in Figure <ref>a.In this specific case, two emission lines stand out: at 656 and676, although the spectra include more features. As in Figure <ref>,these two emission lines vary in strength depending on the excitation wavelength. Specifically,the line at 656 is the brightest under 590illumination, while the 676 line is most efficiently excited at 540. Furthermore, we studied the power dependence of the emission using different excitation wavelengths.This is important, as the count rate for an emitter driven in saturation is directly linked to the quantum efficiency.Much like the excitation spectra, the saturation curves (Figure <ref>b)also show a strong dependence on the excitation wavelength. We fitted the detected counts at the spectral maximaof the two main emission peaks to the expected behavior of the count rate: C_out = R_∞I_in/I_in+I_s+B , where R_∞ is the maximum rate out, I_in is the input power, I_s is thesaturation power and B is the constant background. For the 656(676)line thefits yield R_∞ of 13k (12k), 40k (14k) and 44k (0.6k) counts for530, 570, and 600 excitation, respectively.Furthermore, the saturation powers are140 (520),1250 (650),and 650 (320), respectively. As in a two level system, the value R_∞is only governed by the decay process,the different R_∞ indicate that this system cannot be treated as a simple two level system.However, with these measurements it is not clear whether the two lines in the emitted spectrum stemfrom the same single emitter or two nearby emitters that could not, in contrast to theemitters shown in Figure <ref>, be spatially resolved. In order to discriminate between these two possibilities, we performed antibunchingmeasurements at 300 excitation power (Figure <ref>c) using theHBT setup. Comparing the results with the photoluminsescence spectra in Figure <ref>d,we can conclude that when a single line is dominating (at 600 excitation),the antibunching signature is clear with a peak missing at Δt=0 in the coincidence graphs.Conversely, as on changing the wavelength, the 676 line grows in strength relative to the656 line, a peak in the correlation function at Δt=0 appears and increases in height.The results therefore imply that the two lines in this case comefrom two different emitters located within the same diffraction limited spot, possibly within the same nanoflake. Figure <ref> shows the excitation and emission spectra for several hBN nanoflakesingle photon emitters. Clearly, different emitters have different emissionas well as excitation spectra.No variation of the width or location of the emission lines was found as a functionof the excitation wavelength. Rather, these emitters have stable and reproducibleemission during the scans with narrow lines at a well-defined wavelength. The excitation spectra typically contain a distinct resonance, indicating that theexcitation brings the system into an intermediate energy level, located in the band gap of hBN.From this level, the excitation is transferred to the final excited state,from which photoluminescence occurs. There is also the possibility, that the excitationdecays to the ground without emitting a photon at all, or that the photon emitted is notdetected because it does not fall in the solid angle the microscope objective collects orit falls out of the detection range (see Figure <ref>a). To assess the absolute quantum efficiency of the emitters we compared thedetected photons with the number of photons an ideal emitter would provide. For this,we assume that the ideal emitter would provide one photon per cycle, that is, 80 million counts inour case (here, we neglected re-excitation in the same pulses, as our pulses are muchshorter than the excited state's lifetime). By characterizing the efficiency of our microscope, and taking into account the radiation patterns of dipolar emitters near a silicon/air interface, we deduce the quantum efficiency of the emitters. The emission patterns in Figure <ref>c-d are calculated for an air/silicon interface, at 660nm with n_Si=3.8 and neglecting the losses in the silicon. <cit.>For the emitters in Figure <ref>, the quantum efficiency is estimated to be 0.5-1.0 and 0.2-0.6 for the 656nm and 676nm lines, respectively, for the most efficient excitation wavelengths. Here, we assumed an in-plane dipole (see Figure <ref>b-d) nofurther from the silicon surface than 40nm. The relative spectral dependence of the quantum efficiencies for these emitters is shown in Figure <ref>f (for more data, please see Supporting Information). The large variation observed here clearly shows that when working with single defects in hBN not only the emission should be considered, but also the excitation wavelength is of importance. Here, we emphesize that this is fundamentally different from a change in excitationefficiency as can be observed in many single photon emitters, for example nitrogen vacancy centers in diamond. <cit.>We find that the quantum efficiency has a different spectral dependence compared to therelative excitation efficiency. Naturally, a wavelength-dependent absorption profile will affect the emitted power from a given quantum emitter, but once excited the quantum efficiency from this level is given by the different decay channels. The quantum efficiency is therefore not expected to vary with the excitation wavelength for a simple two-level system. Our results thus imply that there are multiple levels that compete with each other for the excitation energy and that this competition varies in strength over the studied spectral range, ultimately affecting the quantum efficiency of the emission.From these findings, we can now try to get further insight on the level system of the defects. Figure <ref>a show a schematic of possible internal energy levels that includes radiative andnon-radiative decay channels and correctly accounts for the wavelength dependence of theexcitation efficiency. However, due to the varying wavelength dependence of R_∞ and I_satpresentedin Figure <ref>b, it is more likely that there are more (dark)levels involved in the energy dissipation as shown in Figure <ref>b-c.These levels could possibly be excited directly by the laser, and compete with theintermediate level for direct excitation, or be excited via the intermediate level,and compete with the radiative decay channel. Depending on the position of this additional level,one ends up with the different excitation spectra and R_∞, as for the emitters in Figure <ref>. In conclusion, we have shown that the excitation efficiency as well as quantum efficiency ofemitters in hBN are strongly wavelength dependent. This can be used to separate closely spacedemitters as shown in Figure <ref>d-f and Figure <ref>, which together with their photostability makes these emitters a potential candidate for super-resolution imaging techniques. <cit.> Another consequence, for experiments which aim to use defects in hBN as efficient emitters,is that the excitation wavelength has to be tuned to gain the highest quantum efficiency. This is especially important for quantum information processing techniques. Our findings suggest that the level structure of the defects in hBN is muchmore complex than a two or three level system. More details on thelevel structure could be probed using multiple wavelength excitation and pump-probe approaches.§ ACKNOWLEDGEMENT The authors acknowledge financial support from the European Community's Seventh Framework Program under grant QnanoMECA (64790), Fundació Privada Cellex, and the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0522) and grant FIS2016-80293-R, and Swedish Research Council (637-2014-6894). Dr. Fei Ye and Dr. Stephan Steinhauer are respectively acknowledged for their assistance with TEM operation and analysis.10Aharonovich2016 I. Aharonovich, D. Englund, M. Toth. Nat. Photon. 2016, 10, 10, 631.Monroe2002 C. Monroe. Nature 2002, 416, 6877, 238.Obrien2009 J. L. O'Brien, A. Furusawa, J. Vuckovic. Nat. Photon. 2009, 3, 12, 687.Balasubramanian2008 G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, J. Wrachtrup. Nature 2008, 455, 648.Taylor2008 J. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. Hemmer, A. Yacoby, R. Walsworth, M. Lukin. Nat. Phys. 2008, 4, 810.Dolde2011 F. Dolde, H. Fedder, M. Doherty, T. Nöbauer, F. Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L. Hollenberg, F. Jelezko, W. J. Nat. Phys. 2011, 7, 459.Schell2014 A. W. Schell, P. Engel, J. F. Werra, C. Wolff, K. Busch, O. Benson. Nano Lett. 2014, 14, 5, 2623.Fu2007 C.-C. Fu, H.-Y. Lee, K. Chen, T.-S. Lim, H.-Y. Wu, P.-K. Lin, P.-K. Wei, P.-H. Tsao, H.-C. Chang, W. Fann. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 3, 727.Chakraborty2015 C. Chakraborty, L. Kinnischtzke, K. M. Goodfellow, R. Beams, A. N. Vamivakas. Nat. Nanotechnol. 2015, 10, 6, 507.Srivastava2015 A. Srivastava, M. Sidler, A. Allain, D. Lembke, A. Kis, A. Imamoğlu. Nat. Nanotechnol. 2015, 10, 6, 491.He2015 Y.-M. He, G. Clark, J. R. Schaibley, Y. He, M.-C. Chen, Y.-J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, C.-Y. Lu, P. Jian-Wei. Nat. Nanotechnol. 2015, 10, 6.Koperski2015 M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J. Veuillen, J. Marcus, P. Kossacki, M. Potemski. Nat. Nanotechnol. 2015, 10, 6, 503.Tonndorf2015 P. Tonndorf, R. Schmidt, R. Schneider, J. Kern, M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. van der Zant, S. M. de Vasconcellos, R. Bratschitsch. Optica 2015, 2, 4, 347.Tran2015 T. T. Tran, K. Bray, M. J. Ford, M. Toth, I. Aharonovich. Nat. Nanotechnol. 2015, 11, 1 37.Tsai2013 D.-S. Tsai, K.-K. Liu, D.-H. Lien, M.-L. Tsai, C.-F. Kang, C.-A. Lin, L.-J. Li, J.-H. He. Acs Nano 2013, 7, 5, 3905.Chen2015 Y. Chen, C. Tan, H. Zhang, L. Wang. Chemical Society Reviews 2015, 44, 9, 2681.Schell2016 A. W. Schell, T. T. Tran, H. Takashima, S. Takeuchi, I. Aharonovich. APL Photonics 2016, 1, 9, 091302.Sontheimer2017 B. Sontheimer, M. Braun, N. Nikolay, N. Sadzak, I. Aharonovich, O. Benson. arXiv preprint arXiv:1704.06881 2017.Schell2017 A. W. Schell, H. Takashima, T. T. Tran, I. Aharonovich, S. Takeuchi. ACS Photonics 2017, 4, 4, 761.Kianinia2017 M. Kianinia, B. Regan, S. A. Tawfik, T. T. Tran, M. J. Ford, I. Aharonovich, M. Toth. ACS Photonics 2017, 4, 4, 768.Bourrellier2016 R. Bourrellier, S. Meuret, A. Tararan, O. Stéphan, M. Kociak, L. H. Tizei, A. Zobelli. Nano Lett. 2016, 16, 7, 4317.Tran2016 T. T. Tran, C. Elbadawi, D. Totonjian, C. J. Lobo, G. Grosso, H. Moon, D. R. Englund, M. J. Ford, I. Aharonovich, M. Toth. ACS Nano 2016, 10, 8, 7331.Shotan2016 Z. Shotan, H. Jayakumar, C. R. Considine, M. Mackoit, H. Fedder, J. Wrachtrup, A. Alkauskas, M. W. Doherty, V. M. Menon, C. A. Meriles. ACS Photonics 2016, 3, 12, 2490.Li2017 X. Li, G. D. Shepard, A. Cupo, N. Camporeale, K. Shayan, Y. Luo, V. Meunier, S. Strauf. ACS Nano 2017, 10.1021/acsnano.7b00638.Tawfik2017 S. A. Tawfik, S. Ali, M. Fronzi, M. Kianinia, T. T. Tran, C. Stampfl, I. Aharonovich, M. Toth, M. J. Ford. arXiv preprint arXiv:1705.05753 2017.Abdi2017 M. Abdi, J.-P. Chou, A. Gali, M. B. Plenio arXiv preprint arXiv:1709.05414 2017.Jungwirth2017 N. R. Jungwirth, G. D. Fuchs. arXiv preprint arXiv:1704.05536 2017.Chejanovsky2016 N. Chejanovsky, M. Rezai, F. Paolucci, Y. Kim, T. Rendler, W. Rouabeh, F. Favaro de Oliveira, P. Herlinger, A. Denisenko, S. Yang, I. Gerhard, A. Finkler, J. H. Smet, W. J. Nano Lett. 2016, 16, 11, 7037.Lieb2004 M. A. Lieb, J. M. Zavislan, L. Novotny. Journal of the Optical Society of America B 2004, 21, 6, 1210.Davies285 G. Davies, M. F. Hamer. Proc. R. Soc. A 1976, 348, 1653, 285.huang2009super B. Huang, M. Bates, X. Zhuang. Annu. Rev. Biochem. 2009, 78 993.Quantum Emitters in Hexagonal Boron NitrideHave Spectrally Tunable Quantum Efficiency:Supporting Information § TEM AND RAMAN CHARACTERIZATION The transmission electron microscope images where collected using a Jeol JEM 2100F microscope. The hBN nanoflakes were drop casted on a TEM grid (copper with carbon/formvar, 200 mesh, TedPella) and left to evaporate overnight. The diffraction pattern was processed using the CrysTBox software, <cit.> which matched the collected pattern with reported values of the hBN diffraction lines. We note that the (100) and (110) lines are typically the orientations found in these systems. <cit.>The Raman spectra were collected using an In Via Raman microscope (Renishaw), using an 100X objective to focus a 532, 12, optical beam. Various sizes of hBN nanoflake ensembles were interrogated and all showed the typical E_2g1367^-1 line, with nanoflakes with fewer layers yielding a slightly broader peak. <cit.>10Klinger2015 M. Klinger, A. Jäger. J. Appl. Crystallogr. 2015, 48, 6 2012–2018.Tran2015s T. T. Tran, K. Bray, M. J. Ford, M. Toth, I. Aharonovich. Nat. Nanotechnol. 2015, 11, 1 37–41.Guo2012 N. Guo, J. Wei, L. Fan, Y. Jia, D. Liang, H. Zhu, K. Wang, D. Wu. Nanotechnology 2012, 23, 41 415605.Nemanich1981 R. J. Nemanich, S. A. Solin, , R. M. Martin.Phys. Rev. B 1981, 23, 12 6348–6356. | http://arxiv.org/abs/1706.08303v2 | {
"authors": [
"Andreas W. Schell",
"Mikael Svedendahl",
"Romain Quidant"
],
"categories": [
"cond-mat.mes-hall",
"physics.optics",
"quant-ph"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170626095951",
"title": "Quantum Emitters in Hexagonal Boron Nitride Have Spectrally Tunable Quantum Efficiency"
} |
AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krakow, Poland Al. Mickiewicza 30, 30-059 Krakow, [email protected] AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krakow, Poland Al. Mickiewicza 30, 30-059 Krakow, PolandThe orbital effect on the Fulde-Ferrell (FF) phase is investigated in superconducting core/shell nanowires subjected to the axial magnetic field. The confinement in the radial direction results in the quantization of the electron motion with energies determined by the radial j and orbital m quantum numbers. In the external magnetic field the twofold degeneracy with respect to the orbital magnetic quantum number m is lifted which leads to the Fermi wave vector mismatch between the paired electrons (k, j,m,↑) ↔ (-k, j,-m,↓). This mismatch is transfered to the nonzero total momentum of the Cooper pairs which results in the formation of FF phase occurring sequentially with increasing magnetic field.By changing the nanowire radius R and the superconducting shell thickness d, we discuss the role of the orbital effect in the FF phase formation in both the nanowire-like (R/d ≪ 1) and nanofilm-like (R/d ≫ 1) regime. We have found that the irregular pattern of the FF phase, which appears for the case of the nanowire-like regime, evolves towards the regular distribution, in which the FF phase stability regionsappear periodically between the BCS state, for the nanofilm-like geometry.The crossover between these two different phase diagrams is explained as resulting from the orbital effect and the multigap character of superconductivity in core/shell nanowires. 74.78.Na, 84.71.MnFulde-Ferrell state in superconducting core/shell nanowires: role of the orbital effect Paweł Wójcik 03.02.2015 ======================================================================================= § INTRODUCTIONIn the last decade, unconventional superconductivity with a nontrivial Cooper pairinghas attracted the growing interest due to fascinating superconducting properties which arenot observed for the standard BCS state. Among the wide class of unconventional superconductors includinghigh-T_c cuprates,<cit.> iron-pnictides,<cit.> or heavy fermion materials<cit.>recently, the special attention isdrawn to systems with aspatially varying energy gap.<cit.> The existence of such a superconducting phase with the order parameter oscillating in real space was proposed in the mid-1960s by Fulde and Ferrell<cit.> (FF phase) as well as independently by Larkin andOvchinnikov (LO phase). <cit.> According to their original concept, superconductivity can survivein the magnetic field substantially higher than the critical field H_c, due to the creationof an inhomogeneous paired state with a non-zero total momentum of the Cooper pairs(𝐤↑,-𝐤+𝐪↓). This so called FFLO state results from the pairing between electrons from Zeeman splitted parts of the Fermi surface.In spite of a straightforward nature of the theoretical prediction and many ongoing theoretical investigations regarding the appearance of the FFLO state in different materials,<cit.> the experimental evidence of the non-zero momentum pairing has been reported only recently in heavy fermionsystems<cit.> and two dimensional organic superconductors.<cit.> Both of these material classes are characterized by areduction of the orbital pair breaking mechanism which is a crucial physical limitation for the experimental realization of the FFLOphase. The significance of the orbital pair braking is described by the Maki parameter<cit.>defined asα=√(2) H_c2^orb/H_c2^P, where H_c2^orb is the upper critical fieldcalculated without Zeeman splitting and H_c2^P is the critical field in paramagnetic limit.<cit.>It has been established that the FFLO phase can exist at finite temperature if α>1.8.<cit.>This criterion can be met inultrathin metallic nanofilms in which the confinement in the direction perpendicularto the film strongly reduces the orbital effect for the in-plane magnetic field. The theoretical model describing the FFLO phase in metallic nanofilms, besides the possibility of the non-zeromomentum pairing, should also contain the multiband character of supercondcutivity in these systems. In metallic nanostructures with size comparable to the electron wave length, the Fermi surface splits into a set of discrete subbands leading to many interesting effects which are not observed in the bulk limit e.g, the formation of Andreev states<cit.> or oscillations of superconducting properties.<cit.> As reported in our recent paper,<cit.> due to the multiband nature, the Fulde-Ferrell (FF) phase in metallic nanofilms splits intosubphases, number of which corresponds to the number of subbands participating in the formation of thepaired state. Similar behavior has been also reported for a Pauli-limiting two-band superconductors. <cit.>In both of these reports the FF phase has been induced by the Zeeman effect for the magnetic field H>H_c. The multiband character of superconductivity is even more pronounced in metallic nanowires. Very interestingphenomenon has been recently found when studying the superconducting to normal metal transition in nanowires, drivenby the axial magnetic field.<cit.> It turned out that the magnetic field does not destroy superconductivity simultaneouslyin all subbands participating in the paired phase but the transition to the normal state occurs gradually.The magnetic field suppress superconducting correlations step by step in subsequent subbands. It reveals itself as a cascade of jumps in the orderparameter with increasing magnetic field. Such anomalous behavior has inspired our recent study<cit.> in which,surprisingly, we have found that in cylindrical nanowires subjected to the axial magnetic field, the orbital effect, whichso far has been regarded as detrimental to the FFLO phase formation, can in fact induce the non-zero momentum paired state.As shown in Ref. Wojcik2015, the Fermi wave vector mismatch induced by the orbital effect between the subbands with opposite orbital momenta is transfered to the nonzero total momentum of the Cooper pairs which results in the formation of sequentially occurring Fulde-Ferrell (FF) and BCS phases with increasing magnetic field. In this context, understanding the physical mechanism standing behind the change of phase diagrams from the Pauli-limit, in which FF phase occurs in the vicinity of H_c as for nanofilms, to the orbital limit, in which the stable FF phases appear between BCS-paired states for H<H_c, still remains an unexplored issue. This can be done by considering superconducting core/shell nanowires, in which, by the control of the ratio R/d, where R is the core radius and d is the shell thickness, we can switch from the nanowire-like (R/d ≪ 1) to the nanofilm-like (R/d ≫ 1) scenario.<cit.>In the present paper, by controlling the ratio R/d,we discuss the role of the orbital effect in the FF phase formation, in both the nanowire-like and nanofilm-like regime. We have found that the phase diagrams differ considerably in both of these regimes. The irregular pattern of the FF phase occurrence in the nanowire-like regime evolves, with increasing the R/d ratio, towards the regular one, in which the FF phases appear periodically between the BCS states. The crossover between these two different phase diagrams is explained as resulting from the orbital effect and the multigap character of superconductivity in the considered nanostructures.The paper is organized as follows. In the next section we introduce the basic concepts of the theoretical model based on the modified BCS theory, in which the superconducting gap acquires the non-zero total momentum of the Cooper pairs. We explain in detail how the angular-momentum-induced Fermi-surface splitting generates the Fulde-Ferrell phase. In Sec. <ref> we discuss our results considering the contributions of both the orbital and Zeeman effect to the FF state. Finally, Sec. <ref> is devoted to conclusions and short discussion on the possibility of the experimental verification of the phenomena presented in the paper. § THEORETICAL MODEL Let us consider the core/shell nanowire consisting of a core of radius R, surrounded by a superconducting shell of thickness d [Fig. <ref>(a)]. Recently, analogous systems of semiconductor nanowires covered by a superconducting layer, have attractedgrowing interest due to their potential application in topologically protected quantum computing using Majorana zero modes.<cit.>For simplicity, let us assume that the core is an ideal insulator and electrons cannot penetrate the region of the core which allows us to neglect the proximity effect at the superconductor/insulator interface. We start from the general form of the BCS Hamiltonian ℋ̂ =∑ _σ∫ d^3 r Ψ̂^† (𝐫,σ) Ĥ_0 Ψ̂(𝐫,σ) + ∫ d^3 r[ Δ (𝐫)Ψ̂^†(𝐫,↑) Ψ̂^†(𝐫,↓) + h.c.] +∫ d^3r |Δ(𝐫)|^2/g,where σ denotes the spin state (↑,↓), g is the phonon-mediated electron-electron coupling constant and the gap parameter in real space is given byΔ(𝐫)=-g< Ψ̂ (𝐫,↓) Ψ̂ (𝐫,↑) >.Choosing the gauge for the vector potential as 𝐀=(0,eHr / 2,0 ), where the magnetic field H is directed along the nanowire axis, the single-electron Hamiltonian Ĥ_0 in the cylindrical coordinates (r,φ,z) is given byĤ_0=ħ ^2/2m_e [ -1/r∂/∂ r r ∂/∂ r +( - i/r∂/∂φ + eHr/2ħ ) ^2 - ∂ ^2/∂ z ^2 ] + σμ _B H - μ ,where σ=± 1 for spin-up and spin-down electrons, μ is the chemical potential and e, m_e is the electron charge and mass,respectively. If we assume azimuthal invariance and neglect the diamagnetic term ∼𝐀^2, whose energy for nanowiresis one order of magnitude lower than the order parameter, Ĥ_0 can be reduced to the one-dimensional form Ĥ_0,1D=ħ ^2/2m_e [ -1/r∂/∂ r r ∂/∂ r + m^2/r^2 ] + ħ^2 k^2 /2m_e + (m+σ) μ _B H- μ ,with the corresponding single-electron wave functionsψ _k,j,m(r,φ,z)=1/√(2 π L)ϕ _j,m(r)e^imφe^ikz,where L is the nanowire length, j is the radial quantum number, m is the orbital magnetic quantum number andk is the wave vector along the nanowire axis z. By assuming the hard-wall boundary conditions in the shell, ϕ _jm(R)=ϕ _jm(R+d)=0, the radial wave function ϕ _jm(r) can be written as<cit.>ϕ _j,m(r)=1/√(ℳ) [ Y_m(χ _jm R) J_m(χ _jm r) -J_m(χ _jm R) Y_m(χ _jm r)],where J_m(r) and Y_m(r) are the Bessel functions of the first and second kind of m-th order and ℳ is the normalization constant.The parameter χ _jm, related to the single-electron energy ξ _k,j,m,σ by ξ _k,j,m,σ=ħ ^2/2m_e ( χ _jm^2 + k^2 ) + (m+σ) μ _B H - μ,is a solution of the equationY_m(χ _jm R) J_m[χ _jm (R+d)] -J_m[χ _jm (R+d)] Y_m(χ _jm R)=0.From Eq. (<ref>) we can see that for H=0, each single-electron state is fourfold degenerate - two-fold degeneracy with respect to the orbital magnetic quantum number m and two-fold degeneracy with respect to the spin σ. In the presence of external magnetic field both these degeneracies are lifted resulting in a shift between the subbands corresponding to m and -m as well as ↑ and ↓. Since in the superconducting state the pairing appears between particles with opposite spins, momenta and orbital momenta: (k, j,m,↑) ↔ (-k, j,-m,↓), the Fermi-wave vector mismatch induced in the magnetic field can be transferred into the non-zero momentum of the Cooper pairs (q0 along the z axis) giving raise to the FF phase. Schematic illustration of this process is sketched in Fig. <ref>(b).Using the field operators in the form Ψ̂(r,φ,z,σ)=∑_k,j,mψ_k,j,m(r,φ,z) ĉ_k,j,m,σ, Ψ̂^†(r,φ,z,σ)=∑_k,j,mψ^*_k,j,m(r,φ,z) ĉ^†_k,j,m,σ,where ĉ_k,j,m,σ (ĉ^†_k,j,m,σ) is the annihilation (creation) operator, the BCS Hamiltonian with the possibility of non-zero momentum pairing is given by Ĥ = ∑_kmj𝐟̂^†_k,j,m,q𝐇_k,j,m,q𝐟̂_k,j,m,q+∑_k,j,mξ_-k+q,j,-m,σ̅+∑_j,m|Δ_j,m,q|^2/g,where 𝐟̂^†_k,j,m,q=(ĉ^†_k,j,m,↑, ĉ_-k+q,j,-m,↓) is the composite vector operators and𝐇_k,j,m,q=([ ξ_k,j,m,σ Δ_j,m,q; Δ^*_j,m,q -ξ_-k+q,j,-m,σ̅; ]).In the above, for simplicity, we limit to the situation in which all the Cooper pairs have a single momentum q. This assumption corresponds to the Fulde-Ferrel phase. In Eq. (<ref>), Δ_j,m,q is the superconducting energy gap in the subband (j,m) defined asΔ _j,m,q=g/4 π^2∑ _k,j',m' C_j,m,j',m'⟨ĉ_-k+q,j, -m,↓ĉ_k,j,m,↑⟩ ,with the interaction matrix C_j,m,j',m' = ∫_R^R+d dr rϕ _j,m^2(r) ϕ_j',m'^2(r) .Hamiltonian (<ref>) can be diagonalized by the Bogoliubov-de Gennes transformation ([ ĉ_k,j,m,↑; ĉ^†_-k+q,j,-m,↓; ])=([U_k,j,m,qV_k,j,m,q; -V_k,j,m,qU_k,j,m,q;])([ α̂_k,j,m,q; β̂^†_k,j,m,q;]),whereU^2_k,j,m,q=1/2(1+ξ_k,j,m,σ+ξ_-k+q,j,-m,σ̅/√((ξ_k,j,m,σ+ξ_-k+q,j,-m,σ̅)^2+4Δ_j,m,q^2)),V^2_k,j,m,q=1/2(1-ξ_k,j,m,σ+ξ_-k+q,j,-m,σ̅/√((ξ_k,j,m,σ+ξ_-k+q,j,-m,σ̅)^2+4Δ_j,m,q^2)),are the Bogoliubov coherence factors. As a result, one obtains the following form of the quasiparticle energiesE^±_k,j,m,q =1/2( ξ _k,j,m,σ-ξ _-k+q,j,-m,σ̅)±√(1/4( ξ _k,j,m,σ + ξ _-k+q,j,-m,σ̅)^2 +Δ_j,m,q ^2) +(m+σ)μ_B H.By substituting Eq. (<ref>) into Eq. (<ref>) we derive the self-consistent equations for the superconducting gapsΔ _j',m',q = g/4 π^2∫ dk ∑ _j,m C_j,m,j',m'× Δ _j,m,q [ 1- f(E^+_k,j,m,q) - f(E^-_k,j,m,q)]/√(( ξ _k,j,m,σ + ξ _-k+q,j,-m,σ̅)^2 +4 Δ_j,m,q ^2),where f(E) is the Fermi-Dirac distribution. The summation in Eq.(<ref>) is carried out only over the single-energy states ξ _k,j,m,σ inside the Debye widow |ξ _k,j,m,σ|<ħω _D, where ω _D is the Debye frequency.Since the chemical potential in nanostructures strongly deviates from that assumed in the bulk, for each shell thickness we determine μ keeping a constant electron concentration n_e = 1/π^2[(R+d)^2-R^2]∫ dk '∑_j,m∫^R+d_R dr r × { |U_k,j,m,qϕ _jm(r)|^2 f(E^+_k,j,m,q) + |V_k,j,m,qϕ _jm(r)|^2[1-f(E^-_kmjq)]} .In the considered nanowires, the spatial dependence of the superconducting gap results not only from the creation of the FF phase [Δ(r,φ,z)=Δ(r,φ)e^iqz] but it is also induced by the quantum confinement. The spatial dependence of the order parameter in the radial direction can be expressed asΔ_q(r) = g/4π ^2∫ dk '∑_jm |ϕ _jm(r)|^2 × Δ _j,m,q/√(( ξ _k,j,m,σ + ξ _-k+q,j,-m,σ̅)^2 +4 Δ_j,m,q ^2)×[ 1- f(E^+_k,j,m,q) - f(E^-_k,j,m,q)].To obtain the phase diagram, the superconducting gaps Δ_j,m,q and the chemical potential are calculated by solving Eqs. (<ref>) and (<ref>) self-consistently. The wave-vector q is determined by minimizing the free energy of the system.<cit.> Calculations presented in the paper have been carried out for the material parameters typical of aluminum: ħω_D=32.31 meV, gN(0)=0.18, where N(0)=mk_F / 2π^2 ħ ^2 is the bulk density of states at the Fermi level, Δ _bulk=0.25 meV and the chemical potential μ _bulk=0.9 eV which corresponds to theelectron density n_e=3.88 × 10^21 cm^-3. The assumed low value of the chemical potential, in relative to that measured in the bulk, results from the parabolic band approximation (for more details, see Ref. Shanenko2006). Its value has been determined to obtain a good agreement with the experimental data reported inRef. Shanenko2006_exp. The self-consistent procedure has been carried out for a constant electronconcentration which implies a gradual increase of the chemical potential with decreasing shell thickness.Moreover, we do not include a thickness-dependent change in the electron-phonon coupling,<cit.> as it can only result in the quantitativeeffects and do not alter the qualitative picture of the FF phase creation presented in the paper.§ RESULTS AND DISCUSSION To determine geometrical parameters appropriate for the analysis of the non-zero total momentum pairing, we have calculated the spatially averaged superconducting order parameter Δ̅, defined asΔ̅ = 2/d(2R+d)∫ _R ^R+d dr r Δ (r),as a function of the shell thickness for different core radii (Fig. <ref>).The Δ̅(d) oscillations presented in Fig. <ref> are due to the quantum size effect which arises when the system size becomes comparable to the electron Fermi wave length.<cit.> In core/shell nanowires, a reduction of an electron motion in the radial direction implies the energy quantization with energies determined by the quantum numbers j, m, k and σ [Eq. (<ref>)]. Subsequent peaks in Δ̅(d) correspond to subsequent subbands (j,m) passing through the Fermi level while increasing the shell thickness. As seen, the Δ̅(d) oscillations presented in Fig. <ref>(a-d) differ significantly from each other. The irregular oscillations for R=1 nm [Fig. <ref>(a)], reminiscent of these predicted for superconducting nanowires,<cit.> evolves with increasing R towards the regular oscillations characteristic for superconducting nanofilms [Fig. <ref>(d)].<cit.>The crossover from an irregular pattern to the regular regime was explained in details in Ref. Shanenko2006. It is related to the centrifugal term, ħ ^2 m^2 / 2m_e r^2, which for R /d ≤ 1 contributes significantly to the single electron energy leading to the energetically well separated states for different |m|. The irregular oscillations of Δ̅(d) presented in Fig. <ref>(a) reflect the irregular distribution of states (j,m) on the energy scale. For R/d ≫ 1 [Fig. <ref>(d)] the centrifugal term is negligibly small which causes the single electron states with different |m| to be almost degenerate. These energetically close subbands create bands labeled by the radial quantum number j. Each time when the bottom of such a band passes through the Fermi level we observe a resonant increase in Δ̅(d). Equal distant between bands on the energy scale results in regular oscillations presented in Fig. <ref>(d). Therefore, by an appropriate choice of the geometrical parameters, we can strengthen or suppress the centrifugal energy term, which allows for a smooth transition from the nanowire-like to the nanofilm-like regime. Now, let us analyze in detail the contribution of the orbital effect, mμ _B H, to the FF paired phase in both of the considered regimes. We start our study from the nanowires with R=1 nm. In Fig. <ref>, the magnetic field dependence of the averaged superconducting order parameter is presented for values of d marked by red squares in Fig. <ref>(a) which correspond to the resonant (d=1.11 nm and d=1.19 nm) and off-resonant (d=1.39 nm) thicknesses, respectively. While increasing the magnetic field, electrons in different states (j,m) acquire different energies which depend on the orbital magnetic quantum number m and the spin σ [Eq. (<ref>)].As a result, the superconductor to normal metal transition occurs as a cascade of jumps<cit.> (Fig. <ref>), each of which is related to depairing in one of the subbands contributing to the superconducting state. In Fig. <ref>(a), the critical fields H_c^j,m=Δ _jm(H=0)/(|m+1|)μ_B for particular subbands (j,m) are labeled and marked by arrows. Each time the magnetic field H becomes slightly larger than H_c^j,m, to sustain superconductivity in the subband (j,m), the Fermi wave-vector mismatch between the paired electrons (k,j,m,↑)↔ (-k,j,-m,↓) is partially compensated by the non-zero total momentum of the Cooper pairs (k,j,m,↑)↔ (-k+q,j,-m,↓). The formation of the FF phase minimizes the free energy of the system, as shown in Fig. <ref>(b). The further increase of H, well above H_c^j,m, causes that the paired state with the non-zero total momentum becomes energetically less favorable and the standard BCS pairing is restored. This leads to the phase diagramin which the FF phase stability regimes alternate with the BCS state [Fig. <ref>(a,b)]. Now, we discuss in detail the phase diagram for d=1.11 nm presented in Fig. <ref>(a). For the chosen resonant thickness, the full spectrum of q in the whole range of the magnetic field is plotted in Fig. <ref>(a). For completeness, in Fig. <ref>(a) and (e) we present the quasiparticle dispersions E_kjm vs k and the superconducting order parameter Δ (r) calculated for H=0. As one can see, there are twenty two relevant subbands participating in the superconducting state: (0,0) - (0,± 8) and (1,0)-(1,± 2). Their contributions P_j,m(r) to Δ (r,H=0) are displayed in Fig. <ref>(c), whereP_j,m(r) = g/4π ^2∫dk |ϕ _jm(r)|^2 × Δ _j,m,q/√(( ξ _k,j,m,σ + ξ _-k+q,j,-m,σ̅)^2 +4 Δ_j,m,q ^2)×[ 1- f(E^+_k,j,m,q) - f(E^-_k,j,m,q)]. Note, that the states (1,-1) and (0,-1), even though they contribute to the superconductivity [see Fig. <ref>(a)], are not labeled in Fig. <ref>(a) as the Cooper pairs (k,j,-1,↑)↔ (-k,j,1,↓) are unaffected by the magnetic field i.e., for both the electrons from the Cooper pair (m+σ)μ_B H=0. Although, their critical fields H^j,-1_c seem to be infinite, in fact, the magnetic field causes the Cooper pair breaking in these states indirectly, by the reduction of the superconducting correlation insubbands with m-1 [see Eq. (<ref>)]. Finally, the states (1,-1) and (0,-1) become depaired in H_c determined for the whole nanowire.As presented in Fig. <ref>(c), the contributions of the individual subbands to the superconducting order parameter, P_j,m(r), vary significantly. Due to the enhanced density of states, they are the largest for subbands situated in the vicinity of the Fermi surface. For d=1.11 nm, the major contribution to Δ(r) comes from the states (0,± 8) and (1, ± 2). The rest of the subbands play less important role but the most significant contribution is due to the states (0,± 7), (1, 0) and (1, ± 1). With increasing magnetic field the superconducting correlations are suppressed successively in individual subbands. For the subband (j,m), the critical field H_c^j,m, in which the superconductivity is destroyed, depends not only on the orbital magnetic quantum number m but also on the energy gap of excitation Δ _j,m [Eq. (<ref>)]. The latter is considerably affected by the quantum confinement and the Andreev mechanism, which appears due to the spatial variation of the superconducting order parameter.<cit.> One should note that, in nanowire-like regime, Δ _j,m may be different for different quantum numbers leading to the multigap superconductivity. Therefore, the condition H_c^j,m_1<H_c^j,m_2 for |m_1|>|m_2| do not have to be satisfied. This expectation agrees with our numerical results showing that H _c ^0,8 > H _c^0,7 [see Fig. <ref>(a)]. Consequently, the subband (0,7) is the first one in which the superconducting phase is destroyed as the magnetic field increases. The Cooper pair breaking in this single branch entails the formation of the FF phase with the total momentum q which increases with increasing magnetic field [Fig. <ref>(a)]. This FF phase region, shown in Fig. <ref>(a) by the gray area, is stable up to the magnetic field value at which the Cooper pair breaking takes place in the next two states (0,6) and (0,8). Their critical magnetic fields H_c^0, 6 and H_c ^0, 8 are almost equal leading to the substantial jump in Δ̅(H). Preservation of superconductivity in these branches requires to adjust a new value of the Cooper pair momentum q which is shown as a sharp dip in q(H) [Fig. <ref>(a)], after which q starts to increase again up to the magnetic field value at which the ordinary BCS phase is restored. Note that in the presence of magnetic field the Fermi vector mismatch for each of the subbands (j,m) is different which means that each of them has its own favorable total momentum q_j,m. However, the situation where several values of q_j,m appear in the system is impossible due to the coupling between all the branches participating in the superconducting state (cf. Eq. <ref>). Hence, the value of q which minimizes the free energy is usually a result of the Cooper pair breaking processes occurring in several subbands and we can not distinguish between individual contributions to the total momentum q coming from each of them. From Fig. <ref>(a) we can see that all of the FF phase stability regions are extended over the magnetic field range in which the superconductivity is destroyed in several consecutive subbands.The widest one, starting with depairing in the subbands (0, -4) and (0, 2) extends up to H=11.5 T which is the critical field for the states: (0,-3), (0,1), (1,1) and (1,2). The Cooper pair breaking occurring simultaneously in the four subbands is accompanied by the highest jump in Δ̅(H) which is largely caused by the fact that (1,2) is the resonant state with the highest contribution to the superconducting order parameter [see Fig. <ref>(c)]. As shown in Fig. <ref>(a), the formation of the FF phase for this particular case, requires to adjust the Cooper pair momentum q which is almost four times greater than that observed in other FF stability regions. Its high value is mainly determined by the Fermi wave vector mismatch in the resonant subband (1,2). The last FF phase stability region presented in Fig. <ref>(a) is related to the onset of depairing in the states (0,-2) and (0,0). Note that the orbital effect does not exists for states with m=0 and so the FF phase related to depairing in the subband(0,0) is solely induced by the Zeeman effect. For this reason, the Cooper pair momentum q in this region is twice smaller than in the regions with the dominant role of the orbital effect [Fig. <ref>(a)].Calculations carried out for different values of d show that the similar phase diagram, in which the FF phase stability regions are sandwiched betweenthe standard BCS state stability ranges, is characteristic for each resonant thickness. As an example, in Fig. <ref>(b) we present Δ̅ (H)for the neighboring resonant point d=1.19 nm, for which the enhancement of the energy gap Δ̅ is due to the Cooper pairing in the state (1,± 3) whose bottom passes though the Fermi level. Interestingly, the FF phase is not formed for the non-resonant thickness, d=1.39 nm [Fig. <ref>(c)], when the superconductor to normal metal transition has a more BCS-like character without noticeable jumps in Δ̅(H). For d=1.39 nm, the spatially averaged value of the superconducting order parameter Δ̅=0.142 meV< Δ _bulk [Fig. <ref>(f)]. All subbands are far away from the Fermi level having almost equal contributions to the superconducting state. All this causes that deparing in an individual subbands is less energy-consuming and consequently, the formation of the FF phase is unfavorable.Now, let us discuss how will the phase diagram change if we increase the nanowire radius R up to the nanofilm-like regime, where R /d ≫ 1. In Fig. <ref> (right panels) we present the magnetic field dependence of the spatially averaged superconducting gap Δ̅(H) for the resonant shell thickness d=1.09 nm (see Fig. <ref>) and nanowire radii (a) R=4 nm, (c) R=8 nm and (e) R=15 nm. As previously,the FF phase stability ranges are displayed by gray areas while the corresponding values of the Cooper pair momentum q are plotted in the left panels (b,d,f).The phase diagrams in Fig. <ref> differ considerably from that calculated for the nanowire-like regime, for R=1 nm (Fig. <ref>). The irregular pattern of the FF phase occurrence from Fig. <ref>evolves towards the regular distribution, in which the FF phases appear periodically between the BCS state stability ranges. As discussed, with increasing nanowire radius, the centrifugal term of the single electron energy is suppressed which, in turn, leads to the formation of bundles of subbands with the same radial quantum number j and different |m|. Therefore, the number of subbands N_s taking part in the superconducting phase increases significantly. For R=4 nm, shown in Fig. <ref>(a), N_s=58 and the subbands (0,0)-(0,± 23) and (1,0)-(1,± 5) make a contribution to the paired state [see dispersion E_kjm vs k in Fig. <ref>(a)].Such a large number of states N_s makes the contribution of an individual subband to the superconducting order parameter less significant. Consequently, the magnetically-induced depairing in a single subband is not so energy-consuming and it is not accompanied with the jump in Δ̅(H) as in the nanowire-like regime. Contrarily, as show in Fig. <ref>, Δ̅ decreases rather smoothly with increasing magnetic field up to H_c, at which the superconductor to normal metal transition is of the first order. Note that for R=4 nm we can still observe the single small jump [marked by arrow in Fig. <ref>(a)] which gradually disappears for larger R and for R=15 nm it is not observed any longer. Such residual jumps can occur in the intermediate regime (R/d ≈ 1), where a single subband contribution to the paired state can be still substantial (compare scales in the right panels, in Fig. <ref>). In this particular case, the jump is due to the simultaneous Cooper pair breaking in the subbands (0,-5), (0,3), (1,4) with the total contribution to the superconducting order parameter at 9 %. As presented in Fig. <ref>, the regions of the smooth decrease of Δ̅(H) are divided into FF phases which appear quasi-periodically alternating with the ordinary BCS paired state stability ranges. This periodicity is the more noticeable, the closer to the nanofilm-like regime we approach - compare Fig. <ref>(a) and (e). In the intermediate regime, for R=4,8 nm, the quasi-periodic pattern is disturbed in the vicinity of the jump where the corresponding total momentum of the Cooper pair q shows a distinct peak - see Fig. <ref>(b). The reason for this is the simultaneous Cooper pair breaking in three subbands which requires to adjust the Cooper pair momentum q which is almost six times larger than those obtained in the other FF stability regions. Similarly as the jump in Δ̅(H), the peak in q(H) disappears with increasing R and for R=15 nm it is not observed any longer.More detailed analysis of the FF phase formation in the nanofilm-like regime can be made based on Fig. <ref>(e,f) for R=15 nm, where R/d ≫ 1. The regular occurrence of the FF phases presented in Fig. <ref>(f) can be explained based on the same arguments as used in the nanowire-like regime. Namely, each of the FF stability regions is due to the Cooper pair breaking in the individual subbands while increasing magnetic field. Since in the nanofilm-like regime Δ _j,m(H=0) do not depend on the quantum numbers (in contrary to the nanowire-like regime), H_c^j,m=H_c^j,-m-2 and H_c^j,m_1>H_c^j,m_2 for any two states with positive m_1<m_2. It means that the Cooper pair breaking starts from the states (0,M) and (0,-M-2), where M is the highest positive orbital magnetic quantum number and subsequently, it takes place in the subbands (j,m) and (j,-m-2) with m=M,M-1,…,0. It is of interest that, regardless of the number of states N_s, the first FF phase region is derived by the Cooper pair breaking in the subbands (0,38) and (0,-40). The FF phase corresponding to depairing in the states with higher |m| do not occur. As an example, for R=15 nm, N_s=200, the subbands (0,0)-(0,±79) and (1,0)-(1,±20) make a contribution to the paring state [see Fig. <ref>(e)] and although depairing in the states with high |m| starts at H≈ 0.34 T, at which Δ̅ starts decreasing [see Fig. <ref>(e)], the first FF phase occurs for H≈ 0.47 T where the Cooper pair breaking takes place in the subbands (0,38) and (0,-40). It explains the gradual shift of the region where the FF phase stability regions occur, towards H_c for larger R. We expect that in the limit R/d →∞ this region moves to the close vicinity of H_c and, due to the induced degeneration with respect to m, all FF phase stability regions will merge into one. This picture is consistent with the ordinary FF phase diagram predicted for nanofilms.<cit.>In Fig. <ref>(f) we can also observe that the subsequent FF phase regions become narrower with decreasing magnetic field up to value at which the Cooper pair breaking occurs in the subbands (0,38) and (0,-40). Below this critical value H_0, the FF phases do not occur. Simultaneously,the corresponding value of q tends to zero for H=H_0 [see Fig. <ref>(f)]. This characteristic behavior can be explained as resulting from the difference in the orbital energy acquired from the magnetic field by the states with different orbital magnetic quantum number. Since the orbital term m μ _B H is proportional to the quantum number m, in the presence of the magnetic field, the states with higher |m| acquire the orbital energy much greater than the states with lower |m|. As a result the Fermi wave vector mismatch q_j,m between the paired electrons (k,j,m,↑)↔ (-k,j,-m,↓) is larger for states with higher |m|. If the Cooper pairs are broken in the state with high |m|, in a certain magnetic field, the wave vector mismatch q_jm in the states with low |m| is still very small. Since the value of q is a result of the Fermi vector mismatches q_j,m in all states contributing to the superconductivity, the formation of phase with a nonzero q is energetically unfavorable.The critical is depairing in the subbands (0,38) and (0,-40), when the Fermi vector mismatches in all superconducting states become sufficiently large to gain the small value of q by all these subbands. The FF phase with such a small value of q is very susceptible to the magnetic field and even slight increase of H causes that this phase is destroyed and the system switches back to the BCS paring - note that the first FF phase stability region corresponding to depairing in the states (0,38) and (0,-40) is extremely narrow. The same behavior is repeated each time when the Cooper pairs are broken in subsequent subbands while increasing magnetic field. However, for the states with lower |m|, the Fermi wave vector mismatches become larger. Consequently, the reduction of the so-called depairing region on the Fermi sphere requires larger momentum vector q, as shown in Fig. <ref>(f). Since larger q requires a higher magnetic field needed to destroy the FF phase, we observe the gradual extension of the FF phase stability regions for higher magnetic field. § CONCLUSIONS AND OUTLOOK The orbital effect on the FF phase has been investigated in superconducting core/shell nanowires subjected to the axial magnetic field. The energy quantization induced by the confinement of the electron motion in the radial direction leads to the multiband superconductivity, similarly as found in novel superconductors, e.g., MgB_2 or iron pnictides. It reveals in the form of the quantum size oscillations, i.e. the spatially averaged energy gap Δ̅ varies with d at a fixed R. The character of Δ̅ variations changes considerably with increasing R. From irregular pattern typical for nanowires, it evolves to the regular oscillations characteristic for nanofilms, while the crossover between both the regimes is determined by the centrifugal energy. As discussed above, in superconducting core/shell nanowires, the orbital effectwhich so far has been considered as detrimental to the FF phase formation, can in fact induce the non-zero momentum paired state. In the presence of magnetic field, the degeneracy with respect to the orbital magnetic quantum number m is lifted which leads to the Fermi wave vector mismatch between the subbands with opposite orbital momenta in the paired state. Therefore, as the magnetic field increases, the superconductivity is destroyed in subsequent subbands which manifests itself as a cascade of jumps in Δ̅(H). To sustain the Cooper pairing (k,j,m,↑)↔ (-k,j,-m,↓) in the corresponding subband, the non-zero total momentum state (FF phase) is formed which, in turn, leads to the phase diagram of alternating FF and BCS stability regions.In thepresent paper, by controlling of the ratio R/d,we have switched from the nanowire-like (R/d ≪ 1) to the nanofilm-like (R/d ≫ 1) scenario, strengthening or suppressing the centrifugal energy, respectively. We have found that the phase diagrams differ considerably in both regimes. The irregular pattern of the FF phase occurrence in the nanowire-like regime evolves towards the regular distribution, in which the FF phase stability regions appear periodically between the BCS state stability regions,in the nanofilm-like regime. As presented, the crossover between these two different phase diagrams can be explained as resulting from the orbital effect and the multigap character of superconductivity. In the nanowire-like regime, the centrifugal term, ħ ^2 m^2/2m_er^2, contributes significantly to the single electron energy, which leads to the well separated states for different |m|. Due to the quantum confinement and Andreev mechanism, induced by the spatially dependent superconducting order parameter, the system exhibits multiband and multigap superconductivity, in which Δ _j,m may vary for different quantum numbers. In the presence of magnetic field, the orbital effect leads to a situation in which the Fermi wave vector mismatch between the paired electrons (k,j,m,↑)↔ (-k,j,-m,↓) varies with m. The critical magnetic field for thesubband (j,m) is given by H_c^j,m=Δ _jm/(|m+1|)μ_B. As shown, the Cooper pair breaking in each of the subbands entails the formation of the FF phase. Therefore, due to the multigap character of superconductivity andirregular position of states with different m on the energy scale, the FF phase occurrence shows irregular pattern in the nanowire-like regime. This picture changes considerably if we increase R up to the limit R/d ≫ 1.In the nanofilm-like regime the centrifugal term is negligibly small and the states with different |m| for H=0 are almost degenerate forming the bands labeled by the radial quantum number j. The multigap character of the superconductivity vanishes, i.e. all subbands (j,m) have the same value of Δ _jm. Consequently, FF phases start to occur quasi-periodically each of which is related to the Cooper pair breaking in subsequent subbands with decreasing |m|=M,M-1,.... In the limit R/d →∞, our explanation leads to the phase diagram consistent with that predicted for nanofilms.Althoughthe presented transitions between FF and BCS phases seem to be verifiable by the use of standard experimental techniques, e.g., the specific heat measurements or detection of a supercurrent induced by the magnetic field, the FF phase appearance in the realistic core/shell nanowires requires some additional remarks, especially with regard to the assumptions made in the theoretical model.First, for the sake of simplicity, in this work we consider the Fulde-Ferrell phase which assumes the single Cooper pair momentum q for all subbands contributing to the superconducting state. Since, due to the orbital effect, the magnetically-induced Fermi wave vector mismatch is different for different subbands, it would be interesting to study the case when the superconducting order parameter is a combination of many components with different q vectors. In this manner, we would be able to reduce the so-calleddepairing region on the Fermi sphere to a greater extent what, in effect, would minimize the free energy of the system to the value much lower than that obtained for the FF phase. In fact, this energetically more favorable multi-momentum Cooper pair state could be observed in experiment but its appearance does not alter the qualitative picture of the non-zero momentum phase creation presented in the paper. The serious limitation in the experimental studies of the FF phase in metallic nanostructures is the appearance of impurities as the FF state could be readily destroyed by scattering. Even in ultra-clean nanowires, the surface scattering can destroy the FF state. However, as discussed above, the phase diagram of alternating FF and BCS stability regions emerges only for the resonant thicknesses when the superconducting order parameter is controlled by thesingle-electron subbands whose bottom is situated in the vicinity of the Fermi level. Its characteristic location on the energy scale causes that the corresponding longitudinal wave vector k<0.5 nm^-1. It corresponds to the electron wave length grater than 50 nm. Propagation of such long waves should be insensitive to the local surface imperfection with the size of the unit cell. This argument is even stronger if we consider the wave length corresponding to the FF phase for which the total momentum of the Cooper pairs q<0.01 nm^-1 gives the wave length greater than 600 nm.Finally, we would also like to address fluctuations which may appear in superconducting low dimensional structures. Thermally activated phase slip and quantum phase slip are known to play a serious role in superconducting nanowires making the use of the mean field theory questionable. However, as shown by recent experimental studies of superconducting Pb nanofilms, the use of the mean-field theory seems to be justified giving a surprisingly good agreement with experiment for the nanofilm thickness down to 2-5 monolayers.<cit.> For superconducting nanowires, the predicted diameter limit is 5-8 nm,<cit.> below which the quantum-phase slip can suppress the superconductivity. However, recent experiments for nanowires with diameter 5-6 nm<cit.> do not show any signature of phase fluctuations. Thus, we may expect that the mean-field approach used in the paper is reasonable for considered geometry of the core/shell nanowires for which phase fluctuations are assumed not to occur. § ACKNOWLEGEMENTThis research was supported in part by PL-Grid Infrastructure. The authors acknowledge valuable discussions with M. Zegrodnik.50 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Tabis et al.(2014)Tabis, Li, Le Tacon, Braicovich, Kreyssig, Minola, Dellea, Weschke, Veit, Ramazanoglu, Goldman, Schmitt, Ghiringhelli, Barišić, Chan, Dorow, Yu, Zhao, Keimer, and Greven]Tabis2014 author author W. Tabis, author Y. Li, author M. Le Tacon, author L. Braicovich, author A. Kreyssig, author M. Minola, author G. Dellea, author E. Weschke, author M. J.Veit, author A. Ramazanoglu, author A. I. Goldman, author T. Schmitt, author G. Ghiringhelli, author N. Barišić, author M. K. Chan, author C. J. Dorow, author G. Yu, author X. Zhao, author B. Keimer,and author M. Greven, @noopjournal journal Nature Communications volume 5, pages 5875 (year 2014)NoStop [Badoux et al.(2016)Badoux, Tabis, Laliberté, Grissonnanche, Vignolle, Vignolles, Béard, Bonn, Hardy, Liang, Doiron-Leyraud, Taillefer, and Proust]Badoux2016 author author S. Badoux, author W. Tabis, author F. Laliberté, author G. Grissonnanche, author B. Vignolle, author D. Vignolles, author J. Béard, author D. A. Bonn, author W. N. Hardy, author R. Liang, author N. Doiron-Leyraud, author L. Taillefer,and author C. Proust, @noopjournal journal Nature volume 531, pages 210 (year 2016)NoStop [Zegrodnik and J(2017)]Zegrodnik2017 author author M. Zegrodnik and author S. J,@noopjournal journal Phys. Rev. Bvolume 95, pages 024507 (year 2017)NoStop [Paglione and Greene(2010)]Paglione2010 author author J. Paglione and author R. L. Greene, @noopjournal journal Nature Phys. volume 6, pages 645 (year 2010)NoStop [Matsuda and Shimahara(2007)]Matsuda2007 author author Y. Matsuda and author H. Shimahara, @noopjournal journal J. Phys. Soc. Jpn. volume 76, pages 051005 (year 2007)NoStop [Hamidian et al.(2016)Hamidian, Edkins, Hyun Joo, Kostin, Eisaki, Uchida, Lawler, Kim, Mackenzie, Fujita, Lee, and Davis]Hamidian2017 author author M. H. Hamidian, author S. D. Edkins, author S. Hyun Joo, author A. Kostin, author H. Eisaki, author S. Uchida, author M. J. Lawler, author E. A.Kim, author A. P. Mackenzie, author K. Fujita, author J. Lee,and author S. Davis, @noopjournal journal Nature volume 532, pages 343 (year 2016)NoStop [Fulde and Ferrell(1964)]Fulde1964 author author P. Fulde and author A. Ferrell, @noopjournal journal Phys. Rev. volume 135, pages A550 (year 1964)NoStop [Larkin and Ovchinnikov(1964)]Larkin1964 author author A. I. Larkin and author Y. N. Ovchinnikov, @noopjournal journal Zh. Eksp. Teor. Fiz. volume 47, pages 1136 (year 1964)NoStop [Kaczmarczyk and Spałek(2010)]Kaczmarczyk2010 author author J. Kaczmarczyk and author J. Spałek, @noopjournal journal J. Phys. Condens. Matter volume 22, pages 355702 (year 2010)NoStop [Maśka et al.(2010)Maśka, Mierzejewski, Kaczmarczyk, andSpałek]Maska2010 author author M. M. Maśka, author M. Mierzejewski, author J. Kaczmarczyk,and author J. Spałek, @noopjournal journal Phys. Rev. B volume 82, pages 054509 (year 2010)NoStop [Ptok and Crivelli(2013)]Ptok2013_1 author author A. Ptok and author D. Crivelli, @noopjournal journal J. Low Temp. Phys. volume 172, pages 226 (year 2013)NoStop [Ptok(2014)]Ptok2014_1 author author A. Ptok, @noopjournal journal Eur. Phys. J. B volume 87, pages 2 (year 2014)NoStop [Zegrodnik and Spałek(2014)]Zegrodnik2014 author author M. Zegrodnik and author J. Spałek, @noopjournal journal Phys. Rev. B volume 90, pages 174597 (year 2014)NoStop [Blanchi et al.(2003)Blanchi, Movshovich, Capan, Paglluso, and Sarrao]Blanchi2003 author author A. Blanchi, author R. Movshovich, author C. Capan, author P. G. Paglluso,and author J. L. Sarrao, @noopjournal journal Phys. Rev. Lett. volume 91, pages 187004 (year 2003)NoStop [Kumagai et al.(2006)Kumagai, Saitoh, Oyaizu, Furukawa, Takashima, Nohara, Takagi, and Matsuda]Kumagai2006 author author K. Kumagai, author M. Saitoh, author T. Oyaizu, author Y. Furukawa, author S. Takashima, author M. Nohara, author H. Takagi,and author Y. Matsuda, @noopjournal journal Phys. Rev. Lett. volume 97, pages 227002 (year 2006)NoStop [Correa et al.(2007)Correa, Murphy, Martin, Purcell, Palm, Schmiedenshoff, Cooley, and Tozer]Correa2007 author author V. F. Correa, author T. P. Murphy, author C. Martin, author K. M. Purcell, author E. C. Palm, author G. M. Schmiedenshoff, author J. C. Cooley,and author W. Tozer, @noopjournal journal Phys. Rev. Lett. volume 98, pages 087001 (year 2007)NoStop [Beyer and Wosnitza(2013)]Beyer2013 author author R. Beyer and author J. Wosnitza, @noopjournal journal Low Temp. Phys. volume 39, pages 225 (year 2013)NoStop [Singleton et al.(2000)Singleton, Symington, Nam, Ardavan, Kurmoo, and Day]Singleton2000 author author J. Singleton, author J. A. Symington, author M. S. Nam, author A. Ardavan, author M. Kurmoo,and author P. Day, @noopjournal journal J. Phys. Condens. Matter volume 12, pages L641 (year 2000)NoStop [Tanatar et al.(2002)Tanatar, Ishiguro, Tanaka, andKobayashi]Tanatar2002 author author M. A. Tanatar, author T. Ishiguro, author H. Tanaka,and author H. Kobayashi, @noopjournal journal Phys. Rev. B volume 66, pages 134503 (year 2002)NoStop [Shinagawa et al.(2002)Shinagawa, Kurosaki, Zhang, Parker, Brown, Jerome, Christensen, and Bechgaard]Shinagawa2007 author author J. Shinagawa, author Y. Kurosaki, author F. Zhang, author C. Parker, author S. E. Brown, author D. Jerome, author J. B. Christensen,and author K. Bechgaard, @noopjournal journal Phys. Rev. Lett. volume 98,pages 147002 (year 2002)NoStop [Maki(1966)]Maki1966 author author K. Maki, @noopjournal journal Physical Review volume 148, pages 362 (year 1966)NoStop [Maki(1964)]Maki1964 author author K. Maki, @noopjournal journal Prog. Theor. Phys. volume 31, pages 945 (year 1964)NoStop [Clogstone(1962)]Clogston1962 author author A. M. Clogstone, @noopjournal journal Phys. Rev. Lett. volume 9, pages 266 (year 1962)NoStop [Chandrasekhar(1962)]Chandrasekhar1962 author author B. S. Chandrasekhar, @noopjournal journal Appl. Phys. Lett. volume 1, pages 7 (year 1962)NoStop [Gruenberg and Gunther(1966)]Gruenberg1966 author author L. W. Gruenberg and author L. Gunther, @noopjournal journal Phys. Rev. Lett. volume 16, pages 996 (year 1966)NoStop [Shanenko et al.(2007a)Shanenko, Croitoru, Mints, and Peeters]Shanenko_andreev author author A. A. Shanenko, author M. D. Croitoru, author R. G. Mints,and author F. M. Peeters,@noopjournal journal Phys. Rev. Lett.volume 99, pages 067007 (year 2007a)NoStop [Zhang et al.(2010)Zhang, Cheng, Li, Sun, Wang G, He, Wang, Ma, Chen, Wang, Liu, Q, Jia, and Xue]Zhang2010 author author T. Zhang, author P. Cheng, author W. J. Li, author Y. J. Sun, author X. G. Wang G, Zhu, author K. He, author L. L. Wang, author X. C. Ma, author X. Chen, author Y. Y.Wang, author Y. Liu, author L. H. Q, author J. F. Jia,and author Q. K. Xue, @noopjournal journal Nat. Phys. volume 6, pages 104 (year 2010)NoStop [Uchihashi et al.(2011)Uchihashi, Mishra, Aono, andNakayama]Uchihashi2011 author author T. Uchihashi, author P. Mishra, author M. Aono,and author T. Nakayama, @noopjournal journal Phys. Rev. Lett. volume 107, pages 207001 (year 2011)NoStop [Qin et al.(2009)Qin, Kim, Niu, and Shih]Qin2009 author author S. Qin, author J. Kim, author Q. Niu,and author C. K. Shih, @noopjournal journal Science volume 324, pages 1314 (year 2009)NoStop [Özer et al.(2006)Özer, Thompson, and Weitering]Ozer2006_b author author M. M. Özer, author J. R. Thompson,and author H. H. Weitering, @noopjournal journal Phys. Rev. B volume 74, pages 235427 (year 2006)NoStop [Özer et al.(2007)Özer, Jia, Zhang, Thompson, and Weitering]Ozer2007 author author M. M. Özer, author Y. Jia, author Z. Zhang, author J. R. Thompson,and author H. H. Weitering, @noopjournal journal Science volume 316, pages 1594 (year 2007)NoStop [Guo et al.(2004)Guo, Zhang, Bao, Han, Tang, Zhang, Zhu, Wang, Niu, Qiu, Jia, Zhao, and Xue]Guo2004 author author Y. Guo, author Y. F. Zhang, author X. Y. Bao, author T. Z. Han, author Z. Tang, author L. X. Zhang, author W. G.Zhu, author E. G. Wang, author Q. Niu, author Z. Q. Qiu, author J. F. Jia, author Z. X. Zhao,and author Q. K. Xue, @noopjournal journal Science volume 306, pages 1915 (year 2004)NoStop [Wójcik and Zegrodnik(2014a)]Wojcik2014_1 author author P. Wójcik and author M. Zegrodnik, @noopjournal journal phys. stat. sol. (b) volume 251, pages 1069 (year 2014a)NoStop [Wójcik and Zegrodnik(2014b)]Wojcik2014_3 author author P. Wójcik and author M. Zegrodnik, @noopjournal journal J. Phys. Condens. Matter volume 26, pages 455302 (year 2014b)NoStop [Shanenko and Croitoru(2010)]Shanenko2010 author author A. A. Shanenko and author M. D. Croitoru, @noopjournal journal Phys. Rev. B volume 73, pages 012510 (year 2010)NoStop [Shanenko et al.(2008)Shanenko, Croitoru, and Peeters]Shanenko2008 author author A. A. Shanenko, author M. D. Croitoru,and author F. M. Peeters, @noopjournal journal Phys. Rev. B volume 78, pages 024505 (year 2008)NoStop [Wójcik and Zegrodnik(2016)]Wojcik2016 author author P. Wójcik and author M. Zegrodnik, @noopjournal journal Physica E volume 83, pages 442 (year 2016)NoStop [Takahashi et al.(2014)Takahashi, Mizushima, and Machida]Takahashi2014 author author M. Takahashi, author T. Mizushima,and author K. Machida, @noopjournal journal PRBvolume 89, pages 064505 (year 2014)NoStop [Wójcik et al.(2015)Wójcik, Zegrodnik, and Spałek]Wojcik2015 author author P. Wójcik, author M. Zegrodnik,and author J. Spałek, @noopjournal journal Phys. Rev. B volume 91, pages 224511 (year 2015)NoStop [Chen et al.(2012)Chen, Shanenko, and Peeters]Chen2012 author author Y. Chen, author A. A. Shanenko,and author F. M. Peeters,@noopjournal journal Phys. Rev. Bvolume 85, pages 224517 (year 2012)NoStop [Chen et al.(2010)Chen, Shanenko, and Peeters]Chen2010 author author Y. Chen, author A. A. Shanenko,and author F. M. Peeters,@noopjournal journal Phys. Rev. Bvolume 81, pages 134523 (year 2010)NoStop [Mourik et al.(2012)Mourik, Zuo, Frolov, Plissard, Bakkers, and Kouwenhoven]Mourik2012 author author V. Mourik, author K. Zuo, author S. M. Frolov, author S. R. Plissard, author E. P. A. M. Bakkers,and author L. P. Kouwenhoven, @noopjournal journal Science volume 336, pages 1003 (year 2012)NoStop [Krogstrup et al.(2015)Krogstrup, Ziino, Chang, Albrecht, Madsen, Johnson, Nygåard, Marcus, and Jespersen]Krogstrup2015 author author P. Krogstrup, author N. L. B. Ziino, author W. Chang, author S. M. Albrecht, author M. H. Madsen, author E. Johnson, author J. Nygåard, author C. M.Marcus,and author T. S.Jespersen, @noopjournal journal Nature Mater. volume 14, pages 400 (year 2015)NoStop [Kosztin et al.(1998)Kosztin, Kos, Stone, and Legget]Kosztin1998 author author I. Kosztin, author S. Kos, author M. Stone,and author A. J. Legget, @noopjournal journal Phys. Rev. B volume 58, pages 9365 (year 1998)NoStop [Shanenko and Croitoru(2006)]Shanenko2006 author author A. A. Shanenko and author M. D. Croitoru, @noopjournal journal Phys. Rev. B volume 73, pages 012510 (year 2006)NoStop [Shanenko et al.(2006)Shanenko, Croitoru, Zgirski, andPeeters]Shanenko2006_exp author author A. A. Shanenko, author M. D. Croitoru, author M. Zgirski, and author F. M. Peeters,@noopjournal journal Phys. Rev. Bvolume 74, pages 052502 (year 2006)NoStop [Hwang et al.(2000)Hwang, Das Sarma, and Stroscio]Hwang2000 author author E. H. Hwang, author S. Das Sarma, and author M. A. Stroscio,@noopjournal journal Phys. Rev. Bvolume 61, pages 8659 (year 2000)NoStop [Shanenko et al.(2007b)Shanenko, Croitoru, and Peeters]Shanenko2007 author author A. A. Shanenko, author M. D. Croitoru,and author F. M. Peeters, @noopjournal journal Phys. Rev. B volume 75, pages 014519 (year 2007b)NoStop [Arutyunov et al.(2008)Arutyunov, Golubev, and Zaikin]Arutyunov2008 author author K. Y. Arutyunov, author D. S. Golubev,and author A. D. Zaikin, @noopjournal journal Phys. Rep. volume 464, pages 1 (year 2008)NoStop [Altomare et al.(2006)Altomare, Chang, Melloch, Hong, and Tu]Altomare2006 author author F. Altomare, author A. M. Chang, author M. R. Melloch, author Y. Hong,and author C. W. Tu, @noopjournal journal Phys. Rev. Lett. volume 97, pages 017001 (year 2006)NoStop | http://arxiv.org/abs/1707.09057v1 | {
"authors": [
"M. Mika",
"P. Wójcik"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170727220105",
"title": "Fulde-Ferrell state in superconducting core/shell nanowires: role of the orbital effect"
} |
Counting Pseudo Landau Levels in Spatially Modulated Dirac Systems Toshikaze Kariyado December 30, 2023 ==================================================================§ INTRODUCTIONThe Large Hadron Collider (LHC) is operated at very high instantaneous luminosities to achieve the large statistics required to search for exotic Standard Model (SM) or beyond the SM processes as well as for precision SM measurements. At a hadron collider, protons are grouped together in bunches; as the luminosity increases for a fixed bunch spacing, the number of protons within each bunch that collide inelastically increases as well. Most of these inelastic collisions are soft, with the protons dissolving into mostly low-energy pions that disperse throughout the detector. A typical collision of this sort at the LHC will contribute about 0.6 GeV/rad^2 of energy <cit.>. Occasionally, one pair of protons within a bunch crossing collides head-on, producing hard (high-energy) radiation of interest. At high luminosity, this hard collision, or leading vertex (LV), is always accompanied by soft proton-proton collisions called pileup. The data collected thus far by ATLAS and CMS have approximately 20 pileup collisions per bunch crossing on average (⟨NPU⟩∼ 20); the data in Run 3 are expected to contain ⟨NPU⟩∼ 80; and the HL-LHC in Runs 4-5 will have ⟨NPU⟩∼ 200. Mitigating the impact of this extra energy on physical observables is one of the biggest challenges for data analysis at the LHC.Using precision measurements, the charged particles coming from the pileup interactions can mostly be traced to collision points (primary vertices) different from that of the leading vertex. Indeed, due the to the excellent vertex resolution at ATLAS and CMS <cit.> the charged particle tracks from pileup can almost completely be identified and removed.[Some detector systems have an integration time that is (much) longer than the bunch spacing of 25ns, so there is also a contribution from pileup collisions happening before or after the collision of interest (out-of-time pileup).This contribution will not have charged particle tracks and can be at least partially mitigated with calorimeter timing information. Out-of-time pileup is not considered further in this analysis.] This is the simplest pileup removal technique, called charged-hadron subtraction. The challenge with pileup removal is therefore how to distinguish neutral radiation associated with the hard collision from neutral pileup radiation.[Charged-hadron subtraction follows a particle-flow technique that removes calorimeter energy from pileup tracks.Due to the calorimeter energy resolution, there will be a residual contribution from charged-hadron pileup.This contribution is ignored but could in principle be added to the neutral pileup contribution.] Since radiation from pileup is fairly uniform[This work will not explicitly discuss identification of real high energy jets resulting from pileup collisions.The ATLAS and CMS pileup jet identification techniques are documented in Ref. <cit.> and <cit.>, respectively. ], it can be removed on average, for example, using thejet areas technique <cit.>. The jet areas technique focuses on correcting the overall energy of collimated sprays of particles known as jets. Indeed, both the ATLAS and CMS experiments apply jet areas or similar techniques to calibrate the energy of their jets <cit.>. Unfortunately, for many measurements, such as those involving jet substructure or the full radiation patterns within the jet, removing the radiation on average is not enough.Rather than calibrating only the energy or net 4-momentum of a jet, it is possible to correct the constituents of the jet.By removing the pileup contamination from each constituent, it should be possible to reconstruct more subtle jet observables. We can coarsely classify constituent pileup mitigation strategies into several categories: constituent preprocessing, jet/event grooming, subjet corrections, and constituent corrections.Grooming refers to algorithms that remove objects and corrections describe scale factors applied to individual objects. Both ATLAS and CMS perform preprocessing to all of their constituents before jet clustering.For ATLAS, pileup-dependent noise thresholds in topoclustering <cit.> suppresses low energy calorimeter deposits that are characteristic of pileup.In CMS, charged-hadron subtraction removes all of the pileup particle-flow candidates <cit.>.Jet grooming techniques are not necessarily designed to exclusively mitigate pileup but since they remove constituents or subjets in a jet (or event) that are soft and/or at wide angles to the jet axis, pileup particles are preferentially removed <cit.>.Explicitly tagging and removing pileup subjets often performs comparably to algorithms without explicit pileup subjet removal <cit.>.A popular event-level grooming algorithm called SoftKiller <cit.> removes radiation below some cutoff on transverse momentum, p_T^cut chosen on an event-by-event basis so that half of a set of pileup-only patches are radiation free.While grooming algorithms remove constituents and subjets, there are also techniques that try to reconstruct the exact energy distribution from the primary collision. One of the first such methods introduced was Jet Cleansing <cit.>. Cleansing works at the subjet level, clustering and declustering jets to correct each subjet separately based on its local energy information. Furthermore, Cleansingexploits the fact that the relative size of pileup fluctuations decreases as ⟨NPU⟩→∞ so that the neutral pileup-energy content of subjets can be estimated from the charged pileup-energy content.A series of related techniques operate on the constituents themselves <cit.>.One such technique called PUPPI also uses local charged track information but works at the particle level rather than subjet level. PUPPI computes a scale factor for each particle, using a local estimate inspired by the jets-without-jets paradigm <cit.>. In this paper, we will be comparing our method to PUPPI and SoftKiller. In this paper, we present a new approach to pileup removal based on machine learning. The basic idea is to view the energy distribution of particles as the intensity of pixels in an image <cit.>.Convolutional neural networks applied to jet images <cit.> have found widespread applications in both classification <cit.> and generation <cit.>. Previous jet-images applications have included boosted W-boson tagging <cit.>, boosted top quark identification <cit.>, and quark/gluon jet discrimination <cit.>.Most of these previous applications were all classification tasks: extracting a single binary classifier (quark or gluon, W jet or background jet, etc.) from a highly-correlated multidimensional input. The application to pileup removal is a more complicated regression task, as the output (a cleaned-up image) should be of similar dimensionality to the input. PUMML is among the first applications of modern machine learning tools to regression problems in high energy physics.To apply the convolutional neural network paradigm to cleaning an image itself, we exploit the finer angular resolution of the tracking detectors relative to the calorimeters of ATLAS and CMS. Building on the use of multichannel inputs in <cit.>, we give as input to our network three-channel jet images: one channel for the charged LV particles, one channel for the charged pileup particles, and one channel, at slightly lower resolution, for the total neutral particles. We then ask the network to reconstruct the unknown image for LV neutral particles. Thus our inputs are like those of Jet Cleansing but binned into a regular grid (as images) rather than single numbers for each subjet <cit.>. Further, the architecture is designed to be local (as with Cleansing or PUPPI), with the correction of a pixel only using information in a region around it. The details of our network architecture are described in Section <ref>. Section <ref> documents its performance in comparison to other state-of-the-art techniques. The remainder of the paper contains some robustness checks and a discussion in Section <ref> of the challenges and opportunities for this approach. § PUMML ALGORITHM The goal of the PUMML algorithm is to reconstruct the neutral leading vertex radiation from the charged leading vertex, charged pileup, and total neutral information. Since neutral particles do not have tracking information available, the challenge is to determine what fraction of the total neutral energy in each direction came from the leading vertex and what fraction came from pileup. To assist this discrimination, we take as inputs into our network the energy distribution of charged particles, separated into leading vertex and pileup contributions, in addition to the total neutral energy distribution[Both ATLAS <cit.> and CMS <cit.> are proposing precision timing detectors are part of their upgrades for the HL-LHC; such information could naturally be incorporated into another layer of the network.]. A natural way to combine these observables is using the multichannel images approach introduced in <cit.> based on color-image recognition technology. We apply this machine learning technique to R=0.4 anti-k_t jets. The jet image inputs are square grids in pseudorapidity-azimuth (η,ϕ) space of size 0.9 × 0.9 centered on the charged leading vertex transverse momentum (p_T)-weighted centroid of the jet.One could combine all layers to determine the jet axis, but in practice the axis determined from the charged leading vertex captures dominates because of its superior angular resolution and pileup robustness. To simulate the detector resolutions of charged and neutral calorimeters, charged images are discretized into Δη×Δϕ = 0.025× 0.025 pixels and neutral images are discretized into Δη×Δϕ = 0.1× 0.1 pixels[These dimensions are representative of typical tracking and calorimeter resolutions, but would be adapted to the particular detector in practice.We ignore other detector effects in this algorithm demonstration, as has also been done also for PUPPI and SoftKiller.In principle, additional complications due to the detector response can be naturally incorporated into the algorithm during training.]. We use the following three input channels:red = the transverse momenta of all neutral particles green =the transverse momenta of charged pileup particles blue = transverse momenta of charged leading vertex particlesThe output of our network is also an image:output = the transverse momenta of neutral leading vertex particles. Only charged particles with p_T > 500 MeV were included in the green or blue channels. Charged particles not passing this charged reconstruction cut were treated as if they were neutral particles. Otherwise, the separation into channels is assumed perfect. No image normalization or standardization was applied to the jet images, allowing the network to make use of the overall transverse momentum scale in each pixel. The different resolutions for charged and neutral particles initially present a challenge, since standard architectures assume identical resolution for each color channel. To avoid this issue, we perform a direct upsampling of each neutral pixel to 4× 4 pixels of size Δη×Δϕ = 0.025× 0.025 and divide each pixel value by 16 such that the total momentum in the image is unchanged.In summary, the following processing was applied to produce the pileup images: * Center: Center the jet image by translating in (η,ϕ) so that the total charged leading vertex p_T-weighted centroid pixel is at (η,ϕ) = (0,0). This operation corresponds to rotating and boosting along the beam direction to center the jet.* Pixelate: Crop to a 0.9× 0.9 region centered at (η,ϕ) = (0,0). Create jet images from the transverse momenta of all neutral particles, the charged leading vertex particles, the charged pileup particles, and the neutral leading vertex particles. Pixelizations of Δη×Δϕ = 0.025× 0.025 and Δη×Δϕ = 0.1× 0.1 are used for the charged and neutral jet images, respectively.* Upsample: Upsample each neutral pixel to sixteen Δη×Δϕ = 0.025× 0.025 pixels, keeping the total transverse momentum in the image unchanged. The convolutional neural net architecture used in this study took as input 36× 36 pixel, three-channel pileup images. Two convolutional layers, each with 10 filters of size 6× 6 with 2×2 strides, were used after zero-padding the input images and first convolutional layer with a 2-pixel buffer on all sides. The output of the second layer has size 9×9×10, with the 9×9 part corresponding to the size of the target output and the 10 corresponding to the number of filters in the second layer. In order to project down to a 9×9×1 output, a third convolution layer with filter size 1×1 is used.This last 1× 1 convolutional layer is a standard scheme for dimensionality reduction.A rectified linear unit (ReLU) activation function was applied at each stage. A schematic of the framework and architecture is shown in Fig. <ref>.All neural network implementation and training was performed with the python deep learning libraries Keras <cit.> and Theano <cit.>. The dataset consisted of 56k pileup images, with a 90%/10% train/test split. He-uniform initialization <cit.> was used to initialize the model weights. The neural network was trained using the Adam <cit.> algorithm with a batch size of 50 over 25 epochs with a learning rate of 0.001. The choice of loss function implicitly determines a preference for accuracy on harder pixels or softer pixels. To that end, the loss function used to train PUMML was a modified per-pixel logarithmic squared loss:ℓ = < log(p_T^(pred) + p̅/p_T^(true) + p̅)^2 >,where p̅ is a hyperparameter that controls the choice between favoring all p_T equally (p̅→∞) or favoring soft pixels (p̅→ 0). After mild optimization, a value of p̅ = 10 GeV was chosen, though the performance of the model as measured by correlations between reconstructed and true observables is relatively robust to this choice. PUMML was found to give good performance even with a standard loss function such as the mean squared error, which favors all p_T equally.The PUMML architecture is local in that the rescaling of a neutral pixel is a function solely of the information in a patch in (η,ϕ)-space around that pixel. The size of this patch can be controlled by tuning the filter sizes and number of layers in the architecture. Further, due to weight-sharing in convolutional layers, the same function is applied for all pixels. Building this locality and translation invariance into the architecture ensures that the algorithm learns a universal pileup mitigation technique, while carrying the benefit of drastically reducing the number of model parameters. Indeed, the PUMML architecture used in this study has only 4,711 parameters, which is small on the scale of deep learning architectures, but serves to highlight the effectiveness of using modern machine learning techniques (such as convolutional layers) in high energy physics without necessarily using large or deep networks.While we considered jets and jet images in this study, the PUMML architecture using convolutional nets readily generalizes to event-level applications. The locality of the algorithm implies that the trained model can be applied to any desired region of the event using only the surrounding pixels. To train the model on the event level, either the existing PUMML architecture could be generalized to larger inputs and outputs or the event could be sliced into smaller images and the model trained as in the present study. The parameters of the PUMML architecture are the convolutional filter sizes, the number of filters per layer, and the number of convolutional layers, which may be optimized for a specific application. Here, we have presented an architecture optimized for simplicity and performance for jet-level pileup subtraction. PUMML is designed to be applicable at both jet- and event-level.§ PERFORMANCE To test the PUMML algorithm, we consider qq̅ light-quark-initiated jets coming from the decay of a scalar with mass m_ϕ = 500 GeV. Events were generated using Pythia 8.183 <cit.> with the default tune for pp collisions at √(s) = 13 TeV.Pileup was generated by overlaying soft QCD processes onto each event. Final state particles except muons and neutrinos were kept. The events were clustered with FastJet 3.1.3 <cit.> using the anti-k_t algorithm <cit.> with a jet radius of R = 0.4. A parton-level p_T cut of 95 GeV was applied and up to two leading jets with p_T > 100 GeV and η∈[-2.5,2.5] were selected from each event. All particles were taken to be massless.Samples were generated with the number of pileup vertices ranging from 0 to 180. Since the model must be trained to fix its parameters, the learned model depends on the pileup distribution used for training. For our pileup simulations, we trained on a Poisson distribution of NPUs with mean ⟨NPU⟩ = 140. For robustness studies, we also tried training with NPU=140 for each event or NPU=20 for each event. The average jet image inputs for this sample are shown in Fig. <ref>. For comparison, we show the performance of two powerful and widely used constituent-based pileup mitigation methods: PUPPI <cit.> and SoftKiller <cit.>. In both cases, default parameter values were used: R_0=0.3, R_min=0.02, w_cut = 0.1, p_T^cut(NPU) = 0.1+0.007×NPU (PUPPI), grid size = 0.4 (SoftKiller). Variations in the PUPPI parameters did not yield a large difference in performance. Both PUPPI and SoftKiller were implemented at the particle level and then discretized for comparison with PUMML. We show the action of the various pileup mitigation methods on a random selection of events in Fig. <ref>. On these examples, PUMML more effectively removes moderately soft energy deposits that are retained by PUPPI and SoftKiller.To evaluate the performance of different pileup mitigation techniques, we compute several observables and compare the true values to the corrected values of the observables. To facilitate a comparison with PUMML, which outputs corrected neutral calorimeter cells rather than lists of particles, a detector discretization is applied to the true and reconstructed events. Our comparisons focus on the following six jet observables: * Jet Mass:Invariant mass of the leading jet.* Dijet Mass: Invariant mass of the two leading jets.* Jet Transverse Momentum: The total transverse momentum of the jet.* Neutral Image Activity, N_95 <cit.>: The number of neutral calorimeter cells which account for 95% of the total neutral transverse momentum.* Energy Correlation Functions, ECF_N^(β) <cit.>: Specifically, we consider the logarithm of the two- and three-point ECFs with β = 4. Fig. <ref> illustrates the distributions of several of these jet observables after applying the different pileup subtraction methods. While these plots are standard, they do not give a per-event indication of performance. A more useful comparison is to show the distributions of the per-event percent error in reconstructing the true values of the observables, which are shown in Fig. <ref>. To numerically explore the event-by-event effectiveness, we can look at the Pearson linear correlation coefficient between the true and corrected values or the interquartile range (IQR) of the percent errors. Table <ref> summarizes the event-by-event correlation coefficients of the distributions shown in Fig. <ref>. Table <ref> summarizes the IQRs of the distributions shown in Fig. <ref>. PUMML outperforms the other pileup mitigation techniques on both of these metrics, withimprovements for jet substructure observables such as the jet mass and the energy correlation functions.§ ROBUSTNESSIt is important to verify that PUMML learns a pileup mitigation function which is not overly sensitive to the NPU distribution of its training sample. Robustness to the NPU on which it is trained would indicate that PUMML is learning a universal subtraction strategy. To evaluate this robustness, PUMML was trained on 50k events with either NPU=20 or NPU=140 and then tested on samples with different NPUs. Fig. <ref> shows the jet mass correlation coefficients as a function of the test sample NPU. PUMML learns a strategy that is surprisingly performant outside of the NPU range on which it was trained. Further, we see that by this measure of performance, PUMML consistently outperforms both PUPPI and SoftKiller.A related robustness test is to probe how the performance of PUMML depends on thep_T spectrum of the training sample. To explore this, we generated two large training samples (50k events): one with a scalar mass of 200 GeV and one with a scalar mass of 2 TeV; we did not impose any parton-level p_T cuts on these samples.After training these two networks, we tested them on a set of samples generated from scalars with intermediate masses, from 300 GeV to 900 GeV. As can be seen in Fig. <ref>, the performance of PUMML is very robust to the p_T distribution of the jets in the training sample: the networks trained on the 200 GeV resonance and the 2 TeV resonance have identical performance. The figure also shows that the performance of PUMML is less sensitive to of the p_T of the testing sample than eitherPUPPI or Soft-Killer. This robustness test speaks to the PUMML algorithm's ability to learn universal aspects of pileup mitigation. A number of modifications of PUMML were also tried. Locally connected layers were tried instead of convolutional layers and were found to perform worse due to a large increase in the number of parameters of the model, while losing the translation invariance that makes PUMML powerful. We tried training without various combinations of the input channels; the model was found to perform moderately worse without either of the charged channels but suffered severe degradation without the total neutral channel. We tried using simpler models with only one layer or fewer filters per layer. Remarkably, even with only a single layer and a single 4× 4 filter (a model that has just 49 parameters), PUMML performed only moderately worse than the version presented in this study, which was allowed to be more complicated in order to achieve even better performance. § WHAT IS PUMML LEARNING?While it is generally very difficult to determine what a network is learning, one possible probe is to examine the weights of the filter layers in the convolutional network. For our full network, these weights are complicated and the subtractor that the network learns is difficult to probe analytically. Instead, we trained a simplified PUMML network with a single 12×12 pixel filter, which spans 3× 3 neutral pixels, with no bias term. The different channels of this filter are shown in Fig. <ref>. The neutral filter clearly selects the relevant neutral pixel for subtraction, while the charged pileup filter is approximately uniform (with the value dependent on the specific choice of loss function and activation function), and the charged leading vertex filter does not significantly contribute. The filter values motivate the following parameterization of what PUMML is learning:p_T^N,LV = 1.0p_T^N,total - βp_T^C,PU + 0.0 p_T^C,LV ,for some 𝒪(1)constant β, where p_T^N,LV, p_T^N,total,p_T^C,PU, and p_T^C,LV are the neutral-pixel-level transverse momenta of the neutral leading-vertex particles, all neutral particles, charged pileup particles, and charged leading-vertex particles, respectively. The values 1.0 and 0.0 in Eq. (<ref>) are stable (to the 0.05 level) under variations in the loss and activation functions. This is reassuring as the learned subtractor is thereby robust in the NPU → 0 limit despite begin trained on ⟨NPU⟩ = 140.Eq. (<ref>) is remarkably similar to the physically-motivated formula used in Jet Cleansing <cit.>. Cleansing is built on the observation that since pileup is the incoherent sum of many separate scattering events, its variance is smaller than the variance of the radiation from the leading-vertex. Thus, it is better to estimate p_T^N,PU from p_T^C,PU than to estimate p_T^N,LV from p_T^C,LV. The simplest form of Cleansing (Linear Cleansing) gives the formula:p_T^N,LV =p_T^N,tot -(1/γ_0-1)p_T^C,PU,where γ_0 is the average ratio of charged p_T to total p_T in a subjet. Thus this simple one 12×12 filter PUMML network is learning a subtractor of precisely the same parametric form as Linear Cleansing!The value of γ_0 in Linear Cleansing and the value ofβ that is learned in Eq. (<ref>) depend on how soft radiation is handled. For example, if no reconstruction threshold is applied, γ_0≈ 2/3 (since 2/3 of pions are charged). In addition, the value of β depends on the loss function used. For example, if the loss function is minimized when the means of the true and predicted neutral transverse momenta are equal:ℓ=|⟨ p_T^(true)⟩-⟨ p_T^(pred)⟩| =| ⟨ p_T^N,LV⟩ - ⟨ p_T^N,total⟩+ β⟨p_T^C,PU⟩|,then we find that the optimal β is:β = ⟨ p_T^N,PU⟩/⟨ p_T^C,PU⟩.Training the 12× 12 PUMML filter without a ReLU or bias term, using the loss function of Eq. (<ref>) with the average taken pixel-wise over the batch, we find β = 0.59 with no charged reconstruction cut and β = 1.18 with the cut. These values are consistent with those predicted by Eq. (<ref>) of 0.62 and 1.26, respectively.On the other hand, if we take a mean squared error loss function:ℓ=⟨(p_T^(true)-p_T^(pred))^2 ⟩,then the minimum occurs at:β = ⟨ p_T^N,PUp_T^C,PU⟩/⟨ p_T^C,PUp_T^C,PU⟩.This still depends only on the pileup properties, as with Linear Cleansing, but also depends on correlations between neutral and charged radiation. For example, training the 12×12 PUMML filter without a ReLU or bias term using a mean squared error loss function, we find β=0.56 with no charged reconstruction cut and β =0.97 with the cut. These numbers are in general agreement (within 10-20%) with a direct evaluation of the right-hand side of Eq. (<ref>). In the limit that neutral and charged pileup radiation are constant, Eq. (<ref>) reduces to Eq. (<ref>).Whether the loss function of Eq. (<ref>) or Eq. (<ref>) (or something else entirely) is better is not simple to establish. The inclusion of the ReLU activation function further complicates the analysis since the model is equally penalized for all non-positive predictions. We find with the single 12× 12 filter, using the loss function of Eq. (<ref>) and including a ReLU and bias term, PUMML achieves a jet mass correlation coefficient of 90.4%. This is competitive with the values listed in Table. <ref>, as we might expect since Linear Cleansing has comparable performance to PUPPI and SoftKiller. The full network improves on Linear Cleansing by exploiting additional correlations that are hard to disentangle by looking at the filters.§ CONCLUSIONSIn this paper, we have introduced the first application of machine learning to the critically important problem of pileup mitigation at hadron colliders. We have phrased the problem of pileup mitigation in the language of a machine learning regression problem. The method we introduced, PUMML, takes as input the transverse momentum distribution of charged leading-vertex, charged pileup, and all neutral particles, and outputs the corrected leading vertex neutral energy distribution. We demonstrated that PUMML works at least as well as, and often better than, the competing algorithms PUPPI and SoftKiller in their default implementations.It will be exciting to see these algorithms compared with a full detector simulation, where it will be possible to test the sensitivity to important experimental effects such as resolutions and inefficiencies.There are several extensions and additional applications of the PUMML framework beyond the scope of this study. As mentioned in Section <ref>, PUMML can very naturally be extended from jet images to entire events. Applying this event-level PUMML to the problem of missing transverse energy would be a natural next step.While the filter sizes can be the same for the event and jet images, the network training will likely require modification.Furthermore, the inhomogeneity of the detector response with |η| will require attention.Another potentially useful modification to PUMML would be to train to predict the neutral pileup p_T rather than the neutral leading vertex p_T in order to increase out-of-sample robustness of the learned pileup mitigation algorithm. Additionally, using larger-R jets may be of interest, thereby necessitating a resizing of the local patch or other PUMML parameters, all of which is easily achieved.An important consideration when using machine learning for particle physics applications is how the method can be used with data and whether or not the systematic uncertainties are under control. Unlike a purely physically-motivated algorithm, such as PUPPI or SoftKiller, machine learning runs the risk of being a “black-box” which can be difficult to understand. Nevertheless, machine learning is powerful, scaleable, and capable of complementing physical insight to solve complicated or otherwise intractable problems.To prevent the model from learning simulation artifacts, it is preferable to train on actual data rather than simulation. In many machine learning applications in collider physics, obtaining truth-level training samples in data is a substantial challenge. To overcome this challenge in classification tasks, <cit.> introduces an approach to train from impure samples using class proportion information. For PUMML and pileup mitigation more broadly, a more direct method to train on data is possible. To simulate pileup, we overlay soft QCD events on top of a hard scattering process, both generated with Pythia. Experimentally, there are large samples of minimum bias and zero-bias (i.e. randomly triggered) data. There are also samples of relatively pileup-free events from low luminosity runs. Thus we can construct high-pileup samples using purely data. This kind of data overlay approach, which has already been used by experimental groups in other contexts <cit.>, could be perfect for training PUMML with data. Therefore, an implementation of ML-based pileup mitigation in an actual experimental setting could avoid mis-modeling artifacts during training, thus adding more robustness and power to this new tool.The authors would like to thank Philip Harris, Francesco Rubbo, Ariel Schwartzman and Nhan Tran for stimulating conversations, in particular for suggesting some of the extensions mentioned in the conclusions. We would also like to thank Jesse Thaler for helpful discussions. PTK and EMM would like to thank the MIT Physics Department for its support. Computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University. This work was supported by the Office of Science of the U.S. Department of Energy (DOE) under contracts DE-AC02-05CH11231 and DE-SC0013607, the DOE Office of Nuclear Physics under contract DE-SC0011090, and the DOE Office of High Energy Physics under contract DE-SC0012567. Cloud computing resources were provided through a Microsoft Azure for Research award. Additional support was provided by the Harvard Data Science Initiative. JHEP-2 | http://arxiv.org/abs/1707.08600v3 | {
"authors": [
"Patrick T. Komiske",
"Eric M. Metodiev",
"Benjamin Nachman",
"Matthew D. Schwartz"
],
"categories": [
"hep-ph",
"hep-ex",
"stat.ML"
],
"primary_category": "hep-ph",
"published": "20170726182743",
"title": "Pileup Mitigation with Machine Learning (PUMML)"
} |
An Evolutionary SLS Framework for One-Dimensional Cutting-Stock Problems Software Competence Center Hagenberg GmbH Softwarepark 21 4232 Hagenberg, Austria Tel.: +43 7236 3343 857Fax: +43 7236 3343 888{georgios.chasparis,michael.rossbory,verena.haunschmid}@scch.atAn Evolutionary Stochastic-Local-Search Framework for One-Dimensional Cutting-Stock Problems Preliminary versions of part of this paper appeared in <cit.>, <cit.>. The research reported in this article has been supported by the Austrian Ministry for Transport, Innovation and Technology, the Federal Ministry of Science, Research and Economy, and the Province of Upper Austria in the frame of the COMET center SCCH. Partially this work has also been supported by the European Union grant EU H2020-ICT-2014-1 project RePhrase (No. 644235)Georgios C. Chasparis Michael Rossbory Verena Haunschmid Received: December 30, 2023/ Accepted: date ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== We introduce an evolutionary stochastic-local-search (SLS) algorithm for addressing a generalized version of the so-called 1/V/D/R cutting-stock problem. Cutting-stock problems are encountered often in industrial environments and the ability to address them efficiently usually results in large economic benefits. Traditionally linear-programming-based techniques have been utilized to address such problems, however their flexibility might be limited when nonlinear constraints and objective functions are introduced. To this end, this paper proposes an evolutionary SLS algorithm for addressing one-dimensional cutting-stock problems. The contribution lies in the introduction of a flexible structural framework of the optimization that may accommodate a large family of diversification strategies including a novel parallel pattern appropriate for SLS algorithms (not necessarily restricted to cutting-stock problems). We finally demonstrate through experiments in a real-world manufacturing problem the benefit in cost reduction of the considered diversification strategies.§ INTRODUCTIONCutting-stock problems formulate economic optimization problems aiming at the minimization of stock use so that certain job requirements are satisfied. They are concerned with the efficient slitting of bands of material out of a set of given rolls of material. Apparently, due to the immediate economic incentives, cutting-stock problems have attracted considerable attention, especially with respect to the development of optimization techniques for more efficient solutions. Cutting-stock problems have a long history, starting with the first known formulation of <cit.>, and the first advanced solution procedures based on linear programming proposed in <cit.>. In industrial environments, such as in the electrical transformers industry <cit.>, linear-programming techniques may not be appropriate due to the large number of constraints, some of which may be nonlinear in the parameters. Examples of such nonlinear constraints may include constraints imposed through eco-design measures (cf., <cit.>). Due to these nonlinear constraints, heuristic-based optimization techniques seem more appropriate for addressing this family of problems. To this end, this paper introduces a stochastic-local-search (SLS) algorithm for addressing a class of one-dimensional cutting-stock problems, namely the category 1/V/D/R according to the typology of <cit.>.In the remainder of this section, Section <ref> presents related work and Section <ref> states the main contribution of this paper. In the remainder of this paper, Section <ref> presents the class of the cutting-stock problems addressed by this paper and Section <ref> presents the proposed evolutionary SLS algorithm. In Sections <ref>–<ref>, we provide a more detailed description of the building blocks of the proposed algorithm, namely the set of possible operations for local search in Section <ref> and the set of diversification strategies in Section <ref>. Finally, in Section <ref>, we provide an experimental evaluation of the proposed algorithm, and Section <ref> presents concluding remarks. §.§ Related workThe one- or two-dimensional cutting stock problem is the problem of optimizing the slitting of material into smaller pieces of predefined sizes subject to several constraints. Almost 80 years ago the cutting stock problem was first addressed by Kantorovich in 1939 (translated to English in 1960 <cit.>) and Brooks in 1940 (reprinted in 1987 <cit.>) (but not yet using that name). This problem first arose in the field of cutting paper rolls <cit.> and was later also used for the processing of metal sheets <cit.>, wood <cit.>, glass, plastics, textiles and leather <cit.>. More recent publications also deal with stent manufacturing <cit.> and cutting liquid crystal display panels from a glass substrate sheet <cit.>.The cutting stock problem is closely related to bin packing, strip packing, knapsack, vehicle loading and several other problems <cit.>. A very commonly used objective is trim-loss or waste minimization. Other potential objectives that are mentioned in <cit.> can be the minimization of a generalized total cost function (consisting of material inputs, number of setups, labour hours and overdue time), subject to material availability, overtime availability and date constraints. Another interesting characteristic is the possibility to have multiple stock lengths <cit.> or the possibility of using leftovers <cit.>. An additional difficulty is when both the master rolls and the customer order can have multiple quality gradations <cit.>. Due to the fact that many different types of cutting stock problems can be found in the literature, e.g. with respect to dimensionality, characteristics of large and small objects, shape of figures and so forth, it is highly desirable that the scientific community is using the same language and terminology when describing problems and the corresponding solution approaches. In <cit.> a typology that covered the most important four characteristics at the time to classify optimization problems like the cutting stock problems was introduced. This typology was improved by <cit.>. An overview of characteristics how to classify the problems in the literature can be found in <cit.>. The problem can further be distinguished whether it is off-line (full knowledge of the input) or on-line (no knowledge of the next items) <cit.>. The cutting stock problem can be expressed as an integer programming problem with the downside that the large number of variables involved makes the computation of an optimal solution infeasible <cit.>. To solve such integer optimization problems a broad range of algorithms exist. First advanced solution procedures based on linear programming were proposed by <cit.>.A review and classification of approaches into heuristics, item-oriented or cutting pattern-oriented can be found in <cit.>.Most publications in this field employ exact algorithms which guarantee to find the optimal solution <cit.>. Exact solution approaches comprise column generation <cit.>, branch-and-bound methods <cit.> or a combination of them, e.g., the so-called branch-and-price algorithm <cit.>. A review of several linear programming formulations for the one-dimensional cutting stock and bin packing problem can be found in <cit.>.Another group of publications is dealing with heuristic solution approaches. Contrary to exact algorithms, heuristic approaches do not guarantee to find the optimal solution <cit.>. Heuristic methods can be split into three categories: sequential heuristic procedures, linear programming based methods and metaheuristics <cit.>. A procedure that is often mentioned in the literature is the sequential heuristic procedure <cit.>. Many other publications deal with heuristics as well. In <cit.> a very simple greedy heuristic is used. Some authors use heuristic-based column generation <cit.> or reduction <cit.> techniques. An example for an approach which combines an exact (branch-and-bound) with a heuristic (sequential heuristic procedure) method can also be found in the literature <cit.>.It appears that the first mention of using genetic algorithms for the cutting stock problem can be found in <cit.>. The authors state that the earliest use of heuristic search techniques in the field of operations research was published in 1985. Whereas reference <cit.> uses an improved version of the traditional chromosome representation, reference <cit.> represents cutting patterns as tree graphs. Genetic algorithms have also been used for related combinatorical problems, e.g. integrating processing planning and scheduling (usually those problems are considered separately) <cit.> and solving the variable sized bin-packing problem <cit.>. Further genetic algorithm approaches have been published in <cit.>. An overview over previous genetic algorithm applications can be found in <cit.>. The authors also provide a genetic algorithm model which was used to study three real-world case studies. Recently, an efficient genetic algorithm for the cutting stock problem is proposed by <cit.>. While most publications describe the crossover and mutation operators implemented <cit.>, none of them elaborate on diversification strategies which are an essential part of genetic algorithms.§.§ Contribution The main objective of this paper is the development of an evolutionary SLS algorithm to efficiently address 1/V/D/R cutting-stock problems including nonlinear constraints. In particular, our goal is to provide an optimization framework that a) exploits the capabilities of modern computing systems through parallelization, and b) provides a flexible integration and tuning of a large family of diversification strategies. In fact, the proposed optimization framework led to the development of a novel high-level parallel pattern specifically tailored for evolutionary SLS algorithms, an earlier version of which presented by the authors in <cit.>. The proposed diversification strategies are easily incorporated to the structural characteristics of the parallel pattern and demonstrate the flexibility of this framework to accommodate a great variety of settings depending on both user demands and computational capabilities. Finally, we present an experimental evaluation demonstrating the relative cost reduction incurred due to the introduction of the proposed diversification strategies.A preliminary version of the proposed methodology first introduced by the authors in <cit.>, however, without providing the details of the optimization algorithm. The present paper provides a ready-to-use comprehensive treatment of one-dimensional cutting-stock problems usually encountered in the industry.§.§ Notation For convenience, we summarize here some generic notation that is used throughout the paper.* 𝕀_A denotes the index function such that 𝕀_A≐ 1 A=𝚝𝚛𝚞𝚎,0 * e_i∈ℝ^n denotes the unit vector of size n, such that its ith entry is equal to 1 and the rest are equal to 0;* For a matrix X=[x_ij]_i∈ℐ,j∈𝒥, we denote the row vector corresponding to the ith row as X_i: = [x_ij]_j∈𝒥 and the column vector corresponding to the jth column as X_:j = [x_ij]_i∈ℐ.* For a finite set A, A denotes its cardinality. § PROBLEM FORMULATION We consider a general class of one-dimensional (1/V/D/R) cutting stock problems. In particular, we are given a set of items (or bands) of certain types ℐ={1,2,...,n}, each of which is characterized by its width b_i and its desired weight w_i. The set of desired items ℐ need to be placed to a set of objects (or rolls) of material, denoted by 𝒥={1,2,...,m} each of which is characterized by its width b_j, its length ℓ_j and its specific weight d_j, j∈𝒥. Let also w_j denote the overall weight of roll j.We define an assignment of bands into rolls, denoted x_ij∈ℤ_+, so that x_ij denotes the number of bands of type i assigned to roll j. The objective is to find an assignment X≐{x_ij}_i,j of bands into the available rolls, so that: * the overall weight of each band type i exceeds its desired weight w_i, i.e.,b_i∑_j=1^mx_ijℓ_j d_j ≥ w_i.To simplify notation, let us denote α_j≐ℓ_jd_j, in which case the constraint above may equivalently be written asy_i(X)w_i - b_i∑_j=1^mx_ijα_j≤ 0,where y_i(X) corresponds to the rest weight of item i. * the overall width assigned to each roll i does not exceed the width of the roll, b_i, while at the same time the residual band, denoted r_j(X) ≐ b_j - ∑_i=1^nx_ijb_i ∈ℛ_j,where ℛ_j⊆[0,∞) is a union of closed intervals of allowable residual bands, which might be different for each roll j∈𝒥. Both of the above constraints are hard constraints and need to be satisfied for any assignment. Unfortunately, in most but trivial cases, there might be a multiplicity of admissible assignments, each of which might be utilizing a different subset of rolls 𝒥. We wish to minimize the overall weight of the rolls utilized by an assignment, i.e., we wish to address the following optimization problem:min_X∈ℤ_+^m×n g(X),whereg(X) ∑_j∈𝒥w_j𝕀_{∃ i∈ℐ:x_ij>0}is the total weight of rolls used by assignment X.Then, the overall optimization problem may be written in the following form:min_Xg(X) y_i(X) ≤ 0 r_j(X) ∈ℛ_j X∈ℤ_+^n×mNote that we allow for ℛ_j to depend on the roll j∈𝒥. This is due to the fact that some extra processing of the rolls may be necessary for each item which depends on the roll itself (e.g., seaming of the edges). Furthermore, we do not allow all possible residual bands in ℛ_j, since not all residual bands can be stored in stock. This constraint introduces a nonlinearity. The details of the optimization problem (<ref>) (i.e., the characteristics of items ℐ and objects 𝒥 and the set of allowable rest-widths ℛ_j, j∈𝒥), define an instance of the optimization problem, denoted π. Let also Π be the family of such optimization instances.We have currently chosen a rather small number of constraints, namely the job-admissibility and rest-width constraint. However, it is important to point out that additional constraints may also be added, depending on the application of interest, such as the eco-design measures mentioned in <cit.>. Such constraints may be nonlinear in the parameters. Furthermore, alternative objective criteria may be considered (e.g., minimization of trim loss, as considered in <cit.>, or minimization of rolls needed by the assignment when the objects are identical, as considered in <cit.>). It is important to note that the forthcoming algorithm does not depend on the specifics of the considered constraints in optimization (<ref>). In fact, there is no restriction on the type and number of constraints imposed. Thus, the reader should consider optimization problem (<ref>) more as an example that will convenience the discussion throughout the paper, rather than as a restriction. § EVOLUTIONARY STOCHASTIC-LOCAL-SEARCH (SLS) ALGORITHM The proposed algorithm consists of the following phases: * Initialization ();* Optimization ();* Filtering ();* Selection (). The role of the initialization () phase is the establishment of candidate solutions (not necessarily satisfying all the constraints). The performance of the initialization function is rather important to the performance of the overall optimization, since a good head-start may significantly improve the performance of the SLS algorithm. The role of the optimization () phase is the execution of admissible local improvement steps, accompanied with admissible perturbations. Responsible for the execution of these improvement steps and/or perturbations of the existing candidate solutions are the working units, namely , each of which plays a distinctive role (explained in detail in the forthcoming Section <ref>). We may argue that the core of the proposed architecture lies in the design of the working units.The role of the filtering () phase is the observation of the current status of the optimization algorithm, the storage and/or re-processing of candidate solutions. In other words, it supervises the state of all candidate solutions and decides on their future use. Finally, the role of the selection () phase is to decide on whether the optimization algorithm should be terminated according to some criteria (e.g., the size of the pool or time constraints) and, if yes, what should be the best estimate X̂ of the optimal solution. The details of the (SLS) optimization algorithm are presented in Figure <ref>, and they will be explained progressively in the forthcoming sections. §.§ Initialization () The initialization phase discovers initial candidate solutions (i.e., assignments X which satisfy the job-admissibility constraint). The benefit of discovering such candidate solutions is rather important since it may fasten significantly the following optimization phase. Furthermore, if there exist a large number of local optima[We use the term local optima to refer to assignments X at which any local search (based on some predefined operations/modifications) may not result in a strictly better assignment with respect to the objective function.] (as it is often the case), the starting point may influence significantly the resulting performance. To this end, it is necessary that we consider alternative fitness functions for the generation of candidate solutions.The goal of the initialization routine is to create an initial set of job-admissible assignments X, denoted 𝔓 (main pool) of bounded maximum size (𝔓_max). This pool of candidate solutions will then be used for optimization.The possibilities for designing such a routine are, in fact, open ended. Here, we would like to provide one general structure, the specifics of which are provided in Table <ref>. As a first step, we sort both items ℐ and objects 𝒥 in descending order with respect to their width, similarly to a standard First-Fit-Decreasing (FFD) algorithm <cit.>. This is usually necessary, at least in cutting-stock problems, since large items may only fit in large objects. Then, for each item i (with positive rest weight), we go through each available object j (which may fit i), and we check whether an assignment of i to j maximizes a fitness criterion. In particular, we introduce here three candidate fitness functions:* negative weight of residual band of object j, i.e., f_i(X;j) ≐ - r_j(X)α_j.This fitness function is maximized when the weight of the residual band takes values that are close to zero. In other words, we try to fit as many objects as possible in each object j∈𝒥. * negative absolute rest weight of item i minus the weight of the residual band of object j∈𝒥, i.e., f_i(X;j) ≐ - |y_i(X)| - r_j(X)α_j.Note that this function takes large values when both a) the rest weight of item i and b) the residual weight of object j are close to zero. In other words, under this criterion, we try to fit as many items as possible within object j without though generating large rest weight for the selected items. * band i's weight minus the weight of object j's residual band, i.e.,f_i(X;j) ≐α_jb_i - α_jr_j(X).Note that this function takes large values when the weight of item i is large compared to the residual band's weight, i.e., we try to fit as many bands as possible in each single object j. Alternative fitness functions may be defined. Note that all the above fitness criteria employ a form of weight optimization (contrary to standard First-Fit-Decreasing algorithms). Such form of weight optimization is motivated primarily by two practical reasons: a) the difficulty in finding admissible assignments when the stock material is limited (something that cannot be guaranteed by standard First-Fit-Decreasing algorithms), and b) the limited time offered to solve the optimization problem, which makes the initial assignment a rather important factor over the final performance. In the forthcoming experimental evaluation (Section <ref>), we will demonstrate the impact of this good head-start in the final performance.In the remainder of this paper, we will assume that an initial candidate solution X that satisfies the job-admissibility constraint is available as a result of this initialization phase. Note that we do not impose the rest-width constraint during this initialization phase, due to the fact that in several cases it is rather difficult to meet this constraint. A special treatment of the rest-width constraint is suggested as part of the following optimization phase.§.§ Optimization () As described in Figure <ref>, the results of theroutine are used to populate an initial set of candidate solutions (main pool 𝔓) that will later be processed withto generate admissible solutions for the optimization instance π∈Π.In particular, the following steps are executed. First, the available candidate solutions in 𝔓 are sequentially processed through a sequence of working units. They are responsible for performing the necessary steps for either establishing admissible solutions and/or for further improving the performance of a candidate solution. We introduce three types of working units, namely: * * * where ℭ{𝒞_ job,𝒞_ rw} is the family of constraints considered, corresponding to the job-admissibility (<ref>) and rest-width (<ref>) constraints. The role of theis to address the rest-width admissibility constraint, since it might not easily be satisfied forthe job-admissible assignments generated at . In case additional constraints were considered in the optimization problem (<ref>), then additional such workers would have to be introduced.The role of theis to generate improved assignments X with respect to the objective function in (<ref>).Finally, the role of theis to perform a sequence of perturbations in the currently admissible solutions in order to avoid convergence to local optima.All these classes of working/processing units are based upon appropriately introduced modifications of the candidate solution X, so that either a constraint is satisfied or an improvement is observed in the cost function. These modifications, namely operations, constitute the core of the proposed optimization scheme and will be described in detail in the forthcoming Section <ref>. A set of available operations will be considered, denoted 𝒪. Let also o∈𝒪 denote a representative element of this set. For now, we will only assume that each operation o∈𝒪 is equipped with three alternative modes, denoted: * ,* ,* .The above introduced modes designate three alternative possibilities for searching and executing operations in the candidate solution X. The better reply of an operation o∈𝒪, denoted, is based upon a semi-stochastic search of neighboring assignments to X which improves the cost of the current assignment X, i.e., it generates an improved candidate solution with respect to the objective g() that also satisfies the constraints in ℭ. Themode performs a similar type of search, however the objective is to guarantee that the resulting assignment satisfies the constraints within ℭ of the optimization. The set ℬ⊂𝒥 is the set of “bad” objects that currently do not satisfy the constraint. Such a mode may be particularly beneficial when X does not currently satisfy some of the constraints. Finally, themode of an operation o∈𝒪 is based upon a random generation of local to X assignments that satisfy the constraints ℭ, however they may not necessarily improve the cost of X.In the remainder of this section, we describe in detail the three introduced working units of the optimization routine which utilize some or all of the three alternative operation modes stated above. §.§.§ Rest-width worker () Note that the initially generated assignments X in 𝔓 define candidate solutions that satisfy only the job-admissibility constraint (<ref>). The role of the rest-width worker is to operate on the candidate solutions on the main pool 𝔓, so that they satisfy the rest-width admissibility constraint for all objects j∈𝒥, without violating the job-admissibility constraint. The rest-width constraint is usually the most difficult constraint to be satisfied, and this is the reason it cannot be treated during the initialization phase of Section <ref>. Usually, a sequence of operations might be necessary before rest-width admissibility is achieved. The rest-width worker is governed by the parameter N_ con which corresponds to the total number of processing steps executed per each run of this worker. The architecture of the rest-width worker is presented in Table <ref>. As we can see, it simply executes a number of operations under themode until a rest-width admissible solution is found.It is important to point out that if a candidate solution X satisfies this constraint (after passing through this worker), then it will satisfy this constraint for the remaining of its processing history, i.e., no further processing will be requested for X within this working unit. This is due to the fact that any subsequent working unit will never violate any of the constraints in ℭ.§.§.§ Local optimization worker () The local optimization worker receives a candidate solution, X, which satisfies both constraints (<ref>)–(<ref>), i.e., job-admissibility and rest-width constraint. The objective is to perform local operations in the candidate solution X under themode, so that the resulting assignment reduces the cost (<ref>), while maintaining admissibility. The operation of the local-optimization worker is governed by N_ loc∈ℕ which is the total number of operations executed per run of this worker. A description of this worker is provided in Table <ref>. Note that the number of steps executed within this worker is fixed and independent on whether a cost reduction is found or not.§.§.§ Perturbation worker () With probability λ>0, a number of operations under themode are executed while maintaining admissibility and without necessarily reducing the cost (<ref>). A fixed number of N_ per operations is executed per each run of this worker. The goal of such operations is to escape from local optima by temporarily accepting worse performances. A description of the architecture of the perturbation worker is provided in Table <ref>. §.§ Filtering () The role of theis to assess the quality of the produced candidate solutions and to decide which of the candidate solutions need to be reprocessed and which should be reserved for later use. The operation of theis based upon the notions of reprocessing and reservation. In particular, the filter is responsible for updating two sets of candidate solutions, namely the main pool 𝔓 and the reserve pool ℜ. The pool, 𝔓, is the set of candidate solutions that are currently getting processed (i.e., they go through the optimization steps in ). The reserve pool, ℜ, maintains candidate solutions from earlier stages of the optimization of high potential that may replace candidate solutions within the main pool 𝔓 upon request.More specifically, the main steps of theare shown in Figure <ref>. In words, for each candidate solution X_i of the main pool 𝔓, we first assess its good standing and decide on whether it should further be processed based on prior performances summarized in memory 𝔐(X_i). In particular, if X_i is not of a good standing, then it is removed from the main pool (i.e., ) and replaced by another candidate solution from the reserve pool (i.e.,and ). If, instead, X_i is of a good standing, then it remains part of the main pool 𝔓 and continues on the assessment of its high potential, based again on prior performances summarized in memory 𝔐(X_i). If X_i is identified as having high potential, then it is saved into the reserve pool ℜ (i.e., ). When all candidate solutions have been filtered (i.e., assessed with respect to their good standing and high potential), then the main pool 𝔓 exits the .The reserve pool ℜ is initially empty and it is gradually populated by solutions of high-potential. However, we assume that ℜ has a bounded maximum size ℜ_max, which means that when its capacity is reached, the newly entered solution will replace the oldest one in the pool. The reason for selecting a bounded maximum size is to allow for dynamic restarts within a bounded fitness-distance from the current best. In many practical scenarios, performing dynamic restarts from very early stages may delay significantly the optimization speed. However, by properly selecting the maximum size of the reserve pool, we may find an appropriate balance between earlier dynamic restarts and optimization speed. It is also important to note that the maximum size of the main pool 𝔓 is also bounded by 𝔓_max (as discussed in Section <ref>). When a reserved solution needs to replace another one from the main pool, the oldest one in ℜ is always picked. Also, when a reserved solution is requested to replace X∈𝔓 but ℜ is currently empty, then a replacement is not possible and the size of the main pool is going to reduce by one. When the main pool 𝔓 becomes empty, then the optimization should terminate.§.§.§ Good standing () The functionassesses the potential of a candidate solution X to continue providing reductions in the cost value. It depends on the current version of the candidate solution X, its prior memory 𝔐(X), and the currently best candidate solution X̂. Informally, the role of the good-standing assessment is to capture whether a candidate solution X is currently able to follow the path of the currently best candidate X̂, and this is achieved by evaluating a) the fitness-distance of X from the current best X̂, b) the steps elapsed since the last time X provided an improvement to X̂, and c) the gradient of the cost reduction during the most recent processing steps of X. In fact, we would like that the candidate solution X is sufficiently close to the current best, and its cost gradient is sufficiently large.To accomplish this assessment, we first introduce the following parameters:* N_ gs^* denotes the number of processing steps over which the good standing of a candidate solution is evaluated.* D(X,X̂) denotes the normalized fitness-distance between X and the current best X̂, defined as follows:D(X,X̂) g(X)-g(X̂)/g(X̂). * D_ gs^* denotes the maximum allowed normalized fitness distance from the current best for which the good standing is maintained.* G(X,𝔐(X),N_ gs^*) denotes the gradient of cost reduction observed in the candidate solution X, defined as follows:G(X,𝔐(X),N_ gs^*) g(X[N_ ps-N_ gs^*]) - g(X[N_ ps])/N_ gs^* · g(X[N_ ps-N_ gs^*])≥ 0,for N_ ps > N_ gs^*, where N_ gs^* denotes the number of recent processing steps over which we estimate the gradient change of the cost function.* G_ gs^* denotes a threshold based on which the gradient G of cost reduction is being evaluated.* N_ rb(X) denotes the number of processing steps since the candidate solution path leading to X has provided a recent best (i.e., a reduction to the cost function).In particular, the exact steps for assessing the good standing of X are provided in Table <ref>. First, we check whether enough processing steps have elapsed for the good standing criterion to be evaluated (i.e., N_ ps(X) ≥ N_ gs^*). This initial check is required due to the absence of any prior knowledge regarding the potential of a candidate solution X. Then, we evaluate a) the normalized fitness-distance of X from the current best X̂, D(X,X̂), and b) the processing steps elapsed since the last improvement offered by X, N_ rb(X). Both have to be small enough to maintain a good standing. Lastly, we check whether X exhibits a sufficiently large progress rate. If its progress rate is larger than G_ gs^*, then X will retain its good standing. In other words, for maintaining a good standing, it is not sufficient that X is close enough (with respect to fitness) to the current best, rather it should also provide a progress rate that is promising for improving the current best.The parameters D_ gs^*, G^*, and N_ gs^* need to be determined by the user. §.§.§ High potential () The notion of high potential of a candidate solution X captures its ability to provide significant reductions to the objective function (<ref>). For example, an indicative criterion of a high potential is an extremely high progress rate G(X,𝔐(X)). We wish to store candidate solutions exhibiting high potential at different stages of their processing paths, so that we are able to restart the processing from these stages. This form of restarts may serve as a mechanism for escaping from local optima, while at the same time it may increase the probability of converging to the optimal solution.To assess the high potential of a candidate solution, we first introduce the following notation: * N_ hp^* denotes the number of processing steps over which the high potential of a candidate solution is evaluated.* D_ hp^* denotes the maximum allowed normalized fitness-distance from the current best for which the high potential property can be assessed. * G_ hp^* denotes a threshold based on which the gradient of cost reduction G is being evaluated.* N_ lr(X) denotes the elapsed processing steps since the candidate solution X was last reserved into ℜ.The exact steps for assessing the high potential of a candidate solution X are provided in Table <ref>. First we check whether enough processing steps have elapsed for the high-potential criterion to be evaluated. Then, we evaluate a) the normalized fitness-distance of X from the current best X̂, D(X,X̂), and b) the processing steps elapsed since the last reservation offered by the processing path of X. We would like X to be close enough to the current best, however we would also like X not to have provided recently any reservations (thus not creating subsequent reservations from the same processing paths). Lastly, we check whether X exhibits a sufficiently large progress rate, i.e., larger than G_ hp^*. We should expect that D_ gs^*>D_ hp^* and that G_ gs^*<G_ hp^*, since high-potential should also imply good-standing but not the other way around. §.§ Selection () The last part of the optimization algorithm is the selection phase (). In this phase, the main pool 𝔓 of candidate solutions has already been filtered, and the question is whether we should terminate processing and select the current best estimate X̂ or continue processing 𝔓. The specifics of this phase are shown in Figure <ref>. The terminate decision is taken upon the current size of the main pool 𝔓 as well as the maximum allowed processing time t_max imposed by the user. In particular, when the size of the main pool is zero, or when the elapsed processing time has exceeded t_max, then the optimization is terminated and the current best estimate X̂ is provided as an output.§ OPERATIONS As we mentioned in the description of the optimization phase (), the means by which local modifications of candidate solutions are created/searched is through a set of available operations 𝒪. In this section, we would like to provide a description of the operations considered, as well as a description of their alternative modes (as initially introduced in Section <ref>), namely ,and . The operations implemented are the following: * , * ,* ,* ,* ,* ,§.§operation The move operation moves a single item from one object (or roll) to another object (see Figure <ref>). The details of themode of this operation are described in Table <ref>.Note that this mode implements a form of nested search over the source object (), an item from this object () and the destination object (). Then, it operates a move of thefromto . Theshould be selected from the set of objects that are currently used (), i.e., from the set of objects with at least one item assigned to them. Given that the number of objects might be large, we can bound the size of this nested search via the parameter N_ br_trials. The selection of this parameter depends on the number of available objects 𝒥, and the desired time of optimization t_max determined by the user.Similar in architecture is themode of this operation, however the objective is to simply generate admissible assignments for those objects that do not satisfy the constraint. The details are shown in Table <ref>. In particular, for N_ con_trials times, we go through the set of “bad” objects that do not satisfy the constraint. For each one of the items of this object, we go through a number of destination objects (different than the “bad” object), and we check whether admissibility of the modified assignment has been resolved with respect to the “bad” object (without violating the job-admissibility requirements for the moved item). Similarly to themode, this mode is controlled by appropriately selecting the parameter N_ con_trials.Themode of this operation is described in Table <ref>. It simply relies on a random selection of two (distinct) objects, one of which should have a non-zero number of items assigned to it. Then, an item is randomly picked from one of the objects and is assigned to the second object. The objective is to generate a new assignment X' that maintains ℭ-admissibility. The effect of this mode into the cost is not considered. Finally, note that there is not a unique way to initialize the objects/items involved in these modes, and this is also true for all the operations that follow. For example, inmode, we have selected a nested type of search to initialize the objects/items involved, while inmode, we implement a completely random initialization. This is not restrictive. In practice, a nested type of initialization of the parameters provided a faster discovery of better replies, however formulating the most efficient architecture for these modes requires a separate investigation. §.§operation Theoperation swaps one or more items from one object (or roll) with one or more items from another object (see Figure <ref>). In particular, theoperation is based upon the selection of two objects (among all objects with non-zero items) and the selection of a combination of items from these objects. Then, the selected items from both objects are swapped with each other. Figure <ref> provides an example, where two items from object 1 are swapped with one item from object 2.Themode of this operation, for the case of 2 objects, is described in Table <ref>. This mode incorporates the swap of a combination of items from object 1 with another combination of items from object 2 (different than object 1). This mode can be augmented by alternative swap types, where more objects are involved. Theandmodes of theoperation are similar to the corresponding modes of theoperation, however following the structure of the swap operation. Accordingly, themode is based upon a random selection of objects, and a random selection of a combination of items from these objects.§.§operation Theoperation splits one item currently allocated in one object into two other distinct objects (see Figure <ref>).Themode of this operation is described in Table <ref>.This mode randomly selects an object () from the set ofand a combination of 2 distinct objects ( and ) that are different from . Then it assigns a randomly selected item fromto bothand . This selection of objects and an item is repeated for a fixed number of times controlled by N_ br_trials or until a better reply is found. Theandmodes of theoperation are similar to the random mode of theoperation, however following the structure of theoperation. §.§operation Theoperation involves an object with two or more items and moves a combination of its items to new distinct objects (which may or may not have items assigned to them), see Figure <ref>. In a way, theoperation compliments theoperation, since it involves more than one item and more than one destination objects. Themode of this operation is described in Table <ref>. According to this mode, we select a) a source object, b) a combination of items from the source object, and c) a combination of distinct destination objects equal in number to the selected items. Then, each one of the items moves to one of the destination objects, so that the cost of the optimization is reduced (e.g., see example of Figure <ref>).Similarly, we may define theandmodels of this operations (the same way we did for theoperation in Section <ref>). §.§operation Theoperation performs a reverse form of theoperation. In particular, the goal of this operation is to combine items from several source objects and move them into a new destination object, see Figure <ref>. Themode of this operation is described in Table <ref>. According to this mode, we find a) a combination of source objects, b) an item from each source object, and c) a destination object. We, then, move the selected items to the destination object and we check whether the cost has been reduced. Theandmodes of this operation can be defined in a similar way, as we did for theoperation in Section <ref>.§.§operation Theoperation checks whether an item assigned to an object can be removed without violating any of the constraints (i.e., job-admissibility and rest-width constraints). Obviously if any such item can be removed without violating the constraints, then it is likely that the objective function will be reduced. For example, this might be the case when there are objects with a single item assigned to them. Themode of this operation is described in Table <ref>. Similar is the reasoning inandmodes of this operation. § DIVERSIFICATION STRATEGIES The proposed structure of the optimization algorithm may naturally incorporate several diversification strategies. Diversification strategies aim at preventing the search process getting trapped in local optima. In this section, we present a set of possible diversification strategies that can be exploited.Given that optimization problems of the form (<ref>) may differ significantly from each other, both in the number of feasible solutions as well as in the search space, the development of a unified selection and tuning of diversification strategies becomes almost impossible. The goal of this paper is to demonstrate that the diversification strategies considered here can make a difference at least on average.We may group the set of diversification strategies considered here in the following categories.* Diverse initialization strategies* Bounded-distance dynamic restarts* Random perturbations* Parallelization* Cost-free operations§.§ Diverse initialization strategies As we discussed in Section <ref>, we introduced three alternative criteria in the formulation of initial candidate solutions. Our objective is to generate initial admissible solutions that a) are close enough to the optimal solution, and b) are diverse enough to increase the possibility for convergence to the optimal solution. §.§ Bounded-distance dynamic restarts A candidate solution may currently be located at search spaces with no significant potential in reducing the current cost. Thus, in order to avoid stagnation, we allow such candidate solution to be replaced by another one from an earlier stage but within a bounded fitness distance from the current best. There are two operations that indirectly control the process of dynamic restarts, namely theandfunctions. As already described in Section <ref>, thefunction and its parameters (i.e., D_ gs^*, G_ gs^* and N_ gs^*) control the criteria for dropping a candidate solution, while thefunction and its parameters (i.e., D_ hp^*, G_ hp^* and N_ hp^*) control the criteria for reserving a candidate solution. Apart from these parameters, the size of the main and reserve pool also affect the evolution of the optimization algorithm. These parameters need to be carefully tuned so that the bounded-distance dynamic restarts increase the probability of escaping from a local optimum without though delaying significantly the optimization process. The response to these parameters may differ significantly between application scenarios.§.§ Random perturbations Random perturbations constitute an alternative way to avoid stagnation, through the introduction of a sequence of randomly generated operations. In particular, with a small positive probability λ>0, a candidate solution goes through the , in which a sequence of operations is randomly selected, and a random initialization of this operation is set ( mode). If the perturbation leads to an admissible solution, then the modification is accepted independently of the resulting objective value. This is essentially similar in spirit with the random-walk extension introduced in <cit.>. §.§ Parallelization The implementation of the optimization algorithm depicted in Figure <ref> suggests a high-level (domain-specific) parallel architecture that does not fit to standard parallel patterns. Simple for-loop parallelization, using, e.g.,a parallel-for pattern, is not possible, since the iteration space is not known a-priori. In fact, we do not know how many times a candidate solution should pass through the optimization worker () before we stop processing it. To this end, first we partition the main pool 𝔓 into K sets of candidate solutions (where K is our planned parallel degree). We then process these sets in parallel through thefunction. Finally, we decide on whether a candidate solution should continue processing through the implementation of the . The proposed parallelization architecture is depicted in Figure <ref>.The proposed parallel pattern is a modification of the so-called pool-pattern, first implemented in <cit.> and incorporated in the FastFlow parallel programming framework in <cit.>. The modification lies in the introduction of the reserve pool ℜ that simplifies the implementation of the bounded-distance dynamic restarts, which is an essential part of several SLS algorithms. As probably expected, the larger the parallel degree, the larger the probability of getting closer to the optimal solution. This will be demonstrated through experiments in the forthcoming Section <ref>, but it may also be supported by the statistical literature <cit.> as pointed out in <cit.>. §.§ Cost-free operations By cost-free operations we mean operations that may result in (practically) the same cost. The role of such operations may be proven rather important mainly because they introduce perturbations with no impact to the objective. Thus, through such operations, we may diversify the search of optimal solutions without necessarily restarting from worse assignments. More formally, let X' be a modification over an initial candidate solution X. A modification becomes acceptable subject to a better-reply criterion if the following smooth condition is satisfied:g(X') < g(X) + ζ(X,X'),for some real-valued function ζ. Function ζ applies a form of extra allowance in the acceptance of an improvement. For example, we may assume that ζ=ϵ for some small positive constant ϵ>0. Alternative criteria may be defined.§ EXPERIMENTAL EVALUATION The goal in this section is to evaluate the behavior of the evolutionary algorithm and understand the role of the alternative diversification strategies introduced. In particular, we are going to demonstrate the effect of each one of the proposed diversification strategies in the overall performance of the algorithm. To this end, we consider a number of optimization problems of the form (<ref>) borrowed from the industry of manufacturing transformer cores (cf., <cit.>). Each optimization problem is fully described by a set of available rolls and a set of required items to be produced (characterized by their width and weight). The optimization problems considered span from jobs of small number of itemsto medium or large number of items, thus covering a range of possibilities and enough variety to test the effect of the proposed algorithm and the diversification strategies.The optimization problem (<ref>) minimizes the used objects' weight. Since the performance of the optimization might be affected by the size of the problem (i.e., the weight of the produced items), we introduce a performance metric normalized by the size of the problem. Let us assume that a set Π of different optimization problems is considered. Let X̂_π denote the best estimate discovered by the algorithm, and W_π be the total weight of the required items, i.e., W_π∑_i∈ℐw_i. We introduce the following evaluation metric, 𝒢:Π→ℝ_+, such that 𝒢(π) g(X̂_π)/W_πg(X̂_π)/∑_π∈Πg(X̂_π).The first part of the performance metric corresponds to the ratio of objects' weight over the items' weight. Naturally the higher this ratio is, the less efficient the solution would be. However, an inefficient solution at a small-size problem does not have the same impact with an inefficient solution at a large-size problem. To this end, the role of the second ratio is to normalize the effect of the solution in the overall cost. Furthermore, note that ∑_π∈Π𝒢(π) is the weighted average ratio of the used objects' weight over the required items' weight, i.e., it captures the efficiency of the optimization. In the following subsections, we will investigate the effect of the alternative diversification strategies proposed here (as well as some of the optimization parameters) and their impact in the overall performance of the algorithm via metric (<ref>).§.§.§ Diverse initialization strategies We start our investigation by varying the set of strategies considered in the initialization phase of the algorithm. In Figure <ref>, we see the impact of the initially considered candidate solutions when we restrict the set of strategies used to generate them. In the first scenario (dashed line), we employ the initialization criterion of Equation (<ref>), while in the second scenario (solid line), we consider all available initialization criteria of Equations (<ref>)–(<ref>). In either case, the initial set of candidate solutions (main pool 𝔓) has the same size and it is populated equally by each one of the considered initialization strategies.It is important to point out that the benefit of the larger variety of initialization strategies is also due to the fact that the considered strategies employ a form of weight optimization therein. We should not expect that the same improvement occurs when we consider initialization strategies that do not apply a form of weight optimization. In Figure <ref>, as well as in the following experiments, we also demonstrate how the performance varies as we increase the parallel degree (K). This corresponds to the effect of parallelization following the architecture of Figure <ref>. Furthermore, in all these experiments, t_max=5min, unless otherwise specified.§.§.§ Random perturbations In this set of experiments, we would like to test the effect of the random perturbations controlled by the parameter λ>0. Figure <ref> shows the increase in performance when λ increases to λ=0.1 (i.e., with probability 0.1, a candidate solution goes through the . Note that the random perturbations have a positive impact in the optimization performance.§.§.§ Optimization time We would also like to investigate how the optimization algorithm responds when the maximum allowed optimization time (t_max) increases. It is natural to expect that the performance in this case should increase, since some of the candidate solutions will be processed for a longer period of time. Note further that the increase in performance seems to reduce as we increase the parallel degree. One explanation for this behavior is the fact that an increased parallel degree already allows for a faster processing, thus it is highly likely that we are already sufficiently close to the optimal solution. §.§.§ High-potential threshold Recall that D_ hp^* and G_ hp^* are two thresholds that control the reservation of candidate solutions into the reserve pool ℜ. We would like here to investigate the effect of D_ hp^* which corresponds to the normalized fitness-distance from the current best within which a high-potential candidate solution can be stored into the reserve pool ℜ. In other words, we would like to investigate how the performance of the optimization varies when we move the reservation threshold towards earlier stages.When candidate solutions from earlier stages are stored into the reserve pool ℜ, we should expect that the escape probability from local optima should increase. However, the size of the reserve pool ℜ is fixed, which means that the oldest candidate solutions may be dropped to create space for more recent reservations. Thus, when we just increase the threshold D_ hp^* the benefit is not guaranteed (Figure <ref>). However, if we also increase the size of the reserve pool, then the benefit for increasing the threshold can be materialized (Figure <ref>).§.§.§ Good-standing threshold Recall that D_ gs^* and G_ gs^* are two thresholds that control the assessment over allowing a candidate solution to continue processing (Section <ref>). In particular, D_ gs^* denotes the normalized fitness distance from the current best within which a candidate solution is allowed to continue processing. We wish to investigate how the performance of the optimization will change as we increase D_ gs^*, i.e., if we allow candidate solutions to remain longer within the main pool 𝔓. Figure <ref> demonstrates how the performance changes as we progressively increase D_ gs^*. As expected, the longer we allow a candidate solution to remain within the main pool, the slower the rate of using reserved candidate solutions, and the greater the probability that the process is trapped in a local optimum. Thus, we should expect that the performance may degrade as we increase D_ gs^* more than necessary (Figure <ref>). However, if we also increase the optimization time, then we may achieve better performance (since we exploit better the fact that solutions remain in the pool for longer periods of time), Figure <ref>.§.§.§ Cost-free operations In the last experiment, we wish to investigate the effect of cost-free operations in the performance of the optimization. Cost-free operations essentially allow for local operations that do not have any significant impact in the overall performance. As probably expected, this allows for a candidate solution to escape from a local optimum with no impact in the performance. Figure <ref> demonstrates the fact that cost-free operations can indeed increase the overall performance rather constantly over the parallel degree. § CONCLUSIONS This paper presented an evolutionary SLS algorithm specifically tailored for one-dimensional cutting-stock optimization problems. The novelty of the proposed algorithm lies in its architecture that allowed for the introduction of a novel parallel pattern and the easy integration of a large family of diversification strategies. Although the proposed SLS algorithm was presented within the context of one-dimensional cutting-stock problems, its architecture is generic enough to be integrated into a larger family of combinatorial optimization problems. Finally, the presented experimental evaluation demonstrated a significant improvement in efficiency due to both the proposed parallel pattern as well as the additional diversification strategies presented. splncs03 | http://arxiv.org/abs/1707.08776v1 | {
"authors": [
"Georgios C. Chasparis",
"Michael Rossbory",
"Verena Haunschmid"
],
"categories": [
"cs.NE"
],
"primary_category": "cs.NE",
"published": "20170727083117",
"title": "An Evolutionary Stochastic-Local-Search Framework for One-Dimensional Cutting-Stock Problems"
} |
Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM_110 mode for ultrafast electron microscopy [ December 30, 2023 ======================================================================================================================================== Convolutional Neural Network (CNN) models have become the state-of-the-art for most computer vision tasks with natural images. However, these are not best suited for multi-gigapixel resolution Whole Slide Images (WSIs) of histology slides due to large size of these images. Current approaches construct smaller patches from WSIs which results in the loss of contextual information. We propose to capture the spatial context using novel Representation-Aggregation Network (RAN) for segmentation purposes, wherein the first network learns patch-level representation and the second network aggregates context from a grid of neighbouring patches. We can use any CNN for representation learning, and can utilize CNN or 2D-Long Short Term Memory (2D-LSTM) for context-aggregation. Our method significantly outperformed conventional patch-based CNN approaches on segmentation of tumour in WSIs of breast cancer tissue sections.§ INTRODUCTION Recent technological developments in digital imaging solutions have led to wide-spread adoption of whole slide imaging (WSI) in digital pathology which offers unique opportunities to quantify and improve cancer treatment procedures. Stained tissue slides are digitally scanned to produce digital slides <cit.> at different resolutions till 40× as shown in Figure <ref>. These digital slides result in an explosion of data which leads to new avenues of research for computer vision, machine learning and deep learning communities. Moreover, these multi-gigapixel histopathological WSIs can be excellently absorbed by data hungry deep learning methods to tackle digital pathology problems.Convolutional Neural Network (CNN) models have significantly improved the state-of-the-art in many natural image based problems such as visual object detection and recognition <cit.> and scene labelling <cit.>. However, classification of WSIs through a CNN raises serious challenges due to multi-gigapixel nature of images. Feeding the complete WSI or resizing WSI either leads to computationally unfeasible methods or loss of crucial cell level features essential for segmentation. This results in processing WSIs which are typically 200K ×100K pixels in size in a patch-by-patch manner. Since patch based approaches face difficulties in handling images larger than a few thousand pixels, therefore using larger patches to capture maximum context is not a solution. A huge difference between patch size and WSI size results in loss of global context information which is extremely important for many tumour classification tasks <cit.>. We propose Representation-Aggregation Networks (RANs) to efficiently model spatial context in multi-gigapixel histology images. RANs employ a representation learning network as a CNN which encodes the appearance and structure of a patch as a high dimensional feature vector. This network can be any state-of-the-art network such as AlexNet <cit.>, GoogLeNet <cit.>, VGGNet <cit.> or ResNet <cit.>. A 2D-grid of features is generated by packing feature vectors for neighbouring patches in the WSI as encoded by the representation learning network. The first variant of context-aggregation network (RAN-CNN) in RAN utilizes a CNN with only convolutional and dropout layers. RAN-CNN takes input as a 2D-grid and outputs a tumour probability for each cell in the 2D-grid. Recurrent Neural Networks (RNNs) along with their variants Long Short Term Memory (LSTM) <cit.> and Gated Recurrent Units (GRUs) <cit.> have excelled at modelling sequences in challenging tasks like machine translation and speech recognition. We build the second variant of RAN (RAN-LSTM) by combining CNNs with 2D-LSTMs. RAN-LSTM captures the context information by treating WSIs as a two-dimensional sequence of patches. RAN-LSTM extends 2D-LSTMs for tumour segmentation task in multi-gigapixel histology images by using learned representations of neighbouring patches from represenation learning network as a context for tumour classification of a single patch. RAN-LSTM is constituted by four 2D-LSTMs running diagonally, one from each corner. Tumour predictions across all the dimensions are averaged together to get the final tumour classification. The complete workflow of the proposed architecture is shown in Figure <ref>.We demonstrate the effectiveness of modelling context using RANs for tumour segmentation. RANs significantly outperform traditional methods on the dataset from Camelyon'16 challenge <cit.> on all metrics. Our main contributions can be summarized as follows: * We propose RANs as a generic architecture for context modelling in multi-gigapixel images.* We utilize both CNNs and 2D-LSTMs for context-aggregation network.* We show the effectiveness of the addition of context-aggregation network on top of a representation network for segmentation of tumour areas in multi-gigapixel histology images. § RELATED WORKWith large memory storage and fast computational power available in modern machines, processing WSIs has become feasible. Recent studies have exploited WSIs for cell detection and classification <cit.>, nuclei segmentation <cit.> and tumour segmentation <cit.>. Both these approaches follow a patch based approach to process a WSI which significantly limits the available context information. Bejnordi et al. <cit.> proposed a similar approach for breast tissue classification by using large input patches and stacking CNNs together. To deal with large input patches, the network is trained in two steps. On the other hand, RANs generalize the segmentation task through context-aggregation from encoded representations of a 2D-grid of small patches. RANs can incorporate CNNs, 2D-LSTMs or a combination of both for modelling spatial context in WSIs.Multi-dimensional RNNs <cit.> have been employed to model sequences in both temporal and spatial dimensions. Recent approaches <cit.> <cit.> model spatial sequences in an image to accomplish dense output for semantic segmentation tasks. Byeon <cit.> utilized four 2D-LSTMs running in each direction, whereas Visin . <cit.> employed two bi-directional RNNs as two layers for up-down and left-right spatial modelling. The key difference between these two and our approach is that we try to model spatial context by aggregating multiple patches as a 2D-grid of patches instead of modelling spatial context within a single patch. Both <cit.> and <cit.> model spatial context for natural images, whereas RANs can model much larger context in multi-gigapixel images. § REPRESENTATION-AGGREGATION NETWORKSThe proposed Representation-Aggregation Networks (RANs) have a two-network architecture wherein the first network learns patch-level representation, which is passed on to the second context-aggregation network. RAN is able to incorporate context from a large region by aggregating the learned features from the first network as 2D-grid of patches. RANs analyze a 2D-grid of patch-level features at once, and predict tumour probabilities for each cell in the grid by feeding representation of neighbouring cells as context.The first network is essentially a representation learning network. It takes in input patches of size n× m×3 and yields a D-dimensional representation. One can use any state-of-the-art image classifier for this purpose. For our experiments, we train AlexNet <cit.> on our dataset to classify patches as tumour or non-tumour. D-dimensional representations, denoted by p_t are obtained by extracting features from an intermediate layer of a trained network. We experiment with various intermediate layers with later layers being more task specific. We discuss the proposed variants for context-aggregation network in the following subsections. §.§ RAN-CNNConvolutional Neural Networks are good at learning the spatial relations from the input. RAN-CNN is designed to capture spatial context from the neighbouring patches. It consists of five 3×3 convolutional layers. The first convolutional layer takes in 2D-grid feature as an input and subsequent layers operate on the output from the previous layer. One can control the context region by varying the convolutional filter size. Last three convolutional layers are followed by dropout layers to avoid overfitting. 𝐲_i = 𝐟_conv(𝐩_t,𝐖_i) ∘𝐟_a(·)𝐲_j = 𝐟_conv(𝐩_i,𝐖_j) ∘𝐟_a(·)∘𝐟_d(·)𝐲 = 𝐟_conv(𝐩_j,𝐖) ∘𝐟_a(·) where f_conv, f_a and f_d are the convolution, activation and dropout functions respectively; W_i, W_j and W are the trainable weights; the operator (∘) provides the output of preceding function to the superseding function and operator (·) represents the output of the preceding function; y_i and y_j are the outputs of i^th and j^th layers, i ∈{1,2,…,C}, j ∈{1,2,…,D} and C, D are the number of convolutional layers with and without dropout layers. y represents the output of final prediction layer which maps the number of feature maps from y_d^th layer to the total number of classes. §.§ RAN-LSTMFor modelling image sequences through standard 1D-LSTM, a sequence of D-dimensional representation is used as an input to the LSTM. On the other hand, 2D-LSTMs take two-dimensional inputs represented as a sequence of two D-dimensional vectors and generate either sparse or dense output predictions as required by the task. RAN-LSTM extends 2D-LSTM to model the context information along a 2D-grid of patches. Each 2D-LSTM unit (i, j) has one input gate (i_t), two forget gates (f^x_t, f^y_t), two cell memory gates (c̃^x_t, c̃^y_t) and one output gate (o_t) for neighbouring patches in x and y direction respectively. The hidden states and cell states for current unit are denoted by h_t and c_t respectively. h^x_t-1 and h^y_t-1 denote hidden states for the neighbouring unit on left and top respectively. Similarly, c^x_t-1 and c^y_t-1 denote cell states for the neighbouring units. Unit (i, j) is pairwise connected to its 4 neighbours i.e. [(i-1, j), (i, j-1)], [(i-1, j), (i, j+1)], [(i+1, j), (i, j-1)], [(i+1, j), (i, j+1)] where each relation is exploited by an independent 2D-LSTM as shown in Figure <ref>. These four 2D-LSTMs run in different directions, one from each corner to the diagonally opposite corner. Final predictions are obtained by aggregating results from 2D-LSTM from all directions. The governing equations for 2D-LSTM are given below where p_t, W_*, U_*, b_* denote input vector and weights matrices for hidden states, inputs and constants respectively. σ, tanh and ⊙ denote sigmoid activation, hyperbolic tangent activation function and dot product respectively. 2D-LSTM can be treated as a layer which accepts input of size N×M×D and outputs predictions of size N×M×D, where H indicates the hidden dimension of 2D-LSTM layer. Multiple 2D-LSTM layers can be stacked one after another to form RAN-LSTM just as convolutional layers for RAN-CNN. 𝐢_t = σ(𝐖_i{𝐡^x_t-1, 𝐡^y_t-1} + 𝐔_i p_t + 𝐛_i){𝐟^x_t, 𝐟^y_t}= σ(𝐖_f{𝐡^x_t-1, 𝐡^y_t-1} + 𝐔_f p_t + 𝐛_f){𝐜̃^𝐱_𝐭, 𝐜̃^𝐲_𝐭}= tanh(𝐖_c{𝐡^x_t-1, 𝐡^y_t-1} + 𝐔_c p_t + 𝐛_c)𝐨_t = σ(𝐖_o{𝐡^x_t-1, 𝐡^y_t-1} + 𝐔_o p_t + 𝐛_o)𝐜_t =𝐢_t ⊙𝐜̃_𝐭 + 𝐟^x_t ⊙𝐜^x_t-1+ 𝐟^y_t ⊙𝐜^y_t-1 𝐡_t = 𝐨_t ⊙𝐜_t Both variants are trained to minimize cross-entropy loss L for 2D-grid as given below, where y'_i,j, P(y_i,j) denote the ground truth label and the predicted tumour probability respectively.L = 1/NM∑^N_i = 1∑^M_j = 1 y'_i,jlogP(y_i,j) § RESULTS AND DISCUSSION The Dataset. We evaluate our proposed method on the Camelyon'16 dataset <cit.> which consists of 110 tumour and 160 normal WSIs. For all WSIs, we extract the tissue region using a simple 2-layer Fully Convolution Network (FCN). We used 80% WSIs for training and remaining for validation. Then, we randomly crop 188K patches of size 224×224 to form the training set for patch-level network, out of which around 90K are tumour patches. For training of context-aggregation networks, we extract a total of 190K 2D-grids by aggregating 64 (N=8, M=8) patches together. For validation, 20 complete WSI images are processed yielding a total of 99K 2D-grids or 6 million patches. Because of the relatively large size of training as well as validation dataset, we are fairly confident with the obtained scores and the generalization ability of our method.Model Specification. For representation learning, we train AlexNet on the training set and experiment with FC6 and FC7 features as input to the context-aggregation networks. We fixed the context depth as 8 and aggregated 64 (N=8, M=8) patches together as a 2D-grid to be processed by context-aggregation network, RAN-CNN or RAN-LSTM. We experimented with different network architectures of RAN-CNN to find a suitable one. First, we compared the impact of different number of convolutional layers and found that network with more convolutional layers performed better. Finally, we kept 5 convolution layers along with dropout for the last 3 convolutional layers, denoted by RAN-CNN-FC6-5L-D in Table <ref>. After comparing the performance of FC6 and FC7 features, we decided to stick to FC6 features because of its superior performance. We experimented with the number of 2D-LSTM layers with 512 dimensional hidden state in each layers. Finally, we utilized two 2D-LSTM layers followed by a convolution layer to reduce the hidden state dimensions to the number of classes, which is 2 in our case.Training Details. RAN-CNN model was trained using Adam optimizer with a batch size of 64. RAN-CNN converged after four epochs with total training time of 6 hours. For training RAN-LSTM, we used Adam optimizer with learning rate and decay rate as 0.0001 and 0.5 after every 2 epochs respectively. The model is trained with a batch-size of 10 for a total of 25 epochs which took a total of 45 hours to train. All the codes were implemented in Tensorflow, and trained on a single NVIDIA GeForce GTX TitanX GPU. Out of 190K training 2D-grids which is equivalent to 12 million patches, only 6% patches were tumorous. To tackle this class imbalance problem, we sample all 2D-grids that had at least one tumour patch along with the same number of non-tumour patches. This resulted in 28K training 2D-grids for training the context-aggregation network in RANs. We evaluate several variants of RAN using precision, recall and F1-score. We select F1-score as a metric for model performance instead of accuracy because of class imbalance in our data. Figure <ref> shows model performance through Precision-Recall curve and F1-scores at various thresholds. RANs lead to significant increase in F1-scores from 0.40 for AlexNet to 0.82, 0.83 for RAN-CNN and RAN-LSTM respectively. Since AlexNet classifies only a single patch at a time, the resultant predictions consist of several discontinuous blobs over the tumour region as shown in Figure <ref>. This demonstrates the importance of context information while segmenting tumour region in multi-gigapixel histology images. The RAN-CNN and RAN-LSTM improve the prediction by incorporating the spatial context, and output smoother continuous regions. Thus, these are able to identify the global structure of the tumour region as opposed to AlexNet which only captures the local information from a single patch.Both variants of RAN achieve competitive results as summarized in Table <ref>. From the various different architectural variants of RAN-CNN using FC6 features with 5 convolution layers with dropout performs the best. RAN-LSTM with a single layer (RAN-LSTM-1L) is not able to perform well due to underfitting. A two layered RAN-LSTM (RAN-LSTM-2L) gives much better performance than RAN-LSTM-1L. We refer to RAN-CNN-FC6-5L-D as RAN-CNN and RAN-LSTM-2L as RAN-LSTM for convenience. RAN-CNN gives better recall of 0.83 as compared to 0.81 with RAN-LSTM but loses on precision with 0.81 and 0.85 for RAN-CNN and RAN-LSTM respectively. RAN-LSTM outperforms all the approaches yielding the best F1-score of 0.83. The superior performance of RAN-LSTMs may be attributed to its ability to capture global context of the complete 2D-grid at once, where as RAN-CNN generates output predictions largely from local context. From Figure <ref>, we see that RAN-LSTM succeeds in modelling the entire tumour region as a single component whereas RAN-CNN has few discontinuities within the tumour region.§ CONCLUSIONS Technical advances in digital scanning of tissue slides are posing unique challenges to the researchers in the area of digital pathology. These gigapixel tissue sides open the way for automated analysis of cancerous tissues by deep learning algorithms. We demonstrated how segmentation demands sophisticated deep learning approaches when dealing with multi-gigapixel histology images. We proposed Representation-Aggregation Network (RAN) as a generic network that can incorporate the context from the neighbouring patches to make global decisions on a task involving multi-gigapixel images. RANs can be easily modified by varying representation learning network and context-aggregation network with networks suited for a particular task. We evaluate the performance of RANs for the task of tumour segmentation where it outperforms standard CNN approaches by a large margin. | http://arxiv.org/abs/1707.08814v1 | {
"authors": [
"Abhinav Agarwalla",
"Muhammad Shaban",
"Nasir M. Rajpoot"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727105658",
"title": "Representation-Aggregation Networks for Segmentation of Multi-Gigapixel Histology Images"
} |
EXTERNAL LITTELMANN PATHS OF TYPE A]EXTERNAL LITTELMANN PATHS FOR CRYSTALSOF TYPE A Bar-Ilan University, Ramat-Gan, Israel, [email protected] Bar-Ilan University, Ramat-Gan, Israel,[email protected][2010] 17B10, 17B35 ^1Partially supported by Ministry of Science, Technology and Space fellowship , at Bar-Ilan University.[ Ola Amara-Omari^1 andMary Schaps Received: date / Accepted: date ======================================For the Kashiwara crystal of a highest weight representation of an affine Lie algebra oftype A and rank e, with highest weight Λ, there is a labeling by multipartitions and by piecewise linear paths in the real weight space called Littelmann paths.Both labelings are constructed recursively, but since Kashiwara demonstrated that the crystals are isomorphic, there is a bijection between the labels.We choose a multicharge (k_1,…,k_r), with 0 ≤ k_1≤k_2....≤ k_r ≤ e-1. We put k_i in the node at the upper left corner of partition i of the multipartition and let the residues from ℤ/ e ℤ increase across rows and decrease down columns.For e=2, we call a multipartition residue-homogeneous if all nonzero rows end in nodes of the same residue and partitions with the same corner residue have first rows of the same parity. It is strongly residue homogeneous if each partition ends in a triangle of whose side has length one less than the first row of the next partition.In this paper we show that each such multipartition corresponds to a Littelmann path which is unidirectional in the sense that the projection of the the main part of the path to the coordinates of the fundamental weights consists of long paths all lying in either the second or fourth quadrant, separated by oscillating paths with a fixed integer oscillator. The path corresponding to such a multipartition can be constructed non-recursively using only integers describing the structure of the multipartition.§ INTRODUCTION Let e be the rank of an affine Lie algebra𝔤 of type A and let r be the level of a highest weight module with highest weight Λ. For type A, the level ris the sum of the coefficients of the fundamental weights in the dominant integral weightΛ. There is a labelling of the elements of the basis of the highest weight module by multipartitions and those multipartitions which are recursively constructed as labels of the basis elements are called e-regular.The level r is the common number of partitions in these multipartitions. The problem of determining the e-regular multipartions of type A in a non-recursive manner has shown only slow progress. It is known for r=1.Mathas settled it for e=2 in <cit.>, and Ariki, Kreiman and Tsuchioka for r=2 in <cit.>.A few other results extending these are also available.We are trying to attack the problem through Littelmann paths, by trying to find a direct way to translate between the multipartition labellinga crystal basis element and the corresponding Littelmann path. The results in this paper are for the e=2 case, for which, as we mentioned, Mathas already solved the problem of finding a criterion for e-regular multipartitions, but we hope that finding a way to translate from Littelmann paths to multipartitions will be relevant to the problem of e-regularity, since only e-regular multipartitions will correspond to a Littelmann path.In 2 and 3, we give needed background, and the definition of a unidirectional Littelmann path.In 4, we define a strongly residue homogeneous multipartition and prove our main result, that a strongly residue-homogeneous multipartition corresponds to a unidirectional Littelman path whose invariants can be determined from the structure of the multipartition. In 5, we give examples to illustrate the limitations of the theorem and explain what remains open.§ DEFINITIONS AND NOTATION Let 𝔤 be the affine Lie algebra A^(1)_e-1. Let C be the Cartan matrix, and δ the null root. Let Λ be a dominant integral weight, let V(Λ) be the highest weight module with thathighest weight, and let P(Λ) be the set of weights of V(Λ). Let I be the set of residues ℤ / e ℤ and letQ be the ℤ-lattice generated by the simple roots,_0,…,_e-1.LetQ_+ be the subset of Q in which all coefficients are non-negative. The weight space P of the affine Lie algebra has two different bases.One is given by the fundamental weights together with the null root, Λ_0,…, Λ_e-1, δ,and one is given by Λ_0, _0,…,_e-1.We will usually use the first basis for our weights.A highest weight module V(Λ) is integrable.Every weight λ has the form Λ-α, for α∈ Q_+.The vector of nonnegative integers giving the coefficients of α is called the content of λ. We will not repeat all the standard material about the symmetric form (- | -), which can be found in <cit.>. We follow <cit.> in defining the defect of a weight by (λ)=1/2((Λ|Λ)-(λ|λ)). Since we are in a highest weight module, we always have (Λ|Λ) ≥ (λ|λ), and the defect is in fact an integer for the affine Lie algebras of type A treated in this paper. The weights of defect 0 are those lying in the Weyl group orbit of Λ and will play an important role in the definition of the Littelmann paths.Definemax P(Λ)={λ∈ P(Λ) |λ + δ∉P(Λ)}, and by <cit.>, every element ofP(Λ) is of the form {y-kδ| y ∈max P(Λ), k ∈ℤ_≥ 0}. Let W denote the Weyl group, generated by reflections s_0,…, s_e-1.By the ground-breaking work of Chuang and Rouquier <cit.>, the highest weight module V(Λ) can be categorified.The weight spaces lift to categories of representations ofblocks of cyclotomic Hecke algebras, the basis vectors lift to simple modules,the Chevalley generators e_i,f_i lift to restriction and induction functors E_i,F_i, and the simple reflections in the Weyl group lift to derived equivalences, which in a few important cases are actually Morita equivalences.We will not be using the categorified version, but it provides the underlying motivation for trying to understand the multipartitions, which label the simple modules of the cyclotomic Hecke algebras. For r=1, in what is called the degenerate case, these are the group algebras of symmetric groups.There are three distinct labellings for an element of a Kashiwara crystal B(Λ), <cit.>,<cit.>, of type A.We will not directly need the definition of the crystal or of its crystal base. An element of the crystal base can be labelledby its Littelmann path,by its multipartitions, and by its canonical basis.In this paper we are concerned only with the first two, looking for cases where there is a direct connection between them, so that we can read off the multipartition from the Littelmann path or the Littelmann path from themultipartition.In a different paper, <cit.>, we make a similar direct passage from a multipartition to a canonical basis element for a symmetric crystal. The theory originated with work of Lakshmibai and Seshadri for type A, but it was Littelmann in <cit.> who extended to other types andwho proved many of the most important properties of the path model of the highest weight representations.A path π is a continuous, piecewise linear function from the closed real interval [0,1] into the real space ℝ⊗_ℤ P(Λ) such that π(0)={0} and π(1) ∈ P(Λ).The weight π(1) will be called the weight of the path.For any residue ϵ in the set of residues I, we define a function H_ϵ^π(t)=π(t),h_ϵ, which is simply the projection of the path onto the coefficient of Λ_ϵ.We then setm_ϵ=min_t (H^π_ϵ(t)). This minimum is always achieved at one of the finite set of corner weights and is always non-positive, sinceH_ϵ^π(0)=0.We let 𝒫_int be the set of paths for which thism_ϵ is an integer for all ϵ∈ I. Littelmann proves in <cit.>, Lemma 4.5(d), that all the Littelmann paths in the crystal for a dominant weight Λ lie in P_int. Note that since π(1) ∈ P(Λ), all the numbers H_ϵ^π(1) are integers.Littelmann's function f_ϵ is given on 𝒫_int as follows: * If H^π_ϵ(1)=m_ϵ, then f_ϵ(π)=0.* Sett_0= max_t { t ∈ [0,1] | H^π_ϵ (t)=m_ϵ}t_1= min_t { t ∈ [t_0,1] | H^π_ϵ (t)=m_ϵ+1}then if H^π_ϵ(t) is monotonically increasing on the interval [t_0,t_1], we definef_ϵ(π)(t)= π(t)t ∈ [0,t_0] π(t_0)+s_ϵ(π(t)-π(t_0))t ∈ [t_0,t_1] π(t)-α_ϵ t ∈ [t_1,1]In the more complicated case, where the pathH^π_ϵ(t) is not monotonicfrom t_0 to t_1, but contains some segments which oscillate, can be found in <cit.>. We will have oscillating sections of the path, but they will all occur before t_0 or after t_1. The definition of e_ϵ is dual and reverses the action of f_ϵ. The definition is given in <cit.>. We will be using primarily f_ϵ. Let 𝒫 denote the set of paths. For ν,μ defect 0 weights, we define ν≥μ if there is a sequence of defect 0 weightsν=ν_0,...,ν_s=μ and positive real roots β_1,…,β_s such thatν_i =s_β_i(v_i-1)and ν_i-1,β_i^∨ <0, i=1,2,…,s. Following a suggestion of Kashiwara, Littelmann then defines a distance function between ν and μ.We do not need the entire function and its properties, only the case dist(μ,ν)=1, which means that no other weight can be inserted between them preserving the order. <cit.> An a-chain for (μ,ν) is a sequence μ=λ_0>λ_1>…>λ_s=ν of defect 0 weights such that either s=0 and μ=ν or λ_i=s_β_i(λ_i-1) for some positive real rootsβ_1,…,β_swith dist(λ_i,λ_i-1)=1 andaλ_i ,h_β_i∈ℤ<cit.> Arational pathof class Λ is an ordered set of sequences (ν; h) for whichfollowing hold: * ν_1>…>ν_s is a linearly ordered set of defect 0 weights in WΛ. * 0<b_1<b_2<…<b_s=1 are rational numbers.We will denote 0 by b_0.We identify π with the parameterized path π(t)=∑^j-1_ℓ=1 (b_ℓ-b_ℓ -1)ν_ℓ +(t-b_j-1)ν_j,b_j-1≤ t ≤ b_j A rational path (ν, b) is called an LS-path if there is an b_i-chain connecting ν_i to ν_i+1. We will generally write the rational path in the form(ν_1,…, ν_s ;b_1,b_2,…,b_s=1).and call itan LS-representation of the path. In this paper, the numbers b_i will be endpoints of subintervals of [0,1] parameterizing straight subpaths and will be called parameter endpoints.In Example <ref>, we give an example of a rational path with non-trivial b_i-chains.In <cit.> the authors refer to ν_1 as the ceiling and ν_s as thefloor.Since there is a unique multipartition for a defect 0 weight, they sometimes use the same notation for the corresponding multipartitions but in this article the ceiling and floor will always be weights.Let B=B(Λ) be the crystal of a dominant integral weight Λ. For any v∈ B, we let θ(v)=(θ_0,…, θ_e-1) be the hub of v, where θ_i= wt(v),α_i^∨The hub is the projection of the weight of v onto the subspace of the weight space generated by the fundamental weights <cit.>. By doing numerous calculations of Littelmann paths, we found one particular class, which we call unidirectional,that was particularly easy to handle.We consider the monotone sequence of weights of our rank 2 affine Lie algebra, ψ_1=s_0Λ, ψ_2=s_1s_0Λ, …ψ_m=… s_0s_1s_0Λ, where m is the number of reflections. For each m we define an integer d_m=rm-b, which will be the number of rows in the multipartion corresponding to that weight. We will demonstrate later that the hub θ_m of ψ_m is [-d_m, d_m+1] if m is odd and [d_m+1,-d_m] if m is even. A Littelmann path for type A and rank 2 will be called unidirectionalif there is an integer k ≥ 0 and a strictly descending sequence of positive integers of the same parity, m_1,m_2,… m_k,a sequence of positive integers, I_m_1, I_m_2,… I_m_k, an integer m_k+1 with 0 ≤ m_k+1<m_k,and a non-negative sequence of integers C_1,C_2,…, C_k+1, such thatthe projection of the path onto the hubs consists of two parts: * The main part, which is entirely contained in the second or fourth quadrant.Let ϵ be the residue corresponding to the positive coordinate of the quadrant.Its hubs are θ_m for m_k+1≤ m ≤ m_1. so that ϵ =1 in the second quadrant and ϵ=0 in the fourth quadrant. Starting at the origin, the main part consists of k long paths ofpositive integral ϵ-length I_m_i, which are multiples of θ_m_i, i ≤ k, followed by a sequence of paths of the form C_i/d_md_m+1θ_m, for m=m_i,m_i-1,…,m_i+1+1, which we call oscillating paths. Note that, if C_i ≠ 0 so that the set of oscillating paths is non-empty, the first of the oscillating paths is a continuation of the long path with index i.* The seed part, whose hub may contain or not contain any of the following:* A transit path1/d_m_k+1+1θ_m_k+1, which may be embedded in a longer path.* A tail, which is a multiple of θ_0 or of θ_1* A long path which is a multiple of θ_1 or θ_2, , together with an oscillating path.* An oscillating set of straight paths, multiples of θ_u for u ≤ m_k+1 with coefficient C_k+1/d_ud_u+1, where C_k=C_k+1+d_m_k+1.In our main theorem, we will determine the exact significance of these elements of the seed part, but the intention of the name is that the transit path and the tail are the seeds from which new long pathscan grow.§ MULTIPARTITIONS We now turn to the second labelling which will concern us in this paper, the multipartitions.Unlike the Littelmann paths, this labelling is available only is the important case of type A.Also unlike the Littelmann paths, it requires some non-canonical choices, a choice between two dual versions of the theory and the choiceof an object called the multicharge. Once these choices have been made, the aim of the paper will be to find conditions on the multipartition which insure that the corresponding Littelmann path will be unidirectional. We will then be able to read off the structure of the Littelmann path directly from the multipartition.A multicharge will be a sequence s=(k_1,k_2,…,k_r) of integers with 0 ≤ k_i ≤ e-1, for some natural number r which will be called the level of the multicharge. A multicharge s determines a dominant integral weightΛ=Λ_k_1+…+Λ_k_r.In the abelian weight space the order of the summands is of course irrelevant; it is simply a notational reminder of the multicharge. We will follow Mathas in <cit.> in requiring k_1 ≤ k_2 ≤…≤ k_r.We can then summarize by setting Λ=a_0 Λ_0+…+a_e-1Λ_e-1We can regard the integers k_i aselements of ℤ / e ℤ, and with a slight abuse of notation we will identify k_i with this residue.We have to assume that we know the ceiling and floor of an multipartition, constructedin <cit.> from an LS-representation by letting the ceiling be ν_1 and the floor be ν_s. The Young diagram of the multipartition of a defect 0 weight λ will be represented by Y(λ). If the t-th partition λ^tof λ is nonempty, then we associate to each node in the Young diagram a residue, where the node (i,j) is given residuek_t +j-iThis will be called a k_t-corner partition.Due to the Mathas condition on the ordering of the k_i, this means that the multipartition will consist of a_o 0-corner partitions, followed by a_1 1-corner partitions and so forth.§.§ Standard LS-paths The main theorem of this paper will be a proof that a strongly residue homogeneous multipartition corresponds to a unidirectional Littelmann path. However, we do more:we show that the LS-repesentation of the Littelmann path can be written downdirectly from the Young diagram of the multipartition. This close connection between the multipartiton and the Littelmann path will be crucial in carrying out the induction inour main theorem. An LS-representation for which both the defect 0 weights and the parameter endpoints can be determined from the multipartition in the way we describewill be called standard. We will give the definition in general for any rank, but will apply it only in the case e=2. We must first prepare considerable notation. An important part of the description of each partition λ^t in the multipartition λ is the length a(λ^t) of the first row. Since these lengths decrease from the ceiling to the floor, the ascending numbering from ceiling to floor used in<cit.> or <cit.> is not compatible with our notation, so we will reverse the numbering.Weassume thatwe have fixed areduced word s_i_q… s_i_1 and the sequence of defect 0 weights μ=μ_q, μ_q-1,…, μ_0, with μ_j=s_i_jμ_j-1,going back to Λ. By the partial ordering of defect 0 weights which we introduced above, we thus have μ_j > μ_j-1 and in fact, since we are using simple roots, theywill have dist(μ_j , μ_j-1)=1. That will be important, because our a-chains will be strings of these μ_j.If we let d_j be the number of times we must operate on μ_j-1 with f_i_j to produce the reflection s_i_j, then in terms of the multipartitions, we have d_j=#(Y(μ_j)-Y(μ_j-1))that is to say, the number of nodes in the difference between the two Young diagrams. The d_j increase as j increases.Since all the added nodes have the same residue, we never add more than one to any given row. Now we consider the set((Y(μ_j)-Y(μ_j-1))∩ Y(λ))which is called a ladder in the article by Fayers <cit.> generalizing the LLT-algorithm <cit.> for canonical bases. If the ladder intersects the top row of the multipartition at node m of residue i, we will call this the m-ladder, whereas for Fayers it was an i-ladder.Not only the proof but even the statement of our main theorem will depend on the lengths of these ladders, which we denote byc_m=#(((Y(μ_j)-Y(μ_j-1))∩ Y(λ)).Clearly c_j ≤ d_j for all j.However, the c_i do not necessarily increase as j increases.Given a choice of reduced word and, as above,a sequenceμ_q, μ_q-1,…, μ_0 of defect 0 weightssete_j=c_j/d_jfor j>0 and set e_0=1. Let j_p, j_p-1,…,j_0be the subsequence of q, q-1,…,0 of j for whiche_j ≠ e_j+1so the μ_j_0 will be the floor.Then the LS-representation(μ_j_p,…, μ_j_0;e_j_p,…,e_j_0)will be called standard with respect to the multipartition λ if the sequence of e_i is increasing and the Littelmann path determined by this LS-representation corresponds in the crystal to the multipartition. (A standard LS-representation with gaps) Let e=3. let Λ=3Λ_0and set λ =[(8,6,1),(),()]. From our computer calculations, we know that the ceiling is the defect zeromultipartition withweight-12Λ_1 +15Λ_2-24 δ. Thiscorresponds to three copies of the partition (8,6,4,2), and equals three times the weight of a single copy of that partition in B(Λ_0). This ceiling has a periodic representation as s_1s_0s_2s_1s_0s_2s_1s_0Λ. The first ladder is the 8-ladder with residue 1, consisting of positions 8 and 6 in the first two rows.For m=7, we also get an m-ladder of length 2, consisting of positions 7 and 5 in the first two rows, and similarly for m=6,we get a ladder of length 2, so that c_8=c_7=c_6=2. Now, however, with the 5-ladder of residue 1 we get c_5=3.If we calculate the d_j, descending from 8 to 1, we get 12, 12,9,9,6,6,3,3.Thec_j in the same order are 2,2,2,3,2,2,1,1. The quotients are then 1/6,1/6, 2/9, 1/3, 1/3, 1/3, 1/3, 1/3. Finally, we get e_0=1 with weight Λ. Since some of the adjacent fractions are identical, this means that we do not need the entire sequence μ_8,μ_7,…,μ_1 but that, in writing a standard LS-representations, the μ_j corresponding to differences which are equal to zero should be omitted.When we eliminate all j for which e_j=e_j+1,we are then left with μ_8,μ_6,μ_5,Λ as the weights, and quotients1/6, 2/9, 1/3,1 as the fractions. Since we take difference between the fractions, the actual Littelman path is1/6μ_8, 1/18μ_6, 1/9μ_5,2/3Λ,with the coefficients summing to 1 as they should. We are dealing, in this paper, with the e-regular case, in which addable node are added from the top right down to the bottom left, as distinguished from thee-restricted case in <cit.> and <cit.> where the addable nodes are added from the bottom left to upper right.In trying to reconstruct the multipartition in a non-recursive fashion from the Littelmann path in the form we have written it, all we have is the number of nodes of Y(λ) intersecting (Y(μ_j)-Y(μ_j-1)), not their location.However, if they are fact located from the top down, then we could indeed reconstruct λ if we knew the reduced word, giving us the d_m,and the fractions e_m correponding to the various μ_j, which would give us the c_m as d_me_m. § THE CASE Λ=AΛ_0+BΛ_1 We consider the case e=2, with multicharge s=(0,0,…,0,1,1,…,1). If there are a copies of 0 and b copies of 1, with level r=a+b, then when we fill in residues in the nodes of the Young diagram of a multipartition with this multicharge, there will be a of the0-corner partitions followed by b of the 1-corner partitions.A partition in which the partsalternate between odd and even will be called alternating.A multipartition for e=2 will be residue-homogeneous if* Every nonempty partition is alternating.* All 0-corner partitions have the first line of the same parity, and the 1-corner partitions have the first line of the opposite parity. The last row of every non-zero partition, except possibly the last, is odd. A multipartition for e=2 will be strongly residue-homogeneous if* It is residue homogeneous, and* For each partition λ^i after the first partition, if the top row has length n, the previous partition ends with a triangle of n-1 rows and columns if both have the same corner residue, and if it is the top partition of thethe 1-corner partitions, then the previous partition ends with a triangle of length n-2 if it has fewer rows thanλ^i, and of n if it has n or more rows. We recall that, by the dual of Mathas' resultProp. 4.9in <cit.>, a multipartition is 2-regular if and only ifthe length a(λ^t)of the first row of a partition isless that or equal to the number of rowsℓ(λ^t-1) in the previous partition, except on the boundary between 0 and 1, where, for words beginning with s_1, it can be greater by 1.If the top row of the first 1-corner partition is not greater by 1, we say that the multipartition is non-increasing.A strongly residue-homogeneous multipartition is 2-regular.We begin with two adjacent partitions λ^t-1 and λ^t in the multipartition which have the same corner residue, either 0 or 1. By the definition of residue homogeneous, the top rows have the same parity.Letting n=a(λ^t), the length of the top row of the lower partition, then by the condition that the multipartition be strongly residue homogeneous, the upper partition ends in a triangle with at least n-1 rows.Thus ℓ(λ^t-1) ≥ n-1. Because λ^t-1 is a partition with no two rows identical, a(λ^t-1) ≥ℓ(λ^t-1). By the condition that the parities are the same, we get that ℓ(λ^t-1) ≥ n, which is the first condition needed for the dual Mathas result quoted above the lemma. At the boundary between 0-corner partitions and 1-corner partitions, we no longer have the parities equal, but we get as before thatℓ(λ^t-1) ≥ n-1, and this is enough.For the case of e=2, there are only two families of reduced words.The words with s_0 first on the right, i.e., … s_1s_0,give the defect 0 weight whose multipartitions have the upper triangular partitions largerwill be denoted by ψ^-_m, m>0. The words with s_1 on the right, i.e. … s_0s_1, give the defect 0 weightsψ^+_m, m≥ 0 whose upper triangular partitions are smaller.All the 0-corner partitions have the samefirst row m, and all the 1-corner partitions all have the same length first row, which is either m-1 or m+1, respectively. When b=0, we will use ψ_m^- by default.The floor and ceiling in the case r=1 are determined by the lengthℓ(λ) anda(λ) of the first column and first row, respectively.The situation for r>1 is considerably more complicated.In the case of general e for which we defined the standard LS-representation, the denominator d_m depended on the choice of reduced word. We now define d^±_n to be the number of rows in a defect 0 multipartition corresponding to a word beginning on the rightwith s_1 or s_0, respectively, where n is the length of the first row.Since all partitions with 0 in the corner have n rows, and all partitions with 1 in the corner have n ± 1 rows, we getd^±_n=an+b(n ± 1)=rn ± b=r(n ± 1)∓ aThe corresponding projection of the weight to the hub is θ^±_2s=[d^±_2s+1,-d^±_2s] θ^±_2s+1=[-d^±_2s+1,d^±_2s+2]In order to avoid writing every hub twice for the odd and even cases, we let ϕ be the operation of interchanging the 0 and 1 coordinates and then can write θ^±_m=ϕ^m[d^±_m+1,-d^±_m]Note that when a=1 and b=0, d_m^-=m. The segments are defined by putting together rows whose lengths drop by only one, with the following exception: if the length of the first row of the partition equals the number ℓof rows in the previous partition, or, at the boundary between 0 and 1, equals ℓ± 1, then it does not start a new segment. The segment boundary isthe line between the last row of the old segment and the first of the new. For strongly residue-homogeneous multipartitions, the segments will correspond to the long paths of the corresponding unidirectional Littelmann path.We number the segments from 1 to k, and for segment i * We let v_i be the number of partitions before the end partition of the segment, where v_i=0 if the segment ends in the first partition.* Inside the last partition intersecting the segment, we let t_i be the number ofrows inside the partition down to the last row of the segment.* We let z_i be a Boolean parameter equal to 1 if the segment ends in a singleton and 0 if not. * Let n_i be the length of the first row of the segment as before.If the segment starts at the top of a 0-corner partition, then set n_i'=n_i, and at the top of a1-corner partition, n_i'=n_i ∓ 1.If it starts in the middle ofa 0-corner partition, set n_i'=n_i +t_i-1 and in the middle of a 1-corner partition, n_i'=n_i+t_i-1∓ 1. Because the partitions are alternating and the first rows of 0-corner partitions all have the same parity opposite to that of the 1-corner partitions, all the n_i' will have the same parity.We also have thatn_1'>n_2'> …> n_k'. At each segment boundary, we extend the bottom row of segment i upward to the top of the partition, which it will hit at n_i', and this will be less than the point n_i-1' at which the segment i-1 will hit the top of the partition, since otherwise the two segment would have been combined into one.The segment corresponds toa number of adjacent rows in the multipartition with weight ψ_n_i'^- orψ_n_i'^+. If λ is a non-increasing partition, we will use “-” and if it is an increasing partition at the boundary between 0-corner and 1 corner partitions, we use ψ_n_i'^+.The default, when the multipartition does not contain non-empty 1-corner partitions, will be “-”.The proof of our main theorem, that a strongly homogeneous multipartitions has a unidirectional Littelmann path, involves a double induction.The main induction is on the length n of the top row of the multipartition, but there will be a further internal induction on the segments.In the course of this second induction we will need to know not only the lengths of various ladders, but also the lengths of the intersections of the ladders with the various segments. In order to calculate this precisely, we need more notation. Letting m be a number satisfying n_i+1'<m ≤ n_i',we set v^m=v_i, the number of partitions above the partition containing the last row of the segment. Our aim is to find a formula for thelength of the m-ladder that contains themth-node in the top row of the multipartition.The integer v^m will allow us to find the length of the intersection of the m-ladder with the partitions above the end partition, and now we must define a new quantity to measure the length of the part of the m-ladder in the last partition of the segment.If the m-ladder ends on the last row of the segment, then this length is t_i, and particularly ifm=n_i',the number of rows in the last partition intersection the segment.If the m-ladder ends in the first column, then the length of its intersection is m or m ± 1, depending on whether we are in a 0-corner partition or a 1-corner partition.In either case we get a number less than or equal to t_i.Therefore we definet^m= min(t_i,m),v_i < a min(t_i, m ± 1),v_i ≥ afor each such m, where we use the option “-" for non-increasing multipartitions. We can combine the two cases as min(t_i, m ± 1_v_i ≥ a). The integer t^m was defined to give exactly the number of nodes in the intersection of the ladder with the last partition. We can calculate d^±_m as above, and we can now give a algebraic formula for the lengths of the m-ladders.c_m=v^m· m +t^m, v^m ≤ aa · m +(v^m-a) · (m ± 1) +t^mv^m>a If we use the standard function x_+=max(x,0), then we can unite the two cases into a single formulac_m=v^mm±(v^m-a)_++t^m. If this is in segment i, we getc_m=v_im ±(v_i-a)_++min(t_i, m ± 1_v_i ≥ a).The value of c_m will change in the middle of the segment i whenever v_i>0 and also if t^m depends on m.We always have t^m=t_i when m=n_i', but if the segment ends in a singleton or if i=k and we pass to the seed part of the path, we can havet^m<t_i.As in the introduction, we sete_m=c_m/d_m In theinduction step of the main theorem, we will be required to prove that each segment corresponds to a straight path of integral length in the positive coordinate of the hub, whose length equals the number of addable nodes in that segment.We will be required to express this integralnumber in terms of various numbers associated with the multipartition, which we now define. In order to deal with cases n_i+1'<m<n_i', we extend the Boolean parameter z_i from Def. <ref>by setting z^m equal to 0 if the intersection of the segment with ψ^±_m does notend with a singleton, and equal to 1 if it does.To deal with the last segment, we let n'_k+1=0 if it ends in a singleton, i.e., z_k=1.if the last row is longer than 1, we prepare the way for starting a new segment at the end of the current partition by setting z_k+1=1 if the last row is odd andz_k+1=0 if the last row is even, and set n'_k+1=t_k ∓ 1_v_i ≥ a,n_k+2=0. When m=n_k+1', let z^m =z_k+1.Finally,defineI_m=c_m-c_m+1+v^m+z^m. We divide into cases: * Ifm=n_i' for i ≤ k, then I_m=c_m-c_m+1+v^m+z^m is the number of addable rows in the segment i, withI_n_i' =(v_i-v_i-1)(n_i'+1) ± ((v_i-a)_+-(v_i-1-a)_+)+(t_i+z_i-min(t_i-1,n_i'+1 ± 1_v_i ≥ a)) * Ifn_i+1'<m<n_i', for 1 ≤ i ≤ k+1, then I_m=0. * If m=n_k+1'>0, then I_m=z_k+1. If m=n_i' for i ≤ k, then t^m=t_i, v^m=v_i, z^m=z_i, andc_m+1 belongs to segment i-1.We then get I_m =c_m-c_m+1+v^m+z^m=(v_im ±(v_i-a)_++t_i)-(v_i-1(m+1) ±(v_i-1-a)_+)+min(t_i-1,m+1 ± 1_v_i-1≥ a))+v_i+z_i.=(v_i-v_i-1)(n_i'+1) ±((v_i-a)_±-(v_i-1-a)_+)+(t_i+z_i-min(t_i-1,n_i'+1 ± 1_v_i-1≥ a))It remains to show that this is the number of addable nodes in segment i. All the addable nodes of segment i lie in the n_i'+1 ladder. The number of possible internal partitions of the segment isv_i-v_i-1 .This would give us(v_i-v_i-1)(n_i'+1) if they were all 0-corner partitions, but to adust for the possibility that some of the partitions are 1-corner partitions and are thus larger or smaller by 1, we must add an adjustment term ±((v_i-a)_+-(v_i-1-a)_+). Now from the last partition, there are an additional t_i+z_i addable nodes in the segment.If the segment i-1 ends in middle of a partition, we have to remove t_i-1, since those nodes belong to the previous segment.If the segment ends at a segment boundary, we need to remove only the nodes in the n_i'+1-ladder, of which there are n_i'+1 ± 1_v_i-1≥ a.If n_i+1' <m<n_i', then c_m+1 belongs tothe same segment i, so c_m-c_m+1=-v_i-z_i, which means that I_m=0.Ifn_k+2' <m ≤ n_k+1, then we are not in a real segment, but in a virtual segment, which will only become a real segment if we decide to add the one addable node in the next row when z_k+1=1 .We havec_m-c_m+1=-v_k-z^m+1, so I_m=z^m-z^m+1.If m<n_k+1' , then z_m+1=z^m=1 so again I_m=0 in that case. If m=n_k+1'>0, then z^m+1=0 and z^m=z_k+1, and so I_m=z_k+1 and thus is either 0 or 1, depending on whether there is or is not an addable node.The integer I_n_i' is the number of times we must operate by the appropriate f_ϵ in order to fill in the segment to the next level, and we are interested in having this as our denominator for the long paths.For e=2, Λ=3Λ_0+2Λ_1, the multipartition[(11,10,7,6,5,4,3,2,1),(7,6,5,4,3,2,1),(7,6,3,2,1),(2,1), ∅]has five segments, containing, 2, 7,9,3,2 rows where the respective segment pairs (v_i,t_i)are(v_1,t_1)=(0,2), (v_2,t_2)=(0,7), (v_3,t_3)=(2,2),(v_4,t_4)=(2,5), (v_5,t_5)=(3,2).Since c_12=0 and v^11=z^11=0, we get I_11=c_11=2. For the second segment, we have n_2'=9, so I_9=c_9-c_10+v^9+z_1=9-2+0+1=8. For the third segment, we use c_7=16, c_8=8, v^8=2, so I_7=16-8+2=10 .The fourth segment has 3 rows, and we getI_5=c_5-c_6+v^5+z_4=15-14+2+1=3. In the fifth and last segment we have by I_2=c_2-c_3+v^2+z_5=8-9+3+1=3. All the remaining I_m are zero. The last row ends in a singleton, so there is no virtual segment k+1.Every 2-regular non-increasing strongly residue-homogeneous multipartion λ can be built up from the empty partition by a sequence of operationsof the following two types:* (Widening) Adding every addable node down to the segment boundary of the last non-zerosegment, giving λ^*.* (Deepening) Adding every addable node down to the end of λ and then one singleton, in the same partition if the last row is odd and in the next partition if the last row is even and not at the boundary from 0-corner to 1 corner residues, to get λ̅ and then, if desired, adding singletons of the same cornerresidue to some of the addable nodes inpreviously empty partitions to get λ̃. Adeepening operation will always be followed by a widening or by a deepening which does not add a new partition with the same residue in the corner. Induction on the number mof nodes in the first line of the multipartition λ. Assume that the lemma is true for everystrongly residue-homogeneous multipartition with first line less than m. If the last rowof the last non-zero segmentisnot a singleton, then removinganode from each row above the segment boundary is a reversible operation whose inverse is a widening, producing a strongly residue-homogeneous multipartition with first row m-1, to which we can apply the induction hypothesis. The second condition in the definition ofstrongly residue homogeneous is stable because, though we reduce the triangles at the ends of each partitionexcept the last by removing one node from each row, we also reduce the first row of each partition by one node. If the last non-zero row is a singleton, then we have two cases.If it is the last row of a larger partition, we apply a reversible operation whose inverse is a deepening, removing one node from each non-zero row. The last non-zero row will vanish. The number of rows in the triangular tails will be reduced by 1, but so will the first row of each partition.We are reduced to a strongly residue-homogeneous multipartition with top row m-1 and we apply the induction hypothesis.We will call the partition (1) a solitary partition. If the singleton is a solitary partition, then we must remove all the solitary partitions above it.If it is in the 1-corner partitions, then this process must stop at or before the boundary between 0-corner and 1-corner partitions, because at that point the residue of a solitary partition changes from 1 to 0, and our multipartition is residue homogeneous. If it is a 0-corner partition, then if all the partitions are solitary, we have gotten back to the empty partitions. We now consider two subcases.* Suppose that the corner of the upper solitary partition and the last non-solitary partition have the same corner residue. If the last row of this lowest non-solitary partition is a singleton, then we remove one node from each row and have a deepening.If it is not a singleton, then by the definition of stronglyresidue-homogeneous, it must be odd and removing one node from each rowwill give an even last row and we will have a deepening which adds partitions. * Suppose the corner of thesolitary partition has corner residue 1 andthe last non-solitary partition has corner residue 0.Then by the definition of strongly residue-homogeneous the last 0-corner partition must end in a singleton, and we continue by the reverse of a deepening.Our aim is to show that we havelong paths alternating with oscillating short paths. The lengths of the long path corresponding to segment i will correspond to the number of ϵ-addable nodes in the segment.The hub of the long path corresponding to segment i is a multiple of θ_m, where m=n_i'. The positive coordinate of θ_m has value d_m+1 and, in a standard Littelmann path, the coefficient is e_m-e_m+1. We begin with a technical lemma, which, when combined with Lemma <ref>, will do most of the work needed to demonstrate unidirectionality.Let λ be a 2-regular strongly residue-homogeneous multipartition with standard LS-representation.Let m be an integer such that n_i+1'<m ≤ n_i'. The coefficient q^±_m=e_m-e_m+1 of the hubθ^±_m in the Littelmann path can be written in the form q_m^±=I_m/d^-_m+1+C_i/d^-_md^-_m+1where I_m =0 unless m=n_i', in which caseI_n_i'=(v_i-v_i-1)(n_i'+1)-((v_i-a)_+-(v_i-1-a)_+)+(t_i+z_i-min(t_i-1,n_i'+1-1_v_i-1≤ a))andC_i= ∓ v_ib± r(v_i-a)_++rt_i,z_i=0, i ≤ k,∓ (v_i+1)b± r(v_i-a)_+± r1_v_i ≥ a,z_i=1, or i=k+1Before deriving the formulae, we note now that we will substitute d^±_m+1=d^±_m+rin the numeratorbut not in the denominator.We will also use the formula d_m^±=rm ± b. By definition,q^±_m= e_m-e_m+1 = c_m/d^±_m-c_m+1/d^±_m+1= c_md^±_m+1-c_m+1d^±_m/d^±_md^±_m+1.= (c_m-c_m+1)d^±_m+rc_m/d^±_md^±_m+1.NowI_m=c_m-c_m+1+v^m+z^m is the numerator we want for d_m+1^±.In order to get it, we must add and subtract(v^m+z^m)d_m^±, gettingq_m^±= (c_m-c_m+1+v^m+z^m)d^±_m/d^±_md^±_m+1+-(v^m+z^m)d^±_m+rc_m/d^±_md^±_m+1= I_m/d^±_m+1+-(v^m+z^m)(rm ± b)+rc_m/d^±_md^±_m+1,= I_m/d^±_m+1+-v^mrm-z^m rm ∓ v^mb ∓ z^m b+r(v^mm+t^m±(v^m-a)_+)/d^±_md^±_m+1= I_m/d^±_m+1+∓ v_ib± r(v_i-a)_+ ∓ z^mb+r(t^m-z^m m)/d^±_md^±_m+1,where in the last step we cancelled two copies ofv^mrmwith opposing signs and substitute v_i for v^m. We now divide into cases, since the values of z^m and t^m may depend on m.Case 1:z_i=0, i<k.Then z^m=z_i=0 for all m satisfying n_i+1'<m ≤ n_i', and t^m=t_i, so by substitution we get C_i=∓ v_ib± r(v_i-a)_++rt_iCase 2:z_i=1.Then z^m=z_i=1 for all m satisfying n_i+1'<m ≤ n_i', and t^m=m ± 1_v_i ≥ a, where the additional 1 is added or subtracted only in the case v_i ≥ a.Then t^m-z^mm=± 1_v_i ≥ a, so by substitution we get C_i=∓ (v_i+1)b± r((v_i-a)_+ +1_v_i ≥ a)Case 3:i=k+1: If m≤ n_k+1', we have z^m=1, t^m=m±1_v_i ≥ a so then we get, as in Case 2,C_i=∓ (v_i+1)b± r((v_i-a)_++1_v_i ≥ a)For i<k, or i=k, and either z_i=1 orz_k=0 and m ± 1_v_i ≥ a> t_i we have C_i=∓ (v_i+z_i)b± r(v_i-a)_++r((1-z_i)t_i ± z_i1_v_i ≥ a)),In the listed cases,the choice of formula depends only on the value of z_i, so we take a weighted average with weights z_i and 1-z_i. The number C_i will be called the oscillator, and the important point is that there are two kind of oscillator. The first, called the widening oscillator, is for oscillating paths following a segment not ending in a singleton, or between the last segment and the transit path, while the second, which is independent of t_i, is only used for after segments which end in a singleton, where it is called a deepening oscillatoror forthe oscillating paths after the (possibly empty)transit path connecting to the (possibly empty) tail, where it will be called the terminal oscillator.When a=1 and b=0, so that we have a single partition, then C_i=t_i.What makes the general case so much more complicated is that the addable nodes of segment i lie not only at the ends of rows, but also at boundaries between partitions.This segment boundary adjustment will occur frequently in our calculations, so denote it by s_i, wheres_i=v_i+z_i-v_i-1-z_i-1For i ≤ k, we have C_i-C_i-1=rI_n_i'-d_n_i'+1s_i,For i=k+1, we have C_k+1-C_k=-d_n_k+1',If i=k+2, with w non-zero partitions andy added partitions, then C_k+2-C_k+1=yb, w≤ a,-ya,r>w>a,Thus the terminal oscillator rises in steps of b from 0 to ab until the primarytail a-w/aθ_0 is used up, then descends from ab to 0 is steps of a=d_1, in such a way thatthe secondary tail, whichcoincides with the oscillating path C_k+1/d_1d_2θ_1=r-w/d_2θ_1in λ̅, becomes C_k+2/d_1d_2θ_1=r-w-y/d_2θ_1 inλ̃. The right side of the equationcontains a large term equal to r(m+1)(v_i-v_i-1) which occurs twice with opposite signs and must be cancelled, so we will begin with the right side of the equation. rI_m-d_m+1s_i =r [(v_i-v_i-1)(m+1)-((v_i-a)_+-(v_i-1-a)_+) ]+ [(t_i+z_i-min(t_i-1,m+1-1_v_i-1≥ a)]-(r(m+1)-b)s_i=r [-((v_i-a)_+-(v_i-1-a)_+) ]+ [t_i+z_i-min(t_i-1,m+1-1_v_i-1≥ a)]-(r(m+1)(z_i-z_i-1))+bs_i=r[-(v_i-a)_++(1-z_i)t_i+z_i1_v_i ≥ a ]+(v_i+z_i)b+r [(v_i-1-a)_+-(1-z_i-1)t_i-1 -z_i1_v_i-1≥ a ]-(v_i-1+z_i-1)b.=C_i-C_i-1 In the i=k+1 case, we have v_k=v_k+1, and t_k=n_k+1± -1_v_k+1≥ a, soC_k+1-C_k =[(v_k+1+1)b ± r(v_k+1-a)_+± r1_v_k+1≥ a ]+[ v_k b ± r(v_k-a)_++rt_k].=b± r1_v_k+1≥ a-rt_k =b± r1_v_k+1≥ a-r(n_k+1± 1_v_k+1≥ a)=b-r(n_k+1')=-d_n_k+1' If w is the number of non-empty partitions in λ, then from the formula for the second oscillator, substituting w for v_k+1, we have that the terminal oscillator C_k+1 isC_k+1=wb, w≤ a,(r-w)a,r>w>a,The new terminal oscillator C_k+2 will be the same with w replaced by w+y, all within the 0-corner or 1-corner partitions, since we are adding the y solitary partitions with a single residue. Consider e=2, a=3, b=2. We calculate the q_m in this format for a sequence of multipartitions. The first has a transit path at θ_3, with d_3=13 and 17=C_1=C_2+d_3. The fourth has a transit path at 5, with d_5 =23 and 27=C_2= C_3+d_5. We will continue to have C_k+1=4 until we open a new partition and get terminal oscillator 6 or 3. * [(5,4,3,2,1)(5,4,3), ∅,∅,∅]e: 8/23, 7/18, 6/13, 4/8, 2/3,1,(9/28+17/28· 23)θ_5, 17/23· 18θ_4,17/18· 13θ_3, 4/13 · 8θ_2, 4/8· 3θ_1,1/3θ_0* [(6,5,4,3,2,1)(6,5,4,1), ∅,∅,∅]e: 9/28, 8/23, 8/18, 6/13, 4/8, 2/3,1,(10/33+17/33· 28)θ_6, 17/28· 23θ_5,(2/23+4/23· 18)θ_4,4/18· 13θ_3, 4/13 · 8θ_2, 4/8· 3θ_1,1/3θ_0* [(7,6,5,4,3,2,1)(7,6,5,2,1), ∅,∅,∅]e: 10/33,9/28, 10/23, 8/18, 6/13, 4/8, 2/3,1,(11/38+17/38· 33)θ_7, 17/33· 28θ_6, (3/28+4/28· 23)θ_54/23· 18θ_4, 4/13 · 8θ_3, 4/13 · 8θ_2, 4/8· 3θ_1,1/3θ_0* [(8,7,6,5,4,3,2,1)(8,7,6,3,2), ∅,∅,∅]e:11/38,10/33,11/28, 10/23, 8/18, 6/13, 4/8, 2/3,1,(12/43+17/43· 38)θ_8,17/38· 33θ_7,(2/33+ 27/33· 28)θ_6,27/28· 23θ_5,4/23· 18θ_4, 4/13 · 8θ_3, 4/13 · 8θ_2, 4/8· 3θ_1,1/3θ_0* λ= [(9,8,7,6,5,4,3,2,1)(9,8,7,4,3), (1),∅,∅]e:12/43,11/38,12/33,11/28, 10/23, 8/18, 6/13, 4/8, 3/3(13/48+17/48· 43)θ_9,17/43· 38θ_8,(2/38+27/38· 33)θ_7,27/33· 28θ_6,(1/28+4/28· 23)θ_5,4/23· 18θ_4, 4/13 · 8θ_3, 4/13 · 8θ_2,(1/8+1/8+6/8· 3)θ_1* λ^*= [(10,9,8,7,6,5,4,3,2,1)(10,9,8,5,4,1), (2),∅,∅]e: 13/48,12/43,13/38,12/33,12/28, 10/23, 8/18, 6/13, 5/8, 3/3( 14/53+17/53· 48)θ_10, 17/48· 43θ_9,(2/43+27/43· 38)θ_8,27/38· 33θ_7,(2/33+4/33· 28)θ_6,+4/28· 23θ_5,4/23· 18θ_4, 4/13 · 8θ_3, ( 1/13+9/13 · 8)θ_2,(1/8+6/8· 3)θ_1* λ̅= [(10,9,8,7,6,5,4,3,2,1)(10,9,8,5,4,1), (2,1),∅,∅]e: 13/48,12/43,13/38,12/33,12/28, 10/23, 8/18, 6/13, 6/8, 3/3( 14/53+17/53· 48)θ_10, 17/48· 43θ_9,(2/43+27/43· 38)θ_8,27/38· 33θ_7,(2/33+4/33· 28)θ_6,+4/28· 23θ_5,4/23· 18θ_4, 4/13 · 8θ_3, ( 3/13+6/13 · 8)θ_2,6/8· 3θ_1* λ̃= [(10,9,8,7,6,5,4,3,2,1)(10,9,8,5,4,1), (2,1),(1),∅]e: 13/48,12/43,13/38,12/33,12/28, 10/23, 8/18, 6/13, 7/8, 3/3( 14/53+17/53· 48)θ_10, 17/48· 43θ_9,(2/43+27/43· 38)θ_8,27/38· 33θ_7,(2/33+4/33· 28)θ_6,+4/28· 23θ_5,4/23· 18θ_4, 4/13 · 8θ_3, ( 5/13+3/13 · 8)θ_2,3/8· 3θ_1Note the changes in the oscillators caused by transit paths, from 4 to 17, from 4 to 27 and, in λ^*, from 6 to 9.Our aim is to prove thatevery non-increasing strongly residue-homogeneous multipartition λ has a unidirectionalLittelmann path.This Littelmann path will have a standard LS-representation corresponding to λ. The proof will proceed by induction.The assumption will imply that the Littlemann path consists of long paths separated by oscillating paths. Given a strongly residue-homogeneous multipartition λ whose addable nodes are all of residue ϵ, we recall that we defined λ^* to be the multipartition obtained by adding ϵ-addable nodes down to the last non-zero row, and if the last row of λ is odd, we defined λ̅to be the multipartition obtained by adding nodes of residue ϵ in place of all the ϵ-addable nodes down to the end of the last non-empty partition, including a singleton at the end. By applying the Littelmann algorithm for constructing f_ϵ, we will want to show that λ^*, and if possibleλ̅, both have standard Littelmann paths, and that if we continue to fill up empty partitions with singletons of residue ϵ, the new multipartition λ̃ also has standard Littelmann path. In rank e=2, for highest weight aΛ_0+bΛ_1, and a>0, any 2-regular non-increasing strongly residue-homogeneous multipartitionλhas a Littelmann path π_λwitha standard LS-representation for λ which satisfies in addition that * If some partitions are empty, there is a primarytail whose length in the 0-direction is the number of 0-corner ∅, or, if every 0-corner partition is non-empty, there is a secondarytail whose length in the 1-direction is the number of 1-corner ∅ in λ. * The Littelmann path is unidirectional. We are trying to prove this theorem only in the case of -, since, as we will show in the last section, it is not generally true in the case of +, so we will simplify the notation by letting d_m=d^-_m and θ_m=θ^-_m. We will show by induction on the length of the top row n that the Littelmann path is standard for its multipartition. Using Lemmas 4.3 and <ref>, this will give us a unidirectional Littelmann path, where we will take the m_i to be the numbers n_i' for i ≤ k, the integral length of the long path with index i will be I_n_i', the oscillators will be the C_i from Lemma <ref>. If the last segment, which has indexk, ends in a singleton there will be no transit path and m_k+1 will be the index of the floor, which will be 0 or 1 if there are empty partitions, but may be larger if there are no empty partitions. If the last segment does not end in a singleton, there will be a transit path.Relying on Lemma <ref>, we assume that we have a λ with standard LS-representation which is unidirectional, and using the operations given there we will do either a widening to get λ^*, a deepening appending a singletonto get λ̅,or a deepening which adds solitary partitions (that is, with a single part of length 1) to get λ̃. Furthermore, our multipartitions can have several segments, so we must do an induction on the segments as well.Our hypothesis on this inner induction is that all the segments previously treated consist of long paths with oscillators,with coefficients as in a standard LS-representation. For all segments i with i<k, the multipartitions and the Littelmann path for all three operations will be identical, so to condense the notation, we will take the multipartition λ and add the ϵ-addable nodes of that segment, of which there areI_n_i', to get the new segment in λ^*.We then perform the operation f_ϵ to the corresponding long path of π_λ, and check that the various invariantse_m, m_i, and C_i of the segment imatch up to the new invariantse_m^*, m_i^* and C_i^*. Then for i=k, we must takee_m, I_m,m_k, and C_k and use them to find the newe_m^*, I_m^*, m_k^*, andC_k^* in the case of widening, the new e̅_m, I̅_m,m̅_k, and C̅_k in the case of deepening, as well as the new ẽ_m, Ĩ_m, m̃_k and C̃_k' in the case of a deepening which adds extra solitary partitions.In all cases, for 1 ≤ i ≤ k, the addition of all the ϵ- addablenodes will add nodes to each row andone node to each of the segment boundaries, so that we will have(n_i')^*=n_i'+1. In a segment which is entirely internal to a partition and does not end in a singleton,the addable nodes will be added to existing rows, and we will haveI_n_i'=I_(n_i')^*^*.However, if the segment is not entirely internal to a partition, the number of addable nodes not added to previous rows is v_i+z_i from the bottom of the segment to the top of the multipartition, and similarly from the bottom of segment i-1 to the top of the multipartion we getv_i-1+z_i-1 addable nodes at the boundaries between partitions. Thus in fact we must add the segment boundary adjustment s_i=(v_i+z_i)-(v_i-1+z_i-1) defined just before Corollary <ref>.I_(n_i')^*^*=I_n_i'+s_iWe divide intofour cases: for all i, continuing to widen or deepen the segment i, and for i=k, to widen after deepening, to deepen after widening and finally to addingnew solitary partitions. * 1 ≤i≤ k, where if the segment ends in a singleton, we deepen and if does not end in a singleton, we widen. For i<k, there is no difference between widening and deepening, and what we do is add all the addable nodes ineach segment.Thus it will suffice to calculatethe newe_m^*, m_i^*, I_m_i^* andC_i^*, with some special treatment for the case i=k.Each segmenthas its own long path, of which the length is the number of addable rows in the segment. If we are at m=n_i', then we reflect the entire long path.As in Lemma <ref>, this produces a new long path in the new direction, that of θ_m+1. By the induction hypothesis, λ has a standard LS representation, which depends on the sequence e_m=c_m/d_m, 1 ≤ m ≤ n. In order to prove that the widened multipartition λ^* has a standard LS-representation, we have to show that the new Littelmann path we get by the various reflections and translations given by Littelmann has an LS-representation given by a sequencee_m^*=c_m^*/d_m, for 1 ≤ m ≤ n+ 1. Now in fact, c_m is only changed where nodes are added in the m-ladder.For each segment i, the nodes are added in one particular ladder, that corresponding to n_i'+1. Thus for all remaining m in segment i, we have c_m=c_m^*, which means that for all q_m which do not involve n_i'+1, q_m^*=q_m. Furthermore, since we already showed that I_n_i' is the number of addable nodes in segment i, we have, for m=n_i', thatc_m+1^*=c_m+1+I_m We now calculate the new q_m^* andq_m+1^*.q^*_m= e_m^*-e_m+1^* = c_m^*/d_m-c_m+1^*/d_m+1= c_m/d_m-c_m+1+I_m/d_m+1= q_m-I_m/d_m+1=(I_m/d_m+1+C_i/d_md_m+1 )-I_m/d_m+1= C_i/d_md_m+1This is the same oscillator that occurs in q_m-1, so we still have an oscillating part,the new oscillator C_i^* is identical to C_i and I_m^* becomes 0 as we expect in the oscillating part.We now calculate the coefficicient of θ_m+1 in the new Littelmann path obtained by reflecting the long path.Thepart which is reflected is the subpath I_m/d_m+1θ_m, so after I_m reflections it becomes I_m/d_m+1θ_m+1.It joins the small oscillating part from the previous segment,C_i-1/d_m+1d_m+2θ_m+1 so that the coefficient of θ_m+1 will be I_m/d_m+1+C_i-1/d_m+1d_m+2.q^*_m+1= e_m+1^*-e_m+2^* = c_m+1^*/d_m+1-c_m+2^*/d_m+2= c_m+1+I_m/d_m+1-c_m+2/d_m+2= I_m/d_m+1+q_m+1= I_m/d_m+1+C_i-1/d_m+1d_m+2The number of addable nodes in the new segment should be the number of addable nodes in the segment i of λ plus the number of s_i of partition boundaries in thesegment, which is s_i=v_i-v_i-1+z_i-z_i-1. so that I_m+1^*=I_m+s_iWe now rewrite q_m+1^* in terms of the invariants of λ^*, where at the end we apply Corollary <ref> and then the result we just proved that C_i=C_i^*. q^*_m+1= I_m/d_m+1+C_i-1/d_m+1d_m+2= I_m/d_m+2+C_i-1+(d_m+2-d_m+1)I_m/d_m+1d_m+2= I_m+s_i/d_m+2+C_i-1+(d_m+2-d_m+1)I_m-d_m+1s_i/d_m+1d_m+2= I_m+1^*/d_m+2+C_i-1+rI_m-d_m+1s_i/d_m+1d_m+2= I_m+1^*/d_m+2+C_i/d_m+1d_m+2= I_m+1^*/d_m+2+C_i^*/d_m+1d_m+2 The little left-over piece at the end after we reflected most of the long path of segment i will be C_i/d_md_m+1θ_m.Whereas the oscillating paths used to pair in a way which always returned them to the same integer value of the positive coordinate, they will now pair in a way which always returns them to the same integer value of the other coordinate.If i<k, then the last piece will now join the next long path. For the case of deepening a segmentwith i=k ending in a singleton, the entire calculation is valid if we just replaceq_m^* with q̅_m and similarly for m+1.Note that if i=1, there was no previous small path, but in that case we also have C_i-1=0. In summary, the long path of segment i has shifted up one index from m to m+1 as we passed from λ to λ^*. * i=k, where the segment ends in a singleton and we want to widen.The case of i=k is considerably more complicated than the case of i<k because our induction must take into account five separate sections of the Littelmann path, which we now list in order going away from the origin:The long path of segment k, the oscillating part of segment k, the possibly empty transit path, the possibly empty terminal oscillator and the possibly empty tail.The transit path can only be used as a seed to start a new segment when the last row is odd. We are assuming that λhas a standard LS-representation and is unidirectional. We now take as our inner induction hypothesis that i=k, andthat segment k ends in a singleton. We wishto do a widening and to compute the new oscillator.Let m=n_k'. We already showed in the previous section of this proof that if we were to do a deepening to get λ̅, adding all I_m addable nodes, we would get a new long pathwith standard LS-representation, whereq̅_m=C_k/d_md_m+1 q̅_m+1=I_m/d_m+1+C_k/d_m+1d_m+2 Instead of adding all the I_n addable nodes, we add only I_n-z_kaddable nodes.In terms of the Littelmann path, this leaves a path of length1 in the positive direction which will be the transit path and a continuation with the same terminal oscillator C_k as before.( z_k/d_m+1+C_k/d_md_m+1 )θ_m=( C_k+z_kd_m/d_md_m+1 )θ_mNow we check that this widening will still give a standard LS-representation withC_k^*= C_k+z_kd_m as oscillator of segment k.We have c_m+1^*=c_m+1+I_m-z_k and c_ℓ=c_ℓ^* forall other c_ℓ in the segment, so we need check only q_ℓ for ℓ=m, m+1. q^*_m= e_m^*-e_m+1^* = c_m^*/d_m-c_m+1^*/d_m+1= c_m/d_m-c_m+1+I_m-z_k/d_m+1= q_m-I_m-z_k/d_m+1=(I_m/d_m+1+C_k/d_md_m+1 )-I_m-z_k/d_m+1= C_k+z_kd_m/d_md_m+1 This is precisely the formula for the oscillator C_k^* in Lemma <ref>, where now z_i^*=0, and the path after widening is still unidirectional.Thus we have a short path which is ambiguous:it can either be thought of as the length 1 transit path with a continuation which belongs to the terminal ocillator, or as the first step in a new oscillating part. With m=n_k',q^*_m+1= e_m+1^*-e_m+2^* = c_m+1^*/d_m+1-c_m+2^*/d_m+2= c_m+1+I_m-z_k/d_m+1-c_m+2/d_m+2= I_m-z_k/d_m+1+q_m+1= I_m-z_k/d_m+1+C_k-1/d_m+1d_m+2= I_m-z_k/d_m+2+(d_m+2-d_m+1)(I_m-z_k)+C_k-1/d_m+1d_m+2= I_m-z_k+s_k^*/d_m+2+r(I_m-z_k)-d_m+1s_k^*+C_k-1/d_m+1d_m+2= I_m+1^*/d_m+2+C_k^*/d_m+1d_m+2where the last step is a result of Lemma <ref> applied to λ^*, C_k^*=rI_n_k'^*^*-d_n_k'^*+1s_k^*+C_k-1^*,using the fact that C_k-1=C_k-1^*since k-1<k, that I_m+1^*=I_m+s_k-2z_k, that s_k^*=s_k-z_k and that d_m+2 = d_m+1+r. Further widening, as we saw in the first part of the proof, will move the long paths up by one, and will add new short paths with the same oscillator.The position of the transit path will be unaffected, so we will have m_k+1=m_k+1^*and the section from the transit path to the tail will only be translated, so it will memain unchanged.* i=k.Deepening after widening without adding a new partition. The partition ended in a singleton when we first started to widen at m=m_k+1 and the transit path has a hub which is a multiple of θ_m_k+1. Thus, after widening to get λ^*, if the last row is oddwe can continue one more step and reflect the part of length 1 in the transit path to get λ̅.This becomes the new long path of segment k+1. There will not be a transit path in the new seed part until after the next widening. The tail and the oscillating part before it remain as they were, so the new path is unidirectional.It remains to show that the new path has a standard LS-representation.The ladders c̅_m are identical to those of c_m^*, until we come to m=n_k+1'. The last row being odd, this is not the beginning of a segment for ℓ=m,m+1, so we have c_ℓ^*=c_ℓ.In λ̅, we add a new node in them+1-ladder, so that c̅_m+1= c_m+1+1.We have already checked the LS-representation for ladders outside of m for λ^*, so now we check q̅_m, where I_m, the number of addable nodes in the new segment k+1, is 2. q̅_m= c_m/d_m-c_m+1+1/d_m+1= q_m-1/d_m+1=(1/d_m+1+C_k+1/d_md_m+1 )-1/d_m+1=(C_k+1/d_md_m+1 ) q̅_m+1= c_m+1+1/d_m+1-c_m+2/d_m+2= q_m+1+1/d_m+1=(1/d_m+1+C_k/d_m+1d_m+2 )=(d_m+2/d_m+1d_m+2+d_m+C_k+1/d_m+1d_m+2 )=(2/d_m+2+C_k+1/d_m+1d_m+2 ) If the last row is even, we must either widen again or proceed to the next possibility, adding a new partition.* i=k.Deepening to add new partitions. The new partitions must come from the tail.In order to get to the tail, we must add all addable nodes between the last long path and the tail. There will be exactly one such addable node if there is a transit path, and none otherwise. If there is no transit path, then the solitary partitions will open a new segment and the addable node will belong to segment k.If there is a transit path, and the addable node belongs to the same ladder as the solitary partitions, then the addable node will belong to the same segment as the new solitary partition. Under all other conditions, we will have to add two segments.In order to treat all cases at once, if there is a transit path, we will call the segment with the solitary partitions k+2, and, when we finish deepening, rename it as k+1 if necessary.It is only possible to add y solitary partitions is there are empty partitions, which is to say, if there is a tail,which will beas always at θ_m form= 1_v_i-1≥ a.Only c_m+1 will be changed by the addition of these addable nodes. We have already shown that λ̅ has a standard LS-representation and is unidirectional,so as in the earlier parts of the proof, it only necessary to check q_m and q_m+1, and see that they correspond to the Littelmann path obtained by reflecting a piece of the tail of length y in the positive coordinate.It is also true that e_m=1.This is by definition if m=0, and we can only have m=1 if c_1=d_1=a. We have c̃_m+1=c_m+1+y, and for the remaining relevant indices u we have c̃_u=c_u. q̃_m= ẽ_m-ẽ_m+1 = c̃_m/d_m-c̃_m+1/d_m+1= 1-c_m+1+y/d_m+1= d_m+1-c_m+1-y/d_m+1 which gives the new tail, possibly zero, and coincides with the tail calculated in Lemma <ref>.Now, in order to find the new terminal oscillator, we calculate q̃_m+1= c_m+1+y/d_m+1-c_m+2/d_m+1d_m+2= q_m+1+y/d_m+1= q_m+1+2y/d_m+2+y(d_m+2-2d_m+1)/d_m+1d_m+2= q_m+1+2y/d_m+2+y(-d_m)/d_m+1d_m+2 where the added piece2y/d_m+2 gives two addable nodes for each of the added solitary partitions. If m=1,since d_1=a,then the quantity y(-a)/d_m+1d_m+2 is precisely the difference C_k+2-C_k+1 from Lemma <ref>. In the case of m=0,then d_0=-b and again the quantity y(b)/d_m+1d_m+2 is the difference C_k+2-C_k+1 from Lemma <ref>. To illustrate the theorem, we givecase where the partition crosses the boundary between the 0-corner and 1-corner partitions.Consider e=2, a=2, b=3. We calculate the q_m in this format for a sequence of multipartitions. The second has a transit path at θ_2, with d_2=7 and 13=C_1=C_2+d_2. The first four multipartitions have tails; the last does not, but it does have three long paths, and terminal oscillator 0. * [(2,1),(2,1),∅,∅,∅]e:4/7, 2/2,(6/12+ 6/12 · 7)θ_2, 6/7· 2θ_1* [(3,2,1),(3,2),∅,∅,∅]e:5/12, 4/7, 2/2,(6/17+ 13/17· 12)θ_3, 13/12 · 7θ_2, 6/7· 2θ_1* [(4,3,2,1),(4,3),(1),(1),∅]e:6/17, 5/12, 6/7, 2/2,(7/22+ 13/22· 17)θ_4, 13/17· 12θ_3, (5/12+ 2/12 · 7)θ_2, 2/7· 2θ_1* [(5,4,3,2,1)(5,4,1), (2,1),(2,1),∅]e: 7/22, 6/17, 10/12, 6/7, 2/2,(8/27+13/27· 22)θ_5, 13/22· 17θ_4, (8/17+ 2/17· 12)θ_3, 2/12 · 7θ_2, 2/7· 2θ_1* [(6,5,4,3,2,1)(6,5,2,1), (3,2,1),(3,2,1),(1)]e: 8/27, 7/22, 14/17, 10/12, 7/7,(9/32+13/32· 27)θ_6, 13/27· 22θ_5,( 11/ 22+ 2/22· 17)θ_4,2/17· 12θ_3, (2/12+0/12 · 7)θ_2. § MOTIVATION AND OPEN PROBLEMS To motivate the definition of the residue-homogeneous multipartitions, we use the block reduced crystal graph from <cit.>, furtherdeveloped in <cit.>, in which the vertices are the weightsΛ-α in P(Λ), where two weights connected by an edge if we can find a basis element from the weight space of each such that they are connected by f_i for some i. In Figure 1, we have drawn the caseof Λ=2Λ_0+Λ_1, truncated at degree 17. Vertices on the same horizontal line correspond to multipartitions of fixed degree n. The diagonal edges down to the left, which are called 0-strings,represent action by f_0 and the edges going down to the right, called 1-strings,represent f_1.The defects can all be determined in this case from the fact that acting on a stringby a simple reflection preserves defect, the highest weight element has defect 0, adding the null root δ adds the level r=3 to the defect, and (Λ-α_0)=1.The number of multipartitions corresponding to a vertex is fixed for each defect.In this case there is one multipartition for each vertex of defect 0 or 1, there are no vertices of defect 2, and there are 2 multipartitions for a vertex of defect 3.We have written in the multipartitions for the external vertices of defect 0,1 and 3.We originally began to study the unidirectional Littelmann path in the context of external vertices.It follows from the work of Scopes <cit.> and generalizations by Ariki and Koike <cit.>, Kleshchev and Brundan <cit.>, that if we have a weight all of whose crystal elements are external, then acting on the block of the cyclotomic Hecke algebraby the Weyl group in the direction increasing the degree will produce a Morita equivalence. For every defect, there is a degree after which all weights of that defect are external, so the external vertices are a sort of limit case.The set of strongly residue homogeneous multipartitions contained almost all of the external vertices, as well as a number of other vertices which are definitely not external. We definedstrongly residue homogeneous multipartitions here only for the case e=2, though it is possible to make an extension to larger ranks. We also consider unidirectional Littelmann paths only for defect 0 weights for which the reflection multiplying Λ is s_0, the reflection through the 0-th simple root α_0. Since we are in the case e=2, the dominant integral weight Λ is of the form Λ=aΛ_0+bΛ_1 and we set r=a+b, which in type A is the level.A vertex of the crystalfor which all multipartitions are residue-homogeneous multipartitions is external.Let λ be a residue-homogeneous multipartition.Since each partition is alternating, the rows in a partition all end with the same residue, and by the conditions on the parity of the first rows, this residue is the same for each partition.Let 1-ϵ be the common residue at the end of all non-zero rows, so that the addable nodes at the ends are all of residue ϵ. Since there are no ϵ removable rows, we must have that e_ϵ is zero. The multipartitions labelling external vertices in Figure 1 are all strongly residue-homogeneous. Among them, the multipartition [(3),(1),(2,1)] isstrongly residue-homogeneousbecause a singleton not at the boundary between 0- and 1-corner partitions makes no restriction on the previous partition. The main theorem is severely limited, in that it only applies to the case e=2 and even there, only to the non-increasing partitions.We can infact get some results for e=3 and perhaps higher, but we must restrict ourselves to sequences of defect 0 weights which are periodic, that is to say, generated by words which are periodic n the Weyl generators.The case of 3-periodic words is addressed in the original Ph.D. thesis, <cit.>. We again get long paths and short oscillating sections between them.To move in the other direction, from a unidirectional Littelmann path to its multipartition is harder, since the I_m and the C_i are nonlinear functions of the multipartition parameters v_i,t_i, and z_i. However, the nonlinear function, as we showed in Lemma <ref>, is a two part piecewise-linear function.Furthermore, the same lemma allows us to determine the s_i, since, in the definition of unidirectional, we specify the C_i andI_n_i'. In our two main examples, we usedtwo distinct primes as a and b, which made it easy to identify the terminal oscillators, as small multiples of a or b, while the widening oscillators were much larger. In order to explain what interferes with getting a standard LS-representation in the “+” case, where the first 1-corner partition has first row larger than the number of rows in the previous partition, as in [(1),(1),(2,1)], we look an easy caseA pseudo-floor of λ is a multipartition which begins like a defect 0 multipartition with weightψ^± _ℓ, but istruncatedafter w partitions. If it is increasing at the bundary between 0-corner and 1-corner partition, it will not have a standard LS-representation. Suppose we have a pseudo-floor λ and do a deepening to λ̅ We need to show that e̅_n ≥e̅_n+1.This is equivalent to showing the q_n≥ 0 . Since n<n_1' =n+1, we know from Lemma <ref> for λ̅ that I_n=0.Thus we need to determine if C_1≥ 0.We have i=1=k since there is only one segment and z_i=1 because we have a pseudo-floor, so we are in case the second case in Lemma <ref>.C_1= ∓(v_1+1)b± r((v_1-a)_++1_v_1 ≥ a)In the + case, a<w<r.We then have (v_1-a) ≥ 0, so we can drop the +.Substituting v_1+1=w and using b=(r-w)+(w-a), we have C_1= -wb+ r(w-a)=(r-w)(w-a-w)=-(r-w)a<0. Thus only the “-” case will give us an LS-representation.Since the formulae for a pseudo-floor are simple, we compute both the oscillator and the tail: C_1=wb, w≤ a,(r-w)a,r>w>a,Also for a pseudo-floor, we can calculate the hub of the tail hub(T)= a-w/aθ_0, w< a, r-w/bθ_1,r>w ≥ aLet Λ=2Λ_0+Λ_1 and let λ be the multipartition [(5,2,1),(1),∅].* λ= [(5,2,1),(1),∅]e: 1/14,1/11,3/8,2/5,2/2(1/17+ 3/17· 14)θ_5, 3/14 · 11θ_4, (3/11+ 1/11· 8)θ_3, 1/8 · 5θ_2, 4+2/5· 2θ_1* λ^*=[(6,3,2,1),(2),∅]e: 1/17,1/14,4/11,3/8,3/5,2/2) (1/20+ 3/20· 17)θ_6, 3/17· 14θ_5 ,(4/14+ 1/14 · 11)θ_4, 1/11· 8θ_3, (1/8+ 4/8 · 5)θ_2, 4/5· 2θ_1* λ̅=[(6,3,2,1),(2,1),∅]e: 1/17,1/14,4/11,3/8,4/5,2/2,(1/20+ 3/20· 17)θ_6, 3/17· 14θ_5 ,(4/14+ 1/14 · 11)θ_4, 1/11· 8θ_3, (3/8+ 2/8 · 5)θ_2, 2/5· 2θ_1* λ̃=[(6,3,2,1),(2,1),(1)]e: 1/17,1/14,4/11,3/8,5/5,(1/20+ 3/20· 17)θ_6, 3/17· 14θ_5 ,(4/14+ 1/14 · 11)θ_4, 1/11· 8θ_3, (5/8+ 0/8 · 5)θ_2. In Figure 2, we draw the hubs of the Littelmann paths in the case of deepening.There are three long paths, but in the last path there is a piece of 1-length b=1, which is actually the tail, and another piece of1-length 2 which will be needed if we wish to add two nodes to the partitions (1).If we choose to widen, then we get λ^*=[(6,3,2,1),(2),∅] in which only the first third of the last long path is reflected. If we choose to continue deepening to λ̅, without opening another partition, then we will reflect the second third get a multipartitionλ̃=[(6,3,2,1),(2,1),∅] in which the tail is obvious. Finally, if we continue to fill in the last non-zero partition, the tail will straighten out in the direction of the last long path, giving the Littelmann path ofλ̃. SchHap AKT S. Ariki, V. Kreiman, & S. Tsuchioka, On the tensor product of two basic representations of U_v(ŝl_e), Advances in Mathematics 218 (2008), 28-86.AK S. Ariki & K. Koike, A Hecke algebra of (Z/rZ) ≀ S_n and construction of its irreducible representations, Adv. Math. 106 (1994), 216–243.AS H. Arisha & M. Schaps Maximal Strings in the crystal graph of spin representations of symmetric and alternating groups, Comm. in Alg., Vol 37, no. 11 (2009), 3779-3795. BFS O. Barshavsky, M. Fayers & M. Schaps,A non-recursive criterion for weights of highest-weight modules for affine Lie algebras, Israel Jour. of Mathematics, vol.197(1) (2013), 237-261.CR J. Chuang & R. Rouquier, Derived equivalences for symmetric groups and sl_2 categorifications, Ann. of Math. (2) 167 (2008), no. 1, 245-298. Fa M. Fayers, Weights of multipartitions and representations of Ariki–Koike algebras, Adv. Math. 206 (2006), 112–144. Fa2 M. Fayers, An LLT-type algorithm for calculating higher type canonical bases, Journal of Pure and Applied Algebra,Volume 214, Issue 12, December 2010, Pages 2186-2198. Ka V. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press (1990). K1 M. Kashiwara, The crystal base and Littelmann's refined Demazure character formula, Duke Math. J. 1 (1993) 839-858. K2 M. Kashiwara, Bases Cristallines des Groupes Quantiques, Cours Specialises, Collection SMF, No. 9, (2002). Kl A. Kleshchev, Representation theory of symmetric groups and related Hecke algebras, Bull. Amer. Math. Soc. 47 (2010), 419–481. L P. Littelmann, Paths and root operators in representation theory, Annals of Mathematics, 2nd Ser. Vol. 142, No. e (Nov., 1995), 499-525. LLT A. Lascoux,B. Leclerc, J.-Y. Thibon,Hecke algebras at roots of unity and crystal basis of quantum affine algebras, Comm. Math. Physics 181 (19960, 205=263. MA. Mathas, Simple modules of Ariki-Koike algebras, Proc. Sym. Pure Math(1997), 383-396.O O. Amara-OmariLabelling Algorithms for the Irreducible Modules of Cyclotomic Hecke Algebrasof type A, Ph.D. thesis, Bar-Ilan University, 2019. OS O. Amara-Omari & M. Schaps Non-recursive canonical basis computations for rank 2 Kashiwara crystals of type A, preprint. Sc J. Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric group, J. Algebra 142 (1991), 441–455. | http://arxiv.org/abs/1707.08720v3 | {
"authors": [
"Ola Amara-Omari",
"Mary Schaps"
],
"categories": [
"math.RT",
"17B10, 17B65"
],
"primary_category": "math.RT",
"published": "20170727065504",
"title": "External Littelmann paths for crystals of type A"
} |
Wave breaking for the Stochastic Camassa-Holm equation Paper submitted to the Physica D special issue Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics dedicated to Prof. Edriss S. Titi on the occasion of his 60th birthday. Work partially supported by the EPSRC Standard Grant EP/N023781/1. Dan Crisan Department of Mathematics, Imperial College, London SW7 2AZ, UK. Email: [email protected] Darryl D Holm Department of Mathematics, Imperial College, London SW7 2AZ, UK. Email: [email protected] ======================================================================================================================================================================================================================================================================================================================== We show that wave breaking occurs with positive probability for the Stochastic Camassa-Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space-time paths. § THE DETERMINISTIC CAMASSA-HOLM (CH) EQUATION The deterministic CH equation, derived in <cit.>, is a nonlinear shallow water wave equation for a fluid velocity solution whose profile u(x,t) and its gradient both decay to zero at spatial infinity, |x|→∞, on the real line ℝ. Namely, u_t-u_xxt+3uu_x=2u_xu_xx+uu_xxx ,where subscripts t (resp. x) denote partial derivatives in time (resp. space). This nonlinear, nonlocal, completely integrable PDE may be written in Hamiltonian form for a momentum density m:=u-u_xx undergoing coadjoint motion, as <cit.> m_t = {m,h(m)} =- (∂ _xm+m∂ _x)δ h/δ m ,which is generated by the Lie-Poisson bracket {f,h} (m) =- ∫δ f/δ m(∂ _xm+m∂ _x)δ h/δ m dxand Hamiltonian function h(m)=1/2∫_ℝmK∗ m dx=1/2∫_ℝu^2+u_x^2 dx=1/2‖ u‖ _H^1^2=const.Here, K∗ m:=∫ K(x,y) m(y,t)dy denotes convolution of the momentum density m with the Green's function of the Helmholtz operator L=1-∂ _x^2, so that δ h/δ m=K∗ m=uwith K(x-y)=1/2exp (-|x-y|) .Alternatively, the CH equation (<ref>) may be written in advective form as u_t+uu_x =- ∂ _x(K∗(u^2+1/2u_x^2) ) =- ∂ _x∫_ℝ1/2exp (-|x-y|)( u^2(y,t)+1/2u_y^2(y,t))dy . The deterministic CH equation admits signature solutions representing a wave train of peaked solitons, called peakons, given by u(x,t)=1/2∑_a=1^Mp_a(t)e^-|x-q_a(t)| = K*m,which emerge from smooth confined initial conditions for the velocity profile. Such a sum is an exact solution of the CH equation (<ref>) provided the time-dependent parameters {p_a} and {q_a}, a=1,…,M, satisfy certain canonical Hamiltonian equations, to be discussed later. In fact, the peakon velocity wave train in(<ref>) is the asymptotic solution of the CH equation for any spatially confined C^1 initial condition, u(x,0).The peakon-train solutions of CH represent an emergent phenomenon. A wave train of peakons emerges in solving the initial-value problem for the CH equation (<ref>) for essentially any spatially confined initial condition. An example of the emergence of a wave train of peakons from a Gaussian initial condition is shown in Figure <ref>.emergent phenomenon! peakon wave trainBy equation (<ref>), the momentum density corresponding to the peakon wave train (<ref>) in velocity is given by a sum over delta functions in momentum density, representing the singular solution, m(x,t) = ∑_a=1^M p_a(t) δ(x-q_a(t)),in which the delta function δ(x-q) is defined by f(q) = ∫ f(x)δ(x-q) dx,for an arbitrary smooth function f. Physically, the relationship(<ref>) represents the dynamical coalescence of the CH momentum density into particle-like coherent structures (Young measures) which undergo elastic collisions as a result of their nonlinear interactions. Mathematically, the singular solutions of CH are captured by recognizing that the singular solution ansatz (<ref>) itself is an equivariant momentum map from the canonical phase space of M points embedded on the real line, to the dual of the vector fields on the real line. Namely, m: T ^∗ Emb(ℤ, ℝ) →𝔛(ℝ )^∗.This momentum map property explains, for example, why the singular solutions(<ref>) form an invariant manifold for any value of M and why their dynamics form a canonical Hamiltonian system, <cit.>. The complete integrability of the CH equation as a Hamiltonian system follows from its isospectral problem. The CH equation in (<ref>) follows from the compatibility conditions for the following CH isospectral eigenvalue problem and evolution equation for the real eigenfunction ψ(x,t), ψ_xx = (1/4 - m/2λ)ψ ,∂_tψ = -(λ + u) ψ_x + 1/2 u_xψ ,with real isospectral parameter, λ.By direct calculation, equating cross derivatives ∂_tψ_xx = ∂_x^2∂_tψ using equations (<ref>) and (<ref>) implies the CH equation in (<ref>), provided dλ/dt=0. The complete integrability of the CH equation as a Hamiltonian system and its soliton paradigm explain the emergence of peakons in the CH dynamics. Namely, their emergence reveals the initial condition's soliton (peakon) content.§.§ Steepening Lemma: the mechanism for peakon formation In the following we will continue working on the entire real line ℝ, although similar results are also available for a periodic domain with only minimal effort. We use the notation u_2, u_1,2 and u_∞ to denote, respectively, u_2:=∫_-∞^∞(u^2)d y,u_1,2:=∫_-∞^∞(u^2+1/2u_y^2) d y,andu_∞ := sup_x∈ℝu(x) .As reviewed in <cit.>, the deterministic CH equation (<ref>) is locally well posed on ℝ, for initial conditions in H^s with s>3/2. In particular, with such initial data, CH solutions are C^∞ in time and the Hamiltonian h(m) in (<ref>) is bounded for all time, h:=‖ u(· ,t)‖ _1,2<∞ .In fact, CH solutions preserve the Hamiltonian in (<ref>) given by the ‖ u(· ,t)‖ _1,2 norm ‖ u(· ,t)‖ _1,2=h=constant, forallx∈ℝ.By a standard Sobolev embedding theorem, (<ref>) also implies the useful relation that M:=sup_t∈ 0,∞ )‖ u(· ,t)‖ _∞<∞ .The mechanism for the emergent formation of the peakons seen in Figure <ref> may also be understood as a variant of classical formations of weak solutions in fluid dynamics by showing that initial conditions exist for which the solution of the CH equation (<ref>) can develop a vertical slope in its velocity u(t,x), in finite time. The mechanism turns out to be associated with inflection points of negative slope, such as occur on the leading edge of a rightward-propagating, spatially-confined velocity profile. In particular,Suppose the initial profile of velocity u(x,0) has an inflection point at x=x to the right of its maximum, and otherwise it decays to zero in each direction and that u(·,0)_1,2<∞. Moreover we assume that u_x(x̅,0)<-√(2M), where M is the constant defined in(<ref>). Then, the negative slope at the inflection point will become vertical in finite time.steepening lemma! CH equation in 1DConsider the evolution of the slope at the inflection point t↦x(t) that starts at time 0 from an inflection point x= x of u(x,0) to the right of its maximum so that s_0:=u_x(x(0),0)<∞ .Define s_t:=u_x(x(t),t),t≥ 0. From the spatial derivative of the advective form of the CH equation (<ref>) one obtains ∂ _x(u_t+uu_x)=-∂ _x^2K∗ (u^2+1/2 u_x^2)=u^2+1/2u_x^2-K∗ (u^2+1/2u_x^2),which leads to ∂ _tu_x=-uu_xx+u^2-1/2u_x^2-K∗ (u^2+1/ 2u_x^2).This, in turn, yields an equation for the evolution of t↦ s_t. Namely, by using u_xx(x(t),t)=0 and (<ref>) one finds ds/dt = - 1/2s^2+u^2(x (t),t)-1/2∫_-∞^∞ e^-|x (t)-y|( u^2+1/2u_y^2) dy≤ - 1/2s^2+M . Let s̃ be the solution of the equation ds̃/dt=- 1/2s̃^2+M,s̃_0=s_0 .Observe that d/dt((s_t-s̃_t)e^1/2∫_0^t(s_p+s̃_p)dp)≤ 0,s_0-s̃_0=0,therefore, s_t≤s̃_t for all t>0 (as long as both are well defined). However, equation (<ref>) admits the explicit solution s̅=√(2M)( σ +t/2√(2M)) , σ = ^-1( s_0/√(2M)) <0 .Since lim_t↦ -2σ /√(2M)s̅_t=-∞ it follows that there exists a time τ≤ -2σ /√(2M) by which the slope s_t=u_x(x(t),t) becomes negative and vertical, i.e. lim_t↦τs̅_t=-∞. Suppose the initial condition is anti-symmetric, so the inflection point at u=0 is fixed and dx/dt=0, due to the symmetry (u,x)→(-u,-x) admitted by equation (<ref>). In this case, the total momentum vanishes, i.e. M=0, and no matter how small |s(0)| (with s(0)<0), the verticality s→-∞ develops at x in finite time. The Steepening Lemma of <cit.> proves that in one dimension any initial velocity distribution whose spatial profile has an inflection point with negative slope (for example, any antisymmetric smooth initial distribution of velocity on the real line) will develop a vertical slope in finite time. Note that the peakon solution (<ref>) has no inflection points, so it is not subject to the steepening lemma.The Steepening Lemma underlies the mechanism for forming these singular solutions, which are continuous but have discontinuous spatial derivatives. Indeed, the numerical simulations in Figure <ref> show that the presence of an inflection point of negative slope in any confined initial velocity distribution triggers the steepening lemma as themechanism for the formation of the peakons. Namely, according to Figure <ref>, the initial (positive) velocity profile “leans” to the right and steepens, then produces a peakon that is taller than the initial profile, so it propagates away to the right, since the peakon moves at a speed equal to its height. This process leaves a profile behind with an inflection point of negative slope; so it repeats, thereby producing a wave train of peakons with the tallest and fastest ones moving rightward in order of height. In fact, Figure <ref> shows that this recurrent process produces only peakon solutions, as in (<ref>). This is a result of the isospectral property of CH as a completely integrable Hamiltonian system, <cit.>. Namely, the eigenvalues of the initial profile u(x,0) for the associated CH isospectral problem are equal to the asymptotic speeds of the peakons in the wave train (<ref>).The peakon solutions lie in H ^1 and have finite energy. We conclude that solutions with initial conditions in H^s with s > 3/2 go to infinity in the H^s norm in finite time, but they remain in H^1 and presumably continue to exist in a weak sense for all time in H^1. § ADVECTIVE FORM OF THE STOCHASTIC CAMASSA-HOLM (SCH) EQUATION Following <cit.>, we derive the SCH equation by introducing the stochastic Hamiltonian function, h(m)=1/2∫_ℝm(x,t)K∗ m(x,t) dx dt+∫_ℝm(x,t)∑_i=1^Nξ ^i(x)∘ dW_t^i dx .The second term generates spatially correlated random displacements, by pairing the momentum density with the Stratonovich noise in (<ref>) via a set of time-independent prescribed functions ξ ^i(x), i=1,2,… ,N, representing the spatial correlations. Thus, the resulting SCH equation is given by 0=𝖽m+(∂ _xm+m∂ _x)δh(m)/δ m=𝖽m+(∂ _xm+m∂ _x)v ,where m:=u-u_xx and the stochastic vector field v, defined by v(x,t):=u(x,t) dt+∑_i=1^Nξ ^i(x)∘ dW_t^i ,represents random spatially correlated shifts in the velocity, <cit.>. Thus, the noise introduced in(<ref>) and (<ref>) represents an additional stochastic perturbation in the momentum transport velocity. §.§ Peakon solutions and isospectrality for the SCH equationThe SCH equation (<ref>) with the stochastic vector field v in(<ref>) admits the singular momentum solution for CH in(<ref>) for peakon wave trains.Substituting the singular momentum relation (<ref>) into the stochastic Hamiltonian h(m) in (<ref>) and performing the integrals yields the Hamiltonian for the stochastic peakon trajectories as h(q,p):= 1/4∑_a,b=1^M p_a(t)p_b(t) e^-|q_a(t) - q_b(t)| + ∑_a,b=1^M p_a(t)∑_i=1^N ξ ^i(q_a(t))∘ dW_t^i .The canonical Hamiltonian equations for the stochastic peakon trajectories and momenta are thus given by𝖽q_a= ∂h/∂ p_a = 1/2∑_b=1^M p_b(t) e^-|q_a(t) - q_b(t)| dt + ∑_i=1^N ξ ^i(q_a(t))∘ dW_t^i= u(q_a(t)) dt + ∑_i=1^N ξ ^i(q_a(t))∘ dW_t^i = v(q_a(t)),and 𝖽p_a = - ∂h/∂ q_a = -p_a(t) ∂ u/∂ q_a dt - p_a(t) ∑_i=1^N ∂ξ ^i/∂ q_a∘ dW_t^i = -p_a(t) ∂ v(q_a(t))/∂ q_a .Substituting these stochastic canonical Hamiltonian equations for q_a(t) and p_a(t) into the singular momentum solution for CH in (<ref>) recovers the SCH equation (<ref>) and the stochastic vector field v in (<ref>). Thus, the SCH equation (<ref>) admits peakon wave train solutions whose peaks in velocity follow the stochastic trajectories given by the stochastic vector field v in (<ref>) and satisfy stochastic canonical Hamiltonian equations. The corresponding canonical Hamiltonian equations in the absence of noise describe the trajectories and momenta of CH wave trains. For numerical studies of the interactions of stochastic peakon solutions, see <cit.>.Remarkably, a certain amount of the isospectral structure for the deterministic CH equation is preserved by the addition of the stochastic transport perturbation we have introduced in (<ref>) and(<ref>).The SCH equation in (<ref>) follows from the compatibility condition for the deterministic CH isospectral eigenvalue problem (<ref>), and a stochastic evolution equation for the real eigenfunction ψ, ψ_xx = (1/4 - m/2λ)ψ ,𝖽ψ = -(λ + v) ψ_x + 1/2 v_xψ ,with v: = u dt + ∑_i=1^Nξ ^i(x) ∘ dW^i_t,and real isospectral parameter, λ, provided 𝖽λ=0 and ξ^i(x) = C^i+A^ie^x+B^ie^-x, for constants A^i,B^i and C^i.By direct calculation, equating cross derivatives 𝖽ψ_xx = ∂_x^2𝖽ψ using equations (<ref>) and (<ref>) implies, when 𝖽λ=0, that𝖽m+(∂ _xm+m∂ _x)v + λ(m_x - (v_x-v_xxx)) = 0 .Consequently, the compatibility condition for equations (<ref>) and (<ref>) implies the SCH equation in (<ref>), provided 𝖽λ=0 and ξ_x^i(x)-ξ_xxx^i(x) = 0. The latter means that ξ^i(x) is either constant, or exponential. Theorem <ref> means that the SCH equation (<ref>) with stochastic vector field v (<ref>) with ξ_x^i(x)- ξ_xxx^i(x) = 0 has the same countably infinite set of conservation laws as for the deterministic CH equation (<ref>). However, the SCH Hamiltonian h(m) in (<ref>) is not conserved by the SCH equation (<ref>), because it depends explicitly on time. Consequently, for those choices of ξ^i(x), the SCH equation is equivalent to consistency of the linear equations (<ref>) and(<ref>). Therefore, SCH is solvable by the isospectral method for each realisation of the stochastic process in (<ref>). However, SCH is almost certainly not completely integrable as a Hamiltonian system, even for constant ξ^i. See <cit.> for an example of an integrable stochastic deformation of the CH equation. The issue now and for the remainder of the paper is to find out whether the wave breaking property which is the mechanism for the creation of peakon wave trains in the deterministic case also survives the introduction of stochasticity. §.§ Wave breaking estimates for SCH In the following we will assume the conditions under which the stochastic integrals appearing in equation (<ref>) for u as well as the equation for u_x are well defined and summable. In particular, we assume that the vector fields ξ _i are smooth and bounded and that ∑_i>0((‖ξ ^i‖ _∞)^2+(‖ξ _x^i‖ _∞)^2+(‖ξ ^i‖ _2,1)^2)<∞ .Let A^i,∂ _xA^i, i∈ℤ_+ be the following set of operators A^i(u)= u_xξ ^i-K∗( u_xξ _xx^i(x)+2uξ _x^i(x)) ,∂ _xA^i(u)= u_xxξ ^i+u_xξ _x^i-∂ _xK∗( u_xξ _xx^i(x)+2uξ _x^i(x)) ,( ∂ _xA^i is obtained by formally differentiating A^i). In the following we will assume that there is a local solution of equation(<ref>) such that the operators A^i,∂ _xA^i, i∈ℤ_+ are well defined.Let us deduce first the equation for the velocity slope, u_x. We have the following Lemma: Under the above conditions, we have 𝖽u_x=-1/2( u_x^2+2uu_xx-u^2)dt-K∗( u^2+1/2u_x^2) dt-∑_iA_x^i(u)∘ dW_t^i.Expanding out the SCH equation in terms of u and v gives 0=𝖽m+(∂ _xm+m∂ _x)v=(1-∂ _x^2)𝖽u+2uv_x+u_xv-2v_xu_xx-vu_xxx =(1-∂ _x^2)(𝖽u+vu_x)+u_xv_xx+2uv_x =(1-∂ _x^2)(𝖽u+vu_x)+∂ _x( u^2+ 1/2u_x^2)dt +∑( u_xξ _xx^i(x)+2uξ _x^i(x)) ∘ dW_t^i .Therefore, applying the smoothing operator K∗ :=(1-∂ _x^2)^-1, given by the convolution with the Green's function K(x,y) in (<ref>) for the Helmholtz operator (1-∂ _x^2), to both sides of the previous equation yields 𝖽u+uu_x dt= -u_x( ∑ξ ^i∘ dW_t^i) -∂ _xK∗( u^2+1/2u_x^2)dt+∑ K∗( u_xξ _xx^i+2uξ _x^i) ∘ dW_t^i=-∂ _xK∗( u^2+1/2u_x^2) dt-∑ (u_xξ ^i-K∗( u_xξ _xx^i(x)+2uξ _x^i(x)))∘ dW_t^i= -∂ _xK∗( u^2+1/2u_x^2) dt-∑ A^i(u)∘ dW_t^i,in which the derivative ∂_x is understood to act on everything standing to its right.Consequently, we have 𝖽u_x=-( u_x^2+uu_xx)dt-∂ _xxK∗( u^2+1/2u_x^2) dt-∑ A_x^i(u)∘ dW_t^i.Then, since ∂ _xxK∗( u^2+1/2u_x^2)=-( u^2+1/2u_x^2) +K∗( u^2+1/2u_x^2),we deduce (<ref>).Observe that ∂ _xK∗( u_xξ _xx^i(x)+2uξ _x^i(x))= ∂ _xxK∗ (uξ _xx^i(x))-∂ _xK∗( uξ _xxx^i(x)+2uξ _x^i(x))=-uξ _xx^i(x)+K∗ (uξ _xx^i(x))-∂ _xK∗( uξ _xxx^i(x)+2uξ _x^i(x)).Hence, the last term in the expression of (<ref>) can be controlled by the supremum norm of u. Just as in the deterministic case, we define next the process t↦ν _t as the inflection point of u to the right of its maximum so that u_xx( ν _t,t) =0,ands_t=u_x( ν _t,t) <0.In what follows, we will assume, without proof, that the process t↦ν _t is a semi-martingale. The argument to show that validity of this property is based on the implicit function theorem. Indeed one can show thatν satisfies the equation dν _t=- 1/u_xxx(ν _t,t)(𝖽u_xx)(ν _t,t) ,provided the equation is well defined. That is, assume we have an inflection point, not an inflection interval; which means we have assumed u_xxx(ν _t,t)≠ 0. Using the semimartingale property of ν and the Itô -Ventzell formula (see, e.g., <cit.>), we deduce that 𝖽( u_x(ν _t,t)) =( 𝖽u_x) ( ν _t,t) +u_xx( ν _t,t) ∘𝖽ν _t=( 𝖽u_x) ( ν _t,t) .Hence, by (<ref>) and (<ref>), we find that ds_t = -( 1/2s_t^2-u^2( ν _t,t) ) dt-K∗( u^2+1/2u_x^2) ( ν _t) -∑_i( s_tξ _ν _t^i+B^i(u)|_ν _t) ∘ dW_t^i,where the operators B^i are given by B^i(u)=-uξ _xx^i(x)+K∗ (uξ _xx^i(x))-∂ _xK∗( uξ _xxx^i(x)+2uξ _x^i(x)) ,i∈ℤ_+We will henceforth consider the particular case when the vector fields ξ ^i are spatially homogeneous, so that B^i(u)=0. (This is also the isospectral case, which we discussed in the previous section.) In this case, just as in the deterministic case, we have ‖ u(· ,t)‖ _1,2=‖ u(· ,0)‖ _1,2,forallx∈ℝ.and, again, (<ref>) implies that M:=sup_t∈ 0,∞ )‖ u(· ,t)‖ _∞<∞ .This bound arises because the stochastic term vanishes when computing d‖ u(· ,t)‖ _1,2^2 . More precisely, the stochastic term is given by the expression ∑ (2uu_x+u_xu_xx) ξ ^i∘ W_t^i,whose spatial integral over the real line vanishes for constant ξ ^i, for the class of solutions u(· ,t) which vanish at infinity and whose gradient also vanishes at infinity. Note that the constant M in (<ref>) is independent of the realization of the Brownian motions W^i, i∈ℤ_+. By a standard Sobolev embedding theorem,(<ref>) also implies the useful relation that M:=sup_t∈ 0,∞ )‖ u(· ,t)‖ _∞<∞ .As in the deterministic case, suppose the initial profile of velocity u(x,0) has an inflection point at x=x to the right of its maximum, and it decays to zero in each direction; so that u(·,0)_1,2<∞. Consider the expectation of the slope at the inflection point, s̅_t=E[ s_t]. If u_x(x̅,0) is sufficiently small, then there exists τ<∞ such that lim_t↦τs̅_t=-∞. By changing from Stratonovitch to Itô integration, we obtain from(<ref>) that ds_t=-( 1/2s_t^2-u^2( ν _t,t) ) dt-K∗( u^2+1/2u_x^2) ( ν _t) -∑_is_tξ ^idW_t^i+1/2∑_is_t( ξ ^i) ^2and, by taking expectation, we deduce that dE[ s_t] ≤ -1/2( E[ s_t^2] -E [ s_t] ^2) -1/2( E[ s_t] ^2) +||ξ||/2E [ s_t] +MConsequently, ds̅_t≤ -1/2( s̅_t) ^2+||ξ||/2s̅_t+M≤ - 1-ε/2( s̅_t) ^2+( M+||ξ|| ^2/2ε)from which we deduce that the magnitude |s̅_t| blows up in finite time, just as in the deterministic case.Note that Proposition <ref> does not guarantee pathwise blow up of the process for the magnitude of the negative slope at the inflection point |s_t|, only the blow up of its mean, |s̅_t|. The following theorem shows that, indeed the pathwise negative slope s_t blows up in finite time with positive probability, albeit not with probability 1. Under the same assumptions as those introduced in Proposition <ref>, with positive probability, the negative slope at the inflection point s_t=u_x(ν_t,t) will become vertical in finite time. We define a new Brownian motion W, as follows W_t=-∑_iξ ^iW_t^i/||ξ||, t>0.Then the equation for s_t becomesds_t=-( 1/2s_t^2-||ξ|| ^2/2s_t-u^2( ν _t,t) ) dt-K∗( u^2+1/2u_x^2) ( ν _t) dt+||ξ|| s_tdW_t .We introduce a Brownian motion B such that the stochastic integral ∫_0^ts_pdW_p can be represented as ∫_0^ts_pdW_p=B_∫_0^ts_p^2dp .Then, as above, s_t ≤ s_0+1/2∫_0^t( ( M+||ξ|| ^2/2ε) +ε s_p^2/2) dp+( -1-2ε/2∫_0^ts_p^2dp+||ξ|| B_∫_0^ts_p^2dp)≤ s_0+1/2∫_0^t( M+||ξ|| ^2/2ε) dp+X_∫_0^ts_p^2dpwhere Xis a Brownian motion with negative drift[ If X be a Brownian motion with negative drift, X(t)=σ B(t)+μ t, μ <0, lim_t↦∞X( t) =-∞ . Let M=max_s≥ 0X( t) . Then P( M≥ a) =exp( -a( 2|μ|/σ ^2) ) and we conclude that M has an exponential distribution with mean 2|μ|/σ ^2. Put it differently, no matter where we start the Brownian motion with drift there is a positive probability that it will reach any level, before it drifts off to -∞ . Vice versa, it never hits level a with positive probability, see e.g.<cit.>.] X( t) =-1-3ε/2t+||ξ|| B_t.With positive probability (though not 1!), the process t↦ X_∫_0^ts_p^2dpremains smaller than, say, s_0/2<0 for all t>0. If the magnitude of the negative slope at the inflection point |s_0| is sufficiently large, then with positive probabilitythe term 1/2∫_0^t( ( M+||ξ|| ^2/2ε) -ε s_p^2/2) dswill always stay negative. It follows that, with positive probability, we have lim_t↦∞∫_0^ts_p^2dp=∞ ,andlim_t↦∞s_t≤lim_t↦ 0(s_0+X_∫_0^ts_p^2dp)=-∞ .Moreover, for sufficiently large t, X_∫_0^ts_p^2dp=-1-2ε/2∫_0^ts_p^2dp+||ξ|| B_∫_0^ts_p^2dp≤ -1-3ε/2∫_0^ts_p^2dp .Hence, s_t≤ s_0-1-3ε/2∫_0^ts_p^2dp.which, in turn, implies, as in the deterministic case, that the negative slope s_t at the inflection point must become vertical in finite time. In a similar manner we can show that s_t≥s̃_t, t≥ 0, wheres̃_t=-( 1/2s̃_t^2-||ξ|| ^2/2s̃_t+M) dt+||ξ||s̃ _tdW_t , s̃_0=s_0.To show this one proceeds, as in the proof of the steepening lemma, by firstjustifying the inequalityd/dt((s_t-s̃_t)e^1/2∫_0^t(s_p+s̃_p)dp+||ξ|| ^2t/2+||ξ|| W_t)≥ 0,s_0-s̃_0=0,so that, s_t≥s̃_t for all t>0 (as long as both are well defined). In turn, s̃_t, and therefore s_t, may achieve positive values with positive probability, which could, in principle, lead to a violation of the conditions under which a peakon may emerge in finite time due to the presence of an inflection point with slope s_t. Future work is planned by the authors to further investigate the emergence of peakons as well as the local well-posedness of the stochastic CH equation. 9 Ar2015 Alexis Arnaudon [2015], The stochastic integrable AKNS hierarchy, arXiv:1511.07080 [nlin.SI].CH1993 Roberto Camassa and Darryl D. Holm [1993] An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71: 1661–1664.Holm2015 Darryl D. Holm [2015], Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471: 20140963.HoMa2005 Darryl D. Holm and Jerrold E. Marsden [2005], Momentum maps and measure valued solutions (peakons, filaments, and sheets) of the Euler-Poincaré equations for the diffeomorphism group. In The Breadth of Symplectic and Poisson Geometry, A Festshrift for Alan Weinstein , 203-235, Progr. Math., 232, J.E. Marsden and T.S. Ratiu, Editors, Birkhäuser Boston, Boston, MA, 2005. Preprint at<arxiv.org/abs/nlin.CD/0312048>HoMaRa1998 Darryl D. Holm, Jerrold E. Marsden, and Tudor S. Ratiu [1998], The Euler–Poincaré equations and semidirect products with applications to continuum theories.Advances in Mathematics, 137(1):1–81.HoTyr2016 Darryl D. Holm and Tomasz M. Tyranowski [2016], Variational Principles for Stochastic Soliton Dynamics, Proc Roy Soc A 2016 472 20150827.Preprint available at arXiv:1503.03127 <http://arxiv.org/pdf/1503.03127.pdf>HoTyr2017 Darryl D. Holm and Tomasz M. Tyranowski [2017], Stochastic Discrete Hamiltonian Variational Integrators, Preprint available at arXiv:1609.00463 <http://arxiv.org/pdf/1609.00463.pdf>op Daniel Ocone, Etienne Pardoux, A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 1, 39-71.KS Karl, Sigman, Notes on Brownian Motion,< http://www.columbia.edu/ks20/FE-Notes/4700-07-Notes-BM.pdf> | http://arxiv.org/abs/1707.09000v1 | {
"authors": [
"Dan O. Crisan",
"Darryl D. Holm"
],
"categories": [
"math-ph",
"math.MP",
"nlin.SI",
"physics.flu-dyn"
],
"primary_category": "math-ph",
"published": "20170727185642",
"title": "Wave breaking for the Stochastic Camassa-Holm equation"
} |
empty Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM_110 mode for ultrafast electron microscopy [ December 30, 2023 ======================================================================================================================================== We introduce the concept of a morphism from the set of Butson Hadamard matrices over k roots of unity to the set of Butson matrices over ℓ roots of unity.As concrete examples of such morphisms, we describe tensor-product-like maps which reduce the order of the roots of unity appearing in a Butson matrix at the cost of increasing the dimension. Such maps can be constructed from Butson matrices with eigenvalues satisfying certain natural conditions.Our work unifies and generalises Turyn's construction of real Hadamard matrices from Butson matrices over the 4 roots and the work of Compton, Craigen and de Launey on `unreal' Butson matrices over the 6 roots.As a case study, we classify all morphisms from the set of n × n Butson matrices over k roots of unity to the set of 2n× 2n Butson matrices over ℓ roots of unity where ℓ < k. [2010 Mathematics Subject Classification: 05B20, 05B30, 05C50][Keywords: Butson Hadamard matrix, morphism, eigenvalues] § INTRODUCTION Let M be an n × n matrix with entries in the complex numbers ℂ. If every entry m_ij of M has modulus bounded by 1, then Hadamard's theorem states that |(M)| ≤ n^n/2. Hadamard himself observed that a matrix M meets this bound with equality if and only if every entry in M has modulus 1, and every pair of distinct rows of M are orthogonal (with respect to the usual Hermitian inner product) <cit.>. While Hadamard's name has become associated with real matrices meeting the bound, his original paper is not restricted to this case.It is well known that a real Hadamard matrix of order n (i.e. a matrix with entries in {± 1 } meeting the Hadamard bound) can exist only if n = 1, 2 or n ≡ 04. The Hadamard conjecture states that these necessary conditions are also sufficient. Since the discovery in 2005 of a real Hadamard matrix of order 428, the smallest open case is n = 668 <cit.>. Asymptotic existence results are also available in the real case: building on pioneering work of Seberry, Craigen has shown that for any odd integer m, there exists a Hadamard matrix of order 2^αlog_2(m) + 2m where α < 1 is a constant <cit.>. Despite the existence of many constructions for Hadamard matrices, the density of integers n for which it is known that there exists a Hadamard matrix of order n is essentially given by the density of the Paley matrices (i.e. the density of the primes, all other constructions contribute a higher order correction term) <cit.>.There is an analogue of the Hadamard conjecture for any number field: the usual Hadamard conjecture concerns the existence of Hadamard matrices over ℚ. Examples of matrices are known in which the entries are described as roots of modulus 1 of certain polynomial equations, see matrix A_8^(0) of the database <cit.> for example. We will be exclusively interested in matrices whose entries are roots of unity in this paper. We write M^∗ for the Hermitian transpose of M. A matrix H of order n with entries in ⟨ζ_k⟩ is Butson Hadamard if HH^∗ = n I_n. We write (n,k) for the set of such matrices. There exists a large literature on real Hadamard matrices, which have close ties to symmetric designs, certain binary codes and difference sets <cit.>. The Butson classes (n,4) and (n,6) have received some attention in the literature, due to applications in signal processing and connections to real Hadamard matrices <cit.>. In fact, if (n,4) is non-empty, then (2n,2) is necessarily non-empty. This motivates the complex Hadamard conjecture: that there exists a Hadamard matrix of order n with entries in {± 1, ± i} whenever n is even.More generally, de Launey and Winterhof have described necessary conditions for the existence of Hadamard matrices whose entries are roots of unity <cit.>. A typical application of these results shows that (n, 3^a) is empty whenever the square-free part of n is divisible by a prime p ≡ 56. In contrast to (n,2) and (n,4), there does not appear to be any consensus in the literature on sufficient conditions for (n,k) to be non-empty for general values of k. Apart from the obvious examples of character tables of abelian groups, relatively few constructions for matrices in (n,k) are known. An early result of Butson shows that for all primes p, the set (n,p) is empty unless n ≡ 0p, and that (2p,p) is non-empty <cit.>. Further constructions and a survey of known results are given by Agaian <cit.>. § MORPHISMS OF BUTSON MATRICES We define a (complete) morphism of Butson matrices to be a function (n,k) →(r,ℓ). The Kronecker product is perhaps the best-known example of a morphism: if M ∈(m,d) and H is any matrix in (n,k) then H ⊗ M ∈(mn, ℓ) where ℓ is the least common multiple of k and d. To be explicit, the map - ⊗ M: (n,k) →(mn,ℓ) is a Butson morphism.We will restrict our attention to morphisms which come from embeddings of matrix algebras (as the tensor product does), such morphisms can be considered generalised plug-in constructions. In this case, we say a morphism (n,k) →(mn,ℓ) is of degree m.We will be particularly interested in the construction of morphisms where k does not divide ℓ. We will also relax our conditions to allow partial morphisms, where the domain is a proper subset of (n,k); typically we impose a restriction on matrix entries of H. We introduce the concept of a sound pair to collect necessary and sufficient conditions for our main existence theorem.We define ζ_k = e^2π i/k, and set G_k = ⟨ζ_k⟩.We define H^ϕ to be the entrywise application of ϕ to H whenever ϕ is a function defined on the entries of H, and we write H^(r) for the function which replaces each entry of H by its r power.Let X,Y ⊆ G_k be fixed. Suppose that H ∈(n,k) such that every entry of H is contained in X, and that M ∈(m,ℓ) such that every eigenvalue of √(m)^-1M is contained in Y.Then the pair (H, M) is (X,Y)-sound if * For each ζ_k^i∈ X, we have √(m)^1-i M^i∈(m, ℓ).* For each ζ_k^j∈ Y, we have H^(j)∈(n,k).We will often say that (H,M) is a sound pair if there exist sets X and Y for which (H, M) is (X, Y)-sound.Often the first condition of a sound pair is satisfied only when i is odd: this occurs in particular when ℚ[ζ_k] contains no elements of absolute value √(m). But there do exist matrices satisfying this condition where m is not a perfect square: one example is given by the square of the matrixM_24 = [ [11;i -i ]] ,√(2)^-1M_24^2 =[ [ ζ_8 ζ_8^7; ζ_8 ζ_8^3 ]]which belongs to (2,8). This example also shows that one should distinguish carefully between ℓ and the degree of the cyclotomic field generated by the entries of M in Definition <ref>. A natural question arises about the smallest field containing the entries of √(m)^1-iM^i for all i. Since we are interested exclusively in Butson matrices, this will be the largest cyclotomic field contained in ℚ[ζ_ℓ, √(m)], which is at most a quadratic extension of ℚ[ζ_ℓ], so its torsion units will be at most 2ℓ roots of unity if ℓ is even and 4ℓ roots of unity if ℓ is odd.Before we prove our main theorem, we recall that for an n× n matrix A, and an m × m matrix B, the Kronecker products A ⊗ B and B ⊗ A are similar matrices. There exists a permutation matrix P_mn such that P_mn (A⊗ B) P_mn^-1 = B ⊗ A. We call this matrix the Kronecker shuffle. The locations of the non-zero entries in P_mn can be precisely described.For any m,n ∈ℕ, the nm × nm Kronecker shuffle matrix P_mn isP_mn = [ δ^im - ⌊(i-1)/n⌋(mn-1)_j + m - 1]_1≤ i, j ≤ mn .If M is an mn × mn matrix consisting of n× n diagonal blocks, then P_mnM P_mn^-1 has m× m blocks down the diagonal and is zero elsewhere. We now show that the conditions of a sound pair are sufficient to guarantee that a plug-in construction gives a Hadamard matrix. Let H ∈(n,k) and M ∈(m,ℓ) be Hadamard matrices. Define a map ϕ: ζ_k^i↦√(m)^1-i M^i, and write H^ϕ for the entrywise application of ϕ to H. If (H, M) is a sound pair then H^ϕ∈(mn,ℓ).Fix sets X,Y ⊆ G_k such that (H, M) is (X, Y)-sound. By the first property of a sound pair, for every ζ_k^i∈ X, the image ϕ(ζ_k^i) = √(m)^1-i M^i∈(m, ℓ). Hence every entry in H^ϕ is in ⟨ζ_ℓ⟩. To show that H^ϕ is a Hadamard matrix, it will suffice to show that every eigenvalue of H^ϕ has absolute value √(mn).Every Hadamard matrix is diagonalisable. Write A for a matrix such that AMA^-1 is diagonal, and define ψ(ζ_k^i) = √(m)^1-i AM^iA^-1. Observe that if ζ_k^α is the t diagonal entry of AMA^-1, then ψ(ζ_k^i)_t,t = ζ_k^iα, where by hypothesis ζ_k^α∈ Y. Writing H^ψ for the entrywise application of ψ to H, it follows easily thatH^ψ = (I_n⊗ A) H^ϕ (I_n⊗ A^-1) .Now, H^ψ is an mn × mn block matrix in which each m× m block is diagonal. Applying Proposition <ref>, we obtain a block diagonal matrix,P_mn H^ψ P_mn^-1 = diag [ B_1, B_2, …, B_k]. To conclude, observe that B_t[i,j] = √(m)ψ(h_i,j)_t,t = √(m)h_i,j^α with the notation chosen above. So B_t = √(m)H^(α), and by the second property of a sound pair, each eigenvalue of B_t has absolute value √(mn). This argument holds for each block B_t, so since P_mnH^ψP_mn^-1 and H^ϕ are similar, we have H^ϕ∈(mn,ℓ) as required. In the special case that every eigenvalue of M is a primitive k root of unity, the second condition of Definition <ref> is vacuous: raising each entry to its α power for α coprime to k is a field automorphism. Such an operation preserves both the modulus of entries of H and the orthogonality of rows. These are examples of global equivalence operations as considered by de Launey and Flannery <cit.>. The first condition of Definition <ref> still places restrictions on the entries of H: it can never contain 1, for example; since the image of ζ_k^0 = 1 in H^ϕ never has entries of modulus 1. Next we identify some sound pairs of special interest, since they correspond to complete morphisms. Suppose that M ∈(m,ℓ), that all eigenvalues of M are primitive k roots of unity, and that √(m)^1-i M^i is Hadamard for all i coprime to k. Let d = ∏_i=1^r p_i^α_i such that p_i^α_i+1| k for 1 ≤ i ≤ r. Then there exists a complete Butson morphism (n,d) →(mn,ℓ).Under the hypotheses, the primitive k roots of unity contain a translate of the d roots of unity. If H ∈(n,d) then every entry of ζ_kH is a primitive k root. The conditions of Definition <ref> are satisfied with X and Y both taken to be the primitive k roots of unity. The claim follows from Theorem <ref>.The following construction of Turyn illustrates Corollary <ref>.[Turyn, <cit.>]Let M_8 be the matrixM_8 = [ [11; -11 ]] .It is easily verified that the eigenvalues of M_8 are √(2)ζ_8 and √(2)ζ_8^7. Likewise, it can be verified that1/2 M_8^3 =[ [ -11; -1 -1 ]]that 2^-2M_8^5 = -M_8 and that 2^-3M_8^7 = -M_8^3. (In fact M_8^8 = 16I_2.) Thus M_8 satisfies all the conditions of Corollary <ref> with m = 2 and ℓ = 8. So for any H ∈(n,4), the pair (ζ_8 H, M_8) is sound. We recover Turyn's famous morphism: (n,4) →(2n,2). Next, we illustrate the full generality of Theorem <ref>.[Compton, Craigen, de Launey <cit.>]Let M_6 be the following matrix (which is in fact similar to the matrix given by Compton, Craigen and de Launey):M_6 = [ [1111; -11 -11; -111 -1; -1 -111 ]] .One computes that the eigenvalues of 2^-1M_6 are the primitive sixth roots of unity, each with multiplicity 2. Likewise, one can check that 2^-1M_6^2 is Hadamard and that M_6^3 = -8I_4. As a result, 2^-3M_6^4 and 2^-4M_6^5 are Hadamard.In the definition of a sound pair, we can take X = {ζ_6, ζ_6^2, ζ_6^4, ζ_6^5} and Y = {ζ_6, ζ_6^5}. Since raising each entry to its first power is the identity map on ⟨ζ_6⟩ and raising an entry to its 5 power is complex conjugation, the restrictions placed on H by Y are vacuous. Since M_6^3 is a scalar matrix, we cannot allow -1 as an entry in H. Compton, Craigen and de Launey call a matrix in (n,6) with entries in X unreal. We have constructed a partial morphism: (n,6) →(4n,2) with domain the unreal matrices. We conclude this section with a question which we feel should have a positive answer. If the eigenvalues of M ∈(m, ℓ) are all primitive k roots of unity, is it true that √(m)^1-iM^i ∈(m, ℓ) for all i coprime to k?§ CONSTRUCTION OF BUTSON MORPHISMS§.§ Partial morphisms We begin by describing a relationship between certain sets of mutually unbiased bases and partial morphisms of Hadamard matrices. Let V be an n-dimensional vector space over ℂ carrying the usual Hermitian inner product. Two orthonormal bases B_1 and B_2 of V are unbiased if |⟨ u,v ⟩| = n^-1/2 for all u ∈ B_1 and v ∈ B_2. A set of bases is mutually unbiased if each pair is unbiased. We refer to sets of mutually unbiased bases as MUBs. Such objects are of substantial interest in quantum physics, and are well studied in the literature. Normalising so that B_0 is the standard normal basis, every other B_i is necessarily represented by a Hadamard matrix. Being mutually unbiased means that B_iB_j^∗ is again Hadamard for all i ≠ j. The largest possible number of MUBs in ℂ^d is d+1. Maximal sets of MUBs are known to exist in every prime power dimension, and in no other dimensions. It is believed that the maximal number of MUBs in dimension 6 is three, but this is an outstanding open problem <cit.>.To construct sound pairs, we require a Hadamard matrix M for which a specified set of its powers are Hadamard. Since M^i (M^j)^∗ = M^i-j, a subset ℬ of the powers of M is a set of MUBs if and only if M^i-j is Hadamard for all M^i, M^j∈ℬ. A remarkable construction of Gow yields Butson matrices with the property that every power is either Hadamard or scalar. The construction uses the representation theory of extra-special p-groups of exponent p over ℂ for odd p, these are the so-called discrete Heisenberg groups. (For p = 2, Gow uses generalised quaternion groups.) Let q = 2^a such that q+1 is prime. There exists M_q∈(q, 4) and M'_q∈(q^2, 2) such that M_q has only primitive (q+1)^st roots of unity as eigenvalues.Since M'_q∈(q^2, 2), every eigenvalue of q^-1M'_q is a complex number of modulus 1. Since q+1 is prime, the eigenvalues are even (q+1)^st roots of unity (though not necessarily all primitive). Gow proves that the trace of q^-1M'_q is -q. Together with the irreducibility of the cyclotomic polynomials over ℚ, this shows that every eigenvalue of q^-1M'_q is a primitive (q+1)^st root of unity.The proof is similar in the complex case: again, the matrix √(q)^-1M_q is of multiplicative order q+1. So the eigenvalues are (q+1)^st roots of unity, and the minimal polynomial divides the cyclotomic polynomial Φ_q+1(x). Factoring q+1 over the Gaussian integers and applying a generalisation of Eisenstein's criterion, shows that Φ_q+1(x) remains irreducible over ℚ[i], so every eigenvalue of √(q)^-1M_q is again a (q+1)^st root of unity. The smallest example of Gow's theorem is in dimension 2, where Gow gives the matrix2^-1/2ζ_8M = 1+i/2[ [ -1i;1i ]].Some care is necessary in interpreting the matrices of Theorem <ref> as partial morphisms, however. While Gow's unitary matrix is defined over ℚ[i] and has order 3, the corresponding Hadamard matrix (denoted M above) has order 24. In fact, if there exists M ∈(2, ℓ) such that 2^-3/2M^3 = I_2 then 8 |ℓ. Interestingly, applying the Turyn morphism to ζ^5_8M yields a matrix in (4, 2) which is similar to the Compton-Craigen-de Launey matrix of Example <ref>. For a Fermat prime p > 3, it is well known that p-1 is a perfect square, and we do not need to pass to a larger field. Suppose that p = 2^a + 1 is a Fermat prime, p> 3. Then there exists M_p∈(2^a, 4) and M'_p∈(2^2a, 2) with only primitive p roots as eigenvalues. Hence, for any H ∈(n,p) with no entries equal to 1, the pairs (H, M_p) and (H, M'_p) are sound. Gow has also constructed large sets of MUBs as powers of a single matrix in odd prime power dimensions. But when q+1 is not prime, the conditions imposed by Definition<ref> on H may be impossible to satisfy. Gow's 5 × 5 matrix has eigenvalues {-1, ±ω, ±ω^2},which require that H^(2) and H^(3) both be Hadamard. But this implies that H is generalised Hadamard overthe cyclic group of order 6: generalised Hadamard matrices have been constructed only over p-groups, and it has been conjectured that no such matrices exist over groups of composite order. (See Section 2.10 of <cit.> for the definition of generalised Hadamard matrices, and <cit.> for some non-existence results.) It seems rather difficult to realise Gow's work as an explicit construction for partial morphisms. We constructed several examples computationally. The follows matrix has as eigenvalues the primitive fifth roots of unity.M_5 = [ [ -1 -1 -1 -1;1 -11 -1;ii -i -i;i -i -ii ]] .Equivalently, M_5 induces a partial morphism (n, 5)→(4n, 4) where the domain consists of matrices have no entries equal to 1. Allowing negations, we obtain a map (n, 10) →(4n, 4) with domain the unreal matrices (i.e. those containing no real entries). These morphisms generalise the Compton-Craigen-de Launey result which is the case a = 1 of Corollary <ref>.§.§ New morphisms from old Suppose that there is a complete morphism (n,d) →(nm,ℓ) where M, d and k are defined as in Corollary <ref>. The next result allows us to construct new morphisms over larger roots of unity. If (ζ_kH,M) is (X,Y)-sound for all H ∈(n,d) where Y consists only of primitive k roots of unity, then (ζ_ktH, ζ_tM) is sound for any t coprime to k.The eigenvalues of ζ_tM are primitive kt roots of unity. Let T ⊂{1,…,k-1} be the set of all i such that √(m)^1-iM^i ∈(m, ℓ). By hypothesis, M induces a complete morphism (n,d) →(nm,ℓ); so T contains an arithmetic progression of length d, say D ⊆ T.Furthermore, √(m)^1-kt(ζ_tM)^kt = I_m, and √(m)^1-a(ζ_tM)^a ∈(m, ℓ t) whenever a ≡ ik and i ∈ T. Let T' be the set of all a such that √(m)^1-a(ζ_tM)^a ∈(m, ℓ t). Define D' ⊆ T' to be the set exponents such that at is in D. Then D' is an arithmetic progression of length dt.Thus (ζ_ktH, ζ_tM) is (X',Y')-sound for all H ∈(n,d), where X' = {ζ_kt^i: i ∈ T'} and Y' is the set of primitive kt roots of unity.If there exists a complete morphism (n,d) →(nm,ℓ), there exists a complete morphism (n,dt) →(nm,(t,ℓ)). As an illustration of Theorem <ref> we generalise Turyn's morphism. Let M_8 be the matrix of Example <ref>. The eigenvalues of ζ_3M_8 are primitive 24 roots of unity, and √(2)^1-i(ζ_3M_8)^i is Hadamard for all odd i. So for any H ∈(n,12), the pair (ζ_24H,ζ_3M_8) is sound, and thus there is a complete morphism (n,12) →(2n,6). We can also generalize to partial morphisms.For example, the matrixM_12 = 2^-1[ [1111;1 -1 -11;11 -1 -1; -11 -11 ]]is of order 12, has primitive 12 roots as its eigenvalues, and 2^-iM_12^i is Hadamard for all i ∈ T = {1,2,4, 5,7,8,10,11}. Then 2^1-i(ζ_5M_12)^i is Hadamard for all i ∈ T' = {1,…,60-1}∖{3,6,…,57}, and the eigenvalues of ζ_52^-1M_12 are primitive 60 roots of unity.So T' contains two arithmetic progressions of length 20.Thus, with the appropriate choice of X and Y, (ζ_60H,ζ_5M_12) is sound for any H ∈(n,20), and (K,ζ_5M_12) is sound for any K ∈(n,60) such that no entry in K is a 20 roots of unity.So there is a complete morphism (n,20) →(4n,10) and a partial morphism (n,60) →(4n,10).§.§ Construction of complete morphisms The construction of large sets of MUBs is a challenging open problem, even without the added restriction that all of the bases arise as powers of a single matrix. As a result, we do not expect in general to find subgroups of ⟨ m^-1/2 M ⟩ which contain many Hadamard matrices. On the other hand, there seem to be few restrictions on cosets containing many Hadamard powers. This is precisely the intuition behind Corollary <ref>.It seems natural (though not strictly necessary) to study matrices M such that M ∈(m, ℓ) and the unitary matrix m^-1/2M has as eigenvalues only primitive k roots of unity. The characteristic polynomial of m^-1/2M is a divisor of a power of the cyclotomic polynomial Φ_k(x). So the possible characteristic polynomials then depend on the factorisation of Φ_k(x) in ℚ[ζ_ℓ, √(m)]. The following Proposition gives an easy construction of larger complete morphisms from small ones. Suppose that 4 | k and that the eigenvalues of M ∈(m, ℓ) are primitive k roots of unity. If H ∈(n, ℓ) is Hermitian, then the eigenvalues of H ⊗ M ∈(mn, ℓ) are all primitive k roots of unity.The eigenvalues of H are real, and so all lie in {±√(n)}. The eigenvalues of a tensor product are the products of the eigenvalues of the component matrices, and by hypothesis, ζ_k^i is primitive if and only if ζ_k^i + n/2 is. The first interesting examples of complete morphisms occur when M is real and its eigenvalues are primitive 8 roots of unity. The matrix M_8 in Example <ref> is one such. It is somewhat unusual in that the size of the matrix is smaller than the degree of Φ_8(x); in fact its characteristic polynomial comes from an exceptional factorisation of cyclotomic polynomials related to Sophie Germain's identity. Larger examples of complete morphisms can be constructed from the Turyn example via tensor products. We combine Proposition <ref> and Example <ref>: since real symmetric Hadamard matrices are easily constructed at orders 1, 2 and 4t for all t ≤ 10, we will focus on the cases n ≡ 48. The classification of (10, 4) and (14, 4) has been completed up to Hadamard equivalence by Lampio, Szöllősi and Östergård: Hermitian matrices exist at both orders <cit.>. We have not yet managed to construct a real Hadamard matrix of order 36 with eigenvalues in the set {6ζ_8, 6ζ_8^3, 6ζ_8^5, 6ζ_8^7}. There exists a complete morphism (n, 4) →(2mn,2) whenever there exists a Hermitian matrix in (m, 4). In particular, there exist such morphisms for m = 1 and for all even m ≤ 8. It seems more challenging to construct complete morphisms (n,k) →(mn, ℓ) for which k and ℓ are coprime. What is the smallest m for which there is a complete morphism (n,3) →(mn,2)? If M yields a complete morphism as in the question, then the eigenvalues of M are primitive 9k^th roots of unity for some k ∈ℕ. If k = 4, the characteristic polynomial of (12t)^-1/2M is necessarily (x^12 - x^6 + 1)^t, where M is a matrix of order 12t. The characteristic polynomial of M can be obtained from that of the unitary matrix via the substitution x^k↦ (12t)^(n-k)/2x^k.§.§ Eigenvalues of (2,ℓ) In this section we classify all M ∈(2,ℓ) for which the corresponding unitary matrix has finite multiplicative order, and study morphisms derived from these matrices. We begin with a lemma on roots of unity which will constrain the eigenvalues of M.Suppose that α and λ are roots of unity such that (α) = √(2)(λ). Then up to negation and complex conjugation, [α, λ] is one of [i, i], [1,ζ_8], [ζ_8, ζ_6].Recall that (ζ) = 1/2(ζ + ζ^-1). So we require the solutions of the identityα + α^-1/λ + λ^-1 = √(2) .Expanding as a quadratic in α and solving, we have that√(2)α = -(λ + λ^-1) ±√(λ^2 + λ^-2) .Applying field automorphisms, we may assume that λ = ζ_k for some suitable integer k. Suppose that k ≥ 8, then λ + λ^-1 = 2(λ) ≥√(2), while |√(λ^2 + λ^-2)| ≤√(2). Since k ≥ 8, we have that (λ^2) > 0, hence the right hand side of the equation is real and negative. The unique solution of absolute value √(2) occurs when k = 8. The solutions with k < 8 can be found by inspection. Suppose that M ∈(2, ℓ) such that 2^-1/2M has finite multiplicative order. Then λ_1λ_2^-1∈{-1, ± i, ±ζ_3}.For an arbitrary 2 × 2 Butson matrixM = [ [ α β; γ δ ]] ∈(2, ℓ),we observe that (α^-1/2δ^-1/2) M has real trace, and hence its eigenvalues are conjugate. Furthermore, the ratio of the eigenvalues of H is preserved by scalar multiplication. Orthogonality of rows forces β = -γ^∗. In fact, different choices of β yield similar matrices. So to compute the ratio of the eigenvalues of M, it suffices to consider matrices of the formM_α = [ [αα; -α^*α^* ]] .We compute the eigenvalues explicitly:λ = (α) + i √( 2 - (α)^2),λ^∗ = (α) - i √( 2 - (α)^2) .Since clearly (α) ≤ 1, the second term is always purely imaginary and we have (α) = (λ). The matrix 2^-1/2M_α has finite order if and only if its eigenvalues are roots of unity. Setting 2^-1/2λ = λ', we require the classification ofthe pairs of roots of unity (α, λ') for which (α) = √(2)(λ'). Now apply Proposition <ref>. Corollary <ref> reduces the analysis of the 2 × 2 Butson matrices to three cases. We deal with the traceless matrices separately, since the proof is short, and we never obtain morphisms for which k ≥ℓ. Suppose that M ∈(2, ℓ) such that 2^-1/2M has finite multiplicative order with k root of unity eigenvalues. Then if λ_1 = -λ_2, k ≤ℓ.Let a = m_11, and so m_22 = -a. Orthogonality of the rows of M implies that m_12 = ab and m_21 = ab^∗ for arbitrary b of modulus 1. It is easily verified that different choices of b produce similar matrices and that the eigenvalues of M are ±√(2) a. Hence k ≤ℓ. Now we turn our attention to the remaining cases of Corollary <ref>. We will restrict attention to matrices M in (2, ℓ) for which the eigenvalues of 2^-1/2M are primitive k roots. From this information, we easily obtain a classification of all complete morphisms of order 2. Suppose that M ∈(2, ℓ), and that the eigenvalues of M are both primitive k roots of unity for some k > ℓ. Then M is one of the following, where a,b are ℓ roots of unity. * M_1 = ( [ aab; -ab^∗ a ]), and ℓ = 2^αt where t is odd. Then k > ℓ when α≤ 2, and both eigenvalues have the same order when α≠ 3. * M_2 = ( [a ab; -iab^∗ ia ]), and ℓ = 2^α3^βt where t is coprime to 6. Then k > ℓ when β≤ 1 or α≤ 3, and both eigenvalues have the same order when β≠ 1 and α≠ 3. Throughout we write λ_1,λ_2 for the eigenvalues of M, and m_ij for the entry in row i and column j. By Corollary <ref> and Lemma <ref>, we may assume that (up to relabelling of eigenvalues and negation of M) that λ_1λ_2^-1∈{i, ζ_3}. 1: Suppose λ_1 = iλ_2. Then writing λ_1 = √(2)ω for some ω∈ℂ of modulus 1, we have that m_11 + m_22 = 2ζ_8ω, where ζ_8ω. By the triangle inequality, a := m_11 = m_22.Enforcing the orthogonality of the rows of M, we find that m_12 = ab and m_21 = -ab^∗ for arbitrary b of modulus 1.Thus under the assumption that M ∈(2, ℓ), we have that M = M_1. It is easily verified that different choices of b produce similar matrices, and the eigenvalues of M_1 are √(2)ζ_8a and √(2)ζ_8^7 a.For convenience, we write M_1(a) for the unitary matrix with 2^-1/2a on the diagonal, and up to similarity we can take b = 1. The group generated by M_1(a) is finite and cyclic, and so a direct product of a cyclic 2-group and group of odd order. Since M_1(a)^4 = -a^4I_2 a maximal subgroup of odd order is scalar. Squaring is an automorphism on the roots of unity of odd order, hence there exists some root of unity of odd order ζ∈⟨ζ_ℓ⟩ such that M_1(a) = ζ M_1(a'), where a' is a root of unity of order 2^α.It is easily verified that when a' ∈{± 1, ± i} that the eigenvalues of M_1(a') are primitive 8 roots of unity. But when a' is a primitive 8 root, then one eigenvalue of M_1(a') is real while the other is purely imaginary. Finally, a' is a primitive root of unity of order 2^α for α≥ 3, then both eigenvalues of M_1(a') are again of order 2^α.We write ℚ[ζ_ℓ] for the coefficient field of √(2)M_1(a), where ℓ = 2^αt and t is odd. Orthogonality of the rows of M_1(a) implies that α≥ 1. When α∈{1, 2}, the eigenvalues of M_1(a) are both primitive roots of unity of order 8t, and so k > ℓ. When α = 3 the eigenvalues have different orders, and in all cases, M_1(a)^4t = I_2, while when α≥ 4 we always have k ≤ℓ. 2: Suppose that λ_1 = ζ_3λ_2. Then the sum of the eigenvalues has modulus √(2), from which we conclude that a := m_11 = im_22. As before, we can solve for the off diagonal entries in terms of a single unknown, obtaining m_12 = ab and m_21 = -iab^∗. Thus under the assumption that M ∈(2, ℓ), we have that M = M_2. The eigenvalues of M_2 are λ_1 = √(2)ζ_24^7a and λ_2 = √(2)ζ_24^23a.For convenience, we write M_2(a) for the unitary matrix with 2^-1/2a and 2^-1/2ia on the diagonal, again up to similarity we can take b = 1.The group generated by M_2(a) is finite and cyclic when a is a root of unity, and contains a scalar subgroup of index 3 generated by (2^-3/2a^3(2-2i))I_2.Suppose now that a is a primitive ℓ root of unity where ℓ = 2^α3^βt, with (t, 6) = 1. For each choice of α, β, t, it is routine to compute the orders of the eigenvalues, though there are a large number of cases to consider. Since -ia is an entry of √(2)M_2(a), we can assume that α≥ 2. Next we will show that the eigenvalues of M_2(a) have distinct orders if and only if β = 1 or α = 3.Suppose that β = 1: then the eigenvalues of M_2(a) are ζ_24ℓ^24i + 7ℓ and ζ_24ℓ^24i + 23ℓ where i ≡ 1, 23 and ℓ≡ 3, 69 (since a is a primitive ℓ root). Suppose that i ≡ 13 and ℓ≡ 39, then 24i + 7 ℓ≡ 09 while 24 i + 23 ℓ≡ 39. But since 9 | 24 ℓ, the eigenvalues do not have the same order. The remaining three cases are similar; in no case do the eigenvalues of M have the same order. When a is a primitive 8 root, M_2(a) has order 3, 6 or 12 (depending on the choice of primitive root). When ℓ = 2^3t with (t, 6) = 1 we obtain matrices of orders 3t, 6t, 12t. And by a similar argument, when ℓ = 2^3 3^β t with β≥ 2 we obtain matrices of order 3^βt, 2 · 3^βt,4 · 3^βt.It remains to examine the cases α = 2 and β≠ 1, and α≥ 4 with β≠ 1. In the first case, when α = 2 and β = 0, we have that a = ζ_24^jζ_t for j ∈{6, 18}. So the eigenvalues of M(a) are ζ_24^j+7ζ_t and ζ_24^j -1ζ_t. For each choice of j, one finds that both exponents are coprime to 24, and hence the eigenvalues have multiplicative order 24t = 6ℓ. The remaining computations are all similar, in each case one finds that both eigenvalues are primitive roots of order (24, ℓ). Suppose that M = M_1 where a, b are chosen to be 2t roots of unity for odd t. Then the eigenvalues of M_1 are Y = {ζ_8a, ζ_8^7a} and √(2)^1-iM^i∈(2, 2t) whenever i is odd. Hence the pair (H, M) is sound whenever the entries of H lie in X = {ζ_8t^2i+1| i ∈ℕ}. We have recovered the generalisation of Turyn's morphism described in Theorem <ref>. When α≥ 3, then k ≤ℓ, and we never obtain an interesting morphism. When α = 2, we obtain a partial morphism with the same domain as when α = 1; but the image lies over a larger field. These are the only non-trivial morphisms obtained from M_1.We can also examine the morphisms obtained from M_2: an interesting case occurs when a = b = 1, denote this matrix by M_24. Its entries are fourth roots, and its eigenvalues are primitive 24 roots. It is easily verified that X = {ζ_24^3i+j| i ∈ℕ, j = 1, 2} and Y = {ζ_24^7, ζ_24^23}. The set X consists of all 24 roots which are not 8 roots. We obtain the obvious partial morphism (n, 24)→(n, 4). Since X contains an arithmetic progression of length 8, we have that the pair (ζ_24H, M_24) is sound for any H ∈(n, 8). Hence we obtain a complete morphism (n, 8) →(2n, 4) which does not seem to have appeared previously in the literature. §.§ Acknowledgements The first author has been fully supported by Croatian Science Foundation under the project 1637. The argument for Proposition <ref> was provided by Richard Stanley in response to a question posted on MathOverflow. abbrv | http://arxiv.org/abs/1707.08815v3 | {
"authors": [
"Ronan Egan",
"Padraig Ó Catháin"
],
"categories": [
"math.CO",
"05B20, 05B30, 05C50"
],
"primary_category": "math.CO",
"published": "20170727110118",
"title": "Morphisms of Butson classes"
} |
International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science, Tsukuba 305-0044, Japan [email protected] a system with Dirac cones, spatial modulation in material parameters induces a pseudo magnetic field, which acts like an external magnetic field. Here, we derive a concise formula to count the pseudo Landau levels in the simplest setup for having a pseudo magnetic field. The formula is so concise that it is helpful in seeing the essence of the phenomenon, and in considering the experimental design for the pseudo magnetic field. Furthermore, it is revealed that anisotropic Dirac cones are advantageous in pseudo Landau level formation in general. The proposed setup is relatively easy to be realized by spatial modulation in the chemical composition, and we perform an estimation of the pseudo magnetic field in an existing material (an antiperovskite material), by following the composition dependence with the help of the ab-initio method.Counting Pseudo Landau Levels in Spatially Modulated Dirac Systems Toshikaze Kariyado December 30, 2023 ================================================================== The external magnetic field is not the only source of the Landaulevels. For instance, a certain strain on graphene leads to theso-called pseudo magnetic field and the resultant Landaulevel structures<cit.>. Thisphenomenon is tied to the most intriguingproperty of graphene, i.e., the emergent relativistic electron, or theexistence of Dirac cone in the band structure<cit.>. Having a Dirac cone,the key toward the finite pseudo magnetic field is the resemblancebetween the shift of the Dirac cone in the Brillouin zone and theminimal coupling in the U(1) gauge theory. Since the essence is simplythe Dirac cone shift, the idea is not limited to graphene, but isapplicable to any system with emergent linear dispersion, such asthree-dimensional Dirac/Weyl semimetals <cit.>. The study of the pseudo magnetic field has several importantaspects. Obviously, it is conceptually interesting to observe magneticphenomena like chiral magnetic effect<cit.> or quantum oscillations<cit.> without actually applying magneticfield. Furthermore, the field strength can possibly exceed the maximumavailable strength for the real magnetic field. As an extreme case,even for a system inert to the real magnetic field (e.g. neutralparticle systems, photonic or phononic crystals<cit.>), the pseudo magnetic field canbe influential as far as there are Diraccones. Typically, Dirac cones come with pairs,resulting in multiple Diracnodes in the Brillouin zone, and the direction of the pseudo magneticfield depends on the nodes. (So, a pseudo magnetic field is regardedas an axial magnetic field.) Then, if the real and pseudo magneticfield coexist,they enhance or cancel with each other depending on the nodes, whichinduces the valley imbalance <cit.>. In that sense, the study of the pseudomagnetic field also has potential importance in valleytronics as nextfunctionalization of materials. As we have noted, the essence of the pseudo magnetic field is the similarity between the Dirac/Weyl node shift and the minimal coupling.The shift of Dirac/Weyl nodes can be induced in many ways <cit.>, and the strain has been frequently used in the context of the pseudo magnetic field in the literature. However, the strain is not the only choice: the spatial dependence of the magnetic moment <cit.>, or the spatially modulating chemical composition should be equally sufficient. In this paper, we consider a simple setup for the pseudo magnetic field generation, which is expected to be relatively easy to realize with the spatial modulation of the chemical composition. We first give a notably concise formula [Eq. (<ref>)] for the number of observable pseudo Landau levels. The formula requires only two dimensionless parameters N and R, where N characterizes the length scale of the spatial modulation, while R characterizes the size of the Dirac/Weyl node shift. The conciseness of the formula makes the essence of the pseudo magnetic field transparent, and helps to considerexperimental designs. Furthermore, inspired by the simple formula, it is pointed out that anisotropic Dirac cones are better than the isotropic ones to appreciate the pseudo Landau level structure.In the latter half of this paper, we perform semi ab-initio estimation of R in a real material, antiperovskite A_3SnO (A=Ca,Sr) <cit.>, to make a quantitative argument on the pseudo magnetic field generation.Let us start with the formulation. The setup in this paper is illustrated in Fig. <ref>(a). The considered system consists of three regions, bulk 1, bulk 2, and the buffer region. For having a transparent discussion, we assume that bulk 1 (bulk 2) extends to y=+∞ (y=-∞), which excludes free surfaces from our consideration. Bulk 1 and bulk 2 are similar to each other, having Dirac cones in the band structure, but with slightly different node positions. Namely, the effective model for each region is H^(±)_k=ħv(k±k_0)·σ,where k_0, which denotes the center of Dirac cones, takes different values for the two regions [see Fig. <ref>(a)]. In the buffer region, we assume that the Dirac cones are smoothly shifted from the position in bulk 1 to the one in bulk 2. For simplicity, our focus is limited to the case that the Dirac cones are shifted in k_x direction. As we can see from Fig. <ref>(a), the system is periodic in x and z direction, and we have a band structure as a function of k_x and k_z. (If a two-dimensional Dirac/Weyl system is our target, we simply neglect any structure along z-axis, and omit k_z.) Then, the bulk contribution to the band structure typically looks like Fig <ref>(b) reflecting the shifted node positions. The question is, how about the contribution from the state at the buffer region.Within the buffer region, the position of the Dirac cones depends on y. Then, in a naive treatment, the effective model is assumed to beH^(±)=ħv(-i∇±k_0(y))·σ,which is obtained by replacing k by -i∇ and taking account of the y dependence of k_0. The latter arguments reveal that this naive treatment works well. By comparing Eq. (<ref>) with the standard minimal coupling (-i∇-eA), we haveA^(±)=∓ħ/ek_0(y).Once A is given, the pseudo magnetic field is given by B=∇×A. In our setup, the node shift is only in k_x direction, and depends only on y, which means that∂ A_x/∂ y is the only relevant component. Assuming that k_0 linearly interpolates bulk 1 and bulk 2, the pseudo magnetic field strength B=|B^(±)| is estimated asB∼ħ/eΔ k/L,where L is the thickness of the buffer region, and Δ k is the size of the node shift [See Fig. <ref>(b)]. To make the formulation concise, we introduce two dimensionless parameters N and R respectively as L=Na and Δ k=2π R/a, where a is the lattice constant. (It is implicitly assumed that the lattice constants in x and y are the same, but the extension toanisotropic cases is trivial.) N represents the length scale of the spatial modulation, while R measures the size of the node shift in the unit of the Brillouin zone size. As a rough estimation, using a typical atomic scale a∼ 5 Å, Eq. (<ref>) leads toB ∼ 1.6× 10^4×R/N [T].That is, we potentially have 16 thousand Tesla, and the available strength is reduced by a factor of R/N.The obtained pseudo magnetic field induces the Landau levels. Plugging Eq. (<ref>) into a textbook formula, the energy of nth Landau level is E_n=±√(4π v^2ħ^2R|n|/Na^2).[For three-dimensional systems, Eq. (<ref>) corresponds to the energy at k_z=0.] However, we should note that not the all Landau levels are observable in the energy spectrum. That is, since we have the bulk regions as well as the buffer region, the Landau levels in the buffer region can be masked by the bulk contribution in the energy spectrum. It turns out that the diamond region with height Δ E and width Δ k in Fig. <ref>(b) is available for the Landau levels (see also the latter arguments on the numerical results in Fig. <ref>). Since Δ E is estimated as Δ E=ħ vΔ k, the condition that the nth Landau level falls into this diamond region becomes√(4π v^2ħ^2R|n|/Na^2)<ħ vΔ k/2,which leads to a concise expression|n|<π/4NR.Here we make a short summary: (i) to make the pseudo magnetic field strong, N should be small [Eq. (<ref>)], (ii) to observe the large number of Landau levels, N should be large [Eq. (<ref>)], and (iii) large R is always beneficial. Let us move on to the numerical validation of the derived formula. For this purpose, we introduce a two-dimensional square lattice tight-binding model with mobile Dirac nodes. Specifically, the Hamiltonian isH_k=[1+δ+2(cos k_x+cos k_y)]σ_z + 2αsin k_y σ_ywhere the lattice constant a is set to 1.By expanding the Hamiltonian with respect to k̃_x≡ k_x-2π/3 and k_y up to the first order in each of the parameters, we end up withH_k∼ -√(3)[ (k̃_x-δ̃)σ_z+α̃ k_yσ_y ],where δ̃=δ/√(3) and α̃=2α/√(3). That is, δ̃ behaves as A_x, and α̃ is essentially anisotropy of the Fermi velocity, v_y/v_x. Therefore, if we assign δ̃/2π=0.05 for bulk 1, δ̃/2π=-0.05 for bulk 2, and the linearly interpolated value for the buffer region, R=0.1 is achieved.In the actual calculation, we make the system periodic also in y direction not to have free edges. Namely, the system consists of the repetition of the chunk of bulk 1 – buffer – bulk 2 – buffer. Figures <ref>(a)–<ref>(d) summarizes the results for α̃=1. If there appears some flat sections in the band structure, they can be regarded as the Landau levels. (A flat section gives a peak in the density of state.) For N=0 [Fig. <ref>(a)], i.e., if the change between bulk 1 and bulk 2 is sharp, only the zeroth Landau level is identified in the band structure. For N=1 [Fig. <ref>(b)], which results in π/4NR∼ 0.8, we still only see the zeroth Landau level. If N is further increased to π/4NR∼ 2.4 [Fig. <ref>(c)], the n=1 Landau level becomes clearly visible, and we see a small signature of the n=2 Landau level as well. For N=50 leading to π/4NR=3.9 [Fig. <ref>(d)], the clear identification of the Landau levels up to n=3 is possible. All of these observations confirm the formula Eq. (<ref>).It is worth noting that the expected peak structure in the (local) density of state should be a key to experimental detection of the pseudo Landau levels. Any measurements capable of detecting the density of state, such as STM/STS as a direct measurement or optical conductivity, might be useful. So far, we have been treating the isotropic Dirac cone. Actually, the anisotropy of the Dirac cone gives significant influence on the observable Landau levels. If the Dirac cone becomes anisotropic, v^2 in the left hand side of Eq. (<ref>) is replaced by v_xv_y, and v in the right hand side by v_x. Consequently, the formula Eq. (<ref>) is rewritten as |n|<π/4v_x/v_yNR.This implies that if we have v_x>v_y, the number of observable Landau levels increases compared with the isotropic case with the same NR.Physically, this is because larger v_x means larger Δ E, and smaller v_y means larger density of states, both of which is advantageous to observe more Landau levels.The formula Eq. (<ref>) is again confirmed using the toy model by modifying α̃. Figures <ref>(e) and <ref>(f) shows that for v_x/v_y∼ 2 (α̃=2), the number of the Landau levels is doubled comparing with the isotropic case, while for v_x/v_y∼ 0.5 (α̃=0.5), only the n=0 and n=1 Landau levels are clearly seen. Therefore, if one attempts to observe large number of pseudo Landau levels, it is better to focus on the system with anisotropic Dirac cones. Hereafter, we work on the quantitative estimation of the pseudo magnetic field in existing materials. Having formulae Eqs. (<ref>) and (<ref>), the estimation of R is essential, and we derive R in semi ab-initio way. Here, semi ab-initio means that we apply the first-principles density functional theory <cit.> with a small assumption in the crystal structure to calculate the electronic band structure.Since the two-dimensional cases have been studied in graphene extensively, our focus is on the three-dimensional cases – we take the cubic antiperovskite family A_3SnO (A=Ca,Sr), where the three-dimensional Dirac cones are found on k_x, k_y, and k_z axes in the first-principles calculation. (Note that the experimental studies on this materials are now in progress <cit.>.) The great advantage of the antiperovskite family is that we already know the way to tune the electronic structure near the Fermi energy at the qualitative level <cit.>. Namely, it is natural to expect that the Dirac cone shift is realized by preparing Ca_3(1-x)Sr_3xSnO and adjusting x <cit.>. Before we proceed, we would like to point out a minor disadvantage of the antiperovskite family. Strictly speaking, there is a tiny mass gap at the Fermi energy, and therefore, we have to achieve R such that Δ E is sufficiently larger than the mass gap.Here, the electronic structure for 0<x<1 is obtained in the following way. Firstly, we apply the first-principles calculation for x=0 and x=1. (The computational details are in parallel with Ref. twin.) Using those results, we construct effective models for x=0 and x=1, and then, the model for arbitrary x is obtained by interpolating the parameters in the effective model. To be quantitative, the construction of the effective model is conducted using the maximally localized Wannier function method implemented in Wannier90 package <cit.>. In practice, we construct a 12 orbital model where 12 comes from (2 spins) × (3 p-orbitals on Sn atom + d_x^2-y^2/d_y^2-z^2/d_z^2-x^2 on 3 crystallographically equivalent Ca/Sr atoms) =2×(3+3) <cit.>. Since we intend to focus on the small variation around x=0.5, we fix the lattice constant as the average of the lattice constants for x=0 and x=1, which are experimentally known, throughout the calculation. The band structures along k_x axis for x=4/9 and x=5/9 obtained by the interpolation are shown in Fig. <ref>. The crossing points at the Fermi energy are the Dirac cones in this system. The inspection of the band structure reveals that the Dirac cone locates at k_x∼ 0.0975 for x=4/9, while k_x∼ 0.1037 for x=5/9, where the momentum is measured in the unit of 2π/a. At the end, R is estimated to be approximately 0.006.As a study complementary to the Wannier orbital based interpolation, we perform the calculation with a superstructure, containing Ca and Sr in a ratio x=4/9 and x=5/9. In specific, we consider the superstructure in z direction as shown in the right panels of Fig. <ref>. Since the unit cell is not enlarged in x-y plane, the node shift in k_x direction can be discussed in exactly the same footing as the previous paragraph (no Brillouin zone folding in k_x and k_y direction). Again, assuming that x=4/9 and x=5/9 are sufficiently close to x=0.5, we use the averaged lattice constant.The band structures for x=4/9 and x=5/9 on the k_x axis are shown in Fig. <ref>. From this result, we extract the node position for x=4/9 as k_x=0.092, while the position for x=5/9 as k_x=0.114, which leads to R=0.022. To summarize, we derive a compact formula to count the number of the observable Landau levels induced by the pseudo magnetic field.The formula is so simple that the essence of the pseudo magnetic field becomes evident. Having a concise formula is also beneficial in designing experiments on the pseudo Landau levels in real materials. In fact, we show an explicit estimation of the pseudo magnetic field in an antiperovskite Dirac material in an ab-initio manner. For future developments, it is pointed out that anisotropic Dirac cones are more advantageous than isotropic ones in appreciating the pseudo Landau level structure. The author thanks M. Ogata, H. Takagi and A. Vishwanath for stimulatingdiscussions. The computation in this work has been done using thefacilities of the Supercomputer Center, the Institute for Solid StatePhysics, the University of Tokyo. This work was supported by aGrant-in-Aid for Scientific Research No. 17K14358. 36 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Guinea et al.(2010)Guinea, Katsnelson, and Geim]Guinea:2010aa author author F. Guinea, author M. I. Katsnelson,and author A. K. Geim, http://dx.doi.org/10.1038/nphys1420 journal journal Nat Phys volume 6, pages 30 (year 2010)NoStop [Low and Guinea(2010)]doi:10.1021/nl1018063 author author T. Low and author F. Guinea,10.1021/nl1018063 journal journal Nano Lett. volume 10, pages 3551 (year 2010)NoStop [Levy et al.(2010)Levy, Burke, Meaker, Panlasigui, Zettl, Guinea, Neto, andCrommie]Levy544 author author N. Levy, author S. A. Burke, author K. L. Meaker, author M. Panlasigui, author A. Zettl, author F. Guinea, author A. H. C.Neto,and author M. F.Crommie, 10.1126/science.1191700 journal journal Science volume 329, pages 544 (year 2010)NoStop [Yeh et al.(2011)Yeh, Teague, Yeom, Standley, Wu, Boyd, and Bockrath]YEH20111649 author author N.-C. Yeh, author M.-L. Teague, author S. Yeom, author B. Standley, author R.-P. Wu, author D. Boyd,and author M. Bockrath, http://dx.doi.org/10.1016/j.susc.2011.03.025 journal journal Surface Science volume 605, pages 1649(year 2011)NoStop [Castro Neto et al.(2009)Castro Neto, Guinea, Peres, Novoselov, and Geim]RevModPhys.81.109 author author A. H. Castro Neto, author F. Guinea, author N. M. R. Peres, author K. S. Novoselov,and author A. K. Geim, 10.1103/RevModPhys.81.109 journal journal Rev. Mod. Phys. volume 81, pages 109 (year 2009)NoStop [Liu et al.(2013)Liu, Ye, and Qi]PhysRevB.87.235306 author author C.-X. Liu, author P. Ye,andauthor X.-L. Qi, 10.1103/PhysRevB.87.235306 journal journal Phys. Rev. B volume 87, pages 235306 (year 2013)NoStop [Chen et al.(2013)Chen, Wu, and Burkov]PhysRevB.88.125105 author author Y. Chen, author S. Wu,andauthor A. A. Burkov, 10.1103/PhysRevB.88.125105 journal journal Phys. Rev. B volume 88, pages 125105 (year 2013)NoStop [Chernodub et al.(2014)Chernodub, Cortijo, Grushin, Landsteiner, and Vozmediano]PhysRevB.89.081407 author author M. N. Chernodub, author A. Cortijo, author A. G. Grushin, author K. Landsteiner,and author M. A. H. Vozmediano, 10.1103/PhysRevB.89.081407 journal journal Phys. Rev. B volume 89, pages 081407 (year 2014)NoStop [Cortijo et al.(2015)Cortijo, Ferreirós, Landsteiner, andVozmediano]PhysRevLett.115.177202 author author A. Cortijo, author Y. Ferreirós, author K. Landsteiner,and author M. A. H.Vozmediano, 10.1103/PhysRevLett.115.177202 journal journal Phys. Rev. Lett. volume 115, pages 177202 (year 2015)NoStop [Pikulin et al.(2016)Pikulin, Chen, and Franz]PhysRevX.6.041021 author author D. I. Pikulin, author A. Chen, and author M. Franz, 10.1103/PhysRevX.6.041021 journal journal Phys. Rev. X volume 6, pages 041021 (year 2016)NoStop [Grushin et al.(2016)Grushin, Venderbos, Vishwanath, andIlan]PhysRevX.6.041046 author author A. G. Grushin, author J. W. F. Venderbos, author A. Vishwanath,and author R. Ilan, 10.1103/PhysRevX.6.041046 journal journal Phys. Rev. X volume 6,pages 041046 (year 2016)NoStop [Cortijo et al.(2016)Cortijo, Kharzeev, Landsteiner, andVozmediano]PhysRevB.94.241405 author author A. Cortijo, author D. Kharzeev, author K. Landsteiner,andauthor M. A. H. Vozmediano,10.1103/PhysRevB.94.241405 journal journal Phys. Rev. B volume 94, pages 241405 (year 2016)NoStop [Liu et al.(2017)Liu, Pikulin, and Franz]PhysRevB.95.041201 author author T. Liu, author D. I. Pikulin, and author M. Franz, 10.1103/PhysRevB.95.041201 journal journal Phys. Rev. B volume 95, pages 041201 (year 2017)NoStop [Gorbar et al.(2017)Gorbar, Miransky, Shovkovy, and Sukhachov]PhysRevB.95.115422 author author E. V. Gorbar, author V. A. Miransky, author I. A. Shovkovy,and author P. O. Sukhachov, 10.1103/PhysRevB.95.115422 journal journal Phys. Rev. B volume 95, pages 115422 (year 2017)NoStop [Tchoumakov et al.(2017)Tchoumakov, Civelli, and Goerbig]PhysRevB.95.125306 author author S. Tchoumakov, author M. Civelli,and author M. O. Goerbig,10.1103/PhysRevB.95.125306 journal journal Phys. Rev. B volume 95, pages 125306 (year 2017)NoStop [Brendel et al.(2017)Brendel, Peano, Painter, andMarquardt]Brendel25042017 author author C. Brendel, author V. Peano, author O. J. Painter,andauthor F. Marquardt, 10.1073/pnas.1615503114 journal journal Proc. Natl. Acad. Sci. USA volume 114, pages E3390 (year 2017)NoStop [Abbaszadeh et al.()Abbaszadeh, Souslov, Paulose, Schomerus, and Vitelli]arXiv:1610.06406 author author H. Abbaszadeh, author A. Souslov, author J. Paulose, author H. Schomerus,and author V. Vitelli, @noopjournal arXiv:1610.06406 NoStop [Chaves et al.(2010)Chaves, Covaci, Rakhimov, Farias,and Peeters]PhysRevB.82.205430 journal author author A. Chaves, author L. Covaci, author K. Y.Rakhimov, author G. A.Farias,and author F. M.Peeters, 10.1103/PhysRevB.82.205430 journal journal Phys. Rev. B volume 82, pages 205430 (year 2010)NoStop [Roy et al.(2013)Roy, Hu, and Yang]PhysRevB.87.121408 author author B. Roy, author Z.-X. Hu,andauthor K. Yang, 10.1103/PhysRevB.87.121408 journal journal Phys. Rev. B volume 87, pages 121408 (year 2013)NoStop [Settnes et al.()Settnes, García, and Roche]1705.09085 author author M. Settnes, author J. H. García,and author S. Roche, @noopjournal arXiv:1705.09085NoStop [Montambaux et al.(2009)Montambaux, Piéchon, Fuchs, andGoerbig]Montambaux2009 journal author author G. Montambaux, author F. Piéchon, author J.-N.Fuchs,and author M. O.Goerbig, 10.1140/epjb/e2009-00383-0 journal journal Eur. Phys. J. B volume 72, pages 509 (year 2009)NoStop [Kariyado and Hatsugai(2013)]PhysRevB.88.245126 author author T. Kariyado and author Y. Hatsugai, 10.1103/PhysRevB.88.245126 journal journal Phys. Rev. B volume 88, pages 245126 (year 2013)NoStop [Kariyado and Hatsugai(2015)]Kariyado:2015aa author author T. Kariyado and author Y. Hatsugai, http://dx.doi.org/10.1038/srep18107 journal journal Sci. Rep. volume 5, pages 18107 (year 2015)NoStop [Araki et al.(2016)Araki, Yoshida, and Nomura]PhysRevB.94.115312 author author Y. Araki, author A. Yoshida, and author K. Nomura, 10.1103/PhysRevB.94.115312 journal journal Phys. Rev. B volume 94, pages 115312 (year 2016)NoStop [Kariyado and Ogata(2011)]JPSJ.80.083704 author author T. Kariyado and author M. Ogata, 10.1143/JPSJ.80.083704 journal journal J. Phys. Soc. Jpn. volume 80, pages 083704 (year 2011)NoStop [Kariyado and Ogata(2012)]JPSJ.81.064701 author author T. Kariyado and author M. Ogata, 10.1143/JPSJ.81.064701 journal journal J. Phys. Soc. Jpn. volume 81, pages 064701 (year 2012)NoStop [Klintenberg et al.(2014)Klintenberg, Haraldsen, and Balatsky]klintenberg author author M. Klintenberg, author J. T. Haraldsen,and author A. V. Balatsky, @noopjournal journal Appl. Phys. Res. volume 6, pages 31 (year 2014)NoStop [Hsieh et al.(2014)Hsieh, Liu, and Fu]PhysRevB.90.081112 author author T. H. Hsieh, author J. Liu,andauthor L. Fu, 10.1103/PhysRevB.90.081112 journal journal Phys. Rev. B volume 90, pages 081112 (year 2014)NoStop [Giannozzi et al.(2009)Giannozzi, Baroni, Bonini, Calandra, Car, Cavazzoni, Ceresoli, Chiarotti, Cococcioni, Dabo, Dal Corso, de Gironcoli, Fabris, Fratesi, Gebauer, Gerstmann, Gougoussis, Kokalj, Lazzeri, Martin-Samos, Marzari, Mauri, Mazzarello, Paolini, Pasquarello, Paulatto, Sbraccia, Scandolo, Sclauzero, Seitsonen, Smogunov, Umari, and Wentzcovitch]QE-2009 author author P. Giannozzi, author S. Baroni, author N. Bonini, author M. Calandra, author R. Car, author C. Cavazzoni, author D. Ceresoli, author G. L.Chiarotti, author M. Cococcioni, author I. Dabo, author A. Dal Corso, author S. de Gironcoli, author S. Fabris, author G. Fratesi, author R. Gebauer, author U. Gerstmann, author C. Gougoussis, author A. Kokalj, author M. Lazzeri, author L. Martin-Samos, author N. Marzari, author F. Mauri, author R. Mazzarello, author S. Paolini, author A. Pasquarello, author L. Paulatto, author C. Sbraccia, author S. Scandolo, author G. Sclauzero, author A. P.Seitsonen, author A. Smogunov, author P. Umari, and author R. M. Wentzcovitch,http://www.quantum-espresso.org journal journal J. Phys.: Condens. Matter volume 21,pages 395502 (year 2009)NoStop [Lee et al.(2013)Lee, Wu, Kumar, Hunte, Schwartz, and Narayan]doi:10.1063/1.4820770 author author Y. F. Lee, author F. Wu, author R. Kumar, author F. Hunte, author J. Schwartz,and author J. Narayan, 10.1063/1.4820770 journal journal Appl. Phys. Lett. volume 103, pages 112101 (year 2013)NoStop [Nuss et al.(2015)Nuss, Mühle, Hayama, Abdolazimi, and Takagi]Nuss:dk5032 author author J. Nuss, author C. Mühle, author K. Hayama, author V. Abdolazimi,and author H. Takagi, 10.1107/S2052520615006150 journal journal Acta Cryst. B volume 71, pages 300 (year 2015)NoStop [Samal et al.(2016)Samal, Nakamura, and Takagi]doi:10.1063/1.4955213 author author D. Samal, author H. Nakamura, and author H. Takagi, 10.1063/1.4955213 journal journal APL Materials volume 4, pages 076101 (year 2016)NoStop [Okamoto et al.(2016)Okamoto, Sakamaki, and Takenaka]doi:10.1063/1.4952393 author author Y. Okamoto, author A. Sakamaki, and author K. Takenaka, 10.1063/1.4952393 journal journal J. Appl. Phys. volume 119, pages 205106 (year 2016)NoStop [Oudah et al.(2016)Oudah, Ikeda, Hausmann, Yonezawa, Fukumoto, Kobayashi, Sato,and Maeno]Oudah:2016aa author author M. Oudah, author A. Ikeda, author J. N. Hausmann, author S. Yonezawa, author T. Fukumoto, author S. Kobayashi, author M. Sato,and author Y. Maeno, http://dx.doi.org/10.1038/ncomms13617 journal journal Nature Commun. volume 7, pages 13617 EP(year 2016)NoStop [Kariyado and Ogata()]twin author author T. Kariyado and author M. Ogata, @noopjournal arXiv:1705.08934NoStop [Mostofi et al.(2008)Mostofi, Yates, Lee, Souza, Vanderbilt, and Marzari]MOSTOFI2008685 journal author author A. A. Mostofi, author J. R.Yates, author Y.-S. Lee, author I. Souza, author D. Vanderbilt,and author N. Marzari, http://dx.doi.org/10.1016/j.cpc.2007.11.016 journal journal Comp. Phys. Commun. volume 178,pages 685(year 2008)NoStop | http://arxiv.org/abs/1707.08601v1 | {
"authors": [
"Toshikaze Kariyado"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726183009",
"title": "Counting Pseudo Landau Levels in Spatially Modulated Dirac Systems"
} |
[email protected]@icfo.es ^1Consiglio Nazionale delle Ricerche, CNR-SPIN, Via Vetoio 10, 67100 L'Aquila, Italy ^2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain ^3Department of Industrial and Information Engineering and Economics, Via G. Gronchi 18, University of L'Aquila, I-67100 L'Aquila, Italy ^4Institute for Complex Systems (ISC-CNR), Via dei Taurini 19, 00185, Rome, Italy ^5Department of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185, Rome, ItalyOptical parametric oscillators are widely-used pulsed and continuous-wave tunable sources for innumerable applications, as in quantum technologies, imaging and biophysics. A key drawback is material dispersion imposing the phase-matching condition that generally entails a complex setup design, thus hindering tunability and miniaturization. Here we show that the burden of phase-matching is surprisingly absent in parametric micro-resonators adopting monolayer transition-metal dichalcogenides as quadratic nonlinear materials. By the exact solution of nonlinear Maxwell equations and first-principle calculation of the semiconductor nonlinear response, we devise a novel kind of phase-matching-free miniaturized parametric oscillator operating at conventional pump intensities. We find that different two-dimensional semiconductors yield degenerate and non-degenerate emission at various spectral regions thanks todoubly-resonant mode excitation, which can be tuned through the incidence angle of the external pump laser. In addition we show that high-frequency electrical modulation can be achieved by doping through electrical gating that efficiently shifts the parametric oscillation threshold. Our results pave the way for new ultra-fast tunable micron-sized sources of entangled photons, a key device underpinning any quantum protocol. Highly-miniaturized optical parametric oscillators may also be employed in lab-on-chip technologies for biophysics, environmental pollution detection and security. Phase-matching-free parametric oscillators based on two dimensional semiconductors C. Conti^4,5 December 30, 2023 ==================================================================================KEYWORDS: two-dimensional materials, nonlinear response, parametric oscillators, sensors, quantum sources, entangled photons, micro-resonators.§ INTRODUCTION Optical nonlinearity in photonic materials enables an enormous amount of applications such as frequency conversion <cit.>, all-optical signal processing <cit.>, and non-classical sources <cit.>. Parametric down-conversion (PDC) furnishes tunable sources of coherent radiation <cit.> and generators of entangled photons and squeezed states of light <cit.>. In traditional configurations, a nonlinear crystal with broken centrosymmetry and second-order nonlinearity sustains PDC <cit.>; more recently, effective PDC was reported in centrosymmetric crystals with third-order nonlinearity <cit.> and semiconductor microcavities <cit.>.Since three-wave parametric coupling is intrinsically weak, one can achieve low oscillation thresholds only by doubly or triply resonant optical cavities. In addition, parametric effects are severely hampered by the destructive interference among the three waves propagating with different wavenumbers k_1,2,3 in the dispersive nonlinear medium because the momentum mismatch Δ k = k_3-k_2-k_1 does not generally vanish (see Fig.1a). To avoid this highly detrimental effect, the use of phase-matching (PM) strategies is imperative. The commonly-adopted birefringence-PM method <cit.> is critically sensible to the nonlinear medium orientation. Quasi-PM <cit.> exploits the momentum due to a manufactured long-scale periodic reversal of the sign of the nonlinear susceptibility and cannot be easily applied in miniaturized system. In semiconductors, PM is achieved by the S-shaped energy-momentum polariton dispersion in the strong coupling of excitons and photons <cit.>, only accessible at low temperatures and large pump angles. Cavity PM <cit.>, also denoted “relaxed” PM <cit.>, occurs in Fabry-Perot microcavities with cavity length ℓ shorter than the coherence length π / Δ k; this technique drastically reduces the effective quadratic susceptibility χ_ eff^(2) (see Fig.1a). Any of the above mentioned PM techniques entails a non-trivial setup design that is further constrained by the need of resonance operation.In this manuscript, we show that emerging two-dimensional (2D) materials with high quadratic nonlinearity open unprecedented possibilities for tunable parametric micro-sources. Very remarkably, when illuminated with different visible and infrared waves, these novel 2D materials provide a negligible dispersive dephasing owing to their atomic-scale thickness (i.e ℓ≃ 0, see Fig.1b). Due to the lack of destructive interference, 2D materials support PDC without any need of satisfying a PM condition. Furthermore, these “phase-matching-free” devices turn out to be very versatile and compact, with the additional tunability offered by electrical gating of 2D materials, which provides ultrafast electrical-modulation functionality.The most famous 2D material, graphene, is not the best candidate for PDC owing to the centrosymmetric structure. In principle, a static external field may break centrosymmetry and induce a χ_ eff^(2), but the spectrally-flat absorption of graphene is severely detrimental for PDC. Recent years have witnessed the rise of transition metal dichalcogenides (TMDs) as promising photonic 2D materials. TMDs possess several unusual optical properties dependent on the number of layers. Bulk TMDs are semiconductors with an indirect bandgap, but the optical properties of their monolayer (ML) counterpart are characterized by a direct bandgap ranging from ∼ 1.55 eV to ∼ 1.9 eV <cit.> that is beneficial for several optoelectronic applications <cit.>. In addition, ML-TMDs have broken centrosymmetry and thus undergo second-order nonlinear processes <cit.>. Here we study PDC in micro-cavities embedding ML-TMDs; we find that the cavity design is extremely flexible if compared to standard parametric oscillators thanks to their phase-matching-free operation (see Figs.1a,1b). We demonstrate that, at conventional infrared pump intensity, parametric oscillation occurs in wavelength-sized micro-cavities with ML-TMDs. We show that the output signal and idler frequencies can be engineers thanks to the mode selectivity of doubly-resonant cavities; these frequencies are tuned by the pump incidence angle and modulated electrically by an external gate voltage. § RESULTSTwo hexagonal lattices of chalcogen atoms embedding a plane of metal atoms arranged at trigonal prismatic sites between the chalcogen neighbors form the structure of ML-TDMs. <cit.> Figure 1c shows the lattice structure of MX_2 ML-TMDs (M = Mo, W, and X = S, Se), and Figs.2a and 2b report the valence and conduction bands of MoS_2 obtained from tight-binding calculations <cit.>. The electronic band structure of other MX_2 materials considered is qualitatively similar. The direct bandgap is about 1.5 eV and implies transparency for infrared radiation; the linear surface conductivity has very small real part (corresponding to absorption) and higher imaginary part at infrared wavelengths. Figure 2c shows the wavelength dependence of the linear surface conductivities of MX_2. In the presence of an external pump field with angular frequency ω_3, the ML-TMD second-order nonlinear processes lead to down-converted signal and idler waves with angular frequencies ω_1 and ω_2, such that ω_3 =ω_1+ω_2. Figure 2e illustrates the PDC mixing surface conductivities for MoS_2. Both linear and nonlinear conductivities are calculated by a perturbative expansion of the tight-binding Hamiltonian of MX_2 [see Methods and Supplementary Information (SI)]. For infrared photons with energy smaller than the bandgap, extrinsic doping by an externally applied gate voltage (see Fig.1c) modifies the optical properties and leads to an increase of absorption due to free-carrier collisions and to smaller PDC mixing conductivities. Figures 2d and 2f show the dependence of linear and nonlinear surface conductivities on the Fermi level E_ F. As detailed below, extrinsic doping generally leads to a decrease of PDC efficiency.Figure 1b shows the parametric oscillator design with ML-TMDs. The cavity consists of a dielectric slab (thickness L) surrounded by two Bragg grating mirrors (BGs); the ML-TMD is placed on the left BG inside the cavity. The cavity is illuminated from the left by an incident (i) pump field (frequency ω_3) and the oscillator produces both reflected (r) and transmitted (t) signal and idler fields with frequencies ω_1 = (ω_3 + Δω)/2 and ω_2 = (ω_3 - Δω)/2, where Δω is the beat-note frequency of the parametric oscillation (PO).As detailed in Methods, the cavity equations for the fields do not contain the momentum mismatch Δ k. Indeed, due to their atomic thickness, ML-TMDs are not optically characterized by a refractive index but rather by a surface conductivity. Hence, the parametric coupling produced by the quadratic surface current of ML-TMDs is not hampered by dispersion and no PM condition is required accordingly. In order to observe signal and idler generation, only the PO condition is required along with the signal resonance (SR) and idler resonance (IR) conditions leading to a dramatic reduction of the intensity threshold (see Methods). Since there is no PM requirement, such requirements can be met by adjusting either the cavity length L or the pump incidence angle θ as tuning parameters. For SR and IR, one needs highly reflective mirrors for both signal and idler (see Methods), as obtained by locating stop band of the micron-sized BGs at the half of the pump frequency ω_3/2 <cit.>. Figure 3 shows the PO analysis for a cavity composed of two BGs with polymethyl methacrylate (PMMA) and MoS_2 deposited on the left mirror. The infrared pump has wavelength λ_3 = 780 nm in the spectral region where the nonlinear properties of MoS_2 are very pronounced (see Fig.2e). The BGs are tuned with their stop bands centered at 1560 nm (=2 λ_3) <cit.>. In Fig.3a we consider the case of normal incidence θ = 0 and we plot the PO (black), SR (red), and IR (green) curves in the (L/λ_3,Δω / ω_3)plane. Doubly resonant POs (DRPOs) corresponding to the intersection points of these three curves <cit.> are labeled by dashed circles. Therefore, at pump normal incidence, degenerate (Δω =0) and non-degenerate (Δω≠ 0) DRPOs exist at specific cavity lengths. Note that such oscillations also occur for sub-wavelength cavity lengths (L < λ_3). Each oscillation starts when the incident pump intensity I_3^(i) is larger than a threshold I_3Th^(i) (see Methods) <cit.>. Figures 3a2 and 3a3 show the threshold for two specific degenerate and non-denegenerate DRPOs.Panels a2.1 and a3.1 of Fig.3 report the thresholds (black curves on the shadowed vertical planes) corresponding to the PO (black) curves; one can observe that the minimum thresholds occur at SR and IR (identified by the intersection between red and green curves). The minimum intensity thresholds are of the order of GW/cm^2 and the non-degenerate DRPO threshold is greater than the degenerate DRPO one because the reflectivity of the Bragg mirror is maximum at Δω =0 (i.e. at half the pump frequency, as discussed above). In panels a2.2, a2.3 and a2.4 of Fig. 3 (and, seemingly, panels a3.2, a3.3 and a3.4) we report the basic DRPO features by plotting the intensities I_1^(t),I_2^(t),I_3^(t) of the transmitted signal, idler and pump fields as functions of the scaled cavity length L/λ_3 and the incident pump intensity. Note that, in the considered example, the range of L/λ_3 where the oscillation actually occurs is rather narrow owing to the adopted BGs high reflectivity.We emphasize that tuning of the PO may be realized by the pump incidence angle θ, which negligibly affects the oscillation thresholds. In Fig.3b and 3c, we analyze the DRPOs by using θ as tuning parameter for a given cavity length. In particular, in Fig.3b we consider a cavity with fixed length as in Fig.3a2. The PO, SR and IR curves of Fig.3b1 intersect at a degenerate DRPO point at θ≃ 6 deg.In Fig.3b2 we plot the transmitted signal intensity I_1^(t) as a function of the pump incidence angle and intensity I_3^(i); one can observe that the intensity threshold is comparable to the case in Fig.3a2 and the range of angles θ where PO occurs is of the order of a hundredth of degree and experimentally feasible. We show similar results in Figs.3c1 and 3c2, where the non-degenerate DRPO of Fig.3a3 is investigated in a cavity with slightly different length, and achieved at a finite incident angle with unchanged note-beat frequency Δω. A more accurate analysis of Fig.3c1 also reveals that, for a given L, the cavity sustains multiple DRPOs (both degenerate and non-degenerate) at different incidence angles θ. In Fig.3c3 we plot the transmitted intensity of a degenerate DRPO that grows with the pump intensity above the ignition threshold.The novel PO with ML-TMDs as nonlinear media are PM-free because of the atomic size of ML-TMDs. The reported several examples of POs with MoS_2 can be also designed by other families of ML-TMDs leading to qualitatively similar results. In the Supplementary Material, we compare the calculated dependence of the pump intensity threshold as function of wavelength λ_3 for parametric oscillators embedding MoS_2, WS_2 and MoSe_2, WSe_2; we find that the chosen material affects the minimal threshold intensity in a given spectral range. One can optimize the choice of the material for a desired spectral content and threshold level.A further degree of freedom offered by ML-TMDs lies in the electrical tunability through an external gate voltage, as depicted in Fig.1c. The gate voltage increases the Fermi level, and hence affects nonlinearity and absorption because of the electron-electron collision in the conduction band (see Figs.2d,f). Although electrical tunability of MX_2 has not been hitherto experimentally demonstrated, to the best of our knowledge, we emphasize that such a further degree of freedom is absent in traditional parametric oscillators. In the Supplementary Material, we report the pump intensity threshold as a function of the Fermi level of MoS_2, and we show that the threshold may increase by one order of magnitude. The external gate voltage can switch-off PO at fixed optical pump, and fast electrical modulation of the output signal and idler fields can be achieved.§ CONCLUSIONS POs can be excited in micron-sized cavities embedding ML-TMDs as nonlinear media at conventional pump intensities in a PM-free regime. The cavity design remains inherently free of the complexity imposed by the need for PM and may result into doubly resonant PDC of signal and idler waves. The flexibility offered by such a novel oscillator design enables the engineering of selective degenerate or non-degenerate down-converted excitations by simply modifying the incident angle of the pump field. Furthermore, the electrical tunability of ML-TMDs can modulate fast output signal and idler waves by bringing POs below threshold. Based on our calculations, we envisage that novel parametric oscillators embedding ML-TMDs are a new technology for all the applications in which highly-miniaturized tunable source are relevant, including enviromental detection, security, biophyics, imaging and spectroscopy. PM-free ML-TMD microresonators may also potentially boost the realization of micrometric sources of entangled photons when pumped slightly below threshold, thus paving the way for the development of integrated quantum processors. AM acknowledges useful discussions with F. Javier García de Abajo, Emanuele Distante and Ugo Marzolino. AC and CR thank the U.S. Army International Technology Center Atlantic for financial support (Grant No. W911NF-14-1-0315). CC acknowledges funding from the Templeton foundation (Grant number 58277 ) andPRIN NEMO (reference 2015KEZNYM). Author Contributions A.C. and A.M. conceived the idea and worked out the theory. All the authors discussed the results and wrote the paper. § METHODS Parametric down-conversion of MX_2. We calculate the linear and PDC mixing surface conductivities of MX_2 starting from the tight-binding (TB) Hamiltonian of the electronic band structure <cit.>. Since the properties of infrared photons with energies smaller than the bandgap are determined by small electron momenta around the K and K' valleys, we approximate the full TB Hamiltonian as a sum of k· p Hamiltonians of first and second order H_0( k,τ,s) <cit.>, where k is the electron wavenumber and τ and s are the valley and spin indexes, respectively. We then derive the light-driven electron dynamics through the minimal coupling prescription leading to the time-dependent Hamiltonian H_0[ k+(e/ħ) A(t),τ,s], where -e is the electron charge, ħ is the reduced Planck constant, and A(t) is the radiation potential vector, and we obtain Bloch equations for the interband coherence and the population inversion. Finally, we solve perturbatively the Bloch equations of ML-TMDs in the weak excitation limit, obtaining the surface current density K(t) after integration over the reciprocal spaceK(t) = Re{∑_j = 1^3[ σ̂^ L (ω_j)E_je^-iω_j t] + σ̂^(1,2) E_1E_2e^-iω_3 t + ..+ σ̂^(1,3) E_1^*E_3e^-iω_2 t + σ̂^(2,3) E_2^*E_3e^-iω_1 t},where σ̂^ L(ω_j) (j=1,2,3) and σ̂^(l,m) (l,m=1,2,3) are the linear and PDC surface conductivity tensors, respectively. Note that our approach is based on the independent-electron approximation and is fully justified only for infrared photons far from exciton resonances occurring at photon energies higher than 1.5 eV <cit.>. Parametric oscillations. The signal, idler and pump fields, labelled with subscripts 1,2,3 respectively, have frequencies ω_n satisfying ω_1+ω_2 = ω_3. By the Transfer Matrix approach, the full electromagnetic analysis of the cavity (see Supplementary Material) yields the equations Δ _1 Q_1+ σ̃_23 Q_2^* Q_3 =0,Δ _2 Q_2+ σ̃_13 Q_1^* Q_3 =0,Δ _3 Q_3+ σ̃_12 Q_1 Q_2 =P_3,where Q_1,Q_2,Q_3 are complex amplitudes proportional to the output fields produced by the pump field which is proportional to the amplitude P_3. Here σ̃_nm are scaled quadratic conductivities of the MX_2 ML-TMD and Δ _n = σ̃_n- c /ω_n q_n( r_n^( R )- 1/r_n^( R )+ 1 + r_n^( R ) e^iq_n L- e^ - iq_n L/r_n^( R ) e^iq_n L+ e^ - iq_n L). are parameters characterizing the linear cavity where σ̃_n are scaled linear surface conductivities, q_n= ω_n/c√(ε( ω _n ) - sin ^2 θ) are the longitudinal wavenumbers inside the dielectric slab, ϵ(ω) is the relative permittivity of the dielectric slab, θ is the pump incidence angle whereas r_n^( R ) are the complex reflectivities for right illumination of the left Bragg mirror (with vacuum and the dielectric slab on its left and right sides, respectively). It is worth stressing that the phase-mismatch Δ k = k_3 - k_1 - k_2 does not appear in the basic cavity equations (<ref>). Hence parametric coupling is here not affected by the fields destructive interference and the phase-matching constraint is strictly avoided. Paramateric oscillations (POs) are solutions of Eqs.(<ref>) with Q_1 ≠ 0 and Q_2 ≠ 0 and in this case the compatibility of the first two equations yields (see Supplementary Material) | P_3 | ^2 ≥Δ _1 Δ _2^* /σ̃_23σ̃_13^* | Δ _3 |^2, which is the leading PO condition. As the right hand side of Eq. (<ref>) is generally a complex number, for the PO we have the condition arg(Δ _1/σ̃_23) =arg(Δ _2/σ̃_13). Eq. (<ref>) can be physically interpreted as a locking of the phase difference Q_1 -Q_2^* allowing the signal and idler to oscillate. Once Eq.(<ref>) is satisfied, Eq.(<ref>) provides the pump threshold for the onset of PO. Due to the absolute smallness of the nonlinear surface conductivities, in order to have a feasible threshold, the cavity parameters |Δ_n| must be minimized. This can be obtained by choosing the doubly resonant condition for signal and idler corresponding to the minima of |Δ_1| and |Δ_2|, respectively. In order for this minima to be very small, we need that|r_1^( R )| and |r_2^( R )| are very close to one. One can satisfy such a constraint by a suitable Bragg mirror design to have the stop-band centered at the half of the pump frequency ω_3/2 since, in this case, signal and idler fields experience large mirror reflectance. Here we provide additional information on technical aspects of the theoretical methods used to model parametric down-conversion and the resulting phase-matching free resonant oscillations within the micro-cavities described in the main paper.§ PARAMETRIC DOWN-CONVERSION OF MX_2 We calculate the linear and parametric down-conversion (PDC) mixing surface conductivities of monolayer (ML) transition metal dichalcogenides (TMDs) MX_2 (M = Mo, S, and X = S, Se) starting from a tight-binding (TB) description of the electronic band structure of these materials <cit.> and studying the light-driven electron dynamics by means of Bloch equations for the valence and conduction bands. Note that our approach is based on the independent-electron approximation and is fully justified at the frequencies considered in the main paper since they are far from exciton resonances happening at photon energies of ≃ 1.5 eV or higher <cit.>. In addition, for the infrared photon energies considered, the relevant valence and conduction band regions affecting the infrared response are the ones closer to the band gap around the K and K' band edges, for which the full TB Hamiltonian can be approximated by a two-band k· p Hamiltonian H_0( k,τ,s) =H_1( k,τ,s) +H_2( k,τ) <cit.>, whereH_1( k,τ,s) = [ [ Δ/2 t_0 a (τ k_x - i k_y); t_0 a (τ k_x + i k_y) τ s Λ - Δ/2 ]], H_2( k,τ,s) = [ [ γ_1a^2k^2 γ_3a^2(τ k_x + i k_y)^2; γ_3a^2(τ k_x - i k_y)^2 γ_2a^2k^2 ]],and τ,s=± 1 label non-degenerate valleys and spins, while k = (k_x k_y) indicates the electron wave-vector.The physical parameters of H_0( k,τ,s) are obtained by fitting the k· p valence and conduction energy bands with the ones obtained from first-principles GW simulations <cit.> accounting for both non-degenerate valleys and spin-orbit coupling, and are listed in Table S1. We calculate the linear and PDC mixing conductivities of ML-TMDs by introducing the time-dependent Hamiltonian H_0((t),τ,s), where we have replaced the electron wave-vector with the minimum coupling prescription for the electron quasi-momentum ħ(t) = ħ k + eA(t), where e is the electron charge and A(t) is the electromagnetic potential vector accounting for pump, signal, and idler waves. With this prescription, we define unperturbed and interacting Hamiltonians H_0( k,τ,s) and H_ I( k,τ,s,t), respectively, and write the total Hamiltonian as H_ T( k,τ,s,t) =H_0[(t),τ,s] =H_0( k,τ,s) +H_ I( k,τ,s,t), whereH_ I( k,τ,s,t) =e/ħ [D_x A_x(t) + D_y A_y(t)] + e^2/ħ^2 [D_xx A_x^2(t) + D_xy A_x(t)A_y(t) + D_yy A_y^2(t)],and the interaction operators are explicitly given byD_x = t_0aτ[ |ψ_ V⟩⟨ψ_ C| + |ψ_ C⟩⟨ψ_ V| ], D_y = i t_0 a[ |ψ_ V⟩⟨ψ_ C| - |ψ_ C⟩⟨ψ_ V| ], D_xx= γ_1a^2 |ψ_ C⟩⟨ψ_ C| + γ_2a^2 |ψ_ V⟩⟨ψ_ V| + γ_3a^2 [ |ψ_ V⟩⟨ψ_ C| + |ψ_ C⟩⟨ψ_ V| ], D_yy= γ_1a^2 |ψ_ C⟩⟨ψ_ C| + γ_2a^2 |ψ_ V⟩⟨ψ_ V| - γ_3a^2 [ |ψ_ V⟩⟨ψ_ C| + |ψ_ C⟩⟨ψ_ V| ], D_xy= 2iγ_3a^2τ[ |ψ_ C⟩⟨ψ_ V| - |ψ_ V⟩⟨ψ_ C| ].In the expressions above we use the Dirac notation for the conduction |ψ_ C⟩ and valence |ψ_ V⟩ band eigenstates, and we approximate the matrix elements by their values at the band edges (k = 0). Inserting the Ansatz |ψ⟩ = c_-|ψ_ V⟩ + c_+ |ψ_ C⟩ in the time-dependent Schrödinger equation iħ∂_t |ψ⟩ =H_ T |ψ⟩, and defining the inversion population n_ k=|c_+|^2-|c_-|^2 and the interband coherence ρ_ k=c_+c_-^*, one getsρ̇_ k= -i/ħ (E_ C - E_ V)ρ_ k - γρ_ k + ie/ħ^2 n_ k{ D_x^ CV A_x(t) + D_y^ CV A_y(t) + e/ħ [ D_xx^ CV A_x^2(t) + D_xy^ CV A_x(t)A_y(t) + D_yy^ CV A_y^2(t) ] } ++ie^2/ħ^3[ (D_xx^ VV - D_xx^ CC)A_x^2(t) + (D_yy^ VV - D_yy^ CC)A_y^2(t) ] ρ_ k,ṅ_ k= - 4e/ħ^2 Im{ρ_ k[ D_x^ VCA_x(t) + D_y^ VCA_y(t) + e/ħ [ D_xx^ VCA_x^2(t) + D_xy^ VCA_x(t)A_y(t) + D_yy^ VCA_y^2(t)] ]},where E_ C ( k) and E_ V ( k) are the conduction and valence energy bands of the unperturbed Hamiltonian H_0, D^ CV_j=⟨ψ_ C | D_j | ψ_ V⟩ are the interaction matrix elements, and we have introduced a phenomenological relaxation rate γ = 10 ps^-1 accounting for coherence dephasing <cit.>. In order to obtain the PDC surface conductivities, we consider a coherent superposition of three monochromatic fields E(t) =Re{ E_1 e^-iω_1 t+ E_2 e^-iω_2 t+ E_3 e^-iω_3 t} with amplitudes E_1, E_2, and E_3, and with angular frequencies ω_1, ω_2, and ω_3, respectively. In our notation, the field E_3 indicates the external pump field, while E_1, E_2 label the down-converted signal and idler fields, respectively. The down-converted angular frequencies ω_1 and ω_2 are not independent, but are such that ω_1+ω_2=ω_3 owing to energy conservation. The electromagnetic potential vector related to the coherent superposition of pump, signal, and idler waves is thus given byA(t) =Re{( E_1/iω_1) e^-iω_1 t+( E_2/iω_2) e^-iω_2 t+( E_3/iω_3) e^-iω_3 t}.We then solve perturbatively the equations above in the vanishing temperature T→ 0 and weak excitation limits such that n_ k≈ - Θ[ E_ C( k) - E_ F], where Θ(x) indicates the Heaviside step function and E_ F is the Fermi energy. Taking the Ansatz ρ_ k = ∑_j = ±1,±2,±3ρ_|j|^(j/|j|) e^i(j/|j|)ω_|j| t and disregarding generation of higher harmonics we obtain analytical expressions for the coefficients ρ_|j|^(j/|j|), finding that the macroscopic surface current density given byK(t) = - e/4π^2ħ∑_τ,s=-1,1∫_-∞^+∞ dk_x ∫_-∞^+∞ dk_y [ ⟨ψ(t)| ∇_ k H_ T (t) | ψ (t) ⟩ - ⟨ψ_ V| ∇_ k H_ T (t) | ψ_ V⟩] == - e/2π^2ħ∑_τ,s=-1,1∫_-∞^+∞ dk_x ∫_-∞^+∞ dk_yRe{ρ_ k (t)[ ∇_ k H_0^ VC + ∇_ k D_x^ VCe/ħ A_x(t) + ∇_ k D_y^ VCe/ħ A_y(t) + . .. . + ∇_ k D_xx^ VCe^2/ħ^2 A_x^2(t) + ∇_ k D_xy^ VCe^2/ħ^2 A_x(t)A_y(t) + ∇_ k D_yy^ VCe^2/ħ^2 A_y^2(t) ] + .. + Θ[ E_ F - E_ C( k) ] [D_xx^ CCe/ħA_x(t)+D_yy^ CCe/ħA_y(t)]},can be recast intoK(t) =Re{∑_j = 1^3[ σ̂^ L (ω_j)E_je^-iω_j t] + σ̂^(1,2) E_1E_2e^-iω_3 t + σ̂^(1,3) E_1^*E_3e^-iω_2 t + σ̂^(2,3) E_2^*E_3e^-iω_1 t},where σ̂^ L(ω_j) (j=1,2,3) and σ̂^(l,m) (l,m=1,2,3) are the linear and PDC surface conductivity tensors, respectively, and we have neglected again generation of higher harmonics. Since centrosymmetry is broken along the y-direction in the notation used, the relevant components of the surface conductivity tensors for PDC are the ones such that pump, signal, and idler fields are polarized along the y-direction, for whichσ^ L_yy (ω) = ie^2D_yy^ CC/2π^2ħ^2(ω+iγ)∑_τ,s = ± 1∫_-∞^+∞ d k_x ∫_-∞^+∞ d k_y Θ[ E_ F - E_ C( k) ] + + e^2/4iπ^2ħ^2ω∑_τ,s = ± 1∫_-∞^+∞ d k_x ∫_-∞^+∞ d k_y Θ[ E_ C( k) - E_ F] {|D_y^ CV|^2/[(E_ C - E_ V) - ħ (ω + iγ)] + |D_y^ CV|^2/[(E_ C - E_ V) + ħ (ω + iγ)]},σ^ (1,2)_yyy (ω_1,ω_2,ω_3) = -e^3/4π^2ħ^3ω_1ω_2∑_τ,s = ± 1∑_j = 1^3 ∫_-∞^+∞ d k_x ∫_-∞^+∞ d k_y Θ[ E_ C( k) - E_ F] {D_y^ CVD_yy^ VC/[(E_ C - E_ V) - ħ (ω_j + iγ)] + .. + D_y^ VCD_yy^ CV/[(E_ C - E_ V) + ħ (ω_j + iγ)]},σ^ (1,3)_yyy (ω_1,ω_2,ω_3) = e^3/4π^2ħ^3ω_1ω_3∑_τ,s = ± 1∫_-∞^+∞ d k_x ∫_-∞^+∞ d k_y Θ[ E_ C( k) - E_ F] {D_y^ CVD_yy^ VC/[(E_ C - E_ V) + ħ (ω_1 - iγ)] + .. + D_y^ VCD_yy^ CV/[(E_ C - E_ V) - ħ (ω_1 - iγ)] + D_y^ CVD_yy^ VC/[(E_ C - E_ V) + ħ (ω_2 + iγ)] + D_y^ VCD_yy^ CV/[(E_ C - E_ V) - ħ (ω_2 + iγ)] + .. + D_y^ CVD_yy^ VC/[(E_ C - E_ V) - ħ (ω_3 + iγ)] + D_y^ VCD_yy^ CV/[(E_ C - E_ V) + ħ (ω_3 + iγ)]},σ^ (2,3)_yyy (ω_1,ω_2,ω_3) = e^3/4π^2ħ^3ω_2ω_3∑_τ,s = ± 1∫_-∞^+∞ d k_x ∫_-∞^+∞ d k_y Θ[ E_ C( k) - E_ F] {D_y^ CVD_yy^ VC/[(E_ C - E_ V) + ħ (ω_1 + iγ)] + .. + D_y^ VCD_yy^ CV/[(E_ C - E_ V) - ħ (ω_1 + iγ)] + D_y^ CVD_yy^ VC/[(E_ C - E_ V) + ħ (ω_2 - iγ)] + D_y^ VCD_yy^ CV/[(E_ C - E_ V) - ħ (ω_2 - iγ)] + D_y^ CVD_yy^ VC/[(E_ C - E_ V) - ħ (ω_3 + iγ)] + .. + D_y^ VCD_yy^ CV/[(E_ C - E_ V) + ħ (ω_3 + iγ)]}.Data reported in the main paper are obtained through the expressions above. In what follows, for convenience we will assume the simplified notation σ_n = σ^ L_yy(ω_n) and σ_nm = σ^(n,m)_yyy(ω_1,ω_2,ω_3) since the pump, signal, and idler electric fields are polarized in the y-direction for maximizing PDC within the micro-cavity.§ EQUATIONS FOR THE OUTPUT FIELDS In Fig.<ref> we sketch the geometry of the parametric oscillator (PO) considered in our calculations. A dielectric (PMMA) slab of thickness L with a MX_2 monolayer lying on its left side (at z=0) is placed between two Bragg mirrors of thickness d (for convenience we choose the right mirror to be the reflected z → -z copy of the left one). The left side of the cavity is illuminated with an incident (i) pump field which is a monochromatic Transverse Electric (TE) plane wave of frequency ω_3 with incidence angle θ. In addition to the reflected (r) and transmitted (t) pump fields, due to PDC, the cavity also produces (r) and (t) TE plane waves at the frequencies ω_1 (signal) and ω_2 (idler) such that ω_3 = ω_1 + ω_2. It is convenient to set ω_1= 1/2( ω_3 +Δω),ω_2= 1/2( ω_3 -Δω),since the note-beat frequency Δω = ω_1 - ω_2 is sufficient to label the signal and idler frequencies produced by a pump field of frequency ω_3. Conservation of transverse momentum of the three fields implies that the their complex amplitudes (∼ e^-iω_n t, n=1,2,3) are E_n =e^iω_n/c xsinθ[ A_ny( z )ê_y ],H_n =e^iω_n/c xsinθ√(ε _0 /μ _0 )[ A_nx( z )ê_x+ A_ny( z ) sinθê_z ]. Accordingly the three two-component column vectors (A_nx(z) A_ny(z))^T fully describe the field and in vacuum, i.e. outside the cavity, they are [ A_nx; A_ny;] = E_n^( i )[ - cosθ;1;]e^iω_n/c( z + d)cosθ+ E_n^( r )[ cosθ;1;]e^ - iω_n/c( z + d)cosθ, z <- d, [ A_nx; A_ny;] = E_n^( t )[ - cosθ;1;]e^iω_n/c( z - L - d)cosθ,z> L+d, where E_n^( i ), E_n^( r ) and E_n^( t ) are field amplitudes with E_1^( i ) = E_2^( i ) =0. Resorting to the transfer matrix approach, the fields at the left (z=0^-) and right (z=0^+) sides of the MX_2 monolayer are [ A_nx; A_ny;]_z = 0^ - =F_n [ A_nx; A_ny;]_z =- d ,[ A_nx; A_ny;]_z = 0^ + =B_n B'_n [ A_nx; A_ny;]_z = L + d, where F_n, B_n and B'_n are the transfer matrix describing the forward, backward and backward propagations through the left Bragg mirror, the dielectric slab and the right Bragg mirror, respectively. The transfer matrix F_n of the left Bragg mirror is the (ordered) product of the transfer matrices of the slabs composing the mirror. For later convenience it is useful to represent this matrix as <cit.> F_n = [ c q_n /ω_n cosθ[ t_n^( L ) t_n^( R )+ ( 1 + r_n^( L ))( 1 - r_n^( R ))/2t_n^( R )]c q_n /ω_n [- t_n^( L ) t_n^( R )+ ( 1 - r_n^( L ))( 1 - r_n^( R ))/2t_n^( R )]; 1/cosθ[- t_n^( R ) t_n^( L )+ ( 1 + r_n^( L ))( 1 + r_n^( R ))/2t_n^( R )][ t_n^( L ) t_n^( R )+ ( 1 - r_n^( R ))( 1 + r_n^( R ))/2t_n^( R )];], where q_n= ω_n/c√(ε( ω _n ) - sin ^2 θ) are the longitudinal wavenumbers inside the dielectric slab, ϵ(ω) is the relative permittivity of the dielectric slab whereas r_n^( L ),t_n^( L ),r_n^( R ),t_n^( R ) are the complex reflectivities r and transmittivities t for left (L) and right (R) illumination of the left Bragg mirror (with vacuum and the dielectric on its left and right sides, respectively). The other relevant transfer matrices are <cit.> B_n = [ cos( q_n L) ic q_n /ω_n sin( q_n L); iω_n/c q_nsin( q_n L) cos( q_n L); ], B'_n= [ 1 0; 0 - 1; ] F_n [ 1 0; 0 - 1; ], where the last of Eqs. (<ref>) is a consequence of the fact that the right Bragg mirror is the reflected image of the left one. Using Eqs. (<ref>), Eqs. (<ref>) yield [ A_nx; A_ny;]_z = 0^ - = [ V_nx^( i ); V_ny^( i );]E_n^( i )+[ V_nx^( r ); V_ny^( r );]E_n^( r ) , [ A_nx; A_ny;]_z = 0^ + = [ V_nx^( t ); V_ny^( t );]E_n^( t ), where [ V_nx^( i ); V_ny^( i );] = 1/t_n^( R )[ c q_n /ω_n [- t_n^( L ) t_n^( R )+ r_n^( L )( r_n^( R )- 1)];t_n^( L ) t_n^( R )- r_n^( L )( r_n^( R )+ 1);], [ V_nx^( r ); V_ny^( r );] = 1/t_n^( R )[ - c q_n /ω_n ( r_n^( R )- 1); r_n^( R )+ 1;],[ V_nx^( t ); V_ny^( t );] = 1/t_n^( R )[ c q_n /ω_n ( r_n^( R ) e^iq_n L- e^ - iq_n L);( r_n^( R ) e^iq_n L+ e^ - iq_n L) ]. The monolayer of MX_2 in the presence of the above TE electromagnetic field hosts a surface current whose harmonic complex amplitudes are K_n = K_n ê_ywhere K_1 = [ σ _1 E_1y+ σ _23 E_2y^* E_3y]_z=0,K_2 = [ σ _2 E_2y+ σ _13 E_1y^* E_3y]_z=0,K_3 = [ σ _3 E_3y+ σ _12 E_1y E_2y]_z=0, showing both a linear and a quadratic response to the electric field. The effect of such surface current on the field is provided by the electromagnetic boundary conditions at z=0, namely ê_z×{[ E_n ]_z = 0^ +- [ E_n ]_z = 0^ -} = 0 and ê_z×{[ H_n ]_z = 0^ +- [ H_n ]_z = 0^ -} = K_n which, using Eqs. (<ref>), can be casted within the two-component column vector description as [ A_1x; A_1y;]_z = 0^ +- [ A_1x; A_1y;]_z = 0^ -= [ σ̃_1 A_1y+ σ̃_23 A_2y^* A_3y;0;]_z = 0^ +,[ A_2x; A_2y;]_z = 0^ +- [ A_2x; A_2y;]_z = 0^ -= [ σ̃_2 A_2y+ σ̃_13 A_1y^* A_3y;0;]_z = 0^ +,[ A_3x; A_3y;]_z = 0^ +- [ A_3x; A_3y;]_z = 0^ -= [ σ̃_3 A_3y+ σ̃_12 A_1y A_2y;0;]_z = 0^ +, where, for each component, we have set σ̃= √(μ _0 /ε _0 )σ, where ε _0 and μ _0 indicate the dielectric permittivity and magnetic permeability of vacuum, respectively. After inserting Eqs. (<ref>) along with E_1^( i ) = E_2^( i ) =0 into Eqs. (<ref>) we obtain V_1x^( t ) E_1^( t )- V_1x^( r ) E_1^( r ) = σ̃_1 V_1y^( t ) E_1^( t )+ σ̃_23 V_2y^( t )* V_3y^( t ) E_2^( t )* E_3^( t ) ,V_1y^( t ) E_1^( t )- V_1y^( r ) E_1^( r ) =0,V_2x^( t ) E_2^( t )- V_2x^( r ) E_2^( r ) = σ̃_2 V_2y^( t ) E_2^( t )+ σ̃_13 V_1y^( t )* V_3y^( t ) E_1^( t )* E_3^( t ) ,V_2y^( t ) E_2^( t )- V_2y^( r ) E_2^( r ) =0,V_3x^( t ) E_3^( t )- V_3x^( i ) E_3^( i )- V_3x^( r ) E_3^( r ) = σ̃_3 V_3y^( t ) E_3^( t )+ σ̃_12 V_1y^( t ) V_2y^( t ) E_1^( t ) E_2^( t ) ,V_3y^( t ) E_3^( t )- V_3y^( i ) E_3^( i )- V_3y^( r ) E_3^( r ) =0, which are six equations for the six unknown amplitudes E_n^( t ), E_n^( r ) of the signal, idler and pump fields transmitted and reflected by the cavity illuminated by incident pump field of amplitude E_3^( i ). The second, fourth and sixth of Eqs. (<ref>) can be written as E_1^( r ) = V_1y^( t )/V_1y^( r )E_1^( t ),E_2^( r ) = V_2y^( t )/V_2y^( r )E_2^( t ),E_3^( r ) = V_3y^( t )/V_3y^( r )E_3^( t )- V_3y^( i )/V_3y^( r )E_3^( i ), showing that the reflected fields can be evaluated once the transmitted fields are known. Substituting the reflected fields from Eqs. (<ref>) into Eqs. (<ref>) and using Eqs. (<ref>) we eventually get Δ _1 Q_1+ σ̃_23 Q_2^* Q_3 =0,Δ _2 Q_2+ σ̃_13 Q_1^* Q_3 =0,Δ _3 Q_3+ σ̃_12 Q_1 Q_2 =P_3, where Q_n = ( r_n^( R ) e^iq_n L+ e^ - iq_n L/t_n^( R ))E_n^( t ),P_3 = ( 2t_3^( R )cosθ/r_3^( R )+ 1)E_3^( i ),Δ _n = σ̃_n- c q_n /ω_n( r_n^( R )- 1/r_n^( R )+ 1 + r_n^( R ) e^iq_n L- e^ - iq_n L/r_n^( R ) e^iq_n L+ e^ - iq_n L). Equations (<ref>) are the basic equations for the output fields. Note that Eqs. (<ref>) have been derived without resorting to any electromagnetic approximation commonly used in cavity nonlinear optics (e.g. slowly varying amplitude approximation, decoupled approximation for counter-propagating waves, etc.) and this is a consequence of the fact the overall nonlinear response of the MX_2 monolayer is confined to a single plane.§ DOUBLY RESONANT PARAMETRIC OSCILLATION CONDITIONS Equations (<ref>) provide the amplitudes Q_n (proportional to the amplitudes of the transmitted signal, idler and pump fields) for a given amplitude P_3 (proportional to the amplitude of the incident pump field). Note that they always admit the solution Q_1 = Q_2 = 0,Q_3= P_3/Δ_3, which describes the linear response of the cavity to the pump field without parametric oscillations (POs) in turn characterized by Q_1 ≠ 0 and Q_2 ≠ 0. On the other hand, the first and the complex conjugate of the second of Eqs. (<ref>) are a linear system for Q_1 and Q_2^* and it admits nontrivial solutions only if its determinant vanishes (see Section IV below) or | Q_3 |^2= Δ _1 Δ _2^* /σ̃_23σ̃_13^* . This condition entails the occurrence of POs since if it is fulfilled, Eqs. (<ref>) have solutions with Q_1 ≠ 0 and Q_2 ≠ 0 which are stable whereas the linear one (Q_1 = Q_2 = 0) becomes unstable. Since the right hand side of Eq. (<ref>) is generally a complex number, it is evident that POs can occur only if such complex number is real and positive or Δ _1 Δ _2^* /σ̃_23σ̃_13^* = | Δ _1 Δ _2^* /σ̃_23σ̃_13^* |, which, due to Eqs. (<ref>) and (<ref>), is a constraint joining the cavity length L, the note-beat frequency Δω, and the incident angle θ. Geometrically, Eq. (<ref>) represents a surface Σ of the three-dimensional cavity state space (L,Δω,θ), and its typical slices (θ = 0 or L = L_0) are illustrated in Figs.3a1, 3b1 and 3c1 of the main paper (green curves). At each point of the surface Σ, PO ignites if |Q_3| is sufficiently large to fulfill Eq. (<ref>). Therefore, considering an experiment where the incident pump |P_3| is gradually increased starting from the linear regime where Q_1=Q_2=0, the PO thresholdis obtained by inserting the linear solution Q_3= P_3/Δ _3 into Eq. (<ref>), thus obtaining ( | P_3 |^2 )_th = Δ _1 Δ _2^* /σ̃_23σ̃_13^* | Δ _3 |^2, which, through the second of Eqs. (<ref>) and the relation I_3^(i) = 1/2√(ϵ_0/μ 0)| E_3^( i )|^2, entails the intensity threshold for the incident pump. Note that the denominator of the right hand side of Eq. (<ref>) contains the nonlinear conductivities σ̃_23 and σ̃_13 whose moduli are so small to generally yield exceedingly large and unfeasible intensity thresholds. A viable way for observing POs thus necessitates the identification of the points (L,Δω,θ) of the surface Σ for which |Δ_1| and |Δ _2| are very close to zero. An inspection of the third of Eqs. (<ref>) reveals that |Δ _n| can not be small if |r_n^( R )| is not close to 1. Therefore, by choosing Bragg mirrors with high reflectivities, the third of Eqs. (<ref>) can be expanded up to the first order in the parameter 1 - | r_n^( R )| ≪ 1 thus getting Δ _n= [ σ̃_n- i( 2 c q_n /ω_n )sinΓ _n /cosΓ _n+ cos( q_n L)] + {( 2 c q_n /ω_n )1 + ( cosΓ _n+ 2isinΓ _n )cos( q_n L)/[ cosΓ _n+ cos( q_n L)]^2 }( 1 - | r_n^( R )|), where Γ_n= q_n L + r_n^( R ). The minima of |Δ_n| are easily seen to occur for Γ_n = m π (where m is any integer) which is exactly the cavity resonance condition for the frequency ω_n <cit.> and where, up to the first order of 1 - | r_n^( R )|, Δ _n= σ̃_n+ [ ( 2 c q_n /ω_n )1/1 + cos(r_n^( R ))] ( 1 - | r_n^( R )|). Therefore, as for standard POs based on bulk nonlinear media, the pump intensity threshold is here minimum when one or more of the three fields meet the cavity resonant condition. Designing a Bragg mirror with high reflectivity for both signal and idler fields is relatively simple (see the Section III below) and therefore in this paper we consider only doubly resonant (DR) states where the signal resonance (SR) and idler resonance (IR) conditions q_1 L +r_1^( R ) =m_1 π,q_2 L +r_2^( R ) =m_2 π, are both achieved whereas the pump is non-resonant. Such two equations represents two surfaces Σ_1 and Σ_2 of the space (L,Δω,θ) whose typical slices are reported in Figs.3a1, 3b1 and 3c1 of the main text (black and red lines respectively) and whose intersection describes the DR cavity states where |Δ_1 Δ_2| is minimum. The (nontrivial) intersection among the three surfaces Σ, Σ_1 and Σ_2 is the set of the DRPO cavity states with feasible intensity threshold. Note that if Σ_1 and Σ_2 intersect at a specific (L,0,θ) point this point also belongs to the surface Σ since for Δω = 0 Eqs. (<ref>) is trivially satisfied since evidently ω_1 = ω_2, Δ_1 = Δ_2 and σ_23 = σ_13. In other words a degenerate (ω_1 = ω_2) DR state always supports a PO which we refer to as a degenerate DRPO. In addition, note that if | Re σ̃_n| ≪ | Im σ̃_n| and | Re σ̃_nm| ≪ | Im σ̃_nm|, if both Eqs. (<ref>) are satisfied with Δω≠ 0, Eq. (<ref>) implies that |Im( Δ _1 Δ _2^* /σ̃_23σ̃_13^* ) | ≪| Re( Δ _1 Δ _2^* /σ̃_23σ̃_13^* ) | so that, remarkably, if both linear and nonlinear absorption are small the non-degenerate DR states are always very close to PO states. As a consequence the non-degenerate DRPOs with feasible intensity threshold are associated to those points of the surface Σ which are as close as possible to points of the intersection between the surface Σ_1 and Σ_2. Both degenerate and non-degenerate DRPO states are labelled with a dashed disk in Figs.3a1, 3b1 and 3c1 of the main text.§ BRAGG MIRROR DESIGN The Bragg mirror is a periodic structure composed of N bi-layers whose dielectric materials have refractive indexes n^(a) and n^(b) and thicknesses a and b. If the layers' thicknesses are chosen to satisfy the Bragg interference condition a n^(a) = b n^(b) = π c/2 ω̅, the mirror has (for normal incidence θ = 0), a spectral stop-band centered at ω̅ whose width is proportional to the refractive index contrast |n^(a) - n^(b)| <cit.>. Within the stop-band the mirror reflectivity is very large, the larger N the closer |r^(R) (ω)| to 1. As explained above in Section II, in order for one of the three fields (pump, signal and idler) to be resonant, it is necessary a very large reflectivity of the Bragg mirror at the field angular frequency, or in other words the frequency ω_n has to lie within the mirror stop-band. As noted above, the degenerate DR states with Δω = 0 (where, from Eqs. (<ref>), ω_1= ω_2 =ω_3/2) rigorously supports POs so that it is convenient to set the center of the mirror stop-bad at ω̅= ω_3/2. Due to the refractive index contrast |n^(a) -n^(b)|, this condition assures that both signal and idler fields experience very large mirror reflectivity in a range of Δω and can accordingly be resonant at the same time. On the other hand the pump frequency ω_3 is twice the central mirror frequency ω̅ and requiring also the pump to resonate would require very large refractive contrast. To avoid this difficulty we have chosen to leave the pump out of resonance.In the analysis reported in Fig.3 of the main text, we have set as pump wavelength λ_3 = 780 nm. For the Bragg Mirror we have chosen the refractive indexes n_a = 1.2 and n_b = 2.5 so that, in order to have the center of the stop-band at ω̅= ω_3/2 we have chosen the thicknesses a = λ_3/(2 n^(a)) = 325 nm and b = λ_3/(2 n^(b)) = 156 nm. We have also set N=8 for dealing with an efficient, feasible and compact Bragg mirror of length d = N(a+b) =3848 nm. Using the transfer matrix approach, the complex reflectivity r^(R) (for normal incidence θ=0) of the Bragg mirror which has vacuum and the dielectric at its left and right sides, respectively, is easily evaluated and we plot its absolute value and argument in panel (a) and (b), respectively, of Fig.<ref>. Accordingly, the Bragg mirror stop-band is centered at ω_3/2 and its spectral width is ≃ 0.22 ω_3. As a consequence, if ω_1 and ω_2 lie within this stop-band, signal and idler waves can resonate simultaneously since r_1^(B) = r^(B) (ω_1) and r_2^(B) = r^(B) (ω_2) have moduli very close to 1. The mirror stop-band width therefore yields the note-beat frequency range 0 ≤Δω < 0.22 ω, which is the one considered in the analysis reported in Fig.3 of the main text. Note that ω_3 lies outside the mirror stop-band and thus the pump field does not resonate.§ OUTPUT FIELDS In order to evaluate the PO output fields E_n^( t ), Eqs. (<ref>) have to be solved for a given incident pump field E_3^( i). POs are characterized by Q_1 ≠ 0 and Q_2 ≠ 0 for a given Q_3 ≠ 0. First note that Eqs. (<ref>) are left invariant by the gauge transformation Q_1 →Q_1 e^i θ,Q_2 →Q_2 e^ - i θ, and this implies that for a given P_3 there are infinite pairs (Q_1,Q_2), all with the same Ψ =Q_1 +Q_2. In other words, the phase difference Φ =Q_1 -Q_2 is not set by the pump field P_3. Evidently, such symmetry is spontaneously broken in actual experiments where a single pair (Q_1,Q_2) (i.e. a single value of Φ) is selected by the specific way chosen to trigger POs.In the case of POs, the first and the complex conjugate of the second of Eqs. (<ref>) can be casted as Q_1 /Q_2^*= - σ̃_23 Q_3/Δ _1 ,Q_1 /Q_2^*=- Δ _2^* /σ̃_13^* Q_3^* , whose consistency requires their right and left hand sides to coincide or | Q_3 |^2= Δ _1 Δ _2^* /σ̃_23σ̃_13^* which is the PO oscillation condition of Section II. [see Eq. (<ref>)]. The right hand side of Eq. (<ref>) is a positive real number if and only if the complex numbers Δ _1 /σ̃_23 and Δ _2 /σ̃_13 have the same argument φ or Δ _1 = σ̃_23| Δ _1 /σ̃_23|e^iφ,Δ _2 = σ̃_13| Δ _2 /σ̃_13|e^iφ, which are equivalent to Eq. (<ref>) of Section II so that, considering only those states for which Eqs. (<ref>) are satisfied, Eqs. (<ref>) yield | Δ _1 /σ̃_23|e^iφ Q_1+ Q_2^* Q_3 =0,| Δ _2 /σ̃_13|e^iφ Q_2+ Q_1^* Q_3 =0,Δ _3 Q_3+ σ̃_12 Q_1 Q_2 =P_3. To solve this equation, after noting that Eqs. (<ref>) require that |Q_1 /Q_2 |^2= |Δ _2σ̃_23/Δ _1 σ̃_13| and exploiting the above discussed gauge symmetry, we set Q_1 = √(| Δ _2 /σ̃_13|)| Q |e^i1/2( Ψ+ Φ), Q_2 = √(| Δ _1 /σ̃_23|)| Q |e^i1/2( Ψ- Φ), where we have used the symbol |Q| to stress than this quantity is real. Inserting Eqs. (<ref>) into Eqs. (<ref>) yield Q_3 =- √(| Δ _1 Δ _2 /σ̃_23σ̃_13|) e^i( Ψ+ φ) ,| Q |^2+Δ _3 /σ̃_12√(| σ̃_23σ̃_13/Δ _1 Δ _2 |) e^ - iΨ Q_3 = √(| σ̃_23σ̃_13/Δ _1 Δ _2 |)P_3 /σ̃_12e^ - iΨ. Note that the first two of Eqs. (<ref>) both reduces to the first of Eqs. (<ref>) through the change of variables given by Eqs. (<ref>) and this is a consequence of the necessary Eq. (<ref>). Substituting Q_3 from the first of Eqs. (<ref>) into the second we get | Q |^2- Δ _3 /σ̃_12e^iφ= √(| σ̃_23σ̃_13/Δ _1 Δ _2 |)P_3 /σ̃_12 e^ - iΨ, which is a single complex equation for |Q| and Ψ. After equating the square-moduli of the left and right sides of this equation we get | Q |^4- 2 Re( Δ _3 /σ̃_12e^iφ)| Q |^2+ ( | Δ _3 /σ̃_12e^iφ|^2- | σ̃_23σ̃_13/Δ _1 Δ _2 | | P_3 /σ̃_12|^2 ) = 0, which is a biquadratic equation for | Q | whose solutions are | Q | = √( Re( Δ _3 /σ̃_12e^iφ) +ξ√(| σ̃_23σ̃_13/Δ _1 Δ _2 | | P_3 /σ̃_12|^2-Im^2 ( Δ _3 /σ̃_12e^iφ) )), where ξ = ± 1. Note that, due to the ξ factor, generally there are two |Q| corresponding to a |P_3| and hence bistable POs can in principle occur. In addition, it is fundamental stressing that |Q| is real and hence Eq. (<ref>) provides its value only if the arguments of the square roots are positive. Before discussing the range of |P_3| where this is the case (see below), we assume |Q| real and we deduce the output field amplitudes. Equation (<ref>) yields e^iΨ= √(| σ̃_23σ̃_13/Δ _1 Δ _2 |)P_3 /σ̃_12( | Q |^2- Δ _3 /σ̃_12e^iφ), which is consistent since the modulus of its right hand side, due to Eq. (<ref>), is equal to 1. Hence Eqs. (<ref>) and the first of Eqs. (<ref>) eventually yield Q_1 = ζ| Δ _2 σ̃_23/Δ _1 σ̃_13|^1/4√(P_3 /σ̃_12( | Q |^2- Δ _3 /σ̃_12e^iφ)) e^iΦ/2| Q |,Q_2 = ζ| Δ _1 σ̃_13/Δ _2 σ̃_23|^1/4√(P_3 /σ̃_12( | Q |^2- Δ _3 /σ̃_12e^iφ)) e^ - iΦ/2| Q |,Q_3 = P_3 /σ̃_12( | Q |^2- Δ _3 /σ̃_12e^iφ)e^i( φ+ π), where ζ = ± 1 and the principal branch is assumed for all the complex square-roots.Note that the output fields of Eqs. (<ref>) satisfy Eq. (<ref>) so that they describe all the possible cavity POs whenever they exist or, in other words, whenever |Q| of Eq. (<ref>) is a positive real number. Such requirement evidently sets a range for the input pump intensity |P_3|^2 and there are four different cases corresponding to the two values of ξ and of the two signs of Re( Δ _3 /σ̃_12e^iφ). The results of this analysis are reported in the following table. [Re( Δ _3 /σ̃_12e^iφ)>0 Re( Δ _3 /σ̃_12e^iφ) <0;; ξ=1| P_3 |^2> ( | P_3 |^2 )_- | P_3 |^2> ( | P_3 |^2 )_th;;;ξ=-1 ( | P_3 |^2 )_- < | P_3 |^2< ( | P_3 |^2 )_th no|P_3|^2;; ] Here we have set ( | P_3 |^2 )_- = | Δ _1 Δ _2 /σ̃_23σ̃_13| Im ^2 ( Δ _3 | σ̃_12|/σ̃_12e^iφ) < | Δ _1 Δ _2 /σ̃_23σ̃_13|| Δ _3 |^2= ( | P_3 |^2 )_th. Therefore at each state where PO can occur (i.e. at a each point of the surface Σ) the scenario is the following one. If Re( Δ _3 /σ̃_12e^iφ) < 0, there is only one PO (with ξ =1) that effectively starts when Eq. (<ref>) is satisfied, thus confirming the analysis of Section III. On the other hand, if Re( Δ _3 /σ̃_12e^iφ) > 0 the scenario changes qualitatively since in this case there are two allowed POs (with ξ=1 and ξ=-1) when( | P_3 |^2 )_- < | P_3 |^2< ( | P_3 |^2 )_thand a single PO (with ξ=1) when Eq. (<ref>) is satisfied. As a consequence in this case POs also exist below the threshold.In order to grasp the reason why the sub-threshold POs have not been entailed in Section III, note that in the case Re( Δ _3 /σ̃_12e^iφ) > 0, if | P_3 |^2 = ( | P_3 |^2 )_-, Eq. (<ref>) implies that | Q | = √( Re( Δ _3 /σ̃_12e^iφ)) 0 so that Q_1 ≠ 0 and Q_2 ≠ 0. In other words in this situation the signal and idler fields do not vanish at the threshold and accordingly this case is ruled out from the reasoning of Section III where the threshold has been obtained for PO oscillation starting from the linear regime. In a realistic experiment PO is switched on starting from the linear regime, with the intensity threshold given by Eq. (<ref>). However, once PO ignites, by changing the pump intensity, incidence angle θ, or the cavity length, we argue that one can in principle access sub-threshold PO states. In every design considered in the main paper we have focused on the case Re( Δ _3 /σ̃_12e^iφ) < 0, where sub-threshold PO does not occur.§ PUMP INTENSITY THRESHOLDSIn Fig. <ref> we compare the calculated pump intensity thresholds versus the pump wavelength λ_3 for parametric oscillators embedding MoS_2, WS_2 (Fig. <ref>a) and MoSe_2, WSe_2 (Fig. <ref>b). Note that, while the minimal pump intensity threshold occurs at λ_3 ≈ 780 nm for MoS_2 and WS_2, it shifts to λ_3 ≈ 940 nm for MoSe_2 and WSe_2. None of the ML-TMDs examined enables feasible PO with low pump intensity threshold at optical frequencies owing to the enlarged absorption in this frequency range, which is the main responsible for oscillation quenching. In addition, at optical frequencies such materials exhibit exciton resonances <cit.> (not taken into account in our theoretical approach) that are also detrimental for POs owing to the enhanced absorption they are accompanied with. In Fig. <ref>c we plot the pump intensity threshold as a function of the Fermi level of MoS_2, showing that it can be increased efficiently. Thus, the external gate voltage quenches POs when the optical pump is fixed and fast modulation of the output signal and idler fields can be achieved with novel parametric oscillators embedding ML-TMDs.10 Franken1961 P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,´´ Phys. Rev. Lett. 7, 118 - 119 (1961).Birks2000 T. A. Birks, W. J. Wadsworth, P. St.J. Russell, “Supercontinuum generation in tapered fibers,´´ Opt. Lett. 25, 1415 - 1417 (2000).Stegeman1996 G. I. Stegeman, D. J. Hagan, and L. Torner, “χ^(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,´´ Opt. Quant. Electron. 28, 1691 - 1740 (1996).Koos2009 C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,´´ Nat. Photon. 3, 216 - 219 (2009).Kwiat1995 P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New High-Intensity Source of Polarization-Entangled Photon Pairs,´´ Phys. Rev. Lett. 75, 4337 - 4341(1995).Stevenson2006 R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,´´ Nature 439, 179 - 182 (2006).Giordmaine1965J. A. Giordmaine and R. C. Miller, “Tunable Coherent Parametric Oscillation in LiNbO3 at Optical Frequencies,´´ Phys. Rev. Lett. 14, 973 - 976 (1965).Yariv1966 A. Yariv and W. H, Louisell, “Theory of the Optical Parametric Oscillator,´´ IEEE J. Quant. Electron. 9, 418 - 424 (1966).Brosnan1979 S. J. Brosnan and R. L. Byer, “Optical Parametric Oscillator Threshold and Linewidth Studies,´´ IEEE J. Quant. Electron. 15, 415 - 431 (1979).Eckardt1991 R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky, and R. L. Byer, “Optical parametric oscillator frequency tuning and control,´´ J. Opt. Soc. Am. B 8, 646 - 667 (1991).Debuisschert1993 T. Debuisschert, A. Sizmann, E. Giacobino, and C. Fabre, “Type-II continuous-wave optical parametric oscillators: oscillation and frequency-tuning characteristics,´´ J. Opt. Soc. Am. B 10, 1668 - 1680 (1993).Fabre1997 C. Fabre, P. F. Cohadon, and C. Schwob, “CW optical parametric oscillators: single mode operation and frequency tuning properties,´´ Quantum Semiclass. Opt. 9, 165 - 172 (1997).Levy2010 J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,´´ Nat. Photon. 4, 37 - 40 (2010).Razzari2010 L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyperparametric oscillator,´´ Nat. Photon. 4, 41 - 45 (2010).Wu1987 L.-A. Wu, M. Xiao, and H. J. Kimble, “Squeezed states of light from an optical parametric oscillator,´´ J. Opt. Soc. Am. B 4, 1465 - 1475 (1987).Lu2000 Y. J. Lu and Z. Y. Ou, “Optical parametric oscillator far below threshold: Experiment versus theory,´´ Phys. Rev. A 62, 033804 (2000).Ciuti2003 C. Ciuti, P. Schwendimann and A. Quattropani, “Theory of polariton parametric interactions in semiconductor microcavities,´´ Semicond. Sci. Technol. 18, S279 – S293 (2003).Diederichs2006 C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, Ph. Roussignol and C. Delalande, “Parametric oscillation in vertical triple microcavities´´ Nature 440, 904 - 907 (2006).Abbarchi2011 M. Abbarchi, V. Ardizzone, T. Lecomte, A. Lemaıtre, I. Sagnes, P. Senellart, J. Bloch, P. Roussignol, and J. Tignon, “One-dimensional microcavity-based optical parametric oscillator: Generation of balanced twin beams in strong and weak coupling regime,´´ Phys. Rev. B 83, 201310(R) (2011).Giordmaine1962 J. A. Giordmaine, “Mixing of Light Beams in Crystals,´´ Phys. Rev. Lett. 8, 19 – 20 (1962).Vodopyanov2004 K. L. Vodopyanov, O. Levi, P. S. Kuo, T. J. Pinguet, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Optical parametric oscillation in quasi-phase-matched GaAs,´´ Opt. Lett. 29, 1912 – 1914 (2004).Canalias2007 C. Canalias, and V. Pasiskevicius, “Mirrorless optical parametric oscillator,´´ Nature Photonics 1 459 - 462 (2007).Savvidis2000 P. G. Savvidis, J. J. Baumberg, R. M. Stevenson, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts, “Angle-Resonant Stimulated Polariton Amplifier,´´ Phys. Rev. Lett. 84, 1547 – 1550 (2000).Ciuti2000 C. Ciuti, P. Schwendimann, B. Deveaud, and A. Quattropani, ´´Theory of the angle-resonant polariton amplifier,´´ Phys. Rev. B 62, R4825–-R4828 (2000).Xie2011 Z. D. Xie, X. J. Lv, Y.H. Liu, W. Ling, Z. L. Wang, Y. X. Fan, and S. N. Zhu, ´´Cavity Phase Matching via an Optical Parametric Oscillator Consisting of a Dielectric Nonlinear Crystal Sheet,´´ Phys. Rev. Lett. 106, 083901 (2011).Clément2015 Q. Clément, J.-M. Melkonian, M. Raybaut, J.-B. Dherbecourt, A. Godard, B. Boulanger, and M. Lefebvre, ´´Ultrawidely tunable optical parametric oscillators based on relaxed phase matching: theoretical analysis,´´ J. Opt. Soc. Am. B 32, 52 - 68 (2015).Mak2010 K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz,“Atomically Thin MoS_2: A New Direct-Gap Semiconductor,´´ Phys. Rev. Lett. 105, 136805 (2010).Splendiani2010 A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C. Y. Chim, G. Galli, and F. Wang, “Emerging photoluminescence in monolayer MoS_2,´´Nano Lett. 10 1271 - 1275 (2010).Wang2012 Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, “Electronics and optoelectronics of two-dimensional transition metal dichalcogenides,´´ Nat. Nanotechnol. 7, 699 - 712 (2012).Sun2016 Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,´´ Nat. Photon. 10, 227 - 238 (2016).Kumar2013 N. Kumar, S. Najmaei, Q. Cui, F. Ceballos, P. M. Ajayan, J. Lou, and H. Zhao, “Second harmonic microscopy of monolayer MoS_2,´´Phys. Rev. B 87, 161403(R) (2013).Li2013 Y. Li, Y. Rao, K. F. Mak, Y. You, S. Wang, C. R. Dean, T. F. Heinz, “Probing symmetry properties of few-layer MoS_2 and h-BN by optical second-harmonic generation,´´ Nano Lett. 13, 3329 - 3333 (2013).Malard2013 L. M. Malard, T. V. Alencar, A. P. M. Barboza, K. F. Mak, and A. M. de Paula, “Observation of intense second harmonic generation from MoS_2 atomic crystals,´´ Phys. Rev. B 87, 201401(R) (2013).Janisch2014 C. Janisch, Y. Wang, D. Ma, N. Mehta, A. L. Elias, N. Perea-Lopez, M. Terrones, V. Crespi, and Z. Liu, Sci. Rep. 4, 5530 (2014).Le2016 C. T. Le, D. J. Clark, F. Ullah, V. Senthilkumar, J. I. Jang, Y. Sim, M.-J. Seong, K.-H. Chung, H. Park, and Y. S. Kim, “Nonlinear optical characteristics of monolayer MoSe_2,´´ Ann. Phys. 528, 551 - 559 (2016).LWY2013G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao, “Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides,´´ Physical Review B 88, 085433 (2013).EPAPS See Supplemental Material for more details on the technical aspects of the theory.Ugeda2014 M. M. Ugeda, A. J. Bradley, S.-F. Shi, F. H. da Jornada, Y. Zhang, D. Y. Qiu, W. Ruan, S.-K. Mo, Z. Hussain, Z.-X. Shen, F. Wang, S. G. Louie, and M. F. Crommie, “Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor´´, Nat. Mater. 13, 1091-1095 (2014).Selig2016 M. Selig, G. Berghäuser, A. Raja, P. Nagler, C. Schüller, T. F. Heinz, T. Korn, A. Chernikov, E. Malic, and A. Knorr, “Excitonic linewidth and coherence lifetime in monolayer transition metal dichalcogenides´´, Nat. Commun 7, 13279 (2016). LWY2013G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao, “Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides,´´ Physical Review B 88, 085433 (2013). UBS2014 M. M. Ugeda, A. J. Bradley, S.-F. Shi, F. H. da Jornada, Y. Zhang, D. Y. Qiu, W. Ruan, S.-K. Mo, Z. Hussain, Z.-X. Shen, F. Wang, S. G. Louie, and M. F. Crommie, “Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor,´´ Nat. Mater. 13, 1091 - 1095 (2014). CCB2016 F. Ceballos,Q. Cui, M. Z. Bellusa, and H. Zhao, “Exciton formation in monolayer transition metal dichalcogenides,´´ Nanoscale 8, 11681 - 11688 (2016). YYC2014 X. Yin, Z. Ye, D. A. Chenet, Y. Ye, K. O'Brien, J. C. Hone, X. Zhang, “Edge Nonlinear Optics on a MoS2 Atomic Monolayer´´, Science 344, 488–490 (2014). Kavok A. V. Kavokin, J. J. Baumberg, G. Malpuech and F. P. Laussy, Microcavities (Oxford University Press, USA 2008) | http://arxiv.org/abs/1707.08843v1 | {
"authors": [
"A. Ciattoni",
"A. Marini",
"C. Rizza",
"C. Conti"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170727125254",
"title": "Phase-matching-free parametric oscillators based on two dimensional semiconductors"
} |
National Science Center "Kharkiv Institute of Physics and Technology", Akademichna Street 1, 61108 Kharkiv, Ukraine [email protected] the formalism of the Classical Nucleation Theory beyond the dilute solution approximation, this paper considers a difference between the actual solute supersaturation (given by the present-to-saturated solute activity ratio) and the nominal supersaturation (given by the present-to-saturated solute concentration ratio) due to formation of subcritical transient solute clusters, called heterophase fluctuations. Based on their distribution function, we introduce an algebraic equation of supersaturation that couples the nominal supersaturation of a binary metastable solution with its actual supersaturation and a function of the specific interface energy and temperature. The applicability of this approach is validated by comparison to simulation data [E. Clouet et al., Phys. Rev. B 69, 064109 (2004)] on nucleation of Al_3Zr and Al_3Sc in model binary Al alloys.A1. Supersaturated solutions A1. Nucleation A2. Growth from solutions A1. Solid solutions B1. Alloys § INTRODUCTION Because decomposition of supersaturated solutions occurs in many natural phenomena and important technological processes, this problemtraditionally attracts much attention (for some recent results see, e.g., Refs. <cit.> and references therein).From the thermodynamic point of view, the solution properties, including the nucleation driving force (see Eq. (<ref>) below), are determined by the actual supersaturation:S_act=a / a^sat, where a is the solute thermodynamic activity and a^sat corresponds to the solubility (saturation) limit. On the other hand, the only experimentally available value is the nominalsupersaturation:S_nom=c_tot/c_tot^sat, where c_tot is the total solute concentration, calculated as the total number of solute atoms N, divided by the solution volume V:c_tot=N/V, and c_tot^sat corresponds to the solubility (saturation) limit.A very popular dilute solution (DS) model (see, e.g., Ref. <cit.>) does not distinguish between the nominal supersaturation and the actual supersaturation: S=S_nom=S_act (cf., e.g., Eqs. (2.13) and (2.14) of Kashchiev <cit.>), which appears to be a good approximation for essentially weak solutions, where the vast majority of solute atoms is in the monomer state. According to the Frenkel's concept <cit.> of heterophase fluctuations (HF), pretransition processes near the saturation limit lead to formation of dimers, trimers and larger transient solute clusters.Recent studies demonstrate, that HF in solutions play an important role in the nucleation kinetics <cit.>. Here we reveal a crucial role of HF in the thermodynamic treatment of supersaturated metastable solutions.Based on the distribution function of HF, in Section <ref> we derive an algebraic equation of supersaturation of a binary metastable solution, coupling the values of the nominal (<ref>) and the actual (<ref>) supersaturation with a function of the specific cluster-solution interface energy σ and temperature T (see Eq. (<ref>) below). In Section <ref> we apply this equation to compare its results to some simulation data <cit.> on nucleation of Al_3Zr and Al_3Sc in model binary Al alloys (see Figs. <ref>–<ref> below) and demonstrate that a good agreement is achieved in some cases and in all cases Eq. (<ref>) gives a much better agreement than the dilute solution model does. We leave Section <ref> for conclusions and outlook of future tasks.§ EQUATION OF SUPERSATURATION OF A BINARY METASTABLE SOLUTION In this paper we consider solutions with a single type of solute molecules which do not dissociate.Within the Classical Nucleation Theory (CNT), the Gibbs free energy change on forming acluster of n solute monomers (at constant pressure and temperature)is (see, e.g., Eq. (3.39) of Kashchiev <cit.>):Δ G(n)= Δμ· n + k_BT ·α_n· n^2/3. The first term in the right-hand side of Eq. (<ref>) contains the chemical potential change on clusterization, equal to the nucleation driving force taken with the opposite sign (see, e.g., Eq. (2.13) of Kashchiev <cit.>):Δμ= - k_BT ·ln( S_act), where k_B is the Boltzmann's constant. The value given by Eq. (<ref>) can change sign depending on the value of S_act. The second term in the right-hand side of Eq. (<ref>) is the free energy of the cluster-solution interface.For a spherical cluster, the dimensionless size-dependent specific interface energy α_n isα_n= 3·√(4πω^2/3)·σ_n /k_BT, where ω is a volume per solute monomer in the cluster and σ_n is a size-dependent specific cluster-solution interface energy. The value given by Eq. (<ref>) is positively defined. By utilizing Eqs. (<ref>) and (<ref>), the equilibrium distribution function of HF of different sizes, introduced by Frenkel <cit.>, can be presented as follows (see, e.g., Eq. (7.17) of Kashchiev <cit.>):c_n^0=c_1^sat·exp[ln( S_act) · n - α_n·(n^2/3 - 1 )], where c_1^sat is a saturated concentration of solute monomers. Eq. (<ref>) is a particular form of the Boltzmann distribution for the clusters with the energy spectrum (<ref>),specially normalized to give the concentration (number density) of clusters of size n.For n=1, to get a trivial identity c_1^0=c_1 from Eq. (<ref>), one has to setS_act=a / a^sat=c_1 / c_1^sat. Eq. (<ref>) is a direct result of Eq. (<ref>) and, therefore, originates from the choice of the work of clusterization in the form (<ref>) and (<ref>). It means that, within CNT, the solute thermodynamic activity is determined by the concentration of solute monomers only. This approximation seems natural, provided that the diffusivity of solute monomers greatly exceeds that of solute clusters. From Eq. (<ref>) one can see that, in the supersaturated case S_act > 1, the equilibrium distribution of clusters diverges rapidly as n →∞. To avoid this unphysical behavior, one has to take into account a nonzero value of the net cluster flux along the size axis:J_n, n+1(t)= w^(+)_n, n+1 c_n(t) - w^(-)_n+1, n c_n+1(t), where w^(+)_n, n+1 andw^(-)_n+1, n are, respectively, the rates of attachmentand detachment of solute monomers at the cluster-solutioninterface. The special case, when the net cluster flux (<ref>) is zero for any n, corresponds to the state of detailed balance, when the HF distribution function takes its equilibrium form (<ref>).In the steady-state nucleation regime, the net cluster flux (<ref>) is assumed to be a step function of size:J_n, n+1^st=J, 1 ≤ n ≤ n_max;0, n > n_max, where the steady-state nucleation rate is (see, e.g., Eq. (13.30) of Kashchiev <cit.>):J={∑_n=1^n_max[w^(+)_n, n+1c_n^0]^-1}^-1, and the rate of attachment of solute monomers at the interface in the diffusion-limited case is (see, e.g., Eq. (10.23) of Kashchiev <cit.>): w^(+)_n, n+1=4π√(3ω/ 4π)(1+n^1/3)· D(1+n^-1/3) · c_1 , D being the solute diffusion coefficient. In the steady-state nucleation regime, the HF distribution function is (see, e.g., Eq. (13.17) of Kashchiev <cit.>): c_n^J=c_n^0· J ·∑_m=n^n_max[w^(+)_m, m+1c_m^0]^-1 for 1 ≤ n ≤ n_max and c_n^J=0 for n > n_max.In the undersaturated and saturated solutions S_act≤ 1 and from Eq. (<ref>) one getslim_n →∞ c_n^0=0 and, therefore, formally setting n_max→∞, from Eqs. (<ref>) and (<ref>) one gets J=0 and c_n^J=c_n^0. With the above speculations in mind, one can represent the total solute concentration (<ref>) as a sum of HF contributions:c_tot=∑_n=1^n_max n · c_n^J. Eq. (<ref>) is also valid for the undersaturated and saturated cases, where c_n^J=c_n^0 and n_max→∞. The saturated monomer concentration c_1^sat can be extracted from the total solubility c_tot^sat, using Eqs. (<ref>) and (<ref>), as follows:c_1^sat=c_tot^sat/∑_n=1^∞ n ·exp[ - α_n·(n^2/3 - 1 )]. With Eqs. (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) in mind, after some algebra one can express the nominal supersaturation (<ref>) as follows: S_nom=∑_n=1^n_max{n · S_act^n·exp[- α_n·(n^2/3 - 1 )] ·∑_m=n^n_maxexp[ α_m·(m^2/3 - 1 )]/(1+m^-1/3)(1+m^1/3)· S_act^m}/{∑_n=1^n_maxexp[α_n·(n^2/3 - 1 )]/(1+n^-1/3)(1+n^1/3)· S_act^n}{∑_n=1^∞ n ·exp[ - α_n·(n^2/3 - 1 )]}. Eq. (<ref>) can be called an equation of supersaturation (ES) of a binary metastable solution, because it couples the nominal solute supersaturation S_nom with the actual supersaturation S_act and a function {α_n} of the specific cluster-solution interface energy and temperature. It is also valid for undersaturated and saturated thermodynamically equilibrium solutions, where one has to assume formally n_max→∞.§ RESULTS AND DISCUSSION To obtain a practically valuable result, ES (<ref>) can be resolved to find the actual supersaturation S_act (or the thermodynamic driving force k_BT ·ln( S_act)) as a function ofthe nominal supersaturation S_nom and {α_n}. From ES (<ref>) one can see that the difference between the actual supersaturation and the nominal supersaturation becomes sizeable when HF are energetically cheap, i.e. when either S_act is large or {α_n} is small.On the other hand, for small S_act or large {α_n}, when HF are energetically expensive, one can retain only the first terms in the sums in both the numerator and the denominator of ES (<ref>) to obtain the DS result S_nom=S_act. To illustrate the practical applicability of ES (<ref>), below we compare the results of the present approach to the data of the Monte Carlo (MC) simulations <cit.> of the model Al-Zr and Al-Sc alloys, which exhibit homogeneous nucleation of the Al_3Zr and Al_3Sc phases, respectively. In Figs. <ref>–<ref> below we demonstrate the simulated data together with the calculated ones, using size-dependent specific interface energies σ_n shown in Fig. <ref> and calculated from Eq. (20) and Table III of Ref. <cit.> for 1 ≤ n ≤ 9. For n>9 we take σ_n=σ_9.For the face-centered cubic lattice used in the MC simulations <cit.>, we calculate the volume per Al_3Zr or Al_3Sc formula unit (4 lattice sites) as ω=4 · a^3/4=a^3, taking the lattice constant to be equal to that of the aluminium a=4.05·10^-10 m. The values of the upper summation limit n_max are chosen to satisfy the condition c_n_max^J/c_n_max^0≤ 10^-2. The values of other parameters are adopted from Ref. <cit.> and collected in Table <ref>.In Fig. <ref> we plot the chemical potential change on clusterization (per Al_3Zr formula unit) for the Al-Zr alloy as a function of the Zr volume fraction (bottom axis) or the nominal supersaturation (top axis) at T=723 K, obtained by the cluster variation method (CVM) and the direct calculation method (DCM) from the MC simulations <cit.>[In Ref. <cit.> these values are calculated per lattice site. To rescale them per Al_3Zr formula unit, we multiply them by 4.] together with the results of Eq. (<ref>), with the actual supersaturation calculated from ES (<ref>). One can see that the calculated dependence is close to the CVM one and is practically identical to the DCM one, while the DS approximation is valid only in the low-supersaturation regime. In Fig. <ref> we plot the cluster size distributions in the Al-Zr and Al-Sc alloys, obtained from the MC simulations <cit.>, performed at different temperatures and nominal supersaturations, together with the results of Eq. (<ref>), with the actual supersaturation calculated from ES (<ref>). From Fig. <ref> one can see that the present theory is able to reproduce the simulation data on the cluster distribution functions with a good accuracy in all cases except those corresponding to the highest supersaturations of Zr at 723 K in Fig. <ref> (a). For a possible explanation of this discrepancy see discussion of the nucleation rate data in Fig. <ref> below.In Fig. <ref> we display the nucleation rates for the Al_3Zr and Al_3Sc clusters, obtained from the MC simulations <cit.>, performed at different temperatures and nominal supersaturations, together with the results of Eq. (<ref>), with the actual supersaturation given byES (<ref>). The DS result with S_act=S_nom is also given as a reference. One can see that the present theory gives a quantitative agreement with the MC simulations for both systems at all temperatures and not very high nominal supersaturations.Fig. <ref> shows a general tendency of divergence between the simulation and theoretical data with the increase of supersaturation. The same tendency can be observed for the cluster size distribution data in Fig. <ref> (a). A probable reason for this discrepancy is the effect of frustration <cit.>, which increases with the increase of the solute volume fraction. For the sake of brevity, this effect is neglected here. One can see that the values of the nucleation rate, calculated in the DS approximation, considerably overestimate the simulation data. § CONCLUSIONS AND OUTLOOK In conclusion, the CNT-based Frenkel's model of heterophase structure of supersaturated solutions is used to construct an algebraic equation of supersaturation of a binary metastable solution, coupling the values of the nominal and the actual supersaturation with a function of the specific interface energy and temperature. This equation is also valid for saturated and undersaturated solutions. Application of this equation is shown to result in a much better (compared to the dilute solution model) agreement with MC simulation <cit.> data on nucleation of Al_3Zr and Al_3Sc in model binary Al alloys. This approach may be further advanced by taking into account the effect of frustration <cit.>. § ACKNOWLEDGMENTSThe author is grateful to Dr. A. Turkin for numerous discussions and commenting the manuscript. I thank an anonymous reviewer for a suggestion to use the exact form of Eq. (<ref>) instead of the approximate one. This work has been funded by the National Academy of Science of Ukraine, grant # X-4-4/2017.§ REFERENCES99Lin Chen Lin,Yang Zhang, Jing J Liu, and Xue Z Wang,J. Cryst. Growth 469, 59-64 (2017).Dhivya R. Dhivya, R. Ezhil Vizhi, and D. Rajan Babu,J. Cryst. Growth 468, 84-87 (2017).Binder Kurt Binder and Peter Virnau,J. Chem. Phys. 145, 211701 (2016).Greer A. L. Greer,J. Chem. Phys. 145, 211704 (2016).Warrier Pramod Warrier, M. Naveed Khan, Vishal Srivastava, C. Mark Maupin, and Carolyn A. Koh,J. Chem. Phys. 145, 211705 (2016).Bi Yuanfei Bi, Anna Porras, and Tianshu Li,J. Chem. Phys. 145, 211909 (2016).Lifanov Yuri Lifanov, Bart Vorselaars, and David Quigley,J. Chem. Phys. 145, 211912 (2016).Legg Benjamin A. Legg and James J. De Yoreo,J. Chem. Phys. 145, 211921 (2016).Peters Baron Peters,J. Cryst. Growth 317, 79-83 (2011).Mangere M. Mangere, J. Nathoo, and A.E. Lewis,J. Cryst. Growth 312, 3178-3182 (2010).Landau L.D. Landau and E.M. Lifshitz, Statistical Physics. Part 1. 3rd edition (Butterworth-Heinemann, Oxford, 1980).Kashchiev D. Kashchiev, Nucleation: Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000).Frenkel J. Frenkel,J. Chem. Phys. 7, 538 (1939). B_T A. A. Turkin and A. S. Bakai,Problems of Atomic Science and Technology #3(2), 394 (2007).Lepinoux2009 J. Lepinoux, Acta Materialia 57, 1086 (2009).Lepinoux2010 J. Lepinoux, Philosophical Magazine 90, 3261 (2010).S_O R. V. Shapovalov and O. A. Osmayev,Problems of Atomic Science and Technology #1, 273 (2012).Borisenko1 Oleksandr Borysenko,Condensed Matter Physics 18, 23603 (2015).Borisenko2 Alexander Borisenko,Phys. Rev. E 93, 052807 (2016) Clouet Emmanuel Clouet, Maylise Nastar, and Christophe Sigli,Phys. Rev. B 69, 064109 (2004). Lepinoux2006 J. Lepinoux, Philosophical Magazine 86, 5053 (2006). | http://arxiv.org/abs/1707.08777v3 | {
"authors": [
"Alexander Borisenko"
],
"categories": [
"physics.chem-ph",
"cond-mat.mtrl-sci"
],
"primary_category": "physics.chem-ph",
"published": "20170727083509",
"title": "Nominal vs. actual supersaturation of solutions"
} |
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China Key Laboratory of Dark Matter and Space Astronomy, Nanjing 210008, ChinaPurple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China Key Laboratory of Dark Matter and Space Astronomy, Nanjing 210008, China Institute of space physics, Luoyang Normal University, Luoyang, 471934, China Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China College of Optoelectronic Engineering, Chongqing University, Chongqing, 400044 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008, China Fast radio bursts are radio transients observed mainly around 1.5 GHz. Their peak frequency decreases at a rate of 100∼500MHz/s and some of them have a broader pulse with an exponentially decaying tail. Common assumptions for fast radio bursts include a dispersion effect resulting in the peak frequency drifting and a scattering effect resulting in pulse broadening. These assumptions attribute the abnormally large dispersion measure and scattering measure to the environmental medium of the host galaxy. Here we show that the radiation of fast radio bursts can be explained as an undulator radiation and the large dispersion measure can be due to a motion effect mainly from the rotation of the source which is probably variable stars. In our scenario, the pulse broadening is near-field effects and the pulse itself represents a Fresnel diffraction pattern sweeping the observer. Our work is the first analysis of properties of fast radio bursts in the context of a special mechanism of the radiation instead of a special propagation environment of the radiation. § INTRODUCTION The spectrum diagram of a fast radio burst (FRB) is similar to that of a radio pulsar <cit.>. Both of them show a negative frequency sweep, i.e., the higher frequency radiation arrives earlier. Observational differences between them are: (1) an FRB usually has a larger dispersion measure (DM), i.e., the peak frequency decreases slower than that of a radio pulsar and (2) for most FRBs, the frequency sweep is not repeatable while for a radio pulsar, it repeats periodically.Interpretations of these two phenomena are different very much. While no one believes that the short pulse of a radio pulsar is a burst lasting only a few milliseconds on a neutron star, FRB is often thought of a transient burst occurring like a delta function <cit.>. Taking into consideration that there are neutron stars emitting intermittently <cit.> and occasionally <cit.>, and there is a repeatable FRB <cit.> besides, it is not reasonable for the difference in repeatability leading to such a large discrepancy in interpretation. The difference in DM is obviously not a reason for a different timescale either. We believe that an observationally self-consistent explanation of both phenomena exists. The pulse from either of them represents a radiation cone sweeping the observer. In this paper, we interpret the peak frequency drift of FRBs with motion effects instead of propagation effects. We achieve the goal by introducing the undulator radiation <cit.> as the radiation mechanism and a model with a rotation source. We find that the mechanism is also able to explain the pulse broadening of FRBs. The main contributions of this letter are:* The monochromaticity variation of FRBs can be explained by an undulator radiation observed from a different angle.We introduce this radiation mechanism in Section <ref>.* To explain the frequency drift of FRBs, we introduce a model with a rotation source in section <ref>. Further analysis shows that the source of FRBs should be variable stars, which is consistent with the proposals made by <cit.> and <cit.>.* The pulse broadening<cit.> is a signature of near-field effects <cit.>. We identify and discuss these effects in section <ref>.We show that the pulse profile is a Fresnel diffraction pattern.Although our arguments are self-consistent, we note that other mechanism may also be responsible for the same observational phenomenon.We propose a way to justify our explanation in Section <ref>. § UNDULATOR RADIATION AND MONOCHROMATICITY The behavior of FRBs can be compared similarly to a rotating slit in optics. We begin with a qualitative discussion of two different kinds of FRBs and their association with a single slit experiment: * The intensity of FRB 150807 <cit.> is the strongest among observed FRBs. Its radiation is not monochromatic. Many sparkles exist outside the burst width at each frequency that low-frequency and high-frequency emission are received simultaneously.* The intensity of FRB 110523 <cit.> is weaker than that of FRB 150807. Its radiation is monochromatic. At any instance, the radiation is concentrated in a narrow bandwidth ∼ 5MHz. The compromise between chromaticity and intensity in the above cases has an analogy in optics. In a white light diffraction pattern from a single slit, the light is the strongest at the central maximum, but not monochrome. It is a superposition of light with different wavelengths. Near the second maximum, the light is weak, but in any position are monochrome. In radio band, a similar compromise between intensity and monochromaticity occurs when observing an undulator radiation from a different angle.An undulator is a periodic array of dipole magnets with alternating polarity. It is commonly used to produce quasi-monochromatic synchrotron radiation with relativistic particles. The frequency of radiation in the direction of angle θ is determined by the equation <cit.>:f≈2γ^2/1+γ^2θ^2(c/λ_u) Where γ is Lorenz factor, c the speed of light, λ_u the period length of magnetic field structure. This equation is essentially describing a Doppler effect. The Doppler shift is angle dependent; the highest frequency is in the direction θ=0; the frequency gets lower and lower as θ increases. The magnetic field of undulator device should be sufficiently weak so that particles moving in it are not going to be deflected away from observer's beam. The radiation received by the observer is a coherent superposition of radiation from all the periods and is monochromatic if the number of periods is large. For a typical undulator λ_u4 cm and γ 10^3, the radiation is in X-ray wavelengths. But if weseparate those magnets as shown in Figure <ref> with thousand kilometers away from each other, the output would be in the radio band, in which FRBs are observed. The spectral width of radiation received at a given position is about 1/N of the observing frequency, where N is the number of periods in the undulator. The dynamic spectrum width of FRBs is 1 MHz; it takes 10^3 magnetic periods to produce such a narrow spectrum. Consider the electric field emitted in a weak plane undulator in Figure <ref> by a beam of electrons traveling along the z-axis. What's the difference between the emission observed by observer A and B from a different angle? Assume the observer is at a large distance and we only consider the first harmonic:* At point A, as θ =0, f≈ 2γ^2(c/λ_u), the emission frequency is proportional to γ^2.γ usually has a small spread in distribution, because particles often have an energy spread. Hence, the radiation is not monochrome, while the intensity of radiation reaches its maximum at θ = 0.This case is like FRB 150807. * At point B, where θ≫ 1/γ, f≈ 2/θ^2(c/λ_u), the emission frequency is solely determined by θ; Particles traveling at the same angle radiate at the same frequency even at different speed. This radiation is monochromatic. If it is in the near field, the peak frequency is in the off-axis direction <cit.>. The intensity in large angle will be still large <cit.>, but due to spectrum broadening the intensity is lower than that of A. This case is like FRB 110523. Not only is the chromatic property of undulator radiation similar to the diffraction of a plane wave by a narrow slit there is also an interesting analogy between the intensity distribution produced by a slit diffraction and the off-axis undulator radiation. We will discuss this in Section <ref>.§ DISPERSION MEASURE ANDROTATION OF THE SOURCE Although it is not certain that the overly large DMs of FRBs partly originate from the rotation of the source itself, we wish to propose a model which assumes this. The rotation is not only important for the production of the ultrarelativistic particle beam in this model, but also for the generation of large-scale periodic magnetic structures. The rotation period predicted by this model is in the same order of magnitude of the periods of variable stars, which are found by <cit.> and <cit.> and proposed of the sources of FRBs. The model is illustrated in Figure <ref>. Charged particles are accelerated to a relativistic speed in a rotating source. The outgoing direction of particles is changing at the same angular speed as that of the source. The magnetic field in acceleration region is strong. Undulator radiation can not be produced here. After traveling some distance, the particles enter a weak and axial symmetric magnetic field with many periods, as illustrated by the solid curves in Figure <ref>. The field variation of each period is not necessarily sinusoidal. Kick-like fields from magnetic discontinuities also work as long as they cause periodic deflections of particles. Undulator radiation is produced here. Generated slight earlier, beam A will produce an emission earlier than beam B. Radiating at a smaller angle (θ_1 < θ_2) to the observer, beam A will produce an emission with higher frequency than that of beam B. This is how the time delay from the high-frequency radiation to the low-frequency is produced. Note this is an ultra-relativistic case; we must distinguish the emitter time from observer time. Because particles are chasing the wave front, a pulse lasting one second is produced by the source in ∼ 2γ^2 seconds. However, a simple frequency decreasing due to the uniform rotation is not enough to produce a dispersive like time delay. Our simulation of FRB 110703 (shown in Figure <ref>) shows that an angular acceleration speed should exist. This type of none-uniform angular speed usually exists in orbital motion.With the approximation f≈ 2/θ^2(c/λ_u), an estimator of the rotation period of the source can be given by: P̂=2π/ω̅=4148.808s×DM/(pc cm^-3)×π√(%s/%s)2λ_uc(√(%s/%s)1ν_l+√(%s/%s)1ν_u)(1/ν_l+1/ν_u) Where ν_l denotes the lower limit of the observation frequency in the unit of MHz,ν_u the upper limit, λ_u the length of a magnetic period.If DM is 1103 pc cm^-3(as of FRB 110703) and λ_u is between 10 km and 10000 km, then the rotation period of the source is between 2 hours and 3.4 days. This range is coincident with the period range of the variable stars observed by <cit.> and <cit.>. They found three variable stars near the location of FRB 110703, FRB 110627 and FRB 010621, two of which are low mass contact binaries, the other one is a slowly pulsating B-star <cit.>. The periods of these stars are from 7.8 hours to 2.5 days. They proposed that these variable stars are the sources of the corresponding FRBs. Without a priori knowledge of λ_u, we are not able to verify their proposal by estimating period from λ_u, but with a known period of the star, we can estimate λ_ufrom Equation <ref>. The λ_u of FRB 110703 is about 90 km. The observation angle θ increases from 7.22 arcmin to 7.56 arcmin.This model can explain why the DMs of FRBs are so large. The reason is that the rotation also contributes to the time delay between the high-frequency and low-frequency radiation. However, it can not explain why the time delay is dispersive like. A selection effect seems to exist that only when the orbital motion produces a time delay ∼ν^-2, the event is identified as an FRB.§ PULSE BROADENING AND NEAR FIELD EFFECTS The pulse shape of FRBs carries the most important clues to the radiation process. Several bursts of the repeating fast radio burst FRB 121102 <cit.> have multi peaks. FRB 130729 <cit.> has a right peak lower than the left one. And there is a variety of pulses from different FRBs with different asymmetries in shapes. From the spectrum point of view, the dynamic spectra of these FRBs are variable.We will never be able to give an explanation to them unless we find a mechanism capable of producing a diversity of spectra in this radio band.The suggestive clue regarding the radiation mechanism comes from the double-peaked pulse of FRB 121002 <cit.>. Its pulse contains two peaks. The dynamic spectrum also consists of two peaks separated each other from the central frequency. For undulator radiation, such a splitting of spectrum suggests near-field effects <cit.>. Walker modeled the effects analytically and numerically. The double-peaked spectrum appears when the light from different parts of the source develop a path length difference of ∼ 3/4 wavelength (W=3 in Figure <ref>) at the observer position.Near-field effects will broaden the spectrum, split the spectrum into several peaks. It will affect both the spectral width and the angular spread of spectrum. Our explanation is the first work applies it to explain an astrophysical phenomenon. We prefer to give a brief explanation to this effect in the context of the radio radiation produced in stellar magnetic structures. Near field effects are described by the parameter:W=L^2θ^2/2λ Din which L is the total length of the periodic magnetic structure. λ is the wavelength of radiation. D is the distance from the source to observer and θ is the observation angle. Lθ is the apparent size of the magnetic structure. It is equivalent to the width of a single slit. When the width (Lθ) is larger than the geometric average of wavelength and distance, near-field effects will occur. The observer could not be thought of at infinity and the electromagnetic wave should be treated as a spherical wave.If W=1, a π/2phase difference is among the output radiation.Walker's analysis in the phase space shows that both spectrum and angular distribution in a near field case can be reduced to a dimensionless intensity distribution of Fresnel diffraction. We replot his result in Figure <ref>, where ω_1 is the central frequency given by Equation <ref> for a given observation angle θ, ω is the small deviation from ω_1 . For the spectrum distribution, the dimensionless X coordinate is N/√(W)ω/ω_1, it is equal to the distance from a screen point to the center of a single slit diffraction pattern. The reduced width of a slit is 2√(W). Different W means different slit width. The double-peaked spectrum of FRB 121002 is equivalent to an interference fringe we see in a slit diffraction pattern when W=3. All curves in Figure <ref> are symmetric because Walker's analysis in phase space didn't consider the angular variation of intensity. With this simplification, the angular distribution and the spectrum distribution are identical, i.e., the spectral shape in Figure <ref> is also the shape of the corresponding pulse, which is true for FRB 121002. His numeric simulation taking the variation of intensity into account shows that the spectrum is generally asymmetric. It depends on the observation angle and harmonics. The pulse shapes ofthe No. 5, 7 and 10 burst of FRB 121102 <cit.> are comparable to some angular distributions given by the numeric simulation for W>4.With large angle approximation, W=N^2λ_u/D. where N is the number of magnetic periods. W is mainly determined by N, because it is proportional to the square of N. If N is sufficiently large, we will observe near-field effects even from the source at a cosmological distance. If the repeating FRB 121102 is really at a distance of 972 MPc <cit.>. For W=4 and λ_u=90 km, N is ∼ 3.65×10^10. The size of the source is ∼ 0.35 light year.Near field effects provide a competing interpretation to the pulse profile of FRBs, other than the scattering broadening. No matter which process is really behind FRB 121102, the mathematical form of the process likely includes a Fresnel integral. These effects also add a new mechanism to explaining the spectral structure inthe GHz and MHz band astronomical observation.§ DISCUSSION§.§ verification of the mechanismA simple way to verify the mechanism proposed by us is to check whether there is a positive frequency sweep ahead of the negative frequency sweep. The intensity distribution of undulator radiation is symmetric to the beam <cit.>. We will be swept firstly by a cone whose low-frequency radiation reaches us first. §.§ density modulation of the beamNot only a periodic magnetic field distributioncan increase the spectral flux, but a periodic density distribution of particles can do. The latter one is called a density modulation. We'll use the sparkle structure observed in the strongest FRB 150807 <cit.> as an example to explain the idea. The observation ofFRB 150807 shows many 100 kHz sparkles. For a GHz radiation, 100 kHz bandwidth means the coherence length is about ∼ 10^4 λ. Two processes will produce the sparkles: * A beam of particles passes througha magnetic structure with ∼ 10^4 periods. This is a process in a usual undulator device, which we've already discussed. * A beam of particles, divided into 10^4 bunches and separated each other by λ, passes through a magnetic structure with only a few periods. This is a process with a density modulation involved, which hasn't been discussed.The bandwidths of the radiation produced byboth processes above are equal to each other. The output energy of the latter one will greatly increase by the order of particle number. The latter scheme is used in a free-electron laser (FEL) experiment to increase pulse energy by 10^10 times <cit.>. The density modulation produced by a natural process may be much less perfect than the modulation achieved by an FEL experiment. But as long as the modulation exist, it will increase the spectral flux. Some giant pulses from the Crab pulsar can exceed 2MJy <cit.>. The intensity of the FRB 150807 sparkles also exceeds 1 KJy. Such a variation of intensity by orders implies a density modulation exist. The magnitudes of events are decided by how much particles are modulated and involved in the coherent process. The low occurrence rate of FRBs is similar to the giant pulses from young pulsars. They are conjectured to be the same things <cit.>. From the density modulation point of view, the low occurrence rate reflects the difficulty for a natural process in binary stars or a neutron star to produce a perfectly modulated beam like the one produced ina man-made FEL. These two events are not necessarily the same things. Their requirements for a perfectly density modulated beam are same to each other. So both the low occurrence rate and the uncommonly high intensity of radiation support the existence of a density modulation of the beam.§.§ periodic magnetic structureThe large-scale periodic magnetic structure will be easily destroyed by turbulence.Then what is the periodic magnetic structure of our model in practice? Could it be generated in stellar space?In our opinion,for a magnetic structure with λ_u ∼ 90 km,even if it isaround the Sun, the closest star to us, we are not able to observe it with current technology and can't prove the existence of the structure by observation. On the contrary, the mechanism proposed in this letter provides a method to infer it. The information we can tell from the polarization of radiation is: * Some FRBs are produced inplanar magnetic structures so that their polarizations are linear. In interplanetary space, several sinusoidal magnetic periods areusually ahead of a shock <cit.> driven by a magnetic cloud. The magnetic field there is planar. Combining with a density modulated beam, a monochromatic and linear polarized radiation can be produced.* Some periodic structures are not planar. So the radiation produced in them are not polarized. They may be periodically distributed discontinuities or plasma oscillations frequently generated everywhere.The magnetic structure is not going to be too long if only a density modulation exists.This research is supported by the Opening Project of Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences. It is also supported by NSFC 11533009, U1631135 and 11203083. [Champion et al.(2016)]cha16 Champion, D. J., et al. 2016, Monthly Notices of the Royal Astronomical Society, 460, L30 [Chatterjee et al.(2017)]cha17 Chatterjee, S., et al. 2017, Nature, 541, 58[Dai et al.(2016)]dai16 Dai, Z. G., Wang, J. S., Wu, X. F., & Huang, Y. F. 2016, , 829, 27 [De Cat(2007)]de2007 De Cat, P. 2007, Communications in Asteroseismology, 150, 167 [Feng et al.(2008)]Feng08 Feng, H. Q., Lin, C. C., Chao, J. K., et al. 2008, Journal of Geophysical Research (Space Physics), 113, A05216 [Geloni et al.(2010)]Gel10 Geloni, G., Saldin, E., Samoylova, L., et al. 2010, New Journal of Physics, 12, 035021 [Hankins & Eilek(2007)]Han07 Hankins, T. H., & Eilek, J. A. 2007, , 670, 693 [Hirai et al.(1984)]hir84Hirai, Y., Luccio, A., & Yu, L.-H. 1984, Journal of Applied Physics, 55, 25 [Jackson(1999)]jac99Jackson, J. D., 1999, Classical Electrodynamics (3rd ed.: Wiley, New York) [Katz(2016)]kat16 Katz, J. I. 2016, , 818, 19 [Keane et al.(2012)]kea12 Keane, E. F., Stappers, B. W., Kramer, M., & Lyne, A. G. 2012, , 425, L71[Kramer et al.(2006)]kram06 Kramer, M., Lyne, A. G., O'Brien, J. T., Jordan, C. A., & Lorimer, D. R. 2006, Science, 312, 549 [Loeb et al.(2014)]loe14Loeb, A., Shvartzvald, Y., & Maoz, D. 2014, Monthly Notices of the Royal Astronomical Society, 439, L46 [Lorimer et al.(2007)]lor07Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777 [Maoz et al.(2015)]mao15 Maoz, D., et al. 2015, Monthly Notices of the Royal Astronomical Society, 454, 2183 [Masui et al.(2015)]mas15 Masui, K., et al. 2015, Nature, 528, 523 [McLaughlin et al.(2006)]mcl06 McLaughlin, M. A., Lyne, A. G., Lorimer, D. R., et al. 2006, , 439, 817 [Mossessian & Heimann(1995)]mos95Mossessian, D. A., & Heimann, P. A. 1995, Review of Scientific Instruments, 66, 5153 [Motz(1951)]mot51 Motz, H. 1951, J. Appl. Phys. 22, 527 [Petroff et al.(2016)]pet16Petroff, E., et al. 2016, Publications of the Astronomical Society of Australia, 33 [Ravi et al.(2016)]rav16Ravi, V., et al. 2016, Science, 354, 1249 [Spitler et al.(2016)]spi16Spitler, L. G., et al. 2016, Nature, 531, 202 [Tendulkar et al.(2017)]ten17Tendulkar, S. P., et al. 2017, The Astrophysical Journal Letters, 834 [Thornton et al.(2013)]tho13Thornton, D., et al. 2013, Science, 341, 53 [Walker(1988)]wal88Walker, R. P. 1988, Nuclear Instruments and Methods in Physics Research A, 267, 537 | http://arxiv.org/abs/1707.08979v2 | {
"authors": [
"Qiwu Song",
"Yu Huang",
"Hengqiang Feng",
"Lei Yang",
"Tuanhui Zhou",
"Qingyu Luo",
"Tengfei Song",
"Xuefei Zhang",
"Yu Liu",
"Guangli Huang"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170727180223",
"title": "The radiation mechanism of fast radio bursts"
} |
α β̱ ε̧ δ̣ ϵ ϕ γ θ κ̨ łλ μ ν ψ ∂ ρ̊ σ τ ῠ φ̌ ω ξ η ζ Δ Γ Θ̋ ŁΛ Φ Ψ Σødiag Spin SOØ O SU U Sp SL trM_ PlInt. J. Mod. Phys. Mod. Phys. Lett. Nucl. Phys. Phys. Lett. Phys. Rev. Phys. Rev. Lett. Prog. Theor. Phys. Z. Phys. df BHinf inf evap eq smM_ Pl GeVDepartment of Physics, University of California, Berkeley, California 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA In supersymmetric theories, the gravitino is abundantly produced in the early Universe from thermal scattering, resulting in a strong upper bound on the reheat temperature after inflation. We point out that the gravitino problem may be absent or very mild due to the early dynamics of a supersymmetry breaking field, i.e. a sgoldstino.In models of low scale mediation, the field value of the sgoldstino determines the mediation scale and is in general different in the early Universe from the present one. A large initial field value since the era of the inflationary reheating suppresses the gravitino production significantly. We investigate in detail the cosmological evolution of the sgoldstino and show that the reheat temperature may be much higher than the conventional upper bound, restoring the compatibility with thermal leptogenesis.Gravitino Production Suppressed by Dynamics of SgoldstinoKeisuke Harigaya December 30, 2023 =========================================================== § INTRODUCTION One of the most challenging puzzles in the standard model is the hierarchy problem, in which the Higgs mass is unstable against quantum corrections at high energy scales. As one of the most motivated solutions, supersymmetry (SUSY) ensures the cancellation of quantum corrections between the SM particles and their superpartners, which considerably relaxes the hierarchy problem <cit.>. On the other hand, gauge coupling unification at a high energy scale gives strong hints to the Grand Unified Theories (GUTs). Remarkably, supersymmetric GUTs do not suffer from the proton decay problem faced by the standard model GUTs, and further improve the precision of gauge coupling unification <cit.>. Despite all the successes in particle physics, supersymmetry is known to create cosmological difficulties. In the case of low scale mediation of supersymmetry breaking such as gauge mediation, the gravitino is much lighter than the weak scale and is often the lightest supersymmetric particle. The gravitino is abundantly produced from the scattering of the thermalized particles in the early Universe <cit.>. In order not to overproduce gravitino dark matter, the reheat temperature after inflation T_R must be sufficiently low, T_R 10^6 GeV(m_3/2/ GeV) <cit.>, where m_3/2 is the gravitino mass, and this bound strongly restricts the cosmological history including inflation models and baryogenesis. Especially, T_ R < 10^9 GeV is in conflict with thermal leptogenesis <cit.>. This is known as the gravitino problem in low scale mediation of supersymmetry breaking.Several solutions have been considered so far. One may assume a non-conventional cosmology model with a large amount of dilution from the decay of a long-lived particle <cit.>. The large entropy production needed to reproduce the observed dark matter abundance also dilutes away the baryon asymmetry created previously, which calls for an efficient mechanism of baryogenesis. For example, the observed baryon asymmetry can be explained by thermal leptogenesis only if the reheat temperature is extremely high, T_R 10^16 GeV (m_3/2 /GeV). Refs. <cit.> introduce a low messenger scale and a small coupling between the goldstino component of the gravitino and the messenger. The gravitino production is then suppressed at a temperature higher than the messenger scale. The suppressed production helps reduce the dilution factor neededand thus relaxes the stringent lower bound on the reheat temperature from thermal leptogenesis. A different solution in Ref. <cit.> involves an additional field whose field value determines the coupling between the messenger and the goldstino. By a smaller field value and thus a smaller coupling in the early Universe, the upper bound on the reheat temperature is relaxed.The interaction rate between the thermal bath and the gravitino is suppressed by the mediation scale, which is given by the field value of the scalar component of the SUSY breaking field (sgoldstino). We point out that if the sgoldstino potential is flat enough, the field value may be large in the early Universe, suppressing the gravitino production. We study the dynamics of the sgoldstino including thermal effects, and find that the reheat temperature may be much higher than the conventional upper bound. The compatibility of our scenario with thermal leptogenesis is also investigated. We emphasize that this suppression mechanism is a result of a thorough analysis of the dynamics of the existing fields necessary for low scale mediation, and can be applicable to a broad class of models with a sufficiently flat sgoldstino potential. § REVIEW OF THE GRAVITINO PROBLEM IN GAUGE MEDIATION We first review gauge mediation and the production of gravitinos from the thermal bath. The SUSY breaking field S is coupled to the messenger field Q and Q̅ via the superpotential termW = y S Q Q̅,which in turn generates the following term in the Lagrangian when Q is integrated outℒ = ∑_i ∫ dθ^2 α_i/4πS/v_S W_i^α W_iα,where i is summed over (U(1), SU(2), SU(3)) and v_S is the vev of the scalar component of S. Here we assume that Q and Q̅ form a complete multiplet of SU(5) GUT. We parametrize the F term of S asF_S = k √(3) m_3/2 M_Pl,where k ≤ 1 parametrizes the fractional contribution to SUSY breaking and M_Pl = 2.4 × 10^18 GeV is the reduced Planck mass. The gaugino mass is then given bym_i = α_i/4πF_S/v_S = √(3)α_i/4πk m_3/2 M_Pl/v_S . The viable parameter space is as follows. To prevent Q from being tachyonic, we require y ≥( 4π/α_i)^2 m_i^2/k √(3) m_3/2 M_Pl,while to ensure that the quantum corrections to the S mass do not exceed its vacuum massΔ m_S^2 = y^2/16π^2k^2 m_3/2^2 M_Pl^2/v_S^2 < m_S^2,we impose the condition that y ≤α_i m_S/m_i.The consistency between Eqs. (<ref>) and (<ref>) holds only if k > 16π^2/α_i^3m_i^3/√(3) m_3/2 M_Pl m_S .In the class of models where the low energy effective superpotential of S is given by W ≃√(3) k m_3/2 M_PlS, [This is not the case, for example, in a model of indirect gauge mediation with a superpotential W = λ S ϕ_1 ϕ_2 and the fields ϕ_1 and ϕ_2 obtain negative soft masses by a coupling with a SUSY breaking sector. The masses of ϕ_1 and ϕ_2 are as large as that of S, and we may not integrate them out.] the supergravity effect generates a tadpole term of S, V(S) = - √(3) k m_3/2^2 M_Pl S. The tadpole term places a minimum on the vev today, v_S, unless the vev is fine tuned. This translates into a lower bound on the S massm_S10 ( m_3 m_3/2)^1.011/2≃ 300( m_3/ TeV)^1.011/2( m_3/2/)^1.011/2. The functional form of the gravitino abundance producedat a temperature T is derived as follows;ρ_3/2/s≃m_3/2 n_i^2 σ_i v/H s≃m_3/2 k^2 M_Pl/v_S^2 T ≃(4π/α_i)^2 m_i^2/3 m_3/2M_Pl T ,where ρ and s are the energy and entropy density respectively, while σ_i refers to the scattering cross section between the gravitino and the gaugino/gauge boson, which follows the thermal equilibrium number density n_i. Here we assume that the temperature is sufficiently small so that the gravitino is not thermalized. As can be seen in Eq. (<ref>), the production mode by thermal scattering is dominated at higher temperature, which we call “UV dominated," and peaked at the reheat temperature after inflation T_R. The precise result of Eq. (<ref>) is derived e.g. in Refs. <cit.>, which translates into the constraint on T_R T_R ≤ 5 × 10^6( m_3/2/) (TeV/m_3)^2 ≡ T_ co.For m_3/2 1 MeV the upper bound is smaller than the typical gaugino mass, invalidating Eq. (<ref>).The spin-3/2 component of the gravitino is also produced from the thermal bath via Planck-scale suppressed interactions. Using the result in Ref. <cit.>, we obtain an upper bound on T_R, T_R ≤ 2× 10^12 GeV(GeV/m_3/2).Although the constraint is much weaker than the one in Eq. (<ref>), it will be important in our mechanism where the production of the spin-1/2 component is suppressed. § SGOLDSTINO DYNAMICS AS A SOLUTION We propose a new cosmological scenario of gauge mediation where the gravitino problem is much milder. In Eq. (<ref>), it is assumed that v_S has been a constant from the inflationary reheating until today. This is, however, not necessarily the case. In this section, we explore the possibility that the field value v_S(T) of the sgoldstino evolves with the temperature according to its potential energy V(S). In particular, we consider the case where the initial field value of the sgoldstino, v_S0, is much larger than today's vev v_S. Based on Eq. (<ref>), a large initial field value results in the suppression of the gravitino interaction with the thermal bath in the early Universe.We refer readers to Ref. <cit.> and the references therein for discussions on the evolution of a scalar field in the early Universe including thermal effects.We can parametrize the temperature dependence of the sgoldstino oscillation amplitude as v_S(T) ∝ T^n. It is striking that the gravitino production from thermal scattering given in Eq. (<ref>) is dominated at a lower temperature, which we call “IR dominated," for any n>1/2, which is easily satisfied by the typical polynomial and logarithmic potentials. As a result, the gravitino production is insensitive to the reheat temperature. In the case with no dilution from entropy production, the conventional constraint on the reheat temperature, T_ co, can be evaded as long as the combination T/v_S^2(T) in Eq. (<ref>) never exceeds T_ co/v_S^2 for any T. In general, the constraint with dilution ismax(T/v_S^2(T)) 1/D≤T_ co/v_S^2 ,where max(f(T)) refers to the maximum value of f(T) throughout the cosmological evolution. This is more likely the case for quadratic and logarithmic potentials because steeper potentials lead to a smaller initial field value of S as well as an earlier onset of the oscillation.§.§ Evolution of the Sgoldstino FieldWe first consider the case where the sgoldstino field begins to oscillate via thermal effects. Through the coupling with S in Eq. (<ref>), Q obtains a large mass from the large field value of the sgoldstino and further generates the thermal logarithmic potential for S V_ th(S) = a_0α_3(T)^2 T^4 ln(y^2 S^2/T^2) ,where a_0 is a constant of order unity <cit.> and the logarithmic temperature dependence of α_3(T) will be neglected for simplicity. Here it is assumed that the messenger mass is larger than the temperature and we verify that this is true in the entire allowed parameter space. The condition for the onset of the oscillations during inflationary reheating is given by V_ th” (v_S0)H^2, which leads toα_3 T^2/v_S0√(π^2 g_*/90)T^4/T_R^2 M_Pl,where g_* is the effective number of relativistic species. [Here it is assumed that the radiation produced by the decay of the inflaton is thermalized and follows thermodynamics. See <cit.> and the references therein for discussion on the thermalization process.] The oscillation temperature readsT_ osc≃ T_R ( 90/π^2 g_*)^1.011/4√(α_3 M_Pl/v_S0) .We defineδ≡(π^2 g_*/90)^1.013/8v_S0/α_3 M_Plto parametrize the initial field value and this particular definition of δ simplifies the numerical pre-factors in the following derivations. Here it is implicitly assumed that v_S0α_3 M_Pl√(90/π^2g_*) and T_R > √(m_S v_S0/α_3),so that the sgoldstino begins its oscillation by the thermal logarithmic potential. Ifone of these conditions is violated, the sgoldstino begins its oscillation via its temperature independent potential. The evolution of the sgoldstino for that case is discussed later. The amplitude of the oscillation, v_S(T), evolves as follows. The mass of S is given by α_3 T^2 / v_S(T). Then the number density of S is proportional to T^2 v_S(T), which decreases with a^-3. During the inflaton dominated era and the radiation dominated era a^-3∝ T^8 and T^3, and hence v_S(T) ∝ T^6 and T, respectively. The field value of the sgoldstino at the reheat temperature is then given byv_S(T_R) = v_S0( T_R/T_ osc)^6 ≃δ^4 α_3 M_Pl.After reheating, the field value evolves asv_S(T) = v_S(T_R) T/T_R≃δ^4 α_3 M_PlT/T_R. In the above analysis, we assume that reheating is caused by a perturbative decay of the inflaton. It is also possible that the reheating is caused by other dynamics such as the scattering with the thermal bath. In this case the relation between the initial field value of the sgoldstino and the field value at T_R is different from Eq. (<ref>). It is also possible that a large Hubble induced mass term of the sgoldstino causes non-trivial dynamics of the sgoldstino before the completion of reheating. For those cases, one may still use δ^4 to parametrize v_S(T_R) without changing the discussion below. As the temperature drops, the thermal logarithmic potential in Eq. (<ref>) becomes less effective and eventually becomes subdominant to the vacuum potential. To be concrete, we assume that the vacuum potential is given by a simple quadratic one,V_ vac(S) = m_S^2 |S- v_S|^2 (≃ m_S^2 |S|^2 for v_S(T) ≫ v_S).The transition to the quadratic potential occurs at the temperature T_2 defined by V_ th(v_S(T_2)) = V_ vac(v_S(T_2)),T_2 ≃δ^4 m_S M_Pl/T_R .Note that T_2 < T_R as long as the conditions in Eq. (<ref>) are satisfied. We now quantify v_S(T_2) in relation to v_S. This will tell us whether the gravitino production actually becomes enhanced instead by v_S(T) < v_S because the sgoldstino oscillates around the minimum at the origin set by V_ th(v_S(T)). v_S(T_2)/v_S = v_S(T_R)/v_ST_2/T_R = δ^8 4π/√(3)M_Pl m_S m_3/k T_R^2 m_3/2 ≃ 5 δ^8/k( 10^12 /T_R)^2 (m_S/300) ( m_3/ TeV)^1.011/2( /m_3/2).When this ratio is larger than unity, which is the case for the most of the allowed parameter space, before v_S(T) drops to v_S, S starts to follow V_ vac(S) and oscillates around the minimum today v_S. After T_2, v_S(T) continues to decrease as T^3/2 until the temperature T_S, at which the amplitude is as large as the vev v_S. Using v_S(T_S) = v_S, one obtainsT_S = T_2 ( v_S/v_S(T_2))^1.012/3≃ 2 × 10^8( k/δ^2)^1.012/3( T_R/10^12 )^1.011/3(m_S/300)^1.011/3(TeV/m_3)^1.012/3( m_3/2/)^1.012/3 .When the ratio is smaller than unity, v_S (T) drops below v_S. After T_2, S follows V_ vac(S) and oscillates around the minimum today v_S. After the few oscillations by V_ vac(S), v_S(T) increases and quickly becomes as large as v_S.When the initial field value of the sgoldstino is large or the reheat temperature is small, the sgoldstino begins its oscillation by the quadratic potential V_ vac(S), rather than the thermal potential V_ th(S). This occurs if the condition in Eq. (<ref>) is violated, namely,v_S0 > α_3 M_Pl√(90/π^2g_*) or T_R <√(m_S v_S0/α_3).The oscillation temperature becomes independent of the initial amplitude and readsT_ osc = ( 90/π^2 g_*)^1.011/8( m_S M_Pl T_R^2 )^1.011/4,where T_ osc > T_R is assumed. The field value of the sgoldstino at the reheat temperature is given byv_S(T_R) = δ( π^2 g_*/90)^1.011/8α_3 T_R^2/m_S.After reheating, the field value evolves asv_S(T) = v_S(T_R) (T/T_R)^1.013/2=δ( π^2 g_*/90)^1.011/8α_3 T^3/2 T_R^1/2/m_Sand reaches v_S at the temperatureT_S = T_R ( v_S/v_S(T_R))^1.012/3≃ 8 × 10^7( k/δ)^1.012/3( 10^10 /T_R)^1.011/3(m_S/300)^1.012/3(TeV/m_3)^1.012/3( m_3/2/)^1.012/3 . The sgoldstino eventually delivers all its energy to radiation by scattering with (decaying to) thermal particles at the destruction (decay) temperature T_ des (T_ dec), which is discussed in Sec. <ref> (<ref>).Fig. <ref> summarizes the sgoldstino evolution and serves as the schematic picture of the suppression mechanism. The vertical axis shows the gravitino abundance produced per Hubble time at a given temperature in the horizontal axis. The conventional production is UV dominated and thus we need T_R ≤ T_ co to avoid overproduction. In the case where v_S(T) is decreasing faster than T^1/2, the suppressed production becomes IR dominated, as seen in high temperature behavior. The kinks of the colored lines occur at T_2, as the temperature dependence of v_S(T) changes from Eq. (<ref>) to Eq. (<ref>) at the transition from the thermal logarithmic potential to the quadratic one. The abrupt change of the lines at T_ des orginates from the fact that the field value of the sgoldstino is suddenly set to the vev today when the condensate is destroyed by thermal scattering. In the case of the orange line, the observed dark matter abundance is reproduced. On the other hand, the blue line represents underproduction. §.§ Destruction of Sgoldstinos by Thermal Scattering The discussion in Sec. <ref> assumes that the sgoldstino condensate is intact throughout its evolution. However, due to its coupling with the messenger Q, the sgoldstino scatters with thermalized particles at the following rate given in Refs. <cit.> Γ_ scatt = ( T(Q)/16π^2)^2 (12π)^2/ln(α_3^-1)α_3^2 T^3/v_S^2(T)≡ b α_3^2 T^3/v_S^2(T) ,where T(Q) is the index of Q's representation of SU(3) and we take T(Q) = 1/2. The condensate is destroyed whenever the scattering rate becomes larger than the Hubble rate. The temperature at which such destruction occurs is called T_ des. §.§.§ Sgoldstino Oscillations Driven by Thermal EffectsWe first explore the case where the sgoldstino begins to oscillate via the thermal logarithmic potential. Overproduction of gravitinos excludes the possibility where the sgoldstino condensate is destroyed before the quadratic potential dominates, i.e. T_ des > T_2. This is because for such a case the field S is trapped at the origin, making Q massless and thermalized and greatly enhancing the gravitino production rate. [If the Yukawa coupling y is sufficiently small, the produced gravitino abundance is not necessarily very large and thus the entropy production by the sgoldstino trapped at the origin may have sufficient dilution for the gravitino abundance <cit.>. We however do not consider this scenario in this paper.] Requiring Γ_ scatt(T_2) < H(T_2) givesT_R 10^14δ^4 (m_S/300)^1.011/3 .As the condensate is destroyed, the sgoldstino is driven to the local minimum of the potential. In order for S=v_S to be the local minimum at the temperature T_ des, the thermal mass from the messenger should be small enough,y T_ des<m_S .This upper bound on y should be consistent with the lower bound in Eq. (<ref>). Below T_2, on the other hand, Γ_ scatt(T)/H is IR dominated only before T_S. This implies either that T_ des > T_S or that there is no destruction by scattering. In order to distinguish our mechanism from other solutions of the gravitino problem, we first explore the parameter space where the sgoldstino condensate does not produce entropy. We take v_S(T) = v_S (T/T_S)^3/2 because T_2 > T_ des > T_S and derive the destruction temperatureT_ des≃6 × 10^5 δ^-2( T_R/10^12 )^1.011/2(m_S/300)^1.011/2 .To be consistent with T_ des > T_S so that the sgoldstino is successfully destroyed, one requiresk10^-4 δ^-1( T_R/10^12 )^1.011/4(m_S/300)^1.011/4( m_3/ TeV)( /m_3/2) .In the case where k < 1, there should be another SUSY breaking field. If the scalar component of that SUSY breaking field is excited in the early Universe, its decay may also produce gravitinos. To avoid cosmological complications, we assume that this scalar component has a positive Hubble-induced mass and/or efficiently decays into hidden sector fields other than the gravitino.According to Eq. (<ref>), it is required that T_ des≤ T_ co to avoid overproduction of gravitinos. Thiscondition is satisfied whenT_R 10^13δ^4 (300/m_S) ( m_3/2/)^2 (TeV/m_3)^4.Furthermore, to identify the parameter space with no dilution, we need to ensure that the sgoldstino condensate is destroyed before its energy density dominates over radiation. We can estimate the temperature T_M^ (th) at which the matter energy density dominates over that of radiationπ^2/30 g_* (T_M^ (th))^4 = m_S^2 v_S^2(T_2) ( T_M^ (th)/T_2)^3and the result readsT_M^ (th) = 30/π^2α_3^2/g_* T_2 = δ^430/π^2α_3^2/g_*m_S M_Pl/T_R≃10^5 δ^4 ( 10^12 /T_R) (m_S/300) ,where we use α_3^2 T_2^4 = m_S^2 v_S^2(T_2) (the definition of T_2). No entropy is produced when T_ des > T_M^ (th), which is the case forT_R10^11δ^4 (m_S/300)^1.011/3. If the scattering is inefficient, the sgoldstino dominates the energy density of the Universe. After the sgoldstino dominates, destruction occurs via scattering with the thermal bath created from the condensation of the sgoldstino. The destruction temperature is derived in App. <ref> and reads T_ des≃ 3 × 10^5( m_S/300)^1.012/3.The dilution factor D via entropy production is given byD = T_M^ (th)/T_ des≃ 3δ^4( m_S/300)^1.011/3( 10^11 /T_R).The condition from Eq. (<ref>) is T_ des /D ≤ T_ co so the upper bound on T_R is relaxed toT_R10^12δ^4 ( m_3/2/) ( 300/m_S)^1.011/3(TeV/m_3)^2.§.§.§ Sgoldstino Oscillations Driven by Vacuum Potential We next explore the case where the sgoldstino begins to oscillate via the vacuum mass term. In the case where the condensate is a subdominant component, the destruction temperature is given byT_ des = 3^3/4 10^3/8√(b)/π ^3/4δ g_*^3/8m_S √(M_Pl)/√(T_R)≃ 3 × 10^5δ^-1( m_S/300) ( 10^9/T_R)^1.011/2 .One needs to require T_ des > T_S to ensure successful destruction, which limits k10^-4 δ^-1/2( m_S/300)^1.011/2( m_3/ TeV) ( /m_3/2) ( 10^9/T_R)^1.011/4.When the scattering rate is inefficient, the Universe enters the matter-dominated era at the temperature T_M^ (vac) = v_S0^2 T_R/3 M_Pl^2 = ( 90/π^2g_*)^1.013/4δ^2 α_3^2 T_R/3 .With the destruction temperature given in Eq. (<ref>), the dilution factor can be computedD = T_M^ (vac)/T_ des =10^1/4α _3^4/3δ^2 / 3^1/6√(π) g_*^1/4b^1/3T_R/( M_Pl m_S^2 )^1/3≃ 9δ^2 ( T_R/10^10 ) ( 300/m_S)^1.012/3 .Therefore, the sgoldstino does not produce entropy when T_R10^9δ^-2( m_S/300)^1.012/3 . The condition from Eq. (<ref>) is T_ des /D ≤ T_ co and places the following upper bound on T_R T_R2 × 10^10(0.6/δ)^2 ( 200 MeV/m_3/2) ( m_S/ TeV)^1.014/3( m_3/2 TeV)^2.Let us discuss the compatibility with thermal leptogenesis. The maximal baryon asymmetry Y_B,max that can be obtained from thermal leptogenesis in the units of that observed today Y_B,obs is given in Refs. <cit.> Y_B,max≃T_R/10^9 Y_B,obs.This baryon asymmetry may be subject to dilution from subsequent entropy production, which leads to a more stringent lower bound on T_R,T_R/D(T_R)10^9.There is a further constraint from the production of the fermion component of S, ψ_S for the following reason. Since we are currently concerned with the case where the sgoldstino condensate is destroyed by thermal scattering, a small k given in Eq. (<ref>) is assumed. For a small k, the production of ψ_S is enhanced by 1/k^2 compared to that of the gravitino. To avoid the gravitino overproduction from the decay of ψ_S, we require that the mass of ψ_S is larger than that of the lightest observable supersymmetric partner (LOSP) so that ψ_S can decay into the LOSP. [This requires a direct coupling between the SUSY breaking sector containing S and other SUSY breaking sector. Otherwise, ψ_S is a pseudo-goldstino and obtains a mass only as large as the gravitino.] We find that the LOSP from the decay of ψ_S immediately annihilate to the SM particles with a negligible amount decaying to the gravitino.The mass of the sgoldstino would not be much smaller than that of ψ_S, and thus we fix m_S = 1 TeV. We summarize the above discussions in Fig. <ref>. The light gray region is excluded by Eq. (<ref>) because the sgoldstino is destroyed by thermal scattering when the potential is still governed by the thermal logarithmic potential, whose minimum is at the origin of S. This results in a vanishing messenger scale and the gravitino is overproduced by the scattering of thermalized messengers. The orange region is excluded by Eqs. (<ref>) and (<ref>) because the sgoldstino destruction occurs too early and the mechanism fails to suppress the messenger scale. The dilution factor D is labeled by the black contours in the right panel, whereas the left panel does not have dilution for the chosen value of δ. The maximal value δ_ max≃ 0.67 is inferred from Eqs. (<ref>) and (<ref>). Above (below) the sharp kink of the orange boundary, the onset of the sgoldstino oscillation is driven by the thermal logarithmic (vacuum) potential, discussed in Secs. <ref> and <ref> respectively. The yellow region is excluded by Eq. (<ref>) because, at the time of sgoldstino destruction, the thermal mass dominates and the destruction will set the field value to the origin, resulting in the gravitino overproduction. Here k_c refers to the critical value in Eq. (<ref>) for successful destruction. The purple region is excluded by Eq. (<ref>) for overproduction of spin-3/2 gravitinos through supergravity interactions. We find that in the allowed parameter region the thermal leptogenesis can create an enough amount of the lepton asymmetry. §.§ Destruction of Sgoldstinos by Decay It is pointed out in Sec. <ref> that k has to be smaller than the critical value k_c given in Eqs. (<ref>), (<ref>) or (<ref>) in order for the sgoldstino condensate to be destroyed by thermal scattering. In this section, we assume a sufficiently large k, meaning that thermal scattering is never effective enough and instead the sgoldstino condensate eventually decays to particles in the thermal bath. The real and imaginary parts of S may have different decay modes <cit.>.In the phase convention where v_S is real, both the real and imaginary components of S can decay to a pair of gluons at a rateΓ_ dec^gg≃( α_3/4π)^2 m_S^3/8π v_S^2.The real component of S can also decay to Higgs/electroweak (EW) gauge bosons if kinematically allowed at a rateΓ_ dec^ h,W,Z≃1/8πm_H^4/m_S v_S^2 .Assuming m_S∼ TeV, this decay mode is more efficient than the one into gluons. The relative abundance of the real and imaginary parts depends on the phase of the initial field value v_S0. As the decay to Higgs is more efficient, the real component will decay before the imaginary one. As a result, the final decay temperature is mainly governed by the decay to gluons if the initial relative abundance is comparable or dominated by the imaginary component.To find the temperature T_ dec when the sgoldstino decays to gluons, one equates the decay rate with the Hubble rate and obtainsT_ dec^gg≃√(Γ_ dec^gg M_Pl)≃4 MeVk^-1( m_S/300)^1.013/2(100 MeV/m_3/2) ( m_3/ TeV) .The decay temperature then allows us to compute the dilution factor using Eq. (<ref>)D = T_M^ (th)/T_ dec^gg≃8 × 10^6 kδ^4 ( 10^12 /T_R) ( 300/m_S)^1.011/2( m_3/2/ 100 MeV) (TeV/m_3) .Since the decay occurs well after the field value of S settles to the minimum today v_S, the gravitino production peaks at T_S. With dilution, the constraint from the gravitino abundance using Eq. (<ref>) becomes T_S /D ≤ T_ co and givesT_R 5 × 10^15 k^1/4 δ^4 ( 300/m_S)^1.015/8( m_3/2/ 100 MeV)(TeV/m_3)^1.017/4 .With accidental suppression of the imaginary part of the initial field value or the presence of a CP violating mixing between the real and imaginary components of S, the decay temperature T_ dec is now determined by the larger of the rate into gluons and that into EW bosons. For the decays to H, W^±, and Z, we obtainT_ dec^ h,W,Z≃√(Γ_ dec^ h,W,Z M_Pl)≃5k^-1( m_H/ TeV)^2 ( 300/m_S)^1.011/2(100 MeV/m_3/2) ( m_3/ TeV). The dilution factor becomesD = T_M^ (th)/T_ dec^ h,W,Z≃6 × 10^3 kδ^4 ( 10^12 /T_R) (TeV/m_H)^2 ( m_S/300)^1.013/2( m_3/2/ 100 MeV) (TeV/m_3) .Finally, the constraint of the gravitino abundance from Eq. (<ref>) requires T_S /D ≤ T_ co, givingT_R 3 × 10^13 k^1/4 δ^4 (TeV/m_H)^1.013/2( m_S/300)^1.017/8( m_3/2/ 100 MeV)(TeV/m_3)^1.017/4.In the case where the condition in Eq. (<ref>) is violated, the sgoldstino starts oscillating via the vacuum potential V_ vac(S) rather than the thermal potential V_ th(S). The decay temperatures T_ dec calculated above do not change but the matter-domination temperature should become T_M^ (vac) in Eq. (<ref>). The dilution factors are modified as D = T_M^ (vac)/T_ dec^ gg≃ 8 × 10^8 kδ^2 ( T_R/10^10 ) ( 300/m_S)^1.013/2( m_3/2/ 100 MeV) (TeV/m_3) , D = T_M^ (vac)/T_ dec^ h,W,Z≃ 6 × 10^5 kδ^2 ( T_R/10^10 ) ( m_S/300)^1.011/2( m_3/2/ 100 MeV) (TeV/m_3) (TeV/m_H)^2 ,for the decays to into gluons and into Higgs and EW bosons, respectively.In addition, the sgoldstino can decay to a pair of gravitinos at a rate given by Ref. <cit.> Γ_ dec^3/2 = k^2 m_S^5/96 π^2 m_3/2^2 M_Pl^2,where k accounts for the mixing of the gravitino with ψ_S. Based on Eqs. (<ref>) and (<ref>), the sgoldstino dominates the energy density in the parameter space of interest. This implies that the sgoldstino decay can give a sizable contribution to the gravitino despite the small branching ratio B_3/2≡Γ_ dec^3/2/ Γ_ tot. For the decays to gluinos and to H, W^±, and Z respectively, the non-thermal gravitino abundance is estimated asρ_3/2/s =2 m_3/2ρ_S B_3/2/m_S s = k^2 m_S^4/48 π m_3/2 M_Pl T_ dec ≃0.6 eV( k/0.1)^3 ( m_S/300)^1.015/2(TeV/m_3) 0.04 eVk^3 (TeV/m_H)^2 ( m_S/300)^1.019/2(TeV/m_3) .The various constraints discussed in this section are shown in Fig. <ref>. In the left panels, we take m_S = m_Smin, which refers to the theoretical minimum given in Eq. (<ref>). k_ min is the theoretical lower bound in Eq. (<ref>), which excludes the cyan region. In the right panels, we take k = k_c, where k_c stands for the critical value of k in Eqs. (<ref>), (<ref>), and (<ref>) applicable for different ranges of T_R. In the light (dark) gray region, the sgolstino is necessarily destroyed by scattering at high temperatures because Γ_ scatt(T_2) > H(T_2) (k_c > 1), whose result is previously shown in Fig. <ref>. We use the maximal value δ_ max≃ 0.67 inferred from Eqs. (<ref>) and (<ref>). The dilution factors in Eqs. (<ref>) and (<ref>) are marked with the black contours. The green contours separate different cosmological evolutions, where v_S(T) in Eq. (<ref>) does (not) drop below v_S above (below) the contours. The orange region is excluded by Eqs. (<ref>) and (<ref>) because the sgoldstino field values drops to the today's value v_S too quickly such that the mechanism fails to suppress the gravitino production until the conventional constraint temperature T_ co. The brown region is excluded by Eq. (<ref>) because the gravitino produced from the sgoldstino decay overcloses the Universe. The purple region is excluded by Eq. (<ref>) due to overproduction of spin-3/2 gravitinos via supergravity interactions. The red regions are excluded as the decay of sgoldstinos occurs after and thus spoils Big Bang Nucleosynthesis (BBN) <cit.>. Regions below the blue contours are incompatible with thermal leptogenesis because the baryon asymmetry in Eq. (<ref>) is depleted by too large of a dilution factor. The blue contours do not extend into the light gray regions, where the dilution factor is unity. For a smaller δ, the orange region as well as the blue line shift downward. The lower bound on T_R from thermal leptogenesis is then relaxed, until the orange region catches up with the blue line.§ CONCLUSIONSWe have investigated the possibility that the sgoldstino has a large field value in the early Universe. This suppresses the early production of the gravitino and is expected to relax the upper bound on the reheat temperature after inflation. As a proof of principle, we analyze a specific case where the supersymmetry breaking field S and the messenger fields couple minimally via Eq. (<ref>) and the mass term governs the zero-temperature potential of the sgoldstino. The constraints on the gravitino mass and the reheat temperature are summarized in Figs. <ref>-<ref>. When the field S provides sufficiently subdominant supersymmetry breaking, the sgoldstino condensate is destroyed by thermal scattering without producing (much) entropy. The reheat temperature may be as large as 10^12 GeV, and thermal leptogenesis is viable as long as the reheat temperature is larger than 10^9 GeV. On the contrary, if thermal scattering is inefficient, the sgoldstino condensate decays late with entropy production. The gravitino problem is then solved both by the suppression of the gravitino production and by dilution from entropy production. For a given reheat temperature, the dilution factor required to obtain a small enough gravitino abundance is smaller in our mechanism than the conventional scenario with dilution but not suppression. As a result, the reheat temperature can be as high as 3× 10^13 GeV. When the sgoldstino field breaks supersymmetry subdominantly and later decays, thermal leptogenesis is possible with a reheat temperature T_R10^12`-13 GeV. Hence, there exist regions in the parameter space where thermal leptogenesis is viable and the gravitino problem is absent or much milder than previously claimed.§ ACKNOWLEDGEMENTThe authors thank Yiannis Dalianis, Lawrence Hall, and Aaron Pierce for fruitful discussions. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DE-AC02-05CH11231 and by the National Science Foundation under grants PHY-1316783 and PHY-1521446. The work of R.C. was in part supported by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number DE-SC0014664. The work of K.H. was in part performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. § DESTRUCTION TEMPERATURE AFTER MATTER-DOMINATED ERA In this section, we derive the destruction temperature in the case where the sgoldstino condensate dominates the energy of the Universe. First of all, we need to require again that v_S(T) is still decreasing before the condensate is destroyed, i.e. T_S < T_ des. Otherwise, the scattering rate is insufficient to destroy the sgoldstino condensate. Under this assumption, we first study the temperature dependence of the Hubble rate during the non-adiabatic era when the dominant source of the thermal bath is the scattering products of the sgoldstino as opposed to existing radiation. We define the temperature at the beginning of the non-adiabatic era as T_ NA. By conservation of energy transferred from the sgoldstino condensate to radiation, we writeρ_S Γ_ scatt/H = 3 M_Pl^2 H Γ_ scatt = π^2/30 g_* T^4 ,Γ_ scatt = b α_3^2T^3/v_S^2(T) = b α_3^2 m_S^2 T^3/3 H^2 M_Pl^2 ,where we repeatedly use the fact that the total energy density is dominated by the sgoldstino vacuum potential given in Eq. (<ref>), H = √(ρ_S/3) / M_Pl = m_S v_S(T) / √(3)M_Pl. The Hubble rate during this non-adiabatic phase is then given byH(T) = 30 bα_3^2 /π^2 g_* m_S^2/T.This demonstrates that the Hubble rate is inversely proportional to the temperature and that the temperature during the non-adiabatic phase is increasing over time. Asρ_S ∝ H^2(T) ∝ T^-2, one can compare this new scaling with the usual temperature dependence ρ_S ∝ T^3 during a radiation-dominated epoch and argue that the dilution factor is D = T_M / T_ des = ( T_ des / T_ NA)^5. We consider T_ des as the temperature at which H(T_ des) is determined by the radiation energy density ρ_R(T_ des) ∝ T_ des^4, which leads toT_ des =3^2/3√(10)α _3^2/3 b^1/3/π√(g_*)( M_Pl m_S^2 )^1.011/3≃ 3 × 10^5( m_S/300)^1.012/3.In fact, at the destruction temperature, the energy densities of the sgoldstino and radiation are comparable within a factor of a few, which allows us to compute the field value at T_ des v_S(T_ des) = √(π^2 g_*/30)T_ des^2/m_S = 3^5/6√(10)α _3^4/3 b^2/3/π√(g_*)( m_S M_Pl^2 )^1.011/3.The earlier assumption, T_S < T_ des, is equivalent to v_S(T_ des) > v_S, which places an upper bound on k k ≤4 × 3^2/3√(10)α _3^1/3 b^2/3/√(g_*)m_3/m_3/2( m_S/M_Pl)^1.011/3≃ 10^-4( m_S/300)^1.011/3( m_3/ TeV) ( /m_3/2).§ NON-PERTURBATIVE EFFECTS As the sgoldstino field oscillates with a large amplitude, v_S(T)>v_S, the messenger field may be produced in a non-perturbative way because of the rapid change of its mass. In the main sections, we assume the non-perturbative effect is negligible, which we will now justify.The mass of the messenger is given by m_Q^2 ≃ y^2 S^2 + g^2 T^2,where g is the gauge coupling constant. The adiabaticity of the mass of the messenger is characterized by the following quantity, q≡|ṁ_Q|/m_Q^2≃y^2 |S Ṡ|/(y^2 S^2 + g^2 T^2)^3/2 . When the sgoldstino oscillates with the thermal logarithmic potential, |Ṡ| ≃α T^2, and q is maximized around yS ∼ gT,qy / (4π) .As long as y< 𝒪(1), the non-perturbative effect is negligible.When the sgoldstino oscillates with the vacuum mass term, |Ṡ|≃ m_S v_S(T). We first consider the case where the sgoldstino is destroyed by scattering. As v_S(T) > v_S, S may vary and q is maximized around y S = gT,q≲y m_S v_S(T)/g^2 T^2.As long as the sgoldstino is the subdominant component of the energy density of the Universe, q<y<1. After the sgoldstino dominates, q grows until the thermal bath is dominated by the radiation produced from the sgoldstino at the temperature of T_ NA. Using the formulae in App. <ref>, we obtain the maximal q, q =√(π^2 g_*/30)y/g^2 D^3/5≲ D^3/5m_S/m_3 and 10^-2 D^3/5( m_S/1 TeV)^1.011/3 ,where in the inequality we use the upper bound on y in Eq. (<ref>) andy T_ des < m_S. We find that q is smaller than unity for the parameter space considered in Fig. <ref>.We next consider the case where the sgoldstino decays. For v_S(T) >v_S, Eq. (<ref>) is applicable, and q<1 as long as the sgoldstino is subdominant. We find that T_S>T_M in the parameter space where the dilution factor is small enough that thermal leptogenesis is viable. For the parameter region, q<1 for v_S(T) > v_S. Once v_S(T) < v_S, q is given byq = y^2 m_S v_S v_S(T)/ max(y^3 v_S^3, g^3 T^3) < m_S/ y v_Sv_S(T)/v_S,which is smaller than unity as long as m_S < y v_S.99MaianiLecture L. Maiani. in Proceedings: Summer School on Particle Physics, Paris, France (1979).Veltman:1980mjM. J. G. Veltman,Acta Phys. Polon. B12, 437 (1981). Witten:1981nfE. Witten,Nucl. Phys. B188, 513 (1981). Kaul:1981wpR. K. Kaul,Phys. Lett. B109, 19 (1982). Ellis:1990wkJ. R. Ellis, S. Kelley and D. V. Nanopoulos,Phys. Lett. B260, 131 (1991).Amaldi:1991cnU. Amaldi, W. de Boer and H. Furstenau,Phys. Lett. B260, 447 (1991).Giunti:1991taC. Giunti, C. W. Kim and U. W. Lee,Mod. Phys. Lett. A6, 1745 (1991).Pagels:1981keH. Pagels and J. R. Primack,Phys. Rev. Lett.48, 223 (1982).Weinberg:1982zqS. Weinberg,Phys. Rev. Lett.48, 1303 (1982).Khlopov:1984pfM. Y. Khlopov and A. D. Linde,Phys. Lett.138B, 265 (1984).Ellis:1984erJ. R. Ellis, D. V. Nanopoulos and S. Sarkar,Nucl. Phys. B259, 175 (1985).Moroi:1993mbT. Moroi, H. Murayama and M. Yamaguchi,Phys. Lett. B303, 289 (1993).Fukugita:1986hrM. Fukugita and T. Yanagida,Phys. Lett. B174, 45 (1986).Ibe:2006rcM. Ibe and R. Kitano,Phys. Rev. D75, 055003 (2007)[hep-ph/0611111]. Hamaguchi:2009hyK. Hamaguchi, R. Kitano and F. Takahashi,JHEP0909, 127 (2009)[arXiv:0908.0115 [hep-ph]]. Fukushima:2012raH. Fukushima, R. Kitano and F. Takahashi,JHEP1302, 140 (2013)[arXiv:1209.1531 [hep-ph]]. Kim:1992euJ. E. Kim,Phys. Rev. Lett.67, 3465 (1991).Lyth:1993zwD. H. Lyth,Phys. Rev. D48, 4523 (1993)[hep-ph/9306293]. Kawasaki:2008jcM. Kawasaki and K. Nakayama,Phys. Rev. D77, 123524 (2008)[arXiv:0802.2487 [hep-ph]]. Co:2016flnR. T. Co, F. D'Eramo and L. J. Hall,JHEP1703, 005 (2017)[arXiv:1611.05028 [hep-ph]]. Co:2017orlR. T. Co, F. D'Eramo, L. J. Hall and K. Harigaya,arXiv:1703.09796 [hep-ph]. Fujii:2002fvM. Fujii and T. Yanagida,Phys. Lett. B549, 273 (2002)[hep-ph/0208191]. Ibe:2010ymM. Ibe, R. Sato, T. T. Yanagida and K. Yonekura,JHEP1104, 077 (2011)[arXiv:1012.5466 [hep-ph]]. Choi:1999xmK. Choi, K. Hwang, H. B. Kim and T. Lee,Phys. Lett. B467, 211 (1999)[hep-ph/9902291]. Fukushima:2013vxaH. Fukushima and R. Kitano,JHEP1401, 081 (2014)[arXiv:1311.6228 [hep-ph]]. Badziak:2015dycM. Badziak, I. Dalianis and Z. Lalak,JHEP1602, 173 (2016)[arXiv:1512.06795 [hep-ph]]. Rychkov:2007uqV. S. Rychkov and A. Strumia,Phys. Rev. D75, 075011 (2007)[hep-ph/0701104]. Kawasaki:2008qeM. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi,Phys. Rev. D78, 065011 (2008)[arXiv:0804.3745 [hep-ph]]. Mukaida:2012qnK. Mukaida and K. Nakayama,JCAP1301, 017 (2013)[arXiv:1208.3399 [hep-ph]]. Anisimov:2000wxA. Anisimov and M. Dine,Nucl. Phys. B619, 729 (2001)[hep-ph/0008058]. Harigaya:2013vwaK. Harigaya and K. Mukaida,JHEP1405, 006 (2014)[arXiv:1312.3097 [hep-ph]]. Bodeker:2006ijD. Bodeker,JCAP0606, 027 (2006)[hep-ph/0605030]. Giudice:2003jhG. F. Giudice, A. Notari, M. Raidal, A. Riotto and A. Strumia,Nucl. Phys. B685, 89 (2004)[hep-ph/0310123]. Buchmuller:2004nzW. Buchmuller, P. Di Bari and M. Plumacher,Annals Phys.315, 305 (2005)[hep-ph/0401240]. Kawasaki:2000enM. Kawasaki, K. Kohri and N. Sugiyama,Phys. Rev. D62, 023506 (2000) doi:10.1103/PhysRevD.62.023506 [astro-ph/0002127]. | http://arxiv.org/abs/1707.08965v2 | {
"authors": [
"Raymond T. Co",
"Keisuke Harigaya"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170727180001",
"title": "Gravitino Production Suppressed by Dynamics of Sgoldstino"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.