A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
Yuan Yao Masaki Oshikawa





smartyao@issp.u-tokyo.ac.jp
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan



We introduce a new boundary condition which renders the flux-insertion argument for the Lieb-Schultz-Mattis type theorems in two or higher dimensions
free from the specific choice of system sizes.
It also enables a formulation of the Lieb-Schultz-Mattis type theorems in arbitrary dimensions in terms of the anomaly in field theories in 1+1 dimensions
with a bulk correspondence as a BF-theory in 2+1 dimensions.
Furthermore, we apply the anomaly-based formulation to the constraints on a half-filled spinless fermion on a square lattice with π flux, utilizing a time-reversal, magnetic translations and an on-site internal U(N) symmetries.
This demonstrates the role of the time-reversal anomaly on the ingappabilities of a lattice model.
Moreover, by our new boundary condition, we show that the many-body Chern number of this lattice model is non-vanishing as N2N in the presence of U(N) and magnetic translations.
 
This can be a general mechanism of anomaly-based constraints on quantized Hall conductance, which generally depends on high-energy
physics, from field theory.


A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
    Masaki Oshikawa
    December 30, 2023
================================================================================================================





§ INTRODUCTION

Quantifying various phases for quantum many-body systems is a central task in condensed matter and statistical physics. Recent decades have witnessed several significant phase classifications, e.g. topological ordered phases <cit.>
and symmetry-protected topological (SPT) phases beyond Landau's symmetry-breaking pattern of strongly-correlated systems with non-perturbative interactions <cit.>.
Furthermore, symmetries together with filling fractions also constrain low-energy spectrums when critical phases are gapped, which induce the concept of symmetry-protected critical phases <cit.> that nontrivial critical phases are ingappable with a unique ground state if the symmetries are respected by Hamiltonians.

One of the most important general principles in quantum many-body systems is Lieb-Schultz-Mattis (LSM) theorem <cit.> and
its generalizations <cit.>.
They show the interplay between the global U(1)_Q charge and translation symmetries.
The theorem states, under certain conditions,
an “ingappability” of the system, that is, either the presence of
gapless excitations above the ground states or a ground-state degeneracy
in the limit of the large system size.
It is valid for Hamiltonians with appropriate symmetries for
arbitrary strong interactions, and thus is non-perturbative in nature.

Generally in physics, we expect that the bulk property would not depend
on the choices of boundary conditions. If that is the case, we can
use a boundary condition which is convenient for the calculation and
infer physical results which would be valid independent of the boundary
condition.
In many cases, the periodic boundary condition is chosen as a boundary
condition. A typical example is the band theory of electronic structures.

The original proof <cit.> of the LSM theorem was also based
on the periodic boundary condition.
The LSM theorem in higher dimensions turns out to be more subtle.
While it is easy to see the failure of the original proof in higher
dimensions, the original proof would still work <cit.>
in an “anisotropic” thermodynamic limit in which the ratio of the system
sizes in each direction diverges.
However, one might worry that the system is essentially one-dimensional
in such a limit.
An alternative argument based on an adiabatic flux insertion
and gauge invariance, which does not need such an anisotropic limit,
was proposed later <cit.>.
It still depends on several nontrivial assumptions including the
stability of the gap against the flux insertion, and special
choices of system size as we will discuss later.
It was followed by a more rigorous proof <cit.> of the LSM
theorem in higher dimensions, and its further
mathematical refinements <cit.>.

Nevertheless, the flux insertion argument is still attractive in
its simplicity, intuitiveness, and connections to other
concepts in physics.
Indeed, some of the recent extensions <cit.>
of the LSM theorem is based on the flux insertion argument.
Because of this, it would be valuable to improve the flux insertion
approach to the LSM-type theorems.
One of the subtleties in the flux insertion argument was that
the system sizes must meet a special condition: the lengths
in all but one direction must be coprime with the denominator
of the filling fraction.
Although one may consider the “thermodynamic limit” with a series
of finite-size systems satisfying this condition keeping the
ratio of length to be of order of 1, it is desirable to remove
such a rather artificial condition.



In this work, we introduce a new class of boundary conditions, which
we call tilted boundary conditions (TLBC), which is useful for
derivation of the LSM-type theorems in dimensions higher than one.
As we will demonstrate, with TLBC, the flux insertion argument can be
applied without the artificial condition on the system sizes.
This is also the case for the higher symmetry
(SU(N)) generalizations <cit.>.

Furthermore, the TLBC reveals previously unnoticed
relations between anomaly in field theory with the LSM-type theorems.
While the LSM theorem has been well understood in the context of
Tomonaga-Luttinger Liquids in one spatial dimension, field-theoretical
understanding of LSM-type theorems in higher dimensions has been rather
limited.
The TLBC allows us to understand the LSM theorem as an anomaly manifestation.
As a futher application of the anomaly-based approach,
we will discuss the implications of the time-reversal anomaly <cit.> with a lattice interpretation.
Taking the advantage of the bulk-boundary correspondence between SPT phase and the time-reversal anomaly on its boundary, the anomaly leads to an ingappability constraint
for the half-filled π-flux system when time-reversal, on-site U(N) and magnetic translational symmetries are respected.
 Moreover, the TLBC enables us to obtain a constraint on integer quantum Hall conductances. We show that the π-flux system must have non-vanishing N2N many-body Chern number in the presence of U(N) and magnetic translational symmetries.
Our constraint generalizes those derived for lattice models
known for N=1 <cit.>.
Furthermore, our proposal can be a general mechanism for non-perturbative restrictions on quantize Hall conductivity from field theory.
It reflects a deep relation between such phenomena with symmetry-broken surface of SPT phases. 

This paper is organized as follows. In Sec. <ref>, we introduce our new boundary condition TLBC and apply it to the LSM theorem in arbitrary dimensions. In Sec. <ref>, the TLBC is used to generalize the LSM theorem for spin systems. In Sec. <ref>, we discuss the LSM-type ingappabilities on the half-filled π-flux square lattice with an onsite U(N) symmetry, magnetic translational symmetries and a time-reversal symmetry by a time-reversal anomaly. As a further application, a constraint on the integer quantum Hall conductances of the gapped π-flux system in the presence of U(N) symmetry and magnetic translations is obtained by a bulk-boundary correspondence with the geometry of TLBC in Sec. <ref>. In the Appendix, we present a detailed derivation of the constraint.



§ TLBC AND LSM THEOREM





 §.§ Flux insertion with PBC



Let us first review the flux insertion argument <cit.>
for the LSM theorem in d ≥ 2 dimensions, and some of the problems
in it.
We are interested in the energy spectrum of a quantum
many-particle system on a periodic lattice.
We assume that the number of particles is exactly conserved, and
consider the the limit of
the large system size (thermodynamic limit) with a fixed
“filling fraction” (number of perticles per unit cell) ν.

The many-body gap is defined as the energy gap between
the (possibly degenerate, multiple) ground state and
the continuum of excited states, in the thermodynamic limit.
It is not to be confused with the finite-size gap defined by the
gap between the ground state and the lowest excited state in
a finite system of a fixed size.

When the many-body gap vanishes, namely if the continuum of excited
states starts at an infinitesimal energy above the ground state,
the many-particle system is said to be gapless.
On contrary, when the many-body gap is non-vanishing, the system
is said to be gapped.
The LSM theorem is a constraint on the energy spectrum,
in particular the ground-state degeneracy if the system is gapped,
when the filling fraction ν is not an integer.






For simplicity, let us consider
the (hyper)cubic lattice in d dimensions with PBC

    r⃗∼r⃗+L_ix̂_i,  i=1,2,⋯,d,

where L_i is the length along i-th unit vector x̂_i, and the Hamiltonian is required to possess translational symmetry and U(1)_Q symmetry.

We set the filling factor as a rational number ν=p/q,
where p and q are coprime.
There is an additional charge quantization condition due to the fundamental degrees of freedom being charge one:

    ν V∈ℕ,

where V≡∏_iL_i is the total volume, and this condition is also implied by the compactness of global U(1)_Q symmetry.

The PBC means that the system has topologically non-trvial loops,
which may be visualized with “holes” in a higher-dimensional
embedding.
Each hole can contain a magnetic flux (Aharanov-Bohm flux, AB flux),
which affects the system through the Aharonov-Bohm effect.
Here we will utilize the AB flux Φ which is enclosed by
the closed loop in 1-direction.
Such a flux is represented by the vector potential in 1-direction
A_1 = Φ /L_1 .


The many-particle system may be gapless or gapped.
If it is gapless, there is nothing more to say in the LSM theorem.
Therefore, we can assume that the system is gapped, namely the
many-body gap separating the continuum of the excited states
from the ground state(s) is non-vanishing (at Φ=0).
Starting from a ground state, which is also a momentum eigenstate,
as the initial state,
we consider an adiabatic insertion of unit flux quantum Φ=2π.
Namely, we consider the
application of a time-dependent U(1)_Q gauge field

    A_1= 2π t/TL_1,

with T→+∞.
After the insertion of the unit flux quantum, the Hamiltonian is
equivalent (under a large gauge transformation) to the original one without
the flux.
Here we make a crucial assumption that the many-body gap
(which was non-vanishing at zero flux, by the initial assumption)
does not vanish during the flux insertion.
It then follows from the adiabaticity
that the final state must be a ground state below the gap.
One might expect the final state after the large gauge transformation
would be identical to the original ground state.

However, for an incommensurate filling, there is
a nontrivial momentum shift caused by the flux insertion.
For instance, the lattice momentum change along x̂_1 is

    Δ P^PBC_1=2πp/qL_2L_3⋯ L_d2π,

which is nonzero if (L_2⋯ L_d) is nondivisible by q.
In this case, the final state cannot be identical to the initial
ground state.
Restricting the system sizes L_2,3,…,d to coprime with q,
the statement of the LSM theorem is derived.
Namely, the system is either gapless or has at least q degenerate
ground states below the gap.
However, this artificial restriction on the system size is clearly
undesired. Physically we would expect the same statement to hold
for generic (large) system sizes, even though the above argument
does not lead to a nontrivial restriction if
L_2L_3⋯ L_d is divisible by q.



Here let us also comment on the nontrivial assumption
that the non-zero many-body gap does not collapse during the flux insersion.
While this assumption, which is crucial for “flux insertion” arguments,
has not proven,
it is rather natural, as the change of the hopping terms is only
of the order of O(1/L_1) in the uniform gauge.
In fact, there are some numerical <cit.> and
analytical <cit.> supporting evidences.
Furthermore, some  versions of the LSM theorem, that follow from
the assumption, have been proven
rigorously <cit.>.
Although a rigorous justification (or a clarification on the range of
validity) of the assumption is an interesting problem,
it is outside the scope of the present paper.








 §.§ Flux insertion with tilted boundary condition


Here we propose to use, instead of the standard PBC,
the following tilted boundary condition (TLBC)

    {[ r⃗+L_ix̂_i∼r⃗+x̂_i+1,            i=1,⋯,d-1;;        r⃗+L_ix̂_i∼r⃗,                  i=d, ].

where x̂_i is the unit vector along L_i.
We sketch the two-dimensional case of TLBC in FIG. (<ref>).

Geometrically, under such identifications, the space is a d-dimensional torus.
Combined with the flux insertion and momentum counting, this
leads to the LSM theorem without the artificial restriction on the system
sizes, as we will discuss below.
The TLBC is consistent with all the imposed symmetries—U(1)_Q and translations.
Now let us consider the following flux insertion process (t∈[0,T]):

    A_i(t)=2π t/T∏_k=i^dL_k,  i=1,⋯,d.

During the flux insertion,
the gauge (<ref>) still respects the translational symmetries since it is site-independent.
Thus the canonical momentum remains exactly unchanged during the
time evolution under the flux insertion.

Similarly to the case of PBC,
after completing the flux insertion,
we perform a large gauge transformation
to bring the Hamiltonian back to its original form.
Reflecting the different boundary condition and the different way
of inserting the flux, the large gauge transformation
used for PBC <cit.> does not work as is.
Nevertheless, we find that,
the following large gauge transformation exactly eliminates
the vector potential (<ref>)
at t=T, after the flux insertion process:

    Û_0=exp(∑_r⃗ i 2π r_1n̂_r⃗/V),

where we fix the range of r⃗ as

    r⃗=r_1x̂_1,   (r_1=1,⋯,V),

exhausting all the lattice points.

By applying the large gauge transformation (<ref>) on the final state, which maps the Hamiltonian back to its original form, we can compare the lattice momenta along x̂_i of the state before and after this insertion process in the same U(1)_Q gauge. The momentum difference is


    Δ P_i   =   2π n/∏_k=i^dL_k(2πV/∏_k=i^dL_k)
       =   2πν(V/∏_k=i^dL_k)(2πV/∏_k=i^dL_k),

where i=1,2,⋯,d, the total number of particles is denoted by n and n/V=ν is the filling fraction by definition, and the different periods of Δ P_i's results from the identification of translations as T_i+1=(T_i)^L_i due to TLBC.


As in the case of the PBC reviewed in Sec. <ref>,
the momentum shift (<ref>) implies the LSM
theorem: if ν=p/q with p and (q>1) coprimes,
the ground states in a gapped phase must be at least q-fold degenerate,
as long as the Hamiltonian respects the
U(1)_Q and the lattice translation symmetries.
That is, a trivial insulating phase with a unique ground state is excluded.
We note that, in the present argument, each component of
ΔP⃗ gives exactly the same constraint on the ground-state
degeneracies of gapped phases, in contrast to the previous derivation
in Eq. (<ref>).  More importantly, there is no artificial
requirement on the system size beyond the charge quantization condition
of Eq. (<ref>).


It should be noted, however, the present argument still relies on the
nontrivial assumption that the excitation gap does not collapse
under the adiabatic flux insertion, discussed at the end of
Sec. <ref>.



 §.§ Anomaly manifestation of LSM theorem


The TLBC also reveals a connection between the LSM theorem in
higher dimensions and anomaly in field theory in 1+1 dimensions.

Let us consider an electronic system, without loss of generality, on a square lattice with TLBC, the lattice point of which can be thought as that of unit cells on a general lattice with translational symmetries.
First, we consider a non-interacting, tight-binding model:

    H_T-B=-t∑_⟨r⃗,r⃗'⟩c^†_r⃗c_r⃗'-μ∑_r⃗c^†_r⃗c_r⃗,

where t is a real positive number with chemical potential μ and c^†_r⃗ the creation operator of spinless fermion at lattice site r⃗ with ⟨⋯⟩ denoting summations only over nearest-neighboring sites.

By a generalized 't Hooft anomaly matching <cit.>, nontrivial symmetry anomalies in a low-energy effective field theory of an arbitrarily fine-tuned lattice model implies an ingappability of general lattice models, constrained only by filling and symmetry structures. Therefore, the following analysis for the non-interacting model should be also valid for more general, interacting systems.

Under the TLBC,
we can exhaust all the sites by translating a single site to one direction only, e.g. r⃗=jx̂_1 with j=1,2,⋯,V. With this coordinate convention, we further define Ψ_j≡ c_jx̂_1 which implies

    H_T-B   =   -t∑_j=1^V[∑_i=1^dΨ^†_j+V_iΨ_j+h.c.]-μ∑_j=1^VΨ^†_jΨ_j;
    
    V_i   ≡   V/∏_j=i^dL_j,

with the boundary condition as Ψ_j=Ψ_j+V if the range of j is extended to all integers. Then we can express the Hamiltonian in the momentum space by

    Ψ_j=1/√(V)∑_k=0^V-1Ψ(k)exp(- i 2π k/Vj),

as

    H_T-B   =   ∑_k=0^V-1{-2t[∑_i=1^dcos(2π V_i/Vk)]-μ}Ψ^†(k)Ψ(k)
       =   ∑_k∈B.Z.{-2t[∑_i=1^dcos(2π V_i/Vk)]-μ}Ψ^†(k)Ψ(k)
       ≡   ∑_k∈B.Z.ϵ(k)Ψ^†(k)Ψ(k),

where we have chosen the Brillouin zone which is symmetric at k=0 so that the zeros of ϵ(k) is symmetric around the origin. Thus we can label these zeros as {± K_c}_c=1,⋯,N_0 due to the even parity: ϵ(k)=ϵ(-k).
The low-energy Hamiltonian, which describes excitations near
the zeros of ϵ(k), is given as

    ℋ=∑_c=1^N_0∫_-Λ<k<Λv_ck(Ψ^†_c(k)Ψ_c(k)-Ψ^†_c̅(k)Ψ_c̅(k)).

Thus N_0 ∼ L_2 × L_3 …× L_d
represents the number of the one-dimensional channels (1+1-dimensional Dirac fermions).
Here the ultraviolet cut-off Λ≪2π/(N_0a), where a is the lattice constant,
and v_f is the value of fermi velocity not necessarily positive. The fermionic operator Ψ_c(k)=Ψ(K_c+k) and Ψ_c̅(k)=Ψ(-K_c+k).
It is essential to remark the role played by the charge quantization condition in Eq. (<ref>).
To obtain a free system in the continuum limit where k is taken to be a continuum variable on the whole real axis, it is necessary that the k=0 modes or Ψ(± K_c) modes exist in the Hilbert space under the continuum limit, as we will see later.
In d=1, Eq. (<ref>) implies precisely such existence of Ψ(± K_c) modes. Therefore, we expect this conclusion generalizes to arbitrary d≥1 and the charge quantization indeed permits us to do this continuum limit.

We absorb the |v_c| into the following definition of two-component Dirac operator:

    ψ_c(k)={[ ([  |v_c|Ψ_c(k); |v_c|Ψ_c̅(k) ]),                            v_c>0;; ([ |v_c|Ψ_c̅(k);  |v_c|Ψ_c(k) ]),                            v_c<0. ].

With the definition above, the real space formulation of Lagrangian density takes a compact form as

    ℒ_U(1)=∑_c=1^N_0∑_μ=0^1ψ̅_c(t,x)iγ^μ∂_μψ_c(t,x),

where γ^0=σ_2 and γ^1=σ_1 with σ⃗ Pauli matrices, and the chirality is γ_3=σ_3.

The role played by the existence of the zero mode ψ_c(k=0) or Ψ(± K_c) can be understood in the real space formulation (<ref>): if we did not have these zero modes, we would discard the zero mode of the Fourier expansion of ψ_c(t,x) by hand when we quantize it, which makes (<ref>) not free Dirac fermions.


Then we minimally couple it with an external gauge field as

    A_1(t,x)= 2π t/VT ,

and A_0(t,x)=0: ∂_μ→∂_μ-ieA_μ,
which exactly corresponds to the flux insertion defined by Eq. (<ref>) except for that we are left by only one spatial dimension since we have applied the coordinate convention r⃗=jx̂_1 with j=1,⋯,V before on the lattice.

Combining Eqs. (<ref>,<ref>), we obtain the lattice translation T_1 in x̂_1 direction representation of ψ_c(t,x) as:

    ψ^T_1_c(t,x)   ≡    T_1ψ_c(t,x)T_1^-1
       =   exp[ i sgn(v_c)πγ_32K_c/V]ψ_c(t,x),

which is simply the chiral symmetry transformation, at low energy, appearing on-site. It is straightforward to apply Fujikawa's method <cit.> to calculate the global symmetry anomaly as the phase ambiguity of the fermionic partition function responding to the chiral transformation as Eq. (<ref>) in the background gauge field configuration A_μ(t,x).

    Z_U(1)
       ≡   ∫𝒟(ψ̅^T_1,ψ^T_1)exp(-∫ℒ_U(1)[ψ̅^T_1_c(τ,x),ψ^T_1_c(τ,x),A_μ])/∫𝒟(ψ̅,ψ)exp(-∫ℒ_U(1)[ψ̅_c(τ,x),ψ_c(τ,x),A_μ])
       =   exp[ i ∑_c=1^N_02sgn(v_c)2π K_c/V]
       =   exp( i 2πν),

where the fermionic measure 𝒟(ψ̅^T_1,ψ^T_1)=J_T_1·𝒟(ψ̅,ψ) is calculated by Fujikawa's U(1) gauge-invariant regularization <cit.> and we have used the fact the filling fraction is ν and ν=∑_c2sgn(v_c)K_c/V, and evaluate the formal path integral in the Euclidean signature: τ=it.

We define a lattice partition function ratio as:

    Z^_U(1)   ≡   Tr_G.S.[T_1Û_0Û_fluxT_1^-1]/Tr_G.S.[Û_0Û_flux]
       =   ⟨G.S.|T_1Û_0Û_fluxT_1^-1|G.S.⟩/⟨G.S.|Û_0Û_flux|G.S.⟩,

where we denote by “Tr_G.S.” taking the trace only within lowest energy states since the Wick rotation “τ=it” in the definition of Z_U(1) implies that we should project out excited states when we define Z^_U(1). We also assume a unique gapped lattice ground state |G.S.⟩ and Û_flux denotes the unitary time evolution by flux insertion followed by the large gauge transformation Û_0.
The continuum-limit form of Z'_U(1) in Eq. (<ref>) exactly coincides with the form of Z_U(1) in Eq. (<ref>), where the large gauge transformation Û_0 is implicitly presented in the path integrals within Z_U(1) since the inner product of wave functionals |{ψ̅,ψ}⟩_A_μ's at the last time slice can be done only after fixing the gauge by the (large) gauge transformation Û_0, which is in the same situation as the lattice model.
The necessity of Û_0 can be seen once one notices that the wave functional {ψ} is an associated complex line sector of the underlying U(1)_Q principal bundle and Û_0 in Eq. (<ref>) is exactly the gluing transition function between the initial and the final time slices.
It implies that Z_U(1) is the low-energy limit of Z^_U(1).


Then we can rephrase the consequence of anomaly Z_U(1) that the (discrete) chiral symmetry (<ref>) will be broken once U(1)_Q is gauged, in the lattice language.
Since the anomalous chiral symmetry means the chiral charge is not conserved and here the chiral symmetry corresponds to the lattice translation, we expect the lattice momentum is not conserved
under the gauge field.
Indeed, this is a well known phenomenon that the particles are
accelerated by electric field and acquire a momentum.
However, it should be noted that the momentum is a gauge-dependent
quantity.
Under the time-dependent uniform gauge field (<ref>), the
momentum is exactly conserved thanks to the translation invariance.
However, after a finite time, the gauge field (vector potential) is nonzero
and the momentum cannot be directly compared to the initial value.
Nevertheless, when the system encloses an integral multiple of the
unit flux quantum (2π), the gauge field can be exactly
eliminated by the large gauge transformation Û_0, so that
the momentum can be compared with its initial value.
It is this large gauge transformation Û_0 that induces
the change in the momentum, as we can see from the flux-insertion
argument for the LSM theorem <cit.>.
This observation also supports the identification of Z_U(1)
as the low-energy limit of Z^_U(1).
Additionally, the role of the gauge invariance of Fujikawa's regularization applied in the calculation of Eq. (<ref>) should be noted since a gauge non-invariant regularization method can give a vanishing chiral anomaly <cit.> corresponding to the conservation of the lattice momentum in the fixed initial gauge, which can be also explicitly seen by the translational symmetry of the lattice Hamiltonian.
It exactly reflects the mixing nature of the chiral anomaly, which characterizes the conflicting between U(1) and T_1. 






Since Z_U(1) is a topological invariant, e.g. invariant along any symmetry-respecting renormalization-group flow by the generalized 't Hooft anomaly-matching, we can evaluate Z^_U(1) by its low-energy limit Z_U(1):

    Z^_U(1)=Z_U(1).

However, the unique ground state must be featureless hence a T_1-eigenstate, which implies T_1|G.S.⟩=exp(iP)|G.S.⟩ thereby Z_U(1)=1. This contradicts with Z_U(1)=Z^_U(1)=exp(i2πν) if q≠1, unless

    Û_flux|G.S.⟩⊥|G.S.⟩,

which still conflicts the unique gapped ground state. Thus the ground states must be degenerate for fractional fillings.
Then we arrive at the LSM theorem with a well-defined anomaly-manifestation in a general thermodynamic limit.
Furthermore, the physical interpretation of Z_U(1) can be understood by its lattice partner Z^_U(1) which exactly measures the momentum changes after the charge pumping. In this sense, we call that the chiral anomaly derived from the Dirac field theory is lattice-realized.

The anomaly we have considered is not an emergent anomaly at low energy <cit.> which can be seen as follows.

Let us first tune the hoppings in (<ref>) nonzero only along x̂_1.
At low energy under TLBC, U(1) and translation symmetries (<ref>) are reduced to U(1)×ℤ_q <cit.>, where ν=p/q defined before.
Since the translation is actually represented by ℤ on the lattice with TLBC in the thermodynamic limit, we should do a symmetry extension,
which is a quantitative treatment of the translation symmetry at low-energy honestly as ℤ rather than its emergent expression ℤ_q <cit.>. The symmetry extension is a mapping from the [U(1)×ℤ_q]-anomaly classes to the [U(1)×ℤ]-anomaly classes, associated with the dual of the third mapping in the short exact sequence 0→ℤℤ→ℤ_q→0 of symmetries [Here the associated dual mapping is Ω^3_Spin^c(Bℤ_q)→Ω^3_Spin^c(Bℤ), where Ω is the spin^c cobordism of the classifying spaces Bℤ_q and Bℤ. U(1) gauge contribution has been taken into account by the spin^c structure.],
This symmetry extension does not trivialize
the anomaly classes generating by (<ref>) [The mixed anomaly classes generated by the anomaly factor exp(i2πν) forms ℤ_q⊂Ω^3_Spin^c(Bℤ_q) and the symmetry extension maps this ℤ_q injectively to Ω^3_Spin^c(Bℤ)≅ U(1) by the natural inclusion ℤ_qU(1). ], which means the anomaly classes generated by (<ref>) are still there even when we treat translation faithfully.
The general cases where the translation is represented by (<ref>) can be argued by anomaly-matching to the fine-tuned point above.
In contrast, if we orbifolded theory by the ℤ_q symmetry using its finite cyclicity, which no longer holds at the lattice scale, more anomalies would emerge, but eventually be trivialized after the symmetry extension <cit.>.
Furthermore, we calculate the anomaly at a critical point with a special translation representation (<ref>), which can be expected as an intrinsic anomaly at the lattice level even if the lattice model is tuned away from this point, in the following way. Let us assume the lattice degrees of freedom are realized by a higher dimensional lattice bulk B with the corresponding U(1)×ℤ symmetry at the lattice scale.
Then the intrinsic anomaly is unambiguously determined by the (U(1)×ℤ)-crystalline-SPT (cSPT) class that this bulk B belongs to.
It has been shown that cSPT one-to-one corresponds to an SPT where U(1)×ℤ is realized purely on-site and the continuum limit here is proposed to give such a correspondence <cit.>.
It means this (onsite) SPT class corresponds to the anomaly of our critical point in the continuum limit, calculated by the representation (<ref>) in a ℤ sense.
Then the anomaly (<ref>) is associated to that cSPT, thereby intrinsic and applicable away from the chosen critical point.


Finally, for d=0, due to Eq. (<ref>), the filling ν_0∈ℕ. Then, the low-energy effective response theory is simply the (0+1)-dimensional Chern-Simons theory with level ν_0:

    s_U(1)=ν_0∫ dtA(t),

by a minimal coupling observation. Such an effective action is well-defined since ν_0 is an integer, and the zero-dimensional theory is U(1)_Q anomaly-free thereby well-defined. It is applicable even for bosonic theory since there is no spatial coordinate to give one a choice of spin structures while the boundary condition along t or τ for field operator is already fixed as, respectively, periodic and anti-periodic for bosonic and fermionic situations. Thus, d=0 cases can be always trivially gapped if we only have a U(1)_Q symmetry.
This agrees with the trivial d=0 version of the LSM theorem.
While the statement is rather trivial, this is still a useful exercise to
check the consistency of the anomaly argument.
In the next section, we will see somewhat more non-trivial consistency check in d=0 for a higher symmetry.



 §.§ Bulk-boundary correspondences: LSM theorem with U(1)_Q


In this Subsection, based on our field-theory formulation related to the TLBC, we discuss the LSM theorem from the point of view
of anomaly inflow.
This enables us to construct
a higher dimensional SPT bulk theory where the massless theory ℒ_U(1) in Eq. (<ref>) can be seen as a boundary theory attached to
the SPT bulk.

To simplify the discussion, in the following we first assume N_0=1.
We will generalize the argument to arbitrary N_0 later.
The translation symmetry is reduced to:

    ψ^T_1   =   exp(iγ_3πν)ψ=exp(iγ_3πp/q)ψ.


Our physical degrees of freedom is the free Dirac fermion composed by ψ=[ψ_L,ψ_R] where ψ_L,R are the left- and right-moving chiral fermions, respectively.
It is well-known that a single left-moving chiral fermion can be generated on the edge of an integer quantum Hall system with σ_H=+1 and a right mover by σ_H=-1.
Thus the (bulk) matter field, which is able to support ψ_L and ψ_R fermions on its edge, consists of two (2+1)-dimensional massive Dirac fermion with opposite masses, which realizes σ_H=±1, separately [The Pauli-Villars regulator of each massive Dirac fermion has an opposite mass to the regulated fermion.].
Then we couple the background U(1)_Q gauge field A_U(1) and the gauge field aγ^3/2 of the translational symmetry in (<ref>) to the bulk fermions.
After the bulk matter field is integrated out,

    S_bulk,U(1)=∫[
    -i/4πA_L∧ dA_L+
    i/4πA_R∧ dA_R],

where

    A_L   =   A_U(1)+a/2;   A_R=A_U(1)-a/2.

where we have taken a “1/2” normalization convention so that a has the same constraint as a ℤ_q-gauge field:

    da=0   and ∮_closed loopa∈2πp/qℤ.

Then,


    S_bulk,U(1)=-∫i/2πa ∧ dA_U(1),


which is exactly a BF-theory <cit.> coupling the ℤ_q-gauge field and the U(1)_Q-gauge field. To detect the nontrivial aspect of such an SPT bulk, we evaluate the bulk partition function on a compact closed manifold, e.g. three-torus T^3=T^2× S^1, with the following background gauge field:

    ∫_T^2dA_U(1)/2π=1,  ∫_S^1a=2πν,

where a is a pull-back from a flat gauge field on S^1 and A_U(1) is a pull-back from a gauge field on T^2 with a unit Chern number.
With this background gauge field configuration, we obtain the bulk partition function as

    Z_bulk,U(1)   =   exp(-S_bulk,U(1))
       =   exp(i2πν).

To understand the physical meaning of this bulk partition function evaluated on T^2× S^1, we briefly review the useful tool called mapping torus <cit.> below.
Given a potentially anomalous theory defined on a physical spacetime ℳ, its phase ambiguity of partition function by a transformation, e.g. a gauge or global symmetry transformation, can be calculated by the partition function of corresponding bulk on a manifold ℳ× S^1 <cit.>  constructed in the following way.
We take a “cylinder” ℳ× [0,1] where the bulk field stays and it reproduces the pre-transformed boundary theory at ℳ×{0} and the transformed theory at the other end ℳ×{1}.
Then we paste these two ends by the certain transformation twisting, which corresponds to a twisted boundary condition of the bulk field on ℳ× I, to form the mapping “torus” ℳ× S^1.
In our case (<ref>) above, we take S^1 as the extra dimension and T^2 represents the physical space-time of boundary theory.
The holonomy ∫_S^1a precisely realizes the twisted T_1 transformation of the bulk field and the instanton ∫_T^2dA_U(1)/2π implies the flux insertion process by a unit flux.
Therefore, this mapping-torus bulk partition function is equals to the phase ambiguity, brought by a T_1 transformation, of the boundary partition function with a flux insertion in physical spacetime, which is precisely the quantity calculated in (<ref>). 




This bulk construction can be generalized to arbitrary N_0 in ℒ_U(1) by replacements of “1/2” in Eq. (<ref>) by color-dependent coefficients “sgn(v_c)K_c/(Vν)” due to Eq. (<ref>). Then, after the color indices “c” is summed up and by ν=∑_c2sgn(v_c)K_c/V, we arrive at the same effective bulk response theory as Eq. (<ref>).


The bulk description of the anomaly proposes an alternative way to detect the anomaly by the boundary partition function (See e.g. Appendix D of <cit.>).
Let us take the T^2 in (<ref>) as the manifold spanned by the physical spatial dimension and the extra dimension. S^1 is taken as the physical dimension of time.
Then the holonomy (∫_S^1a) means we translate the system by a lattice distance during the time period S^1 and thus the resultant partition function is an exponential of the lattice momentum.
The instanton in T^2 means that we are comparing this momentum obtained in two gauge choices of the gauge field along the physical spatial dimension,
which are connected by a large gauge transformation.

Therefore, the bulk-response approach (<ref>) provides a unified way to describe its boundary anomaly.




§ LSM THEOREM FOR SU(N) SPIN-ROTATION SYMMETRY



In the following, we will use the methods developed in the previous section
 to investigate the higher symmetry generalization of LSM theorem, e.g. replacement of the global onsite symmetry by SU(N) spin-rotation symmetry, or more precisely a PSU(N)=SU(N)/ℤ_N global symmetry <cit.> by the spin operator satisfying the following su(N) Lie algebra commutation relations:

    [S^α_r⃗,β,S^γ_r⃗',δ]=δ_r⃗,r⃗'(δ^α_δ S^γ_r⃗,β-δ^γ_β S^α_r⃗,δ),

where α and β are the “spin” indices that take values among 1 to N.
In particular, we focus on the quantum-anomaly manifestation of the LSM theorem.
Its advantage is that the lattice ingappabilities can be detected at any fine-tuned critical point which simplifies the calculation, thanks to the 't Hooft anomaly matching.



 §.§ “Spin”-quantization condition

Let us consider the most general situation that the total number of Young-tableaux boxes per unit cell is b. Analogous to Eq. (<ref>), we also take the following “spin”-quantization condition:

    bV/N∈ℕ.

The reason we assume this condition is different from the U(1)_Q LSM case where charge quantization is naturally imposed by fundamental degrees of freedom. To clarify this seemingly unnatural requirement, let us see the situation where Eq. (<ref>) is not satisfied, namely bV/N∉ℕ. Then the total Young-tableaux boxes bV of the system is not divisible by N. By the knowledge from representation theory, there is no SU(N)-singlet sector in the Hilbert space and by the global PSU(N) symmetry, the system is exactly degenerate, even at all excited states since the states within any non-singlet irreducible representation must have the same energy due to Schur's lemma applied within any of these nontrivial irreducible sectors.

Such a rather trivial type of ingappabilities is essential to understand finite-system spectrum. Nevertheless, we are not interested in it since these ingappabilities are not a many-body effect and they depend on a specific choice of system sizes. A many-body ingappability is commonly considered as almost degeneracies, e.g. the ingappabilities make sense only at general thermodynamic limits.
Therefore, in the following discussion for (d≥1)-dimensional system, we avoid such an exact degeneracy explicitly exposed above by imposing Eq. (<ref>). Nevertheless, when stating the final theorem without loss of generality, we will also include the cases that bV/N∉ℕ for which the ground states are exactly degenerate.



 §.§ Generalized LSM theorem and anomaly manifestation

Since we are only interested in the low-energy spectrum of the lattice model, we can equivalently reconstruct its low-energy physical properties by coupling b copies of the lattice models each of which has one fundamental SU(N) degree of freedom within each unit cell. It is possible due to the group-theoretical knowledge that any SU(N) irreducible representation with b_0 Young-tableaux boxes is contained in the tensor products of b_0 of fundamental representations. Then we can project out all the undesired degrees of freedom in the analog of Affleck-Kennedy-Lieb-Tasaki chain construction from spin-1/2 degrees of freedom <cit.>. Moreover, such a projection can be realized by a strong interaction dynamically, hence the quantum anomaly factor of the original system can be obtained by a summation of the anomalies of these b of fundamental lattices. Then the problem is reduced to the anomaly related to fundamental lattices we will calculate below.

Similarly to the generalized LSM theorem in one dimension, we do the following N-flavor fermionization representing spin degrees of freedom:

    S^α_r⃗,β=Ψ^α†(r⃗)Ψ_β(r⃗)-1/Nδ^α_β

with the restriction of total particle number on every site “r⃗”

    ∑_α=1^NΨ^α†(r⃗)Ψ_α(r⃗)=1,

so that S^α_r⃗,β's defined in Eq. (<ref>) satisfy Eq. (<ref>).

Again, we impose the TLBC defined as Eq. (<ref>). In the analog of SU(2) cases, a fine-tuned critical model can be the Hubbard model in the conductive phase:

    H_SU(N)   =   -t∑_⟨r⃗,r⃗'⟩[∑_α=1^NΨ^α†(r⃗)Ψ_α(r⃗')+h.c.]
          -U[∑_α=1^NΨ^α†(r⃗)Ψ_α(r⃗')-1]^2,

where the Hund's rule coupling U≫|t| realizes the particle number restriction per unit cell in Eq. (<ref>). By an observation on Eq. (<ref>) and Eq. (<ref>), we can see that: 1) SU(N) case has a stronger particle number restriction per unit cell; 2) each flavor has a filling fraction 1/N within a unit cell. With this comparison, it is straightforward to derive the effective theory of the current SU(N) model after we apply the coordinate system r⃗=jx̂_1 with j=1,⋯,V again:

    ℒ_SU(N)[𝒜]   =   ∑_c=1^N_0∑_α=1^Nψ̅^α_c(t,x)iγ^μ(∂_μ-i𝒜_μ)ψ_c,α(t,x), 
    ν_eff   =   1/N,

where the inclusion of a dynamical fluctuating U(1)_Q gauge field is used to eliminate U(1)_Q phase degrees of freedom. It is because U(1)_Q is unphysical, e.g. the fundamental observable S^α_r⃗,β is invariant under U(1)_Q.
It can be also viewed as a realization of the particle number constraint as Eq. (<ref>) regularized by the Grassmann-number ordering ambiguity to make the Lagrangian explicitly U(1)_Q gauge invariant:

    [                           ∑_α=1^NΨ^α†(r⃗)Ψ_α(r⃗)=1,;                                                   ↓; ∑_α=1^N(1-1/N)Ψ^α†(r⃗)Ψ_α(r⃗)-1/NΨ_α(r⃗)Ψ^α†(r⃗)=0,;                                       ↓_Grassmanian; ∑_α(1-1/N)ψ^α†(t,x)ψ_α(t,x)-1/Nψ_α(t,x)ψ^α†(t,x)=0,;                                        ↓_Reordering;                             ∑_αψ^α†(t,x)ψ_α(t,x)=0, ]

which can be done by a Lagrangian multiplier ∑_α𝒜_0ψ̅^αγ^0ψ_α consistent with gauge invariance.
The zeros of Hamiltonian ϵ_α(k) for each flavor in momentum space in the same notations satisfy:

    ∑_c=1^N_02sgn(v_c)K_c/V=ν_eff.

Similarly to Eq. (<ref>), the U(1)_Q-LSM case, Eq. (<ref>) also guarantees the existence of the continuum limit of k variables in Ψ_α(± K_c+k).
Then we couple the theory to a background PSU(N) gauge field A_PSU(N) which locally takes value in su(N) algebra.
In the following discussion, t_N^2-1≡diag[1,1,⋯,1,-(N-1)]_N× N is denoted as the matrix representation of the last su(N) generator.
First, let us analyze the current gauge group, which actually is not U(1)× SU(N)={(g_1,g_N)|g_1∈ U(1),g_N∈ SU(N)} since its center ℤ_N generated by (exp(i2π/N),exp(-i2π t_N^2-1/N))∈ U(1)× SU(N) does not transform the matter field {ψ_α}.
It implies that the gauge group is [U(1)× SU(N)]/ℤ_N≅ U(N), and

thus only A_PSU(N)+𝒜, NA_PSU(N) and N𝒜 or their integer multiples can be lifted to canonical 1-form u(N) connections thereby expressible by well-defined u(N) connections.

We consider the following flux insertion which generalizes the U(1) case and the connection can be written down by a u(N) connection:

    A_PSU(N)+𝒜=2π t/TVN(1-t_N^2-1)_N× N+δ𝒜,

where δ𝒜 is a dynamical canonical U(1) connection since [2π t(1-t_N^2-1)/(TVN)] is globally well-defined, and then the functional integration ∫𝒟𝒜 over u(1)/ℤ_N connection can be converted to ∫𝒟δ𝒜 over u(1) connection.
The physical interpretation of such a PSU(N) “flux” insertion is that we adiabatically rotate the spin along x̂_1 direction and this spatial dependent rotation matrix projectively represented by SU(N) matrix is multi-valued since it is identified up to a center of SU(N) due to the local U(1) gauge degrees of freedom by 𝒜. Therefore, the accompany U(1) twisting is simply to compensate this artificial ambiguity in the SU(N) rotation matrix.

In a similar sense, we can calculate the anomaly factor by Fujikawa's gauge-invariant regularization <cit.> on the fermionic measure ∏_c,α𝒟(ψ̅_c,α^T_1,ψ_c,α^T_1)=J(T_1)∏_c,α𝒟(ψ̅_c,α,ψ_c,α) below:

    Z_SU(N)
       ≡   ∫𝒟𝒜𝒟(ψ̅_c,α^T_1,ψ_c,α^T_1)exp(-∫ℒ^T_1_SU(N)[𝒜+A_PSU(N)])/∫𝒟𝒜𝒟(ψ̅_c,α,ψ_c,α)exp(-∫ℒ_SU(N)[𝒜+A_PSU(N)])
       =   exp[ i ∑_c=1^N_02sgn(v_c)2π K_c/V]
       =   exp( i2π/N),

where we have made use of the same notation for the transformation rule of ψ→ψ^T_1 as Eq. (<ref>) and ℒ^T_1_SU(N)≡ℒ_SU(N)[ψ̅^T_1(τ,x),ψ^T_1(τ,x)] except that here we have a specified filling factor ν_eff=1/N for each flavor.

Thus for the general case of b of Young-tableaux boxes per unit cell:

    Z^(b)_SU(N)   =   (Z_SU(N))^b=exp( i 2πb/N).

Similarly to the U(1) case in LSM theorem, we can also define the lattice-realization of Z^(b)_SU(N) by Z'^(b)_SU(N) analogous to Eq. (<ref>), and by anomaly-matching: Z'^(b)_SU(N)=Z^(b)_SU(N).



 §.§ LSM-type anomaly in 0 dimension



To complete the SU(N) generalization of LSM theorem, we also study the case of d=0, which is a problem in single-body quantum mechanics.
Unlike in d ≥ 1, no quasi-degeneracy (asymptotic degeneracy in the
thermodynamic limit) can be defined from the energy spectrum.
Exact ground-state degeneracy can still be defined for d=0.
However, the exact degeneracy is generally a direct consequence of
Schur's lemma.
Namely, an SU(N) spin in any nontrivial irreducible representation
must have exact degeneracy for a SU(N)-symmetric Hamiltonian. 


Here we consider the spectrum of Hamiltonian imposing only the discrete
subgroup ℤ_N×ℤ_N of PSU(N), instead
of the full PSU(N) symmetry.
In this case, an irreducible representation of PSU(N) can become
reducible and the degeneracy of the spectrum may be lifted.
Whether the ground state can be made unique by
such a level splitting is a question analogous to
the “ingappability” discussed for d ≥ 1 dimensions
in the context of the LSM theorem.


Before the general discussion for SU(N) case, we give several explicit examples for N=2,3 cases. If N=2, the ℤ_2×ℤ_2 is generated by exp(iπ S_x/2) and exp(iπ S_z/2), which, in the fundamental representation ρ_f, satisfy

    ρ_f[exp(iπ S_x/2)]ρ_f[exp(iπ S_z/2)]
          ·ρ_f^-1[exp(iπ S_x/2)]ρ_f^-1[exp(iπ S_z/2)]=-1.

If N=3, ℤ_3×ℤ_3 is generated by exp[iπ (t_8-t_3)/3] and exp[-i2π(t_2-t_5+t_7)/(3√(3))] where, in the fundamental representation, ρ_f[t_8]=diag[1,1,-2], ρ_f[t_3]=diag[1,-1,0], ρ_f[t_2]_lm=iδ_l,2δ_m,1-iδ_l,1δm,2, ρ_f[t_5]_lm=iδ_l,3δ_m,1-iδ_l,1δ_m,3 and ρ_f[t_7]_lm=iδ_l,3δ_m,2-iδ_l,2δ_m,3, and, by <cit.>,

    ρ_f[exp[iπt_8-t_3/3]]ρ_f[exp[-i2πt_2-t_5+t_7/3√(3))]]·
          ·ρ_f^-1[exp[iπt_8-t_3/3]]ρ_f^-1[exp[-i2πt_2-t_5+t_7/3√(3)]]
       =   exp(i2π/3).

The same commutators in Eqs. (<ref>,<ref>), for the adjoint representation, are trivial <cit.> and the constructions of the ℤ_N×ℤ_N generators for other SU(N) cases can be found in <cit.>. A general representation with b Young-tableaux boxes for SU(N) cases yields a commutator as exp(i2π b/N), which reflects the H^2(ℤ_N×ℤ_N,U(1))≅ℤ_N classification of the projective representation of ℤ_N×ℤ_N, where H^2(·,U(1)) denotes the second group cohomology with a U(1) coefficient ring. Moreover, two transformations with a nontrivial commutator as Eqs. (<ref>,<ref>) cannot share a single one-dimensional invariant sub-Hilbert-space. These results exactly imply that any b Young-tableaux representation SU(N) “spin” can be gapped with a unique ground state respecting ℤ_N×ℤ_N if and only if N divides b.


Let us discuss and derive such (0+1)-dimensional (in)gappabilities in the framework of anomaly.
The partition function would have a phase ambiguity from the projective representation by an SU(N) spin.
Indeed, the ungauged Dijkgraaf-Witten theory provides a non-Lagrangian construction to expose this relation between the projective representation and (0+1)-dimensional anomaly for finite symmetries <cit.>.
As follows, we will also manifest this relation in our concrete fermionic representation of spins.

For general SU(N) cases, we first study the fundamental representation case or b=1 at a single point.
The low-energy effective theory for the (fine-tuned) lattice Hamiltonian H_SU(N)=0 is simply:

    L_SU(N)[𝒜]   =   ∑_α=1^Nψ̅^α(τ)i(∂_τ-i𝒜_τ)ψ_α(τ),

defined on the temporal circle τ∈ S^1=ℝ/(2πℤ) around which
ψ(τ+2π)=-ψ(τ), where the minus sign results from the fermionic path integral.
Hence, we do not have ℤ_N×ℤ_N twisting around S^1.
{ψ_α}_α=1^N constitutes the projective representation of ℤ_N×ℤ_N whose generators V_N and W_N represented by matrices ρ(V_N) and ρ(W_N) satisfying

    ρ(V_N)ρ(W_N)ρ^-1(V_N)ρ^-1(W_N)=exp(i2π/N).


The U(1)_Q dynamical gauge field 𝒜_τ is again used to project out the unphysical Hilbert subspace, and the U(1)_Q charge at the site is restricted to be 1 the same as Eq. (<ref>).
Due to this charge number restriction, the response theory of (<ref>) to the gauge field 𝒜_τ takes the form as

    z_SU(N)[𝒜]   =   |z_SU(N)[𝒜]|exp(-i∫_S^1𝒜_τ),

by the minimal coupling argument.
The partition function is obtained after the dynamical 𝒜 is integrated out:

    z_SU(N)   =   ∫𝒟𝒜z_SU(N)[𝒜].

Since the anomaly can be detected by different bulk extensions over which the spin structure and gauge fields extend.
We rewrite z_SU(N)[𝒜] into two different bulk extensions:

    z_SU(N)[𝒜,X]   =   |z_SU(N)[𝒜]|exp(-i∫_Xℱ),
    
    z_SU(N)[𝒜,X']   =   |z_SU(N)[𝒜]|exp(-i∫_X'ℱ),

where ∂ X=∂ X'=S^1 and they are indeed the extension of z_SU(N)[𝒜] in (<ref>) since by the Stokes' theorem

    ∫_Xℱ=∫_X'ℱ=∫_S^1𝒜_τ

in the absence of ℤ_N×ℤ_N background gauge field.
The absolute value of the partition function is well-defined so it does not depend on the extension <cit.>.
z_SU(N)[X] and z_SU(N)[X'] are still defined by integrating out 𝒜.
Although we do not include any ℤ_N×ℤ_N twisting along the physical dimension S^1, we can still insert ℤ_N×ℤ_N domain walls into the gauge bundle on X and X', which does not affect the trivial ℤ_N×ℤ_N bundle on their edge circle ∂ X=∂ X'=S^1 as follows.
Due to this anti-periodic boundary condition on S^1, we can extend it onto the disk X=D^2 (See e.g. Chapter 3 of <cit.>) and onto X' such that X̃≡ X∪(-X')=T^2.
Then the bulk dependence z_SU(N)[𝒜,X]/z_SU(N)[𝒜,X']=exp(-i∫_X̃ℱ).

The ℤ_N×ℤ_N bundle on X̃=T^2 is sketched in FIG. (<ref>) where along one cycle of T^2 there is a V_N twisting and along the other cycle a W_N twisting. 

To obtain a consistent gauge bundle for the fermionic degrees of freedom on T^2, we need to insert a U(1)_Q flux:

    ∫_X̃ℱ=-2π/N 2π,

to induce an Aharonov-Bohm phase exactly eliminating the phase ambiguity brought by the commutator (<ref>).


































Thus the phase ambiguity of z_SU(N)[𝒜] due to the different extension X and X' is exp(i2π/N). Furthermore, since such a phase is independent on the fluctuating gauge field 𝒜 which is integrated within one gauge sector connected by small gauge transformations, we can extract this phase out of the integration ∫𝒟𝒜. Thus this phase ambiguity denoted by Z_SU(N) is also shared by z_SU(N) after such a functional integration:

    Z_SU(N)≡z_SU(N)[X]/z_SU(N)[X']=exp(i2π/N).

By additivity of anomaly, we have for general representations with total b of Young-tableaux boxes:

    Z^(b)_SU(N)   =   (Z_SU(N))^b=exp( i 2πb/N).

This precisely implies that the partition function is enjoying an anomaly resultant from a projective representation of ℤ_N×ℤ_N.
Indeed, when b is divisible by N, such as spin-1 (adjoint) representation of su(2), the interaction S_z^2 can gap the ground state uniquely preserving ℤ_2×ℤ_2 generated by exp(iπ S_x/2) and exp(iπ S_z/2). For the adjoint representation of su(3), which is made of three Young-tableaux boxes, the ground state can be also gapped uniquely since the generators of ℤ_N×ℤ_N share at least one, actually eight, one-dimensional invariant sub-Hilbert-spaces <cit.>.

Including the exact degeneracies in d≥1 when bV/N∉ℕ as well, Eq. (<ref>) together with Eq. (<ref>) exactly gives the following generalized LSM theorem in any dimension of d≥0, which has a well-defined thermodynamic limit independent on the system size specification:

If a (d>0)-dimensional Hamiltonian possesses SU(N) spin-rotation and translation symmetries, or if a (d=0)-dimensional Hamiltonian possesses ℤ_N×ℤ_N, the system does not permit a trivially gapped phase when the total number b of Young-tableaux boxes per unit cell is not divisible by N. The converse is true if d=0. 

It is an appropriate point to remark that the role played by filling constraint is different between d=0 and d>0. In the former case, the filling condition restricts the theta term of a possible bulk extension, while filling restricts the form of translation symmetry at the low-energy continuum limit in d>0.
Moreover, unlike LSM theorem in arbitrary dimensions discussed in Sec. <ref>, our generalization of LSM does not have a compact form when d=0 is included, because of the higher symmetry PSU(N) than U(1)_Q.




 §.§ Bulk-boundary correspondences: LSM theorem with PSU(N)


In this Subsection, we will construct the SPT “bulk” theory with PSU(N)
symmetry
in one higher dimensions, which has the boundary theory with the LSM-type anomaly, in two different ways.




  §.§.§ A weak SPT point of view

To check whether the lattice system has any other LSM-type anomaly beyond Eq. (<ref>), we make use of a weak-SPT viewpoint on quantum anomaly calculated before. Since the quantum anomaly of a spatially d-dimensional system can be understood as that of the boundary of a ((d+1)+1)-dimensional bulk SPT phase. Therefore, the relevant SPT bulk that the LSM-type anomaly of system corresponds to is assumed to lie in the classes of H^d+2(PSU(N)×ℤ^d,U(1)) which is the PSU(N) (group) cohomology group with U(1) as a (trivial) group-module <cit.>, where ℤ^d is a direct product of d of lattice translations corresponding to the translations of its d-dimensional spatial boundary in the thermodynamical limit <cit.>. By Künneth formula,

    H^d+2(PSU(N)×ℤ^d)
       =   H^d+2(PSU(N)×ℤ^d-1)⊕ H^d+1(PSU(N)×ℤ^d-1)
    
       =   ⋯
       =   [⊕_k=3^d+2H^k(PSU(N)×ℤ^k-3)]⊕ H^2(PSU(N)),

where, for clearness, the coefficient ring U(1) is suppressed. Let us explain the first line of the equation above. H^d+2(PSU(N)×ℤ^d-1) represents the ((d+1)+1)-dimensional SPT phases protected by PSU(N)×ℤ^d-1 where the d-th ℤ is neglected, which is consistent with the projection H^d+2(PSU(N)×ℤ^d)→ H^d+2(PSU(N)×ℤ^d-1) where the group structure of d-th ℤ is forgotten.
H^d+1(PSU(N)×ℤ^d-1) corresponds to the (d+1)-dimensional SPT phases constructed by stacking, uniformly in the d-th direction, d-dimensional SPT's protected by PSU(N)×ℤ^d-1.
This correspondence can be derived in a nonlinear sigma model (with target space as classifying spaces of groups <cit.>) approach (See e.g. Appendix C. in <cit.>). Intuitively speaking, the d-th translation ℤ introduces an obstruction for the phase to be trivialized with unit cells along that direction.

The stacking constructions of SPT classes in H^d+2(PSU(N)×ℤ^d) can be also justified after we continue to apply the Künneth formula to (<ref>) to obtain

    H^d+2(PSU(N)×ℤ^d)=⊕_r=0^d[H^r+2(PSU(N))]^d!/(d-r)!r!,

where the copy number d!/[(d-r)!r!] of H^r+2(PSU(N)) is exactly the total number of perpendicular directions to stack ((r+1)+1)-dimensional PSU(N)-SPT phase by the translations ℤ^d.
Therefore, the last term H^2(PSU(N)) means ((d+1)+1)-dimensional SPT phases stacked by copies of a one-dimensional SPT phase (or chain) protected by PSU(N) uniformly in all the d spatial dimensions while all the d of translations are obstruction for it to be trivialized.
One characterizing property of phases by H^2(PSU(N)) is that lifting any of the d translations and PSU(N) out will trivialize the phase, while other components do not have this property.
In this sense, H^2(PSU(N)) is the maximally mixed anomaly of PSU(N)×ℤ^d.

On the other hand of our current system, the generalized LSM also has the same “mixing” characteristic since any N of unit cells can be SU(N) singlet if any of the translations are permitted to be broken explicitly, and PSU(N) symmetry is obviously essential for the ingappability as well.
Thus the LSM-type anomaly of our system is a class in H^2(PSU(N)).
Conversely, the anomaly factor in Eq. (<ref>) exactly implies a ℤ_N group structure and the generator is the fundamental case: b=1. Therefore, b=1,⋯,N represents the ℤ_N elements. Since H^2(PSU(N))≅ℤ_N thereby saturated by b=1,⋯,N, we can come to the conclusion that all possible LSM-type anomalies have been extracted out by Eq. (<ref>) with b=1,2,⋯,N, which are exactly classified by H^2(PSU(N)).



  §.§.§ A BF-theory approach

Similarly to the U(1)_Q case studied in details in Sec. <ref>, we can also construct the bulk massive fermions with opposite masses for the two opposite chiralities of Dirac fermions and derive the response theory for the fundamental chain (b=1) as the following BF theory by replacing A_U(1) in (<ref>) by A_PSU(N)+𝒜≡ A_U(N)+𝒜' with a dynamical U(1)-gauge field 𝒜' properly quantized as ∫ d𝒜'∈2πℤ:

    Z^(b=1)_bulk,PSU(N)
       =   ∫𝒟𝒜exp{∫i/2πTr[a ∧ d(A_PSU(N)+𝒜)]}
       =   ∫𝒟𝒜'exp{∫i/2πTr[a ∧ d(A_U(N)+𝒜')]}
       =   exp{∫i/2πTr[a ∧ dA_U(N)]},

where the smooth sector of the dynamical field 𝒜' plays the role as a Lagrangian multiplier and its topologically nontrivial sectors restrict the translation-gauge field a to satisfy (See e.g. Appendix C.3. of <cit.>):

    da=0   and ∮_closed loopa∈2π/Nℤ.

Comparing (<ref>) with (<ref>) justifies the effective filling factor (<ref>).
We can take the following gauge bundle on the mapping torus T^2× S^1:

    ∫_S^1a=2π/N   and  ∫_T^2Tr[dA_U(N)/2π]=1,

where a is a pull-back from a flat gauge field on the extra dimension S^1 and A_U(N) is a pull-back from a gauge field on the physical spacetime T^2 representing a flux insertion.
The translation-symmetry holonomy ∫_S^1a in (<ref>) realizes the translation-symmetry twisting of bulk field and thus mapping-torus partition function Z^(b=1)_bulk,PSU(N) exactly reproduces the phase ambiguity (<ref>) brought by a translation transformation. 

Therefore, due to the additivity of the bulk,

    Z^(b)_bulk,PSU(N)   =   (Z^(b=1)_bulk,PSU(N))^b
       =   exp(i2πb/N),

which is the same anomaly factor as Eq. (<ref>).



§ TLBC APPLIED TO THE INGAPPABILITIES BY TIME REVERSAL


In this Section, we will discuss the application of TLBC to the
ingappability constrained by a time reversal symmetry.
As an example, let us consider the following half-filled N-flavor spinless fermion on a square lattice with π-flux per plaquette:

    H_π   =   ∑_f=1^Nt(∑_r⃗c^†_(r⃗+x̂)fc_r⃗f+h.c.)
          +∑_f=1^Nt[∑_r⃗c^†_(r⃗+ŷ)fc_r⃗f(-1)^r_1+h.c.]
       ≡   H_π1+H_π2,

where we have chosen a gauge such that H_π is still lattice-translation symmetric along ŷ. Nevertheless, the translation symmetry along x̂ seems to be broken to two sites. However, physically, the system should be still symmetric along x̂ with one site, and such a translation symmetry and the original translation along ŷ in the current gauge choice, called magnetic translations <cit.> equally for all flavors f's satisfies:

    T_1c_r⃗fT_1^-1   =   c_(r⃗+x̂)fexp( i π r_2), 
    
    T_2c_r⃗fT_2^-1   =   c_(r⃗+ŷ)f.

We will use T_1,2 to denote the magnetic translations in the following discussion, instead of original lattice translations.
To make T_1 a well-defined unitary transformation representing an exact symmetry, we need to impose that L_y∈2ℕ.
Then the charge quantization in Eq. (<ref>) is automatically fullfilled.
It should be noted that the gauge-invariant nature of T_1,2 is encoded in their commutator:

    T_1T_2T_1^-1T_2^-1=-1.

Here “half-filled” implies that the particle number per physical 1×1 unit cell is 1/2.
However, the LSM theorem, even after generalized to U(N) cases, cannot say anything nontrivial for the present case with the magnetic translation symmetry.
Indeed, the following interaction, respecting U(N) and T_1,2, can open a gap with a unique ground state:

    Δ H   =   ∑_r⃗,ft(-1)^r_1[ i c_r⃗f^†c_(r⃗+x̂+ŷ)f+c_r⃗f^†c_(r⃗+3x̂+ŷ)f]+h.c..

It should be noted, however, that Δ H breaks the time-reversal symmetry explicitly.
Thus we may expect that imposing time-reversal symmetry in addition to the
magnetic translation symmetry potentially obstructs the trivially gapping.

Indeed, it has been proposed that, in N=1 case, the Hall conductance of the system when trivially gapped by a U(1)_Q and T_1,2 symmetric interaction, must be odd <cit.>.
This implies that, there cannot be a unique ground state with a non-vanishing excitation gap, when the time-reversal symmetry is additionally imposed.
However, the arguments in Refs. <cit.> again rely on the special choice of the system sizes.
Here we apply the TLBC to the systems invariant under
the magnetic translation and the time reversal, which
reveals an anomaly manifestation of the ingappability constraint
due to the time-reversal symmetry in the context of field theory.




 §.§ Low-energy effective theory and symmetries


Our strategy is to apply the TLBC to the low-energy effective field theory.
As a preparation, we first use the PBC to define the momentum
representation (Fourier components)

    c_(2n_1,r_2)   =   ∑_k⃗a_k⃗exp( i k_1n_1+ i k_2r_2),
    
    c_(2n_1+1,r_2)   =   ∑_k⃗b_k⃗exp( i k_1n_1+ i k_2r_2),

where k⃗∈(-π,π]×(-π,π].
We will introduce the TLBC in the next subsection.
The momentum representation of the tight-binding Hamiltonian (<ref>)
reads

    H_π1   =   t∑ a^†_kb_k[1+exp(- i k_1)]+b^†_ka_k[1+exp( i k_1)],
    
    H_π2   =   t∑ 2(a^†_ka_k-b^†_kb_k)cos k_2.

Therefore,

    H_π=t∑ (a^†_k,b^†_k)([       2cos k_2 1+exp(- i k_1);  1+exp( i k_1)      -2cos k_2 ])([ a_k; b_k ]).

The low-energy excitation momentum points are localized around

    K   =   (π,-π/2) and K'=(π,+π/2).

The low-energy effective theory can be obtained as,

    H_π≈ t∑_l={1,2};kψ_k^†(l)(-2σ_3k_2-σ_2k_1)ψ^(l)_k,

where

    ψ_k^(1)   =   σ_2([ a_k+K; b_k+K ])=([ - i b_k+K;   i a_k+K ]),
    ψ_k^(2)   =   ([ a_k+K'; b_k+K' ]).

We have the following symmetry representation:


  * Flavor U(N) symmetry U_{ϕ}

Conventionally, the global U(N) symmetry is defined as

    U_{ϕ} c_r⃗U^-1_{ϕ}=exp(∑_k=0^N^2-1-iϕ_kt_k)c_r⃗,

where t_k's: t^ff'_0=δ_f,f' the U(1) generator while {t_k:k=1,2,⋯,N^2-1} are SU(N) generators in the fundamental representation with a renormalization such that ϕ_k's are each compactified by 2π.
Equivalently,

    U_{ϕ}=exp(∑_r⃗,f,f'∑_k=0^N^2-1ic^†_r⃗ft_k^ff'ϕ_kc_r⃗f').

Since U(1) and SU(N) share a center, the definition above is not faithful and a more systematic approach is to impose a global structure both on U(1) and SU(N) parameters {ϕ_k} by a quotient over ℤ_N since U(N)≅[U(1)× SU(N)]/ℤ_N.

For the low-energy degrees of freedom,

    U_{ϕ}ψ U^-1_{ϕ}=exp(∑_k=0 ^N^2-1-iϕ_kt_k)ψ.

In the following discussion, the flavor indices “f” will be suppressed for simplicity.


  * Magnetic translation T_1

The lattice Hamiltonian is invariant under the magnetic translation along x̂-axis:

    T_1c_r⃗T_1^-1=c_r⃗+x̂exp( i π r_2),

which means

    T_1a_kT_1^-1   =   b_k+Q,
    
    T_1b_kT_1^-1   =   a_k+Qexp( i k_1),

where Q≡ K'-K=(0,π). 

In the low-energy field theory,

    T_1ψ T_1^-1= i τ_1ψ,

where τ matrices act on the valley components.


  * Translation T_2

The Hamiltonian is also invariant under the conventional translation along ŷ-axis:

    T_2c_r⃗T_2^-1=c_r⃗+ŷ,

which gives

    T_2a_kT_2^-1   =   a_kexp( i k_2),
    
    T_2b_kT_2^-1   =   b_kexp( i k_2),

and

    T_2ψ T_2^-1=- i τ_3ψ.



  * Time-reversal symmetry ℤ_2^T

Since we are interested in the constraint brought by time-reversal symmetry ℤ_2^T, we define the following (anti-unitary) time-reversal symmetry on our spinless fermions as

    Θ_0 i Θ_0^-1=- i ,
    Θ_0c_r⃗Θ_0^-1=c_r⃗.

Thus

    Θ_0a_kΘ_0^-1   =   a_-k,
    Θ_0b_kΘ_0^-1   =   b_-k,

and

    Θ_0ψ_k⃗Θ_0^-1=- i τ_2⊗σ_2ψ_-k⃗,

or

    Θ_0ψ_1,kΘ_0^-1   =   -σ_2ψ_2,-k,
    Θ_0ψ_2,kΘ_0^-1   =   σ_2ψ_1,-k.



Let us do a trivial rescaling k_2→ k_2/2 and t→1. Back to the real space, the Hamiltonian density becomes:

    ℋ   =   ∑_f=1^N∑_l=1^2 i ψ_f^†(l)(σ_3∂_2+σ_2∂_1)ψ^(l)_f
       =   - i ψ̅(-γ^0σ_2∂_1-γ^0σ_3∂_2)ψ
       =   - i ψ̅(γ^1∂_1+γ^2∂_2)ψ,

where, for clearness, we have suppressed the summation over flavor “f” and valley “l” indices. ψ̅≡ψ^†γ^0 and {γ^μ,γ^ν}=2η^μν=2·diag(+,-,-), the Dirac algebra in (1+2) dimensions. We can choose the following basis:

    {[     γ^0=σ_1,; γ^1=- i σ_3,;  γ^2= i σ_2. ].

Then the Lagrangian density is ℒ_Dirac=ψ̅ i γ^i∂_iψ with the following symmetry representations:

    T̂_1ψ(t,x,y)   =    i τ_1ψ(t,x,y),
    T̂_2ψ(t,x,y)   =   - i τ_3ψ(t,x,y),
    Θ̂_0ψ(t,x,y)   =   -τ_2⊗γ^2ψ^*(-t,x,y)
       =   -τ_2(Θ_Eψ)(t,x,y)

where Θ_rel=γ^2K is the emergent relativistic time-reversal of Dirac spinor with K the antilinear operator K i =- i K.
After a Wick rotation ∂_t→ i∂_τ, the eigenvalue problem of the corresponding Euclidean Dirac operator reads

    Γ^i∂_iψ(τ,x,y)=λψ(τ,x,y),

in which {Γ^j,Γ^k}=-2δ^jk=-2·diag(+,+,+) and

    {[  Γ^0= i σ_1,; Γ^1=- i σ_3,;  Γ^2= i σ_2. ].


The symmetry field-configuration representations is analytically continued to <cit.>

    T̂_1ψ(τ,x,y)   =    i τ_1ψ(τ,x,y),
    T̂_2ψ(τ,x,y)   =   - i τ_3ψ(τ,x,y),
    Θ̂_0ψ(τ,x,y)   =   - i τ_2⊗Γ^2Γ^0ψ^*(-τ,x,y)
       =   -τ_2⊗ (CR_τψ)(τ,x,y),

with Cψ(τ,x,y)=Γ^2ψ^*(τ,x,y) the charge conjugation and R_τψ(τ,x,y)= i Γ^0ψ(-τ,x,y) the reflection about the temporal direction.
The analytical continuation Θ_rel↦ CR_τ above can be understood from their behaviors on the gauge field <cit.>: Θ_relA_μ(t,x⃗)=(-1)^1+δ_μ,tA_μ(-t,x⃗), CA_i(τ,x⃗)=(-1)A_i(τ,x⃗) and R_τ A_i(τ,x⃗)=(-1)^δ_i,τA_i(-τ,x⃗), which implies that CR_τ reproduces Θ_rel. 




 §.§ TLBC for the effective field theory


Let us now impose the TLBC for the low-energy degrees of
freedom ψ(τ,x,y) as

    {[ ψ(τ,x+L_x,y)=T_2ψ(τ,x,y);;    ψ(τ,x,y+L_y)=ψ(τ,x,y). ].

Then we do a flux insertion into the hole rounded by L_x:

    {[                       A_τ(τ,x,y)=0;;           A_x(τ,x,y)=π/L_xτ/L_τt_0;; A^θ_y(τ,x,y)=θ/L_y1/N(t_0-t_N^2-1), ].

where t_0 is an (N× N) identity matrix and t_N^2-1=diag[1,1,⋯,1,-(N-1)]. For later use, we also formally introduce a static flux into the hole of L_y, but we take θ=02π later, which is gauge equivalent to A_y=0 by a large U(N)-gauge transformation as exp[i2π y(t_0-t_N^2-1)/(NL_y)]. Afterwards, we further do a T_1 transformation.
Then the Euclidean path-integral partition function related to this flux insertion followed by T_1 transformation takes the form as:

    z(θ)   ≡   ∫𝒟(ψ̅_α,ψ_α)exp(-∫ℒ_Dirac[A_x(τ),A_y^θ])

with the space-time manifold as a four-dimensional torus T^4 and 𝒯 the time-ordering operator with the boundary condition combined with Eq. (<ref>) as

    {[ ψ(τ+L_τ,x,y)=-T_1ψ(τ,x,y);;  ψ(τ,x+L_x,y)=T_2ψ(τ,x,y);;     ψ(τ,x,y+L_y)=ψ(τ,x,y), ].

where the minus sign in the boundary condition along τ is due to the fermionic nature of the path integral.

TLBC by T_1,2 is obviously consistent with Θ_0 since [T_1,2,Θ_0]=0.
Nevertheless, one might have noticed that the flux-insertion by A_x is only π flux through the τ-x plane rather than 2π. At the first glance, it gives an ill-defined transition function across the cut between  τ=L_τ and τ=0. However, it is actually canonical because the latter T_1 transformation at τ=L_τ anticommutes with T_2 imposed in x̂ direction: T_1T_2T_1^-1T_2^-1=-1, and it makes the boundary condition Eq. (<ref>) appear inconsistent, either. Such an extra “-1” sign inconsistency in the boundary condition exactly compensates the transition-function sign ambiguity brought by a π-flux insertion by A_x(τ,x,y) in Eq. (<ref>). A systematic construction of the gauge bundle above is given in <cit.>.

Then let us take a look at the following ratio

    Z_π≡z(θ=2π)/z(θ=0),


and the phase of Z_π denotes the difference between the momentum transfers in the presence of A_y^θ=2π and A_y^θ=0 (c.f. Eqs. (<ref>,<ref>)).
Thus, due to the gauge equivalence of A_y^θ=2π and A_y^θ=0, Z_π is 1 if the theory is anomaly-free.
Thus the discrepancy between Z_π and 1 signals the gauge anomaly of our Dirac theory (<ref>).

To evaluate this anomaly factor Z_π, we can first introduce a Pauli-Villars regulator Lagrangian ∑_α,f i ξ̅_α,f[ i Γ^i(∂_i- i A_i)-m_P-V]ξ_α,f to regularize the 3D partition function, where ξ_α is a two-flavor bosonic spinor and the diagonal mass term (m_P-Vδ^αβδ^ff') preserves U(N) and T_1,2.
The partition function is regularized as |Z_3D|exp(-iπη_3D/2) explicitly breaking ℤ_2^T due to the regulator mass, where |Z_3D| is the absolute value of our massless theory partition function and η_3D≡lim_ε→0^+∑_ksgn(λ_k)exp(-ελ_k^2) with {λ_k} the spectrum of 3D two-flavor Dirac operator.
However, ℤ_2^T can be restored when we attach a 4D bulk on it, or equivalently, add a 4-D counter-term supported by X':∂ X'=T^3 whose boundary is our 3D manifold T^3: 2[(2/48)∫_X'trR∧ R/(2π)^2+∫_X'(1/2)TrF∧ F/(2π)^2], where R is the 2-form curvature tensor and “tr” is taken over the four spacetime indices and “Tr” over the flavors.
Thus the regularized partition function on X' by Atiyah-Patodi-Singer (APS) index theorem <cit.> is

    Z_3D(X')   =   |Z_3D|exp(-iπℐ_4D(X')), 
    ℐ_4D(X')   =   η_3D(∂ X')/2-[∫_X'N/24trR∧ R/(2π)^2+TrF∧ F/(2π)^2],

in which Z_3D(X') explicitly respects time-reversal symmetry since ℐ_4D of a 4-D Dirac operator with N flavors and 2 valleys, the number difference of zero modes between positive and negative chiralities, is proportional to the effective action of a (3+1)-dimensional massive fermion and it is an integer calculated under APS boundary condition.
Z_3D(X') potentially depends on the extension X', which disables our massless system to be purely a 3D model.

As shown below, Z_π in (<ref>) can be calculated by a phase ambiguity induced by two different extensions, with and without, respectively, a 2π(t_0-t_N^2-1)/N-flux in the orientable D_X' spanned by the extra dimension and the ŷ-direction circle S^1_(ŷ) (non-orientable extensions to be discussed later).
Thus X'=T^2× D_X' and we also set ∂ D_X'=S^1_(ŷ) so that ∂ X'=T^3 is the spacetime of our (2+1)-dimensional system.
We take X'_1,2=T^2× D_X'_1,2 to extend both the space-time and the gauge field, where D_X'_1 and D_X'_2 are not necessarily the same.
We insert 2π(t_0-t_N^2-1)/N-flux through D_X_1' as in FIG. (<ref>) and we obtain A_y=A_y^θ=2π on its boundary S^1_(ŷ) by the Stokes' theorem up to a small gauge transformation.
On the other side, D_X'_2 has no flux through it.
Then A_y=A_y^θ=0 on its boundary S^1_(ŷ) up to a small gauge transformation.
Therefore, the bulk-extension dependence here precisely characterizes the gauge anomaly (<ref>) of the boundary theory between the two different gauge choices A_y^θ=2π and A_y^θ=0:

    Z_π   =   Z_3D(X'_1)/Z_3D(X'_2)=(-1)^ℐ_4D(X'_1∪-X'_2)=-1,

where we have used the index pasting rule ℐ_4D(X'_1)-ℐ_4D(X'_2)=ℐ_4D(X'_1∪-X'_2) with X'_1,2 glued as FIG. (<ref>) through their common boundary T^3 along which the gauge fields are pasted by a gauge transformation, and the index for the closed manifold X'_1∪-X'_2:

    ℐ_4D(X'_1∪-X'_2)
       =   2[-N/48∫_X'_1∪-X'_2trR∧ R/(2π)^2-1/2TrF∧ F/(2π)^2]
       =   -2/32π^2∫_X'_1∪-X'_2d^4xϵ^ijklTr(F_ijF_kl)4N
       =   14N,

with the gravitational contribution on general closed orientable manifolds valued in 2×2× Nℤ=4Nℤ thereby irrelevant to Z_π, and a unit U(N) instanton inside D_X_1'∪-D_X_2' as shown in FIG. (<ref>).
Equation (<ref>) also means that the SPT bulk attached is nontrivial and the low-energy field theory (<ref>) cannot be regulated by local regulators respecting onsite U(N), T_1,2 and ℤ_2^T without the bulk attachment (<ref>).































The anomaly we have obtained is fully a mixed type since it will be trivialized once we discard any of the required symmetries.
This property is consistent with the ingappability on the lattice level, e.g. the magnetic-translation and U(N) symmetric gapping term but time-reversal breaking Δ H in Eq. (<ref>) trivially gaps the system.
Thus we expect such a non-abelian symmetry anomaly is the desired LSM type. Moreover, two layers (or copies) of lattice systems can be trivially gapped respecting all the required symmetries since we can always set opposite chemical-potential term, e.g. ∑_f∑_Z=1,2(-1)^Z∑_r⃗δμ c^† Z_r⃗,fc^Z_r⃗,f with Z labelling the layers so that one of two is fully-filled while the other empty, without breaking (total) filling fraction and symmetries. This aspect is also implied by and consistent with our ℤ_2 anomaly classification. In addition, there is no purely U(N)⋊ℤ_2^T anomaly, which implies that without T_1,2 we can trivially gap the system, consistent with the lattice situation as well.
Furthermore, we do not use the emergent properties of T_1,2 at low-energy limit, such as their finite cyclicality, so the corresponding emergent anomalies, if any, are not included.
Instead, we only make use of the gauge-invariant nontrivial commutator T_1T_2T_1^-1T_2^-1=-1 already held at the lattice level.
Additionally, due to the time-reversal symmetry, we should also consider non-orientable manifold in Euclidean signature twisted by CR_τ (c.f. Eq. (<ref>)). However, the anomaly class detected in this way is still ℤ_2 identical to the ℤ_2-class in our orientable cases which implies the absence of pure time-reversal anomaly (See e.g. Sec. IV. G. of <cit.>).
Combining these observations above, this mixed anomaly in field theory suggests the ingappability in the presence of the time reversal symmetry together with the flavor and magnetic translation symmetries.
This will be further confirmed in the next subsection, by defining a corresponding quantity on the lattice.




 §.§ Lattice-realization of ℤ_2-classifying anomaly



Z_π=-1 defined for the effective field theory in the previous subsection
is a topological invariant, e.g. it cannot be changed along renormalization flow (RG) from the critical point as long as the interaction perturbing the system respects all the required symmetries.
Here we look for a lattice quantity Z'_π whose low-energy and continuum limit is Z_π. The 't Hooft anomaly matching then suggests it would be a robust topological invariant which can be evaluated exactly by its infrared counterpart Z_π.


Let us assume that the system is trivially gapped under all the symmetries.
It will lead to a contradiction, as we will demonstrate below.
To do so, we first impose the following TLBC:

    {[ c_x+L_x,y=c_x,y+1,;   c_x,y+L_y=c_x,y. ].

The lattice translation T_2 is well-defined, but the magnetic translation T_1, which is defined on the lattice before modded by the relation (<ref>), is incompatible with (<ref>) since it is y-dependent and y coordinate has a freedom to be adjusted by Eq. (<ref>).
A sensible form of T_1 in the modded space can be obtained by fixing the assignment of lattice coordinates, e.g. c_x,0 with x=1,⋯,L_xL_y similar to Eq. (<ref>). By Eq. (<ref>), T_1 reduces to T'_1 which is the lattice translation.
The existence of T_1 on the lattice with periodic boundary conditions implies that such a symmetry has a different form in our tilted case especially in the low-energy sense.
It follows straightforwardly from the assumption of the gapped unique ground state that the ground state should go back to itself after an adiabatic insertion (T→+∞) of π flux into the hole rounded by L_x: 

    A_x(t,r⃗)=π/L_xt/Tt_0

followed by the T'_1 transformation:

    T'_1c_x,yT'_1^-1=c_x+1,y.

It is due to that the following large gauge transformation V:

    V'c_x,yV'^-1=exp( i π/L_xx)exp( i π y)c_x,y,

with (x,y)∈[1,L_x]×[1,L_y] is well-defined since the original magnetic translation T_1 requires L_y∈2ℤ as mentioned before, and it is able to restore the initial Hamiltonian after the preceding π-flux insertion and T_1'.

Let us consider the following quantity: (𝒯 is time-ordering)

    z^(θ)
       ≡   Tr_G.S.{T̂'_1V'𝒯exp{- i ∫ H[A_x(t),A_y^θ]dt}}
       =   ⟨G.S.|T̂'_1V'𝒯exp{- i ∫ H[A_x(t),A_y^θ]dt}|G.S.⟩,

where H is the time-dependent lattice Hamiltonian and |G.S.⟩ is the certain presumed unique gapped ground state.
We also set an artificial flat U(1)_Q gauge field in y direction by A_y=θ(t_0-t_N^2-1)/(NL_y) where our case corresponds to θ=0.
Then z^latt(θ) satisfies:

    z^(θ)=z^(θ+2π),

due to the large gauge transformation mentioned before.

Let us imagine a series of model with θ changing from 0 to 2π
and consider

    Z_π^latt   ≡   z^[θ=2π]/z^[θ=0].


Before evaluating this ratio of partition functions, we first notice that T_1'V' acting on c_x,y with (x,y)∈[1,L_x]×[1,L_y]:

    T_1'V'c_x,y(T_1'V')^-1
       =   exp( i π/L_xx)exp( i π y)c_x+1,y
       =   T_1Vc_x,y(T_1V)^-1,

which has exactly the same effect as the T_1 on periodic lattice followed by a “half-large” gauge transformation Vc_x,yV^-1≡exp( i π x/L_x) with (x,y)∈[1,L_x]×[1,L_y], which is well-defined according to the discussion below Eq. (<ref>).

Therefore, the continuum limit of z' is

    z^   =   Tr_G.S.{T̂'_1V'𝒯exp{-∫ H[A_x(τ),A_y^θ]dτ}}
       →   Tr_G.S.{T̂_1V𝒯exp{-∫ H[A_x(τ),A_y^θ]dτ}},

which precisely reproduces the form of z in Eq. (<ref>) with the boundary condition as Eq. (<ref>).
Similarly as the U(1)_Q case, the role played by V in the path integral of z is implicitly an assignment of a transition function between the last two time slices preceding the T̂_1 gluing.
Thus, we can make use of 't Hooft anomaly matching to evaluate Z_π^latt by Z_π:

    Z^latt_π=Z_π=-1.

Therefore, we have

    z^latt(θ=0)=z^latt(θ=2π)=-z^latt(θ=0),

which is impossible unless

    z^latt(θ=0)=0.

It implies the following two equal-energy states are actually orthogonal:

    ..T̂'_1V'𝒯exp{- i ∫_0^TH[A_x(t),A_y=0]dt}|G.S.⟩⊥|G.S.⟩,

thereby contradicting the assumption of the unique |G.S.⟩. It means the ground-state degeneracy must be nontrivial and we arrive at an LSM-type ingappabillity.



§ TLBC APPLIED TO MANY-BODY CHERN NUMBERS


In this Section, we will apply TLBC to obtain a nontrivial constraint on integer quantum Hall conductances. The well-known Thouless-Kohmoto-Nightingale-Nijs
formula <cit.>
identifies the Hall conductance of a band insulator
with the Chern number defined by the Berry connection of
eigenstates in the Brillouin zone.
It was then generalized to “many-body Chern number” for
more general interacting systems <cit.>.
It should be noted that, by definition, evaluation of
the Chern number generally requires an integral over the
entire Brillouin zone or the entire parameter space.
Such integrations can be time-consuming
even for free electrons (band insulator) with generic band structures,
especially when Chern numbers n_ch are large, since the critical mesh size of the discretized Brillouin zone is of order 𝒪(√(n_ch)) <cit.>.
Evaluation of many-body Chern numbers of interacting
systems can be even more difficult
(however, see also Ref. <cit.>).


 
On the other hand, since the low-energy physics of lattice models
can be often described by field theory, such as Dirac fermions,
it is natural to attempt to determine the Hall conductance (Chern number)
by the low-energy effective field theory.
If this works, the Chern number appears to be determined with
only the information in the low-energy limit.
However, the most naive expectation fails
in general <cit.>.
This is not surprising, considering the global nature of the Chern number.
In terms of field theory, the discrepancy can be attributed
to so-called “spectator” fermions which may exist at higher
energies and are invisible in low-energy
physics <cit.>.


 
Nevertheless,
anomaly, which is a central concept utilized in the present paper,
is a essential property of field theory which is robust even
at higher energies.
We can therefore derive a nontrivial constraint on the Hall conductance
from field theory, as we demonstrate below.
In other words, the anomaly controls possible behaviors of the spectator
fermions.



 
Let us consider the half-filled π-flux system above. Since the system can be gapped if we explicitly break the time-reversal symmetry, e.g. by Δ H in Eq. (<ref>), it is gappable with a unique ground state in the presence of U(N) and magnetic translations. In the following discussion, we will investigate the constraint on the many-body Chern number
in such a gapped phase
when U(N) and magnetic translations are respected by the half-filled spinless lattice Hamiltonians.


 
To derive the constraint, however, it is convenient to start from
the gapless system which is obtained by imposing the time-reversal
symmetry.
By TLBC in Eq. (<ref>) together with its lattice realization in Eq. (<ref>), we have obtained the partition function Eq. (<ref>) for the critical Hamiltonian H_π in Eq. (<ref>).

It can be rewritten as

    Z_3D(X')   =   Z_3D:P-Vexp[iπ∫_X':∂ X'=T^3TrF∧ F/(2π)^2],

where Z_3D:P-V≡|Z_3D|exp(-i πη_3D/2) is regularized by the Pauli-Villars regulator defined before, which is automatically gauge invariant, and T^3 is the (2+1)-dimensional spacetime where our square lattice is defined. At the first sight, the partition function depends on the choice of the 4-dimensional spacetime X'. However, as we will see, the physical observables, such as Hall conductances, when we gap it, is independent of this dimension extension. In addition, the X'-bulk regulator is the only sensible way to regularize the partition function up to a non-universal part as summarized by <cit.>.


 
By the discussion following (<ref>), the partition function (<ref>) is, up to a non-universal factor, equal to the partition function of a (3+1)-dimensional Dirac fermion in ℝ^4⊃ X' whose mass is opposite of the P-V regulator within X' and is the same as the P-V regulator mass in (ℝ^4-X'), which can be identified as the vacuum [c.f. (<ref>) and (<ref>)] <cit.> as in FIG. (<ref>).

The (3+1)-dimensional Dirac mass inevitably vanishes on the interface T^3=∂ X'=-∂(ℝ^4-X'), along which the (2+1)-dimensional gapless mode flows, because the masses have different signs on the two sides of the interface and the mass is restricted to be real throughout the whole bulk by the time-reversal symmetry.
Then we introduce a general local interaction, not necessarily perturbative, to the Hamiltonian H_π in Eq. (<ref>), and the interaction gaps the system with a unique ground state, breaking time-reversal symmetry while respecting U(N) and magnetic translations.
In the equivalent (3+1)-dimensional bulk-interface-bulk picture described above, the locality of this gapping interaction in (2+1) dimensions is translated as follows: the interaction only takes place locally around the interface ∂ X'=T^3 (since it is local on ∂ X') and thus the time reversal is still preserved deeply in the X' and the vacuum (ℝ^4-X') on its two sides.
Then the partition function for this gapped system takes the following general universal form by the interface interpretation above (c.f. Appendix):

    Z_3D-gapped(X')=exp[i∫_ℝ^4θ(x)TrF∧ F/(2π)^2],

with

    θ(x)={[                 -2nπ, x deep in the vacuum;;                    π,         x deep in X', ].

where the universal quantity n∈ℤ since the theta-angle of the vacuum is 02π and the varying θ(x) only breaks time reversal near ∂ X'=T^3 [Here we fix the theta-angle deep in X' but relax that in the vacuum without loss of generality since only their difference is physical.].
(A non-local gapping interaction is able to change the θ-angle arbitrarily deeply inside X' or the vacuum.)
Thus, by an integration by part for the exponential in Eq. (<ref>) and taking a U(1)_Q connection A=A_u(1)𝕀_N× N, we obtain

    Z_3D-gapped(X')=exp[i∫_T^3(2n+1)N/4πA_u(1)∧ dA_u(1)],

which is explicitly a pure (2+1)-dimensional partition function on T^3 independent of the “bulk” X', as expected since our system is purely supported by a 2-dimensional spatial lattice.
The physical meaning of the winding number “n” of θ(x) is the stack of n layers of gapped U(N) and magnetic-translation symmetric systems, e.g. n layers of 2 copies of our boundary theories (<ref>) with mass terms, which are able to non-anomalously respect the time reversal in addition to U(N) and T_1,2 at the low energy in their own dimensions. 


We can directly, by the level of the Chern-Simons term in Eq. (<ref>), identify the Hall conductance or in terms of many-body Chern number as:

    σ_H=N2N.


The permitted Hall conductances by the constraint above are symmetric about the zero value, which is not a coincidence. Let us assume that, preserving the U(N) and magnetic translations, we have realized an integer quantum Hall state with σ^(0)_H. Then we do a time-reversal transformation Θ_0 on this state (or its parent Hamiltonian) and the resultant state has the Hall conductance -σ^(0)_H, which still respects the original U(N) since, the group elements of Θ_0^-1U(N)Θ_0 are the same as U(N), which can be seen in a basis where all the U(N) generators are purely real or imaginary [This can be always done for general N's since the generators are Hermitian.]. This state is also symmetric under the original magnetic translations since the gauge-invariant commutator (<ref>) is time-reversal invariant (namely, π-flux is time-reversal symmetric). Thus, the permitted Hall conductances must be symmetric about the zero. For general flux cases out of the scope, the above argument does not work, since if we want to map the magnetic translations back to its initial form, we need to further do a spatial reflection which, together with the time reversal, trivially transforms σ_H^(0) back to σ_H^(0).


This generalizes the constraint for N=1
obtained from the lattice argument <cit.>.
It is notable that this generalized
constraint can be derived within the field theory.
We can see that the contribution from the spectator fermion is 02N which is controllable in the presence of U(N) and magnetic translations.

In a short summary, TLBC is useful in the bulk SPT identification of the gapless point Eq. (<ref>) since it constitutes a gauge bundle in Eq. (<ref>) to detect the anomaly Eq. (<ref>). This bulk SPT correspondence is helpful since it provides us a visualizable way to treat our (2+1)-dimensional system as a domain wall in FIG. (<ref>) and, furthermore, to restrict the integer Hall conductances. 


 
We can further argue that such a constraint on many-body Chern numbers is part of a more general framework.
An observation of Eq. (<ref>) implies that we can imagine a general system as a ℤ_2^T-broken surface of an SPT bulk protected by (G×ℤ^2)⋊ℤ_2^T where G is an onsite symmetry and ℤ^2 can be translations or magnetic translations. Then the bulk regularization can induce a natural constraint on the G-response, e.g. Hall conductances. The surface theory is trivial in the sense that the bulk is trivial if ℤ_2^T is broken, while the nontrivial property, e.g. constraints of Hall conductances, results from that the surface theory can be attached to nontrivial SPT bulk where ℤ_2^T is preserved deeply in bulk. In addition, the method is also directly applicable to arbitrary even spatial dimensions where Chern-Simons actions can be defined.





§ CONCLUSIONS AND DISCUSSIONS

In this work, we introduced a generalized boundary condition TLBC for discussion of the LSM theorem and related problems.
Under the TLBC, each lattice site can be reached by repeating the lattice translation in a particular direction.
As an advantage of the TLBC, the LSM theorem in two or more dimensions can be derived by the flux insertion argument without an artificial restriction on the system sizes, which renders the thermodynamic limit more natural.
Moreover, with the TLBC, the LSM theorem in arbitrary dimensions is related to the global chiral anomaly of Dirac fermion in (1+1) dimensions in a unified manner.
This is also extended to the LSM theorem with the larger SU(N) symmetry, and to the similar ingappability constraint under the time reversal and the magnetic translation symmetries with the on-site U(N) symmetry.
 Furthermore, we also utilized the TLBC to derive a nontrivial constraint
on many-body Chern numbers from field theory, which generalizes
the known result from the lattice.
It shows the power of anomaly which universally dictates higher energies,
within the universal field theory formulation and independently of the
regularization/realization.
Although similar phenomena have been
proposed on the surface of topological insulators <cit.>, our
result is the first to relate the Hall conductance of a pure
2-dimensional lattice system to SPT phases.


Despite of the usefulness of TLBC as we have demonstrated, it cannot be
applied to exploit consequences of spatial reflection symmetries, which
are inconsistent with the TLBC as introduced in this paper.
Nevertheless, we believe that the idea of modifying the boundary
condition from the standard PBC in the LSM-type argument,
and combining it with anomaly in field theory, is opening up a new
direction in quantum many-body theory.
 In fact, after the appearance of the early version of the present
paper in arXiv, Furuya and Horinouchi <cit.>
generalized our work by introducing a twisted boundary condition
and deriving a nontrivial constraint for systems on the checkerboard lattice.
We hope that more developments will follow in the future.











Both authors thank Yohei Fuji, Shunsuke Furuya, Hans Hansson, Chang-Tse Hsieh, Yuan-Ming Lu, Ying Ran, Shinsei Ryu, Xiaolin Sun, Yuji Tachikawa, Yasuhiro Tada, Cenke Xu, and Xu Yang for useful discussions.
Y. Y. was supported by JSPS fellowship.
This work was supported in part by MEXT/JSPS KAKENHI Grant Nos. JP19J13783 (Y. Y.), JP17H06462 (M. O.) and JP19H01808 (M. O.).
A part of the present work was performed at Kavli Institute for Theoretical Physics, University of California at Santa Barbara, supported by
US National Science Foundation Grant No. NSF PHY-1748958.


*


§ DERIVATIONS OF (<REF>), (<REF>) AND (<REF>)


In this part, we will give a rigorous proof of (<ref>) and (<ref>).

Due to the discussion around Eq. (<ref>), the low-energy effective theory can be thought as the interface theory between the vacuum and a bulk in the Euclidean signature as

    S_bulk=∑_α,f∫_X' iψ̅_α,f(-i𝒟_4-m_0)ψ_α,f+S_reg,

where S_reg≡∑_α∫_ℝ^4 iχ̅_α,f(-i𝒟_4+m_0)χ_α,f is the bosonically statistical spinor regulator and 𝒟_4=∑_i=0^3Γ̃^i(∂_i-𝕀A_i) is the 4-D Dirac operator where {Γ̃^i,Γ̃^j}=-2δ^ij, i,j∈{0,1,2,3}. The vacuum action is simply

    S_vac=∑_α,f∫_ℝ^4-X' iψ̅_α,f(-i𝒟_4+m_0)ψ_α,f+S_reg.

The symmetries U(N) and T_1,2 all extends in a natural way to both bulks. Gapping our (2+1)-dimensional system by a U(N) and T_1,2 gapping term can be seen as physically equivalent to gapping the interface between the artificial bulks S_bulk and S_vac by a U(N) and T_1,2 respecting boundary interaction within the interface. Therefore, we can resolve the former question in the approach to the later one.

Since X' is embedded in ℝ^4 in a natural way, we take a local geometry near T^3=∂ X' as T^3×ℝ where ℝ:x^3∈(-∞,+∞).
We introduce a spatial dependence U(N)-preserving mass term m^αβ(x^3) to gap the boundary states where α and β are valley indices, namely the whole material to be gapped throughout x_3∈(-∞,∞):

    S_full   =   ∑_α,β∫_ℝ^4 iψ̅_α,f[-i𝒟_4δ^αβ-m^α,f;β,f'(x^3)]ψ_β,f'
          +S_reg,

where m^α,f;β,f'(x^3) interpolates the nontrivial bulk S_bulk asymptotical at x^3→+∞ and the vacuum at x^3→-∞:

    m^α,f;β,f'(x^3)={[  m_0δ^α,βδ^f,f',          x^3→+∞;; -m_0δ^α,βδ^f,f',          x^3→-∞. ].

Moreover, the gapping is required to respect U(N) and T_1,2 proportional to τ_3,1, and then

    m^α,f;β,f'(x^3)∝δ^α,βδ^f,f'

since any matrix in SL(2,ℂ) commuting with τ_1,3 must be proportional to identity.
Combining this result and that a general non-vanishing mass term can only take a Dirac-mass or a chiral-mass form, then the interpolating action is

    S_full   =   ∑_α,f∫_ℝ^4 iψ̅_α,f{-i𝒟_4+m_0exp[iΓ̃^5θ(x^3)]}ψ_α,f
          +S_reg,

where we have, without loss of generality, normalized the mass amplitude to be a constant m_0 which can be done canonically, since the mass gap never closes for any x^3, and Γ̃^5≡-Γ̃^0Γ̃^1Γ̃^2Γ̃^3 is Hermitian and the continuous function θ(x^3) satisfies

    θ(x^3)={[    π2π, x^3→+∞;;    02π, x^3→-∞. ].

To evaluate the partition function of S_full, we do the following x^3-dependent chiral rotation

    ψ_α,f   →   exp[iΓ̃^5θ(x^3)/2]ψ_α,f;
    ψ̅_α,f   →   ψ̅_α,fexp[iΓ̃^5θ(x^3)/2]

to eliminate the chiral phase of the mass term. However, it is inevitable to introduce a theta term due to 4-D global chiral anomaly although the matter partition function is regularized to be unity by S_reg. Therefore, after a U(1)_Q connection A≡ At_0 is taken,

    Z_full   =   exp[i∫_ℝ^4θ(x^3)TrF∧ F/(2π)^2]
       =   exp{iN/(2π)^2∫_ℝ^4{d[θ∧ d(A∧ dA)-dθ∧ A∧ dA]}}
       =   exp[i∫_T^3(∫ dθ/π)N/4πA∧ dA],

where we have made use of Bianchi identity d(dA)=0 for U(1)_Q connections and the assumption that A_μ is nearly x^3-independent within the interface of two bulks.

The response exactly implies that, at the interface, namely our interested (2+1)-dimensional system on T^3,

    σ_H   =   N∫ dx^3∂_x^3θ/π
       =   N2N,

where we have used the winding property of θ(x^3) in Eq. (<ref>). Thus we reach the final conclusion of the constraint on the Hall conductance of the uniquely gapped U(N) and T_1,2 respecting system: σ_H=N2N.
The role played by T_1,2 is also obvious from the derivation above, e.g. without T_1 or T_2, we can gap the boundary, respectively, by a mass as m_0exp[τ_3⊗ iΓ̃^5θ(x^3)]δ^f,f' or m_0exp[τ_1⊗ iΓ̃^5θ(x^3)]δ^f,f', resulting an unrestricted σ_H.














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