We consider charge transport for interacting many-body systems with a gapped ground state subspace that is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary that preserves the ground state space, we associate an index that is an integer multiple of 1/p, where p is the ground state degeneracy. We prove that the index is additive under composition of unitaries. This formalism gives rise to several applications: fractional quantum Hall conductance, a fractional Lieb–Schultz–Mattis (LSM) theorem that generalizes the standard LSM to systems where the translation-invariance is broken, and the interacting generalization of the Avron–Dana–Zak relation between the Hall conductance and the filling factor.

The use of topology to study condensed matter systems is among the most influential developments of late 20th century theoretical physics.1,2 The first major application of topology appeared in the context of the quantum Hall effect3–5 in the early 1980s, and topological concepts have since been applied systematically to discover and classify phases of matter.6–12 The full classification for independent fermions is well developed, in particular by K-theory,13–15 but a framework of similar scope is lacking for interacting systems, except possibly in one dimension where there is a classification of matrix product states16–18 and cellular automata.19,20

For non-interacting systems, several topological indices can be formulated as Fredholm–Noether indices21–23 or, equivalently, as transport through a Thouless pump.24 These formulations have been influential and insightful, in particular for non-translation-invariant systems.25 For example, the quantum Hall conductance,26 the Z2-Kane–Mele index,27,28 and the particle density can be expressed as (integer-valued) Fredholm indices. Let us briefly recall the setting in the easiest case of one-dimensional systems, playing on the Hilbert space 2(Z). It takes as input a self-adjoint projection P, which we think of as a Fermi projection of a local and gapped one-particle Hamiltonian, and a unitary U that commutes with P, [P, U] = 0. The index is then defined as Tr[P(U*1NU1N)], provided that [U,1N] is trace-class, with 1N being the orthogonal projection on the subspace 2(N). The upshot is then that, on the one hand, this index is integer-valued, and on the other hand, it can be identified with the average charge transported by U across the origin, when starting from the fermionic state defined by the projection P, i.e., the filled Fermi sea; see Refs. 22, 29, and 30 for proofs and details.

In this paper, we develop an interacting analog to this formalism. It is similar to the non-interacting theory in that it is very modular. Instead of a Fermi projection, it takes as input the ground state projection of a gapped many-body Hamiltonian. Instead of the one-particle unitary, we consider a many-body unitary U commuting with P. We also need a notion of locally conserved charge that was absent in the single-particle setting as the charge there was implicitly defined as the number of fermions. The index is then constructed out of these data (P, U, and the charge) under appropriate regularity conditions. A striking difference is the possibility that the projector P has a higher rank p>1, corresponding to degenerate or quasi-degenerate ground states. In this situation, the possible values of the index are now in 1pZ. There are two generic situations that lead to a finite p > 1, namely, spontaneous breaking of a discrete symmetry and topological order. The first case can morally speaking be reduced to the case of p = 1 by restricting to superselection sectors, in which case U can fail to be a symmetry and only some power Un survives as a symmetry in the superselection sector. This leads to a rational index in a straightforward way. We cover this case explicitly in Sec. VI. The case of a topological order31,32 is more interesting. It means that the different vectors in the range of P cannot be distinguished by local observables. Our framework is tailored toward this case, allowing, for example, for fractional quantum Hall conductance. In Ref. 33, fractional quantization is discussed in the same setting as here, but the proof sketched there relies on a different more involved strategy. We would however like to point out that, at present, we do not know any model where we can rigorously confirm that this index takes a non-integer value in a topologically ordered subspace ran(P), as one expects to happen in fractional quantum Hall systems.

The present paper generalizes our previous work34 that was restricted to rank-1 projections. Just like there, different choices for U correspond to different physical situations: Our index is the (fractional) quantum Hall conductance when U is associated with an adiabatic increase in flux; it is the ground state filling factor when U is a discrete translation, and the theorem is a new, fractional and multidimensional version of the Lieb–Schultz–Mattis (LSM) theorem; see Refs. 35 and 36 for the integral case. We go further and also prove that the index is additive under composition of unitaries, as it should be if it has the interpretation of a charge transport. Applied to the specific situation of a family of covariant Hamiltonians, the additivity yields a relation between two indices. In the context of the quantum Hall effect, it relates the filling factor to the quantum Hall conductance. This is well known in the non-interacting setting.37 Here, the relation is shown to hold in the interacting, possibly fractional setting; see also Ref. 38 for similar results as well as39 for a geometric perspective.

Finally, we mention a technical difference with the non-interacting theory briefly discussed above. We do not formulate our theory in infinite volume from the start, mainly because the above concepts “ground state projector” and “many-body unitary” are in general not available in a Hilbert space setting if the volume, or more precisely the number of particles, is infinite. When fixing a reference state, one can consider a Hilbert space given by the GNS representation, but that is not versatile enough for our purposes, except possibly in the nondegenerate setting, p=1, in general in one dimension and for some two-dimensional systems.40 Rather, we work here in large but finite volumes and all bounds are uniform in the volume. Strictly speaking, the index is therefore associated with sequences of operators rather than just the three operators P, U, and the charge.

Let Λ be a graph equipped with the graph distance and having diameter diam(Λ) = L. We write |Λ| for the number of vertices. With each vertex, we associate a copy of the Hilbert space Cn. We denote by HΛ the total Hilbert space of the system of dimension n|Λ|. We treat simultaneously spin systems, where n is the number of components of the spin at each site, and fermionic systems, where n = 2f, with f being the number of flavors of fermions.

The spatial structure of Λ is reflected in the algebra of observables A. To any element OA, we associate a spatial support supp(O) ⊂ Λ. The crucial property is the following: If X, Y ⊂ Λ are disjoint, and if supp(OX) ⊂ X, supp(OY) ⊂ Y, then [OX, OY] = 0. These notions of locality are completely standard and probably well known to most of our readers. For convenience, a short exposition is provided in  Appendix B.

We will consider sequences of models indexed by LN and be interested in their asymptotic properties as L. Writing the index L everywhere would clutter the text, and we choose not to do so. We will always have such a family in mind, and the upshot of our results is that, unless otherwise specified, all constants and parameters can be chosen independently of L. In particular, the parameters RH,RQ,mH,mQ,γ,p to be introduced below are assumed to be independent of L. Furthermore, we will use constants c, C, whose value can change between equations, but they are also always independent of L. We will often say A=O(L), which means that the sequence of operators A = AL satisfies ALCkLk pointwise for all kN. For completeness, we explain the large L setting in more detail in  Appendix C.

With these considerations in mind, we now set up the main objects of our work. We keep this section abstract on purpose and refer the reader to Sec. V or to Ref. 34 for specific examples.

The Hamiltonian is a sum of local, finite range terms of the form

HZΛhZ,
(2.1)

where

supp(hZ)=Z,hZ=0 unless diam(Z)RH,

which are uniformly bounded: supZ⊂ΛhZ‖ ≤ mH. Note that RH stands here for the interaction range, not the Hall conductance. As Λ is assumed to be finite dimensional (see Sec. II C), it follows that ‖H‖ ≤ C|Λ|.

We consider local charge operators qZ with supp(qZ) = Z satisfying

  1. qZ = 0 unless diam(Z) ≤ RQ,

  2. supZ⊂ΛqZ‖ ≤ mQ,

  3. σ(qZ)Z for all Z, where σ(A) denotes the spectrum of A, and

  4. [qZ, qZ] = 0 for all Z, Z′.

The total charge in S ⊂ Λ is defined as

QSZSqZ.
(2.2)

Finally, we assume that the Hamiltonian conserves this charge, namely,

[QΛ,H]=0.

Using the properties of qZ, it follows that we can choose the decomposition H = ZhZ such that

[QΛ,hZ]=0
(2.3)

for all Z ⊂ Λ. This implies in particular that for any S ⊂ Λ, the commutator [QS, H] is supported in a strip along the boundary of S.

For a set S, we define S(r) to be its r-fattening, namely,

S(r){xΛ:dist(x,S)r},
(2.4)

and its boundary to be

SS(1)(Λ\S)(1).

We can now state the two conditions imposed on the graph Λ:

  1. Λ has a finite spatial dimension in the sense of supx∈Λ|{x}(r)| ≤ C(1 + r)d for all r ≥ 0, i.e., the size of balls grows at most polynomially with the radius.

  2. There is a set Γ ⊂ Λ such that

Γ=+,dist(,+)cL.
(2.5)

These assumptions are illustrated in Fig. 1, in the case where Λ is a discrete 2-torus. We will consider the transport of the charge across one of Γ’s boundaries.

1. Almost local operators and quasi-local unitaries

We will often need to localize sequences of operators approximately, in a looser sense than by their support. To explain this, and only in this section, we keep the L-dependence explicit in order to be maximally clear; see also  Appendix C. A sequence of operators A = AL is almost supported in a sequence of sets Z = ZL if there are sequences Ar=AL,r,rN with supp(AL,r)(ZL)(r) [see (2.4)] such that

ALAL,r=AL|ZL|O(r).

We denote57 the set of sequences of operators that are almost supported in Z by AZ. With this notation, a sequence of unitaries U is called locality-preserving if

U*AZUAZ

for all sequences of sets Z.

FIG. 1.

A typical realization of the assumed global spatial structure. Here, Λ is a two-dimensional discrete torus (Z/LZ)2 and Γ is one half of the torus with two disjoint boundaries ±.

FIG. 1.

A typical realization of the assumed global spatial structure. Here, Λ is a two-dimensional discrete torus (Z/LZ)2 and Γ is one half of the torus with two disjoint boundaries ±.

Close modal

With these general properties set up, we can now state the results announced above and the assumptions they require.

Assumption 1
(gap).Let E1E2En|Λ| be the eigenvalues of H, counted with multiplicities. There are L-independent constants γ > 0, Δ > 0, and p such that
Ep+1EpγandEpE1Δ,
and γ > 2Δ.

We refer to the rank-p spectral projector corresponding to the spectral patch {E1, …, Ep} as P, and its range is called the “the ground state space.”

One consequence of the gap assumption is the following exponential clustering result.

Proposition 3.1.
For anyAAX,BAYwithXY = ∅ and for any normalized Ω ∈ ran(P),
|Ω,ABΩΩ,APBΩ|=AB|X||Y|O(d(X,Y)).
(3.1)

This is a slightly weaker statement than41,42 in that the decay is only superpolynomial but also under weaker assumptions: it holds in finite volumes and with an energy width of the ground state patch Δ that may remain bounded away from 0 in the infinite volume limit. We refer to  Appendix A for a proof of this finite volume clustering theorem. As seen there, the condition γ > 2Δ is of technical nature.

The second assumption is about a locality-preserving unitary; see Sec. II C 1. As discussed in the Introduction, this U is the unitary implementing the process transporting charge, whether by translation, flux insertion, or else.

Assumption 2
(charge and locality preserving U). There is a locality-preserving unitary U that leaves ran(P) invariant
[U,P]=O(L)
(3.2)
and that conserves the total charge
[U,QΛ]=0.

Since U is locality-preserving and the charge is a sum of local terms, this leads to a continuity equation: for any spatial set Z,

U*QZUQZAZ.

In words, the net charge transported by U in or out of any set is supported near the boundary of the set. Applying this assumption to Z = Γ and using the spatial structure introduced in Sec. II C, we get

U*QΓUQΓ=:T+T+
(3.3)

with T±A± and ‖T±‖ ≤ C|Λ|. We naturally interpret T± as the operators of charge transport across the boundaries ±. This defines T± only up to vanishing tails and an arbitrary additive constant. We fix a choice such that

e2πi(Q+T±)=1+O(L),T±C|Λ|,
(3.4)

where we denoted Q = QΓ as we shall do from here onward. Such a choice exists. Indeed, for any T̃± satisfying (3.3), we have, by integrality of the spectrum of Q and the assumption (2.5) about the spatial structure, that

1=e2πiU*QU=e2πi(Q+T̃+T̃+)+O(L)=e2πi(Q+T̃)e2πi(Q+T̃+)+O(L)
(3.5)

with exp(2πi(Q+T̃±))A±. Hence, there exists ν such that

exp(2πi(Q+T̃±))=e±iν+O(L).

Then, T±T̃±ν2π satisfies both (3.3) and (3.4). We can now state our main result. For any ϵ > 0, we denote

Z(ϵ){xR:dist(x,Z)<ϵ}.

Theorem 3.2.
If Assumptions 1 and 2 hold, then
Tr(PT)Z(O(L)).

While the trace is an integer, the physically relevant quantity is the expectation value of charge transport in the state given by the density matrix p−1P, which makes the above into a rational index indeed. We denote it by

IndP(U)T(U)P,

where APp1Tr(PA), further emphasizing the mathematical fact that it is a general index associated with the pair (U, P) of a locality-preserving unitary and a finite-dimensional projection that commutes with the unitary.

There are two natural settings where p = rk(P) > 1: topologically ordered ground states and spontaneous symmetry breaking with a local order parameter. In both cases, a value of the index can be attributed to the individual ground states themselves. We cover here the case of a topological order, and we postpone symmetry breaking to Sec. VI.

Assumption 3
(topological order). For any Z such that diam(Z) < C and for any operator AAZ of norm 1,
PAPAPP=O(L).

This assumption prevents the local order since the restriction of any local observable to the ground state space is trivial. No local observable can be used to distinguish between the different states in the range of P. We note that this assumption implies that the splitting Δ in Assumption 1 vanishes in the infinite volume limit because Ω,HΩ=HP+O(L) by the topological order for any normalized Ω ∈ran(P).

Corollary 3.3.
If, in addition to Assumptions 1 and 2, Assumption 3 also holds, then
pΨ,TΨZ(O(L))
for any normalized Ψ ∈ ran(P).

Proof.

By (3.3) and the locality-preserving property of U, T can be approximated by sums of local terms, to which Assumption 3 applies.□

We now proceed with the proof of the theorem, postponing discussions of further properties of the index and of applications.

From now on, we denote by =L equality up to terms of O(L), in operator norm.

Recalling that Q is the charge in the half space, let us define

KW(t)eitHi[H,Q]eitHdt,
(4.1)

with W being a real-valued, bounded, integrable function satisfying W(t)=O(|t|) and Ŵ(ω)=1iω for all |ω|≥ γ, with γ being the spectral gap as in Assumption 1. Since, by functional calculus for adH = [H, ·],

K=Ŵ(adH)(iadH(Q)),

the properties of W yield that [K, P] = [Q, P]. By charge conservation (2.3) and spatial structure (2.5), we see that

i[H,Q]=J+J+,J±A±.

Plugging this decomposition into (4.1) and by the Lieb–Robinson bound, we conclude that there are K±A± such that ‖K±‖ < C|Λ| and such that

Q¯:=QKK+
(4.2)

leaves ran(P) invariant,

[Q¯,P]=0.
(4.3)

Note that while (4.1) is also the generator of the “quasi-adiabatic” flow (see Refs. 43 and 44), its use in the present context was introduced in Ref. 34.

We now present three lemmas before heading to the main argument. While the first and third ones are general and purely technical, the second one refers explicitly to the spatial structure of the problem (see Sec. II C) and plays an essential role in the following.

Lemma 4.1.

LetVbe a unitary, and letPbe an orthogonal projection. Then,

  • ‖[V, P]‖ ≤ ϵimpliesPVP2 ≥ 1 −ϵ2and

  • PVP2 ≥ 1 −ϵ2implies ‖[V, P]‖ ≤ 2ϵ.

Proof.
We first note that
1=VP2=PVP2+(1P)VP2.
Therefore, if ‖[V, P]‖ ≤ ϵ, then
PVP2=1(1P)[V,P]21ϵ2,
proving (i). The same identity implies that if ‖PVP2 ≥ 1 −ϵ2, then
(1P)VP2ϵ2,
which yields the second claim since [V, P] = (1 − P)VPPV(1 − P).□

Lemma 4.2.

LetV±A±be unitary operators, and letVVV+. If Assumption 1 holds, then[V,P]=L0implies[V±,P]=L0.

Proof.
By clustering (3.1),
PVPPVPV+P=L0.
(4.4)
Moreover,
PVPV+PPVPV+P=PVP
since V+ is unitary. Thus, from (4.4), the assumption [V,P]=L0 and Lemma 4.1(i) yield
PVP=L1.
An application of Lemma 4.1(ii) concludes the proof.□

Lemma 4.3.
LetA = A*andPbe an orthogonal projection. If ‖[A, P]‖ < ϵ, then
[eiϕA,P]ϕϵ.

Proof.
The proof follows immediately from the identity
[eiϕA,P]=i0ϕeisA[A,P]ei(ϕs)Ads
and the unitarity of the exponentials.□

We recall that the spatial setup of Sec. II C (see also Fig. 1) is such that the boundaries ± are separated by cL. Let now Λ±=(±)(cL)Γ be regions such that dist(Λ+, Λ) ≥ cL (recall that c can change the value from equation to equation). We denote

Q±QΛ±,QmQQQ+.

For ϕ ∈ [0, 2π], we set

Z(ϕ)U*eiϕQ¯UeiϕQ¯=eiϕQ¯UeiϕQ¯,
(4.5)

as well as

Z±(ϕ)eiϕQ¯±UeiϕQ¯±.
(4.6)

Here, we defined Q¯UU*Q¯U, K±UU*K±U, and

Q¯±Q±K±,Q¯±UQ±+T±K±U.
(4.7)

To avoid later confusion, we point out that Q¯±UU*QU. With these definitions, the following identities hold:

Q¯=Q¯+Qm+Q¯+,Q¯U=LQ¯U+Qm+Q¯+U.
(4.8)

The crucial point of these definitions is the commutation property

[Qm,A±]=L0,[A,B+]=L0,
(4.9)

with A±, B± being any of the above objects carrying the subscript ±. This is immediate for operators in A±, but it also holds for Q±,Q¯±,Q¯±UAΛ±, and hence for their exponentials by Lemma 4.3, because these operators reduce to the charge away from ±. This immediately leads to

Z(ϕ)=LZ(ϕ)Z+(ϕ).

By Assumption 2 and by construction of Q¯ [see (4.3)], all four factors of Z(ϕ)=U*eiϕQ¯UeiϕQ¯ commute with P so that

[P,Z(ϕ)]=L0.

This and Lemma 4.2 now yield the first essential observation,

[P,Z±(ϕ)]=L0.
(4.10)

The second one follows by recalling the integrality of the spectrum of Qm, which implies that

[P,e2πiQ¯±]=L0,
(4.11)

again by Lemma 4.2 applied to e2πiQ¯=e2πiQ¯e2πiQ¯+ [see (4.8)].

We now consider the function ϕZ(ϕ)PZ(ϕ)P and let DQ¯UQ¯. Then,

iddϕZ(ϕ)=PeiϕQ¯UDeiϕQ¯P=PZ(ϕ)eiϕQ¯DeiϕQ¯P=LZ(ϕ)PeiϕQ¯DeiϕQ¯P=LZ(ϕ)PeiϕQ¯DeiϕQ¯P=LZ(ϕ)eiϕQ¯PDPeiϕQ¯.

The first two equalities are immediate calculations, the third one follows from (4.10), the fourth one uses the commutations (4.9), and the fifth one is by property (4.3) of Q¯. The unique solution of this differential equation with Z(0)=1 is

Z(ϕ)=Leiϕ(PDP+Q¯)eiϕQ¯.
(4.12)

Note that both unitary factors on the right independently commute with P. It remains to study Z(2π) to conclude. By the integrality of the spectrum of Qm, Q+,

U*e2πiQ¯U=U*e2πi(Q¯+Qm+Q+)U=e2πi(U*QUKU),

where we used definition (4.7) of Q¯. Since U*QU=LQ+T+T+, from the integrality of the spectrum of charge and commutation property (4.9), we conclude that

U*e2πiQ¯U=Le2πi(Q++T+)e2πi(Q+TKU)=Le2πi(Q+TKU),
(4.13)

where the second equality uses choice (3.4). Since the exponent is precisely Q¯U [see again (4.7)], we multiply from the right by e2πiQ¯ to obtain

Z(2π)=LPU*e2πiQ¯Ue2πiQ¯P.
(4.14)

All four unitaries on the right-hand side commute with P [see (4.11)]. Moreover, since PV*PVPP = PV*[P, V]P for any unitary V, the condition [P,V]=L0 implies that PVP is invertible on ran(P) with (PVP)1=LPV*P. By the continuity of the determinant, we conclude that

detP(Z(2π))=L1,

where detP(A) ≔  det(PAP + (1 − P)). On the other hand, (4.12) and the relation detP(eA) = eTr(PA) yield

detP(Z(2π))=Le2πiTr(PDP+Q¯)e2πiTr(Q¯)=e2πiTr(PD).

We further observe that

D=TU*KU+K

[see (4.7)] so that Tr(PD)=LTr(PT) by the unitary invariance of the trace. Hence, Tr(PT)Z(O(L)), which concludes the Proof of Theorem 3.2.

Remark 1.
Had we not enforced choice (3.4) of T±, (4.14) would read as
Z(2π)=LPe2πi(Q++T+)U*e2πiQ¯Ue2πiQ¯P
(4.15)
[see (4.13)]. From (3.5), we know that e2πi(Q++T+)=Le2πi(Q+T) and that they are multiples of the identity. Taking the determinant of (4.15), we conclude that
e2πipTP=Le2πip(Q+T).
(4.16)
This means that to check that T± satisfies (4.14), it suffices to verify that pTPZ(O(L)). We will use this in the proof of additivity below.

If U1, U2 both satisfy Assumption 2, then so does their product by Leibniz’ rule for the commutators and preservation of locality. It is then natural to expect that the charge transported by the composed action of U1, U2 is equal to the sum of the charges transported by the action of each of them individually; see Ref. 22 for the non-interacting case. This is indeed true if we make the choice

T(U2U1)U2*T(U1)U2+T(U2).
(5.1)

This is the content of the following proposition.

Proposition 5.1.
Suppose thatU1, U2both satisfy Assumption 2. LetT(U2U1)be defined by(5.1). Then,(3.3)and(3.4)hold forT(U2U1)and
IndP(U2U1)=LIndP(U2)+IndP(U1).
(5.2)

Proof.
The definitions of T(U1)± and T(U2)± yield
U2*U1*QU1U2Q=U2*(T(U1)+T(U1)+)U2+T(U2)+T(U2)+.
Since U2 is locality-preserving, choice (5.1) indeed satisfies (3.3). Using [U2,P]=L0, we get
Tr(PT(U2U1))=LTr(PT(U1))+Tr(PT(U2))Z(O(L)).
(5.3)
By Remark 1, this shows that T(U2U1) indeed satisfies (3.4). The additivity (5.2) is then precisely (5.3).□

Here, we constrain the choice of graph Λ to have a good notion of translation. Let Λ′ be a d − 1 dimensional graph in the sense of Sec. II C. Let TZ/L1Z be a discrete circle. Then, Λ is of the form

ΛTΛ,

the Cartesian product of these graphs. We write (x1, x′) ∈ Λ with x1T,xΛ, and we let Θ be a unitary shift along T. Finally, let

Γ={(x1,x):0x1<L1/2,xΛ}.

This choice is consistent with the setup of Sec. II C, provided that we let L1cL, with L being the diameter of Λ. We assume that the Hamiltonian is translation invariant, i.e., [H, Θ] = 0. Then, so is its ground state space, namely, [Θ, P] = 0, and since translation clearly preserves locality, Assumption 2 holds for Θ. Moreover,

Θ*QΘQ=Q[0]+Q[L1/2],
(5.4)

where [x1] = {(x1, x′), x′ ∈ Λ′} and ⌈L1/2⌉ is the smallest integer not smaller than L1/2. We make the natural choice T = −Q[0], for which (3.4) holds. By translation invariance, the total charge per slab Λ′ is

Q[0]P=1L1QΛP.

We then obtain the following fractional Lieb-Schultz-Mattis (LSM) theorem.

Corollary 5.2.
If Assumption 1 holds, then
pQ[0]PZ(O(L)).

Note that ⟨Q[0]P in the statement cannot be expected to be convergent as the volume |Λ| grows, which renders the claim somewhat unfamiliar. This corollary becomes, in particular, useful when one has full translation invariance, i.e., not only in the x1-direction. To be very specific, we consider two-dimensional L1 × L2-tori Λ, that is, we specify now Λ=Z/L2Z with diameter L=L1+L22. We specify a particular sequence of tori by picking a function L ↦ (L1, L2), satisfying L=L1+L22. We choose this function such that L2 runs through all positive integers and that both L1, L2 > cL. Let us now assume that the total charge density ρL1|Λ|QΛP converges to a limiting density ρ fast enough, namely, L(ρρL) → 0. This is a natural assumption because for gapped systems, we would expect to be able to choose the boundary conditions so that local observables in the bulk approach their thermodynamic values (almost) exponentially fast (see Refs. 44 and 45). With this, Corollary 5.2 implies

pρZ.
(5.5)

Indeed, from the index theorem, we obtain that pL2ρLZ(O(L)). Writing L2ρL = L2ρ + L2(ρLρ) and using the assumption of fast convergence and L2 > cL, we get (pL2ρ) mod  1 = o(1). If L2 runs through N, the rational non-integer is ruled out directly and irrational ρ is ruled out by the ergodicity of irrational rotations on the torus.

It is widely accepted and proved in some specific settings16,46,47 that there is no intrinsic58 topological order in one dimension. The present paper proves a particular version of this statement. Namely, in one dimension and with a topological order as in Assumption 3, the index IndP(U) is integer-valued, even when p > 1. For example, in the case of the Lieb–Schultz–Mattis theorem just discussed, this means that there is no topological charge fractionalization59 in the ground state sector.

Indeed, in one dimension, the region is finite so that Pe2πiQ¯P=LzP for some zC,|z|=1, by Assumption 3 (note that [e2πiQ¯,P]=L0 in all dimensions, but the exponential acts, in general, non-trivially on ran(P)). But then,

Z(2π)=LPU*e2πiQ¯Ue2πiQ¯P=LPU*PUP|z|2=LP
(5.6)

since U commutes with P by Assumption 2. With a topological order, (4.12) reads

Z(ϕ)=LeiϕPDP=LeiϕPTP=LeiϕpTr(PT)P,

which, with (5.6), yields

IndP(U)Z(O(L))

as claimed.

We now discuss the case of systems with magnetic fields. Our main result is an interacting version of the Avron–Dana–Zak (ADZ) relation37 between the Hall conductance and the filling factor. We start with an extensive introduction of the setup and a presentation of the relation, leaving the general rigorous result for Sec. V D 5. See also Refs. 38 and 39 for another view on the same results.

1. Harper/Hubbard model

We consider again Λ=(Z/L1Z)(Z/L2Z), i.e., the L1 × L2 discrete torus with coordinates 1 ≤ x1,2L1,2 and unit vectors ê1=(1,0),ê2=(0,1). We describe spinless fermions in a uniform magnetic field. Let

ϕ=2πmn,
(5.7)

with m, n being coprime integers, be the magnetic flux piercing through the unit cell, and let L2Φ be the magnetic flux threaded through the torus (see Fig. 2). Then, in the Landau gauge, the Hamiltonian is

HΦ=txΛei(ϕx1Φ)cx+ê2*cx+cx+ê1*cx+h.c.μxΛqx+x,yΛu(xy)qxqy.
(5.8)

We have written qx=cx*cx for the occupation operators, and the parameters tμ, u(·) are, respectively, the hopping strength, the chemical potential, and the interaction potential. We impose L1ϕ2πZ in order that the Hamiltonian is well defined on the torus.

FIG. 2.

The parameter ϕ is the magnetic flux piercing each unit cell on the torus, leading to a constant magnetic field. In contrast, L2Φ is the total flux threaded through a hole of the torus, and it does not result in any magnetic field on the surface of the torus.

FIG. 2.

The parameter ϕ is the magnetic flux piercing each unit cell on the torus, leading to a constant magnetic field. In contrast, L2Φ is the total flux threaded through a hole of the torus, and it does not result in any magnetic field on the surface of the torus.

Close modal

Let us comment on how this model fits our setup. The on-site operator qx is a concrete example of the general charge introduced in Sec. II B. For u = 0, the model is non-interacting and it is the Harper model in its second quantized version, also known as the Hofstadter model.48 By choosing μ to lie in one of the gaps of the corresponding one-particle Schrödinger operator, one obtains a gapped many-body Hamiltonian. In that case, fermionic perturbation theory49–51 yields that the gap remains open for sufficiently weak u, and so, Assumption 1 is satisfied. This case corresponds to p = 1. Although rigorous results are absent, it is believed that at strong interaction, the system exhibits a topological order; hence, p > 1, and it is a fractional quantum Hall insulator (see, for example, Ref. 52). The result below applies to that case as well, but we cannot establish the validity of Assumption 1. Below, we introduce two concrete unitaries that fit the general framework, namely, translation and magnetic flux insertion.

2. Translation and magnetic translation

Hamiltonian (5.8) is translation-invariant in the x2-direction, but it is more interesting to consider translation in the x1-direction,

Θ*cxΘ=cx+ê1,Θ*cx*Θ=cx+ê1*,

as well as the magnetic translation defined by

U*cxU=cx+ê1eix2ϕ,U*cx*U=cx+ê1*eix2ϕ.

At ϕ ≠ 0, the ordinary translation is not a symmetry, but the magnetic translation is a symmetry provided that L2ϕ2πZ,

[HΦ,Θ]0,[HΦ,U]=0.

In that case, one can apply the index theorem with U being magnetic translation and the conserved charge being the fermion number. Since, in the notation used before Corollary 5.2, we still have T = Q[0], we recover the result of the corollary. However, conclusion (5.5) fails for a non-integer flux ϕ/(2π) because L2 must be a multiple of n. Therefore, the strongest result that can be obtained on the density ρ is simply (pn)ρZ, which also follows from an application of the theorem with Θn. A simple check with free fermions confirms indeed that no sharper result is possible, at least not for p = 1. In other words, in the case of magnetic systems, the density that satisfies (5.5) is not the charge per unit cell but the charge per magnetic unit cell, which is n times larger.

3. Flux threading and Hall conductance

As illustrated in Fig. 2, the parameter Φ is the flux threaded through the torus, per unit length in the x2-direction. If the gap of HΦ remains bounded away from 0 as the parameter Φ goes from Φ to Φ′, then the ground state projector PΦ can be parallel transported to PΦ′ by a locality-preserving unitary F(Φ, Φ′) (see Sec. V D 5 for details), namely,

F(Φ,Φ)*PΦF(Φ,Φ)=PΦ.
(5.9)

If the total threaded flux is an integer number of elementary flux quanta, i.e., L2(ΦΦ)2πZ, then the effect of parallel transport on PΦ is the same as that of a gauge transformation60 implemented by the unitary

FΔΦ=eiΔΦxΛx2nx,ΔΦ=ΦΦ.

This follows because in that case

FΔΦHΦFΔΦ*=HΦ.
(5.10)

Combining (5.9) and (5.10), we obtain indeed

[PΦ,F(Φ,Φ)FΔΦ]=0.

Therefore, the unitary F(Φ,Φ)FΔΦ satisfies Assumption 2 and it defines an associated index of HΦ. By the Laughlin argument, this index is the number of threaded flux quanta L2ΔΦ/(2π) times the quantum Hall conductance. This is discussed in Ref. 34, where we also provide an explicit proof relating this index (more precisely, an equivalent one) to the adiabatic curvature, and Ref. 53 details the relation of the adiabatic curvature to other expressions for Hall conductance in a many-body setting.

We conclude by noting that the convenient choice ΔΦ2πZ yields FΔΦ=1, in which case F(Φ, Φ′) itself is a symmetry of PΦ. This will be exploited below.

4. The Avron–Dana–Zak relation

We are now equipped to obtain a relation between Hall conductance and charge density, taking advantage of the fact that flux threading can be intertwined with translations to yield a new symmetry, quite similar to the case of magnetic translations. Indeed, Hamiltonian (5.8) is covariant in the sense that HΦ+ϕ = Θ*HΦΘ. Using this, we obtain

PΦ=ΘPΦ+ϕΘ*=ΘF(Φ+ϕ,Φ)PΦF(Φ+ϕ,Φ)*Θ*

so that U: = ΘF(Φ + ϕ, Φ) is a symmetry satisfying Assumption 2. We shall later establish that

Un=ΘnF(Φ+nϕ,Φ)

(see Lemma 5.5). By the above-mentioned remarks and (5.7), the two unitaries on the right-hand side are now symmetries in their own right. By the additivity of the index, Proposition 5.1, we conclude that

IndP(Θn)+IndP(F(Φ+nϕ,Φ))npZ(O(L)),P=PΦ.

The first term on the left-hand side is Q[0,n1]P=x:0x2<nnxP, which is n times the average density per slab, namely, nL2ρ, and the second term is nϕL2 times the Hall conductance σL (see again Ref. 34). Since L2 is arbitrary and if the convergence of ρL, σL to their infinite-volume limits ρ, σ satisfies L2(ρρL), L2(σσL) → 0, then the argument of Sec. V B implies that

ρ+ϕσ1pZ,

which is the fractional ADZ relation.

5. A many-body Avron–Dana–Zak theorem

In this section, we repeat the above discussion under detailed assumptions and complete the proofs. The setting is as in Sec. V B with, in particular, the graph product Λ=T Λ′, where T is the discrete circle with L1 sites, and we again assume that L1 > cL. The unitary Θ is a translation along T. The operator Q continues to be the charge in a half (along T) system. We consider a family of local Hamiltonians {HΦ:ΦR} of form (2.1), satisfying the following.

Assumption 4.

  1. The parameters RH, mH can be chosen to be uniform in Φ and the local terms hZ are themselves C1 functions of Φ such that ‖ΦhZ(Φ)‖ is bounded uniformly in Z and L.

  2. Periodicity:
    HΦ+2π=HΦ.
  3. Covariance:
    Θ*HΦΘ=HΦ+ϕ,
    (5.11)
    where
    ϕ=2πmn,mZ,nN,m,n coprime.
  4. Assumption 1 holds uniformly in Φ, where P = PΦ are the ground state projectors. This implies, in particular, that the rank p=rk(PΦ) is independent of Φ.

While the assumption is obviously motivated by Hamiltonian (5.8) and the corresponding physical phenomenology, HΦ below is not restricted to that specific form; we only impose that Assumption 4 holds.

Items (i) and (iv) imply that Φ ↦ PΦ is itself differentiable and periodic,

PΦ+2π=PΦ
(5.12)

for any ΦR. Items (ii) and (iii) lead to

[Θn,PΦ]=0,

and hence, Θn satisfies Assumption 2. Furthermore, it follows that there exists a locality-preserving, charge conserving unitary propagator F(Φ, Φ′) [see Refs. 43 and 44 and the paragraph above (5.15)] such that

PΦF(Φ,Φ)=F(Φ,Φ)PΦ
(5.13)

exactly for any Φ,ΦR. In particular, we obtain for PP0 that

[F,P]=0,

where FF(, 0). In other words, the unitary F also satisfies Assumption 2.

Theorem 5.3.
Let Assumption 4 hold. Then,
pnIndP(Θn)+IndP(F)ZO(L).

We now turn to the Proof of Theorem 5.3. By the gap assumption, the self-adjoint family AΦ of so-called quasi-adiabatic generators defined by

AΦW(t)eitHΦḢΦeitHΦdt,
(5.14)

where ̇=Φ and W was introduced after (4.1), are such that

iṖΦ=[AΦ,PΦ].

The corresponding unitary propagator F(Φ, Φ′) is the unique solution of

iḞ(Φ,Φ)=AΦF(Φ,Φ),F(Φ,Φ)=1,

and it is an intertwiner between PΦ′ and PΦ [see (5.13)]. It satisfies the cocycle relation

F(Φ,Φ)F(Φ,Φ)=F(Φ,Φ).
(5.15)

The covariance assumption and (5.14) imply the following relation between translations and quasi-adiabatic propagator.

Lemma 5.4.
For anykZand anyΦ,ΦR,
F(Φ,Φ)Θk=ΘkF(Φ+kϕ,Φ+kϕ).

Proof.
Assumption 4 implies that Θ*ḢΦΘ=ḢΦ+ϕ, and with (5.14),
Θ*AΦΘ=AΦ+ϕ.
Hence, the operator FΘ(Φ) ≔ Θ*F(Φ, Φ′)Θ is a solution of
iḞΘ(Φ)=AΦ+ϕFΘ(Φ),FΘ(Φ)=1.
By uniqueness of the solution of the differential equation, we conclude that FΘ(Φ) = F(Φ + ϕ, Φ′ + ϕ). In other words,
F(Φ,Φ)Θ=ΘF(Φ+ϕ,Φ+ϕ),
a k-fold application of which yields the claim.□

Lemma 5.5.
LetU(Φ) ≔ ΘF(Φ + ϕ, Φ). Then, for anyΦR,
[U(Φ),PΦ]=0.
(5.16)
Moreover, [U(Φ)k, PΦ] = 0 and
U(Φ)k=ΘkF(Φ+kϕ,Φ)
(5.17)
for anykZ.

Proof.
Assumption 4 implies ΘPΦ+ϕΘ* = PΦ, and hence, by (5.13), we have
ΘF(Φ+ϕ,Φ)PΦ=ΘPΦ+ϕF(Φ+ϕ,Φ)=PΦΘF(Φ+ϕ,Φ).
This proves (5.16), and U(Φ)kPΦ = PΦU(Φ)k follows by k-fold application. We show that (5.17) holds by induction. Assuming that the formula holds for k − 1, we have
U(Φ)k=U(Φ)Θk1F(Φ+(k1)ϕ,Φ).
By Lemma 5.4, we have
Θk1F(Φ+(k1)ϕ,Φ)=F(Φ,Φ(k1)ϕ)Θk1,
and hence, by the cocycle property (5.15) of the propagator,
U(Φ)k=ΘF(Φ+ϕ,Φ(k1)ϕ)Θk1.
Another application of Lemma 5.4 yields the claim.□

We note that by Lemma 5.4,

U(Φ)=F(Φ,Φϕ)Θ,U(Φ)k=F(Φ,Φkϕ)Θk.

Proof of Theorem 5.3.
Since nϕZ, we have that
F(Φ+nϕ,Φ)PΦ=PΦF(Φ+nϕ,Φ)
by (5.12). Setting k = n in Lemma 5.5, we further conclude that PΦ is also invariant under Θn. It follows from Theorem 3.2 that both indices IndPΦ(Θn),IndPΦ(F) are well defined with denominator p. Finally, the same applies to U(Φ) and U(Φ)n by Lemma 5.5. Altogether,
IndPΦ(Θn)+IndPΦ(F)=LIndPΦ(ΘnF)=LIndPΦ(U(Φ)n)=LnIndPΦ(U(Φ)),
where the first equality is by Proposition 5.1, the second one is by Lemma 5.5, and the last one is additivity again. Since IndPΦ(U(Φ))p1ZO(L), the theorem follows.□

In many physically relevant situations, the topological order assumption is violated, and ground states can de distinguished by a local order parameter. The prime example thereof is dimerization and, more generally, the breaking of a discrete symmetry. The projection P decomposes as

P=m=1MPm,
(6.1)

where Pm=Pm*=Pm2 and ran(Pm) are extremal in a sense to be made clear below. We refer to ran(Pm) as the superselection sectors of the system, and let

pmrk(Pm).

As usual, M,pm are finite and fixed, independent of L for L large enough. We denote mpm1Tr(Pm).

Assumption 5.

Decomposition (6.1) is such that

  1. for any OAZ of norm 1 and with diam(Z) ≤ C,
    PmOPm=L0
    (6.2)
    whenever mm′;
  2. there are self-adjoint observables A1AX,A2AY with diam(X) ≤ C, diam(Y) ≤ C, and dist(X, Y) > cL such that
    A1m=A2m
    for all m, while
    |A1mA1m|c>0,mm;
  3. each Pm satisfies the topological order Assumption 3.

Remark 2.

Assumption (ii) postulates the existence of observables that detect the local order. In the case of a translation-invariant system, A2 is a translate of A1. In other situations, A2 might be a linear combination of more natural physical observables that achieves the equality of expectation values between A2 and A1. Points (i) and (iii) of the assumptions imply that Pm are extremal in the sense that a finer decomposition of P also satisfying (i) and (iii) cannot exist.

These assumptions imply the following specific form of the unitary U. Let SM be the symmetric group on {1, …, M}.

Lemma 6.1.
LetUm,lPmUPlandUPUP. There is a permutationπUSMsuch that
U=Lm=1MUπU(m),m.
In particular,
(i)Ul,m=L0 unless l=πU(m),(ii)rk(PπU(m))=rk(Pm),(iii)UPmU*=LPπU(m).

Proof.
Since [U,P]=L0 and Pm are subprojections of P,
Tr(UPmU*O)=LTr(UPmU*POP).
Since P=l=1MPl, Assumption 5(i) now implies that for any O satisfying the assumption,
Tr(UPmU*O)=Ll=1MTr(UPmU*PlOPl).
By (iii) of the assumption, we conclude that
U*OUm=Ll=1Mρm(l)Ol,
(6.3)
where, for any 1 ≤ m, lM,
ρm(l)U*PlUm=(Ul,m)*Ul,mm.
For any fixed m, ρm is a probability distribution on {1, …, M}, and for O = Aj as in Assumption 5(ii), we interpret (6.3) as the expectation value in ρm of a random variable aj:{1,,M}R given by aj(l) = ⟨Ajl. Therefore,
U*AjUm=Em(aj),
where Em is the expectation value associated with ρm. In terms of the random variables a1, a2, Assumption 5(ii) is rephrased as the statement that a1 = a2 and that a1 is injective. Clustering (see Lemma 6.3) implies that
Em(a1a2)=LEm(a1)Em(a2),
but since a1 = a2, we simply obtain
Em(a12)=LEm(a1)2.
We conclude that a1 is constant on the intersection of its support with the support of ρm. Since a1 is injective, the support of ρm is a singleton. This means that there is a unique π(m) such that Uπ(m),m is non-vanishing. The invertibility of U on ran(P) and (U*)l,m=(Um,l)* imply that π is a bijection, proving (i). With this,
UPmU*=PπU(m)UPmU*PπU(m)
so that UPmU* is a subprojection of PπU(m). By the invertibility of U, the two must have the same rank, proving (ii), and hence, they are equal, proving (iii).□

For any 1 ≤ mM, let (πU · m) denote the cycle of the permutation πU containing m, and let U(m) be its length. We then have the following generalizations of Theorem 3.2.

Proposition 6.2.

Let Assumptions 1, 2, and 5 hold. Then,

  • for any normalized Ψm ∈ran(Pm),
    pmΨm,TUU(m)ΨmZ(O(L));
  • for any normalizedΨm(πUm)ran(Pm),
    U(m)pmΨ,TUΨZ(O(L)).

We point out that the Proof of Theorem 3.2 presented in Sec. IV does not require P to satisfy Assumption 1 per se. It uses only two consequences thereof, namely, (i) the clustering property (3.1) and (ii) and the invariance of ran(P) under Q¯. Lemmas 6.3 and 6.4 show that Assumption 5 implies both properties for the subprojections Pm.

Lemma 6.3.
Let Ψm ∈ Ran(Pm) be normalized, and letAAX,BAYbe of norm 1 withd(X, Y) > cL. If one ofA, Bis an observable as in Assumption 5(i), then
Ψm,A(1Pm)BΨm=L0.

Proof.

Pm being a subprojection of P, clustering (3.1) implies that Ψm,A(1P)BΨm=L0. But then, P can be replaced by Pm by (6.1) and (6.2).□

Lemma 6.4.

For anym,[Q¯,Pm]=L0.

Proof.
The equality [Q¯,P]=L0 holds by construction. However, Pm = PPmP implies
[Q¯,Pm]=P[Q¯,Pm]P=LPm[Q¯,Pm]Pm=L0,
where the second equality is by Assumption 5(i) and the fact that Q¯ is a sum of local terms.□

Proof of Proposition 6.2.
By the above lemmas, Pm satisfies the clustering property and it is invariant under Q¯. Furthermore, the unitary UU(m) keeps Pm invariant by definition of U(m), namely,
[UU(m),Pm]=L0.
The proof and hence the result of Theorem 3.2 carry step by step through with P replaced by Pm and U replaced by UU(m). This establishes (i).

Since (πU · m) is a cycle, m(πUm)ran(P) is invariant under U. By Lemma 6.1(ii), all factors have rank pm so that the dimension of m(πUm)ran(P) is U(m)pm and the claim follows as mentioned above.□

The authors would like to thank an anonymous referee for his comments on the clustering theorem that led us to provide Proposition 3.1. The work of S.B. was supported by the NSERC of Canada. M.F. was supported, in part, by the NSF under Grant No. DMS-1907435. W.D.R. thanks the Flemish Research Fund (FWO) for support via Grant Nos. G076216N and G098919N. A.B. was supported by VILLIUM FONDEN through the QMATH Center of Excellence (Grant No. 10059).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

We prove Proposition 3.1 under Assumption 1. First of all, γ > 2Δ implies − γ + Δ < −Δ. We consider a smooth, real-valued function g that satisfies

g(ω)=0,ωΔ1,ωγ+Δ.
(A1)

For concreteness, we take

g=ϕθ(γ/2),

where, for any aR, 1 −θa is the shifted Heaviside function (with discontinuity at a), −γ/2 lies in the interval (−γ + Δ, −Δ), the product ⋆ denotes the convolution, and ϕCc((δ,δ)) is a non-negative function such that ∫ϕ = 1.

Let

Q(O)=g(adH)(O),

where adH(O) = [H, O]. This is well defined by functional calculus for self-adjoint matrices, given that the operator O ↦ [H, O] is self-adjoint with respect to the Hilbert–Schmidt inner product. We claim, and shall prove below, that the Fourier transform of g is a tempered distribution whose singular support is {0} and whose regular part vanishes at infinity as ĝ(t)=O(|t|), and so,

Q(O)=12πĝ(t)eitHOeitHdt
(A2)

is well defined. This expression further implies by the Lieb–Robinson bound that Q(O) is almost supported in the support of O; see again below.

With this,

Q(O)P=0andPQ(O)=PO(1P).

This follows from (A2), namely,

Q(O)=j,kg(EjEk)PjOPk,

where Pl is the eigenprojection corresponding to the eigenvalue El and (A1). Hence, for any AAX,BAY,

AB=APB+A(1P)B=APB+Q(A)B=APB+[Q(A),B].

It remains to note that Q(A)AX implies [Q(A),B]=O(d(X,Y)). This yields the claimed clustering result.

We finally turn to the technical questions left open above. The Fourier transform of g is the tempered distribution ĝ given by

ĝ=ϕ̂θ̂(γ/2),θ̂(γ/2)=eitγ/2θ̂0.

The function ϕ being smooth, we have that ϕ̂(t)=O(|t|). Moreover, the Plemelj–Sochotcki formula implies that

θ̂0=π2δ+i2πP1t,

where P denotes the principle value, and we recall that

|θ̂0[ψ]|π/2ψ+2/πψ
(A3)

for any Schwartz function ψ.

The smooth operator-valued function O(t) = eitHOe−itH is bounded since ‖O(t)‖ = ‖O‖, and its derivative O′(t) = eitHi[H, O]e−itH is similarly bounded: ‖O′(t)‖ ≤ CO‖|supp(O)|. It follows, in particular, that eitγ/2ϕ̂(t)O(t) is a Schwartz operator-valued function. The expression on the right-hand side of (A2) is well defined for any local observable O, with

ĝ(t)eitHOeitHdt=θ̂0[eitγ/2ϕ̂(t)O(t)],

and the bound (A3) holds in the operator norm.

We can finally address the locality of Q(O). We assume that O is a local operator, and the general case of an almost local one follows by the approximation. The δ contribution in θ̂0 is supported on the support of O (it is the normalized partial trace in the case of a quantum spin system, while a similar projection can be constructed for lattice fermions; see Ref. 54). As for the P1t contribution, the integral is split as |t| ≥ T and |t| < T. The fast decay of ϕ̂ yields a bound OO(T) for the long time part. For the second, short time part, we write O(t) = ΠR(O(t)) + (O(t) − ΠR(O(t))), where ΠR is the projection onto an R-fattening of the support of O. Since ΠR(O(t)) is smooth, the first contributions are well defined, bounded uniformly in T, and local by construction. For the second one, the Lieb–Robinson bound yields

1tO(t)ΠR(O(t))CO|supp(O)|eξRecv|t|1t,

which is integrable at 0 with [T,T]t1ecv|t|1dt2(ecvT1). It remains to pick T=ξ2cvR to conclude that both terms are O(R). Altogether, we conclude that Q(O) is almost localized on the support of O.

We consider first the case of spin systems. Then, HΛxΛCn and AxΛCn2, and so, there is a natural tensor product representation AxSCn2xScCn2 for any S ⊂ Λ. The support of OA is then the smallest set S such that OOS1Sc in this natural representation. We refer to Sec. 2.6 of Ref. 55; see also Secs. 5.2.2.1 and 6.2.1 of Ref. 56 for further detailed discussions.

For fermionic systems, a bit more setup is needed. The total algebra of operators is generated by cx,α,cx,α* and the identity, with cx,α,cx,α* being the annihilation/creation operator of a fermion at site x ∈Λ and with label α ∈ {1, …, f}. The label can correspond to spin states or something else. These operators satisfy the CAR

{cx,α,cx,α*}=δx,xδα,α,{cx,α,cx,α}={cx,α*,cx,α*}=0.

The Hilbert space HΛ is generated by acting with these operators on the vacuum state Ω, which is the common eigenvector of all annihilation operators. The total algebra of observables is graded by fermion parity: monomials in the cx,α,cx,α* of even/odd degree are called even/odd and the even monomials generate an algebra, which we call A. The support of OA is then the smallest set S such that O is in the linear span of even monomials with x restricted to S.

In the main text, we have systematically omitted the index L but at the same time used symbols such as O(L) and AZ that make sense only for a sequence of operators labeled by L. We believe that the chance of misunderstanding coming from this convention is small, but we now provide more details for the sake of completeness.

Let AL be a sequence of operators defined on a sequence of operator algebras A=AL associated with a sequence of graphs ΛL with diameter L. Neither the graphs nor the operators AL are a priori related in any way. We say that AL=O(L) if for any kN, there exists a constant Ck such that ‖AL‖ ≤ CkLk for all L. Furthermore, let ZL ⊂ΛL be a sequence of sets. The sequence AL belongs to the set AZL if for all r there exists a sequence of operators AL,r supported in (ZL)(r) such that for any kN, there exists a constant Ck with

ALAL,rCkAL|ZL|rk

for all L.

In the setting of Sec. II C, we postulate the existence of a sequence of sets ΓL ⊂ ΛL with boundaries −,L and +,L whose distance satisfies (2.5). Assumption 2, in particular, (3.3), then says that T±,LA±,L. Assumption 1 postulates the existence of a sequence of ground state projections PLAL of L-independent rank p.

Theorem 3.2 states that there exists a sequence of integers nL such that

Tr(PLT,L)nL=O(L).

The projection PL may be further constrained by Assumption 3: for any sequence of operators ALAZL such that diam(ZL) < C and ‖AL‖ = 1, we have PLALPL = p−1Tr(PLAL)PL + O(L). In the topologically ordered situation, Corollary 3.3 states that for any sequence ΨL ∈ ran(PL), we have pΨL,T,LΨLnL=O(L).

As a final note, we point out that for an L × L torus, ±,L have 2L sites. If U is a translation by one site in the x1-direction, then T±,L is equal to the charge in the boundary [see (5.4)], and nL is the total charge in this circle of length L. This shows that in a physically interesting setting, all these sets, operators, and algebras may indeed have a non-trivial dependence on L.

1.
F. D. M.
Haldane
and
N.
Lecture
, “
Topological quantum matter
,”
Rev. Mod. Phys.
89
(
4
),
040502
(
2017
).
2.
R. B.
Laughlin
, “
Nobel lecture: Fractional quantization
,”
Rev. Mod. Phys.
71
(
4
),
863
(
1999
).
3.
D. J.
Thouless
,
M.
Kohmoto
,
M. P.
Nightingale
, and
M.
den Nijs
, “
Quantized Hall conductance in a two-dimensional periodic potential
,”
Phys. Rev. Lett.
49
(
6
),
405
408
(
1982
).
4.
Q.
Niu
,
D. J.
Thouless
, and
Y.-S.
Wu
, “
Quantized Hall conductance as a topological invariant
,”
Phys. Rev. B
31
(
6
),
3372
(
1985
).
5.
J. E.
Avron
,
R.
Seiler
, and
B.
Simon
, “
Homotopy and quantization in condensed matter physics
,”
Phys. Rev. Lett.
51
(
1
),
51
53
(
1983
).
6.
R. B.
Laughlin
, “
Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations
,”
Phys. Rev. Lett.
50
(
18
),
1395
(
1983
).
7.
N.
Read
and
E.
Rezayi
, “
Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level
,”
Phys. Rev. B
59
(
12
),
8084
(
1999
).
8.
F. D. M.
Haldane
, “
Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states
,”
Phys. Rev. Lett.
51
(
7
),
605
(
1983
).
9.
X. -G.
Wen
, “
Vacuum degeneracy of chiral spin states in compactified space
,”
Phys. Rev. B
40
(
10
),
7387
(
1989
).
10.
X. -G.
Wen
,
F.
Wilczek
, and
A.
Zee
, “
Chiral spin states and superconductivity
,”
Phys. Rev. B
39
(
16
),
11413
(
1989
).
11.
G.
Moore
and
N.
Read
, “
Nonabelions in the fractional quantum hall effect
,”
Nucl. Phys. B
360
(
2-3
),
362
396
(
1991
).
12.
J.
Fröhlich
,
U. M.
Studer
, and
E.
Thiran
, “
A classification of quantum Hall fluids
,”
J. Stat. Phys.
86
(
3
),
821
897
(
1997
).
13.
A.
Kitaev
, “
Periodic table for topological insulators and superconductors
,”
AIP Conf. Proc.
1134
,
22
(
2009
).
14.
A. P.
Schnyder
,
S.
Ryu
,
A.
Furusaki
, and
A. W. W.
Ludwig
, “
Classification of topological insulators and superconductors in three spatial dimensions
,”
Phys. Rev. B
78
(
19
),
195125
(
2008
).
15.
P.
Heinzner
,
A.
Huckleberry
, and
M. R.
Zirnbauer
, “
Symmetry classes of disordered fermions
,”
Commun. Math. Phys.
257
(
3
),
725
771
(
2005
).
16.
X.
Chen
,
Z.-C.
Gu
, and
X.-G.
Wen
, “
Classification of gapped symmetric phases in one-dimensional spin systems
,”
Phys. Rev. B
83
(
3
),
035107
(
2011
).
17.
A. M.
Turner
,
F.
Pollmann
, and
E.
Berg
, “
Topological phases of one-dimensional fermions: An entanglement point of view
,”
Phys. Rev. B
83
(
7
),
075102
(
2011
).
18.
L.
Fidkowski
and
A.
Kitaev
, “
Topological phases of fermions in one dimension
,”
Phys. Rev. B
83
(
7
),
075103
(
2011
).
19.
D.
Gross
,
V.
Nesme
,
H.
Vogts
, and
R. F.
Werner
, “
Index theory of one dimensional quantum walks and cellular automata
,”
Commun. Math. Phys.
310
(
2
),
419
454
(
2012
).
20.
J. I.
Cirac
,
D.
Perez-Garcia
,
N.
Schuch
, and
F.
Verstraete
, “
Matrix product unitaries: Structure, symmetries, and topological invariants
,”
J. Stat. Mech.
2017
(
8
),
083105
.
21.
F.
Noether
, “
Über eine Klasse singulärer Integralgleichungen
,”
Math. Ann.
82
(
1
),
42
63
(
1920
).
22.
J.
Avron
,
R.
Seiler
, and
B.
Simon
, “
The index of a pair of projections
,”
J. Funct. Anal.
120
,
220
237
(
1994
).
23.
J.
Bellissard
,
A.
van Elst
, and
H.
Schulz‐ Baldes
, “
The noncommutative geometry of the quantum Hall effect
,”
J. Math. Phys.
35
(
10
),
5373
5451
(
1994
).
24.
D. J.
Thouless
, “
Quantization of particle transport
,”
Phys. Rev. B
27
(
10
),
6083
6087
(
1983
).
25.
E.
Prodan
and
H.
Schulz-Baldes
,
Bulk and Boundary Invariants for Complex Topological Insulators
(
Springer
,
2016
).
26.
J. E.
Avron
,
R.
Seiler
, and
B.
Simon
, “
Quantum Hall effect and the relative index for projections
,”
Phys. Rev. Lett.
65
(
17
),
2185
2188
(
1990
).
27.
H.
Katsura
and
T.
Koma
, “
The Z2 index of disordered topological insulators with time reversal symmetry
,”
J. Math. Phys.
57
(
2
),
021903
(
2016
).
28.
G.
De Nittis
and
H.
Schulz-Baldes
, “
Spectral flows of dilations of Fredholm operators
,”
Can. Math. Bull.
58
(
1
),
51
68
(
2015
).
29.
A.
Kitaev
, “
Anyons in an exactly solved model and beyond
,”
Ann. Phys.
321
,
2
111
(
2006
).
30.
S.
Bachmann
,
A.
Bols
,
W.
De Roeck
, and
M.
Fraas
, “
A many-body Fredholm index for ground state spaces and Abelian anyons
,”
Phys. Rev. B
101
,
085138
(
2020
).
31.
X. G.
Wen
, “
Topological orders in rigid states
,”
Int. J. Mod. Phys. B
04
(
02
),
239
271
(
1990
).
32.
S.
Bravyi
and
M. B.
Hastings
, “
A short proof of stability of topological order under local perturbations
,”
Commun. Math. Phys.
307
(
3
),
609
627
(
2011
).
33.
M. B.
Hastings
and
S.
Michalakis
, “
Quantization of Hall conductance for interacting electrons on a torus
,”
Commun. Math. Phys.
334
,
433
471
(
2015
).
34.
S.
Bachmann
,
A.
Bols
,
W.
De Roeck
, and
M.
Fraas
, “
A many-body index for quantum charge transport
,”
Commun. Math. Phys.
375
(
2
),
1249
1272
(
2020
).
35.
M. B.
Hastings
, “
Lieb-Schultz-Mattis in higher dimensions
,”
Phys. Rev. B
69
,
104431
(
2004
).
36.
B.
Nachtergaele
and
R.
Sims
, “
A multi-dimensional Lieb-Schultz-Mattis theorem
,”
Commun. Math. Phys.
276
,
437
472
(
2007
).
37.
I.
Dana
,
Y.
Avron
, and
J.
Zak
, “
Quantised Hall conductance in a perfect crystal
,”
J. Phys. C: Solid State Phys.
18
(
22
),
L679
(
1985
).
38.
Y.-M.
Lu
,
Y.
Ran
, and
M.
Oshikawa
, “
Filling-Enforced constraint on the quantized Hall conductivity on a periodic lattice
,”
Ann. Phys.
413
,
168060
(
2020
).
39.
A.
Matsugatani
,
Y.
Ishiguro
,
K.
Shiozaki
, and
H.
Watanabe
, “
Universal relation among the many-body Chern number, rotation symmetry, and filling
,”
Phys. Rev. Lett.
120
(
9
),
096601
(
2018
).
40.
S.
Bachmann
and
Y.
Ogata
, “
C*-algebraic index with application to the quantum Hall effect in the plane
” (unpublished) (
2020
).
41.
B.
Nachtergaele
and
R.
Sims
, “
Lieb-Robinson bounds and the exponential clustering theorem
,”
Commun. Math. Phys.
265
(
1
),
119
130
(
2006
).
42.
M. B.
Hastings
and
T.
Koma
, “
Spectral gap and exponential decay of correlations
,”
Commun. Math. Phys.
265
(
3
),
781
804
(
2006
).
43.
M. B.
Hastings
and
X.-G.
Wen
, “
Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance
,”
Phys. Rev. B
72
(
4
),
045141
(
2005
).
44.
S.
Bachmann
,
S.
Michalakis
,
B.
Nachtergaele
, and
R.
Sims
, “
Automorphic equivalence within gapped phases of quantum lattice systems
,”
Commun. Math. Phys.
309
(
3
),
835
871
(
2012
).
45.
W.
De Roeck
and
M.
Schütz
, “
Local perturbations perturb—Exponentially–locally
,”
J. Math. Phys.
56
(
6
),
061901
(
2015
).
46.
N.
Schuch
,
D.
Pérez-García
, and
I.
Cirac
, “
Classifying quantum phases using matrix product states and projected entangled pair states
,”
Phys. Rev. B
84
(
16
),
165139
(
2011
).
47.
Y.
Ogata
, “
A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III
,”
Commun. Math. Phys.
352
(
3
),
1205
1263
(
2017
).
48.
D. R.
Hofstadter
, “
Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields
,”
Phys. Rev. B
14
(
6
),
2239
(
1976
).
49.
A.
Giuliani
,
V.
Mastropietro
, and
M.
Porta
, “
Universality of the Hall conductivity in interacting electron systems
,”
Commun. Math. Phys.
349
(
3
),
1107
1161
(
2017
).
50.
M. B.
Hastings
, “
The stability of free fermi Hamiltonians
,”
J. Math. Phys.
60
(
4
),
042201
(
2019
).
51.
W.
De Roeck
and
M.
Salmhofer
, “
Persistence of exponential decay and spectral gaps for interacting fermions
,”
Commun. Math. Phys.
365
,
773
796
(
2019
).
52.
The Quantum Hall Effect
, Graduate Text in Contemporary Physics, edited by
S. M.
Girvin
and
R.
Prange
(
Springer-Verlag
,
Berlin, Heidelberg, New York
,
1987
).
53.
S.
Bachmann
,
A.
Bols
,
W.
De Roeck
, and
M.
Fraas
, “
Note on linear response for interacting Hall insulators
,” in
Analytical Trends in Mathematical Physics
, Contemporary Mathematics Vol. 741 (
American Mathematical Society
,
2020
), pp.
23
58
.
54.
B.
Nachtergaele
,
R.
Sims
, and
A.
Young
, “
Lieb–Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems
,” in
Mathematical Problems in Quantum Physics
, Contemporary Mathematics Vol. 717 (
American Mathematical Society
,
2018
), pp.
93
116
.
55.
O.
Bratteli
and
D. W.
Robinson
,
Operator Algebras and Quantum Statistical Mechanics 1: C*- and W*-Algebras, Symmetry Groups, Decomposition of States
, 2nd ed. (
Springer-Verlag
,
Berlin, Heidelberg, New York
,
1987
).
56.
O.
Bratteli
and
D. W.
Robinson
,
Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States
, Models in Quantum Statistical Mechanics, 2nd ed. (
Springer-Verlag
,
Berlin, Heidelberg, New York
,
1997
).
57.

Our notation differs here from the tradition of reserving this symbol for the smaller algebra of observables that are strictly supported in Z.

58.

As opposed to the “symmetry protected” topological order, which is not considered here.

59.

The adjective “topological” cannot be omitted here. In the case of spontaneous symmetry breaking with a local order parameter, the index can be rational; see Sec. VI.

60.

It is often stressed that this is a “large” gauge transformation, referring to the fact that it is not connected to the identity within the gauge group. However, on the lattice, it is not straightforward to make this distinction precise.