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.

## I. INTRODUCTION

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 effect^{3–5} 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 states^{16–18} and cellular automata.^{19,20}

For non-interacting systems, several topological indices can be formulated as *Fredholm–Noether indices*^{21–23} 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 $\u21132(Z)$. 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 $Tr[P(U*1NU\u22121N)]$, provided that $[U,1N]$ is trace-class, with $1N$ being the orthogonal projection on the subspace $\u21132(N)$. 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 *U*^{n} 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 order^{31,32} 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 work^{34} 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 as^{39} 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.

## II. SETTING

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 $H\Lambda $ 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* = 2^{f}, 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 $O\u2208A$, we associate a spatial support supp(*O*) ⊂ Λ. The crucial property is the following: If *X*, *Y* ⊂ Λ are disjoint, and if supp(*O*_{X}) ⊂ *X*, supp(*O*_{Y}) ⊂ *Y*, then [*O*_{X}, *O*_{Y}] = 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 $L\u2208N$ 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 $RH,RQ,mH,mQ,\gamma ,p$ 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 $A=O(L\u2212\u221e)$, which means that the sequence of operators *A* = *A*_{L} satisfies $\Vert AL\Vert \u2264CkLk$ pointwise for all $k\u2208N$. For completeness, we explain the large *L* setting in more detail in Appendix C.

### A. Hamiltonian

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

where

which are uniformly bounded: sup_{Z⊂Λ}‖*h*_{Z}‖ ≤ *m*_{H}. Note that *R*_{H} 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|.

### B. Charge

We consider local charge operators *q*_{Z} with supp(*q*_{Z}) = *Z* satisfying

*q*_{Z}= 0 unless diam(*Z*) ≤*R*_{Q},sup

_{Z⊂Λ}‖*q*_{Z}‖ ≤*m*_{Q},$\sigma (qZ)\u2282Z$ for all

*Z*, where*σ*(*A*) denotes the spectrum of*A*, and[

*q*_{Z},*q*_{Z′}] = 0 for all*Z*,*Z*′.

The total charge in *S* ⊂ Λ is defined as

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

Using the properties of *q*_{Z}, it follows that we can choose the decomposition *H* = *∑*_{Z}*h*_{Z} such that

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

### C. Spatial structure

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

and its boundary to be

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

Λ has a finite spatial dimension in the sense of sup

_{x∈Λ}|{*x*}_{(r)}| ≤*C*(1 +*r*)^{d}for all*r*≥ 0, i.e., the size of balls grows at most polynomially with the radius.There is a set Γ ⊂ Λ such that

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* = *A*_{L} is *almost supported* in a sequence of sets *Z* = *Z*_{L} if there are sequences $Ar=AL,r,r\u2208N$ with $supp(AL,r)\u2282(ZL)(r)$ [see (2.4)] such that

We denote^{57} 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

for all sequences of sets *Z*.

## III. THE INDEX THEOREM

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

*H*, counted with multiplicities. There are

*L*-independent constants

*γ*> 0, Δ > 0, and

*p*such that

*γ*> 2Δ.

We refer to the rank-*p* spectral projector corresponding to the spectral patch {*E*_{1}, …, *E*_{p}} as *P*, and its range is called the “the ground state space.”

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

*For any*$A\u2208AX,B\u2208AY$

*with*

*X*∩

*Y*= ∅

*and for any normalized*Ω ∈ ran(

*P*)

*,*

This is a slightly weaker statement than^{41,42} 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.

*U*). There is a locality-preserving unitary

*U*that leaves ran(

*P*) invariant

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*,

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

with $T\xb1\u2208A\u2202\xb1$ 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

where we denoted *Q* = *Q*_{Γ} as we shall do from here onward. Such a choice exists. Indeed, for any $T\u0303\xb1$ satisfying (3.3), we have, by integrality of the spectrum of *Q* and the assumption (2.5) about the spatial structure, that

with $exp(2\pi i(Q+T\u0303\xb1))\u2208A\u2202\xb1$. Hence, there exists *ν* such that

Then, $T\xb1\u2254T\u0303\xb1\u2213\nu 2\pi $ satisfies both (3.3) and (3.4). We can now state our main result. For any *ϵ* > 0, we denote

*If Assumptions*1

*and*2

*hold, then*

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*^{−1}*P*, which makes the above into a rational index indeed. We denote it by

where $\u3008A\u3009P\u2254p\u22121Tr(PA)$, 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.

*Z*such that diam(

*Z*) <

*C*and for any operator $A\u2208AZ$ of norm 1,

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 $\u3008\Omega ,H\Omega \u3009=\u3008H\u3009P+O(L\u2212\u221e)$ by the topological order for any normalized Ω ∈ran(*P*).

*If, in addition to Assumptions 1 and 2, Assumption 3 also holds, then*

*for any normalized*Ψ ∈ ran(

*P*)

*.*

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.

## IV. PROOF OF THEOREM 3.2

From now on, we denote by $=L$ equality up to terms of $O(L\u2212\u221e)$, in operator norm.

### A. Preliminaries

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

with *W* being a real-valued, bounded, integrable function satisfying $W(t)=O(|t|\u2212\u221e)$ and $W\u0302(\omega )=\u22121i\omega $ for all |*ω*|≥ *γ*, with *γ* being the spectral gap as in Assumption 1. Since, by functional calculus for ad_{H} = [*H*, ·],

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

Plugging this decomposition into (4.1) and by the Lieb–Robinson bound, we conclude that there are $K\xb1\u2208A\u2202\xb1$ such that ‖*K*_{±}‖ < *C*|Λ| and such that

leaves ran(*P*) invariant,

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.

*Let* *V* *be a unitary, and let* *P* *be an orthogonal projection. Then,*

‖[

*V*,*P*]‖ ≤*ϵ**implies*‖*PVP*‖^{2}≥ 1 −*ϵ*^{2}*and*‖

*PVP*‖^{2}≥ 1 −*ϵ*^{2}*implies*‖[*V*,*P*]‖ ≤ 2*ϵ**.*

*V*,

*P*]‖ ≤

*ϵ*, then

*PVP*‖

^{2}≥ 1 −

*ϵ*

^{2}, then

*V*,

*P*] = (1 −

*P*)

*VP*–

*PV*(1 −

*P*).□

*Let* $V\xb1\u2208A\u2202\xb1$ *be unitary operators, and let* *V* ≔ *V*_{–}*V*_{+}*. If Assumption* 1 *holds, then* $[V,P]=L0$ *implies* $[V\xb1,P]=L0$*.*

*V*

_{+}is unitary. Thus, from (4.4), the assumption $[V,P]=L0$ and Lemma 4.1(i) yield

*Let*

*A*=

*A*

^{*}

*and*

*P*

*be an orthogonal projection. If*‖[

*A*,

*P*]‖ <

*ϵ*

*, then*

### B. The main argument

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 $\Lambda \xb1=(\u2202\xb1)(cL)\u2229\Gamma $ be regions such that dist(Λ_{+}, Λ_{−}) ≥ *cL* (recall that *c* can change the value from equation to equation). We denote

For *ϕ* ∈ [0, 2*π*], we set

as well as

Here, we defined $Q\xafU\u2254U*Q\xafU$, $K\xb1U\u2254U*K\xb1U$, and

To avoid later confusion, we point out that $Q\xaf\xb1U\u2260U*Q\u2212U$. With these definitions, the following identities hold:

The crucial point of these definitions is the commutation property

with *A*_{±}, *B*_{±} being any of the above objects carrying the subscript ±. This is immediate for operators in $A\u2202\xb1$, but it also holds for $Q\xb1,Q\xaf\xb1,Q\xaf\xb1U\u2208A\Lambda \xb1$, and hence for their exponentials by Lemma 4.3, because these operators reduce to the charge away from *∂*_{±}. This immediately leads to

By Assumption 2 and by construction of $Q\xaf$ [see (4.3)], all four factors of $Z(\varphi )=U*ei\varphi Q\xafUe\u2212i\varphi Q\xaf$ commute with *P* so that

This and Lemma 4.2 now yield the first essential observation,

The second one follows by recalling the integrality of the spectrum of *Q*_{m}, which implies that

again by Lemma 4.2 applied to $e2\pi iQ\xaf=e2\pi iQ\xaf\u2212e2\pi iQ\xaf+$ [see (4.8)].

We now consider the function $\varphi \u21a6Z\u2212(\varphi )\u2254PZ\u2212(\varphi )P$ and let $D\u2212\u2254Q\xaf\u2212U\u2212Q\xaf\u2212$. Then,

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 $Q\xaf$. The unique solution of this differential equation with $Z\u2212(0)=1$ is

Note that both unitary factors on the right independently commute with *P*. It remains to study $Z\u2212(2\pi )$ to conclude. By the integrality of the spectrum of *Q*_{m}, *Q*_{+},

where we used definition (4.7) of $Q\xaf\u2212$. Since $U*QU=LQ+T\u2212+T+$, from the integrality of the spectrum of charge and commutation property (4.9), we conclude that

where the second equality uses choice (3.4). Since the exponent is precisely $Q\xaf\u2212U$ [see again (4.7)], we multiply from the right by $e\u22122\pi iQ\xaf\u2212$ to obtain

All four unitaries on the right-hand side commute with *P* [see (4.11)]. Moreover, since *PV*^{*}*PVP* − *P* = *PV*^{*}[*P*, *V*]*P* for any unitary *V*, the condition $[P,V]=L0$ implies that *PVP* is invertible on ran(*P*) with $(PVP)\u22121=LPV*P$. By the continuity of the determinant, we conclude that

where det_{P}(*A*) ≔ det(*PAP* + (1 − *P*)). On the other hand, (4.12) and the relation det_{P}(e^{A}) = e^{Tr(PA)} yield

We further observe that

[see (4.7)] so that $Tr(PD\u2212)=LTr(PT\u2212)$ by the unitary invariance of the trace. Hence, $Tr(PT\u2212)\u2208Z(O(L\u2212\u221e))$, which concludes the Proof of Theorem 3.2.

*T*

_{±}, (4.14) would read as

*T*

_{±}satisfies (4.14), it suffices to verify that $p\u3008T\u2212\u3009P\u2208Z(O(L\u2212\u221e))$. We will use this in the proof of additivity below.

## V. ADDITIVITY, FILLING, AND AVRON–DANA–ZAK RELATIONS

### A. Additivity of the index

If *U*_{1}, *U*_{2} 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 *U*_{1}, *U*_{2} 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

This is the content of the following proposition.

*U*

_{2}is locality-preserving, choice (5.1) indeed satisfies (3.3). Using $[U2,P]=L0$, we get

### B. The fractional Lieb–Schultz–Mattis theorem

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 $T\u2254Z/L1Z$ be a discrete circle. Then, Λ is of the form

the Cartesian product of these graphs. We write (*x*_{1}, *x*′) ∈ Λ with $x1\u2208T,x\u2032\u2208\Lambda \u2032$, and we let Θ be a unitary shift along $T$. Finally, let

This choice is consistent with the setup of Sec. II C, provided that we let *L*_{1} ≥ *cL*, 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,

where [*x*_{1}] = {(*x*_{1}, *x*′), *x*′ ∈ Λ′} and ⌈*L*_{1}/2⌉ is the smallest integer not smaller than *L*_{1}/2. We make the natural choice *T*_{−} = −*Q*_{[0]}, for which (3.4) holds. By translation invariance, the total charge per slab Λ′ is

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

*If Assumption 1 holds, then*

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 *x*_{1}-direction. To be very specific, we consider two-dimensional *L*_{1} × *L*_{2}-tori Λ, that is, we specify now $\Lambda \u2032=Z/L2Z$ with diameter $L=\u230aL1+L22\u230b$. We specify a particular sequence of tori by picking a function *L* ↦ (*L*_{1}, *L*_{2}), satisfying $L=\u230aL1+L22\u230b$. We choose this function such that *L*_{2} runs through all positive integers and that both *L*_{1}, *L*_{2} > *cL*. Let us now assume that the total charge density $\rho L\u22541|\Lambda |\u3008Q\Lambda \u3009P$ converges to a limiting density *ρ* fast enough, namely, *L*(*ρ* − *ρ*_{L}) → 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

Indeed, from the index theorem, we obtain that $pL2\rho L\u2208Z(O(L\u2212\u221e))$. Writing *L*_{2}*ρ*_{L} = *L*_{2}*ρ* + *L*_{2}(*ρ*_{L} − *ρ*) and using the assumption of fast convergence and *L*_{2} > *cL*, we get (*pL*_{2}*ρ*) mod 1 = *o*(1). If *L*_{2} runs through $N$, the rational non-integer *pρ* is ruled out directly and irrational *ρ* is ruled out by the ergodicity of irrational rotations on the torus.

### C. On the absence of topological order in one dimension

It is widely accepted and proved in some specific settings^{16,46,47} that there is no intrinsic^{58} 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 Ind_{P}(*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 fractionalization^{59} in the ground state sector.

Indeed, in one dimension, the region *∂*_{−} is finite so that $Pe2\pi iQ\xaf\u2212P=LzP$ for some $z\u2208C,|z|=1$, by Assumption 3 (note that $[e2\pi iQ\xaf\u2212,P]=L0$ in all dimensions, but the exponential acts, in general, non-trivially on ran(*P*)). But then,

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

which, with (5.6), yields

as claimed.

### D. Magnetic systems

We now discuss the case of systems with magnetic fields. Our main result is an interacting version of the Avron–Dana–Zak (ADZ) relation^{37} 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 $\Lambda =(Z/L1Z)\u25a1(Z/L2Z)$, i.e., the *L*_{1} × *L*_{2} discrete torus with coordinates 1 ≤ *x*_{1,2} ≤ *L*_{1,2} and unit vectors $e\u03021=(1,0),e\u03022=(0,1)$. We describe spinless fermions in a uniform magnetic field. Let

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

We have written $qx=cx*cx$ for the occupation operators, and the parameters *t*, *μ*, *u*(·) are, respectively, the hopping strength, the chemical potential, and the interaction potential. We impose $L1\varphi \u22082\pi Z$ in order that the Hamiltonian is well defined on the torus.

Let us comment on how this model fits our setup. The on-site operator *q*_{x} 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 theory^{49–51} 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 *x*_{2}-direction, but it is more interesting to consider translation in the *x*_{1}-direction,

as well as the magnetic translation defined by

At *ϕ* ≠ 0, the ordinary translation is not a symmetry, but the magnetic translation is a symmetry provided that $L2\varphi \u22082\pi Z$,

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*/(2*π*) because *L*_{2} must be a multiple of *n*. Therefore, the strongest result that can be obtained on the density *ρ* is simply $(pn)\rho \u2208Z$, which also follows from an application of the theorem with Θ^{n}. 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 *x*_{2}-direction. If the gap of *H*_{Φ} remains bounded away from 0 as the parameter Φ goes from Φ to Φ′, then the ground state projector *P*_{Φ} can be parallel transported to *P*_{Φ′} by a locality-preserving unitary *F*(Φ, Φ′) (see Sec. V D 5 for details), namely,

If the total threaded flux is an integer number of elementary flux quanta, i.e., $L2(\Phi \u2032\u2212\Phi )\u22082\pi Z$, then the effect of parallel transport on *P*_{Φ} is the same as that of a gauge transformation^{60} implemented by the unitary

This follows because in that case

Therefore, the unitary $F(\Phi ,\Phi \u2032)F\Delta \Phi $ satisfies Assumption 2 and it defines an associated index of *H*_{Φ}. By the Laughlin argument, this index is the number of threaded flux quanta *L*_{2}ΔΦ/(2*π*) 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 $\Delta \Phi \u22082\pi Z$ yields $F\Delta \Phi =1$, in which case *F*(Φ, Φ′) itself is a symmetry of *P*_{Φ}. 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 *H*_{Φ+ϕ} = Θ^{*}*H*_{Φ}Θ. Using this, we obtain

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

(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

The first term on the left-hand side is $\u3008Q[0,n\u22121]\u3009P=\u2211x:0\u2264x2<n\u3008nx\u3009P$, which is *n* times the average density per slab, namely, *nL*_{2}*ρ*, and the second term is *nϕL*_{2} times the Hall conductance *σ*_{L} (see again Ref. 34). Since *L*_{2} is arbitrary and if the convergence of *ρ*_{L}, *σ*_{L} to their infinite-volume limits *ρ*, *σ* satisfies *L*_{2}(*ρ* − *ρ*_{L}), *L*_{2}(*σ* − *σ*_{L}) → 0, then the argument of Sec. V B implies that

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 $\Lambda =T\u25a1$ Λ′, where $T$ is the discrete circle with *L*_{1} sites, and we again assume that *L*_{1} > *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 ${H\Phi :\Phi \u2208R}$ of form (2.1), satisfying the following.

The parameters

*R*_{H},*m*_{H}can be chosen to be uniform in Φ and the local terms*h*_{Z}are themselves*C*^{1}functions of Φ such that ‖*∂*_{Φ}*h*_{Z}(Φ)‖ is bounded uniformly in*Z*and*L*.- Periodicity:$H\Phi +2\pi =H\Phi .$
- Covariance:where(5.11)$\Theta *H\Phi \Theta =H\Phi +\varphi ,$$\varphi =2\pi mn,m\u2208Z,n\u2208N,m,n\u2009coprime.$
Assumption 1 holds uniformly in Φ, where

*P*=*P*_{Φ}are the ground state projectors. This implies, in particular, that the rank $p=rk(P\Phi )$ is independent of Φ.

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

Items (i) and (iv) imply that Φ ↦ *P*_{Φ} is itself differentiable and periodic,

for any $\Phi \u2208R$. Items (ii) and (iii) lead to

and hence, Θ^{n} 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

exactly for any $\Phi ,\Phi \u2032\u2208R$. In particular, we obtain for *P* ≔ *P*_{0} that

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

*Let Assumption*4

*hold. Then,*

We now turn to the Proof of Theorem 5.3. By the gap assumption, the self-adjoint family *A*_{Φ} of so-called quasi-adiabatic generators defined by

where $\u2003\u0307=\u2202\Phi $ and *W* was introduced after (4.1), are such that

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

and it is an intertwiner between *P*_{Φ′} and *P*_{Φ} [see (5.13)]. It satisfies the cocycle relation

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

*For any*$k\u2208Z$

*and any*$\Phi ,\Phi \u2032\u2208R$

*,*

*F*

_{Θ}(Φ) ≔ Θ*

*F*(Φ, Φ′)Θ is a solution of

*F*

_{Θ}(Φ) =

*F*(Φ +

*ϕ*, Φ′ +

*ϕ*). In other words,

*k*-fold application of which yields the claim.□

*Let*

*U*(Φ) ≔ Θ

*F*(Φ +

*ϕ*, Φ)

*. Then, for any*$\Phi \u2208R$

*,*

*Moreover,*[

*U*(Φ)

^{k},

*P*

_{Φ}] = 0

*and*

*for any*$k\u2208Z$

*.*

*P*

_{Φ+ϕ}Θ* =

*P*

_{Φ}, and hence, by (5.13), we have

*U*(Φ)

^{k}

*P*

_{Φ}=

*P*

_{Φ}

*U*(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

We note that by Lemma 5.4,

*k*=

*n*in Lemma 5.5, we further conclude that

*P*

_{Φ}is also invariant under Θ

^{n}. It follows from Theorem 3.2 that both indices $IndP\Phi (\Theta n),IndP\Phi (F)$ are well defined with denominator $p$. Finally, the same applies to

*U*(Φ) and

*U*(Φ)

^{n}by Lemma 5.5. Altogether,

## VI. SUPERSELECTION SECTORS

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

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

As usual, $M,pm$ are finite and fixed, independent of *L* for *L* large enough. We denote $\u3008\u22c5\u3009m\u2254pm\u22121Tr(Pm\u22c5)$.

Decomposition (6.1) is such that

- for any $O\u2208AZ$ of norm 1 and with diam(
*Z*) ≤*C*,whenever(6.2)$Pm\u2032OPm=L0$*m*≠*m*′; - there are self-adjoint observables $A1\u2208AX,A2\u2208AY$ with diam(
*X*) ≤*C*, diam(*Y*) ≤*C*, and dist(*X*,*Y*) >*cL*such thatfor all$\u3008A1\u3009m=\u3008A2\u3009m$*m*, while$|\u3008A1\u3009m\u2212\u3008A1\u3009m\u2032|\u2265c>0,m\u2260m\u2032;$ each

*P*_{m}satisfies the topological order Assumption 3.

Assumption (ii) postulates the existence of observables that detect the local order. In the case of a translation-invariant system, *A*_{2} is a translate of *A*_{1}. In other situations, *A*_{2} might be a linear combination of more natural physical observables that achieves the equality of expectation values between *A*_{2} and *A*_{1}. Points (i) and (iii) of the assumptions imply that *P*_{m} 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 *S*_{M} be the symmetric group on {1, …, *M*}.

*Let*$Um,l\u2254PmUPl$

*and*$U\u2254PUP$

*. There is a permutation*

*π*

_{U}∈

*S*

_{M}

*such that*

*In particular,*

*P*

_{m}are subprojections of

*P*,

*O*satisfying the assumption,

*m*,

*l*≤

*M*,

*m*,

*ρ*

_{m}is a probability distribution on {1, …,

*M*}, and for

*O*=

*A*

_{j}as in Assumption 5(ii), we interpret (6.3) as the expectation value in

*ρ*

_{m}of a random variable $aj:{1,\u2026,M}\u2192R$ given by

*a*

_{j}(

*l*) = ⟨

*A*

_{j}⟩

_{l}. Therefore,

*ρ*

_{m}. In terms of the random variables

*a*

_{1},

*a*

_{2}, Assumption 5(ii) is rephrased as the statement that

*a*

_{1}=

*a*

_{2}and that

*a*

_{1}is injective. Clustering (see Lemma 6.3) implies that

*a*

_{1}=

*a*

_{2}, we simply obtain

*a*

_{1}is constant on the intersection of its support with the support of

*ρ*

_{m}. Since

*a*

_{1}is injective, the support of

*ρ*

_{m}is a singleton. This means that there is a unique

*π*(

*m*) such that $U\pi (m),m$ 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,

*UP*

_{m}

*U** is a subprojection of $P\pi U(m)$. By the invertibility of

*U*, the two must have the same rank, proving (ii), and hence, they are equal, proving (iii).□

For any 1 ≤ *m* ≤ *M*, let (*π*_{U} · *m*) denote the cycle of the permutation *π*_{U} containing *m*, and let *ℓ*_{U}(*m*) be its length. We then have the following generalizations of Theorem 3.2.

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

*for any normalized*Ψ_{m}∈ran(*P*_{m}),$pm\Psi m,TU\u2113U(m)\u2212\Psi m\u2208Z(O(L\u2212\u221e));$*for any normalized*$\Psi \u2208\u2295m\u2032\u2208(\pi U\u22c5m)ran(Pm\u2032)$*,*$\u2113U(m)pm\Psi ,TU\u2212\Psi \u2208Z(O(L\u2212\u221e)).$

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 $Q\xaf$. Lemmas 6.3 and 6.4 show that Assumption 5 implies both properties for the subprojections *P*_{m}.

*Let*Ψ

_{m}∈ Ran(

*P*

_{m})

*be normalized, and let*$A\u2208AX,B\u2208AY$

*be of norm 1 with*

*d*(

*X*,

*Y*) >

*cL*

*. If one of*

*A*,

*B*

*is an observable as in Assumption*5

*(i), then*

*For any* *m**,* $[Q\xaf,Pm]=L0$*.*

*P*

_{m}=

*PP*

_{m}

*P*implies

*P*

_{m}satisfies the clustering property and it is invariant under $Q\xaf$. Furthermore, the unitary $U\u2113U(m)$ keeps

*P*

_{m}invariant by definition of

*ℓ*

_{U}(

*m*), namely,

*P*replaced by

*P*

_{m}and

*U*replaced by $U\u2113U(m)$. This establishes (i).

Since (*π*_{U} · *m*) is a cycle, $\u2295m\u2032\u2208(\pi U\u22c5m)ran(P)$ is invariant under *U*. By Lemma 6.1(ii), all factors have rank $pm$ so that the dimension of $\u2295m\u2032\u2208(\pi U\u22c5m)ran(P)$ is $\u2113U(m)pm$ and the claim follows as mentioned above.□

## ACKNOWLEDGMENTS

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 AVAILABILITY

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

### APPENDIX A: CLUSTERING IN FINITE VOLUME

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

For concreteness, we take

where, for any $a\u2208R$, 1 −*θ*_{a} is the shifted Heaviside function (with discontinuity at *a*), −*γ*/2 lies in the interval (−*γ* + Δ, −Δ), the product ⋆ denotes the convolution, and $\varphi \u2208Cc\u221e((\u2212\delta ,\delta ))$ is a non-negative function such that ∫*ϕ* = 1.

Let

where ad_{H}(*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 $g\u0302(t)=O(|t|\u2212\u221e)$, and so,

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,

This follows from (A2), namely,

where *P*_{l} is the eigenprojection corresponding to the eigenvalue *E*_{l} and (A1). Hence, for any $A\u2208AX,B\u2208AY$,

It remains to note that $Q(A)\u2208AX$ implies $[Q(A),B]=O(d(X,Y)\u2212\u221e)$. 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 $g\u0302$ given by

The function *ϕ* being smooth, we have that $\varphi \u0302(t)=O(|t|\u2212\u221e)$. Moreover, the Plemelj–Sochotcki formula implies that

where $P$ denotes the principle value, and we recall that

for any Schwartz function *ψ*.

The smooth operator-valued function *O*(*t*) = e^{itH}*O*e^{−itH} is bounded since ‖*O*(*t*)‖ = ‖*O*‖, and its derivative *O*′(*t*) = e^{itH}i[*H*, *O*]e^{−itH} is similarly bounded: ‖*O*′(*t*)‖ ≤ *C*‖*O*‖|supp(*O*)|. It follows, in particular, that $e\u2212it\gamma /2\varphi \u0302(t)O(t)$ is a Schwartz operator-valued function. The expression on the right-hand side of (A2) is well defined for any local observable *O*, with

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 $\theta \u03020$ 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 $\varphi \u0302$ yields a bound $\Vert O\Vert O(T\u2212\u221e)$ for the long time part. For the second, short time part, we write *O*(*t*) = Π_{R}(*O*(*t*)) + (*O*(*t*) − Π_{R}(*O*(*t*))), where Π_{R} is the projection onto an *R*-fattening of the support of *O*. Since Π_{R}(*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

which is integrable at 0 with $\u222b[\u2212T,T]t\u22121ecv|t|\u22121dt\u22642(ecvT\u22121)$. It remains to pick $T=\xi 2cvR$ to conclude that both terms are $O(R\u2212\u221e)$. Altogether, we conclude that $Q(O)$ is almost localized on the support of *O*.

### APPENDIX B: NOTIONS OF LOCALITY

We consider first the case of spin systems. Then, $H\Lambda \u2245\u2297x\u2208\Lambda Cn$ and $A\u2245\u2297x\u2208\Lambda Cn2$, and so, there is a natural tensor product representation $A\u2245\u2297x\u2208SCn2\u2297\u2297x\u2208ScCn2$ for any *S* ⊂ Λ. The support of $O\u2208A$ is then the smallest set *S* such that $O\u2245OS\u22971Sc$ 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 $cx,\alpha ,cx,\alpha *$ and the identity, with $cx,\alpha ,cx,\alpha *$ 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

The Hilbert space $H\Lambda $ 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 $cx,\alpha ,cx,\alpha *$ of even/odd degree are called even/odd and the even monomials generate an algebra, which we call $A$. The support of $O\u2208A$ is then the smallest set *S* such that *O* is in the linear span of even monomials with *x* restricted to *S*.

### APPENDIX C: THE LARGE *L* ASYMPTOTIC ANALYSIS

In the main text, we have systematically omitted the index *L* but at the same time used symbols such as $O(L\u2212\u221e)$ 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 *A*_{L} be a sequence of operators defined on a sequence of operator algebras $A=AL$ associated with a sequence of graphs Λ_{L} with diameter *L*. Neither the graphs nor the operators *A*_{L} are *a priori* related in any way. We say that $AL=O(L\u2212\u221e)$ if for any $k\u2208N$, there exists a constant *C*_{k} such that ‖*A*_{L}‖ ≤ *C*_{k}*L*^{−k} for all *L*. Furthermore, let *Z*_{L} ⊂Λ_{L} be a sequence of sets. The sequence *A*_{L} belongs to the set $AZL$ if for all *r* there exists a sequence of operators *A*_{L,r} supported in $(ZL)(r)$ such that for any $k\u2208N$, there exists a constant *C*_{k} with

for all *L*.

In the setting of Sec. II C, we postulate the existence of a sequence of sets Γ_{L} ⊂ Λ_{L} with boundaries *∂*_{−,L} and *∂*_{+,L} whose distance satisfies (2.5). Assumption 2, in particular, (3.3), then says that $T\xb1,L\u2208A\u2202\xb1,L$. Assumption 1 postulates the existence of a sequence of ground state projections $PL\u2208AL$ of *L*-independent rank $p$.

Theorem 3.2 states that there exists a sequence of integers *n*_{L} such that

The projection *P*_{L} may be further constrained by Assumption 3: for any sequence of operators $AL\u2208AZL$ such that diam(*Z*_{L}) < *C* and ‖*A*_{L}‖ = 1, we have *P*_{L}*A*_{L}*P*_{L} = *p*^{−1}Tr(*P*_{L}*A*_{L})*P*_{L} + *O*(*L*^{−∞}). In the topologically ordered situation, Corollary 3.3 states that for any sequence Ψ_{L} ∈ ran(*P*_{L}), we have $p\u3008\Psi L,T\u2212,L\Psi L\u3009\u2212nL=O(L\u2212\u221e).$

As a final note, we point out that for an *L* × *L* torus, *∂*_{±,L} have 2*L* sites. If *U* is a translation by one site in the *x*_{1}-direction, then *T*_{±,L} is equal to the charge in the boundary [see (5.4)], and *n*_{L} 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*.

## REFERENCES

^{*}-algebraic index with application to the quantum Hall effect in the plane

^{*}- and W

^{*}-Algebras, Symmetry Groups, Decomposition of States

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

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

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.

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.