We investigate the persistence of spectral gaps of one-dimensional frustration free quantum lattice systems under weak perturbations and with open boundary conditions. Assuming that the interactions of the system satisfy a form of local topological quantum order, we prove explicit lower bounds on the ground state spectral gap and higher gaps for spin and fermion chains. By adapting previous methods using the spectral flow, we analyze the bulk and edge dependence of lower bounds on spectral gaps.

Dedicated to the memory of Ludwig Faddeev

An important result in the study of gapped ground state phases of quantum lattice systems (with or without topological order) is the stability of the spectral gap(s) under uniformly small extensive perturbations. The stability property implies that the gapped phases are full-dimensional regions in the space of Hamiltonians free of phase transitions.1 In recent years, such results were obtained in increasing generality.2,3,5,7,8,16–18 Our goal here is to extend the existing results applicable in one dimension to Hamiltonians with the so-called “open” boundary conditions, meaning that we consider systems defined on intervals [a,b]Z and not on a cycle Z/(nZ). Specifically, this implies that the neighborhoods of the boundary points a and b may be treated differently than the bulk. There are physical and mathematical situations where one is naturally led to considering open boundary conditions. For example, in the series of recent studies by Ogata,13–15 clarifying the crucial role of boundary states in the classification of quantum spin chains with matrix product ground states required the study of systems with open boundary conditions. Another situation of interest to us is the application of results for quantum spin chains to fermion models in one dimension by making use of the Jordan-Wigner transformation, which in the finite system setup only works well with open boundary conditions. In this way, we obtain explicit bounds on the spectral gaps in the spectrum of perturbed spin and even fermion chains with one or more frustration free ground states that satisfy a local topological order condition. This complements previous results that prove stability of gapped fermion systems by other approaches.4,5,12

Denote by (Z,||) the metric graph of integers. Let Pf(X) denote the finite subsets of XZ. We will use Λ to refer exclusively to nonempty, finite intervals of the form [a,b]={nN:anb}. Let bΛ(x, n) = {m ∈ Λ: |xm| ≤ n} denote the restriction of a metric ball to the interval. For each x ∈ Λ, denote by rx and Rx the following distances to the boundary:


Although rx and Rx depend on the interval [a, b], we omit this dependence from the notation since we will always fix a finite volume [a, b] throughout our arguments.

In the following, we will consider both spin systems and fermion systems on the one-dimensional lattice. Without difficulty, we could also treat systems that include both types of degrees of freedom, but for simplicity of the notations, we will not consider such systems in this paper. It is also possible to consider inhomogeneous systems for which the number of spin or fermion states depends on the site. In order to present the main ideas without overly burdensome notation, we will only consider homogeneous systems in the note.

The algebra of observables of the finite system in Λ, of either spins or fermions, will be denoted by AΛ. If we want to specify that we are specifically considering spins or fermions, we will use the notation AΛs or AΛf, respectively. These algebras, and the associated Hilbert space they are represented on, are defined as follows:

For spin systems, we have


where d is the dimension of the Hilbert space of a single spin, i.e., d = 2S + 1.

For fermions, AΛf denotes the C*-algebra generated by {a(x), a*(x): x ∈ Λ}, the annihilation and creation operators defining a representation of the Canonical Anticommutation Relations (CAR) on the antisymmetric Fock space FΛ=F(2(Λ)). The dimension of FΛ is 2|Λ| and AΛf is *-isomorphic to the matrix algebra M2|Λ|(C).

Given an exhaustive net of CAR or spin algebras {AΛ:ΛPf(Z)}, the inductive limit AZ, the d UHF algebra, is obtained by norm completion,


This algebra is often referred to as the quasi-local algebra, and Aloc=AΛ is referred as the local algebra.

Define by NX = xXa*(x)a(x) the number operator for XPf(Z), and define the parity automorphism by


Say that AAΛf is even if ρΛ(A) = A and odd if ρΛ(A) = −A. The observable A is even if and only if it commutes with the local symmetry operator exp(iπNΛ), which is if and only if A is the sum of even monomials in the generating set {a(x), a*(x): x ∈ Λ}. Unlike the odd observables, the even observables form a *-subalgebra of AΛf, which we denote by AΛ+.

Let I be a subinterval of Z, not necessarily finite. An interaction on I is a function Φ:Pf(I)Aloc such that Φ(X)=Φ(X)*AX for all XPf(I). The corresponding local Hamiltonian of the finite system on Λ ⊂ I is HΛ = X⊂ΛΦ(X). Say that Φ is non-negative if Φ(X) ≥ 0 for all XPf(I). Say that Φ is an even interaction of the CAR algebra if Φ(X)AX+.

The interactions in our perturbative setup will satisfy the following assumptions. First, let η:Pf(Z)Aloc be a non-negative interaction with distinguished local Hamiltonians HΛ. We will refer to η as the unperturbed interaction. We assume that η has the following properties:

  • Finite range: There exists R > 0 such that diam(X) > R implies η(X) = 0.

  • Uniformly bounded: There exists M > 0 such that for all XPf(Z), ∥η(X)∥ < M.

  • Frustration free: For all intervals ΛPf(Z), ker(HΛ) ⊋ {0}.

  • Uniformly locally gapped: There exists γ0 > 0 such that for all intervals [a,b]Pf(Z), with baR, γ0 is lower bound a for non-zero eigenvalues of H[a,b].

  • Local topological quantum order (LTQO) of the ground state projectors.

The concept of LTQO was introduced in Ref. 2. We will need to adapt the definition to take into account parity and boundary conditions, which we do in Sec. II C.

Next, we consider the perturbations. To allow edge effects, we will consider perturbations given in terms of a family of interactions on intervals. For each Λ, let ΦΛ:Pf(Λ)Aloc be an interaction on the interval, and denote by [Φ] the collection of these perturbative interactions,


The perturbed Hamiltonians have the form


and while the Hamiltonians depend on the interval Λ, lower bounds on gaps in the spectrum will be uniform in the volume.

Our main assumption on the interactions ΦΛ in [Φ] is that ΦΛ(X) decays rapidly with the diameter of X. To make this precise, we use F-functions and provide explicit bounds in terms of the F-norm. The definition and properties of F-functions and F-norm can be found in the  Appendix. In our argument, we will use functions of the form


where κ > 2 and L, c > 0. The function h: [0, ) → [0, ) is a monotone increasing, subadditive weight function. At times, it will be necessary to precompose F with a transformation τ:[0,)R, and so we will take as convention Fτ(x) = F(0) for τ(x) < 0. We will denote by ∥⋅∥F the extended norm (A1) induced by F.

Using F-function terminology, we assume for the perturbations:

  • Fast decay: There exists an F-function F(r)=ehΦ(r)L(1+cx)κ, for L, c > 0 and κ > 2, such that supΛ∥ΦΛF < .

  • Metric ball support: For all Λ, ΦΛ(X) ≠ 0 implies X = bΛ(z, n) for some z ∈ Λ and nN.

The assumption that ΦΛ is supported on metric balls is not restrictive since a finite-volume Hamiltonian of any fast-decaying interaction can be rewritten as the finite-volume Hamiltonian of a balled interaction with comparable decay (c.f. the Appendix of Ref. 18).

Consider the unperturbed interaction η and its local Hamiltonians. Denote by PX the orthogonal projection onto ker(HX), and define the state


The unperturbed interaction η satisfies local topological quantum order if there exists a monotone function Ω: [0, ) → [0, ), decreasing to 0, such that for all x ∈ Λ and n,kN satisfying 0 ≤ krx and knRx, the following bound holds:
where zx:NN is the cutoff function defined in terms of distance to the boundary of Λ (2.1),
zx(m)=mif mrxrxelse .
If η:Pf(Z)Alocf is an even interaction and (2.6) holds for the restricted class of observables AAbΛ(x,k)+, then we will say that η has Z2-LTQO.

For example, the AKLT interaction with either periodic or open boundary conditions has LTQO with Ω(r) = (1/3)r. The interaction defined in (5.1) has Z2-LTQO with Ω(r) = 0 for r greater than a cutoff D > 0 defined by the interaction parameters, and Ω(x) = 2 otherwise (Proposition 5.4).

For any finite interval Λ, we consider the local Hamiltonian HΛ(ε) given in (2.4). There exist continuous functions λ1,,λN:[0,1]R such that for all ε ∈ [0, 1], {λ1(ε), …, λN(ε)} are the eigenvalues of HΛ(ε). We partition sp(HΛ(ε)) into two disjoint regions, an upper and a lower part of the spectrum, and call the minimum distance between these two sets the spectral gap above the ground state or the spectral gap,

sp0,Λ(ε)=λi(ε):λi(0)=0,  sp1,Λ(ε)=λj(ε):λj(0)>0,

For a class of sufficiently small perturbations, the main result of this paper establishes a lower bound for the size of the spectral gap which does not depend on Λ, under the assumptions that η has LTQO, the interactions in [Φ], from (2.3), decay sufficiently fast and, in the case of fermions, that the interactions are even. The spectrum may have other gaps which can be defined similarly in terms of eigenvalue splitting, and we also prove an estimate showing how these gaps persist under weak perturbations. To state these results, we define several constants that characterize the effect of the perturbation and the presence of edge effects.

The effect of perturbations near the boundary of Λ is, in general, different and stronger than far away from the boundary. As a consequence, our stability result for open chains features a distance parameter D ≥ 0, in terms of which we distinguish sites near and far away from the boundary. In Sec. III, we write each ΦΛ as the sum of an interaction ΦD(Λ), with a local Hamiltonian ΦΛD supported at the D-boundary, and a bulk interaction ΦInt(Λ). Define the following two finite constants quantifying the strength of the bulk and edge perturbations, respectively:


Then, for constant,


where F0(x) = Fb(x/18 − R − 3/2), we are able to prove the following theorem.

Theorem 3.11
(Ground state gap stability for spin chains). Supposeη:Pf(Z)Alocshas LTQO with Ω(n) ≤ nν, for ν > 4, and there exist K > 0, s ∈ (0, 1] such that hΦsatisfies hΦ(r) ≥ Krs. Then there exists ε(γ0) > 0 such that 0 ≤ ε < ε(γ0) and diam(Λ) > max{2D, R} imply
The constant ε(γ0) can be taken as

As a consequence, if we assume that η:Pf(Z)Aloc+ has Z2-LTQO, and Ω and ΦΛ:Pf(Λ)AΛ+ have the same decay assumptions as in Theorem 3.11, we are also able to prove:

Theorem 4.4.
There exist ε′(γ0) > 0 and constantmDsuch that 0 ≤ ε < ε′(γ0) and diam(Λ) > max{2D, R} implies
The constantsmDand ε′(γ0) can be explicitly determined by the constants m, MD, and ε(γ0).

The proofs of Theorems 3.11 and 4.4 rely on a relative form bound argument. We remark that the proof will depend strongly on the fact that the size of the boundary of Λ can be bounded independently of the size of Λ itself. This is special about one-dimensional systems. The stability of the gap in higher dimensions requires a careful analysis of the locality of perturbations11 and more complicated assumptions.

Additionally, due to the relative form bound, the hypotheses for a stable ground state spectral gap also imply general stability of the spectrum. Precisely, we prove the following statement about the persistence of higher spectral gaps. In the statement, J1, J2, J3 refer to Eqs. (3.12) and (3.15).

Proposition 3.12.
Let T, γ > 0, and denoteres(HΛ)=C\sp(HΛ). Suppose η, [Φ] satisfy the hypotheses of Theorem 3.11. There exists ε(γ, T) > 0 such that for sufficiently large Λ and 0 ≤ ε < ε(γ, T), if ν, μ ∈ sp(HΛ) with (ν, μ) ⊂ res(HΛ) ∩ [0, T] and μν > γ, then the gap between ν and μ is stable. Precisely, if we denote
for 0 ≤ ε < ε(γ, T) and p, q defined as

Here, we make the distinction between a perturbation near the boundary and in the bulk. In this section, unless otherwise noted, we fix an interval Λ = [a, b] and let Φ denote the interaction ΦΛ, with a local Hamiltonian ΦΛ = X⊂ΛΦ(X).

Let DN define a uniform distance parameter, and denote by IntD(Λ) the relative interior [a + D, bD]. The piece of the perturbation associated with x ∈ Λ is Φx=n=1RxΦ(bΛ(x,n)), and the whole perturbation is split by the relative interior, ΦΛ=ΦΛD+ΦΛInt, where

ΦΛD=xΛ\IntD(Λ)Φx,  ΦΛInt=xIntD(Λ)Φx

are the edge and bulk perturbations, respectively. Let ΦD(Λ),ΦInt(Λ):Pf(Λ)AΛs denote the corresponding local interactions.

If x ∈ IntD(Λ), then nrx implies ΦbΛ(x,n)ΦFF(D), and so even though the bulk perturbative interaction contains terms which extend to the boundary, their contribution to the total perturbation is relatively small as a function of D.

Since the Hamiltonian HΛ + εΦΛ is close in the operator norm to the bulk-perturbed Hamiltonian, it will suffice to prove ground state spectral gap stability for HΛ+εΦΛInt. To do this, we will use a unitary decomposition method depending on spectral flow. First proved in Ref. 8, our present formulation of the following theorem using F-functions comes from Ref. 18.

Let Ψ:Pf(I)Alocs be an arbitrary interaction, Λ ⊂ I, and suppose γ ∈ (0, γ0). Let εΛ > 0 be such that 0 ≤ εεΛ implies γ(HΛ(ε)) ≥ γ, where HΛ(ε) = HΛ + εΨΛ. We may take εΛ to be maximal. Because γ(HΛ(ε)) is bounded below by γ and εΨΛ is uniformly bounded on [0, εΛ], we may construct the spectral flow (also known as quasi-adiabatic evolution) α:[0,εΛ]AΛs, whose quasi-local properties are extensively discussed in Refs. 1 and 6. Briefly summarizing, there exists a norm-continuous family U(ε) of unitaries such that, if P(ε) denotes the orthogonal projection onto the kernel of HΛ(ε),

αε(A)=U(ε)*AU(ε)and P(ε)=U(ε)P(0)U(ε)*.

The unitaries are the solution to iddεU(ε)=D(ε)U(ε) with U(0)=1, where the generator D(ε) is given by


for a weight function wgammaL1 with compactly supported Fourier transform (see Lemma 2.3 in Ref. 1). Since the quasi-local properties of its generator are made clear by the expression (3.2), the spectral flow automorphism transforms the perturbed Hamiltonian HΛ(ε) into a unitarily equivalent finite-volume Hamiltonian of a well-behaved, local interaction. Identifying this local interaction is the content of the unitary decomposition theorem:

Theorem 3.1.

SupposeΨ:Pf(I)Alocssatisfies a finiteF-norm for F andhΨ(r)Krtfor someK>0and t ∈ (0, 1]. Then for all 0 ≤ ε ≤ εΛ,

  • there exists an interactionΦ1(ε):Pf(Λ)AΛssuch that αε(HΛ(ε)) = HΛ+ Φ1(ε), and

  • Φ1(ε) is supported on the metric balls of Λ, that is,

whereΦx1(ε)=n=1RxΦ1(bΛ(x,n),ε)and eachΦ1(bΛ(x,n),ε)AbΛ(x,n)s. Furthermore, for all x ∈ Λ,[P(0),Φx1(ε)]=0.There exists a constant C > 0, depending on the uniform bound M, range R, uniform gap γ0, and decay parametersKand t, such that
where Fφis anF-function depending onK,t,γsuch that Fφ(r) decays faster than any polynomial in r.

This reformulated statement of the original decomposition theorem found in Ref. 8 is proved in Theorem 6.3.4 in Ref. 18, and so we record here only the precise form of Fφ. Define
μ(r)=(e/κ)κif reκr/(logr)κelse r>eκ.
Define K0=min{K,2/7}, and denote by νΨ the Lieb-Robinson velocity for the Heisenberg dynamics generated by the interaction Ψ. Denote μ̃(r)=μ(Kγr2νΨ) and
Then the F-function in the statement of the theorem is given by
Fφ(r)=GΨ(0)if r18R+27GΨ(r/18R3/2)else r>18R+27.

For the remainder of this section, let U(ε), αε, and Φ1(ε) be from an application of Theorem 3.1 when Ψ is the bulk perturbative interaction ΦInt(Λ) with a local Hamiltonian ΦΛInt.

Lemma 3.2.
The local operator Φ1(ε) can be rewritten as
for terms defined by


This follows from a direct calculation using the fact that [Φx1(ε),P]=0.

The reason for separating the boundary terms R(ε) from the rest of the transformed perturbation is for notational convenience since the following argument will use the fact that ⌊rx/2⌋ > 0 for x ∈ Int2(Λ).

The argument for relative form boundedness of the transformed perturbation Φ1(ε) will depend on the following two elementary lemmas.

Lemma 3.3.
Suppose x ∈ Λ. For any 1 ≤ m ≤ rx,

Denote AωΛ(A) = A0 and bΛ(x, n) = bx(n), for brevity. For 0 ≤ mrx, by linearity of ωΛ,
We bound the two summands separately. The right summand is bounded by Proposition A.1,
The left summand is bounded by local topological quantum order and the F-norm,
Combining these bounds proves the lemma.

The next lemma uses the cutoff function zx defined in Sec. II C, Eq. (2.7).

Lemma 3.4.
Suppose x ∈ Int2(Λ). If 1 ≤ m ≤ rxand m ≤ n ≤ Rx, then

Suppose AAbx(k)s. The C*-identity and LTQO imply
In the case A=k=1mΦ1(bx(k),ε)0, the above bound and Proposition A.1 imply
By Theorem 3.1, Φx1(ε) commutes with P. So, using Lemma 3.3, we get
Proposition 3.6 uses a finite resolution of identity {Enx} defined at each site x ∈ Int2(Λ) by
Enx=1Pbx(1)if n=1Pbx(n1)Pbx(n)if 1<nrxPbx(rx)Pif n=rx+1Pelse n=rx+2.

Lemma 3.5.
The family { E n x } has the properties
1 . k = 1 r x + 2 E k x = 1 a n d k = 1 m E k x = 1 P b x ( m ) i f 1 m r x 1 P i f m = r x + 1 , 2 . P b x ( k ) E k x = 0 f o r k r x .


We only comment that the second property follows from the frustration free assumption on η.

Proposition 3.6.
Let x ∈ Int2(Λ) and 0 ≤ ε ≤ εΛ. There exist local operatorsΘβx(n,ε), for 3 ≤ n ≤ rx, and operatorΘαx(ε)such that
Furthermore,Pbx(n)Θβx(n,ε)=0, andΘβx(n,ε)andΘαx(ε)decay rapidly,

Fix x ∈ (a, b) and ε ∈ [0, εΛ]. Abbreviate Q=1P and Φk1=Φ1(bx(k),ε)0, i.e.,
Define a “cutoff” parameter nx=rx2 and split Φx1(ε)0 into two sums,
The tail μαx=k=nx+1RxQΦk1Q can be bounded above in operator norm by using LTQO, so we turn our attention to the other summand. Denote by Qbx(l) the complement projection 1Pbx(l). Using the resolution {En} at x, we rewrite QΦk1Q for all 1 ≤ knx as
Define the following terms to organize the summands in (3.7):
so that
For convenience, extend τβx(m) to previously undefined m by declaring τβx(m)=0. The derivation of the Θβx(ε,n),Θαx(ε) operators will result from an interchange of order for the summation of terms in (3.3) over n and k. The following definition for Θβx(n,ε) accounts for the parity of rx,
where Θαx(ε)=μαx+k=1nxναx(k). Next, the frustration free property of HΛ implies that ker(Hbx(n))ker(Hbx(n1)), and so
Furthermore, we have the following bounds on operator norm, for all x ∈ Int2(Λ) and 3 ≤ n < rx, by Lemma 3.4 and Proposition A.1,
Now, we define several quantities which will appear in the derivation of the form bound. Note that the weight function ehφ(x) of Fφ is bounded above by 1 on its domain. So any expression in Fφ is bounded above by the corresponding sum using the shifted base F-function
from (2.5) and (3.4). Define
κ(n, ε) does not depend on either Λ or the lower bound γ on the instantaneous gap, and the inequalities from (3.9) are rewritten as
Last, we see by the assumed decay of Ω that the following sums are finite:
The following argument for concluding form boundedness is essentially due to Ref. 8, modified to work with the boundary terms introduced by Proposition 3.6. We divide a large part of the Hamiltonian with respect to a convenient partition of Int2(Λ). For nN, define the relation xny if and only if xy(2n+1)Z. Index each of the parts Λni of Int2(Λ)/∼n by a representative iI(n) ⊂ Int2(Λ). Note that the cardinality of I(n) is roughly bounded above by 3n. The corresponding parts of the Hamiltonian are defined by
By definition of the Θβx(n,ε) operators, Φ2(ε)=n,iΦni. In order to compare Hni to Φni, we use a resolution of identity from Ref. 8, whose properties we record here.

Lemma 3.7.
For a configuration σ : Λ n i { 0,1 } , define the projection S n i ( σ ) = x Λ n i σ x Q b x ( n ) + ( 1 σ x ) P b x ( n ) . Then
1 . σ : Λ n i 0,1 S n i ( σ ) = 1 , 2 . S n i ( σ ) S n i ( σ ) = δ σ , σ S n i ( σ ) , 3 . f o r a l l x Λ n i , [ Θ β x ( n , ε ) , S n i ( σ ) ] = 0 .


These properties follow immediately from the fact that Pbx(n)Θβx(n,ε)=0 and that xny implies bx(n) ∩ by(n) = .

Proposition 3.8.
Suppose diam(Λ) > max{4, R}. There exist constants δ, β > 0, dependent on ∥ΦInt(Λ)Fsuch that 0 ≤ ε ≤ εΛimplies, for allvHΛ,
Precisely, we may choose
δ=J2(ηF+ΦInt(Λ)F)  and  β=3γ0J1(ηF+ΦInt(Λ)F).

Denote dΛ = diam(Λ). For any x ∈ Int2(Λ), if n > rx, say that Θβx(n,ε)=0. Suppose uHΛ. Then by Proposition 3.6,
The second term xInt2(Λ)κ(rx,ε) is bounded above by the constants in (3.12), so we focus on the first summand. Since [Φni,Sni(σ)]=0,
|u,n=3dΛiI(n)Φniu|n=3dΛ|u,iI(n)ΦniσSni(σ)u|n=3dΛiI(n)σ:Λni0,1xΛniSni(σ)Θβx(n,ε)u,Sni(σ)un=3dΛiI(n)κ(n,ε)γ0xΛni  σ:Λni0,1σx=1γ0u,Sni(σ)u=n=3dΛiI(n)κ(n,ε)γ0xΛniu,γ0Qbx(n)un=3dΛ3nκ(n,ε)γ0u,HΛu.

Corollary 3.9.
There exists a constant α, dependent on ∥ΦInt(Λ)F, such that 0 ≤ ε ≤ εΛand diam(Λ) > max{4, R} imply
Precisely, we may takeα=C(ηF+ΦInt(Λ)F)[J3 + 4] + δ.

Suppose x ∈ Int2(Λ). Set m=rx2 in an application of Lemma 3.3 to show
But by the decay of Ω and F0, we have that the following sum is finite:
And, summing over x ∈ Int2(Λ),
Next, it is straightforward to apply Proposition A.1 to R(ε) to get an upper bound on the norm,
Until now, all estimates have been expressed using a local bound ∥ΦInt(Λ)∥F on the strength of the bulk perturbation for fixed Λ. In order to obtain volume independent lower bounds on the spectral gap, we use the following uniform quantity:

Proposition 3.10.
There exist εInt > 0 and constant m > 0 such that 0 ≤ ε < εIntand diam(Λ) > {4, R} imply
The constants εIntand m can be taken as the following expressions:


Let γ ∈ (0, γ0). For fixed Λ with diam(Λ) > max{4, R}, there exists εΛ > 0 such that for all 0 ≤ εεΛ, γ(HΛ+εΦΛInt)γ. By continuity of the eigenvalue functions, we may assume that εΛ is maximal, i.e., either εΛ = 1 or there exists c > 0 such that for all μ ∈ (εΛ, εΛ + c), γ(HΛ+μΦΛInt)<γ.

Since the gap does not close on [0, εΛ], we use the spectral flow decomposition (3.5) to transform HΛ+εΦΛInt by unitaries and a shift in the spectrum,
But by Corollary 3.9, if εεΛ, then Φ(ε)=Φ2(ε)+Φ3(ε)+R(ε) is HΛ-bounded. Now, by the relation P(ε) = U(ε)P(0)U(ε)* in (3.1), the span of the eigenvectors to the 0-group of HΛ + Φ(ε) is exactly ker(HΛ). So, if λ is in the 0-group, which we will denote by sp(0, ε), then there exists a unit norm u ∈ ker(HΛ) such that
Next, define ε1 > 0 as the solution to h(ε) = γ, where h is defined as
Set εγ = min{ε1, 1}. Combining (3.16) and Corollary 3.9, we see that if 0 ≤ ε < min{εγ, εΛ}, then
By maximality, either εΛ = 1 or γ(HΛ+εΛΦΛInt)=γ. Hence εγεΛ necessarily and γ(HΛ+εΦΛInt)h(ε)>γ for all ε < εγ. But now, γ was arbitrarily smaller than γ0. Set
Evidently εInt does not depend on Λ, and if 0 ≤ ε < εInt, then
where the constant
comes from rewriting the lower bound h(ε) as a linear equation of ε.
Denote by MD the following finite uniform bound on the strength of the edge perturbations:
We remark that MInt and m are defined in terms of F-function decay, while MD is defined in terms of the operator norm.

Theorem 3.11
(Ground state gap stability for spin chains). Supposeη:Pf(Z)Alocshas LTQO with Ω(n) ≤ n−ν, for ν > 4, and there exist K > 0, s ∈ (0, 1] such that hΦsatisfies hΦ(r) ≥ Krs. Then there exists ε(γ0) > 0 such that 0 ≤ ε < ε(γ0) and diam(Λ) > max{2D, R} imply
The constant ε(γ0) can be taken as

Considering εΦΛD as a perturbation of H+εΦΛInt, the spectrum of H+εΦΛD+εΦΛInt must be contained in the compact neighborhood,
That is,

Since the stability theorem guarantees a Λ-independent neighborhood of 0 where a relative form bound of the perturbation will hold, we can also conclude the stability of spectral gaps which are located higher in the spectrum.

Proposition 3.12.
Let T, γ > 0, and denoteres(HΛ)=C\sp(HΛ). Suppose η, [Φ] satisfy the hypotheses of Theorem 3.11. There exists ε(γ, T) > 0 such that for sufficiently large Λ and 0 ≤ ε < ε(γ, T), if ν, μ ∈ sp(HΛ) with (ν, μ) ⊂ res(HΛ) ∩[0,T]and μ − ν > γ, then the gap between ν and μ is stable. Precisely, if we denote
for p, q defined by

Let Φ(ε) be defined as in Proposition 3.10, for 0 ≤ ε < εInt. By Proposition 3.9, for all uHΛ,
Let z=ν+μ2 and denote Rζ(ε)=(ζHΛΦ(ε))1, with Rζ = Rζ(0). Let U denote the polar unitary such that Rz = U|Rz|. Since Rz is self-adjoint, |Rz|U* = U|Rz|, and so for unit norm u,
That is, for sufficiently small ε,
and by the expansion
we derive the lower bound
Hence for sufficiently small ε, independently of sufficiently large Λ,

So far, we have studied finite spin chains and shown that, under a set of general assumptions, the group of eigenvalues continuously connected to the ground state energy of a finite frustration-free Hamiltonian remains separated by a gap from the rest of the spectrum, uniformly in the length of the chain and as long as the perturbations are not too large. We now want to show that the states associated with this group of eigenvalues all converge to a ground state of the model in the thermodynamic limit. The lower bound for the gap of finite chains is then also a lower bound for the gap above those ground states of the infinite chain.

For concreteness, we consider Hamiltonians of the form (2.4), where η satisfies the assumption set out in Sec. II B, and [Φ]={ΦΛΛPf(Z)} is a family of perturbations given in terms of interactions Φ,ΦbBF and a few parameters that define the boundary conditions. Specifically, consider intervals ΛZ of the form [−a, b], a, b ≥ 0, and for any D ≥ 0, let IntD(Λ) = [−a + D, bD]. Let denote the triple of parameters (D1, D2, s), D1, D2 ≥ 0, s ∈ [0, 1], and consider


This form of the Hamiltonian covers a broad range of perturbations and boundary conditions. The dynamics generated by HΛ(ϵ) is the one-parameter group of automorphism τtHΛ(ϵ).

As explained in Subsection 2 of the  Appendix, if we take, for example, Λn = [−an, bn], sn ∈ [0, 1] arbitrary, and D1,n, D2,n such that min(an, bn) − max(D1,n, D2,n) → , then there is a strongly continuous group of automorphisms τtϵ,tR on AZ such that

limnτtHΛnn(ϵ)(A)τtϵ(A)=0,for all AAZloc.

If we take ϵ ∈ [0, ϵ(γ0)), with ϵ(γ0) as in Theorem 3.11, we have a uniform gap separating the lower portion of the spectrum of HΛnn(ϵ), denoted by sp0,Λn(ε) in (2.8), and the rest of the spectrum. The following results provide an estimate of diam(sp0,Λn(ε)). For simplicity, let Λn = [−n, n] for the remainder of the section.

Lemma 3.13.
In the assumptions of above, choose sn= 0 and put D1,n= Dn. Then, there exists a functionG:[0,)[0,)which decreases to 0 as n tends to infinity and, for large enough n,
Precisely, we may take
whereF̃is anF-function depending on ∥η∥Fand MInt.

Suppose n is sufficiently large so that 12Dn/2>R, the range of the interaction η. By the spectral flow decomposition in (3.5),
for A, B defined by complementary regions of the interval Λn = [−n, n],
By applying LTQO and F-norm bounds,
where F0 is the shifted base F-function from (3.10). For the norm bound on B, let ΔX(k) denote the partial trace difference operators from the Proof of Theorem 3.1 (c.f. Theorem 6.3.4 in Ref. 18), defined with respect to an enlargement of X ⊂ Λn. Suppose − nx < − n + ⌊Dn/2⌋or n −⌊Dn/2⌋< xn. Denote dx(n)=d(x,IntDn(Λn)). By the locality assumption on ΦΛn and the fact that dx(n)/2 > R, if k ≤ ⌊dx(n)/2⌋, then, in the notation of the Proof of Theorem 3.1,
and so
Using the quasi-locality of the generator iD(ε) of the spectral flow unitaries,
and there exists an F-function F̃, independent of Λn, such that
Let G(r)=8k=r/2F̃(k/2)+C(MInt+ηF)[Ω(k/2)+F0(k/2)]. Then
Let Pn(ε) denote the spectral projection of HΛnn(ϵ) associated with the isolated portion of the spectrum sp0,Λn(ε) and define the set of states of AΛns, Sn(ε), with support in the range of Pn(ε),
Sn(ε)={ωωis a state on AΛns with ω(Pn(ε))=1}.
We now consider the thermodynamic limits of these states,
S(ε)={ωstate on AZs(nk)increasing and ωkSnk(ε)s.t.limkωk(A)=ω(A),AAZloc}.

Lemma 3.14.

Let c n ( ε ) = d i a m ( sp 0 , Λ n ( ε ) ) . Then

  • for all ω S n ( ε ) and A A Λ n s , we have

R e ω ( A * [ H Λ n n ( ϵ ) , A ] ) c n ( ε ) A 2 , a n d    I m ω ( A * [ H Λ n n ( ϵ ) , A ] ) c n ( ε ) A 2 .
  • (ii)

    If sn= 0 and D1,nis such that limn[n − D1,n] = limnD1,n= ∞, then, for allωS(ε)andAAZloc, we have

lim n ω ( A * [ H Λ n n ( ϵ ) , A ] ) 0 .


The proof of (i) is elementary and the proof of (ii) follows by noting that the additional assumptions imply that the sequence [HΛnn(ϵ),A] converges in norm and that lim cn(ε) = 0 by Lemma 3.13.

In other words, the conditions of part (ii) of the lemma imply that the states in Sn(ϵ) converge to ground states of the infinite system. In Subsection 2 of the  Appendix, it is explained that the spectral flow automorphisms, like the time evolution of the system, converge to the same limit regardless of the choice of boundary condition n. Since we have the relation Pn(0)=αεΛn,n(Pn(ε)), we also have
and as an easy consequence of the convergence (see Ref. 1, Lemma 5.6), we then also have

Since the same αϵ relates limiting states regardless of the boundary conditions, for example, with a constant sequence n = , for any n, these limiting states must be the same and, hence, also ground states of the infinite systems defined by the dynamics τt. The same conclusion then holds for the lower bound on the spectral gap above these ground states (see Ref. 10 for the details).

Suppose AΛ is a local algebra of observables which is *-isomorphic to AΛs. Let ϕ:AΛAΛs denote a possible *-isomorphism. Given a local Hamiltonian HΛ in AΛ, ϕ unitarily transforms HΛ into a Hamiltonian HΛs=ϕ(HΛ) of the spin algebra. Using an exhaustive family of conditional expectations {θXi:XiXi+1}, HΛs can again be realized as the sum of local operators through a telescoping sum,


The Proof of Theorem 3.1 uses this method of decomposition in the setting where ϕ is a quasi-local *-automorphism, and the θXj are normalized partial trace over increasing metric balls Xj = bx(j). The quasi-locality property, defined below, guarantees that the transformed local interaction will have decay comparable to that of the original interaction.

In this section, we prove the stability of the spectral gap for even Hamiltonians in the CAR algebra of fermions satisfying Z2-LTQO. To do this, we will use the Jordan-Wigner isomorphism to transform even fermion interactions into spin interactions in a way that respects the parity symmetry.

Let ΛPf(Z) be a nonempty interval. A linear map α:AΛsAΛs is quasi-local if there exist constants C>0,pN and a decay function g: [0, ) → [0, ) such that if X, Y ⊂ Λ are disjoint subsets, then for all AAXs and BAYs, the following bounds hold:
α(A)C|X|pA  [α(A),B]CAB|X|pg(d(X,Y)).

The local Heisenberg dynamics τΛ:URAut(AΛs) generated by an interaction Ψ with a finite F-norm is a collection of quasi-local maps parametrized by t. Let F be an F-function such that ∥Ψ∥F < , and denote by νΨ the Lieb-Robinson velocity. There exists a constant CΨ > 0 such that for X, YPf(Λ) disjoint sets and AAXs,BAYs, the following Lieb-Robinson bound holds:
But by properties of the F-function,
So take Ct=CΨ(eνΨ|t|1), pt = 1, and
In particular, the spectral flow automorphism αΛ:[0,εΛ]Aut(AΛs) is quasi-local.1 
Last, we specify the normalized partial trace maps. Let
denote an enlargement of XPf(Λ). Denote the normalized partial trace of the state space over Λ∖X(n) by
For convention, we will take the trace over H as the identity map. Then define, for all AAΛs,
ΔX(0)(A)=θX(0)(A),  ΔX(n)(A)=θX(n)(A)θX(n1)(A).

Recall, we denote by AΛ+AΛf the even operators of the CAR algebra over Λ. We say that βAut(AΛ) is even if it preserves the parity. Even interactions are defined similarly. We also denote S±=12(σ1±iσ2). The following definition is the well-known Jordan-Wigner transformation, which gives a C*-isomorphism of CAR and spin algebras.

Consider the case A{x}s=M2(C). Let 𝜗Λ:AΛfAΛs denote the Jordan-Wigner map defined by
The Jordan-Wigner transformation extends the notion of parity to the spin 1/2 algebra. We say that AAΛs is even if 𝜗Λ1(A)AΛ+.

Proposition 4.1.
Let X ⊂ Λ be any subinterval.
1.If  AAX+,then  𝜗Λ(A)AXs.2.If  α:AΛsAΛs  is  an  even  quasilocal  map,then  ΔX(n)α  is  also  even.

Suppose A is a monomial ca#(x1)⋯a#(x2n). By the CAR, we may assume xjxj+1. A direct computation shows that the first part of the lemma holds for the even monomials which generate AX+,
Next, we show that the partial trace is an even map. For any x ∈ Λ, define the following four unitary operators:
Now, let Z ⊂ Λ and BCAZsAΛ\Zs. Denote by IΛ∖Z the set of finite sequences ι: Λ∖Z → {0, 1, 2, 3}. Define
Using elementary properties of trace and locality in the spin algebra,
The relation in (4.2) uniquely defines the partial trace; hence,
The second part of the lemma follows from this formula.

In the following, we will assume the interactions are supported on intervals:


An interaction Φ is supported on intervals if Φ(X) ≠ 0 only if X = [a, b] for some a,bZ.

Any interaction can be “regrouped” into one with interval support, and while the methods to do this are neither new nor canonical, we record here a simple way without changing the local Hamiltonians, at the expense of rate of decay.

Proposition 4.2.
SupposeIZis an interval andΨ:Pf(I)Alocis an interaction. Then there exists an interactionΦ:Pf(I)Aloc, supported on intervals, such that for all finite intervals Λ ⊂ I, the local Hamiltonians are equal,
If Ψ is an unperturbed interaction with uniform bound M, range R, and local gap γ0, then so is Φ, with uniform bound 2RM and the same range and local gap.
Furthermore, if ∥Ψ∥F< ∞, where F is theF-function in (A2), and h(r) ≥ Krsfor some K > 0 and s ∈ (0, 1], then ∥Φ∥G ≤ ∥Ψ∥Ffor theF-function defined by

If IZ, then we may extend Ψ to Z by Ψ(Z) = 0 for ZI, and by construction, Φ defined in terms of the extension will restrict to an interaction on I. So we may assume I=Z. We will define Φ by induction on the diameter n of intervals [k, k + n]. When n = 0, 1, define
Φ(x)=Ψ(x)and Φ(x,x+1)=Ψ(x,x+1).
For larger n, define
By construction, ΦΛ = ΨΛ. Now, suppose Φ is an unperturbed interaction with constants M, R, γ0. Since ΦbΛ(x,n)=ΨbΛ(x,n) for all x and n, Φ and Ψ have the same local gap. Similarly, it is clear that Φ and Ψ have the same range R, and if diam([a, b]) ≤ R,
Now, suppose Φ is some interaction, not necessarily finite range, with ∥Φ∥F. For fixed kZ and n ≥ 0, by Proposition A.1,
So for x,yZ,
That is,

Proposition 4.3.
SupposeΨ:Pf(I)Alocfis an even interaction supported on intervals. Then there exists an even interactionΦ:Pf(I)Alocssuch that for any Λ ⊂ I,
If Ψ satisfies a finiteF-norm for some F of the form (A2), then so does Φ. If Ψ is an unperturbed interaction, then so is Φ for the same constants.

For Λ0 ⊂ Λ, let ιΛ0,Λ denote the inclusion AΛ0fAΛf. If AAΛ0+, then by expanding in an even generating set of monomials, we see
So there exists an injective *-morphism 𝜗:AΛ+Alocs which extends every ϑΛ, from which we define Φ(X) = ϑ (Ψ(X)). By Proposition 4.1, this is a well-defined interaction which is also supported on intervals. Evidently Φ is an even interaction; i.e., ϑ−1(Φ(X)) is even for any X.
ϑ is isometric, and for the F-function F,
Now suppose Ψ is an unperturbed interaction. Then evidently Φ is uniformly bounded. Φ is frustration free and uniformly locally gapped since, for any Λ, there exists a unitary QΛ:HΛFΛ such that for AAΛf,
Since ϑ is an isometry which preserves support for even observables, and QΛ is unitary, Φ has the same uniform bound, range, and local gap as Ψ.

Theorem 4.4
(Ground state gap stability for fermion chains). There existεγ0>0and constantmDsuch that0ε<εγ0and diam(Λ) > max{2D, R} imply
The constantsmDandεγ0can be explicitly determined by the expressions in (3.17).


By Proposition 4.2, we assume that η and ΦΛ = Φ are supported on intervals. Proposition 4.3 implies the existence of spin interactions ηS and ΦS with the same uniform bound, range, local gap γ0, and decay.

Let γ ∈ (0, γ0) and DN be a chosen distance from the boundary, uniform in the volume, and consider fixed Λ with sufficiently large diameter. By Theorem 3.1, the spectral flow decomposes the local Hamiltonian HΛ(ε) of ηS + εΦS,

Since ϑΛ is implemented by some unitary, ηS has Z2-LTQO for the same decay function Ω. So to apply the norm boundedness argument in Sec. III, it suffices to argue that Φ1(bΛ(x, n), ε) is even.

But the Proof of Theorem 3.1 in Ref. 18 guarantees the existence of even interactions Ψi:Pf(Λ)AΛs, i = 1, 2, 3, and quasi-local maps Ki(ε):AΛsAΛs such that
The Ki(ε) are defined in terms of the spectral flow automorphism and are also even maps. Hence, by Lemma 4.1, Φ1(bΛ(x, n), ε) must also be even since the even observables form a subalgebra.

Here we describe an example of an interaction of the CAR algebra which satisfies the stability hypotheses of Theorem 3.11. Let χ={fi:iB} and Y={gj:jB} be two collections of vectors in 2(Z) such that

  • χY is an orthonormal basis for 2(Z);

  • there exist R ≥ 0 and collections {xi:iB}, {yj:jB} such that for all i, j,

supp(fi)b(xi,R), supp(gj)b(yj,R),ijimpliesb(xi,R)b(xj,R)==b(yi,R)b(yj,R).

Furthermore, denote χW={fi:supp(fi)W} and YW={gj:supp(gj)W}. We will also assume the following:

  • (iii)

    There exists a diameter N0 such that for all intervals Λ, diam(Λ) > N0 implies χΛ and YΛ.

Let η:Pf(Z)Alocf be the finite-range interaction defined by

Lemma 5.1.

Suppose Λ is an interval such that diam(Λ) > N0. Then HΛis non-negative, uniformly gapped, and frustration free.

Let (fn1,,fnΛ) and (gm1,,gmΛ) be the collections of vectors whose support is contained in Λ. If necessary, complete the list to an orthonormal basis of 2(Λ) with (h1, …, hp), p = |Λ|− nΛmΛ. Evidently HΛ is uniformly gapped and non-negative. So we prove that
where we define
ϕΛ=fn1fnΛ,   ζX=hik:ikX[1,p],   ψX=ϕΛζX.
By calculation, ψX ∈ ker(HΛ) for any X ⊂ [1, p]. But each term of the interaction HΛ is a projection, the complement projection of some a*(fq)a(fq). So HΛψ = 0 implies ψ ∈ ran(a*(fi)a(fi)) for each i = n1, …, nΛ.

Next, we show that the number of auxiliary orthonormal basis vectors hi needed to complete χΛ and YΛ to a basis of 2(Λ) is uniformly bounded in Λ, and that each hi has support contained toward the edge of Λ.

Lemma 5.2.

Suppose diam(Λ) > N0. LetZ(Λ)={h1(Λ),,hn(Λ)}, n = n(Λ), be a basis for the complement ofspan(χΛYΛ)in ℓ2(Λ). Then

  • for each i ∈ [1, n] ,supp(hi(Λ))Λ\Int3R(Λ);

  • | Z ( Λ ) | 6 R

Let (ξk) denote the orthonormal basis from χY. Suppose supp(f) ⊂ Int3R(Λ). Then xi ∉ Λ implies supp(fi) ∩ supp(f) = ∅, that is, ⟨fi, f⟩ = 0 (respectively, yj and ⟨gj, f⟩ = 0). Hence
Hence fspan(χΛYΛ). Now, a basis of the orthogonal complement of 2(Int3R(Λ)) in 2(Λ) is necessarily supported on Λ ∖Int3R(Λ), proving (1). Additionally, the dimension of 2(Λ ∖Int3R(Λ)) is an upper bound for |Z(Λ)|, which proves (2).

This lemma has an immediate corollary:

Corollary 5.3.
LetA(W)denote the C*-subalgebra ofAZfgenerated by the operators a*(f), a(f) such thatfW2(W). Then for all intervals Λ with diameter larger than 6R,

To conclude this section, we prove that the interaction defined in (5.1) satisfies Z2-LTQO. Denote D = max{N0, 3R}. Recall that if nD then HbΛ(x,n) is non-negative and frustration free with kernel indexed by Z(bΛ(x,n)).

Define the step-function Ω: [0, ) → [0, ) by
Ω(x)=0if xD2otherwise.

Proposition 5.4.
Suppose diam(Λ) > 2D, and let x ∈ Λ, and(n,k)N2be such that 0 ≤ k ≤ rx, k ≤ n ≤ Rx. Let Pndenote the projection ontoHbΛ(x,n). Then for allAAbΛ(x,k)+,


We handle the two cases of n when diam(bΛ(x, n)) ≥ N0 or diam(bΛ(x, n)) < N0. Suppose the former. Now, there are two subcases for k: either bΛ(x, k) ⊄ IntD(bΛ(x, n)) or bΛ(x, k) is contained in that interior.

Suppose bΛ(x, k) ⊂ IntD(bΛ(x, n)). Then zx(n) − kD, necessarily. Denote
χbΛ(x,n)=χn=fi1,fiM,   Z(bΛ(x,n))=Z(n)=h1,,hp
Let ψXn=fi1fiMhn1hn|X| be a generic unit norm basis vector of the kernel, indexed by XZ(n). A calculation shows
But by the theory of quasi-free states and Corollary 5.3,
Now suppose bΛ(x, k) is not contained in the D-interior of bΛ(x, n). This implies zx(n) − k < D. And by the trivial commutator bound,
Last, suppose diam(bΛ(x, n)) < N0. Then nkn < N0D. Hence zx(n) − k < D as well, and the trivial bound agrees with Ω. Conclude that HΛ satisfies LTQO for Ω.

Based upon work supported by the National Science Foundation under Grant Nos. DMS-1207995 (A.M.), DMS-1515850 (A.M. and B.N.), and DMS-1813149 (B.N.). We would like to thank the referee for helpful comments and suggestions.

1. F-functions and decay of interactions

In addition to LTQO, a critical assumption for our spectral gap stability argument is rapid decay of the perturbations in [Φ]. We choose to describe this decay through F-functions, which have several useful properties, one of which define an extended norm on the real vector space of interactions.


A function F: [0, ) → (0, ) is an F-function for(Z,||) if

  • F=xZF(x)<;

  • there exists CF > 0 such that for all x,yZ,

Furthermore, if Φ is an interaction, then the F-norm of Φ (with respect to F) is defined as

Suppose h: [0, ) → [0, ) is a monotone increasing, subadditive function and κ > 2. The following function F defines an F-function,
The F-function in (A2) and following properties will be used extensively in the proof of spectral gap stability.

Proposition A.1.
Suppose Φ is an interaction with finiteF-norm for some F. Then
1.foranycollectionZ1Z2ZN,k=1NΦ(Zk)ΦFF(diam(Z1));2.ifηisauniformlybounded,finiterangeinteraction,thenηF<   andη+ΦF<.

Let diam(Z1) = n, and choose x, yZ1 such that |xy| = n. Then
Now, denote the range of η by R and uniform bound by M. Suppose x,yZ. If ZPf(Z) contains x, y, and Φ(Z) ≠ 0, then Zb(x, R) ∩ b(y, R). Hence
Then ∥η + Φ∥F < by the triangle inequality.

2. Thermodynamic limit of the spectral flow

There are standard results giving conditions on an interaction Φ under which the finite-volume dynamics defined by






converges to a strongly continuous co-cycle of automorphism, τs,tΦ, of the algebra of quasi-local observables AΓ, which is defined as the norm completion of the (strictly) local observables given by


One sufficient condition is that Φ(⋅, t) is a continuous curve taking values in the space BF, where F defines an F-norm, ∥⋅∥F on interactions as in Subsection 1 of the  Appendix. For any compact interval I, we can define the space BF(I) as the set of all such continuous curves, and for ΦBF(I), the function


is continuous and bounded. Strong convergence of the automorphisms here means that

limΛΓτs,tΦ,Λ(A)τs,tΦ(A)=0,for all AAΓloc.

The limit is taken over any sequence of ΛnP(Γ) increasing to Γ and the limiting dynamics is independent of the choice of sequence.

One can also show that the dynamics depends continuously on the interaction Φ in the following sense:9 


which holds for all AAX and s, tI, and where, for ΦBF(I), and s, tI, the quantity It,s(Φ) is defined by


It is often important to include in the definition of the finite-volume Hamiltonian HΛ terms that correspond to a particular boundary condition. Such terms affect the ground states and equilibrium states of the system, including in the thermodynamic limit but, in general, do not affect the infinite-volume dynamics. In order to express this freedom in the interactions defining the finite-volume dynamics that lead to the same thermodynamic limit, we use another, weaker, notion of convergence of interactions interaction Φ introduced in Ref. 9, where it is called local convergence in F-norm.


Let (Γ,d,AΓ) be a quantum lattice system, F be an F-function for (Γ, d), and IR be an interval. We say that a sequence of interactions {Φn}n1converges locally in F-norm to Φ such that

  • ΦnBF(I) for all n ≥ 1,

  • ΦBF(I), and

  • for any ΛP0(Γ) and each [a, b] ⊂ I, one has

lim n a b ( Φ n Φ ) Λ F ( t ) d t = 0 .

In this appendix, we want to apply this notion to the spectral flow generated by perturbations of the form (2.4) and its thermodynamic limit.

The spectral flow αϵΛ, for the curve of Hamiltonians HΛ(ϵ),ϵ[0,ϵ0), defined in (3.20), also depends on a parameter γ > 0. This parameter is assumed to be a lower bound for the gap of interest in the spectrum of HΛ(ϵ) in the stability argument, but this assumption is not needed for the construction of αϵΛ,. The automorphisms αϵΛ, are generated by the self-adjoint operators DΛ(ϵ), defined by
where the map KϵΛ,:AΛAΛ is given by
Note that KϵΛ, is defined as a linear map, but it depends itself on HΛ(ϵ) and therefore DΛ(ϵ) depends non-linearly on the perturbation. In Ref. 9 [Sec. V D], a detailed study of transformations of the form KϵΛ, is performed. The following proposition follows directly from applying the more general results in that work to the situation here.

Proposition A.2
(Ref. 9). There exists an F-functionF̃of the form (A2) and interactionsΨΛ,BF̃([0,ϵ0])such that
Furthermore, there exists an interactionΨBF̃([0,ϵ0])such that for any sequencesΛn=[an,bn]Z, ∂n= (D1,n, D2,n, sn), the interactionsΨΛn,nhave uniformly boundedF̃-norm and converge locally inF̃-norm to Ψ.

As a consequence, we can apply the following theorem from Ref. 9 to the sequences of interactions ΨΛn,n.

Theorem A.3
(Ref. 9, Theorem 3.8). Let(Φn)n1be a sequence of time-dependent interactions on Γ with Φnconverging locally in the F-norm to Φ with respect to F. Suppose that for every [a, b] ⊂ I,
Then for anyXP0(Γ),
for allAAXand each s, t ∈ I. Moreover, the convergence is uniform for s, t in compact intervals.
Now, consider a sequence Λn=[an,bn]Z, n = (D1,n, D2,n, sn). As the result of applying Proposition A.2 and Theorem A.3, we obtain the strong convergence of the finite-volume spectral flow automorphism generated by ΨΛn,n to one and the same spectral flow for the infinite chain: for ϵ ∈ [0, 1],
limnαϵΛn,n(A)=αϵ(A),for all AAΓloc.

, and
, “
Automorphic equivalence within gapped phases of quantum lattice systems
Commun. Math. Phys.
, and
, “
Topological quantum order: Stability under local perturbations
J. Math. Phys.
, and
, “
Low-temperature phase diagrams of quantum lattice systems. I. stability for quantum perturbations of classical systems with finitely-many ground states
J. Stat. Phys.
de Roeck
, “
Persistence of exponential decay and spectral gaps for interacting fermions
Commun. Math. Phys.
(published online,
); e-print arXiv:1712.00977 (
M. B.
, “
The stability of free Fermi Hamiltonians
,” e-print arXiv:1706.02270 (
M. B.
X. G.
, “
Quasi-adiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance
Phys. Rev. B
, “
Hidden symmetry breaking and the Haldane phase in s = 1 quantum spin chains
Commun. Math. Phys.
J. P.
, “
Stability of frustration-free Hamiltonians
Commun. Math. Phys.
, and
, “
Quasi-locality bounds for quantum lattice systems and perturbations of gapped ground states. Part I
” (unpublished).
, and
, “
Quasi-locality bounds for quantum lattice systems and perturbations of gapped ground states. Part II
” (unpublished).
, and
, “
Stability of gapped phases of fermionic lattice systems
” (unpublished).
, and
, “
Lieb-Robinson bounds, the spectral flow, and stability for lattice fermion systems
,” in
Proceedings of the conference QMATH13
Contemporary Mathematics, American Mathematical Society
, “
A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization. I
Commun. Math. Phys.
), MR 3555356.
, “
A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization. II
Commun. Math. Phys.
), MR 3555357.
, “
A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization. III
Commun. Math. Phys.
M. M.
, “
Perturbation theory for parent Hamiltonians of matrix product states
J. Stat. Phys.
); e-print arXiv:1402.4175.
D. A.
, “
Ground states in relatively bounded quantum perturbations of classical lattice systems
Commun. Math. Phys.
, “
Spectral properties of multi-dimensional quantum spin systems
,” Ph.D. thesis,
University of California