We investigate the persistence of spectral gaps of onedimensional 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
I. INTRODUCTION
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 fulldimensional 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 socalled “open” boundary conditions, meaning that we consider systems defined on intervals $[a,b]\u2282Z$ 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 JordanWigner 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}
II. SETTING AND MAIN RESULT
A. Notations
Denote by $(Z,\u2009\u22c5\u2009)$ the metric graph of integers. Let P_{f}(X) denote the finite subsets of $X\u2282Z$. We will use Λ to refer exclusively to nonempty, finite intervals of the form $[a,b]={n\u2208N:a\u2264n\u2264b}$. Let b_{Λ}(x, n) = {m ∈ Λ: x − m ≤ n} denote the restriction of a metric ball to the interval. For each x ∈ Λ, denote by r_{x} and R_{x} the following distances to the boundary:
Although r_{x} and R_{x} 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 onedimensional 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 $A\Lambda $. If we want to specify that we are specifically considering spins or fermions, we will use the notation $A\Lambda s$ or $A\Lambda f$, 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, $A\Lambda f$ 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 $F\Lambda =F(\u21132(\Lambda ))$. The dimension of $F\Lambda $ is 2^{Λ} and $A\Lambda f$ is *isomorphic to the matrix algebra $M2\Lambda (C)$.
Given an exhaustive net of CAR or spin algebras ${A\Lambda :\Lambda \u2208Pf(Z)}$, the inductive limit $AZ$, the d^{∞} UHF algebra, is obtained by norm completion,
This algebra is often referred to as the quasilocal algebra, and $Aloc=\u22c3A\Lambda $ is referred as the local algebra.
Define by N_{X} = ∑_{x∈X}a^{*}(x)a(x) the number operator for $X\u2208Pf(Z)$, and define the parity automorphism by
Say that $A\u2208A\Lambda f$ is even if ρ_{Λ}(A) = A and odd if ρ_{Λ}(A) = −A. The observable A is even if and only if it commutes with the local symmetry operator exp(iπN_{Λ}), 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 $A\Lambda f$, which we denote by $A\Lambda +$.
B. Assumptions
Let I be a subinterval of $Z$, not necessarily finite. An interaction on I is a function $\Phi :Pf(I)\u2192Aloc$ such that $\Phi (X)=\Phi (X)*\u2208AX$ for all X ∈ P_{f}(I). The corresponding local Hamiltonian of the finite system on Λ ⊂ I is H_{Λ} = ∑_{X⊂Λ}Φ(X). Say that Φ is nonnegative if Φ(X) ≥ 0 for all X ∈ P_{f}(I). Say that Φ is an even interaction of the CAR algebra if $\Phi (X)\u2208AX+$.
The interactions in our perturbative setup will satisfy the following assumptions. First, let $\eta :Pf(Z)\u2192Aloc$ be a nonnegative interaction with distinguished local Hamiltonians H_{Λ}. 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 $X\u2208Pf(Z)$, ∥η(X)∥ < M.
Frustration free: For all intervals $\Lambda \u2208Pf(Z)$, ker(H_{Λ}) ⊋ {0}.
Uniformly locally gapped: There exists γ_{0} > 0 such that for all intervals $[a,b]\u2208Pf(Z)$, with b − a ≥ R, γ_{0} is lower bound a for nonzero 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 $\Phi \Lambda :Pf(\Lambda )\u2192Aloc$ 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 Φ^{Λ} in [Φ] is that Φ^{Λ}(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 $\tau :[0,\u221e)\u2192R$, and so we will take as convention F◦τ(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)=e\u2212h\Phi (r)L(1+cx)\kappa $, for L, c > 0 and κ > 2, such that sup_{Λ}∥Φ^{Λ}∥_{F} < ∞.
Metric ball support: For all Λ, Φ^{Λ}(X) ≠ 0 implies X = b_{Λ}(z, n) for some z ∈ Λ and $n\u2208N$.
The assumption that Φ^{Λ} is supported on metric balls is not restrictive since a finitevolume Hamiltonian of any fastdecaying interaction can be rewritten as the finitevolume Hamiltonian of a balled interaction with comparable decay (c.f. the Appendix of Ref. 18).
C. Local topological quantum order
Consider the unperturbed interaction η and its local Hamiltonians. Denote by P_{X} the orthogonal projection onto ker(H_{X}), and define the state
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).
D. The main result
For any finite interval Λ, we consider the local Hamiltonian H_{Λ}(ε) given in (2.4). There exist continuous functions $\lambda 1,\u2026,\lambda N:[0,1]\u2192R$ such that for all ε ∈ [0, 1], {λ_{1}(ε), …, λ_{N}(ε)} are the eigenvalues of H_{Λ}(ε). We partition sp(H_{Λ}(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,
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 Φ^{Λ} as the sum of an interaction Φ^{D}(Λ), with a local Hamiltonian $\Phi \Lambda D$ supported at the Dboundary, and a bulk interaction Φ^{Int}(Λ). Define the following two finite constants quantifying the strength of the bulk and edge perturbations, respectively:
Then, for constant,
where F_{0}(x) = F^{b}(x/18 − R − 3/2), we are able to prove the following theorem.
As a consequence, if we assume that $\eta :Pf(Z)\u2192Aloc+$ has $Z2$LTQO, and Ω and $\Phi \Lambda :Pf(\Lambda )\u2192A\Lambda +$ have the same decay assumptions as in Theorem 3.11, we are also able to prove:
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 onedimensional systems. The stability of the gap in higher dimensions requires a careful analysis of the locality of perturbations^{11} 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, J_{1}, J_{2}, J_{3} refer to Eqs. (3.12) and (3.15).
III. STABILITY OF SPECTRAL GAP IN SPIN CHAINS
A. Perturbations at the boundary
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 Φ^{Λ}, with a local Hamiltonian Φ_{Λ} = ∑_{X⊂Λ}Φ(X).
Let $D\u2208N$ define a uniform distance parameter, and denote by Int_{D}(Λ) the relative interior [a + D, b − D]. The piece of the perturbation associated with x ∈ Λ is $\Phi x=\u2211n=1Rx\Phi (b\Lambda (x,n))$, and the whole perturbation is split by the relative interior, $\Phi \Lambda =\Phi \Lambda D+\Phi \Lambda Int$, where
are the edge and bulk perturbations, respectively. Let $\Phi D(\Lambda ),\Phi Int(\Lambda ):Pf(\Lambda )\u2192A\Lambda s$ denote the corresponding local interactions.
If x ∈ Int_{D}(Λ), then n ≥ r_{x} implies $\Vert \Phi b\Lambda (x,n)\Vert \u2264\Vert \Phi \Vert FF(D)$, 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 H_{Λ} + εΦ_{Λ} is close in the operator norm to the bulkperturbed Hamiltonian, it will suffice to prove ground state spectral gap stability for $H\Lambda +\epsilon \Phi \Lambda Int$. 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.
B. Spectral flow decomposition
Let $\Psi :Pf(I)\u2192Alocs$ be an arbitrary interaction, Λ ⊂ I, and suppose γ ∈ (0, γ_{0}). Let ε_{Λ} > 0 be such that 0 ≤ ε ≤ ε_{Λ} implies γ(H_{Λ}(ε)) ≥ γ, where H_{Λ}(ε) = H_{Λ} + εΨ_{Λ}. We may take ε_{Λ} to be maximal. Because γ(H_{Λ}(ε)) is bounded below by γ and εΨ_{Λ} is uniformly bounded on [0, ε_{Λ}], we may construct the spectral flow (also known as quasiadiabatic evolution) $\alpha :[0,\epsilon \Lambda ]\u2192A\Lambda s$, whose quasilocal properties are extensively discussed in Refs. 1 and 6. Briefly summarizing, there exists a normcontinuous family U(ε) of unitaries such that, if P(ε) denotes the orthogonal projection onto the kernel of H_{Λ}(ε),
The unitaries are the solution to $\u2212idd\epsilon U(\epsilon )=D(\epsilon )U(\epsilon )$ with $U(0)=1$, where the generator D(ε) is given by
for a weight function $wgamma$ ∈ L^{1} with compactly supported Fourier transform (see Lemma 2.3 in Ref. 1). Since the quasilocal properties of its generator are made clear by the expression (3.2), the spectral flow automorphism transforms the perturbed Hamiltonian H_{Λ}(ε) into a unitarily equivalent finitevolume Hamiltonian of a wellbehaved, local interaction. Identifying this local interaction is the content of the unitary decomposition theorem:
Suppose $\Psi :Pf(I)\u2192Alocs$ satisfies a finite $F$norm for F and $h\Psi (r)\u2265Krt$ for some $K>0$ and t ∈ (0, 1]. Then for all 0 ≤ ε ≤ ε_{Λ},
there exists an interaction $\Phi 1(\epsilon ):Pf(\Lambda )\u2192A\Lambda s$ such that α_{ε}(H_{Λ}(ε)) = H_{Λ} + Φ^{1}(ε), and
Φ^{1}(ε) is supported on the metric balls of Λ, that is,
For the remainder of this section, let U(ε), α_{ε}, and Φ^{1}(ε) be from an application of Theorem 3.1 when Ψ is the bulk perturbative interaction Φ^{Int}(Λ) with a local Hamiltonian $\Phi \Lambda Int$.
This follows from a direct calculation using the fact that $[\Phi x1(\epsilon ),P]=0$.
The reason for separating the boundary terms $R(\epsilon )$ from the rest of the transformed perturbation is for notational convenience since the following argument will use the fact that ⌊r_{x}/2⌋ > 0 for x ∈ Int_{2}(Λ).
C. Relative form boundedness of perturbations
The argument for relative form boundedness of the transformed perturbation Φ^{1}(ε) will depend on the following two elementary lemmas.
The next lemma uses the cutoff function z_{x} defined in Sec. II C, Eq. (2.7).
We only comment that the second property follows from the frustration free assumption on η.
These properties follow immediately from the fact that $Pbx(n)\Theta \beta x(n,\epsilon )=0$ and that x ∼_{n} y implies b_{x}(n) ∩ b_{y}(n) = $\u2205$.
Let γ ∈ (0, γ_{0}). For fixed Λ with diam(Λ) > max{4, R}, there exists ε_{Λ} > 0 such that for all 0 ≤ ε ≤ ε_{Λ}, $\gamma (H\Lambda +\epsilon \Phi \Lambda Int)\u2265\gamma $. By continuity of the eigenvalue functions, we may assume that ε_{Λ} is maximal, i.e., either ε_{Λ} = 1 or there exists c > 0 such that for all μ ∈ (ε_{Λ}, ε_{Λ} + c), $\gamma (H\Lambda +\mu \Phi \Lambda Int)<\gamma $.
Since the stability theorem guarantees a Λindependent 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.
D. The thermodynamic limit
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 frustrationfree 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 $[\Phi ]={\Phi \Lambda \u2223\Lambda \u2208Pf(Z)}$ is a family of perturbations given in terms of interactions $\Phi ,\Phi b\u2208BF$ and a few parameters that define the boundary conditions. Specifically, consider intervals $\Lambda \u2282Z$ of the form [−a, b], a, b ≥ 0, and for any D ≥ 0, let Int_{D}(Λ) = [−a + D, b − D]. Let ∂ denote the triple of parameters (D_{1}, D_{2}, s), D_{1}, D_{2} ≥ 0, s ∈ [0, 1], and consider
This form of the Hamiltonian covers a broad range of perturbations and boundary conditions. The dynamics generated by $H\Lambda \u2202(\u03f5)$ is the oneparameter group of automorphism $\tau tH\Lambda \u2202(\u03f5)$.
As explained in Subsection 2 of the Appendix, if we take, for example, Λ_{n} = [−a_{n}, b_{n}], s_{n} ∈ [0, 1] arbitrary, and D_{1,n}, D_{2,n} such that min(a_{n}, b_{n}) − max(D_{1,n}, D_{2,n}) → ∞, then there is a strongly continuous group of automorphisms $\tau t\u03f5,t\u2208R$ on $AZ$ such that
If we take ϵ ∈ [0, ϵ(γ_{0})), with ϵ(γ_{0}) as in Theorem 3.11, we have a uniform gap separating the lower portion of the spectrum of $H\Lambda n\u2202n(\u03f5)$, denoted by $sp0,\Lambda n(\epsilon )$ in (2.8), and the rest of the spectrum. The following results provide an estimate of $diam(sp0,\Lambda n(\epsilon ))$. For simplicity, let Λ_{n} = [−n, n] for the remainder of the section.
Let $ c n ( \epsilon ) =diam ( sp 0 , \Lambda n ( \epsilon ) ) $ . Then

for all $\omega \u2208 S n ( \epsilon ) $ and $A\u2208 A \Lambda n s $ , we have
 (ii)
If s_{n} = 0 and D_{1,n} is such that lim_{n}[n − D_{1,n}] = lim_{n}D_{1,n} = ∞, then, for all $\omega \u2208S(\epsilon )$ and $A\u2208AZloc$, we have
The proof of (i) is elementary and the proof of (ii) follows by noting that the additional assumptions imply that the sequence $[H\Lambda n\u2202n(\u03f5),A]$ converges in norm and that lim c_{n}(ε) = 0 by Lemma 3.13.
Since the same α_{ϵ} 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 τ_{t}. The same conclusion then holds for the lower bound on the spectral gap above these ground states (see Ref. 10 for the details).
IV. STABILITY OF SPECTRAL GAP IN FERMION CHAINS
A. Quasilocal maps
Suppose $A\Lambda $ is a local algebra of observables which is *isomorphic to $A\Lambda s$. Let $\varphi :A\Lambda \u2192A\Lambda s$ denote a possible *isomorphism. Given a local Hamiltonian H_{Λ} in $A\Lambda $, ϕ unitarily transforms H_{Λ} into a Hamiltonian $H\Lambda s=\varphi (H\Lambda )$ of the spin algebra. Using an exhaustive family of conditional expectations ${\theta Xi:Xi\u2282Xi+1}$, $H\Lambda s$ 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 quasilocal *automorphism, and the $\theta Xj$ are normalized partial trace over increasing metric balls X_{j} = b_{x}(j). The quasilocality 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 JordanWigner isomorphism to transform even fermion interactions into spin interactions in a way that respects the parity symmetry.
B. Transformation of even fermion interactions
Recall, we denote by $A\Lambda +\u2282A\Lambda f$ the even operators of the CAR algebra over Λ. We say that $\beta \u2208Aut(A\Lambda )$ is even if it preserves the parity. Even interactions are defined similarly. We also denote $S\xb1=12(\sigma 1\xb1i\sigma 2)$. The following definition is the wellknown JordanWigner transformation, which gives a C^{*}isomorphism of CAR and spin algebras.
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 $a,b\u2208Z$.
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.
By Proposition 4.2, we assume that η and Φ^{Λ} = Φ are supported on intervals. Proposition 4.3 implies the existence of spin interactions η^{S} and Φ^{S} with the same uniform bound, range, local gap γ_{0}, and decay.
Since ϑ_{Λ} is implemented by some unitary, η^{S} has $Z2$LTQO for the same decay function Ω. So to apply the norm boundedness argument in Sec. III, it suffices to argue that Φ^{1}(b_{Λ}(x, n), ε) is even.
V. EXAMPLE OF EVEN HAMILTONIAN SATISFYING STABILITY HYPOTHESES
Here we describe an example of an interaction of the CAR algebra which satisfies the stability hypotheses of Theorem 3.11. Let $\chi ={fi:i\u2208B}$ and $Y={gj:j\u2208B}$ be two collections of vectors in $\u21132(Z)$ such that
$\chi \u222aY$ is an orthonormal basis for $\u21132(Z)$;
there exist R ≥ 0 and collections ${xi:i\u2208B}$, ${yj:j\u2208B}$ such that for all i, j,
Furthermore, denote $\chi W={fi:supp(fi)\u2282W}$ and $YW={gj:supp(gj)\u2282W}$. We will also assume the following:
 (iii)
There exists a diameter N_{0} such that for all intervals Λ, diam(Λ) > N_{0} implies $\chi \Lambda \u2260\u2205$ and $Y\Lambda \u2260\u2205$.
Suppose Λ is an interval such that diam(Λ) > N_{0}. Then H_{Λ} is nonnegative, uniformly gapped, and frustration free.
Next, we show that the number of auxiliary orthonormal basis vectors h_{i} needed to complete $\chi \Lambda $ and $Y\Lambda $ to a basis of ℓ^{2}(Λ) is uniformly bounded in Λ, and that each h_{i} has support contained toward the edge of Λ.
Suppose diam(Λ) > N_{0}. Let $Z(\Lambda )={h1(\Lambda ),\u2026,hn(\Lambda )}$, n = n(Λ), be a basis for the complement of $span(\chi \Lambda \u222aY\Lambda )$ in ℓ^{2}(Λ). Then

for each i ∈ [1, n] , $supp(hi(\Lambda ))\u2282\Lambda \Int3R(\Lambda )$;

$Z ( \Lambda ) \u22646R$
This lemma has an immediate corollary:
To conclude this section, we prove that the interaction defined in (5.1) satisfies $Z2$LTQO. Denote D = max{N_{0}, 3R}. Recall that if n ≥ D then $Hb\Lambda (x,n)$ is nonnegative and frustration free with kernel indexed by $Z(b\Lambda (x,n))$.
We handle the two cases of n when diam(b_{Λ}(x, n)) ≥ N_{0} or diam(b_{Λ}(x, n)) < N_{0}. Suppose the former. Now, there are two subcases for k: either b_{Λ}(x, k) ⊄ Int_{D}(b_{Λ}(x, n)) or b_{Λ}(x, k) is contained in that interior.
ACKNOWLEDGMENTS
Based upon work supported by the National Science Foundation under Grant Nos. DMS1207995 (A.M.), DMS1515850 (A.M. and B.N.), and DMS1813149 (B.N.). We would like to thank the referee for helpful comments and suggestions.
APPENDIX: DETAILS ABOUT $\mathcal{F}$FUNCTIONS AND SPECTRAL FLOW
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 $(Z,\u22c5)$ if
$\Vert F\Vert =\u2211x\u2208ZF(x)<\u221e$;
there exists C_{F} > 0 such that for all $x,y\u2208Z$,
2. Thermodynamic limit of the spectral flow
There are standard results giving conditions on an interaction Φ under which the finitevolume dynamics defined by
with
and
converges to a strongly continuous cocycle of automorphism, $\tau s,t\Phi $, of the algebra of quasilocal observables $A\Gamma $, 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 Fnorm, ∥⋅∥_{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 $\Phi \u2208BF(I)$, the function
is continuous and bounded. Strong convergence of the automorphisms here means that
The limit is taken over any sequence of $\Lambda n\u2208P(\Gamma )$ 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 $A\u2208AX$ and s, t ∈ I, and where, for $\Phi \u2208BF(I)$, and s, t ∈ I, the quantity I_{t,s}(Φ) is defined by
It is often important to include in the definition of the finitevolume Hamiltonian H_{Λ} 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 infinitevolume dynamics. In order to express this freedom in the interactions defining the finitevolume 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 Fnorm.
Let $(\Gamma ,d,\u2009A\Gamma )$ be a quantum lattice system, F be an Ffunction for (Γ, d), and $I\u2282R$ be an interval. We say that a sequence of interactions ${\Phi n}n\u22651$ converges locally in Fnorm to Φ such that

$\Phi n\u2208BF(I)$ for all n ≥ 1,

$\Phi \u2208BF(I)$, and

for any $\Lambda \u2208P0(\Gamma )$ and each [a, b] ⊂ I, one has
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.
As a consequence, we can apply the following theorem from Ref. 9 to the sequences of interactions $\Psi \Lambda n,\u2202n$.