Stability of gapped ground state phases of spins and fermions in one dimension

Moon, Alvin; Nachtergaele, Bruno

doi:10.1063/1.5036751

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

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 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 $[a, b] \subset Z$ and not on a cycle $Z / (n Z)$ ⁠. 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

II. SETTING AND MAIN RESULT

A. Notations

Denote by $(Z, | \cdot |)$ the metric graph of integers. Let P_f(X) denote the finite subsets of $X \subset Z$ ⁠. We will use Λ to refer exclusively to nonempty, finite intervals of the form $[a, b] = {n \in N : a \leq n \leq b}$ ⁠. 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:

r_{x} = min \{x - a, b - x\}, R_{x} = max \{x - a, b - x\} .

(2.1)

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 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 $A_{Λ}$ ⁠. If we want to specify that we are specifically considering spins or fermions, we will use the notation $A_{Λ}^{s}$ or $A_{Λ}^{f}$ ⁠, respectively. These algebras, and the associated Hilbert space they are represented on, are defined as follows:

For spin systems, we have

A_{Λ}^{s} = M_{d} {(C)}^{\otimes | Λ |}, h_{Λ} = {(C^{d})}^{\otimes | Λ |},

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

For fermions, $A_{Λ}^{f}$ 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 $F_{Λ} = F (ℓ^{2} (Λ))$ ⁠. The dimension of $F_{Λ}$ is 2^|Λ| and $A_{Λ}^{f}$ is *-isomorphic to the matrix algebra $M_{2^{| Λ |}} (C)$ ⁠.

Given an exhaustive net of CAR or spin algebras ${A_{Λ} : Λ \in P_{f} (Z)}$ ⁠, the inductive limit $A_{Z}$ ⁠, the d^∞ UHF algebra, is obtained by norm completion,

A_{Z} = \bar{⋃_{Λ \in P_{f} (Z)} A_{Λ}} .

This algebra is often referred to as the quasi-local algebra, and $A_{l o c} = ⋃ A_{Λ}$ is referred as the local algebra.

Define by N_X = ∑_x∈Xa^*(x)a(x) the number operator for $X \in P_{f} (Z)$ ⁠, and define the parity automorphism by

ρ_{Λ} (A) = \exp (i π N_{Λ}) A \exp (i π N_{Λ}) .

(2.2)

Say that $A \in A_{Λ}^{f}$ 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 $A_{Λ}^{f}$ ⁠, which we denote by $A_{Λ}^{+}$ ⁠.

B. Assumptions

Let I be a subinterval of $Z$ ⁠, not necessarily finite. An interaction on I is a function $Φ : P_{f} (I) \to A_{l o c}$ such that $Φ (X) = Φ {(X)}^{*} \in A_{X}$ for all X ∈ P_f(I). The corresponding local Hamiltonian of the finite system on Λ ⊂ I is H_Λ = ∑_X⊂ΛΦ(X). Say that Φ is non-negative if Φ(X) ≥ 0 for all X ∈ P_f(I). Say that Φ is an even interaction of the CAR algebra if $Φ (X) \in A_{X}^{+}$ ⁠.

The interactions in our perturbative setup will satisfy the following assumptions. First, let $η : P_{f} (Z) \to A_{l o c}$ be a non-negative 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 $X \in P_{f} (Z)$ ⁠, ∥η(X)∥ < M.
Frustration free: For all intervals $Λ \in P_{f} (Z)$ ⁠, ker(H_Λ) ⊋ {0}.
Uniformly locally gapped: There exists γ₀ > 0 such that for all intervals $[a, b] \in P_{f} (Z)$ ⁠, with b − a ≥ R, 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 $Φ^{Λ} : P_{f} (Λ) \to A_{l o c}$ be an interaction on the interval, and denote by [Φ] the collection of these perturbative interactions,

[Φ] = \{Φ^{Λ} : Λ \in P_{f} (Z)\} .

(2.3)

The perturbed Hamiltonians have the form

H_{Λ} (ε) = \sum_{X \subset Λ} η (X) + ε Φ^{Λ} (X), ε \in [0,1],

(2.4)

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

\begin{aligned} F (x) & = e^{- h (x)} F^{b} (x), \\ F^{b} (x) & = \frac{L}{{(1 + c x)}^{κ}}, \end{aligned}

(2.5)

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 $τ : [0, \infty) \to R$ ⁠, 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 $F (r) = e^{- h_{Φ} (r)} \frac{L}{{(1 + c x)}^{κ}}$ ⁠, 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 $n \in N$ ⁠.

The assumption that Φ^Λ 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).

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

ω_{Λ} (A) = \frac{1}{tr (P_{Λ})} tr (P_{Λ} A), A \in A_{Λ} .

Definition.

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, k \in N

satisfying 0 ≤ k ≤ r_x and k ≤ n ≤ R_x, the following bound holds:

\forall A \in A_{b_{Λ} (x, k)} : ‖ P_{b_{Λ} (x, n)} (A - ω_{Λ} (A)) P_{b_{Λ} (x, n)} ‖ \leq Ω (z_{x} (n) - k) ‖ A ‖,

(2.6)

where

z_{x} : N \to N

is the cutoff function defined in terms of distance to the boundary of Λ (),

\begin{aligned} z_{x} (m) = \{\begin{matrix} m & if m \leq r_{x} \\ r_{x} & else . \end{matrix}) \end{aligned}

(2.7)

If

η : P_{f} (Z) \to A_{l o c}^{f}

is an even interaction and () holds for the restricted class of observables

A \in A_{b_{Λ} (x, k)}^{+}

⁠, then we will say that η has

Z_{2}

-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 $Z_{2}$ -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 $λ_{1}, \dots, λ_{N} : [0,1] \to R$ such that for all ε ∈ [0, 1], {λ₁(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,

{sp}_{0, Λ} (ε) = \{λ_{i} (ε) : λ_{i} (0) = 0\}, {sp}_{1, Λ} (ε) = \{λ_{j} (ε) : λ_{j} (0) > 0\},

(2.8)

γ (H_{Λ} (ε)) = min \{λ - μ : λ \in {sp}_{1, Λ} (ε), μ \in {sp}_{0, Λ} (ε)\} .

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 $Φ_{Λ}^{D}$ supported at the D-boundary, and a bulk interaction Φ^Int(Λ). Define the following two finite constants quantifying the strength of the bulk and edge perturbations, respectively:

\begin{array}{l} M_{I n t} = sup_{Λ} \{‖ Φ^{I n t} {(Λ) ‖}_{F} : d i a m (Λ) > max \{2 D, R\}\}, \\ M_{D} = sup_{Λ} \{‖ Φ_{Λ}^{D} ‖ : d i a m (Λ) > max \{2 D, R\}\} . \end{array}

Then, for constant,

\begin{aligned} m = () \sum_{| n | \geq 3} 20 C (3 | n | + 2) [) Ω () \frac{| n | - 1}{2} {))}^{1 / 2} & + F_{0} () \frac{| n | - 3}{2}))]) \\ + C () \sum_{n \in Z} Ω () \frac{| n |}{2})) + 2 F_{0} () ⌊ \frac{| n |}{2} ⌋)) + 8)))) (‖ {η ‖}_{F} + M_{I n t}), \end{aligned}

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

Theorem 3.11

(Ground state gap stability for spin chains). Suppose

η : P_{f} (Z) \to A_{l o c}^{s}

has LTQO with Ω(n) ≤ n^−ν, for ν > 4, and there exist K > 0, s ∈ (0, 1] such that h_Φ satisfies h_Φ(r) ≥ Kr^s. Then there exists ε(γ₀) > 0 such that 0 ≤ ε < ε(γ₀) and diam(Λ) > max{2D, R} imply

γ (H_{Λ} (ε)) \geq γ_{0} - (m + 2 M_{D}) ε > 0 .

The constant ε(γ₀) can be taken as

ε (γ_{0}) = min \{1, \frac{γ_{0}}{m + 2 M_{D}}\} .

As a consequence, if we assume that $η : P_{f} (Z) \to A_{l o c}^{+}$ has $Z_{2}$ -LTQO, and Ω and $Φ^{Λ} : P_{f} (Λ) \to A_{Λ}^{+}$ 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 constant

m_{D}^{'}

such that 0 ≤ ε < ε′(γ₀) and diam(Λ) > max{2D, R} implies

γ (H_{Λ} (ε)) \geq γ_{0} - m_{D}^{'} ε > 0 .

The constants

m_{D}^{'}

and ε′(γ₀) can be explicitly determined by the constants m, M_D, 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 denote

r e s (H_{Λ}) = C \ s p (H_{Λ})

⁠. 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(H_Λ) with (ν, μ) ⊂ res(H_Λ) ∩ [0, T] and μ − ν > γ, then the gap between ν and μ is stable. Precisely, if we denote

γ (ν, μ, ε) = min \{λ (ε) \in s p (H_{Λ} (ε)) : λ (0) \geq μ\} - max \{λ (ε) \in s p (H_{Λ} (ε)) : λ (0) \leq ν\},

then

γ (ν, μ, ε) \geq (1 - p ε) γ - 2 (q + p T + M_{D}) ε > 0

for 0 ≤ ε < ε(γ, T) and p, q defined as

p = \frac{3}{γ_{0}} J_{1} (‖ {η ‖}_{F} + M_{I n t}), q = [C (J_{3} + 4) + J_{2}] (‖ {η ‖}_{F} + M_{I n t}) .

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⊂ΛΦ(X).

Let $D \in N$ 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 $Φ_{x} = \sum_{n = 1}^{R_{x}} Φ (b_{Λ} (x, n))$ ⁠, and the whole perturbation is split by the relative interior, $Φ_{Λ} = Φ_{Λ}^{D} + Φ_{Λ}^{I n t}$ ⁠, where

Φ_{Λ}^{D} = \sum_{x \in Λ \ {I n t}_{D} (Λ)} Φ_{x}, Φ_{Λ}^{I n t} = \sum_{x \in {I n t}_{D} (Λ)} Φ_{x}

are the edge and bulk perturbations, respectively. Let $Φ^{D} (Λ), Φ^{I n t} (Λ) : P_{f} (Λ) \to A_{Λ}^{s}$ denote the corresponding local interactions.

If x ∈ Int_D(Λ), then n ≥ r_x implies $‖ Φ () b_{Λ} (x, n) ‖ \leq ‖ {Φ ‖}_{F} F (D)$ ⁠, 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 bulk-perturbed Hamiltonian, it will suffice to prove ground state spectral gap stability for $H_{Λ} + ε Φ_{Λ}^{I n t}$ ⁠. 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 $Ψ : P_{f} (I) \to A_{l o c}^{s}$ be an arbitrary interaction, Λ ⊂ I, and suppose γ ∈ (0, γ₀). 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 quasi-adiabatic evolution) $α : [0, ε_{Λ}] \to A_{Λ}^{s}$ ⁠, 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 H_Λ(ε),

α_{ε} (A) = U {(ε)}^{*} A U (ε) and P (ε) = U (ε) P (0) U {(ε)}^{*} .

(3.1)

The unitaries are the solution to $- i \frac{d}{d ε} U (ε) = D (ε) U (ε)$ with $U (0) = 1$ ⁠, where the generator D(ε) is given by

D (ε) = \int_{- \infty}^{\infty} w_{γ} (t) \int_{0}^{t} e^{i s H_{Λ} (ε)} Ψ_{Λ} e^{- i s H_{Λ} (ε)} d s d t

(3.2)

for a weight function $w_{g a m m a}$ ∈ L¹ 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 H_Λ(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.

Suppose $Ψ : P_{f} (I) \to A_{l o c}^{s}$ satisfies a finite $F$ -norm for F and $h_{Ψ} (r) \geq K r^{t}$ for some $K > 0$ and t ∈ (0, 1]. Then for all 0 ≤ ε ≤ ε_Λ,

there exists an interaction $Φ^{1} (ε) : P_{f} (Λ) \to A_{Λ}^{s}$ such that α_ε(H_Λ(ε)) = H_Λ + Φ¹(ε), and
Φ¹(ε) is supported on the metric balls of Λ, that is,

Φ_{Λ}^{1} (ε) = \sum_{x \in Λ} Φ_{x}^{1} (ε),

where

Φ_{x}^{1} (ε) = \sum_{n = 1}^{R_{x}} Φ^{1} (b_{Λ} (x, n), ε)

and each

Φ^{1} (b_{Λ} (x, n), ε) \in A_{b_{Λ} (x, n)}^{s}

. Furthermore, for all x ∈ Λ,

[P (0), Φ_{x}^{1} (ε)] = 0

.There exists a constant C > 0, depending on the uniform bound M, range R, uniform gap γ₀, and decay parameters

K

and t, such that

‖ Φ^{1} {(ε) ‖}_{F_{φ}} \leq C ε (‖ {η ‖}_{F_{Ψ}} + ‖ {Ψ ‖}_{F_{Ψ}}),

where F_φ is an

F

-function depending on

K, t, γ

such that F_φ(r) decays faster than any polynomial in r.

Proof.

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 F_φ. Define

\begin{aligned} μ (r) = \{\begin{matrix} {(e / κ)}^{κ} & if r \leq e^{κ} \\ r / {(\log r)}^{κ} & else r > e^{κ} . \end{matrix}) \end{aligned}

(3.3)

Define

K_{0} = min {K, 2 / 7}

⁠, and denote by ν_Ψ the Lieb-Robinson velocity for the Heisenberg dynamics generated by the interaction Ψ. Denote

\tilde{μ} (r) = μ (\frac{K γ r}{2 ν_{Ψ}})

and

G_{Φ} (r) = e^{- \frac{K_{0}}{K} \tilde{μ} ◦ h_{Φ} (r)} F^{b} (r) .

Then the

F

-function in the statement of the theorem is given by

\begin{aligned} F_{φ} (r) = \{) \begin{matrix} G_{Ψ} (0) & if r \leq 18 R + 27 \\ G_{Ψ} (r / 18 - R - 3 / 2) & else r > 18 R + 27 . \end{matrix} \end{aligned}

(3.4)

For the remainder of this section, let U(ε), α_ε, and Φ¹(ε) be from an application of Theorem 3.1 when Ψ is the bulk perturbative interaction Φ^Int(Λ) with a local Hamiltonian $Φ_{Λ}^{I n t}$ ⁠.

Lemma 3.2.

The local operator Φ¹(ε) can be rewritten as

Φ^{1} (ε) = Φ^{2} (ε) + Φ^{3} (ε) + ω_{Λ} (\tilde{Φ^{1}} (ε)) + R (ε)

(3.5)

for terms defined by

\begin{array}{l} \tilde{Φ^{1}} (ε) = \sum_{x \in {I n t}_{2} (Λ)} Φ_{x}^{1} (ε), \\ Φ^{2} (ε) = (1 - P) (\tilde{Φ^{1}} (ε) - ω_{Λ} (\tilde{Φ^{1}} (ε)) 1) (1 - P), \\ Φ^{3} (ε) = P (\tilde{Φ^{1}} (ε) - ω_{Λ} (\tilde{Φ^{1}} (ε)) 1) P, \\ R (ε) = Φ_{a}^{1} (ε) + Φ_{a + 1}^{1} (ε) + Φ_{b}^{1} (ε) + Φ_{b + 1}^{1} (ε) . \end{array}

Proof.

This follows from a direct calculation using the fact that $[Φ_{x}^{1} (ε), P] = 0$ ⁠.

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 ⌊r_x/2⌋ > 0 for x ∈ Int₂(Λ).

C. Relative form boundedness of perturbations

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 ≤ r_x,

‖ P (Φ_{x}^{1} (ε) - ω_{Λ} (Φ_{x}^{1} (ε))) P ‖ \leq ‖ Φ^{1} (ε) ‖ () Ω (r_{x} - m) + 2 F_{φ} (m))) .

Proof.

Denote A − ω_Λ(A) = A₀ and b_Λ(x, n) = b_x(n), for brevity. For 0 ≤ m ≤ r_x, by linearity of ω_Λ,

P () Φ_{x}^{1} (ε) {))}_{0} P = \sum_{k = 1}^{R_{x}} P Φ^{1} {(b_{x} (k), ε)}_{0} P = \sum_{k = 1}^{m} P Φ^{1} {(b_{x} (k), ε)}_{0} P + \sum_{k = m + 1}^{R_{x}} P Φ^{1} {(b_{x} (k), ε)}_{0} P .

We bound the two summands separately. The right summand is bounded by Proposition A.1,

\sum_{k = m + 1}^{R_{x}} ‖ P Φ^{1} {(b_{x} (k), ε)}_{0} P ‖ \leq 2 ‖ Φ^{1} {(ε) ‖}_{F_{φ}} F_{φ} (m) .

The left summand is bounded by local topological quantum order and the

F

-norm,

\begin{aligned} \sum_{k = 1}^{m} ‖ P Φ^{1} {(b_{x} (k), ε)}_{0} P ‖ & \leq \sum_{k = 1}^{m} Ω (r_{x} - k) ‖ Φ^{1} (b_{x} (k), ε) ‖ \\ \leq Ω (r_{x} - m) ‖ Φ^{1} {(ε) ‖}_{F_{φ}} . \end{aligned}

Combining these bounds proves the lemma.

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

Lemma 3.4.

Suppose x ∈ Int₂(Λ). If 1 ≤ m ≤ r_x and m ≤ n ≤ R_x, then

∥) \sum_{k = 1}^{m} () Φ^{1} (b_{x} (k), ε) {))}_{0} P_{b_{x} (n)} ∥) \leq ‖ Φ^{1} {(ε) ‖}_{F_{φ}} [) 5 Ω {(z_{x} (n) - m)}^{1 / 2} + 4 F_{φ} (m)]) .

Proof.

Suppose

A \in A_{b_{x} (k)}^{s}

⁠. The C^*-identity and LTQO imply

{|) ‖ A P_{b_{x} (n)} ‖ - ‖ A P ‖ |)}^{2} \leq |) {‖ A P_{b_{x} (n)} ‖}^{2} - {‖ A P ‖}^{2} |) \leq 2 {‖ A ‖}^{2} Ω (z_{x} (n) - m) .

In the case

A = \sum_{k = 1}^{m} Φ^{1} {(b_{x} (k), ε)}_{0}

⁠, the above bound and Proposition A.1 imply

‖ A P_{b_{x} (n)} ‖ \leq 4 ‖ Φ^{1} {(ε) ‖}_{F_{φ}} Ω {(z_{x} (n) - m)}^{1 / 2} + ‖ A P ‖

By Theorem 3.1,

Φ_{x}^{1} (ε)

commutes with P. So, using Lemma 3.3, we get

\begin{aligned} ‖ A P ‖ & \leq ‖ P Φ_{x}^{1} {(ε)}_{0} P ‖ + 2 \sum_{k = m + 1}^{R_{x}} ‖ Φ^{1} (b_{x} (k), ε) ‖ \\ \leq ‖ Φ^{1} {(ε) ‖}_{F_{φ}} [) Ω (r_{x} - m) + 4 F_{φ} (m)]) . \end{aligned}

Proposition 3.6 uses a finite resolution of identity

{E_{n}^{x}}

defined at each site x ∈ Int₂(Λ) by

\begin{aligned} E_{n}^{x} = \{) \begin{matrix} 1 - P_{b_{x} (1)} & if n = 1 \\ P_{b_{x} (n - 1)} - P_{b_{x} (n)} & if 1 < n \leq r_{x} \\ P_{b_{x} (r_{x})} - P & if n = r_{x} + 1 \\ P & else n = r_{x} + 2 . \end{matrix} \end{aligned}

Lemma 3.5.

The family

{E_{n}^{x}}

has the properties

\begin{array}{l} 1 . \sum_{k = 1}^{r_{x} + 2} E_{k}^{x} = 1 a n d \sum_{k = 1}^{m} E_{k}^{x} = \{\begin{matrix} 1 - P_{b_{x} (m)} & i f 1 \leq m \leq r_{x} \\ 1 - P & i f m = r_{x} + 1, \end{matrix}) \\ 2 . P_{b_{x} (k)} E_{k}^{x} = 0 f o r k \leq r_{x} . \end{array}

Proof.

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 ≤ ε_Λ. There exist local operators

Θ_{β}^{x} (n, ε)

, for 3 ≤ n ≤ r_x, and operator

Θ_{α}^{x} (ε)

such that

Φ_{x}^{1} {(ε)}_{0} = \sum_{n = 3}^{r_{x}} Θ_{β}^{x} (n, ε) + Θ_{α}^{x} (ε) .

Furthermore,

P_{b_{x} (n)} Θ_{β}^{x} (n, ε) = 0

, and

Θ_{β}^{x} (n, ε)

and

Θ_{α}^{x} (ε)

decay rapidly,

\begin{aligned} ‖ Θ_{β}^{x} (n, ε) ‖ & \leq 20 ‖ Φ^{1} {(ε) ‖}_{F_{φ}} [) Ω () \frac{n - 1}{2} {))}^{\frac{1}{2}} + F_{φ} () \frac{n - 3}{2}))]), \\ ‖ Θ_{α}^{x} (ε) ‖ & \leq 20 ‖ Φ^{1} {(ε) ‖}_{F_{φ}} [) Ω () \frac{r_{x} - 1}{2} {))}^{\frac{1}{2}} + F_{φ} () \frac{r_{x} - 3}{2}))]) . \end{aligned}

Proof.

Fix x ∈ (a, b) and ε ∈ [0, ε_Λ]. Abbreviate

Q = 1 - P

and

Φ_{k}^{1} = Φ^{1} {(b_{x} (k), ε)}_{0}

⁠, i.e.,

Φ_{x}^{1} {(ε)}_{0} = \sum_{k = 1}^{R_{x}} Q Φ_{k}^{1} Q .

Define a “cutoff” parameter

n_{x} = ⌊ \frac{r_{x}}{2} ⌋

and split

Φ_{x}^{1} {(ε)}_{0}

into two sums,

Φ_{x}^{1} {(ε)}_{0} = \sum_{k = 1}^{n_{x}} Q Φ_{k}^{1} Q + \sum_{k = n_{x} + 1}^{R_{x}} Q Φ_{x}^{1} Q .

(3.6)

The tail

μ_{α}^{x} = \sum_{k = n_{x} + 1}^{R_{x}} Q Φ_{k}^{1} Q

can be bounded above in operator norm by using LTQO, so we turn our attention to the other summand. Denote by

Q_{b_{x} (l)}

the complement projection

1 - P_{b_{x} (l)}

⁠. Using the resolution {E_n} at x, we rewrite

Q Φ_{k}^{1} Q

for all 1 ≤ k ≤ n_x as

Q Φ_{k}^{1} Q = Q_{b_{x} (2 k)} Φ_{k}^{1} Q_{b_{x} (2 k)} + \sum_{n = 2 k + 1}^{r_{x} + 1} [) E_{n} Φ_{k}^{1} () \sum_{m = 1}^{n - 1} E_{m})) + () \sum_{m = 1}^{n} E_{m})) Φ_{k}^{1} E_{n}]) .

(3.7)

Define the following terms to organize the summands in ():

\begin{array}{l} ν_{α}^{x} (k) = E_{r_{x} + 1} Φ_{k}^{1} Q_{b_{x} (r_{x})} + Q Φ_{k}^{1} E_{r_{x} + 1}, θ_{β}^{x} (n, k) = E_{n} Φ_{k}^{1} Q_{b_{x} (n - 1)} + Q_{b_{x} (n)} Φ_{k}^{1} E_{n}, \\ τ_{β}^{x} (2 k) = Q_{b_{x} (2 k)} Φ_{k}^{1} Q_{b_{x} (2 k)} \end{array}

so that

Q Φ_{k}^{1} Q = ν_{α}^{x} (k) + τ_{β}^{x} (2 k) + \sum_{n = 2 k + 1}^{r_{x}} θ_{β}^{x} (n, k) .

For convenience, extend

τ_{β}^{x} (m)

to previously undefined m by declaring

τ_{β}^{x} (m) = 0

⁠. The derivation of the

Θ_{β}^{x} (ε, n), Θ_{α}^{x} (ε)

operators will result from an interchange of order for the summation of terms in () over n and k. The following definition for

Θ_{β}^{x} (n, ε)

accounts for the parity of r_x,

\begin{aligned} \forall 3 \leq n < r_{x} : Θ_{β}^{x} (n, ε) & = [) \sum_{k = 1}^{⌊ \frac{n - 1}{2} ⌋} θ_{β}^{x} (n, k)]) + τ_{β}^{x} (n), \\ Θ_{β}^{x} (r_{x}, ε) & = \sum_{k = 1}^{⌊ \frac{r_{x}}{2} ⌋} θ_{β}^{x} (r_{x}, k) + τ_{β}^{x} (r_{x}) . \end{aligned}

Then

Q Φ_{x}^{1} {(ε)}_{0} Q = \sum_{k = 1}^{R_{x}} Q Φ_{k}^{1} Q = \sum_{n = 3}^{r_{x}} Θ_{β}^{x} (n, ε) + Θ_{α}^{x} (ε),

where

Θ_{α}^{x} (ε) = μ_{α}^{x} + \sum_{k = 1}^{n_{x}} ν_{α}^{x} (k)

⁠. Next, the frustration free property of H_Λ implies that

\ker (H_{b_{x} (n)}) \subset \ker (H_{b_{x} (n - 1)})

⁠, and so

\forall 3 \leq n \leq r_{x} : P_{b_{x} (n)} Θ_{β}^{x} (n, ε) = Θ_{β}^{x} (n, ε) P_{b_{x} (n)} = 0 .

(3.8)

Furthermore, we have the following bounds on operator norm, for all x ∈ Int₂(Λ) and 3 ≤ n < r_x, by Lemma 3.4 and Proposition A.1,

\begin{aligned} ‖ Θ_{β}^{x} (n, ε) ‖ & \leq ‖ {(\sum_{k = 1}^{⌊ \frac{n - 1}{2} ⌋} Φ_{k}^{1})}^{*} E_{n} ‖ + ‖ \sum_{k = 1}^{⌊ \frac{n - 1}{2} ⌋} Φ_{k}^{1} E_{n} ‖ + ‖ τ_{β}^{x} (n) ‖ \\ \leq 20 ‖ Φ^{1} {(ε) ‖}_{F_{φ}} [) Ω () \frac{n - 1}{2} {))}^{\frac{1}{2}} + F_{φ} () \frac{n - 3}{2}))]), \\ max \{‖ Θ_{β}^{x} (r_{x}, ε) ‖, ‖ Θ_{α}^{x} (ε) ‖\} & \leq 20 ‖ Φ^{1} {(ε) ‖}_{F_{φ}} [) Ω () \frac{r_{x} - 1}{2} {))}^{\frac{1}{2}} + F_{φ} () \frac{r_{x} - 3}{2}))]) . \end{aligned}

(3.9)

Now, we define several quantities which will appear in the derivation of the form bound. Note that the weight function

e^{- h_{φ} (x)}

of F_φ is bounded above by 1 on its domain. So any expression in F_φ is bounded above by the corresponding sum using the shifted base

F

-function

F_{0} (r) = F^{b} (r / 18 - R - 3 / 2)

(3.10)

from () and (). Define

κ (n, ε) = 20 C ε (‖ {η ‖}_{F} + ‖ Φ^{I n t} {(Λ) ‖}_{F}) [) Ω () \frac{n - 1}{2} {))}^{\frac{1}{2}} + F_{0} () \frac{n - 3}{2}))]) .

(3.11)

κ(n, ε) does not depend on either Λ or the lower bound γ on the instantaneous gap, and the inequalities from () are rewritten as

‖ Θ_{β}^{x} (n, ε) ‖ \leq κ (n, ε), ‖ Θ_{α}^{x} (ε) ‖ \leq κ (r_{x}, ε) .

Last, we see by the assumed decay of Ω that the following sums are finite:

\begin{aligned} J_{1} & = \sum_{n \in Z} 20 C | n | [Ω {((| n | - 1) / 2)}^{1 / 2} + F_{0} ((| n | - 3) / 2)], \\ J_{2} & = \sum_{n \in Z} 20 C [Ω {((| n | - 1) / 2)}^{1 / 2} + F_{0} ((| n | - 3) / 2)] . \end{aligned}

(3.12)

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

n \in N

⁠, define the relation x ∼_n y if and only if

x - y \in (2 n + 1) Z

⁠. Index each of the parts

Λ_{n}^{i}

of Int₂(Λ)/∼_n by a representative i ∈ I(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

H_{n}^{i} = \sum_{x \in Λ_{n}^{i}} H_{b_{x} (n)}, Φ_{n}^{i} = \sum_{x \in Λ_{n}^{i}} Θ_{β}^{x} (n, ε) .

By definition of the

Θ_{β}^{x} (n, ε)

operators,

Φ^{2} (ε) = \sum_{n, i} Φ_{n}^{i}

⁠. In order to compare

H_{n}^{i}

to

Φ_{n}^{i}

⁠, we use a resolution of identity from Ref. 8, whose properties we record here.

Lemma 3.7.

For a configuration

σ : Λ_{n}^{i} \to {0,1}

, define the projection

S_{n}^{i} (σ) = \prod_{x \in Λ_{n}^{i}} σ_{x} Q_{b_{x} (n)} + (1 - σ_{x}) P_{b_{x} (n)}

. Then

\begin{aligned} 1 . \sum_{σ : Λ_{n}^{i} \to \{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 \in Λ_{n}^{i}, [Θ_{β}^{x} (n, ε), S_{n}^{i} (σ)] = 0 . \end{aligned}

Proof.

These properties follow immediately from the fact that $P_{b_{x} (n)} Θ_{β}^{x} (n, ε) = 0$ and that x ∼_n y implies b_x(n) ∩ b_y(n) = $\emptyset$ ⁠.

Proposition 3.8.

Suppose diam(Λ) > max{4, R}. There exist constants δ, β > 0, dependent on ∥Φ^Int(Λ)∥_F such that 0 ≤ ε ≤ ε_Λ implies, for all

v \in H_{Λ}

⁠,

| ⟨ v, Φ^{2} (ε) v ⟩ | \leq δ ε ‖ {v ‖}^{2} + β ε ⟨ v, H_{Λ} v ⟩ .

(3.13)

Precisely, we may choose

δ = J_{2} (‖ {η ‖}_{F} + ‖ Φ^{I n t} {(Λ) ‖}_{F}) a n d β = \frac{3}{γ_{0}} J_{1} (‖ {η ‖}_{F} + ‖ Φ^{I n t} {(Λ) ‖}_{F}) .

Proof.

Denote d_Λ = diam(Λ). For any x ∈ Int₂(Λ), if n > r_x, say that

Θ_{β}^{x} (n, ε) = 0

⁠. Suppose

u \in H_{Λ}

⁠. Then by Proposition 3.6,

| ⟨ u, Φ^{2} (ε) u ⟩ | \leq | ⟨ u, \sum_{n = 3}^{d_{Λ}} \sum_{i \in I (n)} Φ_{n}^{i} u ⟩ | + \sum_{x \in {I n t}_{2} (Λ)} κ (r_{x}, ε) ‖ {u ‖}^{2} .

The second term

\sum_{x \in {I n t}_{2} (Λ)} κ (r_{x}, ε)

is bounded above by the constants in (), so we focus on the first summand. Since

[Φ_{n}^{i}, S_{n}^{i} (σ)] = 0

⁠,

\begin{aligned} | ⟨ u, \sum_{n = 3}^{d_{Λ}} \sum_{i \in I (n)} Φ_{n}^{i} u ⟩ | & \leq \sum_{n = 3}^{d_{Λ}} | ⟨ u, \sum_{i \in I (n)} Φ_{n}^{i} [) \sum_{σ} S_{n}^{i} (σ)]) u ⟩ | \\ \leq \sum_{n = 3}^{d_{Λ}} \sum_{i \in I (n)} \sum_{σ : Λ_{n}^{i} \to \{0,1\}} \sum_{x \in Λ_{n}^{i}} ‖ S_{n}^{i} (σ) Θ_{β}^{x} (n, ε) ‖ ⟨ u, S_{n}^{i} (σ) u ⟩ \\ \leq \sum_{n = 3}^{d_{Λ}} \sum_{i \in I (n)} \frac{κ (n, ε)}{γ_{0}} \sum_{x \in Λ_{n}^{i}} \sum_{\begin{array}{c} σ : Λ_{n}^{i} \to \{0,1\} \\ σ_{x} = 1 \end{array}} γ_{0} ⟨ u, S_{n}^{i} (σ) u ⟩ \\ = \sum_{n = 3}^{d_{Λ}} \sum_{i \in I (n)} \frac{κ (n, ε)}{γ_{0}} \sum_{x \in Λ_{n}^{i}} ⟨ u, γ_{0} Q_{b_{x} (n)} u ⟩ \\ \leq \sum_{n = 3}^{d_{Λ}} \frac{3 n κ (n, ε)}{γ_{0}} ⟨ u, H_{Λ} u ⟩ . \end{aligned}

(3.14)

Hence

\begin{aligned} | ⟨ u, Φ^{2} (ε) u ⟩ | & \leq [) \sum_{x \in {I n t}_{2} (Λ)} κ (r_{x}, ε)]) ‖ {u ‖}^{2} + [) \sum_{n = 3}^{d_{Λ}} \frac{3 n κ (n, ε)}{γ_{0}}]) ⟨ u, H_{Λ} u ⟩ \\ \leq J_{2} (‖ {η ‖}_{F} + ‖ Φ^{I n t} {(Λ) ‖}_{F}) ‖ {u ‖}^{2} + \frac{3}{γ_{0}} J_{1} (‖ {η ‖}_{F} + ‖ Φ^{I n t} {(Λ) ‖}_{F}) ε ⟨ u, H_{Λ} u ⟩ . \end{aligned}

Corollary 3.9.

There exists a constant α, dependent on ∥Φ^Int(Λ)∥_F, such that 0 ≤ ε ≤ ε_Λ and diam(Λ) > max{4, R} imply

\forall u \in H_{Λ} : | ⟨ u, (Φ^{2} (ε) + Φ^{3} (ε) + R (ε)) u ⟩ | \leq α ε ‖ {u ‖}^{2} + β ε ⟨ u, H_{Λ} u ⟩ .

Precisely, we may take

α = C ({∥ η ∥}_{F} + {∥ Φ^{Int} (Λ) ∥}_{F})

[J₃ + 4] + δ.

Proof.

Suppose x ∈ Int₂(Λ). Set

m = ⌊ \frac{r_{x}}{2} ⌋

in an application of Lemma 3.3 to show

‖ P {(Φ_{x}^{1} (ε))}_{0} P ‖ \leq ‖ Φ^{1} {(ε) ‖}_{F_{φ}} [Ω (r_{x} / 2) + 2 F_{φ} (⌊ r_{x} / 2 ⌋)] .

But by the decay of Ω and F₀, we have that the following sum is finite:

J_{3} = \sum_{z \in Z} Ω (| z | / 2) + 2 F_{0} (⌊ | z | / 2 ⌋) .

(3.15)

And, summing over x ∈ Int₂(Λ),

‖ Φ^{3} (ε) ‖ \leq \sum_{x \in {I n t}_{2} (Λ)} ‖ P {(Φ_{x}^{1} (ε))}_{0} P ‖ \leq ‖ Φ^{1} {(ε) ‖}_{F_{φ}} J_{3} .

Next, it is straightforward to apply Proposition A.1 to

R (ε)

to get an upper bound on the norm,

‖ R (ε) ‖ \leq 4 ‖ Φ^{1} {(ε) ‖}_{F_{φ}} .

Until now, all estimates have been expressed using a local bound ∥Φ^Int(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:

M_{I n t} = sup_{Λ} \{‖ Φ^{I n t} {(Λ) ‖}_{F} : d i a m (Λ) > max \{2 D, R\}\} .

Proposition 3.10.

There exist ε_Int > 0 and constant m > 0 such that 0 ≤ ε < ε_Int and diam(Λ) > {4, R} imply

γ (H_{Λ} + ε Φ_{Λ}^{I n t}) \geq γ_{0} - m ε > 0 .

The constants ε_Int and m can be taken as the following expressions:

\begin{matrix} m = () 3 J_{1} + 2 J_{2} + C (J_{3} + 8))) (‖ {η ‖}_{F} + M_{I n t}), \\ ε_{I n t} = min \{1, \frac{γ_{0}}{m}\} . \end{matrix}

Proof.

Let γ ∈ (0, γ₀). For fixed Λ with diam(Λ) > max{4, R}, there exists ε_Λ > 0 such that for all 0 ≤ ε ≤ ε_Λ, $γ (H_{Λ} + ε Φ_{Λ}^{I n t}) \geq γ$ ⁠. 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), $γ (H_{Λ} + μ Φ_{Λ}^{I n t}) < γ$ ⁠.

Since the gap does not close on [0, ε_Λ], we use the spectral flow decomposition () to transform

H_{Λ} + ε Φ_{Λ}^{I n t}

by unitaries and a shift in the spectrum,

α_{ε} (H_{Λ} + ε Φ_{Λ}^{I n t}) - ω_{Λ} (\tilde{Φ^{1} (ε)}) = H_{Λ} + Φ^{2} (ε) + Φ^{3} (ε) + R (ε) .

But by Corollary 3.9, if ε ≤ ε_Λ, then

Φ (ε) = Φ^{2} (ε) + Φ^{3} (ε) + R (ε)

is H_Λ-bounded. Now, by the relation P(ε) = U(ε)P(0)U(ε)^* in (), the span of the eigenvectors to the 0-group of H_Λ + Φ(ε) is exactly ker(H_Λ). So, if λ is in the 0-group, which we will denote by sp(0, ε), then there exists a unit norm u ∈ ker(H_Λ) such that

| λ | = | ⟨ u, (H_{Λ} + Φ (ε)) u ⟩ | \leq α ε .

(3.16)

Next, define ε₁ > 0 as the solution to h(ε) = γ, where h is defined as

h (ε) = (1 - β ε) γ_{0} - δ ε - 4 C ε (‖ {η ‖}_{F} + M_{I n t}) - α ε .

Set ε_γ = min{ε₁, 1}. Combining () and Corollary 3.9, we see that if 0 ≤ ε < min{ε_γ, ε_Λ}, then

\begin{aligned} γ (H_{Λ} (ε)) & = min_{v \in \ker {(H_{Λ})}^{⊥} : ‖ v ‖ = 1} ⟨ v, [H_{Λ} + Φ^{2} (ε) + R (ε)] v ⟩ - maxsp (0, ε) \\ \geq h (ε) \\ > γ . \end{aligned}

By maximality, either ε_Λ = 1 or

γ (H_{Λ} + ε_{Λ} Φ_{Λ}^{I n t}) = γ

⁠. Hence ε_γ ≤ ε_Λ necessarily and

γ (H_{Λ} + ε Φ_{Λ}^{I n t}) \geq h (ε) > γ

for all ε < ε_γ. But now, γ was arbitrarily smaller than γ₀. Set

ε_{I n t} = sup {ε_{γ} : γ \in (0, γ_{0})} .

Evidently ε_Int does not depend on Λ, and if 0 ≤ ε < ε_Int, then

γ (H_{Λ} + ε Φ_{Λ}^{I n t}) \geq h (ε) = γ_{0} - m ε > 0,

where the constant

m = () 3 J_{1} + 2 J_{2} + C (J_{3} + 8))) (‖ {η ‖}_{F} + M_{I n t})

comes from rewriting the lower bound h(ε) as a linear equation of ε.

Denote by M_D the following finite uniform bound on the strength of the edge perturbations:

M_{D} = sup_{Λ} \{‖ Φ_{Λ}^{D} ‖ : d i a m (Λ) > max \{2 D, R\}\} .

We remark that M_Int and m are defined in terms of

F

-function decay, while M_D is defined in terms of the operator norm.

Theorem 3.11

(Ground state gap stability for spin chains). Suppose

η : P_{f} (Z) \to A_{l o c}^{s}

has LTQO with Ω(n) ≤ n^−ν, for ν > 4, and there exist K > 0, s ∈ (0, 1] such that h_Φ satisfies h_Φ(r) ≥ Kr^s. Then there exists ε(γ₀) > 0 such that 0 ≤ ε < ε(γ₀) and diam(Λ) > max{2D, R} imply

γ (H_{Λ} (ε)) \geq γ_{0} - (m + 2 M_{D}) ε > 0 .

The constant ε(γ₀) can be taken as

ε (γ_{0}) = min \{1, \frac{γ_{0}}{m + 2 M_{D}}\} .

(3.17)

Proof.

Considering

ε Φ_{Λ}^{D}

as a perturbation of

H + ε Φ_{Λ}^{I n t}

⁠, the spectrum of

H + ε Φ_{Λ}^{D} + ε Φ_{Λ}^{I n t}

must be contained in the compact neighborhood,

O_{Λ} (ε) = \{r \in R : d (r, s p (H + ε Φ_{Λ}^{I n t})) \leq ‖ ε Φ_{Λ}^{D} ‖\} .

That is,

γ (H_{Λ} (ε)) \geq γ (H_{Λ} + ε Φ_{Λ}^{I n t}) - 2 ‖ ε Φ_{Λ}^{D} ‖ \geq γ_{0} - (m + 2 M_{D}) ε .

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 denote

r e s (H_{Λ}) = C \ s p (H_{Λ})

. 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(H_Λ) with (ν, μ) ⊂ res(H_Λ) ∩

[0, T]

and μ − ν > γ, then the gap between ν and μ is stable. Precisely, if we denote

γ (ν, μ, ε) = min \{λ (ε) \in s p (H_{Λ} (ε)) : λ (0) \geq μ\} - max \{λ (ε) \in s p (H_{Λ} (ε)) : λ (0) \leq ν\},

then

γ (ν, μ, ε) \geq (1 - p ε) γ - 2 (q + p T + M_{D}) ε > 0

for p, q defined by

p = \frac{3}{γ_{0}} J_{1} (‖ {η ‖}_{F} + M_{I n t}), q = (‖ {η ‖}_{F} + M_{I n t}) [C (J_{3} + 4) + J_{2}] .

(3.18)

Proof.

Let Φ(ε) be defined as in Proposition 3.10, for 0 ≤ ε < ε_Int. By Proposition 3.9, for all

u \in H_{Λ}

⁠,

| ⟨ u, Φ (ε) u ⟩ | \leq p ε ⟨ u, H_{Λ} u ⟩ + q ε ‖ {u ‖}^{2} .

Let

z = \frac{ν + μ}{2}

and denote

R_{ζ} (ε^{'}) = {(ζ - H_{Λ} - Φ (ε^{'}))}^{- 1}

⁠, with R_ζ = R_ζ(0). Let U denote the polar unitary such that R_z = U|R_z|. Since R_z is self-adjoint, |R_z|U^* = U|R_z|, and so for unit norm u,

\begin{aligned} sup_{‖ w ‖ = 1} | ⟨ w, | R_{z} |^{1 / 2} U^{*} Φ (ε) | R_{z} |^{1 / 2} u ⟩ | & \leq ‖ | R_{z} |^{1 / 2} Φ (ε) | R_{z} |^{1 / 2} ‖ \\ \leq sup_{‖ v ‖ = 1} [) q ε ‖ | R_{z} |^{1 / 2} {v ‖}^{2} + p ε ⟨ v, H_{Λ} | R_{z} | v ⟩]) . \end{aligned}

(3.19)

That is, for sufficiently small ε,

‖ | R_{z} |^{1 / 2} U^{*} Φ (ε) | R_{z} |^{1 / 2} ‖ \leq q ε ‖ R_{z} ‖ + p ε (1 + | z | ‖ R_{z} ‖) < 1,

and by the expansion

R_{z} {(ε)}^{- 1} = U | z - H |^{1 / 2} (1 - | R_{z} |^{1 / 2} U^{*} Φ (ε) | R_{z} |^{1 / 2}) | z - H |^{1 / 2},

we derive the lower bound

d (z, s p (H_{Λ} + Φ (ε))) \geq (1 - p ε) γ - 2 (q + p T) ε .

Hence for sufficiently small ε, independently of sufficiently large Λ,

γ (ν, μ, ε) \geq (1 - p ε) γ - 2 (q + p T + M_{D}) ε > 0 .

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 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 $[Φ] = {Φ^{Λ} ∣ Λ \in P_{f} (Z)}$ is a family of perturbations given in terms of interactions $Φ, Φ^{b} \in B_{F}$ and a few parameters that define the boundary conditions. Specifically, consider intervals $Λ \subset Z$ 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 (D₁, D₂, s), D₁, D₂ ≥ 0, s ∈ [0, 1], and consider

H_{Λ}^{\partial} (ϵ) = \sum_{X \subset Λ} η (X) + ϵ (\sum_{X \subset Λ_{D_{1}}} Φ (X) + s \sum_{X \subset (Λ \ Λ_{D_{2}})} Φ^{b} (X)) .

(3.20)

This form of the Hamiltonian covers a broad range of perturbations and boundary conditions. The dynamics generated by $H_{Λ}^{\partial} (ϵ)$ is the one-parameter group of automorphism $τ_{t}^{H_{Λ}^{\partial} (ϵ)}$ ⁠.

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 $τ_{t}^{ϵ}, t \in R$ on $A_{Z}$ such that

lim_{n \to \infty} ‖ τ_{t}^{H_{Λ_{n}}^{\partial_{n}} (ϵ)} (A) - τ_{t}^{ϵ} (A) ‖ = 0, for all A \in A_{Z}^{l o c} .

(3.21)

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 $H_{Λ_{n}}^{\partial_{n}} (ϵ)$ ⁠, denoted by ${sp}_{0, Λ_{n}} (ε)$ in (2.8), and the rest of the spectrum. The following results provide an estimate of $d i a m ({sp}_{0, Λ_{n}} (ε))$ ⁠. For simplicity, let Λ_n = [−n, n] for the remainder of the section.

Lemma 3.13.

In the assumptions of above, choose s_n = 0 and put D_1,n = D_n. Then, there exists a function

G : [0, \infty) \to [0, \infty)

which decreases to 0 as n tends to infinity and, for large enough n,

d i a m ({sp}_{0, Λ_{n}} (ε)) \leq ε G (D_{n}) .

Precisely, we may take

G (r) = 8 \sum_{k = ⌊ r / 2 ⌋}^{\infty} \tilde{F} (k / 2) + C (M_{I n t} + ‖ {η ‖}_{F}) [Ω (k / 2) + F_{0} (⌊ k / 2 ⌋)],

where

\tilde{F}

is an

F

-function depending on ∥η∥_F and M_Int.

Proof.

Suppose n is sufficiently large so that

\frac{1}{2} ⌊ D_{n} / 2 ⌋ > R

⁠, the range of the interaction η. By the spectral flow decomposition in (),

\begin{aligned} d i a m ({sp}_{0, Λ_{n}} (ε)) & \leq 2 ‖ P Φ^{1} {(ε)}_{0} P ‖ \\ = 2 ‖ \sum {P Φ_{x}^{1} {(ε)}_{0} P : x \in Λ_{n}} ‖ \\ \leq 2 (A + B) \end{aligned}

for A, B defined by complementary regions of the interval Λ_n = [−n, n],

\begin{aligned} A & = \sum \{‖ P Φ_{x}^{1} {(ε)}_{0} P ‖ | \forall x \in Λ_{n} : - n + ⌊ D_{n} / 2 ⌋ \leq x \leq n - ⌊ D_{n} / 2 ⌋\}, \\ B & = \sum \{‖ P Φ_{x}^{1} {(ε)}_{0} P ‖ | \forall x \in Λ_{n} : - n \leq x < - n + ⌊ D_{n} / 2 ⌋ or n - ⌊ D_{n} / 2 ⌋ < x \leq n\} . \end{aligned}

By applying LTQO and

F

-norm bounds,

‖ A ‖ \leq 4 C (M_{I n t} + ‖ {η ‖}_{F}) ε \sum_{k = ⌊ D_{n} / 2 ⌋}^{\infty} [Ω (k / 2) + F_{0} (⌊ k / 2 ⌋)],

where F₀ is the shifted base

F

-function from (). For the norm bound on B, let Δ_X(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 ⊂ Λ_n. Suppose − n ≤ x < − n + ⌊D_n/2⌋or n −⌊D_n/2⌋< x ≤ n. Denote

d_{x} (n) = d (x, {I n t}_{D_{n}} (Λ_{n}))

⁠. By the locality assumption on

Φ^{Λ_{n}}

and the fact that d_x(n)/2 > R, if k ≤ ⌊d_x(n)/2⌋, then, in the notation of the Proof of Theorem 3.1,

Φ^{1} (b_{x} (k), ε) = Δ_{b_{x} (k)} ((α_{ε} - i d) ◦ F_{w_{γ_{0}, ε}} (h_{x})),

and so

\begin{aligned} ‖ P Φ_{x}^{1} {(ε)}_{0} P ‖ & \leq ‖ \sum_{k = 1}^{⌊ d_{x} (n) / 2 ⌋} P Φ^{1} {(b_{x} (k), ε)}_{0} P ‖ + \sum_{k = ⌊ d_{x} (n) / 2 ⌋ + 1}^{R_{x}} ‖ P Φ^{1} {(b_{x} (k), ε)}_{0} P ‖ \\ \leq 4 ‖ (α_{ε} - i d) ◦ F_{w_{γ_{0}, ε}} (h_{x}) ‖ + 2 C (M_{I n t} + ‖ {η ‖}_{F}) ε F_{0} (⌊ d_{x} (n) / 2 ⌋) . \end{aligned}

Using the quasi-locality of the generator iD(ε) of the spectral flow unitaries,

(α_{ε}^{Λ_{n}} - i d) ◦ F_{w_{γ_{0}}, ε} (h_{x}) = \int_{0}^{ε} i α_{s}^{Λ_{n}} () [D (s), \sum_{k = 1}^{R_{x}} Δ_{b_{x} (k)} (F_{w_{γ_{0}, ε}} (h_{x}))])) d s,

and there exists an

F

-function

\tilde{F}

⁠, independent of Λ_n, such that

2 ‖ (α^{Λ_{n}} - i d) ◦ F_{w_{γ_{0}, ε}} (h_{x}) ‖ \leq ε \tilde{F} (⌊ d_{x} (n) / 2 ⌋) .

Hence

‖ B ‖ \leq \sum_{k = ⌊ D_{n} / 2 ⌋}^{\infty} ε \tilde{F} (k / 2) + 2 C (M_{I n t} + ‖ {η ‖}_{F}) ε F_{0} (k / 2) .

Let

G (r) = 8 \sum_{k = ⌊ r / 2 ⌋}^{\infty} \tilde{F} (k / 2) + C (M_{I n t} + ‖ {η ‖}_{F}) [Ω (k / 2) + F_{0} (⌊ k / 2 ⌋)]

⁠. Then

d i a m ({sp}_{0, Λ_{n}} (ε)) \leq ε G (D_{n}) .

Let P_n(ε) denote the spectral projection of

H_{Λ_{n}}^{\partial_{n}} (ϵ)

associated with the isolated portion of the spectrum

{sp}_{0, Λ_{n}} (ε)

and define the set of states of

A_{Λ_{n}}^{s}

⁠,

S_{n} (ε)

⁠, with support in the range of P_n(ε),

S_{n} (ε) = {ω ∣ ω is a state on A_{Λ_{n}}^{s} with ω (P_{n} (ε)) = 1} .

We now consider the thermodynamic limits of these states,

S (ε) = {ω state on A_{Z}^{s} ∣ \exists (n_{k}) increasing and ω_{k} \in S_{n_{k}} (ε) s.t. lim_{k} ω_{k} (A) = ω (A), \forall A \in A_{Z}^{l o c}} .

Lemma 3.14.

Let $c_{n} (ε) = d i a m ({sp}_{0, Λ_{n}} (ε))$ . Then

for all $ω \in S_{n} (ε)$ and $A \in A_{Λ_{n}}^{s}$ , we have

R e ω (A^{*} [H_{Λ_{n}}^{\partial_{n}} (ϵ), A]) \geq - c_{n} (ε) ‖ {A ‖}^{2}, a n d |I m ω (A^{*} [H_{Λ_{n}}^{\partial_{n}} (ϵ), A])| \leq c_{n} (ε) ‖ {A ‖}^{2} .

(ii)
If s_n = 0 and D_1,n is such that lim_n[n − D_1,n] = lim_nD_1,n = ∞, then, for all $ω \in S (ε)$ and $A \in A_{Z}^{l o c}$ , we have

lim_{n \to \infty} ω (A^{*} [H_{Λ_{n}}^{\partial_{n}} (ϵ), A]) \geq 0 .

Proof.

The proof of (i) is elementary and the proof of (ii) follows by noting that the additional assumptions imply that the sequence $[H_{Λ_{n}}^{\partial_{n}} (ϵ), A]$ converges in norm and that lim c_n(ε) = 0 by Lemma 3.13.

In other words, the conditions of part (ii) of the lemma imply that the states in

S_{n} (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

P_{n} (0) = α_{ε}^{Λ_{n}, \partial_{n}} (P_{n} (ε))

⁠, we also have

S_{n} (ε) = S_{n} (0) ◦ α_{ε}^{Λ_{n}, \partial_{n}},

and as an easy consequence of the convergence (see Ref. 1, Lemma 5.6), we then also have

S (ε) = S (0) ◦ α_{ε} .

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. Quasi-local maps

Suppose $A_{Λ}$ is a local algebra of observables which is *-isomorphic to $A_{Λ}^{s}$ ⁠. Let $ϕ : A_{Λ} \to A_{Λ}^{s}$ denote a possible *-isomorphism. Given a local Hamiltonian H_Λ in $A_{Λ}$ ⁠, ϕ unitarily transforms H_Λ into a Hamiltonian $H_{Λ}^{s} = ϕ (H_{Λ})$ of the spin algebra. Using an exhaustive family of conditional expectations ${θ_{X_{i}} : X_{i} \subset X_{i + 1}}$ ⁠, $H_{Λ}^{s}$ can again be realized as the sum of local operators through a telescoping sum,

\forall B \in A_{Λ} : ϕ (B) = θ_{X_{1}} (ϕ (B)) + \sum_{j = 1}^{N - 1} θ_{X_{j + 1}} (ϕ (B)) - θ_{X_{j}} (ϕ (B)) .

The Proof of Theorem 3.1 uses this method of decomposition in the setting where ϕ is a quasi-local *-automorphism, and the $θ_{X_{j}}$ are normalized partial trace over increasing metric balls X_j = b_x(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 $Z_{2}$ -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.

Definition.

Let

Λ \in P_{f} (Z)

be a nonempty interval. A linear map

α : A_{Λ}^{s} \to A_{Λ}^{s}

is quasi-local if there exist constants

C > 0, p \in N

and a decay function g: [0, ∞) → [0, ∞) such that if X, Y ⊂ Λ are disjoint subsets, then for all

A \in A_{X}^{s}

and

B \in A_{Y}^{s}

⁠, the following bounds hold:

‖ α (A) ‖ \leq C | X |^{p} ‖ A ‖ ‖ [α (A), B] ‖ \leq C ‖ A ‖ ‖ B ‖ | X |^{p} g (d (X, Y)) .

(4.1)

Example.

The local Heisenberg dynamics

τ^{Λ} : U \subseteq R \to Aut (A_{Λ}^{s})

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 ν_Ψ the Lieb-Robinson velocity. There exists a constant C_Ψ > 0 such that for X, Y ∈ P_f(Λ) disjoint sets and

A \in A_{X}^{s}, B \in A_{Y}^{s}

⁠, the following Lieb-Robinson bound holds:

‖ [τ_{t}^{Λ} (A), B] ‖ \leq C_{Ψ} (e^{ν_{Ψ} | t |} - 1) ‖ A ‖ ‖ B ‖ \sum_{x \in X, y \in Y} F (| x - y |) .

But by properties of the

F

-function,

\begin{aligned} \sum_{x \in X, y \in Y} F (| x - y |) \leq | X | sup \{\sum_{\begin{array}{c} y \in Z \\ | x - y | \geq d (X, Y) \end{array}} F (| x - y |) : x \in Z\} < \infty . \end{aligned}

So take

C_{t} = C_{Ψ} (e^{ν_{Ψ} | t |} - 1)

⁠, p_t = 1, and

g_{t} (n) = sup \{\sum_{\begin{array}{c} y \in Z \\ | x - y | \geq n \end{array}} F (| x - y |) : x \in Z\} .

In particular, the spectral flow automorphism

α^{Λ} : [0, ε_{Λ}] \to Aut (A_{Λ}^{s})

is quasi-local.¹

Last, we specify the normalized partial trace maps. Let

X (n) = \{z \in Λ : \exists x \in X, | z - x | \leq n\}

denote an enlargement of X ∈ P_f(Λ). Denote the normalized partial trace of the state space over Λ∖X(n) by

θ_{X (n)} = \frac{1}{\dim H_{Λ \ X (n)}} {tr}_{H_{Λ \ X (n)}} .

For convention, we will take the trace over

H_{\emptyset}

as the identity map. Then define, for all

A \in A_{Λ}^{s}

⁠,

Δ_{X (0)} (A) = θ_{X (0)} (A), Δ_{X (n)} (A) = θ_{X (n)} (A) - θ_{X (n - 1)} (A) .

B. Transformation of even fermion interactions

Recall, we denote by $A_{Λ}^{+} \subset A_{Λ}^{f}$ the even operators of the CAR algebra over Λ. We say that $β \in Aut (A_{Λ})$ is even if it preserves the parity. Even interactions are defined similarly. We also denote $S^{\pm} = \frac{1}{2} (σ^{1} \pm i σ^{2})$ ⁠. The following definition is the well-known Jordan-Wigner transformation, which gives a C^*-isomorphism of CAR and spin algebras.

Definition.

Consider the case

A_{{x}}^{s} = M_{2} (C)

⁠. Let

𝜗_{Λ} : A_{Λ}^{f} \to A_{Λ}^{s}

denote the Jordan-Wigner map defined by

a (x) \mapsto \exp () - i π \sum_{j < x} S_{j}^{+} S_{j}^{-})) S_{x}^{-}, a^{*} (x) \mapsto \exp () i π \sum_{j < x} S_{j}^{+} S_{j}^{-})) S_{x}^{+} .

The Jordan-Wigner transformation extends the notion of parity to the spin 1/2 algebra. We say that

A \in A_{Λ}^{s}

is even if

𝜗_{Λ}^{- 1} (A) \in A_{Λ}^{+}

⁠.

Proposition 4.1.

Let X ⊂ Λ be any subinterval.

\begin{aligned} 1 . I f A \in A_{X}^{+}, t h e n 𝜗_{Λ} (A) \in A_{X}^{s} . \\ 2 . I f α : A_{Λ}^{s} \to A_{Λ}^{s} i s a n e v e n q u a s i - l o c a l m a p, t h e n Δ_{X (n)} ◦ α i s a l s o e v e n . \end{aligned}

Proof.

Suppose A is a monomial ca^#(x₁)⋯a^#(x_2n). By the CAR, we may assume x_j ≤ x_j+1. A direct computation shows that the first part of the lemma holds for the even monomials which generate

A_{X}^{+}

⁠,

𝜗_{Λ} (A) = c \prod_{k = 2}^{2 n} S_{x_{k}}^{♭} S_{x_{k - 1}}^{♭} \exp () \pm i π \sum_{j = x_{k - 1}}^{x_{k} - 1} S_{j}^{+} S_{j}^{-})) \in A_{X}^{s} .

Next, we show that the partial trace is an even map. For any x ∈ Λ, define the following four unitary operators:

u_{x}^{(0)} = 1, u_{x}^{(1)} = σ_{x}^{1}, u_{x}^{(2)} = i σ_{x}^{2}, u_{x}^{(3)} = σ_{x}^{3} .

Now, let Z ⊂ Λ and

B \otimes C \in A_{Z}^{s} \otimes A_{Λ \ Z}^{s}

⁠. Denote by I_Λ∖Z the set of finite sequences ι: Λ∖Z → {0, 1, 2, 3}. Define

u (ι) = \prod_{z \in Z} u_{z}^{(ι_{z})} .

Using elementary properties of trace and locality in the spin algebra,

\frac{1}{\dim (H_{Λ \ Z})} B \otimes tr (C) 1 = \frac{1}{4^{| Λ \ Z |}} \sum_{ι \in I_{Λ \ Z}} u {(ι)}^{*} [B \otimes C] u (ι) \in A_{Z}^{s} .

(4.2)

The relation in () uniquely defines the partial trace; hence,

θ_{Z} (\cdot) = \frac{1}{4^{| Λ \ Z |}} \sum_{ι \in I_{Λ \ Z}} u {(ι)}^{*} [\cdot] u (ι) .

The second part of the lemma follows from this formula.

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

Definition.

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

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.

Suppose

I \subset Z

is an interval and

Ψ : P_{f} (I) \to A_{l o c}

is an interaction. Then there exists an interaction

Φ : P_{f} (I) \to A_{l o c}

, supported on intervals, such that for all finite intervals Λ ⊂ I, the local Hamiltonians are equal,

Φ_{Λ} = \sum_{X \subset Λ} Φ (X) = Ψ_{Λ} .

If Ψ is an unperturbed interaction with uniform bound M, range R, and local gap γ₀, then so is Φ, with uniform bound 2^RM and the same range and local gap.

Furthermore, if ∥Ψ∥_F < ∞, where F is the

F

-function in (), and h(r) ≥ Kr^s for some K > 0 and s ∈ (0, 1], then ∥Φ∥_G ≤ ∥Ψ∥_F for the

F

-function defined by

\begin{aligned} G (r) & = e^{- \frac{1}{2} h (r)} \frac{C_{Φ}}{{(1 + c r)}^{κ}}, \\ C_{Φ} & = L \sum_{n = 1}^{\infty} n e^{- \frac{1}{2} h (n)} . \end{aligned}

Proof.

If

I ⊊ Z

⁠, then we may extend Ψ to

Z

by Ψ(Z) = 0 for Z ⊄ I, 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

Φ ([k, k + n]) = \sum \{Ψ (X) : X \subset [k, k + n], d i a m (X) = n\} .

By construction, Φ_Λ = Ψ_Λ. Now, suppose Φ is an unperturbed interaction with constants M, R, γ₀. Since

Φ_{b_{Λ} (x, n)} = Ψ_{b_{Λ} (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,

‖ Φ ([a, b]) ‖ \leq 2^{R} M .

Now, suppose Φ is some interaction, not necessarily finite range, with ∥Φ∥_F. For fixed

k \in Z

and n ≥ 0, by Proposition A.1,

\begin{aligned} ‖ Φ ([k, k + n]) ‖ \leq \sum_{\begin{array}{c} X \in P_{f} (Z) \\ k, k + n \in X \end{array}} ‖ Ψ (X) ‖ \leq ‖ {Ψ ‖}_{F} F (n) . \end{aligned}

So for

x, y \in Z

⁠,

\begin{aligned} \sum_{\begin{array}{c} k, n \\ x, y \in [k, k + n] \end{array}} ‖ Φ ([k, k + n]) ‖ & = \sum_{n \geq | x - y |} \sum_{\begin{array}{c} k \\ x, y \in [k, k + n] \end{array}} ‖ Φ ([k, k + n]) ‖ \\ \leq \sum_{n \geq | x - y |} \sum_{\begin{array}{c} k \\ x, y \in [k, k + n] \end{array}} ‖ {Ψ ‖}_{F} F (n) \\ \leq ‖ {Ψ ‖}_{F} \sum_{n \geq | x - y |} (n + 1 - | x - y |) e^{- h (n)} \frac{L}{{(1 + c n)}^{κ}} \\ \leq ‖ {Ψ ‖}_{F} () \sum_{n = 1}^{\infty} n e^{- \frac{1}{2} h (n)})) e^{- \frac{1}{2} h (| x - y |)} \frac{L}{{(1 + c | x - y |)}^{κ}} . \end{aligned}

That is,

\begin{aligned} ‖ {Φ ‖}_{G} = sup_{x, y \in Z} \{\sum_{\begin{array}{c} X \in P_{f} (Z) \\ x, y \in X \end{array}} \frac{‖ Φ (X) ‖}{G (| x - y |)}\} & \leq ‖ {Ψ ‖}_{F} . \end{aligned}

Proposition 4.3.

Suppose

Ψ : P_{f} (I) \to A_{l o c}^{f}

is an even interaction supported on intervals. Then there exists an even interaction

Φ : P_{f} (I) \to A_{l o c}^{s}

such that for any Λ ⊂ I,

𝜗_{Λ} (Ψ_{Λ}) = Φ_{Λ} .

If Ψ satisfies a finite

F

-norm for some F of the form (), then so does Φ. If Ψ is an unperturbed interaction, then so is Φ for the same constants.

Proof.

For Λ₀ ⊂ Λ, let

ι_{Λ_{0}, Λ}

denote the inclusion

A_{Λ_{0}}^{f} ↪ A_{Λ}^{f}

⁠. If

A \in A_{Λ_{0}}^{+}

⁠, then by expanding in an even generating set of monomials, we see

ι_{Λ_{0}, Λ} ◦ 𝜗_{Λ_{0}} (A) = 𝜗_{Λ} ◦ ι_{Λ_{0}, Λ} (A) .

So there exists an injective *-morphism

𝜗 : ⋃ A_{Λ}^{+} \to A_{l o c}^{s}

which extends every ϑ_Λ, 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,

\begin{aligned} ‖ {Ψ ‖}_{F} = sup_{x, y} \sum_{\begin{array}{c} X \in P_{f} (Z) \\ x, y \in X \end{array}} \frac{‖ 𝜗 (Ψ (X)) ‖}{F (| x - y |)} = ‖ {Φ ‖}_{F} . \end{aligned}

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

Q_{Λ} : H_{Λ} \to F_{Λ}

such that for

A \in A_{Λ}^{f}

⁠,

𝜗 (A) = 𝜗_{Λ} (A) = Q_{Λ}^{*} A Q_{Λ} .

Since ϑ is an isometry which preserves support for even observables, and Q_Λ 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 exist

ε_{γ_{0}}^{'} > 0

and constant

m_{D}^{'}

such that

0 \leq ε < ε_{γ_{0}}^{'}

and diam(Λ) > max{2D, R} imply

γ (H_{Λ} (ε)) \geq γ_{0} - m_{D}^{'} ε > 0 .

The constants

m_{D}^{'}

and

ε_{γ_{0}}^{'}

can be explicitly determined by the expressions in ().

Proof.

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 γ₀, and decay.

Let γ ∈ (0, γ₀) and

D \in N

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 H_Λ(ε) of η^S + εΦ^S,

α_{ε}^{Λ} (H_{Λ} + ε Φ_{Λ}) = H_{Λ} + \sum_{x \in Λ} Φ_{x}^{1} (ε) = H_{Λ} + Φ^{2} (ε) + Φ^{3} (ε) + R (ε) + ω_{Λ} (\tilde{Φ^{1} (ε)}) .

Since ϑ_Λ is implemented by some unitary, η^S has $Z_{2}$ -LTQO for the same decay function Ω. So to apply the norm boundedness argument in Sec. III, it suffices to argue that Φ¹(b_Λ(x, n), ε) is even.

But the Proof of Theorem 3.1 in Ref. 18 guarantees the existence of even interactions

Ψ_{i} : P_{f} (Λ) \to A_{Λ}^{s}

⁠, i = 1, 2, 3, and quasi-local maps

K_{i}^{(ε)} : A_{Λ}^{s} \to A_{Λ}^{s}

such that

\begin{aligned} Φ^{1} (b_{Λ} (x, n), ε) = Δ_{b_{Λ} (x, n)} ◦ K_{1}^{(ε)} & (Ψ_{1} (\{x\})) + \sum_{k = 1}^{n} ε Δ_{b_{Λ} (x, n)} ◦ K_{2}^{(ε)} (Ψ_{2} (b_{Λ} (x, k))) \\ + Δ_{b_{Λ} (x, n)} ◦ K_{3}^{(ε)} (Ψ_{3} (b_{Λ} (x, k))) . \end{aligned}

The

K_{i}^{(ε)}

are defined in terms of the spectral flow automorphism and are also even maps. Hence, by Lemma 4.1, Φ¹(b_Λ(x, n), ε) must also be even since the even observables form a subalgebra.

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 $χ = {f_{i} : i \in B}$ and $Y = {g_{j} : j \in B}$ be two collections of vectors in $ℓ^{2} (Z)$ such that

$χ \cup Y$ is an orthonormal basis for $ℓ^{2} (Z)$ ⁠;
there exist R ≥ 0 and collections ${x_{i} : i \in B}$ ⁠, ${y_{j} : j \in B}$ such that for all i, j,

\begin{array}{l} supp (f_{i}) \subset b (x_{i}, R), supp (g_{j}) \subset b (y_{j}, R), \\ i \neq j implies b (x_{i}, R) \cap b (x_{j}, R) = \emptyset = b (y_{i}, R) \cap b (y_{j}, R) . \end{array}

Furthermore, denote $χ_{W} = {f_{i} : supp (f_{i}) \subset W}$ and $Y_{W} = {g_{j} : supp (g_{j}) \subset W}$ ⁠. We will also assume the following:

(iii)
There exists a diameter N₀ such that for all intervals Λ, diam(Λ) > N₀ implies $χ_{Λ} \neq \emptyset$ and $Y_{Λ} \neq \emptyset$ ⁠.

Definition.

Let

η : P_{f} (Z) \to A_{l o c}^{f}

be the finite-range interaction defined by

η (b (x_{i}, R)) = 1 - a^{*} (f_{i}) a (f_{i}), η (b (y_{j}, R)) = a^{*} (g_{j}) a (g_{j}) .

(5.1)

Lemma 5.1.

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

Proof.

Let

(f_{n_{1}}, \dots, f_{n_{Λ}})

and

(g_{m_{1}}, \dots, g_{m_{Λ}})

be the collections of vectors whose support is contained in Λ. If necessary, complete the list to an orthonormal basis of ℓ²(Λ) with (h₁, …, h_p), p = |Λ|− n_Λ − m_Λ. Evidently H_Λ is uniformly gapped and non-negative. So we prove that

\ker (H_{Λ}) = span (ψ_{X} : X \subset [1, p]),

where we define

ϕ_{Λ} = f_{n_{1}} \land \dots f_{n_{Λ}}, ζ_{X} = ⋀ \{h_{i_{k}} : i_{k} \in X \subset [1, p]\}, ψ_{X} = ϕ_{Λ} \land ζ_{X} .

By calculation, ψ_X ∈ ker(H_Λ) for any X ⊂ [1, p]. But each term of the interaction H_Λ is a projection, the complement projection of some a^*(f_q)a(f_q). So H_Λψ = 0 implies ψ ∈ ran(a^*(f_i)a(f_i)) for each i = n₁, …, n_Λ.

Next, we show that the number of auxiliary orthonormal basis vectors h_i needed to complete $χ_{Λ}$ and $Y_{Λ}$ to a basis of ℓ²(Λ) is uniformly bounded in Λ, and that each h_i has support contained toward the edge of Λ.

Lemma 5.2.

Suppose diam(Λ) > N₀. Let $Z (Λ) = {h_{1}^{(Λ)}, \dots, h_{n}^{(Λ)}}$ , n = n(Λ), be a basis for the complement of $s p a n (χ_{Λ} \cup Y_{Λ})$ in ℓ²(Λ). Then

for each i ∈ [1, n] , $supp (h_{i}^{(Λ)}) \subset Λ \ {Int}_{3 R} (Λ)$ ⁠;
$| Z (Λ) | \leq 6 R$

Proof.

Let (ξ_k) denote the orthonormal basis from

χ \cup Y

⁠. Suppose supp(f) ⊂ Int_3R(Λ). Then x_i ∉ Λ implies supp(f_i) ∩ supp(f) = ∅, that is, ⟨f_i, f⟩ = 0 (respectively, y_j and ⟨g_j, f⟩ = 0). Hence

f = \sum ⟨ ξ_{k}, f ⟩ ξ_{k} = \sum_{ξ \in χ_{Λ} \cup Y_{Λ}} ⟨ ξ, f ⟩ ξ .

Hence

f \in span (χ_{Λ} \cup Y_{Λ})

⁠. Now, a basis of the orthogonal complement of ℓ²(Int_3R(Λ)) in ℓ²(Λ) is necessarily supported on Λ ∖Int_3R(Λ), proving (1). Additionally, the dimension of ℓ²(Λ ∖Int_3R(Λ)) is an upper bound for |Z(Λ)|, which proves (2).

This lemma has an immediate corollary:

Corollary 5.3.

Let

A (W)

denote the C^*-subalgebra of

A_{Z}^{f}

generated by the operators a^*(f), a(f) such that

f \in W \subset ℓ^{2} (W)

. Then for all intervals Λ with diameter larger than 6R,

A_{{I n t}_{3 R} (Λ)} \subset A (χ_{Λ} \cup Y_{Λ}) .

To conclude this section, we prove that the interaction defined in (5.1) satisfies $Z_{2}$ -LTQO. Denote D = max{N₀, 3R}. Recall that if n ≥ D then $H_{b_{Λ} (x, n)}$ is non-negative and frustration free with kernel indexed by $Z (b_{Λ} (x, n))$ ⁠.

Define the step-function Ω: [0, ∞) → [0, ∞) by

\begin{aligned} Ω (x) = \{\begin{matrix} 0 & if x \geq D \\ 2 & otherwise . \end{matrix}) \end{aligned}

Proposition 5.4.

Suppose diam(Λ) > 2D, and let x ∈ Λ, and

(n, k) \in N^{2}

be such that 0 ≤ k ≤ r_x, k ≤ n ≤ R_x. Let P_n denote the projection onto

H_{b_{Λ} (x, n)}

. Then for all

A \in A_{b_{Λ} (x, k)}^{+}

,

‖ P_{n} (A - ω_{Λ} (A)) P_{n} ‖ \leq Ω (z_{x} (n) - k) ‖ A ‖ .

Proof.

We handle the two cases of n when diam(b_Λ(x, n)) ≥ N₀ or diam(b_Λ(x, n)) < N₀. 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.

Suppose b_Λ(x, k) ⊂ Int_D(b_Λ(x, n)). Then z_x(n) − k ≥ D, necessarily. Denote

χ_{b_{Λ} (x, n)} = χ_{n} = \{f_{i_{1}} \dots, f_{i_{M}}\}, Z (b_{Λ} (x, n)) = Z (n) = \{h_{1}, \dots, h_{p}\}

Let

ψ_{X}^{n} = f_{i_{1}} \land \dots f_{i_{M}} \land h_{n_{1}} \land \dots h_{n_{| X |}}

be a generic unit norm basis vector of the kernel, indexed by

X \subset Z (n)

⁠. A calculation shows

‖ P_{n} (A - ω_{Λ} (A)) P_{n} ‖ \leq 6 R sup_{X \subset Z (n)} | ⟨ ψ_{X}^{n}, A ψ_{X}^{n} ⟩ - ω_{Λ} (A) | + 2^{6 R} sup_{X \neq Y} | ⟨ ψ_{X}^{n}, A ψ_{Y}^{n} ⟩ | .

But by the theory of quasi-free states and Corollary 5.3,

sup_{X \subset Z (n)} | ⟨ ψ_{X}^{n}, A ψ_{X}^{n} ⟩ - ω_{Λ} (A) | = sup_{X \neq Y} | ⟨ ψ_{X}^{n}, A ψ_{Y}^{n} ⟩ | = 0 .

Now suppose b_Λ(x, k) is not contained in the D-interior of b_Λ(x, n). This implies z_x(n) − k < D. And by the trivial commutator bound,

‖ P_{n} (A - ω_{Λ} (A)) P_{n} ‖ \leq 2 ‖ A ‖ = Ω (z_{x} (n) - k) ‖ A ‖ .

Last, suppose diam(b_Λ(x, n)) < N₀. Then n − k ≤ n < N₀ ≤ D. Hence z_x(n) − k < D as well, and the trivial bound agrees with Ω. Conclude that H_Λ satisfies LTQO for Ω.

ACKNOWLEDGMENTS

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.

APPENDIX: DETAILS ABOUT $ℱ$ -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.

Definition.

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

$‖ F ‖ = \sum_{x \in Z} F (x) < \infty$ ⁠;
there exists C_F > 0 such that for all $x, y \in Z$ ⁠,

\sum_{z \in Z} F (| x - z |) F (| z - y |) \leq C_{F} F (| x - y |) .

Furthermore, if Φ is an interaction, then the

F

-norm of Φ (with respect to F) is defined as

\begin{aligned} ‖ {Φ ‖}_{F} = sup \{\sum_{\begin{array}{c} Z \in P_{f} (Z) \\ x, y \in Z \end{array}} \frac{‖ Φ (Z) ‖}{F (| x - y |)}\} \in [0, \infty] . \end{aligned}

(A1)

Example.

Suppose h: [0, ∞) → [0, ∞) is a monotone increasing, subadditive function and κ > 2. The following function F defines an

F

-function,

F (r) = e^{- h (r)} \frac{L}{{(1 + c r)}^{κ}}, L, c > 0 .

(A2)

The

F

-function in () and following properties will be used extensively in the proof of spectral gap stability.

Proposition A.1.

Suppose Φ is an interaction with finite

F

-norm for some F. Then

\begin{aligned} 1 . f o r a n y c o l l e c t i o n Z_{1} \subset Z_{2} \subset \dots \subset Z_{N}, \sum_{k = 1}^{N} ‖ Φ (Z_{k}) ‖ \leq ‖ {Φ ‖}_{F} F (d i a m (Z_{1})); \\ 2 . i f η i s a u n i f o r m l y b o u n d e d, f i n i t e r a n g e i n t e r a c t i o n, t h e n ‖ {η ‖}_{F} < \infty \\ a n d ‖ η + {Φ ‖}_{F} < \infty . \end{aligned}

Proof.

Let diam(Z₁) = n, and choose x, y ∈ Z₁ such that |x − y| = n. Then

\begin{aligned} \sum_{k = 1}^{N} ‖ Φ (Z_{k}) ‖ \leq \sum_{\begin{array}{c} X \in P_{f} (Z) \\ x, y \in X \end{array}} ‖ Φ (X) ‖ \leq ‖ {Φ ‖}_{F} F (n) . \end{aligned}

Now, denote the range of η by R and uniform bound by M. Suppose

x, y \in Z

⁠. If

Z \in P_{f} (Z)

contains x, y, and Φ(Z) ≠ 0, then Z ⊂ b(x, R) ∩ b(y, R). Hence

\begin{aligned} \sum_{\begin{array}{c} X \in P_{f} (Z) \\ x, y \in X \end{array}} ‖ η (X) ‖ \leq 2^{3 R} M . \end{aligned}

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

τ_{s, t}^{Φ, Λ} (A) = U {(s, t)}^{*} A U (s, t), A \in A_{Λ},

with

H_{Λ} (s) = \sum_{X \subset Λ} Φ (X, s)

and

\frac{d}{d s} U (s, t) = i H_{Λ} (s) U (s, t), U (t, t) = 1, s, t \in I \subset R,

converges to a strongly continuous co-cycle of automorphism, $τ_{s, t}^{Φ}$ ⁠, of the algebra of quasi-local observables $A_{Γ}$ ⁠, which is defined as the norm completion of the (strictly) local observables given by

A_{Γ}^{l o c} = ⋃_{Λ \in P_{f} (Γ)} A_{Λ} .

One sufficient condition is that Φ(⋅, t) is a continuous curve taking values in the space $B_{F}$ ⁠, 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 $B_{F} (I)$ as the set of all such continuous curves, and for $Φ \in B_{F} (I)$ ⁠, the function

‖ {Φ ‖}_{F} (t) = sup_{x, y \in Γ} \frac{1}{F (d (x, y))} \sum_{\overset{X \in P_{f} (Γ) :}{x, y \in X}} ‖ Φ (X, \cdot) ‖ (t)

(A3)

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

lim_{Λ ↑ Γ} ‖ τ_{s, t}^{Φ, Λ} (A) - τ_{s, t}^{Φ} (A) ‖ = 0, for all A \in A_{Γ}^{l o c} .

The limit is taken over any sequence of $Λ_{n} \in P (Γ)$ 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

‖ τ_{t, s}^{Φ} (A) - τ_{t, s}^{Ψ} (A) ‖ \leq \frac{2 ‖ A ‖}{C_{F}} ‖ F ‖ | X | e^{2 min (I_{t, s} (Φ), I_{t, s} (Ψ))} I_{t, s} (Φ - Ψ) .

(A4)

which holds for all $A \in A_{X}$ and s, t ∈ I, and where, for $Φ \in B_{F} (I)$ ⁠, and s, t ∈ I, the quantity I_t,s(Φ) is defined by

I_{t, s} (Φ) = C_{F} \int_{min (t, s)}^{max (t, s)} ‖ {Φ ‖}_{F} (r) d r .

(A5)

It is often important to include in the definition of the finite-volume 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 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.

Definition.

Let $(Γ, d, A_{Γ})$ be a quantum lattice system, F be an F-function for (Γ, d), and $I \subset R$ be an interval. We say that a sequence of interactions ${Φ_{n}}_{n \geq 1}$ converges locally in F-norm to Φ such that

$Φ_{n} \in B_{F} (I)$ for all n ≥ 1,
$Φ \in B_{F} (I)$ ⁠, and
for any $Λ \in P_{0} (Γ)$ and each [a, b] ⊂ I, one has

lim_{n \to \infty} \int_{a}^{b} ‖ (Φ_{n} - Φ) {↾_{Λ} ‖}_{F} (t) d t = 0 .

(A6)

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

α_{ϵ}^{Λ, \partial}

for the curve of Hamiltonians

H_{Λ}^{\partial} (ϵ), ϵ \in [0, ϵ_{0})

⁠, defined in (), 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

H_{Λ}^{\partial} (ϵ)

in the stability argument, but this assumption is not needed for the construction of

α_{ϵ}^{Λ, \partial}

⁠. The automorphisms

α_{ϵ}^{Λ, \partial}

are generated by the self-adjoint operators

D_{Λ}^{\partial} (ϵ)

⁠, defined by

D_{Λ}^{\partial} (ϵ) = K_{ϵ}^{Λ, \partial} (\sum_{X \subset Λ_{D_{1}}} Φ (X) + s \sum_{X \subset (Λ \ Λ_{D_{2}})} Φ (X)),

(A7)

where the map

K_{ϵ}^{Λ, \partial} : A_{Λ} \to A_{Λ}

is given by

K_{ϵ}^{Λ, \partial} (A) = \int_{- \infty}^{\infty} τ^{H_{Λ}^{\partial} (ϵ)} (A) W_{γ} (t) d t .

(A8)

Note that

K_{ϵ}^{Λ, \partial}

is defined as a linear map, but it depends itself on

H_{Λ}^{\partial} (ϵ)

and therefore

D_{Λ}^{\partial} (ϵ)

depends non-linearly on the perturbation. In Ref. 9 [Sec. V D], a detailed study of transformations of the form

K_{ϵ}^{Λ, \partial}

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-function

\tilde{F}

of the form () and interactions

Ψ^{Λ, \partial} \in B_{\tilde{F}} ([0, ϵ_{0}])

such that

D_{Λ}^{\partial} (ϵ) = \sum_{X \subset Λ} Ψ^{Λ, \partial} (X, ϵ) .

Furthermore, there exists an interaction

Ψ \in B_{\tilde{F}} ([0, ϵ_{0}])

such that for any sequences

Λ_{n} = [a_{n}, b_{n}] \subset Z

, ∂_n = (D_1,n, D_2,n, s_n), the interactions

Ψ^{Λ_{n}, \partial_{n}}

have uniformly bounded

\tilde{F}

-norm and converge locally in

\tilde{F}

-norm to Ψ.

As a consequence, we can apply the following theorem from Ref. 9 to the sequences of interactions $Ψ^{Λ_{n}, \partial_{n}}$ ⁠.

Theorem A.3

(Ref. 9, Theorem 3.8). Let

{(Φ_{n})}_{n \geq 1}

be a sequence of time-dependent interactions on Γ with Φ_n converging locally in the F-norm to Φ with respect to F. Suppose that for every [a, b] ⊂ I,

sup_{n \geq 1} \int_{a}^{b} ‖ {Φ_{n} ‖}_{F} (t) d t < \infty .

(A9)

Then for any

X \in P_{0} (Γ)

,

lim_{n \to \infty} ‖ τ_{t, s}^{Φ_{n}} (A) - τ_{t, s}^{Φ} (A) ‖ = 0

(A10)

for all

A \in A_{X}

and each s, t ∈ I. Moreover, the convergence is uniform for s, t in compact intervals.

Now, consider a sequence

Λ_{n} = [a_{n}, b_{n}] \subset Z

⁠, ∂_n = (D_1,n, D_2,n, s_n). 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

Ψ^{Λ_{n}, \partial_{n}}

to one and the same spectral flow for the infinite chain: for ϵ ∈ [0, 1],

lim_{n \to \infty} α_{ϵ}^{Λ_{n}, \partial_{n}} (A) = α_{ϵ} (A), for all A \in A_{Γ}^{loc} .

(A11)

REFERENCES

1.

Bachmann

,

S.

,

Michalakis

,

S.

,

Nachtergaele

,

B.

, and

Sims

,

R.

, “

Automorphic equivalence within gapped phases of quantum lattice systems

,”

Commun. Math. Phys.

309

,

835

–

871

(

2012

).

https://doi.org/10.1007/s00220-011-1380-0

Google Scholar

Crossref

2.

Bravyi

,

S.

,

Hastings

,

M.

, and

Michalakis

,

S.

, “

Topological quantum order: Stability under local perturbations

,”

J. Math. Phys.

51

,

093512

(

2010

).

https://doi.org/10.1063/1.3490195

Google Scholar

Crossref

3.

Datta

,

N.

,

Fernández

,

R.

, and

Fröhlich

,

J.

, “

Low-temperature phase diagrams of quantum lattice systems. I. stability for quantum perturbations of classical systems with finitely-many ground states

,”

J. Stat. Phys.

84

,

455

(

1996

).

https://doi.org/10.1007/bf02179651

Google Scholar

Crossref

4.

de Roeck

,

W.

and

Salmhofer

,

M.

, “

Persistence of exponential decay and spectral gaps for interacting fermions

,”

Commun. Math. Phys.

(published online,

2018

); e-print arXiv:1712.00977 (

2017

).

https://doi.org/10.1007/s00220-018-3211-z

Google Scholar

5.

Hastings

,

M. B.

, “

The stability of free Fermi Hamiltonians

,” e-print arXiv:1706.02270 (

2017

).

Google Scholar

6.

Hastings

,

M. B.

and

Wen

,

X. G.

, “

Quasi-adiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance

,”

Phys. Rev. B

72

,

045141

(

2005

).

https://doi.org/10.1103/physrevb.72.045141

Google Scholar

Crossref

7.

Kennedy

,

T.

and

Tasaki

,

H.

, “

Hidden symmetry breaking and the Haldane phase in s = 1 quantum spin chains

,”

Commun. Math. Phys.

147

,

431

–

484

(

1992

).

https://doi.org/10.1007/bf02097239

Google Scholar

Crossref

8.

Michalakis

,

S.

and

Zwolak

,

J. P.

, “

Stability of frustration-free Hamiltonians

,”

Commun. Math. Phys.

322

,

277

–

302

(

2013

).

https://doi.org/10.1007/s00220-013-1762-6

Google Scholar

Crossref

9.

Nachtergaele

,

B.

,

Sims

,

R.

, and

Young

,

A.

, “

Quasi-locality bounds for quantum lattice systems and perturbations of gapped ground states. Part I

” (unpublished).

10.

Nachtergaele

,

B.

,

Sims

,

R.

, and

Young

,

A.

, “

Quasi-locality bounds for quantum lattice systems and perturbations of gapped ground states. Part II

” (unpublished).

11.

Nachtergaele

,

B.

,

Sims

,

R.

, and

Young

,

A.

, “

Stability of gapped phases of fermionic lattice systems

” (unpublished).

12.

Nachtergaele

,

B.

,

Sims

,

R.

, and

Young

,

A.

, “

Lieb-Robinson bounds, the spectral flow, and stability for lattice fermion systems

,” in

Proceedings of the conference QMATH13

(

Contemporary Mathematics, American Mathematical Society

,

2018

).

Google Scholar

Crossref

13.

Ogata

,

Y.

, “

A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization. I

,”

Commun. Math. Phys.

348

(

3

),

847

–

895

(

2016

), MR 3555356.

https://doi.org/10.1007/s00220-016-2696-6

Google Scholar

Crossref

14.

Ogata

,

Y.

, “

A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization. II

,”

Commun. Math. Phys.

348

(

3

),

897

–

957

(

2016

), MR 3555357.

https://doi.org/10.1007/s00220-016-2697-5

Google Scholar

Crossref

15.

Ogata

,

Y.

, “

A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization. III

,”

Commun. Math. Phys.

352

(

3

),

1205

–

1263

(

2017

).

https://doi.org/10.1007/s00220-016-2810-9

Google Scholar

Crossref

16.

Szehr

,

O.

and

Wolf

,

M. M.

, “

Perturbation theory for parent Hamiltonians of matrix product states

,”

J. Stat. Phys.

159

,

752

–

771

(

2015

); e-print arXiv:1402.4175.

https://doi.org/10.1007/s10955-015-1204-2

Google Scholar

Crossref

17.

Yarotsky

,

D. A.

, “

Ground states in relatively bounded quantum perturbations of classical lattice systems

,”

Commun. Math. Phys.

261

,

799

–

819

(

2006

).

https://doi.org/10.1007/s00220-005-1456-9

Google Scholar

Crossref

18.

Young

,

A.

, “

Spectral properties of multi-dimensional quantum spin systems

,” Ph.D. thesis,

University of California

,

Davis

,

2016

.

Google Scholar

2018

Author(s)

Stability of gapped ground state phases of spins and fermions in one dimension

I. INTRODUCTION

II. SETTING AND MAIN RESULT

A. Notations

B. Assumptions

C. Local topological quantum order

D. The main result

III. STABILITY OF SPECTRAL GAP IN SPIN CHAINS

A. Perturbations at the boundary

B. Spectral flow decomposition

C. Relative form boundedness of perturbations

D. The thermodynamic limit

IV. STABILITY OF SPECTRAL GAP IN FERMION CHAINS

A. Quasi-local maps

B. Transformation of even fermion interactions

V. EXAMPLE OF EVEN HAMILTONIAN SATISFYING STABILITY HYPOTHESES

ACKNOWLEDGMENTS

APPENDIX: DETAILS ABOUT $ℱ$ -FUNCTIONS AND SPECTRAL FLOW

1. $F$ -functions and decay of interactions

2. Thermodynamic limit of the spectral flow

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Stability of gapped ground state phases of spins and fermions in one dimension

I. INTRODUCTION

II. SETTING AND MAIN RESULT

A. Notations

B. Assumptions

C. Local topological quantum order

D. The main result

III. STABILITY OF SPECTRAL GAP IN SPIN CHAINS

A. Perturbations at the boundary

B. Spectral flow decomposition

C. Relative form boundedness of perturbations

D. The thermodynamic limit

IV. STABILITY OF SPECTRAL GAP IN FERMION CHAINS

A. Quasi-local maps

B. Transformation of even fermion interactions

V. EXAMPLE OF EVEN HAMILTONIAN SATISFYING STABILITY HYPOTHESES

ACKNOWLEDGMENTS

APPENDIX: DETAILS ABOUT ℱ-FUNCTIONS AND SPECTRAL FLOW

1. F-functions and decay of interactions

2. Thermodynamic limit of the spectral flow

REFERENCES

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APPENDIX: DETAILS ABOUT $ℱ$ -FUNCTIONS AND SPECTRAL FLOW

1. $F$ -functions and decay of interactions