Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems

We prove that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finite-system Hamiltonians uniform in the system size. To obtain this result, we adapt the Bravyi-Hastings-Michalakis strategy to the GNS representation of the infinite-system ground state.


Introduction
One of the characteristic properties of gapped topologically ordered ground state phases of quantum many-body systems is the stability of the spectral gap above the ground state with respect to small perturbations of the Hamiltonian.Stability results for the ground state gap have a long history.The first result that included the stability of the gap of the AKLT chain is due to Yarotsky [48].The approach we follow in this paper has a much broader range of applicability; it was introduced by Bravyi, Hastings, and Michalakis [10] and further developed in [11,21,32,38].Other approaches have been introduced in recent years [14][15][16][17].These new approaches can also treat some cases of models with unbounded on-site Hamiltonians, see [38, Section 1] for a more detailed discussion.The Bravyi-Hastings-Michalakis strategy, however, is the only approach that handles general cases with non-trivial topological order.
One obstacle to proving spectral gaps for topological insulators is the common occurrence of gapless edge states.Spectral analysis for interacting many-body systems is usually carried out for finite systems for which edge states typically imply that there is no spectral gap uniform in the system size.Nevertheless, there may be a bulk gap, meaning excitations away from the boundary of the system have energy bounded below uniformly in the system size.The goal of this work is to prove stability for the bulk gap in a way that does not require the assumption of a uniform positive lower bound in the spectrum of finite systems.Previously, it was shown how certain cases can be handled by considering sequences of finite systems with suitable boundary conditions.For example, such an approach may work if the edge states are absent in the model considered with periodic boundary conditions [32,38].In general, however, there may not be a suitable boundary condition that 'gaps out' the boundary modes or we may not know whether such a boundary condition exists.Systems defined on a quasicrystal structure, for example, may be an instance where no simple way of removing gapless edge modes is available [30].In our approach here, we only assume that the infinite system described in the GNS representation of the ground state has a gap.Under natural assumptions consistent with the current state-of-the-art for ground state stability, we prove that sufficiently small but extensive perturbations do not close the gap.
We adapt the strategy of Bravyi, Hastings, and Michalakis [10,11,32] and use the techniques we developed in [37,38] to handle the infinite system setting.From a certain perspective, and apart from the technical aspects to deal with unbounded Hamiltonians, the infinite system setting allows for a simplification in the statement of conditions and the main result.In particular, the local topological quantum order (LTQO) condition is simpler to state directly for the infinite system.The LTQO property is well know to hold for one-dimensional systems with MPS ground states [38,Appendix B].It is also well-established for Kitaev's quantum double models [13,26] and the Levin-Wen string-net model [28,44].LTQO was recently also shown to hold for the AKLT model on a decorated hexagonal lattice [31].
For concreteness, we work in the quantum spin system setting but, using the arguments of [36], our approach is applicable to lattice fermion systems too.
The assumption that the bulk Hamiltonian has a gap in the spectrum above the ground state appears in several important recent works.For example, the construction of an index for the classification of symmetry-protected topological phases in the works of Ogata and co-authors makes use of this assumption [33,[39][40][41].Other examples are in the recent work on adiabatic theorems for infinite many-body systems [2,3,23].All of these works use the same general setting as described here in Section 2. In addition to the main stability result, we also prove Theorem 3.6, which shows that a differentiability assumption introduced in [33] and also used in later work [39,[41][42][43] is always satisfied.

Setup and statement of the main results
2.1.Setup and notation.The models considered in this work are defined on a ν-regular discrete metric space (Γ, d), for some ν > 0. This means that there exists κ > 0 so that for all x ∈ Γ, n ≥ 1, |b x (n)| ≤ κn ν , where b x (n) = {y ∈ Γ | d(x, y) ≤ n}.For Λ ∈ P 0 (Γ), the finite subsets of Γ, and n ≥ 0, we also define the sets Λ(n) by The algebra of local observables of the system is the usual A loc = Λ∈P 0 (Γ) A Λ .Here, A Λ is the matrix algebra x∈Λ M dx with d x the dimension of the spin at x.The C * -algebra of quasi-local observables A is the completion of A loc with respect to the operator norm.For A ∈ A loc , the support of A, denoted by supp A, is the smallest X ⊂ Γ such that A ∈ A X .For any X ⊂ Γ, Π X : A → A X is the conditional expectation with respect to the tracial state ρ on A: In particular, for local A, Π X (A) is a normalized partial trace.We are specifically interested in systems defined on infinite Γ and often want to consider approximations A n ∈ A Λn of A ∈ A, where Λ n ∈ P 0 (Γ) is an increasing sequence of finite volumes such that n Λ n = Γ.We call such a sequence (Λ n ) an increasing and absorbing sequence (IAS).It will often be important to have an estimate for the speed of convergence of A n → A, in terms of a non-increasing function g : [0, ∞) → (0, ∞) that vanishes at infinity, which we call a decay function.
In this paper we will only use decay functions that satisfy a moment condition of the form (2.3) n≥0 (n + 1) µ g(n) < ∞, for some µ ≥ ν.
For a proof that A (Λn),g is the Banach space of all A ∈ A for which A (Λn),g < ∞, see [33].In fact, A (Λn),g is a Banach * -algebra.
For any decay function g satisfying (2.5), any two norms from { • x,g | x ∈ Γ} are equivalent.
In this case, A g is a Banach * -algebra.Elements A ∈ A g are called g-local.
We will often also assume that a decay function g is uniformly summable over Γ, i.e., (2.9) and additionally, that there is a constant C > 0 such that (2.10) z∈Γ g(d(x, z))g(d(z, y)) ≤ Cg(d(x, y)), for all x, y ∈ Γ.
Assumption 2.1 (Initial Interaction).We assume the initial model is defined by a finite-range, uniformly bounded, frustration-free interaction h given in terms of a family h = {h x } x∈Γ which satisfies: i.There is a number R ≥ 0, called the interaction radius, for which h * ii.These terms are uniformly bounded in the sense that (2.12) iii.The interaction is frustration-free, meaning that h x ≥ 0 for all x ∈ Γ and for any Λ ∈ P 0 (Γ), (2.13) min spec(H Λ ) = 0 where The frustration-free condition implies that the ground state space is ker(H Λ ) for any finite volume Λ.Moreover, ψ ∈ ker(H Λ ) if and only if ψ ∈ ker(h x ) for each x ∈ Γ with supp(h x ) ⊂ Λ.Thus, denoting by P Λ the orthogonal projection onto ker(H Λ ), for any Λ 0 ⊂ Λ, one has (2.14) For such a model, the derivation δ 0 determining the infinite system dynamics is given by It is a standard result that there is a closed derivation extending δ 0 , which we also denote by δ 0 , with domain dom(δ 0 ) for which A loc is a core [9, Theorem 6.2.4] (note that the factor i is absorbed in the definition of the derivation in this reference).The system dynamics is then the strongly continuous one-parameter group of C * -automorphisms {τ In fact, this differential equation holds for all A ∈ dom(δ 0 ).Two other general properties are: More generally, quantum spin models can be defined by an interaction on Γ which, by definition, is a map Φ : P 0 (Γ) → A loc , with the property that Φ(X) * = Φ(X) ∈ A X for all X ∈ P 0 (Γ).For any decay function g, an interaction norm is defined by When the above quantity is finite for some interaction Φ, the function g is said to measure the decay of Φ.If g is an F -function, the norm • g is called an F-norm.If g is summable, in the sense of (2.9), and Φ g < ∞, then a closable derivation on A loc can be defined by setting One can prove conditions that guarantee that the derivation δ defined on A loc is a generator of a strongly continuous dynamics given by automorphisms of A [8,9].In practice, however, one usually directly proves the existence of the thermodynamic limit of the Heisenberg dynamics {τ t | t ∈ R}.Standard results along these lines prove the existence of the dynamics for Φ in a suitable Banach space of interactions [9,46,47] starting from a convergent series for small |t|.An alternative approach, based on Lieb-Robinson bounds [29], was introduced by Robinson [45].Lieb-Robinson bounds can be derived for any interaction Φ with a finite F -norm [34], and this allows one to extend the results for existence of the dynamics beyond the Banach spaces of interactions B λ introduced by Ruelle [46].These ideas are important for the construction of the spectral flow automorphisms [5].This and some other generalizations relevant for the present work are discussed in detail in [37].
Recall that infinite-volume ground states associated to δ are those states ω on A that satisfy In the case of a frustration-free model as in (2.13), a state ω is called a zero-energy ground state, or a frustration free ground state, if ω(h x ) = 0 for all x ∈ Γ.It is easy to see that a zero-energy ground state satisfies (2.19).
Let (H, π, Ω) be the GNS triple of ω.For the GNS representation of a ground state, as in (2.19), there exists a unique, non-negative self-adjoint operator H on H, with dense domain dom H, satisfying HΩ = 0 and (2.20) π(τ t (A)) = e itH π(A)e −itH for all A ∈ A and t ∈ R.
The full domain of H is seldom described explicitly.However, for all systems we consider in this paper, π(A loc )Ω is a core for H.
The (GNS) gap of the model in the state ω is defined as If the set on the RHS is empty, one defines gap(H) = 0. We say that a ground state ω is gapped if gap(H) > 0.
The equivalence of the following two conditions is easy to verify: i.For some γ > 0, ω satisfies ii.The ground state of the GNS Hamiltonian H is unique and gap(H) ≥ γ.A case of special interest is when Γ is infinite and describes the bulk of a physical model while the same system on a subset of Γ with a boundary would describe an edge.In the first situation we will refer to the GNS gap as the bulk gap of the system.A model with the same interaction restricted to a subspace of Γ describing an edge, may have a vanishing gap while the bulk gap is positive.This is precisely the situation of interest here.
We will use that the GNS representation π is an isometry.This follows from the fact that A is simple [18,Theorem 5.1], which implies that ker π = {0}.

Main results
. We now state the assumptions for the main results.Assumption 2.2 (Bulk gap).We assume γ 0 := gap(H 0 ) > 0, where H 0 is the GNS Hamiltonian of an infinite-volume, zero-energy ground state ω 0 of a finite-range, uniformly bounded, frustration free interaction {h x } as in Assumption 2.1.
We also need to impose a condition that the local gaps do not close too fast.There generally is some freedom in choosing the family of finite volumes on which to impose this condition.We will assume that there is a family Λ(x, n) ∩ Λ(y, n) = ∅ for all x, y ∈ T i n with x = y.In such cases, we say that T is of (c, ζ)-polynomial growth.
As an example, in the case of Γ = Z ν , we may take for Λ(x, n) the ∞ -ball of radius n centered at x, define I n = Λ(0, n) and, for each i ∈ I n , set (2.24) T i n = {x ∈ Z ν |x j = i j mod 2n + 1, i = 1, . . ., ν}.Assumption 2.3 (Local gaps).For an interaction {h x } of range R, we assume there exist families S and T , such that T separates S and is of ζ-polynomial growth, and an exponent α ≥ 0 and constant γ 1 > 0, and such that the finite-volume Hamiltonians satisfy: It is important here that the local gaps are allowed to vanish in the limit of infinite system size.For example, certain types of topologically ordered two-dimensional systems are expected to have chiral edge modes with an energy of order L −1 on a finite volume of diameter L. Whether or not such edge modes occur in frustration-free systems, however, is not clear.For the class of systems studied in [27], the authors find that finite-volume gaps of a system with gapless edge modes in the thermodynamic limit would have to decay at least as fast as L −3/2 .Other results of this type are in [1,19,24].This is consistent with the gapless boundary modes found in a class of toy models called Product Vacua with Boundary States which are of order L −2 [4,6].In any case, regardless of the possible values of the exponent α, we will prove stability of the bulk gap.
The next assumption was introduced in the form we use here in [32] where it is called Local Topological Quantum Order (LTQO).

Assumption 2.4 (LTQO). There is a decay function G
and such that for all m ≥ k ≥ 0, x ∈ Γ, and A ∈ A bx(k) , the ground state projections satisfy (2.27) As explained in detail in [38,Section 8], if both the initial Hamiltonian and the perturbation (see below) have a local gauge symmetry, only observables A that commute with this symmetry need to satisfy (2.27).Other discrete symmetries can be treated similarly (see [38,Section 8]).Therefore, the stability results proved here (Theorems 2.8 and 2.9) will also hold for symmetry-protected topological phases.
It is an interesting observation that the GNS Hamiltonians associated to frustration free models which satisfy Assumption 2.4 automatically have a unique ground state.This is the content of the following propostion.
Proposition 2.5.Let ω 0 be an infinite-volume, zero-energy ground state of a frustration-free model satisfying Assumption 2.4.The kernel of the GNS Hamiltonian H 0 is one-dimensional.
Proof.By way of contradiction, let us assume there is a unit vector ψ ∈ ker(H 0 ) with ψ, Ω = 0. To simplify notation let us denote by Ã := π 0 (A) the representative of A ∈ A in the GNS space.The LTQO condition (2.27) can be restated in B(H) as follows: for all m ≥ k ≥ 0, x ∈ Γ, and First, since π 0 (A loc )Ω is dense in H, there is x ∈ Γ and operators A n ∈ A bx(n) for which Next, we turn to the perturbations of the Hamiltonian H 0 .We consider Φ(x, n) * = Φ(x, n) ∈ A bx(n) for all x ∈ Γ and n ≥ 0. These define what we call an anchored interaction Φ.By regrouping, we need only consider those terms with n ≥ R.
Remark 2.7.Assume Φ satisfies Assumption 2.6.As indicated in (2.11), for any 0 < a < a and ξ > ν, the function F : [0, ∞) → (0, ∞) given by (2.34) F (r) = e −a r θ (1 + r) ξ for all r ≥ 0, is an F -function on Γ.Let δ = a − a > 0 and note that for any x, y ∈ Γ with d(x, y) ≥ R, we have (2.34).The focus of this work is to analyze the stability of the bulk gap under the presence of perturbations given by an anchored interaction Φ satisfying Assumption 2.6.We consider perturbed Hamiltonians of the form where, for any finite volume Λ ∈ P 0 (Γ), (2.37) Clearly, V Λ ∈ A Λ is bounded and self-adjoint, and so H(Λ, s) defines for all s ∈ R a self-adjoint Hamiltonian on H with the same dense domain as H 0 .
In the next several sections we will prove the following theorem, which establishes that the spectral gap of H(Λ, s) remains open for small |s| uniformly in the finite volume Λ.
We remark that the quantity s 0 (γ) only depends on the values of κ and ν of the lattice, h , the gap γ 0 , the parameters in Assumption 2.3, the decay function in Assumption 2.4, and a suitable F -norm of the perturbation Φ.From the arguments in this paper, one can derive an explicit lower bound for s 0 (γ) in terms of these quantities, see Section 5.2.
We also investigate the situation where the perturbation region Λ tends to all of Γ.Consider any IAS (Λ n ).We will denote by τ (respectively, δ s ) being the a priori well-defined strongly continuous dynamics on A (respectively, the closure of the derivation restricted to A loc ) generated by the interaction h + sΦ.Neither of these limits depend on the choice of IAS sequence Λ n .
Our second result is then concerned with the ground state and its gap for a family of extensive perturbations.In particular, the uniformity of the stability result in Theorem 2.8 allows one to prove, almost as a corollary, that for all |s| ≤ s 0 (γ) there is a gapped ground state ω s of δ s in the sense of (2.22).To make this precise, we introduce the limiting spectral flow.For any γ > 0 and IAS (Λ n ), take where the spectral flows α Λn s will be introduced in more detail in the next section, see (3.23).For now, it suffices to observe that this limit exists and is independent of the choice of IAS.In fact, the interactions defining the spectral flows α Λn s converge locally in F -norm by arguments as in [37, Section VI.E.2].This limiting spectral flow α s defines a strongly continuous co-cycle of automorphisms of A, and moreover, under the assumptions we have made, for A ∈ A loc , s → α s (A) is differentiable to all orders.We prove bounded differentiability for A ∈ A g , for suitable g in Theorem 3.6.
Theorem 2.9 (Stability of the bulk gap).Under the assumptions of Theorem 2.8, let γ ∈ (0, γ 0 ) and take s with |s| < s 0 (γ).The state ω s = ω 0 • α s is a gapped ground state of the perturbed infinite dynamics δ s , i.e. (2.44) In particular, the GNS Hamiltonian H s of ω s has a one-dimensional kernel and spec H s has a gap above its ground state bounded below by γ.

Quasi-locality, Domains and Local Decompositions
The strategy used here for proving spectral gap stability of infinite systems relies in an essential way on quasi-locality properties of the observables, the dynamics, and several transformations defined in terms of the dynamics.Quasi-locality of observables is the topic of Section 3.1.In Section 3.1.1,we recall general methods for making strictly local approximations of both quasilocal observables and maps.The specific quasi-local maps and estimates used in the stability proof are discussed in Sections 3.1.2-3.1.4.Their corresponding counterparts in the GNS representation are treated in Section 3.2.In Section 3.3, we prove how the action of certain unbounded operators on a dense domain can expressed as limits of sequences of bounded operators with finite support.In Section 3.4 we prove a local bound on the generator of the quasi-adiabatic evolution which, while not used in this work, is important for applications.
3.1.Quasi-Locality.We first recall some general features of quasi-locality estimates and then turn to some important examples relevant for this work.[A, B] ≤ B , whenever B ∈ A loc Γ\X , the following estimate was shown in [12,35]: A linear map K : A → A is said to be quasi-local with constant C ≥ 0, power p ≥ 0, and decay function G if Using (3.2), for such a map K and A ∈ A bx(k) , we have When the corresponding decay function G is summable, this estimate guarantees the absolute convergence of telescopic sums, i.e. for any n 0 ≥ 0, since the terms satisfy A common choice is n 0 = 0 and we adopt the notation We now review four quasi-local maps as well as a few of their important properties that are used in the stability argument.For more details of these maps see [37].Throughout Sections 3.1.2-3.2 we work under the assumptions of Theorem 2.8.

Dynamics. It is well-known that the unperturbed dynamics τ
(0) t defined as in (2.16) satisfies an exponential Lieb-Robinson bound [29].Namely, for every µ > 0 there exists C µ > 0 and v µ > 0 such that the bound (3.8) [τ It is easy to check that the perturbed interaction h + sΦ Λ has a finite F -norm for the same F as Φ, and that this F -norm is uniformly bounded in |s| ≤ 1 and Λ.As a consequence, there are C F > 0 and v F > 0, independent of s and Λ, such that for any choice of X, Y ∈ P 0 (Γ), (3.9) [τ Since each sV Λ is bounded and self-adjoint, [9, Proposition 5.4.1]implies that | t ∈ R} is a one-parameter family of unitaries on A which are uniquely defined as the A-valued solution of These unitaries are quasi-local as, for any A ∈ A loc and t > 0, An application of (3.8) then shows that for any µ > 0 and A ∈ A X with X ∈ P 0 (Γ \ Λ), (3.13) [K for any s, t ∈ R. Thus, K (Λ,s) t ∈ A g for any exponential g, by (3.2).
3.1.3.Weighted Integral Operators.Fix γ > 0. For each Λ ∈ P 0 (Γ) and s ∈ R, we define two weighted integral operators where the real-valued functions w γ , W γ ∈ L 1 (R), are defined in [37, Section VI.B].In particular, they decay faster than any stretched exponential.Both of these maps depend on the choice of γ through their weight functions, w γ and W γ respectively, but we suppress this in the notation.
Arguing as in [37, Section VI.E.1], see also [38,Section 4.3.2],we find that for all A ∈ A e. these maps are bounded uniformly with respect to s ∈ R and Λ ∈ P 0 (Γ).Moreover, they are uniformly quasi-local in the sense that for each K ∈ {F, G} there is a decay function G K such that: for any choice of X, Y ∈ P 0 (Γ), we have (3.16)sup for all A ∈ A X and B ∈ A Y .As shown in [37, Lemma 6.10-6.11], the decay functions G K can be made explicit.For our purposes here, we need only stress that they can be taken independent of Λ ∈ P 0 (Γ) and s ∈ [−1, 1], and with decay faster than any power.Thus, for any µ ≥ 0, (3.17) 3.1.4.The Spectral Flow.Fix γ > 0. For each Λ ∈ P 0 (Γ) and s ∈ R, denote by with G Λ s as defined in (3.14).Clearly, D(Λ, s) is self-adjoint and s → D(Λ, s) is uniformly bounded by (3.15).
For t ∈ R fixed, the strong derivative of s → τ is given by the Duhamel formula [37, Proposition 2.7]: Using (3.19), one obtains the norm continuity of s → D(Λ, s) from the following estimate: Given these properties of D(Λ, s), there is a unique solution of which is given by unitaries in A. Using similar arguments as in (3.13) with (3.16) and (3.18), one can show that for s > 0 for any A ∈ A X with X ∈ P 0 (Γ \ Λ).Thus, U (Λ, s) ∈ A g for some g with finite moments of all orders by (3.17).The spectral flow is then the family of inner automorphisms on A induced by U (Λ, s): Quasi-locality of this map is then a consequnece of a Lieb-Robinson bound.To this end, first rewrite the generator as using (2.37).Applying the conditional expectations and telescopic sum from (3.7), we further write Arguing as in [38, Appendix A], there is a decay function G Ψ and a positive number m≥k One can be explicit about estimates for G Ψ , see [38,Corollary A.3], but for our purposes, we only need that is has finite moments of all orders.Given (3.25) and (3.26), well-known Lieb-Robinson bounds imply the existence of a decay function G α so that for all X, Y ∈ P 0 (Γ), and s ∈ R. G α is independent of Λ ∈ P 0 (Γ) and has finite moments of all orders.

3.2.
In the GNS space.The spectral perturbation arguments are carried out in the GNS representation of the reference state ω 0 .The quasi-local maps discussed in the previous subsection can be lifted to the GNS representation and we now present the necessary properties we will need in this setting.
3.2.1.Dynamics.As discussed in Section 2.1, the unperturbed dynamics τ (0) t is implemented in the GNS representation of ω 0 by the GNS Hamiltonian H 0 , as in (2.20).We further show that the perturbed dynamics τ (Λ,s) t is implemented in the GNS representation of ω 0 by the Hamiltonian Applying the GNS representation to the interaction picture representation (3.10) gives ) .
Then, (3.29) follows by observing that as by (3.11) K(Λ,s) ) is the unique, unitary solution of 3.2.2.Weighted Integral Operators.For any γ > 0, Λ ∈ P 0 (Γ), and s ∈ R we map the weighted integral operators of (3.14) to the GNS space by defining FΛ s and GΛ s by 3. The Spectral Flow.For fixed Λ ∈ P 0 (Γ), following [37, Section VI.A] we define a normcontinuous family of unitaries Ũ (Λ, s) ∈ B(H) as the unique solution of The spectral flow associated with H(Λ, s) is the family of automorphisms of B(H) defined by . This is Hastings' quasi-adiabatic evolution [20,22].
For any Λ ∈ P 0 (Γ) and s ∈ R, the state ω Λ s given by (3.41) ) for all A ∈ A is a vector state in the GNS space: Finally, we recall that with the parameters γ and s as above that the weighted integral operator FΛ s from (3.33) satisfies the relation See, e.g.[37, Lemma 6.8], for a proof of this property.
3.3.On Domains.Recall that (Γ, d) is a ν-regular metric space.Let F be an F -function on (Γ, d), and Φ an interaction with Φ F < ∞.As in Section 2, let δ Φ be the closed derivation with dense domain dom(δ Φ ) ⊂ A, and which satisfies Although the sum on the right-hand-side above may be infinite, it is absolutely convergent when Φ has a finite F -norm.In fact, δ Φ is locally bounded: For any interaction Φ on Γ with Φ F < ∞, we have that A g ⊂ dom(δ Φ ).
Proof.For n ≥ 1, and A ∈ A g , for some x ∈ Γ, and observables In this case, the bound A n+1 − A n ≤ 2 A x,g g(n) is clear.Using (3.46) and νregularity of Γ, we conclude Thus, for all m < n, Since we assumed that g has a finite ν-moment, this implies that δ Φ (A n ) is a Cauchy sequence.Since A n → A and A loc is a core for δ Φ , it follows that A ∈ dom(δ Φ ).
It will be important that on an appropriate dense domain, the action of the unbounded Hamiltonians can be expressed as a limit of finite-volume quantities.This is the content of the next lemma.
Lemma 3.2.Let (H, π 0 , Ω) be the GNS representation of ω 0 , an infinite-volume, zero energy, ground state of a frustration free model as in Assumption 2.1.For any decay function g with a finite ν-moment and any IAS (Λ n ), (3.53) lim where H Λn ∈ A Λn is as in (2.13) and H 0 is the GNS Hamiltonian.
Proof.Note that (3.53) is trivially satisfied for ψ = π 0 (A)Ω, for A ∈ A loc since For the first equality we used π 0 (H Λn )Ω = 0, which is a consequence of the frustration-free property.
Then, by the finite-range condition on the unperturbed model, [H Λn , A] becomes constant for n sufficiently large.Take ψ = π 0 (A)Ω for any A ∈ A g .By the definition of A g , there exists x ∈ Γ and observables A m ∈ A bx(m) so that A − A m ≤ A x,g g(m) for all m ≥ 1, and so the vectors ψ m := π 0 (A m )Ω satisfy (3.55) ψ − ψ m ≤ A x,g g(m).
Moreover, since the interaction h is uniformly bounded with range R, it follows from (3.45) and ν-regularity that for any k ≥ 1, Then, by the last equality of (3.54), one finds that (H 0 ψ m ) m∈N is Cauchy as where we set D = 4κ 2 R ν h ∞ A x,g .Since H 0 is closed, and ψ ∈ dom(H 0 ) by Lemma 3.1 and the subsequent discussion, the bound follows immediately from (3.55)-(3.56).
In the case of a local Hamiltonian, using again the first equality in (3.54), a similar argument shows that for all n ≥ 1, Putting all of this together, one finds that for any n ≥ 1 and each k ≥ 1, For k ≥ 1 sufficiently large, (3.57) and (3.58) guarantee that the first and last term above can be made arbitrarily small.Given such a k, the middle term vanishes for n sufficiently large, see the comment following (3.54).This completes the proof.Lemma 3.2 also trivially applies to the perturbed system in the GNS space.In fact, for Λ ∈ P 0 (Γ) and s ∈ R, under assumptions as above, a direct application of Lemma 3.2 shows that we also have (3.60)lim Remark 3.
For fixed k, the above is the sum of finitely many 'tails' of the uniformly summable function F .
We now investigate how the weighted integral operator FΛ s from (3.33) can be applied to the unbounded Hamiltonian H(Λ, s).To begin, we prove an analogue of the desired statement for the unperturbed dynamics; this is Lemma 3.4 below.To this end, assume w : R → R satisfies To simplify notation, let us also write (3.64) τ (0) u (A) = e iuH 0 Ae −iuH 0 .Our first result is as follows.
Proof.Fix an IAS (Λ n ) and take ψ = π 0 (A)Ω for some A ∈ A g .We can rewrite the convergence claimed in (3.65) as the convergence of integrals of a sequence of functions f n : R → H given by (3.66) f n (u) = w(u)τ (0) u (π 0 (H Λn ))ψ for all u ∈ R .Since H 0 Ω = 0, the above can be re-written as −u (A))Ω using (2.20).We claim that there is a decay function g τ with a finite ν-moment such that τ (0) −u (A) ∈ A gτ for all u ∈ R. Given this, Lemma 3.2 applies and we find that By (3.62), the integral of this limit coincides with the right-hand-side of (3.65).Therefore, to complete the proof we only need to justify an application of dominated convergence.Let us first prove the existence of a decay function g τ as claimed.Fix A ∈ A g .In this case, there is x ∈ Γ, C ≥ 0, and observables A m ∈ A bx(m) for which A − A m ≤ Cg(m) for all m ∈ N. Let u ∈ R and for any n ∈ N, set where, to ease notation, we have written Π n = Π bx(n) , for the conditional expectation from Section 3.1.1.A straightforward estimate shows that for any µ > 0, where we used (3.8) and (3.4) for the final bound.The existence of the decay function g τ follows from the moment condition on g and the decay of the exponential term..We now turn to finding a dominating function for f n .Recall that for any m 0 ∈ N, A can be written as an absolutely convergent, telescopic sum: Inserting this decomposition of A into (3.67),we find that for any n ∈ N and each u ∈ R: Now, by the zero-energy property of the ground state we find the bound which we stress is uniform in n.This suggests a mechanism for bounding the first term in (3.72).Let 0 ≥ m 0 and write where we have used the short-hand ∆ m for ∆ bx(m) as in (3.7).For = 0 , the bound follows from (3.73).For ≥ 0 , the estimate follows from another application of (3.73) and the quasi-locality estimate for the unperturbed dynamics in combination with (3.6).We conclude that π 0 (H Λn )π 0 (τ If we now take 0 = v µ |u| + m 0 , then we have found that there is K ≥ 0 for which (3.78) π 0 (H Λn )π 0 (τ The terms B k in (3.72) can be estimated similarly.Regarding k as m 0 and arguing as in (3.74) -(3.77) with some 0 ≥ k, a bound analogous to (3.78) can be found.Of course, here one replaces A m 0 with B k .Since B k ≤ 2Cg(k − 1) and g has a finite 2ν-moment, we have obtained a bound on the right-hand-side of (3.72) of the form: By the assumption on w, i.e. (3.62), the above is a dominating function for the sequence f n .This justifies dominated convergence and completes the proof.
We will also need a version of Lemma 3.4 for the perturbed system.Recall that for any γ > 0, s ∈ R, and Λ ∈ P 0 (Γ), the weighted integral operator FΛ s : B(H) → B(H) are defined by We note that w γ from [37, Section VI.B] satisfies (3.62).It is clear that (3.81) FΛ s (e iuH(Λ,s) ) = e iuH(Λ,s) for all u, s ∈ R , since the dynamics leaves this bounded operator invariant and w γ integrates to 1. Lemma 3.5 provides a differential version of this fact.Lemma 3.5.Let (Γ, d) be ν-regular, g be a decay function with a finite 2ν-moment.Let Λ ∈ P 0 (Γ) and take s ∈ R. For each choice of IAS (Λ n ), consider the weighted integral operator FΛ s , as in (3.80), with arbitrary w : R → R satisfying (3.62).Then with H Λn ∈ A Λn as in (2.13) and V Λ as in (2.37).
Proof.Fix an IAS (Λ n ) where we assume for convenience that Λ ⊂ Λ 1 .As in the proof of Lemma 3.4, take ψ = π 0 (A)Ω with A ∈ A g , and for each n ∈ N, consider f n : R → H given by where, in analogy to (3.64), we have set (3.84) τ (Λ,s) t (A) = e itH(Λ,s) Ae −itH(Λ,s) for all A ∈ B(H) and t ∈ R .
Using (3.29), (3.10), and (2.20), we may write for all u ∈ R. In this case, we find that Following a similar argument and using (3.13), one shows that there is a decay function g with a finite ν-moment such that τ As a result, the point-wise limit is clear from properties of the interaction picture dynamics, see the discussion following (3.10).
The argument demonstrating that we can apply the dominated convergence theorem also proceeds as in the proof of Lemma 3.4.Since the differences stemming from the presence of the u-dependence in the operators A m 0 and B k are minor, we leave the details to the reader.

3.4.
Proof of bounded differentiability of the spectral flow.At this point we pause to prove a result in the setting of this paper that is useful for related considerations concerning gapped insulators and symmetry protected phases.For this purpose we consider differentiable curves of interactions [0, 1] s → Φ(s), for which we assume a finite norm of the following form: where for each finite X ⊂ Γ and 0 ≤ s ≤ 1, Φ(X, s) * = Φ(X, s) ∈ A X , Φ(X, •) : [0, 1] → A X is differentiable, and F is an F -function of stretched exponential decay as in (2.11).For any differentiable curves of interactions Φ with finite norm Φ 1 F , there is an F -function F , and an s-dependent interaction Ψ(s) with that generates the infinite volume spectral flow automorphisms α s [37, Section VI.E.2].We further know by [37,Theorem 3.9], that this infinite volume spectral flow is differentiable on A loc Γ with Our aim is the show that is a bounded map when defined on a suitable Banach algebra of g-local obervables.
Theorem 3.6.Suppose g is a decay function and (Λ n ) an IAS that satisfy Then, the derivative α s of the spectral flow α s is a well-defined bounded linear map A (Λn),g → A, satisfying with F 1 as defined in (2.9).
Proof.We start by noting that for any local observable A ∈ A Y with Y ⊂ Γ finite, we have the following estimate: which implies that the sequence {Π n (A)} ∞ n=1 converges to A (in norm) at a rate governed by g.Define a map α s : A (Λn),g → A by setting (3.94) α s (A) = lim n→∞ d ds α s (Π n (A)) for all A ∈ A (Λn),g .
Note that for any strictly local observable A, Π n (A) = A if n is sufficiently large, and thus this definition agrees with the standard definition of the derivative of α s (A) for A ∈ A loc .Now, consider integers 1 ≤ M < N < ∞.For any observable A, and for A ∈ A g , we also have that for any n ≥ 1, In the above, we have used (3.93) and the fact that g is non-increasing.We conclude that Here, for the second inequality above we used (3.92), and the final bound comes from (3.96).We conclude that whenever g satisfies (3.90), the sequence {α s (Π n (A))} ∞ n=1 is norm Cauchy; hence, norm convergent.This shows that α s as given in (3.94) is well-defined and equals the derivative of α s (A), for A ∈ A (Λn),g .Moreover, for any N ≥ 1, as claimed.
This proves that Assumption 1.2 (vii) in [33] (see also [7,41]) is always satisfied under the assumptions made in that reference.We note that in the same way one obtains the extension of the invariants constructed by Ogata in [39,41,42] from models with finite-range interactions to the setting with interactions of stretched exponential decay.

Construction of a unitarily equivalent perturbed system
The crux of the stability strategy introduced in [10], is the construction of a unitarily equivalent perturbed system using the spectral flow (aka quasi-adiabatic evolution) for which one can prove a relative form bound using quasi-locality estimates and LTQO.In the infinite-system setting, this means we need to prove that by transforming the unbounded Hamiltonian H(Λ, s) from (2.36 by the spectral flow one arrives at an equivalent Hamiltonian of the form with W (Λ, s) a bounded operator with an explicit, Λ-independent form-bound with respect to H 0 and E(Λ, s) the ground state energy of H(Λ, s) from (2.39).That W (Λ, s) is well-defined is a consequence of the fact that Ũ (Λ, s) ∈ π 0 (A g ) for a function g of the form (2.11), which guarantees that A g is an algebra.Hence Ũ (Λ, s)π 0 (A)Ω ∈ dom H(Λ, s) for A ∈ A loc , by Lemma 3.2.The proof of Theorem 2.8 is a consequence of two results.The first,Theorem 4.1, establishes that W (Λ, s) is indeed bounded and can be decomposed in way that is suitable for deriving a relative form bound.The second, Theorem 5.1 in Section 5, is the relative form bound itself.
The decomposition from Theorem 4.1 is proved in two steps.The first uses quasi-locality and conditional expectations to prove that for all |s| ≤ s Λ 0 (γ), the action of the spectral flow on the GNS Hamiltonian H(Λ, s) can be again realized as a perturbation of H 0 .Namely, we show that for all ψ ∈ π 0 (A loc )Ω where the perturbation terms Φ(1) (x, m, s) ∈ π 0 (A bx(m) ) are self-adjoint, satisfy a norm bound that is linear in s, and are absolutely summable over x ∈ Γ and m ≥ R.This is accomplished in Theorem 4.2 of Section 4.1 below.
In the second step, carried out in Section 4.2, the final form of (4.3) from Theorem 4.1 is proved using the frustration-free and LTQO ground state properties to produce a refined decomposition of the perturbation terms from (4.4).

4.1.
Quasilocal decomposition of the transformed perturbation.We now turn to establishing the first decomposition (4.4), which is the content of the following theorem.The global term Φ(1) (x, s) above will result from applying quasi-local maps K i,Λ s , i = 1, 2, to the interaction and perturbation terms associated to the site x.These maps are defined in terms of the examples introduced in Section 3.1, and emerge from fixing any IAS (Λ n ) and then applying Lemmas 3.4-3.5 to rewrite where we choose F = FΛ 0 .As the argument in the above limit is a finite sum of bounded operators, the various relationships (3.34)- (3.38) between the quasi-local maps in the GNS representation to those on the C * -algebra implies that for each n: Given this, for i = 1, 2 the map K i,Λ s : A → A are defined by (4.9) ).It was proved, e.g. in [38,Lemma 4.4], that both of these maps satisfy a local bound and quasilocal estimate that is independent of the finite volume Λ.Specifically, for each i = 1, 2 there are non-negative numbers p i , q i and C i , and a decay function G K i (all independent of Λ) such that hold for any X, Y ∈ P 0 (Γ), A ∈ A X , B ∈ A Y , and s ∈ R. In fact, one can take p 1 = q 1 = 2, p 2 = 0 and q 2 = 1 and make explicit estimates on the decay function, see e.g.[38,Remark 4.7].However, it suffices to note that each G K i have finite moments of all orders in the sense of (3.17).
However, as Λ n ↑ Γ when n → ∞, to prove that the decomposition in (4.6) is absolutely summable, we will need refinements of (4.10)-(4.11)for K 1,Λ  s that also decay in the distance d(X, Λ).Both of these bounds will be a consequence of the perturbation V Λ being locally supported, which implies that the spectral flow α Λ s is approximately the identity far from Λ.The necessary bounds are the content of Lemmas 4.3 and 4.4 below.

Lemma 4.3 (Distance Locality Bound for K 1 s
).There exists a decay function F K 1 , with finite moments of all orders for which, given any X, Λ ∈ P 0 (Γ) with d(X, Λ) > 0, A ∈ A X , and any s ∈ R, the following local bound holds: It is easy to check that for fixed ∈ (0, 1) and any decay function F with finite ν-moment, the function is also a decay function.The proof of Lemma 4.3 shows that one may take (4.14) where G F and G Ψ are the decay functions previously discussed in (3.16) and (3.26).Since G F and G Ψ both have finite moments of all orders, the same is true for The proof of Lemma 4.3 will also make use of the following bound, which holds for any F and as in (4.13), and Λ, X ∈ P 0 (Γ) such that d(X, Λ) > 0: This follows from the following simple calculation z∈Λ where the last inequality uses that |X(n)| ≤ κn ν |X| for any n ≥ 1 by ν-regularity, see (2.1).
Proof of Lemma 4.3: Fix X, Λ ∈ P 0 (Γ) such that X ∩ Λ = ∅, and let A ∈ A X be arbitrary.Recall that K 1,Λ s is as defined in (4.9), and that D(Λ, s) from (3.18) is the generator of the spectral flow.Then, since α 0 = id and F = F Λ 0 , it follows that (4.17) where one uses (3.19) and [37,Equation (6.37)] to obtain Here, the final two equalities follow from integrated by parts, and the fact that the supports of V Λ and A are disjoint.
Returning to (4.17), we expand the generator as in (3.25) to write Fix ∈ (0, 1), and for each z ∈ Λ, set k z ( ) = d(z, X).For each term in (4.19), we approximate F Λ r (A) with a strictly local approximation: where one uses conditional expectation associated with the inflated set X(k z ( )), see (2.1)-(2.2).For the second term, one can apply the quasi-local bound for F Λ r from (3.16) coupled with (3.4) to produce Then, summing over z ∈ Λ and n ≥ R, and applying (3.26) and (4.15) gives the final estimate (4.22) z∈Λ n≥R To estimate the remaining terms in (4.20), note that for each z ∈ Λ, b z (n) ∩ X(k z ( )) = ∅ only when n ≥ k z (1 − ).As a result, arguments similar to the prior estimate produce the bound Recalling the specific decay function from (4.14), the bound claimed in (4.12) now follows by inserting (4.19) into (4.17) and using the estimates found in (4.22) and (4.23) above.
By combining the estimate in Lemma 4.3 and the original quasi-locality bound from (4.11), one arrives that the following quasi-locality bound for [K 1,Λ s (A), B] , which decays in both the distance between X = supp(A) and Y = supp(B) as well as the distance between Λ and X.This is the content of the next lemma.
)) holds for all A ∈ A X , B ∈ A Y , and s ∈ R.More precisely, for any δ ∈ (0, 1), one may choose where F K 1 and G K 1 are the decay functions from Lemma 4.3 and (4.11), and G δ (m) := (G(m)) δ .
In applications, it can be convenient to bound G(n, m) by a function that separates over the two arguments.In this case, taking δ as in (4.25), In the case that d(X, Λ) ≤ d(X, Y ), the quasi-locality estimate (4.11) shows that The bound (4.24) is then a consequence of (4.27) and (4.28).
We conclude this section by using Lemmas 4.3-4.4 to prove Theorem 4.2.
Let χ Λ be the characteristic function of Λ ⊂ Γ.Then, for each x ∈ Γ and s ∈ R such that |s| ≤ s Λ 0 (γ), the self-adjoint operator Φ(1) (x, s) = π 0 (Φ (1) To show that each Φ(1) (x, s) commutes with the ground state projection |Ω Ω|, recall that the ground state of the perturbed system is Ω(Λ, s) = Ũ (Λ, s)Ω if |s| ≤ s Λ γ .Then, recalling the relations (3.34)-(3.38), a simple calculation shows that for all A ∈ A where the final equality uses that (3.44) holds since |s| ≤ s Λ 0 (γ).Since (4.30) trivially holds for s = 0, considering (4.9), the above implies that (4.31) Hence, [ Φ(1) (x, s), |Ω Ω|] = 0 for all x ∈ Γ and |s| ≤ s Λ 0 (γ) as claimed.To establish (4.6), use the condition expectations from (3.7) to decompose each Φ(1) (x, s) as With respect to this notation, ( Since this π 0 (A loc )Ω ⊆ H is dense, the equality in (4.6) follows from establishing absolute summability of the terms Φ(1) (x, m, s).This is achieved by defining a function function which bounds the norms of these terms and satisfies (4.5).Here, we note that Λ(R) is as in (2.1), and the functions As R ≥ 0 is the finite range of the unperturbed interaction, (4.33) simplifies to (4.36) Φ (1) (x, m, s) = ∆ m bx(R) (K 1,Λ s (h x )) .Then, applying Lemmas 4. where for any fixed δ ∈ (0, 1), the function G 2 can be taken to be Here, C = κ 2 R 2ν h ∞ , F K 1 is the function from Lemma 4.3, and (4.39) More specifically, the bound in (4.37) for m = R is a direct application of Lemma 4. Given (4.35), the summability of G Λ over the sites x ∈ Γ \ Λ(R) follows from observing that as both F K 1 and G K 1 (and, thus, F δ ) have finite moments of all orders.In particular, for any decay function We now turn to the sites x ∈ Λ(R), for which we demonstrate that (4.41) Φ (1) where G 1 is a summable function.First consider (4.33) when m = R. Combining the local bounds (4.10), the uniform bound (2.12), and the interaction bound in Assumption 2.6, one produces the x-independent bound Alternatively, for m ≥ R + 1, (4.33) can be estimated as Φ (1) where one uses the quasi-local estimates from (4.11) and the local approximation bound in (3.6).Given Assumption 2.6, the final sum above can be further estimated as To simplify notation, let M Φ (r) := k≥r k ν e −ak θ denote the ν-th moment of the decay function associated with the perturbation Φ from Assumption 2.6.Then, in summary, one has that for x ∈ Λ(R), (4.41) holds for the decay function G 1 defined by Since each of the decay functions in (4.46) has finite moments of all orders, it is clear that m≥R G 1 (m) < ∞.As a consequence, G Λ as in (4.35) satisfies This demonstrates absolute summability of the terms in (4.6), and hence, completes the proof of Theorem 4.2.

4.2.
The final decomposition of the transformed perturbation via LTQO.We now turn our attention to proving Theorem 4.1, which is a consequence of one last decomposition of the transformed perturbation from Theorem 4.2, i.e.
The key component for proving the desired norm bounds for this final decomposition is Lemma 4.5 below, and it is in the proof of this result where one needs the LTQO property from Assumption 2.4.
To this end, we first shift the transformed perturbation terms by their expectation in the ground state Ω, as this will put us in the appropriate setting to apply LTQO.Throughout this section, we assume γ ∈ (0, γ 0 ) is fixed and that s ∈ R is such that |s| ≤ s Λ 0 (γ).As such, Ω(Λ, s) = Ũ (Λ, s)Ω is the ground state of H(Λ, s), and one finds that W (Λ, s)Ω = 0 from considering (4.1) in the case ψ = Ω.Thus, Theorem 4.2 implies that for any ψ ∈ π 0 (A loc )Ω (4.47) where the (self-adjoint) observables Φ(1) ω (x, m, s) ∈ π 0 (A bx(m) ) are defined by holds where G 0 is the decay function from Assumption 2.4, and G where P Ω = |Ω Ω| and Ã = π 0 (A).To see this, first note that the inequality |a − b| Recalling that (H, π 0 , Ω) is the GNS representation of the unperturbed ground state ω 0 , the second term on the right-hand-side above is simply As π 0 is norm-preserving, it also follows that Given these observations, Assumption 2.4 then implies that where the last inequality follows from Theorem 4.2 as Φ( 1) First, note that Pn converges strongly to P Ω for all ψ ∈ H by the frustration-free and LTQO properties.As a consequence, the collection of operators Using (4.49), the above properties imply that for all ψ, φ ∈ H, φ, (1l We note that the triple sum of operators actually converges absolutely in norm, and so the operator equality holds in the norm sense. Each term Φ (2) (x, m, s) ∈ A bx(m) will be defined as a sum of two self-adjoint terms For each m ≥ R, define Θ 1 (x, m, s) ∈ π 0 (A bx(m) ), by These operators are self-adjoint, satisfy Θ 1 (x, m, s) Pm = 0, and Theorem 4.2 implies that their norm is bounded from above by 2sG Λ (x, m/2) as for m even: For the Θ 2 terms, one sums the remaining terms m>2k Φk,m over k, and then uses the indicator function χ m>2k to exchange the summations as follows: Finally, applying Lemma 4.5, Moreover, these operators satisfy that for all m ≥ R and x ∈ Γ, Φ Λ (x, m/2 ) + 2G Λ (x, R) m/2 ν G 0 (m/2).The absolute summability of the series in (4.67) is a direct consequence of G 0 being summable as well as that both G Λ and G Λ satisfy (4.5).For G Λ this can easily be seen from the fact that G Λ is a combination of functions with finite moments of all orders, see specifically (4.35), (4.38) and (4.46).

Proving Theorems 2.8 and 2.9 via a form bound for the GNS Hamiltonian
In this section, we generalize [38, Theorem 3.8], which was itself based off [32, Proposition 2], so that it is applicable to the setting of infinite systems in their GNS representation.Afterwards, we apply the form bound in conjunction with Theorem 4.1 to prove Theorem 2.8.We then conclude with the proof of Theorem 2.9, which follows as a consequence of Theorem 2.8.
Theorem 5.1 (Michalakis-Zwolak [32]).Let H 0 be the GNS Hamiltonian associated with a zeroenergy ground state of an initial system satisfying Assumptions 2.1 and 2.3, and let V ∈ A be a perturbation associated with an absolutely-summable, anchored interaction on (Γ, d).That is, there exist In addition, assume that terms of V annihilate the finite-volume ground state projections of the initial system, i.e.
and that there is a decay function G with finite Then, for all ψ ∈ dom H 0 , In keeping with the notation from the previous sections, denote by Ã = π 0 (A) the image of any observable A ∈ A under the GNS representation (H, π 0 , Ω) of the zero-energy ground state ω 0 .The proof of Theorem 5.1 follows closely the argument proving [38,Theorem 3.8], with the proviso that one must check that the infinite operator sums replacing the finite operator sums from [38] are well-defined.
To this end, let n ≥ R, and consider the n-th separating partition T n = {T i n : i ∈ I n }.For each i ∈ I n and any choice of x, y ∈ T i n , one has that (5.5) [ HΛ(x,n) , HΛ(y,n) ] = 0 and [ PΛ(x,n) , PΛ(y,n) ] = 0 .
This follows since the corresponding algebra elements H Λ(x,n) , H Λ(y,n) ∈ A are supported on disjoint sets (and similarly for the ground state projections), which carries over to the GNS space by the homomorphism property.With n ≥ R and i ∈ I n fixed, denote by C i n the collection of all configurations associated to T i n .More precisely, (5.6) For each σ ∈ C i n , we define |σ| by Recall that π 0 (A loc )Ω is a dense subspace of H. Let ψ = ÃΩ for some A ∈ A X , and set QΛ(x,n) = 1l − PΛ(x,n) .In this case, one has that (5.8) PΛ(x,n) ψ = ψ and QΛ(x,n) ψ = 0 whenever Λ(x, n) satisfies Λ(x, n) ∩ X = ∅.It is also clear that for any X ∈ P 0 (Γ), the set of sites {x ∈ T i n : Λ(x, n) ∩ X = ∅} is finite.Thus, for any ψ ∈ π 0 (A loc )Ω, define the operator S(σ) by (5.9) S(σ)ψ = Note that, if |σ| < ∞, then at most finitely many of these factors act non-trivially, and moreover, by (5.5), all factors above commute.Since π 0 (A loc )Ω is dense, there is a unique extension of S(σ) to an element of B(H ω ) for each σ ∈ C i n .One checks that these operators satisfy: (5.10) S(σ) * = S(σ), S(σ)S(σ ) = δ σ,σ S(σ), and for all σ, σ ∈ C i n and each x ∈ T i n .We use these families of orthogonal projections in the following proof.
Proof of Theorem 5.1.In the GNS representation, the terms of Ṽ can be rearranged using the family of separating partitions from Assumption 2.3 as and so by (5.3) it follows that that for each x ∈ T i n , the term Φ(x, n) satisfies (5.17) [ Φ(x, n), PΛ(y,n) ] = 0 for all y ∈ T i n .Arguing as in (5.12)-(5.13)above, one then finds that for all x ∈ T i n and σ, σ ∈ C i n (5. Λ (x, m, s))ψ, where E(Λ, s) is the ground state energy of H(Λ, s), and Φ Λ (x, m, s) is a balled interaction satisfying the conditions of Theorem 5.1 with norm bounds that are linear in |s| and given in terms of a Λ-dependent decay function.However, we will show below that the constant β from Theorem 5.1 can be taken independent of Λ. Theorem 2.8 will then follow from applying [38,Corollary 3.3].In our context, the latter result states the following.Suppose that H 0 is a self-adjoint, positive operator on a Hilbert space H with min spec H 0 = 0 and (0, γ 0 ) ∩ spec(H 0 ) = ∅.Then, for any V = V * ∈ B(H) such that there exists 0 ≤ β < 1 for which | ψ, V ψ | ≤ β ψ, H 0 ψ ∀ψ ∈ dom(H 0 ), one has that (5.24) spec(H 0 + V ) ∩ (0, (1 − β)γ 0 ) = ∅.
We conclude with using the uniform estimate from Theorem 2.8 to establish the claimed lower bound estimate on the gap of the extensively perturbed system from Theorem 2.9.
Proof of Theorem 2.9.Let 0 < γ < γ 0 and consider |s| ≤ s 0 (γ).Recall that for any IAS (Λ n ), the following limits hold in A as n → ∞: ) and the local uniform convergence of α Λn s , the limit n → ∞ can be taken on both sides of (5.28) to obtain (2.44).The remaining claims follow as in (2.22).
all x and n, and an associated family of partitions of Γ which separates S and has at most polynomial growth.Concretely, this means there is a family of sets T = {T n | n ≥ 0} and positive numbers c and ζ, such that for each n ≥ 0, T n = {T i n : i ∈ I n } is a partition of Γ satisfying |I n | ≤ cn ζ and (2.23)

α
Λn s (A) → α s (A), for all A ∈ A (5.26) δ Λn s (A) → δ s (A), for all A ∈ A loc , (5.27) see (2.42) and (2.43).As a consequence, ω Λn s (A) = ω 0 • α Λn s (A) → ω s (A) for all A ∈ A and, moreover, ω s is a ground state of δ s .Now consider A ∈ A loc for which ω s (A) = 0. Given Theorem 2.8, the GNS Hamiltonian H(Λ n , s) along any IAS (Λ n ) has a gap above its unique ground state lower bounded by γ for all |s| ≤ s 0 (γ).Therefore, ω Λn s satisfies (2.22) and, in particular, the inequality (5.28) ω Λn s (B * n δ Λn s (B n )) ≥ γω Λn s (B * n B n ) holds for the observable B n = A − ω Λn s (A)1l ∈ A loc .Combining (5.27 3. An analogue of Lemma 3.2 holds more generally.In fact, if F is an F -function with a finite ν-moment, then for any frustration free interaction Φ with Φ F < ∞, the GNS Hamiltonian again satisfies (3.53).The argument is identical to the above except that one uses the more general estimate in Lemma 3.1 and bounds the middle term in (3.59) by X∈P 0 (Γ): (x, k, s) ∈ π 0 (A bx(m) ), one finds that for any n ≥ m, ω m k=R k).