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

Bruno Nachtergaele  Department of Mathematics and Center for Quantum
Mathematics and Physics
University of California, Davis
Davis, CA 95616, USA<EMAIL_ADDRESS>,  Robert Sims  Department of
Mathematics
University of Arizona
Tuscon, AZ 85721, USA<EMAIL_ADDRESS>and  Amanda Young  Munich Center
for Quantum Science and Technology, and
Zentrum Mathematik, TU München
85747 Garching, Germany<EMAIL_ADDRESS>
###### Abstract.

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.

##  1\. 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.
Over the past decade, several results proving the stability of the spectral
gap for more or less general classes of models have appeared in the literature
[7, 8, 21, 14, 10, 11]. One obstacle has been the common occurrence of gapless
edge states in topological insulators. Spectral analysis for interacting many-
body systems is usually carried out for finite systems and 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 [21, 25]. In
general, however, we may not know a suitable boundary condition that ‘gaps
out’ the boundary modes. In our approach here we only assume that the infinite
system described in the GNS representation of the ground state has a gap above
the ground state. Under natural assumptions similar to those in previous
works, we prove that sufficiently small but extensive perturbations do not
close the gap.

We follow the strategy of Bravyi, Hastings, and Michalakis [7, 8, 21] and use
the techniques we developed in [26, 25] 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. For
concreteness, we work in the quantum spin system setting, but our approach is
applicable to lattice fermion systems too.

We do not provide new applications of gap stability in this work and just
mention that 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 [30, 27, 28, 29]. Other examples are in [22] and the recent
work on adiabatic theorems for infinite many-body systems [16].

##  2\. Setup and statement of the main results

###  2.1. Setup and notation

As the reference system (before perturbation), we consider a frustration-free,
finite-range model defined on a discrete metric space $(\Gamma,d)$. For
$x\in\Gamma$ and $n\geq 0$, define $b_{x}(n)=\\{y\in\Gamma\mid d(x,y)\leq
n\\}$. We assume that $(\Gamma,d)$ is $\nu$-regular for some $\nu>0$, meaning
there is a constant $\kappa$ such that $|b_{x}(n)|\leq\kappa n^{\nu}$, for all
$x\in\Gamma$ and $n\geq 1$. For $\Lambda\in{\mathcal{P}}_{0}(\Gamma)$, the
finite subsets of $\Gamma$, and $n\geq 0$, we define the sets $\Lambda(n)$ by

(2.1) $\Lambda(n)=\bigcup_{x\in\Lambda}b_{x}(n).$

The algebra of local observables of the system is the usual
${\mathcal{A}}^{\rm
loc}=\bigcup_{\Lambda\in{\mathcal{P}}_{0}(\Gamma)}{\mathcal{A}}_{\Lambda}$.
Here, ${\mathcal{A}}_{\Lambda}$ is the matrix algebra
$\bigotimes_{x\in\Lambda}M_{d_{x}}$ with $d_{x}$ the dimension of the spin at
$x$. The $C^{*}$-algebra of quasi-local observables ${\mathcal{A}}$ is the
completion of ${\mathcal{A}}^{\rm loc}$ with respect to the operator norm. For
$A\in{\mathcal{A}}^{\rm loc}$, the support of $A$, denoted by
$\operatorname{supp}A$, is the smallest $X\subset\Gamma$ such that
$A\in{\mathcal{A}}_{X}$.

We are specifically interested in systems defined on infinite $\Gamma$ and
will frequently consider $A\in{\mathcal{A}}$ that are limits of
$A_{n}\in{\mathcal{A}}_{\Lambda_{n}}$, where
$\Lambda_{n}\in{\mathcal{P}}_{0}(\Gamma)$ is an increasing sequence of finite
volumes such that $\bigcup_{n}\Lambda_{n}=\Gamma$. We call such a sequence
$(\Lambda_{n})$ an increasing and absorbing sequence (IAS). An often used
example is $\Lambda_{n}=b_{x}(n)$, the sequence of balls centered at
$x\in\Gamma$ of radius $n=0,1,2,\ldots$.

For any $A\in{\mathcal{A}}$ and any IAS $(\Lambda_{n})$, there are
$A_{n}\in{\mathcal{A}}_{\Lambda_{n}}$ such that $\lim_{n}A_{n}=A$. It will
often be important to have an estimate for the speed of this convergence in
terms of a non-increasing function $g:[0,\infty)\to(0,\infty)$ that vanishes
at infinity. We will call any such function $g$ a decay function. Given a
decay function $g$, we will call $A\in{\mathcal{A}}$ $g$-local if for all
$x\in\Gamma$, there exists $C>0$ such that there are
$A_{n}\in{\mathcal{A}}_{b_{x}(n)}$, for all $n\geq 1$, with the property that

(2.2) $\|A-A_{n}\|\leq Cg(n).$

Define ${\mathcal{A}}^{g}$ to be the set of all $g$-local elements of
${\mathcal{A}}$. Since we assumed $g(r)>0$ for all $r\geq 0$, it is easy to
see that ${\mathcal{A}}^{\rm
loc}\subset{\mathcal{A}}^{g}\subset{\mathcal{A}}$. In fact,
${\mathcal{A}}^{g}$ is a dense $*$-subalgebra of ${\mathcal{A}}$.

We will often also assume that a decay function $g$ is uniformly summable over
$\Gamma$, i.e.,

(2.3) $\sup_{x\in\Gamma}\sum_{y\in\Gamma}g(d(x,y))<\infty\,,$

and additionally, that there is a constant $C>0$ such that

(2.4) $\sum_{z\in\Gamma}g(d(x,z))g(d(z,y))\leq Cg(d(x,y)),\mbox{ for all
}x,y\in\Gamma.$

Any decay function $g$ satisfying (2.3) and (2.4) will be called an
$F$-function. For example, with $\Gamma={\mathbb{Z}}^{\nu}$, the following is
a family of examples of $F$-functions appearing in this work:

(2.5) $F(r)=\frac{1}{(1+r)^{\zeta}}e^{-ar^{\theta}},\quad\zeta>\nu,a\geq
0,\theta\in(0,1].$

For a discussion of these examples and some basic inequalities, see [26,
Appendix].

We assume the initial model is defined by a finite-range, uniformly bounded,
frustration-free interaction. We denote this interaction by $\eta$ and
describe the corresponding interaction terms by a family
$\\{h_{x}\\}_{x\in\Gamma}$ satisfying:

  1. i.

There is a number $R\geq 0$, called the interaction radius, for which
$h_{x}^{*}=h_{x}\in\mathcal{A}_{b_{x}(R)}$ for all $x\in\Gamma$.

  2. ii.

These terms are uniformly bounded in the sense that

(2.6) $\|\eta\|_{\infty}=\sup_{x\in\Gamma}\|h_{x}\|<\infty\,.$

  3. iii.

The interaction $\eta$ is frustration-free in the sense that $h_{x}\geq 0$ for
all $x\in\Gamma$ and for any $\Lambda\in\mathcal{P}_{0}(\Gamma)$,

(2.7) ${\rm min\,spec}(H_{\Lambda})=0\quad\mbox{where}\quad
H_{\Lambda}=\sum_{\stackrel{{\scriptstyle x\in\Lambda:}}{{{\rm
supp}(h_{x})\subset\Lambda}}}h_{x}\,.$

It follows from this frustration-free condition that the ground state is ${\rm
ker}(H_{\Lambda})\subseteq\bigotimes_{x\in\Lambda}{\mathbb{C}}^{d_{x}}$ for
any finite volume $\Lambda\in{\mathcal{P}}_{0}(\Gamma)$. Moreover,
$\psi\in{\rm ker}(H_{\Lambda})$ if and only if $\psi\in{\rm ker}(h_{x})$ for
each $x\in\Gamma$ with ${\rm supp}(h_{x})\subset\Lambda$. Thus, if we denote
by $P_{\Lambda}$ the orthogonal projection onto ${\rm ker}(H_{\Lambda})$, then
for any finite subsets $\Lambda_{0}\subset\Lambda\subset\Gamma$ we have that

(2.8) $P_{\Lambda}P_{\Lambda_{0}}=P_{\Lambda_{0}}P_{\Lambda}=P_{\Lambda}\,.$

In the above, we have identified $H_{\Lambda_{0}}$ with
$H_{\Lambda_{0}}\otimes{\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}_{\Lambda\setminus\Lambda_{0}}\in\mathcal{A}_{\Lambda}$
whenever $\Lambda_{0}\subset\Lambda$.

Given such an interaction $\eta$, the infinite system dynamics on $\Gamma$ is
obtained through the derivation $\delta_{0}$ defined for
$A\in{\mathcal{A}}^{\rm loc}$ by

(2.9) $\delta_{0}(A)=\sum_{x\in\Lambda(R)}[h_{x},A]\quad\mbox{for any
}A\in{\mathcal{A}}_{\Lambda}\mbox{ and }\Lambda\in{\mathcal{P}}_{0}(\Gamma).$

It is a standard result that there is a closed derivation extending
$\delta_{0}$, which we also denote by $\delta_{0}$, with domain $\mathop{\rm
dom}(\delta_{0})$ for which ${\mathcal{A}}^{\rm loc}$ is a core [6, 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 $\\{\tau^{(0)}_{t}\mid
t\in{\mathbb{R}}\\}$ satisfying

(2.10)
$\frac{d}{dt}\tau_{t}^{(0)}(A)=i\tau_{t}^{(0)}(\delta_{0}(A))\quad\mbox{ for
all }A\in{\mathcal{A}}^{\rm loc}.$

In fact, this differential equation holds for all $A\in\mathop{\rm
dom}(\delta_{0})$. Two other general properties are:

  1. i.

$\tau_{t}^{(0)}(\mathop{\rm dom}(\delta_{0}))\subset\mathop{\rm
dom}(\delta_{0})$ for all $t\in\mathbb{R}$;

  2. ii.

$\tau_{t}^{(0)}(\delta_{0}(A))=\delta_{0}(\tau_{t}^{(0)}(A))$ for all
$A\in\mathop{\rm dom}(\delta_{0})$ and $t\in\mathbb{R}$.

More generally, quantum spin models can be defined by an interaction on
$\Gamma$ which, by definition, is a map
$\Phi:{\mathcal{P}}_{0}(\Gamma)\to{\mathcal{A}}^{\rm loc}$, with the property
that $\Phi(X)^{*}=\Phi(X)\in{\mathcal{A}}_{X}$ for all
$X\in{\mathcal{P}}_{0}(\Gamma)$. For any decay function $g$, an interaction
norm is defined by

(2.11)
$\|\Phi\|_{g}=\sup_{x,y\in\Gamma}g(d(x,y))^{-1}\sum_{\begin{subarray}{c}X\in{\mathcal{P}}_{0}(\Gamma):\\\
x,y\in X\end{subarray}}\|\Phi(X)\|\,.$

When the above quantity is finite for some interaction $\Phi$, the function
$g$ is said to measure the decay of $\Phi$. If $g$ is an $F$-function, the
norm $\|\cdot\|_{g}$ is called an F-norm. If $g$ is summable, in the sense of
(2.3), and $\|\Phi\|_{g}<\infty$, then a closable derivation on
${\mathcal{A}}^{\rm loc}$ can be defined by setting

$\delta(A)=\sum_{Y,Y\cap X\neq\emptyset}[\Phi(Y),A]\quad\mbox{for
}A\in{\mathcal{A}}^{\rm loc}\mbox{ with }{\rm supp}(A)\subset
X\in\mathcal{P}_{0}(\Gamma)\,.$

One can prove conditions that guarantee that the derivation $\delta$ defined
on ${\mathcal{A}}^{\rm loc}$ is a generator of a strongly continuous dynamics
given by automorphisms of ${\mathcal{A}}$ [5, 6]. In practice, however, one
usually directly proves the existence of the thermodynamic limit of the
Heisenberg dynamics. Standard results along these lines prove the existence of
the dynamics for $\Phi$ in a suitable Banach space of interactions [32, 6, 33]
starting from a convergent series for small $|t|$. An alternative approach,
based on Lieb-Robinson bounds [20], was introduced by Robinson [31]. Lieb-
Robinson bounds can be derived for any interaction $\Phi$ with a finite
$F$-norm [23], and this allows one to extend the results for existence of the
dynamics beyond the Banach spaces of interactions ${\mathcal{B}}_{\lambda}$
introduced by Ruelle [32]. These ideas are important for the construction of
the spectral flow automorphisms [3]. This and some other generalizations
relevant for the present work are discussed in detail in [26].

In this work, we consider an infinite-volume, zero-energy, ground state of the
frustration free model defined by $\delta_{0}$, denoted by $\omega_{0}$. Let
$({\mathcal{H}},\pi_{0},\Omega)$ be the GNS triple of $\omega_{0}$. This means
that $\pi_{0}:{\mathcal{A}}\to{\mathcal{B}}({\mathcal{H}})$ is a
representation of the $C^{*}$-algebra ${\mathcal{A}}$ on a Hilbert space
${\mathcal{H}}$ for which $\\{\pi_{0}(A)\Omega|A\in{\mathcal{A}}^{\rm loc}\\}$
is dense in ${\mathcal{H}}$. Moreover, the normalized vector
$\Omega\in{\mathcal{H}}$ is such that
$\omega_{0}(A)=\langle\Omega,\pi_{0}(A)\Omega\rangle$ for all
$A\in{\mathcal{A}}$. Recall that infinite-volume ground states associated to
$\delta_{0}$ are defined as those states $\omega$ on ${\mathcal{A}}$ which
satisfy

(2.12) $\omega(A^{*}\delta_{0}(A))\geq 0\quad\mbox{for all
}A\in{\mathcal{A}}^{\rm loc}.$

For the GNS representation of a ground state, as in (2.12), there exists a
unique, non-negative self-adjoint operator $H_{0}$ on ${\mathcal{H}}$, with
dense domain $\mathop{\rm dom}H_{0}$, satisfying $H_{0}\Omega=0$ and

(2.13)
$\pi_{0}(\tau_{t}^{(0)}(A))=e^{itH_{0}}\pi_{0}(A)e^{-itH_{0}}\quad\mbox{for
all }A\in{\mathcal{A}}\mbox{ and }t\in{\mathbb{R}}.$

The full domain of $H_{0}$ is seldom described explicitly. However, for all
systems we consider in this paper, $\pi_{0}({\mathcal{A}}^{\rm loc})\Omega$ is
a core for $H_{0}$.

Frustration-free models are characterized by the property that they have a
zero-energy ground state, meaning there is a state $\omega_{0}$ on
${\mathcal{A}}$ such that $\omega_{0}(h_{x})=0$ for all $x\in\Gamma$. It is
easy to see that a zero-energy ground state is a ground state in the sense of
(2.12). Frustration-free models may have other ground states but thermodynamic
limits of ground states of

(2.14) $H_{\Lambda_{n}}=\sum_{\stackrel{{\scriptstyle x\in\Lambda_{n}:}}{{{\rm
supp}(h_{x})\subset\Lambda_{n}}}}h_{x}$

automatically have the zero-energy property. Conversely, any zero-energy
infinite volume ground state is a thermodynamic limit of finite-volume zero-
energy ground states. We will refer to the GNS representation of a zero-energy
ground state of a frustration-free model as a frustration-free representation.

###  2.2. Main results

We now state the assumptions for the main results in this paper. We say that
the ground state $\omega_{0}$ is gapped if there exists $\gamma>0$ such that
$(0,\gamma)\cap\mathop{\rm spec}(H_{0})=\emptyset$. The bulk gap of the model
in the state $\omega_{0}$ is defined as

(2.15) $\gamma_{0}=\sup\\{\gamma>0\mid(0,\gamma)\cap\mathop{\rm
spec}(H_{0})=\emptyset\\}.$

If the set on the RHS is empty, one defines $\gamma_{0}=0$. Given that
$\omega_{0}$ is time-invariant since it is a ground state, the equivalence of
the following two conditions is easy to verify:

  1. i.

For some $\gamma>0$, $\omega_{0}$ satisfies

(2.16)
$\omega_{0}(A^{*}\delta_{0}(A))\geq\gamma\omega_{0}(A^{*}A)\quad\mbox{for all
}A\in{\mathcal{A}}^{\rm loc}\mbox{ with }\omega_{0}(A)=0;$

  2. ii.

The ground state of the GNS Hamiltonian $H_{0}$ is unique and the bulk gap
satisfies $\gamma_{0}\geq\gamma$.

In this work, we assume positivity of the bulk gap for the reference model.

######  Assumption 2.1 (Bulk gap).

Let $\omega_{0}$ be an infinite-volume, zero-energy ground state of a finite-
range, uniformly bounded, frustration free model as above. The bulk gap of the
associated GNS Hamiltonian $H_{0}$, see (2.15), is strictly positive.

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 we have a family
${\mathcal{S}}=\\{\Lambda(x,n)\mid n\geq
0,x\in\Gamma\\}\subset{\mathcal{P}}_{0}(\Gamma)$, with
$b_{x}(n)\subset\Lambda(x,n)$ for all $x$ and $n$, and moreover, there is an
associated family of partitions of $\Gamma$ which _separates ${\mathcal{S}}$
and has at most polynomial growth_. Concretely, this means there is a family
of sets ${\mathcal{T}}=\\{{\mathcal{T}}_{n}\mid n\geq 0\\}$ and positive
numbers $c$ and $\zeta$, such that for each $n\geq 0$,
${\mathcal{T}}_{n}=\\{{\mathcal{T}}_{n}^{i}:i\in{\mathcal{I}}_{n}\\}$ is a
partition of $\Gamma$ satisfying $|{\mathcal{I}}_{n}|\leq cn^{\zeta}$ and

(2.17) $\Lambda(x,n)\cap\Lambda(y,n)=\emptyset\;\text{ for all
}\;x,y\in{\mathcal{T}}_{n}^{i}\;\text{with}\;x\neq y.$

In such cases, we say that ${\mathcal{T}}$ is of $(c,\zeta)$-polynomial
growth.

As an example, in the case of $\Gamma={\mathbb{Z}}^{\nu}$, we may take for
$\Lambda(x,n)$ the $\ell^{\infty}$-ball of radius $n$ centered at $x$, define
${\mathcal{I}}_{n}=\Lambda(0,n)$ and, for each $i\in{\mathcal{I}}_{n}$, set

(2.18) ${\mathcal{T}}^{i}_{n}=\\{x\in{\mathbb{Z}}^{\nu}|x_{j}=i_{j}\mod
2n+1,i=1,\ldots,\nu\\}.$

######  Assumption 2.2 (Local gaps).

Let $\eta$ be a finite-range, uniformly bounded frustration-free interaction
on $\Gamma$. Associated to $\eta$, there exists a family
${\mathcal{S}}=\\{\Lambda(x,n)\mid n\geq
0,x\in\Gamma\\}\subset{\mathcal{P}}_{0}(\Gamma)$ with
$b_{x}(n)\subset\Lambda(x,n)$ for all $x$ and $n$, a family
${\mathcal{T}}=\\{{\mathcal{T}}_{n}:n\geq 0\\}$ of partitions of $\Gamma$
which separates ${\mathcal{S}}$ and is of $\zeta$-polynomial growth, and an
exponent $\alpha\geq 0$ and constant $\gamma_{1}>0$, such that the finite-
volume Hamiltonians associated to $\eta$ satisfy:

(2.19) $\mathop{\rm
spec}(H_{\Lambda(x,n)})\subset\\{0\\}\cup[n^{-\alpha}\gamma_{1},\infty)\quad\mbox{for
all }x\in\Gamma\mbox{ and }n\geq R,$

where $R$ is the interaction radius of $\eta$.

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 $1/L$ 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 [19], 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 [12, 17, 1].
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 $L^{-2}$
[2, 4]. In any case, regardless of the possible values of the exponent
$\alpha$, we will prove stability of the bulk gap.

The next assumption was introduced in the form we use here in [21] where it is
called Local Topological Quantum Order (LTQO).

######  Assumption 2.3 (LTQO).

Let $\omega_{0}$ be an infinite-volume, zero-energy ground state of a finite-
range, uniformly bounded, frustration free model. There is a non-increasing
function $G_{0}:[0,\infty)\to[0,\infty)$, for which

(2.20) $\sum_{n\geq 1}n^{\zeta+\alpha+\nu/2}\sqrt{G_{0}(n)}<\infty,$

and such that for all $m\geq k\geq 0,x\in\Gamma$, and
$A\in{\mathcal{A}}_{b_{x}(k)}$, we have

(2.21)
$\|P_{b_{x}(m)}AP_{b_{x}(m)}-\omega_{0}(A)P_{b_{x}(m)}\|\leq\|A\|(1+k)^{\nu}G_{0}(m-k).$

As explained in detail in [25, 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 (2.21). Other discrete
symmetries can be treated similarly (see [25, Section 8]). Therefore, the
stability results proved here (Theorems 2.7 and 2.8) 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 2.3 automatically have a
unique ground state. This is the content of the following propostion.

######  Proposition 2.4.

Let $\omega_{0}$ be an infinite-volume, zero-energy ground state of a
frustration-free model satisfying Assumption 2.3. The kernel of the GNS
Hamiltonian $H_{0}$ is one-dimensional.

###### Proof.

By way of contradiction, let us assume there is a unit vector $\psi\in{\rm
ker}(H_{0})$ with $\langle\psi,\Omega\rangle=0$. To simplify notation, for any
$A\in{\mathcal{A}}$, let us denote by $\tilde{A}:=\pi_{0}(A)$ the
representative of $A$ in the GNS space. The LTQO condition (2.21), in
${\mathcal{B}}({\mathcal{H}})$, can be stated as: for all $m\geq k\geq 0$,
$x\in\Gamma$, and $A\in{\mathcal{A}}_{b_{x}(k)}$,

(2.22)
$\|\tilde{P}_{b_{x}(m)}\tilde{A}\tilde{P}_{b_{x}(m)}-\langle\Omega,\tilde{A}\Omega\rangle\tilde{P}_{b_{x}(m)}\|\leq\|A\|(1+k)^{\nu}G_{0}(m-k)\,.$

First, since $\pi_{0}({\mathcal{A}}^{\rm loc})\Omega$ is dense in
${\mathcal{H}}$, there is $x\in\Gamma$ and operators
$A_{n}\in{\mathcal{A}}_{b_{x}(n)}$ for which
$\psi_{n}=\tilde{A}_{n}\Omega\to\psi$. In this case,

(2.23)
$\lim_{n\to\infty}\langle\Omega,\tilde{A}_{n}\Omega\rangle=\lim_{n\to\infty}\langle\Omega,\psi_{n}\rangle=\langle\Omega,\psi\rangle=0.$

Thus, there is $n_{1}\in\mathbb{N}$ so that
$|\langle\Omega,\tilde{A}_{n}\Omega\rangle|\leq 1/3$ for all $n\geq n_{1}$.

Next, since ${\rm ker}(H_{0})\subset{\rm ker}(\tilde{H}_{b_{x}(m)})$ for all
$m\geq 0$ and $x\in\Gamma$, we have

(2.24)
$\langle\psi,\tilde{P}_{b_{x}(m)}\tilde{A}_{n}\tilde{P}_{b_{x}(m)}\Omega\rangle=\langle\psi,\tilde{A}_{n}\Omega\rangle=\langle\psi,\psi_{n}\rangle\,$

for all $m\geq 0$. Since $1=\lim_{n\to\infty}\langle\psi,\psi_{n}\rangle$,
there is some $n_{2}\in\mathbb{N}$ for which
$|\langle\psi,\psi_{n}\rangle|\geq 2/3$ for all $n\geq n_{2}$. Combining this
with (2.24), we find that for any $m\geq n\geq n_{2}$,

(2.25) $2/3\leq\|\tilde{P}_{b_{x}(m)}\tilde{A}_{n}\tilde{P}_{b_{x}(m)}\|.$

It follows that for $n\geq\max\\{n_{1},n_{2}\\}$,

(2.26)
$1/3\leq\|\tilde{P}_{b_{x}(m)}\tilde{A}_{n}\tilde{P}_{b_{x}(m)}-\langle\Omega,\tilde{A}_{n}\Omega\rangle\tilde{P}_{b_{x}(m)}\|,$

a contradiction to (2.22), as the decay function $G_{0}$ satisfies
$G_{0}(m)\to 0$ as $m\to\infty$. ∎

Next, we turn to the perturbations of the Hamiltonian $H_{0}$. We consider
perturbations defined by an anchored interaction $\Phi$ on $\Gamma$ with terms
given by $\\{\Phi(x,n)\,|\,x\in\Gamma,\,n\geq 0\,\\}$. This means that
$\Phi(x,n)^{*}=\Phi(x,n)\in\mathcal{A}_{b_{x}(n)}$ for all $x\in\Gamma$ and
$n\geq 0$. By regrouping, we need only consider those terms with $n\geq R$ the
interaction radius of $\eta$.

######  Assumption 2.5 (Short-range perturbation).

There is a constant $\|\Phi\|\geq 0$, $a>0$, and $\theta\in(0,1]$ such that
for all $x\in\Gamma$

(2.27) $\|\Phi(x,n)\|\leq\|\Phi\|e^{-an^{\theta}}\quad\mbox{ for all }n\geq
R.$

######  Remark 2.6.

Assume $\Phi$ satisfies Assumption 2.5. As indicated in (2.5), for any
$0<a^{\prime}<a$ and $\zeta>\nu$, the function $F:[0,\infty)\to(0,\infty)$
given by

(2.28) $F(r)=\frac{e^{-a^{\prime}r^{\theta}}}{(1+r)^{\zeta}}\quad\mbox{for all
}r\geq 0,$

is an $F$-function on $\Gamma$. Let $\delta=a-a^{\prime}>0$ and note that for
any $x,y\in\Gamma$ with $d(x,y)\geq R$, we have

(2.29) $\displaystyle\sum_{z\in\Gamma}\sum_{\stackrel{{\scriptstyle n\geq
R:}}{{x,y\in b_{z}(n)}}}\|\Phi(z,n)\|$ $\displaystyle=$
$\displaystyle\sum_{n\geq d(x,y)}\sum_{z\in b_{y}(n)\cap
b_{x}(n)}\|\Phi(z,n)\|$ $\displaystyle\leq$
$\displaystyle\kappa\|\Phi\|\sum_{n\geq
d(x,y)}n^{\nu}e^{-an^{\theta}}\leq\|\Phi\|_{F}F(d(x,y))$

with $\|\Phi\|_{F}=\kappa\|\Phi\|\sum_{n\geq 0}(1+n)^{\nu+\zeta}e^{-\delta
n^{\theta}}<\infty$. Thus any such $\Phi$ has a finite $F$-norm with $F$ as in
(2.28).

The focus of this work is to analyze the stability of the bulk gap under the
presence of perturbations whose terms decay as in (2.27). Given an anchored
interaction $\Phi$ satisfying Assumption 2.5, the main focus of this work will
be perturbed Hamiltonians of the form

(2.30) $H(\Lambda,s)=H_{0}+s\pi_{0}(V_{\Lambda}),\quad s\in{\mathbb{R}}$

where, for any finite volume $\Lambda\in{\mathcal{P}}_{0}(\Gamma)$,

(2.31) $V_{\Lambda}=\sum_{x\in\Lambda}\sum_{\stackrel{{\scriptstyle n\geq
R:}}{{b_{x}(n)\subset\Lambda}}}\Phi(x,n).$

Clearly, $V_{\Lambda}\in{\mathcal{A}}_{\Lambda}$ is bounded and self-adjoint,
and so $H(\Lambda,s)$ defines for all $s\in{\mathbb{R}}$ a self-adjoint
Hamiltonian on ${\mathcal{H}}$ with the same dense domain as $H_{0}$.

The core result we will prove in the next several sections is the following
theorem which establishes that the spectral gap of $H(\Lambda,s)$ remains open
for small $|s|$ uniform in the finite volume $\Lambda$.

######  Theorem 2.7 (Stability of the gap uniformly in the perturbation
region).

Let $\omega_{0}$ be an infinite-volume, zero-energy ground state of a finite-
range, uniformly bounded, frustration free model defined by an interaction
$\eta$. Suppose Assumptions 2.1, 2.2, and 2.3 hold, and let $\Phi$ be an
anchored interaction for which Assumption 2.5 holds. Then, for all
$\gamma\in(0,\gamma_{0})$, there exists $s_{0}(\gamma)>0$, such that for all
real $s$, $|s|<s_{0}(\gamma)$, and $\Lambda\in{\mathcal{P}}_{0}(\Gamma)$, we
have

(2.32) $\mathop{\rm
spec}H(\Lambda,s)\subset\\{E(\Lambda,s)\\}\cup[E(\Lambda,s)+\gamma,\infty)$

with $H(\Lambda,s)$ as in (2.30) and

(2.33) $E(\Lambda,s)=\inf\mathop{\rm spec}H(\Lambda,s).$

We remark that the quantity $s_{0}(\gamma)$ only depends on the values of
$\kappa$ and $\nu$ of the lattice, $\|\eta\|_{\infty}$, the gap $\gamma_{0}$,
the parameters in Assumption 2.2, the decay function in Assumption 2.3, and a
suitable $F$-norm of the perturbation $\Phi$. From the arguments in this
paper, one can derive an explicit lower bound for $s_{0}(\gamma)$ in terms of
these quantities, see Section 6

We also investigate the situation where the perturbation region $\Lambda$
tends to all of $\Gamma$. Consider any IAS $(\Lambda_{n})$, and let $\Phi$ be
an anchored interaction satisfying Assumption 2.5. We will denote by
$\tau_{t}^{(\Lambda_{n},s)}$ the dynamics on ${\mathcal{A}}$ generated by the
interaction $\eta+s\Phi\restriction_{\Lambda_{n}}$, which is generated by the
derivation

(2.34)
$\delta_{s}^{\Lambda_{n}}(A)=\delta_{0}(A)+[sV_{\Lambda_{n}},A]\quad\mbox{for
}A\in{\mathcal{A}}^{\rm loc}\,.$

As discussed in [26, Definition 3.7], the sequence of interactions
$\eta+s\Phi\restriction_{\Lambda_{n}}$ converges locally in $F$-norm to the
interaction $\eta+s\Phi$; here one may use any $F$-function with the form
(2.28). Using [26, Theorem 3.8], we conclude local convergence in the sense
that

(2.35)
$\lim_{n\to\infty}\tau_{t}^{(\Lambda_{n},s)}(A)=\tau_{t}^{(s)}(A)\quad\mbox{
for all }A\in{\mathcal{A}}^{\rm loc}$

as well as

(2.36) $\lim_{n\to\infty}\delta_{s}^{\Lambda_{n}}(A)=\delta_{s}(A)\quad\mbox{
for all }A\in{\mathcal{A}}^{\rm loc}$

with $\tau_{t}^{(s)}$ (respectively, $\delta_{s}$) being the a priori well-
defined strongly continuous dynamics on ${\mathcal{A}}$ (respectively, the
closure of the derivation restricted to ${\mathcal{A}}^{\rm loc}$) generated
by the interaction $\eta+s\Phi$. Neither of these limits depend on the choice
of IAS sequence $\Lambda_{n}$.

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 2.7 allows one to prove, almost as a corollary,
that for all $|s|\leq s_{0}(\gamma)$ there is a gapped ground state
$\omega_{s}$ of $\delta_{s}$ in the sense of (2.16). To make these precise, we
introduce the limiting spectral flow. For any $\gamma>0$ and IAS
$(\Lambda_{n})$, take

(2.37) $\alpha_{s}(A)=\lim_{n}\alpha^{\Lambda_{n}}_{s}(A)\quad\mbox{ for all
}A\in{\mathcal{A}}\,,$

where the spectral flows $\alpha_{s}^{\Lambda_{n}}$ will be introduced in more
detail in the next section, see (3.25). 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 $\alpha_{s}^{\Lambda_{n}}$ converge
locally in $F$-norm by arguments as in [26, Section VI.E.2]. This limiting
spectral flow $\alpha_{s}$ defines a strongly continuous co-cycle of
automorphisms of ${\mathcal{A}}$, and moreover, under the assumptions we have
made, for $A\in{\mathcal{A}}^{\rm loc}$, $s\mapsto\alpha_{s}(A)$ is
differentiable to all orders.

######  Theorem 2.8 (Stability of the bulk gap).

Under the assumptions of Theorem 2.7, let $\gamma\in(0,\gamma_{0})$ and take
$s$ with $|s|<s_{0}(\gamma)$; the quantity whose existence is guaranteed by
Theorem 2.7. For the limiting spectral flow $\alpha_{s}$, as in (2.37) above,
the state $\omega_{s}=\omega_{0}\circ\alpha_{s}$ is a gapped ground state of
the perturbed infinite dynamics $\delta_{s}$, i.e.

(2.38) $\omega_{s}(A^{*}\delta_{s}(A))\geq\gamma\omega_{s}(A^{*}A)\quad\mbox{
for all }A\in{\mathcal{A}}^{\rm loc}\mbox{ with }\omega_{s}(A)=0\,.$

In particular, the GNS Hamiltonian $H_{s}$ of $\omega_{s}$ has a one-
dimensional kernel and $\mathop{\rm spec}H_{s}$ has a gap above its ground
state bounded below by $\gamma$.

##  3\. Quasi-locality, Domains and Local Decompositions

In this section, we set the stage for proving stability in the infinite-system
setting. Quasi-locality on ${\mathcal{A}}$ is the topic of Section 3.1. In
Section 3.1.1, we recall general methods for making strictly local
approximations of both quasi-local observables and maps. Useful examples, as
well as explicit estimates, are contained in Sections 3.1.2 \- 3.1.5. We
prepare results about frustration-free GNS representations we will need in
Section 3.2. Lastly, to deal with unbounded operators, we must have some
control over certain vectors in their domain, which is dealt with in Section
3.3.

###  3.1. Quasi-Locality

We will frequently need controlled strictly local approximations of global
quantities, which we refer to in general as quasi-locality estimates. We first
recall some general features of quasi-locality estimates. Important examples
which will be relevant in this work are described in detail in Sections 3.1.2
\- 3.1.5.

####  3.1.1. Quasi-locality estimates.

Let ${\mathcal{A}}$ be a $C^{*}$-algebra of quasi-local observables over
$\Gamma$ as discussed in Section 2. For any $X\in\mathcal{P}_{0}(\Gamma)$, a
conditional expectation $\Pi_{X}:{\mathcal{A}}\to{\mathcal{A}}$ is defined
with respect to the tracial state $\rho$ on ${\mathcal{A}}$ by setting

(3.1) $\Pi_{X}=\rho\restriction_{\Gamma\setminus X}\otimes{\rm
id}_{{\mathcal{A}}_{X}}.$

Here we identify elements of the sub-algebra ${\mathcal{A}}_{X}$ with their
representatives in ${\mathcal{A}}^{\rm loc}\subset{\mathcal{A}}$. Thus, in
particular, if $A\in{\mathcal{A}}_{\Lambda}$ and $X\subset\Lambda$, then
$\Pi_{X}(A)$ is just given by the normalized partial trace.

It is often the case that for some $A\in{\mathcal{A}}$, there are
$X\in\mathcal{P}_{0}(\Gamma)$ and $\epsilon>0$ for which

(3.2) $\|[A,B]\|\leq\epsilon\|B\|\quad\mbox{whenever
}B\in{\mathcal{A}}_{\Gamma\setminus X}^{\rm loc}\,.$

We refer to such an observable $A$ as quasi-local. In fact, a general result
from [9, 24] shows that any such quasi-local observable $A$ also satisfies

(3.3) $\|A-\Pi_{X}(A)\|\leq\epsilon\,,$

i.e. $A$ has a quantifiable, strictly local approximation; namely,
$\Pi_{X}(A)$.

Similar notions hold for maps. A linear map
${\mathcal{K}}:{\mathcal{A}}\to{\mathcal{A}}$ is said to be quasi-local if
there are $C\geq 0$, $p\geq 0$, and a decay function $G$ for which

(3.4) $\|[{\mathcal{K}}(A),B]\|\leq C|X|^{p}\|A\|\|B\|G(d(X,Y))\mbox{ for all
}A\in{\mathcal{A}}_{X}\mbox{ and }B\in{\mathcal{A}}_{Y}\,.$

For a quasi-local map ${\mathcal{K}}$, the image ${\mathcal{K}}(A)$ of a
strictly local observable $A$ can be approximated by a strictly local
observable using (3.3). In the present context, it suffices to localize
observables on balls. For any quasi-local map ${\mathcal{K}}$ on
${\mathcal{A}}$ satisfying (3.4), (3.3) shows that for any $n\geq 0$ and
$A\in{\mathcal{A}}_{b_{x}(k)}$,

(3.5) $\|{\mathcal{K}}(A)-\Pi_{b_{x}(k+n)}({\mathcal{K}}(A))\|\leq
C|b_{x}(k)|^{p}\|A\|G(n).$

When the corresponding decay function $G$ is summable, such an estimate
guarantees the (absolute) convergence of telescopic sums, i.e. for any
$n_{0}\geq 0$,

(3.6)
${\mathcal{K}}(A)=\Pi_{b_{x}(k+n_{0})}({\mathcal{K}}(A))+\sum_{n=n_{0}+1}^{\infty}\left(\Pi_{b_{x}(k+n)}-\Pi_{b_{x}(k+n-1)}\right)({\mathcal{K}}(A))$

since the terms are easily seen to satisfy

(3.7)
$\|\left(\Pi_{b_{x}(k+n)}-\Pi_{b_{x}(k+n-1)}\right)({\mathcal{K}}(A))\|\leq
2C|b_{x}(k)|^{p}\|A\|G(n-1)\quad\mbox{for }n\geq 1.$

A common choice is $n_{0}=0$ and we adopt the notation

(3.8) ${\mathcal{K}}(A)=\sum_{n\geq
k}\Delta_{b_{x}(k)}^{n}({\mathcal{K}}(A))\quad\text{where}\quad\Delta_{b_{x}(k)}^{n}=\begin{cases}\Pi_{b_{x}(k)},&n=k\\\
\Pi_{{b_{x}(n)}}-\Pi_{{b_{x}(n-1)}},&n\geq k+1.\end{cases}$

We now briefly review four quasi-local maps as well as a few of their
important, well-known properties that are necessary for our later arguments.

####  3.1.2. The Unperturbed Dynamics

Let $\tau_{t}^{(0)}$ be the dynamics generated by the finite-range, uniformly
bounded, frustration-free interaction $\eta$ as discussed in Section 2. As is
well-known, $\tau_{t}^{(0)}$ satisfies an exponential Lieb-Robinson bound,
i.e. for every $\mu>0$ there exists $C_{\mu}>0$ and $v_{\mu}>0$ such that for
any choice of $X,Y\in\mathcal{P}_{0}(\Gamma)$, we have that

(3.9) $\|[\tau_{t}^{(0)}(A),B]\|\leq
C_{\mu}\min(|X|,|Y|)\|A\|\|B\|e^{-\mu(d(X,Y)-v_{\mu}|t|)}$

for all $A\in\mathcal{A}_{X}$, $B\in\mathcal{A}_{Y}$, and $t\in\mathbb{R}$.

####  3.1.3. The Perturbed Dynamics

Let $\eta$ be a finite-range, uniformly bounded, frustration free interaction
and $\Phi$ an anchored interaction satisfying Assumption 2.5. For each
$\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in\mathbb{R}$, we have denoted by
$\tau_{t}^{(\Lambda,s)}$ the strongly continuous dynamics generated by
$\eta+s\Phi{\restriction_{\Lambda}}$. Since $\eta$ is finite-range and
uniformly bounded and $\Phi$ has a finite $F$-norm for any $F$-function of the
form (2.28), it is clear that $\eta+s\Phi_{\restriction_{\Lambda}}$ has a
finite $F$-norm as well. In fact, the $F$-norm of this interaction has an
upper bound which is uniform with respect to
$\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s$ in compact sets. We conclude that
there are $C_{F}>0$ and $v_{F}>0$ such that for any choice of
$X,Y,\Lambda\in\mathcal{P}_{0}(\Gamma)$,

(3.10) $\|[\tau_{t}^{(\Lambda,s)}(A),B]\|\leq
C_{F}\|A\|\|B\|e^{v_{F}|t|}\sum_{x\in X}\sum_{y\in Y}F(d(x,y))$

for all $A\in\mathcal{A}_{X}$, $B\in\mathcal{A}_{Y}$, and $t\in\mathbb{R}$.

It will also be useful for us to observe that this perturbed dynamics
$\tau_{t}^{(\Lambda,s)}$ may equivalently be viewed as the dynamics generated
by the closure of the derivation $\delta_{s}^{\Lambda}$ defined by
${\mathcal{A}}^{\rm loc}$ by

(3.11) $\delta_{s}^{\Lambda}(A)=\delta_{0}(A)+[sV_{\Lambda},A]\quad\mbox{for
all }A\in{\mathcal{A}}^{\rm loc}.$

Since each $sV_{\Lambda}$ is bounded and self-adjoint, [6, Proposition 5.4.1]
implies that

(3.12)
$\tau_{t}^{(\Lambda,s)}(A)=(\Gamma_{t}^{(\Lambda,s)})^{*}\tau_{t}^{(0)}(A)\Gamma_{t}^{(\Lambda,s)}\quad\mbox{for
all }A\in{\mathcal{A}}\mbox{ and }t\in\mathbb{R}$

where $\\{\Gamma_{t}^{(\Lambda,s)}\,|\,t\in\mathbb{R}\\}$ is a one-parameter
family of unitaries on ${\mathcal{A}}$ which are uniquely defined as the
${\mathcal{A}}$-valued solution of

(3.13)
$\frac{d}{dt}\Gamma_{t}^{(\Lambda,s)}=-i\tau_{t}^{(0)}(sV_{\Lambda})\Gamma_{t}^{(\Lambda,s)}\quad\mbox{with}\quad\Gamma_{0}^{(\Lambda,s)}={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}\,.$

Additionally, these unitaries are quasi-local as, for any
$A\in{\mathcal{A}}^{\rm loc}$ and $t>0$,

(3.14)
$\displaystyle\|[\Gamma_{t}^{(\Lambda,s)},A]\|=\|(\Gamma_{t}^{(\Lambda,s)})^{*}A\Gamma_{t}^{(\Lambda,s)}-A\|$
$\displaystyle=$
$\displaystyle\left\|\int_{0}^{t}\frac{d}{du}\left((\Gamma_{u}^{(\Lambda,s)})^{*}A\Gamma_{u}^{(\Lambda,s)}\right)\,du\right\|\,$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\|[\tau_{u}^{(0)}(sV_{\Lambda}),A]\|\,du\,.$

An application of (3.9) then shows that for any $\mu>0$ and
$A\in{\mathcal{A}}_{X}$ with $X\in\mathcal{P}_{0}(\Gamma\setminus\Lambda)$,

(3.15) $\|[\Gamma_{t}^{(\Lambda,s)},A]\|\leq
C_{\mu}|s||\Lambda|\|V_{\Lambda}\|\|A\||t|e^{\mu v_{\mu}|t|}e^{-\mu
d(\Lambda,X)}$

for any $s,t\in\mathbb{R}$. We conclude, using e.g. (3.3), that
$\Gamma_{t}^{(\Lambda,s)}\in{\mathcal{A}}^{g}$ for some $g$ of exponential
decay.

####  3.1.4. Weighted Integral Operators

Under the assumptions of Theorem 2.7, fix $\gamma>0$. For each
$\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in{\mathbb{R}}$, we define two
weighted integral operators
${\mathcal{F}}_{s}^{\Lambda}:{\mathcal{A}}\to{\mathcal{A}}$ and
${\mathcal{G}}_{s}^{\Lambda}:{\mathcal{A}}\to{\mathcal{A}}$ by setting

(3.16)
${\mathcal{F}}_{s}^{\Lambda}(A)=\int_{-\infty}^{\infty}\tau_{t}^{(\Lambda,s)}(A)w_{\gamma}(t)\,dt\quad\mbox{and}\quad{\mathcal{G}}_{s}^{\Lambda}(A)=\int_{-\infty}^{\infty}\tau_{t}^{(\Lambda,s)}(A)W_{\gamma}(t)\,dt\,,$

for all $A\in{\mathcal{A}}$, where the real-valued functions $w_{\gamma}$ and
$W_{\gamma}$, both of which are in $L^{1}(\mathbb{R})$, are as defined in [26,
Section VI.B]. By strong-continuity of the perturbed dynamics, both of the
integrals appearing in (3.16) are well-defined in the sense of Bochner, see
e.g. [34]. Both these maps depend on the choice of $\gamma$ through their
weight functions, $w_{\gamma}$ and $W_{\gamma}$ respectively, but we suppress
this in the notation. Arguing as in [26, Section VI.E.1], see also [25,
Section 4.3.2], we find that for all $A\in{\mathcal{A}}$

(3.17)
$\|{\mathcal{F}}_{s}^{\Lambda}(A)\|\leq\|A\|\quad\mbox{and}\quad\|{\mathcal{G}}_{s}^{\Lambda}(A)\|\leq\|W_{\gamma}\|_{1}\|A\|\,,$

i.e. these maps are bounded uniformly with respect to $s\in\mathbb{R}$ and
$\Lambda\in\mathcal{P}_{0}(\Gamma)$. Moreover, they are uniformly quasi-local
in the sense that for each ${\mathcal{K}}\in\\{{\mathcal{F}},{\mathcal{G}}\\}$
there is a decay function $G_{{\mathcal{K}}}$ such that: for any choice of
$X,Y\in\mathcal{P}_{0}(\Gamma)$, we have

(3.18) $\sup_{s\in[-1,1]}\|[{\mathcal{K}}_{s}^{\Lambda}(A),B]\|\leq
2\|A\|\|B\||X|G_{{\mathcal{K}}}(d(X,Y))$

for all $A\in\mathcal{A}_{X}$ and $B\in\mathcal{A}_{Y}$. As in Lemma 6.10 and
Lemma 6.11 of [26], one can be quite explicit in estimating the decay of these
functions $G_{{\mathcal{K}}}$. For our purposes here, we need only stress that
these decay functions are independent of $\Lambda\in\mathcal{P}_{0}(\Gamma)$
and $s\in[-1,1]$. In addition, these decay functions decay faster than any
polynomial and, thus, have finite moments of all orders, i.e. for any $\mu\geq
0$,

(3.19) $\sum_{n=1}^{\infty}(n+1)^{\mu}G_{{\mathcal{K}}}(n)<\infty\,.$

####  3.1.5. The Spectral Flow

Under the assumptions of Theorem 2.7, fix $\gamma>0$. For each
$\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in{\mathbb{R}}$, denote by

(3.20)
$D(\Lambda,s)={\mathcal{G}}_{s}^{\Lambda}(V_{\Lambda})=\int_{-\infty}^{\infty}\tau^{(\Lambda,s)}_{t}(V_{\Lambda})W_{\gamma}(t)dt$

with ${\mathcal{G}}^{\Lambda}_{s}$ the weighted integral operator defined in
(3.16) above. From (3.17), it is clear that $s\mapsto D(\Lambda,s)$ is
uniformly bounded. One further checks that for each $s\in\mathbb{R}$,
$D(\Lambda,s)$ is self-adjoint. We now show that $s\mapsto D(\Lambda,s)$ is
norm continuous. For $t\in\mathbb{R}$ fixed, the strong differentiability of
$s\mapsto\tau_{t}^{(\Lambda,s)}$ is proven in [26, Proposition 2.7], in
particular, it is shown that

(3.21)
$\frac{d}{ds}\tau_{t}^{(\Lambda,s)}(A)=i\int_{0}^{t}\tau_{r}^{(\Lambda,s)}([V_{\Lambda},\tau_{t-r}^{(\Lambda,s)}(A)])\,dr\,.$

Using (3.21), we find that

(3.22) $\displaystyle\|D(\Lambda,s)-D(\Lambda,s_{0})\|$ $\displaystyle\leq$
$\displaystyle\int_{-\infty}^{\infty}\|\tau^{(\Lambda,s)}_{t}(V_{\Lambda})-\tau^{(\Lambda,s_{0})}_{t}(V_{\Lambda})\||W_{\gamma}(t)|dt$
$\displaystyle\leq$ $\displaystyle
2\|V_{\Lambda}\|^{2}|s-s_{0}|\int_{-\infty}^{\infty}|t||W_{\gamma}(t)|\,dt.$

The norm-continuity follows from this estimate. Again we refer to [26, Section
VI.B] for details.

Given these properties of $D(\Lambda,s)$, there is a unique solution of

(3.23)
$\frac{d}{ds}u(\Lambda,s)=-iD(\Lambda,s)u(\Lambda,s)\quad\mbox{with}\quad
u(\Lambda,0)={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}\,,$

which is given by unitaries in ${\mathcal{A}}$.

Two important consequences of (3.23) follow. First, as we saw with the
unitaries defined by (3.13), this solution is quasi-local. In fact, similar
arguments show that for $s>0$

(3.24)
$\|[u(\Lambda,s),A]\|\leq\int_{0}^{s}\|[{\mathcal{G}}_{r}^{\Lambda}(V_{\Lambda}),A]\|\,dr\leq
2s\|A\||\Lambda|\|V_{\Lambda}\|G_{{\mathcal{G}}}(d(X,\Lambda))$

for any $A\in{\mathcal{A}}_{X}$ with
$X\in\mathcal{P}_{0}(\Gamma\setminus\Lambda)$. For the above we have used
(3.20), i.e. that $D(\Lambda,s)={\mathcal{G}}_{s}^{\Lambda}(V_{\Lambda})$, and
(3.18). The function $G_{{\mathcal{G}}}$ decays faster than any stretched
exponential. Again, we conclude that $u(\Lambda,s)\in{\mathcal{A}}^{g}$ for
some $g$ with finite moments of all orders.

Using (3.23) we define the spectral flow as the inner automorphisms on
${\mathcal{A}}$ given by

(3.25) $\alpha^{\Lambda}_{s}(A)=u(\Lambda,s)^{*}Au(\Lambda,s)\quad\mbox{ for
all }A\in{\mathcal{A}}.$

Quasi-locality of this map follows by re-writing the generator as

(3.26)
$D(\Lambda,s)={\mathcal{G}}_{s}^{\Lambda}(V_{\Lambda})=\sum_{x\in\Lambda}\sum_{\stackrel{{\scriptstyle
n\geq R:}}{{b_{x}(n)\subset\Lambda}}}{\mathcal{G}}_{s}^{\Lambda}(\Phi(x,n))\,$

using (2.31). Applying the conditional expectations and telescopic sum from
(3.8), we further write

(3.27) $D(\Lambda,s)=\sum_{x\in\Lambda}\sum_{m\geq
R}\Psi_{\Lambda}(x,m,s)\quad\mbox{where}\quad\Psi_{\Lambda}(x,m,s)=\sum_{\stackrel{{\scriptstyle
R\leq n\leq
m:}}{{b_{x}(n)\subset\Lambda}}}\Delta_{b_{x}(n)}^{m}({\mathcal{G}}_{s}^{\Lambda}(\Phi(x,n)))\,.$

Arguing as in [25, Appendix A], there is a decay function $G_{\Psi}$ and a
positive number $\|\Psi\|_{G_{\Psi}}$ such that for all
$\Lambda\in\mathcal{P}_{0}(\Gamma)$, $s\in[-1,1]$, $x\in\Lambda$ and $k\geq
R$,

(3.28) $\sum_{m\geq
k}\|\Psi_{\Lambda}(x,m,s)\|\leq\|\Psi\|_{G_{\Psi}}G_{\Psi}(k)\,.$

Again, as in [25, Corollary A.3], one can be explicit about estimates for
$G_{\Psi}$, but for our purposes here, let us only remark that is has finite
moments of all orders, as in (3.19). Given (3.27) and (3.28), well-known Lieb-
Robinson bounds (for time-dependent interaction) apply to the spectral flow.
As a result, there is a decay function $G_{\alpha}$ so that for any choice of
$X,Y\in\mathcal{P}_{0}(\Gamma)$,

(3.29) $\|[\alpha^{\Lambda}_{s}(A),B]\|\leq s\|A\|\|B\||X|G_{\alpha}(d(X,Y))$

for all $A\in\mathcal{A}_{X}$, $B\in\mathcal{A}_{Y}$, and $s\in\mathbb{R}$.
$G_{\alpha}$ is independent of $\Lambda\in\mathcal{P}_{0}(\Gamma)$ and has
finite moments of all orders.

###  3.2. In the GNS space

The proofs of our main results will be primarily carried out in the GNS
representation of the reference state $\omega_{0}$.

As discussed in Section 2, let $\eta$ be a finite-range, uniformly bounded,
frustration free interaction on $\Gamma$. Denote by $\delta_{0}$ the
corresponding closed derivation with dense domain ${\rm
dom}(\delta_{0})\subset{\mathcal{A}}$ having ${\mathcal{A}}^{\rm loc}$ as a
core, and let $\\{\tau_{t}^{(0)}|t\in\mathbb{R}\\}$, denote the strongly
continuous, system dynamics. For any infinite-volume, zero-energy ground state
$\omega_{0}$, let $({\mathcal{H}},\pi_{0},\Omega)$ denote the corresponding
GNS triple. The quasi-local maps discussed in the previous subsection can be
represented in the GNS representation and we now present the necessary
properties we will need in this setting.

The Unperturbed Dynamics: Since $\omega_{0}$ is a ground state of
$\delta_{0}$, there is a unique, non-negative, self-adjoint operator $H_{0}$
with dense domain ${\rm dom}(H_{0})\subset{\mathcal{H}}$ satisfying
$H_{0}\Omega=0$ and

(3.30)
$\pi_{0}(\tau_{t}^{(0)}(A))=e^{itH_{0}}\pi_{0}(A)e^{-itH_{0}}\quad\mbox{for
all }A\in{\mathcal{A}}\mbox{ and }t\in\mathbb{R}\,.$

Moreover, $\pi_{0}({\mathcal{A}}^{\rm loc})\Omega$ is a core for $H_{0}$.

The Perturbed Dynamics: Let $\Phi$ be an anchored interaction satisfying
Assumption 2.5. For any $\Lambda\in\mathcal{P}_{0}(\Gamma)$, take
$V_{\Lambda}$ as in (2.31) and recall that for any $s\in\mathbb{R}$,

(3.31) $H(\Lambda,s)=H_{0}+\pi_{0}(sV_{\Lambda})$

defines a self-adjoint operator on ${\mathcal{H}}$ with the same dense domain
as $H_{0}$. As before, let us also introduce the perturbed dynamics
$\tau_{t}^{(\Lambda,s)}$ on ${\mathcal{A}}$, see (3.11). We claim that for all
$\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in\mathbb{R}$, the representation
of this perturbed dynamics in the GNS space coincides with the Heisenberg
dynamics, generated by the Hamiltonian $H(\Lambda,s)$, on
$\pi_{0}({\mathcal{A}})$ in the sense that

(3.32)
$\pi_{0}(\tau_{t}^{(\Lambda,s)}(A))=e^{itH(\Lambda,s)}\pi_{0}(A)e^{-itH(\Lambda,s)}\quad\mbox{for
all }A\in{\mathcal{A}}\mbox{ and }t\in\mathbb{R}\,.$

In fact, using the interaction picture representation (3.12), as well as
(3.30), we find that in $\mathcal{B}(\mathcal{H})$,

(3.33)
$\pi_{0}(\tau_{t}^{(\Lambda,s)}(A))=(\pi_{0}(\Gamma_{t}^{(\Lambda,s)}))^{*}e^{itH_{0}}\pi_{0}(A)e^{-itH_{0}}\pi_{0}(\Gamma_{t}^{(\Lambda,s)})\,.$

Then, (3.32) follows by observing that

(3.34) $e^{-itH(\Lambda,s)}=e^{-itH_{0}}\pi_{0}(\Gamma_{t}^{(\Lambda,s)})\,,$

as by (3.13) one readily checks that
$\tilde{\Gamma}_{t}^{(\Lambda,s)}:=\pi_{0}(\Gamma_{t}^{(\Lambda,s)})$ is the
unique, unitary solution of

(3.35)
$\frac{d}{dt}\tilde{\Gamma}_{t}^{(\Lambda,s)}=-ie^{itH_{0}}\pi_{0}(sV_{\Lambda})e^{-itH_{0}}\tilde{\Gamma}_{t}^{(\Lambda,s)}\quad\mbox{with}\quad\tilde{\Gamma}_{0}^{(\Lambda,s)}={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}.$

Weighted Integral Operators: For any $\gamma>0$,
$\Lambda\in\mathcal{P}_{0}(\Gamma)$, and $s\in\mathbb{R}$ we can map the
weighted integral operators, see (3.16), to the GNS space by setting
$\tilde{{\mathcal{F}}}_{s}^{\Lambda}:{\mathcal{B}}({\mathcal{H}})\to{\mathcal{B}}({\mathcal{H}})$
and
$\tilde{{\mathcal{G}}}_{s}^{\Lambda}:{\mathcal{B}}({\mathcal{H}})\to{\mathcal{B}}({\mathcal{H}})$
as

(3.36)
$\tilde{{\mathcal{F}}}_{s}^{\Lambda}(A)=\int_{-\infty}^{\infty}e^{itH(\Lambda,s)}Ae^{-itH(\Lambda,s)}w_{\gamma}(t)\,dt\quad\mbox{and}\quad\tilde{{\mathcal{G}}}_{s}^{\Lambda}(A)=\int_{-\infty}^{\infty}e^{itH(\Lambda,s)}Ae^{-itH(\Lambda,s)}W_{\gamma}(t)\,dt$

for all $A\in{\mathcal{B}}({\mathcal{H}})$. Using (3.32), it is clear that

(3.37)
$\pi_{0}({\mathcal{F}}_{s}^{\Lambda}(A))=\tilde{{\mathcal{F}}}_{s}^{\Lambda}(\pi_{0}(A))\quad\mbox{and}\quad\pi_{0}({\mathcal{G}}_{s}^{\Lambda}(A))=\tilde{{\mathcal{G}}}_{s}^{\Lambda}(\pi_{0}(A))\quad\mbox{for
all }A\in{\mathcal{A}}\,.$

The Spectral Flow: As with the weighted integral operators discussed above, we
can also map the spectral flow, as defined in (3.25), to the GNS space. In
fact, for any $\gamma>0$, $\Lambda\in\mathcal{P}_{0}(\Gamma)$, and
$s\in\mathbb{R}$ take
$U(\Lambda,s)=\pi_{0}(u(\Lambda,s))\in{\mathcal{B}}({\mathcal{H}})$ where
$u(\Lambda,s)$ is the unique, unitary solution of (3.23) in ${\mathcal{A}}$.
Since $\pi_{0}$ is a representation, each $U(\Lambda,s)$ is unitary on
${\mathcal{H}}$ and hence, the automorphisms

(3.38) $\tilde{\alpha}_{s}^{\Lambda}(A)=U(\Lambda,s)^{*}AU(\Lambda,s)$

define the spectral flow for all $A\in{\mathcal{B}}({\mathcal{H}})$. In this
case, of course, we again have that

(3.39)
$\pi_{0}(\alpha_{s}^{\Lambda}(A))=\tilde{\alpha}_{s}^{\Lambda}(\pi_{0}(A))\quad\mbox{for
all }A\in{\mathcal{A}}\,.$

Let us now explain the rationale for calling these automorphisms the spectral
flow. For any $\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in\mathbb{R}$, we are
considering the perturbed Hamiltonian $H(\Lambda,s)$ as in (3.31). As in
(2.33), denote the ground state energy of $H(\Lambda,s)$ by

(3.40) $E(\Lambda,s)=\inf\mathop{\rm spec}(H(\Lambda,s))\,.$

Now regard $\Lambda\in\mathcal{P}_{0}(\Gamma)$ as fixed. It is clear that
$H(\Lambda,0)=H_{0}$ is the unperturbed Hamiltonian with $E(\Lambda,0)=0$.
Under our assumptions, Proposition 2.4 holds and so the kernel of $H_{0}$ is
one-dimensional. Moreover, by Assumption 2.1, the bulk gap associated to
$H_{0}$ is strictly positive. By standard results, see e.g [18], for $|s|$
sufficiently small the kernel of
$H(\Lambda,s)-E(\Lambda,s){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}$
is one-dimensional and the ground state gap does not immediately close. More
precisely, denote the ground state gap of $H(\Lambda,s)$ by

(3.41) ${\rm
gap}(H(\Lambda,s))=\sup\\{\delta>0:(E(\Lambda,s),E(\Lambda,s)+\delta)\cap{\rm
spec}(H(\Lambda,s))=\emptyset\\}.$

By assumption, the bulk gap $\gamma_{0}={\rm gap}(H(\Lambda,0))={\rm
gap}(H_{0})>0$. Fix $0<\gamma<\gamma_{0}$. In this case, for any
$\Lambda\in\mathcal{P}_{0}(\Gamma)$, there is $s_{0}^{\Lambda}(\gamma)>0$ such
that

(3.42) ${\rm gap}(H(\Lambda,s))\geq\gamma\quad\mbox{for all }|s|\leq
s_{0}^{\Lambda}(\gamma)\,.$

As a result, an application of [26, Theorem 6.3] shows that there is a family
of automorphisms of ${\mathcal{B}}({\mathcal{H}})$, which we denote by
$\hat{\alpha}_{s}^{\Lambda}$, that satisfy

(3.43)
$\hat{\alpha}_{s}^{\Lambda}(P^{\Lambda}(s))=P^{\Lambda}(0)\,\quad\mbox{whenever
}|s|\leq s_{0}^{\Lambda}(\gamma)\,,$

where by $P^{\Lambda}(s)$ we denote the orthogonal projection onto the ground
state space of $H(\Lambda,s)$. This is Hastings’ quasi-adiabatic evolution
[13, 15]. In fact, this family of automorphisms is implemented by the unique,
unitary ${\mathcal{B}}({\mathcal{H}})$-valued solution of

(3.44)
$\frac{d}{ds}\tilde{U}(\Lambda,s)=-i\tilde{D}(\Lambda,s)\tilde{U}(\Lambda,s)\quad\mbox{with}\quad\tilde{U}(\Lambda,0)={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}\,,$

with

(3.45)
$\tilde{D}(\Lambda,s)=\tilde{{\mathcal{G}}}_{s}^{\Lambda}(\pi_{0}(V_{\Lambda}))=\int_{-\infty}^{\infty}e^{itH(\Lambda,s)}\pi_{0}(V_{\Lambda})e^{-itH(\Lambda,s)}W_{\gamma}(t)\,dt\,.$

Comparing this unique solution with the representative of the unique solution
of (3.23), we conclude that
$\hat{\alpha}_{s}^{\Lambda}=\tilde{\alpha}_{s}^{\Lambda}$ as defined in
(3.38).

For any $\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in\mathbb{R}$, the state
$\omega_{s}^{\Lambda}:{\mathcal{A}}\to\mathbb{C}$ given by

(3.46)
$\omega_{s}^{\Lambda}(A)=\omega_{0}(\alpha_{s}^{\Lambda}(A))\quad\mbox{for all
}A\in{\mathcal{A}}\,$

is a vector state in the GNS space:

(3.47)
$\omega_{s}^{\Lambda}(A)=\langle\Omega(\Lambda,s),\pi_{0}(A)\Omega(\Lambda,s)\rangle\quad\mbox{for
all }A\in{\mathcal{A}}\,,$

where $\Omega(\Lambda,s)=U(\Lambda,s)\Omega\in{\mathcal{H}}$. By our
assumptions, the ground state of $H_{0}$ is unique, and hence,
$P^{\Lambda}(0)=|\Omega\rangle\langle\Omega|$. An application of (3.43) then
shows that

(3.48)
$P^{\Lambda}(s)=U(\Lambda,s)|\Omega\rangle\langle\Omega|U(\Lambda,s)^{*}=|\Omega(\Lambda,s)\rangle\langle\Omega(\Lambda,s)|\quad\mbox{for
all }|s|\leq s_{0}^{\Lambda}(\gamma),$

and thus $\Omega(\Lambda,s)$ is the ground state of $H(\Lambda,s)$; in
particular, $H(\Lambda,s)\Omega(\Lambda,s)=E(\Lambda,s)\Omega(\Lambda,s)$.
Given this, (3.43) can be written somewhat differently. Let
$0<\gamma<\gamma_{0}$, $\Lambda\in\mathcal{P}_{0}(\Gamma)$, and consider
$|s|\leq s_{0}^{\Lambda}(\gamma)$, then we have that

(3.49)
$\tilde{\alpha}^{\Lambda}_{s}(|\Omega(\Lambda,s)\rangle\langle\Omega(\Lambda,s)|)=|\Omega\rangle\langle\Omega|\,.$

Finally, recall that with the parameters $\gamma$ and $s$ as above that the
weighted integral operator $\tilde{{\mathcal{F}}}_{s}^{\Lambda}$ from (3.36)
satisfies the relation

(3.50)
$\left[\tilde{{\mathcal{F}}}^{\Lambda}_{s}(A),|\Omega(\Lambda,s)\rangle\langle\Omega(\Lambda,s)|\right]=0\quad\mbox{for
all }A\in{\mathcal{B}}({\mathcal{H}})\,.$

See, e.g. [26, Lemma 6.8], for a proof of this property.

###  3.3. On Domains

In this section, we describe the action of certain unbounded Hamiltonians on
specific dense subspaces of vectors in their domain.

We begin with a basic observation about general derivations. Let $(\Gamma,d)$
be a $\nu$-regular metric space, $g$ an $F$-function on $(\Gamma,d)$, and
$\Phi$ an interaction on $\Gamma$ with $\|\Phi\|_{g}<\infty$. As in Section 2,
corresponding to any such $\Phi$ there is well-defined, closed derivation
$\delta^{\Phi}$ with dense domain ${\rm
dom}(\delta^{\Phi})\subset{\mathcal{A}}$ which can be obtained as the closure
of the linear map $\delta^{\Phi}:{\mathcal{A}}^{\rm loc}\to{\mathcal{A}}$
defined by

(3.51) $\delta^{\Phi}(A)=\sum_{\stackrel{{\scriptstyle
Y\in\mathcal{P}_{0}(\Gamma):}}{{Y\cap
X\neq\emptyset}}}[\Phi(Y),A]\quad\mbox{for any }A\in{\mathcal{A}}_{X}\,.$

Although the sum on the right-hand-side above may be infinite, it is
absolutely convergent when $\Phi$ has a finite $F$-norm. In fact,
$\delta^{\Phi}$ is locally bounded:

(3.52) $\|\delta^{\Phi}(A)\|\leq D|X|\|A\|\quad\mbox{for all
}X\in{\mathcal{P}}_{0}(\Gamma),A\in{\mathcal{A}}_{X},$

with $D=2\|F\|\|\Phi\|_{F}$, see Example 4.7 of [26, Section IV.B.1]. We also
have the following lemma.

######  Lemma 3.1.

Let $(\Gamma,d)$ be $\nu$-regular and $g$ be an $F$-function on $(\Gamma,d)$
with a finite $\nu$-moment, i.e.,

(3.53) $\sum_{n=1}^{\infty}(n+1)^{\nu}g(n)<\infty\,.$

For any interaction $\Phi$ on $\Gamma$ with $\|\Phi\|_{g}<\infty$, we have
that ${\mathcal{A}}^{g}\subset\mathop{\rm dom}(\delta^{\Phi})$.

###### Proof.

For $n\geq 1$, and $A\in{\mathcal{A}}^{g}$, there is some $x\in\Gamma$, $C\geq
0$, and observables $A_{n}\in{\mathcal{A}}_{b_{x}(n)}$ satisfying
$\|A-A_{n}\|\leq Cg(n)$. In this case, the bound $\|A_{n+1}-A_{n}\|\leq
2Cg(n)$ is clear. Using (3.52) and $\nu$-regularity of $\Gamma$, we conclude

(3.54) $\|\delta^{\Phi}(A_{n+1})-\delta^{\Phi}(A_{n})\|\leq 2\kappa
CD(n+1)^{\nu}g(n).$

Thus, for all $m<n$,

(3.55) $\|\delta^{\Phi}(A_{n})-\delta^{\Phi}(A_{m})\|\leq 2\kappa
CD\sum_{k=m}^{n-1}(k+1)^{\nu}g(k).$

Since we assumed that $g$ has a finite $\nu$-moment, this implies that
$\delta^{\Phi}(A_{n})$ is a Cauchy sequence. Since $A_{n}\to A$ and
${\mathcal{A}}^{\rm loc}$ is a core for $\delta^{\Phi}$, it follows that
$A\in\mathop{\rm dom}(\delta^{\Phi})$. ∎

Given our assumptions, Lemma 3.1 clearly applies to the derivation
$\delta_{0}$ defined by the unperturbed interaction $\eta$. Using that
$H_{0}\Omega=0$, one readily checks the relation

(3.56) $e^{itH_{0}}\pi_{0}(A)\Omega=\pi_{0}(\tau_{t}^{(0)}(A))\Omega\,,$

from which the inclusion $\pi_{0}(\mathop{\rm
dom}(\delta_{0}))\Omega\subset\mathop{\rm dom}(H_{0})$ is clear. As a result,
if $g$ is an $F$-function with a finite $\nu$-moment, then for any
$A\in{\mathcal{A}}^{g}$, we have that $\pi_{0}(A)\Omega\in\mathop{\rm
dom}(H_{0})=\mathop{\rm dom}(H(\Lambda,s))$ for any
$\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in\mathbb{R}$.

Now, under Assumption 2.1, the bulk gap $\gamma_{0}>0$. For any
$0<\gamma<\gamma_{0}$ and $\Lambda\in\mathcal{P}_{0}(\Gamma)$, the quantity
$s_{0}^{\Lambda}(\gamma)$ is defined so that the gap constraint, see (3.42),
is satisfied. In this case, uniqueness of the corresponding ground state
implies that whenever $|s|\leq s_{0}^{\Lambda}(\gamma)$,

(3.57)
$\langle\psi,(H(\Lambda,s)-E(\Lambda,s){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}})\psi\rangle\geq\gamma\langle\psi,\psi\rangle$

for all $\psi\in{\rm dom}(H_{0})$ with
$\langle\Omega(\Lambda,s),\psi\rangle=0$. Recalling our notation for the
perturbed state $\omega_{s}^{\Lambda}$, see (3.46) and (3.47), we conclude
that for any $A\in{\mathcal{A}}^{\rm loc}$,

(3.58)
$0=\omega_{s}^{\Lambda}(A)=\langle\Omega(\Lambda,s),\psi\rangle\quad\mbox{where}\quad\psi=\pi_{0}(A)\Omega(\Lambda,s).$

Since $\Omega(\Lambda,s)=U(\Lambda,s)\Omega$,
$U(\Lambda,s)=\pi_{0}(u(\Lambda,s))$, and $u(\Lambda,s)$ satisfies the quasi-
locality estimate in (3.24), we conclude that each such
$\psi=\pi_{0}(Au(\Lambda,s))\Omega\in\mathop{\rm dom}(H(\Lambda,s))$ by Lemma
3.1. As a result, whenever $A\in{\mathcal{A}}^{\rm loc}$ and $|s|\leq
s_{0}^{\Lambda}(\gamma)$, if $\omega_{s}^{\Lambda}(A)=0$, then the estimate

(3.59)
$\omega_{s}^{\Lambda}(A^{*}\delta_{s}^{\Lambda}(A))\geq\gamma\omega_{s}^{\Lambda}(A^{*}A)$

follows from (3.57).

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 $(\Gamma,d)$ be $\nu$-regular, $g$ be an $F$-function on $(\Gamma,d)$ with
a finite $\nu$-moment, and $\omega_{0}$ be an infinite-volume, zero energy,
ground state of the frustration free model defined by $\delta_{0}$. Denote by
$({\mathcal{H}},\pi_{0},\Omega)$ the frustration free representation and by
$H_{0}$ the corresponding GNS Hamiltonian. Then, for any IAS $(\Lambda_{n})$,

(3.60) $\lim_{n\to\infty}\pi_{0}(H_{\Lambda_{n}})\psi=H_{0}\psi\quad\mbox{ for
all }\psi\in\pi_{0}({\mathcal{A}}^{g})\Omega\,,$

where $H_{\Lambda_{n}}\in{\mathcal{A}}_{\Lambda_{n}}$ is as in (2.14).

###### Proof.

Let $(\Lambda_{n})$ be an IAS and take
$\psi\in\pi_{0}({\mathcal{A}}^{g})\Omega$. Since $\psi=\pi_{0}(A)\Omega$ for
some $A\in A^{g}$, it is clear that $\psi\in\mathop{\rm dom}(H_{0})$ by Lemma
3.1 and the above discussion. Using that $A\in A^{g}$, there is some
$x\in\Gamma$ and observables $A_{m}\in{\mathcal{A}}_{b_{x}(m)}$ with
$\|A-A_{m}\|\leq Cg(m)$ for all $m\geq 1$. Considering the vectors
$\psi_{m}=\pi_{0}(A_{m})\Omega\in{\mathcal{H}}$, we have that

(3.61) $\|\psi-\psi_{m}\|\leq
Cg(m)\quad\mbox{and}\quad\|\psi_{m}-\psi_{k}\|\leq 2Cg(k)\quad\mbox{whenever
}1\leq k<m\,.$

Since the frustration-free property implies that

(3.62) $H_{0}\pi_{0}(B)\Omega=\pi_{0}(\delta_{0}(B))\Omega\quad\mbox{ for all
}B\in\mathop{\rm dom}\delta_{0}.$

Arguing as in (3.55) above, we conclude that, again for $k<m$,

(3.63)
$\|H_{0}\psi_{m}-H_{0}\psi_{k}\|\leq\|\delta_{0}(A_{m})-\delta_{0}(A_{k})\|\leq
2\kappa CD_{0}\sum_{\ell=k}^{m-1}(\ell+1)^{\nu}g(\ell)\,$

Note that here we may take $D_{0}=2\kappa R^{\nu}\|\eta\|_{\infty}$. The bound

(3.64) $\|H_{0}\psi-H_{0}\psi_{k}\|\leq 2\kappa
CD_{0}\sum_{\ell=k}^{\infty}(\ell+1)^{\nu}g(\ell)\,$

follows. By the zero-energy property of the ground state, we similarly see
that for any $n\geq 1$,

(3.65)
$\|\pi_{0}(H_{\Lambda_{n}})\psi_{m}-\pi_{0}(H_{\Lambda_{n}})\psi_{k}\|\leq\|[H_{\Lambda_{n}},A_{m}-A_{k}]\|\leq
2\kappa CD_{0}\sum_{\ell=k}^{m-1}(\ell+1)^{\nu}g(\ell)$

for all $1\leq k<m$ and therefore, we obtain

(3.66) $\|\pi_{0}(H_{\Lambda_{n}})\psi-\pi_{0}(H_{\Lambda_{n}})\psi_{k}\|\leq
2\kappa CD_{0}\sum_{\ell=k}^{\infty}(\ell+1)^{\nu}g(\ell)\,$

Now for any $n\geq 1$ and each $k\geq 1$, we find that

(3.67) $\displaystyle\|\pi_{0}(H_{\Lambda_{n}})\psi-H_{0}\psi\|$
$\displaystyle\leq$
$\displaystyle\|\pi_{0}(H_{\Lambda_{n}})\psi-\pi_{0}(H_{\Lambda_{n}})\psi_{k}\|+\|\pi_{0}(H_{\Lambda_{n}})\psi_{k}-H_{0}\psi_{k}\|$
$\displaystyle\quad+\|H_{0}\psi_{k}-H_{0}\psi\|$

For $k\geq 1$ sufficiently large, (3.64) and (3.66) 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; here we used the zero energy
property of the ground state. This completes the proof. ∎

Lemma 3.2 also trivially applies to the perturbed system in the GNS space. In
fact, for $\Lambda\in\mathcal{P}_{0}(\Gamma)$ and $s\in\mathbb{R}$, under
assumptions as above, a direct application of Lemma 3.2 shows that we also
have

(3.68)
$\lim_{n}\pi_{0}(H_{\Lambda_{n}}+sV_{\Lambda})\psi=H_{0}\psi+s\pi_{0}(V_{\Lambda})\psi=H(\Lambda,s)\psi\quad\mbox{for
all }\psi\in\pi_{0}({\mathcal{A}}^{g})\Omega.$

######  Remark 3.3.

An analogue of Lemma 3.2 holds more generally. In fact, if $(\Gamma,d)$ is
$\nu$-regular and $g$ is an $F$-function on $(\Gamma,d)$ with a finite
$\nu$-moment, then for any frustration free interaction $\Phi$ with
$\|\Phi\|_{g}<\infty$, the GNS Hamiltonian in any frustration free
representation satisfies (3.60). 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 (3.67) by

(3.69) $\sum_{\stackrel{{\scriptstyle X\in\mathcal{P}_{0}(\Gamma):}}{{X\cap
b_{x}(k)\neq\emptyset,X\cap\Lambda_{n}^{c}\neq\emptyset}}}\|\pi_{0}(\Phi(X))\psi_{k}\|\leq\|\Phi\|_{g}\sup_{m\geq
1}\|A_{m}\|\sum_{y\in b_{x}(k),z\in\Lambda_{n}^{c}}g(d(y,z))$

For fixed $k$, the above is the sum of finitely many ‘tails’ of the uniformly
summable function $g$.

We now investigate how the weighted integral operator
$\tilde{{\mathcal{F}}}_{s}^{\Lambda}$, see (3.36), can be applied to the
unbounded Hamiltonian $H(\Lambda,s)$. Recall that for any $\gamma>0$,
$s\in\mathbb{R}$, and $\Lambda\in\mathcal{P}_{0}(\Gamma)$, we have defined a
weighted integral operator
$\tilde{{\mathcal{F}}}_{s}^{\Lambda}:{\mathcal{B}}({\mathcal{H}})\to{\mathcal{B}}({\mathcal{H}})$
by setting

(3.70)
$\tilde{{\mathcal{F}}}_{s}^{\Lambda}(A)=\int_{-\infty}^{\infty}e^{itH(\Lambda,s)}Ae^{-itH(\Lambda,s)}w_{\gamma}(t)\,dt\quad\mbox{for
all }A\in{\mathcal{B}}({\mathcal{H}})\,.$

It is clear that

(3.71)
$\tilde{{\mathcal{F}}}^{\Lambda}_{s}(e^{iuH(\Lambda,s)})=e^{iuH(\Lambda,s)}\quad\mbox{for
all }\;u,s\in{\mathbb{R}}\,,$

since the dynamics leaves this bounded operator invariant and $w_{\gamma}$
integrates to 1. We will need a differential version of this fact, which we
turn to next.

First, we prove an analogue of the desired statement for the unperturbed
dynamics; this is Lemma 3.4 below. To this end, assume
$w:\mathbb{R}\to\mathbb{R}$ satisfies

(3.72)
$\int_{\mathbb{R}}w(u)\,du=1\quad\mbox{and}\quad\int_{\mathbb{R}}|u|^{\nu}|w(u)|\,du<\infty\,,$

and define a weighted integral operator
$\tilde{{\mathcal{F}}}:\mathcal{B}({\mathcal{H}})\to\mathcal{B}({\mathcal{H}})$
by setting

(3.73)
$\tilde{{\mathcal{F}}}(A)=\int_{-\infty}^{\infty}e^{iuH_{0}}Ae^{-iuH_{0}}w(u)\,du\quad\mbox{for
all }A\in{\mathcal{B}}({\mathcal{H}})\,.$

To simplify notation, let us also write

(3.74) $\tilde{\tau}^{(0)}_{u}(A)=e^{iuH_{0}}Ae^{-iuH_{0}}.$

Our first result is as follows.

######  Lemma 3.4.

Let $(\Gamma,d)$ be $\nu$-regular, $g$ be an $F$-function on $(\Gamma,d)$ with
a finite $2\nu$-moment, and $w:\mathbb{R}\to\mathbb{R}$ satisfy (3.72). For
each choice of IAS $(\Lambda_{n})$, the weighted integral operator
$\tilde{{\mathcal{F}}}$, as in (3.73), satisfies

(3.75)
$\lim_{n\to\infty}\tilde{{\mathcal{F}}}(\pi_{0}(H_{\Lambda_{n}}))\psi=H_{0}\psi\quad\mbox{
for all }\psi\in\pi_{0}({\mathcal{A}}^{g})\Omega\,,$

where $H_{\Lambda_{n}}\in{\mathcal{A}}_{\Lambda_{n}}$ is as in (2.14).

###### Proof.

Fix an IAS $(\Lambda_{n})$ and take $\psi=\pi_{0}(A)\Omega$ for some
$A\in{\mathcal{A}}^{g}$. We can rewrite the convergence claimed in (3.75) as
the convergence of integrals of a sequence of functions
$f_{n}:\mathbb{R}\to{\mathcal{H}}$ given by

(3.76)
$f_{n}(u)=w(u)\tilde{\tau}_{u}^{(0)}(\pi_{0}(H_{\Lambda_{n}}))\psi\quad\mbox{for
all }u\in\mathbb{R}\,.$

Since $H_{0}\Omega=0$, the above can be re-written as

(3.77)
$f_{n}(u)=w(u)e^{iuH_{0}}\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\tau_{-u}^{(0)}(A))\Omega$

using (3.30). We claim that there is an $F$-function $g_{\tau}$ with a finite
$\nu$-moment such that $\tau_{-u}^{(0)}(A)\in{\mathcal{A}}^{g_{\tau}}$ for all
$u\in\mathbb{R}$. Given this, Lemma 3.2 applies and we find that

(3.78)
$\lim_{n\to\infty}f_{n}(u)=w(u)e^{iuH_{0}}H_{0}\pi_{0}(\tau_{-u}^{(0)}(A))\Omega=w(u)H_{0}\psi\quad\mbox{for
all }u\in\mathbb{R}\,.$

By the normalization of $w$, see (3.72), the integral of this limit coincides
with the right-hand-side of (3.75). Therefore, to complete the proof we only
need to justify an application of dominated convergence.

Let us first prove the existence of an $F$-function $g_{\tau}$ as claimed. Fix
$A\in{\mathcal{A}}^{g}$. In this case, there is $x\in\Gamma$, $C\geq 0$, and
observables $A_{m}\in{\mathcal{A}}_{b_{x}(m)}$ for which $\|A-A_{m}\|\leq
Cg(m)$ for all $m\in\mathbb{N}$. Let $u\in\mathbb{R}$ and for any
$n\in\mathbb{N}$, set

(3.79) $A_{n}(u)=\Pi_{n}(\tau_{u}^{(0)}(A_{\lceil
n/2\rceil}))\in{\mathcal{A}}_{b_{x}(n)}$

where, to ease notation, we have written $\Pi_{n}=\Pi_{b_{x}(n)}$, for the
conditional expectation from Section 3.1.1. A straightforward estimate shows
that for any $\mu>0$,

(3.80) $\displaystyle\|\tau_{u}^{(0)}(A)-A_{n}(u)\|$ $\displaystyle\leq$
$\displaystyle\|\tau_{u}^{(0)}(A)-\tau_{u}^{(0)}(A_{\lceil
n/2\rceil})\|+\|\tau_{u}^{(0)}(A_{\lceil n/2\rceil})-A_{n}(u))\|$
$\displaystyle\leq$ $\displaystyle Cg(n/2)+\kappa
C_{\mu}e^{\mu(v_{\mu}|u|+1)}(\sup_{m\geq 1}\|A_{m}\|)(n/2+1)^{\nu}e^{-\mu
n/2}$

where, for the final bound above, we used (3.9) and (3.5). The claimed
$F$-function $g_{\tau}$ can now be obtained by estimating with the minimum of
$g(n/2)$ and $(n/2+1)^{\nu}e^{-\mu n/2}$.

We now turn to finding a dominating function for $f_{n}$. Recall that each
$A\in{\mathcal{A}}^{g}$ corresponds to a sequence of observables $A_{m}$ as
described above. In this case, for any $m_{0}\in\mathbb{N}$, $A$ can be
written as an absolutely convergent, telescopic sum:

(3.81) $A=A_{m_{0}}+\sum_{k=m_{0}+1}^{\infty}B_{k}\quad\mbox{where}\quad
B_{k}=A_{k}-A_{k-1}\quad\mbox{ and }\quad\|B_{k}\|\leq 2Cg(k-1).$

Inserting this decomposition of $A$ into (3.77), we find that for any
$n\in\mathbb{N}$ and each $u\in\mathbb{R}$:

(3.82)
$\|f_{n}(u)\|\leq|w(u)|\left(\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\tau_{-u}^{(0)}(A_{m_{0}}))\Omega\|+\sum_{k=m_{0}+1}^{\infty}\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\tau_{-u}^{(0)}(B_{k}))\Omega\|\right)\,.$

Now, by the zero-energy property of the ground state, it is clear that

(3.83)
$\pi_{0}(H_{\Lambda_{n}})\pi_{0}(A)\Omega=\pi_{0}([H_{\Lambda_{n}},A])\Omega\quad\mbox{for
all }A\in{\mathcal{A}}^{\rm loc},$

and thus, in particular, we have the bound

(3.84) $\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(A)\Omega\|\leq 2\kappa
D_{0}(k+R)^{\nu}\|A\|\quad\mbox{for all }A\in{\mathcal{A}}_{b_{x}(k)}\,,$

which we stress is uniform in $n$. This suggests a mechanism for bounding the
first term in (3.82) above. In fact, let $\ell_{0}\geq m_{0}$ and use
conditional expectations, as in (3.8), to write

(3.85)
$\tau_{-u}^{(0)}(A_{m_{0}})=\sum_{\ell\geq\ell_{0}}\Delta_{\ell_{0}}^{\ell}(\tau_{-u}^{(0)}(A_{m_{0}}))\,,$

where again we have used a short-hand $\Delta_{m}^{\ell}$ for
$\Delta_{b_{x}(m)}^{\ell}$ as in (3.8). For $\ell=\ell_{0}$, the bound

(3.86)
$\displaystyle\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\Delta_{\ell_{0}}^{\ell_{0}}(\tau_{-u}^{(0)}(A_{m_{0}})))\Omega\|$
$\displaystyle=$
$\displaystyle\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\Pi_{\ell_{0}}(\tau_{-u}^{(0)}(A_{m_{0}})))\Omega\|$
$\displaystyle\leq$ $\displaystyle 2\kappa
D_{0}(\ell_{0}+R)^{\nu}\|A_{m_{0}}\|$

follows from (3.84). For $\ell\geq\ell_{0}$, the estimate

(3.87)
$\displaystyle\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\Delta_{\ell_{0}}^{\ell}(\tau_{-u}^{(0)}(A_{m_{0}})))\Omega\|$
$\displaystyle\leq$ $\displaystyle 2\kappa
D_{0}(\ell+R)^{\nu}\|\Delta_{\ell_{0}}^{\ell}(\tau_{-u}^{(0)}(A_{m_{0}}))\|$
$\displaystyle\leq$ $\displaystyle
4\kappa^{2}D_{0}C_{\mu}m_{0}^{\nu}(\ell+R)^{\nu}\|A_{m_{0}}\|e^{-\mu(\ell-
m_{0}-1-v_{\mu}|u|)}$

follows from another application of (3.84), the general estimate (3.7) for
these differences, and the quasi-locality estimate for the unperturbed
dynamics. We conclude that

$\displaystyle\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\tau_{-u}^{(0)}(A_{m_{0}}))\Omega\|$
$\displaystyle\leq\sum_{\ell\geq\ell_{0}}\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\Delta_{\ell_{0}}^{\ell}(\tau_{-u}^{(0)}(A_{m_{0}})))\Omega\|$
(3.88) $\displaystyle\leq 2\kappa
D_{0}\|A_{m_{0}}\|\left((\ell_{0}+R)^{\nu}+2\kappa
C_{\mu}m_{0}^{\nu}\sum_{\ell\geq\ell_{0}+1}(\ell+R)^{\nu}e^{-\mu(\ell-
m_{0}-1-v_{\mu}|u|)}\right).$

If we now take $\ell_{0}=\lceil v_{\mu}|u|+m_{0}\rceil$, then we have found
that there is $K\geq 0$ for which

(3.89)
$\|\pi_{0}(H_{\Lambda_{n}})\pi_{0}(\tau_{-u}^{(0)}(A_{m_{0}}))\Omega\|\leq
K\|A_{m_{0}}\|\left(m_{0}^{2\nu}+|u|^{\nu}+(m_{0}|u|)^{\nu}+1\right)$

and here $K=K(\kappa,\mu,\nu,R)$.

Now, we claim that the terms $B_{k}$ in (3.82) can be estimated similarly. In
fact, regarding $k$ as $m_{0}$, argue as in (3.85) - (3.3) with some
$\ell_{0}\geq k$. A bound analogous to (3.89) can be found; of course, here
one replaces $\|A_{m_{0}}\|$ with $\|B_{k}\|$. Since $\|B_{k}\|\leq 2Cg(k-1)$
and $g$ has a finite $2\nu$-moment, we have obtained a bound on the right-
hand-side of (3.82) with the form:

(3.90) $\|f_{n}(u)\|\leq\tilde{K}(1+|u|^{\nu})|w(u)|\quad\mbox{for all
}u\in\mathbb{R}\,.$

By the assumption on $w$, i.e. (3.72), 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.

######  Lemma 3.5.

Let $(\Gamma,d)$ be $\nu$-regular, $g$ be an $F$-function on $(\Gamma,d)$ with
a finite $2\nu$-moment, and $w:\mathbb{R}\to\mathbb{R}$ satisfy (3.72). Let
$\Lambda\in\mathcal{P}_{0}(\Gamma)$ and take $s\in\mathbb{R}$. For each choice
of IAS $(\Lambda_{n})$, the weighted integral operator
$\tilde{{\mathcal{F}}}_{s}^{\Lambda}$, as in (3.70), satisfies

(3.91)
$\lim_{n\to\infty}\tilde{{\mathcal{F}}}_{s}^{\Lambda}(\pi_{0}(H_{\Lambda_{n}}+sV_{\Lambda}))\psi=H(\Lambda,s)\psi\quad\mbox{
for all }\psi\in\pi_{0}({\mathcal{A}}^{g})\Omega\,,$

with $H_{\Lambda_{n}}\in{\mathcal{A}}_{\Lambda_{n}}$ as in (2.14) and
$V_{\Lambda}$ as in (2.31).

###### Proof.

Fix an IAS $(\Lambda_{n})$ where we assume for convenience that
$\Lambda\subset\Lambda_{1}$. As in the proof of Lemma 3.4, take
$\psi=\pi_{0}(A)\Omega$ with $A\in{\mathcal{A}}^{g}$, and for each
$n\in\mathbb{N}$, consider $f_{n}:\mathbb{R}\to{\mathcal{H}}$ given by

(3.92)
$f_{n}(u)=w(u)\tilde{\tau}_{u}^{(\Lambda,s)}(\pi_{0}(H_{\Lambda_{n}}+sV_{\Lambda}))\pi_{0}(A)\Omega\quad\mbox{for
all }u\in\mathbb{R}\,,$

where, in analogy to (3.74), we have set

(3.93)
$\tilde{\tau}_{t}^{(\Lambda,s)}(A)=e^{itH(\Lambda,s)}Ae^{-itH(\Lambda,s)}\quad\mbox{for
all }A\in{\mathcal{B}}({\mathcal{H}})\mbox{ and }t\in\mathbb{R}\,.$

Using (3.32), (3.12), and (3.30), we may write

(3.94)
$\tilde{\tau}_{u}^{(\Lambda,s)}(\pi_{0}(H_{\Lambda_{n}}+sV_{\Lambda}))=\pi_{0}(\Gamma_{u}^{(\Lambda,s)})^{*}e^{iuH_{0}}\pi_{0}(H_{\Lambda_{n}}+sV_{\Lambda})e^{-iuH_{0}}\pi_{0}(\Gamma_{u}^{(\Lambda,s)})$

for all $u\in\mathbb{R}$. In this case, we find that

(3.95)
$f_{n}(u)=w(u)\pi_{0}(\Gamma_{u}^{(\Lambda,s)})^{*}e^{iuH_{0}}\pi_{0}(H_{\Lambda_{n}}+sV_{\Lambda})\pi_{0}(\tau^{(0)}_{-u}(\Gamma_{u}^{(\Lambda,s)}A))\Omega.$

Arguing as before, here we will also need (3.15), one shows that there is an
$F$-function $g^{\prime}$ with a finite $\nu$-moment such that
$\tau^{(0)}_{-u}(\Gamma_{u}^{(\Lambda,s)}A)\in{\mathcal{A}}^{g^{\prime}}$. As
a result, the point-wise limit

(3.96)
$\lim_{n}f_{n}(u)=w(u)\pi_{0}(\Gamma_{u}^{(\Lambda,s)})^{*}e^{iuH_{0}}(H_{0}+s\pi_{0}(V_{\Lambda}))\pi_{0}(\tau^{(0)}_{-u}(\Gamma_{u}^{(\Lambda,s)}A))\Omega=w(u)H(\Lambda,s)\psi$

is clear using properties of the interaction picture dynamics, see discussion
following (3.12).

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. ∎

##  4\. Construction of a unitarily equivalent perturbed system

###  4.1. Construction of a relatively bounded perturbation

The crux of the proof of stability introduced in [7], 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. In the
infinite-system setting, this means transforming the unbounded Hamiltonian
$H(\Lambda,s)$, see (2.30), by the spectral flow to achieve an equivalent
Hamiltonian of the form

(4.1)
$U(\Lambda,s)^{*}H(\Lambda,s)U(\Lambda,s)\psi=H_{0}\psi+W(\Lambda,s)\psi+E(\Lambda,s)\psi\quad\mbox{for
all }\psi\in\pi_{0}({\mathcal{A}}_{\rm loc})\Omega\,,$

with $W(\Lambda,s)$ a bounded operator with an explicit, $\Lambda$-independent
form-bound with respect to $H_{0}$ and $E(\Lambda,s)$ the ground state energy
of $H(\Lambda,s)$ as in (2.33). In previous work, we established quasi-
locality estimates for this spectral flow when the perturbation is of
sufficiently rapid decay. Given Assumption 2.5, the resulting quasi-locality
can be used to establish that each $\psi\in\pi_{0}({\mathcal{A}}_{\rm
loc})\Omega$ is in the domain of the unitarily evolved Hamiltonian in (4.1)
whenever $\Lambda\in\mathcal{P}_{0}(\Gamma)$ as
$U(\Lambda,s)\psi\in{\mathcal{A}}^{g}$ for an appropriate choice of $g$.

The proof of Theorem 2.7 will be obtained as a direct consequence of two
results. The first result establishes that $W(\Lambda,s)$ can be expressed in
a suitable form for deriving a relative form bound (Theorem 4.1). The second
result is the relative form bound itself (Theorem 5.1 in Section 5).

######  Theorem 4.1.

If Assumptions 2.1-2.5 are satisfied, for any $\gamma\in(0,\gamma_{0})$ and
$|s|\leq s_{0}^{\Lambda}(\gamma)$, there is a familiy of self-adjoint
observables $\Phi^{(2)}(x,m,s)\in{\mathcal{A}}_{b_{x}(m)}$, for each
$x\in\Gamma$ and $m\geq R$, with the following properties:

  1. (i)

$\Phi^{(2)}(x,m,s)P_{b_{x}(m)}=P_{b_{x}(m)}\Phi^{(2)}(x,m,s)=0$;

  2. (ii)

$\|\Phi^{(2)}(x,m,s)\|\leq 2sG^{(2)}_{\Lambda}(x,m)$ with

(4.2)
$G^{(2)}_{\Lambda}(x,m)=G_{\Lambda}(x,m/2)+2G_{\Lambda}^{(1)}(x,m+1)+2G_{\Lambda}^{(1)}(x,R)\sqrt{(1+m)^{\nu}G_{0}(m/2)}$

where $G_{\Lambda}(x,m)$ is as in Theorem 4.2 below and
$G_{\Lambda}^{(1)}(x,m)=\sum_{n\geq m}G_{\Lambda}(x,n)$. Furthermore,
$W(\Lambda,s)$ is given by the absolutely convergent sum

(4.3) $W(\Lambda,s)=\sum_{x\in\Gamma}\sum_{m\geq
R}\pi_{0}(\Phi^{(2)}(x,m,s)).$

The operator $W(\Lambda,s)$ is a priori defined in the GNS representation and
the identity (4.1) only makes sense in that representation. A posteriori, we
find via (4.3) that $W(\Lambda,s)$, in fact, corresponds to a quasi-local
observable in ${\mathcal{A}}$.

We prove the decomposition from Theorem 4.1 in two steps. First, we use quasi-
locality and conditional expectations to prove that for all $|s|\leq
s_{0}^{\Lambda}(\gamma)$, the action of the spectral flow on the GNS
Hamiltonian $H(\Lambda,s)$ can be again realized as a perturbation of $H_{0}$.
Namely, we show that for all $\psi\in\pi_{0}({\mathcal{A}}_{\rm loc})\Omega$

(4.4)
$U(\Lambda,s)^{*}H(\Lambda,s)U(\Lambda,s)\psi=H_{0}\psi+\sum_{x\in\Gamma}\sum_{m\geq
R}\tilde{\Phi}^{(1)}(x,m,s)\psi$

where the perturbation terms
$\tilde{\Phi}^{(1)}(x,m,s)\in\pi_{0}({\mathcal{A}}_{b_{x}(m)})$ are self-
adjoint, absolutely summable, and satisfy a norm bound that is linear in $s$,
see Theorem 4.2 below. In the second step, the final form of (4.3) from
Theorem 4.1 follows from using the frustration-free and LTQO ground state
properties to produce a refined decomposition of the perturbation terms from
(4.4). We now turn to establishing the first step in this decomposition, and
prove Theorem 4.1 in Section 4.2.

######  Theorem 4.2.

Fix any finite volume $\Lambda\subset\Gamma$. Under the conditions of Theorem
4.1, there exists a function $G_{\Lambda}:\Gamma\times[0,\infty)\to[0,\infty)$
for which

(4.5) $\sum_{x\in\Gamma}\sum_{m\geq R}G_{\Lambda}\left(x,m\right)<\infty$

and a self-adjoint operator
$\tilde{\Phi}^{(1)}(x,m,s)^{*}=\tilde{\Phi}^{(1)}(x,m,s)\in\pi_{0}({\mathcal{A}}_{b_{x}(m)})$
for each $x\in\Gamma$ and $m\geq R$, such that
$\|\tilde{\Phi}^{(1)}(x,m,s)\|\leq sG_{\Lambda}(x,m)$ and

(4.6)
$W(\Lambda,s)+E(\Lambda,s){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}=\sum_{x\in\Gamma}\sum_{m\geq
R}\tilde{\Phi}^{(1)}(x,m,s).$

Moreover, for each $x\in\Gamma$, the global operator
$\tilde{\Phi}^{(1)}(x,s)=\sum_{m\geq R}\tilde{\Phi}^{(1)}(x,m,s)$ is quasi-
local, i.e. belongs to $\pi_{0}({\mathcal{A}})$, and commutes with the ground
state projection $|\Omega\rangle\langle\Omega|$.

To prove this result, we will use two quasi-local maps
${\mathcal{K}}_{s}^{i,\Lambda}$, $i=1,2$, which are defined in terms of the
examples introduced in Section 3.1. To see how these maps emerge, we show how
to relate (4.1) to a sequence of finite-volume Hamiltonians
$H_{\Lambda_{n}}+sV_{\Lambda}$. This will also be the initial step for
defining the global operators $\tilde{\Phi}^{(1)}(x,s)$ from Theorem 4.2.

By Lemma 3.2, it is clear that

(4.7)
$(W(\Lambda,s)+E(\Lambda,s){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}})\psi:=U(\Lambda,s)^{*}H(\Lambda,s)U(\Lambda,s)\psi-
H_{0}\psi$

is well-defined for all $\psi\in\pi_{0}({\mathcal{A}}^{\rm loc})\Omega$. Here,
we use that $U(\Lambda,s)\psi\in\pi_{0}({\mathcal{A}}^{g})\Omega$ for a
function $g$ of the form (2.5) with $\zeta>\nu+1$. Fixing any IAS
$(\Lambda_{n})$, Lemmas 3.4 and 3.5 then imply

(4.8)
$(W(\Lambda,s)+E(\Lambda,s){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}})\psi=\lim_{n\to\infty}\left(U(\Lambda,s)^{*}\tilde{{\mathcal{F}}}_{s}^{\Lambda}(\pi_{0}(H_{\Lambda_{n}}+sV_{\Lambda}))U(\Lambda,s)\psi-\tilde{{\mathcal{F}}}(\pi_{0}(H_{\Lambda_{n}}))\psi\right)$

where we choose $\tilde{{\mathcal{F}}}=\tilde{{\mathcal{F}}}_{0}^{\Lambda}$.

Each term in the limit above is the sum of bounded operators. Using the
relationship between the quasi-local maps in the GNS representation to those
on the $C^{*}$-algebra, i.e. (3.37)-(3.39), we can rewrite each of these terms
as follows:

(4.9)
$\displaystyle\tilde{\alpha}_{s}^{\Lambda}\circ\tilde{{\mathcal{F}}}_{s}^{\Lambda}(\tilde{H}_{\Lambda_{n}}+s\tilde{V}_{\Lambda})-\tilde{{\mathcal{F}}}(\tilde{H}_{\Lambda_{n}})$
$\displaystyle=$
$\displaystyle\sum_{x\in\Lambda_{n}}\pi_{0}\left(\alpha_{s}^{\Lambda}\circ{\mathcal{F}}_{s}^{\Lambda}(h_{x})-{\mathcal{F}}(h_{x})\right)$
$\displaystyle\quad+\sum_{x\in\Lambda}\sum_{\begin{subarray}{c}k\geq R:\\\
b_{x}(k)\subseteq\Lambda\end{subarray}}\pi_{0}\left(s\alpha_{s}^{\Lambda}\circ{\mathcal{F}}_{s}^{\Lambda}(\Phi(x,k))\right).$

Given (4.9), for $i=1,2$ we define the map
${\mathcal{K}}^{i,\Lambda}_{s}:{\mathcal{A}}\to{\mathcal{A}}$ by

(4.10)
${\mathcal{K}}_{s}^{1,\Lambda}(A)=\alpha_{s}^{\Lambda}({\mathcal{F}}_{s}^{\Lambda}(A))-{\mathcal{F}}(A)\quad\mbox{and}\quad{\mathcal{K}}^{2,\Lambda}_{s}(A)=s\alpha_{s}^{\Lambda}({\mathcal{F}}_{s}^{\Lambda}(A)).$

It was proved, e.g. in [25, Lemma 4.4], that both of these maps satisfy a
local bound and quasi-local estimate that is independent of the finite volume
$\Lambda$. Specifically, for each $i=1,2$ there are numbers $p_{i}$, $q_{i}$
and $C_{i}$, and a decay function $G_{{\mathcal{K}}^{i}}$ (all independent of
$\Lambda$) for which

(4.11) $\displaystyle\|{\mathcal{K}}_{s}^{i,\Lambda}(A)\|$ $\displaystyle\leq$
$\displaystyle sC_{i}|X|^{p_{i}}\|A\|$ (4.12)
$\displaystyle\|[{\mathcal{K}}_{s}^{i,\Lambda}(A),B]\|$ $\displaystyle\leq$
$\displaystyle s|X|^{q_{i}}\|A\|\|B\|G_{{\mathcal{K}}^{i}}(d(X,Y))$

hold for any $X,Y\in\mathcal{P}_{0}(\Gamma)$, $A\in{\mathcal{A}}_{X}$,
$B\in{\mathcal{A}}_{Y}$, and $s\in\mathbb{R}$. In fact, $p_{1}=q_{1}=2$,
$p_{2}=0$ and $q_{2}=1$. One can make explicit estimates on the decay of
$G_{{\mathcal{K}}^{i}}$, see e.g. [25, Remark 4.7]. However, it suffices to
note that they each have finite moments of all orders in the sense of (3.19).

Given the limit $n\to\infty$ in (4.8), to prove that the decomposition in
(4.6) is absolutely summable, we will need local bound and quasi-local
estimates for ${\mathcal{K}}_{s}^{i,\Lambda}$ as in (4.11)-(4.12) that also
decay in the distance $d(X,\Lambda)$. This is the content of Lemmas 4.3 and
4.4 below. We proceed with the proof of Theorem 4.2 with the knowledge that
such bounds do hold.

###### Proof of Theorem 4.2.

Fix $\gamma\in(0,\gamma_{0})$, $\Lambda\in\mathcal{P}_{0}(\Gamma)$, and take
any IAS $(\Lambda_{n})$ for which $\Lambda\subseteq\Lambda_{n}$ for all $n$.
Define the spectral flow $\alpha_{s}^{\Lambda}$ and the weighted integral
operators ${\mathcal{F}}_{s}^{\Lambda}$,
${\mathcal{F}}={\mathcal{F}}_{0}^{\Lambda}$ with respect to the specific
choices of $\gamma$ and $\Lambda$, see (3.25) and (3.16).

Given ${\mathcal{K}}_{s}^{i,\Lambda}$, $i=1,2$, as in (4.10), for each
$x\in\Gamma$ and $|s|\leq s_{0}^{\Lambda}(\gamma)$ we then define the self-
adjoint operator
$\tilde{\Phi}^{(1)}(x,s)=\pi_{0}(\Phi^{(1)}(x,s))\in{\mathcal{B}}({\mathcal{H}})$
by

(4.13)
$\Phi^{(1)}(x,s)={\mathcal{K}}_{s}^{1,\Lambda}(h_{x})+\chi_{\Lambda}(x)\sum_{\stackrel{{\scriptstyle
k\geq
R:}}{{b_{x}(k)\subset\Lambda}}}{\mathcal{K}}_{s}^{2,\Lambda}(\Phi(x,k))\in{\mathcal{A}}$

where $\chi_{\Lambda}$ is the characteristic function of
$\Lambda\subset\Gamma$. We first show that each term $\tilde{\Phi}^{(1)}(x,s)$
commutes with the ground state projector $|\Omega\rangle\langle\Omega|$, and
then turn to the decomposition from (4.6).

Fix $|s|\leq s_{0}^{\Lambda}(\gamma)$ and recall that
$\Omega(\Lambda,s)=U(\Lambda,s)\Omega$. Combining this with the relations
(3.37)-(3.39) shows that for all $A\in{\mathcal{A}}$

(4.14)
$\displaystyle\left[\pi_{0}(\alpha_{s}^{\Lambda}({\mathcal{F}}_{s}^{\Lambda}(A))),|\Omega\rangle\langle\Omega|\right]$
$\displaystyle=$
$\displaystyle\left[U(\Lambda,s)^{*}\tilde{{\mathcal{F}}}_{s}^{\Lambda}(\pi_{0}(A))U(\Lambda,s),|\Omega\rangle\langle\Omega|\right]$
$\displaystyle=$ $\displaystyle
U(\Lambda,s)^{*}\left[\tilde{{\mathcal{F}}}_{s}^{\Lambda}(\pi_{0}(A)),|\Omega(\Lambda,s)\rangle\langle\Omega(\Lambda,s)|\right]U(\Lambda,s)=0,$

where for the final equality, we used (3.50). Considering (4.10), this
relation implies that

(4.15)
$\left[\pi_{0}({\mathcal{K}}_{s}^{1,\Lambda}(A)),|\Omega\rangle\langle\Omega|\right]=\left[\pi_{0}({\mathcal{K}}_{s}^{2,\Lambda}(A)),|\Omega\rangle\langle\Omega|\right]=0\,.$

from which it is immediate that
$[\tilde{\Phi}^{(1)}(x,s),|\Omega\rangle\langle\Omega|]=0$ for all
$x\in\Gamma$ and $|s|\leq s_{0}^{\Lambda}(\gamma)$ as claimed.

We now establish (4.6). Using conditional expectations, see Section 3.1, each
term $\Phi^{(1)}(x,s)$ may be further decomposed as

(4.16) $\Phi^{(1)}(x,s)=\sum_{m\geq R}\Phi^{(1)}(x,m,s)$

where, for each $m\geq R$, we have introduced the self-adjoint term
$\Phi^{(1)}(x,m,s)\in{\mathcal{A}}_{b_{x}(m)}$ given by

(4.17)
$\Phi^{(1)}(x,m,s)=\Delta_{b_{x}(R)}^{m}({\mathcal{K}}_{s}^{1,\Lambda}(h_{x}))+\chi_{\Lambda}(x)\sum_{\begin{subarray}{c}R\leq
k\leq m:\\\
b_{x}(k)\subseteq\Lambda\end{subarray}}\Delta_{b_{x}(k)}^{m}({\mathcal{K}}_{s}^{2,\Lambda}(\Phi(x,k))).$

With respect to this notation, (4.8), (4.9) and (4.13) show that

(4.18)
$\displaystyle\left(W(\Lambda,s)+E(\Lambda,s){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}\right)\psi$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\sum_{x\in\Lambda_{n}}\tilde{\Phi}^{(1)}(x,s)\psi$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\sum_{x\in\Lambda_{n}}\sum_{m\geq
R}\tilde{\Phi}^{(1)}(x,m,s)\psi.$

for all $\psi\in\pi_{0}({\mathcal{A}}^{\rm loc})\Omega$. Since this set is
dense in ${\mathcal{H}}$, the equality in (4.6) will follow once we establish
absolute summability of these terms. We do so by demonstrating the existence
of a function

(4.19)
$G_{\Lambda}(x,m)=\chi_{\Lambda(R)}(x)G_{1}(m)+\chi_{\Gamma\setminus\Lambda(R)}(x)G_{2}\left(d(b_{x}(R),\Lambda),m\right)$

which bounds the norms of these terms and satisfies (4.5). Here, we note that
$\Lambda(R)$ is as in (2.1), and the functions $G_{1}:[0,\infty)\to[0,\infty)$
and $G_{2}:[0,\infty)\times[0,\infty)\to[0,\infty)$ will be independent of
$\Lambda$.

First, suppose $x\in\Gamma\setminus\Lambda(R)$ where $R\geq 0$ is the finite
range of the unperturbed interaction. In this case, (4.17) simplifies to

(4.20)
$\Phi^{(1)}(x,m,s)=\Delta_{b_{x}(R)}^{m}({\mathcal{K}}_{s}^{1,\Lambda}(h_{x}))\,.$

Then, applying Lemmas 4.3 and 4.4 with the local approximation bound (3.7) one
finds

(4.21) $\|\tilde{\Phi}^{(1)}(x,m,s)\|=\|\Phi^{(1)}(x,m,s)\|\leq
sG_{2}\left(d(b_{x}(R),\Lambda),m\right)$

where for any fixed $\delta\in(0,1)$, the function $G_{2}$ is

(4.22)
$G_{2}(l,m)=C\cdot\left\\{\begin{array}[]{cl}F_{{\mathcal{K}}^{1}}(l),&\mbox{if
}m=R,\\\ 4F_{\delta}(l)\cdot F_{1-\delta}(m-R-1),&\mbox{if }m\geq
R+1.\end{array}\right.$

Here, $C=\kappa^{2}R^{2\nu}\|\eta\|_{\infty}$,
${\mathcal{F}}_{{\mathcal{K}}^{1}}$ is the function from Lemma 4.3, and

(4.23)
$F_{\delta}(l)=\max\left\\{(F_{{\mathcal{K}}^{1}}(l))^{\delta},(G_{{\mathcal{K}}^{1}}(l))^{\delta}\right\\}.$

More specifically, the bound in (4.21) for $m=R$ is a direct application of
Lemma 4.3 while the bound for $m\geq R+1$ follows from the quasi-local
estimate in Lemma 4.4 and the subsequent bound (4.46) coupled with
(3.4)-(3.7).

Summability of $G_{\Lambda}$, see (4.19), over the sites
$x\in\Gamma\setminus\Lambda(R)$ follows from observing that

$\displaystyle\sum_{x\in\Lambda(R)^{c}}\sum_{m\geq
R}G_{2}\left(d(b_{x}(R),\Lambda),m\right)$
$\displaystyle=C\sum_{x\in\Lambda(R)^{c}}F_{{\mathcal{K}}^{1}}(d(b_{x}(R),\Lambda))$
(4.24)
$\displaystyle+4C\sum_{x\in\Lambda(R)^{c}}F_{\delta}(d(b_{x}(R),\Lambda))\sum_{m\geq
R+1}F_{1-\delta}(m-R-1)<\infty$

as both $F_{{\mathcal{K}}^{1}}$ and $G_{{\mathcal{K}}^{1}}$ (and, thus,
$F_{\delta}$ for all $0<\delta<1$) have finite moments of all orders. In
particular, for any decay function $F:[0,\infty)\to[0,\infty)$ with a finite
$\nu$-moment,

$\sum_{x\in\Lambda(R)^{c}}F(d(b_{x}(R),\Lambda))\leq\sum_{n\geq
R}\sum_{x\in\Lambda(n+1)\setminus\Lambda(n)}F(n-R)\leq\kappa|\Lambda|\sum_{n\geq
R}(n+1)^{\nu}F(n-R)<\infty.$

We now turn to the sites $x\in\Lambda(R)$, for which we demonstrate that

(4.25) $\|\Phi^{(1)}(x,m,s)\|\leq sG_{1}(m)$

where $G_{1}$ is a summable function. Consider (4.17) when $m=R$. Clearly,

(4.26) $\displaystyle\|\Phi^{(1)}(x,R,s)\|$ $\displaystyle\leq$
$\displaystyle\|{\mathcal{K}}^{1,\Lambda}(h_{x})\|+\|{\mathcal{K}}^{2,\Lambda}(\Phi(x,R))\|$
$\displaystyle\leq$ $\displaystyle
sC_{1}\kappa^{2}R^{2\nu}\|\eta\|_{\infty}+sC_{2}\|\Phi\|e^{-aR^{\theta}}$

where we have used the local bounds (4.11), the uniform bound (2.6), and the
interaction bound in Assumption 2.5.

For $m\geq R+1$, (4.17) can be estimated as:

(4.27) $\displaystyle\|\Phi^{(1)}(x,m,s)\|$ $\displaystyle\leq$
$\displaystyle\|\Delta_{b_{x}(R)}^{m}({\mathcal{K}}_{s}^{1,\Lambda}(h_{x}))\|+\chi_{\Lambda}(x)\sum_{\begin{subarray}{c}R\leq
k\leq m:\\\
b_{x}(k)\subseteq\Lambda\end{subarray}}\|\Delta_{b_{x}(k)}^{m}({\mathcal{K}}_{s}^{2,\Lambda}(\Phi(x,k)))\|$
$\displaystyle\leq$ $\displaystyle
2s\kappa^{2}R^{2\nu}\|\eta\|_{\infty}G_{{\mathcal{K}}^{1}}(m-R-1)+2s\kappa\sum_{\begin{subarray}{c}R\leq
k\leq m:\\\
b_{x}(k)\subseteq\Lambda\end{subarray}}k^{\nu}\|\Phi(x,k)\|G_{{\mathcal{K}}^{2}}(m-k-1)$

using the quasi-local estimates in (4.12) and the local approximation bound in
(3.7). Given Assumption 2.5, the final sum above can be further estimated as

$\displaystyle\sum_{\begin{subarray}{c}R\leq k\leq m:\\\
b_{x}(k)\subseteq\Lambda\end{subarray}}\\!\\!\\!k^{\nu}\|\Phi(x,k)\|G_{{\mathcal{K}}^{2}}(m-k-1)$
$\displaystyle\leq\|\Phi\|\sum_{k=R}^{m}k^{\nu}e^{-ak^{\theta}}G_{{\mathcal{K}}^{2}}(m-k-1)$
(4.28)
$\displaystyle\leq\|\Phi\|\left(G_{{\mathcal{K}}^{2}}(m/2)\sum_{k=R}^{m/2-1}k^{\nu}e^{-ak^{\theta}}+G_{{\mathcal{K}}^{2}}(0)\sum_{k\geq
m/2}k^{\nu}e^{-ak^{\theta}}\right).$

To summarize, for $x\in\Lambda(R)$ we have proven (4.25) with the decay
function $G_{1}$ defined as

(4.29)
$G_{1}(R)=C_{1}\kappa^{2}R^{2\nu}\|\eta\|_{\infty}+C_{2}\|\Phi\|e^{-aR^{\theta}}$

and for $m\geq R+1$,

(4.30)
$G_{1}(m)=2CG_{{\mathcal{K}}^{1}}(m-R-1)+2\kappa\|\Phi\|\left(M_{\Phi}(R)G_{{\mathcal{K}}^{2}}(m/2)+G_{{\mathcal{K}}^{2}}(0)M_{\Phi}(m/2)\right).$

where we have used the notation

(4.31) $M_{\Phi}(r):=\sum_{k\geq r}k^{\nu}e^{-ak^{\theta}}$

to denote the moments associated to the decay of $\Phi$. Since each of the
decay functions in (4.30) has finite moments of all orders, it is clear that
$\sum_{m\geq R}G_{1}(m)<\infty$. As a consequence, $G_{\Lambda}$ as in (4.19)
satisfies

$\sum_{x\in\Lambda(R)}\sum_{m\geq R}G_{\Lambda}(x,m)\leq\kappa
R^{\nu}|\Lambda|\sum_{m\geq R}G_{1}(m)<\infty.$

This demonstrates absolute summability of the terms in (4.6), and hence,
completes the proof of Theorem 4.2. ∎

Given that we are working in the infinite volume setting, in the case of
${\mathcal{K}}_{s}^{1,\Lambda}$ one needs a more refined version of the
estimates in (4.11) - (4.12) to obtain the absolute summability of the terms
in (4.6). In the situation under consideration, these refined bounds are a
consequence of perturbation $V_{\Lambda}$ being locally supported as this
implies that the spectral flow $\alpha_{s}^{\Lambda}$ is approximately the
identity far from $\Lambda$. We now prove the necessary distance-dependent
locality bound and quasi-local estimates (namely, Lemma 4.3 and Lemma 4.4
below) which were used in the proof of Theorem 4.2.

######  Lemma 4.3 (Distance Locality Bound for ${\mathcal{K}}_{s}^{1}$).

There exists a decay function $F_{{\mathcal{K}}^{1}}$, with finite moments of
all orders, for which: given any $\Lambda\in\mathcal{P}_{0}(\Gamma)$,
$X\in\mathcal{P}_{0}(\Gamma)$ with $d(X,\Lambda)>0$, $A\in{\mathcal{A}}_{X}$,
and any $s\in{\mathbb{R}}$, we have the following local bound

(4.32) $\|{\mathcal{K}}_{s}^{1,\Lambda}(A)\|\leq
s|X|^{2}\|A\|F_{{\mathcal{K}}^{1}}\left(d(X,\Lambda)\right).$

Before we prove this lemma, let $F$ be any decay function with finite
$\nu$-moment and take $\epsilon\in(0,1)$. In this case, a new decay function
$M_{F}^{\epsilon}$ is given by

(4.33) $M_{F}^{\epsilon}(r)=\sum_{n\geq r}(n+1)^{\nu}F(\epsilon
n)\quad\mbox{for any }r\geq 0\,.$

The proof below shows that one may take

(4.34)
$F_{{\mathcal{K}}^{1}}(r)=2\kappa\|\Psi\|_{G_{\Psi}}\left(M_{G_{\Psi}}^{1-\epsilon}(r)+2G_{\Psi}(R)M_{G_{\mathcal{F}}}^{\epsilon}(r)\right)$

where $G_{{\mathcal{F}}}$ and $G_{\Psi}$ are the decay functions previously
discussed in (3.18) and (3.28). Since $G_{{\mathcal{F}}}$ and $G_{\Psi}$ both
have finite moments of all orders, the same is true for $G_{1}$.

Let us also consider any non-negative, non-increasing function $F$ with a
finite $\nu$-moment. We will use the following bound in our proof of Lemma
4.3. Let $\epsilon\in(0,1)$, take $\Lambda\in\mathcal{P}_{0}(\Gamma)$, and let
$X\in\mathcal{P}_{0}(\Gamma)$ with $d(X,\Lambda)>0$. The following estimate
holds:

(4.35) $\sum_{z\in\Lambda}F(\epsilon
d(z,X))\leq\kappa|X|M_{F}^{\epsilon}(d(X,\Lambda)).$

A simple argument for this is:

(4.36) $\displaystyle\sum_{z\in\Lambda}F(\epsilon
d(z,X))\leq\sum_{\begin{subarray}{c}z\in\Gamma:\\\ d(z,X)\geq
d(X,\Lambda)\end{subarray}}F(\epsilon d(z,X))$ $\displaystyle\leq$
$\displaystyle\sum_{n\geq d(X,\Lambda)}\sum_{\begin{subarray}{c}z\in\Gamma:\\\
n\leq d(z,X)<n+1\end{subarray}}F(\epsilon d(z,X))$ $\displaystyle\leq$
$\displaystyle\kappa|X|\sum_{n\geq d(X,\Lambda)}(n+1)^{\nu}F(\epsilon n)$

where for the last inequality we use that for any $n\geq 1$, the set $X(n)$
satisfies $|X(n)|\leq\kappa n^{\nu}|X|$ by $\nu$-regularity, see (2.1).

###### Proof of Lemma 4.3:.

Recall that ${\mathcal{K}}_{s}^{1,\Lambda}$ is as defined in (4.10). Since
$\alpha_{0}=\rm{id}$ and ${\mathcal{F}}={\mathcal{F}}_{0}^{\Lambda}$, one
finds that

(4.37)
${\mathcal{K}}_{s}^{1,\Lambda}(A)=\int_{0}^{s}\frac{d}{dr}\alpha_{r}^{\Lambda}({\mathcal{F}}^{\Lambda}_{r}(A))dr=i\int_{0}^{s}\alpha^{\Lambda}_{r}([D(\Lambda,r),{\mathcal{F}}^{\Lambda}_{r}(A)])dr$

for all $A\in{\mathcal{A}}_{X}$ with $X\cap\Lambda=\emptyset$. Here
$D(\Lambda,s)$ is the generator of the spectral flow, see (3.20), and
moreover, we have used (3.21) and [26, (6.37)] to obtain

(4.38) $\displaystyle\frac{d}{ds}{\mathcal{F}}_{s}^{\Lambda}(A)$
$\displaystyle=$ $\displaystyle
i\int_{-\infty}^{\infty}\int_{0}^{t}\tau_{r}^{(\Lambda,s)}([V_{\Lambda},\tau_{t-r}^{(\Lambda,s)}(A)])\,dr\,w_{\gamma}(t)\,dt$
$\displaystyle=$ $\displaystyle
i\int_{-\infty}^{\infty}\int_{0}^{t}\tau_{r}^{(\Lambda,s)}([V_{\Lambda},\tau_{t-r}^{(\Lambda,s)}(A)])\,dr\,\left(-\frac{d}{dt}W_{\gamma}(t)+\delta_{0}(t)\right)\,dt$
$\displaystyle=$ $\displaystyle
i\int_{-\infty}^{\infty}\tau_{t}^{(\Lambda,s)}([V_{\Lambda},A])W_{\gamma}(t)\,dt=0\,,$

where we integrated by parts and used that $[V_{\Lambda},A]=0$ since the
support of these observables is disjoint.

Returning to (4.37), we expand the generator as in (3.27) to write

(4.39)
$[D(\Lambda,r),{\mathcal{F}}_{r}^{\Lambda}(A)]=\sum_{z\in\Lambda}\sum_{n\geq
R}[\Psi_{\Lambda}(z,n,r),{\mathcal{F}}_{r}^{\Lambda}(A)]$

Fix $\epsilon\in(0,1)$, and for each $z\in\Lambda$, set
$k_{z}(\epsilon)=\epsilon d(z,X)$. For each term in (4.39), we approximate
${\mathcal{F}}_{r}^{\Lambda}(A)$ with a strictly local approximation:

(4.40)
$[\Psi_{\Lambda}(z,n,r),{\mathcal{F}}_{r}^{\Lambda}(A)]=[\Psi_{\Lambda}(z,n,r),\Pi_{X(k_{z}(\epsilon))}({\mathcal{F}}_{r}^{\Lambda}(A))+\left({\mathcal{F}}_{r}^{\Lambda}(A)-\Pi_{X(k_{z}(\epsilon))}({\mathcal{F}}_{r}^{\Lambda}(A))\right)]$

where we use the conditional expectations as described in Section 3.1.1 with
respect to the inflated set $X(k_{z}(\epsilon))$. Using the quasi-locality
bounds for ${\mathcal{F}}_{r}^{\Lambda}$, see (3.18), coupled with (3.5), we
find that

(4.41)
$\|[\Psi_{\Lambda}(z,n,r),{\mathcal{F}}_{r}^{\Lambda}(A)-\Pi_{X(k_{z}(\epsilon))}({\mathcal{F}}_{r}^{\Lambda}(A))]\|\leq
4\|A\||X|\|\Psi_{\Lambda}(z,n,r)\|G_{{\mathcal{F}}}(k_{z}(\epsilon))\,.$

As a result, we find that

(4.42) $\sum_{z\in\Lambda}\sum_{n\geq
R}\|[\Psi_{\Lambda}(z,n,r),{\mathcal{F}}_{r}^{\Lambda}(A)-\Pi_{X(k_{z}(\epsilon))}({\mathcal{F}}_{r}^{\Lambda}(A))]\|\leq
4\kappa\|A\||X|^{2}\|\Psi\|_{G_{\Psi}}G_{\Psi}(R)M_{G_{{\mathcal{F}}}}^{\epsilon}(d(X,\Lambda))$

by using (3.28) and (4.35).

To estimate the remaining terms, note that for each $z\in\Lambda$,
$b_{z}(n)\cap X(k_{z}(\epsilon))\neq\emptyset$ only when $n\geq
k_{z}(1-\epsilon)$. As a result,

(4.43) $\displaystyle\sum_{z\in\Lambda}\sum_{n\geq
R}\|[\Psi_{\Lambda}(z,n,r),\Pi_{X(k_{z}(\epsilon))}({\mathcal{F}}^{\Lambda}_{r}(A))]\|$
$\displaystyle=$ $\displaystyle\sum_{z\in\Lambda}\sum_{n\geq
k_{z}(1-\epsilon)}\|[\Psi_{\Lambda}(z,n,r),\Pi_{X(k_{z}(\epsilon))}({\mathcal{F}}^{\Lambda}_{r}(A))]\|$
$\displaystyle\leq$ $\displaystyle
2\|A\|\|\Psi\|_{G_{\Psi}}\sum_{z\in\Lambda}G_{\Psi}(k_{z}(1-\epsilon))$
$\displaystyle\leq$ $\displaystyle
2\kappa\|A\||X|\|\Psi\|_{G_{\Psi}}M_{G_{\Psi}}^{1-\epsilon}(d(X,\Lambda))$

with arguments similar to the prior estimate.

The bound claimed in (4.32), with the specific decay function in (4.34), now
follows by inserting (4.39) into (4.37) and using the estimates found in
(4.42) and (4.43) above. ∎

An improvement on the quasi-locality bound for
$\|[{\mathcal{K}}_{s}^{1,\Lambda}(A),B]\|$ which decays in both the distance
between the supports of the observables $A$ and $B$, as well as the distance
between $\Lambda$ and the support of $A$ follows by combining the estimates in
Lemma 4.3 and the original quasi-locality bound in (4.12). We now state and
prove this.

######  Lemma 4.4 (Distance Quasi-Locality for ${\mathcal{K}}_{1}$).

There exists a decay function $G:[0,\infty)\times[0,\infty)\to[0,\infty)$ for
which: given any $\Lambda\in\mathcal{P}_{0}(\Gamma)$,
$X\in\mathcal{P}_{0}(\Gamma)$ with $d(X,\Lambda)>0$, and
$Y\in\mathcal{P}_{0}(\Gamma)$, the bound

(4.44) $\|[{\mathcal{K}}_{s}^{1,\Lambda}(A),B]\|\leq
s\|A\|\|B\||X|^{2}G\left(d(X,\Lambda),d(X,Y)\right)$

holds for all $A\in{\mathcal{A}}_{X}$, $B\in{\mathcal{A}}_{Y}$, and
$s\in\mathbb{R}$. More precisely, for any $\delta\in(0,1)$, one may choose

(4.45)
$G(m,n)=\max\left\\{2F_{{\mathcal{K}}^{1}}^{\delta}(m)F_{{\mathcal{K}}^{1}}^{1-\delta}(n),\,G_{{\mathcal{K}}^{1}}^{\delta}(m)G_{{\mathcal{K}}^{1}}^{1-\delta}(n)\right\\}$

where $F_{{\mathcal{K}}^{1}}$ and $G_{{\mathcal{K}}^{1}}$ are the decay
functions from Lemma 4.3 and (4.12), respectively. Here, we use the notation
$G^{\delta}(m)=(G(m))^{\delta}.$

In applications, it is often convenient to have a factorized version of the
bound in (4.44). This is easily achieved. In fact, with $\delta\in(0,1)$ as in
(4.45),

(4.46) $G(m,n)\leq 2F_{\delta}(m)\cdot F_{1-\delta}(n)\quad\mbox{with}\quad
F_{\delta}(m)=\max\\{F_{{\mathcal{K}}^{1}}^{\delta}(m),G_{{\mathcal{K}}^{1}}^{\delta}(m)\\}.\,$

###### Proof.

Fix $0<\delta<1$. We consider two cases. First, suppose $d(X,\Lambda)\leq
d(X,Y)$. The quasi-locality estimate (4.12) shows that

(4.47) $\|[{\mathcal{K}}_{s}^{1,\Lambda}(A),B]\|\leq
s|X|^{2}\|A\|\|B\|G_{{\mathcal{K}}^{1}}^{\delta}(d(X,\Lambda))G_{{\mathcal{K}}^{1}}^{1-\delta}(d(X,Y))$

where we have used that $G_{{\mathcal{K}}^{1}}$ is non-increasing.

Now, suppose $d(X,\Lambda)>d(X,Y)$. The local bound from Lemma 4.3, i.e.
(4.32), implies

(4.48) $\|[{\mathcal{K}}_{s}^{1,\Lambda}(A),B]\|\leq
2\|{\mathcal{K}}_{s}^{1,\Lambda}(A)\|\|B\|\leq
2s|X|^{2}\|A\|\|B\|F_{{\mathcal{K}}^{1}}\left(d(X,\Lambda)\right).$

Since $F_{{\mathcal{K}}^{1}}$ is also non-increasing, the bound
$F_{{\mathcal{K}}^{1}}(d(X,\Lambda))\leq
F_{{\mathcal{K}}^{1}}^{\delta}\left(d(X,\Lambda)\right)F_{{\mathcal{K}}^{1}}^{1-\delta}\left(d(X,Y)\right)$
follows in this case. The bound claimed in (4.44) with the decay function from
(4.45) is then a consequence of (4.47) and (4.48). ∎

###  4.2. Final step in the proof of Theorem 4.1

In the previous section, we proved Theorem 4.2. A consequence of the equality
in (4.7) is that $\langle\Omega,W(\Lambda,s)\Omega\rangle=0$. Then, denoting
by $\tilde{\Phi}^{(1)}(x,m,s)=\pi_{0}(\Phi^{(1)}(x,m,s))$ is clear from
Theorem 4.2 that

(4.49)
$U(\Lambda,s)^{*}H(\Lambda,s)U(\Lambda,s)\psi-E(\Lambda,s)\psi=H_{0}\psi+\sum_{x\in\Gamma}\sum_{m\geq
R}\tilde{\Phi}^{(1)}_{\omega}(x,m,s)\psi,$

for all $\psi\in\pi_{0}({\mathcal{A}}^{\rm loc})\Omega$ where we have set

(4.50)
$\tilde{\Phi}^{(1)}_{\omega}(x,m,s)=\tilde{\Phi}^{(1)}(x,m,s)-\langle\Omega,\tilde{\Phi}^{(1)}(x,m,s)\Omega\rangle{\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}.$

The analogous relation in ${\mathcal{A}}$ holds, namely

(4.51)
$\Phi^{(1)}_{\omega}(x,m,s)=\Phi^{(1)}(x,m,s)-\omega_{0}(\Phi^{(1)}(x,m,s)){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}.$

It will also be convenient for us to introduce the notation

(4.52) $\tilde{\Phi}^{(1)}_{\omega}(x,s)=\sum_{m\geq
R}\tilde{\Phi}_{\omega}^{(1)}(x,m,s)$

which should be compared with (4.16). We begin with a lemma which makes use of
Assumption 2.3.

######  Lemma 4.5.

Denote by
$\tilde{P}_{b_{x}(n)}=\pi_{0}(P_{b_{x}(n)})\in{\mathcal{B}}({\mathcal{H}})$
the representative of the ground state projection $P_{b_{x}(n)}$ in the GNS
space. If (2.21) holds, then

(4.53)
$\left\|\sum_{k=R}^{m}\tilde{\Phi}_{\omega}(x,k,s)\tilde{P}_{b_{x}(n)}\right\|\leq
2s\left(G_{\Lambda}^{(1)}(x,m+1)+G_{\Lambda}^{(1)}(x,R)\sqrt{(1+m)^{\nu}G_{0}(n-m)}\right)$

where $G_{\Lambda}^{(1)}(x,m)=\sum_{k\geq m}G_{\Lambda}(x,k)$ for
$G_{\Lambda}$ as in Theorem 4.2, and $G_{0}$ is the decay function from
Assumption 2.3.

###### Proof.

We first begin by using (2.21) to establish the following inequality: for any
$A\in{\mathcal{A}}_{b_{x}(m)}$

(4.54)
$\left|\|\tilde{A}\tilde{P}_{b_{x}(n)}\|-\|\tilde{A}P_{\Omega}\|\right|\leq\|A\|\sqrt{(1+m)^{\nu}G_{0}(n-m)}.$

where we have set $P_{\Omega}=|\Omega\rangle\langle\Omega|$ and
$\tilde{A}=\pi_{0}(A)$. From the simple inequality
$|a-b|^{2}\leq|a^{2}-b^{2}|$ for $a,b\geq 0,$ it follows that

$\left|\|\tilde{A}\tilde{P}_{b_{x}(n)}\|-\|\tilde{A}P_{\Omega}\|\right|^{2}\leq\left|\|\tilde{A}\tilde{P}_{b_{x}(n)}\|^{2}-\|\tilde{A}P_{\Omega}\|^{2}\right|.$

The second term on the right-hand-side above is

$\|\tilde{A}P_{\Omega}\|^{2}=\left\langle\Omega,\pi_{0}(A^{*}A)\Omega\right\rangle\|P_{\Omega}\|=\omega_{0}(A^{*}A)\|P_{b_{x}(n)}\|.$

As $\pi_{0}$ is norm-preserving, we also have that

$\|\tilde{A}\tilde{P}_{b_{x}(n)}\|^{2}=\|\pi_{0}(AP_{b_{x}(n)})\|^{2}=\|P_{b_{x}(n)}A^{*}AP_{b_{x}(n)}\|.$

Given these observations, we conclude that for $A\in{\mathcal{A}}_{b_{x}(m)}$

$\displaystyle\left|\|\tilde{A}\tilde{P}_{b_{x}(n)}\|-\|\tilde{A}P_{\Omega}\|\right|^{2}$
$\displaystyle\leq\left|\|P_{b_{x}(n)}A^{*}AP_{b_{x}(n)}\|-\omega_{0}(A^{*}A)\|P_{b_{x}(n)}\|\right|$
$\displaystyle\leq\|P_{b_{x}(n)}A^{*}AP_{b_{x}(n)}-\omega_{0}(A^{*}A)P_{b_{x}(n)}\|$
(4.55) $\displaystyle\leq\|A\|^{2}(1+m)^{\nu}G_{0}(n-m),$

as a consequence of (2.21); this is (4.54).

Now using (4.54) with
$\tilde{A}=\sum_{k=R}^{m}\tilde{\Phi}_{\omega}^{(1)}(x,k,s)\in\pi_{0}({\mathcal{A}}_{b_{x}(m)})$,
we find that

(4.56)
$\displaystyle\left\|\sum_{k=R}^{m}\tilde{\Phi}_{\omega}^{(1)}(x,k,s)\tilde{P}_{b_{x}(n)}\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\sum_{k=R}^{m}\tilde{\Phi}_{\omega}^{(1)}(x,k,s)P_{\Omega}\right\|$
$\displaystyle+\left\|\sum_{k=R}^{m}\tilde{\Phi}_{\omega}^{(1)}(x,k,s)\right\|\sqrt{(1+m)^{\nu}G_{0}(n-m)}.$

Since $\|\tilde{\Phi}_{\omega}^{(1)}(x,k,s)\|\leq
2\|\tilde{\Phi}^{(1)}(x,k,s)\|\leq 2sG(x,k)$, the final bound following from
Theorem 4.2, the second term in (4.56) can be estimated as in (4.53).

For the first norm in (4.56), we bound with

(4.57)
$\left\|\sum_{k=R}^{m}\tilde{\Phi}_{\omega}^{(1)}(x,k,s)P_{\Omega}\right\|\leq\left\|\tilde{\Phi}_{\omega}^{(1)}(x,s)P_{\Omega}\right\|+\sum_{k=m+1}^{\infty}\left\|\tilde{\Phi}_{\omega}^{(1)}(x,k,s)P_{\Omega}\right\|.$

where we have used (4.52). Clearly,

(4.58)
$\sum_{k=m+1}^{\infty}\left\|\tilde{\Phi}_{\omega}^{(1)}(x,k,s)P_{\Omega}\right\|\leq
2\sum_{k=m+1}^{\infty}\left\|\tilde{\Phi}^{(1)}(x,k,s)\right\|\leq
2sG_{\Lambda}^{(1)}(x,m+1).$

We now claim that the remaining term is zero. As a result of Theorem 4.2, we
have that

$[\tilde{\Phi}_{\omega}^{(1)}(x,s),P_{\Omega}]=[\tilde{\Phi}^{(1)}(x,s),P_{\Omega}]=0.$

As a result, considering the definition of $\tilde{\Phi}_{\omega}^{(1)}(x,s)$,
see (4.52) and also (4.50),

(4.59)
$\displaystyle\tilde{\Phi}_{\omega}^{(1)}(x,s)P_{\Omega}=P_{\Omega}\tilde{\Phi}_{\omega}^{(1)}(x,s)P_{\Omega}=\left\langle\Omega,\tilde{\Phi}_{\omega}^{(1)}(x,s)\Omega\right\rangle
P_{\Omega}=0,$

where we use (4.52). This completes the proof of (4.53). ∎

We conclude this section with the proof of Theorem 4.1.

###### Proof of Theorem 4.1.

Fix $x\in\Gamma$. Since $[\tilde{\Phi}_{\omega}^{(1)}(x,s),P_{\Omega}]=0$, we
can write

$\tilde{\Phi}_{\omega}^{(1)}(x,s)=P_{\Omega}\tilde{\Phi}_{\omega}^{(1)}(x,s)P_{\Omega}+({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega})\tilde{\Phi}_{\omega}^{(1)}(x,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega})=({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega})\tilde{\Phi}_{\omega}^{(1)}(x,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega})$

where we have applied (4.59). The terms $\tilde{\Phi}^{(2)}(x,m,s)$ will be
defined by decomposing
${\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega}$
in terms of the finite volume ground state projections
$\tilde{P}_{n}:=\tilde{P}_{b_{x}(n)}\in\pi_{0}({\mathcal{A}}_{b_{x}(n)})$.

The finite volume ground state projections $\tilde{P}_{n}$ converge strongly
to $P_{\Omega}$ for all $\psi\in{\mathcal{H}}$. As a consequence, the
collection of operators

(4.60)
$\displaystyle\tilde{E}_{n}=\begin{cases}{\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{R},&n=R\\\
\tilde{P}_{n-1}-\tilde{P}_{n},&n\geq R+1\end{cases}$

forms a family of orthogonal projections that are mutually orthogonal and sum
to
${\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega}$.
That is,

(4.61)
$\tilde{E}_{n}^{*}=\tilde{E}_{n},\quad\tilde{E}_{n}\tilde{E}_{m}=\delta_{m,n}\tilde{E}_{n},\quad\text{and}\quad({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega})\psi=\sum_{n\geq
R}\tilde{E}_{n}\psi\quad\forall\;\psi\in{\mathcal{H}}.$

In fact, the frustration-free property yields
$\tilde{P}_{n}\tilde{P}_{m}=\tilde{P}_{m}$ for $m\geq n$, and moreover,

(4.62)
${\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m}=\sum_{n=R}^{m}\tilde{E}_{n}$

is clear. Using (4.52), the above properties imply that for all
$\psi,\phi\in{\mathcal{H}}$,

(4.63)
$\displaystyle\left\langle\phi,({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega})\tilde{\Phi}_{\omega}^{(1)}(x,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Omega})\psi\right\rangle=\sum_{k,m,n\geq
R}\left\langle\phi,\tilde{E_{n}}\tilde{\Phi}_{\omega}^{(1)}(x,k,s)\tilde{E}_{m}\psi\right\rangle$

We note that the triple sum of operators converges absolutely in norm, and so
the operator equality actually holds in the norm sense.

Each term $\Phi^{(2)}(x,m,s)$ will be defined as a sum of two self-adjoint
terms, i.e.

$\Phi^{(2)}(x,m,s)=\Theta_{1}(x,m,s)+\Theta_{2}(x,m,s),$

each of which is annihilated by the ground state projection $P_{m}$. Fix
$k\geq R$. Using the properties in (4.61)-(4.62) we write

(4.64) $\sum_{n,m\geq
R}\tilde{E}_{n}\tilde{\Phi}^{(1)}_{\omega}(x,k,s)\tilde{E}_{m}=({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{2k})\tilde{\Phi}^{(1)}_{\omega}(x,k,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{2k})+\sum_{m>2k}\tilde{\Phi}_{k,m},$

where $\tilde{\Phi}_{k,m}=\tilde{\Phi}_{k,m}^{*}$ is defined by

(4.65)
$\tilde{\Phi}_{k,m}=\tilde{E}_{m}\tilde{\Phi}^{(1)}_{\omega}(x,k,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m-1})+({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m})\tilde{\Phi}^{(1)}_{\omega}(x,k,s)\tilde{E}_{m}\in\pi_{0}({\mathcal{A}}_{b_{x}(m)}).$

Self-adjointness follows from noting
${\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m}={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m-1}+\tilde{E}_{m}$.

For each $k\geq R$, we define $\Theta_{1}(x,2k,s)$

(4.66)
$\Theta_{1}(x,2k,s)=({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{2k})\tilde{\Phi}^{(1)}_{\omega}(x,k,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{2k})\in\pi_{0}({\mathcal{A}}_{b_{x}(2k)}),$

and define $\Theta_{1}(x,m,s)=0$ for all other values of $m$. Clearly, these
operators satisfy
$\Theta_{1}(x,m,s)\tilde{P}_{m}=\tilde{P}_{m}\Theta_{1}(x,m,s)=0$, and
moreover, by Theorem 4.2 that $\|\Theta_{1}(x,m,s)\|\leq 2sG_{\Lambda}(x,m/2)$
as

(4.67) $\|\Theta_{1}(x,2k,s)\|\leq\|\Phi^{(1)}_{\omega}(x,k,s)\|\leq
2sG_{\Lambda}(x,k).$

Summing the remaining terms $\tilde{\Phi}_{k,m}$ over $k$, we define the
$\Theta_{2}$ terms by exchanging summations as follows:

$\displaystyle\sum_{k\geq R}\sum_{m>2k}\tilde{\Phi}_{k,m}$
$\displaystyle=\sum_{k\geq R}\sum_{m>2R}\tilde{\Phi}_{k,m}\chi_{\\{m>2k\\}}$
$\displaystyle=\sum_{m>2R}\sum_{R\leq k<m/2}\tilde{\Phi}_{k,m}$
$\displaystyle=\sum_{m>2R}\Theta_{2}(x,m,s),$

where, using (4.65), for $m>2R$ we define

(4.68) $\Theta_{2}(x,m,s)=\sum_{k=R}^{\lceil
m/2\rceil-1}\tilde{E}_{m}\tilde{\Phi}^{(1)}_{\omega}(x,k,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m-1})+({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m})\tilde{\Phi}^{(1)}_{\omega}(x,k,s)\tilde{E}_{m}.$

Moreover, we extend this definition to $\Theta_{2}(x,m,s)=0$ for all $m\leq
2R$. To see that $\Theta_{2}(x,m,s)\tilde{P}_{m}=0$, apply the frustration
free property $\tilde{P}_{m}\tilde{P}_{m-1}=\tilde{P}_{m}$ to verify

$0=\tilde{E}_{m}\tilde{P}_{m}=({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m-1})\tilde{P}_{m}.$

Finally, applying Lemma 4.5,

$\displaystyle\|\Theta_{2}(x,m,s)\|$ $\displaystyle\leq
2\left\|\sum_{k=R}^{\lceil
m/2\rceil-1}\tilde{\Phi}^{(1)}_{\omega}(x,k,s)({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{m-1})\right\|$
(4.69) $\displaystyle\leq
4s\left(G_{\Lambda}^{(1)}(x,m+1)+G_{\Lambda}^{(1)}(x,R)\sqrt{(1+m)^{\nu}G_{0}(m/2)}\right)$

where $G_{\Lambda}^{(1)}(x,m)=\sum_{n\geq m}G_{\Lambda}(x,n)$.

Therefore, setting
$\tilde{\Phi}^{(2)}(x,m,s)=\Theta_{1}(x,m,s)+\Theta_{2}(x,m,s)$ for all $m$,
we have constructed self-adjoint operators such that

(4.70) $W(\Lambda,s)=\sum_{x\in\Gamma}\sum_{m\geq R}\tilde{\Phi}^{(2)}(x,m,s)$

such that for all $m\geq R$ and $x\in\Gamma$,

$\tilde{\Phi}^{(2)}(x,m,s)\tilde{P}_{m}=\tilde{P}_{m}\tilde{\Phi}^{(2)}(x,m,s)=0$

and, moreover, combining (4.67) and (4.69) shows

(4.71) $\|\tilde{\Phi}^{(2)}(x,m,s)\|\leq 2sG_{\Lambda}^{(2)}(x,m)$

where

(4.72)
$G_{\Lambda}^{(2)}(x,m)=G_{\Lambda}(x,m/2)+2G_{\Lambda}^{(1)}(x,m+1)+2G_{\Lambda}^{(1)}(x,R)\sqrt{(1+m)^{\nu}G_{0}(m/2)}.$

The absolute summability of the series in (4.70) is a direct consequence of
$G_{0}$ being summable as well as that both $G_{\Lambda}$ and
$G_{\Lambda}^{(1)}$ satisfy (4.5). For $G_{\Lambda}^{(1)}$ this can easily be
seen from the fact that $G_{\Lambda}$ is a combination of functions with
finite moments of all orders, see specifically (4.19), (4.22) and (4.30). ∎

##  5\. Proof of the form bound for the GNS Hamiltonian

In this section, we generalize the statement and proof of Theorem 3.8 in [25]
(which, in turn, is based on a theorem in [21]), so that it is applicable to
our setting for infinite systems.

We work in the setting described in Section 2. In particular, under Assumption
2.2, there is a family of sets $\mathcal{S}=\\{\Lambda(x,n)|x\in\Gamma,n\geq
0\\}$, accompanied with a family $\mathcal{T}=\\{\mathcal{T}_{n}:n\geq 0\\}$
of separating partitions of $(c,\zeta)$-polynomial growth, for which the
associated finite volume Hamiltonians satisfy

(5.1) $H_{\Lambda(x,n)}\geq\gamma(n)P_{\Lambda(x,n)},\mbox{ for all }n\geq
R\,.$

Recall also that the uniform bound on the local gaps, i.e. $\gamma(n)$ above,
has been further assumed to satisfy $\gamma(n)\geq\gamma_{1}/n^{\alpha}$ for
some $\alpha\geq 0$.

######  Theorem 5.1.

Let $V$ be an anchored interaction on $(\Gamma,d)$ with the form:

(5.2) $V=\sum_{x\in\Gamma}\sum_{n\geq
R}\Phi(x,n)\quad\mbox{with}\quad\Phi(x,n)^{*}=\Phi(x,n)\in\mathcal{A}_{b_{x}(n)}\quad\mbox{for
all }x\in\Gamma\mbox{ and }n\geq R\,.$

Suppose that $V$ is absolutely summable, i.e.

(5.3) $\sum_{x\in\Gamma}\sum_{n\geq R}\|\Phi(x,n)\|<\infty,$

that the terms of $V$ annihilate the local ground states, i.e.

(5.4) $\Phi(x,n)P_{b_{x}(n)}=P_{b_{x}(n)}\Phi(x,n)=0\quad\mbox{for all
}x\in\Gamma\mbox{ and }n\geq R\,,$

and let $G:[0,\infty)\to[0,\infty)$ denote any uniform bound on these terms,
i.e.

(5.5) $\sup_{x\in\Gamma}\|\Phi(x,n)\|\leq G(n)\,.$

Then, for all $\psi\in\mathop{\rm dom}{H_{0}}$,

(5.6)
$\left|\left\langle\psi,\pi_{\omega}(V)\,\psi\right\rangle\right|\;\leq\;\beta\left\langle\psi,H_{0}\,\psi\right\rangle$

where

(5.7) $\beta=c\sum_{n\geq R}\frac{n^{\zeta}G(n)}{\gamma(n)}.$

In keeping with the notation from the previous sections, and under Assumption
2.2, for each $\Lambda(x,n)\in\mathcal{S}$, let us denote by
$\tilde{H}_{\Lambda(x,n)},\tilde{P}_{\Lambda(x,n)},$ and
$\tilde{Q}_{\Lambda(x,n)}$ the representations of
$H_{\Lambda(x,n)},P_{\Lambda(x,n)}$, and
$Q_{\Lambda(x,n)}={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\Lambda(x,n)}$
as elements of $\mathcal{B}(\mathcal{H}_{\omega})$; similarly we write
$\tilde{\Phi}(x,n)=\pi_{\omega}(\Phi(x,n))$. The proof of Theorem 5.1 will
follow closely the argument which proves Theorem 3.8 in [25], with the proviso
that we must take care to ensure that the infinite sums replacing the finite
ones in [25] are well-defined.

To this end, let us define some convenient families of orthogonal projections
which act on $\mathcal{H}_{\omega}$ and are associated to the unperturbed
Hamiltonian. Let $n\geq R$, and consider the $n$-th separating partition
$\mathcal{T}_{n}=\\{\mathcal{T}_{n}^{i}:i\in\mathcal{I}_{n}\\}$. For each
$i\in\mathcal{I}_{n}$ and any choice of $x,y\in\mathcal{T}_{n}^{i}$, it is
clear that

(5.8)
$[H_{\Lambda(x,n)},H_{\Lambda(y,n)}]=0\quad\mbox{and}\quad[P_{\Lambda(x,n)},P_{\Lambda(y,n)}]=0$

as these are operators with disjoint support in the algebra. Moreover, the GNS
representation preserves this commutativity:

(5.9)
$[\tilde{H}_{\Lambda(x,n)},\tilde{H}_{\Lambda(y,n)}]=0\quad\mbox{and}\quad[\tilde{P}_{\Lambda(x,n)},\tilde{P}_{\Lambda(y,n)}]=0\,.$

With $n\geq R$ and $i\in\mathcal{I}_{n}$ fixed, let us denote by $C_{n}^{i}$
the collection of all configurations associated to $\mathcal{T}_{n}^{i}$, more
precisely,

(5.10)
$C_{n}^{i}=\\{0,1\\}^{\mathcal{T}_{n}^{i}}=\left\\{\sigma:\sigma=\\{\sigma_{x}\\}\mbox{
where }\sigma_{x}\in\\{0,1\\}\mbox{ for all
}x\in\mathcal{T}_{n}^{i}\right\\}\,.$

For each $\sigma\in C_{n}^{i}$

(5.11) $|\sigma|=\sum_{x\in\mathcal{T}_{n}^{i}}\sigma_{x}$

and note that

(5.12) $|\sigma|<\infty\quad\mbox{if and only
if}\quad|\\{x\in\mathcal{T}_{n}^{i}:\sigma_{x}=1\\}|<\infty\,.$

Recall that
$\mathcal{D}_{\omega}=\\{\pi_{\omega}(A)\Omega_{\omega}:A\in\mathcal{A}^{\rm
loc}\\}$ is a dense subspace of $\mathcal{H}_{\omega}$. Let
$\psi\in\mathcal{D}_{\omega}$. Then $\psi=\tilde{A}\Omega_{\omega}$ for some
$\tilde{A}=\pi_{\omega}(A)$ with $A\in\mathcal{A}_{X}$. In this case, we have
that

(5.13)
$\tilde{P}_{\Lambda(x,n)}\psi=\psi\quad\mbox{and}\quad\tilde{Q}_{\Lambda(x,n)}\psi=0\,$

whenever $\Lambda(x,n)$ satisfies $\Lambda(x,n)\cap X=\emptyset$. It is also
clear that for any $X\in\mathcal{P}_{0}(\Gamma)$,
$\\{x\in\mathcal{T}_{n}^{i}:\Lambda(x,n)\cap X\neq\emptyset\\}$ is finite.
Thus, for any $\psi\in\mathcal{D}_{\omega}$, we define

(5.14)
$S(\sigma)\psi=\left\\{\begin{array}[]{cl}\prod_{x\in\mathcal{T}_{n}^{i}}\left[\sigma_{x}\tilde{Q}_{\Lambda(x,n)}+(1-\sigma_{x})\tilde{P}_{\Lambda(x,n)}\right]\psi&\mbox{if
}|\sigma|<\infty,\\\ 0&\mbox{otherwise}.\end{array}\right.$

Note that, if $|\sigma|<\infty$, then at most finitely many of these factors
act non-trivially, and moreover, by (5.9), all factors above commute. Since
$\mathcal{D}_{\omega}$ is dense, there is a unique extension of $S(\sigma)$ to
an element of $\mathcal{B}(\mathcal{H}_{\omega})$ for each $\sigma\in
C_{n}^{i}$. One checks that these operators satisfy:

(5.15) $S(\sigma)^{*}=S(\sigma),\quad
S(\sigma)S(\sigma^{\prime})=\delta_{\sigma,\sigma^{\prime}}S(\sigma),\quad\mbox{and}\quad\sum_{\sigma\in
C_{n}^{i}}S(\sigma)={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}\,.$

Now since $P_{\Lambda(x,n)}$ is the orthogonal projection onto the kernel of
$H_{\Lambda(x,n)}$, one clearly has that

(5.16) $[\tilde{Q}_{\Lambda(x,n)},\tilde{P}_{\Lambda(y,n)}]=0\quad\mbox{for
all }\;x,y\in\mathcal{T}_{n}^{i}\,.$

As a result, we find that

(5.17) $[\tilde{Q}_{\Lambda(x,n)},S(\sigma)]=0\quad\mbox{for each
}x\in\mathcal{T}_{n}^{i}\mbox{ and all }\;\sigma\in C_{n}^{i}.$

In fact, one need only consider $\sigma\in C_{n}^{i}$ with $|\sigma|<\infty$,
as otherwise the result is trivial. For such configurations, one readily
checks that (5.16) implies $0=[\tilde{Q}_{\Lambda(x,n)},S(\sigma)]\psi$ for
all $\psi\in\mathcal{D}_{\omega}$, and this relation uniquely defines this
bounded commutator. Combining (5.15) and (5.17), we find that

(5.18)
$S(\sigma)\tilde{Q}_{\Lambda(x,n)}S(\sigma^{\prime})=\delta_{\sigma,\sigma^{\prime}}\tilde{Q}_{\Lambda(x,n)}S(\sigma)=\delta_{\sigma,\sigma^{\prime}}({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-\tilde{P}_{\Lambda(x,n)})S(\sigma)=\delta_{\sigma,\sigma^{\prime}}\sigma_{x}S(\sigma)$

for all $\sigma,\sigma^{\prime}\in C_{n}^{i}$ and each
$x\in\mathcal{T}_{n}^{i}$.

Given that these families of orthogonal projections exist, we can now present
our proof.

###### Proof of Theorem 5.1:.

Since $V$ is absolutely summable, see (5.3), its terms can be rearranged. More
precisely, we can write

(5.19) $\tilde{V}=\sum_{x\in\Gamma}\sum_{n\geq R}\tilde{\Phi}(x,n)=\sum_{n\geq
R}\sum_{i\in\mathcal{I}_{n}}\tilde{V}_{n}^{i}$

as a bounded operator acting on the GNS space. Here we have used the family of
separating partitions $\mathcal{T}=\\{\mathcal{T}_{n}:n\geq R\\}$, as in
Assumption 2.2, and denoted by

(5.20) $\tilde{V}_{n}^{i}=\sum_{x\in\mathcal{T}_{n}^{i}}\tilde{\Phi}(x,n)\,.$

As a result, for any $\psi\in\mathcal{H}_{\omega}$, the following naive bound

(5.21) $|\langle\psi,\tilde{V}\psi\rangle|\leq\sum_{n\geq
R}\sum_{i\in\mathcal{I}_{n}}|\langle\psi,\tilde{V}_{n}^{i}\psi\rangle|\quad\mbox{with}\quad|\langle\psi,\tilde{V}_{n}^{i}\psi\rangle|\leq\sum_{x\in\mathcal{T}_{n}^{i}}|\langle\psi,\tilde{\Phi}(x,n)\psi\rangle|$

certainly holds.

Using now that we have assumed $b_{x}(n)\subset\Lambda(x,n)$ for all
$\Lambda(x,n)\in\mathcal{S}$, the frustration-free property of the ground-
state projections implies

(5.22)
$P_{\Lambda(x,n)}=P_{b_{x}(n)}P_{\Lambda(x,n)}=P_{\Lambda(x,n)}P_{b_{x}(n)}.$

By our assumptions on the terms $\Phi(x,n)$, see (5.2) and (5.4), we find that

(5.23) $[\tilde{\Phi}(x,n),\tilde{P}_{\Lambda(y,n)}]=0\quad\mbox{for all
}x,y\in\mathcal{T}_{n}^{i}.$

Arguing as above, we find then that

(5.24) $[\tilde{\Phi}(x,n),S(\sigma)]=0\quad\mbox{for each
}x\in\mathcal{T}_{n}^{i}\mbox{ and all }\;\sigma\in C_{n}^{i},$

and moreover, for all $\sigma,\sigma^{\prime}\in C_{n}^{i}$ and each
$x\in\mathcal{T}_{n}^{i}$, we have that

(5.25)
$S(\sigma)\tilde{\Phi}(x,n)S(\sigma^{\prime})=\delta_{\sigma,\sigma^{\prime}}S(\sigma)\tilde{\Phi}(x,n)=\delta_{\sigma,\sigma^{\prime}}\sigma_{x}S(\sigma)\tilde{\Phi}(x,n)\,.$

In fact, using (5.18), the above may also be re-written as

(5.26)
$S(\sigma)\tilde{\Phi}(x,n)S(\sigma^{\prime})=\delta_{\sigma,\sigma^{\prime}}S(\sigma)\tilde{Q}_{\Lambda(x,n)}\tilde{\Phi}(x,n)\tilde{Q}_{\Lambda(x,n)}S(\sigma)\,.$

As a consequence, the bound

(5.27)
$\displaystyle|\langle\psi,\tilde{\Phi}(x,n)\psi\rangle|\leq\sum_{\sigma,\sigma^{\prime}}|\langle\psi,S(\sigma)\tilde{\Phi}(x,n)S(\sigma^{\prime})\psi\rangle|$
$\displaystyle=$
$\displaystyle\sum_{\sigma}|\langle\psi,S(\sigma)\tilde{Q}_{\Lambda(x,n)}\tilde{\Phi}(x,n)\tilde{Q}_{\Lambda(x,n)}S(\sigma)\psi\rangle|$
$\displaystyle\leq$ $\displaystyle
G(n)\sum_{\sigma}\|\tilde{Q}_{\Lambda(x,n)}S(\sigma)\psi\|^{2}$

follows from (5.15) and (5.26). Summing these orthogonal projections again, we
find that

(5.28)
$\displaystyle\sum_{\sigma}\|\tilde{Q}_{\Lambda(x,n)}S(\sigma)\psi\|^{2}=\sum_{\sigma}\langle\psi,\tilde{Q}_{\Lambda(x,n)}S(\sigma)\psi\rangle$
$\displaystyle=$
$\displaystyle\langle\psi,\tilde{Q}_{\Lambda(x,n)}\psi\rangle$
$\displaystyle\leq$
$\displaystyle\frac{1}{\gamma(n)}\langle\psi,\tilde{H}_{\Lambda(x,n)}\psi\rangle$

where, for the final bound above we introduced the positive local gaps
$\gamma(n)$, used that $\gamma(n)Q_{\Lambda(x,n)}\leq H_{\Lambda(x,n)}$ holds
in the algebra, and that the representation preserves positivity.

Given the above, we conclude that for any $\psi\in{\rm dom}(H_{0})$,

(5.29)
$|\langle\psi,\tilde{V}_{n}^{i}\psi\rangle|\leq\sum_{x\in\mathcal{T}_{n}^{i}}|\langle\psi,\tilde{\Phi}(x,n)\psi\rangle|\leq\frac{G(n)}{\gamma(n)}\sum_{x\in\mathcal{T}_{n}^{i}}\langle\psi,\tilde{H}_{\Lambda(x,n)}\psi\rangle\leq\frac{G(n)}{\gamma(n)}\langle\psi,H_{0}\psi\rangle$

where we have used that $\Lambda(x,n)\cap\Lambda(y,n)=\emptyset$ for
$x,y\in\mathcal{T}_{n}^{i}$ with $x\neq y$. Returning again to (5.21), it is
now clear that

(5.30) $|\langle\psi,\tilde{V}\psi\rangle|\leq\sum_{n\geq
R}\sum_{i\in\mathcal{I}_{n}}|\langle\psi,\tilde{V}_{n}^{i}\psi\rangle|\leq\beta\langle\psi,H_{0}\psi\rangle\,,$

with $\beta$ as in (5.7), since these partitions satisfy the
$(c,\zeta)$-polynomial growth bound. This completes the proof. ∎

##  6\. Proofs of Theorems 2.7 and 2.8

In Section 4 we showed in Theorem 4.1 that, for every finite
$\Lambda\subset\Gamma$, the Hamiltonian

$H(\Lambda,s)=H_{0}+s\pi_{0}(V_{\Lambda})=H_{0}+s\sum_{b_{x}(n)\subset\Lambda}\pi_{0}(\Phi(x,n))$

transforms under the spectral flow unitary as follows

$U(\Lambda,s)^{*}H(\Lambda,s)U(\Lambda,s)\psi-E(\Lambda,s)\psi=H_{0}\psi+\sum_{x\in\Gamma}\sum_{m\geq
R}\pi_{0}(\Phi_{\Lambda}^{(2)}(x,m,s))\psi,$

where $E(\Lambda,s)$ is the ground state energy of $H(\Lambda,s)$ and
$\Phi_{\Lambda}^{(2)}(x,m,s)$ is a rapidly decaying interaction of order $s$.
It follows from Theorem 4.2 that the new perturbation of $H_{0}$ in the RHS is
still bounded and that it is mainly supported in a neighborhood of $\Lambda$.
We now show that Theorem 5.1 applies to $\Phi^{(2)}(x,m,s)$ with a constant
$\beta$ that is independent of $\Lambda$, from which Theorem 2.7 will follow.

###### Proof of Theorem 2.7.

Fix $\gamma\in(0,\gamma_{0})$ where $\gamma_{0}$ is as in Assumption 2.1, and
let $\Lambda\in{\mathcal{P}}_{0}$ be arbitrary. Given Theorem 4.1, it is clear
that Theorem 5.1 applies and produces a non-trivial form bound (see (5.6)) so
long as

$G_{\Lambda}^{(2)}(m):=\sup_{x\in\Gamma}G_{\Lambda}^{(2)}(x,m)$

has a finite $(\zeta+\alpha)$-moment where, as stated in Assumption 2.2,
$\zeta$ is the polynomial growth of the separating partitions and
$\gamma(n)\geq\gamma_{1}n^{-\alpha}$ is the lower bound on the local gaps.
Moreover, to show that

(6.1)
$s_{0}(\gamma):=\inf_{\Lambda\in{\mathcal{P}}_{0}}s_{0}^{\Lambda}(\gamma)>0,$

see (3.42), we wish to show this form bound is uniform in $\Lambda$.

To this end, recall that $G_{\Lambda}$ from (4.19) is defined in terms of two
functions $G_{1}$, $G_{2}$ that are independent of $\Lambda$ and decay faster
than any polynomial. It is then clear that for all
$\Lambda\in{\mathcal{P}}_{0}$, $x\in\Gamma$ and $m\geq R$,

$G_{\Lambda}(x,m)\leq G_{1}(m)+G_{2}(0,m)=:G(m),$

and similarly, $G_{\Lambda}^{(1)}(x,m)\leq G^{(1)}(m)$ where
$G^{(1)}(m)=\sum_{m\geq R}G(m)$. It follows immediately that

$\sup_{\Lambda\in{\mathcal{P}}_{0}}G_{\Lambda}^{(2)}(m)\leq
G^{(2)}(m):=G(m/2)+2G^{(1)}(m/2)+2G^{(1)}(R)\sqrt{(1+m)^{\nu}G_{0}(m/2)}$

and that $G^{(2)}$ has a finite $(\zeta+\alpha)$-moment as long as $G_{0}$
satisfies (2.20).

Given the norm bound from Theorem 4.1, it follows from Theorem 5.1 that for
all $\psi\in\mathop{\rm dom}(H_{0})$

$\left\langle\psi,W(\Lambda,s)\right\rangle\leq
s\beta\left\langle\psi,H_{0}\psi\right\rangle\quad\text{where}\quad\beta=2c\sum_{m\geq
R}\frac{n^{\zeta}G^{(2)}(m)}{\gamma(m)}.$

Applying [25, Corollary 3.3] proves that

${\rm gap}(H(\Lambda,s))\geq\gamma_{0}-s\beta\gamma_{0}.$

This shows that

$s_{0}^{\Lambda}(\gamma)\geq\frac{\gamma_{0}-\gamma}{\beta\gamma_{0}},$

and establishes (6.1) as claimed. ∎

We conclude with using the uniform estimate from Theorem 2.7 to establish the
claimed lower bound estimate on the gap of the extensively perturbed system
from Theorem 2.8.

###### Proof of Theorem 2.8.

Let $0<\gamma<\gamma_{0}$ and consider $|s|\leq s_{0}(\gamma)$. Recall that
for any IAS $(\Lambda_{n})$, the following limits hold in ${\mathcal{A}}$ as
$n\to\infty$:

(6.2) $\displaystyle\alpha_{s}^{\Lambda_{n}}(A)$
$\displaystyle\to\alpha_{s}(A),\quad\text{for all}\quad A\in{\mathcal{A}}$
(6.3) $\displaystyle\delta_{s}^{\Lambda_{n}}(A)$
$\displaystyle\to\delta_{s}(A),\quad\text{for all}\quad A\in{\mathcal{A}}_{\rm
loc},$

see (2.36) and (2.37). As a consequence,
$\omega_{s}^{\Lambda_{n}}(A)=\omega_{0}\circ\alpha_{s}^{\Lambda_{n}}(A)\to\omega_{s}(A)$
for all $A\in{\mathcal{A}}$ and, moreover, $\omega_{s}$ is a ground state of
$\delta_{s}$.

Now consider $A\in{\mathcal{A}}^{\rm loc}$ for which $\omega_{s}(A)=0$. Given
Theorem 2.7, we know that along any IAS $(\Lambda_{n})$, the GNS Hamiltonian
$H(\Lambda_{n},s)$ has a gap above its unique ground state lower bounded by
$\gamma$ for all $|s|\leq s_{0}(\gamma)$. Therefore,
$\omega_{s}^{\Lambda_{n}}$ satisfies (2.16) and, in particular, the inequality

(6.4)
$\omega_{s}^{\Lambda_{n}}(B_{n}^{*}\delta_{s}^{\Lambda_{n}}(B_{n}))\geq\gamma\omega_{s}^{\Lambda_{n}}(B_{n}^{*}B_{n})$

holds for the observable
$B_{n}=A-\omega_{s}^{\Lambda_{n}}(A){\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}\in{\mathcal{A}}^{\rm
loc}$. Combining (6.3) and the locally uniform convergence of
$\alpha_{s}^{\Lambda_{n}}$, we can take the limit $n\to\infty$ on both sides
of (6.4) to obtain (2.38). The remaining claims follow as in (2.16). ∎

## Acknowledgements

Based upon work supported by the National Science Foundation under grant
DMS-1813149 (BN), and the DFG under EXC-2111–390814868 (AY).

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