# Stability of invertible, frustration-free ground states against large
perturbations

Sven Bachmann  Department of Mathematics, University of British Columbia,
Vancouver, BC V6T 1Z2, Canada  Wojciech De Roeck  Institute of Theoretical
Physics, K.U. Leuven, 3001 Leuven, Belgium  Brecht Donvil  Department of
Mathematics and Statistics, University of Helsinki, Helsinki, Finland  Martin
Fraas  Department of Mathematics, University of California, Davis, Davis, CA,
95616, USA

###### Abstract

A gapped ground state of a quantum spin system has a natural length scale set
by the gap. This length scale governs the decay of correlations. A common
intuition is that this length scale also controls the spatial relaxation
towards the ground state away from impurities or boundaries. The aim of this
article is to take a step towards a proof of this intuition. We assume that
the ground state is frustration-free and invertible, i.e. it has no long-range
entanglement. Moreover, we assume the property that we are aiming to prove for
one specific kind of boundary condition; namely open boundary conditions. This
assumption is also known as the ”local topological quantum order” (LTQO)
condition. With these assumptions we can prove stretched exponential decay
away from boundaries or impurities, for any of the ground states of the
perturbed system. In contrast to most earlier results, we do not assume that
the perturbations at the boundary or the impurity are small. In particular,
the perturbed system itself can have long-range entanglement.

##  1 Informal statement of the result

Since a full ab initio statement of our assumptions and theorem requires quite
some setup and definitions, we first state some simplified assumptions and the
result, freely using terminology that is probably known to most of our
readers. We consider a spin system on a finite discrete set $\Gamma$, say, a
subset of $\mathbb{Z}^{d}$. We define a Hamiltonian $H+J$ where both terms $H$
and $J$ are local Hamiltonians, i.e. sums of local terms
$H=\sum_{X\subset\Gamma}h_{X},J=\sum_{X\subset\Gamma}j_{X}$ with
$||h_{X}||,||j_{X}||$ decaying rapidly in $\mathrm{diam}(X)$. We assume
moreover the following properties:

  1. 1.

The spatial support of $J$ is confined to a region $\Gamma_{j}$ with arbitrary
size. Crucially, we do not assume that the terms $j_{X}$ are small.

  2. 2.

The Hamiltonian $H$ has a ground state $\Omega$ that is _invertible_. This
means there is an auxiliary state $\Omega^{\prime}$ such that
$\Omega\otimes\Omega^{\prime}$ is connected to a product state by a locally
generated unitary, i.e. it is automorphically equivalent to a product state.

  3. 3.

$H$ is frustration-free, i.e. the local terms $h_{X}$ are all minimized by the
state $\Omega$.

  4. 4.

The open boundary restrictions (OBC) $H_{Z}=\sum_{X\subset Z}h_{Z}$ for balls
$Z$ with radius $r$ have a spectral gap $\gamma(r)$ above the ground state
sector, that decays no faster than an inverse polynomial in $r$, as
$r\to\infty$.

  5. 5.

The ground states of the open boundary restrictions $H_{Z}$ for balls $Z$
satisfy the so-called ‘local topological quantum order’ (LTQO) condition:
Their local restrictions to a set $X$ approach the local restriction of the
state $\Omega$, quasi-exponentially fast, as
$\mathrm{dist}(X,Z^{c})\to\infty$.

The result is that local restrictions of any ground state of $H+J$ approach
the local restriction of $\Omega$ fast, as a function of the distance to the
impurity or boundary region $\Gamma_{j}$, see also Figure 1.

Informal claim on local stability: _There is a stretching exponent $\beta>0$
and finite constant $C$ such that, for any ground state $\Phi$ of $H+J$, and
for any local observable $O_{X}$ supported in $X\subset\Gamma$ with
$\mathrm{diam}(X)\leq\mathrm{dist}(X,\Gamma_{j})^{\beta}$,_

$|\langle\Phi,O_{X}\Phi\rangle-\langle\Omega,O_{X}\Omega\rangle|\leq
C||O_{X}||e^{-(\mathrm{dist}(X,\Gamma_{j}))^{\beta}}$

Figure 1: The informal claim on local stability concerns ground state
expectation values of observables supported in the region $X$, far away from
$\Gamma_{j}$, the support of $J$.

As far as we can see, there is no natural sense in which $\Phi$ resembles
$\Omega$, other than what is expressed in this claim. In particular, $\Phi$ is
not necessarily the unique ground state, it has in general no gap111Actually,
our assumptions do not explicitly require a global gap for $H$ either, but
such a gap is there in spirit because of the invertibility assumption, and we
have left it in the abstract to fix thoughts., no short range entanglement
properties and it is not frustration-free. To illustrate this, in particular
the possibility of long-range entanglement, we consider
$\Gamma=\\{1,2,\ldots,L\\}$ and we imagine that at each site there is a qubit,
i.e. a two-dimensional Hilbert space, with base states
$\ket{\downarrow},\ket{\uparrow}$. The Hamiltonian is
$H=\sum_{i\in\Gamma}\sigma^{z}_{i}$ where $\sigma^{z}_{i}$ acts at site $i$,
as $\sigma^{z}_{i}\ket{\uparrow}_{i}=\ket{\uparrow}_{i}$ and
$\sigma^{z}_{i}\ket{\downarrow}_{i}=-\ket{\downarrow}_{i}$. This Hamiltonian
has a unique ground state that is a product, namely
$\otimes_{i}\ket{\downarrow}_{i}$ . The perturbation $J$ is chosen as
$J=-\sigma^{z}_{1}-\sigma^{z}_{L}+e^{-cL}j_{1,L}$ with some $c>0$ and
$j_{1,L}$ a two-qubit operator acting on sites $1$ and $L$. Because of the
very small prefactor $e^{-cL}$, this perturbation $J$ satisfies any reasonable
locality requirement. By taking $j_{1,L}=0$, the perturbed system $H+J$ has
4-fold ground state degenerary. By taking $j_{1,L}$ to be minus the rank-1
projector on the entangled Bell state
$\tfrac{1}{\sqrt{2}}\left(\ket{\uparrow}_{1}\otimes\ket{\uparrow}_{L}+\ket{\downarrow}_{1}\otimes\ket{\downarrow}_{L}\right)$,
the perturbed system $H+J$ has a unique ground state that has maximal
entanglement between qubits $1$ and $L$.

###  1.1 Discussion

####  1.1.1 Spectral stability of the ground state sector

The above result should be contrasted with recent work on the stability of the
spectral gap against locally small perturbations. That recent work can be
split in two classes. The first class concerns systems where the unperturbed
ground state is a product, see [1, 2, 3, 4, 5, 6, 7]. The second class is
based on the so-called Bravyi-Hastings-Michalakis (BHM) argument which applies
to frustration-free ground states, see [8, 9, 10, 11]. Both classes are
eventually based on some form of spectral perturbation theory applied to the
ground state sector. In all of the above cases, one also obtains some form of
locality for the action of small perturbations, captured by the slogan ‘local
perturbations perturb locally’, see also [12, 13].

Our result is different in spirit, as our perturbations are not assumed to be
small, but we obtain only information about the perturbed ground states in
regions far away from the region $\Gamma_{j}$ where the perturbation acts. On
the other hand, and in contrast to the quoted works, our result relies
crucially on the variational principle and so it holds only for ground states
(or ceiling states). One should realize that it is not only the proof of the
results in [8, 9, 10] that does not apply in our case, but also the results
should not be expected, as already stressed above through an example.

####  1.1.2 Role of the invertibility assumption

If one drops the invertibility assumption, then the claim on local stability
in Section 1 can no longer hold. To see this, one can consider the two-
dimensional toric code model [14, 15] on a large square $\Gamma$. This model
is frustration-free and with appropriate boundary conditions, there is a
unique groundstate $\Omega$. The OBC restrictions always have a spectral gap
that is bounded below by a positive constant, and the topological order
condition holds. However, the state $\Omega$ is not invertible, regardless of
boundary conditions. The lack of invertibility is associated with anyonic
excitations that are created in pairs. By choosing a boundary condition that
fixes a single anyon excitation at the boundary of a region $\Gamma$, one
forces a partner anyon to be present somewhere in the interior of $\Gamma$,
see Figure 2. In this case, there will be multiple ground states, labelled by
the position $x$ of the partner anyon. Since $x$ can be arbitrarily far from
the boundary, the claim on local stability does not hold.

Figure 2: Left: a pair of anyons at sites $x,x^{\prime}$ connected by a
(locally non-observable) string. This is an excited state of the toric code
Hamiltonian. Right: one of the anyons is fixed at the boundary by the boundary
condition. This constrains the partner anyon to be present in the ground
state, at some arbitrary site $x$.

Of course, the large degeneracy of ground states is not a robust feature of
the model. Indeed, if one adds a small global perturbation to the model, which
will in general give the hitherto static anyons some mobility (see e.g. [16]),
then the degeneracy will be lifted and the unique ground state will correspond
to a delocalized anyon. In this case, the claim on local stability is still
violated because the ground state now differs locally from $\Omega$ by an
amount that is an inverse polynomial in the volume $|\Gamma|$.

####  1.1.3 Remaining assumptions, extensions, and related results

As argued above, the invertibility assumption can not be dropped. We can also
not drop the LTQO condition. Consider for example $H=H_{0}+B$ acting on the
chain of qubits (as the example immediately below the informal claim). We take
$H_{0}$ the classical Ising Hamiltonian
$H_{0}=-\sum_{i=1}^{L-1}\sigma_{i}^{z}\sigma_{i+1}^{z}$ such that its ground
states exhibits spontaneous symmetry breaking, with ground states
$\otimes_{i}\ket{\downarrow}_{i}$ and $\otimes_{i}\ket{\uparrow}_{i}$. The
term $B=\sigma^{z}_{1}$ breaks the symmetry at the boundary, so that $H$ has a
unique ground state $\Omega=\otimes_{i}\ket{\downarrow}_{i}$. In that case our
result does not hold, as we see by taking $J=-B$. The LTQO assumption fails to
hold since in the interior of $\Gamma$, the symmetry remains unbroken and the
OBC restrictions in the interior also have $\otimes_{i}\ket{\uparrow}_{i}$ as
ground states. Of course, one could imagine replacing the LTQO condition by
some form of translation invariance of the Hamiltonian $H$, ruling out the
above example.

Apart from this, we do not see that the other assumptions are fundamentally
necessary, though they are necessary for our proof. However, a mild extension
of our result seems within reach: we hope that techniques introduced within
the BHM argument, see Section 1.1.1, could allow to generalize our result in
the following way: If we were to tighten our assumptions so that the local gap
$\gamma(r)\geq\gamma>0$, i.e. the local gap is uniformly bounded from below,
then the result should also hold for Hamiltonians of the type
$H^{\prime}=H+\epsilon K$ where $H$ and its ground state satisfy the tightened
assumptions, $K$ is a Hamiltonian with the same locality restrictions as $H$
and $J$, and $\epsilon\ll 1$ is a small parameter.

Results similar to ours have recently been obtained independently in [17], as
a corollary to [4]. The result of [17, 4] yields full exponential decay and
requires no explicit frustration freeness, but it is restricted to weakly
interacting spins, i.e. perturbations of products. Instead, our result relies
on the automorphic equivalence of the unperturbed ground state to a product
state, possibly upon adjoining an auxiliary state. Both results need hence an
underlying product structure. Another line of research that is loosely
connected to ours, concerns stability at nonzero temperature in spatial
dimension 1, see [18, 19].

### Acknowledgements

W.D.R thanks Marius Schütz and Simone Warzel for collaboration and discussions
of many years ago that proved crucial for the present paper. We thank two
anonymous referees for very constructive remarks that allowed us to
substantially improve the paper, in particular concerning the necessity of the
assumptions. M.F. was supported in part by the NSF under grant DMS-1907435.
W.D.R. was supported in part by the FWO under grant G098919N. S.B. was
upported by NSERC of Canada.

##  2 Setup

###  2.1 Preliminaries

####  2.1.1 Spatial structure

We consider a finite graph $\Gamma$, equipped with the graph distance
$\mathrm{dist}(\cdot,\cdot)$. Let $B_{r}(x)=\\{y:\mathrm{dist}(x,y)\leq r\\}$
be the ball of radius $r$ centered at $x\in\Gamma$. The graph is assumed to
have a finite dimension $d<\infty$, i.e. there is a $C_{\Gamma}<\infty$ such
that

$\sup_{x\in\Gamma}|B_{r}(x)|\leq 1+C_{\Gamma}r^{d}.$

For reasons of recognizability, we refer to vertices as ‘sites’. To every site
$x$ is associated a finite-dimensional Hilbert space $\mathcal{H}_{x}$ and we
define the total Hilbert space
$\mathcal{H}=\mathcal{H}_{\Gamma}=\otimes_{x\in\Gamma}\mathcal{H}_{x}$ and the
algebra $\mathcal{A}=\mathcal{A}_{\Gamma}=\mathcal{B}(\mathcal{H})$ of bounded
operators on $\mathcal{H}$. For any $X\subset\Gamma$, we have the Hilbert
space $\mathcal{H}_{X}=\otimes_{x\in X}\mathcal{H}_{x}$ and the algebra
$\mathcal{A}_{X}=\mathcal{B}(\mathcal{H}_{X})$. We use the usual embedding
$\mathcal{A}_{X}\to\mathcal{A}$ given by $O_{X}\to
O_{X}\otimes\mathds{1}_{X^{c}}$. As is customary, we will identify
$O_{X}\otimes\mathds{1}_{X^{c}}$ with $O_{X}$ and say that $O_{X}$ is
supported in $X$.

####  2.1.2 Locality

We write $\mathbb{N}^{+}$ for the strictly positive naturals and we let
$\mathcal{M}$ be a class of functions $m:\mathbb{N}^{+}\to\mathbb{R}^{+}$ of
quasi-exponential decay. The class $\mathcal{M}$ is defined by the following
two conditions.

  1. 1.

$m$ is non-increasing.

  2. 2.

For every $0<\alpha<1$, there exists $C_{\alpha},c_{\alpha}>0$ such that
$m(r)\leq C_{\alpha}e^{-c_{\alpha}r^{\alpha}}$.

Note that the same class of functions is obtained by setting $c_{\alpha}=1$,
which is used often in the proofs. To express the locality properties of
Hamiltonians, we consider collections $q$ of local terms
$(q_{X})_{X\subset\Gamma},q_{X}\in\mathcal{A}_{X}$, sometimes called
’interactions’, and we endow them with a family of norms, parametrized by
$m\in\mathcal{M}$:

$||q||_{m}:=\sup_{x\in\Gamma}\sum_{X\ni
x}\frac{||q_{X}||}{m(1+\mathrm{diam}(X))}$

The locality property is then expressed by the finiteness of $||q||_{m}$ for
some $m\in\mathcal{M}$.

####  2.1.3 Trace norms

We denote by $\tr^{(X)}$ the trace on the Hilbert space $\mathcal{H}_{X}$, and
we abbreviate $\tr=\tr^{(\Gamma)}$, i.e. the trace on the global Hilbert space
$\mathcal{H}$. We recall the trace norm $||\cdot||_{1,X}$ on
$\mathcal{A}_{X}=\mathcal{B}(\mathcal{H}_{X})$, defined by

$||O||_{1,X}=\tr^{(X)}\sqrt{OO^{*}},\qquad O\in\mathcal{A}_{X}$

We then denote by $\tr_{X}$ the partial trace
$\tr_{X}:\mathcal{A}\mapsto\mathcal{A}_{X_{c}}$, satisfying
$\tr^{(X)}\tr_{X^{c}}O=\tr O$ for any $O\in\mathcal{A}$. Since we will often
use trace norms of operators resulting from the partial trace we introduce a
short-hand notation:

$|O|_{X}:=||\tr_{X^{c}}O||_{1,X}.$

Nonnegative operators $\rho$ on $\mathcal{H}$ that have unit trace
$\tr\rho=1$, are called density matrices and we write $\langle
O\rangle_{\rho}=\tr(\rho O)$ for $O\in\mathcal{A}$. For $\Psi\in\mathcal{H}$,
we write $\rho_{\Psi}$ for the pure density matrix equal to the orthogonal
projector on $\mathbb{C}\Psi$.

###  2.2 Spaces and Hamiltonian

The Hamiltonian $H$ and the perturbation $J$ are written as

$H=\sum_{X\subset\Gamma}h_{X},\qquad J=\sum_{X\subset\Gamma}j_{X}\qquad
h_{X}=h^{*}_{X}\in\mathcal{A}_{X},\quad j_{X}=j^{*}_{X}\in\mathcal{A}_{X}$ (1)

The perturbation $J$ is spatially restricted to a region
$\Gamma_{j}\subset\Gamma$ in a mild sense

$j_{X}=0\quad\text{unless}\quad X\cap\Gamma_{j}\neq\emptyset$

Then, we assume, for the collections $h,j$,

$||h||_{m_{h}},||j||_{m_{j}}<\infty$

for some $m_{h},m_{j}\in\mathcal{M}$.

###  2.3 Assumptions

Our first assumption states that the Hamiltonian $H$ has a frustration free
ground state $\Omega$, i.e. $\Omega$ is an eigenvector of $H$ with eigenvalue
equal to $\min(\mathrm{spec}(H))$ (${\mathrm{spec}}(\cdot)$ is the spectrum).
Our conventions are such that this eigenvalue is $0$.

######  Assumption 1 (Frustration-free ground state).

All the terms $h_{X}$ are nonnegative: $h_{X}\geq 0$. There is a
$\Omega\in\mathcal{H},||\Omega||=1$ such that $\Omega\in\mathrm{Ker}(h_{X})$
for any $X$. We denote by $\mu=\rho_{\Omega}$ the corresponding density
matrix.

We note that the frustration free property depends on the way the Hamiltonian
is written as a sum of local terms, i.e. on the interaction. Next, we
introduce ‘open boundary condition’ (OBC) restrictions of $H$,

$H_{Z}=\mathop{\sum}\limits_{X\subset Z}h_{X}.$

Just as the frustration free property, the notion of OBC restriction depends
on the interaction. We let $P_{Z}$ be the orthogonal projector on
$\mathrm{Ker}(H_{Z})$. Since $h_{X}\geq 0$, it holds that

$\mathrm{Ker}(H_{Z})=\mathop{\cap}\limits_{X\subset Z}\mathrm{Ker}(h_{X})$

and hence also

$P_{Z}P_{Z^{\prime}}=P_{Z^{\prime}},\qquad Z\subset Z^{\prime}.$

The following assumption has come to be known as ‘Local Topological Quantum
Order’ (LTQO) but, in our case, it is better described as the property that
the density matrix of any ground state of the OBC Hamiltonian $H_{Z}$ looks
similar to the global ground state $\mu$ in the deep interior of $Z$. Recall
that $B_{r}(x)\subset\Gamma$ is the ball or radius $r$ centered at $x$.

######  Assumption 2 (OBC-regularity).

There is $m_{O}\in\mathcal{M}$ and $d_{O}<\infty$ such that, for any
$x\in\Gamma$ and $\Psi\in\mathrm{Ker}(H_{B_{r}(x)})$,

$|\rho_{\Psi}-\mu|_{B_{r-k}(x)}\leq r^{d_{O}}m_{O}(k),\qquad k<r$

Let us briefly comment on the precise form of the above bound. We have in
mind, roughly, that any $\Psi\in\mathrm{Ker}(H_{B_{r}(x)})$ differs from
$\Omega$ only through the presence of boundary modes. One realization of this
would be that any such $\Psi$ is of the form $e^{iF}\Omega$ with $F$ a sum of
terms supported near the boundary of $B_{r}(x)$. This would indeed lead to the
bound above with ${d_{O}}$ chosen such that $|\partial B_{r}(x)|\leq
Cr^{d_{O}}$.

The next assumption concerns the local gap of OBC Hamiltonians. Let

$\gamma(Z)=\min({\mathrm{spec}}(H_{Z})\setminus\\{0\\})$

be the spectral gap of $H_{Z}$.

######  Assumption 3 (Local Gap).

There are $C_{\gamma},d_{\gamma}<\infty$ such that, for any $x\in\Gamma$

$\frac{1}{\gamma(B_{r}(x))}\leq C_{\gamma}r^{d_{\gamma}}.$

This assumption 3 might be misleading. In most examples of models with gapped
frustration-free ground states, that we are aware of, OBC restrictions
$H_{Z}=\sum_{X\subset Z}h_{Z}$ have a gap that is actually bounded below in
$Z$, and the physical edge modes are found as ground states of $H_{Z}$, i.e.
the eigenvalue at $0$ will typically be degenerate. In that case, the genuine
restriction is not so much assumption 3 but rather assumption 2 which
describes the kernel $\mathrm{Ker}(H_{B_{r}(x)})$.

####  2.3.1 Invertibility

Let $\mathcal{H}^{\prime}$ be
$\mathcal{H}^{\prime}=\mathcal{H}^{\prime}_{\Gamma}=\otimes_{x\in\Gamma}\mathcal{H}^{\prime}_{x}$
with $\mathcal{H}^{\prime}_{x}$ finite-dimensional Hilbert spaces. We will now
consider the tensor product $\mathcal{H}\otimes\mathcal{H}^{\prime}$ which we
denote by

$\widetilde{\mathcal{H}}=\mathcal{H}\otimes^{\mathrm{aux}}\mathcal{H}^{\prime}.$

The superscript $\otimes^{\mathrm{aux}}$ reminds us of the fact that this is
not a tensor product between disjoint spatial regions, but between the
‘original’ Hilbert space and an ‘auxiliary’ Hilbert space. This is helpful
because we also view $\widetilde{\mathcal{H}}$ again as a tensor product over
sites

$\widetilde{\mathcal{H}}=\otimes_{x\in\Gamma}\widetilde{\mathcal{H}}_{x},\qquad\widetilde{\mathcal{H}}_{x}=\mathcal{H}_{x}\otimes^{\mathrm{aux}}\mathcal{H}^{\prime}_{x}$

Figure 3: The invertibility assumption uses an auxiliary Hilbert space
$\mathcal{H}^{\prime}$ that has the same spatial structure as the original
space $\mathcal{H}$, i.e. $\Gamma^{\prime}$ is a copy of $\Gamma$.

The local structure of the space $\widetilde{\mathcal{H}}$ is analogous to the
one of $\mathcal{H}$, except that now the dimension of the on-site spaces is
larger, see Figure 3. We consistently put a prime on algebras derived from
$\mathcal{H}^{\prime}$,
$\mathcal{A}^{\prime}=\mathcal{B}(\mathcal{H}^{\prime})$ and we put a tilde on
algebras derived from $\widetilde{\mathcal{H}}$, i.e.
$\widetilde{\mathcal{A}}=\mathcal{B}(\widetilde{\mathcal{H}})$ and
$\mathcal{A}_{X}^{\prime}\subset\mathcal{A}^{\prime},\widetilde{\mathcal{A}}_{X}\subset\widetilde{\mathcal{A}}$
are subalgebras of operators supported in $X$. We also copy the definition of
collections $q$ of local operators $(q_{X})_{X}$ and their norms
$||\cdot||_{m}$, simply replacing $\mathcal{A}_{X}$ by
$\widetilde{\mathcal{A}}_{X}$. The next assumption expresses that the state
$\Omega$ has no _long-range entanglement_. Loosely speaking, one could
describe this assumption as ‘upon adjoining an auxiliary state
$\Omega^{\prime}$, $\Omega\otimes\Omega^{\prime}$ is automorphically
equivalent to a product state’ [12, 20].

######  Assumption 4 (Invertibility).

There are collections $q(s)$ of local operators $(q_{X}(s))_{X\subset\Gamma}$,
indexed by $s\in[0,1]$, such that

  1. 1.

$q_{X}(s)=q^{*}_{X}(s)\in\widetilde{\mathcal{A}}_{X}$ and $s\mapsto q_{X}(s)$
is measurable, for any $X$.

  2. 2.

$\sup_{s\in[0,1]}||q(s)||_{m_{I}}\leq C_{I}<\infty$ for some
$m_{I}\in\mathcal{M}$.

  3. 3.

There is a product state
$\Pi=\otimes_{x\in\Gamma}\Pi_{x}\in\widetilde{\mathcal{H}}$ and a state
$\Omega^{\prime}\in\mathcal{H}^{\prime}$ such that

$\Omega\otimes^{\mathrm{aux}}\Omega^{\prime}=U(1)\Pi,\qquad
U(s)=\mathds{1}+i\int_{0}^{s}du\,H_{q}(u)U(u)$

where $H_{q}(s)=\sum_{X\subset\Gamma}q_{X}(s)$.

We say that $\Omega^{\prime}$ is an ‘inverse’ to $\Omega$.

The ‘invertibility’ assumption roughly corresponds to states that do not have
anyonic excitations. Examples of invertible states are: symmetry protected
topological states (SPT’s) [21, 22, 23], states characterized by an integer
quantum Hall effect [24], etc.. We refer to the extensive literature for a
more thorough discussion. The motivation for considering the type of spatial
decay expressed by the class $\mathcal{M}$ is because this corresponds to the
decay one can prove for the spectral flow connecting ground states of a
uniformly gapped Hamiltonian, by the technique of (quasi-)adiabatic
continuation, see [20, 12].

###  2.4 Result

We say that a unit vector $\Phi\in\mathcal{H}_{\Gamma}$ is a ground state of
the perturbed Hamiltonian $H+J$ if $\Phi$ is an eigenvector of $H+J$ with
eigenvalue equal to $\min{\mathrm{spec}(H+J)}$. Recall that $\mu$ is the
density matrix associated to $\Omega$, the distinghuished ground state of the
unperturbed Hamiltonian $H$, and $\rho_{\Phi}$ is the density matrix
associated to $\Phi$. By a ‘constant’, we mean a quantity that can depend only
on $m_{h},m_{j},m_{I}\in\mathcal{M}$, the numbers $C_{\gamma},C_{\Gamma}$ and
$d,d_{O},d_{\gamma}$. In particular, constants can not depend on the sizes
$|\Gamma|$ and $|\Gamma_{j}|$.

######  Theorem 1.

For any $w>0$, there is a constant $C(w)$ such that, for any ground state
$\Phi$ of $H+J$, and any $x\in\Gamma$, with $R:=\mathrm{dist}(x,\Gamma_{j})$,

$|\rho_{\Phi}-\mu|_{B_{(R^{p}/2)}(x)}\leq C(w)e^{-R^{p-w}},\qquad
p=\frac{1}{d+d_{\gamma}+2}$

To connect this theorem to the informal claim on stability in Section 1, note
that the left hand side is equal to
$\sup_{O\in\mathcal{A}_{X},||O||=1}|\langle\Phi,O\Phi\rangle-\langle\Omega,O\Omega\rangle|$
with $X=B_{(R^{p}/2)}(x)$.

##  3 Proof of Theorem 1

We define some additional notation to be used in this section. For regions
$Z\subset\Gamma$, we define the $r$ fattening

$(Z)_{r}:=\\{x\in\Gamma,\mathrm{dist}(x,Z)\leq r\\}$

(we will simply write $(Z)_{r}=Z_{r}$ when no confusion is possible) and the
boundary

$\partial Z=(Z)_{1}\cap(Z^{c})_{1}$

We use constants $C,c$ as introduced just before Theorem 1, i.e. depending on
a number of fixed parameters. Sometimes we write $C(a)$ to indicate that $C$
can additionally depend on a parameter $a$. We also use the generic notation
$m$ for a function in $\mathcal{M}$ that depends possibly on those same fixed
parameters. Just as the constants $C,c$, the precise function $m$ can change
from line to line. In the same vein, we also use $p$ to denote a polynomial
that depends only on the fixed parameters.

###  3.1 Stitching maps

Recall that $\mu$ is the global ground state density matrix of the unperturbed
Hamiltonian $H$. We say that a family of maps $\Sigma_{Z}$, indexed by
$Z\subset\Gamma$, are stitching maps if the they satisfy the following
properties.

######  Definition 1.

Stitching maps $\Sigma_{Z}$ with $Z\subset\Gamma$ are trace-preserving
completely positive maps $\mathcal{A}\to\mathcal{A}$ iff. there is an
$m\in\mathcal{M}$ such that, for any $Z,X\subset\Gamma$ and any density
matrices $\rho,\omega$ on $\mathcal{H}$,

  1. 1.

$|\Sigma_{Z}(\rho)-\tr_{Z^{c}}\mu\otimes\tr_{Z}\rho|_{X}\leq|X|m(\mathrm{dist}(\partial
Z,X))$

  2. 2.

$|\Sigma_{Z}(\rho)-\Sigma_{Z}(\omega)|_{X}\leq|\rho-\omega|_{(X)_{r}}+|X|m(r)$

  3. 3.

$\Sigma_{Z}(\mu)=\mu$

To state this in a rough way, stitching maps are such that
$\Sigma_{Z}(\mu)=\mu$ and further

$\Sigma_{Z}(\rho)=\begin{cases}\mu&\text{deep inside $Z$}\\\ \rho&\text{far
outside $Z$}\end{cases}$

The stitching maps will be used for regions $Z$ that are far away from the
perturbation region $\Gamma_{j}$.

The purpose of the invertibility assumption 4 is precisely to ensure the
existence of stitching maps, as we show now. Recall that
$\widetilde{\mathcal{H}}$ is the enlarged Hilbert space.

######  Proposition 1.

For any region $Z\subset\Gamma$, there are unitaries $V=V_{Z}$ acting on
$\widetilde{\mathcal{H}}$ such that the following family of CP maps
$\Sigma=\Sigma_{Z}$ are stitching maps in the sense of definition 1:

$\Sigma(\rho)=\tr_{\mathcal{H}^{\prime}}\left[V^{*}\left(\tr_{Z^{c}}(V\mu\otimes^{\mathrm{aux}}\mu^{\prime}V^{*})\otimes\tr_{Z}(V\rho\otimes^{\mathrm{aux}}\mu^{\prime}V^{*})\right)V\right]$
(2)

where $\mu=\rho_{\Omega}$ and $\mu^{\prime}=\rho_{\Omega^{\prime}}$ are
density matrices on $\mathcal{A}$ and $\mathcal{A}^{\prime}$ and
$\Omega^{\prime}\in\mathcal{H}^{\prime}$ is an ‘inverse state’ to $\Omega$,
see assumption 4.

We first construct the appropriate unitaries $V$ featuring in Proposition 1.
To that end, we introduce the truncated interactions $\hat{q}(s)$ by

$\hat{q}_{X}(s)=\begin{cases}q_{X}(s)&X\subset Z\quad\text{or}\quad X\subset
Z^{c}\\\ 0&\text{otherwise}\end{cases}$

with $q(s)$ as given in assumption 4. We let $\hat{U}(s)$ be the unitary
defined analogously to $U(s)$ in the invertibility assumption 4, but with
$H_{q}(s)$ replaced by $H_{\hat{q}}(s)=\sum_{X}\hat{q}_{X}(s)$. We then set

$V(s)=\hat{U}(s)U^{*}(s),\qquad V=V(1)$

and we observe that

$V\Omega\otimes^{\mathrm{aux}}\Omega^{\prime}=\Omega_{Z}\otimes\Omega_{Z^{c}}$
(3)

for some
$\Omega_{Z}\in\widetilde{\mathcal{H}}_{Z},\Omega_{Z^{c}}\in\widetilde{\mathcal{H}}_{Z^{c}}$.
This shows that $\Sigma_{Z}$ satisfies property 3) of definition 1. The full
proof of Proposition 1, i.e. the verifications of properties 1) and 2) of
definition 1, uses some standard terminology and locality estimates that are
not needed in the rest of our proof, hence we postpone them to the appendices,
but we do state the relevant locality property of the unitary operators $V$.
To that end, let us introduce the conditional expectation, for any
$Z\subset\Gamma$,

$\mathbb{E}_{Z^{c}}:\widetilde{\mathcal{A}}\mapsto\widetilde{\mathcal{A}}_{Z}:O\mapsto\tfrac{1}{N}\tr_{Z^{c}}(O),\qquad
N=\dim(\widetilde{\mathcal{H}}_{Z^{c}}).$

######  Lemma 1.

For any regions $X,Z$ and $O\in\widetilde{\mathcal{A}}_{X}$, with $V=V_{Z}$ as
defined above,

$||V^{*}OV-\mathbb{E}_{(X_{r})^{c}}(V^{*}OV)||\leq||O|||X|m(r)$ (4)

and

$||V^{*}OV-O||\leq||O|||X|m(\mathrm{dist}(X,\partial Z))$ (5)

The same estimates hold _•_ as well if we exchange $V$ and $V^{*}$.

The proof of Lemma 1 follows from well-known considerations based on Lieb-
Robinson bounds, and it is sketched in Appendix B. Finally, the proof of
properties 1) and 2) of definition 1 is an intuitive consequence of Lemma 1.
This proof is given in Appendix A.

####  3.1.1 Action of a stitching map $\Sigma_{Z}$ at the stitch $\partial Z$

We will now establish a property of stitching maps, which we will henceforth
denote by $\Sigma_{Z}$, that relies crucially on the OBC-regularity assumption
2. It makes explicit that stitching maps are ‘seamless’, i.e. they do not
introduce errors at the cut $\partial Z$. Recall that $P_{X}$ is the ground
state projector corresponding to the OBC restriction $H_{X}$ and let
$\bar{P}_{X}=\mathds{1}-P_{X}$.

######  Proposition 2.

Let $\sigma=\rho_{\Psi}$ be the pure density matrix associated to some
$\Psi\in\mathcal{H}$. Consider a set $Z$ and balls $B_{r}=B_{r}(x)$ for some
site $x$. Then (recall that $p(\cdot)$ is a polynomial),

$\langle\bar{P}_{B_{r-k}}\rangle_{\Sigma_{Z}(\sigma)}\leq
3\langle\bar{P}_{B_{r}}\rangle_{\sigma}+p(r)m(k).$

###### Proof.

Since $Z$ will be fixed, we drop it and write $\Sigma=\Sigma_{Z}$. Let us
denote $\epsilon:=\langle\bar{P}_{B_{r}}\rangle_{\sigma}$ and remark that
$||P_{B_{r}}\Psi-\Psi||\leq\sqrt{\epsilon}$. If $\epsilon=1$, then the
required bound is trivial, hence we assume $\epsilon<1$. We split

$\sigma=\sum_{i=1}^{4}\zeta_{i}=P_{B_{r}}\sigma
P_{B_{r}}+P_{B_{r}}\sigma\bar{P}_{B_{r}}+\bar{P}_{B_{r}}\sigma
P_{B_{r}}+\bar{P}_{B_{r}}\sigma\bar{P}_{B_{r}}.$

We will treat these terms separately, i,e. find bounds on

$\tr\bar{P}_{B_{r-k}}\Sigma(\zeta_{i}),\qquad i=1,2,3,4.$

The term $\zeta_{1}=P_{B_{r}}\sigma P_{B_{r}}$
Since $\epsilon<1$, $\tr\zeta_{1}>0$ and we can define the pure density matrix
$\zeta=\zeta_{1}/\tr\zeta_{1}$, satisfying
$\langle\bar{P}_{B}\rangle_{\zeta}=0$. By the properties of definition 1, we
get the equality and first inequality in

$|\Sigma(\zeta)-\mu|_{B_{r-k}}=|\Sigma(\zeta)-\Sigma(\mu)|_{B_{r-k}}\leq|\zeta-\mu|_{B_{r-k/2}}+p(r)m(k)\leq
p(r)m(k)$

whereas the last inequality follows from the OBC regularity assumption 2. By
the frustration-free property, $\tr\bar{P}_{B_{r-k}}\mu=0$, and thus the above
inequality yields $\tr\bar{P}_{B_{r-k}}\Sigma(\zeta)\leq p(r)m(k)$. Since
$\zeta_{1}\leq\zeta$, this implies

$\tr\bar{P}_{B_{r-k}}\Sigma(\zeta_{1})\leq p(r)m(k)$

The term $\zeta_{4}=\bar{P}_{B_{r}}\sigma\bar{P}_{B_{r}}$
Here we have $\tr\zeta_{4}=\tr\bar{P}_{B_{r}}\sigma=\epsilon$ and since
$\Sigma$ is trace-preserving,

$\tr\bar{P}_{B_{r-k}}\Sigma(\zeta_{4})\leq\epsilon.$

The term $\zeta_{2}=P_{B_{r}}\sigma\bar{P}_{B_{r}}$
Since $\sigma$ is pure, we can write

$\zeta_{2}=P_{B_{r}}\sigma\bar{P}_{B_{r}}=\langle\bar{P}_{B_{r}}\rangle_{\sigma}^{1/2}\,|a\rangle\langle
b|,\qquad||a||\leq||b||=1,\qquad P_{B_{r}}a=a$

and we note that $|a\rangle\langle a|=\zeta_{1}$. To estimate
$\tr\bar{P}_{B_{r-k}}\Sigma{(|a\rangle\langle b|)}$, we use Lemma 2 below to
get

$|\tr\bar{P}_{B_{r-k}}\Sigma(\zeta_{2})|\leq\sqrt{\langle\bar{P}_{B}\rangle_{\sigma}}\sqrt{\tr(\bar{P}_{B_{r-k}}\Sigma(\zeta_{1}))}\leq\sqrt{\epsilon}\sqrt{p(r)m(k)}\leq\epsilon+p(r)m(k)$

The second inequality follows from the bound for the term $\zeta_{1}$ above.

######  Lemma 2.

For an orthogonal projector $P$, a trace-conserving CP map $\Gamma$ and
$||a||,||b||\leq 1$, we have

$|\tr(P\Gamma(|a\rangle\langle b|))|\leq\sqrt{\tr(P\Gamma(|a\rangle\langle
a|))\tr(P\Gamma(|b\rangle\langle b|))}\leq\sqrt{\tr(P\Gamma(|a\rangle\langle
a|))}$

###### Proof.

Introducing the adjoint map $\Gamma^{*}$, the first inequality reads

$|\langle b,\Gamma^{*}(P)a\rangle|\leq\sqrt{\langle
a,\Gamma^{*}(P)a\rangle\langle b,\Gamma^{*}(P)b\rangle}.$

It follows from positivity of $\Gamma^{*}(P)$ and the Cauchy-Schwarz
inequality. ∎

The term $\zeta_{3}=\bar{P}_{B_{r}}\sigma P_{B_{r}}$
The bound and its proof are analogous to the case of $\zeta_{2}$. The claim of
the lemma follows by adding the bounds on
$\tr\bar{P}_{B_{r-k}}\Sigma(\zeta_{i})$ for $i=1,2,3,4$. ∎

###  3.2 A bound on the energy increase due to stitching

If we apply the stitching map $\Sigma_{Z}$ to a pure density matrix $\sigma$,
we expect its energy (associated to $H$) to decrease in the far interior of
$Z$ and to remain unchanged far outside $Z$. Around the stitch $\partial Z$,
the energy can of course increase. In the present section we prove that this
energy increase is upper bounded by the energy of $\sigma$ around the stitch.
We make this precise in Lemma 3. The proof is based largely on Proposition 2
that allows us to convert a bound on the energy to a bound on the local
deviation from $\mu$.

Figure 4: The classes of sets $X$ in the proof of Lemma 3.

######  Lemma 3.

Let $\sigma$ be a pure density matrix, i.e. $\sigma=\rho_{\Psi}$ for some
$\Psi\in\mathcal{H}$, and let $\Sigma=\Sigma_{Z}$ for some region
$Z\subset\Gamma$. Then

$\sum_{X\not\subset Z\setminus(\partial Z)_{r}}\left(\langle
h_{X}\rangle_{\Sigma(\sigma)}-\langle h_{X}\rangle_{\sigma}\right)\leq
Cr^{d+d_{\gamma}}\langle H_{(\partial Z)_{r}}\rangle_{\sigma}+m(r)|\partial
Z|$

###### Proof.

We split the class of $X$ contributing the left hand side222The condition
$X\not\subset Z\setminus(\partial Z)_{r}$ appearing in the statement of the
lemma, is actually irrelevant for its validity. in disjoint classes
$a,b,c,d,e$:

  * $a)$

$X\subset Z\setminus(\partial Z)_{r/2}$

  * $b)$

$X\subset Z^{c}\setminus(\partial Z)_{r/2}$

  * $c)$

$X\cap(\partial Z)_{r/2}\neq\emptyset$ and $\mathrm{diam}(X)<r/4$.

  * $d)$

$X\cap(\partial Z)_{r/2}\neq\emptyset$ and $\mathrm{diam}(X)\geq r/4$.

  * $e)$

$X\subset((\partial Z)_{r/2})^{c}$ and $X\cap Z\neq\emptyset$ and $X\cap
Z^{c}\neq\emptyset$ .

We will now estimate, with $x=a,b,c,d,e$,

$\sum_{X\in\text{class}\,x}\left(\langle h_{X}\rangle_{\Sigma(\sigma)}-\langle
h_{X}\rangle_{\sigma}\right)$ (6)

  1. class $a$

$\displaystyle\sum_{X\in\text{class}\,a}\left(\langle
h_{X}\rangle_{\Sigma(\sigma)}-\langle h_{X}\rangle_{\sigma}\right)$
$\displaystyle\leq\sum_{X\subset Z\setminus(\partial Z)_{r/2}}\langle
h_{X}\rangle_{\Sigma(\sigma)}$ $\displaystyle\leq\sum_{X\subset
Z\setminus(\partial Z)_{r/2}}\left(\langle
h_{X}\rangle_{\mu}+||h_{X}|||X|m(\mathrm{dist}(X,\partial Z)\right)$
$\displaystyle\leq\sum_{x:\mathrm{dist}(x,\partial Z)\geq
r/2}m(\mathrm{dist}(x,\partial Z)\sum_{X\ni x}||h_{X}|||X|\leq|\partial
Z|m(r)$

The first inequality follows from non-negativity of $h_{X}$. The second from
property 1) of definition 1 and the third from $\langle h_{X}\rangle_{\mu}=0$.
Then we use $\sup_{x}\sum_{X\ni x}||h_{X}|||X|\leq C$ and the rapid decay of
$m$ to absorb a polynomial factor in $r$.

  2. class $b$

$\displaystyle\sum_{X\in\text{class}\,b}\left(\langle
h_{X}\rangle_{\Sigma(\sigma)}-\langle h_{X}\rangle_{\sigma}\right)$
$\displaystyle\leq\sum_{X\subset Z^{c}\setminus(\partial
Z)_{r/2}}||h_{X}|||X|m(\mathrm{dist}(X,Z))$
$\displaystyle\leq\sum_{x:\mathrm{dist}(x,Z)\geq
r/2}m(\mathrm{dist}(x,Z))\sum_{X\ni x}||h_{X}|||X|\leq|\partial Z|m(r)$

The first inequality is property 1) of definition 1, the rest is analogous to
case $a$ above.

  3. class $c$

$\displaystyle\sum_{X\in\text{class}\,c}\left(\langle
h_{X}\rangle_{\Sigma(\sigma)}-\langle h_{X}\rangle_{\sigma}\right)$
$\displaystyle\leq\sum_{x\in(\partial Z)_{r/2}}\quad\sum_{X\ni
x,\mathrm{diam}(X)\leq r/4}\langle h_{X}\rangle_{\Sigma(\sigma)}$
$\displaystyle\leq\sum_{x\in(\partial Z)_{r/2}}\quad\sum_{X\ni
x,\mathrm{diam}(X)\leq
r/4}||h_{X}||\langle\bar{P}_{X}\rangle_{\Sigma(\sigma)}$
$\displaystyle\leq\sum_{x\in(\partial
Z)_{r/2}}C\langle\bar{P}_{B_{r/4}(x)}\rangle_{\Sigma(\sigma)}$
$\displaystyle\leq\sum_{x\in(\partial
Z)_{r/2}}C\langle\bar{P}_{B_{r/2}(x)}\rangle_{\sigma}+m(r)$
$\displaystyle\leq\sum_{x\in(\partial Z)_{r/2}}Cr^{d_{\gamma}}\langle
H_{B_{r/2}(x)}\rangle_{\sigma}+m(r)$ $\displaystyle\leq
Cr^{d+d_{\gamma}}\langle H_{(\partial Z)_{r}}\rangle_{\sigma}+|(\partial
Z)_{r/2}|m(r)$ $\displaystyle\leq Cr^{d+d_{\gamma}}\langle H_{(\partial
Z)_{r}}\rangle_{\sigma}+|\partial Z|m(r)$

The first and second inequality is by nonnegativity of $h_{X}$. The third
inequality uses $\bar{P}_{X}\leq\bar{P}_{X^{\prime}}$ for $X\subset
X^{\prime}$, i.e. the frustration freeness of assumption 1, and $\sum_{X\ni
x}||h_{X}||<C$. The fourth inequality uses Proposition 2. The fifth inequality
is by the local gap assumption 3. The sixth inequality follows because any
$X\subset(\partial Z)_{r}$ is included in at most a number $Cr^{d}$ of balls
with radius $r/2$. For the last inequality, we use $|(\partial Z)_{r/2}|\leq
p(r)|\partial Z|$ and we absorbed the polynomial in $m$.

  4. class $d$

$\displaystyle\sum_{X\in\text{class}\,d}\left(\langle
h_{X}\rangle_{\Sigma(\sigma)}-\langle h_{X}\rangle_{\sigma}\right)$
$\displaystyle\leq\sum_{x\in(\partial Z)_{r/2}}\sum_{X\ni
x:\mathrm{diam}(X)\geq r/4}2||h_{X}||$ $\displaystyle\leq\sum_{x\in(\partial
Z)_{r/2}}m(r)\leq|\partial Z|m(r)$

The first inequality follows from $\langle
h_{X}\rangle_{\Sigma(\sigma)},\langle h_{X}\rangle_{\sigma}\leq||h_{X}||$.
Then we use analogous reasoning as for class $a$.

  5. class $e$

$\displaystyle\sum_{X\in\text{class}\,e}\left(\langle
h_{X}\rangle_{\Sigma(\sigma)}-\langle
h_{X}\rangle_{\sigma}\right)\leq\quad\sum_{x:\mathrm{dist}(x,\partial Z)\geq
r/2}\quad\sum_{X\ni x:\mathrm{diam}(X)\geq\mathrm{dist}(x,\partial
Z)}2||h_{X}||\leq|\partial Z|m(r)$

The reasoning is analogous to that for class $d$ and class $a$.

∎

The previous Lemma 3 was only concerned with the unperturbed Hamiltonian $H$.
We now state an analogous estimate for the Hamiltonian $J$.

Figure 5: The classes of sets $X$ in the proof of Lemma 4. We only need to
consider $X$ such that $X\cap\Gamma_{j}\neq\emptyset$, by the definition of
$j$.

######  Lemma 4.

Let $\sigma$ be a density matrix and let $Z$ be a set such that
$\mathrm{dist}(Z,\Gamma_{j})\geq r$. Then

$|\langle J\rangle_{\Sigma_{Z}(\sigma)}-\langle J\rangle_{\sigma}|\leq
m(r)|\partial Z|$ (7)

###### Proof.

We estimate

$|\langle J\rangle_{\Sigma_{Z}(\sigma)}-\langle
J\rangle_{\sigma}|\leq\sum_{X}|\langle
j_{X}\rangle_{\Sigma_{Z}(\sigma)}-\langle j_{X}\rangle_{\sigma}|$

. We split the sets $X$ into two disjoint classes:

  * $a)$

$\mathrm{dist}(X,Z)\geq r/2$

  * $b)$

$\mathrm{dist}(X,Z)<r/2$

We estimate the contribution of each class and we get in each case the desired
bound.

  1. class $a$

$\displaystyle\sum_{X\in\text{class a}}|\langle
j_{X}\rangle_{\Sigma_{Z}(\sigma)}-\langle j_{X}\rangle_{\sigma}|$
$\displaystyle\leq\sum_{x:\mathrm{dist}(x,Z)>r/2}m(\mathrm{dist}(x,Z))\sum_{X\ni
x}||j_{X}||$ $\displaystyle\leq
C\sum_{x:\mathrm{dist}(x,Z)>r/2}m(\mathrm{dist}(x,Z))\leq m(r)|\partial Z|$

We use property 1 of definition 1 to get the first inequality. The last
inequality follows by the rapid decay of $m$ and the fact that $\Gamma$ has
finite dimension $d$.

  2. class $b$

$\displaystyle\sum_{X\in\text{class b}}|\langle
j_{X}\rangle_{\Sigma_{Z}(\sigma)}-\langle j_{X}\rangle_{\sigma}|$
$\displaystyle\leq\sum_{k\geq
r/2}\,\sum_{x:\mathrm{dist}(x,\Gamma_{j})=k}\,\sum_{X\ni
x:\mathrm{diam}(X)\geq k}2||j_{X}||$ $\displaystyle\leq\sum_{k\geq
r/2}\,\sum_{x:\mathrm{dist}(x,\Gamma_{j})=k}m(k)\leq\sum_{k\geq
r/2}Ck^{d}|\partial Z|m(k)\leq m(r)|\partial Z|$

The first inequality follows because $j_{X}=0$ unless
$X\cap\Gamma_{j}\neq\emptyset$. In the last inequality, we again absorbed a
polynomial factor in $m$.

∎

###  3.3 Isoperimetry

From now on, we assume that $\sigma=\rho_{\Phi}$ with $\Phi$ a ground state of
$H+J$. In particular, this means that the following inequality holds:

$\langle H+J\rangle_{\Sigma_{Z}(\sigma)}-\langle H+J\rangle_{\sigma}\geq 0.$
(8)

We will use Lemma’s 3 and 4 to relate the energy in a region $Z$ to the energy
around $\partial Z$, which explains the title of this subsection. We again
choose the region $Z$ to be far away from $\Gamma_{j}$:
$\mathrm{dist}(Z,\Gamma_{j})\geq r$ with $r\gg 1$. By Lemma 4, we can discard
$J$ in the above inequality at small cost, and we obtain

$\langle H\rangle_{\Sigma_{Z}(\sigma)}-\langle H\rangle_{\sigma}+m(r)|\partial
Z|\geq 0$ (9)

We split this inequality as

$\langle H_{Z\setminus(\partial Z)_{r}}\rangle_{\Sigma_{Z}(\sigma)}-\langle
H_{Z\setminus(\partial Z)_{r}}\rangle_{\sigma}+\sum_{X\not\subset
Z\setminus(\partial Z)_{r}}\left(\langle
h_{X}\rangle_{\Sigma_{Z}(\sigma)}-\langle
h_{X}\rangle_{\sigma}\right)+m(r)|\partial Z|\geq 0$ (10)

from which we get

######  Lemma 5.

$\langle H_{Z\setminus(\partial Z)_{r}}\rangle_{\sigma}\leq
Cr^{d+d_{\gamma}}\langle H_{(\partial Z)_{r}}\rangle_{\sigma}+m(r)|\partial
Z|$

###### Proof.

We start from inequality (10). By the same easy reasoning as used in the proof
of lemma 3 for class $a$, we bound the term $\langle H_{Z\setminus(\partial
Z)_{r}}\rangle_{\Sigma(\sigma)}$ by $m(r)|\partial Z|$. Then the upper bound
of Lemma 3 gives us immediately the desired claim. ∎

Let us now put Lemma 5 to use. We will consider a sequence of regions $Z_{i}$
chosen as concentric balls $Z_{i}=B_{R_{i}+r}=B_{R_{i}+r}(x)$, with $x$ fixed
and $R_{i}$ to be specified. We note that

$B_{R_{i}}=Z_{i}\setminus(\partial Z_{i})_{r-1}$

and we abbreviate

$E_{i}=\langle H_{B_{R_{i}}}\rangle_{\sigma},\qquad\delta_{i}=\langle
H_{(\partial B_{R_{i}+r})_{r-1}}\rangle_{\sigma}$

Then, the inequality in Lemma 5 reads

$E_{i}\leq Cr^{d+d_{\gamma}}\delta_{i}+m(r)|\partial B_{R_{i}+r}|$ (11)

Note also that $E_{i+1}\geq E_{i}+\delta_{i}$ provided that $R_{i+1}\geq
R_{i}+2r$. Our strategy will be to establish a lower bound on $E_{i}$,
depending on $E_{1}$, and then eventually use the a priori upper bound

$E_{i}=\langle H_{B_{R_{i}}}\rangle_{\sigma}\leq||H_{B_{R_{i}}}||\leq
CR_{i}^{d}$

to get an upper bound on $E_{1}$. Proceeding in this way, we obtain

######  Lemma 6.

Let $R=\mathrm{dist}(x,\Gamma_{j})$ with $x$ the center of the balls above.
Then, for any $w>0$

$\langle H_{B_{R^{p}}}\rangle_{\sigma}\leq C(w)e^{-R^{p-w}},\qquad
p=\frac{1}{d+d_{\gamma}+2}.$

###### Proof.

We choose $R_{i}=(2i-1)r$ for $i=1,\ldots,i_{*}$ with $i_{*}$ the largest
integer that is smaller than $R/(2r)$. Using inequality (11) and $E_{i+1}\geq
E_{i}+\delta_{i}$ we obtain

$E_{i+1}\geq aE_{i}-b_{i},\qquad a=(1+cr^{-d-d_{\gamma}}),\qquad
b_{i}=m(r)|\partial B_{R_{i}+r}|$ (12)

where we also updated the function $m$ compared to (11). We will now use a
Grönwall-type inequality. Multiplying the inequality (12) by $a^{-i}$ and
summing over $i$ yields

$\sum_{i=1}^{k}E_{i+1}a^{-i}\geq\sum_{i=1}^{k}a^{-i+1}E_{i}+\sum_{i=1}^{k}b_{i}a^{-i},$

which implies

$E_{i}a^{-i+1}\geq E_{1}-\sum_{j=1}^{i-1}b_{j}a^{-j}.$

and hence, choosing $i=i_{*}$,

$E_{1}\leq E_{i_{*}}a^{-{i_{*}}+1}+\sum_{j=1}^{{i_{*}}-1}b_{j}a^{-j}.$

To estimate the sum on the right hand side, we bound $|\partial
B_{R_{j}+r}|\leq C(3jr)^{d}$ and estimate

$\sum_{j=1}^{{i_{*}}-1}b_{j}a^{-j}\leq
m(r)r^{d}\sum_{j=1}^{\infty}j^{d}a^{-j}\leq m(r).$

Using in addition the a priori bound $E_{i_{*}}\leq\langle
H_{B_{R}}\rangle_{\sigma}<CR^{d}$, we get

$E_{1}\leq CR^{d}a^{1-\frac{R}{2r}}+m(r)\leq
CR^{d}e^{-cRr^{-(d+d_{\gamma}+1)}}+m(r).$

To optimize the inequality we let $r$ grow with $R$, namely $r=R^{p}$. For
$m\in\mathcal{M}$ and $w>0$, we can find $C(w)$ such that $m(r)\leq
C(w)e^{-r^{1-w/p}}$ and we get hence

$E_{1}\leq C(w)e^{-R^{p-w}}.$

which is the claim of the lemma, since $R_{1}=r=R^{p}$.

∎

###### Proof of Theorem 1.

We recall that $\sigma=\rho_{\Phi}$ with $\Phi$ a ground state of $H+J$. By
the local gap assumption 3, we have

$\langle\bar{P}_{B_{r}}\rangle_{\sigma}\leq Cr^{d_{\gamma}}\langle
H_{B_{r}}\rangle_{\sigma}$

and by Lemma 6, we then get, for any $w>0$,

$\langle\bar{P}_{B_{r}}\rangle_{\sigma}\leq\delta=C(w)r^{d_{\gamma}}e^{-R^{p-w}},\qquad
r=R^{p}$ (13)

From (13) we get, with $P=P_{B_{r}}$ and for $R$ large enough, $P\Phi\neq 0$,
and setting $\Theta=\frac{P\Phi}{||P\Phi||}$, we get
$|\langle\Theta,\Phi\rangle|\geq\sqrt{1-\delta}$. From this one bounds
($|\cdot|_{\Gamma}$ is the global trace norm)

$|\rho_{\Phi}-\rho_{\Theta}|_{\Gamma}\leq 2\sqrt{\delta}$

Since $\Theta\in\mathrm{Ker}(H_{B_{r}})$, the OBC regularity assumption 2
allows us to bound

$|\sigma-\mu|_{B_{r-k}}\leq|\sigma-\rho_{\Theta}|_{B_{r-k}}+|\rho_{\Theta}-\mu|_{B_{r-k}}\leq
2\sqrt{\delta}+p(r)m(k)$

with $p(\cdot)$ a polynomial. We choose $k=r/2$. Then we get, for any $w>0$

$|\sigma-\mu|_{B_{R^{p}/2}}\leq C(w)e^{-R^{p-w}}$

This yields the statement of Theorem 1. ∎

##  Appendix A Appendix: Locality properties of CP maps

We establish some vocabulary that is helpful for the proof of Proposition 1.

###  A.1 Completely positive maps

We consider completely positive (CP) maps acting from
$\mathcal{B}(\mathcal{G})$ to $\mathcal{B}(\mathcal{G}^{\prime})$, with
$\mathcal{G},\mathcal{G}^{\prime}$ finite-dimensional Hilbert spaces. We need
identity-preserving CP maps and trace-preserving CP maps, which are dual to
each other. Let
$\Upsilon:\mathcal{B}(\mathcal{G})\mapsto\mathcal{B}(\mathcal{G}^{\prime})$ be
an identity preserving CP map, then the adjoint trace-preserving CP map
$\Upsilon^{*}:\mathcal{B}(\mathcal{G}^{\prime})\mapsto\mathcal{B}(\mathcal{G})$
is defined by

$\tr^{(\mathcal{G}^{\prime})}(S\Upsilon(O))=\tr^{(\mathcal{G})}(\Upsilon^{*}(S)O),\qquad
O\in\mathcal{B}(\mathcal{G}),S\in\mathcal{B}(\mathcal{G}^{\prime}).$

From the Russo–Dye theorem, it follows that identity preserving CP maps are
contracting w.r.t. operator norm: $||\Upsilon(O)||\leq||O||$ and trace-
preserving CP maps are contracting w.r.t trace norm
$||\Upsilon^{*}(S)||_{1}\leq||S||_{1}$ with $||S||_{1}=\tr(\sqrt{SS^{*}})$,
cf. the notation in Section 2.1.3.

###  A.2 Conditional expectations

Let us consider a bipartite finite-dimensional Hilbert space
$\mathcal{G}=\mathcal{G}_{a}\otimes\mathcal{G}_{b}$. For any density matrix
$\sigma_{b}$ on $\mathcal{G}_{b}$, we can define the conditional expectation

$\mathbb{E}_{\sigma_{b}}:\mathcal{B}(\mathcal{G})\mapsto\mathcal{B}(\mathcal{G}):O\mapsto\tr_{\mathcal{G}_{b}}((\mathds{1}_{a}\otimes\sigma_{b})O)\otimes\mathds{1}_{b}.$

Note that $\mathbb{E}_{\sigma_{b}}$ is an identity-preserving CP map and that
its range is $\mathcal{B}(\mathcal{G}_{a})\otimes\mathds{1}_{b}$ which is
naturally identified with $\mathcal{B}(\mathcal{G}_{a})$. Its adjoint is

$\mathbb{E}^{*}_{\sigma_{b}}:\mathcal{B}(\mathcal{G})\mapsto\mathcal{B}(\mathcal{G}):S\mapsto\tr_{\mathcal{G}_{b}}(S)\otimes\sigma_{b}.$
(14)

Let $\iota_{b}$ be the density matrix
$\tfrac{1}{\dim(\mathcal{G}_{b})}\mathds{1}_{b}$ on $\mathcal{G}_{b}$, i.e.
$\iota_{b}$ corresponds to the tracial state on
$\mathcal{B}(\mathcal{G}_{b})$. The case where $\mathcal{G}_{b}$ is
$\mathcal{H}_{Z}$ or $\widetilde{\mathcal{H}}_{Z}$ for some $Z\subset\Gamma$,
and $\sigma_{b}=\iota_{b}$ is of special signficance to us and we denote the
corresponding conditional expectation simply by $\mathbb{E}_{Z}$. This is
consistent with the usage of the symbol $\mathbb{E}_{Z}$ in Lemma 1.

###  A.3 Almost locality preserving CP maps

We consider identity-preserving CP maps $\Upsilon$, acting between
(subalgebras of) $\widetilde{\mathcal{A}}$, i.e. there is a spatial structure.
We say that $\Upsilon$ preserves almost locality if there exists
$m\in\mathcal{M}$ such that,

$||\Upsilon(O)-\mathbb{E}_{(X_{r})^{c}}\circ\Upsilon(O)||\leq||O|||X|m(r),\qquad\forall
O\in\mathcal{A}_{X},\forall X\subset\Gamma,$ (15)

where $\mathbb{E}_{(X_{r})^{c}}$ is the conditional expectation using the
tracial state on $(X_{r})^{c}$, as discussed above. A useful property is that,
if $\Upsilon,\Upsilon^{\prime}$ preserve almost locality (say, with functions
$m_{1},m_{2}$) then so does $\Upsilon^{\prime}\circ\Upsilon$. Indeed, for
$O\in\mathcal{A}_{X}$,

$\displaystyle\Upsilon^{\prime}\circ\Upsilon(O)$
$\displaystyle=\Upsilon^{\prime}\left(\mathbb{E}_{(X_{r/2})^{c}}\circ\Upsilon(O)+E_{1}\right)$
(16)
$\displaystyle=\mathbb{E}_{(X_{r})^{c}}\circ\Upsilon^{\prime}\circ\mathbb{E}_{(X_{r/2})^{c}}\circ\Upsilon(O)+E_{2}+\Upsilon^{\prime}(E_{1})$
(17)
$\displaystyle=\mathbb{E}_{(X_{r})^{c}}\circ\Upsilon^{\prime}\circ\Upsilon(O)-\mathbb{E}_{(X_{r})^{c}}\circ\Upsilon^{\prime}(E_{1})+E_{2}+\Upsilon^{\prime}(E_{1})$
(18)

Using that $\Upsilon,\Upsilon^{\prime}$ are norm-contracting and almost
locality preserving, and that the image of $\mathbb{E}_{(X_{r/2})^{c}}$ is in
$\mathcal{A}_{X_{r/2}}$, the error terms $E_{1,2}$ are bounded as

$||E_{1}||\leq||O|||X|m_{1}(r/2)),\qquad||E_{2}||\leq||O|||X_{r/2}|m_{2}(r/2)).$

Since $|X_{r/2}|\leq C|X|r^{d}$ and also $\mathbb{E}_{(X_{r})^{c}}$ is norm-
contracting, we verify that the overall error term is of the form
$||O||m(r)|X|$, and hence $\Upsilon^{\prime}\circ\Upsilon$ indeed preserves
almost locality.

If an identity-preserving CP map $\Upsilon$ preserves almost locality, then
its adjoint $\Upsilon^{*}$, satisfies the following property, for any density
matrix $\rho$,

$|\Upsilon^{*}(\rho)-\Upsilon^{*}\circ\mathbb{E}^{*}_{(X_{r})^{c}}(\rho)|_{X}\leq|X|m(r)$
(19)

with
$\mathbb{E}^{*}_{(X_{r})^{c}}(\rho)=\tr_{(X_{r})^{c}}\rho\otimes\iota_{(X_{r})^{c}}$
This follows from (15) by duality, using (14).

###  A.4 Verification of properties 1) and 2) of definition 1

We study the the trace-preserving map $\Sigma=\Sigma_{Z}$, defined in
Proposition 1, via its adjoint $\Sigma^{*}$, an identity-preserving CP map. We
decompose $\Sigma^{*}=\Upsilon_{1}\circ\ldots\circ\Upsilon_{6}$ with
$\Upsilon_{i}$ the identity-preserving CP maps given below.

  * $\Upsilon_{6}$:

$\mathcal{A}\to\widetilde{\mathcal{A}}:O\mapsto
O\otimes^{\mathrm{aux}}\mathds{1}_{\mathcal{A}^{\prime}}$

  * $\Upsilon_{5}$:

$\widetilde{\mathcal{A}}\to\widetilde{\mathcal{A}}:O\mapsto VOV^{*}$

  * $\Upsilon_{4}$:

$\widetilde{\mathcal{A}}\to\widetilde{\mathcal{A}}_{Z^{c}}:O\mapsto\mathbb{E}_{\kappa}(O)$
with the density matrix
$\kappa=\tr_{Z^{c}}(V\mu\otimes^{\mathrm{aux}}\mu^{\prime}V^{*})$ on
$\mathcal{H}_{Z}$.

  * $\Upsilon_{3}$:

$\widetilde{\mathcal{A}}_{Z^{c}}\to\widetilde{\mathcal{A}}:O\mapsto\mathds{1}_{Z}\otimes
O$

  * $\Upsilon_{2}$:

$\widetilde{\mathcal{A}}\to\widetilde{\mathcal{A}}:O\mapsto V^{*}OV$

  * $\Upsilon_{1}$:

$\widetilde{\mathcal{A}}\to\mathcal{A}:O\mapsto\mathbb{E}_{\mu^{\prime}}(O)$
with the density matrix $\mu^{\prime}$ on $\mathcal{H}^{\prime}$.

Each of the maps $\Upsilon_{j}$ preserves almost locality, i.e. satisfies
(15). This is immediate for $\Upsilon_{i}$ with $i=1,3,4,6$, whereas for
$i=2,5$ it is a consequence of the bound (4) in Lemma 1. Hence, by duality, we
obtain that the locality property (19) is satisfied for $\Upsilon^{*}=\Sigma$,
the map from Proposition 1. We then write, for any density matrices
$\rho,\omega$,

$\displaystyle\Sigma(\rho)-\Sigma(\omega)=\left(\Sigma(\rho)-\Sigma\circ\mathbb{E}^{*}_{(X_{r})^{c}}(\rho)\right)-\left(\Sigma(\omega)-\Sigma\circ\mathbb{E}^{*}_{(X_{r})^{c}}(\omega)\right)+\Sigma\circ\mathbb{E}^{*}_{(X_{r})^{c}}(\rho-\omega)$
(20)

The first two terms on the right-hand side are bounded in $|\cdot|_{X}$-norm
by (19) with $\Upsilon^{*}=\Sigma$. The last term is bounded as

$|\Sigma\circ\mathbb{E}^{*}_{(X_{r})^{c}}(\rho-\omega)|_{X}\leq|\tr_{(X_{r})^{c}}(\rho-\omega)\otimes\iota_{(X_{r})^{c}}|_{X}\leq|\rho-\omega|_{X_{r}}$

where we used in particular contractivity of $\Sigma$. This yields property 2)
of definition 1. To check property 1) of definition 1, we compare the action
of $\Upsilon_{2},\Upsilon_{5}$ on $\mathcal{A}_{X}$ with $X$ far from
$\partial Z$ with the action of the identity map. By the bound (5) in Lemma 1,
we have,

$||\Upsilon_{j}(O)-O||\leq||O||m(\mathrm{dist}(X,\partial Z))|X|,\qquad
O\in\mathcal{A}_{X},\qquad j=2,5.$

The verification of property 1) now follows by similar reasoning as for
property 2).

##  Appendix B Appendix: Locality estimates

We review the standard propagation bounds that are necessary for the proof of
Lemma 1. These bounds go back to [25] but we use the recent formulation in
[26]. There are a few differences in our treatment compared to [26]. Most
importantly, we have an underlying finite graph $\Gamma$ equipped with the
graph distance, and we always assume that this graph has a finite spatial
dimension $d$ as defined in Section 2.1.1.

###  B.1 Norms on interactions

In the main text, we defined a family of norms $||\cdot||_{m}$ with
$m\in\mathcal{M}$. In [26], as in many preceding works, another family of
norms is used, which we introduce now. Namely, one defines so-called
$F$-functions $F$ as functions satisfying

  1. 1.

$F:\mathbb{N}\to\mathbb{R}^{+}$ is non-increasing.

  2. 2.

There is $C_{F}<\infty$ such that, for all $x,y\in\Gamma$,

$\sum_{z\in\Gamma}F(\mathrm{dist}(x,z))F(\mathrm{dist}(z,y))\leq
C_{F}F(\mathrm{dist}(x,y)).$

  3. 3.

There is $C^{\prime}_{F}$ such that, for each ${x\in\Gamma}$,
$\sum_{y\in\Gamma}F(\mathrm{dist}(x,y))\leq C_{F}^{\prime}$.

Strictly speaking, the latter two properties are of course empty for a finite
graph $\Gamma$, but we have in mind that these properties holds for constants
$C_{F},C_{F}^{\prime}$ that can be chosen uniform in $|\Gamma|$. This is
consistent with the conventions used in the main text, as explained in Section
2.4.

Then we can define a family of corresponding norms on interactions.

$|||z|||_{F}=\sup_{x,y\in\Gamma}\sum_{S\supset\\{x,y\\}}\frac{||z_{S}||}{F(\mathrm{dist}(x,y))}$

(Note that we use the symbol $|||\cdot|||_{F}$ to distinghuish these norms
from the norms $||\cdot||_{m}$ in the main text). If we work with functions
decaying faster than polynomials, one can switch back and forth between the
norms $|||\cdot|||_{F}$ and $||\cdot||_{m}$, as we state in the next lemma. In
stating the comparison, we write $F\in\mathcal{M}$ meaning that
$F\big{|}_{\mathbb{N}^{+}}\in\mathcal{M}$ (The functions $m$ in the main text
are defined on $\mathbb{N}^{+}=\\{1,2,\ldots\\}$ whereas the functions $F$ are
defined on $\mathbb{N}=\\{0,1,2,\ldots\\}$.)

######  Lemma 7.

For any $m\in\mathcal{M}$, there is an $F$-function $F$ in $\mathcal{M}$ such
that

$|||z|||_{F}\leq||z||_{m}$

For any $F$-function $F\in\mathcal{M}$, there is a $m\in\mathcal{M}$ such that

$||z||_{m}\leq|||z|||_{F}$

To prove this, we need the following observation

######  Lemma 8.

Let $f:\mathbb{N}^{+}\to\mathbb{R}^{+}$ be a function such that
$\lim_{r\to\infty}e^{r^{\alpha}}f(r)=0$ for any $0<\alpha<1$, then there is an
$F$-function $F\in\mathcal{M}$ such that $f(r)\leq F(r)$ for any
$r\in\mathbb{N}^{+}$.

###### Proof.

We say a function $f:\mathbb{N}^{+}\to\mathbb{R}^{+}$ is logarithmically
superadditive if

$f(r_{1})f(r_{2})\leq f(r_{1}+r_{2})\qquad r_{1},r_{2}\in\mathbb{N}^{+}.$ (21)

Following [27], we define the following transformation of functions
$f:\mathbb{N}^{+}\to\mathbb{R}^{+}$:

$\mathrm{S}(f)(r)=\sup_{\ell\in\mathbb{N}^{+}}\,\mathop{\sup}\limits_{(r_{1},\ldots,r_{\ell})\in(\mathbb{N}^{+})^{\ell}:\sum_{i}r_{i}=r}\,\prod_{i}f(r_{i})$

This transformation has the following properties

  1. 1.

$\mathrm{S}(f)$ is logarithmically superadditive.

  2. 2.

If $f_{1}\leq f_{2}$, then $\mathrm{S}(f_{1})\leq\mathrm{S}(f_{2})$. In
particular, $f\leq\mathrm{S}(f)$.

  3. 3.

If $f$ is logarithmically superadditive, then $\mathrm{S}(f)=f$.

Let now $f$ satisfy the assumption of the lemma and put
$f^{\prime}(r)=c_{0}(1+r^{d+2})f(r)$ with $c_{0}$ such that $f^{\prime}<1$.
Then, we put

$F(r)=\frac{1}{c_{0}(1+r^{d+2})}\mathrm{S}(f^{\prime}(r)),\qquad
r\in\mathbb{N}^{+},$

and $F(0)=F(1)$. Then $F$ is an $F$-function (such properties are extensively
discussed in [26]) and, by property 2) above, $F(r)\geq f(r)$. It remains to
show that $F$ has sufficiently rapid decay. Since $f^{\prime}<1$ and
$f^{\prime}e^{r^{\alpha}}\to 0$, we can find $c_{\alpha}>0$ such that
$f^{\prime}(r)\leq f_{\alpha}(r)=e^{-(c_{\alpha}r)^{\alpha}}$. Since
$f_{\alpha}$, is logarithmically superadditive, we have

$\mathrm{S}(f^{\prime})\leq\mathrm{S}(f_{\alpha})=f_{\alpha}.$

Therefore, $\mathrm{S}(f^{\prime})$ and hence $F(r)$ have indeed sufficiently
rapid decay and we conclude that $F\in\mathcal{M}$. ∎

###### Proof of Lemma 7.

To get the first claim we observe

$\displaystyle|||z|||_{F}$
$\displaystyle=\sup_{x,y}\frac{1}{F(\mathrm{dist}(x,y))}\sum_{S:S\supset\\{x,y\\}}||z_{S}||$
(22)
$\displaystyle\leq\sup_{x,y}\sum_{S:S\supset\\{x,y\\}}\frac{||z_{S}||}{F(\mathrm{diam}(S))}$
(23) $\displaystyle\leq\sup_{x}\sum_{S:S\ni
x}\frac{||z_{S}||}{F(\mathrm{diam}(S))}$ (24)

In order to have the last expression dominated by $||z||_{m}$, we need to
choose the $F$-function $F$ such that $F(r)\geq m(r+1)$. The possibility of
doing this was argued in Lemma 8. For the second claim, we write

$\displaystyle||z||_{m}$ $\displaystyle=\sup_{x}\sum_{S:S\ni
x}\frac{1}{m(\mathrm{diam}(S)+1)}||z_{S}||$
$\displaystyle=\sup_{x}\sum_{k=0}\,\sum_{S:S\ni
x,\mathrm{diam}(S)=k}\,\frac{1}{m(k+1)}||z_{S}||$
$\displaystyle\leq\sup_{x}\sum_{k=0}\,\sum_{y,z:\mathrm{dist}(y,x)\leq
k,\mathrm{dist}(y,z)=k}\,\frac{1}{m(k+1)}\sum_{S:S\ni\\{y,z\\}}||z_{S}||$
$\displaystyle\leq\sum_{k=0}\frac{(1+C_{\Gamma}k^{d})^{2}}{m(k+1)}\,\sup_{y,z:\mathrm{dist}(y,z)=k}\,\sum_{S:S\ni\\{y,z\\}}||z_{S}||$
$\displaystyle=\sum_{k=0}\frac{1}{1+k^{2}}\frac{1}{f(k+1)}\,\sup_{y,z:\mathrm{dist}(y,z)=k}\,\sum_{S:S\ni\\{y,z\\}}||z_{S}||,\qquad
f(k+1)=\frac{m(k+1)}{(1+k^{2})(1+C_{\Gamma}k^{d})^{2}}$ $\displaystyle\leq
C_{0}\sup_{y,z}\frac{1}{f(\mathrm{dist}(y,z)+1)}\sum_{S:S\ni\\{y,z\\}}||z_{S}||,\qquad
C_{0}=\sum_{k=0}\frac{1}{1+k^{2}}$

In order to have the last expression dominated by $|||z|||_{F}$, we need to
choose $m\in\mathcal{M}$ such that, for all $r\in\mathbb{N}^{+}$, $f(r)\geq
C_{0}F(r-1)$, and hence,

$m(r)\geq g(r)=C_{0}(1+(r-1)^{2})(1+C_{\Gamma}(r-1)^{d})^{2}F(r-1).$

The function $g$ is not necessarily non-increasing, which is remedied by
setting

$m(r)=\sup_{r^{\prime}\geq r}g(r^{\prime}).$

We can now check that $m\in\mathcal{M}$ as it inherits the rapid decay from
$F$. ∎

We call $\mathfrak{B}_{F}$ the set of interactions with finite
$|||\cdot|||_{F}$-norm and $\mathfrak{B}_{F}(I)$, with $I\subset\mathbb{R}$ a
(possibly infinite) interval, the set of time-dependent interactions $I\ni
s\mapsto z(s)$, with finite norm

$|||z|||_{F}=\sup_{s\in I}|||z(s)|||_{F}$

and such that $s\mapsto z_{S}(s)$ is measurable for all $S\subset\Gamma$.
Pointwise addition of interactions
$(z_{1}+z_{2})_{S}(s)=(z_{1})_{S}(s)+(z_{2})_{S}(s)$ makes $\mathfrak{B}_{F}$
and $\mathfrak{B}_{F}(I)$ into Banach spaces w.r.t. the norm
$|||\cdot|||_{F}$.

###  B.2 Locally generated automorphisms

Let $z\in\mathfrak{B}_{F}(I)$ and let $s,t\in I$ with $s\leq t$. We let
$\alpha_{z}(t,s)[\cdot]$ be the dynamics (a family of automorphisms
$\mathcal{A}\to\mathcal{A}$ or
$\widetilde{\mathcal{A}}\to\widetilde{\mathcal{A}}$ ) generated by the family
$z(\cdot)$ acting from time $s$ to time $t$, in the sense that

$\alpha_{z}(t,s)[A]=A+i\int_{s}^{t}du\,\alpha_{z}(u,s)([H_{z}(u),A]),\qquad
H_{z}(u)=\sum_{S\subset\Gamma}z_{S}(u).$

Since $\Gamma$ is finite, the existence and uniqueness of this dynamics
follows from elementary facts on matrix-valued ODE’s. For $s=0$, one often
abbreviates

$\alpha_{z}(t)=\alpha_{z}(t,0)$

as we did in the main text. We first state a version of the Lieb-Robinson
bound, using the language introduced in Section 2.1.

######  Theorem 2 (Lieb-Robinson bound).

Let $z\in\mathfrak{B}_{F}(I)$, and let $s,t\in I$ with $s\leq t$. Let
$A\in\mathcal{A}_{X_{A}},B\in\mathcal{A}_{X_{B}}$ with $X_{A}\cap
X_{B}=\emptyset$. Then

$||[\alpha_{z}(t,s)[A],B]||\leq\frac{||A||||B||}{C_{F}}e^{C_{F}|t-s||||z|||_{F}}\left(\sum_{x\in
X_{A},y\in X_{B}}F(\mathrm{dist}(x,y))\right)$

This is a slightly weakened form of Theorem 3.1 in [26], except for two
details. Our graph $\Gamma$ is finite and we allow our time-dependent
interaction $z$ to be composed of measurable functions $t\mapsto z_{S}(t)$ for
any $S$. Inspection of the proof in [26] confirms that this is all what is
needed.

The Lieb-Robinson bound does not directly address the question how to write
the evolved observable $\alpha_{z}(t,s)[A]$ as a sum of local terms. A natural
way to do this is to use the conditional expectation $\mathbb{E}_{X}$ as
defined in Section 3.1. If $A\in\mathcal{A}_{X}$, then we can write

$\alpha_{z}(t,s)[A]=\sum_{k=0}^{\infty}\Delta_{X_{k}}(\alpha_{z}(t,s)[A])$
(25)

where the maps $\Delta_{X_{k}}$ are defined as

$\Delta_{X_{k}}=\begin{cases}\mathbb{E}_{X^{c}}&k=0\\\
\mathbb{E}_{(X_{k})^{c}}-\mathbb{E}_{(X_{k-1})^{c}}&k>0\end{cases}$

and we recall that $X_{k}=\\{\mathrm{dist}(X,\cdot)\leq k\\}$. We see that the
terms corresponding to $k$ in the sum in (25) are supported on the fattened
sets $X_{k}$. The Lieb-Robinson bound can then be used to provide bounds on
the right-hand side of (25) via the following standard lemma, whose proof we
omit.

######  Lemma 9.

For any $A\in\mathcal{A}$,

$||A-\mathbb{E}_{Z^{c}}[A]||\leq\sup_{O\in\mathcal{A}_{Z^{c}},||O||=1}||[O,A]||.$

####  B.2.1 Evolution of interactions

Let $g$ be an interaction and recall that we associate to it a Hamiltonian
$H_{g}$

$H_{g}=\sum_{S\subset\Gamma}g_{S}$

Let now $z=(z(s))_{s\in I}$ be as above, i.e. a time-dependent interaction,
determining a family of autormorphisms $\alpha_{z}(t,s)$. We can use these
autormorphisms to obtain a new Hamiltonian by transforming $H_{g}$ into
$\alpha_{z}(t,s)[H_{g}]$. It is very intuitive that this Hamiltonian can again
be expressed as a sum of local terms with good decay properties in the size of
the support, i.e. we can express the Hamiltonian as the Hamiltonian
corresponding to an new interaction that we then view as a time-evolved
interaction $\alpha_{z}(t,s)[g]$. A priori, there are multiple ways to define
$\alpha_{z}(t,s)[g]$ since there are multiple ways to represent an operator as
a sum of local terms. For convenience, we follow [26] and set

$(\alpha_{z}(t,s)[g])_{S}=\sum_{X\subset\Gamma,k\in\mathbb{N}:X_{k}=S}\Delta_{X_{k}}(\alpha_{z}(t,s)[g_{X}]),$

using the operators $\Delta_{X_{k}}$ introduced above. This definition has the
following convenient properties

  1. 1.

$H_{\alpha_{z}(t,s)[g]}=\alpha_{z}(t,s)[H_{g}]$.

  2. 2.

If $g$ is anchored in a set $\Gamma_{g}\subset\Gamma$, then
$\alpha_{z}(t,s)[g]$ is also anchored in $\Gamma_{g}$.

  3. 3.

If the $F$-functions $F_{g},F_{z}$ are in $\mathcal{M}$, then there is an
$F$-function $F^{\prime}\in\mathcal{M}$ depending only on $F_{g},F_{z}$, such
that

$|||\alpha_{z}(t,s)[g]|||_{F^{\prime}}\leq
e^{C_{F_{z}}|t-s||||z|||_{F_{z}}}|||g|||_{F_{g}}.$ (26)

The first two properties follow immediately from the definitions. The third
property is derived starting from the Lieb-Robinsin bound. For details we
refer to [26], where one can find explicit choices for $F^{\prime}$.

This enables us to define many new time-dependent interactions in Lemma 10. To
set the stage, we first define the ”commutator” of two interactions
$z_{1},z_{2}$ as

$(i[z_{1},z_{2}])_{S}=\sum_{S_{1},S_{2}:S_{1}\cup S_{2}=S,S_{1}\cap
S_{2}\neq\emptyset}i[z_{S_{1}},z_{S_{2}}]$

We note that if either one of $z_{1},z_{2}$ is anchored in a set $\Gamma_{z}$,
then the commutator is also anchored in $\Gamma_{z}$. Then we can establish

######  Lemma 10.

Let $z_{0},z_{1},z_{2}$ be time-dependent interactions in
$\mathfrak{B}_{F_{0,1,2}}(I)$, respectively, with $F_{0,1,2}\in\mathcal{M}$.
Then the following interactions are in $\mathfrak{B}_{F}(I)$, with
$F\in\mathcal{M}$ and $F$ depending only on $F_{0}$ (for item 1)) and on
$F_{1},F_{2}$ (for items 2,3),

  1. 1.

$\bar{z}_{0}(s)=-\alpha_{z_{0}}(s)[z_{0}(s)]$

  2. 2.

$i[z_{1},z_{2}](s)$

  3. 3.

$w(s)={z_{2}}(s)-{z_{1}}(s)+i(\alpha_{z_{2}}(s))^{-1}\left[\int_{0}^{s}\alpha_{{z_{2}}}(u)\Big{\\{}\big{[}{z_{2}}(u)-z_{1}(u),\alpha_{z_{1}}(s,u)[z_{1}(s)]\big{]}\Big{\\}}du\right]$

The time-dependent interaction $\bar{z}_{0}$ in item 1) satisfies

$\alpha_{\bar{z}_{0}}(t,s)=(\alpha_{z_{0}}(t,s))^{-1}$ (27)

The time-dependent interaction $w$ in item 3) satisfies

$\alpha_{w}(t,s)=(\alpha_{z_{1}}(t,s))^{-1}\circ\alpha_{z_{2}}(t,s)$ (28)

###### Sketch of proof.

First, one verifies that the newly defined time-dependent interactions satisfy
the measurability condition, using that $z_{0,1,2}$ do so. Then, the bound on
$\bar{z}_{0}$ follows directly from (26). The identities (27) and (28) follow
by direct computation. The bound on $w$ follows by combining all the previous
items, i.e. the fact that commutators, time-evolution and inverse time-
evolution map $\mathfrak{B}_{F}(I)$ into $\mathfrak{B}_{F^{\prime}}(I)$ for
certain $F^{\prime}$. ∎

On our way to prove Lemma 1, we first state the following observation
concerning the unitary family $V_{Z}(s)=V(s)$ introduced in the proof of
Proposition 1 in the main text.

######  Lemma 11.

There is an $F$-function $F\in\mathcal{M}$ and a time-dependent interaction
$l\in\mathfrak{B}_{F}([0,1])$ such that

$V(s)OV^{*}(s)=\alpha_{l}(s)[O],\qquad O\in\widetilde{\mathcal{A}}$ (29)

Moreover, $l$ is anchored in $\partial Z$.

###### Proof.

We recall the time-dependent interactions $q,\hat{q}$. These were defined as
having a finite $||\cdot||_{m}$ norm, with $m\in\mathcal{M}$. By Lemma 7, we
find an $F$-function $F_{0}$ such that $q,\hat{q}$ have a finite
$|||\cdot|||_{F_{0}}$-norm. From the definition of $q,\hat{q}$, we deduce that

$V^{*}(s)OV(s)=(\alpha_{{q}}(s))^{-1}\circ\alpha_{\hat{q}}(s)[O]$

We set

$\tilde{q}(s)=q(s)-\hat{q}(s)$

and we note that $\tilde{q}(s)$ is anchored in $\partial Z$, for any $s$. Now
we invoke item 3) of Lemma 10, with $z_{1}=q,z_{2}=\hat{q}$, to conclude that
there is $l\in\mathfrak{B}([0,1])$, given by

$l(s)=-\tilde{q}(s)-i(\alpha_{\hat{q}}(s))^{-1}\left[\int_{0}^{s}\alpha_{\hat{q}}(u)\Big{\\{}\big{[}\tilde{q}(u),\alpha_{q}(s,u)[q(s)]\big{]}\Big{\\}}du\right],$

such that $V(s)OV^{*}(s)=\alpha_{l}(s)[O]$. Since commutators and time-
evolution preserve the property of being anchored in a set, we conclude that
$l$, just as $\tilde{q}$, is anchored in $\partial Z$. ∎

###  B.3 Proof of Lemma 1

For convenience we restate Lemma 1:

######  Lemma 12 (repetition of Lemma 1).

For any regions $X,Z$ and $O\in\widetilde{\mathcal{A}}_{X}$, with $V=V_{Z}$ as
in Lemma 11 above,

$||V^{*}OV-\mathbb{E}_{(X_{r})^{c}}(V^{*}OV)||\leq||O|||X|m(r)$ (30)

and

$||V^{*}OV-O||\leq||O|||X|m(\mathrm{dist}(X,\partial Z))$ (31)

The same estimates hold as well if we exchange $V$ and $V^{*}$.

To get the bound (30), we invoke Lemma 11 and Lemma 9 to relate conditional
expectations to commutators. Then, we use the Lieb-Robinson bound (Theorem 2)
for the dynamics $\alpha_{l}$:

$\displaystyle||V^{*}OV-\mathbb{E}_{(X_{r})^{c}}(V^{*}OV)||$
$\displaystyle\leq\sup_{A\in\widetilde{\mathcal{A}}_{(X_{r})^{c}},||A||\leq
1}||[A,\alpha_{l}(1)[O]||$
$\displaystyle\leq||O||e^{C_{F_{l}}|||z_{l}|||_{F_{l}}}\left(\sum_{x\in
X,y\in(X_{r})^{c}}F_{l}(\mathrm{dist}(x,y))\right)$

The last expression between brackets is estimated as $|X|m(r)$, since
$F_{l}\in\mathcal{M}$ and the spatial dimension $d$ is finite.

Let us now turn to (31). By the Heisenberg equation,

$V^{*}OV-O=i\int_{0}^{1}\alpha_{l}(u)[H_{l}(u),O]du.$

Therefore, we can bound

$\displaystyle||V^{*}OV-O||$
$\displaystyle\leq\int_{0}^{1}du\sum_{S}||\alpha_{l}(u)[l_{S}(u),O]||$
$\displaystyle=\int_{0}^{1}du\sum_{S}||[l_{S}(u),O]||$ $\displaystyle\leq
2||O||\sup_{u\in[0,1]}\sum_{S:S\cap\partial Z\neq\emptyset,S\cap
X\neq\emptyset}||l_{S}(u)||$ $\displaystyle\leq
2||O||\sup_{u\in[0,1]}\sum_{x\in X,y\in\partial
Z}\sum_{S:S\supset\\{x,y\\}}||l_{S}(u)||$ $\displaystyle\leq
2||O||\sup_{u\in[0,1]}|||l(u)|||_{F_{l}}\sum_{x\in X,y\in\partial
Z}F_{l}(\mathrm{dist}(x,y))$ $\displaystyle=2||O|||||l|||_{F_{l}}\sum_{x\in
X,y\in\partial Z}F_{l}(\mathrm{dist}(x,y))$

The third line follows because $O$ is supported in $X$ and $l$ is anchored in
$\partial Z$. The expression in the last line is dominated by
$||O|||X|m(\mathrm{dist}(X,\partial Z))$ upon using that
$F_{l}\in\mathcal{M}$, that the spatial dimension $d$ is finite, and bounding
$|||l|||_{F_{l}}$ by a constant.

To get the bounds (30) and (31) for $VOV^{*}$ instead of $V^{*}OV$, we note
that

$VOV^{*}=(\alpha_{l}(1))^{-1}[O]=\alpha_{\bar{l}}(1)[O]$

with $\bar{l}$ related to $l$ as in Lemma 10. By Lemma 10,
$\bar{l}\in\mathfrak{B}_{F_{\bar{l}}}([0,1])$ for some $F$-function
$F_{\bar{l}}\in\mathcal{M}$. We can therefore copy the arguments given above
for $V^{*}OV=\alpha_{l}(1)[O]$, replacing $l$ by $\bar{l}$.

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