††thanks: These two authors contributed equally to this work.††thanks: These
two authors contributed equally to this work.

# Entanglement area law for 1D gauge theories and bosonic systems

Nilin Abrahamsen  Simons Institute and Department of Mathematics, University
of California, Berkeley, CA 94720, USA  Yu Tong  Department of Mathematics,
University of California, Berkeley, CA 94720, USA  Ning Bao  Computational
Science Initiative, Brookhaven National Laboratory, Upton, NY 11973 USA  Yuan
Su  Google Quantum AI, Venice, CA 90291, USA  Nathan Wiebe  Department of
Computer Science, University of Toronto, Toronto, ON M5S 3E1, Canada
Pacific Northwest National Laboratory, Richland, WA 99354, USA

###### Abstract

We prove an entanglement area law for a class of 1D quantum systems involving
infinite-dimensional local Hilbert spaces. This class of quantum systems
includes bosonic models and lattice gauge theories in one spatial dimension.
Our proof relies on new results concerning the robustness of the ground state
and spectral gap to the truncation of Hilbert space, applied within the
approximate ground state projector (AGSP) framework. Our result provides
theoretical justification for using tensor networks to study the ground state
properties of quantum systems with infinite local degrees of freedom.

####  Introduction.—

It has long been conjectured that for a wide range of quantum systems
described by gapped local Hamiltonians, the entanglement entropy with respect
to any bipartition of the system scales as the boundary area. Such
entanglement entropy scaling is known as the _entanglement area law_. An
entanglement area law for the ground state of 1D quantum spin systems was
first proved in the seminal paper by Hastings [20], and the scaling with
respect to the spectral gap was improved by later work [5, 6]. Entanglement
area laws have also been proved for degenerate ground states [13, 7] and low-
lying eigenstates [7]. Limited results are also available for higher-
dimensional quantum systems, especially for the case where the Hamiltonian is
frustration-free [2, 4]. For 1D systems, whether a quantum state satisfies an
entanglement area law is an important criterion for determining whether it can
be approximated by a matrix-product state [18, 38], which is a key component
in the density-matrix renormalization group (DMRG) algorithm [40, 19]. Using
theoretical tools constructed for proving the 1D area law, polynomial-time
algorithms for computing the ground state of 1D gapped local Hamiltonians were
given [27, 7].

The aforementioned results are all proved in the setting of quantum spin
systems, i.e., each lattice site is associated with a finite-dimensional local
Hilbert space. However, there are many quantum systems of practical interest
that involve _infinite-dimensional_ local Hilbert spaces. Examples of such
include bosonic systems and gauge theories. When a quantum system involves
bosons, each bosonic mode corresponds to an infinite-dimensional local Hilbert
space representing the occupation number of the mode. A similar situation
arises when we consider lattice gauge theories (LGTs), with the Hamiltonians
constructed according to [25]. Given a fixed lattice discretization of a gauge
theory, each gauge link (an edge of the lattice) has a local Hilbert space
that is spanned by all the elements of the symmetry group, which is infinite-
dimensional when the symmetry group contains infinitely many elements. The
gauge theories that are of the greatest interests, i.e., the U(1), SU(2), and
SU(3) theories, all fall into this category.

Tensor network methods have been extensively applied to studying LGTs to
obtain interesting numerical results [31, 12, 11, 9, 10, 33, 30, 34]. The
entanglement area law is a prerequisite for the ground state to be efficiently
approximable by a tensor network state, and hence our result for LGTs provides
a theoretical foundation for these numerical results from previous work.

There are two ways in which standard area law techniques are insufficient for
our current setting: Firstly, the known area laws exhibit a bound which
depends on the dimension of the local Hilbert space. The state-of-the-art 1D
area law result bounds the entanglement entropy as
$\mathcal{O}(\Delta^{-1}\log^{3}(d))$ where $d$ is the local Hilbert space
dimension and $\Delta$ is the spectral gap [6, 7]. This becomes infinity when
the local Hilbert space is infinite-dimensional. Secondly, in quantum spin
systems all local Hamiltonian terms can be rescaled to have operator norm at
most $1$, whereas in the models we consider in this work the local Hamiltonian
terms can be unbounded which precludes such a normalization.

For certain non-interacting bosonic systems involving infinite-dimensional
local Hilbert spaces the entanglement area law has been proven, for example
when the system is exactly solvable [8, 32, 15, 14, 16, 17]. However, a
general methodology is unavailable for establishing area laws for quantum
systems with infinite local degrees of freedom.

This letter gives an entanglement area law for a class of 1D quantum systems
that involve bosons or arise from gauge theories, using the approximate ground
state projector (AGSP) framework developed in [5]. Our quantum systems of
interest involve infinite-dimensional local Hilbert spaces. Examples of such
quantum systems include the Hubbard-Holstein model, U(1) and SU(2) LGTs, all
of which are defined on a 1D chain. For these models, we can introduce a
notion of local quantum number, which is the occupation number in the bosonic
case, the electric field value in the U(1) LGT, and the total angular momentum
in the SU(2) LGT.

####  The abstract model and main result.—

We consider a system on a line of length $N+1$ with a geometrically local
Hamiltonian of the form $H=H_{1}+H_{2}+\cdots+H_{N}$, where $H_{x}$ acts on
sites $x-1$ and $x$. At each site $x$ we have local observable $\lambda_{x}$,
which we call the local quantum number. The conditions for our results are
stated in terms of the following quantities.

######  Definition 1.

  1. 1.

Let $\Pi_{S}^{(x)}=1_{S}(\lambda_{x})$ be the spectral projector for
$\lambda_{x}$ corresponding to eigenvalues in the set $S$.

  2. 2.

For cutoff $\Lambda\geq 0$ define the _truncated local Hilbert space
dimension_ of a site $x$ to be
$d(\Lambda)=\mathrm{rank}(\Pi_{[-\Lambda,\Lambda]}^{(x)})$.

  3. 3.

For cutoff $\Lambda\geq 0$ define the truncated norm of the local Hamiltonian
$H_{x}$ constrained to $[-\Lambda,\Lambda]$ to be
$\mathcal{N}(\Lambda)=\max_{x}\|H_{x}\Pi^{(x)}_{[-\Lambda,\Lambda]}\|$ where
$\|\cdot\|$ is the spectral norm and the maximum is over sites $x$.

We require that the truncated local norm and Hilbert space dimension of the
local Hamiltonian grow at most polynomially with with the cutoff $\Lambda$.

######  Assumption 2.

The maximum truncated local dimension $d(\Lambda)$ and norm
$\mathcal{N}(\Lambda)$ satisfy $d(\Lambda),\
\mathcal{N}(\Lambda)=\mathcal{O}(\poly(\Lambda)).$ Also
$\braket{}{\lambda_{x}}{}=\mathcal{O}(1)$ where $\braket{\cdot}$ denotes the
ground state expectation value.

Following [36] we also assume a site-dependent decomposition of the global
Hamiltonian into a quantum number-modifying part $H_{W}$ and a quantum number-
preserving part $H_{R}$. As will be explained later, these assumptions are
satisfied by a variety of quantum systems, to which our result applies.

######  Assumption 3 (Growth of local quantum numbers).

There exist non-negative real-valued constants $\chi$ and $r$ such that, for
any $x$, the Hamiltonian can be decomposed as

$H=H^{(x)}_{W}+H^{(x)}_{R},$ (1)

where $H^{(x)}_{W}$ and $H^{(x)}_{R}$ satisfy (for
$\Pi_{\lambda}^{(x)}\equiv\Pi_{\\{\lambda\\}}^{(x)}$)

$\displaystyle\Pi^{(x)}_{\lambda}H_{W}^{(x)}\Pi^{(x)}_{\lambda^{\prime}}=0,\
\text{if }|\lambda-\lambda^{\prime}|>1,$ (2a)
$\displaystyle\|H_{W}^{(x)}\Pi^{(x)}_{[-\Lambda,\Lambda]}\|\leq\chi(\Lambda+1)^{r},$
(2b) $\displaystyle[H_{R}^{(x)},\Pi^{(x)}_{\lambda}]=0\leavevmode\nobreak\
\text{for all\leavevmode\nobreak\ }\lambda,$ (2c) for all $x=1,2,\ldots,N$.

######  Theorem (Main result).

For the gapped ground state of any Hamiltonian $H$ satisfying Assumptions 2
and 3, in particular the 1D Hubbard-Holstein model, or the 1D U(1) or SU(2)
LGTs, the entanglement entropy across a cut scales as
$\mathcal{O}(\poly(\Delta^{-1}))$, where $\Delta$ is the spectral gap,
assuming that all coefficients in the Hamiltonian remain constant. In
particular, this entanglement entropy scaling is independent of the system
size.

####  The proof strategy.—

Our result utilizes a local quantum number tail bound recently obtained in
[36]. This tail bound tells us that for the relevant class of quantum systems,
a spectrally isolated energy eigenstate can be well-approximated by a
truncated state with low local quantum numbers. This suggests that we may
choose an effective local Hilbert space with finite dimensions and thereby
apply the spin chain framework developed in [5, 6]. To implement this proof
strategy, we need to apply truncation to the local Hilbert space and by
extension to the Hamiltonian terms, on each lattice site close to the cut. We
show in Theorem 12 that the truncation changes the ground state only by an
exponentially small amount. This is similar to what is known as the robustness
theorem [6, Theorem 6.1]. Because of this we can show that the AGSP framework
developed in [6] is robust to this truncation. In the case of frustration-free
spin chains Ref. [5] showed that the existence of an AGSP with exact target
space and suitable parameters directly implies an area law. Analogously, the
“off-the-shelf” lemma with approximate target space in [1] incorporates
techniques from [7] to directly conclude an area law from the existence of a
sequence of AGSP with increasingly accurate target spaces. Using this off-the-
shelf lemma the truncation-based AGSP implies the entanglement area law for
our class of unbounded quantum systems.

In the course of proving the area law for unbounded systems we show that the
mean absolute value of the local quantum number can be bounded independently
of the system size for many quantum systems without translation symmetry
(Lemmas 7 and 8), a bound that we consider to be of independent interest.
These systems include the Hubbard-Holstein model (Corollary 10) and U(1) and
SU(2) LGTs (Corollary 9) with open boundary conditions.

In proving area laws for unbounded quantum systems, it is tempting to plug the
eigenstate tail bound from [36] directly into the area law result for spin
systems [6]. However, such a naive strategy does _not_ seem to work here. The
main reason is that the tail bound from [36] only guarantees the proximity of
the quantum states before and after truncation, but not the proximity of the
corresponding entanglement entropies. In the finite-dimensional setting, one
could invoke the Fannes’ inequality to estimate the entanglement entropy
difference [39, 41], but its explicit dependence on the dimension of the
Hilbert space will ruin the area law scaling for unbounded quantum systems. In
fact, the main contribution of our paper is to give a careful truncation of
the local Hilbert space that does not blow up entanglement entropy of the
ground state, overcoming the technical difficulties mentioned above.

####  Application to 1D U(1) LGT.—

Our result applies to a wide range of physically relevant quantum systems,
including the Hubbard-Holstein model and the U(1) and SU(2) LGTs. In [36,
Section I] it is discussed how they all satisfy the Assumption 3. For
concreteness, we will discuss in detail how the U(1) LGT fits into the
framework of our current work.

In the 1D U(1) LGT, the system consists of a chain of $N$ nodes with $N-1$
links between adjacent nodes. We denote each node by $x$, and the links by the
node on its left end. The links are sometimes called gauge links.

On each node $x$ we have a fermionic mode whose annihilation operator is
denoted by $\phi_{x}$. Each link consists of a planar rotor, whose
configuration can be described by an angle $\theta\in[0,2\pi]$. The local
Hilbert space is the vector space of square-integrable functions on $U(1)$. An
orthonormal basis of the local Hilbert space can be chosen to be the Fourier
basis (the electric basis). More specifically we denote by $\ket{k}$ the
Fourier mode $(2\pi)^{-1/2}e^{ik\theta}$, and $\\{\ket{k}:k\in\mathbb{Z}\\}$
form the basis we need.

We further define the operators $E_{x}$ and $U_{x}$, which act on the vector
space of the links, through $E_{x}\ket{k}=k\ket{k},U_{x}\ket{k}=\ket{k-1}.$
The Hamiltonian for $U(1)$ LGT can then be described in terms of these
operators via

$H=H_{M}+H_{GM}+H_{E},$ (3)

where the three terms $H_{M},H_{GM},H_{E}$ describe the fermionic mass (using
staggered fermions [25]), the gauge-matter interaction, and the electric
energy respectively,

$\displaystyle H_{M}$
$\displaystyle=g_{M}\sum_{x}(-1)^{x}\phi_{x}^{\dagger}\phi_{x},$ (4)
$\displaystyle H_{GM}$
$\displaystyle=g_{GM}\sum_{x}(\phi_{x}^{\dagger}U_{x}\phi_{x+1}+\phi_{x+1}^{\dagger}U_{x}^{\dagger}\phi_{x}),$
$\displaystyle H_{E}$ $\displaystyle=g_{E}\sum_{x}E_{x}^{2}.$

For LGTs, physical states need to satisfy Gauss’s law: $G_{x}\ket{\Phi}=0$ for
all physical states $\ket{\phi}$ where $G_{x}=E_{x}-E_{x-1}-\rho_{x}$,
$\rho_{x}=\phi_{x}^{\dagger}\phi_{x}+((-1)^{x}-1)/2$. To ensure that the
ground state of the Hamiltonian satisfy Gauss’s law, we add a penalty term to
the original Hamiltonian so that it becomes
$H=H_{M}+H_{GM}+H_{E}+\lambda_{G}\sum_{x}G_{x}^{2}$, where we assume that
$\lambda_{G}=\mathcal{O}(1)$.

We can first write the Hamiltonian as a sum of local terms
$H=H_{1}+H_{2}+\cdots+H_{N}$, where
$H_{x}=g_{M}(-1)^{x}\phi_{x}^{\dagger}\phi_{x}+g_{GM}(\phi_{x}^{\dagger}U_{x}\phi_{x+1}+\phi_{x+1}^{\dagger}U_{x}^{\dagger}\phi_{x})+g_{E}E_{x}^{2}+\lambda_{G}G_{x+1}^{2}$.
We consider the site $x$ to consist of both the node $x$ and the link $x$. The
truncation is done through projecting in the electric basis: we define
$\Pi^{(x)}_{[-\Lambda,\Lambda]}=\sum_{|k|\leq\Lambda}\ket{k}\bra{k}$. The
truncated local Hilbert space is therefore spanned by
$\\{\ket{k}:|k|\leq\Lambda\\}\otimes\\{\ket{0},\ket{1}\\}$, where
$\ket{0},\ket{1}$ are the states of the fermionic mode on node $x$. We define
the local quantum number $\lambda_{x}=E_{x}$, which is consistent with this
truncation.

Now let us first check Assumption 2. The dimension of the truncated local
Hilbert space is clearly $d(\Lambda)=2(2\Lambda+1)$. For
$\mathcal{N}(\Lambda)=\|H_{x}\Pi^{(x)}_{[-\Lambda,\Lambda]}\|$, direct
calculation shows $\mathcal{N}(\Lambda)=\mathcal{O}(\Lambda^{2})$. Verifying
that $\braket{}{\lambda_{x}}{}=\mathcal{O}(1)$ is a non-trivial task, which we
perform in Section III (for the SU(2) LGT and the Hubbard-Holstein model as
well).

Next we check Assumption 3. To decompose the Hamiltonian into $H^{(x)}_{W}$
and $H^{(x)}_{R}$, we observe that the only term that changes the local
quantum number on site $x$ is
$g_{GM}(\phi_{x}^{\dagger}U_{x}\phi_{x+1}+\phi_{x+1}^{\dagger}U_{x}^{\dagger}\phi_{x})$,
which we define to be $H^{(x)}_{W}$. Because $\braket{k}{U_{x}}{k^{\prime}}=0$
if $|k-k^{\prime}|>1$, (2a) is satisfied. All the other terms are collected
into $H^{(x)}_{R}$. Note that $H^{(x)}_{R}$ acts non-trivially on site $x$
(through the term $g_{E}E_{x}^{2}$), but it does not change the local quantum
number. Therefore by this definition (2c) is satisfied. It only remains to
check (2b), which, because in this example $H^{(x)}_{W}$ is bounded even
without truncation, is true by choosing $\chi=2|g_{GM}|$ and $r=0$.

By checking the Assumptions 2 and 3, we can see that our main result (Theorem
20 from the supplemental material) applies to the 1D U(1) LGT, and therefore
the area law is established provided that the spectral gap remains bounded
away from $0$ as the system size increases. A similar procedure can be applied
to the Hubbard-Holstein model and SU(2) LGT in 1D as well.

####  Conclusion.—

We have rigorously established an entanglement area law for the gapped ground
state of 1D quantum systems with infinite-dimensional local Hilbert spaces
under natural assumptions on the local quantum numbers. In particular, our
entanglement area law applies to U(1) and SU(2) LGTs and the Hubbard-Holstein
model in 1D, and the result may be adapted to handle other bosonic and gauge
theory models of interest. Proving the entanglement area law is an important
step toward justifying the use of tensor network methods in classical
simulation of quantum systems. The proof techniques also provide many useful
tools for designing a rigorous RG algorithm for these quantum systems, which
should be investigated by future work. It is also worth considering how the
result in this work can be generalized to degenerate ground state or low-
energy states as in Ref. [7].

It is interesting that our proof of the area law uses seemingly different
techniques from those of [20]. This hints at a deeper underlying structure
that empowers both proof techniques. We will leave exploration for what this
could be for future work.

Our result relies on the local quantum number tail bound proved in Ref. [36],
which in turn follows from technical tools for analyzing the dynamical
simulation of unbounded Hamiltonians on digital quantum computers. Along this
line, previous work such as [3, 26, 28, 37] have found other applications of
quantum simulation techniques, to solving problems in quantum many-body
physics beyond quantum computing. We consider applications of this kind to be
very interesting, as they demonstrate important byproducts of the study of
quantum algorithms which are of interest independently of the hope of building
scalable quantum computers.

####  Acknowledgments.—

This work is partially supported by the Department of Energy under the Quantum
System Accelerator (QSA) program (Y.T.) and the Spin Chain Bootstrap under BES
(N.B.), by the National Science Foundation under the Quantum Leap Challenge
Institute (QLCI) program through grant number OMA-2016245, NSERC Discovery
Program (N.W.), Google Inc (N.W.) and by the Simons Foundation (N.A.). Y.S.
thanks Andrew M. Childs, Abhinav Deshpande, Adam Ehrenberg, Alexey V.
Gorshkov, Andrew Guo, and Minh Cong Tran for helpful discussions about the
entanglement area law. N.W. thanks Yuta Kikuchi for useful preliminary
discussions about area laws for unbounded operators.

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.

.

Supplement to
Entanglement area law for 1D gauge theories and bosonic systems

Organization of the supplemental materials. In Section I we will discuss
examples of quantum systems that are covered in the framework of this work. In
the remaining parts of the supplemental materials we will provide the detailed
proof of our results. In Section II we will discuss properties of the ground
state that can be used to truncate the infinite-dimensional local Hilbert
space. In Section III we prove that the local quantum numbers have a mean
absolute value that can be bounded independently of the system size. This fact
removes an otherwise extra assumption in the proof of the area law. In Section
IV we will demonstrate that the ground state is robust to such truncations
using properties derived in the previous section. In Section V we prove our
main result of the entanglement area law scaling, using an “off-the-shelf”
lemma slightly tighter than the one from [1].

##  I 1D gauge theories and bosonic systems

We now introduce some specific quantum systems that we will study and extract
the common structure of these systems. These quantum systems are all discussed
in [36, Section I]. We first consider the 1D Hubbard-Holstein model [21], a
model describing electron-phonon interactions.

The Hubbard-Holstein model. This model is defined on a 1D chain of $N$ nodes.
Each node in the lattice, indexed by $x$, contains two fermionic modes (spin
up and down) and a bosonic mode. The Hamiltonian is

$H=H_{f}+H_{fb}+H_{b},$ (5)

where $H_{f}$ is the Hamiltonian of the Fermi-Hubbard model [24] acting on
only the fermionic modes,

$H_{fb}=g\sum_{x=1}^{N}(b_{x}^{\dagger}+b_{x})(n_{x,\uparrow}+n_{x,\downarrow}-1),$
(6)

is the boson-fermion coupling, and

$H_{b}=\omega_{0}\sum_{x=1}^{N}b_{x}^{\dagger}b_{x},$

the purely bosonic parts of the Hamiltonian. Here, $b_{x}$ is the bosonic
annihilation operator on node $x$, and $n_{x,\sigma}$ is the fermionic number
operator for node $x$ and spin $\sigma$.

For gauge theories, we consider the Hamiltonian formulation of the U(1) and
SU(2) LGTs [25] in one dimension. The U(1) LGT is also known as the Schwinger
model.

The U(1) lattice gauge theory. The system consists of a chain of $N$ nodes
with $N-1$ links between adjacent nodes. We denote each node by $x$, and the
links by the node on its left end. The links are sometimes called gauge links.

On each node $x$ we have a fermionic mode whose annihilation operator is
denoted by $\phi_{x}$. Each link consists of a planar rotor, whose
configuration can be described by an angle $\theta\in[0,2\pi]$. The local
Hilbert space is the vector space of square-integrable functions on $U(1)$. An
orthonormal basis of the local Hilbert space can be chosen to be the Fourier
basis. More specifically we denote by $\ket{k}$ the Fourier mode
$(2\pi)^{-1/2}e^{ik\theta}$, and $\\{\ket{k}:k\in\mathbb{Z}\\}$ form the basis
we need.

We further define the operators $E_{x}$ and $U_{x}$, which act on the vector
space of the links, through

$E_{x}\ket{k}=k\ket{k},\qquad U_{x}\ket{k}=\ket{k-1}.$ (7)

The Hamiltonian for $U(1)$ LGT can then be described in terms of these
operators via

$H=H_{M}+H_{GM}+H_{E},$ (8)

where the three terms $H_{M},H_{GM},H_{E}$ describe the fermionic mass (using
staggered fermions [25]), the gauge-matter interaction, and the electric
energy respectively,

$\displaystyle H_{M}$
$\displaystyle=g_{M}\sum_{x}(-1)^{x}\phi_{x}^{\dagger}\phi_{x},$ (9)
$\displaystyle H_{GM}$
$\displaystyle=g_{GM}\sum_{x}(\phi_{x}^{\dagger}U_{x}\phi_{x+1}+\phi_{x+1}^{\dagger}U_{x}^{\dagger}\phi_{x}),$
$\displaystyle H_{E}$ $\displaystyle=g_{E}\sum_{x}E_{x}^{2}.$

In the context of the U(1) LGT, we need to ensure that the quantum state
satisfies Gauss’s law:

$G_{x}\ket{\Phi}=0$ (10)

for all physical states $\ket{\phi}$ where

$G_{x}=E_{x}-E_{x-1}-\rho_{x},\quad\rho_{x}=\phi_{x}^{\dagger}\phi_{x}+((-1)^{x}-1)/2.$
(11)

To ensure that the ground state of the Hamiltonian satisfy Gauss’s law, we add
a penalty term to the original Hamiltonian so that it becomes
$H=H_{M}+H_{GM}+H_{E}+\lambda_{G}\sum_{x}G_{x}^{2}$, where we assume that
$\lambda_{G}=\mathcal{O}(1)$.

The SU(2) lattice gauge theory. We consider the theory using the fundamental
representation of SU(2), as done in [11]. Each node $x$ now contains two
fermionic modes, whose annihilation operators are denoted by $\phi_{x}^{l}$,
$l=1,2$. Each gauge link consists of a rigid rotator whose configuration is
described by an element of the group SU(2) [25].

The Hamiltonian takes the form (8), and is invariant under $\mathrm{SU}(2)$
transformations acting either from the left or from the right, which
correspond to rotations of the rigid rotator with respect to space-fixed or
body-fixed axes respectively.

Physical states in SU(2) LGT also needs to satisfy Gauss’s law, which takes a
form that is similar to the U(1) case. We also ensure that it is satisfied in
the ground state we study by adding a penalty term to the Hamiltonian.

##  II Truncating the local Hilbert space

Although we consider the setting where the local Hilbert spaces are infinite
dimensional, we can approximate spectrally isolated eigenstates with states
containing low local quantum numbers. This fact is made rigorous in [36,
Theorem 12] which we restate here in a slightly modified way:

######  Theorem 4 (Quantum number distribution tail bound).

Let $H$ be a Hamiltonian satisfying Assumption 3. Let $\ket{\Psi}$ be an
eigenstate of $H$ corresponding to an eigenvalue $\varepsilon$ with
multiplicity $1$, and $\varepsilon$ be separated from the rest of the spectrum
of $H$ by a spectral gap $\Delta$. Moreover, for a fixed site $x$ we assume
local quantum number $|\lambda_{x}|$ has a finite expectation value:

$\sum_{\lambda}|\lambda|\braket{\Psi}{\Pi^{(x)}_{\lambda}}{\Psi}=\bar{\lambda}_{x}<\infty,$

we then have that

$\|(I-{\Pi}^{(x)}_{[-\Lambda,\Lambda]})\ket{\Psi}\|\leq
e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda}_{x})^{1-r})}\right)}.$

We will further need to apply this result to bound the error that emerges from
applying the truncation bound to many sites. The following corollary provides
such a result in a convenient form.

######  Corollary 5.

Let $\Pi^{\prime}=\prod_{x=\ell+1}^{\ell+s}\Pi^{(x)}_{[-\Lambda,\Lambda]},$
then under the same assumption as in Theorem 4, we have

$\|(I-{\Pi}^{\prime})\ket{\Psi}\|\leq\sqrt{s}e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)},$

where $\bar{\lambda}=\max_{\ell<x\leq\ell+s}\bar{\lambda}_{x}$.

###### Proof.

Proof follows from Theorem 4 via a straightforward application of the triangle
inequality and the sub-multiplicative property of the spectral norm:

$\displaystyle\|(I-{\Pi}^{\prime})\ket{\Psi}\|$
$\displaystyle=\sqrt{\bra{\Psi}(I-{\Pi}^{\prime})^{2}\ket{\Psi}}=\sqrt{\bra{\Psi}(I-{\Pi}^{\prime})\ket{\Psi}}$
$\displaystyle=\sqrt{\sum_{x=\ell+1}^{\ell+s}\bra{\Psi}\prod_{x^{\prime}=\ell+1}^{x-1}\Pi_{[-\Lambda,\Lambda]}^{(x^{\prime})}(I-\Pi^{(x)}_{[-\Lambda,\Lambda]})\ket{\Psi}}$
$\displaystyle\leq\sqrt{\sum_{x=\ell+1}^{\ell+s}\|(I-\Pi^{(x)}_{[-\Lambda,\Lambda]})\ket{\Psi}\|}\leq\sqrt{s}e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)}.$
(12)

∎

A final important consequence of Theorem 4 is a bound on the quantity
$\|\Pi^{\prime}H(I-{\Pi}^{\prime})\ket{\Psi}\|$ where $\Pi^{\prime}$ is
defined as in Corollary 5. We use the above corollary to derive such a bound
below.

######  Corollary 6.

Let $\Pi^{\prime}=\prod_{x=\ell+1}^{\ell+s}\Pi^{(x)}_{[-\Lambda,\Lambda]},$
then under the same assumption as in Theorem 4, we have

$\|\Pi^{\prime}H(I-{\Pi}^{\prime})\ket{\Psi}\|\leq
s^{3/2}\mathcal{N}(\Lambda)e^{-\Omega\left(\sqrt{\chi^{-1}\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)},$

where $\bar{\lambda}=\max_{\ell<x\leq\ell+s}\bar{\lambda}_{x}$.

###### Proof.

We only need to use Corollary 5 along with a bound for
$\|\Pi^{\prime}H(I-{\Pi}^{\prime})\|$. We have

$\Pi^{\prime}H(I-{\Pi}^{\prime})=\sum_{x=1}^{N}\Pi^{\prime}H_{x}(I-{\Pi}^{\prime})=\sum_{x=\ell+1}^{\ell+s}\Pi^{\prime}H_{x}(I-{\Pi}^{\prime}).$
(13)

The second equality is because
$\Pi^{\prime}H_{x}(I-{\Pi}^{\prime})=\Pi^{\prime}(I-{\Pi}^{\prime})H_{x}=0$ if
$x\notin\\{\ell+1,\ldots,\ell+s\\}$. Therefore

$\|\Pi^{\prime}H(I-{\Pi}^{\prime})\|\leq\sum_{x=\ell+1}^{\ell+s}\|\Pi^{\prime}H_{x}(I-{\Pi}^{\prime})\|\leq\sum_{x=\ell+1}^{\ell+s}\|\Pi^{(x)}_{[-\Lambda,\Lambda]}H_{x}\|\leq
s\mathcal{N}(\Lambda).$ (14)

∎

We remark that the bound in the above corollary does not depend on the system
size $N$. However, there is a dependence on $\bar{\lambda}$, an upper bound on
the mean absolute value of the local quantum numbers on sites
$\ell+1,\ell+2,\ldots,\ell+s$. One might worry that $\bar{\lambda}$ will show
up as an independent parameter in our expression of the entanglement entropy.
However, we will show in the next section that for the ground states of U(1)
and SU(2) LGTs as well as the Hubbard-Holstein model, $\bar{\lambda}$ depends
only on the coefficients in the Hamiltonian, and is thus independent of the
system size.

##  III Bounding the mean absolute value of the local quantum number

In this section we prove bounds on the mean absolute value
$\bar{\lambda}_{x}=\braket{}{\lambda_{x}}{}$ of the local quantum numbers in
the ground states of U(1) and SU(2) LGTs, as well as the Hubbard-Holstein
model. $\lambda_{x}$ here is the local quantum number of site $x$ and
$\braket{\cdot}$ denotes the ground state expectation value. We will drop the
subscript $x$ in this section because we will focus on only a single bosonic
mode or gauge link. These bounds are independent of the system size and only
depend on the coefficients in the Hamiltonians.

In the following discussion, we view our lattice models as a bipartite system,
where a subsystem $A$ is a gauge link or bosonic mode, and a subsystem $B$ is
the rest of the system. We then bound $\braket{}{\lambda}{}$, where $\lambda$
is the local quantum number for $A$, using the variational principle.

######  Lemma 7.

Consider a bipartite system with Hilbert space
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where $\mathcal{H}_{A}$
and $\mathcal{H}_{B}$ are the Hilbert spaces for subsystems $A$ and $B$
respectively. Let $H=H_{A}+H_{AB}+H_{B}$ be the Hamiltonian, where $H_{A}$
acts non-trivially only on $A$, and $H_{B}$ on $B$. Let $\ket{\Psi}$ be the
ground state of $H$. Furthermore, we assume that there exists an operator
$K_{A}$ acting non-trivially only on $A$ such that $|H_{AB}|\preceq K_{A}$.
Then we have

$\braket{\Psi}{\left(H_{A}-K_{A}\right)}{\Psi}\leq\braket{\Psi_{A}}{\left(H_{A}+K_{A}\right)}{\Psi_{A}},$
(15)

for any $\ket{\Psi_{A}}\in\mathcal{H}_{A}$. Here $\preceq$ denotes the partial
order induced by the convex cone of positive semi-definite operators and
$|O|=\sqrt{O^{\dagger}O}$ for operator $O$.

###### Proof.

Let $\ket{\Psi_{P}}=\ket{\Psi_{A}}\ket{\Psi_{B}}$ be a product state, where
$\ket{\Psi_{B}}\in\mathcal{H}_{B}$ is the ground state of $H_{B}$, and
$\ket{\Psi_{A}}\in\mathcal{H}_{A}$ is chosen arbitrarily. Then because
$\ket{\Psi}$ is the ground state of $H$ we have

$\braket{\Psi}{H}{\Psi}\leq\braket{\Psi_{P}}{H}{\Psi_{P}}.$ (16)

For $\braket{\Psi}{H}{\Psi}$ we have

$\displaystyle\braket{\Psi}{H}{\Psi}$
$\displaystyle=\braket{\Psi}{H_{A}}{\Psi}+\braket{\Psi}{H_{AB}}{\Psi}+\braket{\Psi}{H_{B}}{\Psi}$
(17)
$\displaystyle\geq\braket{\Psi}{H_{A}}{\Psi}-\braket{\Psi}{K_{A}}{\Psi}+\braket{\Psi}{H_{B}}{\Psi}$
$\displaystyle\geq\braket{\Psi}{\left(H_{A}-K_{A}\right)}{\Psi}+\braket{\Psi_{B}}{H_{B}}{\Psi_{B}},$

where in the first inequality we have used $|H_{AB}|\preceq K_{A}$ and in the
second inequality we have used
$\braket{\Psi}{H_{B}}{\Psi}\geq\braket{\Psi_{B}}{H_{B}}{\Psi_{B}}$, which is
true because $\ket{\Psi_{B}}$ is chosen to be the ground state of $H_{B}$.

For $\braket{\Psi_{P}}{H}{\Psi_{P}}$ we have

$\displaystyle\braket{\Psi_{P}}{H}{\Psi_{P}}$
$\displaystyle=\braket{\Psi_{P}}{H_{A}}{\Psi_{P}}+\braket{\Psi_{P}}{H_{AB}}{\Psi_{P}}+\braket{\Psi_{B}}{H_{B}}{\Psi_{B}}$
(18)
$\displaystyle\leq\braket{\Psi_{P}}{\left(H_{A}+K_{A}\right)}{\Psi_{P}}+\braket{\Psi_{B}}{H_{B}}{\Psi_{B}}$
$\displaystyle=\braket{\Psi_{A}}{\left(H_{A}+K_{A}\right)}{\Psi_{A}}+\braket{\Psi_{B}}{H_{B}}{\Psi_{B}}.$

Combining (16), (17), (18), we have (15).

∎

######  Lemma 8.

Under the same assumptions as in Lemma 7, we further let $\Xi$ be a Hermitian
operator on $A$. Assume that $H_{A}-K_{A}\succeq L(|\Xi|)$ where $L(x)$ is a
convex non-decreasing function for $x\in\mathbb{R}^{+}$ satisfying
$L(x)\to+\infty$ when $x\to+\infty$. Then we have

$\braket{\Psi}{\Xi}{\Psi}\leq
L^{-1}\left(\braket{\Psi}{\left(H_{A}-K_{A}\right)}{\Psi}\right)\leq
L^{-1}\left(\min_{\ket{\Psi_{A}}\in\mathcal{H}_{A}}\braket{\Psi_{A}}{\left(H_{A}+K_{A}\right)}{\Psi_{A}}\right).$
(19)

###### Proof.

By the assumption that $H_{A}-K_{A}\succeq L(|\Xi|)$ and (15), we have

$\braket{\Psi}{L(}{\Xi|)|\Psi}\leq\braket{\Psi_{A}}{\left(H_{A}+K_{A}\right)}{\Psi_{A}}.$

Because $L$ is convex, by Jensen’s inequality
$L(\braket{\Psi}{\Xi}{\Psi})\leq\braket{\Psi}{L(}{\Xi|)|\Psi}$. Because $L$ is
non-decreasing and $\ket{\Psi_{A}}$ can be arbitrarily chosen, we have (19). ∎

Note that the right-hand side of (19) is independent of the subsystem $B$.
This allows us to bound the mean absolute value of the local quantum number in
the lattice models in a way that is independent of the system size. We will
now apply this lemma to the case of gauge theories and the Hubbard-Holstein
model.

First, for U(1) and SU(2) gauge theories with the Hamiltonian defined in (8),
we take $A$ to be a gauge link $x$. Then we let

$H_{A}=g_{E}E_{x}^{2},\qquad
H_{AB}=g_{GM}(\phi_{x}^{\dagger}U_{x}\phi_{x+1}+\phi_{x+1}^{\dagger}U_{x}^{\dagger}\phi_{x}),\qquad
H_{B}=H-H_{A}-H_{B}.$

A nice feature about this Hamiltonian is that $H_{AB}$ is bounded:
$\|H_{AB}\|\leq 2|g_{GM}|$. Therefore we can simply choose $K_{A}=2|g_{GM}|$.
The local quantum number $\lambda_{x}$ in this case is the electric field
value in the U(1) case and the total angular momentum in the SU(2) case. But
$E_{x}^{2}=\lambda_{x}^{2}$ for U(1) LGT and
$E_{x}^{2}=\lambda_{x}(\lambda_{x}+1)\succeq\lambda_{x}^{2}$ for SU(2) LGT
(for the latter see section III.1 below. Also $\lambda_{x}\succeq 0$ for SU(2)
LGT). Therefore we have

$H_{A}-K_{A}\succeq g_{E}\lambda_{x}^{2}-2|g_{GM}|$

for both cases. For the right-hand side of (19) we have

$\min_{\ket{\Psi_{A}}\in\mathcal{H}_{A}}\braket{\Psi_{A}}{\left(H_{A}+K_{A}\right)}{\Psi_{A}}=2|g_{GM}|,$

where the minimum is attained by $\ket{\Psi_{A}}=\ket{0}$. Combining the above
facts, we apply Lemma 8 to get:

######  Corollary 9.

For the U(1) and SU(2) LGTs with the Hamiltonian defined in (8), let
$\lambda_{x}$ be the local quantum number on gauge link $x$, then
$\braket{}{\lambda_{x}}{}\leq 2\sqrt{|g_{GM}|/g_{E}}$, where $\braket{\cdot}$
denotes the ground state expectation value.

Then let us consider the Hubbard-Holstein model with the Hamiltonian described
in (5). We let $A$ be a bosonic mode $x$, and let $B$ be the rest of the
system. We have

$H_{A}=\omega_{0}b_{x}^{\dagger}b_{x},\qquad
H_{AB}=g(b_{x}^{\dagger}+b_{x})(n_{x,\uparrow}+n_{x,\downarrow}-1),\qquad
H_{B}=H-H_{A}-H_{AB}.$

Here $H_{AB}$ is no longer bounded, but we can still construct a $K_{A}$ such
that $|H_{AB}|\preceq K_{A}$. To simplify the discussion we introduce the
position and momentum operators $X$ and $P$:

$X=\frac{1}{\sqrt{2}}(b_{x}^{\dagger}+b_{x}),\qquad
P=\frac{i}{\sqrt{2}}(b_{x}^{\dagger}-b_{x}).$

Then $H_{AB}=\sqrt{2}gX(n_{x,\uparrow}+n_{x,\downarrow}-1)$. Because
$\|n_{x,\uparrow}+n_{x,\downarrow}-1\|\leq 1$, we can define

$K_{A}=\sqrt{2}|g||X|,$

which satisfies $|H_{AB}|\preceq K_{A}$. With this choice of $K_{A}$ we have

$\displaystyle H_{A}-K_{A}$
$\displaystyle=\frac{\omega_{0}}{2}(X^{2}+P^{2}-1)-\sqrt{2}|g||X|$
$\displaystyle\succeq\frac{\omega_{0}}{2}(X^{2}+P^{2}-1)-\frac{\omega_{0}}{4}X^{2}-\frac{2g^{2}}{\omega_{0}}$
$\displaystyle\succeq\frac{\omega_{0}}{4}(X^{2}+P^{2}-1)-\frac{\omega_{0}}{4}-\frac{2g^{2}}{\omega_{0}}$
$\displaystyle=\frac{\omega_{0}}{2}b_{x}^{\dagger}b_{x}-\frac{\omega_{0}}{4}-\frac{2g^{2}}{\omega_{0}}.$

The local quantum number here is the bosonic occupation number, which has to
be non-negative. Therefore

$H_{A}-K_{A}\succeq\frac{\omega_{0}}{2}|\lambda_{x}|-\frac{\omega_{0}}{4}-\frac{2g^{2}}{\omega_{0}}.$

For the right-hand side of (19), we have

$\min_{\ket{\Psi_{A}}\in\mathcal{H}_{A}}\braket{\Psi_{A}}{\left(H_{A}+K_{A}\right)}{\Psi_{A}}\leq\braket{0}{\left(\omega_{0}b_{x}^{\dagger}b_{x}+2}{g||X|}=\frac{2|g|}{\sqrt{\pi}},$

where we have used the analytic solution of the ground state of the harmonic
oscillator in deriving the equality. Combining these results with Lemma 8 we
have:

######  Corollary 10.

For the Hubbard-Holstein model with the Hamiltonian defined in (5), let
$\lambda_{x}$ be the local quantum number on site $x$, then

$\braket{}{\lambda_{x}}{}\leq\frac{1}{2}+\frac{4|g|}{\omega_{0}\sqrt{\pi}}+\frac{4g^{2}}{\omega_{0}^{2}},$

where $\braket{\cdot}$ denotes the ground state expectation value.

###  III.1 SU(2) LGT

For the SU(2) case the operators $E^{2}_{x}$ and $U_{x}$ are different from
the U(1) case. The operator $E^{2}_{x}$ is defined through

$E^{2}_{x}\ket{jmm^{\prime}}=j(j+1)\ket{jmm^{\prime}}.$ (20)

Because $\phi_{x}$ has two components, where each component is a fermionic
mode, $U_{x}$ is a $2\times 2$ matrix, where each of the 4 matrix entries is
an operator acting on the link space

$U_{x}=\begin{pmatrix}U^{11}_{x}&U^{12}_{x}\\\
U^{21}_{x}&U^{22}_{x}\end{pmatrix}.$ (21)

Given that $U_{x,n_{i}}$ transforms as the $j=1/2$ representation of
$\mathrm{SU}(2)$, the rules for the addition of angular momentum imply

$\displaystyle\braket{j_{1}m_{1}m_{1}^{\prime}}{U^{ll^{\prime}}_{x}}{j_{2}m_{2}m_{2}^{\prime}}$
$\displaystyle=0,\ \text{if }|j_{1}-j_{2}|>1/2,$ (22)
$\displaystyle\|U^{ll^{\prime}}_{x}\|$ $\displaystyle\leq 1.$

Here $\|O\|$ denotes the spectral norm of an operator $O$.

##  IV Robustness of the ground state to truncation

In this section, we show that the ground state, the ground state energy, and
the spectral gap are all robust to the truncation of the Hamiltonian, in a way
that we will specify later. Following [6] we focus on the $s$ sites from
$\ell+1$ to $\ell+s$. For convenience we relabel the sites so that the
original site $x$ is now labelled $x-\ell$. The Hamiltonian can be rewritten
as

$H=H_{L}+H_{1}+\cdots+H_{s}+H_{R},$ (23)

where $H_{L}=\sum_{x\leq 0}H_{x}$ and $H_{R}=\sum_{x\geq s+1}H_{x}$. We need
to shift each Hamiltonian term ($H_{L}$, $H_{1}$, $H_{2},\ldots,H_{s}$,
$H_{R}$) by a constant to ensure that they are all positive semi-definite. As
in [6] we look at the entanglement entropy across a cut between sites $s/2$
and $s/2+1$. If $\|H_{x}\|\leq 1$, the local Hilbert space dimension is $d$
and the spectral gap is $\Delta$, then it is known that the entanglement
entropy scales as $\mathcal{O}(\log^{3}(d)/\Delta)$ [6, Theorem 6.2].

Now we want to consider the case where the local Hilbert space dimension is
infinite. This compels us to truncate the local Hilbert space, and
consequently the local Hamiltonian terms $H_{i}$ as well. We denote the
truncation threshold, defined according to the local quantum number introduced
in Section The abstract model and main result.—, by $\Lambda$, and
correspondingly the truncated Hamiltonian term by $H_{x}^{\prime}$. The
truncated Hilbert space dimension is $d(\Lambda)$ and the truncated local term
has a norm that is upper bounded by $\mathcal{N}(\Lambda)$.

###  IV.1 The two truncations

We first clarify in more detail what we mean by truncating the local Hilbert
space. The original Hilbert space is
$\mathcal{H}=\mathcal{H}_{L}\otimes\mathcal{H}_{1}\otimes\cdots\mathcal{H}_{s}\otimes\mathcal{H}_{R}$.
We consider a subspace
$\mathcal{H}^{\prime}=\mathcal{H}_{L}\otimes\mathcal{H}^{\prime}_{1}\otimes\cdots\mathcal{H}^{\prime}_{s}\otimes\mathcal{H}_{R}\subseteq\mathcal{H}$,
where each $\mathcal{H}_{x}$ has dimension $d(\Lambda)$. We denote by
$\Pi^{\prime}_{x}$ the projection operator onto $\mathcal{H}^{\prime}_{x}$,
and define

$\Pi^{\prime}=I_{L}\otimes\Pi^{\prime}_{1}\otimes\cdots\Pi^{\prime}_{s}\otimes
I_{R},$

which is the projection operator onto $\mathcal{H}^{\prime}$. $I_{L}$ and
$I_{R}$ are the identity operators on $\mathcal{H}_{L}$ and $\mathcal{H}_{R}$
respectively. The truncated Hamiltonian is a Hermitian operator $H^{\prime}$
defined to be the restriction of $\Pi^{\prime}H\Pi^{\prime}$ to the subspace
$\mathcal{H}^{\prime}$, by which we mean that $H^{\prime}$ maps elements from
$\mathcal{H}^{\prime}$ to $\mathcal{H}^{\prime}$. We do not directly define
$H^{\prime}$ to be $\Pi^{\prime}H\Pi^{\prime}$ because that would introduce an
artificial eigenvalue $0$ corresponding to the part of the Hilbert space that
is truncated out. We can write $H^{\prime}$ out as

$H^{\prime}=H_{L}^{\prime}+H_{1}^{\prime}+\cdots+H_{s}^{\prime}+H_{R}^{\prime},$
(24)

where each $H^{\prime}_{x}$ is the restriction of
$\Pi^{\prime}H_{x}\Pi^{\prime}$ to the subspace $\mathcal{H}^{\prime}$, and
the same is true for $H^{\prime}_{L}$ and $H^{\prime}_{R}$. Locality is
preserved in this truncation as $H^{\prime}_{x}$ still acts non-trivially on
sites $x$ and $x+1$, $H^{\prime}_{L}$ on sites to the left of and including
site $1$, and $H^{\prime}_{R}$ on sites to the right of site $s$.

The second truncation we consider comes from Ref. [6]. We adopt the definition
in [6], for a Hermitian operator $A$,

$A^{\leq t}=AP_{t}+\|AP_{t}\|(I-P_{t}),$ (25)

where $P_{t}$ is the projection operator onto the subspace spanned by
eigenvectors of $A$ with eigenvalues at most $t$. Then the second truncation
yields the Hamiltonian

$H^{\prime\prime}=(H_{L}^{\prime}+H_{1}^{\prime})^{\leq
t}+H_{2}^{\prime}+\cdots+H_{s-1}^{\prime}+(H_{s}^{\prime}+H_{R}^{\prime})^{\leq
t}.$ (26)

The goal here is to show that these two truncations (i) preserve the spectral
gap up to a constant factor, and (ii) preserve the ground state up to an error
exponentially small in $\Lambda$ and $t$.

###  IV.2 Truncation robustness of the ground state and energy

######  Definition 11.

For a self-adjoint operator $A$ bounded from below, define the sequence
$\epsilon_{0}(A)\leq\epsilon_{1}(A)\leq\ldots$ as follows. Let
$\sigma_{\operatorname{ess}}(A)$ be its (closed) essential spectrum and let
$K\in\mathbb{N}_{0}\cup\\{\infty\\}$ be the number of eigenvalues in
$[-\infty,\min\sigma_{\operatorname{ess}}(A))$, including multiplicity. For
$k<K$, let $\epsilon_{k}(A)$ be the $k$-th eigenvalue with multiplicity ($K$
eigenvalues as we start from $k=0$), and for $k\geq K$ let
$\epsilon_{k}(A)=\min\sigma_{\operatorname{ess}}(A)$.

Each $\epsilon_{k}(A)$ is in the spectrum111I.e., the set of $\epsilon$ such
that $A-\epsilon I$ has no bounded inverse. of $A$ but is not necessarily an
eigenvalue when $\epsilon_{k}(A)=\min\sigma_{\operatorname{ess}}(A)$. The min-
max principle [35, Theorem 4.10] states that for $k\in\mathbb{N}_{0}$,

$\epsilon_{k}(A)=\inf_{\phi_{0},\ldots,\phi_{k}\in\mathcal{D}(A)}\sup\Big{\\{}\braket{\psi}{A}{\psi}\Big{|}\ket{\psi}\in\operatorname{span}\\{\phi_{0},\ldots,\phi_{k}\\},\|\ket{\psi}\|=1\Big{\\}},$
(27)

where $\mathcal{D}$ is the domain of $A$.

Writing $\epsilon_{k}=\epsilon_{k}(H)$, recall that we assume that the
Hamiltonian $H$ has a non-degenerate lowest eigenvalue $\epsilon_{0}$ (the
ground state energy), with a unique ground state $\ket{\psi_{0}}$. We also
assume that $\epsilon_{0}$ is separated from the rest of the spectrum by a gap
$\Delta=\epsilon_{1}-\epsilon_{0}$.

For truncated Hamiltonians $H^{\prime}$ and $H^{\prime\prime}$, we will prove
that there exist unique ground states $\ket{\psi_{0}^{\prime}}$ and
$\ket{\psi_{0}^{\prime\prime}}$, corresponding to non-degenerate lowest
eigenvalues $\epsilon_{0}^{\prime}$ and $\epsilon_{0}^{\prime\prime}$, for the
two truncated Hamiltonians respectively. Write $\epsilon_{k}=\epsilon_{k}(H)$,
$\epsilon_{k}^{\prime}=\epsilon_{k}(H^{\prime})$, and
$\epsilon_{k}^{\prime\prime}=\epsilon_{k}(H^{\prime\prime})$ and
$\Delta^{\prime}=\epsilon_{1}^{\prime}-\epsilon_{0}^{\prime}$,
$\Delta^{\prime\prime}=\epsilon_{1}^{\prime\prime}-\epsilon_{0}^{\prime\prime}$.

######  Theorem 12 (Robustness to truncations).

Let $\Pi^{\prime}$ be the projection operator onto $\mathcal{H}^{\prime}$. Let
$\delta_{1}=\|(I-\Pi^{\prime})\ket{\psi_{0}}\|$,
$\delta_{2}=\|\Pi^{\prime}H(I-\Pi^{\prime})\ket{\psi_{0}}\|$, and

$\frac{\delta_{2}}{1-\delta_{1}^{2}}\leq\frac{\Delta}{18},$ (28)

then for every cutoff of the local quantum number $\Lambda>0$ there exists

$T=\mathcal{O}\left(\frac{\mathcal{N}(\Lambda)^{2}}{\Delta}\left(\frac{\epsilon_{0}+\delta_{2}/(1-\delta_{1}^{2})}{\Delta}+1\right)\right),$
(29)

such that for all $t\geq T$,

  * (i)

For $H^{\prime\prime}$, there exists a non-degenerate ground state
$\ket{\psi_{0}^{\prime\prime}}$ corresponding to the lowest eigenvalue
$\epsilon_{0}^{\prime\prime}$.

  * (ii)

$\Delta^{\prime\prime}=\Omega(\Delta)$;

  * (iii)

The trace distance between $\ket{\psi_{0}}$ and
$\ket{\psi^{\prime\prime}_{0}}$ can be bounded as follows:

$D(\ket{\psi_{0}^{\prime\prime}},\ket{\psi_{0}})\leq\sqrt{\frac{2\delta_{2}}{\Delta(1-\delta_{1}^{2})}}+e^{-\Omega(t/\mathcal{N}(\Lambda))}.$
(30)

We note that in the above theorem we require the eigenvalue cutoff $t$ for
$H_{L}$ and $H_{R}$ to be above a certain $T$, due to a similar requirement in
[6, Theorem 6.1]. The scaling of $T$ has later been improved in Ref. [23].

Before we proceed with the proof we establish the following lemma, which
follows from a similar reasoning as in [6, Lemma 6.4] except we correct a
minor mistake in their bound due to not taking the global phase into account.

######  Lemma 13 (Markov).

Let $H$ be a Hamiltonian with the lowest eigenvalue $\epsilon_{0}$ and all
other eigenvalues at least $\epsilon_{1}>\epsilon_{0}$, and assume
$\ket{\psi_{0}}$ is its unique ground state. Given a quantum state
$\ket{\phi}$ with expectation value $\braket{\phi}{H}{\phi}=E$, we have

$|\braket{\phi}{\psi_{0}}|^{2}\geq\frac{\epsilon_{1}-E}{\epsilon_{1}-\epsilon_{0}}.$
(31)

###### Proof.

Since the eigenstate $\ket{\psi_{0}}$ is the unique ground state, the
expectation value of $H$ in any other eigenstate must be at least
$\epsilon_{1}$ by assumption. Using the fact that
$\epsilon_{1}-\epsilon_{0}>0$, we have from Markov’s inequality

$E=\bra{\phi}H\ket{\phi}\geq\epsilon_{0}|\braket{\phi}{\psi_{0}}|^{2}+\epsilon_{1}(1-|\braket{\phi}{\psi_{0}}|^{2}).$
(32)

The result then follows by re-arranging the above expression. ∎

The proof of Theorem 12 proceeds as follows: we first show that when we go
from $H$ to $H^{\prime}$, the spectral gap is preserved and the ground state
is changed by a small amount, and then show the same is true when we go from
$H^{\prime}$ to $H^{\prime\prime}$. In the first step we obtain:

######  Lemma 14.

Let $\Pi^{\prime}$ be the projection operator onto $\mathcal{H}^{\prime}$. Let
$\delta_{1}=\|(I-\Pi^{\prime})\ket{\psi_{0}}\|$,
$\delta_{2}=\|\Pi^{\prime}H(I-\Pi^{\prime})\ket{\psi_{0}}\|$. Then if
$\delta_{1}<1$ and

$\frac{\delta_{2}}{1-\delta_{1}^{2}}\leq\frac{\Delta}{4},$ (33)

we have the following

  * (i)

For $H^{\prime}$, there exists a non-degenerate ground state
$\ket{\psi_{0}^{\prime}}$ corresponding to the lowest eigenvalue
$\epsilon_{0}^{\prime}$.

  * (ii)

$\Delta^{\prime}=\Omega(\Delta)$.

  * (iii)

The trace distance between $\ket{\psi_{0}}$ and $\ket{\psi^{\prime}_{0}}$ can
be bounded as

$D(\ket{\psi_{0}^{\prime}},\ket{\psi_{0}})\leq\sqrt{\frac{2\delta_{2}}{\Delta(1-\delta_{1}^{2})}}.$
(34)

  * (iv)

$\epsilon_{0}\leq\epsilon^{\prime}_{0}\leq\epsilon_{0}+\frac{2\delta_{2}}{1-\delta_{1}^{2}}$.

Here $D(\cdot,\cdot)$ denotes the trace distance.

###### Proof.

By the min-max theorem (Equation 27), we have for $k\in\mathbb{N}_{0}$

$\displaystyle\epsilon_{k}=$
$\displaystyle\inf_{\phi_{0},\ldots,\phi_{k}\in\mathcal{H}}\sup\Big{\\{}\braket{\psi}{H}{\psi}\Big{|}\ket{\psi}\in\operatorname{span}\\{\phi_{0},\ldots,\phi_{k}\\},\|\ket{\psi}\|=1\Big{\\}}$
(35) $\displaystyle\leq$
$\displaystyle\inf_{\phi_{0},\ldots,\phi_{k}\in\mathcal{H^{\prime}}}\sup\Big{\\{}\braket{\psi}{H}{\psi}\Big{|}\ket{\psi}\in\operatorname{span}\\{\phi_{0},\ldots,\phi_{k}\\},\|\ket{\psi}\|=1\Big{\\}}=\epsilon_{k}^{\prime}\leq\min\sigma_{\operatorname{ess}}(H^{\prime}).$

In particular we have
$\epsilon_{1}\leq\epsilon_{1}^{\prime}\leq\min\sigma_{\operatorname{ess}}(H^{\prime})$.
To establish a gap $\Delta^{\prime}$ we need an upper bound on
$\epsilon_{0}^{\prime}$:

$\displaystyle\epsilon_{0}$ $\displaystyle=\braket{\psi_{0}}{H}{\psi_{0}}$
(36)
$\displaystyle=\braket{\psi_{0}}{\Pi^{\prime}H\Pi^{\prime}}{\psi_{0}}+\braket{\psi_{0}}{(I-\Pi^{\prime})H\Pi^{\prime}}{\psi_{0}}$
$\displaystyle\quad+\braket{\psi_{0}}{\Pi^{\prime}H(I-\Pi^{\prime})}{\psi_{0}}+\braket{\psi_{0}}{(I-\Pi^{\prime})H(I-\Pi^{\prime})}{\psi_{0}}$
$\displaystyle\geq\epsilon^{\prime}_{0}\|\Pi^{\prime}\ket{\psi_{0}}\|^{2}-2\|\Pi^{\prime}H(I-\Pi^{\prime})\ket{\psi_{0}}\|+\epsilon_{0}\|(I-\Pi^{\prime})\ket{\psi_{0}}\|^{2}$
$\displaystyle=\epsilon^{\prime}_{0}(1-\delta_{1}^{2})-2\delta_{2}+\epsilon_{0}\delta_{1}^{2}.$

As a result,

$\epsilon^{\prime}_{0}\leq\epsilon_{0}+\frac{2\delta_{2}}{1-\delta_{1}^{2}}.$
(37)

The bound $\epsilon_{0}\leq\epsilon_{0}^{\prime}$ is immediate from the
variational principle (or (35)), so (iv) is established. The assumption (33)
then yields

$\epsilon_{0}^{\prime}\leq\epsilon_{0}+\Delta/2\leq\epsilon_{1}-\Delta/2\leq\epsilon_{1}^{\prime}-\Delta/2,$

which implies $\Delta^{\prime}\geq\Delta/2$, hence (ii). In particular it
implies (i), that $\epsilon_{0}^{\prime}$ is a simple eigenvalue with
eigenvector $\ket{\psi_{0}^{\prime}}$.

To establish closeness between $\ket{\psi_{0}^{\prime}}$ and $\ket{\psi_{0}}$
we apply Lemma 13,

$\braket{\psi_{0}^{\prime}}{\psi_{0}}^{2}\geq\frac{\epsilon_{1}-\braket{\psi_{0}^{\prime}}{H}{\psi_{0}^{\prime}}}{\epsilon_{1}-\epsilon_{0}}=\frac{\epsilon_{1}-\epsilon_{0}^{\prime}}{\Delta}=1-\frac{\epsilon_{0}^{\prime}-\epsilon_{0}}{\Delta}\geq
1-\frac{1}{\Delta}\frac{2\delta_{2}}{1-\delta_{1}^{2}},$

where the last inequality follows from (37). Claim (iii) follows since
$D(\ket{\psi^{\prime}_{1}},\ket{\psi_{0}})\leq\sqrt{1-|\braket{\psi^{\prime}_{1}}{\psi_{0}}|^{2}}$.
∎

We are then ready to prove the main result.

###### Proof of Theorem 12.

The existence and uniqueness of the lowest eigenvalue
$\epsilon_{0}^{\prime\prime}$ and the ground state
$\ket{\psi_{0}^{\prime\prime}}$ in (i) can be proved similarly to the proof of
Lemma 14 (i). We rescale the Hamiltonian $H^{\prime}$ by a factor
$\mathcal{N}(\Lambda)$ and then apply [6, Theorem 6.1]. That theorem tells us
that $\Delta^{\prime\prime}=\Omega(\Delta^{\prime})$. Combining this fact with
Lemma 14 (ii), we have (ii). [6, Theorem 6.1] also tells us that there exists

$T=\mathcal{O}\left(\frac{\mathcal{N}(\Lambda)^{2}}{\Delta^{\prime}}\left(\frac{\epsilon_{0}^{\prime}}{\Delta^{\prime}}+1\right)\right)=\mathcal{O}\left(\frac{\mathcal{N}(\Lambda)^{2}}{\Delta}\left(\frac{\epsilon_{0}+\delta_{2}/(1-\delta_{1}^{2})}{\Delta}+1\right)\right),$
(38)

such that
$D(\ket{\psi_{0}^{\prime}},\ket{\psi_{0}^{\prime\prime}})=e^{-\Omega(t/\mathcal{N}(\Lambda))}$
for $t\geq T$, where $\mathcal{N}(\Lambda)$ comes from the rescaling. Here we
have used Lemma 14 (iv). Combining with Lemma 14 (iii) and the triangle
inequality, we have proved (iii). ∎

Now we can use Corollaries 5 and 6 to bound $\delta_{1}$ and $\delta_{2}$ in
Theorem 12, which leads to the following robustness result:

######  Corollary 15 (Robustness to truncations).

Assume that the Hamiltonian satisfies Assumptions 2 and 3, and let
$\bar{\lambda}=\max_{1\leq x\leq s}\braket{}{\lambda_{x}}{}.$ Then the
truncated Hamiltonian $H^{\prime\prime}$ has a lowest eigenvalue
$\epsilon_{0}^{\prime\prime}$ corresponding to a non-degenerate ground state
$\ket{\psi_{0}^{\prime\prime}}$. And there exist constants $C_{1}$ and $C_{2}$
such that for any $\Lambda$ and $t$ satisfying

$\Lambda^{1-r}\geq(2\bar{\lambda})^{1-r}+C_{1}\Delta^{-1}\polylog(s,\Delta^{-1}),\qquad
t\geq\frac{C_{2}\mathcal{N}(\Lambda)^{2}}{\Delta^{2}},$ (39)

we have

  * (i)

$\Delta^{\prime\prime}=\Omega(\Delta)$;

  * (ii)

The trace distance between $\ket{\psi_{0}}$ and
$\ket{\psi^{\prime\prime}_{0}}$ can be bounded as follows:

$D(\ket{\psi_{0}^{\prime\prime}},\ket{\psi_{0}})=\poly(s,\Delta^{-1},\Lambda)e^{-\Omega\left(\sqrt{\Delta(\Lambda^{1-r}-(2\bar{\lambda})^{1-r})}\right)}+e^{-\Omega(t/\mathcal{N}(\Lambda))}.$
(40)

We recall that $r$ is the exponent involved in (2b), and that $r=0$ for LGTs,
$r=1/2$ for the Hubbard-Holstein model. $\Delta$ here is the spectral gap, and
$s$ is the size of the region around the cut that we want to pay special
attention to in (23).

If we assume $\bar{\lambda}=O(1)$, which we proved for the Hubbard-Holstien
model and U(1) and SU(2) LGTs in Section III, then to achieve
$D(\ket{\psi_{0}^{\prime\prime}},\ket{\psi})^{2}\leq\delta$, it suffices to
require, for some constant $C$,

$\sqrt{\Delta(\Lambda^{1-r}-C)}\geq
C\log\frac{s\Lambda}{\delta\Delta}\quad\text{and}\quad t\geq
C\mathcal{N}(\Lambda)\log(1/\delta)\vee\frac{C\mathcal{N}(\Lambda)^{2}}{\Delta^{2}},$

where $\vee$ denotes the maximum, which can be satisfied by choosing

$\Lambda=\poly(\Delta^{-1})\polylog(s/\delta)\quad\text{and}\quad
t=\Theta(\mathcal{N}(\Lambda)\log(1/\delta)\vee{\mathcal{N}(\Lambda)^{2}}/{\Delta^{2}}).$
(41)

##  V Area law

In this section, we establish our main result of an entanglement area law for
unbounded quantum systems. We first recall the notion of AGSP from [6,
Definition 2.1].

######  Definition 16.

$K$ is a $(\sigma,R)$-AGSP of a Hamiltonian $H$ on a bipartite system
consisting of two parts $A$ and $B$ that has a non-degenerate ground state, if

  1. 1.

$K\ket{\Psi}=\ket{\Psi}$, where $\ket{\Psi}$ is the ground state of $H$.

  2. 2.

$\|K\ket{\Phi}\|\leq\sigma$ for any $\ket{\Phi}$ such that
$\braket{\Phi}{\Psi}=0$.

  3. 3.

There exist operators $K_{j}^{A}$ and $K_{j}^{B}$, acting on $A$ and $B$
respectively, $j=1,2,\ldots,R$, such that $K=\sum_{j=1}^{R}K_{j}^{A}\otimes
K_{j}^{B}$.

An AGSP preserves the ground state, suppresses the excited states, and
increases the entanglement by a finite amount. The existence of an AGSP onto
the exact ground state is known to imply a bound on the entanglement of ground
state. More precisely, Corollary III.4 of [5] states that if $\sigma R\leq
1/2$ where $\sigma$ is the shrinking factor and $R$ is the entanglement rank
of the AGSP, then the entanglement entropy of the ground state satisfies a
bound of order $\log R$. For frustrated systems the target space of the AGSP
becomes perturbed away from the exact ground state(s) as its construction
involves spectral truncations. Analyses of such a situation are undertaken in
[6] and [7]. These tools were simplified in [1], where an “off-the-shelf”
lemma was stated which generalizes the one of [5] to perturbed and degenerate
target spaces.

Here we make a further improvement to [1] to obtain the cleaner and slightly
stronger statement of Lemma 17 below. For two subspaces
$\mathcal{Y},\mathcal{Z}\subset\mathcal{H}$, we say $\mathcal{Y}$ is
$\delta$-viable for $\mathcal{Z}$ if
$\braket{z}{P_{\mathcal{Y}^{\perp}}}{z}\leq\delta$ for all unit vectors
$\ket{z}\in\mathcal{Z}$, where $P_{\mathcal{Y}^{\perp}}$ is the projection
onto the orthogonal complement of subspace $\mathcal{Y}$. We write
$\mathcal{Y}\approx_{\delta}\mathcal{Z}$ if
$\bra{z}P_{\mathcal{Y}^{\perp}}\ket{z}\leq\delta$ and
$\bra{y}P_{\mathcal{Z}^{\perp}}\ket{y}\leq\delta$ for unit vectors
$\ket{y}\in\mathcal{Y}$ and $\ket{z}\in\mathcal{Z}$ respectively.

######  Lemma 17.

Let $\mathcal{Z}$ be a subspace of bipartite space
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. Let
$\tilde{\mathcal{Z}}_{n}$ be a sequence of subspaces of $\mathcal{H}$ such
that $\tilde{\mathcal{Z}}_{n}\approx_{\delta_{n}}\mathcal{Z}$ where
$\delta_{n}$ is a sequence such that $\sum_{n=0}^{\infty}n\delta_{n}=O(1)$.

Suppose there exist $K_{1},K_{2},\ldots$ such that $K_{n}$ is an
$(\sigma^{n},R^{n})$-AGSP with target space $\tilde{\mathcal{Z}}_{n}$ where
$\sigma=\frac{1}{2R}$. Then the entanglement entropy of any state in
$\mathcal{Z}$ is $O(\log R+\log\dim\mathcal{Z})$.

###### Proof.

This follows from the proof of [1, Lemma 4.7] and the following strengthening
of [1, Lemma 4.6]. ∎

######  Lemma 18.

Let $\mathcal{Z}$ be a subspace of $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ and
suppose there exists a subspace $\mathcal{V}\subset\mathcal{H}_{1}$ with
$\dim(\mathcal{V})=V$ which is $\delta$-viable for $\mathcal{Z}$. Pick any
normalized state $\ket{\psi}\in\mathcal{Z}$ and write the Schmidt
decomposition $\sum_{i}\sqrt{\lambda_{i}}\ket{x_{i}}\ket{y_{i}}$ with
nonincreasing Schmidt coefficients. Then we have the tail bound
$\sum_{i>V}\lambda_{i}\leq\delta$.

###### Proof.

By the definition of $\delta$-viability, we have

$\langle z^{\prime}|(P_{\mathcal{V}^{\bot}}\otimes
I)|z^{\prime}\rangle\leq\delta$ (42)

for all normalized state $|z^{\prime}\rangle$ in $\mathcal{Z}$. In particular,
this implies

$\langle z|(P_{\mathcal{V}^{\bot}}\otimes
I)|z\rangle=\sum_{i}\lambda_{i}\langle
x_{i}|P_{\mathcal{V}^{\bot}}|x_{i}\rangle=\mathrm{tr}\left(P_{\mathcal{V}^{\bot}}\sum_{i}\lambda_{i}|x_{i}\rangle\langle
x_{i}|\right)\leq\delta.$ (43)

Here, $P_{\mathcal{V}^{\bot}}$ projects onto $\mathcal{V}^{\bot}$ with
$\dim(\mathcal{V}^{\bot})=\dim(\mathcal{H}_{1})-V$. Now use Poincaré’s
inequalities [22, Corollary 4.3.39] to conclude that

$\mathrm{tr}\left(P_{\mathcal{V}^{\bot}}\sum_{i}\lambda_{i}|x_{i}\rangle\langle
x_{i}|\right)\geq\sum_{i=\dim(\mathcal{H}_{1})-\dim(\mathcal{V}^{\bot})+1}^{\dim(\mathcal{H}_{1})}\lambda_{i}=\sum_{i=V+1}^{\dim(\mathcal{H}_{1})}\lambda_{i}.$
(44)

This establishes the claimed bound. ∎

We apply Lemma 17 to the case of a simple ground state. Since
$\operatorname{span}\ket{\psi_{1}}\approx_{\delta}\operatorname{span}\ket{\psi_{2}}$
where $\delta=D(\ket{\psi_{1}},\ket{\psi_{2}})^{2}$ we obtain

######  Corollary 19.

Suppose there exist $K_{1},K_{2},\ldots$ such that $K_{n}$ is an
$(\sigma^{n},R^{n})$-AGSP with target state $\ket{\psi_{n}}$ where
$\sigma=\frac{1}{2R}$. If
$\sum_{n=0}^{\infty}nD(\ket{\psi_{n}},\ket{\psi})^{2}=O(1)$ then the
entanglement entropy of $\ket{\psi}$ is $O(\log R)$.

The Hamiltonian $H^{\prime\prime}$ of (26) has spectral norm
$O(s\mathcal{N}(\Lambda)+T(\Lambda))$. Under the conditions of Corollary 15,
$H^{\prime\prime}$ has a spectral gap $\Delta^{\prime\prime}=\Omega(\Delta)$,
so we have

$\Delta^{\prime\prime}/\|H^{\prime\prime}\|=\Omega\Big{(}\frac{\Delta}{s\mathcal{N}(\Lambda)+T(\Lambda)}\Big{)}.$

The Chebyshev polynomial of degree $\ell$ is bounded by $1$ on the unit
interval but satisfies
$T_{\ell}(1+\Delta)\geq\frac{1}{2}(1+\sqrt{2\Delta})^{\ell}=\frac{1}{2}\exp(\Omega(\ell\sqrt{\Delta}))$,
so composing it with linear transformations yields a polynomial $f$ with
$f_{\ell}(0)=1$ and $|f_{\ell}(\lambda)|\leq
2\exp(-\Omega(\ell\sqrt{\Delta/M}))$ for $\lambda\in[\Delta,M]$. Picking
$M=\|H^{\prime\prime}\|$ we get that $K=f_{\ell}(H^{\prime\prime})$ is an AGSP
with shrinking factor

$\sigma\lesssim\exp\Big{(}-\Omega(\ell\sqrt{\Delta^{\prime\prime}/\|H^{\prime\prime}\|})\Big{)}=\exp\Big{[}-\Omega\Big{(}\frac{\ell\sqrt{\Delta}}{\sqrt{s\mathcal{N}+T}}\Big{)}\Big{]}.$
(45)

Assume $\bar{\lambda}=O(1)$. We may pick parameters as in Eq. (41) so that
$D(\ket{\psi_{0}^{\prime\prime}},\ket{\psi_{0}})^{2}\leq\delta$. Since
$\mathcal{N}(\Lambda)=\poly(\Lambda)$, Eq. (41) implies

$\Lambda,\mathcal{N}(\Lambda),T(\Lambda)=\poly(\Delta^{-1})\polylog(s/\delta).$
(46)

Eq. (45) then becomes

$\sigma(\delta)\lesssim\exp\Big{[}-\Omega\Big{(}\ell\cdot\frac{\Delta^{O(1)}}{\sqrt{s}\cdot\polylog(s/\delta)}\Big{)}\Big{]}.$

$H^{\prime\prime}$ is a spin chain with a length-$s$ segment such that each
qudit on the segment has local dimension $d(\Lambda)=\Lambda^{O(1)}$. The
amortization bound from the proof of [6, Lemma 4.2] yields that $K$ has
entanglement rank:

$R(\delta)=(\ell
d)^{O(\ell/s+s)}\leq\exp(C(\ell/s+s)\log(\ell\Lambda))\leq\exp[C(\ell/s+s)\big{(}\log(\ell/\Delta)+\log\log(s/\delta)\big{)}].$

Let $\ell=s^{2}$:

$R(\delta)=\exp\Big{[}Cs\big{(}\log
s+\log(\Delta^{-1})+\log\log(1/\delta)\big{)}\Big{]},\qquad\sigma(\delta)=\exp\Big{[}-cs^{3/2}\cdot(\tfrac{\Delta}{\log(s/\delta)})^{O(1)}\Big{]}.$
(47)

######  Theorem 20 (Area law).

Under Assumptions 2 and 3 with $\bar{\lambda}=O(1)$, the ground state of $H$
satisfies an area law with entanglement entropy bounded by
$\poly(\Delta^{-1})$.

We remark that $\bar{\lambda}=O(1)$ is proved for the U(1) and SU(2) LGTs
(Corollary 9), as well as for the Hubbard-Holstein model (Corollary 10), in
Section III.

###### Proof.

Pick $\delta_{k}=k^{-3}$ so that $\sum k\delta_{k}<\infty$. Let $r$ be a
parameter depending on $\Delta$ and for $k=1,2,\ldots$ consider
$s=rk/\log(rk)$. Then (47) yields entanglement bounds and shrinking factors

$R_{k}=\exp[C_{0}rk\log(\Delta^{-1})],\qquad\sigma_{k}=\exp[-c(rk)^{3/2}\cdot(\tfrac{\Delta}{\log(rk)})^{O(1)}].$

In particular we have $R_{k}=R^{k}$ where $R=R_{1}$. To apply Corollary 19 it
remains to ensure that $\sigma_{k}\leq(2R)^{-k}$ for all $k\in\mathbb{N}$

$(2R)^{k}\sigma_{k}\leq\exp\Big{[}k+C_{0}rk\log(\Delta^{-1})-c(rk)^{3/2}\cdot(\tfrac{\Delta}{\log(rk)})^{O(1)}\Big{]}.$

So we just need the expression in the square brackets to be negative. Dividing
the exponent by $rk$ we see that it suffices to pick $r$ such that for all
$k$,

$1/r+C_{0}\log(\Delta^{-1})\leq
c\sqrt{rk}\cdot\Big{(}\frac{\Delta}{\log(rk)}\Big{)}^{C_{1}}.$

Pick $r=\tilde{\Theta}(\Delta^{-2C_{1}})$ to achieve this for all $k\geq 1$.
By Lemma 17 the ground state entanglement entropy is bounded by:

$\log R=C_{0}r\log(\Delta^{-1})=\poly(\Delta^{{}^{-1}}).$

This completes our proof. ∎

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