# Entanglement Entropy Bounds in the Higher Spin XXZ Chain

Christoph Fischbacher1 1 Department of Mathematics
University of California, Irvine
Irvine, CA, 92697, USA<EMAIL_ADDRESS>and  Oluwadara Ogunkoya2 2 Department
of Mathematics
University of Alabama at Birmingham
Birmingham, AL, 35294, USA<EMAIL_ADDRESS>
###### Abstract.

We consider the Heisenberg XXZ spin-$J$ chain ($J\in\mathbb{N}/2$) with
anisotropy parameter $\Delta$. Assuming that $\Delta>2J$, and introducing
threshold energies $E_{K}:=K\left(1-\frac{2J}{\Delta}\right)$, we show that
the bipartite entanglement entropy (EE) of states belonging to any spectral
subspace with energy less than $E_{K+1}$ satisfy a logarithmically corrected
area law with prefactor $(2\lfloor K/J\rfloor-2)$.

This generalizes previous results by Beaud and Warzel [4] as well as Abdul-
Rahman, Stolz and one of the authors [1], who covered the spin-$1/2$ case.

##  1\. Introduction

In the following, we are going to prove upper bounds for the bipartite
entanglement entropy of the spin–$J$ XXZ Heisenberg chain, where
$J\in\\{1/2,1,3/2,2\dots\\}$. The spin–$1/2$ Heisenberg XXZ chain under the
presence of a random magnetic field has recently attracted significant
interest in the rigorous study of many–body–localization (MBL), where MBL
phenomena such as exponential clustering of correlations, zero–velocity
Lieb–Robinson bounds and area laws for the entanglement entropy (EE) for
states from the lowest energy regime (“droplet” regime) have been shown
recently in [3, 4, 6, 7], for a recent survey of these results, cf. also [21].

Recall that for one-dimensional models such as the chain, we say that the EE
satisfies an area law with respect to the bipartition of the chain into a
“right” and a “left” subchain if it is uniformly bounded in the size of the
subchain. While an area law is generally considered as an indicator of MBL,
many delocalized systems seem to exhibit a logarithmic correction, which means
that the EE scales like the logarithm of the subchain’s length. See the
results in [4, 9, 12, 13, 15, 16, 18, 23], where such logarithmic corrections
to an area for the EE have been obtained.

In what follows, we will adapt the ideas of Beaud and Warzel from [4], where a
log–corrected area law for droplet states of the spin–$1/2$ Heisenberg XXZ
chain for the generic case and a true area law under the presence of a
disordered magnetic field was shown. With the help of technical refinements,
this result was improved to show a log–corrected energy law also for
higher–energy states in [1].

While our results can certainly be viewed as a technical improvement of the
spin-$1/2$ case, we nevertheless believe them to be of additional interest,
since they demonstrate that given a discrete many–particle Schrödinger-type
operator in one dimension, a logarithmically corrected area law for states in
a given energy range follows if the following four criteria are met: (i) a
suitable relative bound that controls the hopping operator in terms of the
potential, (ii) the potential energetically favoring configurations with a
lower number of “building blocks”, (iii) a sufficiently low dimension of the
space of these building blocks and (iv) a generalized Pauli principle that
limits the amount of particles that can occupy any site.

We will proceed as follows:

In Section 2, we will introduce the model and review previous results. We
recall that the Spin-J XXZ model on the chain with anisotropy parameter
$\Delta$ is unitarily equivalent to a direct sum of discrete $N$–particle
Schrödinger operators of the form $-(2\Delta)^{-1}A_{N}+V_{N}$, where $A_{N}$
is a weighted adjacency operator and $V_{N}$ is an interaction potential
(Prop. 2.16). Compared to the previously studied spin-$1/2$ case, $V_{N}$ is
much more complicated and we determine the set of all its minimizing
configurations for sufficiently large $N$ (Prop. 2.5). Using that the kinetic
term $A_{N}$ is controlled by the potential $V_{N}$ in the sense that
$-4JV_{N}\leq A_{N}\leq 4JV_{N}$ (Prop. 2.17), we use a suitable Combes–Thomas
estimate (Thm. 2.9) previously shown in [1], which allows as to obtain decay
estimates on spectral projections (Thm. 2.10). This together with some
previous results on estimating the entanglement entropy (Lemma 2.68) reduces
the problem to showing bounds for the sum of all possible $N$-particle
configurations in the subsystem weighted by their distance to the nearest
configuration with sufficiently low potential (Cor. 3.2).

Showing these bounds is the main goal of Section 3: We firstly prove a
suitable bound for single–cluster configurations in the spin-$1/2$ case (Lemma
3.3), which we appropriately generalize to certain low–potential
configurations for the spin-$J$ case (Lemma 3.5). We then introduce the notion
of “building blocks” (Def. 3.4) out of which any $N$–particle configuration
can be composed. Exploiting that if an $N$-particle configuration has
potential energy of $K$ or less, then it can be composed out of no more than
$(\lfloor K/J\rfloor-1)$ building blocks, we obtain a suitable bound for
configurations of arbitrary high potential energy (Lemma 3.30). Putting
everything together, we prove our main result (Thm. 3.12), where we show that
states belonging to any spectral subspace with energy strictly less than
$E_{K+1}=(K+1)\left(1-\frac{2J}{\Delta}\right)$ satisfy a logarithmically
corrected area law with prefactor $(2\lfloor K/J\rfloor-2)$.

Acknowledgements: C.F. is grateful to the Insitut Mittag-Leffler in Djursholm,
Sweden, where some of this work was done as part of the program Spectral
Methods in Mathematical Physics in Spring 2019. The authors would also like to
thank Günter Stolz for many useful and insightful discussions and Shannon
Starr for encouragement and support. It is also a pleasure to thank Abel Klein
who suggested the much shorter proof of Theorem 2.10 presented here.

##  2\. Previous Results

For any fixed $J\in\mathbb{N}/2$, we consider the chain of length
$L\in\mathbb{N}\setminus\\{1\\}$, which is described by the Hamiltonian

(2.1) $H_{L}=\sum_{j=1}^{L-1}h_{j,j+1}+J(2J-S_{1}^{3}-S_{L}^{3})$

acting on the Hilbert space
$\mathcal{H}_{L}=\bigotimes_{j=1}^{L}\mathbb{C}^{2J+1}=\bigotimes_{j=1}^{L}\mathcal{H}_{j}$,
with the interpretation of $\mathcal{H}_{j}=\mathbb{C}^{2J+1}$ as being the
local Hilbert space describing a spin–$J$ particle located at site $j$. For
later convenience, we also introduce the notation
$\Lambda_{L}=\\{1,2,\dots,L\\}$. The two–site Hamiltonian $h_{j,j+1}$ is given
by

(2.2)
$h_{j,j+1}=J^{2}-S_{j}^{3}S_{j+1}^{3}-\frac{1}{\Delta}(S_{j}^{1}S_{j+1}^{1}+S_{j}^{2}S_{j+1}^{2})=J^{2}-S_{j}^{3}S_{j+1}^{3}-\frac{1}{2\Delta}(S_{j}^{+}S_{j+1}^{-}+S_{j}^{-}S_{j+1}^{+})$

and the additional term $J(2J-S_{1}^{3}-S_{L}^{3})$ describes a boundary
field; usually referred to as “droplet boundary condition”. Here $S^{1},S^{2}$
and $S^{3}$ are the spin–$J$ matrices and $S^{\pm}=(S^{1}\pm iS^{2})/2$ are
the spin raising and lowering operators, respectively (for the full definition
of these matrices see e.g. [8] or [14]). Labeling an operator
$A\in\mathbb{C}^{(2J+1)\times(2J+1)}$ by a site $j$, like e.g. $A_{j}$, means
that it acts as $A$ on the $j$-th factor of the tensor product and as the
identity on all the other factors. In what follows, we assume that the
anisotropy parameter $\Delta$ satisfies $\Delta>2J$. While we emphasize that
$\Delta>2J$ is not necessary for the operators $h_{j,j+1}$ to be non-negative
(see, e.g. [20, p.16]), it is certainly sufficient (see Proposition 2.17
below).

###  2.1. Equivalence to a direct sum of Schrödinger-type operators

In the following, we are going to discuss the equivalence of the spin-$J$ XXZ
Hamiltonian to a direct sum of discrete many-particle Schödinger-type
operators. To this end, we will firstly review previous work [8], where we
described configurations of particles via functions ${\bf
m}:\Lambda_{L}\rightarrow\\{0,1,\dots,2J\\}$, where for each
$i\in\Lambda_{L}$, the value ${\bf m}(i)$ represents number of particles that
are located at site $i$ when in configuration $\bf{m}$.

Equivalently, a configuration of $N$ particles distributed over $\Lambda_{L}$
can be described using ordered multisets $X$ with elements in $\Lambda_{L}$.
Here, the value of the $i$-th element $x_{i}\in X$ represents the location of
the $i$-th particle of a configuration of $N$ particles.

For our purposes, it will be advantageous to use both points of view as they
have their respective advantages: the first description involving occupation
numbers will be more useful when it comes to analyzing the interaction
potential of the Schrödinger-type operators. On the other hand, once it has
been shown how the entanglement entropy of a state can be estimated using the
Combes-Thomas bound for spectral projections, the problem boils down to a
combinatorial problem estimating exponentially weighted sums over a large set
of many-particle configurations, for which the multiset point of view will be
more convenient.

####  2.1.1. Previous approach using occupation numbers

In [8, Prop. 2.1], it was shown that the Hamiltonian $H_{L}$ is unitarily
equivalent to a direct sum of many–body Schrödinger operators. One firstly
observes that $H_{L}$ preserves the total magnetization/particle number: to
this end, we define the _local particle number operator_
$\mathcal{N}^{loc}:=(J-S^{3})$ acting on $\mathbb{C}^{2J+1}$ which has
spectrum $\sigma(\mathcal{N}^{loc})=\\{0,1,\dots,2J-1,2J\\}$. For any site
$j\in\\{1,\dots,L\\}$, we interpret the eigenvalues of $\mathcal{N}^{loc}_{j}$
as the number of particles located at site $j$. We then define the _total
particle number operator_ $\mathcal{N}_{L}$ as

(2.3) $\mathcal{N}_{L}:=\sum_{j=1}^{L}\mathcal{N}_{j}^{loc}\>,$

where the eigenvalues $\sigma(\mathcal{N}_{L})=\\{0,1,2,\dots,2JL-1,2JL\\}$ of
$\mathcal{N}_{L}$ are consequently interpreted as the total number of
particles. It can now be verified that $[H_{L},\mathcal{N}_{L}]=0$, and thus
we decompose $\mathcal{H}_{L}=\bigoplus_{N=0}^{2JL}\mathcal{H}_{L}^{N}$, where
$\mathcal{H}_{L}^{N}$ denotes the eigenspace of $\mathcal{N}_{L}$
corresponding to the eigenvalue $N$ – the space of all $N$-particle
configurations in $\\{1,2,\dots,L\\}$ with the restriction that no site can be
occupied by more than $2J$ particles. We also define
$H_{L}^{N}:=H_{L}\upharpoonright_{\mathcal{H}_{L}^{N}}$. Now, let

(2.4)
${\bf{M}}^{N}_{L}:=\left\\{{\bf{m}}:\\{1,2,\dots,L\\}\rightarrow\\{0,1,\dots,2J\\}:\sum_{j=1}^{L}{\bf{m}}(j)=N\right\\}\>,$

be the set of all functions from $\\{1,2,\dots,L\\}$ to $\\{0,1,\dots,2J\\}$
whose values add up to $N$. For convenience, we also define ${\bf
M}_{L}:=\bigcup_{N=0}^{2JL}{\bf M}_{L}^{N}$ – the set of _all_ functions from
$\\{1,2,\dots,L\\}$ to $\\{0,1,\dots,2J\\}$. Let $\\{e_{k}\\}_{k=0}^{2J}$
denote a normalized eigenbasis of $\mathcal{N}^{loc}$, such that for any
$k\in\\{0,1,\dots,2J\\}$, we have $\mathcal{N}^{loc}e_{k}=k\cdot e_{k}$. We
then define for any ${\bf m}\in{\bf M}_{L}$

(2.5) $\psi_{\bf m}:=\bigotimes_{j=1}^{L}e_{{\bf m}(j)}\>.$

This means in particular that for any $j\in\\{1,2,\dots,L\\}$ we get
$\mathcal{N}_{j}^{loc}\psi_{\bf m}={\bf m}(j)\psi_{\bf m}$. In other words,
$\psi_{\bf m}$ describes a configuration of particles, where at each site
$j\in\\{1,2,\dots,L\\}$, there are exactly ${\bf m}(j)$ particles. Since

(2.6) $\mathcal{N}_{L}\psi_{\bf m}=\left(\sum_{j=1}^{L}{\bf
m}(j)\right)\psi_{\bf m}\>,$

it immediately follows that

(2.7) $\mathcal{H}_{L}^{N}=\operatorname{span}\left\\{\psi_{\bf m}:{\bf
m}\in{\bf{M}}^{N}_{L}\right\\}\>.$

Now, consider the Hilbert space
$\ell^{2}({\bf{M}}^{N}_{L})=\left\\{f:{\bf{M}}^{N}_{L}\rightarrow\mathbb{C}\right\\}$
equipped with inner product $\langle f,g\rangle=\sum_{{\bf
m}\in{\bf{M}}^{N}_{L}}\overline{f({\bf m})}g({\bf m})$ and let $\\{\phi_{\bf
m}\\}_{{\bf m}\in{\bf{M}}^{N}_{L}}$ denote the canonical basis of
$\ell^{2}({\bf{M}}^{N}_{L})$, i.e.

(2.8) $\phi_{\bf m}({\bf n})=\begin{cases}&1\quad\mbox{if}\quad{\bf m}={\bf
n}\\\ &0\quad\mbox{else.}\end{cases}$

The Hilbert spaces $\mathcal{H}_{L}^{N}$ and $\ell^{2}({\bf{M}}^{N}_{L})$ are
unitarily equivalent via

(2.9)
$U_{L}^{N}:\mathcal{H}_{L}^{N}\rightarrow\ell^{2}({\bf{M}}^{N}_{L}),\quad\psi_{\bf
m}\mapsto\phi_{\bf m}\>.$

For any $f\in\ell^{2}({\bf{M}}^{N}_{L})$, let us now define the adjacency
operator $A_{N}$, given by

(2.10) $(A_{N}f)({\bf m})=\sum_{{\bf n}:{\bf n}\sim{\bf m}}w({\bf m},{\bf
n})f({\bf n})\>,$

where for two configurations ${\bf m},{\bf n}\in{\bf{M}}^{N}_{L}$ to be
adjacent (denoted by ${\bf m}\sim{\bf n}$) is defined as follows:

(2.11) $\displaystyle{\bf m}\sim{\bf n}:\Leftrightarrow$ $\displaystyle\exists
j_{0}$ $\displaystyle\in\\{1,2,\dots,L-1\\}:{\bf m}(j_{0})-{\bf n}(j_{0})=\pm
1\mbox{ and }{\bf m}(j_{0}+1)-{\bf n}(j_{0}+1)=\mp 1$ (2.12)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\mbox{ and for any
}j\in\\{1,2,\dots,L\\}\setminus\\{j_{0},j_{0}+1\\}:{\bf m}(j)={\bf n}(j)\>.$

This definition should be interpreted in the following way: two configurations
${\bf m,n}$ of $N$ particles distributed over $L$ sites (with the requirement
that no site be occupied by more than $2J$ particles) are adjacent if one
configuration can be obtained by moving a single particle from the other
configuration to the right or left (cf. Figure 2.1).

$1$$2$$3$$4$$5$$6$$7$$8$${\bf m}$${\bf n}$

Figure 2.1. An example of two adjacent configurations ${\bf m,n}\in{\bf
M}_{8}^{11}$ (here: $J=2$). The values of the functions at a site are
represented by blue circles corresponding to particles occupying the
respective sites, e.g. ${\bf m}(2)=4$. Since ${\bf m}(2)-{\bf n}(2)=4-3=1$ and
${\bf m}(3)-{\bf n}(3)=1-2=-1$, while ${\bf m}(j)={\bf n}(j)$ for any other
site $j$, we have ${\bf m}\sim{\bf n}$. We interpret configuration ${\bf n}$
as obtained from configuration ${\bf m}$ by a particle hopping to the right
from site $2$ to site $3$. (Represented by an arrow emanating from the hopping
particle in configuration ${\bf m}$.)

For ${\bf m}\sim{\bf n}$, the weight function $w({\bf m,n})=w({\bf n,m})$ in
(2.10) is given by

(2.13) $w({\bf m,n})=\prod_{j:{\bf m}(j)\neq{\bf n}(j)}\left(J({\bf m}(j)+{\bf
n}(j)+1)-{\bf m}(j){\bf n}(j)\right)^{1/2}\>,$

however in what follows, the explicit expression in (2.13) will not play a
particularly important role. Moreover, for any ${\bf m,n}\in{\bf M}_{L}^{N}$,
we define their distance $d^{N}({\bf m,n})$ to be the length of the shortest
path connecting ${\bf m}$ and ${\bf n}$, which we will refer to as graph
distance.

######  Remark 2.1.

For more details concerning the construction of the spin-$J$ Heisenberg XXZ
model on more general underlying graph (in lieu of just the chain), we refer
to [8], where appropriate $N$-particle graphs were introduced. Similar
approaches have been used for the spin-$1/2$ case in [10, 17]. Consider also
[2, 5, 19] and the references therein, where similar constructions have been
studied from a graph-theoretic point of view.

Next, we define the full interaction potential $V_{N}$, which is a
multiplication operator on $\ell^{2}({\bf M}_{L}^{N})$, to be given by

(2.14) $(V_{N}f)({\bf m})=V({{\bf m}})f({\bf m})=\left(\sum_{j=1}^{L-1}v({\bf
m}(j),{\bf m}(j+1))\right)f({\bf m})+J({\bf{m}}(1)+{\bf{m}}(L))f(\bf{m})\>,$

where the two–site potential $v$ is given by

(2.15) $v({\bf m}(j),{\bf m}(j+1))=J({\bf m}(j)+{\bf m}(j+1))-{\bf m}(j){\bf
m}(j+1)$

and we refer to the extra term “$J({\bf m}(1)+{\bf m}(L))$” as the “boundary
field”.

In what follows, the following two facts will be particularly important:
firstly that the operator $H_{L}^{N}$ is unitarily equivalent to the
Schrödinger-type operator $-(2\Delta)^{-1}A_{N}+V_{N}$ (Prop. 2.16) and
moreover that that kinetic term can be controlled in terms of the potential
(Prop. 2.17):

######  Proposition 2.2 ([8, Prop. 2.1]).

We have the following unitary equivalence:

(2.16)
$U_{L}^{N}H_{L}^{N}(U_{L}^{N})^{*}=-\frac{1}{2\Delta}A_{N}+V_{N}=:H_{N}\equiv
H_{N}(L)\>.$

###### Remark on the proof.

The only detail which is not discussed in [8] is the unitary equivalence of
the boundary field term “$J({\bf{m}}(1)+{\bf{m}}(L))f(\bf{m})$” in (2.14) to
the boundary field “$J(2J-S_{1}^{3}-S_{L}^{3})$” in (2.1), which can be
verified by an easy calculation. ∎

######  Remark 2.3.

For the special cases $N=0$ and $N=2JL$, note that

$\dim(\ell^{2}({\bf M}_{L}^{0}))=\dim(\ell^{2}({\bf M}_{L}^{2JL}))=1\>.$

On these one-dimensional spaces, the operators $H_{0}$ and $H_{2JL}$ are just
given by $H_{0}=0$ and $H_{2JL}=4J^{2}$.

Let us now recall a useful relative bound of $A_{N}$ in terms of the potential
$V_{N}$. It is because of this particular feature of the model that we do not
have to worry about the explicit form of the weight function $w$ given in
(2.13).

######  Proposition 2.4 ([8, Lemma 2.9]).

The operators $A_{N}$ and $V_{N}$ satisfy the following relative bound:

(2.17) $-4JV_{N}\leq A_{N}\leq 4JV_{N}\>.$

###### Remark on the proof.

Strictly speaking, in [8, Lemma 2.9], it was only shown that $A_{N}\leq
4JV_{N}$. However, the lower bound $-4JV_{N}\leq A_{N}$ follows from a
completely analogous argument.

∎

Let us now further analyze the interaction potential $V_{N}$ and determine all
the configurations which minimize its value, the proof can be found in
Appendix B.1

######  Proposition 2.5.

Let $N\in\\{4J,4J+1,\dots,2JL\\}$ and define $V_{N,0}:=\min\\{V({\bf m}):{\bf
m}\in{\bf M}_{L}^{N}\\}$. Then

(2.18) $V_{N,0}:=4J^{2}\>.$

Moreover, – up to overall translations – the minimizers of $V_{N}$ are given
by

(2.19) ${\bf m}^{N}_{j}(x)=\begin{cases}j\quad&\mbox{if}\quad x=1\\\
2J&\mbox{if}\quad x=2,\ldots,r\\\ 2J-j&\mbox{if}\quad x=r+1\end{cases}$

for $N=2Jr,\;\;2\leq r\leq L-1$, $j=0,\ldots,2J-1$ and

(2.20) ${\bf m}^{N}_{j}(x)=\begin{cases}j\quad&\mbox{if}\quad x=1\\\
2J&\mbox{if}\quad x=2,\ldots,1+\lfloor\frac{N}{2J}\rfloor\\\ N(\mbox{mod
}2J)-j&\mbox{if}\quad x=2+\lfloor\frac{N}{2J}\rfloor\end{cases}$

if $N$ is not a multiple of $2J$, $j=0,\ldots,N(mod\;2J)-1$.

######  Remark 2.6.

The following figure provides an example of a minimizer of the form (2.19),
i.e. when $N\in\\{4J,6J,8J,\dots,2JL\\}$:

$k$$k+1$…$k+r$$j$$2J$$2J$$2J$$2J$$2J$$2J-j$

Figure 2.2. An example of a minimizer ${\bf m}_{j}^{N}(\cdot-k)$ when $N=2Jr$
as in (2.19). Here, $J=9/2$ and $j=3$.

######  Definition 2.7.

For any $K\in\mathbb{N}$, let ${\bf M}_{L,K}^{N}=\\{{\bf m}\in{\bf
M}_{L}^{N}:V({\bf m})\leq K\\}$, i.e. the set of occupation number functions
for which the potential $V_{N}$ is bounded by $K$.

####  2.1.2. An equivalent description using multisets

When it comes to tracking the positions of $N$ individual particles, it will
be convenient to introduce ordered $N$-tuples that satisfy certain conditions.
For brevity, in what follows, we will refer to these $N$-tuples as multisets.
Thus, for any $L\in\mathbb{N}$ and any $N\in\\{1,\dots,2JL\\}$, let us define

(2.21)
${\mathbb{S}}_{L}^{N}=\left\\{(x_{1},x_{2},\dots,x_{N})\in\Lambda_{L}^{N}:x_{1}\leq
x_{2}\leq\cdots\leq x_{N}\mbox{ and
}\min_{k\in\\{1,2,\dots,N-2J\\}}(x_{k+2J}-x_{k})\geq 1\right\\}\>,$

where for any $X=(x_{1},x_{2},\dots,x_{N})\in{\mathbb{S}}_{L}^{N}$, the value
of each individual $x_{i}$ represents the position of the $i$-th particle. The
condition

(2.22) $\min_{k\in\\{1,2,\dots,N-2J\\}}(x_{k+2J}-x_{k})\geq 1$

reflects the fact that no site can be occupied by more than $2J$ particles and
is therefore automatically satisfied if $N\leq 2J$. For later convenience, we
also introduce the convention ${\mathbb{S}}_{L}^{0}:=\\{\emptyset\\}$.

The correspondence between functions of occupation numbers ${\bf m}\in{\bf
M}_{L}^{N}$ and $N$-particle configurations $X\in{\mathbb{S}}_{L}^{N}$ is of
course straightforward. For a given
$X=(x_{1},x_{2},\dots,x_{N})\in{\mathbb{S}}_{L}^{N}$, the corresponding
function ${\bf m}_{X}\in{\bf M}_{L}^{N}$ is defined as

(2.23) ${\bf m}_{X}(j):=|\\{k\in\\{1,2,\dots,N\\}:x_{k}=j\\}|$

for any $j\in\Lambda_{L}$. It is not hard to see that the mapping
$X\mapsto{\bf m}_{X}$ is a bijection from ${\mathbb{S}}_{L}^{N}$ to ${\bf
M}_{L}^{N}$. We thus denote by $X_{\bf m}$ the image of the inverse of that
mapping applied to an arbitrary ${\bf m}\in{\bf M}_{L}^{N}$.111As an example,
consider the multiset corresponding to the occupation function ${\bf n}$ in
Figure 2.1, which would be given by  (2.24) $X_{\bf
n}=(1,2,2,2,3,3,5,7,7,8,8)\>.$ We will also use
$\\{\phi_{X}\\}_{X\in{\mathbb{S}}_{L}^{N}}$ to denote the canonical basis of
$\ell^{2}({\mathbb{S}}_{L}^{N})$, i.e. $\phi_{X}(Y)=1$ if $X=Y$ and
$\phi_{X}(Y)=0$ if $X\neq Y$. Then, the identification of the Hilbert spaces
$\ell^{2}({\bf M}_{L}^{N})$ with $\ell^{2}(\mathbb{S}_{L}^{N})$ via $\phi_{\bf
m}\mapsto\phi_{X_{\bf{m}}}$ is straightforward. So for any
$X,Y\in\mathbb{S}_{L}^{N}$, we define $X\sim Y:\Leftrightarrow{\bf
m}_{X}\sim{\bf m}_{Y}$ as well as $w(X,Y):=w({\bf m}_{X},{\bf m}_{Y})$ and
$V(X):=V({\bf m}_{X})$. By a slight abuse of notation, we will use the same
symbols $H_{N},A_{N}$ and $V_{N}$ to denote the unitarily equivalent operators
on $\ell^{2}(\mathbb{S}_{L}^{N})$, i.e.

(2.25)
$(H_{N}f)(X)=-\frac{1}{2\Delta}(A_{N}f)(X)+(V_{N}f)(X)=-\frac{1}{2\Delta}\sum_{Y:X\sim
Y}w(X,Y)f(Y)+V(X)f(X)$

for any $f\in\ell^{2}(\mathbb{S}_{L}^{N})$.

For any $X,Y\in{\mathbb{S}}_{L}^{N}$, let $d^{N}(X,Y):=d^{N}({\bf m}_{X},{\bf
m}_{Y})$. The merit of the multiset-point-of-view will now be made more
apparent by the following lemma; its proof can be found in Appendix A.

######  Lemma 2.8.

For any two configurations $X,Y\in{\mathbb{S}}_{L}^{N}$, where
$X=(x_{1},x_{2},\dots,x_{N})$ and $Y=(y_{1},y_{2},\dots,y_{N})$, we have

(2.26) $d^{N}(X,Y)=\sum_{i=1}^{N}|x_{i}-y_{i}|\>.$

###  2.2. Combes–Thomas estimate and bounds on spectral projections

One of the main ingredients of the proof of the bound for the entanglement
entropy will be a bound on spectral projections. To be more specific, let
$\mathcal{A}\subset\mathbb{S}_{L}$ be a set of configurations. For any such
$\mathcal{A}$, we define $P_{\mathcal{A}}$ to be the orthogonal projection
onto the subspace of functions that are supported on $\mathcal{A}$, i.e.

(2.27) $(P_{\mathcal{A}}f)(X)=\begin{cases}f(X)\quad&\mbox{if}\quad
X\in\mathcal{A}\\\ 0\quad&\mbox{if}\quad
X\in\mathbb{S}_{L}\setminus\mathcal{A}\end{cases}\>.$

For later convenience, we also define
$\overline{P}_{\mathcal{A}}:={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-P_{\mathcal{A}}$.

######  Theorem 2.9.

For any $N\in\\{1,2,\dots,2JL\\}$, let $Y_{N}$ be an arbitrary multiplication
operator on $\ell^{2}({\mathbb{S}}_{L}^{N})$ and $z\notin\sigma(H_{N}+Y_{N})$
such that there exists $\kappa_{z}>0$ for which

(2.28)
$\left\|V_{N}^{1/2}(H_{N}+Y_{N}-z)^{-1}V_{N}^{1/2}\right\|\leq\frac{1}{\kappa_{z}}<\infty\>.$

Then for all subsets $\mathcal{A,B}\subseteq{\mathbb{S}}_{L}^{N}$, we have

(2.29)
$\left\|P_{\mathcal{A}}\left(H_{N}+Y_{N}-z\right)^{-1}P_{\mathcal{B}}\right\|\leq\frac{1}{V_{N,0}}\left\|P_{\mathcal{A}}V_{N}^{1/2}\left(H_{N}+Y_{N}-z\right)^{-1}V_{N}^{1/2}P_{\mathcal{B}}\right\|\leq\frac{2}{V_{N,0}\kappa_{z}}\,e^{-\eta_{z}d^{N}(\mathcal{A,B})}\>,$

where

(2.30) $\eta_{z}=\log\left(1+\frac{\Delta\kappa_{z}}{4J}\right).$

###### Proof.

An abstract result of this form for Schrödinger-type operators whose kinetic
term is controlled by the potential such as in Proposition 2.17 was shown in
[1, Prop. 3.1]. The theorem now follows from plugging in the constants
particular to this model. ∎

Now, let $K\in\mathbb{N}$ and for any $\delta\in(0,1)$, define
$E_{K,\delta}:=\left(1-\frac{2J}{\Delta}\right)(K+1-\delta)$. Moreover, for
any non-negative multiplication operator $W_{N}$ on
$\ell^{2}({\mathbb{S}}_{L}^{N})$, let $Q_{K,\delta}^{N}\equiv
Q^{N}_{K,\delta}(L,W_{N}):={\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}_{[0,E_{K,\delta}]}(H_{N}(L)+W_{N})$
be the spectral projection of $H_{N}+W_{N}$ associated to the energy interval
$[0,E_{K,\delta}]$. Lastly, for any $K\in\mathbb{N}$, we introduce the set
$\mathbb{S}_{L,K}^{N}:=\\{X_{\bf m}:{\bf m}\in{\bf M}_{L,K}^{N}\\}$, where the
sets ${\bf M}_{L,K}^{N}$ were introduced in Definition 2.7.

######  Theorem 2.10.

Let $\mathcal{A}\subset{\mathbb{S}}_{L}^{N}$ be a set of configurations. We
then get the following estimate:

(2.31)
$\displaystyle\left\|P_{\mathcal{A}}Q_{K,\delta}^{N}\right\|=\left\|Q_{K,\delta}^{N}P_{\mathcal{A}}\right\|\leq
C_{N,K}e^{-\mu_{K}d^{N}(\mathcal{A},{\mathbb{S}}_{L,K}^{N})},$

where

(2.32) $\displaystyle C_{N,K}$
$\displaystyle=C_{N,K}(\Delta,\delta,J)=\max\left\\{1,\frac{8K(K+1)}{V_{N,0}\delta^{2}}\right\\}$
(2.33) $\displaystyle\quad\mbox{and}\quad\mu_{K}$
$\displaystyle=\mu_{K}(\Delta,\delta,J)=\log\left(1+\frac{\delta(\Delta-2J)}{16J(K+1)}\right).$

###### Proof.

We follow ideas from [6, Proof of Lemma 8.2] and [11]. If
$\mathcal{A}\cap{\mathbb{S}}_{L,K}^{N}\neq\emptyset$, which implies
$d^{N}(\mathcal{A},{\mathbb{S}}_{L,K}^{N})=0$, then (2.31) immediately follows
from $C_{N,K}\geq 1$. Hence, assume
${\mathcal{A}}\subset{\mathbb{S}}_{L}^{N}\setminus{\mathbb{S}}_{L,K}^{N}$ from
now on. Then, let us choose $Y_{N}=W_{N}+\gamma P_{{\mathbb{S}}_{L,K}^{N}}$
with $\gamma=\left(1-\frac{2J}{\Delta}\right)K$ and show that the operator
$H_{N}+W_{N}+\gamma P_{{\mathbb{S}}_{L,K}^{N}}$ satisfies the assumptions of
Theorem 2.9. Observe that for any $E\in[0,E_{K,\delta/2}]$ one has

(2.34) $\displaystyle
V_{N}^{-1/2}(H_{N}+Y_{N}-E)V_{N}^{-1/2}\geq-\frac{1}{2\Delta}V_{N}^{-1/2}A_{N}V_{N}^{-1/2}+{\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}V_{N}^{-1}-EV_{N}^{-1}$ (2.35) $\displaystyle\geq$
$\displaystyle\left(1-\frac{2J}{\Delta}\right)+\left(\left(1-\frac{2J}{\Delta}\right)K-E_{K,\delta/2}\right)V_{N}^{-1}P_{{\mathbb{S}}_{L,K}^{N}}-E_{K,\delta/2}\overline{P}_{{\mathbb{S}}_{L,K}^{N}}V_{N}^{-1}$
(2.36) $\displaystyle=$
$\displaystyle\left(1-\frac{2J}{\Delta}\right)\left({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}-(1-\delta/2)V_{N}^{-1}P_{{\mathbb{S}}_{L,K}^{N}}-(K+1-\delta/2)V_{N}^{-1}\overline{P}_{{\mathbb{S}}_{L,K}^{N}}\right)\>.$

Now, note that

(2.37)
$-(1-\delta/2)V^{-1}_{N}P_{{\mathbb{S}}_{L,K}^{N}}\geq-(1-\delta/2)P_{{\mathbb{S}}_{L,K}^{N}}\quad\mbox{as
well
as}\quad-V^{-1}_{N}\overline{P}_{{\mathbb{S}}_{L,K}^{N}}\geq-\frac{1}{K+1}\overline{P}_{{\mathbb{S}}_{L,K}^{N}}\>,$

which we use to further estimate (2.36):

(2.38)
$\eqref{eq:estimate}\geq\left(1-\frac{2J}{\Delta}\right)\left(\frac{\delta}{2}P_{{\mathbb{S}}_{L,K}^{N}}+\frac{\delta/2}{K+1}\overline{P}_{{\mathbb{S}}_{L,K}^{N}}\right)\geq\left(1-\frac{2J}{\Delta}\right)\frac{\delta}{2(K+1)}\>.$

From this, it can be concluded that for any $E\in[0,E_{K,\delta}]$, we have
$E\notin\sigma(H_{N}+W_{N}+\gamma P_{{\mathbb{S}}_{L,K}^{N}})$ and moreover
that

(2.39) $\|V_{N}^{1/2}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-E)^{-1}V_{N}^{1/2}\|\leq\frac{2(K+1)}{\delta\left(1-\frac{2J}{\Delta}\right)}$

and, by a slight modification (see [6, Lemma 4.3]), one gets for any
$\varepsilon\in\mathbb{R}$:

(2.40) $\|V_{N}^{1/2}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-E+i\varepsilon)^{-1}V_{N}^{1/2}\|\leq\frac{4(K+1)}{\delta\left(1-\frac{2J}{\Delta}\right)}\>.$

From Theorem 2.9, we therefore get for any
$\mathcal{A}\in{\mathbb{S}}_{L}^{N}\setminus{\mathbb{S}}_{L,K}^{N}$

(2.41) $\|P_{{\mathbb{S}}_{L,K}^{N}}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-E+i\varepsilon)^{-1}P_{\mathcal{A}}\|\leq\frac{8(K+1)}{V_{N,0}\cdot\delta\left(1-\frac{2J}{\Delta}\right)}e^{-\mu_{K}d^{N}(\mathcal{A},{\mathbb{S}}_{L,K}^{N})}\>,$

for any $E\in[0,E_{K,\delta/2}]$ and any $\varepsilon\in\mathbb{R}$. The
constant $\mu_{K}=\mu_{K}(\Delta,\delta,J)$ is given by

(2.42) $\mu_{K}=\log\left(1+\frac{\delta(\Delta-2J)}{16J(K+1)}\right)\>.$

Now, let $\Gamma$ be the circle centered at $\frac{1}{2}E_{K,\delta}$ with
radius $R:=\frac{1}{2}\left(1-\frac{2J}{\Delta}\right)(K+1)$. Note that this
implies
$\operatorname{dist}(\Gamma,[0,E_{K,\delta}])=\frac{\delta}{2}\left(1-\frac{2J}{\Delta}\right)=:\delta^{\prime}$.
Moreover, it follows from Proposition 2.17 that

(2.43) $\displaystyle H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}\geq-\frac{1}{2\Delta}A_{N}+V_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}\geq\left(1-\frac{2J}{\Delta}\right)V_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}$ (2.44)
$\displaystyle\geq\left(1-\frac{2J}{\Delta}\right)(V_{N}+K)P_{{\mathbb{S}}_{L,K}^{N}}+\left(1-\frac{2J}{\Delta}\right)V_{N}\overline{P}_{{\mathbb{S}}_{L,K}^{N}}$
(2.45) $\displaystyle\geq\left(1-\frac{2J}{\Delta}\right)(K+1)\>,$

which means that there is no spectrum of $H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}$ inside the circle $\Gamma$, which implies

(2.46) $\oint_{\Gamma}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-z)^{-1}\text{d}z=0\>.$

We therefore get

(2.47) $\displaystyle Q_{K,\delta}^{N}=$
$\displaystyle(Q_{K,\delta}^{N})^{2}=\frac{i}{2\pi}Q_{K,\delta}^{N}\oint_{\Gamma}(H_{N}+W_{N}-z)^{-1}\text{d}z$
(2.48) $\displaystyle=$
$\displaystyle\frac{i}{2\pi}Q_{K,\delta}^{N}\oint_{\Gamma}[(H_{N}+W_{N}-z)^{-1}-(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-z)^{-1}]\text{d}z$ (2.49) $\displaystyle=$
$\displaystyle\frac{i\gamma}{2\pi}Q_{K,\delta}^{N}\oint_{\Gamma}(H_{N}+W_{N}-z)^{-1}P_{{\mathbb{S}}_{L,K}^{N}}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-z)^{-1}\text{d}z$

We then proceed to estimate

(2.50) $\displaystyle\|Q_{K,\delta}^{N}P_{\mathcal{A}}\|$
$\displaystyle\leq\gamma
R\max_{z\in\Gamma}\left[\|Q_{K,\delta}^{N}(H_{N}+W_{N}-z)^{-1}\|\|P_{{\mathbb{S}}_{L,K}^{N}}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-z)^{-1}P_{\mathcal{A}}\|\right]$ (2.51)
$\displaystyle\leq\frac{\gamma
R}{\delta^{\prime}}\max_{z\in\Gamma}|\|P_{{\mathbb{S}}_{L,K}^{N}}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-z)^{-1}P_{\mathcal{A}}\|\>,$

where we have used
$\|Q_{K,\delta}^{N}(H_{N}+W_{N}-z)^{-1}\|\leq(\delta^{\prime})^{-1}$. Note
that for any $z\in\Gamma$, we have

(2.52)
$\operatorname{Re}z\leq\left(1-\frac{2J}{\Delta}\right)\left(K+1-\frac{\delta}{2}\right)=E_{K,\delta/2}\>,$

which means that we can apply (2.41) to estimate
$\|P_{{\mathbb{S}}_{L,K}^{N}}(H_{N}+W_{N}+\gamma
P_{{\mathbb{S}}_{L,K}^{N}}-z)^{-1}P_{\mathcal{A}}\|$ uniformly in
$z\in\Gamma$. We then get

(2.53) $\eqref{eq:specCT}\leq\frac{8\gamma
R(K+1)}{V_{N,0}\delta^{\prime}\delta(1-\frac{2J}{\Delta})}e^{-\mu_{K}d^{N}(\mathcal{A},{\mathbb{S}}_{L,K}^{N})}=\frac{8K(K+1)}{V_{N,0}\delta^{2}}e^{-\mu_{K}d^{N}(\mathcal{A},{\mathbb{S}}_{L,K}^{N})}\>,$

which is the desired result. ∎

######  Remark 2.11.

This result applies in particular to non-negative multiplication operators
$W_{N}$, that are of the form

(2.54) $W_{N}(X)=\sum_{i=1}^{N}\nu(x_{i})$

for any $X=(x_{1},x_{2},\dots,x_{N})\in{\mathbb{S}}_{L}^{N}$, where
$\nu:\Lambda_{L}\rightarrow\mathbb{R}^{+}_{0}$ is an arbitrary non-negative
function with domain $\Lambda_{L}$. In this case, $W_{N}$ corresponds to a
background magnetic field in $3$-direction, whose value at each site $j$ is
given by $\nu(j)$. To be more precise, we have

(2.55)
$W_{N}=U_{L}^{N}\left(\sum_{j=1}^{L}\mathcal{N}_{j}^{loc}\nu(j)\upharpoonright_{\mathcal{H}_{L}^{N}}\right)(U_{L}^{N})^{*}\>,$

where the unitary operator $U_{L}^{N}$ was given in (2.9). See also [8, Remark
2.7].

###  2.3. Entanglement Entropy

In what follows, we will mainly use ideas and previous results from [1, Sec.
5]. Let ${\mathbb{S}}_{L}:=\bigcup_{N=0}^{2JL}{\mathbb{S}}_{L}^{N}$, which
allows us to identify

(2.56)
$\ell^{2}({\mathbb{S}}_{L})=\bigoplus_{N=0}^{2JL}\ell^{2}({\mathbb{S}}_{L}^{N})\>.$

Next, let $\psi\in\ell^{2}({\mathbb{S}}_{L})$ be normalized. We denote the
associated density matrix by $\rho_{\psi}=|\psi\rangle\langle\psi|$.222Using
physicist’s notation, for any $\alpha,\beta$ in a Hilbert space $\mathcal{H}$,
the symbol “$|\alpha\rangle\langle\beta|$” denotes the rank-one operator
$\psi\mapsto\alpha\langle\beta,\psi\rangle$ for any $\psi\in\mathcal{H}$.
Analogous to (2.21), for any subset $\Gamma\subset\Lambda_{L}$, and any
$N\in\\{1,\dots,2J|\Gamma|\\}$, we define

(2.57)
${\mathbb{S}}_{\Gamma}^{N}:=\left\\{(x_{1},x_{2},\dots,x_{N})\in\Gamma^{N}:x_{1}\leq
x_{2}\leq\cdots\leq x_{N}\mbox{ and
}\min_{k\in\\{1,2,\dots,N-2J\\}}(x_{k+2J}-x_{k})\geq 1\right\\}$

and – as before – we set ${\mathbb{S}}_{\Gamma}^{0}:=\\{\emptyset\\}$.
Moreover, $\ell^{2}({\mathbb{S}}_{\Gamma})$ is defined analogously to (2.56).
Now, let $\ell\in\mathbb{N}$ such that $1<\ell<L$. Then,
$\Lambda_{L}=\Lambda_{\ell}\cup\Lambda_{\ell}^{c}$ is a spatial bipartition of
$\Lambda_{L}$ into two disjoint discrete subintervals, with the corresponding
decomposition of the Hilbert space
$\ell^{2}({\mathbb{S}}_{L})=\ell^{2}({\mathbb{S}}_{\Lambda_{\ell}})\otimes\ell^{2}({\mathbb{S}}_{\Lambda_{\ell}^{c}})$.
Moreover, for any $X\in{\mathbb{S}}_{\Lambda_{\ell}}$ and any
$Y\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}$, which implies that $X\cup
Y\in{\mathbb{S}}_{L}$, one naturally identifies $\phi_{X\cup
Y}=\phi_{X}\otimes\phi_{Y}$.333For two multisets
$X=(x_{1},x_{2},\dots,x_{j})\in{\mathbb{S}}_{\Lambda_{\ell}}$ and
$Y=(y_{1},y_{2},\dots,y_{K})\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}$, the
notation “$X\cup Y$” denotes the multiset
$(x_{1},x_{2},\dots,x_{j},y_{1},y_{2},\dots,y_{k})\in{\mathbb{S}}_{L}$. This
means in particular that ${\bf m}_{X\cup Y}={\bf m}_{X}+{\bf m}_{Y}$.

The reduced state
$\rho_{1}:\ell^{2}({\mathbb{S}}_{\Lambda_{\ell}})\rightarrow\ell^{2}({\mathbb{S}}_{\Lambda_{\ell}})$
is the linear operator given by

(2.58)
$\rho_{1}(\psi,\Lambda_{\ell})\equiv\rho_{1}=\mbox{Tr}_{\Lambda_{\ell}^{c}}(\rho_{\psi})=\sum_{X_{1},X_{2}\in{\mathbb{S}}_{\Lambda_{\ell}}}\sum_{Y\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}}\psi(X_{1}\cup
Y)\overline{\psi(X_{2}\cup Y)}|\phi_{X_{1}}\rangle\langle\phi_{X_{2}}|$

where $\mbox{Tr}_{\Lambda_{\ell}^{c}}(\cdot)$ denotes the partial trace over
the subsystem $\Lambda_{\ell}^{c}$. Then, the entanglement entropy of
$\rho_{\psi}$, which we denote by $\mathcal{E}(\rho_{\psi})$ is given by

(2.59)
$\mathcal{E}(\rho_{\psi})=-\mbox{Tr}(\rho_{1}\log\rho_{1})=:\mathcal{S}(\rho_{1})\>,$

where $\mathcal{S}(\rho_{1})$ denotes the von Neumann entropy of the reduced
state $\rho_{1}$.444We adapt the convention $0\log 0=0$. As in [1, 4], we will
actually show estimates for the $\alpha$-Rényi entropies
$\mathcal{S}_{\alpha}$ of $\rho_{1}$, which are given by

(2.60)
$\mathcal{S}_{\alpha}(\rho_{1}):=\frac{1}{1-\alpha}\log\mbox{Tr}[(\rho_{1})^{\alpha}]\>,$

where $\alpha\in(0,1)$. Since for every $\alpha\in(0,1)$ one has
$\mathcal{S}(\rho_{1})\leq\mathcal{S}_{\alpha}(\rho_{1})$, showing suitable
bounds for $\mathcal{S}_{\alpha}(\rho_{1})$ will then readily imply the
desired result for the entanglement entropy $\mathcal{E}(\rho_{\psi})$. Let us
now further analyze (2.58):

(2.61)
$\rho_{1}=\sum_{X_{1},X_{2}\in{\mathbb{S}}_{\Lambda_{\ell}}}\psi(X_{1})\overline{\psi(X_{2})}|\phi_{X_{1}}\rangle\langle\phi_{X_{2}}|+\sum_{X_{1},X_{2}\in{\mathbb{S}}_{\Lambda_{\ell}}}\sum_{Y\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}\setminus\\{\emptyset\\}}\psi(X_{1}\cup
Y)\overline{\psi(X_{2}\cup Y)}|\phi_{X_{1}}\rangle\langle\phi_{X_{2}}|\>,$

where the first sum simply corresponds to the contributions in (2.58), for
which $Y=\emptyset$. Now, let the vector
$\Psi\in\ell^{2}({\mathbb{S}}_{\Lambda_{\ell}})$ be given by
$\Psi:=\sum_{X_{3}\in{\mathbb{S}}_{\Lambda_{\ell}}}\psi(X_{3})\phi_{X_{3}}$,
which clearly satisfies
$\|\Psi\|_{\ell^{2}({\mathbb{S}}_{\Lambda_{\ell}})}\leq\|\psi\|_{\ell^{2}({\mathbb{S}}_{L})}$
and observe that

(2.62)
$\sum_{X_{1},X_{2}\in{\mathbb{S}}_{\Lambda_{\ell}}}\psi(X_{1})\overline{\psi(X_{2})}|\phi_{X_{1}}\rangle\langle\phi_{X_{2}}|=|\Psi\rangle\langle\Psi|\>.$

So, we write $\rho_{1}=|\Psi\rangle\langle\Psi|+\hat{\rho}_{1}$ with
$\hat{\rho}_{1}$ being equal to the second sum in (2.61). Let us now focus on
$\mbox{Tr}[(\rho_{1})^{\alpha}]$:

(2.63) $\displaystyle\mbox{Tr}[(\rho_{1})^{\alpha}]$
$\displaystyle=\mbox{Tr}[(|\Psi\rangle\langle\Psi|+\hat{\rho}_{1})^{\alpha}]\leq
2\mbox{Tr}[(|\Psi\rangle\langle\Psi|)^{\alpha}]+2\mbox{Tr}[(\hat{\rho}_{1})^{\alpha}]\leq
2+2\mbox{Tr}[(\hat{\rho}_{1})^{\alpha}]\>,$

where we have used the quasi-norm property of $\mbox{Tr}|\cdot|^{\alpha}$, cf.
[22, Satz 3.21] and the fact that $|\Psi\rangle\langle\Psi|$ is a non-negative
rank-one operator with norm less than or equal to one. Let us now further
estimate

(2.64) $\displaystyle\mbox{Tr}[(\hat{\rho}_{1})^{\alpha}]$
$\displaystyle=\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}}\langle\phi_{X},(\hat{\rho}_{1})^{\alpha}\phi_{X}\rangle\leq\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}}\langle\phi_{X},\hat{\rho}_{1}\phi_{X}\rangle^{\alpha}=\sum_{j=0}^{2J\ell}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}\langle\phi_{X},\hat{\rho}_{1}\phi_{X}\rangle^{\alpha}$
(2.65)
$\displaystyle\leq\left|\bigcup_{j=0}^{4J-1}{\mathbb{S}}_{\Lambda_{\ell}}^{j}\right|^{1-\alpha}+\sum_{j=4J}^{2J\ell}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}\langle\phi_{X},\hat{\rho}_{1}\phi_{X}\rangle^{\alpha}$

where we have used Jensen’s inequality for the first estimate in (2.64) and
the second estimate (2.65) just follows from maximizing
$\sum\sum\langle\phi_{X},\hat{\rho}_{1}\phi_{X}\rangle^{\alpha}$ under the
constraint $\sum\sum\langle\phi_{X},\hat{\rho}_{1}\phi_{X}\rangle\leq 1$. We
estimate further by noting $|{\mathbb{S}}_{\Lambda_{\ell}}^{j}|\leq\ell^{j}$
and thus – assuming $\ell\geq 4J$ –

(2.66)
$\eqref{eq:teilzwei}\leq\ell^{4J(1-\alpha)}+\sum_{j=4J}^{2J\ell}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}\langle\phi_{X},\hat{\rho}_{1}\phi_{X}\rangle^{\alpha}\>.$

Now, for each $X\in{\mathbb{S}}_{\Lambda_{\ell}}$ with $|X|=j$, we get

(2.67) $\displaystyle\langle\phi_{X},\hat{\rho}_{1}\phi_{X}\rangle$
$\displaystyle=\sum_{Y\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}\setminus\\{\emptyset\\}}|\psi(X\cup
Y)|^{2}=\|P_{\mathcal{A}_{X}}\psi\|^{2}\>,$

where given any $X\in{\mathbb{S}}_{\Lambda_{\ell}}$, we have defined
$\mathcal{A}_{X}:=\\{X\cup
Y:Y\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}\setminus\\{\emptyset\\}\\}$.
Altogether, these considerations show the following

######  Lemma 2.12.

Let $\ell\geq 4J$ and $\psi\in\ell^{2}({\mathbb{S}}_{L})$ be normalized. Then,
for any $\alpha\in(0,1)$ we get

(2.68) $\mbox{\emph{Tr}}[({\rho}_{1})^{\alpha}]\leq
2+2\ell^{4J(1-\alpha)}+2\sum_{j=4J}^{2J\ell}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}\|P_{\mathcal{A}_{X}}\psi\|^{2\alpha}\>.$

##  3\. Proof of the logarithmically corrected entanglement bound

###  3.1. Local distance estimates

The following lemma, whose proof can be found in Appendix B.2, provides us
with an estimate for the distance of a configuration in ${\mathbb{S}}_{L}$ to
the nearest configuration in ${\mathbb{S}}_{L,K}$.

######  Lemma 3.1.

Let $X=(x_{1},x_{2},\dots,x_{j})\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}$ with
$j\geq 4J$ and
$Y=(y_{1},y_{2},\dots,y_{k})\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}^{k}$, where
$k\in\mathbb{N}$. Then, for any $K\geq 4J^{2}$, we have the following
estimate:

(3.1) $d^{j}(X,{\mathbb{S}}_{\Lambda_{\ell},K}^{j})\leq d^{j+k}(X\cup
Y,{\mathbb{S}}_{L,K}^{j+k})\>.$

So, the main observation made in Lemma 3.1 is that there always exists a
configuration in ${\mathbb{S}}^{j}_{\Lambda_{\ell},K}$ which is at least as
close to $X$ as any configuration in ${\mathbb{S}}^{j+k}_{L,K}$ might be to
$X\cup Y$. We now combine Lemmas 2.68, 3.1 and Theorem 2.10. To this end, for
any non-negative function $W:{\mathbb{S}}_{L}\rightarrow\mathbb{R}_{0}^{+}$,
let us define $Q_{K,\delta}\equiv
Q_{K,\delta}(L,W):=\bigoplus_{N=0}^{2JL}Q_{K,\delta}^{N}(L,W_{N})$ – the
spectral projection of the full Hamiltonian onto the energy interval
$[0,E_{K,\delta}]$ with background potential $W$. Here,
$W_{N}:\ell^{2}({\mathbb{S}}_{L}^{N})\rightarrow\ell^{2}({\mathbb{S}}_{L}^{N})$,
denotes the multiplication operator induced by $W$, i.e.
$(W_{N}f)(X):=W(X)f(X)$, for any $f\in\ell^{2}({\mathbb{S}}_{L}^{N})$ and any
$X\in{\mathbb{S}}_{L}^{N}$.

######  Corollary 3.2.

Fix $K\in\mathbb{N}$ and let $\ell\geq 4J$. Let
$\psi\in\mbox{\emph{ran}}(Q_{K,\delta})$ be normalized. Then, for any
$\alpha\in(0,1)$, we have the following estimate

(3.2) $\mbox{\emph{Tr}}[(\rho_{1})^{\alpha}]\leq
2+2\ell^{4J(1-\alpha)}+2C_{K}^{\prime
2\alpha}\sum_{j=4J}^{2J\ell}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-2\alpha\mu_{K}d^{j}(X,{\mathbb{S}}^{j}_{\Lambda_{\ell},K})}\>,$

where

(3.3)
$C_{K}^{\prime}=\max\left\\{1,\frac{2K(K+1)}{J^{2}\delta^{2}}\right\\}\>.$

###### Proof.

Let $X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}$, where $j\geq 4J$. Defining
$\mathcal{A}_{X,k}=\\{X\cup Y:Y\in{\mathbb{S}}^{k}_{\Lambda_{\ell}^{c}}\\}$
and using that

(3.4)
$P_{\mathcal{A}_{X}}Q_{K,\delta}=\bigoplus_{k=1}^{2J(L-\ell)}P_{\mathcal{A}_{X,k}}Q_{K,\delta}^{j+k}$

we get

(3.5) $\displaystyle\|P_{\mathcal{A}_{X}}\psi\|$
$\displaystyle=\|P_{\mathcal{A}_{X}}Q_{K,\delta}\psi\|\leq\|P_{\mathcal{A}_{X}}Q_{K,\delta}\|=\max_{k\in\\{1,\dots,2J(L-\ell)\\}}\|P_{\mathcal{A}_{X,k}}Q_{K,\delta}^{j+k}\|$
(3.6)
$\displaystyle\leq\max_{k\in\\{1,\dots,2J(L-\ell)\\}}\left(C_{j+k,K}\cdot
e^{-\mu_{K}d^{j+k}(\mathcal{A}_{X,k},{\mathbb{S}}_{L,K}^{j+k})}\right)$

where we have used Theorem 2.10 for the last inequality and the constants
$C_{j+k,K}$ and $\mu_{K}$ were given in (2.32). Now, since $j+k\geq 4J$ by
assumption, it follows from Proposition 2.5 that $V_{j+k,0}=4J^{2}$ and thus

(3.7)
$C_{j+k,K}=\max\left\\{1,\frac{2K(K+1)}{J^{2}\delta^{2}}\right\\}=C^{\prime}_{K}\>,$

independently of $j$. Now, for any $k\in\\{1,\dots,2J(L-\ell)\\}$, observe
that by Lemma 3.1, we have

(3.8)
$d^{j+k}(\mathcal{A}_{X,k},{\mathbb{S}}_{L,K}^{j+k})=\min_{Y\in{\mathbb{S}}_{\Lambda_{\ell}^{c}}^{k}}d^{j+k}(X\cup
Y,{\mathbb{S}}_{L,K}^{j+k})\geq d^{j}(X,{\mathbb{S}}_{\Lambda_{\ell},K}^{j})$

and thus we may proceed to estimate (3.6) to get

(3.9) $\|P_{\mathcal{A}_{X}}\psi\|\leq\eqref{eq:findmax}\leq
C^{\prime}_{K}\cdot
e^{-\mu_{K}d^{j}(X,{\mathbb{S}}^{j}_{\Lambda_{\ell},K})}\>.$

Using Lemma 2.68, we consequently find

(3.10) $\mbox{Tr}[(\rho_{1})^{\alpha}]\leq
2+2\ell^{4J(1-\alpha)}+2C_{K}^{\prime
2\alpha}\sum_{j=4J}^{2J\ell}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-2\alpha\mu_{K}d^{j}(X,{\mathbb{S}}^{j}_{\Lambda_{\ell},K})}\>,$

which shows (3.2). ∎

###  3.2. Estimates using geometric series

######  Lemma 3.3.

Let
$\mathcal{X}^{N}=\left\\{X=(x_{1},x_{2},\ldots,x_{N})\in\mathbb{Z}^{N}:x_{1}<x_{2}<\cdots<x_{N}\right\\}$
be the set of spin–1/2 configurations on the infinite chain. Moreover, let
$C=(c_{1},c_{2},\ldots,c_{N})=(c,c+1,\ldots,c+(N-1))$ be an arbitrary
configuration of $N$ consecutive particles ($c\in\mathbb{Z}$). Then, for any
$\gamma>0$, we get

(3.11) $\sum_{X\in\mathcal{X}^{N}}e^{-\gamma
d^{N}(X,C)}\leq\left(\frac{1}{1-e^{-\gamma}}\right)\left(\prod_{k=1}^{\infty}\frac{1}{1-e^{-k\gamma}}\right)^{2}:=\mathcal{L}_{\gamma}<\infty\>.$

###### Proof.

Define

$\mathcal{X}^{N,0}=\left\\{X\in\mathcal{X}^{N}:x_{1}\geq
c_{1}\right\\},\;\;\;\;\mathcal{X}^{N,N}=\left\\{X\in\mathcal{X}^{N}:x_{N}<c_{N}\right\\}$

and for any $j\in\left\\{1,\ldots,N-1\right\\}$,

$\mathcal{X}^{N,j}=\left\\{X\in\mathcal{X}^{N}:x_{j}<c_{j},\;x_{j+1}\geq
c_{j+1}\right\\}\>.$

Then, $\displaystyle\mathcal{X}^{N}=\uplus_{j=0}^{N}\mathcal{X}^{N,j}$, where
“$\uplus$” denotes a disjoint union. For any
$j\in\left\\{1,\ldots,N-1\right\\}$, we get by an argument similar to [9,
Lemma A.3] that

(3.12) $\begin{split}\sum_{X\in\mathcal{X}^{N,j}}e^{-\gamma
d^{N}(X,C)}&=\sum_{x_{1}<x_{2}<\cdots<x_{j}<c_{j}}e^{-\gamma(|x_{1}-c_{1}|+\cdots+|x_{j}-c_{j}|)}\sum_{c_{j+1}\leq
x_{j+1}<x_{j+2}<\cdots<x_{N}}e^{-\gamma(|x_{j+1}-c_{j+1}|+\cdots+|x_{N}-c_{N}|)}\\\
&\leq e^{-\gamma
j}\left(\prod_{k=1}^{j}\frac{1}{1-e^{-k\gamma}}\right)\left(\prod_{k=1}^{N-j}\frac{1}{1-e^{-k\gamma}}\right)\leq
e^{-\gamma
j}\left(\prod_{k=1}^{\infty}\frac{1}{1-e^{-k\gamma}}\right)^{2}\>.\\\
\end{split}$

Analogously, we find

(3.13) $\sum_{X\in\mathcal{X}^{N,0}}e^{-\gamma
d^{N}(X,C)}\leq\prod_{k=1}^{N}\frac{1}{1-e^{-k\gamma}}\leq\left(\prod_{k=1}^{\infty}\frac{1}{1-e^{-k\gamma}}\right)^{2}$
(3.14) $\sum_{X\in\mathcal{X}^{N,N}}e^{-\gamma d^{N}(X,C)}\leq e^{-\gamma
N}\prod_{k=1}^{N}\frac{1}{1-e^{-k\gamma}}\leq e^{-\gamma
N}\left(\prod_{k=1}^{\infty}\frac{1}{1-e^{-k\gamma}}\right)^{2}\>.$

Hence,

(3.15) $\displaystyle\sum_{X\in\mathcal{X}^{N}}e^{-\gamma d^{N}(X,C)}=$
$\displaystyle\sum_{j=0}^{N}\sum_{X\in\mathcal{X}^{N,j}}e^{-\gamma
d^{N}(X,C)}$ (3.16) $\displaystyle\leq$ $\displaystyle\sum_{j=0}^{N}e^{-\gamma
j}\left(\prod_{k=1}^{\infty}\frac{1}{1-e^{-k\gamma}}\right)^{2}\leq\left(\frac{1}{1-e^{-\gamma}}\right)\left(\prod_{k=1}^{\infty}\frac{1}{1-e^{-k\gamma}}\right)^{2}\>,$

which is the desired result. The infinite product’s convergence follows from
elementary facts. ∎

######  Definition 3.4.

For any ${\bf m}\in{\bf M}_{L}^{N}$, let $\operatorname{supp}({\bf
m}):=\\{j\in\\{1,2,\dots,L\\}:{\bf m}(j)\neq 0\\}$ denote the support of ${\bf
m}$. Then, for any $L$, we define the set ${\bf K}_{L}$ of “building bocks” in
$\Lambda_{L}$ as follows

(3.17) $\displaystyle{\bf K}_{L}$ $\displaystyle:=\\{{\bf m}\in{\bf
M}_{L}:|\mbox{supp}({\bf m})|=1\\}$ (3.18) $\displaystyle\cup\\{{\bf m}\in{\bf
M}_{L}:\exists\alpha,\beta\in\Lambda_{L}:{\bf m}(i)=2J\mbox{ if }\alpha\leq
i\leq\beta\mbox{ and }{\bf m}(i)=0\mbox{ else}\\}\>.$

In other words, building blocks are configurations of either a single column
of up to $2J$ particles or of discrete intervals, where each site is occupied
by exactly $2J$ particles (rectangular blocks).

######  Lemma 3.5.

Let $\gamma>0$. Then for any $C\in{\bf K}_{L}$, we have

(3.19) $\sum_{X\in{\mathbb{S}}^{N}_{L}}e^{-\gamma
d^{N}(X,C)}\leq\mathcal{L}^{2J}_{\gamma}$

###### Proof.

Let $C$ be a rectangular block, i.e. $\;C=(\stackrel{{\scriptstyle
2J\;\;\text{times}}}{{\overbrace{c,\ldots,c}}},\stackrel{{\scriptstyle
2J\;\;\text{times}}}{{\overbrace{c+1,\ldots,c+1}}},\ldots,\stackrel{{\scriptstyle
2J\;\;\text{times}}}{{\overbrace{c+m,\ldots,c+m}}})$ where $N=2J(m+1)$. For
any $N$-particle configuration
$X=(x_{1},\ldots,x_{N})\in{\mathbb{S}}^{N}_{L}$, let $Z_{0}(X)$ denote the
constraint

(3.20) $Z_{0}(X):x_{1}\leq x_{2}\leq\cdots\leq x_{N}\>.$

Moreover, define the additional constraints

(3.21) $Z_{i}(X):x_{i}<x_{i+2J}<x_{i+4J}<\cdots<x_{i+2JR_{i}}$

where $i\in\left\\{1,2,\ldots,2J\right\\}$ and
$R_{i}=\max\left\\{R\in\\{1,2,\dots,N\\}:i+2JR\leq N\right\\}$.

Let

(3.22) $\mathbb{M}^{N}_{L}:=\left\\{X\in\Lambda^{N}_{L}:\;Z_{i}(X)\;\text{
holds }\;\forall\;i=0,\ldots,2J\right\\}$

and

(3.23) $\Gamma^{N}:=\left\\{X\in\mathbb{Z}^{N}:\;Z_{k}(X)\;\text{ holds
}\;\forall\;k=1,\ldots,2J\right\\}\>.$

Obviously, $\mathbb{M}^{N}_{L}\subseteq\Gamma^{N}$. To see that
$\mathbb{M}^{N}_{L}={\mathbb{S}}^{N}_{L}$ (defined in (2.21)), note that it is
obvious that $\mathbb{M}^{N}_{L}\subseteq{\mathbb{S}}^{N}_{L}$ since for any
$X\in\mathbb{M}^{N}_{L}$, the constraints $Z_{i}(X)$ imply that
$\min\left\\{(x_{k+2J}-x_{k}):k=1,\ldots,N-2J\right\\}\geq 1$.

Conversely, suppose there exists
$X\in{\mathbb{S}}^{N}_{L}\setminus\mathbb{M}^{N}_{L}$, i.e. $X$ violates at
least one constraint, $Z_{t}(X)$ say, where $t\neq 0$. Then there exists
$r\in\mathbb{N}$ such that $x_{t+2Jr}\geq x_{t+2J(r+1)}$. This is a
contradiction since $X\in{\mathbb{S}}^{N}_{L}$. Hence,
${\mathbb{S}}^{N}_{L}=\mathbb{M}^{N}_{L}$.

Therefore, by Lemma (3.3),

(3.24) $\begin{split}\sum_{X\in{\mathbb{S}}^{N}_{L}}e^{-\gamma
d^{N}(X,C)}&\leq\sum_{X\in\Gamma^{N}}e^{-\gamma d^{N}(X,C)}\\\
&=\sum_{Z_{1}(X)}e^{\displaystyle-\gamma(|x_{1}-c_{1}|+\cdots+|x_{1+2JR_{1}}-c_{1+2JR_{1}}|)}\times\cdots\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\cdots\times\sum_{Z_{2J}(X)}e^{\displaystyle-\gamma(|x_{2J}-c_{2J}|+\cdots+|x_{2J+2JR_{{}_{2J}}}-c_{2J+2JR_{{}_{2J}}}|)}\\\
&\leq\mathcal{L}^{2J}_{\gamma}\>.\end{split}$

If $C=(c_{1},\ldots,c_{N})$ is a column block, i.e. $c_{1}=\cdots=c_{N}=c$,
where $c\in\Lambda_{L}$ and $N\leq 2J$, then

(3.25) $\displaystyle\sum_{X\in{\mathbb{S}}^{N}_{L}}e^{-\gamma
d^{N}(X,C)}\leq$ $\displaystyle\sum_{-\infty<x_{1}\leq x_{2}\leq\cdots\leq
x_{N}<\infty}e^{-\gamma[|x_{1}-c|+\cdots+|x_{N}-c|]}$ (3.26)
$\displaystyle\leq$
$\displaystyle\sum_{x_{1},x_{2},\cdots,x_{N}\in\mathbb{Z}}e^{-\gamma[|x_{1}-c|+\cdots+|x_{N}-c|]}=\left(\frac{1+e^{-\gamma}}{1-e^{-\gamma}}\right)^{N}\leq\mathcal{L}^{N}_{\gamma}\leq\mathcal{L}^{2J}_{\gamma}\>,$

which finishes the proof. ∎

######  Remark 3.6.

Note that for this result, we are making use of the generalized Pauli
principle, which requires that no site be occupied by more than $2J$
particles.

###  3.3. Building blocks and potential

Given a value $K\in\mathbb{N}$ and a particle number $N\geq 4J$, it seems
rather cumbersome to give a full description of all configurations in ${\bf
M}_{L,K}^{N}$. However, it can be shown that any configuration in ${\bf
M}_{L,K}^{N}$ can be composed out of no more than $K$ of the “building blocks”
as described in Definition 3.4 before.

######  Definition 3.7.

Given a configuration ${\bf m}\in{\bf M}_{L}$, we define the quantity $B({\bf
m})$ as follows:

(3.27) $B({\bf m}):=\left|\\{i\in\\{1,2,\dots,L-1\\}:{\bf m}(i)+{\bf
m}(i+1)\notin\\{0,4J\\}\\}\right|+(2-\delta_{{\bf m}(1),0}-\delta_{{\bf
m}(L),0})\>.$

Moreover, for any $R\in\mathbb{N}$, we define

(3.28) ${\bf B}_{L,R}:=\\{{\bf m}\in{\bf M}_{L}:B({\bf m})\leq
R\\}\quad\mbox{as well as}\quad{\bf B}_{L,R}^{N}:={\bf B}_{L,R}\cap{\bf
M}_{L}^{N}$

for any $N\in\\{0,1,\dots,2JL\\}$. Additionally, for multisets, we introduce
$\mathbb{B}_{L,R}=\\{X\in{\mathbb{S}}_{L}:{\bf m}_{X}\in{\bf B}_{L,R}\\}$ and
$\mathbb{B}_{L,R}^{N}=\mathbb{B}_{L,R}\cap{\mathbb{S}}_{L}^{N}$.

######  Remark 3.8.

The purpose of $B({\bf m})$ is to count the number of neighboring sites
$\\{i,i+1\\}$, $i=1,2,\dots,L-1$, which are not both occupied by either $0$ or
$2J$ particles. Each of the two additional terms in $(2-\delta_{{\bf
m}(1),0}-\delta_{{\bf m}(L),0})=(1-\delta_{{\bf m}(1),0})+(1-\delta_{{\bf
m}(L),0})$ increases $B({\bf m})$ by a value of one if the sites $1$ or $L$
are occupied by any particles (which formally corresponds to including the
edges $\\{0,1\\}$ and $\\{L,L+1\\}$ in the count).

######  Remark 3.9.

Observe that $B({\bf m})\leq J^{-1}V({\bf m})$ for every ${\bf m}\in{\bf
M}_{L}$. This follows from the fact that $v({\bf m}(i),{\bf m}(i+1))=0$ if and
only if ${\bf m}(i)={\bf m}(i+1)=0$ or ${\bf m}(i)={\bf m}(i+1)=2J$ and
$v({\bf m}(i),{\bf m}(i+1))\geq J$ else, and comparing the extra term
$(2-\delta_{{\bf m}(1),0}-\delta_{{\bf m}(L),0})$ in (3.27) with the boundary
field term $J({\bf m}(1)+{\bf m}(L))$ in (2.14). For $K\geq 4J^{2}$, define
$\widetilde{K}:=\lfloor K/J\rfloor$, and observe that this implies

(3.29)
${{\mathbb{S}}}_{L,K}\subset{\mathbb{B}}_{L,\widetilde{K}}\quad\mbox{and thus
in
particular}\quad{{\mathbb{S}}}^{N}_{L,{K}}\subset{\mathbb{B}}^{N}_{L,\widetilde{K}}\>.$

It is now crucial to observe that any configuration in ${\bf
B}_{L,\widetilde{K}}$ can always by obtained by composing it out of at most
$(\widetilde{K}-1)$ “building blocks”.

######  Remark 3.10.

Observe that for any ${\bf m}\in{\bf M}_{L}$, there exist $\\{{\bf
k}^{(i)}\\}_{i=1}^{\tau}\subset{\bf K}_{L}$, ($\tau\leq L$), with pairwise
disjoint supports, such that ${\bf m}=\sum_{i=1}^{\tau}{\bf k}_{i}$. If in
addition, we have ${\bf m}\in{\bf B}_{L,\widetilde{K}}$, it follows from
(3.27) that $\tau\leq(\widetilde{K}-1)$, i.e. any configuration in ${\bf
B}_{L,\widetilde{K}}$ can be composed out of no more than $(\widetilde{K}-1)$
building blocks. See Figure 3.1 for a pictorial representation of building
blocks.

$1$$2$$3$$4$$5$$6$$7$

Figure 3.1. A configuration ${\bf m}\in{\bf M}_{L}^{N}$ of $N=37$ particles
(here: $J=9/2$). Moreover, $B({\bf m})=6\leq J^{-1}V({\bf m})=26$ and indeed,
${\bf m}$ can be written as a composition of $4\leq(\lfloor J^{-1}V({\bf
m})\rfloor-1)$ building blocks (represented here by four different colors).

######  Lemma 3.11.

Let $j\in\\{4J,4J+1,\dots,2J\ell\\}$, $K\in\\{4J^{2},4J^{2}+1,\dots,2J\ell\\}$
and $\widetilde{K}:=\lfloor K/J\rfloor$. Then, for any $\gamma>0$, we get the
following estimate

(3.30) $\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma
d^{j}(X,{\mathbb{S}}_{\Lambda_{\ell},K}^{j})}\leq(4Je)^{\widetilde{K}-2}\mathcal{L}_{2\alpha\mu_{K}}^{2J(\widetilde{K}-1)}\ell^{2\widetilde{K}-3}\>.$

###### Proof.

Analogously to before, we define
$\mathbb{B}_{\Lambda_{\ell},\widetilde{K}}^{j}:=\mathbb{B}_{L,\widetilde{K}}^{j}\cap{\mathbb{S}}_{\Lambda_{\ell}}$
and observe that due to (3.29), we have
${\mathbb{S}}_{\Lambda_{\ell},K}^{j}\subset\mathbb{B}_{\Lambda_{\ell},\widetilde{K}}^{j}$
and thus, we get

(3.31) $\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma
d^{j}(X,{\mathbb{S}}_{\Lambda_{\ell},K}^{j})}\leq\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}\sum_{Y\in{\mathbb{S}}_{\Lambda_{\ell},K}^{j}}e^{-\gamma
d^{j}(X,Y)}=\sum_{Y\in{\mathbb{S}}_{\Lambda_{\ell},K}^{j}}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma
d^{j}(X,Y)}\leq\sum_{Y\in\mathbb{B}_{\Lambda_{\ell},\widetilde{K}}^{j}}\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma
d^{j}(X,Y)}\>.$

We now claim that for every
$Y\in\mathbb{B}_{\Lambda_{\ell},\widetilde{K}}^{j}$, we get

(3.32) $\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma
d^{j}(X,Y)}\leq\mathcal{L}_{\gamma}^{2J(\widetilde{K}-1)}\>.$

Since $Y\in\mathbb{B}_{\Lambda_{\ell},\widetilde{K}}^{j}$, observe that by
Remark 3.10, there exist ${\bf k}^{(1)},{\bf k}^{(2)},\dots,{\bf
k}^{(\tau)}\in{\bf K}_{L}$ with $\tau\leq(\widetilde{K}-1)$ and pairwise
disjoint support such that

(3.33) ${\bf k}^{(1)}+{\bf k}^{(2)}+\cdots+{\bf k}^{(\tau)}={\bf m}_{Y}\>.$

Without loss of generality, we may assume that the building blocks are ordered
such that $i<j$ implies $\max\operatorname{supp}({\bf
k}^{(i)})<\min\operatorname{supp}({\bf k}^{(j)})$.

For any $i\in\\{1,2\dots,\tau\\}$, let $K^{(i)}:=X_{{\bf k}^{(i)}}$ denote the
multiset associated with the building block ${\bf k}^{(i)}$ and thus
$Y=K^{(1)}\cup K^{(2)}\cup\dots\cup K^{(\tau)}$. Now, decompose any
$X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}$ analogously into $X=X^{(1)}\cup
X^{(2)}\cup\dots\cup X^{(\tau)}$ such that $|X^{(i)}|=|K^{(i)}|=:k_{i}$ for
every $i\in\\{1,2,\dots,\tau\\}$ and $i<j$ implies that
$\max(X^{(i)})<\min(X^{(j)})$.

We then get

(3.34) $\displaystyle\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma
d^{j}(X,Y)}$ $\displaystyle=\sum_{X^{(1)}\cup\dots\cup
X^{(\tau)}\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma\left(d^{k_{1}}(X^{(1)},K^{(1)})+\cdots+d^{k_{\tau}}(X^{(\tau)},K^{(\tau)})\right)}$
(3.35)
$\displaystyle\leq\prod_{i=1}^{\tau}\left(\sum_{X^{(i)}\in{\mathbb{S}}_{\Lambda_{\ell}}^{k_{i}}}e^{-\gamma
d^{k_{i}}(X^{(i)},K^{(i)})}\right)\leq\mathcal{L}_{\gamma}^{2J\tau}\leq\mathcal{L}_{\gamma}^{2J(\widetilde{K}-1)}\>,$

where we have used Lemma 3.5 for estimating the sum in (3.35). This shows
(3.32) which together with (3.31) proves that

(3.36) $\sum_{X\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}}e^{-\gamma
d^{j}(X,{\mathbb{S}}_{\Lambda_{\ell},K}^{j})}\leq\sum_{Y\in{\mathbb{B}_{\Lambda_{\ell},\widetilde{K}}^{j}}}\mathcal{L}_{\gamma}^{2J(\widetilde{K}-1)}=|\mathbb{B}_{{\Lambda_{\ell},\widetilde{K}}}^{j}|\cdot\mathcal{L}_{\gamma}^{2J(\widetilde{K}-1)}\>,$

which means that we need to further estimate the number of configurations in
$\Lambda_{\ell}$ of $(\widetilde{K}-1)$ or less building blocks of $j$
particles. We do this by a rather rough combinatorial argument: firstly, note
that distributing $j$ particles into up to $(\widetilde{K}-1)$ building blocks
can be estimated by $\binom{j+\widetilde{K}-2}{\widetilde{K}-2}$. This is
clearly an overestimate, since it disregards the constraints in (3.17) which
building blocks have to satisfy. Next, we have to account for all the possible
ways, those $(\widetilde{K}-1)$ or less building blocks can be placed in
$\Lambda_{\ell}$. A trivial upper bound for this is given by
$\ell^{\widetilde{K}-1}$, since there are up to $\ell$ sites one could place
each individual building block (disregarding the fact that the supports of the
building blocks have to be disjoint and can be larger than one and thus again
overestimating). If $\widetilde{K}>2$, we therefore conclude

(3.37)
$|\mathbb{B}_{{\Lambda_{\ell},\widetilde{K}}}^{j}|\leq\binom{j+\widetilde{K}-2}{\widetilde{K}-2}\cdot\ell^{\widetilde{K}-1}\leq\left(\frac{(2J\ell+\widetilde{K}-2)e}{\widetilde{K}-2}\right)^{\widetilde{K}-2}\ell^{\widetilde{K}-1}\leq(4Je)^{\widetilde{K}-2}\ell^{2\widetilde{K}-3}\>,$

where we have used the estimate $\binom{\alpha}{\beta}\leq\left(\frac{\alpha
e}{\beta}\right)^{\beta}$ for the binomial coefficient as well as the fact
that $j\leq 2J\ell$ and $4J\leq\widetilde{K}\leq 2\ell$. For the special case
$\widetilde{K}=2$, which can only occur of $J=1/2$, Equation (3.37) still
follows since $|\mathbb{B}_{{\Lambda_{\ell},\widetilde{K}}}^{j}|\leq\ell$.
This shows the lemma.

∎

###  3.4. Proof of the main result

We are now prepared to show our main result:

######  Theorem 3.12.

For any $K\in\\{4J^{2},4J^{2}+1,\dots\\}$ and any $\delta>0$, we get the
following estimate

(3.38)
$\limsup_{\ell\rightarrow\infty}\>\limsup_{L\rightarrow\infty}\>\frac{\sup_{W}\>\sup_{\psi}\mathcal{E}(\rho_{\psi})}{\log\ell}\leq{2\lfloor
K/J\rfloor-2}\>,$

where the suprema are taken over all non-negative functions
$W:{\mathbb{S}}_{L}\rightarrow\mathbb{R}_{0}^{+}$ and over all normalized
elements $\psi\in\mbox{\emph{ran}}(Q_{K,\delta}(L,W))$, respectively.

###### Proof.

Let $W:{\mathbb{S}}_{L}\rightarrow\mathbb{R}^{+}_{0}$ be an arbitrary non-
negative background potential and let $\psi$ be an arbitrary normalized
element of $\mbox{ran}(Q_{K,\delta}(L,W))$. Combining Corollary 3.2 and Lemma
3.30 (with the choice $\gamma=2\alpha\mu_{K}$) and using that for any
$\alpha\in(0,1)$, the $\alpha$-Rényi entropy is an upper bound for the von
Neumann entropy, we obtain

(3.39)
$\displaystyle\mathcal{E}(\rho_{\psi})\leq\frac{1}{1-\alpha}\log\operatorname{{\rm
Tr}}[(\rho_{1})^{\alpha}]\leq\frac{1}{1-\alpha}\log\left(2+2\ell^{4J(1-\alpha)}+(4Je)^{\widetilde{K}-1}C_{K}^{\prime
2\alpha}\mathcal{L}_{2\alpha\mu_{K}}^{2J(\widetilde{K}-1)}\ell^{2\widetilde{K}-2}\right)\>,$

which does not depend on $L$. We therefore get

(3.40)
$\limsup_{\ell\rightarrow\infty}\>\limsup_{L\rightarrow\infty}\>\frac{\sup_{W}\>\sup_{\psi}\mathcal{E}(\rho_{\psi})}{\log\ell}\leq\frac{2\widetilde{K}-2}{1-\alpha}=\frac{2\lfloor{K}/J\rfloor-2}{1-\alpha}\>.$

Since this is true for all $\alpha\in(0,1)$, one can take $\alpha\rightarrow
0$, which yields the desired result. ∎

######  Remark 3.13.

Note that for $J=1/2$, the constant in (3.38), is given by $(4K-2)$. In [1],
where only the case $J=1/2$ was treated, the better bound $(2K-1)$ was
established. The main reason for this discrepancy is that in this special
case, one can actually show that any configuration ${\bf m}$ with $V({\bf
m})\leq K$ can actually be composed out of no more than $K$ building blocks
rather than out of no more than ${\widetilde{K}}-1=2K-1$ building blocks.

##  Appendix A Proof of the distance formula

######  Lemma A.1.

The graph distance $d^{N}(X,Y)$ from $X$ to $Y$ is given by (2.26).

###### Proof.

Let $\delta^{N}(X,Y):=\sum_{i=1}^{N}|x_{i}-y_{i}|$. To prove that
$\delta^{N}(X,Y)$ is a lower bound of the graph distance $d^{N}(X,Y)$, we
first show that ${\bf m}_{X}\sim{\bf m}_{Y}$ if and only if
$\delta^{N}(X,Y)=1\;\;\forall\;X,Y\in{\mathbb{S}}_{L}^{N}$.

For $X,Y\in{\mathbb{S}}^{N}_{L}$, where $X=(x_{1},x_{2},\dots,x_{N})$,
$Y=(y_{1},y_{2},\dots,y_{N})$, suppose $\delta^{N}(X,Y)=1$. By (2.26), there
exists $k_{0}\in\\{1,2,\dots,N\\}$ such that $|x_{k_{0}}-y_{k_{0}}|=1$ and
$x_{k}=y_{k}\;\;\forall\;k\in\\{1,2,\dots,N\\}\setminus\\{k_{0}\\}$ . Define
$j_{0}:=x_{k_{0}}$. Therefore $y_{k_{0}}=j_{0}\pm 1$. Now, let ${\bf m}_{X}$
and ${\bf m}_{Y}$ be the occupation number functions corresponding to $X$ and
$Y$ respectively. Without loss of generality, let $y_{k_{0}}=j_{0}+1$ and
suppose that ${\bf m}_{X}(j_{0})=r$, $r\in\\{1,\ldots,2J\\}$. Then ${\bf
m}_{Y}(j_{0})=r-1$, since $y_{k_{0}}\neq j_{0}$. Hence, ${\bf
m}_{X}(j_{0})-{\bf m}_{Y}(j_{0})=r-(r-1)=1$. Also, suppose ${\bf
m}_{Y}(j_{0}+1)=s,\;\;s\in\left\\{1,\ldots,2J\right\\}$, then ${\bf
m}_{X}(j_{0}+1)=s-1$, since $y_{k_{0}}=j_{0}+1$. Therefore, ${\bf
m}_{X}(j_{0}+1)-{\bf m}_{Y}(j_{0}+1)=s-1-s=-1$ and ${\bf m}_{X}(j)={\bf
m}_{Y}(j),\;\;\forall\;j\in\\{1,\ldots L\\}\setminus\\{j_{0},j_{0}+1\\}$.
Hence, ${\bf m}_{X}\sim{\bf m}_{Y}$.

Conversely, suppose ${\bf m}_{X}\sim{\bf m}_{Y}$. By (2.12), there exists a
unique $j_{0}\in\\{1,\ldots,L-1\\}$ such that ${\bf m}_{X}(j_{0})-{\bf
m}_{Y}(j_{0})=\pm 1,\;\;\;{\bf m}_{X}(j_{0}+1)-{\bf m}_{Y}(j_{0}+1)=\mp 1$ and
${\bf m}_{X}(j)={\bf
m}_{Y}(j)\;\;\forall\;j\in\\{1,\ldots,L\\}\setminus\\{j_{0},j_{0}+1\\}$.
Without loss of generality, suppose ${\bf m}_{X}(j_{0})-{\bf m}_{Y}(j_{0})=1$
and ${\bf m}_{X}(j_{0}+1)-{\bf m}_{Y}(j_{0}+1)=-1$. Using the same definitions
as before that $j_{0}:=x_{k_{0}}$ and ${\bf m}_{X}(j_{0})=r$, let ${\bf
m}_{X}(j_{0}+1)=s^{\prime}$, i.e. $s^{\prime}=s-1$ where ${\bf
m}_{Y}(j_{0}+1)=s$. Let $i_{1}=\min\left\\{k:x_{k}=j_{0}\right\\}$, then
$x_{k}=j_{0}$ for $k=i_{1},\ldots,i_{1}+(r-1)$ and $x_{k}=j_{0}+1$ for
$k=i_{1}+r,\ldots,i_{1}+r+(s^{\prime}-1)$. Therefore,

(A.1)
$\begin{split}\delta^{N}(X,Y)&=\sum_{j=1}^{N}|x_{j}-y_{j}|=0+\cdots+0+|x_{i_{1}+(r-1)}-y_{i_{1}+(r-1)}|+0+\cdots+0\\\
&=|j_{0}-(j_{0}+1)|=1\end{split}$

$\Longrightarrow\;\;$ $\delta^{N}(X,Y)=1$.
It then follows from the triangle inequality that $\delta^{N}(X,Y)$ is a lower
bound of the graph distance.

To prove equality, it suffices to show that there exists a path from $X$ to
$Y$ such that the length of the path is $\delta^{N}(X,Y)$. Let
$X,Y\in{\mathbb{S}}_{L}^{N}$ and $i_{0}=\min\\{k\in\left\\{1,\ldots
N\right\\}:x_{k}\neq y_{k}\\}$. Without loss of generality, suppose
$x_{i_{0}}<y_{i_{0}}$. Consider the path

(A.2)
$\begin{split}Y&=(\ldots,y_{i_{0}-1},y_{i_{0}},y_{i_{0}+1},\ldots)\longrightarrow
Y_{1}=(\ldots,y_{i_{0}-1},y_{i_{0}}-1,y_{i_{0}+1},\ldots)\\\ &\longrightarrow
Y_{2}=(\ldots,y_{i_{0}-1},y_{i_{0}}-2,y_{i_{0}+1},\ldots)\cdots\longrightarrow
Y_{y_{i_{0}}-x_{i_{0}}}=(\ldots,y_{i_{0}-1},x_{i_{0}},y_{i_{0}+1},\ldots)\>.\end{split}$

The case $x_{i_{0}}>y_{i_{0}}$ is similar by switching the roles of $X$ and
$Y$. Notice that ${\bf m}_{Y}\sim{\bf m}_{Y_{1}}\sim{\bf
m}_{Y_{2}}\sim\cdots\sim{\bf m}_{Y_{y_{i_{0}}-x_{i_{0}}}}$ and the length of
this path is $|x_{i_{0}}-y_{i_{0}}|$.
Define $i_{1}:=\min\\{k\in\left\\{i_{0}+1,\ldots,N\right\\}:x_{k}\neq y_{k}$.
Repeating the above process for $x_{i_{1}}<y_{i_{1}}$ (with a similar case for
$x_{i_{1}}>y_{i_{1}}$), we get another path of length $|x_{i_{1}}-y_{i_{1}}|$.
Since $N<\infty$, the process ends at some $i_{s}\leq n$ such that
$i_{s}:=\min\\{k\in\left\\{i_{0}+s,\ldots,N\right\\}:x_{k}\neq y_{k}\\}$ and
repeating the process for $x_{i_{s}}$ yields a path of length
$|x_{i_{s}}-y_{i_{s}}|$. Therefore, we have a path from $X$ to $Y$ of length

$|x_{i_{0}}-y_{i_{0}}|+|x_{i_{1}}-y_{i_{1}}|+\cdots+|x_{i_{s}}-y_{i_{s}}|$
(A.3) $\begin{split}=\sum_{j:1\leq
j<i_{0}}|x_{j}-y_{j}|+|x_{i_{0}}-y_{i_{0}}|&+\sum_{j:i_{0}<j<i_{1}}|x_{j}-y_{j}|+|x_{i_{1}}-y_{i_{1}}|+\cdots\\\
&+\sum_{j:i_{s-1}<j<i_{s}}|x_{j}-y_{j}|+|x_{i_{s}}-y_{i_{s}}|+\sum_{j:i_{s}<j\leq
N}|x_{j}-y_{j}|\\\
=\sum^{N}_{j=1}|x_{j}-y_{j}|=\delta^{N}(X,Y)\>,&\end{split}$

which finishes the proof. ∎

##  Appendix B Auxiliary results concerning the interaction potential

###  B.1. Proof of Proposition 2.5

###### Proof.

Firstly, note that for any ${\bf m}\in{\bf M}_{L}^{N}$ one gets

(B.1) $\displaystyle V({\bf m})$ $\displaystyle=\sum_{j=1}^{L-1}\left[J({\bf
m}(j)+{\bf m}(j+1))-{\bf m}(j){\bf m}(j+1)\right]+J({\bf m}(1)+{\bf m}(L))$
(B.2) $\displaystyle=2J\sum_{j=1}^{L}{\bf m}(j)-\sum_{j=1}^{L-1}{\bf m}(j){\bf
m}(j+1)=2JN-\sum_{j=1}^{L-1}{\bf m}(j){\bf m}(j+1)\>,$

which shows that finding minimizers of the potential is equivalent to finding
maximizers of $Q_{N}({\bf m}):=\sum_{j=1}^{L-1}{\bf m}(j){\bf m}(j+1)$. From
the explicit form of $Q_{N}$, it obvious that if $\operatorname{supp}({\bf
m})$ is not a discrete interval, i.e. if there exists a
$j_{0}\notin\operatorname{supp}({\bf m})$ with $\min\operatorname{supp}({\bf
m})<j_{0}<\max\operatorname{supp}({\bf m})$, then ${\bf m}$ cannot be a
maximizer of $Q_{N}$. Hence, from now on, we will only consider ${\bf
m}\in{\bf M}_{L}^{N}$ such that $\operatorname{supp}({\bf m})$ is a discrete
interval and w.l.o.g. let us assume $\operatorname{supp}({\bf
m})=\\{1,2,\dots,k\\}$ for some $k\in\mathbb{N}$. For a proof by
contradiction, let us now assume that there is an ${\bf m}\in{\bf M}_{L}^{N}$
which maximizes $Q_{N}$ but is not of the form (2.19) or (2.20). Note that
since $N\geq 4J$ but ${\bf m}(j)\leq 2J$ for every $j\in\\{1,2,\dots,k\\}$,
this implies that $k\geq 3$. Note that for $k=3$, we get

(B.3) $Q_{N}({\bf m})={\bf m}(1){\bf m}(2)+{\bf m}(2){\bf m}(3)={\bf
m}(2)(N-{\bf m}(2))\>,$

where we have used that ${\bf m}(3)=N-{\bf m}(1)-{\bf m}(2)$ for the last
equation. Now, observe that since $N\geq 4J$, the maximal possible value for
$Q_{N}$ is only attained if ${\bf m}(2)=2J$, which shows that for $k=3$, any
maximizer of $Q_{N}$ has to be of the form (2.19) or (2.20). Hence, from now
on, we will only consider the case $k\geq 4$. Let us now distinguish a few
different cases and show that for each of these cases, we can construct
another $\widehat{\bf m}\in{\bf M}_{L}^{N}$ such that $Q_{N}(\widehat{\bf
m})>Q_{N}({\bf m})$ – a contradiction to ${\bf m}$ maximizing $Q_{N}$:

  * First case:

${\bf m}(1)>{\bf m}(2)$. In this case, we set $\widehat{\bf m}(1):={\bf
m}(2),\widehat{\bf m}(2):={\bf m}(1)$ and for every other $j$, we define
$\widehat{\bf m}(j):={\bf m}(j)$. This yields

(B.4) $Q_{N}(\widehat{\bf m})-Q_{N}({\bf m})=({\bf m}(1)-{\bf m}(2))\cdot{\bf
m}(3)>0\>,$

which shows that ${\bf m}$ cannot maximize $Q_{N}$ in this case.

  * Second case:

${\bf m}(1)\leq{\bf m}(2)$, and it is not true that ${\bf m}(2)={\bf
m}(3)=2J$. We split this second case into two subcases:

    * First subcase:

${\bf m}(2)<{\bf m}(3)$. In this case, we define the configuration
$\widehat{\bf m}$ by setting $\widehat{\bf m}(1):={\bf m}(1)-1,\widehat{\bf
m}(2):={\bf m}(2)+1$ and for every other $j$, we define $\widehat{\bf
m}(j):={\bf m}(j)$. This yields

(B.5) $Q_{N}(\widehat{\bf m})-Q_{N}({\bf m})=({\bf m}(1)-1)+{\bf m}(3)-{\bf
m}(2)\geq{\bf m}(3)-{\bf m}(2)>0\>,$

which shows that ${\bf m}$ cannot maximize $Q_{N}$ in this case.

    * Second subcase:

${\bf m}(2)\geq{\bf m}(3)$. Note that since we are in the Second case, it is
not possible that ${\bf m}(3)=2J$. We then define the configuration
$\widehat{\bf m}$ by setting $\widehat{\bf m}(1):={\bf m}(1)-1,\widehat{\bf
m}(3):={\bf m}(3)+1$ and for every other $j$, we define $\widehat{m}(j):={\bf
m}(j)$. We then get

(B.6) $Q_{N}(\widehat{\bf m})-Q_{N}({\bf m})={\bf m}(4)>0\>,$

which shows that ${\bf m}$ is not a maximizer of $Q_{N}$.

  * Third case:

${\bf m}(2)={\bf m}(3)=2J$. Note that in this case, we must have $k\geq 5$,
since otherwise ${\bf m}$ would be of the form (2.19) or (2.20). Let
$k_{0}\in\\{4,\dots,k-1\\}$ be the smallest number for which ${\bf
m}(k_{0})\neq 2J$. (Again, note that such a $k_{0}$ must exist, since
otherwise ${\bf m}$ would be of the form (2.19) or (2.20).) In particular,
this means that ${\bf m}(k_{0}-1)=2J$ and ${\bf{m}}(k_{0}+1)>0$. Now, define
$\widehat{\bf m}(1):={\bf m}(1)-1,\widehat{\bf m}(k_{0}):={\bf m}(k_{0})+1$
and for every other $j$, we set $\widehat{\bf m}(j):={\bf m}(j)$. We then find

(B.7) $Q_{N}(\widehat{\bf m})-Q_{N}({\bf m})={\bf m}(k_{0}+1)>0\>,$

which shows that ${\bf m}$ cannot be a maximizer of $Q_{N}$.

Since these three cases exhaust all possibilities, this finishes the proof. ∎

###  B.2. Proof of Lemma 3.1

###### Proof.

Let $Z=(z_{1},z_{2},\dots,z_{j+k})\in{\mathbb{S}}_{L,K}^{j+k}$ be such that
$d^{j+k}(X\cup Y,{\mathbb{S}}^{j+k}_{L,K})=d^{j+k}(X\cup Y,Z)$. Now, introduce
the configurations
${Z}^{\prime}=(z_{1},z_{2},\dots,z_{j})\in{\mathbb{S}}_{L}^{j}$ and
${Z}^{\prime\prime}=(z_{j+1},\dots,z_{j+k})\in{\mathbb{S}}_{L}^{k}$. Moreover,
let ${\bf m}_{Z^{\prime}}$ and ${\bf m}_{Z^{\prime\prime}}$ denote the
corresponding occupation number functions. Note that ${\bf
m}_{Z^{\prime}}(i)=0$ for $i>z_{j}$ as well as ${\bf
m}_{Z^{\prime\prime}}(i)=0$ for $i<z_{j}$. Moreover, note that ${\bf
m}_{Z^{\prime}}(z_{j})+{\bf m}_{Z^{\prime\prime}}(z_{j})\leq 2J$. It follows
that $V({\bf m}_{Z^{\prime}})\leq V({\bf m}_{Z^{\prime}}+{\bf
m}_{Z^{\prime\prime}})$, i.e. $V({Z}^{\prime})\leq V(Z)$ and thus
$Z^{\prime}\in{\mathbb{S}}_{L,K}^{j}$. To see this, observe that

(B.8) $\displaystyle V({\bf m}_{Z^{\prime}}+{\bf m}_{Z^{\prime\prime}})$
$\displaystyle=V({\bf m}_{Z^{\prime}})+(2J-{\bf m}_{Z^{\prime}}(z_{j}-1)){\bf
m}_{Z^{\prime\prime}}(z_{j})+(2J-({\bf m}_{Z^{\prime}}(z_{j})+{\bf
m}_{Z^{\prime\prime}}(z_{j})){\bf m}_{Z^{\prime\prime}}(z_{j}+1)$ (B.9)
$\displaystyle+\sum_{i=z_{j}+1}^{L-1}(2J-{\bf m}_{Z^{\prime\prime}}(i)){\bf
m}_{Z^{\prime\prime}}(i+1)\geq V({\bf m}_{Z^{\prime}})\>,$

where we used that since ${\bf m}_{Z^{\prime}},{\bf
m}_{Z^{\prime\prime}}\in{\bf M}_{L}$, one has ${\bf m}_{Z^{\prime}}(i),{\bf
m}_{Z^{\prime\prime}}(i)\leq 2J$ for any $i\in\\{1,2,\dots,L\\}$ and moreover
that ${\bf m}_{Z^{\prime}}(z_{j})+{\bf m}_{Z^{\prime\prime}}(z_{j})\leq 2J$.
Now, if in addition $z_{j}\leq\ell$, this would imply
$Z^{\prime}\in{\mathbb{S}}_{\Lambda_{\ell},K}^{j}$ and since – trivially –
$d^{j}(X,{\mathbb{S}}_{\Lambda_{\ell},K}^{j})\leq d^{j}(X,Z^{\prime})\leq
d^{j+k}(X\cup Y,Z^{\prime}\cup Z^{\prime\prime})=d^{j+k}(X\cup
Y,{\mathbb{S}}_{L,K}^{j+k})$ this shows (3.1) for the case $z_{j}\leq\ell$. If
$z_{j}>\ell$, we need to modify our argument. If $z_{1}>\ell$, we construct
the occupation number function ${\bf g}\in{\bf M}^{j}_{\Lambda_{\ell}}$ as
follows:

(B.10) ${\bf
g}(i):=\begin{cases}2J&\mbox{if}\qquad\ell-\lfloor{\frac{j}{2J}}\rfloor+1\leq
i\leq\ell\\\
j(\bmod\;2J)&\mbox{if}\qquad\ell-\lfloor{\frac{j}{2J}}\rfloor\end{cases}\>.$

Let $X_{\bf g}\in{\mathbb{S}}^{j}_{\Lambda_{\ell}}$ denote the multiset
associated with $\bf g$. Note that $X_{\bf g}$ is of the form (2.19) or (2.20)
and thus, by Proposition 2.5 we have $X_{\bf
g}\in{\mathbb{S}}^{j}_{\Lambda_{\ell},K}$. Therefore,
$d^{j}(X,{\mathbb{S}}^{j}_{\Lambda_{\ell},K})\leq d^{j}(X,X_{\bf g})\leq
d^{j}(X,Z^{\prime})\leq d^{j+k}(X\cup Y,{\mathbb{S}}^{j+k}_{L,K})$ (where the
second inequality is due to the fact that $x_{i}\leq
g_{i}\leq\ell<z_{i},\;\>i=1,\ldots,j$). So, from now on, we may assume that
$z_{1}\leq\ell$ and thus let
$j_{0}:=\max\\{i\in\\{1,2,\dots,j\\}:z_{i}\leq\ell\\}$ be the index of the
last particle of $Z^{\prime}$ that still lies in $\Lambda_{\ell}$. We now
split $Z^{\prime}=Z^{\prime}_{\Lambda_{\ell}}\cup
Z^{\prime}_{\Lambda_{\ell}^{c}}$, where
$Z^{\prime}_{\Lambda_{\ell}}=(z_{1},\dots,z_{j_{0}})$ and
$Z^{\prime}_{\Lambda_{\ell}^{c}}=(z_{j_{0}+1},\dots,z_{j})$. Let ${\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}\in{\bf M}_{\Lambda_{\ell}}^{j_{0}}$ be the
occupation number function associated to $Z_{\Lambda_{\ell}}^{\prime}$.
Lastly, let $j_{c}:=j-j_{0}$ be the number of particles in
$Z^{\prime}_{\Lambda_{\ell}^{c}}$. We then distinguish the following two
cases:

  * Case 1:

$j_{c}+{\bf m}_{Z_{\Lambda_{\ell}}^{\prime}}(\ell)+{\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}(\ell-1)\geq 4J$.
Define $r_{0}:=\max\\{k\in\\{1,\ldots,\ell-2\\}:\sum_{s=k}^{\ell}(2J-{\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}(s))\geq j_{c}\\}$ and let
$\mu=j_{c}-\sum^{\ell}_{s=r_{0}+1}(2J-{\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}(s))$. We construct the occupation number
function ${\bf c}\in{\bf M}_{\Lambda_{\ell}}^{j}$ as follows:

(B.11) ${\bf c}(i):=\begin{cases}{\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}(i)&\mbox{if}\qquad 1\leq i\leq r_{0}-1\\\
{\bf m}_{Z_{\Lambda_{\ell}}^{\prime}}(i)+\mu&\mbox{if}\qquad i=r_{0}\\\
2J&\mbox{if}\qquad r_{0}+1\leq i\leq\ell\end{cases}$

In other words, the particles in $Z^{\prime}_{\Lambda_{\ell}^{c}}$ are used to
“fill up” each site of configuration ${\bf m}_{Z_{\Lambda_{\ell}}^{\prime}}$
to its maximal occupation number $2J$ – starting from
$\ell,\ell-1,\mbox{etc.}$ until the $j_{c}$ particles in
$Z^{\prime}_{\Lambda_{\ell}^{c}}$ have been exhausted (cf. Figure B.1). Note
that the choice ${\bf c}(\ell)={\bf c}(\ell-1)=2J$ is following the same
principle and is due to the assumption $j_{c}+{\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}(\ell)+{\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}(\ell-1)\geq 4J$.

12…$\ell$$\ell+1$$\ell+2$$\ell+3$$\ell+4$$\ell+5$

Figure B.1. Picture showing the construction of configuration ${\bf c}$, by
moving the $j_{c}=11$ particles in $\Lambda_{\ell}^{c}$ such that they fill up
each site to its maximum occupation number $2J=6$ starting at $\ell$ to the
left (here: from $\ell$ to $\ell-2$).

We now make the following observations: Firstly, let $X_{\bf
c}\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}$ denote the multiset corresponding to
${\bf c}$. Then, by construction, it is clear that $d^{j}(X,X_{\bf c})\leq
d^{j}(X,Z^{\prime})$. Secondly, let
$s_{0}=\min\\{s\in\\{1,2,\dots,\ell-2\\}:\forall t\in\\{s+1,\dots,\ell\\}:{\bf
c}(t)=2J\\}$. If $s_{0}=1$, then $X_{\bf c}$ is a minimizer configuration with
${\bf c}(i)=2J\quad\forall\;i=2,\ldots,\ell$ and hence, $X_{\bf
c}\in{\mathbb{S}}^{j}_{\Lambda_{\ell},K}$. Since $x_{i}\leq c_{i}\leq
z_{i},\;\;i=1,\ldots,j$, we get

(B.12) $d^{j}(X,X_{\bf
c})=\sum^{j}_{i=1}(c_{i}-x_{i})\leq\sum^{j}_{i=1}(z_{i}-x_{i})\leq
d^{j}(X,Z^{\prime})$

and therefore, $d^{j}(X,{\mathbb{S}}^{j}_{\Lambda_{\ell},K})\leq
d^{j}(X,X_{\bf c})\leq d^{j}(X,Z^{\prime})\leq d^{j+k}(X\cup
Y,{\mathbb{S}}^{j+k}_{L,K})$ for the case $s_{0}=1$.

Now for $s_{0}\neq 1$, define ${\bf c}^{\prime\prime}:={\bf c}\cdot
1_{\\{s_{0},\dots,\ell\\}}$, ${\bf c}^{\prime}:={\bf c}-{\bf
c}^{\prime\prime}$, ${\bf d}^{\prime\prime}:={{\bf m}_{Z^{\prime}}}\cdot
1_{\\{s_{0},\dots,L\\}}$ and ${\bf d}^{\prime}:={\bf m}_{Z^{\prime}}-{\bf
d}^{\prime\prime}$. Then, observe that ${\bf c}(s_{0})\geq{\bf
m}_{\mathbb{Z}^{\prime}_{\Lambda_{\ell}}}(s_{0})$ and moreover that ${\bf
c}(s)={\bf m}_{\mathbb{Z}^{\prime}_{\Lambda_{\ell}}}(s)$ for every $s<s_{0}$.
Consequently, we get ${\bf c}^{\prime}={\bf d}^{\prime}$. Observe that ${\bf
c}^{\prime\prime}$ is of the form (2.19) or (2.20) and thus $V({\bf
c}^{\prime\prime})=4J^{2}$. Moreover, by Proposition 2.5, we have $V({\bf
c^{\prime\prime}})\leq V({\bf d^{\prime\prime}})$. Consequently, we get

(B.13) $V({\bf c})=V({\bf c^{\prime}})-{\bf c}(s_{0}-1){\bf c}(s_{0})+V({\bf
c}^{\prime\prime})\leq V({\bf d}^{\prime})-{\bf
m}_{Z^{\prime}_{\Lambda_{\ell}}}(s_{0}-1){\bf
m}_{Z^{\prime}_{\Lambda_{\ell}}}(s_{0})+V({\bf d}^{\prime\prime})=V({\bf
m}_{Z^{\prime}})\leq K\>.$

This shows that $X_{\bf c}\in{\mathbb{S}}_{\Lambda_{\ell},K}^{j}$. Since
$d^{j}(X,{\mathbb{S}}_{\Lambda_{\ell},K}^{j})\leq d^{j}(X,X_{\bf c})\leq
d^{j}(X,Z^{\prime})\leq d^{j+k}(X\cup Y,{\mathbb{S}}_{L,K}^{j+k})$, this shows
the assertion for Case 1.

  * Case 2:

$j_{c}+{\bf m}_{Z_{\Lambda_{\ell}}^{\prime}}(\ell)+{\bf
m}_{Z_{\Lambda_{\ell}}^{\prime}}(\ell-1)<4J$. Firstly, let us define the
following two quantities:

(B.14) $\displaystyle\xi^{\prime}$ $\displaystyle:={\bf
m}_{Z^{\prime}_{\Lambda_{\ell}}}(\ell-1)+\sum_{k\in\mathbb{N}:\ell+2k-1\leq
L}{\bf m}_{Z^{\prime}_{\Lambda^{c}_{\ell}}}(\ell+2k-1)$ (B.15)
$\displaystyle\xi^{\prime\prime}$ $\displaystyle:={\bf
m}_{Z^{\prime}_{\Lambda_{\ell}}}(\ell)\qquad+\sum_{k\in\mathbb{N}:\ell+2k\leq
L}{\bf m}_{Z^{\prime}_{\Lambda^{c}_{\ell}}}(\ell+2k)\quad\mbox{and observe
that}$ (B.16) $\displaystyle\sum_{i=\ell-1}^{L-1}{\bf m}_{Z^{\prime}}(i){\bf
m}_{Z^{\prime}}(i+1)\leq\xi^{\prime}\xi^{\prime\prime}\>,$

since the left hand side of (B.16) is a sum over a subset of the
(non–negative) cross–terms one obtains when expanding the product
$\xi^{\prime}\xi^{\prime\prime}$. Moreover, note that the assumption for being
in Case 2 is equivalent to $\xi^{\prime}+\xi^{\prime\prime}<4J$. Next, we
define the occupation number function ${\bf
e}\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}$ as follows:

(B.17) ${\bf
e}(i):=\begin{cases}\min\\{2J,\xi^{\prime\prime}\\}+\max\\{\xi^{\prime}-2J,0\\}\qquad&\mbox{if}\qquad
i=\ell\\\
\min\\{2J,\xi^{\prime}\\}+\max\\{\xi^{\prime\prime}-2J,0\\}\qquad&\mbox{if}\qquad
i=\ell-1\\\ {\bf m}_{Z^{\prime}_{\Lambda_{\ell}}}(i)\qquad&\mbox{if}\qquad
i\in\\{1,2,\dots,\ell-2\\}\>.\end{cases}$

Let us now show that $V({\bf e})\leq V({\bf m}_{Z^{\prime}})$. To this end,
observe that

(B.18) $\displaystyle V({\bf m}_{Z^{\prime}})-V({\bf e})=$ $\displaystyle{\bf
m}_{Z^{\prime}}(\ell-2)({\bf e}(\ell-1)-{\bf m}_{Z^{\prime}}(\ell-1))+{\bf
e}(\ell-1){\bf e}(\ell)-\sum_{i=\ell-1}^{L-1}{\bf m}_{Z^{\prime}}(i){\bf
m}_{Z^{\prime}}(i+1)$ (B.19) $\displaystyle\geq$ $\displaystyle{\bf
e}(\ell-1){\bf e}(\ell)-\xi^{\prime}\xi^{\prime\prime}\>,$

where we used ${\bf e}(\ell-1)\geq{\bf m}_{Z_{\Lambda_{\ell}}}(\ell-1)$ and
(B.16). Since $\xi^{\prime}+\xi^{\prime\prime}<4J$, there are only three
possible cases:

    * (i)

$\xi^{\prime\prime}>2J$ and $\xi^{\prime}\leq 2J$

    * (ii)

$\xi^{\prime\prime}\leq 2J$ and $\xi^{\prime}>2J$

    * (iii)

$\xi^{\prime}\leq 2J$ and $\xi^{\prime\prime}\leq 2J$.

We will only discuss Cases (i) and (iii); Case (ii) follows from an argument
similar to Case (i). If we are in Case (i), this means ${\bf e}(\ell)=2J$ and
${\bf e}(\ell-1)=\xi^{\prime}+\xi^{\prime\prime}-2J$. Hence, by (B.19), we get

(B.20) $V({\bf m}_{Z^{\prime}})-V({\bf e})\geq
2J(\xi^{\prime}+\xi^{\prime\prime}-2J)-\xi^{\prime}\xi^{\prime\prime}=(2J-\xi^{\prime})(\xi^{\prime\prime}-2J)\geq
0\>.$

For Case (iii), we have ${\bf e}(\ell)=\xi^{\prime\prime}$ and ${\bf
e}(\ell-1)=\xi^{\prime}$ and thus get – again by (B.19) – the estimate

(B.21) $V({\bf m}_{Z^{\prime}})-V({\bf
e})\geq\xi^{\prime}\xi^{\prime\prime}-\xi^{\prime}\xi^{\prime\prime}=0\>.$

Hence, we have shown $V({\bf e})\leq V({\bf m}_{Z^{\prime}})$. Now, let
$X_{\bf e}\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}$ denote the multiset associated
to ${\bf e}$. If we can show that $d^{j}(X,X_{\bf e})\leq
d^{j}(X,Z^{\prime})$, this will prove (3.1) for the Case 2. Now, to see this,
we firstly define the configuration ${\bf f}\in{\bf M}_{\Lambda_{\ell}}^{j}$
as follows

(B.22) ${\bf f}(i):=\begin{cases}\min\\{2J,{\bf
m}_{Z^{\prime}}(\ell)+j_{c}\\}\quad&\mbox{if}\quad i=\ell\\\ {\bf
m}_{Z^{\prime}}(\ell-1)+\max\\{{\bf
m}_{Z^{\prime}}(\ell)+j_{c}-2J,0\\}\quad&\mbox{if}\quad i=\ell-1\\\ {\bf
m}_{Z^{\prime}}(i)\quad&\mbox{if}\quad
i\in\\{1,2,\dots,\ell-2\\}\>,\end{cases}$

which means that configuration ${\bf f}$ is obtained by adding the particles
in $Z^{\prime}_{\Lambda_{c}}$ to the configuration ${\bf m}_{Z^{\prime}}$ –
starting at site $\ell$ and any possibly remaining particles to site $\ell-1$.
Let $X_{\bf f}=(f_{1},f_{2},\dots,f_{j})\in{\mathbb{S}}_{\Lambda_{\ell}}^{j}$
be the multiset associated to ${\bf f}$ and let $p=\min\\{i:z_{i}>\ell\\}$ and
$\xi=\max\left\\{i:z_{i}\leq\ell-2\right\\}$ (note that $\xi$ exists since if
$z_{1}>\ell-2$ and $j_{c}<4J-{\bf m}_{Z^{\prime}}(\ell-1)-{\bf
m}_{Z^{\prime}}(\ell)$, then we would have $j=j_{0}+j_{c}<4J$, a
contradiction). Again, we distinguish two cases:

    * (a)

If $j_{c}\leq 2J-{\bf m}_{Z^{\prime}}(\ell)$, then $z_{i}=f_{i}$ for $1\leq
i\leq p-1$ and $f_{i}=\ell$ for $p\leq i\leq j$. Then,

(B.23)
$\begin{split}d^{j}(X,Z^{\prime})&=\sum_{i=1}^{p-1}|x_{i}-z_{i}|+\sum_{i=p}^{j}(z_{i}-x_{i})\\\
&=\sum_{i=1}^{p-1}|x_{i}-z_{i}|+\sum_{i=p}^{j}(z_{i}-f_{i})+\sum_{i=p}^{j}(f_{i}-x_{i})\\\
&=\sum_{i=1}^{p-1}|x_{i}-f_{i}|+\sum_{i=p}^{j}|z_{i}-f_{i}|+\sum_{i=p}^{j}|f_{i}-x_{i}|\\\
&=d^{j}(X,X_{\bf f})+\sum_{i=j-j_{c}+1}^{j}|z_{i}-f_{i}|\geq d^{j}(X,X_{\bf
f})+j_{c}\end{split}$

    * (b)

If $j_{c}>2J-{{\bf m}_{Z^{\prime}}}(\ell)$, let $\eta=j_{c}-(2J-\alpha_{1})$,
where $\alpha_{1}={{\bf m}_{Z^{\prime}}}(\ell)$ and $\alpha_{2}={{\bf
m}_{Z^{\prime}}}(\ell-1)$. Then, observe that

(B.24) $\left\\{\begin{split}f_{i}&=z_{i},\;\;\;\;\;1\leq
i\leq\xi+\alpha_{2}\\\ f_{i}&=\ell-1,\;\;\;\;\;\xi+\alpha_{2}+1\leq
i\leq\xi+\alpha_{2}+n\\\ f_{i}&=\ell,\;\;\;\;\;i>\xi+\alpha_{2}+\eta\\\
z_{i}&=\ell,\;\;\;\;\;\xi+\alpha_{2}+1\leq i\leq p-1\end{split}\right.$

Hence,

(B.25)
$\begin{split}d^{j}(X,Z^{\prime})&=\sum_{i=1}^{\xi+\alpha_{2}}|x_{i}-z_{i}|+\sum_{i=\xi+\alpha_{2}+1}^{p-1}|x_{i}-z_{i}|+\sum_{i=p}^{j}(z_{i}-x_{i})\\\
\\\
&=\sum_{i=1}^{\xi+\alpha_{2}}|x_{i}-z_{i}|+\sum_{i=\xi+\alpha_{2}+1}^{\xi+\alpha_{2}+\eta}(z_{i}-x_{i})+\sum_{i=\xi+\alpha_{2}+\eta+1}^{p-1}(z_{i}-x_{i})+\sum_{i=p}^{j}(z_{i}-f_{i})+\sum_{i=p}^{j}(f_{i}-x_{i})\\\
\\\
&=\sum_{i=1}^{\xi+\alpha_{2}}|x_{i}-f_{i}|+\sum_{i=\xi+\alpha_{2}+1}^{\xi+\alpha_{2}+\eta}(z_{i}-f_{i})+\sum_{i=\xi+\alpha_{2}+1}^{\xi+\alpha_{2}+\eta}(f_{i}-x_{i})+\sum_{i=\xi+\alpha_{2}+\eta+1}^{p-1}(z_{i}-f_{i})\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sum_{i=\xi+\alpha_{2}+\eta+1}^{p-1}(f_{i}-x_{i})+\sum_{i=p}^{j}(z_{i}-f_{i})+\sum_{i=p}^{j}(f_{i}-x_{i})\end{split}$
$\begin{split}&=\sum_{i=1}^{\xi+\alpha_{2}}|x_{i}-f_{i}|+\sum_{i=\xi+\alpha_{2}+1}^{\xi+\alpha_{2}+\eta}|f_{i}-x_{i}|+\sum_{i=\xi+\alpha_{2}+\eta+1}^{p-1}|f_{i}-x_{i}|+\sum_{i=p}^{j}|f_{i}-x_{i}|\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\sum_{i=\xi+\alpha_{2}+1}^{\xi+\alpha_{2}+\eta}|z_{i}-f_{i}|+\sum_{i=\xi+\alpha_{2}+\eta+1}^{p-1}|z_{i}-f_{i}|+\sum_{i=p}^{j}|z_{i}-f_{i}|\\\
\\\ &=d^{j}(X,X_{\bf
f})+\sum_{i=\xi+\alpha_{2}+1}^{\xi+\alpha_{2}+\eta}|z_{i}-f_{i}|+\sum_{i=\xi+\alpha_{2}+\eta+1}^{p-1}|z_{i}-f_{i}|+\sum_{i=p}^{j}|z_{i}-f_{i}|\\\
\\\ &\geq d^{j}(X,X_{\bf f})+\sum_{i=p}^{j}|z_{i}-f_{i}|\\\ &\geq
d^{j}(X,X_{\bf f})+(j-p+1)=d^{j}(X,X_{\bf f})+j_{c}\end{split}$

Moreover, since ${\bf e}(i)\neq{\bf f}(i)$ only if $i=\ell-1,\ell$, this
implies

(B.26) $d^{j}(X_{\bf e},X_{\bf f})=|{\bf f}(\ell)-{\bf e}(\ell)|\leq j_{c}\>.$

We therefore get

(B.27) $d^{j}(X,X_{\bf e})\leq d^{j}(X,X_{\bf f})+d^{j}(X_{\bf f},X_{\bf
e})\leq d^{j}(X,X_{\bf f})+j_{c}\leq d^{j}(X,Z^{\prime})\;,$

and thus, we have shown (3.1) for Case 2.

This finishes the proof. ∎

## References

  * [1] H. Abdul-Rahman, C. Fischbacher and G. Stolz: Entanglement bounds in the XXZ quantum spin chain, Ann. Henri Poincaré 21, 2327-2366, (2020).
  * [2] L. Adame, L. M. Rivera and A. Trujillo-Negrete: Hamiltonicity of the Double Vertex Graph and the Complete Double Vertex Graph of some Join Graphs (2020). _arXiv:2007.00115_.
  * [3] V. Beaud and S. Warzel: Low-energy Fock space localization for attractive hard-core particles in disorder, Ann. Henri Poincaré 18, 3143-3166 (2017).
  * [4] V. Beaud and S. Warzel: Bounds on the entanglement entropy of droplet states in the XXZ spin chain, J. Math. Phys. 59, 012109 (2018).
  * [5] W. Carballosa, R. Fabila-Monroy, J. Leaños and L. M. Rivera, Regularity and planarity of token graphs, Discuss. Math. Graph Theory 37 (2017), 573–586
  * [6] A. Elgart, A. Klein and G. Stolz, _Many-body localization in the droplet spectrum of the random XXZ quantum spin chain_ , J. Funct. Anal. 275 (2018), 211–258
  * [7] A. Elgart, A. Klein and G. Stolz, _Manifestations of dynamical localization in the disordered XXZ spin chain_ , Comm. Math. Phys. 361 (2018), 1083–1113
  * [8] C. Fischbacher: A Schrödinger Operator Approach to Higher Spin XXZ Systems on General Graphs, Analytic trends in Mathematical physics, Contemp. Math. 741, 83–103 (2020).
  * [9] C. Fischbacher and R. Schulte: Lower Bound to the Entanglement Entropy of the XXZ Spin Ring, _arXiv:2007.00735_
  * [10] C. Fischbacher and G. Stolz: Droplet states in quantum XXZ spin systems on general graphs, J. Math. Phys. 59, 051901 (2018).
  * [11] A. Klein, _private communication_.
  * [12] H. Leschke, A. V. Sobolev and W. Spitzer, Scaling of Rényi entanglement entropies of the free Fermi-gas ground state: A rigorous proof, Phys. Rev. Lett. 112, 160403–1–5 (2014).
  * [13] H. Leschke, A. V. Sobolev and W. Spitzer, Trace formulas for Wiener-Hopf operators with applications to entropies of free fermionic equilibrium states, J. Funct. Anal. 273, 1049–1094 (2017).
  * [14] J. Mulherkar, B. Nachtergaele, R. Sims and S. Starr, _Isolated Eigenvalues of the Ferromagnetic Spin-J XXZ Chain with Kink Boundary Conditions_ , J. Stat. Mech. (2008) P01016
  * [15] P. Müller, L. Pastur and R. Schulte, How much delocalisation is needed for an enhanced area law of the entanglement entropy?, Commun. Math. Phys. 376, 649–679 (2020).
  * [16] P. Müller and R. Schulte, Stability of the enhanced area law of the entanglement entropy (2020), _arXiv:2004.02700_.
  * [17] Y. Ouyang, Computing spectral bounds of the Heisenberg ferromagnet from geometric consideratios. J. Math. Phys. 60(7), 071901 (2019).
  * [18] B. Pfirsch and A. V. Sobolev, Formulas of Szegő type for the periodic Schrödinger operator, Commun. Math. Phys. 358, 675–704 (2018).
  * [19] L. M. Rivera and A. L. Trujillo-Negrete, Hamiltonicity of token graphs of fan graphs, Art Discr. Appl. Math., 1, #P07 (2018).
  * [20] S. Starr, Some properties for the low-lying spectrum of the ferromagnetic, quantum XXZ spin system, PhD Thesis, UC Davis, 2001.
  * [21] G. Stolz, Aspects of the Mathematical Theory of Disordered Quantum Spin Chains, Analytic trends in Mathematical physics, Contemp. Math. 741, 163–197 (2020).
  * [22] J. Weidmann: Lineare Operatoren in Hilberträumen, Teil I Grundlagen, Verlag B.G. Teubner, Stuttgart/Leipzig/Wiesbaden, 2000 (German).
  * [23] M. M. Wolf, Violation of the entropic area law for fermions, Phys. Rev. Lett. 96, 010404–1–4 (2006).