# Entanglement in the Quantum Spherical Model - a Review

Sascha Wald1,2, Raul Arias3, and Vincenzo Alba4 1Statistical Physics Group,
Centre for Fluid and Complex Systems, Coventry University, Coventry, England
2 $\mathbb{L}^{4}$ Collaboration & Doctoral College for the Statistical
Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, Europe  3Instituto
de Física La Plata - CONICET and Departamento de Física, Universidad Nacional
de La Plata C.C. 67, 1900, La Plata, Argentina  4Dipartimento di Fisica dell’
Università di Pisa and INFN, Sezione di Pisa, I-56127 Pisa, Italy

###### Abstract

We review some recent results on entanglement in the Quantum Spherical Model
(QSM). The focus lays on the physical results rather than the mathematical
details. Specifically, we study several entanglement-related quantities, such
as entanglement entropies, and logarithmic negativity, in the presence of
quantum and classical critical points, and in magnetically ordered phases. We
consider both the short as well as the long-range QSM. The study of
entanglement properties of the QSM is feasible because the model is mappable
to a Gaussian system in any dimension. Despite this fact the QSM is an ideal
theoretical laboratory to investigate a wide variety of physical scenarios,
such as non mean field criticality, the effect of long-range interactions, the
interplay between finite-temperature fluctuations and genuine quantum ones.

Keywords: entanglement; entanglement gap; Schmidt gap; entanglement
negativity; universality; phase transition; quantum phase transition;
classical and quantum fluctuations; long-range interactions

##  1 Introduction

Quantifying entanglement in strongly interacting many-body systems has become
an important research theme in recent years, and has provided useful insights
to understand the structure of quantum correlations [1, 2, 3, 4, 5]. In
principle, quantum states are entangled whenever they cannot be written as a
product state. A plethora of entanglement witnesses has been introduced to
quantify the extent to which quantum states are entangled. Widely used tools
that play an important role in entanglement studies comprise, amongst others:
Rényi entropies, the mutual information, the entanglement negativity, and the
entanglement spectrum. We shall introduce these quantities in Sec. 3 in more
detail. It is important to note that there is not a single entanglement
quantifier that works for all setups and systems, reflecting the intricacy of
quantum entanglement in many-body systems.

Crucially, the study of entanglement in many-body systems heavily relies on
numerical simulations that are quite demanding, even with modern computing
hardware. Thus, obtaining reliable scaling predictions or extracting
qualitative behaviors in the thermodynamic limit is challenging and often not
attainable. Similar to the study of continuous phase transitions, a viable
option to overcome computational limitations is to study simplified systems
that allow for analytical investigations and predictions [6]. The spherical
model [7, 8, 9, 10] has firmly established itself as a reference system
whenever investigations in generic many-body systems prove to be challenging.
The spherical model and its quantum formulation [9, 10, 11, 12, 13] are
analytically solvable in a variety of scenarios, including arbitrary spatial
dimension, temperature and external fields. Moreover, the model possesses a
phase transition separating a paramagnetic from a ferromagnetic phase that is
generally not in the mean-field universality class.

Not surprisingly, the QSM, and closely-related models, proved to be useful to
understand entanglement properties of quantum many-body systems [14, 15, 16,
17, 18, 19, 20, 21]. Crucially, the QSM allows to derive the precise finite-
size scaling of entanglement-related quantities, often analytically. This
happens because the QSM is mappable to a Gaussian bosonic system with a
constraint. This implies that entanglement properties are obtained from the
two point correlation functions [22], which are accessible analytically [11,
13].

The aim of this review is to give a few examples of the wide variety of
physical scenarios where the behavior of entanglement can be addressed in the
controlled setup of the QSM, yet retaining the complexity of a quantum many-
body system. Specifically, here we focus on the main results of Refs. [15, 16,
17, 18].

This review is organized as follows. In Sec. 2 we review relevant properties
of the QSM. In particular, we highlight how the spherical model was conceived,
how a quantum version is formulated, and sketch how to generally solve the
model. Phase diagrams of all scenarios considered in this review are also
discussed. In Sec. 3 we introduce all relevant entanglement-related quantities
that we investigate in this work, such as the entanglement entropy, the
negativity, the entanglement spectrum. We also briefly discuss how
entanglement quantities are calculated in the QSM. Sections 4, 5, 6 and 7 are
dedicated to the main results. In Sec. 4 we focus on the interplay between
quantum and classical fluctuations at finite temperature criticality in the
three-dimensional QSM. In particular, we show that the logarithmic negativity
is able to distinguish genuine quantum correlations from classical ones. In
Sec. 5 we explore the entanglement gap (or Schmidt gap), which is the lowest
laying gap of the so-called entanglement spectrum (ES). We consider the zero-
temperature QSM in two dimensions. By using dimensional reduction, we compute
the scaling behavior of the entanglement gap at criticality. In Sec. 6 we show
that in the ferromagnetic phase the entanglement gap can be written in terms
of standard magnetic correlation functions, due to the presence of a Goldstone
mode. Finally, in Sec. 7 we study the entanglement gap in the one-dimensional
QSM with long-range interactions and at zero temperature. In Sec. 8 we
summarize and conclude our results.

##  2 Quantum spherical model

$++$$--$$-+$$+-$Ising model

$++$$--$$-+$$+-$Spherical model

Figure 1: Illustration of configuration spaces: On the left we show all
possible configurations of two Ising spins (vertices of a square). On the
right we show the extension of the configuration space to two spherical spins.

The Ising model, see Ref. [23] for an overview, has significantly contributed
to our modern understanding of collective phenomena [6]. Despite its apparent
simplicity it finds applications in a wide variety of fields, see, e.g., [24,
25, 26, 27, 28] for a brief list that is by no means exhaustive.

The classical Ising model is analytically solvable in one spatial dimension,
see, e.g., [29], but it does not possess a phase transition. In two dimensions
the model is still exactly solvable and it exhibits a finite temperature
transition. The Ising universality class of the transition is one of the most
studied in statistical physics [30]. Alas, already the three dimensional Ising
model is not solved analytically to date [31].

To overcome this problem, Berlin and Kac in 1952 suggested a generalization of
the Ising model by replacing the discrete Ising spin degrees of freedom with
continuous ones with an additional constraint that enforces some of the
properties of the original Ising degrees of freedom [7]. Specifically, since
each Ising spin on a lattice $\mathcal{L}$ satisfies $\sigma_{i}^{2}=1$, it is
obvious that $\sum_{i\in\mathcal{L}}\sigma_{i}^{2}=V$ with $V$ being the
system volume. Replacing the Ising spins with continuous degrees of freedom,
i.e., $\sigma_{i}\to S_{i}\in\mathbb{R}$, while simultaneously enforcing the
external constraint $\sum_{i\in\mathcal{L}}S_{i}^{2}=V$ yields the original
formulation of the classical spherical model, see Fig. 1. This classical
spherical model is exactly solvable in any dimension and supports a finite
temperature phase transition in more than two dimensions $d>2$. After the
paper of Berlin and Kac, later in the same year it was shown that the strict
spherical constraint can be relaxed to be only satisfied on average, i.e.,
$\sum_{i\in\mathcal{L}}\left<S_{i}^{2}\right>=V$, without affecting the
universal bulk behavior of the model [8]. Interestingly, the spherical model
is related to more realistic spin systems like the $O(N)$ Heisenberg model in
the limit $N\to\infty$ of infinite spin dimensionality [32].

The spherical model with the average constraint admits also a quantum
generalization [9, 10, 11, 12, 13]. The Hamiltonian of this quantum spherical
model (QSM) reads

$\displaystyle
H=\sum_{n\in\mathcal{L}}\left(\frac{g}{2}p_{n}^{2}+\frac{1}{2}\sum_{m\in\mathcal{L}}u_{nm}x_{n}x_{m}\right),\quad\text{with}\quad\sum_{n\in\mathcal{L}}\left<x_{n}^{2}\right>=V.$
(1)

Here the classical spin degrees of freedom $S_{i}$ are replaced by position
operators $x_{i}$, and associate momentum operators $p_{i}$ were introduced
which satisfy $[x_{n},p_{m}]=\mathrm{i}\hbar\delta_{nm}$. We consider
$\mathcal{L}$ as a $d$ dimensional hypercubic lattice.

Figure 2: Phase diagrams of the QSM in different dimension: In $d=3$ (left
panel) the QSM with nearest neighbor interactions exhibits a thermal critical
line and a quantum phase transition at $T=0$. In $d=2$ (center panel) only a
quantum phase transition is present. For long-range interactions in $d=1$
(right panel) the zero-temperature critical behavior depends on the long-range
exponent $\alpha$. For $\alpha<2/3$ the transition is mean-field (purple).

In addition to the finite temperature transitions of the classical spherical
model, the QSM supports a zero-temperature quantum phase transition [11, 13].
If $u_{nm}$ is short-ranged, e.g., nearest neighbor interaction, then the
quantum phase transition exists for $d>1$ only. This transition is generally
in the same universality class as the thermal transition in the $d+1$
dimensional classical spherical model [11]. Conversely, for long-ranged
interactions a quantum phase transition is also present for $d=1$ [11]. The
universality class depends on the long range exponent $\alpha$ (see below),
although in a simple manner. Generally, $u_{nn}=2\mu$ where $\mu$ is a
Lagrange multiplier (chemical potential) that allows to enforce the spherical
constraint. This $\mu$ plays the physical role of a mass for the model, or,
equivalently, of the inverse correlation length. Hence, the critical line of
the model is retrieved from the constraint for $\mu=0$ [11, 13].

Here, we only consider translation invariant systems with periodic boundary
conditions such that the Hamiltonian generally decouples in Fourier space. Let
us introduce the Fourier transformed operators as

$\displaystyle
x_{n}=\frac{1}{\sqrt{V}}\sum_{k\in\mathcal{B}}e^{\mathrm{i}kn}q_{k},\qquad
p_{n}=\frac{1}{\sqrt{V}}\sum_{k\in\mathcal{B}}e^{-\mathrm{i}kn}\pi_{k}$ (2)

with the $d$ dimensional Brioullin zone $\mathcal{B}$. We recast the QSM
Hamiltonian in the form

$\displaystyle
H=\sum_{k\in\mathcal{B}}\frac{g}{2}\pi_{k}\pi_{-k}+\frac{u_{k}}{2}q_{k}q_{-k}=\sum_{k\in\mathcal{B}}E_{k}\left(b_{k}^{\dagger}b_{k}+\frac{1}{2}\right)$
(3)

with $u_{k}$ being the Fourier transform of the interaction potential
$u_{nm}$, $E_{k}=\sqrt{gu_{k}}$ the eigenenergies of the QSM, and $b_{k}$,
$b_{k}^{\dagger}$ are adequately chosen bosonic ladder operators. This allows
us to explicitly write the equilibrium correlation functions for a system at
temperature $T$ as

$\displaystyle\mathbb{X}_{nm}$ $\displaystyle:=\langle
x_{n}x_{m}\rangle=\frac{g}{2V}\sum_{k}e^{\mathrm{i}(n-m)k}\frac{1}{E_{k}}\coth\left(\frac{E_{k}}{2T}\right),$
(4) $\displaystyle\mathbb{P}_{nm}$ $\displaystyle:=\langle
p_{n}p_{m}\rangle=\frac{1/g}{2V}\sum_{k}e^{-\mathrm{i}(n-m)k}E_{k}\coth\left(\frac{E_{k}}{2T}\right).$
(5)

In the following sections we shall focus on entanglement quantities in the
QSM. Specifically we consider the following situations

  * •

QSM at finite temperature $T>0$ for $d=3$ with nearest neighbor interactions,
viz.,

$\displaystyle u(k)=2\mu+2(3-\cos k_{x}-\cos k_{y}-\cos k_{z})$ (6)

  * •

QSM at $T=0$ for $d=2$ with nearest neighbor interactions, viz.,

$\displaystyle u(k)=2\mu+2(2-\cos k_{x}-\cos k_{y})$ (7)

  * •

QSM at $T=0$ for $d=1$ with long-range interactions [33, 34], viz.,

$\displaystyle u(k)=2\mu+\left(2(1-\cos k)\right)^{\alpha/2}.$ (8)

In Eq. (8), $\alpha$ is the exponent governing the decay with distance of the
long-range interactions. The phase diagrams for these systems are depicted in
Fig. 2. An important ingredient for the further analysis is to understand the
finite-size scaling of the spherical parameter $\mu$. Specifically, we use
that $\mu\to 0$ in the ferromagnetic phase and at criticality for
$L\to\infty$. For finite $L$ conversely, $\mu$ is always finite. A variety of
works have considered this scaling in the classical spherical model, see,
e.g., Refs. [35, 36, 37, 38].

##  3 Entanglement in the quantum spherical model

In this section we introduce several relevant quantum-information-motivated
quantities, which have attracted a lot of attention in the statistical and
high energy theory communities in the last few years. Consider a many-body
quantum system described by a Hilbert space ${\cal{H}}$ and a density matrix
$\rho=\ket{\psi_{0}}\bra{\psi_{0}}$ in the corresponding zero-temperature
ground state $|\psi_{0}\rangle$. Upon partitioning the system into two parts
$A$ and $B$, see Fig. 3, with corresponding Hilbert spaces
${\cal{H}}={\cal{H}}_{A}\otimes{\cal{H}}_{B}$ we can define the reduced
density matrix $\rho_{A}$ of subsystem $A$ by tracing out subsystem $B$, viz.,

$\displaystyle\rho_{A}=\mathrm{Tr}_{B}(\rho).$ (9)

Although $\rho$ is pure, $\rho_{A}$ is typically a mixed state because the
zero-temperature ground-state is not separable. In this scenario, the
entanglement entropy

$\displaystyle S_{A}=-{\mathrm{Tr}}\rho_{A}\log\rho_{A}$ (10)

is a measure of entanglement between the two subsystems. In terms of the
entanglement spectrum [4], i.e., the eigenvalues $\lambda_{i}$ of the reduced
density matrix $\rho_{A}$, we can express the entanglement entropy as [1, 2,
4, 39]

$\displaystyle S_{A}=-\sum_{j}\lambda_{j}\ln\lambda_{j}.$ (11)

If conversely, the density matrix $\rho$ is not pure, e.g., at finite
temperature, or if $\rho$ is pure but one is interested in the entanglement
between two non-complementary regions (see Fig. 3 c), then the von Neumann
entropy is not a good entanglement witness. A useful quantifier in these cases
is the logarithmic negativity [40, 41, 42, 43, 44, 45].

The negativity is defined from the so-called partially transposed reduced
density matrix. Given a partition of $A$ as $A=A_{1}\cup A_{2}$ (see Fig. 3
$c)$), the matrix elements of the partial transpose $\rho_{A}^{T_{2}}$ with
respect to the degrees of freedom of $A_{2}$ are defined as

$\displaystyle\langle\varphi_{1}\varphi_{2}|\rho_{A}^{T_{2}}|\varphi_{1}^{\prime}\varphi_{2}^{\prime}\rangle:=\langle\varphi_{1}\varphi_{2}^{\prime}|\rho_{A}|\varphi_{1}^{\prime}\varphi_{2}\rangle.$
(12)

Here, $\\{\varphi_{1}\\}$ and $\\{\varphi_{2}\\}$ are orthonormal bases for
$A_{1}$ and $A_{2}$ respectively. In contrast to the eigenvalues of the
reduced density matrix $\rho_{A}$, the eigenvalues $\zeta_{i}$ of
$\rho_{A}^{T_{2}}$ can be positive or negative. The logarithmic negativity is
then defined as

$\displaystyle{\cal{E}}_{A_{1}:A_{2}}=\ln\mathrm{Tr}|\rho_{A}^{T_{2}}|.$ (13)

The behavior of the logarithmic negativity has been fully characterized in
systems that are described by conformal field theory at zero temperature [46]
and at finite temperature [47]. Generally, the negativity follows an area law
scaling as observed in a variety of systems, see, e.g., Refs. [48, 49, 50, 14,
51, 15].

Finally, we introduce the entanglement spectrum (ES), viz.,
$\\{\xi_{i}=-\ln(\lambda_{i})\ |\ \lambda_{i}\in\mathrm{spec}(\rho_{A})\\}$.
The lowest entanglement gap (Schmidt gap) is defined by

$\displaystyle\delta\xi=\xi_{1}-\xi_{0},$ (14)

where $\xi_{0}$ and $\xi_{1}$ are the lowest and the first excited ES level,
respectively.

The ES has received a lot of attention following the observation that it
contains universal information about the edge modes in fractional quantum Hall
systems [52]. Subsequently, the ES was investigated in a variety of setups,
e.g., in conformal field theory [53, 54, 55, 56], in quantum Hall systems [57,
58, 59, 60, 61, 62], in frustrated and magnetically ordered systems [63, 64,
65, 66, 20, 67, 68, 69, 70, 71, 72, 73, 74, 75] or systems with impurities
[76].

Figure 3: Geometry of partitions: In panel a) a generic three-dimensional
system is depicted. In panel b) the system is bipartitioned into two
subsystems $A$ and $B$ with $B$ the complement of $A$. In panel c) the
subsystems $A_{1}$ and $A_{2}$ are non-complementary.

The main topic of this review is to investigate the entanglement-related
quantities introduced above in the QSM in $d=1$, $d=2$ and $d=3$ spatial
dimensions.

Since the QSM is mappable to a Gaussian system of bosons (cf. Eq. (3)),
entanglement-related quantities can be extracted from the position and
momentum correlators (cf. Eqs. (4) and (5)) ${\mathbb{X}}\equiv\langle
x_{n}x_{m}\rangle$ and ${\mathbb{P}}\equiv\langle p_{n}p_{m}\rangle$ (see [22]
for a review). First, we consider the correlators restricted to subsystem $A$,
denoting them as ${\mathbb{X}}[A]$ and ${\mathbb{P}}[A]$. The single-particle
eigenvalues $\epsilon_{i}$, with $i\in[1,|A|]$, of the entanglement
Hamiltonian $H_{A}$, which is defined as $\rho_{A}=\exp(-H_{A})$, are readily
related to the eigenvalues $e_{i}$ of the matrix product
$\mathbb{C}_{A}={\mathbb{X}}[A]{\mathbb{P}}[A]$, viz.,

$\displaystyle\sqrt{e_{i}}=\frac{1}{2}\coth\left(\frac{\epsilon_{i}}{2}\right).$
(15)

The eigenvalues of the entanglement Hamiltonian $H_{A}$ are constructed by
filling the single-particle levels $\epsilon_{i}$ in all possible ways. This
allows also to obtain the von Neumann entropy $S_{\mathrm{vN}}$ in terms of
the eigenvalues $e_{j}$ as

$S_{\mathrm{vN}}=\sum_{j}\left[\left(\sqrt{e_{j}}+\frac{1}{2}\right)\ln\left(\sqrt{e_{j}}+\frac{1}{2}\right)-\left(\sqrt{e_{j}}-\frac{1}{2}\right)\ln\left(\sqrt{e_{j}}-\frac{1}{2}\right)\right].$
(16)

Hence, diagonalizing $\mathbb{C}_{A}$ allows us to deduce the full
entanglement spectrum. In particular, assuming that the single-particle
entanglement spectrum levels are ordered as
$\epsilon_{1}\leq\epsilon_{2}\leq\cdots\leq\epsilon_{|A|}$, the Schmidt gap is
simply given by $\delta\xi=\epsilon_{1}$. For Gaussian bosonic systems, the
logarithmic negativity can be constructed from the two-point correlation
functions [77]. First, we define the transposed matrix
${\mathbb{P}}[A^{T_{2}}]$ as

$\displaystyle{\mathbb{P}}[A^{T_{2}}]\equiv{\mathbb{R}}[A^{T_{2}}]{\mathbb{P}}[A]{\mathbb{R}}[A^{T_{2}}],$
(17)

where the matrix ${\mathbb{R}}[A^{T_{2}}]$ acts as the identity matrix on
$A_{1}$ and as minus the identity matrix on $A_{2}$. The eigenvalues
$\nu_{i}^{2}$ of ${\mathbb{X}}[A]{\mathbb{P}}[A^{T_{2}}]$ form the single-
particle negativity spectrum. In terms of them the negativity can be written
as [77]

$\displaystyle{\cal{E}}=\sum_{i}{\mathrm{max}}(0,-\ln(2\nu_{i})).$ (18)

##  4 Quantum and classical fluctuations at finite temperature criticality

Figure 4: von Neumann entropy and entanglement spectrum: (Left panel) Volume-
law scaling of the von Neumann entropy across the finite-temperature
transition in the three-dimensional QSM. Singular terms are present but they
vanish at the transition and are overshadowed by the analytic background. Non-
analytic contributions are more visible in the entanglement spectrum (notice
the divergence of the largest negative single-particle ES level in the right
panel). In the inset we show $\delta_{n}$ (cf. Eq. (19)), which measures the
nonanalyticity of the levels across the transition. Here $\delta_{n}\neq 0$
signals nonanalytic behavior.

Understanding the interplay of classical and quantum fluctuations is an
important but challenging task [78, 79, 75]. One way of approaching this
question is by studying entanglement witnesses in the vicinity of a finite
temperature phase transition that is driven by classical fluctuations. It has
been observed that a variety of entanglement witnesses are sensitive to
classical criticality. For instance, it has been shown that the negativity
develops cusp-like singularities [14, 51]. In this section we review our
investigation from Ref. [15] of entanglement-related quantities at the finite-
temperature transition in the $d=3$ dimensional QSM, see Fig. 2 (left panel).

First, we discuss the von Neumann entropy $S_{\rm vN}$ for the bipartition of
the system into two equal parts (see Fig. 3). As we mentioned in Sec. 3,
$S_{\rm vN}$ is not a valid entanglement witness at finite temperature. In
fact, the von Neumann entropy becomes the standard thermal entropy at finite
temperature. Indeed, as shown in Fig. 4 (left panel), $S_{\mathrm{vN}}$
satisfies a standard volume-law scaling. Being sensitive to both quantum and
classical correlations, the von Neumann entropy overestimates the amount of
entanglement, which is expected to scale with the boundary between the two
subsystems. Moreover, the von Neumann entropy does not show any singularity at
the transition. This happens because singular terms, although they are
present, vanish at the critical point, and are overshadowed by the analytic
background. Singularities are more visible in the single-particle entanglement
spectrum, as illustrated in the right panel of Fig. 4. In the figure we show
the entanglement spectrum for two adjacent blocks of linear size $\ell$
embedded in an infinite system. The eigenvalues quickly decay upon increasing
their index and most of them satisfy $e_{n}\approx 1/4$. Clearly, only those
eigenvalues with low index can yield potentially singular contributions. In
the inset we investigate the singularity using the quantity

$\displaystyle\delta_{n}=(e_{n})_{+}^{\prime}-(e_{n})_{-}^{\prime}$ (19)

that measures the difference of the right and left derivatives of $e_{n}$ with
respect to $g$ at $g_{c}$. Clearly $\delta_{n}\neq 0$ indicates a non-
analyticity and we observe this for small $n$.

Figure 5: Entanglement negativity in the three-dimensional QSM: The left panel
is an overview of the negativity in the $g-T$ plane, with $T$ the temperature
and $g$ the quantum coupling (cf. Eq. (1)). Here we always consider the half-
system negativity. The critical line is reported as black dashed line. The
red-dashed line is the death line above which the negativity is exactly zero.
The death line as extracted from the negativity between two adjacent spins is
reported as red solid line. The orange solid line is the behavior as
$\sqrt{g}$ at small $g$. The centre (right) panel shows the negativity across
the phase transition varying the temperature (quantum parameter). In the right
panel the continuous line is the result in the thermodynamic limit. The inset
shows the behavior of the negativity at $T=0.2$, i.e., close to the quantum
phase transition.

Next, we discuss the entanglement negativity. As outlined in Sec. 3, the
negativity is a proper entanglement witness, and as such obeys an area law,
see center and right panel in Fig. 5. Crossing the thermal transition at fixed
finite $T$, i.e., changing the quantum driving parameter $g$ the negativity
decays slowly as $1/g$ for large $g$, see Fig. 5 right panel. This is in
contrast to the behavior when crossing the transition with the temperature $T$
at fixed $g$, as depicted in the center panel of Fig. 5. Here, the negativity
shows a sudden death after and remains exactly zero for increasing $T$. We
also see that the negativity does not show any cusp singularity across the
finite temperature transition but develops a cusp when approaching low
temperatures (see inset right panel in Fig. 5). This signals that
singularities, although present, are strongly suppressed. Furthermore, in Fig.
5 (left panel), we map out the negativity in the full phase diagram. In the
figure the dashed line is the critical line separating the paramagnetic and
the ferromagnetic phase. We observe that the negativity generally attains a
maximum at the quantum phase transition, hinting at a strongly entangled
quantum state. We also observe that the negativity increases upon lowering the
temperature and is largest for $T=0$. We also highlight the numerical death
line in Fig. 5 (dotted line in the left panel) above which the negativity is
zero.

Interestingly, most of these findings can be quantitatively understood
considering two adjacent sites embedded in an infinite system. This setup
allows for analytic investigations as shown in Ref. [15]. For large $g$ and
constant $T$, this approach qualitatively predicts the slow negativity decay
from Fig. 5 (right panel), viz.,

$\displaystyle\mathcal{E}=-\ln\left(\frac{g-2}{g}\right)\stackrel{{\scriptstyle
g\to\infty}}{{\simeq}}\frac{2}{g}.$ (20)

Similarly, it predicts the existence of the death line (continuous line in the
figure), and correctly captures its onset for small $g$ as $\sqrt{g}$, see
Fig. 5 (left panel).

##  5 Entanglement gap at 2D quantum criticality

As we have seen in Sec. 4 the low-laying entanglement spectrum encodes
relevant information about the critical properties of the system. To further
investigate this aspect we review in this section our studies in Refs. [16,
17] of the behavior of the Schmidt gap at quantum criticality and in the
ferromagnetic phase in the two dimensional QSM. Interest in the behavior of
the Schmidt gap has spiked in the last decade.

Figure 6: Overview of the entanglement gap in the two-dimensional QSM: The
left panel shows the gap $\delta\xi$ across the whole phase diagram. We
observe distinct behaviors in the paramagnetic phase (finite gap) and the
ferromagnetic phase (vanishing gap). We always consider the half-system
entanglement spectrum. The continuous line in the left panel is the result in
the thermodynamic limit [16]. In the center panel we show the rescaled gap
$\delta\xi\ln(L)$. At the critical point the gap decays as $1/\ln(L)$, which
implies that the rescaled gap should exhibit a crossing. In the right panel we
plot $\delta\xi\ln(L)$ versus $(g-g_{c})L^{1/\nu}$. Upon approaching the
thermodynamic limit $L\to\infty$ a data collapse is expected. The cross symbol
is $\delta\xi\ln(L)$ at $g=g_{c}$ and $L\to\infty$. Still, subleading terms
are too large to observe the collapse, as confirmed in the inset.

In the left panel of Fig. 6 we show the numerical findings for the behavior of
$\delta\xi$ across the phase diagram, see Fig. 2. Here, we consider a
bipartition of the system into two equal halves. In the paramagnetic regime,
we observe that the gap converges rapidly to a finite number upon increasing
the linear system size $L$. Hence, the gap remains finite in the thermodynamic
limit $L\to\infty$. Conversely, the behavior at the critical point [16] and in
the ferromagnetic phase [17] differs from that in the paramagnetic phase. In
the ferromagnetic phase, the entanglement gap scales as [17]

$\displaystyle\delta\xi\simeq\frac{\Omega}{\sqrt{L\ln(L)}},$ (21)

where the constant $\Omega$ is known analytically [17], and depends on low-
energy properties of the QSM and on the geometry. For instance, $\Omega$ is
sensitive to the presence of corners in the boundary between $A$ and the rest.
Hence, the gap closes algebraically, involving logarithmic corrections. At
criticality we find that the Schmidt gap still closes, i.e., $\delta\xi\to 0$
albeit significantly slower. Precisely, the gap vanishes as [15]

$\displaystyle\delta\xi\simeq\frac{\pi^{2}}{\ln(L)}.$ (22)

This result is obtained for the bipartition into equal halves as follows.
Since we use periodic boundary conditions in both directions, and the
bipartition does not introduce corners, the momentum $k_{y}$ remains a good
quantum number also for the reduced density matrix. This allows to exploit
dimensional reduction [80] mapping the problem to a one-dimensional one.
Hence, we may use the analytical result for a one dimensional massive harmonic
chain [22] in order to obtain Eq. (22). Since the harmonic chain result is
derived using the corner transfer matrix on two infinite halves, whereas we
have periodic boundary conditions also along the $x$ direction, Eq. (22) is
exact only at leading order in $L$. Our results are numerically confirmed in
Fig. 7. In the figure we consider the largest eigenvalue $e_{1}$ of
$\mathbb{C}_{A}$ (see Section 3). This is related to the entanglement gap via
Eq. (15) and Eq. (14). In particular, a diverging $e_{1}$ implies a vanishing
entanglement gap. Fig. 7 shows that the leading behavior of $e_{1}$ at large
$L$ is correctly captured by the analytic result (full line). Again, Fig. 7
confirms that the entanglement gap is finite in the paramagnetic phase,
whereas in the ordered phase a faster divergence is observed (cf. Eq. (21)).

In the right panel of Fig. 7 we subtract the analytic prediction for the
leading behavior of $e_{1}$. The continuous line is a fit to
$A_{0}+A_{1}\ln(L)$. The result of the fit confirms the presence of a
logarithmic correction to the leading behavior.

Figure 7: Scaling of the largest eigenvalue $e_{1}$ of $\mathbb{C}_{A}$: The
left panel shows the scaling of $e_{1}$ in the paramagnetic and the
ferromagnetic phase as well as at criticality in the $2d$ QSM. In the
paramagnetic phase $e_{1}$ converges to a finite value and in the
ferromagnetic phase $e_{1}$ diverges (hence $\delta\xi\to 0$). At criticality
we see a slower divergence than in the ordered phase. The full grey line is a
fit to $1/\pi^{4}\ln(8L/\gamma_{2})+A_{0}+A_{1}\ln(8L/\gamma_{2})$, with
$A_{0},A_{1}$ fitting parameters. The leading part $\sim\ln^{2}(L)$ is
obtained analytically. Here $\gamma_{2}$ is a known constant [16]. In the
right panel we investigate the subleading logarithmic term.

##  6 Entanglement gap and symmetry breaking in the QSM

In the ferromagnetically ordered phase of the QSM, the dispersion develops a
zero mode, which reflects the Goldstone mode associated to symmetry breaking.
This implies that the position correlation function (see Eq. (4)) diverges.
Here we show that this fact is sufficient to fully determine the scaling of
the entanglement gap. First, the divergence in Eq. (4) is reflected in the
fact that the eigenvector of $\mathbb{C}_{A}$ associated with the largest
eigenvalue becomes flat in the thermodynamic limit [16, 17]. Hence, we may
rewrite the position correlator up to leading order as

$\displaystyle\mathbb{X}\simeq\chi^{x}\ket{1}\bra{1},\quad\mathrm{with}\,\,\chi^{x}\propto
L,$ (23)

with $\ket{1}$ being a normalized flat vector. First, we note that
$\bra{1}\mathbb{X}\ket{1}=\chi^{x}$. Furthermore, it is easy to verify that
$\bra{1}\mathbb{P}$ and $\ket{1}$ are left and right eigenvectors of
$\mathbb{C}$. Both correspond to the largest (diverging) eigenvalue $e_{1}$.
Hence, it is straightforward to identify

$\displaystyle
e_{1}=\bra{1}\mathbb{X}\ket{1}\bra{1}\mathbb{P}\ket{1}:=\chi^{x}\chi^{t},$
(24)

where $\chi^{t}:=\langle 1|\mathbb{P}|1\rangle$. Here we should observe that
$\chi^{x}$ resembles a susceptibility for the position variables $x_{i}$,
whereas $\chi^{t}$ is the susceptibility of the $p_{i}$. Eq. (24) establishes
a remarkable correspondence between the entanglement gap and standard
quantities such as $\chi^{x}$ and $\chi^{t}$. A similar decomposition as in
Eq. (24) was employed in Ref. [81] to treat the zero-mode contribution to
entanglement in the harmonic chain.

We now verify numerically that, as expected, the eigenvector of
$\mathbb{C}_{A}$ corresponding to its largest eigenvalue $e_{1}$ becomes flat
in the ferromagnetic phase, which ensures the validity of Eq. (24). Our
results are reported in Fig. 8 where we show the overlap $\varphi$ between the
flat vector and the exact eigenvector of the correlation matrix. At
criticality, the eigenvector is not flat in the thermodynamic limit
$L\to\infty$ (see Fig. 8). On the other hand, below the critical point the
eigenvector becomes flat in the thermodynamic limit.

###  6.1 Entanglement gap in the ferromagnetic phase of the 2D QSM

Let us now discuss the application of Eq. (24) in the ordered phase of the
two-dimensional QSM. To employ Eq. (24) we have to evaluate the flat vector
expectation values of the position and momentum correlation matrix. The
standard way is to decompose them in a thermodynamic and a finite size part,
i.e.,

$\displaystyle\bra{1}\mathbb{X}\ket{1}=\bra{1}\mathbb{X}^{\rm(th)}\ket{1}+\bra{1}\mathbb{X}^{(L)}\ket{1}$
(25)

and similar for $\bra{1}\mathbb{P}\ket{1}$, using the Poisson summation
formula

$\sum_{n=a}^{b}f(n)=\frac{f(a)+f(b)}{2}+\int_{a}^{b}f(x){\mathrm{d}}x+2\sum_{p=1}^{\infty}\int_{a}^{b}f(x)\cos(2\pi
px){\mathrm{d}}x.$ (26)

The FSS of these contributions is then obtained from the FSS of the spherical
parameter $\mu$ and from standard methods such as stationary phase methods,
Euler-Maclaurin formulas, and Mellin transform techniques. One finds [17]

$\displaystyle\bra{1}\mathbb{X}^{\rm(th)}\ket{1}\sim
L^{2},\quad\bra{1}\mathbb{X}^{(L)}\ket{1}\sim
L^{2},\quad\bra{1}\mathbb{P}^{\rm(th)}\ket{1}\sim\frac{\ln(L)}{L},\quad\bra{1}\mathbb{P}^{(L)}\ket{1}\sim\frac{\ln(L)}{L}.$
(27)

Notice that $\chi^{t}$ vanishes at $L\to\infty$. Thus, the entanglement gap
scales as $\delta\xi\sim 1/\sqrt{L\ln(L)}$. This is in agreement with our
numerical simulations [17]. Furthermore, it has been suggested that for
continuous symmetries, the gap should vanish as [20] $\delta\xi\sim
1/(L\ln(L))$ which differs from our result. This could be specific of the QSM,
although the issue would require further clarification.

Figure 8: Lowest eigenvector of $\mathbb{C}_{A}$: We show the overlap
$\varphi$ between the eigenvector of $\mathbb{C}_{A}$ corresponding to the
largest eigenvalue and the flat vector. In the ordered phase of the QSM one
has that $\varphi\to 1$ in the thermodynamic limit. This is not the case at
the critical point and in the ferromagnetic phase.

Finally, one can assume that the decomposition in Eq. (24) also holds at the
critical point [16]. This gives

$\bra{1}\mathbb{X}^{\rm(th)}\ket{1}\sim
L,\quad\bra{1}\mathbb{X}^{(L)}\ket{1}\sim
L,\quad\bra{1}\mathbb{P}^{\rm(th)}\ket{1}\sim\frac{\ln(L)}{L},\quad\bra{1}\mathbb{P}^{(L)}\ket{1}\sim\frac{\ln(L)}{L}.$
(28)

This implies that the entanglement gap vanishes as $\delta\xi\sim
1/\sqrt{\ln(L)}$. Although this scaling is not correct, reflecting that Eq.
(24) does not hold at criticality, it still captures the logarithmic character
of the entanglement gap.

##  7 Entanglement gap in 1D QSM with long-range interactions

Recently, there has been increasing interest in quantum systems with long-
range interactions [82, 83], also due to significant experimental advances
[84]. Since long-range interactions affect the structure of quantum
correlations between subsystems, it is interesting to study entanglement
witnesses in these systems. Indeed, the study of entanglement in long-range
systems has seen a significant surge of interest [85, 86, 87, 88, 89, 90, 91,
92, 93, 94, 95].

Arguably one of the paradigmatic systems is the long-range QSM at $T=0$ in one
spatial dimension that we introduced in section 2. In terms of the long-range
exponent $\alpha$ the interaction strength between two lattice sites behaves
as $u_{nm}\sim|n-m|^{-(1+\alpha)}$. The parameter $\alpha$ satisfies
$0\leq\alpha<2$ where $\alpha=2$ would correspond to nearest-neighbor
interaction and $\alpha=0$ is essentially an infinite range interaction. The
zero temperature phase diagram is reported in Fig. 2 (right panel). For
$0<\alpha<2/3$ the transition is of the mean-field type, whereas for
$2/3<\alpha<2$ the model shows a non-mean-field transition [11, 18].

Crucially, despite the long-range nature of the model, in the ordered phase
Eq. (24) holds true. Thus, the study of the scaling of the entanglement gap
proceeds as outlined in section 6. The susceptibilities $\chi^{x}$ and
$\chi^{t}$ can be analyzed for $L\to\infty$ with regularization techniques
involving the Mellin transform [96, 18] (details can be found in Ref. [18]).

Figure 9: Entanglement gap in the long-range $1d$ QSM: The left and center
panel show the entanglement gap for different values of the long-range
exponent $\alpha$ ($\alpha=1$ and $\alpha=1.5$ respectively) across the phase
transition. We observe a qualitative difference between the paramagnetic phase
and the ferromagnetic phase. The right panel shows the closure of the
entanglement gap for different values of $\alpha$ in the ferromagnetic phase.
Symbols are numerical results from exact diagonalization and the continuous
line corresponds to Eq. (24). Black dashed lines correspond to the analytical
prediction.

In Fig. 9 we present in the left and center panel a numerical analysis of the
entanglement gap across the phase diagram. We observe that $\delta\xi$ remains
finite in the paramagnetic phase, whereas it shows a vanishing behavior in the
ferromagnetic phase upon increasing $L$. As in the two dimensional QSM it is
useful to decompose the correlation matrices into a thermodynamic and a finite
size part. Using Mellin techniques we obtain [18]

$\displaystyle\bra{1}\mathbb{X}^{\rm(th)}\ket{1}\sim
L,\quad\bra{1}\mathbb{X}^{(L)}\ket{1}\sim
L^{\alpha/2},\quad\bra{1}\mathbb{P}^{\rm(th)}\ket{1}\sim
L^{-\alpha/2},\quad\bra{1}\mathbb{P}^{(L)}\ket{1}\sim L^{-\alpha/2}.$ (29)

Further details such as the precise prefactors, subleading contributions, the
behavior at criticality and the numerical benchmarks for all these results can
be found in Ref. [18]. These results allow us to deduce the entanglement gap
using Eq. (24). We obtain

$\displaystyle\delta\xi\sim L^{-(1/2-\alpha/4)}.$ (30)

In the right panel of Fig. 9 we show a comparison of exact diagonalization
results and Eq. (30), finding perfect agreement. Note that the entanglement
gap decays algebraically with $L$, and multiplicative logarithmic corrections
are absent. This differs strictly from the logarithmic behavior encountered in
the previous section.

##  8 Summary and Conclusions

We provided an overview of several results on the entanglement scaling in the
QSM. In particular, we investigated a variety of scenarios comprising
$d=1,2,3$ dimensional quantum systems with long and short range interactions.

Precisely, we discussed the interplay of classical and quantum fluctuations at
thermal transitions in the $3d$ QSM. We presented results for the entanglement
entropy, the entanglement negativity and the entanglement spectrum in Sec. 4.
In particular, we mapped out the negativity across the whole phase diagram
(see Fig. 5). A more detailed analysis can be found in our work in Ref. [15].
In Sec. 5 and Sec. 6 we focussed on the quantum phase transition and on the
ferromagnetic phase at zero temperature in two spatial dimensions. We
discussed the behavior of the entanglement gap in the different quantum phases
and at criticality. We showed that the entanglement gap is capable of
detecting criticality in the QSM, although logarithmic corrections are
present. We refer to Refs. [16, 17] for more details on the entanglement
spectrum, the precise finite-size scaling, and for the study of the effect of
corners in the entanglement spectrum. Finally, we reviewed entanglement
properties of a one dimensional spherical quantum chain with long-range
interactions [18]. Again, we considered the behavior of the entanglement gap
across the zero temperature quantum phase transition for different long-range
interactions. Remarkably, the QSM allows for a detailed analytical
investigation of the entanglement gap in the ordered phase of the model. In
particular, it is possible to understand how the entanglement gap is affected
by the long-range nature of the interactions.

The spherical model - here in its quantum formulation - has again proven
itself as a remarkably useful tool to study collective phenomena in strongly
interacting many-body systems. Clearly, the QSM will continue to serve as a
reference system for future studies of entanglement-related quantities. For
example it would be interesting to further investigate the influence of
corners on the entanglement patterns. Furthermore, it would be enlightening to
consider the QSM on quasi two-dimensional structures, such as ladders. It
would also be interesting to study the influence of disorder, or explore the
full entanglement Hamiltonian explicitly. Moreover, it has been shown that the
criticality in the QSM (in and out of equilibrium) can be exploited as a
resource in quantum metrology [97]. It would be interesting to explore to
which extend entanglement might affect and support quantum metrology
protocols. Furthermore, non-equilibrium and relaxational quantum dynamics has
been extensively studied in the QSM in the past years [98, 99, 100, 101, 102,
103]. It would be interesting to derive the spreading of entanglement in such
scenarios in the QSM, possibly exploiting the results from Refs. [104, 105,
106, 103]. An intriguing idea is to study the entanglement using the Kibble-
Zurek dynamics [107, 108].

## Acknowledgement

It is our pleasure to dedicate this work to our friend and colleague Malte
Henkel on the occasion of his $60^{\rm th}$ birthday. SW would like to further
thank Malte for years of excellent and interesting collaborations, exchanges
and guidance.

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