Symmetry-Resolved Entanglement Entropy for Local and Non-local QFTs

Reza Pirmoradian a,b and M Reza Tanhayi c

a School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM)
P.O. Box 19395-5531, Tehran, Iran

b Ershad Damavand, Institute of Higher Education (EDI)
P.O. Box 14168-34311, Tehran, Iran

c Department of Physics, Central Tehran Branch, Islamic Azad University (IAU)
P.O. Box 14676-86831, Tehran, Iran

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
In this paper, we study symmetry-resolved entanglement entropy in free bosonic
quantum many-body systems. Precisely, by making use of the lattice
regularization scheme, we compute symmetry-resolved Rényi entropies for free
complex scalar fields as well as for a simple class of non-local field
theories in which entanglement entropy exhibits volume-law scaling. We present
effective and approximate eigenvalues for the correlation matrix used to
compute the symmetry-resolved entanglement entropy and show that they are
consistent with the numerical results. Furthermore, we explore the
equipartition of entanglement entropy and verify an effective equipartition in
the massless limit. Finally, we make a comment on the entanglement entropy in
the non-local quantum field theories and write down an explicit expression for
the symmetry-resolved Rényi entropies.

###### Contents

  1. 1 Introduction
  2. 2 SREE and correlator method
    1. 2.1 Correlator method
      1. 2.1.1  Free bosonic quantum many-body systems
  3. 3 SREE for complex bosonic theory
    1. 3.1 Complex harmonic chain
    2. 3.2 The lattice model
  4. 4 Numerical analysis
    1. 4.1 massless and massive limit
  5. 5 Non-local quantum fields
    1. 5.1 Numerical analysis for SREE
    2. 5.2 Comment on the entanglement entropy
  6. 6 Conclusions
  7. A Comments on a Single Site

##  1 Introduction

Entanglement, which describes non-local correlations between quantum systems,
is one of the most fundamental topics in physics, and its concept emerged
during the evolution of quantum mechanics [1]. Entanglement plays an important
role in the study of various branches from theoretical to experimental
physics, e.g., quantum information theory [2, 3, 4], condensed matter
physics[5, 6, 7, 8], gravity especially black holes in high energy physics [9,
10, 11] and the holographic principle[12, 13, 14, 15, 16, 17]. In this way,
measuring entanglement remains a challenge and so far, various quantities have
been introduced to quantify entanglement measurements, the most well-known of
which for pure quantum states is the entanglement entropy (EE) [18].
In an arbitrary quantum system, EE can be considered as a universal measure of
the system’s effective degrees of freedom, moreover, it provides important
information about a given state, in particular, quantum correlations in a
ground state between two spatially separated regions. For a system that
decomposes into two subsystems, $\mathcal{A}$ and $\mathcal{B}$ with Hilbert
space ${\cal H}={\cal H}_{\mathcal{A}}\otimes{\cal H}_{\mathcal{B}}$, the EE
for a given quantum state $|\psi\rangle$ (e.g. the ground state), is
identified by the von Neumann entropy of the reduced density matrix [8, 19]

$S_{EE}=-\text{Tr}(\rho_{\mathcal{A}}\log\rho_{\mathcal{A}}),$ (1.1)

where $\rho_{\mathcal{A}}=\text{Tr}_{\mathcal{B}}(|\psi\rangle\langle\psi|)$
is the reduced density matrix of the subsystem $\mathcal{A}$. Despite the
crucial role of EE in the study of quantum systems, by making use of the
conventional methods in quantum field theories (QFTs), in general, there is no
analytic solution for EE, though computing a reduced density matrix is an
incredibly difficult task in many cases. Besides some standard methods for
quantifying EE 111For example, the numerical methods which are mostly based on
Gaussian states [22, 23, 24], conformal field theory (CFT) methods [25, 26,
27] and holographic CFT [12, 28]., for a quantum system described by a pure
state, one may use the replica trick and make $n$ copies of system and define
the Rényi entropies as follows [20]

$S_{n}=\frac{1}{n-1}(n\log Z_{1}-\log Z_{n}),$ (1.2)

where in the path integral formalism, $Z_{1}$ and
$Z_{n}=Tr(\rho_{\mathcal{A}})^{n}$ are the partition functions on the original
Riemann surface and on $n$-sheeted Riemann surfaces, respectively. In this
way, $S_{n}$ which carries all the information of the eigenvalues of the
reduced density matrix can be computed [21]. The $S_{EE}$ is then obtained,

$S_{EE}=\lim_{n\rightarrow 1}\,S_{n}\,\,,$ (1.3)

thus the computation of EE reduces to finding the partition functions. On the
other hand, the question of whether/how a certain symmetry of the system is
related to the structure of entanglement has attracted attention in recent
years [29, 30, 31, 32, 33, 34, 35, 36, 37]. In fact, besides the theoretical
motivation, there is also experimental motivation to investigate such a
relation between entanglement and underlying symmetry of a given system, e.g.,
in Ref. [38], the authors explored the relation between the dynamics of many-
body disordered systems and the entanglement that is formed from various
symmetry sectors. The idea of making a connection between a certain amount of
entanglement and specific individual symmetry sectors of a quantum theory is
known as the symmetry resolution of entanglement. Particularly, symmetry-
resolved entanglement entropy (SREE) is a theoretical framework to understand
the role of the contribution of different symmetry sectors in computing the
entanglement entropy [29, 30]. It should be noted that besides the method of
SREE, many advances have been made to investigate other features of the effect
of symmetry in computing entanglement, for example, symmetry-resolved relative
entropy [39, 40], symmetry decomposition of entanglement negativity [41, 42,
43, 44], symmetry-resolved entanglement entropy in holographic settings [45,
46, 47] in thermal states [49, 48] and excited states [50, 51].
In this paper, for free complex scalar quantum fields and for a simple class
of non-local QFTs introduced in [52], we use the correlator method to read the
eigenvalues of the correlation matrix and then compute the symmetry-resolved
Rényi entropies. To study the SREE, we show that by determining the four (two)
largest eigenvalues for the massless (massive) limit, it is possible to find
an effective expression for the eigenvalues of the correlation matrix in both
massive and massless limits that makes the numerical analysis more effective.
We also investigate the equipartition of entanglement for different charge
sectors and verify the equipartition of entanglement for massless cases
(effectively) and for non-local QFTs, as well.
The organization of this paper is as follows. In section 2, we briefly recall
the SREE and the correlator method, which is used in this study for the
numerical analysis. The SREE and the related equations for complex scalar
fields are studied in section 3. Section 4 is devoted to the numerical
analysis of the expressions obtained in section 3. For non-local QFTs, we also
study the EE and SREE in section 5. The concluding remarks are given in
section 6, and, finally, in Appendix A, we give some details of computing
symmetry-resolved partition functions for a single site.

##  2 SREE and correlator method

It is argued that in the presence of global symmetry, the total EE can be
decomposed into the symmetry sectors, and to investigate this, let us assume a
system with a global internal symmetry decomposes into two complementary
spatial subsystems $\mathcal{A}$ and $\mathcal{B}$. In a given state described
by the density matrix $\rho$, the symmetry is supposed to be generated by a
charge operator ${Q}={Q}_{\mathcal{A}}\oplus{Q}_{\mathcal{B}}$. If the
proposed state is the eigenstate of the symmetry generator ${Q}$, then one has
$[\,{\rho}\,,\,{Q}\,]=0$ and, by tracing over $\mathcal{B}$, one obtains
$[\,{\rho}_{\mathcal{A}},\,{Q}_{\mathcal{A}}]=0$. This means that
${\rho}_{\mathcal{A}}$ is block-diagonal, in such a way that each block
corresponds to a different charge sector and is labeled according to
eigenvalue $q$ of charge operator ${Q}_{\mathcal{A}}$, namely one has

$\rho_{\mathcal{A}}=\bigoplus_{q}p(q)\rho_{\mathcal{A}}(q),$ (2.1)

where $p(q)=\text{Tr}(\Pi_{q}\rho_{\mathcal{A}})$ is the probability that the
measurement of $Q_{\mathcal{A}}$ in region $\mathcal{A}$ is $q$, and $\Pi_{q}$
stands for the projector on the sector of $Q_{\mathcal{A}}$ with eigenvalue
$q$. The entanglement entropy at charge $q$ which is defined according to
$\rho_{\mathcal{A}}$ is then given by

$S(q)=-\mathrm{Tr}\big{(}\rho_{\mathcal{A}}(q)\log\rho_{\mathcal{A}}(q)\big{)},$
(2.2)

that is known as the SREE. According to (2.1), the total entanglement entropy
associated with the density matrix $\rho_{A}$ can be decomposed as,

$S_{EE}=\displaystyle\sum_{q}p(q)S(q)-\displaystyle\sum_{q}p(q)\log p(q),$
(2.3)

where the first part of the right-hand side of the above equation is known as
the configurational entropy that denotes the average entropy per sector; the
second part is the number entropy stands for the charge fluctuations within
the subsystem $\mathcal{A}$ [38]. Similarly, one may use the replica trick and
define the symmetry-resolved Rényi entropies

$S_{n}(q)=\dfrac{1}{n-1}\Big{(}n\log\mathcal{Z}_{1}(q)-\log\mathcal{Z}_{n}(q)\Big{)},$
(2.4)

where $\mathcal{Z}_{n}(q)=\mathrm{Tr}(\Pi_{q}\,\rho^{n}_{\mathcal{A}})$ is the
symmetry-resolved partition function. In order to find these partition
functions one needs to diagonalize the reduced density matrix and
$Q_{\mathcal{A}}$, simultaneously, which in general is a difficult task. The
key idea of the symmetry resolution of entanglement is to use the path
integral approach to facilitate the computation of $\mathcal{Z}_{n}(q)$ [29].
In this formalism, in each sheet of the Riemann geometry an Aharonov-Bohm flux
$\alpha$ is introduced, and consequently, the field acquires a total phase
$\alpha$ after taking into account all the Riemann surface $\mathcal{R}_{n}$.
This implies a new quantity in calculating the path integral which is known as
the charged moments of the reduced density matrix defined by

$Z_{n}(\alpha)=\mathrm{Tr}(\rho_{\mathcal{A}}^{n}\,\,e^{iQ_{\mathcal{A}}\alpha}).$
(2.5)

It is worth emphasizing that the path integral approach provides a framework
to find $Z_{n}(\alpha)$ which is based on identifying the evolution of the
partition function of a $n$ copy of the quantum field according to the
specific boundary condition [53]. The Fourier transforms of the charged
moments give us the symmetry-resolved partition function

$\mathcal{Z}_{n}(q)=\displaystyle\int_{-\pi}^{\pi}\dfrac{d\alpha}{2\pi}e^{-iq\alpha}\,Z_{n}(\alpha).$
(2.6)

Finally, the SREE can be obtained as follows

$\displaystyle S(q)=\lim_{n\rightarrow
1}S_{n}(q)=\log\mathcal{Z}_{1}(q)-\frac{\left.\big{(}\partial_{n}\mathcal{Z}_{n}(q)\big{)}\right|_{n=1}}{\mathcal{Z}_{1}(q)},$
(2.7)

it should be noted that the probability $p(q)$ is simply related to the
moments $\mathcal{Z}_{n}(q)$ as follows

$p(q)=\mathcal{Z}_{1}(q).$ (2.8)

In this paper, we study a free and quadratic theory and for the numerical
computation, we follow the correlator method which was first introduced in
[54] and for more details see [56, 57, 58, 55]. We use this method to find the
charged moments and symmetry-resolved partition functions.

###  2.1 Correlator method

For the Gaussian state, the numerical computation of the EE is mostly based on
the two-point function of degrees of freedom in a given region of the
subsystem. While a similar approach is held for the SREE, however, one should
first evaluate charged moments by using the correlation or two-point functions
this is followed by computing the symmetry-resolved moments, and finally, one
can construct symmetry-resolved Rényi and entanglement entropies. For a given
free theory, the correlator method which works for Gaussian states provides a
powerful numerical framework to find the eigenvalues of the correlation matrix
e.g., for free scalar field theories one should compute the following two-
point functions

$\frac{1}{2}\mathbf{X}_{ij}=\langle\Phi_{i}\Phi_{j}\rangle\,,\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\mathbf{P}_{ij}=\langle\Pi_{i}\Pi_{j}\rangle.$
(2.9)

where $\Pi(x)$ is the momentum conjugated of $\Phi(x)$. In this method, for a
subregion say as $\mathcal{A}$, the correlation matrix
$\Lambda_{\mathcal{A}}=\sqrt{\mathbf{X}_{\mathcal{A}}\cdot\mathbf{P}_{\mathcal{A}}}$
should be built where its eigenvalues are used in finding entanglement
entropy. Precisely, it is shown that in the case of bipartition where the
subsystem $\mathcal{A}$ is given by $\ell$ adjacent lattice sites, the
entanglement entropy for the Gaussian states is given by,

$S_{EE}=\sum_{i=1}^{\ell}\bigg{[}\left(\frac{\nu_{i}+1}{2}\right)\log\left(\frac{\nu_{i}+1}{2}\right)-\left(\frac{\nu_{i}-1}{2}\right)\log\left(\hskip
1.42262pt\frac{\nu_{i}-1}{2}\hskip 1.42262pt\right)\bigg{]},$ (2.10)

where each $\nu_{i}$ are in fact the eigenvalues of the matrix
$\Lambda_{\mathcal{A}}$. This method is general and can be used in quadratic
free quantum field theories, and as mentioned, we elaborate this method to
find SREE for complex scalar fields.

####  2.1.1  Free bosonic quantum many-body systems

It is known that the Gaussian states are completely determined by the
correlation function of the canonical variables, where these functions can be
utilized in calculating quantum entanglement. On the other hand, the harmonic
lattice systems can be considered as the discrete versions of Klein-Gordon
fields, and, to lattice discretization of a given Hamiltonian with periodic
conditions, the continuous space coordinates $x$ should be replaced by a
lattice of discrete points with $\mathcal{N}$ sites and lattice spacing
$\epsilon$ where for the sake of simplicity, we have set $\epsilon=1$. The
Hamiltonian can be written in the following form

${\cal
H}=\sum_{r=1}^{\mathcal{N}}\dfrac{1}{2}\mathcal{P}_{r}^{2}+\sum_{r,s=1}^{\mathcal{N}}\dfrac{1}{2}\mathcal{Q}_{r}K_{rs}\mathcal{Q}_{s},$
(2.11)

where $K$ is a symmetric and positive-definite matrix, the displacement and
momentum conjugation of the $i$-th oscillator are determined by
$\mathcal{Q}_{i}$ and $\mathcal{P}_{i}$. The Gaussian wave function of the
ground state is given by

$\Psi(\\{\mathcal{Q}\\})=\left(\det\dfrac{W}{\pi}\right)^{1/4}\exp\Big{[}-\frac{1}{2}\sum_{r,s}\mathcal{Q}_{r}W_{rs}\mathcal{Q}_{s}\Big{]},$
(2.12)

where $W\equiv K^{1/2}$. After making use of the discrete Fourier
transformation, the diagonalized $K$ matrix yields as follows

$K_{rs}=\frac{1}{\mathcal{N}}\sum_{k}\Big{[}m^{2}+2(1-\cos\dfrac{2\pi
k}{\mathcal{N}})\Big{]}e^{2\pi ik(r-s)/\mathcal{N}}.$ (2.13)

In order to have the limit of a large chain (with fixed $\epsilon$), we need
to take $\mathcal{N}$ to infinity and $p=2\pi k/\mathcal{N}$. The correlation
functions are given by

$\mathbf{P}_{rs}=\int_{-\pi}^{\pi}\dfrac{dp}{(2\pi)}e^{ip(r-s)}\sqrt{m^{2}+2(1-\cos
p)}$ (2.14)

$\mathbf{X}_{rs}=\int_{-\pi}^{\pi}\dfrac{dp}{(2\pi)}e^{ip(r-s)}\frac{1}{\sqrt{m^{2}+2(1-\cos
p)}}$ (2.15)

By plugging (2.14) and (2.15) into the $\Lambda_{\mathcal{A}}$ one can find
the eigenvalues of the correlation matrix and therefore, equation (2.10) gives
us the EE. As a comment, let us mention that in the literature to calculate
the above integrals, numerical methods are often used, and we found the
following relation that might be useful

$\int_{-\pi}^{\pi}\frac{\cos(\mathfrak{n}p)\Big{[}m^{2}+\big{[}2-2\cos(p)\big{]}\Big{]}^{\mathfrak{j}}}{2\pi}\,dp=\left(m^{2}+4\right)^{\mathfrak{j}}\,_{3}\tilde{F}_{2}\left(\frac{1}{2},1,-\mathfrak{j};1-\mathfrak{n},1+\mathfrak{n};\frac{4}{m^{2}+4}\right),$
(2.16)

where $\mathfrak{n}$ is an integer number and ${{}_{3}\tilde{F}_{2}}$ is the
regularized hypergeometric function.

##  3 SREE for complex bosonic theory

###  3.1 Complex harmonic chain

Field theory provides a better way for studying the symmetry of harmonic chain
and their lattice discretization; and to investigate the features of complex
harmonic chains, the complex scalar field is usually used [59, 60, 61, 51].
The Euclidean action and the corresponding Hamiltonian of a free complex
massive scalar field $\Phi(x)$ are given by

$\displaystyle{\cal I}$ $\displaystyle=$ $\displaystyle\int
d^{2}x\left[\partial_{\mu}\Phi^{\dagger}(x)\partial_{\mu}\Phi(x)+m^{2}\Phi^{\dagger}(x)\Phi(x)\right],$
$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle\int
dx\left[\Pi^{\dagger}(x)\Pi(x)+\partial_{x}\Phi^{\dagger}(x)\partial_{x}\Phi(x)+m^{2}\Phi^{\dagger}(x)\Phi(x)\right],$
(3.1)

it is shown that the fields satisfy the following commutation relations

$[\Phi(x),\Pi(y)]=i\delta(x-y)\,.$ (3.2)

In terms of creation and annihilation operators, one can write

$\Phi=\int\frac{dp}{2\pi}\frac{1}{\sqrt{2\omega_{p}}}(a_{p}e^{ipx}+b^{\dagger}_{p}e^{-ipx}),\,\,\,\,\,\Pi=\int\frac{dp}{2\pi}i\sqrt{\frac{\omega_{p}}{2}}(a^{\dagger}_{p}e^{-ipx}-b_{p}e^{+ipx}),$
(3.3)

note that $\omega^{2}_{p}=m^{2}+p^{2}$. A given symmetry can be understood by
the invariance of the action, for example, by rotating the field $\Phi$ with
an arbitrary phase $\Phi(x)\to e^{i\theta}\Phi(x)$ the action (3.1) remains
unchanged, this implies the $U(1)$ symmetry. To make the $U(1)$ symmetry
manifest, one may write the $\Phi(x)$ in terms of two real scalar fields,
$\Phi_{1}(x)$ and $\Phi_{2}(x)$ as

$\Phi(x)=\frac{1}{\sqrt{2}}(\Phi_{1}(x)+i\Phi_{2}(x))\,,\hskip
11.38109pt\Pi(x)=\frac{1}{\sqrt{2}}(\Pi_{1}(x)+i\Pi_{2}(x)),$ (3.4)

thus the Hamiltonian of the complex scalar fields decomposes into two real
Hamiltonian as follows

${\cal H}={\cal H}(\Phi_{1})+{\cal H}(\Phi_{2})=\frac{1}{2}\sum_{i=1}^{2}\int
dx\left[\Pi_{i}^{2}(x)+(\partial_{x}\Phi_{i}(x))^{2}+m^{2}\Phi_{i}^{2}(x)\right],$
(3.5)

it is then easy to verify that the $U(1)$ turns to the $O(2)$ rotation. The
Hamiltonian and the conserved charge can be rewritten in terms of the creation
and annihilation operators

${\cal
H}=\int\frac{dp}{2\pi}\omega(p)\Big{(}a^{\dagger}(p)a(p)+b^{\dagger}(p)b(p)\Big{)},\,\,\,\,\,\,\,Q=\int\frac{dp}{2\pi}\Big{(}a^{\dagger}(p)a(p)-b^{\dagger}(p)b(p)\Big{)},$
(3.6)

where

$a(p)=\frac{1}{\sqrt{2}}(a_{1}(p)+ia_{2}(p)),\qquad
b(p)=\frac{1}{\sqrt{2}}(a_{1}^{\dagger}(p)+ia_{2}^{\dagger}(p)).$ (3.7)

Now in real space, by restriction of the domain of integration into the
subregion ${\cal A}$, one can obtain the conserved charge in ${\cal A}$ which
is given by

$Q_{\cal A}=\int_{\cal
A}dx\Big{(}a^{\dagger}(x)a(x)-b^{\dagger}(x)b(x)\Big{)}.$ (3.8)

To proceed with the numerical method, the theory should be discretized which
can be done by writing a lattice version of the complex scalar field.

Figure 1:  Numerical results of the real and imaginary parts of the
$F_{n}(\alpha)$ function are shown in terms of $\alpha$, for $\ell=70$ and
$m=10^{-1},10^{-3},10^{-6},10^{-10}$ and $n=1$ (top plot), $n=2$ and $n=3$
(bottom plot).

Figure 2:  Numerical results of the real and imaginary parts of the
$F_{n}(\alpha)$ function are shown in terms of $\ell$, for $m=25\times
10^{-3}$ and $\alpha=0.5$ and $1$ for $n=1,2$ and $3$.  Figure 3: The
numerical values of $S_{n}(q)$ for the complex harmonic chain are shown in
terms of $\ell$ where we set $m=25\times 10^{-3}$ and $\leavevmode\nobreak\
m=25\times 10^{-4}$, $n=1,\,2$ and $q=1,\,2$. For definite given values of $n$
and $q$, it can be seen that $S_{n}(q)$ converges at smaller values of $\ell$
with larger mass.

Figure 4:  The numerical results for $\mathcal{Z}_{n}(q)$ and $S_{n}(q)$ in
terms of $q$, for $m=10^{-1}$, $m=10^{-6}$ and $m=10^{-15}$, we set $\ell=250$
and $n=1$ and $2$. For larger values of mass, the left panel shows the
$q$-dependence of symmetry-resolved moments, while this dependence is relaxed
in the light masses. This behavior also holds for $S_{n}(q)$, as can be seen
from the right panel, the dependence $S_{n}(q)$ on $q$ becomes more relaxed
for light masses and larger value of $q$.

###  3.2 The lattice model

To have a continuous symmetry, we consider a complex bosonic theory which on
the lattice might be supposed as a chain of complex oscillators, or
equivalently, one can consider the sum of two real harmonic chains with the
variables $(\mathcal{P}^{(1)},\mathcal{Q}^{(1)})$ and
$(\mathcal{P}^{(2)},\mathcal{Q}^{(2)})$. The corresponding Hamiltonian is
given by

${\cal
H}_{CB}(\mathcal{P}^{(1)}+i\mathcal{P}^{(2)},\mathcal{Q}^{(1)}+i\mathcal{Q}^{(2)})={\cal
H}_{B}(\mathcal{P}^{(1)},\mathcal{Q}^{(1)})+{\cal
H}_{B}(\mathcal{P}^{(2)},\mathcal{Q}^{(2)}).$ (3.9)

The Hamiltonian (3.9) can be also written in terms of mode operators $a_{k}$
and $b_{k}$ satisfying $[a_{k},a^{\dagger}_{j}]=\delta_{j,k}$,
$[b_{k},b^{\dagger}_{j}]=\delta_{j,k}$, as follows

${\cal
H}_{CB}=\sum_{k=0}^{L-1}\omega_{k}(a^{\dagger}_{k}a_{k}+b_{k}^{\dagger}b_{k}),\qquad\omega_{k}=\sqrt{m^{2}+2(1-\cos\dfrac{2\pi
k}{\mathcal{N}})},$ (3.10)

and similarly, the charge operator reads

$Q=\sum_{k=0}^{L-1}\Big{(}a^{\dagger}_{k}a_{k}-b_{k}^{\dagger}b_{k}\Big{)}.$
(3.11)

The conserved charge can be well written in real space and its value in a
given subsystem $\mathcal{A}$ is the same sum restricted to $\mathcal{A}$,
i.e.

$Q_{\mathcal{A}}=\displaystyle\sum_{j\in\mathcal{A}}\Big{(}a^{\dagger}_{j}a_{j}-b^{\dagger}_{j}b_{j}\Big{)}.$
(3.12)

For the charged moments, we need to compute ${\rm
Tr}[\rho_{\mathcal{A}}^{n}e^{i{Q}_{\mathcal{A}}\alpha}]$, making use of
equation (3.12) for ${Q}_{\mathcal{A}}$, the trace factorizes as

$Z_{n}(\alpha)={\rm
Tr}[\rho_{\mathcal{A}}^{n}e^{i{Q}_{\mathcal{A}}\alpha}]={\rm
Tr}[(\rho_{\mathcal{A}}^{a})^{n}e^{iN_{\mathcal{A}}^{a}\alpha}]\times[{\rm
Tr}(\rho_{\mathcal{A}}^{b})^{n}e^{-iN_{\mathcal{A}}^{b}\alpha}],$ (3.13)

where $N_{\mathcal{A}}^{a}=\sum_{j\in A}a^{\dagger}_{j}a_{j}$ and
$N_{\mathcal{A}}^{b}=\sum_{j\in\mathcal{A}}b^{\dagger}_{j}b_{j}$. Using the
relations between the number operator $N_{\mathcal{A}}$ and the eigenvalues of
the correlation matrix $\nu_{k}$, we find

$Z_{n}(\alpha)=\prod_{k=1}^{\ell}\frac{2^{n}}{\left(\nu_{k}+1\right)^{n}-e^{i\alpha}\left(\nu_{k}-1\right)^{n}}\frac{2^{n}}{\left(\nu_{k}+1\right)^{n}-e^{-i\alpha}\left(\nu_{k}-1\right)^{n}},$
(3.14)

and by the following definition for a single harmonic chain [59, 31]

$F_{n}(\alpha)\equiv\log{\rm Tr}[(\rho_{A}^{a})^{n}e^{iN_{A}^{a}\alpha}].$
(3.15)

one can write

$Z_{n}(\alpha)=\exp\Big{[}F_{n}(\alpha)+F_{n}(-\alpha)\Big{]}.$ (3.16)

In the following section, by numerical analysis, we will use the above formula
to find $\mathcal{Z}_{n}(q)$, and also the SREE can be obtained via the (2.4).

##  4 Numerical analysis

In this section, we use the numerical method to compute EE for complex scalar
fields and investigate its partition into different charge sectors for lattice
discretization. In the next section, we do the same for non-local fields. For
the complex scalar fields, we start with the results of the previous section,
and by elaborating on the $F_{n}(\alpha)$ functions in figure 1, we plot
$F_{n}(\alpha)$ in terms of $\alpha$, and in figure 2, the numerical results
of $F_{n}(\alpha)$ in terms of $\ell$ for two different masses and specific
values of $\alpha$ and $n$ are shown. The figures are consistent with the
similar results reported in Ref. [59], in which a corner transfer matrix
approach was used to study $F_{n}(\alpha)$ and $S_{n}(q)$. In figure 3, we
have plotted $S_{n}(q)$ in terms of $\ell$ for some definite values of $q$ and
$n$ and mass. We observe that the behavior of $S_{n}(q)$ depends on mass, such
that they saturate at a smaller value of $\ell$ for heavier masses. To
investigate more, in figure 4, we have depicted the behavior of
$\mathcal{Z}_{n}(q)$ and $S_{n}(q)$ for some values of mass; the graphs show
that for large masses, the dependence of $q$ is apparent, as both of them
clearly change with $q$. However, the changes (slope) in graphs become less
for light masses, thus one can say that the $q$-dependency of the symmetry-
resolved moments becomes more relaxed for light masses. This behavior also
holds for, $S_{n}(q)$, thus one could say that at this regime, the
equipartition of the entanglement entropy is effectively established.

###  4.1 massless and massive limit

It is known that the universal behavior at one-dimensional conformal quantum
critical points is one of the most remarkable results of entanglement entropy.
This is determined by the central charge of the underlying CFT, where at the
quantum critical point the EE and the entanglement Rényi entropies lead to the
following famous results [6, 10, 25]

$S_{EE}^{CFT}=\frac{c}{3}\log\frac{\ell}{\epsilon}+cnst,\hskip
11.38109ptS_{n}^{CFT}=\frac{c}{6}\frac{n+1}{n}\log\frac{\ell}{\epsilon}+cnst.$
(4.1)

For scalar and fermionic quantum fields, the conformal limit has been done in
[62, 63], moreover, the analytical calculations of charged moments of Dirac
and complex scalar fields, can be found in [31]. Here, we intend to
investigate and analyze the behavior governing charged moments of complex
scalar fields, and, consequently $\mathcal{Z}_{n}(q)$ and also the symmetry-
resolved entropies by performing numerical calculations on the basis of
knowing and specifying the eigenvalues of the matrix $\Lambda_{\mathcal{A}}$.
For free complex scalar fields, considering the (2.14), (2.15) and after
making use of (2.16) in the massless limit222Note by massive and massless
limit we mean $m\ell\gg 1$ and $m\ell\ll 1,$ respectively., we obtain

$\displaystyle\mathbf{P}_{rs}$ $\displaystyle\simeq$
$\displaystyle\frac{-4}{\pi\Big{(}4(r-s)^{2}-1\Big{)}}$ (4.2)
$\displaystyle\mathbf{X}_{rs}$ $\displaystyle\simeq$
$\displaystyle\begin{cases}\frac{1}{\pi}\Big{(}3\log 2-\log
m\Big{)}&\quad\text{for }\hskip 8.53581ptr=s\\\ \frac{1}{\pi}\Big{(}3\log
2-\log((r-s)m)-2\Big{)}&\quad\text{for}\hskip 11.38109ptr\neq s\\\
\end{cases}$ (4.3)

It should be mentioned that we have set lattice spacing to one, therefore the
dimensions of $m\epsilon$ and $\frac{\ell}{\epsilon}$ scale as $m$ and $\ell$,
respectively. As we arrange the eigenvalues from large to small, the numerical
calculations indicate that only a few of the eigenvalues (usually 3 to 4)
become significant and play an essential role in the calculations. Therefore,
by interpolating the figures, we show that the behavior of eigenvalues of the
$\Lambda_{\mathcal{A}}$ matrix can be estimated by

$\nu_{i}\sim c_{i}\Big{(}3\log
2+\log\frac{1}{m}\,(\frac{1}{3}\log\ell+1)\Big{)}^{\frac{1}{2}},\hskip
19.91692pti=1,2,3,4$ (4.4)

$c_{i}$’s are numerical coefficients. In order to do the numerical test, in
the right panel of figure 5, we have plotted the three largest eigenvalues of
the matrix $\Lambda_{\mathcal{A}}$ both through the numerical analysis and the
equation (4.4), as it is clear the results are completely consistent. It is
worth noting that the infrared divergence at $m=0$ is due to the famous zero
mode [64, 65]. In the case of the entanglement entropy, by making use of the
equation (2.10) and also the expression of the eigenvalues (4.4), one finds

$S_{EE}(\ell)\sim\frac{\mathfrak{a}}{3}\log\ell+cnst,$ (4.5)

whereby in numerical analysis, we find $\mathfrak{a}\simeq 1$, and for the
complex scalar fields at the conformal limit one obtains

$S^{C}_{EE}(\ell)\sim\frac{2}{3}\log\ell+cnst,$ (4.6)

note that the 2 factor is introduced for complex fields. We would like to
emphasize that the obtained result of the entanglement entropy at the leading
order is similar and consistent with the previous calculations in this regard,
e.g., the entropic $c$-function becomes
$c=\ell\,\frac{d}{d\ell}S_{EE}=\frac{2}{3}$ [62]. In order to compute the
SREE, one can write

$\displaystyle F_{n}(\alpha)$ $\displaystyle\simeq$
$\displaystyle\log\Big{[}\prod_{i=1}^{4}\frac{2^{n}}{(\nu_{i}+1)^{n}-e^{i\alpha}(\nu_{i}-1)^{n}}\Big{]}$
(4.7) $\displaystyle{Z}_{n}(\alpha)$ $\displaystyle\simeq$
$\displaystyle\prod_{i=1}^{4}\frac{2^{2n}}{(\nu_{i}-1)^{2n}+(\nu_{i}+1)^{2n}-2(\nu_{i}^{2}-1)^{n}\cos\alpha},$
(4.8)

Figure 5:  Left panel: The three largest eigenvalues of the matrix
$\Lambda_{\mathcal{A}}$ are shown in terms of $\ell$ for $m=10^{-20}$. Points
and lines, respectively, represent the numerical values and the predicted by
equation (4.4) for the eigenvalues of the massless case. Right panel: $S_{EE}$
is shown in terms of $\ell$ by two methods, numerical analysis (points) and
directly from equation (4.6) (solid lines), in which, we set $m=10^{-15}$,
$m=10^{-20}$ and $m=10^{-25}$. The figures confirm the agreement between the
numerical and estimated calculations.

where $\nu_{i}$’s are given by (4.4), therefore, by using (2.6) and (2.4),
$\mathcal{Z}_{n}(q)$ and the symmetry-resolved entropies can be obtained. In
figure 6, we have plotted the $F_{n}(\alpha)$ function for $n=1,\,2$ and $3$,
and different certain values of $\alpha$ and check the graphs against the
numerical ones. In figure 7, the graphs of $\mathcal{Z}_{n}(q)$ and $S_{n}(q)$
have been presented where we used the equation (4.8) (solid lines) in terms of
$\ell$ for $n=1$, $q=1$ (upper plots) and $q=1,3$ and $n=1,2$ and $3$ (bottom
plots). The graphs show that considering the four largest eigenvalues would be
sufficient in the numerical analysis.

Figure 6:  Matching the results of numerical analysis (symbols) with the
values predicted by equation (4.7) (solid line) in the plotting real and
imaginary parts of the $F_{n}(\alpha)$ function is shown. Top plots: In terms
of $\alpha$ for $m=10^{-20}$ and $n=1,2$ and $3$ for $\ell=70$. Bottom plots:
In terms of $\ell$ for $\alpha=0.5$ and $1$ (bottom plots) which are displayed
for the massless limit case.

Figure 7: The values of $\mathcal{Z}_{n}(q)$ and $S_{n}(q)$ in conformal
limit, including the numerical results (symbols) against to what obtained from
the equation (4.8) (solid lines) in terms of $\ell$ for $n=1$, $q=1$ and
different masses (upper plots) and $m=10^{-20}$, $q=1,3$ and $n=1,2$ and $3$
(bottom panels) are shown.

For the massive limit, we show that only two eigenvalues play a significant
role in the calculations. If we consider the length of the subsystem large
enough, with a good approximation, the eigenvalues become equal and
independent of the length of the subsystem and we show that they fit well with
the following expression

$\nu\simeq 1+\Delta(m),$ (4.9)

where $\Delta\ll 1$ is a real positive parameter dependent on the mass. In
this case, the entanglement entropy can be obtained as

$S_{EE}^{m}\sim\Delta\big{[}1+\log(\frac{2}{\Delta})\big{]}.$ (4.10)

In order to find SREE, we note that $Z_{n}^{m}(\alpha)$ is given by (3.14)
where we shall consider only the two largest eigenvalues,

Figure 8:  Left panel: The two largest eigenvalues of the matrix
$\Lambda_{\mathcal{A}}$ for the massive limit are shown in terms of $\ell$ for
$m=1\leavevmode\nobreak\ $ and $\leavevmode\nobreak\ 1.2$. Right panel:
Numerical results (points) and predicted results in equation (4.10) (solid
lines), in terms of $\ell$ for $m=1$ and, $m=1.2$ are presented. The agreement
between the numerical results and the predicted values is clearly evident.

at the leading order around $\nu=1$ one obtains

${Z}_{n}^{m}(\alpha)\sim\frac{16^{n}}{\big{[}-2^{n}\cos\alpha(n\Delta+2)\Delta^{n}+\Delta^{2n}+4^{n}n\Delta+4^{n}\big{]}^{2}}\,.$
(4.11)

Figure 8 shows the two largest eigenvalues of the matrix
$\Lambda_{\mathcal{A}}$ in terms of $\ell$ for two different values of mass,
also the numerical graphs of the entanglement entropy against what predicted
from equation (4.10) in terms of $\ell$ are presented which are compatible.
Also, in figure 9, the results of $\mathcal{Z}^{m}_{n}(q)$ and $S^{m}_{n}(q)$
obtained from equation (4.11) are compared with the numerical results, which
are clearly seen to be the same.
As a concluding remark of this section, we used the approximate eigenvalues
for the correlation matrix which makes the numerical calculations more
straightforward than the usual methods and the results obtained for the SREE
are in good agreement with the previous works done in this field [59, 31].

Figure 9: $\mathcal{Z}_{n}(q)$ and $S_{n}(q)$ in the massive limit are shown,
including the numerical results (symbols) against what obtained from the
equation (4.11) (solid lines) in terms of $\ell$ for $n=1,\,2$ and $q=1$ and
$n=1,\,q=3$ for $m=1$, which the results are consistent.

##  5 Non-local quantum fields

It is shown that in local quantum field theories and for a gapped Hamiltonian
with an ultraviolet fixed point since the interactions are short-ranged,
entanglement entropy obeys the area law [66].333Although one-dimensional
critical systems violate this universal rule where the entanglement entropy is
scaled with the logarithm of the length of the subsystem [25, 22]. However, it
is observed that by considering the non-local random interactions and non-
local field theories, the entanglement entropy follows the volume law [67].
For a simple class of the non-local scalar field theories, let us assume the
following Hamiltonian

${\cal H}=\tfrac{1}{2}\int dx\left[(d\phi/dt)^{2}+C\phi
e^{A(-\partial^{2})^{w/2}}\phi\right],$ (5.1)

where $A$ and $w$ are positive constants by which we determine the magnitude
of non-locality, and for simplicity, by re-scaling the time we set $C=1$. In
Ref. [52], it is shown that the ground state EE of this type of non-local
field obeys the volume law when the size of the subsystem $\ell$ is smaller
than $A$, and for $w=1$ and $w=2$ the theory provides analytic estimation
confirms the numerical analysis. In this section for this type of non-local
field theories for $A\gg\ell$, by numerical calculations, we give an
estimation for the SREE, and for two values of $w=1\,,2$ it is checked against
the numerical analysis. We also present expressions for the symmetry-resolved
Rényi entropies and show that they are independent of $q$ and $n$. In this
way, by computing the EE, we also estimate analytic expressions for the
configurational and the number entropies introduced in (2.3).
Technically, in order to calculate EE, it is sufficient to obtain the
representation of the correlation function for this non-local theory. As a
result, $\mathbf{P}_{{\mathcal{NL}}}$ and $\mathbf{X}_{{\mathcal{NL}}}$ turn
to,

$(\mathbf{P}^{w}_{{\mathcal{NL}}})_{rs}=\int_{-\pi}^{\pi}\dfrac{dp}{2\pi}e^{ip(r-s)}\exp[A/2(2-2\cos
p)^{w/2}]$ (5.2)
$(\mathbf{X}^{w}_{{\mathcal{NL}}})_{rs}=\int_{-\pi}^{\pi}\dfrac{dp}{2\pi}e^{ip(r-s)}\exp[-A/2(2-2\cos
p)^{w/2}]$ (5.3)

One can obtain the analytical solution of the above integrals in terms of a
combination of the generalized hypergeometric function
$\,{}_{\mathfrak{p}}F_{\mathfrak{q}}(\mathfrak{a}_{1},...,\mathfrak{a}_{\mathfrak{p}};\mathfrak{b}_{1},...,\mathfrak{b}_{\mathfrak{q}};z)$.
For example, in the case of $(\mathbf{P}^{w_{even}}_{{\mathcal{NL}}})_{rs}$ ,
they will be in the following general form

$\,_{\frac{w}{2}}F_{\frac{w}{2}}\left(\frac{1}{w},\frac{3}{w},...,\frac{w-1}{w};\frac{2}{w},\frac{4}{w},...,1;2^{w-1}A\right)-(r-s)^{2}\,_{\frac{w}{2}}F_{\frac{w}{2}}\left(\frac{3}{w},\frac{5}{w},...,\frac{w+1}{w};\frac{4}{w},...,\frac{w+2}{w};2^{w-1}A\right)\\\
+\cdots+(-1)^{i+1}\mathcal{C}_{i}\,{}_{\frac{w}{2}}F_{\frac{w}{2}}\left(\frac{2i-1}{w},\frac{2i+1}{w},...,\frac{2(i+\frac{w}{2})-3}{w};\frac{2i}{w},\frac{2(i+1)}{w},...,\frac{2(i+\frac{w}{2})-2}{w};2^{w-1}A\right)+\cdots$
(5.4)

where $i=1,2,...,Max(|r-s|)+1$ and $\mathcal{C}_{i}$’s are integer
coefficients. In this case, one should also find the eigenvalues for the
matrix
$\Lambda_{\mathcal{A}}=\sqrt{\mathbf{X}_{\mathcal{A}}\mathbf{P}_{\mathcal{A}}}$.
In the following, we investigate this entropy equation in detail. In order to
generalize calculations to the non-local scalar field, one must put
$\omega^{2}_{p}=e^{Ap^{\omega}}$ in (3.3). In many special cases, the
generalized hypergeometric function is automatically converted to other
specific functions, and in this work for more convenience, we focus on the
cases of $w=1$ and $2$ in (5.4). Thus, one obtains

$\displaystyle(\mathbf{P}^{w=1}_{\mathcal{NL}})_{rs}$ $\displaystyle=$
$\displaystyle J_{2(r-s)}(iA)+i\boldsymbol{E}_{2(r-s)}(iA),$ (5.5)
$\displaystyle(\mathbf{P}^{w=2}_{\mathcal{NL}})_{rs}$ $\displaystyle=$
$\displaystyle(-1)^{(r-s)}e^{A}I_{(r-s)}(A),$ (5.6)

where $J_{2r}$ and $\boldsymbol{E}_{2r}$ are Bessel functions of the first
kind and Weber functions, respectively, and $I_{r}$ is the modified Bessel
function of the first kind. It should be noted that to calculate
$\mathbf{X}_{\mathcal{NL}}$, it is enough to replace $-A$ with $A$.

###  5.1 Numerical analysis for SREE

To study the symmetry-resolved moments and the SREE, we consider $w=1$ and
$w=2$ and suppose the size of the subsystem is much smaller than $A$, in this
regime, the expansion of the correlation functions is obtained as follows

$\displaystyle(\mathbf{P}^{w=1}_{{\mathcal{NL}}})_{rs}$ $\displaystyle\sim$
$\displaystyle(-1)^{(r-s)}e^{A}\sqrt{\dfrac{2}{\pi
A}}\big{[}1-\dfrac{1}{A}(2(r-s)^{2}-\dfrac{1}{8})+\dots\big{]}$ (5.7)
$\displaystyle(\mathbf{X}^{w=1}_{{\mathcal{NL}}})_{rs}$ $\displaystyle\sim$
$\displaystyle\dfrac{2}{\pi A}[1-\dfrac{1}{A^{2}}(4(r-s)^{2}-1)+\dots]$ (5.8)
$\displaystyle(\mathbf{P}^{w=2}_{{\mathcal{NL}}})_{rs}$ $\displaystyle\sim$
$\displaystyle(-1)^{(r-s)}\dfrac{e^{2A}}{\sqrt{2\pi
A}}[1-\dfrac{1}{2A}\Big{(}(r-s)^{2}-\dfrac{1}{4}\Big{)}+\dots]$ (5.9)
$\displaystyle(\mathbf{X}^{w=2}_{{\mathcal{NL}}})_{rs}$ $\displaystyle\sim$
$\displaystyle\dfrac{1}{\sqrt{2\pi
A}}[1-\dfrac{1}{2A}\Big{(}(r-s)^{2}-\dfrac{1}{4}\Big{)}+\dots]$ (5.10)

In this case, the eigenvalues of the matrix are $\nu_{i}\sim e^{wA/2}$ [52].
By making use of (3.14), we can write $Z^{A\gg\ell}_{n}(\alpha)$ as follows

$Z^{A\gg\ell}_{n}(\alpha)\simeq\bigg{(}\frac{2^{2n}}{(e^{\frac{Aw}{2}}-1)^{2n}+(e^{\frac{Aw}{2}}+1)^{2n}-2(e^{Aw}-1)^{n}\cos\alpha}\bigg{)}^{\ell}.$
(5.11)

In the present case, the integral of equation (2.6) can be calculated
analytically, and $\mathcal{Z}^{A\gg\ell}_{n}(q)$ turns to the following form

$\mathcal{Z}^{A\gg\ell}_{n}(q)\simeq\exp\big{[}-\Big{(}2(n-1)\ell+1\Big{)}\frac{Aw}{2}\Big{]}.$
(5.12)

Finally, using the equation (2.4), one can calculate the symmetry-resolved
Rényi entropies as

$S^{A\gg\ell}_{n}(q)=-\frac{1}{2}w\,A+\ell\,w\,A,$ (5.13)

which is independent of $n$. While the same result holds for SREE, one can
conclude that the symmetry-resolved entropies are independent of $n$ as well
as $q$. We also observe that $\mathcal{Z}^{A\gg\ell}_{n}(q)$ and consequently
the probability $p(q)$ have the same value in different sectors. For numerical
verification of this claim, in figure 10, $S^{A\gg\ell}_{n}(q)$ is depicted in
terms of $\ell$ for $w=1$ (left) and $w=2$ (right) for two different values of
$A$. In fact, according to the calculations and figures, it can be concluded
that this model respects the equipartition of the entanglement entropy.

Figure 10:  The numerical results for $S_{n}(q)$ are plotted for $w=1$ (left)
and $w=2$ (right). $w$ and $A$ are the parameters that increase the amount of
non-locality. We set $A=400,\,550$ and $n=1,\,2$ and $q=1,\,2$. It can be seen
that the behavior of $S_{n}(q)$ is independent of $n$ and $q$, which confirms
equation (5.13).

###  5.2 Comment on the entanglement entropy

Technically, all we need to calculate the EE is to use the relations (5.5),
(5.6) and $\mathbf{X}_{\mathcal{NL}}$ instead of (2.14) and (2.15). To do so,
first in figure 11, we have plotted EE for the non-local QFT for $\ell\gg A$
in which we could reproduce the corresponding results of Ref. [52], for
instance, the right panel of figure 11 indicates the scaling of $S_{EE}$ is
$A^{2}$. While for $\ell\ll A$, figure 12 represents $S_{EE}$ in terms of
$\ell$ where the scaling is $S_{EE}=c_{b}\ell A$ in which $c_{b}$ is a
constant that depends on $w$. If $A$ is supposed to be large enough, then with
good approximation, $c_{b}$ equals $\frac{w}{2}$. Moreover, by increasing $A$
and $w$ parameters, $S_{EE}$ increases drastically, and its behavior follows
the volume law as well.
Let us end this section by making a further comment on fixing the $c_{b}$
coefficient and the configurational and the number entropies introduced in
(2.3). We use equation (2.3) and write down the entanglement entropy as

$S^{C\leavevmode\nobreak\
{A\gg\ell}}_{EE}=S^{A\gg\ell}(q)-\log\mathcal{Z}^{A\gg\ell}_{1}(q),$ (5.14)

plunging equations (5.12) and (5.13) into the above equation, one obtains
$S^{C\leavevmode\nobreak\ {A\gg\ell}}_{EE}\simeq\ell\,w\,A$, this fixes
$c_{b}\sim\frac{w}{2}$. Numerical verification is done in figure 13, the right
side of figure 13 shows that the predicted results for
$S^{C\leavevmode\nobreak\ {A\gg\ell}}_{EE}$ in equation (5.14) (solid lines)
against the numerical results (symbols) obtained via the equation (2.10) which
are compatible. Finally, one might use (2.3) and (5.14) and argue that the
equation (5.13) gives us the configurational entropy while, making use of
(5.12), the number entropy becomes $\frac{Aw}{2}$.

Figure 11: Entanglement entropy is displayed for non-local scalar fields in
which $\ell\gg A$. Left panel: $S_{EE}$ is plotted in terms of subsystem
length for $w=1$ and $A=20$ (blue dots), $w=1$ and $A=40$ (red dots),
$w=2\leavevmode\nobreak\ and\leavevmode\nobreak\ A=20$ (green dots) and for
$w=2$ and $A=40$ (purple dots). Right panel: it is shown in terms of $A$ for
$\ell=85$. By performing the fit, one observes that $S_{EE}$ scales by
$A^{2}$.

Figure 12:  Entanglement entropy in terms of subsystem length is displayed in
the non-local QFT for $\ell\ll A$. Left panel: $A=200$ and $w=1$ (blue dots),
$w=2$ (red dots) and $w=3$ (green dots). Right panel: $w=2$ and $A=250$ (blue
dots), $A=400$ (red dots) and $A=550$ (green dots). One observes that the
entanglement entropy follows the volume law in this regime.

Figure 13:  The symmetry resolved entropies (left side) and entanglement
entropy (right side) are shown with respect to subsystem size $\ell$ for $w=1$
and $2$ for $A=1200$. The predicted results (solid lines) with values very
close to $c_{b}=\frac{w}{2}$ are in good agreement with the numerical values
(symbols).

##  6 Conclusions

In this paper, we investigated the symmetry-resolved Rényi entropies and SREE
for free complex scalar fields and a simple class of non-local quantum field
theories whose entanglement entropy satisfies a volume law. We focused on the
behavior of the eigenvalues of the correlation matrix for Klein-Gordon fields
in massive and massless limits and by determining and specifying the most
effective and significant eigenvalues of the correlation matrix, we obtained
the effective expressions for the charged moments, and consequently, showed
that the obtained symmetry-resolved entropies are in good agreement with the
numerical calculations in all cases. By making use of the estimated
eigenvalues, we derived the EE for the complex scalar fields in the massless
limit that is compatible with the CFT results. In addition, the behavior of
symmetry-resolved entropies with respect to the mass as well as the size of
the subsystem, $\ell$, was studied, and it was shown that $S_{n}(q)$ saturate
for larger masses at smaller values of $\ell$. For massive cases, we verified
the $q$-dependence of $\mathcal{Z}_{n}(q)$ and $S_{n}(q)$ whereas in the
massless limit and large $q$, the dependence $S_{n}(q)$ on $q$ becomes more
relaxed. In other words, in the massless limit, $S_{n}(q)$ are approximately
independent of symmetry sectors, $q$ which confirms the equipartition of EE at
least effectively. Here we would like to emphasize that our results are
consistent with the existing literature, for example, it was argued that the
conformal invariance implies the equipartition of EE in a system with $U(1)$\-
symmetry [30], and, for generic gapped systems, Ref. [59] reported an
effective equipartition of EE.
Furthermore, we studied SREE in a simple class of non-local QFT in which the
non-locality of the theory is identified by two parameters $A$ and $w$. In
Ref.[52], it was argued that for $\ell\ll A$, one has $S_{EE}=c_{b}\ell A$
while for $\ell\gg A$ the behavior of the EE is proportional to $A^{2}$. We
studied the EE for these theories and an effective expression for the EE was
presented, we observed that for larger values of $w$ and $A$, the $c_{b}$
coefficient with good approximation tends to $\frac{w}{2}$. In the case of
SREE for $\ell\ll A$ by finding effective oppression for symmetry-resolved
partition function $\mathcal{Z}_{n}(q)$, we derived symmetry-resolved Rényi
entropies $S_{n}(q)$ and by testing against numerical calculation, we showed
that $S_{n}(q)$ do not depend on $n$ and $q$. This confirms that the
considered class of non-local field theories when the size of the subsystem is
smaller than $A$ holds the equipartition of the entanglement entropy.

For further study, in the class of non-local quantum fields, we would like to
compute the other measures of entanglement say as mutual information and
tripartite information, and also investigate the famous inequalities between
them. Furthermore, important insights could also come from investigating the
complexity of this class of non-local QFTs.

### Acknowledgments

We wish to thank M. Alishahiha, A. Naseh, A. Mollabashi, M. Reza Mozaffar and
B. Taghavi for their fruitful comments. We would like to acknowledge M.
Ghasemi and G. Jafari for their cooperation in the first part at earlier
versions of this paper, hereby, we appreciate their efforts. We thank the IPM-
Grid computing group for providing computing and storage facilities. R.
Pirmoradian would also like to extend special thanks to M. Reza Lahooti
Eshkevari dean of Ershad Institute for his kindly support.

##  Appendix A Comments on a Single Site

In this appendix, for a single site of bosonic scalar QFTs, we give an
expression for the eigenvalues of the correlator matrix. For a single degree
of freedom on the finite lattice and large $\mathcal{N}$, the eigenvalues are
obtained as

$\displaystyle\nu=\frac{1}{\mathcal{N}}\sqrt{1+\frac{\mathcal{R}_{2}}{m}+m\mathcal{R}_{1}+\mathcal{R}_{1}\mathcal{R}_{2}}$
(A.1)

where $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are defined by

$\displaystyle\mathcal{R}_{1}=\displaystyle\sum_{k=1}^{\mathcal{N}-1}\frac{1}{\omega_{k}}\,,\hskip
11.38109pt\hskip
11.38109pt\mathcal{R}_{2}=\displaystyle\sum_{k=1}^{\mathcal{N}-1}\omega_{k}.$
(A.2)

The massless and the massive limits can be obtained where in the case of the
massless one obtains

$\hskip 11.38109pt\mathcal{R}_{1}\mathcal{R}_{2}\sim 4\mathcal{N}^{2}\,,\hskip
11.38109pt\mathcal{R}_{2}\sim\mathcal{N}\,,\hskip
11.38109ptm\mathcal{R}_{1}\sim 0$ (A.3)

Therefore, the eigenvalues take the following form

$\nu_{massless}=\frac{1}{\mathcal{N}}\sqrt{1+\frac{\mathcal{N}}{m}+4\mathcal{N}^{2}}\sim\frac{1}{\sqrt{m\mathcal{N}}}.$
(A.4)

For the massive limit, one obtains

$\frac{\mathcal{R}_{2}}{m}=m\mathcal{R}_{1}\sim\mathcal{N},\hskip
11.38109pt\mathcal{R}_{1}\mathcal{R}_{2}=\mathcal{N}^{2},$ (A.5)

leading to the following eigenvalue

$\nu_{massive}=\frac{1}{\mathcal{N}}\sqrt{1+2\mathcal{N}+\mathcal{N}^{2}}\sim
1+\frac{1}{\mathcal{N}}$ (A.6)

The calculations can be extended to the case where $\mathcal{N}$ tends to the
infinity, one obtains

$\nu_{massless}=\frac{2}{\pi}\sqrt{\log\left(\frac{8}{m}\right)}+O\left(m^{2}\right)$
(A.7)

where leads to the following symmetry-resolved partition function

$\mathcal{Z}_{1}(q)=\frac{\pi}{2}\frac{1}{\sqrt{\log\left(\frac{8}{m}\right)}}\left(1-\frac{2\pi}{2\sqrt{\log\left(\frac{8}{m}\right)}+\pi}\right)^{\left|q\right|}$
(A.8)

Figure 14: The values of $\mathcal{Z}_{n}(q)$ of a single site, in terms of
$q$, for $m=10^{-20}$, $10^{-5},10^{-2},1,10$ and $10^{2}$ for $n=1$ (left
plot) and $n=2,3$ and for $m=10^{-8},10^{-5},10^{-2}$ (right plot) are
displayed. As can be seen, the slope of the line becomes less and less as the
mass decreases. So, for $m\ll 1$, the value of $\mathcal{Z}_{1}(q)$ becomes
independent of $q$.

And in the massive limit, one obtains

$\nu_{massive}=1+\frac{1}{4m^{4}}+O\left(\frac{1}{m}\right)^{6}$ (A.9)

thus $\mathcal{Z}_{1}(q)$ is obtained as

$\mathcal{Z}_{1}(q)=\frac{4m^{4}\left(8m^{4}+1\right)^{-\left|q\right|}}{4m^{4}+1}\approx\left(8m^{4}+1\right)^{-\left|q\right|}$
(A.10)

For the non-local quantum fields at $\mathcal{N}\to\infty$ and for $w=1$, one
obtains

$\nu^{\mathcal{NL}\,,w=1}=\sqrt{I_{0}(A){}^{2}-\boldsymbol{L}_{0}(A){}^{2}}$
(A.11)

where $\boldsymbol{L}_{0}$ is modified Struve function, in this case one has

$\mathcal{Z}_{1}(q)^{\mathcal{NL}\,,w=1}=\frac{1}{\sqrt{I_{0}(A){}^{2}-\boldsymbol{L}_{0}(A){}^{2}}}\left(\frac{\sqrt{I_{0}(A){}^{2}-\boldsymbol{L}_{0}(A){}^{2}}-1}{\sqrt{I_{0}(A){}^{2}-\boldsymbol{L}_{0}(A){}^{2}}+1}\right){}^{\left|q\right|}$
(A.12)

And for $w=2$, one obtains

$\nu^{\mathcal{NL}\,,w=2}=I_{0}(A)$ (A.13)

leading to

$\mathcal{Z}_{1}(q)^{\mathcal{NL}\,,w=2}=\frac{1}{I_{0}(A)}\left(1-\frac{2}{I_{0}(A)+1}\right){}^{\left|q\right|}.$
(A.14)

The results are shown in figures 14 and 15 (for large $A$ and $w=1$ , $2$), we
have plotted $\mathcal{Z}_{1}(q)$ in terms of $q$. The single site graphs also
confirm that the symmetry-resolved moment for the non-local fields and
$A\gg\ell$ is $q$-independent, and it holds approximately for the massless
limit of complex scalar fields.

Figure 15: The values of $\mathcal{Z}_{1}(q)^{\mathcal{NL}}$ in terms of $q$,
for $A=1,4,12,50,100$ and $200$ for $w=1$ (left plot) and $w=2$ (right plot)
are displayed. As can be seen in the figures, the slope of the line becomes
less and less with the increase of non-locality parameters ($A$ and $w$). For
$A\gg 1$, the value of $\mathcal{Z}_{1}(q)^{\mathcal{NL}}$ becomes independent
of $q$.

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