Odd Entanglement Entropy and Logarithmic Negativity for Thermofield Double
States

Mostafa Ghasemia, Ali Naseha and Reza Pirmoradiana,b

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

b Department of physics, Islamic Azad University Central Tehran Branch,Tehran,
Iran

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
We investigate the time evolution of odd entanglement entropy (OEE) and
logarithmic negativity (LN) for the thermofield double (TFD) states in free
scalar quantum field theories using the covariance matrix approach. To have
mixed states, we choose non-complementary subsystems, either adjacent or
disjoint intervals on each side of the TFD. We find that the time evolution
pattern of OEE is a linear growth followed by saturation. On a circular
lattice, for longer times the finite size effect demonstrates itself as
oscillatory behavior. In the limit of vanishing mass, for a subsystem
containing a single degree of freedom on each side of the TFD, we analytically
find the effect of zero-mode on the time evolution of OEE which leads to
logarithmic growth in the intermediate times. Moreover, for adjacent intervals
we find that the LN is zero for times $t<\beta/2$ (half of the inverse
temperature) and after that, it begins to grow linearly. For disjoint
intervals at fixed temperature, the vanishing of LN is observed for times
$t<d/2$ (half of the distance between intervals). We also find a similar delay
to see linear growth of $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$. All these
results show that the dynamics of these measures are consistent with the
quasi-particle picture, of course apart from the logarithmic growth.

##  1 Introduction

Understanding the non-equilibrium dynamics of the isolated many-body quantum
systems has been one of the major research avenues, both theoretical and
experimental, in the last few decades[1, 2, 3, 4, 5]. One of the protocols for
preparing the out of equilibrium systems is the global quench setting that
enables us to study thermalization in isolated quantum systems and can be best
understood by phenomenological quasi-particle picture[6, 7, 8, 9, 10]. In this
setup, an isolated system is initially prepared at $t=0$ in a given ground
state $|\psi_{0}\rangle$ of a certain Hamiltonian $H_{0}$ and undergoes a
sudden change in a parameter of the Hamiltonian at $t=0$. The system evolves
with a new Hamiltonian $H$ in such a way that $[H,H_{0}]\neq 0$ and the time
evolved state at time $t$ is given by
$|\psi(t)\rangle=e^{-iHt}|\psi_{0}\rangle$. Since the evolution is unitary,
the final state of the system is described by a pure state and one of the
appropriate concepts for understanding the thermalization is quantum
entanglement, $S_{\text{EE}}$ [11]. When considering this time evolution,
there is a regime in which the initial growth is linear and it occurs when the
lattice spacing $\delta$ is much smaller than the correlation length, $\beta$,
which itself is much smaller than the subsystem size $l$, i.e.,

$\displaystyle\delta\ll\beta\ll l.$ (1.1)

In the quasi-particle picture, the quench creates independent entangled pairs
which propagate on a circle (with circumference $\mathcal{L}$) in a ballistic
fashion with an effective velocity

$\displaystyle v_{n}=\frac{\mathcal{L}}{2\pi}\partial_{n}\omega_{n},$ (1.2)

where $n$ is an integer that runs from $0$ to the total number of lattice
sites and $\omega_{n}$ is given by (2.6). The upper bound of velocity is known
as Lieb-Robinson bound [12]. When the quasi-particles and their partners
laying in the interval and it is complementary, respectively(or vise versa),
we observe the linear growth in the entanglement entropy where Alba and
Calabrese [9, 10] provided a formula for it. By carefully tracking quasi-
particles on a circle leaving and re-entering the interval, a phenomenological
relation for $S^{\text{TFD}}_{\text{EE}}(t)-S^{\text{TFD}}_{\text{EE}}(t=0)$
in TFD state111In the following we will define this state., is proposed [9,
10, 13]

$\displaystyle S^{\text{TFD}}_{\text{EE}}(t)-S^{\text{TFD}}_{\text{EE}}(t=0)=$
$\displaystyle\sum_{n}\mathbf{s}_{n}^{\text{TFD}}\mathcal{L}\hskip
2.84544pt\text{frac}\big{(}\frac{v_{n}t}{\mathcal{L}}\big{)}\hskip
71.13188pt\text{if}\hskip 5.69046pt\mathcal{L}\hskip
2.84544pt\text{frac}\big{(}\frac{v_{n}t}{\mathcal{L}}\big{)}<l,$ (1.3)
$\displaystyle\sum_{n}\mathbf{s}_{n}^{\text{TFD}}l\hskip
120.92421pt\text{if}\hskip 5.69046ptl\leq\mathcal{L}\hskip
2.84544pt\text{frac}\big{(}\frac{v_{n}t}{\mathcal{L}}\big{)}<\mathcal{L}-l,$
(1.4)
$\displaystyle\sum_{n}\mathbf{s}_{n}^{\text{TFD}}\mathcal{L}\left(1-\hskip
2.84544pt\text{frac}\big{(}\frac{v_{n}t}{\mathcal{L}}\big{)}\right)\hskip
28.45274pt\text{if}\hskip 5.69046pt\mathcal{L}-l\leq\mathcal{L}\hskip
2.84544pt\text{frac}\big{(}\frac{v_{n}t}{\mathcal{L}}\big{)},$ (1.5)

where frac denotes the fractional part and $\mathbf{s}^{\text{TFD}}_{n}$
is222The part within the large parentheses represents a contribution to the
thermodynamic entropy of the free boson system at the inverse temperature
$\beta$ from the mode $n$.

$\displaystyle\mathbf{s}_{n}^{\text{TFD}}=\frac{2}{\mathcal{L}}\left(\frac{\beta\omega_{n}}{e^{\beta\omega_{n}}-1}-\log(1-e^{-\beta\omega_{n}})\right).$
(1.6)

However, the entanglement entropy is only a proper measure to capture the
dynamics of entanglement for pure quantum states and to have a deeper insight
into the spreading of information in the out of equilibrium situations, we
need to investigate the dynamics of the mixed states correlation measures. One
of those measures is mutual information (MI) which is defined for disjoint
subregions $A$ and $B$. It is given as a linear combination of entanglement
entropies of that regions,

$I(A,B)=S_{\text{EE},A}+S_{\text{EE},B}-S_{\text{EE},A\cup B}.$ (1.7)

However, MI includes both classical and quantum correlations between the
subregions $A$ and $B$ and since it does not vanish for separable states it
would not be a good measure of entanglement [14]. Besides the MI, in the
quantum information context, there are various measures for capturing the
entanglement dynamics of mixed states. But, these measures are usually based
on optimization procedures [11], which are intractable approaches and we need
a way to have explicit computational prescriptions. Among all these proposed
entanglement measures for mixed states, the logarithmic negativity (LN) is the
best computable measure which is expected to capture only quantum correlations
[15, 17, 16]. Recently, another information theoretic quantity is introduced
which is called the odd entanglement entropy (OEE) [18]. In the context of the
AdS/CFT duality[19], the LN [20, 21] and OEE[18] are dual to the same
geometric object which is called the entanglement wedge cross-section
(EWC)[22, 23]. The dynamics of EWC has been recently studied in [24, 25, 26]
and the authors of [28, 27] have denoted that these two measures are
proportional in the case of integrable systems.

The LN and OEE are defined by taking partial transposing of the state which it
is obtained as follows. Let $|e_{i}^{(1)}\rangle$ and $|e_{i}^{(2)}\rangle$
denote the orthogonal basis for states of $\mathcal{H}_{A_{1}}$ and
$\mathcal{H}_{A_{2}}$, respectively. A density matrix acting on the bipartite
Hilbert space $\mathcal{H}=\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}$,
with the basis
$|e_{i}^{(1)}e_{i}^{(2)}\rangle=|e_{i}^{(1)}\rangle\otimes|e_{i}^{(2)}\rangle$,
is denoted by $\rho_{A_{1}A_{2}}$. The $\rho_{A_{1}A_{2}}$ can be expanded in
a basis $|e_{i}^{(1)}e_{j}^{(2)}\rangle$ of $\mathcal{H}$ as follows

$\rho_{A_{1}A_{2}}=\sum_{ijkl}\big{\langle}e_{i}^{(1)}e_{j}^{(2)}\big{|}\rho_{A_{1}A_{2}}\big{|}e_{k}^{(1)}e_{l}^{(2)}\big{\rangle}\hskip
2.84544pt\big{|}e_{i}^{(1)}e_{j}^{(2)}\big{\rangle}\big{\langle}e_{k}^{(1)}e_{l}^{(2)}\big{|}.$
(1.8)

The partial transposition of the density matrix $\rho_{A_{1}A_{2}}$ with
respect to the subsystem $A_{2}$ is given by swapping the matrix elements in
the subsystem $A_{2}$,

$\big{\langle}e_{i}^{(1)}e_{j}^{(2)}\big{|}\rho_{A_{1}A_{2}}^{T_{A_{2}}}\big{|}e_{k}^{(1)}e_{l}^{(2)}\big{\rangle}=\big{\langle}e_{i}^{(1)}e_{l}^{(2)}\big{|}\rho_{A_{1}A_{2}}\big{|}e_{k}^{(1)}e_{j}^{(2)}\big{\rangle}.$
(1.9)

Accordingly, with respect to the original basis of $\mathcal{H}$ one has

$\rho_{A_{1}A_{2}}^{T_{A_{2}}}=\sum_{ijkl}\big{\langle}e_{i}^{(1)}e_{l}^{(2)}\big{|}\rho_{A_{1}A_{2}}\big{|}e_{k}^{(1)}e_{j}^{(2)}\big{\rangle}\hskip
2.84544pt\big{|}e_{i}^{(1)}e_{j}^{(2)}\big{\rangle}\big{\langle}e_{k}^{(1)}e_{l}^{(2)}\big{|}.$
(1.10)

By defining $S_{\text{OEE}}^{(n_{\circ})}(\rho_{A_{1}A_{2}})$ as follows

$\displaystyle
S_{\text{OEE}}^{(n_{\circ})}(\rho_{A_{1}A_{2}})=\frac{1}{1-n_{\circ}}\Big{(}\text{Tr}(\rho_{A_{1}A_{2}}^{T_{A_{2}}})^{n_{\circ}}-1\Big{)},$
(1.11)

where $n_{\circ}$ is an odd positive integer, the $S_{\text{OEE}}$ is given
by[18]

$S_{\text{OEE}}(\rho_{A_{1}A_{2}})=\lim_{n_{\circ}\rightarrow
1}S_{\text{OEE}}^{(n_{\circ})}.$ (1.12)

Since, $\rho_{A_{1}A_{2}}^{T_{A_{2}}}$ is a Hermitian operator and the partial
transposition is not a completely positive map, then
$\rho_{A_{1}A_{2}}^{T_{A_{2}}}$, in general, can have negative eigenvalues.
Accordingly, the $S_{\text{OEE}}$ can be written as

$S_{\text{OEE}}(\rho_{A_{1}A_{2}})=-\sum_{\lambda_{i}>0}\lambda_{i}\log\lambda_{i}+\sum_{\lambda_{i}<0}|\lambda_{i}|\log|\lambda_{i}|,$
(1.13)

where $\lambda_{i}$’s are the eigenvalues of the
$\rho_{A_{1}A_{2}}^{T_{A_{2}}}$ matrix. It is worth mentioning that the OEE
reduces to the entanglement entropy for pure states. Moreover, if we subtract
the von Neumann entropy $S_{\text{EE}}(\rho_{A_{1}A_{2}})$ from OEE, it has
been suggested that the obtained quantity might be dual to the entanglement
wedge cross-section for holographic theories333Although, it has been recently
argued that there might be counterexamples for this duality; for more details
see [29].[18]

$E_{W}(\rho_{A_{1}A_{2}})=S_{\text{OEE}}(\rho_{A_{1}A_{2}})-S_{\text{EE}}(\rho_{A_{1}A_{2}}).$
(1.14)

For holographic theories this quantity is positive but, in general, it can be
negative444Recently, OEE for Lifshitz scalar theories has been studied in [30]
where it has been shown that it can be negative.. The LN can also be defined
along the same lines:

$\mathcal{E}_{A_{1}A_{2}}\equiv\log||\rho_{A_{1}A_{2}}^{T_{A_{2}}}||=\log\text{Tr}|\rho_{A_{1}A_{2}}^{T_{A_{2}}}|,$
(1.15)

where the trace norm
$||\mathcal{O}||\equiv\text{Tr}\sqrt{\mathcal{O}^{\dagger}\mathcal{O}}$.
According to the eigenvalues $\lambda_{i}$ of the operator
$\rho_{A_{1}A_{2}}^{T_{A_{2}}}$ one has

$\mathcal{E}_{A_{1}A_{2}}=-\sum_{\lambda_{i}>0}\lambda_{i}+\sum_{\lambda_{i}<0}|\lambda_{i}|$
(1.16)

It is easy to see that for pure states, the LN reduces to $n=\frac{1}{2}$
Rényi entropy[16]. It is worth mentioning again that LN is expected to capture
only quantum correlations and has been studied previously in several works[31,
32, 33, 34, 35, 36, 39, 37, 38, 40, 41, 42, 43]. Interestingly, it is argued
that in the limit of long times and large subsystems with a fix ratio, the LN
equals half of the Rényi mutual information

$\displaystyle\mathcal{E}(A_{1},A_{2})=\frac{I^{(1/2)}(A_{1},A_{2})}{2}.$
(1.17)

Accordingly, the condition $\mathcal{E}(A_{1},A_{2})=0$ is necessary (but not
sufficient) for the absence of mutual entanglement.

One particular fruitful situation for studying thermalization and global
quench in the context of holography has been to study these phenomena for the
case of thermofield double states (TFD)[44, 45, 46]. Using the dictionary of
the AdS/CFT duality, the TFD state is proposed to be dual to the eternal AdS
black hole[47] and provides a setup to probe various aspects of black holes
from a quantum information theoretic perspective. For example, the TFD state
can be used to study scrambling and quantum chaos[48, 49, 50], dynamics of
entanglement entropy[51], quantum computational complexity[52], and so on.
This state belongs to the product Hilbert space

$\mathcal{H}=\mathcal{H}_{\text{L}}\otimes\mathcal{H}_{\text{R}}.$ (1.18)

By choosing the following Hamiltonian

$H_{\text{TFD}}=H_{L}\otimes I_{R}-I_{L}\otimes H_{R}$ (1.19)

the time-dependent TFD state is given by

$|\text{TFD}(t_{L},t_{R})\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_{n}e^{-\beta
E_{n}/2}e^{-iE_{n}(t_{L}+t_{R})}|E_{n}\rangle_{L}|E_{n}\rangle_{R},$ (1.20)

where $|E_{n}\rangle_{L,R}$ are the energy eigenstates of the left and right
theories (for example two CFTs), respectively, with corresponding times
$t_{L,R}$. Moreover, tracing out either copy leads to a thermal state at the
inverse temperature $\beta$ for the other,
$\rho_{\text{th}}=\frac{1}{Z(\beta)}e^{-\beta H_{i}}$, $i\equiv L,R$ and
$Z(\beta)$ denotes the partition function. According to (1.20) and in the
spirit of [51], our setup to consider the dynamics of LN and OEE will be in a
rather unusual quantum quench scenario[13] in which two decoupled subsystems
are entangled via their initial conditions. Moreover, to have a mixed state,
we will consider two spatial non-complementary regions. Since the underlying
TFD state is a Gaussian state, we can use the covariance matrix approach to
calculate LN and OEE. We will observe that their behaviors under the time
evolution can be summarized as a linear growth followed by saturation which is
similar to the expectations from the quasi-particle picture. Also, we will
observe the oscillatory behavior due to the finite size effect as well as a
logarithmic contribution in the intermediate regime due to the existence of
the zero-mode.

This paper is organized as follows: In section 2, we will briefly review the
reconstruction of the TFD state for two harmonic oscillators and then study
its time dependency. Then, we will generalize this to a $1+1$ dimensional free
real scalar QFT. We will also introduce the covariance matrix formalism and by
using this method in section 3, we will explain how to evaluate the OEE and LN
for subsystems involving an equal number of sites on each spatial side of the
TFD state. The numerical results are presented in section 4. In section 4,
some previously proposed inequalities for OEE are also checked. In section 5,
we will choose a special dividing for subsystems and give an analytical
formula for evaluating the zero-mode contribution. In section 6, we will
conclude and discuss our results and future directions. In appendix A, we will
study the effect of temperature on the dynamics of entanglement. To do so, we
will consider two cases: In the first case, the two intervals are adjacent to
each other and in the second case, the two intervals are separated by a
distance $d$. Further details for evaluation of zero-mode contribution and
logarithmic growth are provided in appendix B.

##  2 Covariance matrix for Gaussian TFD state

As we have mentioned in the introduction, we would like to study the dynamics
of OEE and LN. Accordingly, we will focus on the TFD state of a free real
scalar QFT. This TFD state is a Gaussian state and therefore we can use the
power of the covariance matrix to calculate the OEE and LN555The Covariance
matrix approach has been previously used to probe the dynamics of the
entanglement entropy of bosonic and fermionic Gaussian states[13, 53, 54, 55,
56, 57, 58].. Since, in general, the OEE and LN are divergent quantities in
the continuum, we will regularize the theory by putting it on a lattice. In
the normal mode decomposition, the discretized QFT takes the form of $N$
decoupled simple harmonic oscillators. Accordingly, we will first consider the
construction of time-independent as well as time-dependent TFD states for two
copies of simple harmonic oscillators. Then, the OEE and LN for the TFD state
of discretized QFT will be constructed by a sum on the contribution of each
normal mode (simple harmonic oscillator).

###  2.1 Free Real Scalar QFT on a Lattice

The Hamiltonian of a 1+1 dimensional free real scalar QFT on a circle with a
circumference $\mathcal{L}$ is given by

$\displaystyle
H=\frac{1}{2}\int_{-\frac{\mathcal{L}}{2}}^{\frac{\mathcal{L}}{2}}dx\hskip
1.42262pt\Big{(}\Pi^{2}+(\partial_{x}\hskip
1.42262pt\Phi)^{2}+m^{2}\Phi^{2}\Big{)},$ (2.1)

where $\Pi=\frac{\partial L}{\partial{\dot{\Phi}}}=\dot{\Phi}$ is the
conjugate momentum. The regularized Hamiltonian on a circular lattice with $N$
sites and lattice spacing $\delta=\frac{\mathcal{L}}{N}$ becomes

$\displaystyle H=\sum_{a=1}^{N}\bigg{(}\frac{\delta}{2}\hskip
1.42262pt\mathbf{P}_{a}^{2}+\frac{m^{2}}{2\delta}\hskip
1.42262pt\mathbf{Q}_{a}^{2}+\frac{1}{2\delta^{3}}\left(\mathbf{Q}_{a+1}-\mathbf{Q}_{a}\right)^{2}\bigg{)},$
(2.2)

where redefined canonical variables are

$\displaystyle\mathbf{Q}_{a}=\Phi(x_{a})\delta,\hskip
28.45274pt\mathbf{P}_{a}=\Pi(x_{a}),$ (2.3)

and we have imposed the periodic boundary conditions
$\mathbf{Q}_{N+1}=\mathbf{Q}_{1}$ and $\mathbf{P}_{N+1}=\mathbf{P}_{1}$. By
applying the discrete Fourier transformation666The canonical commutation
relations are given by
$[\tilde{Q}_{k},\tilde{P}^{\dagger}_{l}]=i\delta_{k,l}.$

$\displaystyle\tilde{\mathbf{Q}}_{k}=\frac{1}{\sqrt{N}}\sum_{a=1}^{N}e^{\frac{2\pi
ika}{N}}\hskip 1.42262pt\mathbf{Q}_{a},\hskip
28.45274pt\tilde{\mathbf{P}}_{k}=\frac{1}{\sqrt{N}}\sum_{a=1}^{N}e^{-\frac{2\pi
ika}{N}}\hskip 1.42262pt\mathbf{P}_{a},$ (2.4)

where $\tilde{\mathbf{Q}}^{\dagger}_{k}=\tilde{\mathbf{Q}}_{N-k}$ and
$\tilde{\mathbf{P}}^{\dagger}_{k}=\tilde{\mathbf{P}}_{N-k}$, the Hamiltonian
(2.2) reduces to

$\displaystyle H=\sum_{k=0}^{N-1}\bigg{(}\frac{\delta}{2}\hskip
1.42262pt|\tilde{\mathbf{P}}_{k}|^{2}+\frac{1}{2\delta}\omega_{k}^{2}\hskip
1.42262pt|\tilde{\mathbf{Q}}_{k}|^{2}\bigg{)}.$ (2.5)

In the above formula, the frequencies $\omega_{k}$ are given by

$\displaystyle\omega_{k}^{2}=m^{2}+\frac{4}{\delta^{2}}\sin^{2}(\frac{\pi
k}{N}).$ (2.6)

To construct the corresponding TFD state, one first needs to quantize the
Hamiltonian (2.5) by defining two sets of creation and annihilation operators

$\displaystyle\tilde{\mathbf{Q}}_{k}=\sqrt{\frac{\delta}{2\omega_{k}}}\left(\hat{a}_{N-k}+\hat{a}_{k}^{\dagger}\right),\hskip
28.45274pt\tilde{\mathbf{P}}_{k}=i\sqrt{\frac{\omega_{k}}{2\delta}}\left(\hat{a}_{k}^{\dagger}-\hat{a}_{N-k}\right),$
(2.7)

where $[\hat{a}_{k},\hat{a}^{\dagger}_{k}]=1$. By substituting (2.7) in (2.5)
one gets

$\displaystyle\hat{H}=\sum_{k=0}^{N-1}\omega_{k}\left(\hat{a}_{k}^{\dagger}\hskip
1.42262pt\hat{a}_{k}+\frac{1}{2}\right).$ (2.8)

This is the Hamiltonian of $N$ decoupled simple harmonic oscillators with
equal mass $M=\delta^{-1}$ (not to be confused with the physical mass $m$) and
k-dependent frequencies $\omega_{k}$. Ignoring the constant term, the
$\hat{a}_{k}^{\dagger}\hskip 1.42262pt\hat{a}_{k}\omega_{k}\equiv n\omega_{k}$
denotes the total energy of level ”$n$” for each of these oscillators with
fixed momentum. Since the zero-mode Hamiltonian does not have a normalizable
ground state ($\omega_{0}$ vanishes when $m=0$) one can regularize it by
introducing a very small dimensionless mass, $m\mathcal{L}\ll 1$. Considering
the decoupled form of the Hamiltonian (2.5), the corresponding TFD state of
the free scalar theory will be a product of TFD states for each of the
oscillator modes. Accordingly, in the following we will focus on a single mode
(fixed momentum $k$) and construct its time-dependent TFD state.

###  2.2 TFD state for a simple harmonic oscillator

In this subsection, we will first construct the TFD state for a simple
harmonic oscillator with mass $m$ and frequency $\omega$ at $t=0$ and then
will turn to study its time evolution. The creation and annihilation operators
for a simple harmonic oscillator are given by

$\displaystyle\hat{a}^{\dagger}=\sqrt{\frac{m\omega}{2}}\left(\hat{Q}-i\frac{\hat{P}}{m\omega}\right),\hskip
14.22636pt\hat{a}=\sqrt{\frac{m\omega}{2}}\left(\hat{Q}+i\frac{\hat{P}}{m\omega}\right).$
(2.9)

The $n^{th}$ energy eigenstate is then defined by acting $n$ times with the
creation operator on the vacuum state $\mid 0\rangle$:

$\displaystyle\mid n\rangle=\frac{1}{\sqrt{n!}}\hskip
1.42262pt(\hat{a}^{\dagger})^{n}\mid 0\rangle.$ (2.10)

The action of creation and annihilation operators on this state are given by

$\displaystyle\hat{a}^{\dagger}\mid n\rangle=\sqrt{n+1}\mid n+1\rangle,\hskip
14.22636pt\hat{a}\mid n\rangle=\sqrt{n}\mid n-1\rangle.$ (2.11)

The TFD state at $t=0$ can be constructed as a superposition of a tensor
product of two copies of the energy eigenstate (2.10), which we label by $L$
and $R$, with special wights[59]:

$\displaystyle\mid\text{TFD}\rangle=Z(\beta)^{-\frac{1}{2}}\displaystyle\sum_{n=0}^{\infty}e^{-\frac{\beta}{2}E_{n}}\mid
n\rangle_{L}\hskip 2.84526pt\otimes\mid n\rangle_{R},$ (2.12)

where the normalization factor is $Z(\beta)=(1-e^{-\beta\omega})^{-1}$ and
$E_{n}$ denotes the energy of those eigenstates. Considering this
normalization factor together with (2.10), then the state (2.12) can be
alternatively written as

$\displaystyle\ket{\text{TFD}}=\sqrt{1-e^{-\beta\omega}}\hskip
2.84526pt\exp{\left(e^{-\frac{\beta}{2}\omega}\hskip
2.84526pt\hat{a}^{\dagger}_{L}\hat{a}^{\dagger}_{R}\right)}\mid
0\rangle_{L}\hskip 1.42262pt\otimes\mid 0\rangle_{R}.$ (2.13)

It is worth noting that in (2.13) the operator acting on the total vacuum
state is not a unitary operator. It is convenient to re-express (2.13) by
acting as a unitary operator on the vacuum state $\mid 0\rangle_{L}\hskip
1.42262pt\otimes\mid 0\rangle_{R}$. The result is [60]

$\displaystyle\mid\text{TFD}\rangle=e^{\alpha\left(\hat{a}_{L}^{\dagger}\hat{a}_{R}^{\dagger}-\hat{a}_{L}\hat{a}_{R}\right)}\ket{0}_{L}\otimes\ket{0}_{R},$
(2.14)

with

$\displaystyle\tanh{\alpha}=e^{-\frac{\beta}{2}\omega}.$ (2.15)

The time evolution of the state (2.14) is given by

$\displaystyle\mid\text{TFD}(t)\rangle=e^{-i\hat{H}_{L}t_{L}}e^{-i\hat{H}_{R}t_{R}}\mid\text{TFD}\rangle,$
(2.16)

in which the operators $\hat{H}_{L}$ and $\hat{H}_{R}$ are the corresponding
Hamiltonians for the left and right simple harmonic oscillators. These
quantities are defined as

$\displaystyle\hat{H}_{L}=\hat{a}^{\dagger}_{L}\hskip
1.42262pt\hat{a}_{L}+\frac{1}{2},\hskip
14.22636pt\hat{H}_{R}=\hat{a}^{\dagger}_{R}\hskip
1.42262pt\hat{a}_{R}+\frac{1}{2}.$ (2.17)

By choosing $t_{L}=t_{R}=t/2$, which is the common convention in holography
[61], the time dependent TFD state (2.16) takes the following form

$\displaystyle\mid\text{TFD}(t)\rangle=e^{-\frac{i}{2}\omega t}\hskip
1.42262pt\sqrt{1-e^{-\beta\omega}}\hskip
2.84526pt\exp\bigg{[}e^{-\frac{\beta}{2}\omega}e^{-i\omega t}\hskip
2.84526pta_{L}^{\dagger}a_{R}^{\dagger}\bigg{]}\ket{0}_{L}\otimes\ket{0}_{R}.$
(2.18)

One can write the state (2.18) in a more compact form by acting as a unitary
operator on the vacuum state $\ket{0}_{L}\otimes\ket{0}_{R}$ as follows[60,
13]777Here, we have dropped the global time-dependent phase, since this does
not change the physical state.

$\displaystyle\mid\text{TFD}(t)\rangle=\exp\bigg{[}z\hskip
1.42262pt\hat{a}^{\dagger}_{L}\hskip
1.42262pt\hat{a}^{\dagger}_{R}-z^{*}\hskip 1.42262pt\hat{a}_{L}\hskip
1.42262pt\hat{a}_{R}\bigg{]}\mid 0\rangle_{L}\otimes\mid 0\rangle_{R},$ (2.19)

where

$\displaystyle z=\alpha\hskip 1.42262pte^{-i\omega t},$ (2.20)

and $\alpha$ is the same as (2.15). Now by having the time-dependent TFD state
(2.19), one needs to find the covariance matrix associated with it. That is
the subject of the next subsection.

###  2.3 Covariance matrix formalism

The system (2.5) is described by $2N$ linear observables
$\hat{\xi}=(\hat{q}_{1},\hat{q}_{2},...,\hat{q}_{N},\hat{p}_{1},...,\hat{p}_{N})$.
The $(\hat{q}_{i},\hat{p}_{i})$ are canonical operators where
$[\hat{q}_{i},\hat{p}_{j}]=i\delta_{ij}$. The two-point functions of these
observables in an arbitrary state $\mid\Psi\rangle$ can be decomposed as

$\displaystyle\langle\Psi\mid\hat{\xi}^{a}\hskip
1.42262pt\hat{\xi}^{b}\mid\Psi\rangle=\frac{1}{2}\left(G^{ab}+i\Omega^{ab}\right),$
(2.21)

where $G^{ab}=G^{(ab)}$ and $\Omega^{ab}=\Omega^{[ab]}$ are the symmetric and
the antisymmetric parts of the correlation functions, respectively. For a
bosonic state, $\Omega^{ab}$ is completely fixed by the commutation relations
of $\hat{q}_{i}$ and $\hat{p}_{i}$,

$\displaystyle\Omega^{ab}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$ (2.22)

For a pure Gaussian state with vanishing first moment
$\langle\psi|\hat{\xi}^{a}|\psi\rangle=0$, the covariance matrix is given by
the symmetric part of the two-point function

$G^{ab}=\langle\Psi\mid\hat{\xi}^{a}\hat{\xi}^{b}+\hat{\xi}^{b}\hat{\xi}^{a}\mid\Psi\rangle.$
(2.23)

By using Wick’s theorem, one can compute all of the $n$-point functions from
$G^{ab}$. Hence, it can be used to label the Gaussian states. For a mixed
state $\rho$, when $\langle\Psi|\hat{\xi}^{a}|\Psi\rangle=0$, the covariance
matrix is defined by [62, 63]

$G^{ab}=\text{Tr}\bigg{(}\rho\hskip
1.42262pt\big{(}\hat{\xi}^{a}\hat{\xi}^{b}+\hat{\xi}^{b}\hat{\xi}^{a}\big{)}\bigg{)}.$
(2.24)

To find the covariance matrix associated with the TFD state (2.14), one can
first find the covariance matrix for the vacuum state and then by a unitary
transformation modify it for the TFD state. To do so, we restrict ourselves to
the space of Gaussian states. In this space, the general unitary operator
$\hat{U}(s)$ can be expressed by Hermitian operators which are quadratic in
the canonical operators $\hat{\xi}$,

$\displaystyle\hat{U}(s)=e^{-is\hat{K}},\hskip 28.45274pt\text{with}\hskip
14.22636pt\hat{K}=\frac{1}{2}\hat{\xi}^{a}\hskip 1.42262ptk_{(a,b)}\hskip
1.42262pt\hat{\xi}^{b}\equiv\frac{1}{2}\hat{\xi}\hskip 1.42262ptk\hskip
1.42262pt\hat{\xi}^{T}.$ (2.25)

Accordingly, the transformed state is

$|G_{s}\rangle=\hat{U}(s)|G_{0}\rangle,$ (2.26)

where the subscript “$0$” indicates the vacuum state (for the time-dependent
state (2.19) it refers to the TFD state (2.14)). To find the corresponding
covariance matrix, one needs the operation of $\hat{U}(s)$ on $\hat{\xi}^{a}$
which can be obtained as follows

$\displaystyle\hat{U}^{\dagger}(s)\hskip 1.42262pt\hat{\xi}^{a}\hskip
1.42262pt\hat{U}(s)=\sum_{n=0}^{\infty}\frac{s^{n}}{n!}[i\hat{K},\hat{\xi}^{a}]_{(n)}.$
(2.27)

In the above formula, $[i\hat{K},\hat{\xi}^{a}]_{(n)}$ is defined recursively
by
$[i\hat{K},\hat{\xi}^{a}]_{(n)}=[i\hat{K},[i\hat{K},\hat{\xi}^{a}]_{(n-1)}]$,
and $[i\hat{K},\hat{\xi}^{a}]_{(0)}=[i\hat{K},\hat{\xi}^{a}]$. With respect to
the (2.25) and the commutation relation
$[\hat{\xi}^{a},\hat{\xi}^{b}]=i\Omega^{ab}$, one can find that

$\displaystyle[i\hat{K},\hat{\xi}^{a}]=\Omega^{ab}\hskip
1.42262ptk_{(b,c)}\hskip 1.42262pt\hat{\xi}^{c}.$ (2.28)

By defining $K^{a}_{b}=\Omega^{ac}\hskip 1.42262ptk_{(c,b)}$, the above
formula can be rewritten as

$\displaystyle[i\hat{K},\hat{\xi}^{a}]=K^{a}_{b}\hskip
1.42262pt\hat{\xi}^{b}.$ (2.29)

Hence, the operation of $\hat{U}(s)$ on $\hat{\xi}^{a}$, (2.27), can be
expressed as follows

$\displaystyle\hat{U}^{\dagger}(s)\hskip 1.42262pt\hat{\xi}^{a}\hskip
1.42262pt\hat{U}(s)=(e^{sK})^{a}_{b}\hskip 2.84526pt\hat{\xi}^{b}\equiv
U(s)^{a}_{b}\hskip 2.84526pt\hat{\xi}^{b}.$ (2.30)

Now, the relations (2.25) together with the relation (2.30) implies that the
covariance matrix associated with transformed state $|G_{s}\rangle$ becomes

$\displaystyle G_{s}^{(a,b)}=\langle
G_{s}\mid\left(\hat{\xi}^{a}\hat{\xi}^{b}+\hat{\xi}^{b}\hat{\xi}^{a}\right)|G_{s}\rangle=U(s)^{a}_{c}\hskip
2.84526ptG_{0}^{(c,d)}\hskip 2.84526ptU(s)^{b}_{d},$ (2.31)

where $G_{0}^{(c,d)}$ is its counterpart for the vacuum. Accordingly, in the
compact notation, we have

$|G_{s}\rangle=\hat{U}(s)|G_{0}\rangle=|U(s)\hskip 2.84526ptG_{0}\hskip
2.84526ptU^{\top}(s)\rangle,\qquad G_{s}=U(s)\hskip 2.84526ptG_{0}\hskip
2.84526ptU^{\top}(s).$ (2.32)

In the following, we will demonstrate how to evaluate the covariance matrix
associated with the time-independent as well as time-dependent TFD states.
This is easily achieved by using the above formalism.

####  2.3.1 Covariance matrix for TFD state of the harmonic oscillator

Let us begin with constructing the covariance matrix associated with the time-
independent TFD state (2.14) corresponding to $N=1$ and then extend it to the
time-dependent case (2.19). The Hamiltonian of the system is described by

$\displaystyle\hat{H}=\frac{1}{2m}\Big{(}\hat{p}^{2}_{L}+\hat{p}^{2}_{R}+m^{2}\omega^{2}(\hat{q}^{2}_{L}+\hat{q}^{2}_{R})\Big{)}.$
(2.33)

By rewritting (2.14) in terms of the $(q_{L},p_{L},q_{R},p_{R})$ coordinates
and reading the corresponding $\hat{K}$ and $U(s)$, the covariance matrix of
TFD state (2.14) for a single-mode is given by

$\displaystyle
G^{ab}_{\alpha}=\begin{pmatrix}\frac{\cosh(2\alpha)}{m\omega}&0&-\frac{\sinh(2\alpha)}{m\omega}&0\vspace{.5cm}\par\\\
0&m\omega\cosh(2\alpha)&0&m\omega\sinh(2\alpha)\vspace{.5cm}\par\\\
-\frac{\sinh(2\alpha)}{m\omega}&0&\frac{\cosh(2\alpha)}{m\omega}&0\vspace{.5cm}\par\\\
0&m\omega\sinh(2\alpha)&0&m\omega\cosh(2\alpha)\end{pmatrix}.$ (2.34)

The time-dependent covariance matrix associated with time-dependent TFD state
(2.19) can be derived as follows. The Hamiltonian (2.33) has the form
$\hat{H}=\frac{1}{2}\hat{\xi^{a}}k_{ab}\hat{\xi^{b}}$, where the matrix
representation of $k_{ab}$ with respect to
$\hat{\xi}=(\hat{q}_{L},\hat{p}_{L},\hat{q}_{R},\hat{p}_{R})$ is given by

$\displaystyle k_{ab}=\begin{pmatrix}m\omega^{2}&0&0&0\\\ 0&\frac{1}{m}&0&0\\\
0&0&m\omega^{2}&0\\\ 0&0&0&\frac{1}{m}\end{pmatrix}.$ (2.35)

The simplectic generator $K^{a}_{b}=\Omega^{ac}k_{(c,b)}$ becomes

$\displaystyle K_{b}^{a}=\begin{pmatrix}0&\frac{1}{m}&0&0\\\
-m\omega^{2}&0&0&0\\\ 0&0&0&\frac{1}{m}\\\ 0&0&-m\omega^{2}&0\end{pmatrix},$
(2.36)

and therefore

$\displaystyle U(t)=\exp(tK)=\begin{pmatrix}m\cos(\omega t)&\frac{\sin(\omega
t)}{m}&0&0\vspace{.5cm}\par\\\ -\frac{\sin(\omega t)}{m}&m\cos(\omega
t)&0&0\vspace{.5cm}\par\\\ 0&0&m\cos(\omega t)&\frac{\sin(\omega
t)}{m}\vspace{.5cm}\par\\\ 0&0&-\frac{\sin(\omega t)}{m}&m\cos(\omega
t)\end{pmatrix}.$ (2.37)

Using this and noting to (2.34), the covariance matrix
$G_{\alpha}^{ab}(t)=U(t)\hskip 1.42262ptG_{\alpha}^{ab}\hskip
1.42262ptU^{\top}(t)$ becomes

$\displaystyle
G^{ab}_{\alpha}(t)=\begin{pmatrix}\frac{\cosh(2\alpha)}{m\omega}&0&-\frac{\sinh(2\alpha)\cos(\omega
t)}{m\omega}&\sin(\omega t)\sinh(2\alpha)\vspace{.5cm}\par\\\
0&m\omega\cosh(2\alpha)&\sin(\omega
t)\sinh(2\alpha)&m\omega\sinh(2\alpha)\cos(\omega t)\vspace{.5cm}\par\\\
-\frac{\sinh(2\alpha)\cos(\omega t)}{m\omega}&\sin(\omega
t)\sinh(2\alpha)&\frac{\cosh(2\alpha)}{m\omega}&0\vspace{.5cm}\par\\\
\sin(\omega t)\sinh(2\alpha)&m\omega\sinh(2\alpha)\cos(\omega
t)&0&m\omega\cosh(2\alpha)\end{pmatrix}.$ (2.38)

####  2.3.2 Covariance matrix for TFD state of real scalar QFT

As we explained previously, according to the discretized Hamiltonian (2.5) on
a lattice with $N$ sites, the TFD state (2.19) is described by $2N$ degrees of
freedom on each side. Again, one can choose the following coordinates

$\xi^{a}=\bigg{(}q_{m,L},p_{m,L},q_{m,R},p_{m,R}\bigg{)},$ (2.39)

where $q_{m}=\frac{1}{\sqrt{N}}\Phi_{L}(x_{m})$,
$p_{m}=\frac{\mathcal{L}}{\sqrt{N}}\Pi_{L}(x_{m})$, and $x_{m}=m\delta$ which
$\delta$ is the distance between the two sites. Using the Fourier space
coordinates

$\tilde{q}_{k}=\frac{1}{\sqrt{N}}\sum_{m=1}^{N}e^{\frac{2\pi
ikm}{N}}q_{m},\hskip
14.22636pt\tilde{p}_{k}=\frac{1}{\sqrt{N}}\sum_{m=1}^{N}e^{-\frac{2\pi
ikm}{N}}p_{m},$ (2.40)

and the extension of (2.38), the covariance matrix associated with the time-
dependent TFD state for a real scalar QFT (on the lattice) is real [13, 64,
58] and is given by

$\displaystyle
G(t)=\begin{pmatrix}G_{LL}^{\Phi\Phi}&G_{LL}^{\Phi\Pi}&G_{LR}^{\Phi\Phi}&G_{LR}^{\Phi\Pi}\vspace{.3cm}\par\\\
G_{LL}^{\Pi\Phi}&G_{LL}^{\Pi\Pi}&G_{LR}^{\Pi\Phi}&G_{LR}^{\Pi\Pi}\vspace{.3cm}\par\\\
G_{RL}^{\Phi\Phi}&G_{RL}^{\Phi\Pi}&G_{RR}^{\Phi\Phi}&G_{RR}^{\Phi\Pi}\vspace{.3cm}\par\\\
G_{RL}^{\Pi\Phi}&G_{RL}^{\Pi\Pi}&G_{RR}^{\Pi\Phi}&G_{RR}^{\Pi\Pi}\end{pmatrix},$
(2.41)

where,

$\displaystyle
G_{LL}^{\Phi\Phi}=G_{RR}^{\Phi\Phi}=\frac{1}{N}\sum_{k=0}^{N-1}e^{\frac{2\pi
ik(m-n)}{N}}\hskip 2.84526pt\frac{\cosh(2\alpha_{k})}{\lambda_{k}},$ (2.42)
$\displaystyle
G_{LL}^{\Pi\Pi}=G_{RR}^{\Pi\Pi}=\frac{1}{N}\sum_{k=0}^{N-1}e^{\frac{2\pi
ik(m-n)}{N}}\hskip 2.84526pt\lambda_{k}\cosh(2\alpha_{k}),$ (2.43)
$\displaystyle
G_{LL}^{\Phi\Pi}=G_{LL}^{\Pi\Phi}=G_{RR}^{\Phi\Pi}=G_{RR}^{\Pi\Phi}=0,$ (2.44)
$\displaystyle
G_{LR}^{\Phi\Phi}=G_{RL}^{\Phi\Phi}=-\frac{1}{N}\sum_{k=0}^{N-1}e^{\frac{2\pi
ik(m-n)}{N}}\hskip
2.84526pt\frac{\cos\left(\omega_{k}t\right)\sinh(2\alpha_{k})}{\lambda_{k}},$
(2.45) $\displaystyle
G_{LR}^{\Pi\Pi}=G_{RL}^{\Pi\Pi}=\frac{1}{N}\sum_{k=0}^{N-1}e^{\frac{2\pi
ik(m-n)}{N}}\hskip
2.84526pt\lambda_{k}\cos\left(\omega_{k}t\right)\sinh(2\alpha_{k}),$ (2.46)
$\displaystyle
G_{LR}^{\Phi\Pi}=G_{LR}^{\Pi\Phi}=G_{RL}^{\Phi\Pi}=G_{RL}^{\Pi\Phi}=\frac{1}{N}\sum_{k=0}^{N-1}e^{\frac{2\pi
ik(m-n)}{N}}\hskip 2.84526pt\sin\left(\omega_{k}t\right)\sinh(2\alpha_{k}),$
(2.47)

with $\alpha_{k}=\frac{1}{2}\log\coth(\frac{\beta\hskip
0.28453pt\omega_{k}}{4})$, $\lambda_{k}=\mathcal{L}\omega_{k}$ and
$\mathcal{L}$ denote the circumference of a circle. Note that $L$ or $R$ are
separately described by a thermal state so the associated blocks are time-
independent. But the crossing blocks, $LR$ and $RL$, correspond to
correlations between the left and right sides and are time-dependent. In the
next section, we will see that the time dependence of OEE and LN comes from
these blocks.

##  3 Time evolution of $S_{\text{OEE}}$ and $\mathcal{E}$

In this section, using the mentioned covariance matrix formalism, we will
investigate the time evolution of OEE and LN for the time-dependent $1+1$ TFD
state of a real scalar QFT regularized on a circular lattice. In above it is
shown that this state is just a product of TFD states for each of the
oscillator modes, (2.19). In order to evaluate these quantum information
quantities we will take a subregion of the entire quantum system that contains
a part in both the left and the right QFTs, see figure 1. Let us describe the
whole system as

$\displaystyle\bigg{(}A_{L}\cup B_{L}\bigg{)}\cup\bigg{(}A_{R}\cup
B_{R}\bigg{)},$ (3.1)

in the corresponding TFD state. According to figure 1, the $B_{L(R)}$ are the
complement of $A_{L(R)}$. Moreover, we decompose the region $A_{L(R)}$ into
two subregions $A_{L_{1}(R_{1})}$ and $A_{L_{2}(R_{2})}$ where they describe
two (in general disconnected) subregions on each side

$\displaystyle\bigg{(}(A_{L_{1}}\cup A_{L_{2}})\cup
B_{L}\bigg{)}\cup\bigg{(}(A_{R_{1}}\cup A_{R_{2}})\cup B_{R}\bigg{)}.$ (3.2)

We would like to study $S_{\text{OEE}}(\rho_{12}(t))$ and
$\mathcal{E}(\rho_{12}(t))$ between two non-complementary regions
$A_{1}=(A_{L_{1}}\cup A_{R_{1}})$ and $A_{2}=(A_{L_{2}}\cup A_{R_{2}})$.

$A_{L_{1}}$

$A_{R_{1}}$

$l_{1}$

$l_{1}$

$l_{2}$

$l_{2}$

$A_{L_{2}}$

$A_{R_{2}}$

$d$

$d$

Figure 1: Our setup for the decomposition of the system is depicted in this
figure. We have taken $N$ sites on each side and the total subsystem consists
of four spatially disconnected regions: two intervals on the left ($A_{L}$)
with the separation $d$ and the corresponding (identical) intervals and the
same separation on the right ($A_{R}$) with $N_{A}$ sites on each side.

The reduced covariance matrix $G_{12}$ can be obtained from the covariance
matrix (2.41) where each of the regions $(A_{L_{1}}\cup A_{L_{2}})$ and
$(A_{R_{1}}\cup A_{R_{2}})$ contain $N_{A}$ sites. Accordingly, $G_{12}$ is
decomposed into four blocks corresponding to $LR$ decomposition in position
space such that each block is a $2N_{A}\times 2N_{A}$ matrix. Two of these
blocks, $G_{12}^{LL}$ and $G_{12}^{RR}$, are time-independent and the time
dependence of the covariance matrix comes from mixed blocks, $G_{12}^{LR}(t)$
and $G_{12}^{RL}(t)$:

$G_{12}^{ab}(t)=\left(\begin{array}[]{cc}G_{12}^{LL}&G_{12}^{LR}(t)\par\vspace{.4cm}\\\
G_{12}^{RL}(t)&G_{12}^{RR}\\\ \end{array}\right)_{4N_{A}\times 4N_{A}},$ (3.3)

with

$\displaystyle
G_{12}^{LL}=\left(\begin{array}[]{cc}G_{LL}^{\Phi\Phi}&0\par\vspace{.4cm}\\\
0&G_{LL}^{\Pi\Pi}\\\ \end{array}\right)_{2N_{A}\times 2N_{A}}\hskip
28.45274ptG_{12}^{LR}(t)=\left(\begin{array}[]{cc}G_{LR}^{\Phi\Phi}&G_{LR}^{\Phi\Pi}\par\vspace{.4cm}\\\
G_{LR}^{\Pi\Phi}&G_{LR}^{\Pi\Pi}\\\ \end{array}\right)_{2N_{A}\times 2N_{A}}.$
(3.8)

In the above expression,
$\left\\{G_{LL}^{\Phi\Phi},G_{LL}^{\Pi\Pi},\dots\right\\}$ are given by the
equation (2.42) with $m,n$ restricted to the entangling region. In addition,
$G_{12}^{RR}$, $G_{12}^{RL}(t)$ blocks can be obtained by changing $L$ by $R$
in (3.8). Once the covariance matrix $G_{12}^{ab}(t)$, (3.3), is found one can
express the $S_{\text{EE}}(\rho_{12}(t))$ in terms of eigenvalues $(\nu_{i})$
of the symplectic form

$\displaystyle J=i\hskip 1.42262pt\Omega_{A}^{-1}G_{12}(t).$ (3.9)

Doing so, one obtains [62, 63] 888We are taking the absolute value of the
eigenvalues of the symplectic form. Since these eigenvalues come in pairs, we
should include a factor of $\frac{1}{2}$ in evaluating the entanglement
entropy.:

$S_{\text{EE}}\left(\rho_{12}(t)\right)=\frac{1}{2}\sum_{i=1}^{4N_{A}}s_{\text{EE}}(|\nu_{i}|),$
(3.10)

where,

$s_{\text{EE}}(\nu_{i})=\left(\frac{\nu_{i}+1}{2}\right)\log\left(\frac{\nu_{i}+1}{2}\right)-\left(\frac{\nu_{i}-1}{2}\right)\log\left(\hskip
1.42262pt\frac{\nu_{i}-1}{2}\hskip 1.42262pt\right).$ (3.11)

It is useful to mention that the Rényi entropies $S_{n}$ for $n>0$ can be
computed by replacing the $s_{\text{EE}}(\nu_{i})$ in (3.10) with
$s_{n}(\nu_{i})$ defined as

$s_{n}(\nu_{i})=\frac{1}{n-1}\log\left[\frac{\left(\nu_{i}+1\right)^{n}-\left(\nu_{i}-1\right)^{n}}{2^{n}}\right].$
(3.12)

The entanglement entropy (3.10) can be recovered in the limit $n\rightarrow
1$. In addition, the second Rényi entropy is $S_{2}=\frac{1}{2}\log\det(G)$.
Now, in order to compute $S_{\text{OEE}}$, we must take a partial transpose
with respect to momentum degrees of freedom in the $A_{2}$ subregion [33, 34].
This can be accomplished by acting with the time-reversal operator
$\mathcal{R}_{A_{2}}$ on each block of $G_{12}^{ab}$. For example

$\tilde{G}^{LL}_{12}=\mathcal{R}_{A_{2}}.G_{12}^{LL}.\mathcal{R}_{A_{2}},$
(3.13)

with $\mathcal{R}_{A_{2}}$ given by a $2N_{A}\times 2N_{A}$ square matrix:

$\mathcal{R}_{A_{2}}=\text{diag}\\{1,1,\cdots,1,-1,\cdots,-1\\}.$ (3.14)

The number of “$-1$” elements in the above expression is equal to the length
of the subregion $A_{L_{2}}$. It is worth mentioning that the operator
$\mathcal{R}_{A_{2}}$ is the same for the other blocks,
$G_{12}^{LR}(t),G_{12}^{RL}(t)$ and $G_{12}^{RR}(t)$. This is because we are
considering the simplest case here in which the $L$ and $R$ blocks are exactly
the same. Now, by having $\tilde{G}^{ab}_{12}$, one can compute eigenvalues
$(\tilde{\nu}_{i})$ of partial transposed symplectic form

$\displaystyle\tilde{J}=i\hskip
1.42262pt\Omega_{A}^{-1}\tilde{G}_{A_{1}A_{2}}.$ (3.15)

Having these, the odd entanglement entropy, $S_{\text{OEE}}$, becomes [34]

$S_{\text{OEE}}\left(\rho_{12}(t)\right)=\frac{1}{2}\sum_{i=1}^{4N_{A}}\tilde{s}_{\text{odd}}(|\tilde{\nu}_{i}|),$
(3.16)

with

$\tilde{s}_{\text{odd}}(\tilde{\nu}_{i})=\left(\frac{\tilde{\nu}_{i}+1}{2}\right)\log\left(\frac{\tilde{\nu}_{i}+1}{2}\right)-\text{sgn}\left(\frac{\tilde{\nu}_{i}-1}{2}\right)\bigg{|}\frac{\tilde{\nu}_{i}-1}{2}\bigg{|}\log\left(\hskip
1.42262pt\bigg{|}\frac{\tilde{\nu}_{i}-1}{2}\bigg{|}\hskip 1.42262pt\right).$
(3.17)

It is worth emphasizing that the partial transposition is not a completely
positive map and the appearance of negative eigenvalues is a sign of quantum
entanglement[31]. This is the main reason to consider the absolute value in
(3.17) in comparison with (3.11). Moreover, by knowing the eigenvalues
$\tilde{\nu_{i}}$, the trace norm of the reduced density matrix
$\rho_{12}^{T_{2}}$ becomes [34, 36]999The out-of-equilibrium dynamics of the
negativity after a _different_ quench has been studied previously in several
works[65, 66, 67, 68, 43, 69, 27, 28].

$\displaystyle\text{Tr}|\rho_{12}^{T_{2}}|=\prod_{i=1}^{4N_{A}}\left[|\frac{\tilde{\nu}_{i}+1}{2}|-|\frac{\tilde{\nu}_{i}-1}{2}|\right]^{-1}=\prod_{i=1}^{4N_{A}}\text{max}\left(1,\frac{1}{\tilde{\nu}_{i}}\right),$
(3.18)

which implies that the LN becomes101010The factor $1/2$ comes from the fact
that we are taking the absolute values of eigenvalues which come in pairs.

$\displaystyle\mathcal{E}(A_{1},A_{2})=-\frac{1}{2}\sum_{i=1}^{4N_{A}}\log\left[\text{min}\left(1,\tilde{\nu}_{i}\right)\right].$
(3.19)

According to the above relation, only the symplectic eigenvalues
$\tilde{\nu}<1$ contribute to the LN. Hence, in order to have quantum
correlation we must have at least one symplectic eigenvalue which is less than
one. Note that the LN is a relative entanglement measure, hence, it is
symmetric with respect to the exchange of subsystems.

Before closing this section, let us clarify some points that will be useful
for interpreting the numerical results presented in the next section. The
entanglement entropy enjoys several properties. One of them is known as
subadditivity. With respect to the decomposition

$A=(A_{L_{1}}\cup A_{L_{2}})\cup(A_{R_{1}}\cup A_{R_{2}}),$ (3.20)

the entanglement entropy satisfies

$S_{\text{EE}}(\rho_{A}(t))\leqslant
S_{\text{EE}}(\rho_{A_{L}}(t))+S_{\text{EE}}(\rho_{A_{R}}(t)),$ (3.21)

in which, $A_{L(R)}=A_{L_{1}(R_{1})}\cup A_{L_{2}(R_{2})}$. The subadditivity
provides a time-independent upper bound for the entanglement entropy. The
aforementioned decomposition results in a block structure in the reduced
covariance matrix $G_{A}$ which consists of four blocks corresponding to $LR$
decomposition in the position space. The time-independent blocks,
$G_{12}^{LL}$ and $G_{12}^{RR}$, are related to thermal entropy corresponding
to reduced thermal density matrix of each individual side, $(A_{L_{1}}\cup
A_{L_{2}})$ and $(A_{R_{1}}\cup A_{R_{2}})$. Since intervals are equal and
symmetric, these thermal entropies are equal. Therefore, the time-independent
upper bound is twice the thermal entropy. This means that with respect to the
bound (3.21), the growing behavior of the entanglement entropy over a large
range of times ultimately terminates in twice the thermal entropy.

There is also another inequality that is specific to the Gaussian states [53],

$S_{\text{EE}}(\rho_{A}(t))\leqslant S_{2}(\rho_{A}(t))+2N_{A}(1-\log 2).$
(3.22)

In the above inequality, $N_{A}$ denotes the number of bosonic degrees of
freedom associated with each side of the TFD state in the subregion $A$ and
$S_{2}(\rho_{A_{L}}(t))=\frac{1}{2}\log\det G_{A}(t)$. The inequality (3.22)
can be derived using the below expressions for entanglement entropy and second
Rényi entropy,

$S_{\text{EE}}(\rho_{A}(t))=\frac{1}{2}\sum_{i=1}^{4N_{A}}s(\nu_{i}),\qquad
S_{2}=\frac{1}{2}\sum_{i=1}^{4N_{A}}\log(\nu_{i}),$ (3.23)

where $s({\nu}_{i})$ is given by (3.11). According to (3.11), for all
$\nu\geqslant 1$, we have

$\log(\nu)\leqslant s(\nu)\leqslant\log(\nu)+(1-\log 2).$ (3.24)

Therefore, by summing over all eigenvalues we can conclude that the second
Rényi entropy provides an upper bound for entanglement entropy of Gaussian
states. We will especially use the inequality (3.22) in the analysis of zero-
mode in section 5.

##  4 Numerical results

In this section, we will present our numerical results for the time dependency
of OEE and LN based on equations (3.16) and (3.19). These results are
expressed in terms of two dimensionless parameters $m\mathcal{L}$ and
$\beta/\mathcal{L}$ where $m$ refers to the mass parameter in (2.1),
$\mathcal{L}$ is the length of the circular lattice with lattice spacing
$\delta$, $\mathcal{L}=N\delta$, and $\beta$ is the inverse temperature. Let
us remind that we are decomposing the circular lattice according to (3.2). The
subsystem $A$ is further decomposed into two subsystems $A_{1}$ and $A_{2}$
with length $l_{1}$ and $l_{2}$ respectively:

$\displaystyle N_{A}=N_{A_{1}}+N_{A_{2}}=(l_{1}+l_{2})/\delta=l/\delta.$ (4.1)

Since we are interested in describing the TFD state (for single-mode see
(2.19)), one has two copies of this decomposition: one for the left QFT and
one for the right QFT where on each side we have $N_{A}$ sites (See figure 1).
In what follows, we will study different cases for massless as well as massive
real scalar QFTs. We consider the behavior of OEE and LN by changing $N_{A}$
($N_{A_{1}}$ or $N_{A_{2}}$ or both of them), the separation distance $d$
(between $A_{1}$ and $A_{2}$ on each side), lattice spacing $\delta$ and
inverse temperature $\beta$. The obtained results will also be compared with
their counterparts for the entanglement entropy. To do so, we will discretize
the circular lattice with circumference $\mathcal{L}$ into $N=2n+1$ sites
(almost $1501$) and the length of intervals $l_{1}$ and $l_{2}$ on each side
vary from $0.1\mathcal{L}$ to $0.5\mathcal{L}$. Before going to the details of
numerical results, it is worth to explain why we choose the lattice cites an
odd number. It is clear from (2.3) that $\mathbf{P}_{a}$ and $\mathbf{Q}_{a}$
are real degrees of freedom., i.e. $\mathbf{P}_{a}^{\dagger}=\mathbf{P}_{a}$
and $\mathbf{Q}_{a}^{\dagger}=\mathbf{Q}_{a}$. But when we pass to the normal
mode basis by Fourier transforming, the $\tilde{\mathbf{P}}_{k}$ and
$\tilde{\mathbf{Q}}_{k}$ are no longer real with the exceptions of $k=0$ and
$k=N/2$ (for even $N$). To restore (each) two real degrees of freedom, we have
imposed the constraints
$\tilde{\mathbf{P}}_{k}^{\dagger}=\tilde{\mathbf{P}}_{-k}$ and
$\tilde{\mathbf{Q}}_{k}^{\dagger}=\tilde{\mathbf{Q}}_{-k}$ which imply that
the negative and positive momentum modes will be mixed,
$[\tilde{\mathbf{Q}}_{k},\tilde{\mathbf{P}}_{-l}]=i\delta_{k,l}$. Accordingly,
to avoid over-counting (constraining) those two excepted modes, we choose
$N=2n+1$ for all calculations.

Let us first consider the case in which the two intervals $A_{1}$ and $A_{2}$
on each side are adjacent to each other, i.e., the separation distance
vanishes, $d=0$. In figure 2, we investigate the time dependence of
entanglement entropy $S_{\text{EE}}$(upper plots), odd entanglement entropy
$S_{\text{OEE}}$(middle plots) and logarithmic negativity $\mathcal{E}$
(bottom plots) for short-time (left plots) and long-time (right plots) scales
in which the initial value is subtracted and normalized with respect to the
thermal entropy $S_{\text{th}}$111111For a real scalar QFT, the thermal
entropy in the continuum limit is given by  $\displaystyle
S_{\text{th}}=\mathrm{vol}\int\frac{\mathrm{d}^{d-1}k}{(2\pi)^{d-1}}\bigg{[}\frac{\beta(\omega_{k})}{e^{\beta(\omega_{k})}-1}-\log(1-e^{-\beta(\omega_{k})})\bigg{]}.$
. We take $N=1501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+a$, where
$a=21,31,41$ (dashed blue, orange, dashed green, respectively), and
$m\mathcal{L}=10^{-3}$, $\beta=10^{-2}\mathcal{L}$. For short-times (i.e.
times smaller than $\mathcal{L}/2$ where the system is not sensitive to finite
size effects) the growth of $S_{\text{EE}}$ as well as $S_{\text{OEE}}$ is
linear and lasts until approximately $t\sim l$; this linear growth is then
followed by saturation. For long-times (i.e. times larger than $\mathcal{L}-l$
where the finite size effects can be visible) the time dependency of
$S_{\text{EE}}$ (upper-right panel) as well as $S_{\text{OEE}}$ (middle-right
panel) is periodic with periodicity $\mathcal{L}$. To be more precise, the
pattern of the time evolution of $S_{\text{EE}}$, as well as $S_{\text{OEE}}$,
is consisting of linear growth for early times, $t\sim l$, a quasi-plateau of
width approximately $\mathcal{L}-2l$ in the intermediate times and a linear
decreasing up to order $t\sim l$ and then repeating the same structure121212
Similar revivals of quantum states after _different_ quantum quench has been
already seen [70, 71, 72].. The slope of linear growth, for both of these
quantities, is equal with good accuracy to two times of thermal entropy
density of each copy at inverse temperature $\beta$. This matches with (1.5),
since in massless theory the group velocity is the same for all species of
quasi-particles and the slope becomes the sum of entropy densities (1.6) over
all species. We have said the quasi-plateau since for these flat regimes we
observe a logarithmic growth. In section 5, we explicitly show that this non-
trivial growth is due to the zero momentum mode. It is also worth mentioning
that, upon a closer examination of figure 2, the linear regime does not start
right away and a different behavior can be seen around $t=0$ which we believe
is a manifestation of an expected quadratic growth at early times following a
quantum quench. The time dependence of LN is depicted in the bottom plots for
$a=21,31,41$ (dashed blue, orange, dashed green curves, respectively). In
early times (the bottom-left panel), we observe growth and then decrease
followed by saturation. The oscillatory behavior can be observed for long
times (the bottom-right panel) due to the finite size effects. This can also
be understood by a quasi-particle picture where meeting quasi-particles on the

$\frac{S_{\text{EE}}(t)-S_{\text{EE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{EE}}(t)-S_{\text{EE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{OEE}}(t)-S_{\text{OEE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{OEE}}(t)-S_{\text{OEE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

Figure 2: The time evolution of $S_{\text{EE}}$(upper plot), $S_{\text{OEE}}$
(middle plot) and $\mathcal{E}$ (bottom plot) for various regimes in which the
initial values are subtracted and normalized with respect to the thermal
entropy $S_{\text{th}}$. We take $N=1501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+a$ where
$a=21,31,41$ for dashed blue, orange, dashed green curves respectively and
$m\mathcal{L}=10^{-3}$, $\beta=10^{-2}\mathcal{L}$. The upper-left plot
denotes the growth of $S_{\text{EE}}$ which is linear at times approximately
equal to the size of an interval. The upper-right plot is for longer periods
of time. One can see the finite size effects which induce a periodic behavior
with periodicity $\mathcal{L}$. We see a logarithmic growth due to the
presence of a zero-mode which we study further analytically in the next
section. The middle-left and right panels denote the time dependency of
$S_{\text{OEE}}$. The effective time evolution of $S_{\text{EE}}$ and
$S_{\text{OEE}}$ are the same. The bottom-left plot denotes the short-time
behavior of $\mathcal{E}$ for $a=21$ (dashed blue), $31$(orange), $41$(dashed
green). In early times we see growth and then reduction with respect to the
initial value of LN. The bottom-right plot is for a long time regime, in which
the finite size effect exhibit itself as periodic behavior.

opposite side of the circle leads to reducing the correlations between the
subsystem and its complement.

An important feature of $\mathcal{E}$ is the sudden decreasing of
entanglement. It is worth mentioning that one should not worry about growing
behavior over a large range of times for $S_{\text{EE}}$ since according to
(3.21) we know that this growth ultimately terminates. The similar behavior of
$S_{\text{OEE}}$ over long times is the sign that this quantity also has a
subadditivity characteristic. In figure 3, we consider the time dependence of
$\Delta S=S_{\text{OEE}}-S_{\text{EE}}$ which is normalized with respect to
thermodynamic entropy $S_{\text{th}}$ and subtracted from initial value for
massless theory $m\mathcal{L}=10^{-3}$. Accordingly, $\Delta S(t)-\Delta S(0)$
initially increases and then decreases followed by saturation. This is same as
the behavior (qualitatively) for logarithmic negativity $\mathcal{E}$ (figure
2). In the right panel, the long time behavior is plotted and the finite size
effect can be seen.

$\frac{\Delta S(t)-\Delta S(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{\Delta S(t)-\Delta S(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

Figure 3: Time dependence of $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$
normalized by the thermodynamic entropy, $S_{\text{th}}$, for the free scalar
theory where $N=1501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+a$, with
$a=21$(dashed blue), $a=31$(orange), and $a=41$(dashed green) and
$m\mathcal{L}=10^{-3}$, $\beta=10^{-2}\mathcal{L}$. Left panels correspond to
short-time scales and the right panels are for long-time scales.

In the figure 4, we explore the decompactification limit (equivalently
continuous limit) when theory lives on a line instead of a circle131313We
would like to thank Erik Tonni for discussion about QFT limit.. For this
purpose, we fixing the lattice spacing $\delta$, mass $m$, and inverse
temperature $\beta$ while increasing the total number of lattice sites $N$. We
observe a time delay to see the oscillatory behavior for the larger $N$. In
another words, finite size effects are pushed to later and later times by
increasing the radius of circle on which theory lives. Moreover, we observe
that the coefficient of logarithmic growth becomes smaller by increasing the
value of $N$. This is consistent with the result of next section since for the
fixed mass and fixed lattice spacing and increasing the system size we expect
to only see the initial contribution in (5.29) where the coefficient of
logarithmic growth changes from $1$ to $1/2$. The existence of this long-lived
logarithmic growth comes back to the non-local nature of the zero-mode and it
is related to the periodic boundary condition on a circle. This means that the
logarithmic growth will be absent where the translational invariance is
broken. This happens for example for Dirichlet instead of periodic boundary
condition.

$\frac{S_{\text{EE}}(t)-S_{\text{EE}}(0)}{S_{\text{th}}}$

$t/\beta$

$\frac{S_{\text{OEE}}(t)-S_{\text{OEE}}(0)}{S_{\text{th}}}$

$t/\beta$

$\frac{\Delta S(t)-\Delta S(0)}{S_{\text{th}}}$

$t/\beta$

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}}$

$t/\beta$

Figure 4: The dynamics of $S_{\text{EE}}$(upper-left), $S_{\text{OEE}}$(upper-
right), $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$(bottom-left), and
$\mathcal{E}$(bottom-right) (subtracted from the initial values and normalized
with thermal entropy) extrapolated from circle to a line with fixed lattice
spacing $\delta$ and $m\beta=10^{-5}$. The blue curve denotes the theory on a
circle with the total number of sites $N=1501$ and
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=2+20$. The dashed-
orange curve represents the same theory on a circle with fixing $\delta$ and
increasing the total number of sites (and accordingly subsystem sites) to
$N=3751$. Increasing $N$ causes the effects of finite size to be transferred
to larger times. We also observe the logarithmic growth at the intermediate
times.

In the following, we extend the analysis to the case of two disjoint intervals
on each side with a separation $d$. Moreover, the numerical results about
changing the inverse temperature $\beta$ are provided in appendix A. In figure
5 we investigate the time evolution of the normalized entanglement entropy
$S_{\text{EE}}$ (upper plots) and odd entanglement entropy $S_{\text{OEE}}$
(lower plots) which are subtracted from the initial values for short-times
(left plots) and long-times (right plots) for different values of $d$. We take
$N=1501$, $m\mathcal{L}=10^{-3}$, $\beta=10^{-2}\mathcal{L}$, and
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+31$. For short-
times (times smaller than $\mathcal{L}/2$ and in the absence of finite size
effects), the growth of the $S_{\text{EE}}$, as well as $S_{\text{OEE}}$, is
linear and lasts until approximately $t\sim l$; this is followed by
saturation. For long-times (times larger than $\mathcal{L}-l$ where the finite
size effects can be visible), the time dependence of $S_{\text{EE}}$ (up-right
panel), as well as $S_{\text{OEE}}$ (bottom-right panel), is periodic with
periodicity $\mathcal{L}$. Clearly, the pattern of evolution for
$S_{\text{EE}}$, as well as $S_{\text{OEE}}$, is: linear growth for early
times, $t\sim l$, a plateau of width approximately $\mathcal{L}-2l$ in a
middle timed and then a linear decrease up to order $t\sim l$. The oscillatory
behavior can be observed for long times due to the finite size effect. For the
massive theory, the similar results are presented in figure 6. It is clear
that for large masses, the oscillations are rather irregular due to the
dephasing of the different kinds of quasi-particles with different group
velocities. This dephasing also causes that the slope of linear growth at
early times becomes significantly less than $2S_{\text{th}}/\mathcal{L}$.
Another effect of changing mass can be seen in this figure where both OEE and
EE fluctuate around the same value albeit we have increased the distance $d$.
Of course this can be also understood by the quasi-particle picture where the
entanglement increases, in general, by increasing the distance $d$ while it
decreases by increasing the IR regulator mass $m$.

$\frac{S_{\text{EE}}(t)-S_{\text{EE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{EE}}(t)-S_{\text{EE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{OEE}}(t)-S_{\text{OEE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{OEE}}(t)-S_{\text{OEE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

Figure 5: Time dependence of $S_{\text{EE}}$ (upper panels), $S_{\text{OEE}}$
(bottom panels) is plotted for the free massless scalar theory when $N=1501$ ,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+31$ and
$m\mathcal{L}=10^{-3}$, $\beta=10^{-2}\mathcal{L}$. The curves correspond to
the $d=0$(dashed blue), $d=10$(orange), and $d=100$(dashed green). The upper-
left plot denotes the linear regime for the growth of entanglement entropy.
The linear growth is for times approximately equal to the size of an interval.
The upper-right plot is for a longer period of time for which the finite size
effects induce a periodic behavior with periodicity $\mathcal{L}$. The lower-
left and right panels denote time behavior of odd entanglement entropy. The
effective evolution of both $S_{\text{EE}}$ and $S_{\text{OEE}}$ are the same
in different times.

$\frac{S_{\text{EE}}(t)-S_{\text{EE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{EE}}(t)-S_{\text{EE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{OEE}}(t)-S_{\text{OEE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{S_{\text{OEE}}(t)-S_{\text{OEE}}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

Figure 6: Time dependence of $S_{\text{EE}}$ (upper panels), $S_{\text{OEE}}$
(bottom panels) is plotted for the free massive scalar theory when $N=1501$ ,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+31$ and
$m\mathcal{L}=10^{2}$, $\beta=10^{-2}\mathcal{L}$. The curves correspond to
the $d=0$(dashed blue), $d=10$(orange), and $d=100$(dashed green). The upper-
left plot denotes the linear regime for the growth of entanglement entropy.
The linear growth is for times approximately equal to the size of an interval.
The upper-right plot is for a longer period of time for which the finite size
effects induce a periodic behavior with periodicity $\mathcal{L}$. The lower-
left and right panels denote time behavior of odd entanglement entropy. The
effective evolution of both $S_{\text{EE}}$ and $S_{\text{OEE}}$ are the same
in different times.

Figure 7 illustrates the time evolution of the $\Delta
S=S_{\text{OEE}}-S_{\text{EE}}$ (left panel) and the logarithmic negativity,
$\mathcal{E}(t)$ (right panel) which are normalized with respect to
thermodynamic entropy $S_{\text{th}}$ and subtracted from their initial
values. This figure together with figures 2, 3, 4 and figures 13 and 14 in
appendix A contain an interesting result: _At least for free scalar $QFTs$_,
the $\Delta S$ and $\mathcal{E}$ are qualitatively the same independent of the
mass $m$, lattice spacing $\delta$, separation distance $d$ and inverse
temperature $\beta$141414For the disjoint intervals, we also checked the
entanglement dynamics in the decompactification limit (or equivalently in the
continuum limit): Similar to the adjacent case, the effect of increasing $N$
appears in changing of oscillation period on the circle to larger times. For
$S_{\text{EE}}$ and $S_{\text{OEE}}$, the coefficient of logarithmic growth
also decreases by increasing the number of total sites $N$.. These
observations has an interesting consequence. It is well-known that [65, 73]
the logarithmic negativity is just a measure of entanglement (quantum
correlation) between the sub-systems $A_{1}$ and $A_{2}$ and the quasi-
particles between $(A_{1}+A_{2})$ and $B$ (figure 1) do not contribute to it.
To be more precise, all moments of partial transpose quasi-particles between
$(A_{1}+A_{2})$ and $B$ matter, but they cancel when taking the replica
limit[74]151515We would like to thank Pasquale Calabrese for illuminating this
point and also discussing our results.. In contrast, the entanglement entropy
is just affected by quasi-particles between $(A_{1}+A_{2})$ and $B$. The time
dependency of odd entanglement entropy is _similar_ to the entanglement
entropy but when we subtract it from entanglement entropy, interestingly, it
behaves as logarithmic negativity. This implies that $S_{\text{OEE}}$ not only
has a contribution from the quasi-particles between $A_{1}$ and $A_{2}$ but
also has contribution from quasi-particles between $(A_{1}+A_{2})$ and $B$ in
contrast to logarithmic negativity161616Of course this should be checked for a
generic quantum state.. Another interesting witness for this interpretation
comes from figures 5 and 6. Accordingly, $S_{\text{OEE}}$, in general,
increases by increasing the distance $d$ but the situation is different for
$S_{\text{EE}}$. By increasing $d$, the $S_{\text{EE}}$ firstly decreases but
then increases. According to the above interpretation we can understand these
different behaviors in the following way: For each pair of quasi-particles in
the region outside the $(A_{1}+A_{2})$, two events are possible: one quasi-
particle travels to the region $A_{1}$ (or $A_{2}$) and another one remains in
the outside region or one quasi-particle travels to the region $A_{1}$ and
another one to the region $A_{2}$. The occurrence of the first event increases
both the $S_{\text{OEE}}$ and the $S_{\text{EE}}$. But the occurrence of the
second event increases the $S_{\text{OEE}}$ but it decreases the
$S_{\text{EE}}$. Therefore, $S_{\text{OEE}}$ is almost increasing but the
behavior of $S_{\text{EE}}$ crucially depends on the distance $d$ since the
second event is more likely to occur for shorter distances $d$.

$\frac{\Delta S(t)-\Delta S(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{\Delta S(t)-\Delta S(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

Figure 7: The up-left and up-right plots denote the long-time behavior of
$\Delta S=S_{\text{OEE}}-S_{\text{EE}}$ and $\mathcal{E}(t)$, respectively,
for massless free scalar theory. The two figures in bottom are for the massive
case. These are normalized by the thermodynamic entropy $S_{\text{th}}$ when
$N=1501$, $N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+31$,
$\beta=10^{-2}\mathcal{L}$ and $m\mathcal{L}=10^{-3}$ for massless theory,
$m\mathcal{L}=10^{2}$ for massive one. The lines denote $d=0$ (dashed blue),
$d=10$ (orange), and $d=100$ (dashed green).

It is worth to mention that figure 7 contains an interesting information in
very short times which is provided by figure 8. It is clear that the LN is
zero for times $t<d/2$ and after that, it begins to grow linearly. This is in
agreement with [43, 65]. A similar delay to start linear growth of $\Delta
S=S_{\text{OEE}}-S_{\text{EE}}$ can also be seen in figure 7.

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}}$

$t/\mathcal{L}$

Figure 8: Time dependence of logarithmic negativity normalized by the
thermodynamic entropy $S_{\text{th}}$, for the free scalar theory where
$N=1501$, $N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=20+31$, and
$\beta=10^{-2}\mathcal{L}$ with $d=0$(dashed blue), $d=10$(orange) and
$d=100$(dashed green). Left panel corresponds to massless theory with
$m\mathcal{L}=10^{-3}$ and the right one is for massive theory with
$m\mathcal{L}=10^{2}$.

For further investigation, we would like to study the time evolution of
logarithmic negativity $\mathcal{E}$, mutual information $I$, $1/2$-Rényi
mutual information $I^{(1/2)}$ and explore their similarities and differences.
The numerical results for two adjacent intervals are depicted in figure 9. We
take $\beta=10^{-2}\mathcal{L}$ and $m=10^{-3}/\mathcal{L}$. The top panel
denotes the time evolution of $\mathcal{E}$ and $I$ for long times and
symmetric lengths for two subsystems. The middle panel denotes the same
quantities but with non-symmetric lengths for subsystems. In the symmetric
case, at early times, $\mathcal{E}$ and $I$ have a very similar behavior:
initial linear growth followed by an almost linear decreasing up to time
$t\sim l_{1}(\equiv l_{A_{1}})$. In this case, on time scales of the order of
the system’s size a difference between these quantities appears. In these time
scales, the mutual information reaches a plateau while the logarithmic
negativity decreases monotonically until the appearance of finite size
effects. This observation implies that the decreasing behavior is a
peculiarity of the entanglement and is not reflected by the correlation
measures such as mutual information. Moreover, in the decompactification limit
(increasing the system and subsystem sizes properly) the rate of decreasing of
logarithmic negativity becomes more sharply which it might be the sign of
_sudden death of entanglement_ before the trivial (finite size effect)
revival171717The same phenomenon is explored in different quench setup [65]
and for other entanglement measures [75]. Of course it might be the lattice
effect and will absent in true _continuous_ QFT [76].. Overall, the non-
symmetric case has the same characteristics but with a difference which is the
appearance of a (narrow) plateau after the first linear growth, for both the
logarithmic negativity and mutual information. All these are in agreement with
the results of [65] which is in a different setup. The bottom plot denotes the
time evolution of $\mathcal{E}$(solid curves) and $1/2$-Rényi mutual
information $I^{(1/2)}$ (dotted-dashed curves). The initial growth which is
linear and then decreasing followed by saturation is approximately the same
between them which is in agreement with [43]. This similar behavior implies an
interesting unification of these two seemingly different information-theoretic
quantities in integrable models which it might break down in chaotic theories.

$\mathcal{E}(t)/(l_{1}+l_{2}),I(t)/(l_{1}+l_{2})$

$t/\mathcal{L}$

$\mathcal{E},N=501,l_{1}=l_{2}=15$

$I,N=501,l_{1}=l_{2}=15$

$\mathcal{E},N=1001,l_{1}=l_{2}=30$

$I,N=1001,l_{1}=l_{2}=30$

$\mathcal{E}(t)/(l_{1}+l_{2}),I(t)/(l_{1}+l_{2})$

$t/\mathcal{L}$

$\mathcal{E},N=501,l_{1}+l_{2}=10+20$

$I,N=501,l_{1}+l_{2}=10+20$

$\mathcal{E},N=1001,l_{1}+l_{2}=20+40$

$I,N=1001,l_{1}+l_{2}=20+40$

$\frac{\mathcal{E}(t)-\mathcal{E}(0)}{S_{\text{th}}},\frac{I^{1/2}(t)-I^{1/2}(0)}{2S_{\text{th}}}$

$t/\mathcal{L}$

$\mathcal{E},l_{1}=l_{2}=30$

$I^{1/2},l_{1}=l_{2}=30$

$\mathcal{E},l_{1}=l_{2}=35$

$I^{1/2},l_{1}=l_{2}=35$

Figure 9:  Time dependency of $\mathcal{E}$ and mutual information $I$ are
presented in top and middle panels. The top one is for symmetric subsystem
configuration and the middle one is for non-symmetric case. The bottom panel
demonstrates the time evolution of $\mathcal{E}$(solid curves) and $1/2$-Rényi
mutual information $I^{(1/2)}$ (dotted-dashed curves) for $N=501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=30+30$ (blue) and
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=35+35$ (red) which are
subtracted from zero value and normalized by the thermal entropy
$S_{\text{th}}$.

Before closing this section, it is worth studying several proposed inequality
for odd entanglement entropy $S_{\text{OEE}}$[30]. We can check numerically
some of these inequalities:181818We would like to thank Kotaro Tamaoka for
pointing out the last inequality to us.

  * •

$S_{\text{OEE}}(A:B)\geq 0$ (positive semi-definiteness)

  * •

$S_{\text{OEE}}(A:B_{1}B_{2})\geq S_{\text{OEE}}(A:B_{1})$ (monotonicity)

  * •

$S_{\text{OEE}}(A_{1}:A_{2})\geq\text{max}[S_{\text{EE}}(A_{1}),S_{\text{EE}}(A_{2})]$

According to the results of this section, the positive semi-definiteness
clearly is established. Monotonicity relation means that by enlarging one of
two subsystems, the total amount of correlations between the two subsystems
increases. According to figure 2, it is clear that both of $S_{\text{EE}}$ and
$S_{{\text{OEE}}}$ satisfies the monotonicity relation. In figure 10, we can
also simply see that the $S_{\text{OEE}}$ satisfies the last inequality. In
this figure, $S_{\text{EE}}(t)$ refers to the greater one between
$S_{\text{EE}}(A_{1})$ and $S_{\text{EE}}(A_{2})$.

$S_{\text{OEE}}(t),S_{\text{EE}}(t)$

$t/\mathcal{L}$

$S_{\text{OEE}}(t),S_{\text{EE}}(t)$

$t/\mathcal{L}$

Figure 10: The $S_{\text{EE}}$ (blue) and $S_{\text{OEE}}$ (orange) are
presented for $N=501$, $m\mathcal{L}=10^{-3}/$, $d=0$,
$\beta=10^{-2}\mathcal{L}$. In the left plot,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=35+35$ and in the
right plot $N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=5+65$.

##  5 Logarithmic growth at intermediate times

In the previous section, we have observed the existence of a logarithmic
growth regime for odd entanglement entropy, instead of a sharp saturation, at
intermediate times. In the case of entanglement entropy of a pure state, this
behavior was observed previously[9, 54, 13]. This behavior, as we will discuss
more in the following, is due to the presence of a zero-mode, namely the
momentum mode with $k=0$ in the massless limit. It is worth emphasizing that
this gapless zero-mode does not lead to a ballistic propagation as in the
quasi-particle picture, but it instead has a diffusive nature[54]191919The
authors have considered both a boundary state quench and a global mass quench.
For other relevant zero-mode effect discussions see refs[77, 78].. The authors
of [13] have studied the logarithmic contribution based on analyzing the
entanglement entropy for a single degree of freedom on each side of TFD in the
massless limit, i.e., $m\rightarrow 0$, and found that

$\displaystyle S_{\text{EE}}(t)\hskip
2.84544pt\sim\left\\{\begin{array}[]{rl}\frac{1}{2}\log(t/\mathcal{L})&\hskip
14.22636pt\delta\ll t<\mathcal{L},\\\ \log(t/\mathcal{L})&\hskip
14.22636pt\mathcal{L}<t\ll m^{-1},\\\ \log|\sin(mt)|&\hskip
14.22636pt\mathcal{L}<t.\end{array}\right.$ (5.4)

In order to proceed for odd entanglement entropy, we will follow [13] and
determine the consequences of the existence of the zero-mode analytically. We
will take a single site in each entangling region, and derive the full
asymptotic behavior of odd entanglement entropy in the limit
$\beta,m^{-1}\ll\mathcal{L}$, which will turn out to be the same as (5.4).
When we extend the analysis for logarithmic negativity, no logarithmic
correction will be observed. It is worth noting that in order to make the
study of this logarithmic behavior feasible, we will focus on a subsystem
consisting of a single lattice site on each side of the TFD state.
Accordingly, we will intentionally suppress the linear regime by choosing a
subsystem size that vanishes in the continuum limit. Then, we will extend the
observed asymptotic behavior to the case of larger subsystems due to the fact
that the zero-mode is completely non-local and therefore affects local
subsystems in a similar way, regardless of their size. We should also
emphasize that this expression will fail to describe the entanglement
accurately in the regime $t<\delta$ in which the precise form of oscillations
of various modes will determine the time evolution.

For highly entangled states, the von Neumann entropy
$S_{\text{EE}}(\rho_{A_{12}}(t))$ can be approximated as[53, 55]202020The
error scales as $\exp(-2S_{2})$. Therefore, it decreases exponentially for
highly entangled states[13].

$S_{\text{EE}}(\rho_{A_{12}}(t))\sim
S_{2}(\rho_{A_{12}}(t))+2N_{A_{12}}(1-2\log 2),$ (5.5)

where $S_{2}(\rho_{A_{12}}(t))$ is the second Rényi entropy,

$S_{2}(\rho_{A_{12}}(t))=\frac{1}{2}\log(\det(G_{A_{12}}(t))).$ (5.6)

Using (3.17), we can find a similar upper bound formula for the odd
entanglement entropy. Actually, instead of $G_{A_{12}}(t)$ we use
$\tilde{G}_{A_{12}}(t)$ and approximate the $S_{\text{OEE}}$ for a single-mode
on each side with the modified second Rényi entropy, $\tilde{S_{2}}$. As we
discussed in section 3, in order to find $\tilde{G}^{ab}_{12}$, one first need
to find the covariance matrix, $G_{12}^{ab}$, and then take a partial
transpose with respect to the momentum degrees of freedom in the $A_{2}$
subregion. This can be accomplished by acting with the time-reversal operator
$\mathcal{R}_{A_{2}}$,

$\tilde{G}^{ab}_{12}=\mathcal{R}_{A_{2}}.G_{12}^{ab}.\mathcal{R}_{A_{2}},$
(5.7)

where $\mathcal{R}_{A_{2}}$ is a square matrix of length $2N_{A}\times
2N_{A}$,

$\mathcal{R}_{A_{2}}=\text{diag}\\{1,1,\cdots,1,-1,\cdots,-1\\},$ (5.8)

and the number of $-1$ elements is equal to the length of subregion $A_{2}$.
Here, $N_{A_{1}}\equiv N_{A_{R_{1}}}=1$ and $N_{A_{2}}\equiv N_{A_{L_{1}}}=1$,
so we have

$\det(\tilde{G}^{ab}_{12})=\det(\mathcal{R}_{A_{2}}.G_{12}^{ab}.\mathcal{R}_{A_{2}})=\det(G_{12}^{ab}).$
(5.9)

Note that with respect to this specific choice of subsystems, one obtains
$\det(\tilde{G}^{ab}_{12})=\det(G_{12}^{ab})$. Therefore, we can use the
second Rényi entropy itself for evaluating the zero-mode effect on the
evolution of $S_{\text{OEE}}$. In the following, we explicitly compute it.

The covariance matrix, $G_{A}(t)$, associated to subregion with a single site
at position $x$ on both sides of the TFD, with respect to the dimensionless
basis
$\tilde{\xi}_{k}^{a}=(\tilde{q}_{k}^{L},\tilde{q}_{k}^{R},\tilde{p}_{k}^{L},\tilde{p}_{k}^{R})$212121These
variables are related to variables of momentum space dual to the discretized
field according to $q_{a}=\frac{1}{\sqrt{N}}\Phi(x_{a})$ and
$p_{a}=\frac{\mathcal{L}}{\sqrt{N}}\Pi(x_{a})$., is given as follows:

$\displaystyle\tilde{G}^{ab}_{k}(t)=\langle\text{TFD}(t)\mid\tilde{\xi}^{a}_{k}\tilde{\xi}^{\dagger
b}_{k}+\tilde{\xi}^{\dagger b}_{k}\tilde{\xi}^{a}_{k}\mid\text{TFD}(t)\rangle$
$\displaystyle=\left(\begin{array}[]{cccc}\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})\\\
\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0\\\
0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})\\\
-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})\\\
\end{array}\right),$ (5.14)

where,

$\displaystyle\omega_{k}=\sqrt{m^{2}+\frac{4}{\delta^{2}}\sin^{2}(\frac{\pi
k}{N})},\hskip
14.22636pt\alpha_{k}=\frac{1}{2}\log\coth(\frac{\beta\omega_{k}}{4}),\hskip
14.22636pt\lambda_{k}=\omega_{k}\mathcal{L}.$ (5.15)

By using the inverse Fourier transformation
$\xi_{x}^{a}=\frac{1}{\sqrt{N}}\sum_{k=1}^{N}e^{-\frac{2\pi
ikx}{N}}\tilde{\xi}_{k}^{a}$, in the position basis
$\xi_{x}^{a}=(q_{x}^{L},q_{x}^{R},p_{x}^{L},p_{x}^{R})$, we have

$\displaystyle G^{ab}_{x,y}=\frac{1}{N}\sum_{k}e^{-\frac{2\pi
ik(x-y)}{N}}\tilde{G^{ab}_{k}}=\frac{1}{N}\sum_{k}e^{-\frac{2\pi ik(x-y)}{N}}$
$\displaystyle\times\left(\begin{array}[]{cccc}\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&\frac{\cos(\omega_{\mu
k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})\\\
\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0\\\
0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})\\\
-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})\\\
\end{array}\right).$ (5.20)

By setting $x=y$, we can obtain the odd entanglement entropy of a subsystem
consisting of a single degree of freedom on each left and right side of the
TFD. By studying the asymptotic behavior of odd entanglement entropy, we can
identify the contribution of the zero-mode in the massless limit,
$m\rightarrow 0$; this exhibits itself as the logarithmic term. According to
the above explanations, we can act as follows: setting $x=y$ in the (5) gives:

$\displaystyle G^{ab}_{x,x}(t)=\frac{1}{N}\sum_{k=0}^{N-1}$
$\displaystyle\times\left(\begin{array}[]{cccc}\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})\\\
\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0\\\
0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})\\\
-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})\\\
\end{array}\right).$ (5.25)

Considering (5.5), in order to compute the odd entanglement entropy, one needs
to evaluate the second Rényi entropy which from (5.6) is related to the
determinant in (5), recall $\det(\tilde{G}^{ab}_{12})=\det(G_{12}^{ab})$. The
details of this calculation is postponed to Appendix B and we only report the
final result here. In the range $t\gg\delta=\mathcal{L}/N$ and in the limit
$N\rightarrow\infty$, $m\mathcal{L}\ll 1$, $\beta/\mathcal{L}\ll 1$ and
$m\ll\delta^{-1}$, the odd entanglement entropy becomes:

$S_{\text{OEE}}(\rho_{A}(t))\sim 2(1-2\log 2)+\log\Big{(}\frac{2e_{2}}{N\beta
m}\Big{)}+\frac{1}{2}\log\Bigg{[}\frac{\sin^{2}(mt)+Q}{m^{2}\mathcal{L}^{2}}\Bigg{]},$
(5.26)

where $Q$ is defined by

$\displaystyle
Q=\frac{m^{2}\mathcal{L}^{2}}{\pi^{2}}\sum_{k=1}^{\infty}\frac{1-\cos[(\frac{2\pi
k}{\mathcal{L}})t]}{k^{2}},$ (5.27)

and $e_{2}$ is given in equation (B.14). The logarithmic term in the (5.26)
can be simplified in the three regimes as follows,

$\frac{1}{2}\log\Bigg{[}\frac{\sin^{2}(mt)+Q}{m^{2}\mathcal{L}^{2}}\Bigg{]}\sim\left\\{\begin{array}[]{rl}\frac{1}{2}\log(t/\mathcal{L})&\delta\ll
t<\mathcal{L},\\\ \log(t/\mathcal{L})&\mathcal{L}<t\ll m^{-1},\\\
\log(\frac{|\sin(mt)|}{m\mathcal{L}})&\mathcal{L}\ll t.\end{array}\right.$
(5.28)

Hence, the asymptotic form of the odd entanglement entropy in these regimes is
given by

$S_{\text{OEE}}(\rho_{A}(t))\sim 2(1-2\log 2)+\log\Big{(}\frac{2e_{2}}{N\beta
m}\Big{)}+\left\\{\begin{array}[]{rl}\frac{1}{2}\log(t/\mathcal{L})&\hskip
14.22636pt\delta\ll t<\mathcal{L},\\\ \log(t/\mathcal{L})&\hskip
14.22636pt\mathcal{L}<t\ll m^{-1},\\\
\log(\frac{|\sin(mt)|}{m\mathcal{L}})&\hskip 14.22636pt\mathcal{L}\ll
t,\end{array}\right.$ (5.29)

which matches with our numerical results (see below). Of course, the second
case is simply derived by the third case under the condition $t\ll m^{-1}$. In
the large time asymptotics, the behavior is oscillatory with frequency $m$.
This is expected due to the upper bound for the growth of the entanglement
which is provided by the thermal state. Hence, only for times $t\ll m^{-1}$ we
have the logarithmic behavior and for longer times we have the oscillatory
behavior with frequency $m$. It is worth noting that in order to determine the
entanglement evolution in the regime $t<\delta$ we must account for the
precise form of oscillations of various modes, hence, the above expression
will fail in that regime.

In the following, we present the numerically evaluated $S_{\text{EE}}$,
$S_{\text{OEE}}$ and $\mathcal{E}$ for a single site on each side of TFD in
the limit $m\rightarrow 0$ and find a logarithmic contribution which is due to
the presence of a zero-mode and compare it with analytical result, (5.29). In
figure 11, the analytical (dashed red) and numerical results are illustrated
for the $S_{\text{EE}}$(solid blue) and $S_{\text{OEE}}$ (solid green) in the
top panels. The $\mathcal{E}$ is plotted in the bottom panel. The logarithmic
growth is observed in variant time scales with different coefficients and it
well-matches with the analytical result (5.29). It is worth to mention that we
also observe similar behavior for the case where
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=1+1$, in the massless
limit. The $S_{\text{EE}}$ matches exactly with $S_{\text{OEE}}$ whenever
$\mathcal{E}$ becomes zero. Since the entanglement of two sites does not have
a well-defined continuum limit therefore in this figure we do not divide the
result by the thermodynamic entropy $S_{\text{th}}$.

$S_{\text{EE}}(t)-S_{\text{EE}}(0),S_{\text{OEE}}(t)-S_{\text{OEE}}(0)$

$t/\mathcal{L}$$t\hskip 1.42262ptm$

$\mathcal{E}(t)-\mathcal{E}(0)$

$t/\mathcal{L}$$t\hskip 1.42262ptm$

Figure 11: These plots correspond to the case of a subsystem consisting of a
single site on each side for $m=10^{-3}/\mathcal{L}$,
$\beta=10^{-2}\mathcal{L}$ and total sites $N=1501$. The analytical (dashed
red) and numerical results are illustrated for the $S_{\text{EE}}$(solid blue)
and $S_{\text{OEE}}$(solid green) in the top panels. The $\mathcal{E}$ is
plotted in the bottom panels. In the top-right panel, the periodic behavior
with respect to the parameter $tm$ is clear.

##  6 Conclusion

In this manuscript, we have studied entanglement dynamics of TFD state for
$1+1$-dimensional free scalar theory on a lattice by considering entanglement
entropy, odd entanglement entropy and logarithmic negativity. The scalar field
is discretized on two circles each with length $\mathcal{L}$ and the entangled
subsystems are non-complementary regions each with two adjacent or two
disjoint subregions. To compute desired entanglement measures we have used the
covariance matrix formalism which is proper for Gaussian states. For
evaluating entanglement entropy the eigenvalues of reduced covariance matrix
are needed however for odd entanglement entropy and logarithmic negativity the
eigenvalues of partial transposed reduced covariance matrix are required. It
is worth to mention that in the spirit of [51] our setup is named an unusual
quantum quench scenario[13] in which two decoupled subsystems are entangled
via their initial conditions.

The general perspective for time evaluation of entanglement entropy and odd
entanglement entropy is an early linear growth then saturation for short times
and then oscillatory behavior for longer times with periodicity of order the
circle size $\mathcal{L}$. Also, for massless theory we have observed a
logarithmic growth for the intermediate times due to presence of a zero-mode.
However, this growth is limited by the upper bounds 3.21 and 3.22. The linear
growth and saturation can be understood using the quasi-particle picture of
[7, 8]. According to the quasi-particle picture, the pre-quench initial state
$|\psi_{0}\rangle$ acts as a source for independent entangled pairs each with
an effective group velocity $v_{n}$ (1.2) that move ballistically in opposite
directions through the system. The entanglement can spread when one quasi-
particle is inside the subsystem and its partner is outside. The initially
linear growth arises due to the flux of quasi-particles going out of the
interval while their partners are still inside. This linear growth lasts until
times $t\sim l$ ($l$ is the size of the entangling region) then it saturates
at a value that is proportional to the thermal entropy of the system.
Moreover, the time dependence of quantum correlations exhibits an oscillatory
behavior periodically equal to the circle circumference, $\mathcal{L}$, due to
finite size effects. This can also be understood by a quasi-particle picture
where meeting quasi-particles on the opposite side of the circle leads to
reducing the correlations between the subsystem and its complement. We
observed a regular oscillation in the massless limit since all the quasi-
particles effectively move with the speed of light while for larger masses the
different quasi-particles have different group velocities which lead to
irregular oscillation222222Actually, the quasi-particle picture provide a
universal description for integrable models and it fails to capture dynamics
of entanglement in more generic systems such as chaotic ones.. Besides, we
have found that correlations can be increased by increasing both the
separation $d$ and the temperature. According to the quasi-particle picture,
for the greater separation $d$, more quasi-particle pairs are produced in the
region between the two subsystems which leads to increasing the share of pairs
that contribute to entanglement. Also enhancement by increasing the
temperature indicates that odd entanglement entropy same as entanglement
entropy is a measure of both classical and quantum correlations. To
investigate the entanglement dynamics in the continuum limit, the lattice
spacing $\delta$, mass $m$ and inverse temperature $\beta$ are fixed but the
total number of lattice sites $N$ is increased. Effectively, in this limit the
period of oscillation becomes larger and zero-mode contribution becomes less
important at intermediate times. Furthermore, there are several unproved
inequalities for odd entanglement entropy[30] where we have confirmed some of
them numerically.

It is worth emphasizing again that the logarithmic growth of odd entanglement
entropy (and also entanglement entropy) can not be understood using the quasi-
particle picture. To find a qualitative description and for tractability of
computations, we focused on a subsystem consisting of a single site on each
side of the TFD state. Accordingly, we intentionally suppress the linear
regime by choosing a subsystem size that vanishes in the continuum limit. But
we can restore the original system since zero-mode is non-local and can affect
local subsystems, regardless of their size. In another word, zero-mode
contribution is additive. The analytical results is derived by the approach of
[13], which is based on a relation between the entanglement entropy and Rényi
entropy of order 2 that is held for Gaussian states. By this approach, the
Rényi entropy of order 2 for a configuration with a single site on each side
can be evaluated by the determinant of a 4-by-4 covariance matrix and
therefore its time dependency simply can be analyzed analytically.
Interestingly, in our special decomposition of the subsystem, the determinants
of the modified reduced density matrix $\tilde{G}_{A_{12}}(t)$ and reduced
density matrix $G_{A_{12}}(t)$ became equal therefore evaluation of odd
entanglement entropy become same as entanglement entropy232323For entanglement
entropy, see [13].. The time evolution pattern is composed of three time
regimes: the first logarithmic regime:
$S_{\text{OEE}}(\rho_{A}(t))\sim\frac{1}{2}\log(t/\mathcal{L})$ for
$t<\mathcal{L}$ , a second logarithmic region:
$S_{\text{OEE}}(\rho_{A}(t))\sim\log(t/\mathcal{L})$ for $\mathcal{L}<t\ll
m^{-1}$ and finally an oscillating regime:
$S_{\text{OEE}}(\rho_{A}(t))\sim\log(\sin mt)$ when $t$ is of the same order
as $m^{-1}$.

Apart from the mentioned results, we have also studied logarithmic negativity
for symmetric and non-symmetric subsystems $A_{1}$, $A_{2}$. In the symmetric
case, at early times, it exists initial linear growth followed by an almost
linear decreasing up to revival time. This is in agreement with previous
studies [65, 43] in scaling limit242424The scaling limit means the long times
and large subsystems with their ratio fixed. and different setup. By
comparison with mutual information in the same setup, we have discussed that
this implies that the decreasing behavior is a peculiarity of the entanglement
and is not reflected by the correlation measures. Moreover, in the
decompactification limit the rate of decreasing of logarithmic negativity
becomes more sharply which it might be the sign of _sudden death of
entanglement_ before the trivial (finite size effect) revival. Overall, the
non-symmetric case has the same characteristics but with a difference which is
the appearance of a (narrow) plateau after an early linear growth. We have
also observed that whenever the size of subsystems is greater than separation
distance $d$, logarithmic negativity is non-zero for early times. It is
consistent with a dual holographic picture, in which logarithmic negativity is
dual to the geometric object named as entanglement wedge cross-section. In the
holographic prescription, for two disjoint intervals, whenever the disjoint
separation between two intervals $d$ is less than the size of subsystems, one
has a non-zero entanglement wedge cross-section and when $d$ increases one can
see a phase transition via vanishing the entanglement wedge cross-section. On
the other hand, according to the result of [24], the logarithmic negativity is
proportional to $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$. Therefore, whenever
logarithmic negativity vanishes, the entanglement entropy and odd entanglement
entropy became equal which also matches with our numerical results. Moreover,
for the adjacent case, we observe that by decreasing the temperature the delay
time for starting the growth of logarithmic negativity becomes larger. For the
future direction, it would be very interesting to obtain analytical forms for
evaluation of odd entanglement entropy and logarithmic negativity. It is worth
to mention that no analytic result is found for entanglement entropy of
_bosonic_ QFTs on a lattice even in the ground state. Last but not least, it
is exciting to explore the logarithmic growth at intermediate times for these
measures in the holographic context .

Acknowledgment
Special thanks to Mohsen Alishahiha, Pascuale Calabrese, Lucas Fabian Hackl,
Yuya Kusuki, Ali Mollabashi, Behrad Taghavi, Kotaro Tamaoka and Erik Tonni for
useful comments and fruitful discussions. Authors also thank Behrad Taghavi
and Erik Tonni for carefully reading the draft and IPM-Grid computing group
for providing computing and storage facilities. Part of MG work is supported
by Iran Science Elites Federation (ISEF).

##  Appendix A Temperature effects: various entanglement measures

In this appendix, we consider the effect of changing temperature on
$S_{\text{EE}}$, $S_{\text{OEE}}$, $\mathcal{E}$, and $\Delta
S=S_{\text{OEE}}-S_{\text{EE}}$. As we will see, the behavior of OEE is same
as EE and both of them approach thermal entropy by increasing temperature.
This confirms that OEE is a measure of both classical and quantum
correlations. However, the effect of decreasing temperature on LN behavior is
only a delay in its initial growth which indicates that the LN is a measure of
quantum correlation. The adjacent and disjoint subsystem configurations will
be considered separately.

###  A.1 Two adjacent intervals on each side

In figure 12, left panel, time evolution of the $S_{\text{EE}}$ and
$S_{\text{OEE}}$ are presented for temperatures ranging from
$\beta=10^{-2}\mathcal{L}$ to $10^{2}\mathcal{L}$. The left plot presents
$S_{\text{EE}}$ for $\beta=10^{-2}\mathcal{L}$(solid blue),
$\beta=10^{-1}\mathcal{L}$(solid green), $\beta=10\mathcal{L}$(solid purple)
and $\beta=10^{2}\mathcal{L}$(solid light blue). Also, in this plot
$S_{\text{OEE}}$ is depicted from $\beta=10^{-2}\mathcal{L}$ to $10^{2}$ with
dashed orange, dashed red, dashed brown and dashed yellow, respectively. The
upper two curves present high-temperature limit where the thermal correlation
dominates. For low temperatures, the initial growth happens after a delay. The
right panel denotes the finite size effects for temperatures
$\beta=10\mathcal{L}$ (solid blue, dashed orange), and
$\beta=10^{2}\mathcal{L}$ (solid green and dashed red).

$S_{\text{EE}}(t),S_{\text{OEE}}(t)$

$t/\mathcal{L}$

$S_{\text{EE}}(t),S_{\text{OEE}}(t)$

$t/\mathcal{L}$

Figure 12: The time dependence of $S_{\text{OEE}}$ and $S_{\text{EE}}$ are
presented for $N=1501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=1+1$, and
$m\mathcal{L}=10^{-3}$. The solid and dashed lines show the $S_{\text{EE}}$
and $S_{\text{OEE}}$, respectively for temperatures ranging from
$\beta=10^{-2}\mathcal{L}$ to $10^{2}\mathcal{L}$. The right plot presents a
longer period of time to see the finite size effects.

The time dependence of $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$ is presented in
figure 13, for various ranges of temperatures, $\beta=10^{-2}\mathcal{L}$ (
dark blue), $\beta=10^{-1}\mathcal{L}$ (orange), $\beta=10\mathcal{L}$
(green), and $\beta=10^{2}\mathcal{L}$ (red) curves. The left panel shows the
initial decreasing and the left one shows saturation. In initial times, the
difference is bigger for high temperatures in comparison with low
temperatures.

$\Delta S(t)$

$t/\mathcal{L}$

$\Delta S(t)$

$t/\mathcal{L}$

Figure 13: The time dependence of $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$ is
presented for $N=1501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=1+1$ and
$m\mathcal{L}=10^{-3}$.

$\mathcal{E}(t)$

$t/\mathcal{L}$

$\mathcal{E}(t)$

$t/\mathcal{L}$

$\mathcal{E}(t)$

$t/\mathcal{L}$

$\mathcal{E}(t)$

$t/\mathcal{L}$

Figure 14: The initial growth of logarithmic negativity is presented for
$N=1501$, $N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=1+1$, and
$m\mathcal{L}=10^{-3}$ for temperatures $\beta=10^{-2}\mathcal{L}$ (upper-
left), $\beta=10^{-1}\mathcal{L}$ (upper-right), $\beta=10\mathcal{L}$
(bottom-left), and $10^{2}\mathcal{L}$(bottom-right).

In figures 14 and 15, the time evolution of $\mathcal{E}$ is studied for
temperatures ranging from $\beta=10^{-2}\mathcal{L}$ to $10^{2}\mathcal{L}$.
The temperature decreases by a factor of $10$ from the upper-left panel to the
lower-right panel. In figure 14, the upper panels denote the time evolution
for high temperatures and the bottom panels are related to low temperatures.
Interestingly, the LN is zero for times $t<\beta/2$ and after that, it begins
to grow linearly252525This delay is also reported in [65].. Followed by figure
14, the long time behavior of logarithmic negativity for various temperatures
are presented in figure 15. From upper panels, which are correspond to high
temperatures $\beta=10^{-2}\mathcal{L}$(upper-left) and
$\beta=10^{-1}\mathcal{L}$(upper-right), we see the reduction of logarithmic
negativity for time scales of order $t\sim\mathcal{L}$ and sudden quantum
revival after this time. The finite size effects induce a periodic behavior.
For bottom panels, which correspond to the low temperatures,
$\beta=10\mathcal{L}$(bottom-left) and $\beta=10^{2}\mathcal{L}$(bottom-
right), we observe the growth of logarithmic negativity followed by a plateau
and then an oscillatory behavior due to the finite size effect.

$\mathcal{E}(t)$

$t/\mathcal{L}$

$\mathcal{E}(t)$

$t/\mathcal{L}$

$\mathcal{E}(t)$

$t/\mathcal{L}$

$\mathcal{E}(t)$

$t/\mathcal{L}$

Figure 15: The long time behavior of logarithmic negativity $\mathcal{E}$ is
presented for $N=1501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=1+1$, and
$m\mathcal{L}=10^{-3}$, for temperatures $\beta=10^{-2}\mathcal{L}$ (upper-
left), $\beta=10^{-1}\mathcal{L}$ (upper-right), $\beta=10\mathcal{L}$
(bottom-left), and $10^{2}\mathcal{L}$(bottom-right).

###  A.2 Two disjoint intervals on each side

In this subsection, we will study the effect of changing separation $d$ and
temperature together on the time evolution of $S_{\text{EE}}$,
$S_{\text{OEE}}$ and $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$.

$S_{\text{EE}}(t)$

$t/\mathcal{L}$

$S_{\text{EE}}(t)$

$t/\mathcal{L}$

$S_{\text{OEE}}(t)$

$t/\mathcal{L}$

$S_{\text{OEE}}(t)$

$t/\mathcal{L}$

Figure 16: The time evolution of $S_{\text{EE}}$ and $S_{\text{OEE}}$ for
$N=1501$, $N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=1+1$, and
$m\mathcal{L}=10^{-3}$. {The solid dark blue($\beta=10^{-2}\mathcal{L}$),
solid green($\beta=10^{-1}\mathcal{L}$), solid dark
purple($\beta=10\mathcal{L}$), solid cyan($\beta=10^{2}\mathcal{L}$)}, {dashed
orange($\beta=10^{-2}\mathcal{L}$), dashed light
purple($\beta=10^{-1}\mathcal{L}$), dashed yellow($\beta=10\mathcal{L}$),
dashed red ($10^{2}\mathcal{L}$)}, and {dot-dashed light
green($\beta=10^{-2}\mathcal{L}$), dot-dashed brown
($\beta=10^{-1}\mathcal{L}$), dot-dashed purple ($\beta=10\mathcal{L}$), and
dot-dashed light blue $(\beta=10^{2}\mathcal{L})$} correspond to
{$d=0,10,100$}, respectively.

The figure 16 denotes the time-dependent behavior of $S_{\text{EE}}$(upper
panels) and $S_{\text{OEE}}$(bottom panels). The solid lines correspond to
$d=0$. The dashed and dot-dashed lines correspond to $d=10,100$, respectively.
Effectively, $S_{\text{OEE}}$ and $S_{\text{EE}}$ behave the same and there
exists a competition between increasing the distance $d$ and decreasing the
temperature. By increasing the distance, $(S_{\text{OEE}},S_{\text{EE}})$
increase and by decreasing the temperature both of them decrease. Therefore,
in general, it exists a critical ratio $d_{\text{cr}}/\beta{{}_{\text{cr}}}$
which around it increasing distance or decreasing the temperature wins the
competition. Figure 17 represents the time dependence of $\Delta
S=S_{\text{OEE}}-S_{\text{EE}}$ with initial fluctuation around zero value
followed by saturation. The effects of finite size can be seen as oscillatory
behavior.

$\Delta S(t)$

$t/\mathcal{L}$

$\Delta S(t)$

$t/\mathcal{L}$

$\Delta S(t)$

$t/\mathcal{L}$

$\Delta S(t)$

$t/\mathcal{L}$

Figure 17: Time dependence of $\Delta S=S_{\text{OEE}}-S_{\text{EE}}$ is
presented for $N=1501$,
$N_{A_{L(R)}}=N_{A_{L_{1}(R_{1})}}+N_{A_{L_{2}(R_{2})}}=1+1$,
$m\mathcal{L}=10^{-3}$, $d=10$ (upper panels) and $d=100$ (bottom panels). The
blue, orange, green, and red curves correspond to temperatures
$\beta=10^{-2}\mathcal{L},10^{-1}\mathcal{L},10\mathcal{L},10^{2}\mathcal{L}$,
respectively.

##  Appendix B Zero-mode computations

In this Appendix, we provide details of calculations in section 5. Using (5),
the related covariance matrix in the position basis
$\xi_{x}^{a}=(q_{x}^{L},q_{x}^{R},p_{x}^{L},p_{x}^{R})$ takes the following
form:

$\displaystyle G^{ab}_{x,x}(t)=\frac{1}{N}\sum_{k}^{N-1}$
$\displaystyle\times\left(\begin{array}[]{cccc}\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})\\\
\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}&\frac{\cosh(2\alpha_{k})}{\lambda_{k}}&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0\\\
0&-\sin(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})\\\
-\sin(\omega_{k}t)\sinh(2\alpha_{k})&0&-\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})&\lambda_{k}\cosh(2\alpha_{k})\\\
\end{array}\right).$ (B.5)

The schematic form of this matrix is

$\displaystyle
G^{ab}_{x,x}(t)=\left(\begin{array}[]{cccc}e_{1}&\mathcal{M}_{1}(t)&0&\mathcal{M}_{2}(t)\\\
\mathcal{M}_{1}(t)&e_{1}&\mathcal{M}_{2}(t)&0\\\
0&\mathcal{M}_{2}(t)&e_{2}&\mathcal{M}_{3}(t)\\\
\mathcal{M}_{2}(t)&0&\mathcal{M}_{3}(t)&e_{2}\\\ \end{array}\right),$ (B.10)

whose determinant becomes

$\displaystyle\det\Big{(}G^{ab}_{x,x}(t)\Big{)}=\underset{Leading\hskip
1.70717ptorder}{\underbrace{e_{1}^{2}e_{2}^{2}-e_{2}^{2}F_{1}^{2}(t)}}+\underset{Subleading\hskip
1.70717ptorder}{\underbrace{(\mathcal{M}_{2}^{2}(t)-\mathcal{M}_{1}(t)\mathcal{M}_{2}(t))^{2}-2e_{1}e_{2}\mathcal{M}_{2}^{2}(t)-e_{1}^{2}\mathcal{M}_{3}^{2}(t)}}.$
(B.11)

When $t\gg\delta=\frac{\mathcal{L}}{N}$ and $N\rightarrow\infty$,
$m\mathcal{L}\ll 1$, $\beta/\mathcal{L}\ll 1$ and $m\ll\delta^{-1}$, one can
see that

$\displaystyle\sinh(2\alpha_{k})=1/\sinh(\frac{\beta\omega_{k}}{2})\underset{k\ll
N,\beta\ll\mathcal{L}}{\longrightarrow}\frac{2}{\beta\omega_{k}}$
$\displaystyle\omega_{k}=\omega_{N-k}\sim\frac{2\pi k}{\mathcal{L}}$ (B.12)

which altogether imply that

$\displaystyle
e_{1}=\frac{1}{N}\sum_{k=0}^{N-1}\frac{\cosh(2\alpha_{k})}{\lambda_{k}}=\frac{1}{N}\Big{(}\frac{\cosh(2\alpha_{0})}{\lambda_{0}}+\sum_{k=1}^{N-1}\frac{\cosh(2\alpha_{k})}{\lambda_{k}}\Big{)}=\frac{1}{N}\Big{(}\frac{\cosh(2\alpha_{0})}{\lambda_{0}}+2\sum_{k=1}^{\frac{N}{2}}\frac{\cosh(2\alpha_{k})}{\lambda_{k}}\Big{)}$
$\displaystyle\quad=\frac{2}{N\beta\mathcal{L}m^{2}}\Big{(}1+\frac{m^{2}\mathcal{L}^{2}}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{1}{k^{2}}\Big{)}\sim\frac{2}{N\beta\mathcal{L}m^{2}}\Big{(}1+\frac{m^{2}\mathcal{L}^{2}}{12}\Big{)},$
(B.13) $\displaystyle
e_{2}=\frac{1}{N}\sum_{k=0}^{N-1}\lambda_{k}\cosh(2\alpha_{k})=\frac{1}{N}\Big{(}\lambda_{0}\cosh(2\alpha_{0})+\sum_{k=1}^{N-1}\lambda_{k}\cosh(2\alpha_{k})\Big{)}=\frac{1}{N}\Big{(}\frac{2\mathcal{L}}{\beta}+2\sum_{k=1}^{\frac{N}{2}}\frac{2\mathcal{L}}{\beta}\Big{)}\sim\frac{2\mathcal{L}}{\beta},$
(B.14)

$\displaystyle\mathcal{M}_{1}(t)=\frac{1}{N}\sum_{k=0}^{N-1}\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}=\frac{1}{N}\Big{(}\frac{\cos(\omega_{0}t)\sinh(2\alpha_{0})}{\lambda_{0}}+\sum_{k=1}^{N-1}\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}\Big{)}$
$\displaystyle\qquad=\frac{1}{N}\Big{(}\frac{\cos(\omega_{0}t)\sinh(2\alpha_{0})}{\lambda_{0}}+2\sum_{k=1}^{\frac{N}{2}}\frac{\cos(\omega_{k}t)\sinh(2\alpha_{k})}{\lambda_{k}}\Big{)}$
$\displaystyle\qquad\sim\frac{2}{N\beta\mathcal{L}m}\Big{(}\frac{\cos(\omega_{0}t)}{m}+\frac{m\mathcal{L}^{2}}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{\cos(\omega_{k}t)}{k^{2}}\Big{)}\sim\frac{2}{N\beta\mathcal{L}m}\Big{(}\frac{\cos(mt)}{m}+\frac{m\mathcal{L}^{2}}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{\cos(\omega_{k}t)}{k^{2}}\Big{)},$
(B.15)

$\displaystyle\mathcal{M}_{2}(t)=\frac{-1}{N}\sum_{k=0}^{N-1}\sin(\omega_{k}t)\sinh(2\alpha_{k})=\frac{-1}{N}\Big{(}\sin(\omega_{0}t)\sinh(2\alpha_{0})+\sum_{k=1}^{N-1}\sin(\omega_{k}t)\sinh(2\alpha_{k})\Big{)}$
$\displaystyle\qquad=\frac{-1}{N}\Big{(}\sin(\omega_{0}t)\sinh(2\alpha_{0})+2\sum_{k=1}^{\frac{N}{2}}\sin(\omega_{k}t)\sinh(2\alpha_{k})\Big{)}$
$\displaystyle\qquad\sim\frac{-2}{N\beta}\Big{(}\frac{\sin(\omega_{0}t)}{m}+\frac{\mathcal{L}}{\pi}\sum_{k=1}^{\infty}\frac{\sin(\omega_{k}t)}{k}\Big{)}\sim\frac{-2}{N\beta}\Big{(}\frac{\sin(mt)}{m}+\frac{\mathcal{L}}{\pi}\sum_{k=1}^{\infty}\frac{\sin(\omega_{k}t)}{k}\Big{)},$
(B.16)

$\displaystyle\mathcal{M}_{3}(t)=\frac{-1}{N}\sum_{k}^{N-1}\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})=\frac{-1}{N}\Big{(}\lambda_{0}\cos(\omega_{0}t)\sinh(2\alpha_{0})+\sum_{k=1}^{N-1}\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})\Big{)}$
$\displaystyle\qquad=\frac{-1}{N}\Big{(}\lambda_{0}\cos(\omega_{0}t)\sinh(2\alpha_{0})+2\sum_{k=1}^{\frac{N}{2}}\lambda_{k}\cos(\omega_{k}t)\sinh(2\alpha_{k})\Big{)}$
$\displaystyle\qquad\sim\frac{-2\mathcal{L}m}{N\beta}\Big{(}\frac{\cos(\omega_{0}t)}{m}+\frac{2}{m}\sum_{k=1}^{\infty}\frac{k\cos(\omega_{k}t)}{k}\Big{)}\sim\frac{-2\mathcal{L}m}{N\beta}\Big{(}\frac{\cos(\omega_{0}t)}{m}+\frac{2}{m}\sum_{k=1}^{\infty}\cos(\omega_{k}t)\Big{)},$
(B.17)

According to the above asymptotic results, the leading order contribution for
$\det\Big{(}G^{ab}_{x,x}(t)\Big{)}$ in (B.11) is given by

$\displaystyle
e_{1}^{2}e_{2}^{2}-e_{2}^{2}\mathcal{M}_{1}^{2}(t)=\frac{4e_{2}^{2}}{N^{2}\beta^{2}m^{4}\mathcal{L}^{2}}\Bigg{[}(1+\frac{m^{2}\mathcal{L}^{2}}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{1}{k^{2}}\Big{)}^{2}+\Big{(}\cos(mt)+\frac{m^{2}\mathcal{L}^{2}}{2\pi^{2}}\sum_{k=1}^{\infty}\frac{\cos(\omega_{k}t)}{k^{2}}\Big{)}^{2}\Bigg{]}$
$\displaystyle\hskip
28.45274pt=\frac{4e_{2}^{2}}{N^{2}\beta^{2}m^{2}(m)^{2}\mathcal{L}^{2}}\Bigg{[}\sin^{2}(mt)+\frac{m^{2}\mathcal{L}^{2}}{\pi^{2}}\sum_{k=1}^{\infty}\frac{1-\cos[(\frac{2\pi
k}{\mathcal{L}})t]}{k^{2}}\Bigg{]}$ $\displaystyle\hskip
28.45274pt=\frac{4e_{2}^{2}}{N^{2}\beta^{2}m^{2}}\Bigg{[}\frac{\sin^{2}(mt)+Q}{m^{2}\mathcal{L}^{2}}\Bigg{]},$
(B.18)

where we have defined $Q$ as

$\displaystyle
Q=\frac{m^{2}\mathcal{L}^{2}}{\pi^{2}}\sum_{k=1}^{\infty}\frac{1-\cos[(\frac{2\pi
k}{\mathcal{L}})t]}{k^{2}}.$ (B.19)

Therefore, (5.5) implies that

$S_{\text{OEE}}(\rho_{A}(t))\sim 2(1-2\log 2)+\log\Big{(}\frac{2e_{2}}{N\beta
m}\Big{)}+\frac{1}{2}\log\Bigg{[}\frac{\sin^{2}[mt]+Q}{m^{2}\mathcal{L}^{2}}\Bigg{]}.$
(B.20)

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