# Dynamics of charge imbalance resolved negativity after a global quench in
free scalar field theory

Hui-Huang<EMAIL_ADDRESS>
College of Physics and Communication Electronics, Jiangxi Normal University,
Nanchang 330022, China

In this paper, we consider the time evolution of charge imbalance resolved
negativity after a global quench in the 1+1 dimensional complex Klein-Gordon
theory. We focus on two types of global quenches which are called boundary
state quench and mass quench respectively. We first study the boundary state
quench where the post-quench dynamic is governed by a massless Hamiltonian. In
this case, the temporal evolution of charged imbalance resolved negativity can
be obtained first by evaluating the correlators of the fluxed twist field in
the upper half plane and then applying Fourier transformation. We test our
analytical formulas in the underlying lattice model numerically. We also study
the mass quench in the complex harmonic chain where the system evolves
according to a massive Hamiltonian after the quench. We argue that our results
can be understood in the framework of quasi-particle picture.

###### Contents

  1. 1 Introduction
  2. 2 Charge imbalance resolution of negativity
  3. 3 Boundary state quench
    1. 3.1 The path integral approach to boundary state quench
    2. 3.2 Evolution of charged moments of RDM
  4. 4 Evolution of charged logarithmic negativity in boundary state quench
    1. 4.1 Bipartite system
    2. 4.2 Two adjacent intervals
    3. 4.3 Two disjoint intervals
  5. 5 Charge imbalance resolved negativity
    1. 5.1 Bipartite system
    2. 5.2 Two adjacent intervals
    3. 5.3 Two disjoint intervals
    4. 5.4 Total negativity
  6. 6 Numerical test
  7. 7 Global mass quench of the harmonic chain
    1. 7.1 Quasi-particle picture for entanglement entropy
    2. 7.2 Quasi-particle picture for charged logarithmic negativity
  8. 8 Conclusion

##  1 Introduction

In the past two decades, the investigation of various entanglement measures
has greatly sharpened our understanding of quantum many-body system, quantum
field theory and quantum gravity. In condensed matter theory, entanglement is
a powerful tool to characterize different phases of matter [1, 2, 3]. In the
AdS/CFT correspondence, the Ryu-Takayanagi formula [4, 5] firstly opens the
route of understanding spacetime from entanglement and this idea turns out to
have a key role in the black hole information loss paradox [6, 7, 8].
Entanglement is also an important concept in the studies of equilibration and
thermalization of isolated quantum systems. Among all these progress,
entanglement entropy is the most successful entanglement measure to
characterize the bipartite entanglement of a pure state. When the system is
prepared in a pure state $\ket{\psi}$, the reduced density matrix (RDM) of
subsystem $A$ is defined by tracing out its complement $B$,
$\rho_{A}=\textrm{tr}_{B}\ket{\psi}\bra{\psi}$. From the moments of
$\rho_{A}$, i.e. ${\textrm{Tr}}\rho_{A}^{n}$, one can obtain the Von Neumann
entropy through the replica trick [9]

$S\equiv-{\textrm{Tr}}(\rho_{A}\log\rho_{A})=\lim_{n\rightarrow 1}S_{n}$ (1.1)

where $S_{n}$ is the Rényi entropies

$S_{n}=\frac{1}{1-n}\log{\textrm{Tr}}\rho_{A}^{n}.$ (1.2)

Now suppose we are interested in the entanglement between two subsystems
$A_{1}$ and $A_{2}$, which are not necessarily complementary to each other. In
this situation, $\rho_{A_{1}\cup A_{2}}$ is general a mixed state, and Von
Neumann entropy is no longer a good measure of entanglement. Among different
proposals, a computable measure of mixed state entanglement, entanglement
negativity (or logarithmic negativity equivalently) turns out to be very
useful [10, 11, 12]. The definition is

$\mathcal{N}=\frac{1}{2}(||\rho_{A}^{T_{2}}||-1),$ (1.3)

where $||O||={\textrm{Tr}}\sqrt{O^{\dagger}O}$ denotes the trace norm of the
operator $O$ and $\rho_{A}^{T_{2}}$ is the partial transpose of RDM $\rho_{A}$
with respect to degree of freedom of subsystem $A_{2}$. Let
$\ket{e_{i}^{(1)}}$ and $\ket{e_{j}^{(2)}}$ be two arbitrary bases of the
Hilbert spaces associated to the degree of freedom on $A_{1}$ and $A_{2}$
respectively. The partial transpose (with respect to the second space) of
$\rho_{A}$ is defined as

$\bra{e_{i}^{(1)}e_{j}^{(2)}}\rho_{A}^{T_{2}}\ket{e_{k}^{(1)}e_{l}^{(2)}}=\bra{e_{i}^{(1)}e_{l}^{(2)}}\rho_{A}\ket{e_{k}^{(1)}e_{j}^{(2)}}.$
(1.4)

It’s then useful to define the Rényi negativity

$\mathcal{N}_{n}=\textrm{tr}(\rho_{A}^{T_{2}})^{n}$ (1.5)

which could be analytically continued from an even integer $n_{e}$ to obtain
the negativity, using $||\rho_{A}^{T_{2}}||=\lim_{n_{e}\rightarrow
1}\mathcal{N}_{n_{e}}$. Entanglement negativity has been studied extensively
in both quantum field theories (QFT)[13, 14, 15, 16, 17, 18] and in
holographic theories [19, 20, 21, 22].

In recent years, people are interested in the interplay between symmetries and
entanglement. When our system exhibit a global symmetry, the entanglement will
split into different sectors characterized by eigenvalues of some charge
operator. This symmetry resolution of entanglement attract much attention
recently [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. In the
context of mixed states, the charge imbalance resolved negativity were also
studied in several circumstances [38, 39, 40]. However, much fewer results are
available for its evolution under non-equilibrium setups [41, 42, 43, 44]. It
would be very interesting to further investigate the non-equilibrium dynamics
of these charge resolved measures of entanglement.

In this paper, we will consider the time evolution of the charge imbalance
resolved negativity in a special kind of non-equilibrium state known as global
quantum quenches. A global quantum quench describes a process in which the
sudden change of the Hamiltonian $H_{0}\rightarrow H$ at a given time that we
set as $t=0$, then the initial ground state $\ket{\psi_{0}}$ of the pre-quench
Hamiltonian $H_{0}$ evolves according to post-quench Hamiltonian $H$. Thus, we
have $\ket{\psi(t)}=e^{-iHt}\ket{\psi_{0}}$. The quench dynamics of various
measures of entanglement have been extensively studied in the literature [45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. In this paper, we will focus on two
types of global quenches in 1+1 dimensional complex Klein-Gordon field theory.
The first type is the boundary state quench in which after the quench the mass
of the scalar field is zero, so the time evolution is governed by a conformal
field theory (CFT). The second type we will consider is a mass quench in the
underlying lattice model (complex harmonic chain), where the post-quench
dynamic is governed by a massive Hamiltonian.

The remaining part of this manuscript is organized as follows. In section 2,
we briefly review some basic facts about charge imbalance resolution of
entanglement negativity. In section 3, we discuss how to calculate the
dynamics of charged Rényi negativity in the boundary state quench setup using
CFT techniques. In section 4, we will consider the quench dynamics of charged
(logarithmic) Rényi negativity between two finite intervals in the boundary
state quench protocol. In section 5, by applying Fourier transformation, we
can obtain the charge imbalance resolved negativity from the results derived
in the previous section. In section 6, we check our analytical predictions
against numerical computation in the complex harmonic chain. In section 7, we
further discuss the mass quench in the underlying lattice model numerically
and using quasi-particle interpretation to predict the quench dynamics of
charged logarithmic negativity, finding perfect agreement. Finally, we
conclude in section 8 and discuss some possible interesting extensions of this
work.

##  2 Charge imbalance resolution of negativity

In this section, we will briefly review the decomposition of entanglement
negativity under a global internal symmetry.

Let us first review some basic facts about the symmetry resolution of the
entanglement entropy. We assume that our system exhibit a global $U(1)$
symmetry generated by a local charge $Q$. If $[\rho,Q]=0$ (this can be
achieved if $\rho$ only acts non-trivially on the eigenspace of $Q$), then we
have $[\rho_{A},Q_{A}]=0$, which implies that $\rho_{A}$ admits charge
decomposition according to eigenvalues $q$ of local charge $Q_{A}$

$\rho_{A}=\oplus_{q}\mathcal{P}_{q}\rho_{A}=\oplus_{q}p(q)\rho_{A}(q),\quad
p(q)={\textrm{Tr}}(\mathcal{P}_{q}\rho_{A}),$ (2.1)

where $\mathcal{P}_{q}$ is the projection operator which projects the space to
the eigenspace corresponding to eigenvalue $q$. The symmetry resolved Rényi
entropies are defined as

$S_{n}(q)=\frac{1}{1-n}\log{\textrm{Tr}}[\rho_{A}(q)]^{n}.$ (2.2)

It’s convenient to first introduce the charged moments of $\rho_{A}$,

$Z_{n}(\mu)={\textrm{Tr}}(e^{i\mu Q_{A}}\rho_{A}^{n}).$ (2.3)

Then it’s sufficient to compute its Fourier transform

$Z_{n}(q)=\int_{0}^{2\pi}\frac{d\mu}{2\pi}e^{-iq\mu}Z_{n}(\mu)$ (2.4)

to obtain the Rényi entropies of the sector with charge $q$ as

$S_{n}(q)=\frac{1}{1-n}\log\left[\frac{Z_{n}(q)}{Z_{1}(q)^{n}}\right].$ (2.5)

Finally, the symmetry resolved entanglement entropy can be obtained by taking
the replica limit $S(q)=\lim_{n\rightarrow 1}S_{n}(q)$.

Now we briefly review the symmetry decomposition of entanglement negativity
under the $U(1)$ charge $Q$. Since the charge $Q$ is local, we can write
$Q_{A}=Q_{1}+Q_{2}$, where $Q_{1}$ and $Q_{2}$ are the charges corresponding
to sub subsystem $A_{1}$ and $A_{2}$ respectively. From the relation
$[\rho_{A},Q_{A}]=0$, performing a partial transposition with respect to the
second region $A_{2}$ of subsystem $A$, we obtain

$[\rho_{A}^{T_{2}},\mathcal{Q}_{A}]=0,\quad\mathcal{Q}_{A}\equiv
Q_{1}-Q_{2}^{T_{2}},$ (2.6)

where we have introduced the charge imbalance operator $\mathcal{Q}_{A}$ and
we will denote its eigenvalues as $\mathrm{q}$ to make a distinction with the
eigenvalues of $Q_{A}$. Then $\rho_{A}^{T_{2}}$ has a block matrix form, each
block was characterized by different eigenvalues $\mathrm{q}$ of the imbalance
operator $\mathcal{Q}_{A}$. If we write

$\rho_{A}^{T_{2}}=\oplus_{\mathrm{q}}\mathcal{P}_{\mathrm{q}}\rho_{A}^{T_{2}},\quad\rho_{A}^{T_{2}}({\mathrm{q}})=\frac{\mathcal{P}_{\mathrm{q}}\rho_{A}^{T_{2}}}{{\textrm{Tr}}(\mathcal{P}_{\mathrm{q}}\rho_{A}^{T_{2}})}.$
(2.7)

Then we have

$\rho_{A}^{T_{2}}=\oplus_{\mathrm{q}}p({\mathrm{q}})\rho_{A}^{T_{2}}({\mathrm{q}}),$
(2.8)

where
$p({\mathrm{q}})={\textrm{Tr}}(\mathcal{P}_{\mathrm{q}}\rho_{A}^{T_{2}})$ is
the probability of finding ${\mathrm{q}}$ as the outcome of measurement of
$\mathcal{Q}_{A}$.

The charge imbalance resolved negativity is defined as

$\mathcal{N}({\mathrm{q}})=\frac{1}{2}({\textrm{Tr}}|\rho_{A}^{T_{2}}({\mathrm{q}})|-1).$
(2.9)

The total negativity is given by the sum of charge imbalance resolved
negativity weighted by the corresponding probability

$\mathcal{N}=\sum_{\mathrm{q}}p({\mathrm{q}})\mathcal{N}({\mathrm{q}}).$
(2.10)

It’s useful to define the charge imbalance resolved Rényi negativity

$\mathcal{N}_{n}(\mathrm{q})=\frac{N_{n}(\mathrm{q})-1}{2},\quad
N_{n}({\mathrm{q}})={\textrm{Tr}}[(\rho_{A}^{T_{2}}({\mathrm{q}}))^{n}]=\frac{1}{p({\mathrm{q}})^{n}}{\textrm{Tr}}[\mathcal{P}_{\mathrm{q}}(\rho_{A}^{T_{2}})^{n}].$
(2.11)

Then the charge imbalance entanglement negativity can be obtained by taking
the limit

$\mathcal{N}({\mathrm{q}})=\lim_{n_{e}\rightarrow
1}\mathcal{N}_{n_{e}}(\mathrm{q}).$ (2.12)

The projection operator $\mathcal{P}_{\mathrm{q}}$ has the following integral
representation

$\mathcal{P}_{\mathrm{q}}=\int_{0}^{2\pi}\frac{d\mu}{2\pi}e^{-i\mu{\mathrm{q}}}e^{i\mu\mathcal{Q}_{A}}.$
(2.13)

It’s convenient to first introduce the charged Rényi negativity

$R_{n}(\mu)={\textrm{Tr}}[(\rho_{A}^{T_{2}})^{n}e^{i\mu\mathcal{Q}_{A}}],$
(2.14)

and it’s Fourier transformation

$\mathcal{Z}_{T_{2},n}(\mathrm{q})=\int_{0}^{2\pi}\frac{d\mu}{2\pi}e^{-i\mu\mathrm{q}}R_{n}(\mu)$
(2.15)

then the charge imbalance resolved Rényi negativity are related by Fourier
transform

$\mathcal{N}_{n}({\mathrm{q}})=\frac{\mathcal{Z}_{T_{2},n}(\mathrm{q})}{p(\mathrm{q})^{n}},\qquad
p(\mathrm{q})=\int_{0}^{2\pi}\frac{d\mu}{2\pi}e^{-i\mu{\mathrm{q}}}R_{1}(\mu).$
(2.16)

It’s also useful to introduce the charged Rényi logarithmic negativity
$\mathcal{E}_{n}(\mu)$, defined as

$\mathcal{E}_{n}(\mu)=\log{\textrm{Tr}}[(\rho_{A}^{T_{2}})^{n}e^{i\mu\mathcal{Q}_{A}}].$
(2.17)

The charged logarithmic negativity is defined by taking the following replica
limit

$\mathcal{E}(\mu)=\lim_{n_{e}\rightarrow 1}\mathcal{E}_{n_{e}}(\mu).$ (2.18)

Then the charge imbalance resolved negativity can be obtained as

$\mathcal{N}(\mathrm{q})=\frac{1}{2}\left(\frac{\mathcal{Z}_{T_{2}}(\mathrm{q})}{p(\mathrm{q})}-1\right),\quad\mathcal{Z}_{T_{2}}(\mathrm{q})\equiv\lim_{n_{e}\rightarrow
1}\mathcal{Z}_{T_{2},n_{e}}(\mathrm{q})=\int_{0}^{2\pi}\frac{d\mu}{2\pi}e^{-i\mu\mathrm{q}}e^{\mathcal{E}(\mu)}.$
(2.19)

In section 5, we will use eq. (2.19) and eq. (2.16) to evaluate the charge
imbalance resolved negativity.

##  3 Boundary state quench

In this paper, we will consider the 1+1 dimensional complex free scalar field
theory with the Euclidean action given by

$\mathcal{A}=\int
d^{2}x(\partial_{\mu}\phi^{\dagger}\partial_{\mu}\phi+m^{2}\phi^{\dagger}\phi).$
(3.1)

This action exhibit a $U(1)$ symmetry, i.e. the phase transformation of the
field $\phi\rightarrow e^{i\theta}\phi,\phi^{\dagger}\rightarrow
e^{-i\theta}\phi^{\dagger}$ leaves the action invariant. The Hamiltonian of
this theory is

$H=\int
dx(\pi^{\dagger}\pi+\partial_{x}\phi^{\dagger}\partial_{x}\phi+m^{2}\phi^{\dagger}\phi)$
(3.2)

with $\pi$ is the canonical momentum for $\phi$.

We can rewrite this theory in terms of two real scalar fields $\phi^{(1)}$ and
$\phi^{(2)}$ with $\phi=\frac{1}{\sqrt{2}}(\phi^{(1)}+i\phi^{(2)})$. The
Hamiltonian in terms of these variables becomes

$H=\frac{1}{2}\sum_{\alpha=1,2}\int
d^{2}x[(\pi^{(\alpha)})^{2}+(\partial_{x}\phi^{(\alpha)})^{2}+m^{2}(\phi^{(\alpha)})^{2}].$
(3.3)

The Hamiltonian can be diagonalized in terms of particle and anti-particle
mode operators $a(p),a^{\dagger}(p)$ and $b(p),b^{\dagger}(p)$ with the
commutation relation
$[a(p),a^{\dagger}(q)]=[b(p),b^{\dagger}(q)]=2\pi\delta(p-q)$ and all other
commutators vanishing. We have

$H=\int\frac{dp}{2\pi}e(p)(a^{\dagger}(p)a(p)+b^{\dagger}(p)b(p))$ (3.4)

with $e(p)=\sqrt{m^{2}+p^{2}}$, while the conserved charge corresponding to
the global $U(1)$ symmetry is

$Q=\int\frac{dp}{2\pi}(a^{\dagger}(p)a(p)-b^{\dagger}(p)b(p))=\int
dx(a^{\dagger}(x)a(x)-b^{\dagger}(x)b(x)).$ (3.5)

In the second equal sign of above equation, the conserved charge are expressed
as integral of local density in real space. Thus its value in a given
subsystem $A$ is the same integral restricted to $A$,

$Q_{A}=\int_{A}dx(a^{\dagger}(x)a(x)-b^{\dagger}(x)b(x)).$ (3.6)

The lattice version of the complex Klein-Gordon field theory is the complex
harmonic chain which is equivalent to two decoupled real harmonic chains. The
Hamiltonian of the real harmonic chain made by $L$ sites reads

$H_{HC}=\frac{1}{2}\sum_{j=0}^{L-1}\left[\pi_{j}^{2}+m^{2}\phi_{j}^{2}+(\phi_{j+1}-\phi_{j})^{2}\right],$
(3.7)

where periodic boundary conditions
$\phi_{L}\equiv\phi_{0},\pi_{L}\equiv\pi_{0}$ are imposed and variables
$\pi_{j}$ and $\phi_{j}$ satisfy standard bosonic commutation relations
$[\phi_{i},\phi_{j}]=[\pi_{i},\pi_{j}]=0$ and
$[\phi_{i},\pi_{j}]=\mathrm{i}\delta_{ij}$. The lattice version of the complex
scalar field theory is the sum of two of the above harmonic chain. In terms of
the variables $\phi^{(1)},\pi^{(1)}$ and $\phi^{(2)},\pi^{(2)}$, the
Hamiltonian is

$H_{CHC}=H_{HC}(\pi^{(1)},\phi^{(1)})+H_{HC}(\pi^{(2)},\phi^{(2)}).$ (3.8)

The Hamiltonian eq. (3.8) can be diagonalized by introducing the creation and
annihilation operators $a_{k},a_{k}^{\dagger}$ and $b_{k},b_{k}^{\dagger}$,
satisfying $[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}$ and
$[b_{k},b_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}$. In terms of these
operators, the Hamiltonian eq. (3.8) is diagonal

$H_{CHC}=\sum_{k=0}^{L-1}e_{k}(a_{k}^{\dagger}a_{k}+b_{k}^{\dagger}b_{k}),\quad
e_{k}=\sqrt{m^{2}+4\sin^{2}\left(\frac{\pi k}{L}\right)}.$ (3.9)

While the $U(1)$ charge is

$Q=\sum_{k=0}^{L-1}(a_{k}^{\dagger}a_{k}-b_{k}^{\dagger}b_{k}).$ (3.10)

The conserved charge is local and can also be written in the position space
and for a given subsystem $A$ reads

$Q_{A}=\sum_{j\in A}(a_{j}^{\dagger}a_{j}-b_{j}^{\dagger}b_{j}).$ (3.11)

###  3.1 The path integral approach to boundary state quench

In this section, we will focus on the case in which after the quench the mass
of the scalar field is zero, so time evolution is governed by a CFT. Let’s
first briefly review the imaginary time formalism for the global quenches
where the post-quench dynamic is govern by a CFT. This quench setup is called
a boundary state quench as the correlation measures after quench can be
described by boundary conformal field theory (BCFT), where the gapped initial
state are represent by a spacetime boundary.

The expectation value of a product of equal-time local operators in time
dependent state $\ket{\psi(t)}$ is

$\langle\mathcal{O}(t,\\{x_{i}\\})\rangle=Z^{-1}\bra{\psi_{0}}e^{iHt-\tau_{0}H}\mathcal{O}(\\{x_{i}\\})e^{-iHt-\tau_{0}H}\ket{\psi_{0}}.$
(3.12)

where two factors $e^{-\tau_{0}H}$ have been introduced to make the path
integral representation of this expectation value absolutely convergent and
$Z=\bra{\psi_{0}}e^{-2\tau_{0}H}\ket{\psi_{0}}$ is the normalization factor.
The above prescription is equivalent to assuming that the initial state has
the form $\ket{\psi_{0}}\propto e^{-\tau_{0}H}\ket{B}$, where $\ket{B}$ is
some conformal boundary state. One could interpret $\tau_{0}$ as being
proportional to the correlation length of the initial state. Thus, the
predictions made by this approach are expected to be valid only in the
spacetime scaling limit,
$t\gg\tau_{0},|x_{i}-x_{j}|\gg\tau_{0},t/|x_{i}-x_{j}|=\text{finite}$. The
above correlator admits a path integral represent

$\langle\mathcal{O}(t,\\{x_{i}\\})\rangle=\frac{1}{Z}\int[d\phi(x,\tau)]\mathcal{O}(\\{x_{i}\\},\tau=\tau_{0}+it)e^{-\int_{\tau_{1}}^{\tau_{2}}d\tau
L}\braket{\psi_{0}}{\phi(x,\tau_{2})}\braket{\phi(x,\tau_{1})}{\psi_{0}}.$
(3.13)

where $L$ is the Euclidean Lagrangian corresponding to the post-quench
Hamiltonian $H$. We need to identify $\tau_{1}=0$ and $\tau_{2}=2\tau_{0}$. We
should compute the path integral considering $\tau$ real and only at the end
of the computation to analytically continue it to $\tau=\tau_{0}+it$.

###  3.2 Evolution of charged moments of RDM

In this section, we briefly review the strategy of computing the charged Rényi
negativity in two-dimensional CFT (see [40] for more details). In a generic
two-dimensional QFT, we can view the charged moments $Z_{n}(\mu)$ (defined in
eq. (2.3)) as the partition function on the Riemann surface
$\mathcal{R}_{n,N}$ pierced by an Aharonov-Bohm flux, such that the total
phase accumulated by the field upon going through the entire surface is $\mu$.
The presence of the flux corresponds to impose an additional twist on the
boundary of subsystem $A$. This additional twist fuses with the replica
(ordinary) twist field at the endpoints of subsystem $A$ and can be
implemented by two local fields $\mathcal{T}_{n,\mu}$ and
$\tilde{\mathcal{T}}_{n,\mu}$ named as fluxed twist fields and fluxed anti-
twist fields. These fluxed twist fields take into account not only the
internal permutational symmetry among the replicas but also the presence of
the flux. The partition function on the fluxed Riemann surface is thus
proportional to the $2N$-point function of these fluxed twist operators.

We consider the subsystem $A$ consists of two disjoint intervals on the real
axis, $A=A_{1}\cup A_{2}$ with $A_{1}=[u_{1},u_{2}],A_{2}=[u_{3},u_{4}]$. The
charged moments of RDM of vacuum state $\rho_{A}$, i.e.
$\textrm{tr}\rho_{A}^{n}e^{i\mu Q_{A}}$ is equivalent to a four-point function
of the fluxed twist fields

${\textrm{Tr}}\rho_{A}^{n}e^{i\mu
Q_{A}}=\langle\mathcal{T}_{n,\mu}(w_{1})\tilde{\mathcal{T}}_{n,\mu}(w_{2})\mathcal{T}_{n,\mu}(w_{3})\tilde{\mathcal{T}}_{n,\mu}(w_{4})\rangle_{\text{strip}}~{},\qquad
w_{i}=u_{i}+i\tau,$ (3.14)

where $w=u+i\tau$ with $\tau\in\mathbb{R},\tau\in(0,2\pi)$ is the coordinate
of the strip. The fluxed twist fields $\mathcal{T}_{n,\mu}$ and fluxed anti-
twist field $\tilde{\mathcal{T}}_{n,\mu}$ are primary operators and they have
the same dimension [23, 27]

$\Delta_{n,\mu}=\frac{1}{6}\left(n-\frac{1}{n}\right)-\frac{\mu^{2}}{4\pi^{2}n}+\frac{\mu}{2\pi
n}.$ (3.15)

The charged moments of the partially transposed RDM can be obtained from the
correlator above with the fluxed twist field $\mathcal{T}_{n,\mu}$ and
$\tilde{\mathcal{T}}_{n,\mu}$ at the endpoints of $A_{2}$ exchanged while the
remaining ones keep the same, giving

${\textrm{Tr}}[(\rho_{A}^{T_{2}})^{n}e^{i\mu\mathcal{Q}_{A}}]=\langle\mathcal{T}_{n,\mu}(w_{1})\tilde{\mathcal{T}}_{n,\mu}(w_{2})\tilde{\mathcal{T}}_{n,\mu}(w_{3})\mathcal{T}_{n,\mu}(w_{4})\rangle_{\text{strip}}~{}.$
(3.16)

By conformal map $z(w)=e^{\pi w/2\tau_{0}}$, the four point function on the
strip are mapped to the upper half plane (UHP), then we can write

${\textrm{Tr}}\rho_{A}^{n}e^{i\mu
Q_{A}}=\left(\frac{\pi}{2\tau_{0}}\right)^{4\Delta_{n,\mu}}|z_{1}z_{2}z_{3}z_{4}|^{\Delta_{n,\mu}}\langle\mathcal{T}_{n,\mu}(z_{1})\tilde{\mathcal{T}}_{n,\mu}(z_{2})\mathcal{T}_{n,\mu}(z_{3})\tilde{\mathcal{T}}_{n,\mu}(z_{4})\rangle_{\text{UHP}},$
(3.17)

By global conformal symmetry, four-point functions of primary fields on the
upper half plane depend on 6 cross ratios $\eta_{i,j}$ and have the following
structure

$\langle\mathcal{T}_{n,\mu}(z_{1})\tilde{\mathcal{T}}_{n,\mu}(z_{2})\mathcal{T}_{n,\mu}(z_{3})\tilde{\mathcal{T}}_{n,\mu}(z_{4})\rangle_{\text{UHP}}=c_{n,\mu}^{2}\prod_{a=1}^{4}|z_{a}-\bar{z}_{a}|^{-\Delta_{n,\mu}}\left(\frac{\eta_{1,3}\eta_{2,4}}{\eta_{1,2}\eta_{1,4}\eta_{2,3}\eta_{3,4}}\right)^{\Delta_{n,\mu}}\mathcal{F}_{n}(\\{\eta_{j,k}\\})$
(3.18)

where the cross ratios are defined by
$\eta_{i,j}=\frac{(z_{i}-z_{j})(\bar{z}_{i}-\bar{z}_{j})}{(z_{i}-\bar{z}_{j})(\bar{z}_{i}-z_{j})}$
and $\mathcal{F}(\\{\eta_{j,k}\\})$ depend on the full operator content of the
theory and usually is very complicated. The constants $c_{n,\mu}$ are also
non-universal and are known for some specific theories.

While for the charged Rényi negativity, we have

$\begin{split}{\textrm{Tr}}[(\rho_{A}^{T_{2}})^{n}e^{i\mu\mathcal{Q}_{A}}]=\left(\frac{\pi}{2\tau_{0}}\right)^{4\Delta_{n,\mu}}\prod_{a=1}^{4}|z_{a}|^{\Delta_{n,\mu}}\langle\mathcal{T}_{n,\mu}(z_{1})\tilde{\mathcal{T}}_{n,\mu}(z_{2})\tilde{\mathcal{T}}_{n,\mu}(z_{3})\mathcal{T}_{n,\mu}(z_{4})\rangle_{\text{UHP}}\\\
=c_{n,\mu}^{2}\prod_{a=1}^{4}\Big{|}\frac{z_{a}}{z_{a}-\bar{z}_{a}}\Big{|}^{\Delta_{n,\mu}}\frac{1}{\eta_{1,2}^{\Delta_{n,\mu}}\eta_{3,4}^{\Delta_{n,\mu}}}\left(\frac{\eta_{1,4}\eta_{2,3}}{\eta_{1,3}\eta_{2,4}}\right)^{\Delta^{(2)}_{n,\mu}/2-\Delta_{n,\mu}}\mathcal{G}_{n}(\\{\eta_{j,k}\\})\end{split}$
(3.19)

Fortunately, in the spacetime scaling regime, we have $\eta_{i,j}\rightarrow
0,1$ or $\infty$, and the non-universal functions
$\mathcal{F}_{n}(\\{\eta_{j,k}\\}),\mathcal{G}_{n}(\\{\eta_{j,k}\\})$ are just
constant. In the following section we will just omit these non-universal
functions and just focus on the universal part.

There is one important comment we must address. To compute the four-point
function of fluxed twist fields on the strip, we apply a conformal mapping
from the strip to the UHP. This is somewhat similar to the case of computation
of the negativity at finite temperature, where it is well-known that if the
partial transposition involves an infinite part of an infinite system at
finite temperature, using the conformal map from the cylinder to the complex
plane is naively wrong [56]. This is not the case if one, for example, is
interested in the negativity between two (adjacent or disjoint) finite
intervals [57, 58]. In our situation, we are concentrated on the charged Rényi
negativity between two finite intervals. Thus it’s indeed free of these
troubles.

##  4 Evolution of charged logarithmic negativity in boundary state quench

In this section, we will consider the dynamics of the charged logarithmic
negativity between two intervals after a global quench whose post-quench
evolution is governed by a massless Hamiltonian.

###  4.1 Bipartite system

It’s convenient to first study the case in which $A=A_{1}\cup A_{2}$ is the
entire system. In this case, $\rho_{A}$ corresponds to a pure state. As
explained in previous section, in this case, the temporal evolution of the
charged Rényi negativity is governed by
$\langle\mathcal{T}^{2}_{n,\mu}(w_{1})\tilde{\mathcal{T}}^{2}_{n,\mu}(w_{2})\rangle$
on the strip. The strip two-point function can be computed from the one in the
UHP which has the standard form

$\langle\mathcal{T}^{2}_{n,\mu}(w_{1})\tilde{\mathcal{T}}^{2}_{n,\mu}(w_{2})\rangle_{\text{UHP}}=|(z_{1}-\bar{z}_{1})(z_{2}-\bar{z}_{2})\eta_{1,2}|^{-\Delta_{n,\mu}^{(2)}},$
(4.1)

where

$\Delta_{n,\mu}^{(2)}=\begin{cases}\Delta_{n,2\mu},\quad\text{odd}~{}n\\\
2\Delta_{\frac{n}{2},\mu},\quad\text{even}~{}n\end{cases}$ (4.2)

After mapping the correlator in the UHP to the one in the strip, we find

$\begin{split}{\textrm{Tr}}[(\rho_{A}^{T_{2}})^{n}e^{i\mu\mathcal{Q}_{A}}]=\langle\mathcal{T}^{2}_{n,\mu}(w_{1})\tilde{\mathcal{T}}^{2}_{n,\mu}(w_{2})\rangle_{\text{strip}}\\\
=\left(\frac{\pi}{2\tau_{0}}\right)^{2\Delta_{n,\mu}^{(2)}}\prod_{a=1}^{2}\Big{|}\frac{z_{a}}{z_{a}-\bar{z}_{a}}\frac{1}{\eta_{1,2}}\Big{|}^{\Delta_{n,\mu}^{(2)}}.\end{split}$
(4.3)

In the spacetime scaling limit regime $t\gg\tau_{0},|u_{i}-u_{j}|\gg\tau_{0}$,
we have

$\log\Big{|}\frac{z_{a}}{z_{a}-\bar{z}_{a}}\Big{|}\rightarrow-\frac{\pi
t}{2\tau_{0}},\qquad\log\eta_{i,j}\rightarrow\frac{\pi}{2\tau_{0}}(|u_{i}-u_{j}|-\max(2t,|u_{i}-u_{j}|)).$
(4.4)

Then it’s straightforward to derive the charged Rényi logarithmic negativity

$\mathcal{E}_{n}(\mu)=-\frac{\pi\Delta_{n,\mu}^{(2)}}{2\tau_{0}}\min(2t,l_{1}).$
(4.5)

In particular, for $n=1$, the result is

$\mathcal{E}_{1}(\mu)=-\frac{\pi}{2\tau_{0}}h_{1}(2\mu)\min(2t,l_{1}),\qquad
h_{1}(\mu)=-\frac{\mu^{2}}{4\pi^{2}}+\frac{\mu}{2\pi}.$ (4.6)

The charged logarithmic negativity is obtained by taking replica limit
$n_{e}\rightarrow 1$ in $\mathcal{E}_{n_{e}}(\mu)$

$\mathcal{E}(\mu)=-\frac{\pi}{2\tau_{0}}h(\mu)\min(2t,l_{1}),\qquad
h(\mu)=-\frac{1}{2}-\frac{\mu^{2}}{\pi^{2}}+\frac{2\mu}{\pi}.$ (4.7)

###  4.2 Two adjacent intervals

In this case, the quench dynamics of the charged Rényi negativity between two
adjacent intervals can be obtained by studying the three point function
$\langle\mathcal{T}_{n,\mu}(w_{1})\tilde{\mathcal{T}}^{2}_{n,\mu}(w_{2})\mathcal{T}_{n,\mu}(w_{2})\rangle$
on the strip which can be computed by conformal mapping from the three point
function on the UHP. After dropping the non-universal functions, we finally
have

$\begin{split}{\textrm{Tr}}[(\rho_{A}^{T_{2}})^{n}e^{i\mu\mathcal{Q}_{A}}]=\langle\mathcal{T}_{n,\mu}(w_{1})\tilde{\mathcal{T}}^{2}_{n,\mu}(w_{2})\mathcal{T}_{n,\mu}(w_{3})\rangle_{\text{strip}}\\\
=\left(\frac{\pi}{2\tau_{0}}\right)^{\Delta}\prod_{a=1}^{3}\Big{|}\frac{z_{a}}{z_{a}-\bar{z}_{a}}\Big{|}^{\Delta_{(a)}}\left(\frac{\eta_{1,3}^{\Delta_{n,\mu}^{(2)}-2\Delta_{n,\mu}}}{\eta_{1,2}^{\Delta_{n}^{(2)}}\eta_{2,3}^{\Delta_{n}^{(2)}}}\right)^{1/2}.\end{split}$
(4.8)

where $\Delta=2\Delta_{n,\mu}+\Delta^{(2)}_{n,\mu}$,
$\Delta_{(1)}=\Delta_{(3)}=\Delta_{n,\mu}$ and
$\Delta_{(2)}=\Delta^{(2)}_{n,\mu}$. Then the CFT prediction for the time
evolution of the charged logarithmic Rényi negativity is given by

$\mathcal{E}_{n}(\mu)=-\frac{\pi}{4\tau_{0}}\left[\Delta_{n,\mu}^{(2)}(\min(2t,l_{1})+\min(2t,l_{2}))-(\Delta_{n,\mu}^{(2)}-2\Delta_{n,\mu})\min(2t,l_{1}+l_{2})\right].$
(4.9)

where we have defined $l_{1}=u_{2}-u_{1},l_{2}=u_{3}-u_{2}$.

The temporal evolution of the charged logarithmic negativity is obtained by
taking replica limit $n_{e}\rightarrow 1$ in $\mathcal{E}_{n_{e}}(\mu)$. The
result is

$\mathcal{E}(\mu)=-\frac{\pi}{4\tau_{0}}\left[h(\mu)(\min(2t,l_{1})+\min(2t,l_{2}))-h_{2}(\mu)\min(2t,l_{1}+l_{2})\right].$
(4.10)

with

$h_{2}(\mu)=-\frac{1}{2}-\frac{\mu^{2}}{2\pi^{2}}+\frac{\mu}{\pi}.$ (4.11)

From eq. (4.9), we have

$\mathcal{E}_{1}(\mu)=-\frac{\pi}{4\tau_{0}}\left[h_{1}(2\mu)(\min(2t,l_{1})+\min(2t,l_{2}))-h_{3}(\mu)\min(2t,l_{1}+l_{2})\right],$
(4.12)

where $h_{3}(\mu)=-\frac{\mu^{2}}{2\pi^{2}}$.

###  4.3 Two disjoint intervals

The universal part the charged Rényi negativity is

$\begin{split}&{\textrm{Tr}}\rho_{A}^{n}e^{i\mu\mathcal{Q}_{A}}=\langle\mathcal{T}_{n,\mu}(w_{1})\tilde{\mathcal{T}}_{n,\mu}(w_{2})\tilde{\mathcal{T}}_{n,\mu}(w_{3})\mathcal{T}_{n,\mu}(w_{4})\rangle_{\text{strip}}\\\
&=\left(\frac{\pi}{2\tau_{0}}\right)^{4\Delta_{n,\mu}}\prod_{a=1}^{4}\Big{|}\frac{z_{a}}{z_{a}-\bar{z}_{a}}\Big{|}^{\Delta_{n,\mu}}\frac{1}{\eta_{1,2}^{\Delta_{n,\mu}}\eta_{3,4}^{\Delta_{n,\mu}}}\left(\frac{\eta_{1,4}\eta_{2,3}}{\eta_{1,3}\eta_{2,4}}\right)^{\Delta^{(2)}_{n,\mu}/2-\Delta_{n,\mu}}.\end{split}$
(4.13)

In this case, the CFT prediction for the time evolution of the charged
logarithmic Rényi negativity is given by

$\begin{split}&\mathcal{E}_{n}(\mu)=-\frac{\pi}{2\tau_{0}}\big{[}\Delta_{n,\mu}\left(\min(2t,l_{1})+\min(2t,l_{2})\right)+(\Delta_{n,\mu}^{(2)}/2-\Delta_{n,\mu})\\\
&\times(\max(2t,l_{1}+l_{2}+d)+\max(2t,d)-\max(2t,l_{1}+d)-\max(2t,l_{2}+d))\big{]}.\end{split}$
(4.14)

where $l_{1}=u_{2}-u_{1},l_{2}=u_{4}-u_{3}$ and $d=u_{3}-u_{2}$ are the length
of the subsubsystem $A_{1},A_{2}$ and the distance between them respectively.

The charged logarithmic negativity is easily obtained as

$\begin{split}&\mathcal{E}(\mu)=-\frac{\pi}{2\tau_{0}}\big{[}h_{1}(\mu)\left(\min(2t,l_{1})+\min(2t,l_{2})\right)+\frac{1}{2}h_{2}(\mu)\\\
&\times(\max(2t,l_{1}+l_{2}+d)+\max(2t,d)-\max(2t,l_{1}+d)-\max(2t,l_{2}+d))\big{]}.\end{split}$
(4.15)

As before, we also report the expression of $\mathcal{E}_{1}(\mu)$ here since
it will be useful in the computation of charge imbalance resolved negativity.
The result is

$\begin{split}&\mathcal{E}_{1}(\mu)=-\frac{\pi}{2\tau_{0}}[h_{1}(\mu)(\min(2t,l_{1})+\min(2t,l_{2}))\\\
&+\frac{1}{2}h_{3}(\mu)(\max(2t,l_{1}+l_{2}+d)+\max(2t,d)-\max(2t,l_{1}+d)-\max(t,l_{2}+d))].\end{split}$
(4.16)

We stress that eq. (4.9) and eq. (4.14) are only valid for CFT in which there
is a perfect linear dispersion. However this is not the case for the
underlying lattice model, where the excitation has non-linear dispersion. We
will discuss how to adapt eq. (4.9) and eq. (4.14) to describe the dynamics of
charged Rényi (logarithmic) negativity in the mass quench protocol of the
complex harmonic chain.

##  5 Charge imbalance resolved negativity

In this section, using the results obtained in the last section, we will
compute the dynamics of the charge imbalance resolved negativity between two
intervals after a global quench to a conformal Hamiltonian. According to the
discussion in section 2, our strategy is to first compute
$\mathcal{Z}_{T_{2}}(\mathrm{q})$ and $p(\mathrm{q})$ (cf. eq. (2.19) and eq.
(2.16)) from the charged moments of partially transposed RDM, and then the
charge imbalance resolved negativity is given by eq. (2.19).

###  5.1 Bipartite system

$\boldsymbol{t<l_{1}/2}$

In this time region, we have

$\mathcal{E}(\mu)=-h(\mu)\frac{\pi
t}{\tau_{0}},\qquad\mathcal{E}_{1}(\mu)=-h_{1}(2\mu)\frac{\pi t}{\tau_{0}}.$
(5.1)

After Fourier transformation, we find

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{(-1)^{\mathrm{q}}e^{\frac{\pi\mathrm{q}^{2}}{4t/\tau_{0}}-\frac{\pi
t}{2\tau_{0}}}}{2\sqrt{t/\tau_{0}}}{\textrm{Re}}\left[\mathrm{Erfi}\left(\frac{\sqrt{\pi}(i\mathrm{q}+2t/\tau_{0})}{2\sqrt{t/\tau_{0}}}\right)\right],$
(5.2)

where $\mathrm{Erfi}(x)$ is the imaginary error function

$\mathrm{Erfi}(x)=\frac{-2i}{\sqrt{\pi}}\int_{0}^{ix}dte^{-t^{2}}\xrightarrow{x\rightarrow\infty}\frac{e^{x^{2}}}{\sqrt{\pi}x}.$
(5.3)

A very similar expression exists for $p(\mathrm{q})$ which we omit here. In
the spacetime scaling limit, we have much more concise results

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{2e^{\frac{\pi
t}{2\tau_{0}}}t/\tau_{0}}{\pi\mathrm{q}^{2}+4\pi t^{2}/\tau_{0}^{2}}$ (5.4)

and the probability distribution is given by

$p(\mathrm{q})=\frac{(1+(-1)^{\mathrm{q}})t/\tau_{0}}{\pi\mathrm{q}^{2}+\pi
t^{2}/\tau_{0}^{2}}.$ (5.5)

Thus the charge imbalance resolved negativity is obtained from eq. (2.19)

$\mathcal{N}(\mathrm{q})=\frac{e^{\frac{\pi
t}{2\tau_{0}}}(\mathrm{q}^{2}+t^{2}/\tau_{0}^{2})}{(1+(-1)^{\mathrm{q}})(\mathrm{q}^{2}+4t^{2}/\tau_{0}^{2})}-\frac{1}{2}.$
(5.6)

$\boldsymbol{t>l_{1}/2}$

In this time region, we can simply make the replacement $t\rightarrow l_{1}/2$
to obtain corresponding quantities. In particular, the charge imbalance
resolved negativity is given by

$\mathcal{N}(\mathrm{q})=\frac{e^{\frac{\pi
l_{1}}{4\tau_{0}}}(4\mathrm{q}^{2}+l_{1}^{2}/\tau_{0}^{2})}{4(1+(-1)^{\mathrm{q}})(\mathrm{q}^{2}+l_{1}^{2}/\tau_{0}^{2})}-\frac{1}{2}.$
(5.7)

###  5.2 Two adjacent intervals

Without loss of generality, we will assume $l_{1}<l_{2}$, the other case can
be worked out similarly.

$\boldsymbol{t<l_{1}/2<l_{2}/2<(l_{1}+l_{2})/2}$

For this early time, we have

$\mathcal{E}(\mu)=-(2h(\mu)-h_{2}(\mu))\frac{\pi
t}{2\tau_{0}},\qquad\mathcal{E}_{1}(\mu)=-(2h_{1}(2\mu)-h_{3}(\mu))\frac{\pi
t}{2\tau_{0}}.$ (5.8)

Then after Fourier transformation, we find

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{(-1)^{\mathrm{q}}e^{\frac{\pi\mathrm{q}^{2}}{3t/\tau_{0}}-\frac{\pi
t}{2}}}{\sqrt{3t/\tau_{0}}}{\textrm{Re}}\left[\mathrm{Erfi}\left(\frac{\sqrt{\frac{\pi}{3}}(2i\mathrm{q}+3t/\tau_{0})}{2\sqrt{t/\tau_{0}}}\right)\right],$
(5.9)

Therefore in the spacetime scaling limit ($t\gg\tau_{0}$), we have

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{6e^{\frac{\pi
t}{4\tau_{0}}}t/\tau_{0}}{4\pi\mathrm{q}^{2}+9\pi(t/\tau_{0})^{2}}.$ (5.10)

In the spacetime scaling regime, the probability distribution can be computed
similarly and the finial result is

$p(\mathrm{q})=\frac{t/\tau_{0}}{\pi\mathrm{q}^{2}+\pi t^{2}/\tau_{0}^{2}}.$
(5.11)

Then the charge imbalance resolved negativity is given by

$\mathcal{N}(\mathrm{q})=\frac{3e^{\frac{\pi
t}{4\tau_{0}}}(\mathrm{q}^{2}+t^{2}/\tau_{0}^{2})}{4\mathrm{q}^{2}+9t^{2}/\tau_{0}^{2}}-\frac{1}{2}.$
(5.12)

$\boldsymbol{l_{1}/2<t<l_{2}/2<(l_{1}+l_{2})/2}$

In this case we have

$\mathcal{E}(\mu)=-(h(\mu)-h_{2}(\mu))\frac{\pi t}{2\tau_{0}}-h(\mu)\frac{\pi
l_{1}}{4\tau_{0}}.$ (5.13)

Then

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{(-1)^{\mathrm{q}}e^{\frac{\pi\mathrm{q}^{2}}{(l_{1}+t)/\tau_{0}}-\frac{\pi(l_{1}+2t)}{8\tau_{0}}}}{\sqrt{(l_{1}+t)/\tau_{0}}}{\textrm{Re}}\left[\mathrm{Erfi}\left(\frac{\sqrt{\pi}(2i\mathrm{q}+(l_{1}+t)/\tau_{0})}{2\sqrt{(l_{1}+t)/\tau_{0}}}\right)\right].$
(5.14)

In the spacetime scaling limit

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{2e^{\frac{\pi
l_{1}}{8\tau_{0}}}(l_{1}+t)/\tau_{0}}{4\pi\mathrm{q}^{2}+\pi(l_{1}+t)^{2}/\tau_{0}^{2}}.$
(5.15)

The probability distribution is

$p(\mathrm{q})=\frac{4(l_{1}+2t)/\tau_{0}}{16\pi\mathrm{q}^{2}+\pi(l_{1}+2t)^{2}/\tau_{0}^{2}}.$
(5.16)

Then the charge-imbalance resolved negativity is

$\mathcal{N}(\mathrm{q})=\frac{16\mathrm{q}^{2}\tau_{0}^{2}+(l_{1}+2t)^{2}}{16\mathrm{q}^{2}\tau_{0}^{2}+4(l_{1}+t)^{2}}\frac{(l_{1}+t)e^{\frac{\pi
l_{1}}{8\tau_{0}}}}{l_{1}+2t}-\frac{1}{2}.$ (5.17)

$\boldsymbol{l_{2}/2<t<(l_{1}+l_{2})/2}$

In this time region, we have

$\mathcal{E}(\mu)=h_{2}(\mu)\frac{\pi
t}{2\tau_{0}}-h(\mu)\frac{(l_{1}+l_{2})\pi}{4\tau_{0}},\qquad\mathcal{E}_{1}(\mu)=-\frac{\pi}{4\tau_{0}}(h_{1}(2\mu)(l_{1}+l_{2})-2th_{3}(\mu)).$
(5.18)

Then applying Fourier transformation, we find

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{(-1)^{\mathrm{q}}e^{\frac{\pi\mathrm{q}^{2}}{(l-t)/\tau_{0}}-\frac{\pi
l}{8\tau_{0}}}}{\sqrt{(l-t)/\tau_{0}}}{\textrm{Re}}\left[\mathrm{Erfi}\left(\frac{\sqrt{\pi}(2i\mathrm{q}+(l-t)/\tau_{0})}{2\sqrt{(l-t)/\tau_{0}}}\right)\right].$
(5.19)

Here and in the following sections we will always use the definition $l\equiv
l_{1}+l_{2}$. In the spacetime scaling limit

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{2e^{\frac{\pi(l-2t)}{8\tau_{0}}}(l-t)/\tau_{0}}{4\pi\mathrm{q}^{2}+\pi(l-t)^{2}/\tau_{0}^{2}}.$
(5.20)

the probability distribution is

$p(\mathrm{q})=\frac{4l/\tau_{0}}{16\pi\mathrm{q}^{2}+\pi
l^{2}/\tau_{0}^{2}}.$ (5.21)

the charge imbalance resolved negativity is then easily obtained as

$\mathcal{N}(\mathrm{q})=\frac{16\mathrm{q}^{2}\tau_{0}^{2}+l^{2}}{16\mathrm{q}^{2}\tau_{0}^{2}+4(l-t)^{2}}\frac{(l-t)e^{\frac{\pi(l-2t)}{8\tau_{0}}}}{l}-\frac{1}{2}.$
(5.22)

$\boldsymbol{t>(l_{1}+l_{2})/2}$

In this case, we have

$\begin{split}\mathcal{E}(\mu)=-[h(\mu)-h_{2}(\mu)]\frac{\pi(l_{1}+l_{2})}{4\tau_{0}}=\mathcal{E}_{1}(\mu).\end{split}$
(5.23)

Then the charge imbalance resolved negativity is

$\mathcal{N}(\mathrm{q})=0.$ (5.24)

###  5.3 Two disjoint intervals

The computation of charge imbalance resolved negativity for two disjoint
intervals is similar to previous subsection, here we just report the final
results (the space time scaling limit has been taken). In this circumstance,
the different size relation between $l_{1},l_{2}$ and $d$ may lead to
different behavior. However, for simplicity, we will only consider the case
$d<l_{1}<l_{2}$.

$\boldsymbol{0<t<d/2<l_{1}/2}$

At this early time stage, we have $\mathcal{E}(\mu)=\mathcal{E}_{1}(\mu)$,
therefore

$\mathcal{N}(\mathrm{q})=0.$ (5.25)

$\boldsymbol{d/2<t<l_{1}/2}$

In this time period, we have

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{4e^{\frac{\pi(2t-d)}{8\tau_{0}}}(6t-d)/\tau_{0}}{16\pi\mathrm{q}^{2}+\pi(6t-d)^{2}/\tau_{0}^{2}},\quad
p(\mathrm{q})=\frac{t/\tau_{0}}{\pi\mathrm{q}^{2}+\pi t^{2}/\tau_{0}^{2}}.$
(5.26)

Then the charge imbalance resolved negativity is obtained straightforward as

$\mathcal{N}(\mathrm{q})=\frac{2e^{\frac{\pi(2t-d)}{8\tau_{0}}}(\mathrm{q}^{2}\tau_{0}^{2}+t^{2})}{16\mathrm{q}^{2}\tau_{0}^{2}+(6t-d)^{2}}\frac{6t-d}{t}-\frac{1}{2}.$
(5.27)

$\boldsymbol{l_{1}/2<l_{2}/2<t<(l_{1}+d)/2}$

In this time region, the following results are obtained (in the spacetime
scaling limit)

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{4e^{\frac{\pi(2t-d)}{8\tau_{0}}}(l-d+2t)/\tau_{0}}{16\pi\mathrm{q}^{2}+\pi(l-d+2t)^{2}/\tau_{0}^{2}},\quad
p(\mathrm{q})=\frac{4l/\tau_{0}}{16\pi\mathrm{q}^{2}+\pi l^{2}/\tau_{0}^{2}}.$
(5.28)

The charge imbalance resolved negativity is given by

$\mathcal{N}(\mathrm{q})=\frac{e^{\frac{\pi(2t-d)}{8\tau_{0}}}(16\mathrm{q}^{2}\tau_{0}^{2}+l^{2})}{32\mathrm{q}^{2}\tau_{0}^{2}+2(2t+l-d)^{2}}\frac{2t+l-d}{l}-\frac{1}{2}.$
(5.29)

$\boldsymbol{(l_{2}+d)/2<t<(l_{1}+l_{2}+d)/2}$

In this time period, we have

$\mathcal{Z}_{T_{2}}(\mathrm{q})=\frac{4e^{\frac{\pi(d+l-2t)}{8\tau_{0}}}(d+2l-2t)/\tau_{0}}{16\pi\mathrm{q}^{2}+\pi(d+2l-2t)^{2}/\tau_{0}^{2}},\quad
p(\mathrm{q})=\frac{4l/\tau_{0}}{16\pi\mathrm{q}^{2}+\pi l^{2}/\tau_{0}^{2}}.$
(5.30)

The charge imbalance resolved negativity is

$\mathcal{N}(\mathrm{q})=\frac{e^{\frac{\pi(d+l-2t)}{8\tau_{0}}}(16\mathrm{q}^{2}\tau_{0}^{2}+l^{2})}{32\mathrm{q}^{2}\tau_{0}^{2}+2(d+2l-2t)^{2}}\frac{d+2l-2t}{l}-\frac{1}{2}.$
(5.31)

$\boldsymbol{t>(l_{1}+l_{2}+d)/2}$

In this late time region, we have $\mathcal{E}(\mu)=\mathcal{E}_{1}(\mu)$
again, and charge imbalance resolved negativity is zero

$\mathcal{N}(\mathrm{q})=0.$ (5.32)

(a)

Figure 1: Numerical data of charged Rényi logarithmic negativity
$\mathcal{E}_{2}(\mu)$ as a function of $t/l$ for
$\mu=\frac{\pi}{8},\frac{\pi}{2},\pi$ in the complex harmonic chain. The full
lines are the CFT predictions (cf. eq. (4.9) and eq. (4.14)). Left panel:
Adjacent interval with $L=3000,l_{1}=200,l_{2}=300$. Right panel: Disjoint
interval with $L=3000,l_{1}=200,l_{2}=300,d=100$. We have used the best fitted
value of $\tau_{0}$.

###  5.4 Total negativity

Having obtained the quench dynamics of the charge imbalance resolved
negativity, it’s an easy task to check whether we can recover the known
results of quench dynamics of entanglement negativity. Using the formula

$\sum_{\mathrm{q}=-\infty}^{+\infty}\frac{1}{\pi}\frac{a^{2}}{\mathrm{q}^{2}+a^{2}}=\coth(\pi
a)\xrightarrow{a\rightarrow\infty}1$ (5.33)

and the reconstruction formula of negativity eq. (2.10), it’s easy to check
that indeed in each time period, we can obtain the total entanglement
negativity from the charge imbalance resolved one, i.e. in the spacetime
scaling limit, we have the following relation

$\sum_{\mathrm{q}=-\infty}^{+\infty}p(\mathrm{q})\mathcal{N}(\mathrm{q})=\frac{1}{2}\exp\\{\mathcal{E}(\mu=0)\\}-\frac{1}{2}=\mathcal{N}.$
(5.34)

In this way, we have recovered the known quench dynamics [50] of the total
logarithmic negativity from the charge imbalance resolved ones, providing
evidence of the correctness of our results.

##  6 Numerical test

In this section, we test our analytical predictions against numerical
computation in the complex harmonic chain which is the lattice version of our
complex Klein-Gordon field theory. We will use the correlation matrix
techniques to obtain the charged Rényi (logarithmic) negativity.

In section 3, we have already reported the Hamiltonian and the $U(1)$
conserved charge of the complex harmonic chain (cf. eq. (3.9) and eq. (3.10)).
From the dispersion relation, it’s easy to see that the Hamiltonian has zero
modes for $k=0$ and $m=0$, and the group velocities are obtained from the
dispersion relation as

$v_{p}\equiv\frac{\partial e(p)}{\partial
p}=\frac{\sin(p)}{\sqrt{m^{2}+4\sin^{2}(p/2)}},\qquad p\equiv\frac{2\pi
k}{L}.$ (6.1)

The maximum velocity $v_{\text{max}}\equiv\max_{p}v_{p}$ determine s the
spreading of entanglement and correlations.

In the global quench protocol, for $t<0$, the system is prepared in the ground
state of the Hamiltonian

$H_{CHC}(m_{0})=\sum_{k=0}^{L-1}e_{0,k}(a_{k}^{\dagger}a_{k}+b_{k}^{\dagger}b_{k}),\quad
e_{0,k}=\sqrt{m_{0}^{2}+4\sin^{2}\left(\frac{\pi k}{L}\right)}.$ (6.2)

At $t=0$, the system evolves unitarily according to a new Hamiltonian
$H_{CHC}(m)$ with a different value $m$, namely
$\ket{\psi(t)}=e^{-iH_{CHC}(m)t}\ket{\psi_{0}}$. The dynamics of symmetry
resolved negativity after a global quench described in the previous section
corresponds to set the new parameter $m=0$. In this section, we will focus on
the global quench to a massless Hamiltonian (i.e. $m=0$) to test our CFT
predictions. In the next section we will discuss the case $m\neq 0$.

Since the Hamiltonian of a complex harmonic chain is just two copies of a real
harmonic chain’s Hamiltonian, we could focus on the real harmonic chain first,
and take the double copies into account only at the end of the calculations.
In the correlation matrix method of computing entanglement measures, we first
need to know the following two-point correlators of the real scalars

$\begin{split}\mathbf{X}_{rs}(t)\equiv\bra{\psi_{0}}\phi_{r}(t)\phi_{s}(t)\ket{\psi_{0}},\\\
\mathbf{P}_{rs}(t)\equiv\bra{\psi_{0}}\pi_{r}(t)\pi_{s}(t)\ket{\psi_{0}},\\\
\mathbf{M}_{rs}(t)\equiv\bra{\psi_{0}}\phi_{r}(t)\pi_{s}(t)\ket{\psi_{0}},\\\
\end{split}$ (6.3)

where

$\phi_{r}(t)=e^{iH_{HC}(m)t}\phi_{r}(0)e^{-iH_{HC}(m)t},\qquad\pi_{r}(t)=e^{iH_{HC}(m)t}\pi_{r}(0)e^{-iH_{HC}(m)t}.$
(6.4)

The explicit form of these correlators is given by

$\begin{split}&\mathbf{X}_{rs}(t)=\frac{1}{2L}\sum_{k=0}^{L-1}X_{k}(t)\cos\left[\frac{2\pi
k}{L}(r-s)\right],\\\
&\mathbf{P}_{rs}(t)=\frac{1}{2L}\sum_{k=0}^{L-1}P_{k}(t)\cos\left[\frac{2\pi
k}{L}(r-s)\right],\\\
&\mathbf{M}_{rs}(t)=\frac{i}{2}\delta_{rs}-\frac{1}{2L}\sum_{k=0}^{L-1}M_{k}(t)\cos\left[\frac{2\pi
k}{L}(r-s)\right],\end{split}$ (6.5)

where

$\begin{split}&X_{k}(t)=\frac{1}{e_{k}}\left(\frac{e_{k}}{e_{0,k}}\cos^{2}(e_{k}t)+\frac{e_{0,k}}{e_{k}}\sin^{2}(e_{k}t)\right),\\\
&P_{k}(t)=e_{k}\left(\frac{e_{k}}{e_{0,k}}\sin^{2}(e_{k}t)+\frac{e_{0,k}}{e_{k}}\cos^{2}(e_{k}t)\right),\\\
&M_{k}(t)=\left(\frac{e_{k}}{e_{0,k}}-\frac{e_{0,k}}{e_{k}}\right)\sin(e_{k}t)\cos(e_{k}t).\end{split}$
(6.6)

(a)

Figure 2: Numerical data of charged logarithmic negativity $\mathcal{E}(\mu)$
as a function of $t/l$ for $\mu=\frac{\pi}{6},\frac{\pi}{3},\pi$ in the
complex harmonic chain. The full lines are the CFT predictions (cf. eq. (4.10)
and eq. (4.15)). Left panel: Adjacent interval with
$L=2000,l_{1}=200,l_{2}=250$. Right panel: Disjoint interval with
$L=5000,l_{1}=250,l_{2}=400,d=100$. We have used the best fitted value of
$\tau_{0}$.

From the above expressions, it’s easy to find that the contribution from the
zero mode $m=0$ and $k=0$ is finite

$X_{0}(t)=m_{0}^{-1}+m_{0}t^{2},\quad P_{0}(t)=m_{0},\quad M_{0}(t)=-m_{0}t.$
(6.7)

The evolution of charged Rényi (logarithmic) negativity of any subsystem $A$
containing $l$ lattice sites can be computed from these time-dependent
correlators. Firstly, we should consider the correlation matrices
$\mathbf{X}_{A}(t)$,$\mathbf{P}_{A}(t)$ and $\mathbf{M}_{A}(t)$ obtained by
restricting the the indices of the corresponding correlation matrices to the
sites belonging to $A$. Given $\mathbf{X}_{A}$,$\mathbf{P}_{A}$ and
$\mathbf{M}_{A}$, the covariance matrix $\Gamma_{A}$ and the symplectic matrix
$J_{A}$ associated to the subsystem $A$ are

$\Gamma_{A}(t)=\begin{pmatrix}\mathbf{X}_{A}(t)&\mathbf{M}_{A}(t)\\\
\mathbf{M}_{A}(t)^{\mathrm{t}}&\mathbf{P}_{A}(t)\end{pmatrix},\qquad
J_{A}=\begin{pmatrix}\mathbf{0}_{l}&\mathbf{I}_{l}\\\
-\mathbf{I}_{l}&\mathbf{0}_{l}\end{pmatrix},\qquad$ (6.8)

where $\mathbf{0}_{l}$ and $\mathbf{I}_{l}$ are $l\times l$ zero matrix and
identity matrix respectively. Then find the eigenvalues of the $2l\times 2l$
matrix $iJ_{A}\Gamma_{A}(t)$ which we denoted it by
$\\{\pm\sigma_{1}(t),\cdots,\pm\sigma_{l}(t)\\}$. It’s also convenient to
introduce the Fock space basis
$\ket{\mathbf{n}}\equiv\otimes_{j=1}^{l}\ket{n_{j}}$, defined by products of
eigenstates of the number operator in the subsystem $A$, the reduced density
matrix of $A$ (in a real harmonic chain) can be written as

$\rho_{A}(t)=\sum_{\mathbf{n}}\prod_{j=1}^{l}\frac{1}{\sigma_{j}(t)+1/2}\left(\frac{\sigma_{j}(t)-1/2}{\sigma_{j}(t)+1/2}\right)^{n_{j}}\ket{\mathbf{n}}\bra{\mathbf{n}}.$
(6.9)

In the Fock basis $\\{\ket{\mathbf{n}}\\}$, $Q_{2}^{T_{2}}=Q_{2}$ and the
operator $\mathcal{Q}_{A}=Q_{1}-Q^{T_{2}}_{2}=Q_{1}-Q_{2}$ becomes exactly the
charge imbalance operator. For the complex harmonic chain, the charged Rényi
negativity factorises as

$R_{n}(\mu)={\textrm{Tr}}[(\rho^{T_{2}}_{A})^{n}e^{\mathrm{i}\mu\mathcal{Q}_{A}^{a}}]\times{\textrm{Tr}}[(\rho^{T_{2}}_{A})^{n}e^{-\mathrm{i}\mu\mathcal{Q}_{A}^{b}}].$
(6.10)

In bosonic system, the net effect of partial transposition with respect to
$A_{2}$ is changing the sign of the momenta corresponding to $A_{2}$. Thus the
momenta correlators in the partial transposed density matrix can be obtained
from $\mathbf{P}_{A}$ by simply change the sign of the momenta that in
$A_{2}$, i.e.
$\mathbf{P}_{A}^{T_{2}}=\mathbf{R}_{A_{2}}\mathbf{P}_{A}\mathbf{R}_{A_{2}}$,where
$\mathbf{R}_{A_{2}}$ is the $l_{2}\times l_{2}$ diagonal matrix with elements
$(\mathbf{R}_{A_{2}})_{rs}=(-1)^{\delta_{r\in A_{2}}}\delta_{rs}$. Thus the
partial transformed covariance matrix is

$\Gamma^{T_{2}}_{A}(t)=\begin{pmatrix}\mathbf{I}_{l}&\mathbf{0}_{l}\\\
\mathbf{0}_{l}&\mathbf{R}_{A_{2}}\end{pmatrix}\Gamma_{A}(t)\begin{pmatrix}\mathbf{I}_{l}&\mathbf{0}_{l}\\\
\mathbf{0}_{l}&\mathbf{R}_{A_{2}}.\end{pmatrix}$ (6.11)

If we denote the eigenvalues of $iJ_{A}\Gamma_{A}(t)^{T_{2}}$ by
$\\{\pm\tau_{1}(t),\pm\tau_{2}(t),\cdots,\pm\tau_{l}(t)\\}$, then the charged
Rényi logarithmic negativity is given by222This formula is consistent with the
eq.(6.18) in paper [40].

$\mathcal{E}_{n}(\mu)=-2\sum_{j=1}^{l}\log\Big{|}\left(\tau_{j}(t)+\frac{1}{2}\right)^{n}-e^{\mathrm{i}\mu}\left(\tau_{j}(t)-\frac{1}{2}\right)^{n}\Big{|}$
(6.12)

and the charged logarithmic negativity is

$\mathcal{E}(\mu)=-2\sum_{j=1}^{l}\log\Big{|}|\tau_{j}(t)+\frac{1}{2}|-e^{\mathrm{i}\mu}|\tau_{j}(t)-\frac{1}{2}|\Big{|}.$
(6.13)

The numerical data of the dynamics of the charged Rényi (logarithmic)
negativity are shown in fig. 1 and fig. 2, in which the CFT predictions are
drawn with the full line for comparison.

##  7 Global mass quench of the harmonic chain

In this section, we consider a global quantum quench in which the complex
harmonic chain is initially prepared in the ground state with mass $m_{0}$ and
at time $t=0$ the mass is quenched to a different (non-zero) value $m\neq
m_{0}$.

###  7.1 Quasi-particle picture for entanglement entropy

In free scalar theory, since the particle numbers in each momentum mode are
conserved quantities, the time evolution leads to local and quasi-local
observables converging to their average values in the Generalized Gibbs
Ensemble (GGE) instead of ordinary Gibbs ensemble.

The quasi-particle picture of the time evolution of entanglement after a
global quantum quench has been proposed in [45], The basic idea is that one
can view the initial state as a source of quasi-particle excitations. Pairs of
particles emitted from the same point in space are highly entangled whereas
particles produced from points far apart are incoherent. We assume that the
pairs of quasi-particle are created uniformly with opposite momenta $(p,-p)$.
After the production, these quasi-particles travel ballistically with velocity
$v_{p}=-v_{-p}$. The entanglement entropy and Rényi entropy of subsystem $A$
are proportional to the pairs of entangled quasi-particle shared with its
complement at a given time $t$.

In free models, we have

$S_{n}(t)=\int\frac{dp}{2\pi}s^{(n)}_{GGE}(p)\min(2|v_{p}|t,l),$ (7.1)

with $l$ being the length of subsystem $A$ and $s^{(n)}_{GGE}(p)$ is momentum
space density of the Rényi entropies in the GGE thermodynamic state [59].
Recall that the well-known result of the CFT prediction of the quench to
massless dynamics of Rényi entropies

$S_{n}(t)=\frac{1}{n-1}\frac{\pi\Delta_{n,0}}{2\tau_{0}}\min(2t,l).$ (7.2)

We can formally obtained the formula eq. (7.1) from the CFT prediction eq.
(7.2) by replacing $t\rightarrow|v_{p}|t$,
$-\frac{\pi\Delta_{n,\mu}}{2\tau_{0}}\rightarrow s^{(n)}_{GGE}(p)$ and then
integrating over all possible $p$.

(a)

Figure 3: Charged Rényi logarithmic negativity $\mathcal{E}_{2}(\mu)$ as a
function of $t/l$ for different $\mu$ after the mass quench from $m_{0}=1$ to
$m=2$ in the complex harmonic chain. The full lines are the quasi-particle
predictions (cf. eq. (7.10) and eq. (7.13)). Left panel: Adjacent intervals
with $L=2500,l_{1}=200,l_{2}=300$ and for
$\mu=\frac{\pi}{8},\frac{\pi}{3},\frac{\pi}{2}$. Right panel: Disjoint
intervals with $L=2500,l_{1}=150,l_{2}=200,d=50$ and for
$\mu=\frac{\pi}{10},\frac{\pi}{5},\frac{\pi}{2}$. As shown in the figure, the
agreement is extremely excellent.

###  7.2 Quasi-particle picture for charged logarithmic negativity

The complex harmonic chain is a free models and the stationary behavior of
local and quasi-local observables is described by the GGE

$\rho_{GGE}=Z^{-1}e^{-\sum_{k}\lambda_{k}^{(a)}a_{k}^{\dagger}a_{k}}\otimes
e^{-\sum_{k}\lambda_{k}^{(b)}b_{k}^{\dagger}b_{k}},$ (7.3)

where $\lambda_{k}^{(a)},\lambda_{k}^{(b)}$ are Lagrange mulitpliers and $Z$
is the normalisation constant such that ${\textrm{Tr}}\rho_{GGE}=1$. We have

$Z=\prod_{i=a,b}{\textrm{Tr}}e^{-\sum_{k}\lambda_{k}^{(i)}\hat{n}_{k}^{(i)}}=\prod_{i=a,b}\prod_{k}\sum_{n_{k}^{(i)}=0}^{\infty}e^{-\lambda_{k}^{(i)}n_{k}^{(i)}}=\prod_{i=a,b}\prod_{k}(1-e^{-\lambda_{k}^{(i)}})^{-1}.$
(7.4)

The mode occupation number $\hat{n}_{k}^{(a)}\equiv
a_{k}^{\dagger}a_{k},\hat{n}_{k}^{(b)}\equiv b_{k}^{\dagger}b_{k}$ are
conserved quantities. Therefore we have

$\bra{\psi_{0}}\hat{n}_{k}^{(i)}\ket{\psi_{0}}={\textrm{Tr}}[\hat{n}_{k}^{(i)}\rho_{GGE}]=(e^{\lambda_{k}^{(i)}}-1)^{-1}.$
(7.5)

In the ground state $\ket{\psi_{0}}$, we have
$\bra{\psi_{0}}\hat{n}_{k}^{(a)}\ket{\psi_{0}}=\bra{\psi_{0}}\hat{n}_{k}^{(b)}\ket{\psi_{0}}\equiv
n_{k}$, thus

$\lambda_{k}^{(a)}=\lambda_{k}^{(b)}=\log(1+n_{k}^{-1})\equiv\lambda_{k}.$
(7.6)

Since the quasi-particle picture is not sensitive to $A_{1}\cup A_{2}$ not
being a pure state. For our purpose, it’s sufficient to compute the charged
moments of $\rho_{GGE}$

$\begin{split}&{\textrm{Tr}}[\rho_{GGE}^{n}e^{i\mu
Q_{A}}]=Z^{-n}\prod_{k}\big{|}{\textrm{Tr}}[e^{-(n\lambda_{k}-i\mu)a_{k}^{\dagger}a_{k}}]\big{|}^{2}\\\
&=Z^{-n}\prod_{k}\big{|}\sum_{n_{k}=0}^{\infty}e^{-(n\lambda_{k}-i\mu)n_{k}}\big{|}^{2}=\prod_{k}\Big{|}\frac{(1-e^{-\lambda_{k}})^{n}}{1-e^{-(n\lambda_{k}-i\mu)}}\Big{|}^{2}\\\
&=\prod_{k}|(1+n_{k})^{n}-e^{i\mu}n_{k}^{n}|^{-2}.\end{split}$ (7.7)

We have find the density of the logarithmic charged moment in momentum space

$\epsilon_{n,\mu}(k)=-2\log|(1+n_{k})^{n}-e^{i\mu}n_{k}^{n}|.$ (7.8)

It’s easy to derive that

$n_{k}=\frac{1}{4}\left(\frac{e_{k}}{e_{0,k}}+\frac{e_{0,k}}{e_{k}}\right)-\frac{1}{2}.$
(7.9)

(a)

Figure 4: Charged logarithmic negativity $\mathcal{E}(\mu)$ as a function of
$t/l$ for different $\mu$ after the mass quench from $m_{0}=1$ to $m=2$ in the
complex harmonic chain. The full lines are the quasi-particle predictions (cf.
eq. (7.11) and eq. (7.14)). Left panel: Adjacent intervals with
$L=2500,l_{1}=200,l_{2}=300$ and for $\mu=\frac{\pi}{8},\frac{\pi}{4},\pi$.
Right panel: Disjoint intervals with $L=2500,l_{1}=200,l_{2}=300,d=150$ and
for $\mu=\frac{\pi}{8},\frac{\pi}{4},\pi$. As shown in the figure, the
agreement is perfect.

Then we have all the ingredients to give conjectures about the dynamics of
charged Rényi (logarithmic) negativity after a global mass quench in our free
lattice model.

Adjacent interval

For the adjacent intervals, eq. (4.9) suggest the following formula of quench
dynamics of charged Rényi logarithmic negativity

$\mathcal{E}_{n}(\mu)=\int\frac{dp}{2\pi}\left[\epsilon_{n,\mu}^{(2)}(p)/2(\min(2|v_{p}|t,l_{1})+\min(2|v_{p}|t,l_{2}))-(\epsilon_{n,\mu}^{(2)}(p)/2-\epsilon_{n,\mu}(p))\min(2|v_{p}|t,l)\right].$
(7.10)

The charged logarithmic negativity is obtained by taking the limit
$n_{e}\rightarrow 1$ in $\mathcal{E}_{n_{e}}(\mu)$

$\mathcal{E}(\mu)=\int\frac{dp}{2\pi}\left[\epsilon_{1/2,\mu}(p)(\min(2|v_{p}|t,l_{1})+\min(2|v_{p}|t,l_{2}))-(\epsilon_{1/2,\mu}(p)-\epsilon_{1,\mu}(p))\min(2|v_{p}|t,l)\right].$
(7.11)

where we have defined

$\epsilon_{n,\mu}^{(2)}(p)=\begin{cases}\epsilon_{n,2\mu}(p),\quad\text{odd}~{}n\\\
2\epsilon_{\frac{n}{2},\mu}(p),\quad\text{even}~{}n\end{cases}$ (7.12)

Disjoint intervals

In this case, according to eq. (4.14), we conjecture that the time evolution
of charged Rényi logarithmic negativity after a global mass quench is given by

$\begin{split}&\mathcal{E}_{n}(\mu)=\int\frac{dp}{2\pi}\big{[}\epsilon_{n,\mu}(p)\left(\min(2|v_{p}|t,l_{1})+\min(2|v_{p}|t,l_{2})\right)+(\epsilon_{n,\mu}^{(2)}(p)/2-\epsilon_{n,\mu}(p))\\\
&\times(\max(2|v_{p}|t,l+d)+\max(2|v_{p}|t,d)-\max(2|v_{p}|t,l_{1}+d)-\max(2|v_{p}|t,l_{2}+d))\big{]}.\end{split}$
(7.13)

The charged logarithmic negativity is obtained by taking the limit
$n_{e}\rightarrow 1$ in $\mathcal{E}_{n_{e}}(\mu)$

$\begin{split}&\mathcal{E}(\mu)=\int\frac{dp}{2\pi}\big{[}\epsilon_{1,\mu}(p)\left(\min(2|v_{p}|t,l_{1})+\min(2|v_{p}|t,l_{2})\right)+(\epsilon_{1/2,\mu}(p)-\epsilon_{1,\mu}(p))\\\
&\times(\max(2|v_{p}|t,l+d)+\max(2|v_{p}|t,d)-\max(2|v_{p}|t,l_{1}+d)-\max(2|v_{p}|t,l_{2}+d))\big{]}.\end{split}$
(7.14)

The comparisons between numerical computations and the quasi-particle
predictions are shown in fig. 3 and fig. 4, finding perfect agreement.

##  8 Conclusion

In this paper, we discuss the temporal evolution of charge imbalance resolved
negativity after two types of global quenches. Firstly, we have considered the
boundary state quench in which the post-quench dynamics is governed by a
conformal Hamiltonian. Evaluating the correlators of the fluxed twist fields
in the UHP, we obtained the quench dynamics of charged Rényi negativity. Then
the charge resolved negativity is obtained by Fourier transformation. The
total negativity can be reconstructed from these charge resolved ones. We also
discussed the mass quench in the underlying lattice model and made conjectures
based on the quasi-particle picture.

It would be very interesting to investigate the evolution of other charge
resolved entanglement measures in the same quenches discussed in this paper,
or studying other quench setups such as local joining quench or local operator
quenches in field theories and in holographic theories. One may find the
breakdown of quasi-particle picture in certain circumstance [60, 61]. We will
turn to these problems in the future work.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China,
Grant No. 12005081.

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# Artificial Neural Networks to Recognize Speakers Division from Continuous
Bengali Speech

Hasmot Ali1, Md. Fahad Hossain1, Md. Mehedi Hasan1, Sheikh Abujar1,2, Sheak
Rashed Haider Noori1
1Daffodil International University, Dhaka, Bangladesh
2The University of Alabama at Birmingham, AL, USA
{hasmot15-9632, fahad15-9600<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Voice-based applications are ruling over the era of automation because speech
has a lot of factors that determine a speaker’s information as well as speech.
Modern Automatic Speech Recognition (ASR) is a blessing in the field of Human-
Computer Interaction (HCI) for efficient communication among humans and
devices using Artificial Intelligence technology. Speech is one of the easiest
mediums of communication because it has a lot of identical features for
different speakers. Nowadays it is possible to determine speakers and their
identity using their speech in terms of speaker’s recognition. In this paper,
we presented a method that will provide a speaker’s geographical identity in a
certain region using continuous Bengali speech. We consider eight different
divisions of Bangladesh as the geographical region. We applied the Mel
Frequency Cepstral Coefficient (MFCC) and Delta features on an Artificial
Neural Network to classify speakers’s division. We performed some
preprocessing tasks like noise reduction and 8-10 second segmentation of raw
audio before feature extraction. We used our dataset of more than 45 hours of
audio data from 633 individual male and female speakers. We recorded the
highest accuracy of 85.44%.

###### Index Terms:

Division Classification, Speech Recognition, Convolutional Neural Network

##  I Introduction

Speech is a powerful medium for efficient communication. Having a bunch of
unique features like Qualities, Pitch, Tone, Rhythm, Resonance, Texture, and
others, speech is one of the most useful identifiers to recognize speakers. To
differentiate speakers from the same language and different places speech
features work as a powerful attribute and they work as a unique way of
introducing a group of people who speak the same language. People from a
geographical area or a community have some unique speech properties like they
speak in a specific accent and rhythm. Voice also can be the recognition
apparatus of Gender, Occupation, Height, Age, Division, Weight, or any other
identical parameter of a speaker.

For having perfectly structured multiple-level regularized properties speech
can produce the basic unit of speech which is defined as phonemes. It can
determine the speech clearly and perfectly but for our contribution, speech
features define the nature of speech. By which the identification or
characteristic of a specific speaker could be possible easily. MFCC provides
the necessary feature to train our model so that the model can determine an
unknown speaker’s division. Accents could play an important role in
determining people from specific geographical areas. Accent-based speaker
recognition is one of the emerging topics for ASR researchers. Deep learning
algorithms like RNN, ANN, CNN, and LSTM are performing better for speech
recognition because of their perfectly structured data. For our data, ANN
performs best with the system and configuration we used.

Region or Division recognition is one of the important tasks of classifying
people from a specific geographical area. For our contribution, we consider
eight divisions of Bangladesh named Barisal, Chittagong, Dhaka, Khulna,
Mymensingh, Rajshahi, Rangpur, and Sylhet as eight labels for speakers
Division recognition. The division recognition approach has a lot of
applications including the detection of the speaker’s geographical region,
verifying the area of unknown crime suspects, and speaker recognition.
Nowadays a lot of harassment and crime has been performed using the Voice over
Internet Protocol (VoIP). From the perspective of Bangladesh, a huge number of
fraud calls and voice threads are performed every year.

In this research, we proposed a deep learning model using the Artificial
Neural Network to recognize Bangladeshi speaker’s division from continuous
Bengali Speech. Which will help to recognize the unknown speaker’s division in
terms of crime suspect and voice fraud. It will also contribute to the Bangla
Natural Language Processing and Bangla Automatic Speech Recognition community.

##  II Literature Review

Recent Automatic Speech Recognition (ASR) technology is improving rapidly only
for the huge number of research and development. Researchers are working on
ASR technology and their implementation in real life as a huge amount of
speech data is produced every day but it is not possible to store and use all
this data. Biomechanics technology and applied linguistics are the most
commonly used in early ASR technology. However, the modern ASR is improving
because of the variety of attributes that vary speaker by speaker. Different
factors that affect human speaker recognition application are described by
Amino et al [1]. In terms of detecting unknown speakers a self-learning
speech-controlled system for speaker identification and adaptation using an
Unsupervised Speech Controlled System is performed by Herbig et al [2] -[3].
There are also a lot of speech-based speakers’ gender, height, weight, and age
detection approaches done by different researchers [4] -[7].

Region or Division detection is one of the important tasks of classifying
people from a specific geographical area. We found some work related to
speaker’s region detection using their voice. The most similar work is done by
M. F. Hossain et al [8] where they tried to recognize speakers from different
regions of the United Kingdom using voice based on Accent classification. They
used a traditional Machine Learning algorithm on speech features extracted by
MFCC and showed a maximum accuracy of 99% with k-NN on Crowdsourced high-
quality UK and Ireland English Dialect speech datasets [9]. Different accent
detection is done by Joseph et al [10] for four Indian languages Bengali,
Gujarati, Malayalam, and Marathi. They performed the region detection approach
from different languages spoken in India based on their accent using the
Dynamic Time Warping (DTW) algorithm which shows an impressive result. For the
Tagalog language in the Philippines, Danao et al [11] perform a regional
accent detection approach with an accuracy of 93.33% using a Multi-Layer
Perceptron (MLP) classifier.

Deep learning is widely used in the field of ASR as the speech is the
continuous waveform of an audio signal. So sequential models like LSTM, RNN,
and ANN work better for speech signals. An overview of using Artificial Neural
Network (ANN) approaches used in speech recognition is reviewed by Hennebert
et al [12] where some basic principles of neural networks are briefly
described as well as their current applications and performances in speech
recognition. In most cases, MFCC performs better than any other feature
extraction algorithm in the field of ASR research.

Most of the voice-based region recognition applications work on Accent. We
also found some accent-based voice recognition applications. We found that
Mannepalli et al [13] introduced a method to identify three different accents
namely Coastal Andhra, Rayalaseema, and Telangana of the Telugu language using
the Nearest Neighbor Classifier and achieved 72% accuracy. For Chinese accent
detection, Long et al [14] perform a method based on RASTA – PLP algorithm
extracting features known as the short-time spectrum of each speech segment
and records an accuracy of 80.8% using Naïve Bayes classifier. Zheng et al
[15] propose a new combination of accent discriminative acoustic features,
accent detection, and an acoustic adaptation approach for accented Chinese
speech recognition. We don’t find any related papers for the Bangla Language.

##  III Methodology

We tried to recognize the division of speakers from their voices. We focused
on continuous speech spoken in Bangla. The whole procedure of our work has
been shown in Fig 1 as a workflow.

Figure 1: Overall workflow.

###  III-A Dataset

We spent most of our time in data collection. It took more than 9 months to
collect this data. We collected most of the data onsite. Some of the data were
collected online through Google Forms. We have collected more than 45 hours of
audio data with more than 16 thousand audio samples. Around 633 speakers (both
male and female) contributed to create this dataset. We used more than 350
different scripts. Each script contains a poem, a story, and news. All the
audio files were recorded in WAV format and same configuration.

Among 633 speakers, there were 416 male speakers and 217 female speakers. A
comparison chart of male-female speakers has been provided in Fig 2.

Figure 2: A ratio of male-female speakers.

There are eight divisions of Bangladesh; Barisal, Chittagong, Dhaka, Khulna,
Mymensingh, Rajshahi, Rangpur, and Sylhet. We have collected data from all the
divisions. While labeling the data, we considered speakers’ hometowns as their
divisions. We tried to collect accent-sensitive data by suggesting speakers
speak in their home division accent. We divided the dataset into eight groups
and each group corresponds to one of the eight divisions. A comparison chart
of total data in each division or label has been provided in Fig 3.

Figure 3: Amount of data at each label.

###  III-B Data Preprocessing

After collecting data, we have done some preprocessing. We know that all the
data samples needed to be measured on the same scale. So, we segmented all the
audio samples in the same duration. Each audio sample was segmented into
multiple chunks of 8 to 10 sec duration. Before feature extraction, we also
reduced the noise from each data sample.

###  III-C Feature Extraction

####  III-C1 Mel Frequency Cepstral Coefficients (MFCCs)

MFCC has always been one of the most popular features. After getting
introduced in the late 70s [16], it has overperformed other speech features
for speech recognition. A huge number of experiments and developments have
been made and as a result, different implementations have been proposed. Todor
Ganchev, with his team, have compared the most popular four implementations
for speaker recognition including the first implementation known as MFCC FB-20
implementation [17]. Among the most popular implementations, MFCC FB-40
implementation has been used in this work. The implementation steps have been
discussed below.

  * •

Framing: One of the best ways to get perfect features is to measure the data
on the same scale. But the audio signal is not static, it’s a quasi-stationary
signal. This means, that it behaves like static in a short period but changes
over a long period. That’s why we need to split the whole audio into some
short time frame so that they behave as static and we can measure them on the
same scale. The default time limit for this segmentation is 25 ms. As we said
most of our data samples have a duration of 10 sec, and the sample rate is 16
kHz, after the framing, there are 400 segments or frames of each audio and
each frame has (0.025*16000) = 400 samples.

  * •

Calculating Power Spectrum: Power spectrum gives an estimation of frequencies
present in each frame. To calculate that, the Discrete Fourier Transform has
been applied to each frame. There is an equation [18] for this step.

$P_{i}(k)=\frac{1}{N}\left\lvert\sum^{N}_{n=1}s_{i}(n)h(n)e^{-j(2\div
N)}\right\rvert^{2}\>\>\>\>1\leq k\leq K$ (1)

Here, Pi(k) is the power spectrum of the ith frame, Si(n) is the signal
representation of each sample of the ith frame, h(n) is a hamming window and K
is the length of the Discrete Fourier Transform.

  * •

Applying Mel Filtebank: It is the most important part of MFCC implementation.
In this implementation, a set of 40 equal-area filters have been used. The
mathematical representation [17] of each filter is given below.

$\begin{cases}\hskip 41.25641pt0&for\>\>k<f_{b_{i-1}}\\\
\frac{2(k-f_{b_{i-1}})}{(f_{b_{i}}-f_{b_{i-1}})(f_{b_{i+1}}-f_{b_{i-1}})}&for\>\>f_{b_{i-1}}\leq
k\leq f_{b_{i}}\\\
\frac{2(f_{b_{i+1}}-k)}{(f_{b_{i+1}}-f_{b_{i}})(f_{b_{i+1}}-f_{b_{i-1}})}&for\>\>f_{b_{i}}\leq
k\leq f_{b_{i+1}}\\\ \hskip 41.25641pt0&for\>\>k>f_{b_{i+1}}\end{cases}$ (2)

Here represents the boundary of the ith filter and k represents the k-th
coefficient of the N - point DFT.
These 40 filters are a set of 40 vectors with some values in different
positions of different vectors. The rest of the values of the vectors are
zero. When each of the vectors is multiplied by the power spectrum, the sum of
each vector represents the energy of the corresponding segment of the power
spectrum. After this, there will be 40 energy representations for 40 filters.

  * •

Taking the log of energies: Now the logarithm of the energy estimation is
taken for each filterbank section.

  * •

Taking the DCT of log energies: In the final step, a Discrete Cosine Transform
(DCT) is applied to all the log energies. This represents the 40 features of
the audio as energy estimation.

After calculating the 40 energy features, we have only taken the first 13
features. Because these first 13 features are linear features and mostly
represent the signal. Figure 4 shows a comparison of MFCC feature values
between each label.

Figure 4: Average MFCCs for each label.

####  III-C2 Delta Features

One of the major drawbacks of MFCC is it only contains the static information.
But when we talk about continuous speech, it has both static and dynamic
characteristics depending on the time frame. To retrieve that information, we
need to use dynamic features. It has been proven that dynamic features also
contain some useful information and a combination of static features and
dynamic features can reduce the error rate to half [19] making the model
better. So, we used dynamic features known as delta features which are
calculated from MFCC by the first-order derivation. This first-order
derivation gives us the same number of features we have as MFCC. We have used
a combination of MFCC and delta features.

###  III-D Proposed Model

We have experimented with different models. However, it is found that a
shallow Artificial Neural Network performs better than others. The model has
been described below.

Division Recognition Model
---
1 | : | ADAM (learning rate)
2 | : | For 106 iterations in all batches do:
3 | : | Dense (Node, Activation)
4 | : | Dense (Node, Activation)
5 | : | Dropout (Rate)
6 | : | Dense (Node, Activation)
7 | : | Dense (Node, Activation)
8 | : | Dense (Node, Activation)
9 | : | Dense (Node, Activation)
10 | : | end for

This model is a neural network of fully connected dense layers for classifying
the Bengali Division. In this model, we have also used some regularization
methods like batch normalization [20] and dropout [21].

This model contains five dense layers and one output layer. Each dense layer
has different numbers of hidden units or neurons. The first dense layer has
128 hidden units, the second and third dense layers have 256 hidden units, the
fourth dense layer has 64 hidden units and the fifth dense layer has 32 hidden
units. The ReLU (3) activation function for each dense layer except the output
layer. We have used 20% dropout twice as regularization, after the third layer
and the fourth layer.

$ReLU(y)=max(0,y)$ (3)

The output layer contains eight hidden units, the same amount as the number of
labels. We have used the SoftMax (4) activation function in this layer.

$\sigma(z)_{j}=\frac{e^{z}j}{\sum^{K}_{k=1}e^{Z}k}\;for\;j\;=\;1,...k$ (4)

Figure 5 shows the proposed Division Recognition architecture.

Figure 5: Architecture of Division Recognition Model.

The model architecture summary will help us to determine the model more
clearly. Table 1 shows the summary for a clear visualization of the whole
model.

TABLE I: Age Recognition Model Architecture Summary Layer No(Type) | Output Shape | Parameter | Connected To
---|---|---|---
(Input Layer) | 128 | 3456 - |
1(Dense) | 256 | 33024 | Input Layer
2(Dense) | 256 | 65792 | 1
3(Dropout) | 256 | 0 | 2
4(Dense) | 64 | 16448 | 3
5(Dropout) | 64 | 0 | 4
6(Dense) | 32 | 2080 | 5
7(Dense) | 8 | 264 | 6
Total params: 121,064
Trainable Params:121,064
Non-trainable params: 0

###  III-E Optimizer and Learning Rate

Researchers will minimize neural network algorithm error with the help of the
Optimizer algorithm Proposed Bengali Division Classification model using Adam
Optimizer [22]. Most of the researchers use it to get better performance. This
Optimizer is a reform of the stochastic gradient descent algorithm. The
proposed Bengali Division Classification used Adam optimizer (5) with a
learning rate of 0.001.

$\theta_{t+1}=\theta_{t}-\frac{\eta}{{}^{1}\sqrt{\hat{v}_{t}}+\epsilon}\hat{m}_{t}$
(5)

We calculated the error of our model using a function named categorical cross
entropy (6) which is needed to optimize the algorithm. A study says that
cross-entropy has outperformed other loss-calculating functions such as
classification error and mean squared error etc [23].

$L_{i}=-\sum_{j}t_{i,j}\log(p_{i,j})$ (6)

###  III-F Training the Model

The Bengali Division Classification model was trained on our dataset with a
batch size of 128. After 35 epochs the model got almost 85% accuracy. The
optimizer converged faster by reducing the learning rate with the help of the
automatic learning rate method.

##  IV Model Performance

The model was trained and validated on our dataset and gave a promising result
on a train set, test set, and validation set. We have a total data of 16730
segmented audio as we used our 80% data for training the model, we used 10%
for testing, and 10% for validation.

The following Figure 3 shows the accuracy and loss of the division recognition
model for the training and validation set.

Figure 6: Accuracy and loss during training and validation of division
recognition model.

After 35 epochs, the proposed model got 83.99% accuracy on the training set
and 85.44% accuracy on a validation set of our datasets. Figure 4 shows the
confusion matrix of this model so that the model performance will be clearer
to all.

Figure 7: Confusion matrix of division recognition model.

##  V Conclusion

Bangla is the most precious and beloved language for the Bengali people. In
this research, we tried to recognize the speaker’s division from Continuous
Bengali Speech. We consider Barisal, Chittagong, Dhaka, Khulna, Mymensingh,
Rajshahi, Rangpur, and Sylhet, the eight divisions of Bangladesh as the
Geographical area. We used our dataset of 45 hours of audio containing Story,
News, and Poem scripts from 633 speakers. We performed some preprocessing over
our raw audio like labeling, noise reduction, and audio segmentation. We
perform feature selection and use MFCC for feature extraction. We applied an
Artificial Neural Network to this extracted feature and got the highest
accuracy of 85% for speaker division recognition.

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