\documentclass[amsmath,amssymb,showpacs,dvips,twocolumn,epsfig,aps,prb]{revtex4-2}
\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{bm}
\usepackage{hyperref}
\begin{document}
%\title{Quantum correlation in the spin-1/2 one-dimensional XXZ model with staggered field}
\title{Multipartite entanglement in the XXZ model with Dzyaloshinskii-Moriya interaction and external fields}
%
\author{L. S. Lima}
\affiliation{Department of Physics, Federal Technological Education Center of Minas Gerais, 30510-000, Belo Horizonte, MG, Brazil.  Email: lslima@cefetmg.br/lslima7@yahoo.com.br}
%
\begin{abstract}
 In this paper, we investigate the quantum entanglement on spin-1/2 one-dimensional XXZ chain with Dzyaloshinskii-Moriya interaction (DMI) together with staggered and uniform magnetic fields on $x$ and $z$ directions, respectively. We use the well known mapping of this model on the sine-Gordon theory to calculate the quantum entanglement quantifiers as von Neumann entropy and entanglement negativity, which provides a useful framework for studying the emergence of quantum correlation in this model. How the behavior of the spin velocity implies in a significant impact on energy of elementary spinons excitations carrying spin $S^z=\pm1/2$, since that the DMI interaction and external fields generate a large effect on behavior of the spin velocity, we get a large influence of these coupling on quantum correlation  what  implies in an effect on development of quantum technologies based on this spin system.
\end{abstract}

%\pacs{75.40.Gb, 75.10.Jm}


\maketitle

\section{Introduction}

The sine-Gordon model is a mathematical model that describes a system of interacting particles in one dimension. It is a non-linear field theory that has applications in various fields, such as condensed matter physics and high energy physics. In the context of non-equilibrium dynamics, the sine-Gordon model can exhibit interesting behavior such as soliton formation and the emergence of quantum correlations\cite{R}.

In the self-consistent harmonic approximation (SCHA) for the sine-Gordon model out of equilibrium, one considers the system to be in a thermal state with a non-equilibrium distribution function. The SCHA is an approximation method that assumes that the system can be described by a set of harmonic oscillators that are coupled to each other. The coupling between the oscillators is determined self-consistently by minimizing the free energy of the system. In the context of the sine-Gordon model, the SCHA can be used to study the dynamics of the system and the emergence of quantum correlations. Quantum correlations refer to the non-classical correlations between particles that cannot be explained by classical physics. These correlations can arise due to entanglement between particles or due to the non-linear interactions between particles. In the SCHA, one can calculate the two-point correlation function of the system, which gives information about the quantum correlations in the system. The two-point correlation function can be expressed in terms of the correlation length and the correlation exponent, which are related to the spatial and temporal correlations in the system\cite{AA,IL,EMA,Rub,C,Sachedev,ian,affleck,affleck2,shiomi,mc,Fabian1,Fabian2}.


On the other hand, the study of the effect of Dzyaloshinskii-Moriya interaction (DMI) interaction as well as staggered magnetic fields has attracted many attention in the last years due to relation with topological phase transitions and thermal and spin Hall effect in magnets with frustration in the lattice \cite{Fujimoto,lslima1,lslima2,lslima3,lslima4,Owerre2}.  The transmission of magnons through the system is changed by the Berry curvature  generated by the DMI interaction that acts as an effective field and leads to chiral spin waves with nontrivial topological properties. The thermal and spin Hall effects have been observed in nonmagnetic materials either by using a large magnetic field \cite{JN} or even without field however, using circularly polarized light \cite{Idrish}. It emerges in a magnetic system due to spin-orbit coupling, being different from standard Hall effect which is induced by the Lorentz force\cite{shiomi}.

The Dzyaloshinskii-Moriya interaction is an interaction that arises between spins in magnetic materials. It was first proposed by I.E. Dzyaloshinskii and T. Moriya in 1958\cite{Dzyaloshinskii,Moriya}. It is a relativistic effect that arises due to the lack of inversion symmetry in a crystal lattice where it arises due to the spin-orbit coupling between the electrons in a magnetic material. In a crystal lattice, there are usually two inequivalent sites that are related by inversion symmetry. However, in some crystallin lattices, this symmetry is broken, leading to an effective magnetic field that is perpendicular to the direction of the spin. This field causes the spins to tilt out of the plane of the material, creating a chirality in the spin configuration. The DMI interaction can have a significant impact on the magnetic properties of a material. It can affect the magnetic domain structure, the magnetic anisotropy, and the magnetic dynamics of the material. Moreover, it has been found to be important in a variety of magnetic systems, including multilayer thin films, nanoparticles, and bulk materials. In addition, it  has also been of interest for its potential applications in spintronics, a field that aims to develop devices that use the spin of electrons rather than their charge. Furthermore, the DMI interactions an be used to create skyrmions as well, that are stable magnetic structures that can be used to store and manipulate information. Skyrmions hold promise for future information storage and processing technologies, including magnetic memory and logic devices.

The aim of this paper is to analyze the effect of the parameters of the sine-Gordon model on entanglement negativity. In general, the entanglement negativity in the sine-Gordon model has been studied extensively in recent years, and its behavior as a function of the temperature, coupling constant $\nu$ as well as DMI interaction  depends on the particular entangled state and the size of the subsystems considered. The effect of the DMI interaction on the entanglement negativity in the sine-Gordon model has been a topic of large interest in nowadays. It has been found that the DMI interaction can have a significant impact on the entanglement properties of the system, particularly in the presence of topological defects such as solitons and breathers. For instance, it has been shown that for certain entangled states and subsystem sizes, the entanglement negativity can exhibit a peak as a function of the DMI coupling constant $D$. This peak corresponds to a critical value of $D$ at which the entanglement properties of the system undergo a qualitative change. Beyond this critical value, the entanglement negativity can either increase or decrease depending on the details of the entangled state and the subsystem size. Furthermore, it has been found that the DM interaction can induce new types of entanglement that are absent in the absence of the DM interaction. These include entanglement between topological defects such as solitons and breathers, as well as entanglement between different degrees of freedom of the system.

The paper is organized as follow. In section \ref{2}, we discuss about the model and method. In section \ref{4}, we present the results for the von Neumann entropy and entanglement negativity. In section \ref{6}, we discuss about the SCHA approximation in equilibrium. In section \ref{5}, we present a summary of the results obtained.

\section{Model and Method}{\label{2}}

We consider the spin-1/2 one-dimensional XXZ chain with DMI interaction and uniform and staggered external fields given by the model
\begin{eqnarray}
\hspace{-0.8cm}\mathcal{H}=\mathcal{J}\sum_{j=0}^{N}\mathbf{S}_j\cdot\mathbf{S}_{j+1}+\sum_{j=0}^{N}\mathcal{D}_j\cdot\left(\mathbf{S}_j\times\mathbf{S}_{j+1}\right)\nonumber\\
+h\sum_{j=1}^{N}(-1)^j{\mathbf{S}}_{j}^{x}-H\sum_{j=1}^{N}S^z_j,
\end{eqnarray}
where $H$ and $h$ are the strengths of the uniform and staggered external magnetic fields. The DMI interaction is given by $\mathcal{D}_{j}=D\hat{z}$, where $\mathcal{D}_j$ is pointed on $\hat{z}$ direction.
Performing the rotation of the spin operators\cite{Leone,pires2008,garade}:
%\begin{equation}{\label{rotation}}
$\bar{\mathbf{S}}_{j}^{\pm}=\mathbf{S}_{j}^{\pm}e^{\mp i\alpha_j}$,  %\hspace{0.5cm}
$\bar{\mathbf{S}}_{j}^{z}=\mathbf{S}_{j}^{z}$,
%\end{equation}
being $\bar{\mathbf{S}}_{j}^{\pm}$ satisfying the same commutation relations that $\bar{S}_{j}^{\pm}$, i.e. $[\bar{\mathbf{S}}_{j}^{+},\bar{\mathbf{S}}_{j}^{-}]=i\bar{\mathbf{S}}_{j}^{z}$,
 the above model can be written as
\begin{eqnarray}
\mathcal{H}=\bar{{J}}\sum_{j=1}^{N}(\bar{\mathbf{S}}_{j}^{x}\bar{\mathbf{S}}_{j+1}^{x}+\bar{\mathbf{S}}_{j}^{y}\bar{\mathbf{S}}_{j+1}^{y}+\delta\bar{\mathbf{S}}_{j}^{z}\tilde{\mathbf{S}}_{j+1}^{z})
%\nonumber\\+\sum_{j=1}^{N}\left[h_1+(-1)^jh_2\right]\tilde{\mathbf{S}}_{j}^{z}.
\nonumber\\+h\sum_{j=1}^{N}(-1)^j\left(e^{i\alpha_j j}\bar{\mathbf{S}}_{j}^{+}+e^{-i\alpha_j j}\bar{\mathbf{S}}_{j}^{-}\right)-H\sum_{j=1}^{N}\bar{\mathbf{S}}_{j}^{z},%\nonumber\\
\end{eqnarray}
where $\bar{{J}}=\sqrt{\mathcal{J}^{2}+D^2}$, $\alpha_j=\tan^{-1}(\mathcal{D}/\mathcal{J})$ and $\delta=\mathcal{J}/\bar{{J}}$. We can recast the model above as
\begin{eqnarray}{\label{dz}}
\hspace{-0.4cm}\mathcal{H}=\bar{{J}}\sum_{j=1}^{N}\bigg\{\frac{1}{2}(\bar{\mathbf{S}}_{j}^{+}\bar{\mathbf{S}}_{j+1}^{-}+\bar{\mathbf{S}}_{j}^{-}\bar{\mathbf{S}}_{j+1}^{+})+\delta\bar{\mathbf{S}}_{j}^{z}\bar{\mathbf{S}}_{j+1}^{z} \bigg\}\nonumber\\\hspace{-0.4cm}+h\sum_{j=1}^{N}\sum_{l=0}^{\infty}\left(\frac{\alpha_j^lj^l}{l!}(-1)^{j+l/2}\bar{\mathbf{S}}_{j}^{+}+\frac{\alpha_j^lj^l}{l!}(-1)^{j+l/2+1/2}\bar{\mathbf{S}}_{j}^{-}\right)-H\sum_{j=1}^{N}\bar{\mathbf{S}}_{j}^{z}.\nonumber\\
\end{eqnarray}
%
The continuum limit of the model above is obtained by mapping on field theory that describes the scaling limit of the spin-1/2 Heisenberg chain in a staggered magnetic field on the half-line which is the sine-Gordon model, being its SU(2) symmetric point $\beta=\frac{1}{2}$ given by
\begin{equation}
\hspace{-0.0cm}\mathcal{H}=\int dx\left\{\frac{v}{16\pi}\left[\left(\partial_x\Phi(x)\right)^2+\left(\partial_x\Theta(x)\right)^2\right]+\nu\cos\left(\beta\Theta(x)\right)\right\},
\end{equation}
where $\Phi$ is the dual field. $\Phi$ and $\Theta$ satisfy the commutation relation
\begin{equation}
[\Theta(x),\Phi(x')]=8\pi i\theta(x-x'),
\end{equation}
where $\nu=\nu(h,H)=hc(H)$ and $c(H)$ has been determined numerically\cite{dimitrev,essler}. $\theta(x)$ is the Heaviside step function that is $0$ for $x<0$, $1$ for $x>0$ and $1/2$ for $x=0$. In the regime of interest here, $0\leq\beta<1$, the elementary excitations in the sine-Gordon model are massive solitons and antisolitons possessing a relativistic dispersion relation. In the attractive regime $\beta<1/\sqrt{2}$ there are solitons and antisolitons that can form bound states known as breathers. There are altogether $[1/\xi]$ breathers, where $[x]$ denotes the integer part of $x$ and $\xi=\beta^2/(1-\beta^2)$. %At the point $\beta=1/\sqrt{2}$, the model is equivalent to
% a free massive Dirac theory\cite{Luter}.
 Moreover, the coupling $\beta$ depends on value of the applied uniform field and $x=ja_0$, where $a_0=1$ is the lattice spacing. The repulsive regime is realized as the low-energy limit of the spin-1/2 XYZ chain and $\mathcal{J}>0$, $0<\delta<1$. In the low-energy limit this gives the sine-Gordon model with  $\delta=\cos(\pi\beta^2)$, $\frac{1}{2}<\beta^2<1$, $v=\frac{\bar{{J}}\sin(\beta^2)}{2(1-\beta^2)}$, $\nu\ll 1$. But typically $\beta<\frac{1}{2}$ so that the sine-Gordon modes is in the attractive regime\cite{S,S2,S3}. In addition, one imposes fixed boundary conditions at $x=0$, $\Theta(0)=0$.  Energy and momentum of soliton, antisoliton and breathers are parametrized as
\begin{equation}
E_a(\theta)=\Delta_a\cosh\theta,\hspace{0.25cm}P_a(\theta)=\frac{\Delta_a}{v}\sinh\theta,\hspace{0.25cm}a=s,\bar{s},%B_1,B_2,
\end{equation}
where one introduces the labels $s$ and $\bar{s}$ for solitons and antisolitons, respectively and $b_1,...,b_{[1/\xi]}$ for breathers.  $\Delta=\Delta_s=\Delta$, $\Delta_{k}=2\Delta\sin\left(\frac{\pi\xi k}{2}\right)$ is the soliton gap. $\Delta$ as a function of $H$ and $h$ is given by
\begin{equation}
\frac{\Delta}{\bar{J}}=\frac{2v}{Ja_0\sqrt{\pi}}\frac{\Gamma\left(\frac{\xi}{2}\right)}{\Gamma\left(\frac{1+\xi}{2}\right)}\left[\frac{\bar{J}a_0c(H)\pi}{2v}\frac{\Gamma\left(\frac{1}{1+\xi}\right)}{\Gamma\left(\frac{\xi}{1+\xi}\right)}\frac{h}{\bar{J}}\right]^{(1+\xi)/2}
\end{equation}
where $\Gamma(x)$ is the gamma function. When $H/J=0$, $k=k_F=\pi/2a_0$ and we have that the spin velocity is given by $v=\frac{\bar{J}a_0}{2}\frac{\sin 4\pi\beta^2}{1-4\beta^2}$, where $\beta^2=\arccos(\delta)/4\pi$. Being for $H/J$ non zero, the decreasing  behavior  of $v$ with $H/J$ is determined numerically by Bethe ansatz method.


\section{Results and discussion}{\label{4}}

\textit{Quantum correlation and quantum entanglement:}  Quantum correlation and quantum entanglement are two fundamental concepts in the field of quantum mechanics, describing unique and non-classical relationships between particles that defy our classical intuitions. The former refers to the statistical relationships between the properties or states of quantum systems. In classical physics, correlations between properties of different objects are limited by what's known as local realism, where distant objects have independent properties regardless of measurements performed on one another. However, quantum mechanics challenges this notion by allowing for stronger correlations, known as quantum correlations, that cannot be explained through classical means. Quantum entanglement is a specific form of quantum correlation where two or more particles become interdependent on each other, such that the state of one particle instantly influences the state of the other(s), regardless of the distance between them. When particles become entangled, the individual properties of each particle lose their significance, and their combined state becomes more important. This phenomenon was famously described by Erwin Schrödinger as "entanglement," where the properties of multiple particles are linked and described by a joint, inseparable quantum state\cite{Dan,Michael}.

\textit{Von Neumann entropy}: The von Neumann entropy (VN) is a concept in quantum mechanics that quantifies the amount of uncertainty or information associated with a quantum system's state. Named after physicist John von Neumann, it is a measure of the system's mixedness or the degree of its entanglement with another system. In classical information theory, entropy measures the uncertainty associated with the outcome of a random variable. Similarly, in quantum mechanics, the von Neumann entropy quantifies the uncertainty or lack of knowledge about a quantum system's state. For a quantum system described by a density matrix $\rho$, the von Neumann entropy $S$ is calculated as:
\begin{equation}
S=-\hbox{Tr}\left(\rho\log\rho\right).
\end{equation}
where $\hbox{Tr}$ denotes the trace operation and $\log$ is the logarithm of the density matrix. When a quantum system is in a pure state, meaning that its state can be described by a single wave function, the von Neumann entropy is zero. This implies that there is no uncertainty or lack of information about the system's state. However, if the system is in a mixed state, which occurs when the system is entangled with another system or when there is incomplete information about its state (for example, due to environmental interactions or measurements), the von Neumann entropy is positive. A higher von Neumann entropy indicates greater uncertainty or mixed in the system's state.

%%
%\begin{figure}
%    \centering
%\includegraphics[width=7cm]{Figure_104.ps} \\
%\caption{Von Neumann entropy $(VN)$ as a function of the temperature $T$ for a small value of strength of DMI interaction as $D=0.01$ held fixed. The calculations were performed for a $\delta$ value fixed as $\delta=0.3$ and for values of external magnetic fields: $H\approx0.09(1)$ and $|h|\ll H$.}\label{fig_4}
%\end{figure}
%\noindent
\begin{figure}
    \centering
\includegraphics[width=7cm]{Figure_103.ps} \\
\caption{Von Neumann entropy $(VN)$ as a function of the DMI interaction strength $D$ for a low-temperature value as $T=0.1J$. The calculations were performed for a $\delta$ value fixed as $\delta=0.3$ and for values of external magnetic fields: $H\approx0.09(1)$ and $|h|\ll H$.}\label{fig_3}
\end{figure}
\noindent

%
Close to $\delta\rightarrow 1$, we have the von Neumann entropy is given by \cite{Calabrense}
\begin{eqnarray}{\label{entangl}}
VN=-\mathrm{Tr}(\rho_{A}\ln\rho_{A})=-\mathrm{Tr}\frac{\rho_{A}\ln\rho_{A}}{\mathrm{Tr}\rho_{A}}+\ln\mathrm{Tr}\rho_{A}%=\nonumber\\-\epsilon\frac{\partial\ln Z}{\partial\epsilon}+\ln Z\nonumber\\
\end{eqnarray}
where $\rho\propto e^{-\mathcal{\beta \mathcal{H}}}$ is the Gibbs distribution. $\rho_A=\rho_{GE}=\frac{e^{-\sum_{_\mathbf{k}}\omega_\mathbf{k}\hat{n}_{\mathbf{k}}}}{\mathcal{Z}}$,
where  the partition functions is given by: $\mathcal{Z}=\mathrm{Tr}e^{-\beta\sum_{\mathbf{k}}\omega_\mathbf{k}\hat{n}_{\mathbf{k}}}=\prod_{\mathbf{k}}(1-e^{-\frac{1}{T}\omega_{\mathbf{k}}})^{-1}$, where $\hat{n}_{\mathbf{k}}$ is the number operator and the $\beta$ parameter here is $\beta=\frac{1}{k_BT}$, $k_B=1$. $\omega_{\mathbf{k}}$ is given for the one-particle kinematics as\cite{igor} $\omega_{\mathbf{k}}=\sqrt{\delta_{a}^2+v^2k^2}$, where $v$ is the spin velocity, $\delta_a=\delta$ with $a$ running over all single-particle labels i.e., soliton, antisoliton, and breathers. % Using the form of $\mathcal{Z}$ given above, we obtain
%$S_{GE}=-\mathrm{Tr}(\rho_{GE}\log_2\rho_{GE})$.

In general, the von Neumann entropy of the thermal states is  interpreted
as entanglement when the total state is pure, which is the state for the 1D  Heisenberg antiferromagnet at $T=0$. For the case $T>0$, the thermal state must represent (correspond) to a mixed state in quantum field theory
(at least for any nonzero temperature), and hence, in this case the local entropy cannot
be interpreted as entanglement. The partition against which the local entropy is calculated is $L\rightarrow\infty$ due the fact that the bosonization procedure is valid in this limit\cite{senechal}. Being for pure states is the scaling of the local entropy with
varying of the block-size that
can identify the quantum phase transition.

%In Fig.~\ref{fig_4}, we get von Neumann entropy $(VN)$ as a function of the temperature for a DMI fixed value as $D=0.01$. The decreasing of the entropy with $T$ obtained is due to the fact that the rising of the temperature $T$ leads to an entanglement deterioration. The calculations were performed for a $\delta$ value fixed as $\delta=0.3$ and for values of external magnetic field: $H\approx0.09(1)$ and $|h|\ll H$ also  held fixed. In the limit $|h|\ll H$, the staggered field can be taken into account as a small perturbation to the low-energy limit of the one-dimensional XXZ model in an uniform magnetic field $H$, where the low-energy limit of the model with XY anisotropy $|\delta|<1$ is given by a free bosonic theory\cite{Luther,Haldane}.
In Fig.~\ref{fig_3}, we get $(VN)$ as a function of the DMI strength $D$ and for a small value of temperature as $T=0.1J$. The calculations were performed for $\delta=0.3$ and for values of external magnetic field: $H\approx0.09(1)$ and $|h|\ll H$. We get a rising of $VN$ with the  rising of the strength of $D$ indicating that when the DMI interaction is increased, the system becomes more entangled and hence the von Neumann entropy increases. In general, the behavior of the entropy as a function of  DMI may depend on the specific geometry and topology of the spin system, as well as the values of the coupling constants and other parameters in the Hamiltonian as external fields. For example, in a spin-1/2 chain with a DMI interaction, it has been shown that the entropy exhibits a sharp increase as the DMI interaction is increased, signaling a transition from a topological phase with non-trivial edge modes to a trivial phase with no edge modes\cite{Luther,Haldane}.



\textit{Entanglement Negativity:} Entanglement negativity is a measure of entanglement between two subsystems of a larger quantum system. It is a quantitative way to describe the amount of entanglement present in a quantum state. The entanglement negativity is defined as the logarithm of the trace norm of the partial transpose of the density matrix of the system. Mathematically, if we have a bipartite system $AB$ with a density matrix $\rho_{AB}$, the entanglement negativity is given by\cite{Plenio}
\begin{equation}
E(\rho_{AB}) = \log \|\rho_{AB}^{T_B}\|_1
\end{equation}
where $\rho_{AB}^{T_B}$ is the partial transpose of the density matrix with respect to subsystem $B$, and $\|\cdot\cdot\cdot\|_1$ denotes the trace norm. It has several useful properties, such as being non-negative and monotonically decreasing under local operations and classical communication. It has been widely studied in the context of quantum information theory, quantum computing, and quantum communication. The negativity has been proven to be useful to detect topological order,\cite{C,Y} where one makes $\rho_{AB}=\rho_{GE}$. The ensemble describing the system for long time is expected to be the canonical ensemble, being the reduced density matrix given by\cite{Calabrense,Davide,Vidal,Calabrense2} $\rho_{AB}=\frac{e^{-\beta\sum_{_\mathbf{k}}\omega_\mathbf{k}\hat{n}_{\mathbf{k}}}}{\mathcal{Z}}$,
where $\mathcal{Z}$ is the partition function. In this case, we get that
$\rho_{AB}=\frac{e^{-\beta\sum_{\mathbf{k}}\omega_\mathbf{k}\hat{n}_{\mathbf{k}}}}{\mathcal{Z}}$, where $\mathcal{Z}$ is the partition function, being given by $\mathcal{Z}=\mathrm{Tr}e^{-\beta\sum_{\mathbf{k}}\omega_\mathbf{k}\hat{n}_{\mathbf{k}}}=\prod_{\mathbf{k}}(1+e^{-\beta\omega_{\mathbf{k}}})$.
Using the form of $\mathcal{Z}$, we obtain the entanglement negativity  given by
%\begin{eqnarray}
$E(\rho_{AB})=-\log_2\bigg\|\frac{e^{-\beta\sum_{_\mathbf{k}}\omega_\mathbf{k}\hat{n}_{\mathbf{k}}}}{\mathcal{Z}}\bigg\|%\nonumber\\
=\sum_{\mathbf{k}}\beta\omega_{\mathbf{k}}\hat{n}_{\mathbf{k}}+\log_2\prod_{\mathbf{k}}(1+e^{-\beta\omega_{\mathbf{k}}})$.
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{Figure_100.ps}%\hspace{1cm}
\caption{Behavior of the entanglement negativity  $(E)$ as a function of $T$ for a value of DMI coupling, $D=0.01$. The calculations were performed for a $\delta$ value fixed as $\delta=0.3$ and for values of external magnetic field: $H\approx0.09(1)$ and $|h|\ll H$ held fixed.}\label{fig_12} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}
%\begin{figure}
%    \centering
%\includegraphics[width=7.0cm]{Figure_100.ps}%\hspace{1cm}
%\caption{Behavior of the entanglement negativity  $EN$ as a function of anisotropy $\delta$, for a small DMI value as $D=0.01$ and a low-temperature value as $T=0.1J$. The calculations were performed for values of external magnetic field: $h=1.0$ and $H=1.0$ held fixed.}\label{fig_13} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
%\end{figure}
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{Figure_101.ps}%\hspace{1cm}
\caption{Behavior of the entanglement negativity  $(E)$ as a function of the DMI strength $D$ for a small value of temperature as $T=0.1J$. The calculations were performed for $\delta=0.3$ and for values of external magnetic field: $H\approx0.09(1)$ and $|h|\ll H$.}\label{fig_14} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}

In Fig.~\ref{fig_12}, we get the behavior of the entanglement negativity (EN) as a function of $T$ for a small value of DMI interaction as $D=0.01$. The calculations were performed for a $\delta$ value fixed as $\delta=0.3$ where the dimensionless spin velocity $v/\bar{J}a_0$ and the external field $h$ and $H$ as functions of the magnetization have been determined\cite{igor}. The values of external magnetic field are $H\approx0.0909(3)$, where $|h|\ll H$ is held fixed. As we can see, we get a behavior rising of EN with $T$, as expected, indicating a larger lost of quantum information with the increase of $T$. In a general way, increasing the temperature of the system can lead to a destruction of entanglement, this because thermal fluctuations can cause the system to lose coherence, which in turn can destroy the entanglement. In other cases, increasing the temperature can actually enhance entanglement, as thermal fluctuations can drive the system towards entangled states. However, the precise behavior of entanglement with temperature is an active area of research in quantum information theory and condensed matter physics. It depends on many factors, such as the strength of the interactions between the particles in the system, the geometry of the system, and the nature of the thermal bath with which the system is in contact.


In Fig.~\ref{fig_14}, we present the behavior of the entanglement negativity  as a function of DMI interaction $D$ at $T=0.1J$. The calculations were performed for $\delta=0.3$ and for values of external magnetic field: $H\approx0.0909(3)$ and $|h|\ll H$. As we can see, the $EN$ presents a maximum at $D\sim0.01$. The DMI interaction can enhance the entanglement between two magnetic subsystems. This is because DMI can lead to the formation of chiral magnetic textures, such as skyrmions or chiral domain walls, which are highly entangled. The exact behavior of entanglement negativity as a function of $D$ will depend on the specific geometry and parameters of the system. For example, in the one-dimensional spin chain here considered of magnetic moments with DMI interaction, the entanglement negativity has been shown to exhibit a maximum at a finite value of DMI strength. This maximum occurs when the magnetic subsystems are separated by a distance that matches the length scale of the chiral textures induced by the DMI interaction.
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{EN_vs_field}%\hspace{1cm}
\caption{Behavior of the entanglement negativity  $(E)$ as a function of the $H/\bar{J}$ for $D$ near to zero and $T=0.1J$. The calculations were performed for a $\delta$ value fixed as $\delta=0.3$.}\label{fig_15} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}

In Fig.~\ref{fig_15}, we get the behavior of the entanglement negativity as a function of the external field $H/\bar{J}$ for $D\approx 0.01$  and $T=0.1J$. The calculations were performed for  $\delta=0.3$. Since the energy of spinons in a magnetic field is described using the dispersion relation, It is affected by the Zeeman term introduced by the magnetic fields. However, the precise behavior of spinons in a magnetic field may be highly dependent on the specific details of the system, including the interactions between spins, lattice structure and so on.. As showed in Ref.~\cite{igor}, since that the spin velocity suffers a decreasing with $H/J$,  we get an increasing of the entanglement with $H/J$ and thus a rising of the coherence of the system.

\section{SCHA approach in equilibrium}{\label{6}}

The self-consistent harmonic approximation (SCHA) is a mean-field approach that is commonly used to study the behavior of nonlinear systems\cite{Sak,ABZ}.  The exact breather mass of the sine-Gordon model is given in SCHA approximation as\cite{ABZ}
\begin{equation}
\Delta_2=2\sin\left(\frac{\pi\zeta}{2}\right)\frac{2v}{\sqrt{\pi}\xi}\frac{\Gamma(\zeta/2)}{\Gamma((1+\zeta)/2)}\left(\frac{\pi\xi^2J}{2v}\frac{\Gamma\left(\frac{1}{1+\zeta}\right)}{\Gamma\left(\frac{\zeta}{1+\zeta}\right)}\right)^{(1+\zeta)/2},
\end{equation}
where $\zeta=1/(8K-1)$ and $\xi$ corresponds to a cutoff in momentum space $k_c=2\pi/\xi$. The Hamiltonian is diagonalized by a Bogoliubov transformation
\begin{eqnarray}
\psi_j=\cosh(\theta_j )c_j+\sinh(\theta_j )c^{\dag}_{-j}\\
\theta_j=-\frac{1}{2}\ln\left(\frac{\pi}{2KL|u_j|^2}\left(v^2q_j^2+\Delta^2\right)^{-\frac{1}{2}}\right),
\end{eqnarray}
where $\Delta=\sqrt{\frac{-2h\pi vJ}{K}}$ and $q_j=2\pi j/L$.
$u_j=|\pi/2q_jLK|^{1/2}\hbox{sgn}(q_j)$, if $j\ne 0$, $u_j=\frac{j}{4}\sqrt{\frac{2v}{K}}$, if $j=0$

In terms of the Bogoliubov bosons we get the boson Hamiltonian
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \mathcal{H} =\sum_{j}\sqrt{(vq_j)^2+\Delta^2}\psi_j^{\dag}\psi_j.
\end{eqnarray}
Moreover, we have $\psi_j|0\rangle=0$. The self-consistency condition for $h$ is obtained by calculating $\langle\phi^2\rangle=\langle0|\phi^2(x)|0\rangle$
\begin{equation}
\langle\phi^2\rangle=\frac{\pi v}{2KL}\sum_j\frac{1}{\sqrt{v^2q_j^2+\frac{\pi vJ}{K}e^{-\frac{1}{2}\langle\pi^2\rangle}}},
\end{equation}
where the Bose field can be cast as
\begin{equation}
\phi(x)=\sum_j u_je^{iq_j x}(\psi_j-\psi_{-j}^{\dag}).
\end{equation}
For large enough values of $K$, the SCHA provides accurate approximations. For smaller values of $K$, the SCHA offers a much better prediction of $\Delta$ than the simple harmonic approximation does. However, close to the point $K=1/4$ (Luther-Emely point), which lies at $K=1/4$, the predictions from the SCHA becomes poor as well.
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{sine-gordon-ent.eps}%\hspace{1cm}
\caption{Behavior of the entanglement negativity  $EN$ as a function of $T$ for a value of DMI coupling, $D=0.01$ held fixed. The calculations were performed for a small $\Delta$ value as $\Delta=0.1$ held fixed and for values of external magnetic field: $h=1.0$  held fixed .}\label{fig_12} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}
%


%
To study quantum correlations in the sine-Gordon model out of equilibrium using the SCHA, one typically begins by dividing the system into a number of small subsystems, each of which is modeled as a harmonic oscillator. The dynamics of each subsystem is then described by a set of coupled equations of motion that can be solved numerically. One important aspect of the SCHA is that it assumes that the harmonic oscillator approximation is valid at all times, even when the system is strongly driven out of equilibrium. This assumption can break down in certain regimes, leading to deviations from the predictions of the SCHA. In general, the study of quantum correlations in the sine-Gordon model out of equilibrium using the SCHA is an active area of research that involves a combination of analytical and numerical techniques. The ultimate goal is to develop a better understanding of the non-equilibrium dynamics of quantum systems, with potential applications in areas such as quantum computing and quantum information processing.


%
Using the form of $\mathcal{Z}$, we obtain the entanglement negativity  given by
%\begin{eqnarray}
$EN(\rho)=-\log_2\bigg\|\frac{e^{-\beta\sum_{_\mathbf{k}}\Omega_\mathbf{k}{n}_{\mathbf{k}}}}{\mathcal{Z}}\bigg\|
=\beta\sum_{\mathbf{k}}\Omega_{\mathbf{k}}{n}_{\mathbf{k}}+\sum_{\mathbf{k}}\log_2(1-e^{-\beta\Omega_{\mathbf{k}}})$, where ${n}_{\mathbf{k}}$ is the is the boson occupation number and $\Omega_{\mathbf{k}}=\sqrt{(vk)^2+\Delta^2}$.

In Fig.~\ref{fig_12}, we get the behavior of entanglement negativity (EN) as a function of $T$ for a small value of DMI interaction as $D=0.01$. The calculations were performed for a small value of breather mass of the sine-Gordon model $\Delta=0.1$ and for values of external magnetic field $h_1=h_2=1.0$ held fixed. As we can see, we get a behavior rising for EN with $T$ very close to $T=0$. However, the behavior in higher temperature is only qualitative due to approximation used.  In a general way, increasing the temperature of the system can lead to destruction of entanglement. This is because thermal fluctuations can cause the system to lose coherence, which in turn can destroy the entanglement. In other cases, increasing the temperature can actually enhance entanglement, as thermal fluctuations can drive the system towards entangled states. However, the precise behavior of entanglement with temperature depends on many factors, such as the strength of the interactions between the particles in the system, the geometry of the system, and the nature of the thermal bath with which the system is in contact. Thus, the rising of entanglement with $T$ obtained is due to complex interactions between the system's components and external fields and DMI interaction, that counterbalances the usual thermal decoherence effects.
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{sine-gordon-ent3.eps}%\hspace{1cm}
\caption{Behavior of the entanglement negativity  $EN$ as a function of $\Delta$, for a small DMI value as $D=0.01$ and a low-temperature value as $T=0.005J$ held fixed. The calculations were performed for values of external magnetic field: $h=1.0$ held fixed.}\label{fig_13} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}

In Fig.~\ref{fig_13}, we display the behavior of the entanglement negativity  as a function of $\Delta$, a small DMI value as $D=0.01$ and $T=0.01J$. We a rising of EN with$\Delta$ at limit $\Delta\ll 1.0$, where EN presents a large peak close to $\Delta\sim0.01$. In general, the entanglement negativity has been shown to exhibit a non-analytic behavior as a function of $\Delta$. In particular, it is known to exhibit a discontinuous jump at the critical coupling constant, above which the model undergoes a phase transition. This discontinuous jump is a manifestation of the sudden change in the structure of the ground state entanglement as the model transitions from a disordered to an ordered phase. In addition to the discontinuous jump, the entanglement negativity is also known to exhibit oscillatory behavior as a function of the coupling constant. These oscillations are related to the presence of solitons and instantons in the model, which can lead to long-range entanglement between the two subsystems.

In Fig.~\ref{fig_14}, we present the behavior of the entanglement negativity  as a function of DMI interaction $D$ at $T=0.01J$. The calculations were performed for a small value of $\Delta$ as $\Delta=0.1$. As we can see, the EN present a maximum in $D\sim0.0$, suffering a small decreasing with $D$. The DMI can enhance the entanglement between two magnetic subsystems. This is because DMI can lead to the formation of chiral magnetic textures, such as skyrmions or chiral domain walls, which are highly entangled. The exact behavior of entanglement negativity as a function of $D$ depends on the specific geometry and parameters of the system. For example, in the one-dimensional spin chain here considered of magnetic moments with DMI, the entanglement negativity has been shown to exhibit a maximum at a finite value of DMI strength. This maximum occurs when the magnetic subsystems are separated by a distance that matches the length scale of the chiral textures induced by DMI.
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{sine-gordon-ent2.eps}%\hspace{1cm}
\caption{Behavior of the entanglement negativity  $EN$ as a function of the DMI strength $D$ for a low-temperature value as $T=0.005J$ held fixed. The calculations were performed for a small $\Delta$ value, $\Delta=0.1$ and for values of external field, $h=1.0$.}\label{fig_14} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}

\section{Outlook}{\label{5}}

In summary, we study quantum correlation in the spin-1/2 quantum one-dimensional XXZ chain with Dzyaloshinskii-Moriya interaction (DMI) together with staggered and uniform magnetic fields on $x$ and $z$ directions, respectively. Our study demonstrates the important role of DMI interaction and external fields in determining of the spin velocity, von Neumann entropy and entanglement negativity. The results highlight the complex interplay between magnetic interactions and external fields in determining the magnetic properties of materials, and provide insights into the design of quantum technologies based on spin systems.
Overall, the behavior of entanglement negativity as a function of DMI strength is a rich and complex topic that depends on many factors, including the geometry of the system, the strength and sign of DMI, and the temperature and other environmental parameters. The presence of an external magnetic field can alter the dynamics of the sine-Gordon system, leading to the appearance of new solitonic states and modifying the properties of existing defects. In particular, the application of a magnetic field can shift the positions of kinks and anti-kinks (the simplest solitons in the sine-Gordon model) and modify their energy levels. Furthermore, the effect of the DMI interaction  on the sine-Gordon model is of significant interest, as it can lead to the emergence of chiral solitons, which have unique properties compared to their non-chiral counterparts. It can also modify the interaction between solitons, leading to changes in their scattering properties and affecting the dynamics of the system as a whole.

\appendix{\bf Acknowledgment}

This work was partially supported by National Council for Scientific and Technological Development (CNPq) Brazil.
%
\appendix{\bf Data Availability Statement}:
All data generated or analysed during this study are included in this paper.
%
\bibliography{Bibliography}
\begin{thebibliography}{10}
%
\bibitem{R} R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland Personal Library. Vol. 15. North-Holland 34 (1989).
\bibitem{AA} A. A. Ovchinnikov, Phys. Lett. B 836, 137615 (2023).
\bibitem{IL} I. Lovas, R. Citro, E. Demler, T. Giamarchi, M. Knap, E. Orignac, Phys. Rev. B 106, 075426 (2022).
\bibitem{EMA} E. Wybo, M. Knap, A. Bastianello, Phys. Rev. B 106, 075102 (2022).
\bibitem{Rub} J. Rubinstein,  Journal of Mathematical Physics. 11, 258  (1970).
%\bibitem{C} C. L. Terng, K. Uhlenbeck,  Notices of the AMS. 47, 17 (2000).
\bibitem{Sachedev} S. Sachdev, Quantum Phase Transitions, Second Edition, Cambridge University Press, New York, USA (2011).
\bibitem{ian} G. Mussardo, Statistical Field Theory an introduction to exactly solved models in statistical physics, Oxford University Press, New York, (2014).
\bibitem{affleck} I. Affleck, M. Oshikawa, Phys. Rev. B 60, 1038 (1999).
\bibitem{affleck2} M. Oshikawa, I. Affleck, Phys. Rev. Lett. 79, 2883 (1997).
\bibitem{shiomi} Y. Shiomi, M. Mochizuki, Y. Kaneko, Y. Tokura, Phys. Rev. Lett. 108, 056601 (2012).
\bibitem{mc} J. N. Mc Elearney, S. Merchant, Phys. Rev. B 18, 3612 (1978).
\bibitem{Fabian1} B. Bertini, D. Schuricht, F. H. L. Essler, J. Stat. Mech., P10035 (2014).
\bibitem{Fabian2} Y. D. van Nieuwkerk, F. H. L. Essler, J. Stat. Mech., 084012 (2019).
\bibitem{C} C. Castelnovo, Phys. Rev. A 88, (2013) 042319.
\bibitem{Fujimoto} S. Fujimoto, Phys. Rev. Lett. 103, 047203 (2009).
\bibitem{lslima3} L. S. Lima, J. Magn. Magn. Mater. 323, (2011) 1064.
\bibitem{lslima4} L. S. Lima, J. Magn. Magn. Mater. 454, (2018) 150.
\bibitem{lslima1} L. S. Lima, Eur. Phys. J. D 73, 6 (2019).
\bibitem{lslima2} L. S. Lima, Eur. Phys. J. D 73, 242 (2019).
\bibitem{Owerre2} S. A. Owerre, J. Phys. Condens. Matter. 29, 385801 (2017).
\bibitem{JN} J. -N. Chazalviel, Phys. Rev. B 11, 3918 (1975).
\bibitem{Idrish} M. Idrish Miah. J. Phys. D: Appl. Phys. 40, 1659 (2007).
\bibitem{Dzyaloshinskii} I. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).
\bibitem{Moriya} T. Moriya, Phys. Rev. 120, 91 (1960).
%
\bibitem{Leone} C. J. De Leone, G. T. Zimanyi, Phys. Rev. B 49, 1131 (1994).
\bibitem{pires2008} L. S. Lima, A. S. T. Pires, J. Magn. Magn. Mater. 320, 2316 (2008).
\bibitem{garade} I. Garate, I. Affleck, Phys. Rev. B 81, 144419 (2010).
%\bibitem{Luter} A. Luther, V. J. Emery, Phys. Rev. Lett. 33, 589 (1974).
\bibitem{dimitrev} D. V. Dmitriev, V. Ya. Krivnov, A. A. Ovchinnikov, Phys. Rev. B 65, 172409 (2002).
\bibitem{essler} J. S. Caux, F.H.L. Essler, and U. Loew, Phys. Rev. B 68, 134431 (2003).
\bibitem{S} S. Ghoshal, A. Zamolodchikov, Int. J. Mod. Phys. A 9, 3841 (1994).
\bibitem{S2} S. Ghoshal, Int. J. Mod. Phys. A 9, 4801 (1994).
\bibitem{S3} S. Lukyanov, Mod. Phys. Lett. A 12, 2543 (1997).
%\bibitem{Fabian3} F. H. L. Essler, Phys. Rev. B 59 14376 (1999).
%\bibitem{shultz} H. J. Schulz, Phys. Rev. B 34, 6372 (1986).
\bibitem{Dan} D. C. Marinescu, G. M. Marinescu, {\it{Approaching quantum computing}}, Pearson Prentice Hall, New Jersey, USA (2004).
\bibitem{Michael} M. A. Nielsen, I. L. Chuang,  {\it{Quantum Computing and Quantum Information}}, Cambridge University Press, Cambridge, UK (2000).
%\bibitem{einstein} A. Einstein, E. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935).
\bibitem{senechal} D. Senechal,  arXiv:cond-mat/9908262v1 (1999).
\bibitem{Calabrense} P. Calabrense, J. Cardy, Journal of Statistical Mechanics: Theory and Experiment P06002, (2004).
%\bibitem{Fradkin} E. Fradkin, Field Theories of Condensed Matter Physics, second edition,  Cambridge, U.K (2013).
\bibitem{Luther} A. Luther, I. Peschel, Phys. Rev. B 12, 3908 (1975).
\bibitem{Haldane} F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981).
\bibitem{igor} I. Kuzmenko, F. H. L. Essler, Phys. Rev. B 79, 024402 (2009).
\bibitem{Plenio} M. B. Plenio, Phys. Rev. Lett. 95, (2005) 090503.
%
\bibitem{Y} Y. A. Lee, G. Vidal, Phys. Rev. A 88, (2013) 042318.
%
\bibitem{Davide} D. Bianchini, O. A. Castro-Alvaredo, B. Doyon, E. Levi, F. Ravanini, J. Phys. A: Math. Theor. 48 04FT01 (2014).
\bibitem{Vidal} G. Vidal, J. L. Latorre, E. I. Rico, A. Kitaev, Phys. Rev. Lett. 90, (2003) 227902.
\bibitem{Calabrense2} P. Calabrese, Physica A 504, (2018) 31.
%
\bibitem{Sak} T. Sakaguchi, T. Tamaribuchi, S. Takada, Prog. Theor. Phys. 68, 19 (1982).
\bibitem{ABZ} A. B. Zamolodchikov, Int. J. Mod. Phys. A 10, 1125 (1995).
\end{thebibliography}
%
\end{document}















\textit{Entanglement Negativity:} Entanglement negativity is a measure of entanglement between two subsystems of a larger quantum system. It is a quantitative way to describe the amount of entanglement present in a quantum state. The entanglement negativity is defined as the logarithm of the trace norm of the partial transpose of the density matrix of the system. Mathematically, if we have a bipartite system $AB$ with a density matrix $\rho_{AB}$, the entanglement negativity is given by:
\begin{equation}
EN(\rho_{AB}) = \log \|\rho_{AB}^{T_B}\|_1
\end{equation}
where $\rho_{AB}^{T_B}$ is the partial transpose of the density matrix with respect to subsystem $B$, and $\|\cdot\cdot\cdot\|_1$ denotes the trace norm. It has several useful properties, such as being non-negative and monotonically decreasing under local operations and classical communication. It has been widely studied in the context of quantum information theory, quantum computing, and quantum communication. The negativity has been proven to be useful to detect topological order,\cite{C,Y} where one makes $\rho_{AB}=\rho_{GE}$. The ensemble describing the system for long time is expected to be the canonical ensemble, being the reduced density matrix given by\cite{Calabrense,Davide,Vidal,Calabrense2} $\rho_{AB}=\frac{e^{-\sum_{_\mathbf{k}}\Omega_\mathbf{k}\hat{n}_{\mathbf{k}}}}{\mathcal{Z}}$,
where $\mathcal{Z}$ is the partition function. In this case, we get that
$\rho_{AB}=\frac{e^{-\sum_{\mathbf{k}}\Omega_\mathbf{k}\hat{n}_{\mathbf{k}}}}{\mathcal{Z}}$, where $\mathcal{Z}$ is the partition function.%, being given by $\mathcal{Z}=\mathrm{Tr}e^{-\sum_{\mathbf{k}}\Omega_\mathbf{k}\hat{n}_{\mathbf{k}}}=\prod_{\mathbf{k}}(1+e^{-\beta\Omega_{\mathbf{k}}})$.








 A measure of quantum entanglement of a quantum state is the Von Neumann entanglement entropy. Considering a partition of a physical system $\Phi$ in disjoint subsystems that are labeled by the letters $A$ and $B$ respectively, where $\Phi=A\bigcup B$ and the two subsystems have a vanishing intersection, $A\bigcap B=\emptyset$. Let $\mathcal{H}_A$ and $\mathcal{H}_B$ are the Hilbert spaces of the states with separate support in the system $A$ and in the system $B$ respectively such that the Hilbert space of the states on $\Phi$ is $\mathcal{H}_{\Phi}=\mathcal{H}_A\oplus\mathcal{H}_B$. Let $|\Psi\rangle$  a pure quantum state of the system on $A\bigcup B$. As such, it can be decomposed as $|\Psi\rangle=\sum_{m,n}M_{m,n}|\psi_n^A\rangle\otimes|\psi_m^B\rangle$, where $\{|\psi_n^A\rangle\}$ and $\{|\psi_m^A\rangle\}$ are orthornormal basis states of $\mathcal{H}_A$ and $\mathcal{H}_B$, respectively, and $M_{m,n}$ are matrix elements of an rectangular matrix $\mathbf{M}$.




Quantum entanglement is a phenomenon in quantum mechanics where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the state of the other particle(s). This correlation exists even when the particles are separated by large distances, and it is one of the most striking and counterintuitive features of quantum mechanics. The most famous example of entanglement is the EPR paradox, proposed by Einstein, Podolsky, and Rosen in 1935 \cite{einstein}. In this thought experiment, two particles are created in an entangled state such that their properties are perfectly correlated, even when separated by a large distance. If a measurement is made on one of the particles, the state of the other particle is instantaneously affected, regardless of the distance between them.
Entanglement plays so, a crucial role in various areas of quantum physics, such as quantum computing, quantum cryptography, and quantum teleportation. It also has potential applications in fields such as quantum communication and quantum sensing.
Entanglement is typically quantified using a measure known as entanglement entropy or entanglement entropy and entanglement negativity. There are several different ways to define entanglement entropy, including the von Neumann entropy and the Renyi entropy, and they provide different perspectives on the nature of entanglement. However, all measures of entanglement share the common property that they increase with the amount of entanglement present in a quantum state.









The Von Neumann entanglement entropy $S_A$ for the subsystem $A$ when the total system is in the state $|\Psi\rangle$ is defined as the entropy of the reduced density matrix,
\begin{eqnarray}
S_A\equiv-\mathrm{Tr}\left(\rho_{A}\log_2\rho_{A}\right).
\end{eqnarray}
One considers a field theory with Euclidean action $S\left[\phi\right]=\int_{\Omega}d^Dx\mathcal{L}\left[\phi\right]$, where $\phi$ is some field and $\mathcal{L}$ is the Lagrangian density  and $\Omega$ is some $D$-dimensional space-time manifold. In the Euclidean theory the imaginary-time coordinate $\tau$ is periodic in time with period $\beta=1/T$, while the space directions are arbitrary.

The scaling behavior of the entanglement can be determined by general arguments of conformal field theory (CFT). In the quantum critical region, the entanglement entropy has the behavior
\begin{equation}
S=\frac{c}{3}\ln\left(\frac{L}{a}\right)+\hbox{finite terms}+\mathcal{O}(L^{-1})
\end{equation}
where $c$ is the central charge of the correspondent Virasoro algebra associated with the specific CFT that describes the 1D quantum critical point. The law of areas for the black-hole physics is the expression of Bekenstein-Hawking for the black-hole entropy given as
\begin{equation}
S_{BH}=\frac{1}{4l_P^2}\mathcal{A}
\end{equation}
where $\mathcal{A}$ is the area of the event horizon of the black hole, $l_P=\sqrt{\hbar G/c^3}$ is the Planck length and $G$ is the gravitational constant. Therefore, for $d\rightarrow 1$, the area law can become a logarithm in one dimension. The existence of universal terms in finite-size scaling of the von Neumann entropy is important since they signal the existence of large-scale entanglement. Although the area-law term is the most singular term as function of the size of the observed region, the universal term indicates that it measures the contributions for the entanglement entropy from all length scales. The entanglement entropy generally has a universal term in systems at quantum criticality, i.e. whose effective field theory displays a scale and conformal invariance in the topological phases\cite{Calabrense}.

Using the conformal mapping $w\rightarrow z=(\beta/(2\pi))\ln w$ that maps each sheet in the $w$ plane onto an infinitely long cylinder of circumference $\beta$ that plays the role of inverse temperature, the entropy in a thermally mixed state $T$ is given as\cite{Calabrense,Fradkin}
\begin{eqnarray}
S_A(T,L)=\frac{c}{3}\ln\left(\frac{\beta}{\pi a}\sinh\left(\frac{\pi L}{\beta}\right)\right)+\hbox{constant},
\end{eqnarray}
where at $T=0$ this result reproduces the entanglement entropy in the 1D conformal field theory.



On the other hand,  The self-consistent harmonic approximation (SCHA) is a theoretical method for studying out-of-equilibrium systems that involves approximating the dynamics of the system in terms of a set of coupled harmonic oscillators. Quantum correlations are a fundamental aspect of quantum mechanics that describe the non-classical correlations between quantum systems. In the context of the sine-Gordon model, quantum correlations can manifest themselves in the form of entanglement between different parts of the system.

\textit{Self-Consistent Harmonic Approach:} The self-consistent harmonic approximation (SCHA) is a mean-field approach that is commonly used to study the behavior of nonlinear systems out of equilibrium.  How the sine-Gordon model describes the dynamics of a scalar field $\varphi(x,t)$ in (1+1)-dimensional space-time. The Lagrangian density for the sine-Gordon model is given by
\begin{equation}
\mathcal{L} = \frac{1}{2} (\partial_{\nu}\varphi)^2 - \mu (\cos\varphi - 1),
\end{equation}
where $\mu$ is a coupling constant. The equation of motion for the field is then given by the sine-Gordon equation:
\begin{equation}
(\partial^2/\partial t^2 - \partial^2/\partial x^2)\varphi + \mu\sin\varphi = 0.
\end{equation}
To derive the dispersion relation in the SCHA, one first makes an ansatz for the field $\varphi$ in terms of the classical field $\varphi_{cl}(x,t)$ and quantum fluctuations $\eta(x,t)$
\begin{equation}\label{cl}
\varphi(x,t) = \varphi_{cl}(x,t) + \eta(x,t).
\end{equation}
The classical field $\varphi_{cl}(x,t)$ satisfies the equation of motion in the absence of fluctuations, while the fluctuations $\eta(x,t)$ are assumed to be small and can be treated as a perturbation. The SCHA then proceeds by minimizing the free energy of the system with respect to the fluctuations $\eta(x,t)$. The free energy is given by
\begin{equation}
\mathcal{F}[\varphi] = \frac{1}{2}\int dx [(\partial\varphi/\partial x)^2 + \mu(\cos\varphi - 1)].
\end{equation}
Expanding the free energy to second order in the fluctuations, one obtains:
\begin{equation}
\hspace{-0.8cm}\mathcal{F}[\varphi] = \mathcal{F}[\varphi_{cl}] + \frac{1}{2} \int dx dt [\eta(x,t) (-\partial^2/\partial t^2 + \partial^2/\partial x^2 - \mu \cos\varphi_{cl})],
\end{equation}
where $\mathcal{F}[\varphi_{cl}]$ is the free energy of the classical field.

The fluctuations can then be quantized as harmonic oscillators with frequency $\omega(k)$ given by the dispersion relation:
\begin{equation}
%\omega(k) =\sqrt{k^2 + \mu\cos\varphi_{cl}(k)} ,
\omega(k) =\sqrt{(vk)^2 + \mu^2\cos^2\varphi_{cl}(k)} ,
\end{equation}
where $k$ is the wave vector and $\varphi_{cl}(k)$ is the Fourier transform of the classical field.
%Thus, the dispersion relation in the SCHA for the sine-Gordon model out of equilibrium is given by $\omega^2(k) = k^2 + \mu \cos \varphi_{cl}(k)$, where $\varphi_{cl}(k)$ satisfies the equation of motion the classical field in the absence of fluctuations.
%In addition,
 The SCHA approach is a widely used method in condensed matter physics to study the properties of many-body systems. The sine-Gordon model is a classic model in field theory that describes the dynamics of a scalar field in (1+1) dimensions.
When the sine-Gordon model is out of equilibrium, it can be described by the following Hamiltonian
\begin{equation}{\label{self}}
H = \int dx \left[ \frac{1}{2}(\pi(x))^2 + (\partial_x\varphi(x))^2 - \mu \cos(\beta\varphi(x)) \right],
\end{equation}
where $\varphi(x)$ is the scalar field, $\pi(x)$ is its conjugate momentum, $\mu$ is the coupling constant, $\beta$ is the inverse temperature, and $x$ is the spatial coordinate.


%
To study quantum correlations in the sine-Gordon model out of equilibrium using the SCHA, one typically begins by dividing the system into a number of small subsystems, each of which is modeled as a harmonic oscillator. The dynamics of each subsystem is then described by a set of coupled equations of motion that can be solved numerically. One important aspect of the SCHA is that it assumes that the harmonic oscillator approximation is valid at all times, even when the system is strongly driven out of equilibrium. This assumption can break down in certain regimes, leading to deviations from the predictions of the SCHA. In general, the study of quantum correlations in the sine-Gordon model out of equilibrium using the SCHA is an active area of research that involves a combination of analytical and numerical techniques. The ultimate goal is to develop a better understanding of the non-equilibrium dynamics of quantum systems, with potential applications in areas such as quantum computing and quantum information processing.



Combining the expressions of Eq.~(\ref{boson}), the Hamiltonian can be putted in the form \cite{senechal}
\begin{equation}
\mathcal{H}=\frac{1}{2}v\left[K\Pi^2+\frac{1}{K}\left(\partial_x\phi\right)\right]
\end{equation}
where the constant $K$ (Luttinger parameter) is simply a renormalization of the field $\phi$. In following, is introduced the rescaled operators
\begin{eqnarray}
\phi'=\frac{1}{\sqrt{K}}\phi,\hspace{1cm}\Pi=\sqrt{K}\Pi
\end{eqnarray}
which still obey the canonical commutation relations. The boson Hamiltonian is quadratic in terms of fields $\phi$ and can be written as\cite{senechal}
\begin{eqnarray}
\mathcal{H}=\frac{1}{2}v\left[\Pi'^2+(\partial_x\phi')^2\right], \hspace{0.5cm}\Pi~=\frac{1}{v}\partial_t\phi'
\end{eqnarray}
where the velocity renormalization can be used to write down the exact dispersion relation of charge excitations. However, as the free boson theory (at a certain radius $R=1/\sqrt{4\pi K}$) is equivalent to a free fermion theory, meaning that the spectra of the two theories are identical,\cite{senechal} we can express the Hamiltonian  back in terms of fermions operators as given by\cite{Calabrense}
\begin{equation}
\mathcal{H}=\sum_{j=0}^{\infty}(2j+1)\epsilon n_j
\end{equation}
where
\begin{equation}
\epsilon=\pi\frac{K(\sqrt{1-y^2})}{K(y)}
%\epsilon=\arccos \Delta
\end{equation}
where $K(y)$ is a complete elliptic integral of  first kind, and $y\in\min\left[\Delta,\Delta^{-1}\right]$. In the quantum critical region we have $K(y)\simeq -1/2\log\left(1-y\right)+\mathcal{O}\left((1-y)^0\right)$.

In Fig.~\ref{fig_3}, we present the analytical results for the Von Neumann entropy of quantum entanglement in function of the strength of the Dzyaloshinskii-Moriya interaction $D$ using the bozonization method. Even though we have obtained an increasing of entanglement entropy for large values of ratio $h_x/D$, does not mean necessarily that we had obtained an increasing in the entanglement for large transverse fields because what we intended to analyse here was the effect of variation of boundary among the $LL$ and $N^z$ phases on entanglement. Furthermore, we obtain a decreasing in the entanglement with the increase of the antisymmetric DM coupling $D$ for a value of transverse external magnetic field fixed $h_x=0$. We get that the quantum entanglement decreases with $D$ as expected. The phase boundary between the $LL$ and $N^z$ phases in the phase diagram of Fig.~\ref{fig_1} is determined by the change in the magnetization. In $\Delta=1$ the concurrence and quantum discord for the XXZ  chain have a maximum, where through DMRG calculations the quantum discord turns out to be more precise\cite{Shi}.
In Fig.~\ref{fig_4}, we present the variation of the quantum entanglement with the increase of the ratio between transverse magnetic field and $D$, $h_x/D$. We obtain a strong effect of the variation of the boundary of separation between phases (change of the effective anisotropy in the phase diagram with $h_x/D$)  on quantum entanglement indicating thus, a strong effect of quantum phase transition induced by the DM interaction and transversal magnetic fields on entanglement. How the quantum phase diagram  given in Fig.~\ref{fig_1}, presents a anomalous behavior in the boundary of the  $z$ N\'eel phase to Luttinger liquid phase and the  $y$ N\'eel phase in function of the effective anisotropy and the $x$ component of the external magnetic field, we verify the influence of this anomaly on quantum entanglement.


So far we have discussed the properties of the ground state of the model Eq.~(\ref{model}) and discuss the quantum entanglement in the context of one-dimensional quantum spin chains. However, there are few experimental studies in one-dimensional antiferromagnets with uniform DM interaction. Some superconducting nonostructures can be modeled by Eq.~(\ref{model}) and that may be realizable in experiments. For instance, a one-dimensional array of small superconducting islands separated by Josephson junctions located in close proximity to a bulk superconductor \cite{garate,Fisher,Fisher2}. The applied magnetic field leads to an effective DM interaction in the Josephson-junction array. Moreover, The variations of quantum entanglement should be important in compounds in which the DM interaction  can reach a considerable value such as Cu benzoate \cite{affleck}.







The sine-Gordon model is a nonlinear partial differential equation that arises in both classical and quantum field theory. It is named after the sine and cosine functions that appear in its solutions, and after James Joseph Sylvester and James Clerk Maxwell Gordon, who independently studied the equation in the 19th century. The sine-Gordon model can be described mathematically as:
$\partial^2\Phi/\partial t^2 - \partial^2\Phi/\partial x^2 + \sin(\Phi) = 0$
%$\partial^{\mu}\partial_{\mu}\Phi(x_{\mu})+\sin(\Phi(x_{\mu}))=0$
where %$\Phi(x_{\mu})$
 $\Phi(x,t)$ is a scalar field\cite{R}. This equation can be interpreted as a wave equation that includes a nonlinear term proportional to the sine function. One of the main features of the sine-Gordon model is the existence of soliton solutions, which are localized and stable nonlinear waves that can propagate along the $x$-axis without changing their shape. These solitons are characterized by their amplitude and velocity, and can interact with each other in a complex way. The sine-Gordon model has been used to model a variety of physical systems, including plasma dynamics, superfluidity, and condensed matter physics. In particular, it has been used to study the behavior of one-dimensional quantum magnets, such as the spin-1/2 one-dimensional spin chain and ladder systems\cite{Rub,C,Sachedev,ian,affleck,affleck2,shiomi,mc,Fabian1,Fabian2}.

The spin-1/2 one-dimensional spin chain in a staggered magnetic field is a quantum mechanical system consisting of spin-1/2 particles  as spinons arranged in a line and interacting via a magnetic field\cite{affleck,affleck2,shiomi,mc,Fabian1,Fabian2}. In this model, the magnetic field is not uniform and alternates in strength along the line. This staggered magnetic field causes the spins of the particles to align either parallel or antiparallel to the field, depending on their position.
The Hamiltonian for this system is given by: $\mathcal{H} = \mathcal{J}\sum_{j=1}^{N} \mathbf{S}_j\cdot \mathbf{S}_{j+1} + \sum_{j=1}^{N} h(-1)^jS_j^z$, where $\mathcal{J}$ is the exchange parameter between neighboring spins, $h$ is the strength of the magnetic field at site $j$, and $(-1)^j$ is the parity of site $j$, i.e., $(-1)^j=1$ for even $j$ and $(-1)^j=-1$ for odd $j$. Being, the first term in the Hamiltonian describes the interaction between neighboring spins and is known as the Heisenberg interaction. The second term describes the staggered magnetic field and causes the spins to align in a pattern known as a N\'eel state.
Despite its simplicity, the spin-1/2 one-dimensional Heisenberg model in a staggered magnetic field is a rich and interesting system that exhibits a variety of quantum phenomena, including spin ordering, entanglement, and quantum phase transitions. It has important applications in condensed matter physics and quantum computing.

 The model above can be written for $\mathcal{D}$ along to direction $\hat{\mathbf{k}}$ as
\begin{eqnarray}
\mathcal{H}=\sum_{j=1}^{N}\bigg\{\frac{1}{2}\{\left[\mathcal{J}+i(D_0+(-1)^jD_1)\right]\textbf{S}_{j}^{+}\textbf{S}_{j+1}^{-}\nonumber\\+\left[\mathcal{J}-i(D_0+(-1)^jD_1)\right]\textbf{S}_{j}^{-}\textbf{S}_{j+1}^{+}+\mathcal{J}{\mathbf{S}}_{j}^{z}{\mathbf{S}}_{j+1}^{z}\}\bigg\}%\nonumber\\
%-h\sum_{j=1}^{N}{S}_{j}^{z}.
+h_1\sum_{j=1}^{N}{S}_{j}^{z}\nonumber\\+h_2\sum_{j=1}^{N}(-1)^j{S}_{j}^{z}.\nonumber\\
\end{eqnarray}

The aim is to calculate $m(x)=\langle 0|\cos\left(\beta\Theta(x)\right)|0\rangle$, where $m(x)$ is the staggered magnetization as well as the spin-wave velocity and magnetic susceptibility.
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{Figure_33.ps}%\hspace{1cm}
\caption{Spin-wave velocity  ${v}$ as a function of DMI interaction $D$. The calculations were performed for values of external magnetic fields $h_1=h_2=1.0$.}\label{fig_14} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}
In Fig.~\ref{fig_14}, we plot the spin-wave velocity as a function of DMI interaction. In a general way, the DMI is a fundamental physical phenomenon that arises from the spin-orbit coupling in magnetic materials that plays a crucial role in determining of magnetic properties of materials, including the spin-wave velocity, the staggered magnetization, and the magnetic susceptibility. Thus, the behavior of the spin-wave velocity will generate a large influence on magnetization and magnetic susceptibility $\chi_T$ as well as influence on other thermodynamics quantities as entropy and specific heat as well.

\section{Formalism}{\label{3}}
%\section{Formulation}{\label{3}}

We calculate $m(x)$ using the boundary state formalism\cite{S,S2,S3}
\begin{equation}{\label{mag}}
m(x)=\frac{\langle 0|\cos\left(\beta\Theta(x)\right)|B\rangle}{\langle 0 |B\rangle},
\end{equation}
where $|B\rangle$ is the boundary state corresponding to the fixed boundary conditions at $x=0$. It can be expressed as
\begin{equation}{\label{est}}
|B\rangle=\mathcal{N}\left[1+\frac{1}{2}\int_{-\infty}^{\infty}d\theta K^{ab}(\theta)Z^{\dag}_a(\theta)Z^{\dag}_b(-\theta)+\cdot\cdot\cdot\right]|0\rangle.
\end{equation}
$Z^{\dag}_a(\theta)$ are creation operators of solitons, antisolitons and breathers and $K$ is the boundary $K$-matrix. For $\beta=\frac{1}{2}$ there are two breathers with gaps $\Delta_{B_1}=\Delta$ and  $\Delta_{B_2}=\Delta\sqrt{3}$ respectively, where $\Delta=\Delta_s=\Delta_{\bar{s}}$ is the soliton gap. Energy and momentum of soliton, antisoliton and breathers are parametrized as
\begin{equation}
E_a(\theta)=\Delta_a\cosh\theta,\hspace{0.25cm}P_a(\theta)=\frac{\Delta_a}{v}\sinh\theta,\hspace{0.25cm}a=s,\bar{s},B_1,B_2.
\end{equation}
Inserting (\ref{est}) into (\ref{mag}) gives an expansion
\begin{eqnarray}
\hspace{-0.0cm}m(x)=\langle 0|\cos(\beta\Theta(x))|0\rangle\nonumber\\\hspace{-0.5cm}+\frac{1}{2}\int_{-\infty}^{\infty}d\theta K^{ab}(\theta)\langle 0|\cos\left(\beta\Theta(x)\right)|\theta,-\theta\rangle_{ab}+\cdot\cdot\cdot\nonumber\\
\end{eqnarray}
where all expectation values are now with respect to the bulk sine-Gordon model. In the boundary state picture we have
\begin{equation}
\cos(\beta\Theta(x))=e^{-\mathcal{H}x/v}\cos(\beta\Theta(0))e^{\mathcal{H}x/v},
\end{equation}
so that
\begin{eqnarray}
\hspace{-0.0cm}m(x)=\langle 0|\cos(\beta\Theta(x))|0\rangle\nonumber\\\hspace{-0.0cm}+\frac{1}{2}\int_{-\infty}^{\infty}d\theta K^{ab}(\theta)\langle 0|\cos\left(\beta\Theta(0)\right)|\theta,-\theta\rangle_{ab}e^{\left[\frac{\Delta_a+\Delta_b}{v}\cosh(\theta)x\right]}+\cdot\cdot\cdot\nonumber\\
\end{eqnarray}
Writing out all contributions from two-particle states gives
\begin{eqnarray}{\label{formfactor}}
\hspace{-0.3cm}m(x)=\langle 0|\cos(\beta\Theta(x))|0\rangle\nonumber\\\hspace{-0.3cm}+\frac{1}{2}\int_{-\infty}^{\infty}d\theta K^{B_1B_1}(\theta)\langle 0|\cos\left(\beta\Theta(0)\right)|\theta,-\theta\rangle_{B_1B_1}e^{\left[\frac{2\Delta}{v}\cosh(\theta)x\right]}\nonumber\\
\hspace{-0.3cm}+\frac{1}{2}\int_{-\infty}^{\infty}d\theta K^{s\bar{s}}(\theta)\langle 0|\cos\left(\beta\Theta(0)\right)|\theta,-\theta\rangle_{s\bar{s}}e^{\left[\frac{2\Delta}{v}\cosh(\theta)x\right]}\nonumber\\
\hspace{-0.3cm}+\frac{1}{2}\int_{-\infty}^{\infty}d\theta K^{\bar{s}s}(\theta)\langle 0|\cos\left(\beta\Theta(0)\right)|\theta,-\theta\rangle_{\bar{s}s}e^{\left[\frac{2\Delta}{v}\cosh(\theta)x\right]}\nonumber\\
\hspace{-0.3cm}+\frac{1}{2}\int_{-\infty}^{\infty}d\theta K^{B_2B_2}(\theta)\langle 0|\cos\left(\beta\Theta(0)\right)|\theta,-\theta\rangle_{B_2B_2}e^{\left[\frac{2\sqrt{3}\Delta}{v}\cosh(\theta)x\right]}+\cdot\cdot\cdot\nonumber\\
\end{eqnarray}
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{Figure_11.ps}%\hspace{1cm}
\caption{Behavior of the staggered magnetization  ${m(x)}(\times10^{11})$ as a function of $x$ for a value of DMI interaction as $D=0.1$. The calculations were performed for values of external magnetic fields $h_1=h_2=1.0$.}\label{fig_12} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}

\textit{Two-Particle From Factors}: The soliton-antisoliton form factor is
\begin{eqnarray}
\hspace{-0.25cm}\langle 0|\cos(\beta\Theta(0))|\theta_2,\theta_1\rangle_{\bar{s}s}=\langle 0|\cos(\beta\Theta(0))|\theta_2,\theta_1\rangle_{s\bar{s}}\nonumber\\\hspace{-0.25cm}=\mathcal{G}_{\beta}\frac{G(\theta_{12})}{\mathcal{C}_1}\cot\left(\frac{\pi\xi}{2}\right)\frac{4i\cosh\left(\frac{\theta_{12}}{2}\right)}{\xi\sinh\left(\frac{\theta_{12}+i\pi}{\xi}\right)}\cosh\left(\frac{\theta_{12}+i\pi}{2\xi}\right),\nonumber\\
\end{eqnarray}
where $\xi=1/3$ and
\begin{eqnarray}
\hspace{-0.4cm}{G}(\theta)=i\mathcal{C}_1\sinh\left(\frac{\theta}{2}\right)\exp\left\{-\int_{0}^{\infty}\frac{\sinh((1-\xi)t)\sinh^2(t(1-\frac{i\theta}{\pi}))}{t\sinh(2t)\cosh(t)\sinh(\xi t)}dt\right\},\nonumber\\
\mathcal{G}_a\equiv\langle e^{ia\Theta}\rangle=\left[\frac{a_0\Delta\sqrt{\pi}\Gamma\left(\frac{1}{2-2\beta^2}\right)}{2v\Gamma\left(\frac{\beta^2}{2-2\beta^2}\right)}\right]^{2a^2}\nonumber\\ \hspace{-0.4cm}\times\exp\left\{\int_{0}^{\infty}\frac{dt}{t}\left[-2a^2e^{-2t}+\frac{\sinh^2(2a\beta t)}{2\sinh(t\beta^2)\sinh(t)\cosh(t(1-\beta^2))}\right]\right\},\nonumber\\
\end{eqnarray}
where $a=s,\bar{s},B_1,B_2$. Because of the SU(2) symmetry at $\beta=1/2$ we have
\begin{equation}
\langle 0|\cos\left(\beta\Theta\right)|\theta_2,\theta_1\rangle_{B_1B_1}=\langle 0|\cos\left(\beta\Theta\right)|\theta_2,\theta_1\rangle_{s\bar{s}}.
\end{equation}
The $B_2B_2$ form factor is given by
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\hspace{-1.8cm}  \langle 0|\cos\left(\beta\Theta\right)|\theta_2,\theta_1\rangle_{B_2B_2}=\frac{\mathcal{G}_{\beta}\Lambda^4}{2}\tan(\pi\xi)\left[1+\frac{1}{\cosh(\theta_{12})+\cos(\pi\xi)}\right]\nonumber\\
 \hspace{-1.8cm} \times\frac{R^2(\theta_{12})R(-\theta_{12}-i\pi\xi)R(-\theta_{12}+i\pi\xi)}{R(-i\pi(1+\xi))},\nonumber\\
\hspace{-1.8cm}  R(\theta)=\mathcal{N}\exp\left[8\int_{0}^{\infty}\frac{dt}{t}\frac{\sinh(t\xi)\sinh(t)\sinh((1+\xi)t)}{\sinh^2(2t)}\sinh^2\left(1-\frac{i\theta}{\pi}\right)\right],\nonumber\\
\hspace{-1.8cm}  \mathcal{N}=\exp\left[4\int_{0}^{\infty}\frac{dt}{t}\frac{\sinh(t\xi)\sinh(t)\sinh((1+\xi)t)}{\sinh^2(2t)}\right],\nonumber\\
\hspace{-2.2cm}  \Lambda=2\cos\left(\frac{\pi\xi}{2}\right)\sqrt{2\sin\left(\frac{\pi\xi}{2}\right)}\exp\left[-\int_{0}^{\pi\xi}\frac{dt}{2\pi}\frac{t}{\sin(t)}\right].\nonumber\\
\end{eqnarray}

\textit{Boundary K-Matrix}: The $K$-matrix is given by
\begin{equation}
K^{ab}(\theta)=R^b_{\bar{a}}\left(\frac{i\pi}{2}-\theta\right).
\end{equation}
The boundary reflection factors are
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\hspace{-2.0cm}  R^{s}_s(\theta)= R^{\bar{s}}_{\bar{s}}(\theta)=\frac{\cos\left(\frac{\pi}{12}-\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{4}-\frac{i\theta}{2}\right)\cos\left(\frac{5\pi}{12}-\frac{i\theta}{2}\right)}{\cos\left(\frac{\pi}{12}+\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{4}+\frac{i\theta}{2}\right)\cos\left(\frac{5\pi}{12}+\frac{i\theta}{2}\right)},\nonumber\\
\hspace{-2.0cm}R^{B_1}_{B_1}(\theta)=-\frac{1+i\sinh(\theta)\cos\left(\frac{\pi}{12}-\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{3}+\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{4}+\frac{i\theta}{2}\right)}{1-i\sinh(\theta)\cos\left(\frac{\pi}{12}+\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{3}-\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{4}-\frac{i\theta}{2}\right)},\nonumber\\
\hspace{-2.0cm}R^{B_2}_{B_2}(\theta)=-\left[\frac{\cos\left(\frac{\pi}{6}\right)+i\sinh\theta\cos\left(\frac{\pi}{3}+i\frac{\theta}{2}\right)}{\cos\left(\frac{\pi}{6}\right)-i\sinh\theta\cos\left(\frac{\pi}{3}-i\frac{\theta}{2}\right)}\right]^2\nonumber\\
\hspace{-2.0cm}\times\frac{\cos\left(\frac{\pi}{6}-\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{4}+\frac{i\theta}{2}\right)\cos\left(\frac{5\pi}{12}+\frac{i\theta}{2}\right)}{\cos\left(\frac{\pi}{6}+\frac{i\theta}{2}\right)\cos\left(\frac{\pi}{4}-\frac{i\theta}{2}\right)\cos\left(\frac{5\pi}{12}-\frac{i\theta}{2}\right)}.\nonumber\\
\end{eqnarray}

We solve numerically the integrals for the two-particle soliton-antisoliton form factors of the expansion of Eq.~(\ref{formfactor}) as $\langle 0|\cos\left(\beta\Theta\right)|\theta_2,\theta_1\rangle_{B_1B_1}=\langle 0|\cos\left(\beta\Theta\right)|\theta_2,\theta_1\rangle_{s\bar{s}}=\langle 0|\cos(\beta\Theta(0))|\theta_2,\theta_1\rangle_{\bar{s}s}=6.26011(4)\times 10^{-8}-4.12147(8)\times 10^8i$ and $\langle 0|\cos\left(\beta\Theta\right)|\theta_2,\theta_1\rangle_{B_2B_2}= 6.56150(5)$, where $\Lambda=1.45010(4)$, $\mathcal{N}= 1.491471(1)$ and $R(\theta)= 4.41487(1)$. In addition, $R(\theta)_s^{\bar{s}}= 0.37808(1)+0.92577(2)i$, $R(\theta)_{B_1}^{B_1}= 0.86084(8)+0.50886(3)i$ and
$R(\theta)_{B_2}^{B_2}= -0.74005(7)-0.67254(4)i$ for $\theta=\pi/4$. In Fig.~\ref{fig_12}, we show the behavior of staggered magnetization as a function of $x$, $m(x)$. We get a increase of $m(x)$ for a small value of $D$ as $D=0.1$ held fixed. The precise value of the DMI in spin-1/2 one-dimensional antiferromagnets can be difficult to predict and may depend on the specific material and its crystal structure. The DMI strength can be estimated theoretically using various methods, such as perturbation theory, exact diagonalization, or density functional theory calculations.
For example, in some spin-1/2 one-dimensional antiferromagnets, such as CuGeO$_3$ or LiCu$_2$O$_2$, the DMI strength can be on the order of a few tenths of meV, which is smaller than the exchange interaction energy scale but still significant for the magnetic properties of these materials. In other systems, such as Cs$_2$CuBr$_4$ or Sr$_2$CuO$_3$, the DMI can be weaker or negligible due to the specific crystal structure and the spin configuration of the system. In Fig.~\ref{fig_13}, we show the behavior of staggered magnetization as a function of DMI strength $D$. We obtain a large influence of $m(x)$ with $D$ up to $D=1.0$. In general, the behavior of the staggered magnetization as a function of DMI depends on the geometry of the magnetic system. In a thin film with perpendicular magnetic anisotropy, the DMI interaction can induce a net staggered magnetization, which is the difference between the magnetization of neighboring layers. This staggered magnetization is proportional to the DMI strength and can be tuned by changing the thickness of the film or the magnitude of the magnetic field. Overall, the DMI can have a significant impact on the magnetic properties of materials, and its control and manipulation are important for the development of magnetic devices and technologies.
\begin{figure}
    \centering
\includegraphics[width=7.0cm]{Figure_2.ps}%\hspace{1cm}
\caption{Behavior of the staggered magnetization  ${m(x)}(\times10^{11})$ as a function of DMI strength $D$ at $x=0.0$. The calculations were performed for values of external magnetic fields $h_1=h_2=1.0$.}\label{fig_13} %$J_1=−1$, $J_2=1$, $J_{bq_1}=−3$, $J_{bq_2}=0.5$, (black),
\end{figure}





{\textit{Staggered Susceptibility:}} We discuss the susceptibility of the sine-Gordon
model. Writing the sine-Gordon Lagrangian in the form
\begin{equation}
\mathcal{L}=\frac{1}{2}(\partial_{\nu}\Phi(x))^2+\mu\cos(\beta\Theta(x)),
\end{equation}
where $\Theta$ is the dual field of $\Phi$ and $\beta$ and $\mu$ depends on the value of the applied field, $\mu(h_1,h_2)$ and $\mu$ can not be calculated exactly\cite{Fabian3}. Adopting units where $v=1$ we have the sine-Gordon susceptibility as
\begin{equation}
\chi=-\frac{\partial^2\mathcal{F}}{\partial\mu^2},
\end{equation}
where $\mathcal{F}$ is the free energy. The high-temperature susceptibility is given by
\begin{equation}
\chi_T=\int_{0}^{\beta}\int_{-\infty}^{\infty}d\tau dx\langle 0|\cos(\sqrt{2\pi}\Phi(\tau,x)\cos(\sqrt{2\pi}\Phi(0,0)|0\rangle,
\end{equation}
$\mid 0\rangle$ is the ground state, $\beta=1/T$, being normalized to $1/2r$ at $T=\tau=0$ ($\beta$ infinite). For finite $\tau$ and $T$ we have $(r^2+\tau^2)^{-1/2}\rightarrow\frac{\pi}{\beta}\left[\sin\frac{\pi}{\beta}(\tau+ix)\sin\frac{\pi}{\beta}(\tau-ix)\right]^{-1/2}$. Thus, the susceptibility becomes
\begin{equation}
\hspace{-1.25cm}\chi_T\approx \frac{2\sqrt{2}\pi}{\beta}\int_{0}^{\beta}\int_{-\infty}^{\infty}\frac{dxd\tau}{\sqrt{\cosh(2\pi x/\beta)-\cos(2\pi\tau/\beta)}}\simeq  8.75(4)\beta.
\end{equation}
This result was obtained by analytic continuation in\cite{shultz}. Up to a multiplicative constant and logarithmic corrections, the sine-Gordon susceptibility essentially gives the staggered susceptibility, i.e., its response to a staggered field of the spin-1/2 chain.
In a general way, the DMI can induce a gap in the energy spectrum of the system, which can suppress the low-energy spin-wave excitations and modify the power-law behavior of the magnetic susceptibility. In particular, the magnetic susceptibility can show a non-universal exponent at low temperatures, which depends on the strength of the DMI and the anisotropy of the system. This behavior has been observed in several experimental and theoretical studies of spin-1/2 Heisenberg chains with DMI, such as CuGeO$_3$, LiCu$_2$O$_2$, or Cs$_2$CuBr$_4$.
How the DMI interaction can have a significant impact on the magnetic susceptibility of spin-1/2 one-dimensional Heisenberg models, its control and manipulation can provide new opportunities for the design of magnetic materials and devices.


