Thermal Entanglement and Thermal Discord in two-qubit Heisenberg XYZ Chain with Dzyaloshinskii-Moriya Interactions

TL;DR

This paper analytically investigates how temperature affects quantum correlations, specifically entanglement and discord, in a two-qubit Heisenberg XYZ model with Dzyaloshinskii-Moriya interactions, revealing phase transition phenomena and conditions for their simultaneous disappearance.

Contribution

It provides an analytical study of thermal entanglement and discord in the Heisenberg XYZ model with Dzyaloshinskii-Moriya interactions, identifying critical temperatures and conditions for quantum correlation loss.

Findings

Quantum phase transition occurs at a critical temperature T_c due to sudden death.

T_c increases with |D| in antiferromagnetic cases.

Thermal entanglement and discord vanish simultaneously at specific parameters.

Abstract

In order to explore the effect of external temperature $T$ in quantum correlation we compute thermal entanglement and thermal discord analytically in the Heisenberg $X$ $Y$ $Z$ model with Dzyaloshinskii-Moriya Interaction term ${\bm D} \cdot \left( {\bm \sigma}_1 \times {\bm \sigma}_2 \right)$. For the case of thermal entanglement it is shown that quantum phase transition occurs at $T = T_c$ due to sudden death phenomenon. For antiferromagnetic case the critical temperature $T_c$ increases with increasing $|{\bm D}|$. For ferromagnetic case, however, $T_c$ exhibits different behavior in the regions $|{\bm D}| \geq |{\bm D_*}|$ and $|{\bm D}| < |{\bm D_*}|$, where ${\bm D_*}$ is particular value of ${\bm D}$. It is shown that $T_c$ becomes zero at $|{\bm D}| = |{\bm D_*}|$. We explore the behavior of thermal discord in detail at $T \approx T_c$. For antiferromagnetic case the external temperature makes the thermal discord exhibit exponential damping behavior, but it never reaches to exact zero. For ferromagnetic case the thermal entanglement and thermal discord are shown to be zero simultaneously at $T_c = 0$ and $|{\bm D}| = |{\bm D_*}|$. This is unique condition for simultaneous disappearance of thermal entanglement and thermal discord in this model.

Equations

$$H_N=\sum _{i=1}^{N-1}\left[J_x{\sigma }_i^x\otimes {\sigma }_{i+1}^x+J_y{\sigma }_i^y\otimes {\sigma }_{i+1}^y+J_z{\sigma }_i^z\otimes {\sigma }_{i+1}^z+\mathbf{D}\cdot \left({\mathbf{\sigma }}_i\times {\mathbf{\sigma }}_{i+1}\right)\right],$$

$$H=J_x{\sigma }_1^x\otimes {\sigma }_2^x+J_y{\sigma }_1^y\otimes {\sigma }_2^y+J_z{\sigma }_1^z\otimes {\sigma }_2^z+\mathbf{D}\cdot \left({\mathbf{\sigma }}_1\times {\mathbf{\sigma }}_2\right).$$

$$\displaystyle H_{Z}=\left(\begin{array}[]{cccc}J_{z}&0&0&J_{x}-J_{y}\\ 0&-J_{z}&J_{x}+J_{y}+2iD_{z}&0\\ 0&J_{x}+J_{y}-2iD_{z}&-J_{z}&0\\ J_{x}-J_{y}&0&0&J_{z}\end{array}\right).$$

$$\xi =\sqrt{4D_z^2+(J_x+J_y{)}^2}\theta ={\tan }^{-1}\left(-\frac{2D_z}{J_x+J_y}\right).$$

$$H_Z=\sum _{i=1}^4{E}_{i,z}|z_i⟩⟨z_i|.$$

$${\mathcal{Z}}_Z\equiv \text{mbox}Tr\left[e^{-\beta H_Z}\right]=2e^{-\beta J_z}\cosh \left[\beta (J_x-J_y)\right]+2e^{\beta J_z}\cosh (\beta \xi )$$

$$\displaystyle\rho_{Z}(T)\equiv\frac{1}{{\cal Z}_{Z}}e^{-\beta H_{Z}}=\left(\begin{array}[]{cccc}r&0&0&s\\ 0&u&v&0\\ 0&v^{*}&u&0\\ s&0&0&r\end{array}\right)$$

$$r=\frac{1}{{\mathcal{Z}}_Z}{e}^{-\beta J_z}\cosh \left[\beta (J_x-J_y)\right]s=-\frac{1}{{\mathcal{Z}}_Z}{e}^{-\beta J_z}\sinh \left[\beta (J_x-J_y)\right]$$

$$u=\frac{1}{{\mathcal{Z}}_Z}{e}^{\beta J_z}\cosh (\beta \xi )v=-\frac{1}{{\mathcal{Z}}_Z}\frac{J_x+J_y+2i{D}_z}{\xi }{e}^{\beta J_z}\sinh (\beta \xi ).$$

$$\displaystyle H_{Y}=\left(\begin{array}[]{cccc}J_{z}&D_{y}&-D_{y}&J_{x}-J_{y}\\ D_{y}&-J_{z}&J_{x}+J_{y}&D_{y}\\ -D_{y}&J_{x}+J_{y}&-J_{z}&-D_{y}\\ J_{x}-J_{y}&D_{y}&-D_{y}&J_{z}\end{array}\right).$$

$$\eta =\sqrt{4D_y^2+(J_x+J_z{)}^2}{\varphi }_1={\tan }^{-1}\left[-\frac{2D_y}{\eta -(J_x+J_z)}\right]{\varphi }_2={\tan }^{-1}\left[\frac{2D_y}{\eta +(J_x+J_z)}\right].$$

$${\mathcal{Z}}_Y\equiv \text{mbox}Tr\left[e^{-\beta H_Y}\right]=2e^{-\beta J_y}\cosh \left[\beta (J_x-J_z)\right]+2e^{\beta J_y}\cosh (\beta \eta ).$$

$$\displaystyle\rho_{Y}(T)=\frac{1}{{\cal Z}_{Y}}e^{-\beta H_{Y}}=\left(\begin{array}[]{cccc}r_{1}&-q&q&r_{2}\\ -q&u_{1}&u_{2}&-q\\ q&u_{2}&u_{1}&q\\ r_{2}&-q&q&r_{1}\end{array}\right)$$

$$r_1=\frac{1}{2{\mathcal{Z}}_Y}\left[e^{-\beta E_{2,y}}+{\sin }^2{\varphi }_1{e}^{-\beta E_{3,y}}+{\sin }^2{\varphi }_2{e}^{-\beta E_{4,y}}\right]$$

$$r_2=\frac{1}{2{\mathcal{Z}}_Y}\left[-e^{-\beta E_{2,y}}+{\sin }^2{\varphi }_1{e}^{-\beta E_{3,y}}+{\sin }^2{\varphi }_2{e}^{-\beta E_{4,y}}\right]$$

$$u_1=\frac{1}{2{\mathcal{Z}}_Y}\left[e^{-\beta E_{1,y}}+{\cos }^2{\varphi }_1{e}^{-\beta E_{3,y}}+{\cos }^2{\varphi }_2{e}^{-\beta E_{4,y}}\right]$$

$$u_2=\frac{1}{2{\mathcal{Z}}_Y}\left[e^{-\beta E_{1,y}}-{\cos }^2{\varphi }_1{e}^{-\beta E_{3,y}}-{\cos }^2{\varphi }_2{e}^{-\beta E_{4,y}}\right]$$

$$q=\frac{1}{2{\mathcal{Z}}_Y}\left[\sin {\varphi }_1\cos {\varphi }_1{e}^{-\beta E_{3,y}}+\sin {\varphi }_2\cos {\varphi }_2{e}^{-\beta E_{4,y}}\right].$$

$$\mathcal{C}(\rho )=\text{mbox}max\left({\lambda }_1-{\lambda }_2-{\lambda }_3-{\lambda }_4,0\right),$$

$$\mathcal{C}\left({\rho }_Z\right)=\max \left[|{\xi }_1-{\xi }_3|-{\xi }_2-{\xi }_4,0\right]$$

$${\xi }_1=\frac{1}{{\mathcal{Z}}_Z}{e}^{\beta [|J_x-J_y|-J_z]}{\xi }_2=\frac{1}{{\mathcal{Z}}_Z}{e}^{-\beta [|J_x-J_y|+J_z]}$$

$${\xi }_3=\frac{1}{{\mathcal{Z}}_Z}{e}^{\beta (J_z+\xi )}{\xi }_4=\frac{1}{{\mathcal{Z}}_Z}{e}^{\beta (J_z-\xi )}.$$

$$\mathcal{C}\left({\rho }_Z\right)=\frac{e^{\beta J_z}}{{\mathcal{Z}}_Z}\max \left[|e^{2\beta w}-e^{-2\beta J_z}|-e^{-2\beta w}-e^{-2\beta J_z},0\right]$$

$$e^{2J_z/T_c}\sinh \left(\frac{2w}{T_c}\right)=1.$$

$$e^{2J_z/T_c}\frac{\sinh (\xi /T_c)}{\cosh (|J_x-J_y|/T_c)}=1\text{mbox}when2J_z+\xi \ge |J_x-J_y|$$

$$e^{-2J_z/T_c}\frac{\sinh (|J_x-J_y|/T_c)}{\cosh (\xi /T_c)}=1\text{mbox}when2J_z+\xi <|J_x-J_y|.$$

$$D_{z,\ast }=\sqrt{(J_z-J_{>})(J_z+J_{<})}$$

$$\mathcal{C}\left({\rho }_Y\right)=\max \left[|{\eta }_1-{\eta }_3|-{\eta }_2-{\eta }_4,0\right]$$

$${\eta }_1=\frac{1}{{\mathcal{Z}}_Y}{e}^{\beta [|J_x-J_z|-J_y]}{\eta }_2=\frac{1}{{\mathcal{Z}}_Y}{e}^{-\beta [|J_x-J_z|+J_z]}$$

$${\eta }_3=\frac{1}{{\mathcal{Z}}_Y}{e}^{\beta (J_y+\eta )}{\eta }_4=\frac{1}{{\mathcal{Z}}_Y}{e}^{\beta (J_y-\eta )}.$$

$$e^{2J/T_c}\frac{\sinh (\eta /T_c)}{\cosh (|J-J_z|/T_c)}=1\text{mbox}when2J+\eta \ge |J-J_z|$$

$$e^{-2J/T_c}\frac{\sinh (|J-J_z|/T_c)}{\cosh (\eta /T_c)}=1\text{mbox}when2J+\eta <|J-J_z|.$$

$$e^{2J_y/T_c}\frac{\sinh (\eta /T_c)}{\cosh (|J_x-J_z|/T_c)}=1\text{mbox}when2J_y+\eta \ge |J_x-J_z|$$

$$e^{-2J_y/T_c}\frac{\sinh (|J_x-J_z|/T_c)}{\cosh (\eta /T_c)}=1\text{mbox}when2J_y+\eta <|J_x-J_z|.$$

$$D_{y,\ast }=\sqrt{(J_y-{\tilde{J}}_{>})(J_y+{\tilde{J}}_{<})}$$

Full text

I Introduction

Recently, much attention is paid to quantum technology developed on the foundation of quantum information processing (QIP). The physical resource of QIP is quantum correlation such as quantum entanglementschrodinger-35 ; text ; horodecki09 and quantum discorddiscord1 ; discord2 . Thus, they are at the heart in various QIP such as quantum teleportationteleportation , superdense codingsuperdense , quantum cloningclon , quantum cryptographycryptography ; cryptography2 , quantum metrologymetro17 , and quantum computerqcomputer ; qcreview . In particular, physical realization of quantum cryptography and quantum computer seems to be accomplished in the near future111see Ref. white and web page https://www.computing.co.uk/ctg/news/3065541/european-union-reveals-test-projects-for-first-tranche-of-eur1bn-quantum-computing-fund..

Pure quantum mechanical phenomena can occur in ideally closed system. However, real physical systems inevitably interact with their surroundings. Then, quantum systems undergo decoherencezurek03 and, as a result, lose their quantum properties. Thus, the environment can make various QIP useless due to disappearance of quantum correlation. In order to develop the quantum technology, therefore, it is important to study effect of its surroundings precisely.

Usually, decoherence significantly changes the quantum correlation. For example, decoherence makes degradation of entanglement and discord. For the case of entanglement the entanglement sudden death (ESD) occurs sometimes when the entangled multipartite quantum system is embedded in Markovian environmentsmarkovian ; yu05-1 ; yu06-1 ; yu09-1 ; almeida07 . This means that the entanglement is completely disentangled at finite times. Most typical environment is external temperature. If external temperature induces the ESD phenomenon in a system, this implies that quantum phase transition occurs at critical temperature $T_{c}$. This means that quantum entanglement completely disappears at $T\geq T_{c}$. The purpose of this paper is to examine how quantum entanglement and quantum discord are degraded due to external temperature and to study on the quantum phase transition in detail by introducing the anisotropic Heisenberg $X$ $Y$ $Z$ chain system with Dzyaloshinskii-Moriya (DM) interactiondzya58 ; mori60 . The quantum phase transition of a spin-$3$ Heisenberg model with DM interaction was discussed in Ref. marko17 .

Heisenberg model is a simple spin chain model, which is used to simulate many physical systems such as nuclear spinskane98 , quantum dotsdots , superconductorsuper , and optical latticesoptics . Since spin is two-level, Heisenberg model is ideal for generation of qubit states. Thus, this model attracts attention recently for realization of solid-based quantum computer. In Ref. byrnes17 the DM interaction terms are introduced in this model due to spin-orbit couplings. The general Hamiltonian for $N$-spin Heigenberg model with DM interaction is

[TABLE]

where the last term is called the DM interaction arising from spin-orbit couplings222In addition to DM interaction the spin-orbit coupling induces the second order term called $\Gamma$ tensorshek92 ; suhas08 ; marko18 . This term is neglected in this paper. This means that our paper is valid only in the first order correction of spin-orbit interaction.. The real parameters $J_{\alpha}\hskip 5.69046pt(\alpha=x,y,x)$ denote the symmetric exchange spin-spin interactions, ${\bm{D}}$ is the antisymmetric DM exchange interaction, and $\sigma_{i}^{x,y,z}$ are Pauli spin operators on the site $i$. The negative and positive $J_{\alpha}\hskip 5.69046pt(\alpha=x,y,x)$ correspond to the ferromagnetic and antiferromagnetic nature of the system respectively. If $J_{x}=J_{y}\neq J_{z}$, this system is called $X$ $X$ $Z$ model with DM interaction.

In this paper we study on the effect of external temperature in the entanglement and discord based on the analytic results by introducing $2$-qubit Heisenberg model with DM-interaction. In particular, we focus on the quantum phase transition in entanglement, which occurs due to ESD phenomena. Our motivation is as follows. Although definitions of entanglement and discord are completely different, these are two different measures of quantum correlation. Thus, we guess they should exhibit similar behavior to each other. In order to confirm our conjecture we examine in detail the behavior of thermal discord at $T\approx T_{c}$. It is shown that for antiferromagnetic case ($J_{\alpha}>0$) the temperature dependence of discord exhibits an exponential damping behavior, but it never reaches to zero. This means that thermal discord does not completely vanish in the separable states arising from thermal density matrix at $T\geq T_{c}$. For ferromagnetic case ($J_{\alpha}<0$) it is shown that the critical temperature $T_{c}$ approaches to zero if the DM coupling constant ${\bm{D}}$ approaches to a particular value ${\bm{D}_{*}}$. At ${\bm{D}}={\bm{D}_{*}}$ thermal discord complete vanishes at $T=T_{c}=0$. The point ${\bm{D}}={\bm{D}_{*}}$ and $T=0$ in parameter space is a unique point, where the thermal entanglement and thermal discord simultaneously vanish.

The paper is organized as follows. In section II we derive the thermal density matrices $\rho_{Z}(T)$ when $D_{x}=D_{y}=0$ and $\rho_{Y}(T)$ when $D_{x}=D_{z}=0$. The case of $D_{y}=D_{z}=0$ is not presented in this paper because all calculation is very similar to the case of $D_{x}=D_{z}=0$. If two components of ${\bf D}$ are nonzero, we should rely on numerical analysis for computation of entanglement and discord. Thus, these cases are not considered in this paper. In section III we compute the thermal entanglement of $\rho_{Z}(T)$ and $\rho_{Y}(T)$ analytically. Using our analytical results we examine the critical temperature $T_{c}$, above which the thermal entanglement completely vanishes. It is shown that for antiferromagnetic case ($J_{\alpha}>0,\hskip 5.69046pt\alpha=x,y,z$) the critical temperature is determined by single equation for each thermal density matrix. For ferromagnetic case ($J_{\alpha}<0,\hskip 5.69046pt\alpha=x,y,z$), however, the critical temperature is determined by two different equations in the two separate two regions $|{\bm{D}}|\geq|{\bm{D}_{*}}|$ and $|{\bm{D}}|<|{\bm{D}_{*}}|$. It is shown that $T_{c}$ becomes zero at the boundary of these region $|{\bm{D}}|=|{\bm{D}_{*}}|$. In section IV the thermal discords of $\rho_{Z}(T)$ and $\rho_{Y}(T)$ are analytically computed. Using the analytic results we examine the behavior of thermal discords near the critical temperature in detail. For antiferromagnetic case the $T$-dependence of the discords exhibits exponential damping behavior with increasing $T$ like thermal entanglement, but it does not reach to exact zero. For ferromagnetic case, however, thermal discord becomes exact zero at the point ${\bm{D}}={\bm{D}_{*}}$ and $T_{c}=0$. The behavior of thermal discord at $T=T_{c}$ for arbitrary $|{\bm{D}}|$ is also examined in detail. In section V a brief conclusion is given. In appendix A the thermal discord for $\rho_{Y}(T)$ are explicitly computed. In appendix B we discuss the critical behavior of concurrence when $D_{x}\neq 0$ and $D_{y}\neq 0$ for completeness.

II Thermal Density Matrix

The Hamiltonian for two-spin anisotropic Heisenberg $X$ $Y$ $Z$ chain with DM interaction is given bybyrnes17

[TABLE]

In Ref. guof1 and Ref. cao08 ; langari09 the entanglement of this model with $J_{x}=J_{y}=J_{z}$ and $J_{x}=J_{y}\neq J_{z}$ was discussed respectively. The quantum phase transition with an applied magnetic field was studied in Ref. li08 . Also the thermal entanglement of three-qubit ground state was discussedjafari11 ; marko-5 . Quantum discorddiscord1 ; discord2 , another measure of quantum correlation, of this model was discussed in Ref. zhi10 ; zidan1 when $J_{x}=J_{y}$ and in Ref. zou11 when $J_{z}=0$. Recently, the quantum-memory-assisted entropic uncertaintiesrenes09 ; berta10 were examined in this Heisenberg modelzhang18 .

II.1 $D_{x}=D_{y}=0$ case

In this case the matrix representation of the Hamiltonian in the computational basis $\{\lvert 00\rangle,\lvert 01\rangle,\lvert 10\rangle,\lvert 11\rangle\}$ becomes

[TABLE]

The eigenvalues and corresponding eigenvectors of $H_{Z}$ are summarized in Table I. In this Table $\xi$ and $\theta$ are given by

[TABLE]

[TABLE]

Table I: Eigenvalues and eigenvectors of $H_{Z}$

Thus, the spectral decomposition of $H_{Z}$ can be written as

[TABLE]

The partition function of this system is given by

[TABLE]

where $\beta=1/k_{B}T$, where $k_{B}$ and $T$ are Boltzmann constant and external temperature. Throughout this paper we use $k_{B}=1$ for convenience. Then the thermal density matrix in this case becomes

[TABLE]

where

[TABLE]

It is worthwhile noting $r+u=1/2$, which guarantees $\mbox{Tr}[\rho_{Z}(T)]=1$.

II.2 $D_{x}=D_{z}=0$ case

In this case the matrix representation of the Hamiltonian in the computational basis becomes

[TABLE]

The eigenvalues and corresponding eigenvectors of $H_{Y}$ are summarized in Table II. In this Table $\eta$, $\phi_{1}$ and $\phi_{2}$ are given by

[TABLE]

[TABLE]

Table II: Eigenvalues and eigenvectors of $H_{Y}$

The fact $\cos(\phi_{1}-\phi_{2})=0$ guarantees the orthonormal condition of $\lvert y_{j}\rangle$, i.e., $\langle y_{i}\lvert y_{j}\rangle=\delta_{ij}$.

Thus, the spectral decomposition of $H_{Y}$ is $H_{Y}=\sum_{i=1}^{4}E_{i,y}\lvert y_{i}\rangle\langle y_{i}\lvert$ and the partition function is

[TABLE]

The thermal density matrix in this system can be written in a form

[TABLE]

where

[TABLE]

Since $r_{1}+u_{1}=1/2$, it is easy to show $\mbox{Tr}\left[\rho_{Y}(T)\right]=1$. Since $D_{y}=D_{z}=0$ case is similar to $D_{x}=D_{z}=0$ case, we do not explore this case in this paper.

III Thermal Entanglement

In this section we examine the temperature dependence of entanglement for the thermal density matrices $\rho_{Z}(T)$ and $\rho_{Y}(T)$ by making use of concurrenceform2 ; form3 . Thermal entanglement of $X$ $Y$ $Z$ model was considered in Ref. rigolin04 when there is no DM interaction and in Ref. cao-5 when one component of DM-interaction is nonzero. Thus, our calculation of this section overlaps with that of Ref. cao-5 . However we focus on the critical temperature $T_{c}$ in detail because our main motivation is to examine the quantum discord around $T\sim T_{c}$ Following the procedure presented in Ref. form3 one can compute the concurrence ${\cal C}(\rho)$ for a quantum state $\rho$ by a simple formula

[TABLE]

where the $\lambda_{i}$’s are eigenvalues, in decreasing order, of the Hermitian matrix $\sqrt{\sqrt{\rho}(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})\sqrt{\rho}}$.

Following this procedure it is easy to show that the concurrence of $\rho_{Z}(T)$ is

[TABLE]

where

[TABLE]

Before proceeding further, let us consider $X$ $X$ $Z$ model ($J_{x}=J_{y}=J$) with DM interaction. In this model the concurrence becomes

[TABLE]

where $w=\sqrt{J^{2}+D_{z}^{2}}$. This is plotted in Fig. 1(a) as a function of $T$ and $D_{z}$ with choosing $J=1$ and $J_{z}=0.2$. As this figure shows, this model exhibits a quantum phase transition with the critical temperature $T_{c}$. This means that the entanglement completely vanishes at $T\geq T_{c}$. From Eq. (33) one can show that the critical temperature satisfies

[TABLE]

Now, let us go back to the $X$ $Y$ $Z$ model with DM interaction. In this model the critical temperature $T_{c}$ satisfies

[TABLE]

When $J_{x}=J_{y}$, first equation reduces to Eq. (34) and second equation does not yield any solution.

For antiferromagnetic ($J_{\alpha}>0$) case the second equation does not play any role because the condition $2J_{z}+\xi<|J_{x}-J_{y}|$ cannot be satisfied. Thus, the behavior of the concurrence ${\cal C}(\rho_{Z})$ is similar to that of $X$ $X$ $Z$ model. The $D_{z}$-dependence of $T_{c}$ is plotted in Fig. 1(b) when $J_{x}=1$ and $J_{y}=1.5$ with varying $J_{z}=2$ (black line), $3$ (red dashed line), and $4$ (blue dotted line). Fig. 1(b) shows that $T_{c}$ increases very rapidly with increasing $|D_{z}|$. With fixed $D_{z}$, $T_{c}$ increases with increasing $J_{z}$.

For ferromagnetic ($J_{\alpha}<0$) case, however, the second equation of Eq. (35) provides significant solutions, which result in different behavior of ${\cal C}(\rho_{Z})$. The boundary of two region in Eq. (35) is $D_{z}=D_{z,*}$, where

[TABLE]

with $J_{>}=\max(J_{x},J_{y})$ and $J_{<}=\min(J_{x},J_{y})$. At this point $T_{c}$ becomes exactly zero because $\xi_{1}=\xi_{3}$ at this point. In Fig. 2(a) we plot ${\cal C}(\rho_{Z})$ as a function of $T$ and $D_{z}$ with choosing $J_{x}=-1$, $J_{y}=-1.5$, and $J_{z}=-2$. As this figure shows, the concurrence exhibits different behavior in $0\leq D_{z}\leq\sqrt{7/2}=1.87$ and other regions. This is because of the fact that the second equation of Eq. (35) generates the critical temperature in the region $0\leq D_{z}\leq\sqrt{7/2}$ while the first equation generates it in other region. In order to show more precisely we plot the $D_{z}$-dependence of $T_{c}$ in Fig. 2 (b) when $J_{z}=-2$ (black line), $-3$ (red dashed line), and $-4$ (blue dotted line) with fixing $J_{x}=-1$ and $J_{y}=-1.5$. As this figure shows, $T_{c}$ increases monotonically with increasing $|D_{z}|$ in the region $|D_{z}|\geq D_{z,*}$ as $X$ $X$ $Z$ model. With fixed $D_{z}$, in this region, $T_{c}$ decreases with increasing $|J_{z}|$. However, it behaves differently in the region $|D_{z}|<D_{z,*}$. In this region $T_{c}$ decreases with increasing $|D_{z}|$. With fixed $D_{z}$, in this region, $T_{c}$ increases with increasing $|J_{z}|$. At $D_{z}=D_{z,*}$ this figure confirms $T_{c}=0$.

For neither ferromagnetic nor antiferromagnetic case the behavior of the concurrence is determined whether the second equation of Eq. (35) plays a role or not. For example, let us consider $J_{z}>0$ case. If, in this case, $D_{z}^{2}+J_{x}J_{y}\geq 0$, the second equation of Eq. (35) does not play any role. Thus, in this case the concurrence is similar to Fig. 1 (a). If $D_{z}^{2}+J_{x}J_{y}<0$, the condition for the existence of $D_{z,*}$ is $0<J_{z}<\min(-J_{<},J_{>})$. Thus, both $D_{z}^{2}+J_{x}J_{y}<0$ and $0<J_{z}<\min(-J_{<},J_{>})$ hold, the concurrence of $\rho_{Z}$ is similar to Fig. 2 (a). For negative $J_{z}$ case one can derive $J_{z}<\min(-J_{<},0)$ for $J_{>}>-J_{<}$ and $J_{z}<\min(J_{>},0)$ for $-J_{<}>J_{>}$ for the existence of $D_{z,*}$. In Fig. 3 we plot $D_{z}$- and $T$-dependence of ${\cal C}(\rho_{Z})$ when $J_{x}=3$ and $J_{y}=2$ with $J_{z}=-1$ (Fig. 3 (a)) and $J_{z}=-2.5$ (Fig. 3(b)). As expected Fig. 3(a) and Fig. 3(b) exhibit similar behavior to Fig. 1(a) and Fig. 2(a) respectively.

For $\rho_{Y}(T)$ the concurrence becomes

[TABLE]

where

[TABLE]

Unlike $\rho_{Z}(T)$, even in the $X$ $X$ $Z$ model with DM interaction the critical temperature $T_{c}$ is determined by two different equations as follows

[TABLE]

For the antiferromagnetic case ($J,J_{z}>0$) the critical temperature is determined by only first equation of Eq. (39) because $2J+\eta\geq|J-J_{z}|$ in this case. Thus, the concurrence exhibits similar behavior with that of Fig. 1(a). This is confirmed in Fig. 4(a), where ${\cal C}(\rho_{Y})$ is plotted as a function of $T$ and $D_{y}$ with choosing $J=3$ and $J_{z}=1$. For the ferromagnetic ($J,J_{z}<0$) case, however, both equations of Eq. (39) play significant role for determining the critical temperature. This is also confirmed in Fig. 4(b), where ${\cal C}(\rho_{Y})$ is plotted as a function of $T$ and $D_{y}$ with choosing $J=-3$ and $J_{z}=-1$. In this case the second equation of Eq. (39) determines $T_{c}$ in the region $|D_{y}|<2\sqrt{3}=3.46$ while $T_{c}$ in other region is determined by the first equation of Eq. (39). In Fig. 4(c) the $D_{y}$-dependence of $T_{c}$ is plotted for antiferromagnetic ($J=3,J_{z}=1$) and ferromagnetic ($J=-3,J_{z}=-1$) cases as black and red dashed lines, respectively. As expected, the critical temperature for ferromagnetic case behaves differently from that for antiferromagnetic case.

For $X$ $Y$ $Z$ model with DM interaction Eq. (39) is replaced with

[TABLE]

As in the case of $X$ $X$ $Z$ model the critical temperature $T_{c}$ for the antiferromagnetic case is determined by only first equation of Eq. (40). Thus, the behavior of ${\cal C}(\rho_{Y})$ is similar to Fig. 4(a). Of course, both equations are used to determine $T_{c}$ for ferromagnetic case. The boundary of two regions in Eq. (40) is $D_{y}=D_{y,*}$, where

[TABLE]

with $\tilde{J}_{>}=\max(J_{x},J_{z})$ and $\tilde{J}_{<}=\min(J_{x},J_{z})$. At this point $T_{c}$ becomes zero because $\eta_{1}=\eta_{3}$ as Eq. (38) shows.

In Fig. 5 we plot the $D_{y}$-dependence of $T_{c}$ for antiferromagnetic (Fig. 5(a)) and ferromagnetic (Fig. 5(b)) cases with varying $J_{y}$. In Fig. 5(a) we choose $J_{x}=1$, $J_{z}=2$, and $J_{y}=1.5$ (black line), $2.5$ (red dashed line), $3.5$ (blue dotted line). Like $\rho_{Z}(T)$ $T_{c}$ increases with increasing $|D_{y}|$. The critical temperature $T_{c}$ with fixed $D_{y}$ tends to increase with increasing $J_{y}$. In Fig. 5(b) we choose $J_{x}=-1$, $J_{z}=-2$ and $J_{y}=-1.5$ (black line), $-2.5$ (red dashed line), $-3.5$ (blue dotted line). At the region $|D_{y}|\geq D_{y,*}$ $T_{c}$ increases with increasing $|D_{y}|$. With fixed $D_{y}$, in this region, $T_{c}$ tends to decreases with increasing $|J_{y}|$. At the region $|D_{y}|<D_{y,*}$ $T_{c}$ decreases with increasing $|D_{y}|$. With fixed $D_{y}$, $T_{c}$ increases with increasing $|J_{y}|$.

For nether ferromagnetic and antiferromagnetic case the behavior of the concurrence is determined whether the second equation of Eq. (40) plays a role or not. Since it is similar to that of $D_{x}=D_{y}=0$ case, we will not repeat the analysis.

IV Thermal Discord

In this section we examine the temperature dependence of quantum discorddiscord1 ; discord2 for the thermal density matrices $\rho_{Z}(T)$ and $\rho_{Y}(T)$.

As well as quantum entanglement, quantum discord is another measure of quantum correlation for bipartite quantum system. It is defined through discrepancy between two different quantum analogues of classical mutual information. First analogue is

[TABLE]

where $S$ denotes a von Neumann entropy $S(\rho)=-\mbox{Tr}(\rho\log\rho)$. In our paper, all logarithms are taken to base $2$. This is a quantum analogue of classical mutual information $I_{cl}(A:B)=H(A)+H(B)-H(A,B)$, where $H$ denotes a Shannon entropy. Another expression of classical mutual information is $I_{cl}(A:B)=H(A)-H(A|B)=H(B)-H(B|A)$, where $H(X|Y)$ is a conditional entropy of $X$ given $Y$. The quantum analogue of this representationdiscord1 is

[TABLE]

where333Depending on the specific rules about the local operations and classical communication (LOCC) between parties A and B, one can define several different generalizations of $I_{cl}(A:B)$bro10 ; modi11 . Thus, several different quantum discords can be defined. Our definition (43) corresponds to the optimal efficiency of a one-way purification strategybro10 . $\left\{\Pi_{j}^{B}\right\}$ denotes a complete set of measurement operators prepared by party $B$ and $S(A|\Pi_{j}^{B})$ is a von Neumann entropy of party $A$ after party $B$ obtains a measurement outcome $j$. As obvious, $p_{j}$ is the probability of obtaining outcome $j$ in the quantum measurement. The general quantum mechanical postulatestext imply that

[TABLE]

where $\rho\left(A|\Pi_{j}^{B}\right)=\mbox{Tr}_{B}(\Pi_{j}^{B}\rho_{AB}\Pi_{j}^{B})/p_{j}$. Hence, unlike $I(A:B)$, $J(A:B)$ is dependent on the complete set of measurement operators. The quantum discord is defined as

[TABLE]

where the minimum is taken over all possible choice of the complete set of measurement operators444Although the authors in Ref. discord1 consider the projective measurement, the authors in Ref. discord2 consider the general measurement including positive operator-valued measure (POVM). Thus, the quantum discord in Ref. discord2 is the lower bound of that in discord1 ..

The quantum discord for $\rho_{Z}(T)$ can be easily computed because it belongs to $X$-states. The quantum discord for general $X$-state was computed in Ref.chen11 ; wang11 . For $\rho_{Z}(T)$ the last term of Eq. (45) becomes

[TABLE]

where

[TABLE]

with $p=\frac{1}{2}\left[1+2(|s|+|v|)\right]$. The parameters $r$, $s$, $u$, and $v$ are defined in Eq. (16). Thus, quantum discord for $\rho_{Z}(T)$ is given by

[TABLE]

In Fig. 6 we plot $T$- and $D_{z}$-dependence of the thermal discord ${\cal D}\left(\rho_{Z}\right)$ for (a) antiferromagnetic ($J_{x}=1,J_{y}=1.5,J_{z}=2$) case and (b) ferromagnetic ($J_{x}=-1,J_{y}=-1.5,J_{z}=-2$ case. Although both thermal discords exhibit rapidly damping behavior with increasing $T$, unlike concurrence they do not reach to exact zero. This means that thermal discord does not vanish for separable states. Furthermore, for the ferromagnetic case the thermal discord exhibits an extraordinary behavior at $D_{z}\approx D_{z,*}=\sqrt{7/2}$ like the concurrence. It seems to form a valley in this region. In order to examine this behavior more precisely we plot ${\cal D}\left(\rho_{Z}\right)$ with choosing $D_{z}=0.8$ (red dashed line), $1.8$ (black line), $2.8$ (blue dotted line) in Fig. 6(c) without changing $J_{\alpha}$. This figure shows that ${\cal D}\left(\rho_{Z}\right)$ makes a local minimum at small $T$ region when $D_{z}=1.8$ while other cases exhibit exponential damping behavior without local minimum. At $D_{z}=D_{z,*}$ and $T=0$ one can show $r=u=|v|=1/4$ and $s=\mp 1/4$, which result in ${\cal D}_{z,1}=0$ and ${\cal D}_{z,2}=1$. Thus, the thermal discord of $\rho_{Z}(T)$ is exactly zero when $D_{z}=D_{z,*}$ and $T=0$. Since $T_{c}=0$ at $D_{z}=D_{z,*}$, both thermal entanglement and thermal discord simultaneously vanish at this point.

Since $\rho_{Y}(T)$ does not belong to $X$-states, its thermal discord should be computed explicitly. This is carried out in appendix A, which gives

[TABLE]

where $h(p)=-p\log p-(1-p)\log(1-p)$ and

[TABLE]

In Fig. 7 we plot $T$- and $D_{y}$-dependence of the thermal discord ${\cal D}\left(\rho_{y}\right)$ for (a) antiferromagnetic ($J_{x}=1,J_{y}=1.5,J_{z}=2$) case and (b) ferromagnetic ($J_{x}=-1,J_{y}=-1.5,J_{z}=-2$ case. Like Fig. 6 both thermal discords do not reach to exact zero in spite of exponential damping with increasing $T$. For the ferromagnetic case the thermal discord exhibits an extraordinary behavior at $D_{y}\approx D_{y,*}=\sqrt{7}/2=1.32$ like Fig. 6(b).

Finally, we plot in Fig. 8 that (a) the $D_{z}$-dependence of ${\cal D}(\rho_{Z})$ and (b) $D_{y}$-dependence of ${\cal D}(\rho_{Y})$ at $T=T_{c}$. In both figures red solid and blue dashed lines correspond to ferromagnetic ($J_{x}=-1,J_{y}=-1.5,J_{z}=-2$) and antiferromagnetic ($J_{x}=1,J_{y}=1.5,J_{z}=2$) cases, respectively. Thus, these figures show the discrepancy between thermal discord and thermal entanglement because concurrence at $T=T_{c}$ is exactly zero. Within the range $0\leq D_{z},D_{y}\leq 4$, these figures show ${\cal D}(\rho_{Z})\approx{\cal D}(\rho_{Y})\approx 0.1$ for both antiferromagnetic cases. For ferromagnetic cases ${\cal D}(\rho_{Z})$ and ${\cal D}(\rho_{Y})$ fall to zero at $D_{z}=\sqrt{7/2}$ and $\sqrt{7}/2$ at $T=T_{c}$. This fact implies that for arbitrary ferromagnetic cases ($J_{\alpha}<0$) both thermal discord and thermal entanglement simultaneously vanish at $D_{z}=D_{z,*}$ for $\rho_{Z}(T_{c})$ and $D_{y}=D_{y,*}$ for $\rho_{Y}(T_{c})$. If one extends the range of $D_{z}$ and $D_{y}$ in Fig. 8, both antiferromagnetic and ferromagnetic ${\cal D}(\rho_{Z})$ and ${\cal D}(\rho_{Y})$ approach to one for $D_{z},D_{y}\rightarrow\pm\infty$. Thus, maximal discrepancy between thermal discord and thermal entanglement occurs in this limit at the critical temperature.

V Conclusions

We derive explicitly the thermal density matrices $\rho_{Z}(T)$ and $\rho_{Y}(T)$ for two-qubit Heisenberg $X$ $Y$ $Z$ chain with DM interaction in the $z$- or $y$-direction. For each density matrix the thermal entanglement expressed by ${\cal C}(\rho_{Z})$ or ${\cal C}(\rho_{Y})$ is explicitly computed. Exploiting the explicit expressions of ${\cal C}(\rho_{Z})$ or ${\cal C}(\rho_{Y})$ we discuss on the quantum phase transition in detail. In particular, we focus on the critical temperature $T_{c}$, above which the thermal entanglement completely vanishes. This means that each density matrix becomes separable state at $T\geq T_{c}$555Of course, this critical behavior of entanglement occurs when the DM interaction has arbitrary direction. In order to show this fact more explicitly we discuss the critical behavior in appendix B when $D_{x}\neq 0$ and $D_{y}\neq 0$..

For antiferromagnetic case ($J_{\alpha}>0\hskip 5.69046pt\alpha=x,y,z$) the critical temperature is determined by single equation for each thermal density matrix. As a result, $T_{c}$ monotonically increases with increasing $|D_{z}|$ or $|D_{y}|$. For ferromagnetic case ($J_{\alpha}<0\hskip 5.69046pt\alpha=x,y,z$), however, the situation is different. In this case the critical temperature is determined by two different equations in the two separate regions. For $\rho_{Z}(T)$, for example, these two separate regions are defined by $|D_{z}|<D_{z,*}$ and $|D_{z}|\geq D_{z,*}$. Similarly, the two separate regions for $\rho_{Y}(T)$ are given by $|D_{y}|<D_{y,*}$ and $|D_{y}|\geq D_{y,*}$. At the outer region $|D_{z}|\geq D_{z,*}$ or $|D_{y}|\geq D_{y,*}$ the critical temperature $T_{c}$ monotonically increases with increasing $|D_{z}|$ or $|D_{y}|$ like antiferromagnetic case. At the inner region $|D_{z}|<D_{z,*}$ or $|D_{y}|<D_{y,*}$, however, $T_{c}$ decreases with increasing $|D_{z}|$ or $|D_{y}|$. As a result, the $D_{\alpha}$- $\hskip 2.84544pt(\alpha=z,y)$ and $T$-dependence of thermal entanglements for ferromagnetic case exhibit different behavior from those for antiferromagnetic case.

Thermal discords ${\cal D}(\rho_{Z})$ and ${\cal D}(\rho_{Y})$ for $\rho_{Z}(T)$ and $\rho_{Y}(T)$ are explicitly derived. For antiferromagnetic case with fixed $D_{z}$ or $D_{y}$ both ${\cal D}(\rho_{Z})$ and ${\cal D}(\rho_{Y})$ exhibit exponential damping behavior with increasing $T$, but they do not reach to exact zero. For ferromagnetic case both ${\cal D}(\rho_{Z})$ and ${\cal D}(\rho_{Y})$ exhibit extraordinary behavior at $|D_{z}|\approx D_{z,*}$ and $|D_{y}|\approx D_{y,*}$. In these regions the $T$-dependence of ${\cal D}(\rho_{Z})$ or ${\cal D}(\rho_{Y})$ involves local minimum at small $T$ while in other region they exhibit exponential damping behavior without local minimum. Although definitions of quantum entanglement and quantum discord are completely different, one can infer from our results that they exhibit similar behavior with each other because for ferromagnetic case both exhibit extraordinary behavior at the regions $|D_{z}|\approx D_{z,*}$ and $|D_{y}|\approx D_{y,*}$. At $D_{z}=D_{z,*}$ or $D_{y}=D_{y,*}$ both thermal entanglement and thermal discord for $\rho_{Z}(T)$ or $\rho_{Y}(T)$ simultaneously vanish at $T=T_{c}=0$.

One can apply our analysis when external magnetic field ${\bm{B}}$ is applied to our systemli08 . In this case the Hamiltonian is changed into

[TABLE]

where $H$ is given in Eq. (2). It seems to be of interest to examine the effect of external magnetic field and DM coupling constants in the thermal entanglement and thermal discord. In particular, it is of interest to analyze the behavior of thermal discord near the critical temperature in this model.

Another interesting issue is to examine the quantum phase transition by introducing $3$-spin Heisenberg model with DM interaction, whose Hamiltonian is

[TABLE]

The tripartite entanglement was introduced in Ref.ckw ; ou07-1 . It is of interest to compute the thermal tripartite entanglement and to discuss on the quantum phase transition. It seems to be of interest to examine how the monogamy relation is changed with varying DM coupling constants and external temperature.

One can extend our analysis to continuum system. For example, let us consider two coupled harmonic oscillator system, whose Hamiltonian is

[TABLE]

In this case the thermal density matrix $\rho(x_{1}^{\prime},x_{2}^{\prime}:x_{1},x_{2}:T)$ can be derived exactly by making use of diagonalization of Hamiltonian and Euclidean path-integral techniquefeynman . This thermal density matrix is generally mixed state. For mixed state entanglement is in general defined via a convex-roof methodbenn96 ; uhlmann99-1 ;

[TABLE]

where minimum is taken over all possible pure state decompositions, i.e. $\rho=\sum_{j}p_{j}\lvert\psi_{j}\rangle\langle\psi_{j}\lvert$, with $0\leq p_{j}\leq 1$. However, we do not know how to derive the optimal decomposition for continuum thermal density matrix $\rho(x_{1}^{\prime},x_{2}^{\prime}:x_{1},x_{2}:T)$. We would like to explore this issue in the future.

Acknowledgement: This work was supported by the Kyungnam University Foundation Grant, 2017.