# Thermal coherence of the Heisenberg model with Dzyaloshinsky-Moriya
interactions in an inhomogenous external field

Manikandan Parthasarathy  Department of Physics, Ramakrishna Mission
Vivekananda College, Mylapore, Chennai 600 004, India.  Segar Jambulingam
Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore,
Chennai 600 004, India.  Tim Byrnes  New York University Shanghai, 1555
Century Avenue, Pudong, Shanghai 200122, China.  State Key Laboratory of
Precision Spectroscopy, School of Physical and Material Sciences, East China
Normal University, Shanghai 200062, China  NYU-ECNU Institute of Physics at
NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China.  National
Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430,
Japan  Department of Physics, New York University, New York, NY 100003, USA
Chandrashekar Radhakrishnan<EMAIL_ADDRESS>Laboratoire Systèmes
Complexes et Information Quantify, ESSIE Group, 9 Rue Vèsale, Paris 75005,
France  New York University Shanghai, 1555 Century Avenue, Pudong, Shanghai
200122, China.  NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan
Road North, Shanghai, 200062, China.

###### Abstract

The quantum coherence of the two-site XYZ model with Dzyaloshinsky-Moriya (DM)
interactions in an external inhomogenous magnetic field is studied. The DM
interaction, the magnetic field and the measurement basis can be along
different directions, and we examine the quantum coherence at finite
temperature. With respect to the spin-spin interaction parameter, we find that
the quantum coherence decreases when the direction of measurement basis is the
same as that of the spin-spin interaction. When the spin-lattice interaction
is varied, the coherence always increases irrespective of the relation between
its direction and the measurement basis. Similar analysis of quantum coherence
based on the variation of the external inhomogenous magnetic field is also
carried out, where we find that the coherence decreases when the direction of
the measurement basis is the same as that of the external field.

CAPS Number(s): 05.20.G, 05.70.Ce, 02.30.GP
Keywords: anisotropic spin model, DM interaction, quantum coherence.

##  I Introduction

Quantum coherence has been a central concept in the theory of quantum
mechanics since its earliest days. In the context of quantum optics, coherence
was investigated using phase space distributions and higher order correlation
functions Glauber (1963); Sudarshan (1963). It was only recently that a
rigorous quantum information theoretic framework for measuring coherence was
developed, in the seminal paper by Baumgratz, Cramer and Plenio Baumgratz et
al. (2014). The work introduced a set of axioms which are necessary for a
mathematical function to be a proper quantifier of quantum coherence. In Ref.
Baumgratz et al. (2014), two measures of quantum coherence were introduced
based on these axioms, namely the relative entropy measure of coherence and
the $\ell_{1}$-norm based quantum coherence. The fundamental properties of
quantum coherence have been investigated Yao et al. (2015); Yadin et al.
(2015); Ma et al. (2016); Yadin and Vedral (2016); Bromley et al. (2015); Du
et al. (2015); Pires et al. (2015); Cheng and Hall (2015); Streltsov et al.
(2015) and also their applications in different quantum phenomena Shi et al.
(2017); Hillery (2016); Zhang et al. (2019); Ma et al. (2019); Narasimhachar
and Gour (2015) have been studied. These investigations also gave rise to the
resource theory framework of quantum coherence Winter and Yang (2016); Brandão
and Gour (2015); Chitambar and Hsieh (2015); Chitambar and Gour (2016); del
Rio et al. (2015); Streltsov et al. (2015); Chitambar and Gour (2019);
Radhakrishnan et al. (2016).

Quantum coherence has been investigated with regards to search algorithms Shi
et al. (2017); Hillery (2016), quantum thermodynamics Narasimhachar and Gour
(2015); Lostaglio et al. (2015) and quantum metrology Ma et al. (2019).
Previous studies have established applications of quantum coherence in quantum
information processing tasks like quantum dense coding and teleportation Pan
et al. (2017). Another important application of quantum coherence is in the
characterization of quantum states. Multipartite systems have a large number
of degrees of freedom, and it is of interest to understand what type of
coherences a given state possesses. Such a scheme was introduced in Ref.
Radhakrishnan et al. (2016), where coherence could be subdivided into its
constituent parts. Quantum simulation Georgescu et al. (2014) is a prime
example of where this can be applied, since the naturally arising states of a
condensed matter system can show strikingly different properties depending
upon the physical parameters driving phase transitions in the system. In the
field of condensed matter physics, the measurement and properties of quantum
coherence has been studied in various models Opanchuk et al. (2016); Zheng et
al. (2016); Ishida et al. (2013); Karpat et al. (2014). Heisenberg spin models
Lieb et al. (1961); Niemeijer (1967, 1968); Barouch et al. (1970); Barouch and
McCoy (1971); Giamarchi (2004) are one of the important class of models in
which coherence has been investigated Malvezzi et al. (2016); Li and Lin
(2016); Cheng et al. (2016); Çakmak et al. (2015), where one of the main
themes has been the investigation of quantum phase transitionLi and Lin
(2016); Cheng et al. (2016); Çakmak et al. (2015); Radhakrishnan et al.
(2017a). In the Heisenberg XYZ model, the spins have a direct interaction
between nearby sites. In addition, they can also have a spin-lattice
interaction generally referred to as the Dzyaloshinsky-Moriya (DM) interaction
Dzyaloshinsky (1958); Moriya (1960). Spins are usually aligned parallel or
anti-parallel to each other depending on whether it is in the ferromagnetic
phase or antiferromagnetic phase. Due to the spin-lattice interaction the
spins are tilted at an angle to the parallel, a feature known as spin-canting.
This spin-canting usually leads to weak ferromagnetic behavior even in an
antiferromagnetic system. In addition to these inherent properties, it is also
possible to apply an external magnetic field to the spin system to control and
modify its quantum properties.

A minimal example of the DM model is the two-site model which has been studied
to understand the overall dependence of quantities such as entanglement Wang
(2001); Zhou et al. (2003); Li and Cao (2008); Kheirandish et al. (2008); Li
and Cao (2009); Gurkan and Pashaev (2010), quantum discord Guo et al. (2011);
Zidan (2014); Yi-Xin and Zhi (2010), and coherence Radhakrishnan et al.
(2017b) as a function of the various parameters of the system. In Ref.
Radhakrishnan et al. (2017b), a detailed investigation of quantum coherence in
two-spin models were carried out. In this paper, we extend the investigations
carried out in Ref. Radhakrishnan et al. (2017b) and investigate two-spin
models with DM interactions in the presence of an external field. A detailed
analysis of the dependence of the quantum coherence on various parameters such
as the spin-spin interaction, the spin-orbit interaction, DM interaction, and
external magnetic field is carried out. As quantum coherence is a basis
dependent quantity, in our work we choose different orientations of the DM
interaction parameter and the external fields keeping the measurement
direction in the $\sigma_{z}$ basis for all cases. This will help us to
understand the basis-dependent nature of coherence with respect to the
internal parameters such as spin-spin interaction, as well as the DM
interaction parameter. Finally, the relation between quantum coherence and the
external driving forces such as the homogeneous component and the inhomogenous
component of the magnetic field will be explored.

The article is structured as follows: In Sec. III we discuss the quantum
coherence dependence in the two-spin XYZ model with DM interactions and
external field in the same direction. The features of quantum coherence where
the two-site XYZ model with the DM interaction and external field in different
directions is discussed in Sec. IV. Sec. V gives the simultaneous variation of
the quantum coherence with spin-spin interaction and any one of the other
parameters such as DM interaction, homogeneous component, or the inhomogenous
component of the external field. Finally, we present our conclusions in Sec.
VI.

##  II The Heisenberg model with DM interaction and the coherence measure

The Heisenberg spin model consists of a collection of interacting spins
arranged periodically on a lattice. The Hamiltonian of the Heisenberg spin
chain is written as

$H=\sum_{n}J_{x}\sigma_{n}^{x}\sigma_{n+1}^{x}+J_{y}\sigma_{n}^{y}\sigma_{n+1}^{y}+J_{z}\sigma_{n}^{z}\sigma_{n+1}^{z},$
(1)

where $\sigma_{n}^{x,y,z}$ are the Pauli spin matrices at the site $n$ and
$J_{x,y,z}$ are the spin-spin interaction terms in the $x$,$y$ and $z$
directions. For $J_{x}\neq J_{y}\neq J_{z}$, the spin model is referred to as
XYZ model. The model is antiferromagnetic when $J_{i}>0$ $(i=x,y,x)$ and
ferromagnetic for $J_{i}<0$. Along with the spin-spin interaction we may also
have a spin-lattice interaction which are known as the Dzyaloshinsky-Moriya
(DM) interaction. The Hamiltonian of the XYZ spin chain with the DM
interaction is

$H=\sum_{n}J_{x}\sigma_{n}^{x}\sigma_{n+1}^{x}+J_{y}\sigma_{n}^{y}\sigma_{n+1}^{y}+J_{z}\sigma_{n}^{z}\sigma_{n+1}^{z}+\vec{D}.(\vec{\sigma}_{n}\times\vec{\sigma}_{n+1})$
(2)

The antisymmetric spin-lattice interaction is accounted for by the DM
interaction coefficient $\vec{D}$.

In our work we further consider an inhomogenous external magnetic field. The
Hamiltonian of a two-spin system with DM interaction in an external
inhomogenous field is

$H=J_{x}\sigma_{1}^{x}\sigma_{2}^{x}+J_{y}\sigma_{1}^{y}\sigma_{2}^{y}+J_{z}\sigma_{1}^{z}\sigma_{2}^{z}+\vec{D}\cdot(\vec{\sigma}_{1}\times\vec{\sigma}_{2})+(\vec{B}+\vec{b})\cdot\vec{\sigma}_{1}^{j}+(\vec{B}-\vec{b})\cdot\vec{\sigma}_{2}^{j}.$
(3)

This field can be written as two fields of the form $\vec{B}+\vec{b}$ acting
on the first qubit and the field $\vec{B}-\vec{b}$ acting on the second qubit,
where $B$ is the homogeneous and $b$ is the inhomogenous component of the
external magnetic field.

From the Hamiltonian of this model, the quantum properties can be studied by
examining its eigenstates. In our work, we investigate the quantum coherence
of the system at finite (non-zero) temperature. Quantum coherence is measured
as the distance between the density matrix $\rho$ and the decohered density
matrix $\rho_{d}$. The decohered density matrix is defined according to the
matrix elements

$\langle\sigma_{1}\sigma_{2}|\rho_{d}|\sigma_{1}^{\prime}\sigma_{2}^{\prime}\rangle=\langle\sigma_{1}\sigma_{2}|\rho|\sigma_{1}^{\prime}\sigma_{2}^{\prime}\rangle\delta_{\sigma_{1},\sigma_{1}^{\prime}}\delta_{\sigma_{2},\sigma_{2}^{\prime}}.$
(4)

In Ref. (Baumgratz et al., 2014), two different measures of coherence namely
the relative entropy measure belonging to the entropic class and the
$\ell_{1}$-norm of coherence which is a geometric measure were introduced.
Another measure based on the quantum version of the Jensen-Shannon divergence
Lin (1991); Lamberti et al. (2008); Briët and Harremoës (2009) was introduced
in Ref. Radhakrishnan et al. (2016)

$\mathcal{D}(\rho,\sigma)\equiv\sqrt{J(\rho,\sigma)}=\sqrt{S\left(\frac{\rho+\sigma}{2}\right)-\frac{S(\rho)}{2}-\frac{S(\sigma)}{2}}.$
(5)

Here $\rho,\sigma$ are arbitrary density matrices and $S=-{\rm
Tr}\rho\log\rho$ is the von Neumann entropy. This measure has the advantage
that it has both geometric and entropic properties and is also a metric
satisfying triangle inequality. The coherence is then defined in our case as

${\cal C}(\rho)=\mathcal{D}(\rho,\rho_{d}).$ (6)

The coherence of the two-spin XYZ model in Eq. (3) at finite temperature is
discussed in the rest of the paper. Since quantum coherence is a basis-
dependent quantity we examine the spin model in two different cases namely:
($i$) when the control parameters and measurement basis are in the same
direction; ($ii$) when the control parameters like DM interaction and the
external field are orthogonal to the measurement basis.

##  III DM interaction and the external magnetic field along the same
direction

###  A $\vec{D}\propto\vec{x},\vec{B}\propto\vec{x}$ case

For the case where both the DM interaction and the external magnetic field are
in the $x$ direction, the Hamiltonian reads

$H=J_{x}\sigma_{1}^{x}\sigma_{2}^{x}+J_{y}\sigma_{1}^{y}\sigma_{2}^{y}+J_{z}\sigma_{1}^{z}\sigma_{2}^{z}+D_{x}(\sigma_{1}^{y}\sigma_{2}^{z}-\sigma_{1}^{z}\sigma_{2}^{y})+(B_{x}+b_{x})\sigma_{1}^{x}+(B_{x}-b_{x})\sigma_{2}^{x},$
(7)

where $D_{x}$ is the DM interaction in the $x$-direction, and $B_{x}$ is the
average magnetic field and $b_{x}$ is the inhomogneous magnetic field in the
$x$-direction. The matrix representation of the Hamiltonian in the
$\sigma_{z}$-basis is

$H=\left(\begin{array}[]{cccc}J_{z}&G_{2}&G_{3}&J_{-}\\\
G_{4}&-J_{z}&J_{+}&G_{1}\\\ G_{1}&J_{+}&-J_{z}&G_{4}\\\
J_{-}&G_{3}&G_{2}&J_{z}\\\ \end{array}\right),$ (8)

where $G_{1,2}=iD_{x}+B_{x}\pm b_{x}$, $G_{3,4}=G_{1,2}^{*}$ and
$J_{\pm}=J_{x}\pm J_{y}$. The eigenvalues of the Hamiltonian are

$E_{1,2}=J_{x}\pm\omega_{1},\qquad E_{3,4}=-J_{x}\pm\omega_{2},$ (9)

and their corresponding eigenvectors are

$\displaystyle|\psi_{1,2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\sin\varphi_{1,2}|00\rangle+\cos\varphi_{1,2}|01\rangle+\cos\varphi_{1,2}|10\rangle+\sin\varphi_{1,2}|11\rangle)$
(10) $\displaystyle|\psi_{3,4}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\sin\varphi_{3,4}|00\rangle+\chi\cos\varphi_{3,4}|01\rangle-\chi\cos\varphi_{3,4}|10\rangle-\sin\varphi_{3,4}|11\rangle)$
(11)

where the various factors used are

$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{-iD_{x}-b_{x}}{\sqrt{b_{x}^{2}+D_{x}^{2}}},\quad\omega_{1}=\sqrt{4B_{x}^{2}+(J_{y}-J_{z})^{2}},\quad\omega_{2}=\sqrt{4b_{x}^{2}+4D_{x}^{2}+(J_{y}+J_{z})^{2}}$
$\displaystyle\varphi_{1,2}$ $\displaystyle=$
$\displaystyle\arctan\left(\frac{2B_{x}}{J_{y}-J_{z}\pm\omega_{1}}\right),\quad\varphi_{3,4}=\arctan\left(\frac{2\sqrt{b_{x}^{2}+D_{x}^{2}}}{-J_{y}-J_{z}\pm\omega_{2}}\right).$
(12)

The state of the two-spin spin system at thermal equilibrium is given by the
thermal density matrix $\rho(T)=\exp(-\beta H)/Z$ where
$Z={\hbox{Tr}}[\exp(-\beta H)]$ is the partition function of the system and
$\beta=1/k_{B}T$, and $k_{B}$ and $T$ are the Boltzmann constant and
temperature respectively. For the sake of convenience we assume $k_{B}=1$
throughout our discussion. The matrix form of the thermal density matrix in
the $\sigma_{z}$-basis is

$\rho(T)=\left(\begin{array}[]{cccc}u_{1}&q_{1}^{*}&q_{2}^{*}&u_{2}\\\
q_{1}&v_{1}&v_{2}&q_{2}\\\ q_{2}&v_{2}&v_{1}&q_{1}\\\
u_{2}&q_{2}^{*}&q_{1}^{*}&u_{1}\\\ \end{array}\right).$ (13)

The elements of the thermal density matrix (13) are

$\displaystyle u_{1,2}$ $\displaystyle=$
$\displaystyle\frac{1}{2Z}\left[e^{-\frac{J_{x}+\omega_{1}}{T}}\sin^{2}\varphi_{1}+e^{-\frac{J_{x}-\omega_{1}}{T}}\sin^{2}\varphi_{2}\pm
e^{\frac{J_{x}-\omega_{2}}{T}}\sin^{2}\varphi_{3}\pm
e^{\frac{J_{x}+\omega_{2}}{T}}\sin^{2}\varphi_{4}\right],$ (14) $\displaystyle
v_{1,2}$ $\displaystyle=$
$\displaystyle\frac{1}{2Z}\left[e^{-\frac{J_{x}+\omega_{1}}{T}}\cos^{2}\varphi_{1}+e^{-\frac{J_{x}-\omega_{1}}{T}}\cos^{2}\varphi_{2}\pm
e^{\frac{J_{x}-\omega_{2}}{T}}\cos^{2}\varphi_{3}\pm
e^{\frac{J_{x}+\omega_{2}}{T}}\cos^{2}\varphi_{4}\right],$ (15) $\displaystyle
q_{1,2}$ $\displaystyle=$
$\displaystyle\frac{1}{2Z}\Big{[}e^{-\frac{J_{x}+\omega_{1}}{T}}\sin\varphi_{1}\cos\varphi_{1}+e^{-\frac{J_{x}-\omega_{1}}{T}}\sin\varphi_{2}\cos\varphi_{2}\pm
e^{\frac{J_{x}-\omega_{2}}{T}}\chi\sin\varphi_{3}\cos\varphi_{3}$ (16)
$\displaystyle\pm
e^{\frac{J_{x}+\omega_{2}}{T}}\chi\sin\varphi_{4}\cos\varphi_{4}\Big{]}.$

where the partition function is

$Z=2\left[e^{-\frac{J_{x}}{T}}\cosh\left(\frac{\omega_{1}}{T}\right)+e^{\frac{J_{x}}{T}}\cosh\left(\frac{\omega_{2}}{T}\right)\right].$
(17)

From the density matrix $\rho(T)$, we can write down the decohered density
matrix $\rho_{d}$ using which we can measure coherence introduced through Eq.
(6).

The behavior of quantum coherence is shown in Fig. 1. Since quantum coherence
is a basis dependent quantity, the choice of measurement basis plays an
important role. In the Hamiltonian of Eq. (7), the DM interaction and the
field are in the $x$-direction, and the measurement is carried out in the
$z$-direction. This behavior of coherence is very similar to the entanglement
results reported in Ref. Li and Cao (2008). In Fig. 1(a) the variation of
coherence with temperature is shown and we find that coherence decreases with
temperature. This can be explained by the thermal fluctuations inducing
decoherence in the quantum system, thereby decreasing coherence. The rate of
decrease of coherence is lower for higher values of the DM interaction
parameter. This is because the DM interaction creates an extra spin-lattice
coupling apart from the existing spin-spin coupling. This additional spin
lattice coupling gives rise to additional coherence in the quantum system. The
variation of quantum coherence with the DM interaction parameter is shown in
Fig. 1(b) for different values of temperature. We find that for lower values
of temperature ($T=0.5,T=1.0$), the coherence decreases initially, reaches a
minimum value and then increases to reach a saturation value. Initially at
$D_{x}=0$, the spin-spin interaction is dominant and when it is increased the
spin-lattice interaction is slowly introduced. The initial decrease in the
spin-lattice interaction might be due to the competing effects of the spin-
spin and spin-lattice interaction. The increase later on can be attributed to
the co-operation between the spin-spin and spin-lattice interaction. At higher
values of temperature ($T=1.5$), the coherence increases gradually to a
saturation value.

The variation of the quantum coherence is investigated as a function of the
external magnetic field in Fig. 1(c) and 1(d). We now examine the role played
by the external inhomogenous field. In Fig. 1(c) we show the variation of
quantum coherence with the field $B_{x}$ and we find that for the lower values
of temperature, the coherence initially decreases and reaches a minimum value
and then increases to attain a saturation value. For the higher values of
temperature, the coherence increases with the homogeneous field $B_{x}$. The
overall behavior of the quantum coherence with the inhomogenous field $b_{x}$
resembles the variation of coherence with the field $B_{x}$ as we can see from
a comparison between Figs. 1(c) and 1(d). On comparison with the earlier works
Ref. Li and Cao (2008) we find that quantum coherence exhibits behavior very
similar to entanglement. From concurrence measurements one can see that
entanglement also increases with the DM interaction. The main reason is that
the entire coherence comes from the two-spin correlation in a manner similar
to entanglement.

###  B  $\vec{D}\propto\vec{z},\vec{B}\propto\vec{z}$ case

For the case where the DM interaction and the external magnetic field along
the $z$-direction, the Hamiltonian reads

$H=J_{x}\sigma_{1}^{x}\sigma_{2}^{x}+J_{y}\sigma_{1}^{y}\sigma_{2}^{y}+J_{z}\sigma_{1}^{z}\sigma_{2}^{z}+D_{z}(\sigma_{1}^{x}\sigma_{2}^{y}-\sigma_{1}^{y}\sigma_{2}^{x})+(B_{z}+b_{z})\sigma_{1}^{z}+(B_{z}-b_{z})\sigma_{2}^{z}.$
(18)

Here, $B_{z}$ is the average external magnetic field in the $z$ direction and
$b_{z}$ is the degree of inhomogenity of the field in the $z$-direction. In
the standard measurement basis ($\sigma_{z}$ basis) the Hamiltonian can be
written in matrix form as

$H=\left(\begin{array}[]{cccc}2B_{z}+J_{z}&0&0&J_{x}-J_{y}\\\
0&2b_{z}-J_{z}&2iD_{z}+J_{x}+J_{y}&0\\\
0&-2iD_{z}+J_{x}+J_{y}&-2b_{z}-J_{z}&0\\\ J_{x}-J_{y}&0&0&-2B_{z}+J_{z}\\\
\end{array}\right).$ (19)

The eigenvalues and eigenvectors corresponding to the Hamiltonian are

$\displaystyle E_{1,2}$ $\displaystyle=$ $\displaystyle
J_{z}+\omega_{1}\qquad|\psi_{1,2}\rangle=\sin\theta_{1,2}|00\rangle+\cos\theta_{1,2}|11\rangle,$
(20) $\displaystyle E_{3,4}$ $\displaystyle=$ $\displaystyle-
J_{z}+\omega_{2}\qquad|\psi_{3,4}\rangle=\sin\theta_{3,4}|01\rangle+\chi\cos\theta_{3,4}|10\rangle,$
(21)

where the factors $\omega_{1}$ and $\omega_{2}$ are

$\omega_{1}=\sqrt{4B_{z}^{2}+(J_{x}-J_{y})^{2}},\qquad\omega_{2}=\sqrt{4b_{z}^{2}+4D_{z}^{2}+(J_{x}+J_{y})^{2}}.$
(22)

In the expression for the eigenvectors the parameters used are

$\theta_{1,2}=\arctan\left(\frac{J_{x}-J_{y}}{\pm\omega_{1}-2B_{z}}\right),\qquad\theta_{3,4}=\arctan\left(\frac{\sqrt{(J_{x}+J_{y})^{2}+D_{z}^{2}}}{\pm\omega_{2}-2b_{z}}\right)$
(23)

and $\chi=\frac{J_{x}+J_{y}-2iD_{z}}{\sqrt{(J_{x}+J_{y})^{2}+4D_{z}^{2}}}$.
The thermal density matrix $\rho(T)$ in the $\sigma_{z}$-basis is

$\rho(T)=\frac{1}{2Z}\left(\begin{array}[]{cccc}m_{1}&0&0&m_{3}\\\
0&n_{1}&n_{3}&0\\\ 0&n_{4}&n_{2}&0\\\ m_{3}&0&0&m_{2}\\\ \end{array}\right).$
(24)

The matrix elements of the thermal density matrix (24) are

$\displaystyle m_{1,2}$ $\displaystyle=$ $\displaystyle\mp
e^{\frac{-J_{z}}{T}}\left[\cosh\left(\frac{\mu}{T}\right)\mp\frac{B_{z}}{\mu}\sinh\left(\frac{\mu}{T}\right)\right]$
(25) $\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle-e^{\frac{-J_{z}}{T}}\left[\frac{J_{-}}{\mu}\sinh\left(\frac{\mu}{T}\right)\right]$
(26) $\displaystyle n_{1,2}$ $\displaystyle=$ $\displaystyle
e^{\frac{J_{z}}{T}}\left[\cosh\left(\frac{\nu}{T}\right)\mp\frac{b_{z}}{\nu}\sinh\left(\frac{\nu}{T}\right)\right]$
(27) $\displaystyle n_{3,4}$ $\displaystyle=$
$\displaystyle-e^{\frac{-J_{z}}{T}}\left[\frac{J_{+}\pm
iD_{z}}{\nu}\sinh\left(\frac{\nu}{T}\right)\right]$ (28)

where the partition function

$Z=2\left[e^{-\frac{J_{z}}{T}}\cosh\left(\frac{\mu}{T}\right)+e^{\frac{J_{z}}{T}}\cosh\left(\frac{\nu}{T}\right)\right].$
(29)

The variation of quantum coherence as a function of the DM interaction and
external field in the $\sigma_{z}$ basis is described through the plots in
Fig. 2. The change of coherence with temperature is shown in Fig. 2(a) for
different values of the DM interaction parameter. For larger values of the DM
interaction parameter ($D_{x}=3.0,5.0$), we find that the quantum coherence
decreases with temperature. This is in line with the observation in Ref. Li
and Cao (2008) and the well known effect of thermal decoherene on quantum
systems. For $D_{z}=0$, the coherence decreases but there is an initial
increase. Through Fig. 2(b) we show the evolution of coherence with DM
interaction. Quantum coherence increases with $D_{z}$ and this occurs for all
values of temperature. The influence of magnetic field on quantum coherence is
given in Figs. 2(c) and 2(d) for the homogeneous part and the inhomogenous
part of the field respectively. In Fig. 2(c) we see that the quantum coherence
decreases with the average homogeneous magnetic field $B_{z}$. From Fig. 2(d)
we see that the coherence initially increases with the homogeneous field
$b_{z}$, attains a maximum and then decreases to a saturation value.

In summary, in this section we considered the situation where the DM
interaction and the external magnetic field are along the same direction. In
both the cases the quantum coherence shows a decrease with temperature. When
we look into the variation of quantum coherence with the DM interaction
parameter, the average homogeneous magnetic field $B$, and the inhomogenous
magnetic field $b$ we find that the measurement basis has a definite outcome
on the value of quantum coherence. Hence the qualitative behavior of quantum
coherence is quite different for the situations described in Sec. III A and
Sec. III B and this is a unique feature of quantum coherence which does not
have an analog in entanglement measurements.

##  IV DM interaction and the external field in different directions

In this section we consider the two-site XYZ model in which the spin-lattice
interaction and the external field are in different directions. Under these
conditions we have three different cases as follows: ($i$) The DM interaction
is along the measurement basis; ($ii$) the external inhomogenous field is
along the measurement basis and; ($iii$) both DM interaction and the external
field are in different directions and orthogonal to the measurement basis. The
coherence of all these cases are examined in the discussion below.

###  A $\vec{D}\propto\vec{z},\vec{B}\propto\vec{x}$ case

For the case with the DM interaction along the $z$-axis and the inhomogenous
magnetic field along the $x$-axis, we have the Hamiltonian

$H=J_{x}\sigma_{1}^{x}\sigma_{2}^{x}+J_{y}\sigma_{1}^{y}\sigma_{2}^{y}+J_{z}\sigma_{1}^{z}\sigma_{2}^{z}+D_{z}(\sigma_{1}^{x}\sigma_{2}^{y}-\sigma_{1}^{y}\sigma_{2}^{x})+(B_{x}+b_{x})\sigma_{1}^{x}+(B_{x}-b_{x})\sigma_{2}^{x}.$
(30)

In the $\sigma^{z}$-basis the matrix form of the Hamiltonian is

$H=\left(\begin{array}[]{cccc}J_{z}&-b_{x}+B_{x}&b_{x}+B_{x}&J_{x}-J_{y}\\\
-b_{x}+B_{x}&-J_{z}&2iD_{z}+J_{x}+J_{y}&b_{x}+B_{x}\\\
b_{x}+B_{x}&-2iD_{z}+J_{x}+J_{y}&-J_{z}&-b_{x}+B_{x}\\\
J_{x}-J_{y}&b_{x}+B_{x}&-b_{x}+B_{x}&J_{z}\\\ \end{array}\right).$ (31)

From the Hamiltonian, one can calculate the density matrix $\rho(T)$ at
thermal equilibrium. Using the thermal density matrix we can write down the
diagonal form of the density matrix $\rho_{d}$ and using Eq. (6) we can
calculate coherence in the system.

For this model, the variation of quantum coherence with the different
parameters is shown in Fig. 3 with the measurement being carried out in the
$\sigma_{z}$ basis. Thermal decoherence causes a loss of coherence due to
increase in temperature and this is being observed in Fig. 3(a) for different
strengths of the DM interaction. With respect to the DM interaction parameter,
the coherence increases initially to attain a maximum value and then
decreases. This behavior is shown in Fig. 3(b) for different temperatures.
When the average homogeneous magnetic field $B_{z}$ is varied, the coherence
decreases slightly to reach the minimum value and then increases back to a
saturation value as shown in Fig. 3(c). A similar behavior is observed for the
variation of the inhomogenous component of the field $b_{z}$ as seen in Fig.
3(d).

###  B $\vec{D}\propto\vec{x},\vec{B}\propto\vec{z}$ case

When the magnetic field is oriented along the $z$-axis and the DM interaction
is along the $x$-axis, the Hamiltonian of the system reads:

$H=J_{x}\sigma_{1}^{x}\sigma_{2}^{x}+J_{y}\sigma_{1}^{y}\sigma_{2}^{y}+J_{z}\sigma_{1}^{z}\sigma_{2}^{z}+D_{x}(\sigma_{1}^{y}\sigma_{2}^{z}-\sigma_{1}^{z}\sigma_{2}^{y})+(B_{z}+b_{z})\sigma_{1}^{z}+(B_{z}-b_{z})\sigma_{2}^{z}.$
(32)

The matrix form in the $\sigma^{z}$ basis is

$H=\left(\begin{array}[]{cccc}2B_{z}+J_{z}&iD_{x}&-iD_{x}&J_{x}-J_{y}\\\
-iD_{x}&2b_{z}-J_{z}&J_{x}+J_{y}&iD_{x}\\\
iD_{x}&J_{x}+J_{y}&-2b_{z}-J_{z}&-iD_{x}\\\
J_{x}-J_{y}&-iD_{x}&iD_{x}&-2B_{z}+J_{z}\\\ \end{array}\right).$ (33)

The thermal density matrix $\rho(T)$ can be calculated from the Hamiltonian
(33) using which we can find the quantum coherence of the system from Eq. (6).

The dependence of quantum coherence with temperature is shown in Fig. 4(a). As
before, we find that the coherence decreases with temperature as expected due
to thermal effects removing coherence. With increase in the value of the DM
interaction parameter, we find that quantum coherence increases monotonically
as seen in Fig 4(b). From Fig 4(c) we can see that coherence decreases with
the average homogeneous field $B_{z}$. While this is uniform for higher
temperatures, for lower values of temperature the coherence increases to a
peak value and then decreases. The same behavior is also replicated when we
look into the variation of coherence with the inhomogenous field $b_{z}$ as
seen in Fig. 4(d).

###  C $\vec{D}\propto\vec{x},\vec{B}\propto\vec{y}$ case

An interesting situation arises when the DM interaction and the external field
are perpendicular to each other and neither of them are oriented along the
measurement basis. The Hamiltonian in this case is

$H=J_{x}\sigma_{1}^{x}\sigma_{2}^{x}+J_{y}\sigma_{1}^{y}\sigma_{2}^{y}+J_{z}\sigma_{1}^{z}\sigma_{2}^{z}+D_{x}(\sigma_{1}^{y}\sigma_{2}^{z}-\sigma_{1}^{z}\sigma_{2}^{y})+(B_{y}+b_{y})\sigma_{1}^{y}+(B_{y}-b_{y})\sigma_{2}^{y}.$
(34)

From the knowledge of the Hamiltonian we can calculate $\rho(T)$ the thermal
density matrix of the system and calculate the coherence using Eq. (6). Our
results are shown in Figs. 5(a)-(d).

The quantum coherence decreases with temperature as seen in Fig. 5(a), as
before due to thermal decoherence. The qualitative behavior of coherence with
respect to change in the DM interaction parameter, the external homogeneous
field $B_{z}$, and the inhomogeneous field $b_{z}$ is shown in Figs. 5(b), (c)
and (d) respectively. The presence of spin-lattice interaction gives rise to
new or enhanced contributions to off-diagonal elements and hence the overall
effect is to increase the quantum coherence in the system. However in the XYZ
model with both DM interaction and field in the $x$-direction, there is
initially a decrease in coherence. After reaching the minimum the coherence
again starts to increase. This dip occurs because the coherence arises due to
two different interactions namely the spin-spin and the spin-lattice
interaction. While the amount of coherence due to the spin-spin interaction
decreases, the coherence due to the spin-lattice increases with the DM
parameter $D_{x}$. There is a mismatch between the two rates of change leading
to the formation of a dip in the amount of coherence. For higher values of
temperature no dip is observed. In the case of the other two cases described
in Sec. IV A and Sec. IV C, the coherence increases with $D_{x}$ for higher
values of temperature. For lower values of temperature, the coherence
decreases with DM parameter. This special behavior is also due to the
competing rates of change of coherence of the spin-spin and spin-lattice
interactions. Hence, overall we can conclude that coherence increases with the
DM interaction, even if the effect may not be so easily observed.

In summary, when an external field is applied on a spin system, the field
changes the quantum properties by influencing the spins. In our work, we study
the effect of an inhomogenous external field. The external field can be
divided into an homogeneous component $B$ and an inhomogenous component $b$.
The strength and the direction of the homogeneous component $B$ of the field
is the same for the spins at both the lattice sites. In the case of the
homogeneous component $b$, the strength of the field acting on both the spins
is the same, but their direction is different. From Figs. 1 \- 5 we observe
that the quantum coherence decreases when the direction of the homogeneous
field $B$ is the same as the measurement basis. On the other hand, the
coherence increases when the measurement basis and the external field are
orthogonal to each other. In the case of the inhomogenous component of the
field, the coherence mostly increases except for the isolated case of XYZ
model with DM interaction along the $x$-direction and the field $b$ is along
the $z$-direction.

##  V Coherence variation under two control parameters

Up to this point we have not considered the dependence of the spin-spin
interaction $J_{i}$, which has been considered a constant. In this section, we
examine the variation of coherence with respect to the simultaneous variation
of the spin-spin interaction parameter and one of the other control parameters
such as the DM parameter or the external field.

The quantum coherence in the two-spin XYZ model with DM interaction and the
external field along the $x$-direction is described in the first row of plots
in Figs. 6 (a)-(c). Fig. 6(a) shows the variation of coherence with respect to
the spin-spin interaction parameter $J_{z}$ and the DM interaction parameter
$D_{x}$. From the plot we observe that for a given a constant temperature, the
amount of quantum coherence decreases with the spin-spin interaction parameter
$J_{z}$. Contrarily on increasing $D_{x}$ the amount of coherence increases.
This implies that the spin-spin interaction and the spin-lattice interaction
have counteracting effects on the amount of coherence in the system. Fig. 6(b)
shows the quantum coherence variation with respect to the parameters $J_{z}$
and $B_{x}$ (homogeneous component of the external field). Again we find that
the coherence decreases with $J_{z}$, but increases with the external field.
The external field $B_{x}$ tries to align the spins in a uniform direction and
hence we observe an increase in the amount of quantum coherence. Fig. 6(c)
shows coherence as a function of the spin-spin interaction $J_{z}$ and the
inhomogenous component of the field $b_{x}$. This plot exhibits the features
where we find that coherence decreases with $J_{z}$, whereas it increases with
$b_{x}$. These observations are identical to the results obtained in Fig. 6(a)
and 6(b).

In Figs. 6 (d)-(f) the quantum coherence features of the two-site anisotropic
model with DM interaction and the field in the $z$-direction is shown. We
choose the measurement basis also to be in the $\sigma_{z}$ basis and consider
a constant temperature $T$. From Fig. 6(d) we find that the coherence
increases with $J_{z}$ as well as $D_{z}$. In this situation, there is no
counteracting influence between the spin-spin and spin-lattice interactions.
Rather they exhibit a co-operative nature and enhances the coherence in the
system with increase in these parameters. In Fig. 6(e) we notice that
coherence decreases with increase in $B_{z}$ and it increases with $J_{z}$.
Finally in Fig. 6(f) we observe that coherence increases with the inhomogenity
of the external field $b_{z}$ as well with the increase of the spin-spin
interaction parameter $J_{z}$.

Finally, in Figs. 6 (g)-(i) we look into the coherence dependence of the model
with the spin lattice interaction in the $x$-direction and measurement along
the $\sigma_{z}$ basis. The external inhomogenous field is along the
$y$-direction. When the spin-spin interaction $J$ is increased, the coherence
decreases. But when the spin lattice interaction is increased, the coherence
increases. These results are shown in Fig. 6(g). Also through the plots Fig.
6(h) and Fig. 6(i) we find that coherence decreases with increase in the spin-
spin interaction $J_{z}$. On increasing the homogeneous component $B_{y}$ and
inhomogenous component $b_{y}$ we observe that the coherence increases.

In Table 1, based on the two dimensional plots, we summarize the variation of
quantum coherence with different parameters for all the models. Using this in
conjunction with Fig. 6 we arrive at the following conclusions. We find that
when the direction of the DM interaction and external field is orthogonal to
the measurement, the coherence in the system decreases with the spin-spin
coupling $J$ value. Similarly, the coherence increases with the spin-lattice
interaction parameter $D$. Also in this situation both the homogeneous and
inhomogenous fields helps to increase the coherence in the system. Under the
conditions when the direction of measurement basis is same as the direction of
the spin-lattice, the spin-spin interaction and the DM interaction increases
the quantum coherence in the system. With respect to the external field, the
coherence decreases with increase in the homogeneous component of the field
$B$ and it increases with the inhomogenous component of the field $b$.

Table 1: The variation of the coherences for the different parameters is summarized below. Here $+$ means coherence increases with the variable and $-$ means coherence decrease with the variable. The measurement basis is always in the $z$-direction.  Parameters |  $\vec{D}\propto\vec{x}$ |  $\vec{D}\propto\vec{z}$ |  $\vec{D}\propto\vec{z}$ |  $\vec{D}\propto\vec{x}$ |  $\vec{D}\propto\vec{x}$
---|---|---|---|---|---
|  $\vec{B}\propto\vec{x}$ |  $\vec{B}\propto\vec{z}$ |  $\vec{B}\propto\vec{x}$ |  $\vec{B}\propto\vec{z}$ |  $\vec{B}\propto\vec{y}$
Temperature $T$ |  $-$ |  $-$ |  $-$ |  $-$ |  $-$
DM interaction $D$ |  $+$ |  $+$ |  $+$ |  $+$ |  $+$
Homogeneous magnetic field $B$ |  $+$ |  $-$ |  $+$ |  $-$ |  $+$
Homogeneous magnetic field $b$ |  $+$ |  $+$ |  $+$ |  $-$ |  $+$

##  VI Conclusions

The finite temperature quantum coherence of the two-spin XYZ Heisenberg model
with Dzyaloshinsky-Moriya (DM) interaction subjected to an external
inhomogenous magnetic field was investigated. In the investigation of these
two-site models we consider two major cases namely: ($i$) the DM interaction
and the external magnetic field are in the same direction; ($ii$) when the DM
interaction and the external field are not in the same direction. Since
quantum coherence is a basis-dependent quantity, these two classes are further
divided based on the relation between the measurement basis and the direction
of the DM parameter and the field. We first confirm that quantum coherence
always decreases with temperature, due to the commonly observed effect that
quantum coherence is destroyed by thermal decoherence. Next we find that the
quantum coherence always increases with the strength of the DM interaction
parameter irrespective of the relation between the spin-lattice interaction
and the measurement basis. This result is similar to the one obtained for
entanglement measured using concurrence. The quantum coherence decreases when
the direction of the measurement basis is the same as that of the spin-spin
interaction. A similar behavior of quantum coherence is observed when the
external field is varied.

A spin chain always contains a spin-spin interaction apart from the DM
interaction and the external field as described in our work. We investigated
the change in coherence when the spin-spin interaction is simultaneously
varied with the DM interaction parameter or the external fields. From the
observations we find that the spin-spin interaction and the spin-lattice
interaction have counteracting effects on the quantum coherence of the system
when the direction of the measurement is orthogonal to the direction of the
spin-lattice interaction. We note that this study could be further extended to
larger systems, and studying spin models with staggered DM interactions Ma et
al. (2011); Miyahara et al. (2007) and random field interactions Bruinsma and
Aeppli (1983); Fujinaga and Hatano (2007).

The amount and distribution of quantum coherence is connected to quantum
information processing tasks like dense coding and quantum teleportation Pan
et al. (2017). For example, the optimal dense coding capacity of a bipartite
system is inversely related to the amount of local coherence in the system Pan
et al. (2017). Similarly, there is an inverse relation between the
teleportation fidelity and the local coherence in a system Pan et al. (2017).
From our earlier studies Radhakrishnan et al. (2016) we know that coherence
can exist localized within a qubit or as correlations between the qubits.
Hence a small amount of local coherence must imply a higher amount of global
coherence, which naturally means strongly correlated quantum systems help in
tasks such as dense coding and quantum teleportation. In the case of the spin
qubits considered, we observe that the total coherence in the system is always
global coherence which implies that all the available form of coherence in a
spin qubit can be used for quantum information processing tasks. In addition,
the existence of several control parameters such as the spin-spin interaction
parameter, the spin-lattice interaction parameter and the external parameters
such as magnetic field and temperature helps us to modify the state of a
system to improve the capacity of dense coding and the teleportation fidelity
of quantum states. In Ref. Pan et al. (2017) the authors only demonstrated the
complementarity relations of the local coherence with the dense coding
capacity and the teleportation fidelity. An important development in this
direction would be to directly quantify the dense coding capacity and
teleportation fidelity in terms of the global coherence. Such an improvement
will be directly applicable to the spin systems we have considered in our
work, where we can quantify the ability to perform quantum information
processing tasks using the coherence of the spin system.

## Acknowledgments

TB and RC are supported by the Shanghai Research Challenge Fund; New York
University Global Seed Grants for Collaborative Research; National Natural
Science Foundation of China (61571301,D1210036A); the NSFC Research Fund for
International Young Scientists (11650110425,11850410426); NYU-ECNU Institute
of Physics at NYU Shanghai; the Science and Technology Commission of Shanghai
Municipality (17ZR1443600); the China Science and Technology Exchange Center
(NGA-16-001); and the NSFC-RFBR Collaborative grant (81811530112).

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Figure 1: Quantum coherence measurement in $\sigma_{z}$ basis in the XYZ model
with DM interactions in the $x$-direction and the external magnetic field in
the $x$-direction. (a) Quantum coherence versus temperature $T$ for different
$D_{x}$, for $B_{x}=3.0$, $b_{x}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and
$J_{z}=0.2$. (b) Quantum coherence versus $D_{x}$ for different temperatures
$T$, for $B_{x}=3.0$, $b_{x}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$.
(c) Quantum coherence versus $B_{x}$ for different temperatures $T$, for
$D_{x}=1.0$, $b_{x}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. (d)
Quantum coherence versus $b_{x}$ for different temperatures $T$, for
$D_{x}=1.0$, $B_{x}=3.0$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$.  Figure 2:
Measurement of coherence in the $\sigma_{z}$ basis in the two-spin XYZ model
with DM interactions in the $z$-direction and the external magnetic field in
the $z$-direction. (a) Quantum coherence versus temperature $T$ for different
$D_{z}$, for $B_{z}=3.0$, $b_{z}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and
$J_{z}=0.2$. (b) Quantum coherence versus $D_{z}$ for different temperatures
$T$, for $B_{z}=3.0$, $b_{z}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$.
(c) Quantum coherence versus $B_{z}$ for different temperatures $T$, for
$D_{z}=1.0$, $b_{z}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. (d)
Quantum coherence versus $b_{z}$ for different temperatures $T$, for
$D_{z}=1.0$, $B_{z}=3.0$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. Figure 3:
The change in coherence measured in the $\sigma_{z}$ basis for the two-site
XYZ model with DM interactions in the $z$-direction and the external field in
the $x$-direction. (a) Quantum coherence versus temperature $T$ for different
$D_{z}$ for $B_{x}=3$, $b_{x}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$.
(b) Quantum coherence versus $D_{z}$, for different temperatures $T$, for
$B_{x}=3.0$, $b_{x}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. (c)
Quantum coherence versus $B_{x}$ for different temperatures $T$, for
$D_{z}=1.0$, $b_{x}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. (d)
Quantum coherence versus $b_{x}$ for different temperatures $T$, for
$D_{z}=1.0$, $B_{x}=3.0$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. Figure 4:
The variation in coherence is measured in the $\sigma_{z}$ basis for the two-
spin XYZ model with DM interactions in the $x$-direction and the external
magnetic field in the $z$-direction. (a) Quantum coherence versus temperatures
$T$ for different $D_{x}$ for $B_{z}=3$, $b_{z}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$
and $J_{z}=0.2$. (b) Quantum coherence versus $D_{x}$, for different
temperatures $T$, for $B_{z}=3.0$, $b_{z}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and
$J_{z}=0.2$. (c) Quantum coherence versus $B_{z}$ for different temperatures
$T$, for $D_{x}=1.0$, $b_{z}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$.
(d) Quantum coherence versus $b_{z}$ for different temperatures $T$, for
$D_{x}=1.0$, $B_{z}=3.0$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. Figure 5:
The coherence variation measured in the $\sigma_{z}$ basis for the XYZ model
with DM interactions in the $x$-direction and the external field in the
$y$-direction. (a) Quantum coherence versus temperature $T$ for different
$D_{x}$ for $B_{y}=3$, $b_{y}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$.
(b) Quantum coherence versus $D_{x}$, for different temperatures $T$, for
$B_{y}=3.0$, $b_{y}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. (c)
Quantum coherence versus $B_{y}$ for different temperatures $T$, for
$D_{x}=1.0$, $b_{y}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$. (d)
Quantum coherence versus $b_{y}$ for different temperatures $T$, for
$D_{x}=1.0$, $B_{y}=3.0$, $J_{x}=0.8$, $J_{y}=0.5$ and $J_{z}=0.2$.  Figure 6:
The variation of coherence in the XYZ model with DM interaction and external
field along the $x$-axis is shown through the first row of plots as follows:
(a) coherence Vs $J_{z}$ Vs $D_{x}$ (b) coherence Vs $J_{z}$ Vs $B_{x}$ (c)
coherence Vs $J_{z}$ Vs $b_{x}$. The fixed values are $D_{x}=1.0$,
$B_{x}=3.0$, $b_{x}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $T=0.5$. In the second
row, the XYZ model with DM interaction and the field in the $z$-direction is
given as follows: (d) coherence Vs $J_{z}$ Vs $D_{z}$ (e) coherence Vs $J_{z}$
Vs $B_{z}$ (f) coherence Vs $J_{z}$ Vs $b_{z}$. The fixed values are
$D_{z}=1.0$, $B_{z}=3.0$, $b_{z}=1.5$, $J_{x}=0.8$, $J_{y}=0.5$ and $T=0.5$.
The final row describes the XYZ model with DM interaction in the $x$-direction
and the field along the $y$-direction as follows: (d) coherence Vs $J_{z}$ Vs
$D_{x}$ (e) coherence Vs $J_{z}$ Vs $B_{y}$ (f) coherence Vs $J_{z}$ Vs
$b_{y}$. The fixed values are $D_{x}=1.0$, $B_{y}=3.0$, $b_{y}=1.5$,
$J_{x}=0.8$, $J_{y}=0.5$ and $T=0.5$. All the measurements are made uniformly
in the $\sigma_{z}$ basis.