Download Entanglement in the anisotropic Heisenberg XYZ model with

Entanglement in the anisotropic Heisenberg XYZ model with different Dzyaloshinskii-Moriya
interaction and inhomogeneous magnetic field
Da-Chuang Li1,2∗ , Zhuo-Liang Cao1,2†
1
arXiv:0907.1433v1 [quant-ph] 9 Jul 2009
Department of Physics, Hefei Teachers College, Hefei 230061 P. R. China
2
School of Physics & Material Science, Anhui University, Hefei 230039 P. R. China
We investigate the entanglement in a two-qubit Heisenberg XYZ system with different DzyaloshinskiiMoriya(DM) interaction and inhomogeneous magnetic field. It is found that the control parameters (Dx , Bx
and bx ) are remarkably different with the common control parameters (Dz ,Bz and bz ) in the entanglement and
the critical temperature, and these x-component parameters can increase the entanglement and the critical temperature more efficiently. Furthermore, we show the properties of these x-component parameters for the control
of entanglement. In the ground state, increasing Dx (spin-orbit coupling parameter) can decrease the critical
value bxc and increase the entanglement in the revival region, and adjusting some parameters (increasing bx
and J, decreasing Bx and ∆) can decrease the critical value Dxc to enlarge the revival region. In the thermal
state, increasing Dx can increase the revival region and the entanglement in the revival region (for T or bx ), and
enhance the critical value Bxc to make the region of high entanglement larger. Also, the entanglement and the
revival region will increase with the decrease of Bx (uniform magnetic field). In addition, small bx (nonuniform
magnetic field) has some similar properties to Dx , and with the increase of bx the entanglement also has a
revival phenomenon, so that the entanglement can exist at higher temperature for larger bx .
PACS numbers: 03.67.Mn, 03.65.Ud, 75.10.Jm
I.
INTRODUCTION
Entanglement is one of the most fascinating features of
quantum mechanics and it provides a fundamental resource
in quantum information processing [1]. As a simple system,
Heisenberg model is an ideal candidate for the generation and
the manipulation of entangled states. This model has been
used to simulate many physical systems, such as nuclear spins
[2], quantum dots [3], superconductor [4] and optical lattices
[5], and the Heisenberg interaction alone can be used for quantum computation by suitable coding [6]. In recent years, the
Heisenberg model, including Ising model [7], XY model [8],
XXX model [9], XXZ model [10] and XYZ model [11, 12],
have been intensively studied, especially in thermal entanglement and quantum phase transition. Very recently, F. Kheirandish et al. and Z. N. Gurkan et al. [12] discussed the Heisenberg XYZ model with DM interaction (arising from spin-orbit
coupling) and magnetic field. However, almost all the directions of spin-orbit coupling and the external magnetic field are
fixed in the z-axis in the above papers. The external magnetic
field along other directions is discussed only in some special Heisenberg models, and the spin-orbit coupling parameter
along other directions has never been taken into account. This
motivates us to think about what different phenomena will appear if we choose the parameters along different directions in
the generalized Heisenberg XYZ model. To explore this, here
we choose x-axis as the direction of spin-orbit coupling and
the external magnetic field in the Heisenberg XYZ model.
In this paper, we discuss the differences between the
Heisenberg XYZ models with parameters in different directions, and then analyze the influences of these parameters on
the ground-state entanglement and thermal entanglement. We
find that x-component DM interaction and inhomogeneous
magnetic field are more efficient control parameters, and more
entanglement and higher critical temperature can be obtained
by adjusting the value of these parameters. In order to provide a detailed analytical and numerical analysis, here we take
concurrence as a measure of entanglement [13]. The concurrence C ranges from 0 to 1, C = 0 and C = 1 indicate the vanishing entanglement and the maximal entanglement respectively. For a mixed state ρ, the concurrence of
P4
the state is C(ρ) = max{2λmax − i=1 λi , 0}, where λi s
are the positive
roots
the eigenvalues of the matrix
N ysquare
N of
R = ρ(σ y
σ )ρ∗ (σ y
σ y ), and the asterisk denotes the
complex conjugate. If the state of a system is a pure state,
i.e. ρ = |ΨihΨ|, |Ψi = a|00i + b|01i + c|10i + d|11i, the
concurrence will be reduced to C(|Ψi) = 2|ad − bc|.
Our paper is organized as follows. In Sec. II, we compare the DM coupling parameters along different directions.
In Sec. III, we compare the external magnetic fields (including uniform and nonuniform magnetic field) along different
directions. In Sec. IV, we analyze the entanglement of generalized Heisenberg XYZ model with x-component DM interaction and inhomogeneous magnetic field. Finally, in Sec. V
a discussion concludes the paper.
II. THE COMPARISON BETWEEN THE DM COUPLING
PARAMETERS ALONG DIFFERENT DIRECTIONS
A. DM coupling parameter along z-axis
∗ E-mail:
[email protected]
† E-mail:[email protected]
(Corresponding Author)
The Hamiltonian H for a two-qubit anisotropic Heisenberg XYZ model with z-component DM coupling and external
2
magnetic field is
Dz (σ1x σ2y
−
σ1y σ2x )
|φ1,2 i = sin θ1,2 |00i + cos θ1,2 |11i,
(2a)
|φ3,4 i = sin θ3,4 |01i + χ cos θ3,4 |10i,
(2b)
with corresponding eigenvalues:
−H
)
−Jz r
e T w1
w1
=
w12 cosh2 ( ) − 4Bz2 sinh2 ( )
Zw1 T
T
w1
(4b)
± sinh( )(Jx − Jy ).
T
Thus, the concurrence of this system can be written as [13]:
C = max{|λ1 − λ3 | − λ2 − λ4 , 0},
0.5
0.5
0
0
4
4
0
2
2
D
z
0
2
4
6
D
T
x
0
6
2
4
0
T
FIG. 1: (Color online) (a) Thermal concurrence versus T and Dz
without the external magnetic field. (b) Thermal concurrence versus
T and Dx without the external magnetic field. Here Jx = 1, Jy =
0.5, and Jz = 0.2.
Bz = bz = 0. It shows that the concurrence will decrease
with increasing temperature T and increase with increasing
Dz for a certain temperature. The critical temperature Tc is
dependent on Dz , increasing Dz can increase the critical temperature above which the entanglement vanishes. These results are in accord with those in Ref. [12].
B. DM coupling parameter along x-axis
KB T
ρ(T ) =
, where Z = T r[exp( K−H
)] is the partition
Z
BT
function of the system, H is the system Hamiltonian, T is
the temperature and KB is the Boltzmann costant which we
take equal to 1 for simplicity. Then we can get the analytical
expressions of Z, ρ(T ), and R, but we do not list them
because of the complexity. After straightforward calculations,
the positive square roots of the eigenvalues of R can be
expressed as:
Jz r
e T w2
w2
λ1,2 =
w22 cosh2 ( ) − 4b2z sinh2 ( )
Zw2 T
T
q
w2
± sinh( ) (Jx + Jy )2 + 4Dz2 ,
(4a)
T
λ3,4
1
(3a)
E3,4 = −Jz ± w2 ,
(3b)
p
where w1
=
4Bz2 + (Jx − Jy )2 ,
w2
=
p
J +J −2iD
4b2z + 4Dz2 + (Jx + Jy )2 , and χ = √ x y 2 z 2 ,
√ (Jx +Jy2) +4D2z
(Jx +Jy ) +4Dz
Jx −Jy
).
θ1,2 = arctan( ±w1 −2Bz ), θ3,4 = arctan(
±w2 −2bz
The system state at thermal equilibrium (thermal state) is
exp(
1
(1)
where Ji (i = x, y, z) are the real coupling coefficients, Dz
is the z-component DM coupling parameter, Bz (uniform
external magnetic field) and bz (nonuniform external magnetic field) are the z-component magnetic field parameters,
σ i (i = x, y, z) are the Pauli matrices. The coupling constants
Ji > 0 corresponds to the antiferromagnetic case, and Ji < 0
corresponds to the ferromagnetic case. This model can be reduced to some special Heisenberg models by changing Ji . Parameters Ji , Dz , Bz and bz are dimensionless.
Using the similar process to [12], we can get the eigenstates
of H:
E1,2 = Jz ± w1 ,
(b)
C’
Jz σ1z σ2z +
− bz )σ2z ,
C
H =
(a)
+ Jy σ1y σ2y +
+(Bz + bz )σ1z + (Bz
Jx σ1x σ2x
(5)
Fig. 1(a) demonstrates the concurrence versus temperature
T and Dz , where Jx = 1, Jy = 0.5, Jz = 0.2, and
′
The Hamiltonian H for a two-qubit anisotropic Heisenberg XYZ model with x-component DM coupling and external magnetic field is
′
H = Jx σ1x σ2x + Jy σ1y σ2y + Jz σ1z σ2z + Dx (σ1y σ2z − σ1z σ2y )
+(Bx + bx )σ1x + (Bx − bx )σ2x ,
(6)
similarly, where Ji (i = x, y, z) are the real coupling coefficients, Dx is the x-component DM coupling parameter, Bx
(uniform external magnetic field) and bx (nonuniform external magnetic field) are the x-component magnetic field parameters, and σ i (i = x, y, z) are the Pauli matrices. The coupling constants Ji > 0 corresponds to the antiferromagnetic
case, and Ji < 0 corresponds to the ferromagnetic case. This
model can be reduced to some special Heisenberg models by
changing Ji . Here, all the parameters are dimensionless, and
we introduce the mean coupling parameter J and the partial
J +J
anisotropic parameter ∆ in the YZ-plane, where J = y 2 z
J −J
and ∆ = Jyy +Jzz .
In the standard basis {|00i, |01i, |10i, |11i}, the Hamilto′
nian H can be rewritten as:


Jz
G2
G3
Jx − Jy
′
−Jz Jx + Jy
G1 
 G4
,
(7)
H =
G1
Jx + Jy −Jz
G4 
Jx − Jy
G3
G2
Jz
where G1,2 = iDx + Bx ± bx , G3,4 = −iDx + Bx ± bx . By
′
calculating, we can obtain the eigenstates of H
1
|ψ1,2 i = √ (sin ϕ1,2 |00i + cos ϕ1,2 |01i
2
+ cos ϕ1,2 |10i + sin ϕ1,2 |11i),
(8a)
3
1
0.9
Dz
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
1
2
3
4
5
6
7
8
9
Dx
0
10
y
0
1
2
3
4
U1
′
 Q1
ρ (T ) = 
Q2
U2
5
6
7
8
9
10
T
FIG. 2: (Color online) (a) Thermal concurrence versus DM coupling
parameter Dz (red dotted line), Dx (blue solid line), where T =
3. (b) Thermal concurrence versus temperature T for Dz = 3 (red
dotted line) and Dx = 3 (blue solid line). Here Jx = 1, Jy =
0.5, Jz = 0.2, and Bz,x = bz,x = 0.
′
1
|ψ3,4 i = √ (sin ϕ3,4 |00i + χ cos ϕ3,4 |01i
2
′
′
Jx −w2
T
sin2 ϕ3 ± e
′
Jx +w2
T
sin2 ϕ4 ),
′
Jx −w1
J +w
1
− xT 1
cos2 ϕ1 + e− T cos2 ϕ2
′ (e
2Z
±e
′
(9a)
′
(9b)
p
′
′
′
√ 2x −b2x , w1 =
4Bx2 + (Jy − Jz )2 , w2 =
where χ = −iD
bx +Dx
p
2Bx
4b2x + 4Dx2 + (Jy + Jz )2 , and ϕ1,2 = arctan( J −J
′ ),
±w
z
′
Jx −w2
T
cos2 ϕ3 ± e
′
Jx +w2
T
cos2 ϕ4 ),
′
′
Q1,2 =
E3,4 = −Jx ± w2 ,
(10)
J +w
Jx −w1
1
− xT 1
sin2 ϕ1 + e− T sin2 ϕ2
′ (e
2Z
±e
(8b)
E1,2 = Jx ± w1 ,
λ1,2

U2
Q2 
,
Q1 
U1
′
U1,2 =
with corresponding eigenvalues:
y
Q∗2
V2
V1
Q∗1
′
′
′
Q∗1
V1
V2
Q∗2
where
V1,2 =
−χ cos ϕ3,4 |10i − sin ϕ3,4 |11i),
′
2
density matrix ρ (T ) has the following form:

D
′
z
′
Dz
0.8
C
C
0.9
Dx
0.8
0
√
2
2 b2x +Dx
ϕ3,4 = arctan( −J −J
′ ). In the above standard basis, the
±w
(b)
(a)
1
Jx −w1
Jx +w1
1
(e− T sin ϕ1 cos ϕ1 + e− T sin ϕ2 cos ϕ2
2Z ′
±e
′
Jx −w2
T
′
χ sin ϕ3 cos ϕ3 ± e
′
Jx +w2
T
′
χ sin ϕ4 cos ϕ4 ).
Then the positive square roots of the eigenvalues of the matrix
′
R can be obtained
1
s
1
Jx ′
′
′
′ 2
8b2x
16b2x
2w2
4Dx2 + (Jy + Jz )2
e T 2 2w2
2 w2
4 w2
) − ′ 2 sinh ( ) ±
)
−
)
sinh
(
sinh
(
= ′ cosh(
,
′
′
Z
T
T
T
T w2
w22
w22
′
λ3,4 =
e
−Jx
T
Z′
s
1
′
′
′
′ 2
2
2
2
8B
2w
w
4 w1
x
cosh( 1 ) − ′ x sinh2 ( 1 ) ± (Jy −′ Jz ) sinh2 ( 2w1 ) − 16B
sinh ( ) ,
′
T
T
T
T
w12
w12
w12
w
−Jx
′
′
Jx
w
′
where Z = 2[e T cosh( T1 ) + e T cosh( T2 )]. Thus, according to [13], the corresponding concurrence of this system
can be expressed as:
′
′
′
′
′
C = max{|λ1 − λ3 | − λ2 − λ4 , 0},
′
(12)
In Fig. 1(b), we plot the concurrence C as a function of T and
Dx , where Jx = 1, Jy = 0.5, Jz = 0.2, and Bx = bx = 0. It
shows the similar results to those in the last subsection, but if
we compare the two figures in Fig. 1 in detail, we can easily
find some differences between them. i.e., there is less disentanglement region in Fig. 1(b) than in Fig. 1(a), and increasing x-component parameter Dx can make the entanglement
increase more rapidly, for example, when T = 6 the concurrence increases more rapidly in Fig. 1(b) than in Fig. 1(a).
(11a)
(11b)
These phenomena show that x-component parameter Dx has
some more remarkable influences than the z-component parameter Dz on the thermal entanglement and the critical temperature. To demonstrate the differences more clearly, we plot
Fig. 2, which shows the advantages of using parameter Dx . In
Fig. 2(a), we can easily find that for a certain temperature, Dx
has more entanglement than Dz , and increasing Dx can increase the entanglement more rapidly. Also, in Fig. 2(b), we
can see that for the same Dx and Dz , Dx has a higher critical
temperature. In a word, increasing Dx can enhance the entanglement, and slow down the decrease of the entanglement (by
increasing critical temperature) more efficiently than Dz . So
by changing the direction of DM coupling interaction, we can
get a more efficient control parameter.
4
(a)
(b)
1
0.8
0.7
0.7
0.6
0.6
0.5
0.5
C
C
Bx
0.9
Bz
0.8
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
-5
the concurrence decreases with increasing T for Bz = 1 (Fig.
3(b)) and bz = 1.5 (Fig. 4(b)).
1
Bx
0.9
Bz
B. External magnetic field along x-axis
0.1
-4
-3
-2
-1
0
1
2
3
4
0
5
0
0.5
1
B
1.5
2
2.5
3
T
FIG. 3: (Color online) (a) Thermal concurrence versus uniform magnetic field Bz (red dotted line), Bx (blue solid line), where T = 0.1.
(b) Thermal concurrence versus temperature T for Bz = 1 (red dotted line) and Bx = 1 (blue solid line). Here Jx = 1, Jy = 0.8, Jz =
0.2, and Dz,x = bz,x = 0.
(a)
(b)
1
1
bx
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
-5
bx
0.9
bz
C
C
0.9
bz
0.1
-4
-3
-2
-1
0
b
1
2
3
4
5
0
0
0.5
1
1.5
2
2.5
3
T
FIG. 4: (Color online) (a) Thermal concurrence versus nonuniform
magnetic field bz (red dotted line), bx (blue solid line), where T =
0.2. (b) Thermal concurrence versus temperature T for bz = 1.5 (red
dotted line) and bx = 1.5 (blue solid line). Here Jx = 0.2, Jy =
0.4, Jz = 1, and Dz,x = Bz,x = 0.
III. THE COMPARISON BETWEEN THE EXTERNAL
MAGNETIC FIELDS ALONG DIFFERENT DIRECTIONS
A.
In this case, we use the model in Eq. (6). According Eq.
(12), we analyze similarly the variation of the entanglement
and the roles of the external magnetic field along x-axis, by
fixing other parameters.
In Fig. 3(a), by comparing with |Bz |, we can see that,
though |Bx | has some similar properties to |Bz |, |Bx | has a
larger critical value and a more remarkable revival. In Fig.
4(a), we find that the entanglement decreases more slowly
with the increase of nonuniform magnetic field |bx | than |bz |,
and |bx | has more entanglement when |bz | = |bx |. Also, In
Fig. 3(b), we find that Bx has more entanglement than Bz for
a certain temperature when Bz = Bx . Fig. 4(b) shows that,
for the same bx and bz , bx has a higher critical temperature
and more entanglement (for a certain temperature).
It is evident that for some fixed parameters, using xcomponent magnetic field can increase the entanglement more
efficiently than z-component magnetic field. So we can
change the direction of the external magnetic field to obtain
a more efficient control parameter.
IV. HEISENBERG XYZ MODEL WITH X-COMPONENT
DM INTERACTION AND INHOMOGENEOUS MAGNETIC
FIELD
According to Secs. II and III, we know that x-component
DM interaction and inhomogeneous magnetic field are more
efficient control parameters of entanglement. So in this section, we will analyze the properties of parameters of the model
(Eq. (6)) in detail.
External magnetic field along z-axis
Here, we use the same model expressed in Eq. (1). According to Eq. (4), we can write the expression of concurrence, and then analyze the variation of the entanglement and
the roles of the external magnetic field along z-axis, by fixing
other parameters.
In Fig. 3(a), the thermal concurrence as a function of the
uniform magnetic field is plotted, where T = 0.1. It is shown
that, for small |Bz |, the concurrence has a high entanglement
and does not vary with the variation of |Bz |, and with increasing |Bz |, the entanglement drops to zero suddenly at the critical value of |Bz |, and then undergoes a revival before decreasing to zero. In Fig. 4(a), the thermal concurrence is plotted
versus the nonuniform magnetic field with T = 0.2. It is evident that increasing |bz | will decrease the entanglement. In
Fig. 3(b) and Fig. 4(b), the concurrence as a function of the
temperature T is plotted for the uniform magnetic field and
the nonuniform magnetic field, respectively. It is shown that
A. Ground state entanglement
When T = 0, this system is in its ground state. It is easy to
find that the ground-state energy is equal to
′
′
E2 = Jx − w1 ,
′
′
E4 = −Jx − w2 ,
′
′
if Jx <
′
1 ′
(w1 − w2 ),
2
(13a)
if Jx >
′
1 ′
(w − w2 ),
2 1
(13b)
and for E2 and E4 , the ground state is |ψ2 i and |ψ4 i respec′
′
′
′
tively. At the critical point Jx = 21 (w1 − w2 ) (i.e. E2 = E4 ),
the ground state is |Gi = √12 (|ψ2 i + |ψ4 i) (an equal superposition of |ψ2 i and |ψ4 i). Thus, the concurrence of ground
state can be expressed as
5
′
C (T = 0) =















Jy −Jz ′ ,
w1 J +J
1 Jy −Jz
+ yw′ z )2 +
2 ( w1′
2
1
4Dx2 +(Jy +Jz )2 2
,
′
w
2
2
4Dx
′
w22
−
0.8
0.8
Dx=1.2
0.7
0.7
Dx=1.8
0.6
0.6
C’
C'
0.9
0.5
′
′
(14)
0.5
0.4
0.4
0.3
0.6
′
(b)
1
0.9
0.7
′
if Jx > 12 (w1 − w2 ).
Dx=0.8
C'
0.8
′
, if Jx = 12 (w1 − w2 ),
(a)
1
1
0.9
21
′
2
2Dx
(Jy −Jz )(w2 +Jy +Jz ) 2 )w ′ w ′
(b2x +Dx
1 2
′
if Jx < 12 (w1 − w2 ),
0.3
bx =0
0.5
0.2
bx =0.8
0.2
0.1
bx =1.5
0.1
0
0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
5
' =0.2
' =0.8
' =1.5
0
0.5
1
1.5
2
Dx
2.5
3
3.5
4
4.5
5
Dx
0.3
(c)
1
0
0
1
2
3
4
5
bx
C'
0.1
(d)
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
C'
0.2
0.4
0.5
0.4
0.3
0.3
Bx =0.5
FIG. 5: (Color online) The ground-state concurrence versus bx for
different Dx . Here J = 0.5, ∆ = 0.8, Jx = −1, and Bx = 1.
In Fig. 5, the ground-state concurrence is plotted as a
function of bx for different Dx . With increasing bx , the
′
ground-state concurrence C keeps a constant initially, and
then varies suddenly at the critical value of bx bxc =
p
′
1
(2Jx − w1 )2 − (Jy + Jz )2 − 4Dx2 , at which the quan2
′
tum phase transition occurs. In the region of bx > bxc , C
has a revival before decreasing to zero. By increasing Dx ,
the critical value bxc will decrease, and thus the revival region will become larger. In addition, there will be more entanglement in the revival region for larger Dx . The groundstate concurrence as a function of Dx for different parameters
is shown in Figs. 6(a)-6(d). These figures show that, with
the increasing of Dx , the concurrence keeps a constant initially, which depends onpJ, ∆ and Bx , before reaching the
′
critical value Dxc = 21 (2Jx − w1 )2 − (Jy + Jz )2 − 4b2x ,
then varies suddenly at Dxc , and then undergoes a revival to
′
reach its maximum value (C = 1) finally in the region of
Dx > Dxc . The critical point Dxc decreases as bx and J
increase, and increases as Bx and ∆ increase. Thus, by adjusting the parameters of the system (increasing bx and J, decreasing Bx and ∆), we can decrease the critical point Dxc to
get larger revival region with more entanglement.
B.
Thermal entanglement
As the temperature increases, the thermal fluctuation is
introduced into the system, thus the entanglement will be
0.2
Bx =1
0.2
0.1
Bx =1.5
0.1
0
0
0.5
1
1.5
2
2.5
Dx
3
3.5
4
4.5
5
0
J=0.2
J=0.8
J=1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Dx
FIG. 6: (Color online) The ground-state concurrence versus Dx for
different parameters. (a) bx = 0 (red dotted line), 0.8 (green Dashdot line), and 1.5 (blue solid line), where J = 0.5, ∆ = 0.8, Jx =
−1, and Bx = 1. (b) ∆ = 0.2 (blue solid line), 0.8 (green Dash-dot
line), and 1.5 (red dotted line), where J = 0.5, Jx = −1, Bx = 0.8,
and bx = 1. (c) Bx = 0.5 (blue solid line), 1 (green Dash-dot line),
and 1.5 (red dotted line), where J = 0.5, ∆ = 0.8, Jx = −1, and
bx = 1. (d) J = 0.2 (red dotted line), 0.8 (green Dash-dot line),
and 1.5 (blue solid line), where ∆ = 0.5, Jx = −1, Bx = 0.8, and
bx = 1.
changed due to the mix of the ground state and excited states.
To see the variation the entanglement, we use the thermal state
′
′
ρ (T ) to describe the system state, and the concurrence C to
measure the entanglement. According to Eq. (12), the con′
currence C is plotted in Fig. 7 and Fig. 8 by fixing some
parameters.
Fig. 7 demonstrates the influence of some parameters on
the entanglement and critical temperature. In Fig. 7(A)
and 7(a), when Dx is small, with increasing temperature the
concurrence decreases to zero at Tc1 (the first critical temperature), and then undergoes a revival before decreasing to
zero again at Tc2 (the second critical temperature). When
Dx > Dxc , with increasing temperature the concurrence decreases to zero at Tc2 . Increasing Dx can increase the revival
region (decreasing Tc1 and increasing Tc2 ) and enhance the
entanglement in the revival region, so the entanglement can
exist at higher temperature for larger Dx . In Fig. 7(B) and
7(b), we see that decreasing Bx can enhance the entanglement
6
(a)
1
C’
x
0.8
D =D
x
0.4
xc
D =2
x
0.2
0.5
0.6
C’
C’
(a)
1
1
D =1
0.6
0
5
(A)
Dx=0
0.8
0.5
C’
(A)
1
0
0
4
5
D
T
x
0
1
2
3
0
1
2
3
4
5
D =0
0.4
0
5
6
x
Dx=2
T
0.2
D =5
x
(B)
(b)
1
Dx
B =5
0.5
0
6
2
4
C’
C’
xc
B
6
(b)
D =0
x
D =1
x
0
1
2
x
3
4
5
0.5
T
(C)
Dx=3
0.6
C’
4
5
0.8
C’
5
T
4
0
0
1
2
3
1
x
3
2
1
B =1
0
1
Bx
(B)
B =B
x
0
x
Bx=3
0.5
0
B
x
0
0
0.4
0
5
(c)
1
0.2
b =0
x
0.5
C’
b =1
C’
x
0
5
Dx
b =b
0.5
x
xc
0
6
2
4
b
0
0
0
1
2
3
4
5
6
bx
x
bx=2
0
b
x
5
4
3
T
2
1
0
0
0
1
2
3
4
5
6
T
FIG. 7: (Color online) (A) Thermal concurrence versus Dx and T ,
where Bx = 3, bx = 1.5. (a) Thermal concurrence versus T for
Dx = 0 (red dotted line), 1 (green Dash-dot line), Dxc ≃ 1.575
(blue dashed line), 2 (black solid line). (B) Thermal concurrence
versus Bx and T , where bx = 1.5, Dx = 1. (b) Thermal concurrence versus T for Bx = 5 (red dotted line), 3 (green Dash-dot line),
Bxc ≃ 2.63 (blue dashed line), 1 (black solid line). (C) Thermal
concurrence versus bx and T , where Bx = 3, Dx = 1.6. (c) Thermal concurrence versus T for bx = 0 (red dotted line), 1 (green
Dash-dot line), bxc ≃ 1.47 (blue dashed line), 2 (black solid line).
Here Jx = 0.8, Jy = 0.5, and Jz = 0.2.
and increase the revival region (decreasing Tc1 ), so the entanglement has its maximum value when Bx = 0 and T = 0.
Fig. 7(C) and 7(c) show that small bx has some similar properties to Dx . In addition, Fig. 7(C) shows that the concurrence
has a revival phenomenon as bx increases, so the entanglement
can also exist at higher temperature for larger bx .
On the whole, Fig. 7 can be divided into two regions: (i)
The region with revival. In this region, Dx < Dxc , bx < bxc ,
or Bx > Bxc , where Dxc , bxc and Bxc are
the critical
val′
′
ues of the parameters. When T < Tc1 , λmax = λ3 and thus
′
′
′
′
′
′
′
C = max{λ3 − λ1 − λ2 − λ4 , 0}. When T = Tc1 , λ3 = λ1
′
′
′
′
and thus C = 0. When T > Tc1 , λmax = λ1 and thus C =
′
′
′
′
max{λ1 − λ3 − λ2 − λ4 , 0}. With the increase of temperature,
the entanglement decreases to zero at Tc2 ultimately. (ii) The
region without revival. In this region, Dx > Dxc , bx > bxc ,
′
′
or Bx < Bxc . For arbitrary temperature, λmax = λ1 , so the
′
′
′
′
′
concurrence is C = max{λ1 − λ3 − λ2 − λ4 , 0}. Also, the
entanglement vanishes finally at Tc2 with increasing temperature.
Fig. 8 shows the cross influence of the DM parameter and
the magnetic field on entanglement. In Fig. 8(A) and 8(a),
when Dx is fixed, with increasing Bx the entanglement is a
constant initially, and then dropped to zero suddenly at Bxc .
For Bx > Bxc , a revival phenomenon occurs, and the entanglement becomes another constant. In addition, as Dx increases, the critical value Bxc increases, and the entanglement
FIG. 8: (Color online) (A) Thermal concurrence versus Bx and Dx ,
where bx = 1.5, T = 0.5. (a) Thermal concurrence versus Bx for
Dx = 0 (red dotted line), 2 (green Dash-dot line), 5 (blue solid
line). (B) Thermal concurrence versus bx and Dx , where Bx =
3, T = 0.5. (b) Thermal concurrence versus bx for Dx = 0 (red
dotted line), 1 (green Dash-dot line), 3 (blue solid line). Here Jx =
0.8, Jy = 0.5, and Jz = 0.2.
enhances in the region of Bx < Bxc . In Fig. 8(B) and 8(b),
when Dx is small, with increasing bx the entanglement is also
a constant initially, then dropped to zero suddenly at bxc , and
then undergoes a revival. Furthermore, increasing Dx can increase the revival region (decreasing bxc ), and enhance the
entanglement in the revival region. When Dx is large enough,
with increasing bx the entanglement decreases from its maximum value to zero.
V.
DISCUSSION
We have investigated the entanglement in the generalized
two-qubit Heisenberg XYZ system with different DM interaction and inhomogeneous magnetic field. By comparing with
the common z-component parameters (Dz ,Bz and bz ), we
find that the x-component DM interaction and inhomogeneous
magnetic field (Dx ,Bx and bx ) are more efficient control parameters for the increase of entanglement and critical temperature. Furthermore, we analyze the properties of x-component
parameters for the control of entanglement. In the ground
state, increasing Dx can decrease the critical value bxc to increase to revival region, and increase the entanglement in the
revival region. The critical value Dxc can also be decreased
by increasing bx and J, or decreasing Bx and ∆. In the thermal state, for T or bx , increasing Dx can increase the revival
region and the entanglement in the revival region, for Bx , increasing Dx can increase the critical value Bxc to largen the
region of high entanglement. In addition, decreasing Bx can
also increase the entanglement and the revival region. Small
bx possesses some similar properties to Dx , and with the increase of bx , the entanglement also has a revival phenomenon
7
so that the entanglement can exist at a higher temperature for
larger bx . Our results imply that more efficient control parameters can be gotten by changing the direction of parameters,
and more entanglement and higher critical temperature can be
obtained by adjusting the values of these parameters. Thus, by
using the state with more entanglement as the quantum channel, the ultimate fidelity will be increased in quantum communication. In addition, in practice the need for low temperature
enhances the complexity and difficulty of constructing a quantum computer, however by increasing the critical temperature
the entanglement can exist at a higher temperature, this will
reduce the technical difficulties for the realization of a quantum computer.
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Acknowledgments
This work is supported by National Natural Science Foundation of China (NSFC) under Grant Nos: 60678022 and
10704001, the Specialized Research Fund for the Doctoral
Program of Higher Education under Grant No. 20060357008,
Anhui Provincial Natural Science Foundation under Grant
No. 070412060, the Key Program of the Education Department of Anhui Province under Grant No. KJ2008A28ZC.
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