Thermal Entanglement in a Two-Qutrit Spin-1 Anisotropic Heisenberg Model

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2011 Chinese Phys. Lett. 28 020306

(http://iopscience.iop.org/0256-307X/28/2/020306)

Download details:

IP Address: 130.220.71.25

The article was downloaded on 14/10/2012 at 02:57

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

CHIN. PHYS. LETT. Vol. 28,No. 2 (2011) 020306

Thermal Entanglement in a Two-Qutrit Spin-1 Anisotropic Heisenberg Model

Erhan Albayrak**

Erciyes University, Department of Physics, 38039, Kayseri, Turkey

(Received 10 November 2010)The negativity (𝑁) as a measure of thermal entanglement (TE) is studied for a two-qutrit spin-1 anisotropicHeisenberg 𝑋𝑋𝑍 chain with Dzyaloshinskii–Moriya (DM) interaction in an inhomogeneous magnetic field indetail. The effects of the DM interaction parameter 𝐷𝑧 on the thermal variation of the 𝑁 for given values ofthe external magnetic field 𝐵, a parameter 𝑏 which controls the inhomogeneity of 𝐵 and the bilinear interactionparameters 𝐽𝑥 = 𝐽𝑦 = 𝐽𝑧 are obtained. It is found that 𝑁 persists to higher values and to higher temperaturesfor the higher values of ±𝐷𝑧 and for the higher positive values of 𝐽𝑧, i.e. in the antiferromagnetic (AFM) case.When 𝐽𝑧 < 0, the ferromagnetic (FM) case, and 𝐷𝑧 is small, and if 𝐽𝑧 is strong enough to compete with 𝐷𝑧, 𝑁decreases. In addition, 𝑁 declines with the increasing values of 𝐵 and 𝑏.

PACS: 03.67.Mn, 03.65.Ud, 75.10.Jm DOI: 10.1088/0256-307X/28/2/020306

The TE is extensively investigated lately becauseof its amazing nonclassical nature, i.e. the exis-tence of nonlocal correlation, for quantum systems andthis non-classical quantum property can be used asa key concept such as in quantum computation andquantum-information processing.

The TE for the spin-1/2 case has been studied gen-erally because of its simplicity. Thus, the TE for thetwo-qubit systems were studied in the𝑋𝑋 model,[1] inthe 𝑋𝑌 model,[2] in the 𝑋𝑋𝑋 model,[3] in the 𝑋𝑋𝑍model[4] and in the 𝑋𝑌 𝑍 model.[5]

In the spin-1/2 models the concurrence is usuallyused as a measure of entanglement. For the spin-1models, another measure, i.e. 𝑁 , is used as a mea-sure of entanglement. The TE in spin-1 systems wasconsidered for different cases such as in a uniformmagnetic field,[6] with exchange interaction,[7,8] withboth bilinear and bilinear-biquadratic chains,[9] in thebilinear-biquadratic and anisotropic XXZ model,[10]in ferrimagnetic chains,[11] in the mixed spin-(1,s)systems,[12] for the time evolution of the boundentanglement,[13] in the two-site extended Hubbardmodel,[14] in the supermolecular dimer [Mn4]2

[15] andfor the construction of the entangled states.[16]

The effects of the DM interaction parameter, whichis anisotropic antisymmetric, arising from the spin-orbit coupling were not considered so much: The onlyworks are for spin-1/2 systems, i.e., in a two-qubitHeisenberg 𝑋𝑋𝑍 chain[17] and with DM interactionin an inhomogeneous magnetic field[18] and the 𝑋𝑌 𝑍chain with DM interaction.[19]

So far effects of the DM interaction on the TE for

spin-1 models have not been considered as far as Iknow. Since the addition of one extra term 𝐷𝑧 intro-duces complex quantities into the model which makesthe problem much more difficult to deal with. Thus,we consider the TE of a two-qutrit spin-1 anisotropicHeisenberg 𝑋𝑋𝑍 chain with the DM interaction pa-rameter 𝐷𝑧 in an external magnetic field 𝐵, a param-eter 𝑏 which controls the inhomogeneity of 𝐵 and thebilinear interaction parameters 𝐽𝑥 = 𝐽𝑦 = 𝐽𝑧 on thethermal variation of 𝑁 . Therefore, the Hamiltonianfor the isotropic two-qutrit system including the DMinteraction may be given as

ℋ = 𝐽𝜎1 · 𝜎2 +𝐷𝑧(𝜎𝑥1𝜎

𝑦2 − 𝜎𝑦

1𝜎𝑥2 )

+ (𝐵 + 𝑏)𝜎𝑧1 + (𝐵 − 𝑏)𝜎𝑧

2 , (1)

where 𝐽 is the isotropic bilinear interactions betweenthe spin pair and 𝐵 ≥ 0, 𝜎𝑖 is the spin operator,𝑖 = 1 or 2, and 𝜎𝑧

𝑖 is the 𝑧th component. It becomesthe Hamiltonian for the anisotropic𝑋𝑋𝑍 model when𝐽𝑥 = 𝐽𝑦 = 𝐽 = 𝐽𝑧 and written in terms of the lowering(𝜎−

𝑖 ) and raising operators (𝜎+𝑖 ) as

ℋ = 2𝐽(𝜎+1 𝜎

−2 + 𝜎−

1 𝜎+2 ) + 2𝑖𝐷𝑧(𝜎+

1 𝜎−2 − 𝜎−

1 𝜎+2 )

+ 𝐽𝑧𝜎𝑧1𝜎

𝑧2 + (𝐵 + 𝑏)𝜎𝑧

1 + (𝐵 − 𝑏)𝜎𝑧2 . (2)

In order to obtain the eigenvalues and eigenvectorsfrom Eq. (2), the usual standard basis states, |1, 1⟩,|1, 0⟩, |1,−1⟩, |0, 1⟩, |0, 0⟩, |0,−1⟩, | − 1, 1⟩, | − 1, 0⟩and | − 1,−1⟩ are used. Thus the Hamiltonian is cal-culated in the matrix form as

ℋ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

𝐽𝑧 + 2𝐵 0 0 0 0 0 0 0 00 𝐵 + 𝑏 0 𝛼 0 0 0 0 00 0 −𝐽𝑧 + 2𝑏 0 𝛼 0 0 0 00 𝛼* 0 𝐵 − 𝑏 0 0 0 0 00 0 𝛼* 0 0 0 𝛼 0 00 0 0 0 0 −𝐵 + 𝑏 0 𝛼 00 0 0 0 𝛼* 0 −𝐽𝑧 − 2𝑏 0 00 0 0 0 0 𝛼* 0 −𝐵 − 𝑏 00 0 0 0 0 0 0 0 𝐽𝑧 − 2𝐵

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (3)

**Email: albayrak@erciyes.edu.trc○ 2011 Chinese Physical Society and IOP Publishing Ltd

020306-1

CHIN. PHYS. LETT. Vol. 28,No. 2 (2011) 020306

where 𝛼 = 2(𝐽 + 𝑖𝐷𝑧) and 𝛼* are of the complex con-jugation. The eigenvalues of Eq. (3) can be found tobe

𝐸1 = 𝐽𝑧 + 2𝐵, 𝐸2 = 𝐵 + 𝛾, 𝐸3 = 𝐵 − 𝛾,

𝐸4 = −𝐵 + 𝛾, 𝐸5 = −𝐵 − 𝛾, 𝐸9 = 𝐽𝑧 − 2𝐵, (4)

where 𝛾 =√𝑏2 + 4𝐽2 + 4𝐷2

𝑧 . The other three eigen-values are found from 𝐸3

𝑖 + 𝑎2𝐸2𝑖 + 𝑎1𝐸𝑖 + 𝑎0 = 0

with 𝑎2 = 2𝐽𝑧, 𝑎1 = 𝐽2𝑧 − 4𝑏2 − 8𝐽2 − 8𝐷2

𝑧 and𝑎0 = −8𝐽2𝐽𝑧−8𝐷2

𝑧𝐽𝑧 by using the mathematical iden-tity

𝐸𝑛+6 = 2𝑦 cos[1

3

{arccos

(− 𝐺

3𝑦3

)+ 2𝑛𝜋

}]− 𝑎2

3,

𝑛 = 0, 1, 2. (5)

The eigenvectors corresponding to the above eigenval-ues are found to be

|Ψ1⟩ = |1, 1⟩, |Ψ2⟩ =𝑏+ 𝛾

𝜉|1, 0⟩ +

𝛼*

𝜉|0, 1⟩,

|Ψ3⟩ =𝛼

𝜉|1, 0⟩ − 𝑏+ 𝛾

𝜉|0, 1⟩,

|Ψ4⟩ =𝑏+ 𝛾

𝜉|0,−1⟩ +

𝛼*

𝜉| − 1, 0⟩,

|Ψ5⟩ =𝛼

𝜉|0,−1⟩ − 𝑏+ 𝛾

𝜉| − 1, 0⟩,

|Ψ6⟩ =𝛼𝜁1|1,−1⟩ + 𝜁1𝜁2|0, 0⟩ + 𝛼*𝜁2| − 1, 1⟩√

4(𝐽2 +𝐷2𝑧)(𝜁21 + 𝜁22 ) + 𝜁21𝜁

22

,

|Ψ7⟩ =𝛼𝜂1|1,−1⟩ + 𝜂1𝜂2|0, 0⟩ + 𝛼*𝜂2| − 1, 1⟩√

4(𝐽2 +𝐷2𝑧)(𝜂21 + 𝜂22) + 𝜂21𝜂

22

,

|Ψ8⟩ ={

4[(𝐽2 −𝐷2𝑧)2 + 4𝐽2𝐷2

𝑧 ]1/2|1,−1⟩− [4(𝐽2 +𝐷2

𝑧) − 𝐸8𝜆2]| − 1, 1⟩}

×{

16[(𝐽2 −𝐷2𝑧)2 + 4𝐽2𝐷2

𝑧 ]

+ [4(𝐽2 +𝐷2𝑧) − 𝐸8𝜆2]2

}−1/2,

|Ψ9⟩ = | − 1,−1⟩, (6)

where 𝜁1 = 𝐽𝑧 + 2𝑏 + 𝐸6, 𝜁2 = 𝐽𝑧 − 2𝑏 + 𝐸6,𝜂1 = 𝐽𝑧 +2𝑏+𝐸7, 𝜂2 = 𝐽𝑧−2𝑏+𝐸7, 𝜆2 = 𝐽𝑧−2𝑏+𝐸8

and 𝜉 =√

(𝑏+ 𝛾)2 + 4(𝐽2 +𝐷2𝑧).

The density operator at thermal equilibrium𝜌(𝑇 ) = 1

𝑍 exp(−𝛽ℋ), where the partition function isgiven as 𝑍 = 𝑇𝑟[exp(−𝛽ℋ)] and 𝛽 = 1/(𝑘𝑇 ), 𝑘 is theBoltzmann constant and it is set equal to one. Thedensity operator, 𝜌(𝑇 ) = 1

𝑍

∑9𝑖=1 exp(−𝛽𝐸𝑖)|𝜓𝑖⟩⟨𝜓𝑖|,

can be calculated in the standard basis as

𝜌(𝑇 ) =1

𝑍

{𝑒−𝛽(𝐽𝑧+2𝐵)|1, 1⟩⟨1, 1|+ 𝑒−𝛽(𝐽𝑧−2𝐵)| − 1,−1⟩⟨−1,−1|+ 𝑒−𝛽𝐵

𝜉2

[(𝑏+ 𝛾)2𝑒−𝛽𝛾 + 4(𝐽2 +𝐷2

𝑧)𝑒𝛽𝛾]|1, 0⟩⟨1, 0|

+𝑒−𝛽𝐵

𝜉2

[4(𝐽2 +𝐷2

𝑧)𝑒−𝛽𝛾 + (𝑏+ 𝛾)2𝑒𝛽𝛾

]|0, 1⟩⟨0, 1|+ 𝑒𝛽𝐵

𝜉2

[(𝑏+ 𝛾)2𝑒−𝛽𝛾 + 4(𝐽2 +𝐷2

𝑧)𝑒𝛽𝛾]|0,−1⟩⟨0,−1|

+𝑒𝛽𝐵

𝜉2

[4(𝐽2 +𝐷2

𝑧)𝑒−𝛽𝛾 + (𝑏+ 𝛾)2𝑒𝛽𝛾

]| − 1, 0⟩⟨−1, 0| − 4 sinh(𝛽𝛾)

(𝑏+ 𝛾)

𝜉2

[𝑒−𝛽𝐵{(𝐽 + 𝑖𝐷𝑧)|1, 0⟩⟨0, 1|

+ (𝐽 − 𝑖𝐷𝑧)|0, 1⟩⟨1, 0|}+ 𝑒𝛽𝐵{(𝐽 + 𝑖𝐷𝑧)|0,−1⟩⟨−1, 0|+ (𝐽 − 𝑖𝐷𝑧)| − 1, 0⟩⟨0,−1|}]

+ 𝑒−𝐵𝐸6𝜚6 + 𝑒−𝐵𝐸7𝜚7 + 𝑒−𝐵𝐸8𝜚8

}, (7)

where

𝑍 = 𝑒−𝛽(𝐽𝑧+2𝐵) + 𝑒−𝛽(𝐵+𝛾) + 𝑒−𝛽(𝐵−𝛾) + 𝑒−𝛽(−𝐵+𝛾) + 𝑒−𝛽(−𝐵−𝛾) + 𝑒−𝛽𝐸6 + 𝑒−𝛽𝐸7 + 𝑒−𝛽𝐸8 + 𝑒−𝛽(𝐽𝑧−2𝐵), (8)

𝜚6 = |𝑐3|2|1,−1⟩⟨1,−1| + 𝑐3𝑐*5|1,−1⟩⟨0, 0| + 𝑐3𝑐

*7|1,−1⟩⟨−1, 1| + 𝑐5𝑐

*3|0, 0⟩⟨1,−1| + |𝑐5|2|0, 0⟩⟨0, 0|

+ 𝑐5𝑐*7|0, 0⟩⟨−1, 1| + 𝑐7𝑐

*3| − 1, 1⟩⟨1,−1| + 𝑐7𝑐

*5| − 1, 1⟩⟨0, 0| + |𝑐7|2| − 1, 1⟩⟨−1, 1|. (9)

Also note that (𝑐3, 𝑐5, 𝑐7) is replaced with (𝑐′3, 𝑐′5, 𝑐

′7) and (𝑐′′3 , 𝑐

′′5 = 0, 𝑐′′7) for 𝜚7 and 𝜚8, respectively. Meanwhile,

the coefficients are

𝑐3 =𝛼𝜁1𝜅1

, 𝑐5 =𝜁1𝜁2𝜅1

, 𝑐7 =𝛼*𝜁2𝜅1

, 𝑐′3 =𝛼𝜂1𝜅2

, 𝑐′5 =𝜂1𝜂2𝜅2

, 𝑐′7 =𝛼*𝜂2𝜅2

𝑐′′3 ={

4[(𝐽2 −𝐷2𝑧)2 + 4𝐽2𝐷2

𝑧 ]1/2}/𝜅3, 𝑐′′7 = −

{4(𝐽2 +𝐷2

𝑧) − 𝐸8𝜆2}/𝜅3, (10)

where𝜅1 =

√4(𝐽2 +𝐷2

𝑧)(𝜁21 + 𝜁22 ) + 𝜁21𝜁22 , 𝜅2 =

√4(𝐽2 +𝐷2

𝑧)(𝜂21 + 𝜂22) + 𝜂21𝜂22

𝜅3 =√

16[(𝐽2 −𝐷2𝑧)2 + 4𝐽2𝐷2

𝑧 ] + [4(𝐽2 +𝐷2𝑧) − 𝐸8𝜆2]2.

The 𝑁 as a measure of entanglement[20] is defined as

𝑁 = [‖ 𝜌𝑇𝐴 ‖ −1]/2, (11)

which corresponds to the absolute value of the sum of the negative values of 𝜌𝑇𝐴 , where ‖𝜌𝑇𝐴‖ =√

Tr[(𝜌𝑇𝐴)†𝜌𝑇𝐴 ]denotes the trace norm of the partially transposed density matrix 𝜌𝑇𝐴 . Thus the partially transposed matrix𝜌𝑇𝐴 of the original density matrix 𝜌(𝑇 ) is found in the form as

020306-2

CHIN. PHYS. LETT. Vol. 28,No. 2 (2011) 020306

𝜌𝑇𝐴 =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

𝜌11 0 0 0 𝜌15 0 0 0 𝜌190 𝜌22 0 0 0 𝜌26 0 0 00 0 𝜌33 0 0 0 0 0 00 0 0 𝜌44 0 0 0 𝜌48 0𝜌51 0 0 0 𝜌55 0 0 0 𝜌590 𝜌62 0 0 0 𝜌66 0 0 00 0 0 0 0 0 𝜌77 0 00 0 0 𝜌84 0 0 0 𝜌88 0𝜌91 0 0 0 𝜌95 0 0 0 𝜌99

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

(12)

which is a hermitian matrix whose elements are cal-culated by

𝜌11 =1

𝑍𝑒−𝛽(𝐽𝑧+2𝐵),

𝜌15 = 𝜌*51 =1

𝑍

[− 4 sinh(𝛽𝛾)

(𝑏+ 𝛾)(𝐽 − 𝑖𝐷𝑧)

𝜉2𝑒−𝛽𝐵

],

𝜌19 = 𝜌*91 =1

𝑍

[𝑐7𝑐

*3𝑒

−𝛽𝐸6 + 𝑐′7(𝑐′3)*𝑒−𝛽𝐸7

+ 𝑐′′7(𝑐′′3)*𝑒−𝛽𝐸8],

𝜌22 =1

𝑍

[𝑒−𝛽𝐵

𝜉2

{(𝑏+ 𝛾)2𝑒−𝛽𝛾 + 4(𝐽2 +𝐷2

𝑧)𝑒𝛽𝛾}],

𝜌26 = 𝜌*62 =1

𝑍[𝑐5(𝑐3)*𝑒−𝛽𝐸6 + 𝑐′5(𝑐′3)*𝑒−𝛽𝐸7 ],

𝜌33 =1

𝑍[|𝑐3|2𝑒−𝛽𝐸6 + |𝑐′3|2𝑒−𝛽𝐸7 + |𝑐′′3 |2𝑒−𝛽𝐸8 ],

𝜌44 =1

𝑍

[𝑒−𝛽𝐵

𝜉2

{4(𝐽2 +𝐷2

𝑧)𝑒−𝛽𝛾 + (𝑏+ 𝛾)2𝑒𝛽𝛾}],

𝜌48 = 𝜌*84 =1

𝑍[𝑐7𝑐

*5𝑒

−𝛽𝐸6 + 𝑐′7(𝑐′5)*𝑒−𝛽𝐸7 ],

𝜌55 =1

𝑍[|𝑐5|2𝑒−𝛽𝐸6 + |𝑐′5|2𝑒−𝛽𝐸7 ],

𝜌59 = 𝜌*95 =1

𝑍

[− 4 sinh(𝛽𝛾)

(𝑏+ 𝛾)(𝐽 − 𝑖𝐷𝑧)

𝜉2𝑒𝛽𝐵

],

𝜌66 =1

𝑍

[𝑒𝛽𝐵𝜉2

{(𝑏+ 𝛾)2𝑒−𝛽𝛾 + 4(𝐽2 +𝐷2𝑧)𝑒𝛽𝛾}

],

𝜌77 =1

𝑍[|𝑐7|2𝑒−𝛽𝐸6 + |𝑐′7|2𝑒−𝛽𝐸7 + |𝑐′′7 |2𝑒−𝛽𝐸8 ],

𝜌88 =1

𝑍

[𝑒𝛽𝐵𝜉2

{4(𝐽2 +𝐷2

𝑧)𝑒−𝛽𝛾 + (𝑏+ 𝛾)2𝑒𝛽𝛾}],

𝜌99 =1

𝑍𝑒−𝛽(𝐽𝑧−2𝐵). (13)

After having obtained the partially transposed matrix𝜌𝑇𝐴 , which is also hermitian as the original matrix𝜌(𝑇 ), the 𝑁 can be calculated from Eq. (11), thus itis obtained to be

𝑁 = [√

|𝜌11|2 + |𝜌15|2 + |𝜌19|2 +√|𝜌22|2 + |𝜌26|2

+√|𝜌33|2 +

√|𝜌44|2 + |𝜌48|2

+√

|𝜌51|2 + |𝜌55|2 + |𝜌59|2 +√|𝜌62|2 + |𝜌66|2

+√

|𝜌77|2 +√|𝜌84|2 + |𝜌88|2

+√

|𝜌91|2 + |𝜌95|2 + |𝜌99|2 − 1]/2. (14)

Note that 𝐽 and 𝐷𝑧 come out to be even functionsof 𝑁 . Thus, as long as 𝑁 concerns, the FM case or theAFM case give the same results. The case is also thesame for 𝐷𝑧. 𝑁 is smaller for smaller values of |𝐷𝑧|and vice versa. The sign of 𝐽𝑧, i.e. FM or AFM, doeshave very important effects on 𝑁 . It is well knownthat 𝑁 declines with the increasing values of 𝐵 and𝑏, which are in favor of FM. Our last and maybe themost important parameter is the temperature, i.e. 𝑁decreases when temperature increases.

NN

N

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 42040

60

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 42040

60

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 42040

60

T Dz DzT

T Dz DzT

T Dz DzT

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 42040

60

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2420

4060

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 02 420

4060

(a) (b)

(c) (d)

(e) (f)

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

Fig. 1. Negativity 𝑁 as functions of 𝐷𝑧 and 𝑇 for givenvalues of 𝐽𝑧 = 4.0, 1.0, 0.0, −1.0, −2.0, −6.0 when𝐽 = 1.0, 𝐵 = 1.0 and 𝑏 = 0.0.

Our figures are given in three-dimensional spacespanned with the coordinates (𝑇,𝐷𝑧, 𝑁) by keepingall other parameters unchanged except one of themthat is varied appropriately. Figure 1 illustrates thechanges of𝑁 when 𝐽𝑧 = 4.0, 1.0, 0.0, −1.0, −2.0, −6.0for 𝐽 = 1.0, 𝐵 = 1.0 and 𝑏 = 0.0. When 𝐽𝑧 = 4.0,𝑁 is almost equal to the one about zero temperature,except 𝐷𝑧 is around zero where 𝑁 drops slightly. It isclear that 𝑁 persists to higher temperatures for highervalues of |𝐷𝑧|, see the wings when one looks directlyto the 𝐷𝑧-axes. For 𝐽𝑧 = 1.0, 𝑁 shows very smallchanges. However, when 𝐽𝑧 is dropped to zero, 𝑁falls to about 0.5 for 𝐷𝑧 in the interval −0.6–0.6 at𝑇 = 0. The symmetrical two parts still have 𝑁 ∼ 1but 𝑁 is less than the previous two figures. It dropsfurther to about 0.2 for 𝐽𝑧 = −1.0 around 𝐷𝑧 = 0.0and 𝑇 = 0.0. This part is embedded between the twosymmetrical parts with 𝑁 of about 0.5, and in turnthey are embedded between the parts about 𝑁 = 0.95.Next figure for 𝐽𝑧 = −2.0 is similar to the previousfigure, except that 𝑁 is zero now for values of 𝐷𝑧 be-tween ±1.0. Finally for 𝐽𝑧 = −6.0, the symmetrical

020306-3

CHIN. PHYS. LETT. Vol. 28,No. 2 (2011) 020306

part around 1 vanishes, only the symmetrical partsabout 0.5 remains, and the number of zeros for 𝑁 in-creases, i.e. they span in the region with 𝐷𝑧 = ±3.2.It is clear that as 𝐽𝑧 becomes more negative, the 𝑁 ’sstart from zero at 𝑇 = 0 for increasing values of |𝐷𝑧|.This may be interpreted as follows: 𝐷𝑧 supports AFMand negative values of 𝐽𝑧 support FM, thus when 𝐷𝑧

cannot compete with 𝐽𝑧, 𝑁 falls to zero at 𝑇 = 0. 𝑁drops to lower temperatures for lower 𝐽𝑧 values andthe figures are symmetrical with respect to 𝐷𝑧 = 0.0lines as mentioned above.

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 420 4060

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 420 4060

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2420 40

60

0.0

0.2

0.4

0.6

0.8

1.0

-4 -20 2 420 40

60

(a) (b)

(c) (d)

N

T DzT Dz

T Dz

N

T Dz

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

Fig. 2. Negativity 𝑁 as functions of 𝐷𝑧 and 𝑇 for givenvalues of 𝐵 = 0.0, 1.5, 3.0, 4.0 when 𝐽 = 1.0, 𝐽𝑧 = 1.0and 𝑏 = 1.0.

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 420 4060

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 420 4060

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 420 40 60

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 420 4060

(a) (b)

(c) (d)

NN

T Dz T Dz

T DzT Dz

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

J/

⊲

J

z/

⊲

B/

⊲

b/

⊲

Fig. 3. Negativity 𝑁 as functions of 𝐷𝑧 and 𝑇 for givenvalues of 𝑏 = 0.5, 4.0, 10.0, 30.0 when 𝐽 = 1.0, 𝐽𝑧 = 1.0and 𝐵 = 1.0.

In Fig. 2, the effect of varying 𝐵 on 𝑁 is illustratedfor 𝐽 = 𝐽𝑧 = 𝑏 = 1.0. As seen, the tips of 𝑁 lines at𝑇 = 0 make nice wings, their center has the lowest𝑁 at 𝐷𝑧 = 0.0 for 𝐵 = 0.0. Figure 2(b) shows that𝑁 drops down to lower values and lower temperatureswhen 𝐵 is set equal to 1.5. When 𝐵 = 3.0, one portionof 𝑁 about 0.5 and two symmetrical portions about1.0 at 𝑇 = 0 is seen. With the further decrease of 𝐵,i.e. to 4.0, reduces 𝑁 to zero for 𝐷𝑧 values between±0.8 and two symmetrical parts about 0.5. Thus theincreasing values of 𝐵, decreases the values and tem-peratures of 𝑁 . 𝐵 supports FM as negative 𝐽𝑧, thus𝑁 drops down to lower values for lower |𝐷𝑧|.

Figure 3 illustrates the dependence of 𝑁 on 𝑏. Asseen, the 𝑁 lines preserve their shapes for all the val-ues of 𝑏, but their values and temperatures decreasewith the increasing values of 𝑏. The change of 𝑁 oc-curs continuously here in comparison with the previ-ous figures. The inhomogeneity and 𝐷𝑧 are in com-petition similarly to 𝐽𝑧 or 𝐵, whereas the values of 𝑏must be much higher to win its fight against 𝐷𝑧 todecrease the values of 𝑁 .

In conclusion, the 𝑁 as a measure of TE has beenstudied for a two-qutrit spin-1 anisotropic Heisenberg𝑋𝑋𝑍 chain with DM interaction in an inhomogeneousmagnetic field in detail. The effects of the DM inter-action parameter 𝐷𝑧 on the thermal variation of the𝑁 for given values of the external magnetic field 𝐵,a parameter 𝑏 which controls the inhomogeneity of 𝐵and the bilinear interaction parameters 𝐽𝑥 = 𝐽𝑦 = 𝐽𝑧are obtained. It is found that 𝑁 persists to higher val-ues and to higher temperatures for the higher valuesof ±𝐷𝑧 and for the higher positive values of 𝐽𝑧, i.e.in the AFM case. When 𝐽𝑧 < 0, the FM case, and𝐷𝑧 is small, and if 𝐽𝑧 is strong enough to computewith 𝐷𝑧, 𝑁 decreases. In addition, 𝑁 declines withthe increasing values of 𝐵 and 𝑏.

References[1] Yeo Y 2002 Phys. Rev. A 66 062312[2] Wang X 2001 Phys. Rev. A 64 012313

Kamta G L and Starace A F 2002 Phys. Rev. Lett. 88107901Kamta G L, Istomin A Y and Starace A F 2007 Eur. Phys.J. D 44 389Sun Y, Chen Y and Chen H 2003 Phys. Rev. A 68 044301Zhang R and Zhu S 2006 Phys. Lett. A 348 110

[3] Zhang Y et al 2007 Commun. Theor. Phys. 47 787Asoudeh M and Karimipour V 2005 Phys. Rev. A 71022308

[4] Zhao X Y and Zhou L 2007 Int. J. Theor. Phys. 46 2437Wang X, Fu H and Solomon A I 2001 J. Phys. A: Math.Gen. 34 11307Santos L F 2003 Phys. Rev. A 67 062306Gu S J, Lin H Q and Li Y Q 2003 Phys. Rev. A 68 042330Zhang G F and Li S S 2005 Phys. Rev. A 72 034302

[5] Zhou L et al 2003 Phys. Rev. A 68 024301[6] Yan Z, Zhu S Q and Xiang H 2007 Chin. Phys. 16 2229[7] Zhang G F et al 2005 Opt. Commun. 245 457[8] Zhang G F et al 2005 Eur. Phys. J. D 32 409[9] Wang X, Li H B, Sun Z and Li Y Q 2005 J. Phys. A: Math.

Gen. 38 8703[10] Wang X and Gu S J 2007 J. Phys. A: Math. Theor. 40

10759[11] Wang X and Wang Z D 2006 Phys. Rev. A 73 064302[12] Huang H, Wang X, Sun Z and Yang G 2008 Physica A 387

2736[13] Zhu G Q and Wang X G 2008 Commun. Theor. Phys. 49

343[14] Yang Z and Ning W Q 2008 Chin. Phys. Lett. 25 31[15] He M M, Xu C T and Liang J Q 2006 Phys. Lett. A 358

381[16] Ha K C, Kye S H and Park Y S 2003 Phys. Lett. A 313

163[17] Zhang G F 2007 Phys. Rev. A 75 034304

Li D C et al 2008 J. Phys.: Condens. Matter. 20 325229Wang X 2001 Phys. Lett. A 281 101

[18] Albayrak E 2009 Eur. Phys. B 72 491[19] Li D C and Cao Z L 2009 Chin. Phys. Lett. 26 020309[20] Vidal G and Werner R F 2002 Phys. Rev. A 65 032314

020306-4