Effects of Different Spin-Spin Couplings and Magnetic Fields on Thermal Entanglement in Heisenberg XY Z Chain

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1 Commun. heor. Phys. (Beijing China 53 (00 pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53 No. 4 April 5 00 Effects of Different Spin-Spin Couplings and Magnetic Fields on hermal Entanglement in Heisenberg XY Z Chain LI Da-Chuang (Ó Å and CAO Zhuo-Liang ( Ê Department of Physics and Electronic Engineering Hefei Normal University Hefei 3006 China School of Physics & Material Science Anhui University Hefei China (Received February 6 009; revised manuscript received June Abstract he effects of spin-spin interaction on thermal entanglement of a two-qubit Heisenberg XYZ model with different inhomogeneous magnetic fields are investigated. It is shown that the entanglement is dependent on the spin-spin interaction and the inhomogeneous magnetic fields. he larger the J i (i-axis spin-spin interaction the higher critical value the B i (i-axis uniform magnetic field has. Moreover in the weak-field regime the larger J i corresponds to more entanglement while in the strong-field regime different J i correspond to the same entanglement. In addition it is found that with the increase of J i the concurrence can approach the maximum value more rapidly for the smaller B i and can reach a larger value for the smaller b i (i-axis nonuniform magnetic field. So we can get more entanglement by increasing the spin-spin interaction J i or by decreasing the uniform magnetic field B i and the nonuniform magnetic field b i. PACS numbers: Mn 75.0.Jm Lx Key words: Heisenberg XY Z model inhomogeneous magnetic field spin-spin coupling thermal entanglement Introduction Entanglement is the most fascinating nonclassical feature of quantum mechanics and plays a central role in quantum information processing. [ 3] Heisenberg model as a simple model is the ideal candidate for the generation and the manipulation of entangled states and it can be used for quantum computation by suitable coding. [4 6] In solid-state physics the one-dimensional Heisenberg models have been extensively investigated such as nuclear spins [7] quantum dots [8 0] superconductor [ ] and optical lattices. [3] Recently the concept of the thermal entanglement in solid was introduced and investigated in Heisenberg model. [4 3] Wang studied the thermal entanglement in the Heidenberg XY model. [3] Under the circumstance of an inhomogeneous magnetic field Kheirandish investigated the effect of spin-orbit interaction on entanglement of two-qubit Heisenberg model. [33] Very recently we investigated the thermal entanglement in Heisenberg models with Dzyaloshinskii-Moriya interaction. [34 36] In those studies the influences of spin-orbit interaction or z-axis external magnetic field on thermal entanglement have been discussed but the effect of spin-spin coupling on thermal entanglement under a certain magnetic field has not been considered. In this paper we consider the influences of different spin-spin interactions and inhomogeneous magnetic fields on thermal entanglement in the generalized Heisenberg XY Z model. Especially we discuss the effects of J i on the entanglement and the critical value of magnetic field in the presence of B i and b i respectively. In order to provide a detailed analytical and numerical analysis here we take concurrence as a measure of entanglement. [37 38] he concurrence C ranges from 0 to C = 0 and C = indicate the vanishing entanglement and the maximal entanglement respectively. For a mixed state ρ the concurrence of the state is C(ρ = max{λ max 4 i= λ i 0} λ i s are the positive square roots of the eigenvalues of the matrix R = ρ(σ y σ y ρ (σ y σ y and the asterisk denotes the complex conjugate. his paper is organized as follows. In Sec. we introduce the Heisenberg XY Z models with different magnetic fields and give the expressions of the concurrence. In Sec. 3 we analyze the influences of spin-spin interaction on thermal entanglement under different inhomogeneous magnetic fields. Finally in Sec. 4 a discussion concludes the paper. Model and Hamiltonian. Heisenberg XY Z Model with x-axis Inhomogeneous Magnetic Field he Hamiltonian H for a two-qubit anisotropic Heisenberg XY Z model with x-axis inhomogeneous magnetic field is H = J x σ x σ x + J y σ y σy + J zσ z σ z + (B x + b x σ x + (B x b x σ x ( σ i (i = x y z are the Pauli matrices J x J y and J z are the spin-spin coupling coefficients B x and b x are the Supported by National Natural Science Foundation of China under Grant No Anhui Provincial Natural Science Foundation under Grant No the Major Program of the Education Department of Anhui Province under Grant No. KJ00ZD08 and the Key Program of the Education Department of Anhui Province under Grant No. KJ00A87 dachuang@ahu.edu.cn zhuoliangcao@gmail.com (Corresponding Author

2 660 LI Da-Chuang and CAO Zhuo-Liang Vol. 53 x-axis uniform and nonuniform external magnetic fields respectively. he coupling constant J i > 0 (i = x y z corresponds to the antiferromagnetic case and J i < 0 corresponds to the ferromagnetic case. We only consider the J i > 0 case for simplicity. Parameters J i B x and b x are dimensionless. In the standard basis { } the above Hamiltonian can be rewritten as J z B x b x B x + b x J x J y B x b x J z J x + J y B x + b x H = B x + b x J x + J y J z B x b x. ( J x J y B x + b x B x b x J z By a straightforward calculation we can get the eigenstates of H: Φ = (sin θ 00 + cosθ 0 + cosθ 0 + sin θ Φ = (sin θ 00 + cosθ 0 + cosθ 0 + sin θ Φ 3 = (sin θ 3 00 cosθ cosθ 3 0 sin θ 3 Φ 4 = (sin θ cosθ 4 0 with corresponding eigenvalues: cosθ 4 0 sin θ 4 (3 E = J x ± w E 34 = J x ± w (4 ( B x θ = arctan J y J z ± w ( b x θ 34 = arctan w J y J z w = 4Bx + (J y J z w = 4b x + (J y + J z. he system state at thermal equilibrium (thermal state is ρ( = exp( H/K B Z Z = r[exp( H/K B ] is the partition function of the system H is the system Hamiltonian is the temperature and K B is the Boltzmann constant which we take equal to for simplicity. hus in the above standard basis we can obtain the following analytical expression of the density matrix ρ(: m q q m q n n q ρ( = q n n q m q q m (5 m = ( e E / sin θ + e E/ sin θ Z ± e E3/ sin θ 3 ± e E4/ sin θ 4 n = ( e E / cos θ + e E/ cos θ Z ± e E3/ cos θ 3 ± e E4/ cos θ 4 q = ( e E / sin θ cosθ + e E/ sinθ cosθ Z e E3/ sinθ 3 cosθ 3 ± e E4/ sin θ 4 cosθ 4. After straightforward calculations the positive square roots of the eigenvalues of the matrix R = ρ(σ y σ y ρ (σ y σ y can be expressed as: λ = ejx/ Zw w cosh 4b x sinh ± (J y + J z sinh (6a w cosh 4Bx sinh ± (J y J z sinh (6b λ 34 = e Jx/ Zw Z = e Jx/ cosh + e Jx/ cosh hus the concurrence of this system can be written as: [37 38]. C ( ρ( = { max{ λ λ 3 λ λ 4 0} if J y > J z max{ λ λ 4 λ λ 3 0} if J y J z. (7. Heisenberg XY Z Model with y-axis Inhomogeneous Magnetic Field Here we consider the case of a two-qubit anisotropic Heisenberg XY Z chain with y-axis inhomogeneous magnetic field. he Hamiltonian is H = J x σ x σ x + J y σ y σy + J zσ z σ z + (B y + b y σ y + (B y b y σ y (8 similarly B y and b y are the y-axis uniform and nonuniform magnetic fields respectively and we assume J i > 0 in this paper. Using the same process as the above subsection by calculating we can get the eigenstates and eigenvalues of H then the density matrix ρ ( (we do not list them for simplicity and then the positive square roots of the

3 No. 4 Effects of Different Spin-Spin Couplings and Magnetic Fields on hermal Entanglement in Heisenberg XY Z Chain 66 eigenvalues of the matrix R = ρ (σ y σ y ρ (σ y σ y : λ = ejy/ Z w w cosh 4b y sinh ± (J x + J z sinh (9a λ 34 = e Jy/ Z w w cosh 4By sinh ± (J x J z sinh (9b w = 4By + (J x J z w 4b = y + (J x + J z Z = e Jy/ ( cosh w + e Jy/ cosh. hus the concurrence of this system can be expressed as: C ( ρ ( = { max{ λ λ 3 λ λ 4 0} if J x > J z max{ λ λ 4 λ λ 3 0} if J x J z. (0.3 Heisenberg XY Z Model with z-axis Inhomogeneous Magnetic Field he Hamiltonian for a two-qubit anisotropic Heisenberg XY Z chain with z-axis inhomogeneous magnetic field is H = J x σ x σ x + J y σ y σy + J zσ z σ z + (B z + b z σ z + (B z b z σ z ( J i > 0 B z and b z are the z-axis uniform and nonuniform magnetic fields respectively. Similarly by straightforward calculations we can also get the eigenstates and eigenvalues of H and the density matrix ρ ( (we do not list them for simplicity then we can obtain the positive square roots of the eigenvalues of the matrix R = ρ (σ y σ y ρ (σ y σ y : λ = ejz/ Z w λ 34 = e Jz/ Z w w cosh w cosh 4b z sinh 4B z sinh ± (J x + J y sinh ± (J x J y sinh w = 4B z + (J x J y w = 4b z + (J x + J y Z = e Jz/ cosh hus the concurrence of this system can be expressed as: C ( ρ ( { max{ λ λ 3 λ λ 4 0} = if J x > J y max{ λ λ 4 λ λ 3 0} if J x J y. ( + e Jz/ cosh. (3 3 Effects of Spin-Spin Interaction on Entanglement Under Different Inhomogeneous Magnetic Fields 3. Case for Uniform External Magnetic Fields In the antiferromagnetic Heisenberg XY Z model for x-axis parameters we have plotted Fig. to demonstrate the properties of parameters J x and B x. In Fig. it is easy to find that with the increase of J x or the decrease of B x the entanglement increases as a whole. o show the characteristics of parameters in detail Fig. has been plotted. In Fig. (a with the increase of B x the concurrence C decreases then drops suddenly at the critical value of B x (the Quantum Phase ransition (QP occurs at this critical point [39] and then the concurrence undergoes a revival before decreasing to zero. In addition corresponding to the larger J x there is a higher critical value of B x. For small B x the larger J x corresponds to more entanglement while for large B x (after the QP there is the same value of entanglement for different J x. In Fig. (b with the increase of J x the concurrence increases directly to the maximum value for small B x but undergoes a revival before increasing to the maximum value for large B x. Furthermore the smaller the B x the more rapidly the concurrence approaches the maximum value with increasing J x. Fig. hermal concurrence is plotted versus B x and J x J y = 0.5 J z = b x = 0 and = 0.5.

4 66 LI Da-Chuang and CAO Zhuo-Liang Vol. 53 For y-axis and z-axis parameters Figs. 3 and 4 have respectively been plotted to show the properties of the parameters. In Figs. 3(a and 4(a one can also see the effects of spin-spin coupling and uniform magnetic field on entanglement i.e. as a whole the entanglement increases with increasing spin-spin coupling or decreasing uniform magnetic field. Similarly to Fig. (a in Figs. 3(b and 4(b with the increase of B yz the concurrence C also decreases then drops suddenly at the critical value of B yz and then undergoes a revival before decreasing to zero. Additionally the larger the spin-spin coupling J yz the higher critical value the uniform magnetic field B yz has. Furthermore there is more entanglement for the larger J yz in the presence of small B yz but the same entanglement for different J yz in the presence of large B yz. Fig. (a he concurrence is plotted versus B x for J x = 0. (solid line J x = 0.8 (dashed line and J x =.5 (dashdotted line. (b he concurrence is plotted as a function of J x for B x = 0. (solid line B x =.8 (dashed line and B x = 3 (dash-dotted line. Here J y = 0.5 J z = b x = 0 and = 0.5. Fig. 3 (a hermal concurrence versus B y and J y. (b hermal concurrence versus B y for different values of J y J x = 0. J z = 0.5 b y = 0 and = 0.3. Fig. 4 (a hermal concurrence versus B z and J z. (b hermal concurrence versus B z for different values of J z J x = 0.8 J y = 0. b z = 0 and = 0.5.

5 No. 4 Effects of Different Spin-Spin Couplings and Magnetic Fields on hermal Entanglement in Heisenberg XY Z Chain Case for Nonuniform External Magnetic Fields In the antiferromagnetic Heisenberg XY Z model with nonuniform magnetic fields we have plotted Figs. 5 and 6 to demonstrate the properties of x-axis spin-spin interaction J x and nonuniform magnetic field b x. In Fig. 6(a it is easy to see that for large J x the concurrence decreases with the increase of b x. In addition when b x is small the larger J x corresponds to more entanglement while when b x is large there is the same entanglement for different J x. In Fig. 6(b one can see that with the increase of J x the concurrence can reach a larger value for the smaller b x and then keep invariant however the concurrence will keep a constant all along when b x is large to a certain degree. hese phenomena can also be seen in Fig. 5. Fig. 5 hermal concurrence is plotted versus b x and J x J y = 0.5 J z =.8 B x = 0 and =. Fig. 6 (a he concurrence is plotted versus b x for J x = 0. (solid line J x = 0.8 (dashed line and J x = 3 (dash-dotted line. (b he concurrence is plotted as a function of J x for b x = 0. (solid line b x = (dashed line and b x = 3.5 (dash-dotted line. Here J y = 0.5 J z =.8 B x = 0 and =. Fig. 7 (a hermal concurrence versus b y and J y. (b hermal concurrence versus b y for different values of J y J x = 0.3 J z = 0.8 B y = 0 and =. Similarly Figs. 7 and 8 have also been plotted to analyze the properties of y-axis and z-axis parameters respectively. In the two figures we can find that with the increase of b yz the concurrence first increases and then decreases (i.e. the concurrence has a maximum for small J yz but decreases monotonously for large J yz. In addition there is more entanglement for larger J yz in the presence of small b yz but the same entanglement for different J yz in the presence of large b yz.

6 664 LI Da-Chuang and CAO Zhuo-Liang Vol. 53 Fig. 8 (a hermal concurrence versus b z and J z. (b hermal concurrence versus b z for different values of J z J x = J y = 0.3 B z = 0 and =. 4 Conclusions In summary we have investigated the effects of spin-spin interaction on thermal entanglement of a two-qubit Heisenberg XY Z model with different inhomogeneous magnetic fields. We have found that the larger the spin-spin interaction J i the higher critical value the B i has. Furthermore for small B i or small b i the larger J i corresponds to more entanglement while for large B i or large b i there is the same entanglement in the presence of different J i. In addition we have shown that with the increase of J i the concurrence can approach the maximum value more rapidly for the smaller B i and can reach a larger value for the smaller b i. hus in the weak-field regime increasing the spin-spin interaction J i can increase the entanglement and the critical value of magnetic field. Moreover the thermal entanglement can also be enhanced by decreasing the inhomogeneous magnetic fields. References [] Phys. World ( special issue on quantum information. [] C.H. Bennett and S.J. Wiesner Phys. Rev. Lett. 69 ( [3] A.K. Ekert Phys. Rev. Lett. 67 ( [4] D.A. Lidar D. Bacon and K.B. Whaley Phys. Rev. Lett. 8 ( [5] D.P. DiVincenzo et al. Nature (London 408 ( [6] L.F. Santos Phys. Rev. A 67 ( [7] B.E. Kane Nature (London 393 ( [8] D. Loss and D.P. DiVincenzo Phys. Rev. A 57 ( [9] G. Burkard D. Loss and D.P. DiVincenzo Phys. Rev. B 59 ( [0] B. rauzettel et al. Nature Phys. 3 ( []. Senthil et al. Phys. Rev. B 60 ( [] M. Nishiyama et al. Phys. Rev. Lett. 98 ( [3] A. Sørensen and K. Mølmer Phys. Rev. Lett. 83 ( [4] C. Akyüz E. Aydiner and Ö.E. Müstecaplioǧlu Optics Communications 8 ( [5] M.C. Arnesen S. Bose and V. Vedral Phys. Rev. Lett. 87 ( [6] Y. Yeo. Liu Y.E. Lu and Q.Z. Yang J. Phys. A: Math. Gen. 38 ( [7] D. Gunlycke et al. Phys. Rev. A 64 ( [8] G.L. Kamta and Anthony F. Starace Phys. Rev. Lett. 88 ( [9] Y. Sun et al. Phys. Rev. A 68 ( [0] M. Asoudeh and V. Karimipour Phys. Rev. A 7 ( [] G.F. Zhang Phys. Rev. A 75 ( [] G.F. Zhang and S.S. Li Phys. Rev. A 7 ( [3] L. Zhou et al. Phys. Rev. A 68 ( [4] Z.N. Gurkan and O.K. Pashaev e-print arxiv:quantph/ and arxiv:quant-ph/ [5] X. Wang Phys. Rev. A 66 ( [6] X. Wang H. Fu and A.I. Solomon J. Phys. A: Math. Gen. 34 ( [7] M. Cao and S. Zhu Phys. Rev. A 7 ( [8] X. Hao and S. Zhu Phys. Rev. A 7 ( [9] M. Cao and S. Zhu Chin. Phys. Lett. 3 ( [30] Y. Zhu S. Zhu and X. Hao Chin. Phys. 6 ( [3] R. Zhang and S. Zhu Phys. Lett. A 348 ( [3] X. Wang Phys. Rev. A 64 ( [33] F. Kheirandish S.J. Akhtarshenas and H. Mohammadi Phys. Rev. A 77 ( [34] D.C. Li X.P. Wang and Z.L. Cao J. Phys.: Condens. Matter 0 ( [35] D.C. Li and Z.L. Cao Eur. Phys. J. D 50 ( [36] D.C. Li and Z.L. Cao International Journal of Quantum Information 7 ( [37] S. Hill and W.K. Wootters Phys. Rev. Lett. 78 ( [38] W.K. Wootters Phys. Rev. Lett. 80 ( [39] G.H. Yang W.B. Gao L. Zhou and H.S. Song arxiv:quant-ph/06005.