钻石型等级晶格上量子自旋系统热纠缠的研究
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摘要
量子纠缠具有奇特的相干性与非定域性，既是量子力学中的重要概念之一，也成为实现量子信息处理的关键。固态自旋系统展现出丰富的纠缠特性，引起了量子信息学与凝聚态物理领域的共同关注。由于精确求解的困难，对于自旋系统上热纠缠问题的研究大都集中在小尺度一维体系上。本文利用统计物理中的实空间重整化群方法，研究了分形晶格上多体（大尺度）自旋系统的热纠缠性质。
研究了有限温度情况下分形维数d f=2的钻石型等级晶格上量子Heisenberg自旋系统的纠缠特性。采用格点消约重整化群方法并结合纠缠负值度（negativity）的定义计算了晶格上相距较远的外端点间的热纠缠。结果表明温度T、各向异性参数Δ和系统尺度L对端点热纠缠具有重要影响。各向异性参数确定后，在温度为零或极低时端点纠缠最大；随着温度的升高纠缠单调减小，而且存在临界温度TC，温度高于此值时纠缠消失；随着各向异性参数的减小，纠缠最大值降低而临界温度上升。温度一定时，端点纠缠在各向异性参数大于等于零时不存在；随着各向异性参数在小于零的范围内增大时纠缠先增大，在Δ=0附近纠缠迅速减小为零。当系统尺度增大（自旋数目增多）时，纠缠的最大值以及系统临界温度均降低，纠缠减小的越缓慢。当系统尺度很大时纠缠依然存在，这表明系统的热纠缠具有一定稳定性。
研究了特殊的多分支分形维数d f=2.32以及d f=2.58的钻石型等级晶格上量子Heisenberg自旋系统的热纠缠。计算了等级晶格上两外端点自旋间的热纠缠。研究发现纠缠表现出一定的共性：当各向异性参数确定时，端点纠缠随温度升高单调减小，温度高于临界值时端点之间不存在纠缠。当温度一定时，随着各向异性参数增大时端点纠缠先增大后迅速降为零。研究还发现了这两种晶格与d f=2的晶格的端点纠缠的不同之处，随系统尺度增大时下降更缓慢，而且当各向异性参数取一定值时，随系统尺度的增大，纠缠出现“交叉”现象，系统的临界温度TC没有降低反而升高并趋于定值，这些结果表明，对于温度的引起的退相干，纠缠表现出更强的鲁棒性。系统的分形结构不同可以影响系统的能级结构，从而导致热纠缠的特性不同。
Quantum entanglement, which shows amazing coherence and nonlocality, is not only one ofthe important basic concept in quantum mechanics but also the key resource to realize quantuminformation process. Solid spin system, which displays rich entanglement properties, hasreceived much attention both in quantum information and condensed matter physics recently.Due to the mathematical difficulty in exact solution, previous studies of the thermalentanglement on spin systems have mostly focused on the small size one-dimensional systems.In this thesis we study the thermal entanglement of many-body (large size) spin system on thediamond-type hierarchical lattices by using renormalization group (RG) methods.
The thermal entanglement of quantum Heisenberg spin system on the diamond-typehierarchical lattice with fractal dimension d f=2is studied. By using the decimation RGmethod and the negativity, we calculate the entanglement of two distant terminal spins on thelattices, and it is found that temperature T the anisotropic parameter Δ and the size of systemL have important effect on the entanglement. When Δ is fixed and the temperatureapproaching zero or very low, the entanglement of terminal spins is the larges and with thetemperature increasing, the entanglement decreases monotonically. There exists the criticaltemperatureTC above which the entanglement vanishes. With the decrease of Δ, the maximumvalue of entanglement decrease andTC increases. At certain temperature, there exist noentanglement when the anisotropic parameter Δ≥0. In the range of Δ <0, the entanglementincreases with Δ increasing and sharply decreases to zero in the vicinity of Δ=0. The maximalvalue of thermal entanglement and the critical temperatureTC decrease as the size of the systembecomes large. The entanglement shows stability and exists when L is very large.
By using the same method, we calculate the thermal entanglement of quantum Heisenbergspin system on the diamond-type hierarchical lattice with fractal dimension d f=2.32andd f=2.58. It is found that some similar properties of the entanglement on these two lattices andthe above lattice. The entanglement decreases with T increasing when Δ is certain and itincreases firstly and then decreases with Δ increasing at certain temperature. We also find thatthe entanglement of these two lattices compared with that of d f=2lattice exhibit differentfeatures that there occurs “entanglement crossing”. With the size of the system becoming large,the entanglement decrease much more slowly and the critical temperatureTC doesn’t decreasesbut increases and tends to the stable value. These results indicate that the entanglement is muchmore robust against the decoherence caused by temperature. The different fractal can affect different the energy level structure of system, which can lead to the different entanglementproperties.
研究了有限温度情况下分形维数d f=2的钻石型等级晶格上量子Heisenberg自旋系统的纠缠特性。采用格点消约重整化群方法并结合纠缠负值度（negativity）的定义计算了晶格上相距较远的外端点间的热纠缠。结果表明温度T、各向异性参数Δ和系统尺度L对端点热纠缠具有重要影响。各向异性参数确定后，在温度为零或极低时端点纠缠最大；随着温度的升高纠缠单调减小，而且存在临界温度TC，温度高于此值时纠缠消失；随着各向异性参数的减小，纠缠最大值降低而临界温度上升。温度一定时，端点纠缠在各向异性参数大于等于零时不存在；随着各向异性参数在小于零的范围内增大时纠缠先增大，在Δ=0附近纠缠迅速减小为零。当系统尺度增大（自旋数目增多）时，纠缠的最大值以及系统临界温度均降低，纠缠减小的越缓慢。当系统尺度很大时纠缠依然存在，这表明系统的热纠缠具有一定稳定性。
研究了特殊的多分支分形维数d f=2.32以及d f=2.58的钻石型等级晶格上量子Heisenberg自旋系统的热纠缠。计算了等级晶格上两外端点自旋间的热纠缠。研究发现纠缠表现出一定的共性：当各向异性参数确定时，端点纠缠随温度升高单调减小，温度高于临界值时端点之间不存在纠缠。当温度一定时，随着各向异性参数增大时端点纠缠先增大后迅速降为零。研究还发现了这两种晶格与d f=2的晶格的端点纠缠的不同之处，随系统尺度增大时下降更缓慢，而且当各向异性参数取一定值时，随系统尺度的增大，纠缠出现“交叉”现象，系统的临界温度TC没有降低反而升高并趋于定值，这些结果表明，对于温度的引起的退相干，纠缠表现出更强的鲁棒性。系统的分形结构不同可以影响系统的能级结构，从而导致热纠缠的特性不同。
Quantum entanglement, which shows amazing coherence and nonlocality, is not only one ofthe important basic concept in quantum mechanics but also the key resource to realize quantuminformation process. Solid spin system, which displays rich entanglement properties, hasreceived much attention both in quantum information and condensed matter physics recently.Due to the mathematical difficulty in exact solution, previous studies of the thermalentanglement on spin systems have mostly focused on the small size one-dimensional systems.In this thesis we study the thermal entanglement of many-body (large size) spin system on thediamond-type hierarchical lattices by using renormalization group (RG) methods.
The thermal entanglement of quantum Heisenberg spin system on the diamond-typehierarchical lattice with fractal dimension d f=2is studied. By using the decimation RGmethod and the negativity, we calculate the entanglement of two distant terminal spins on thelattices, and it is found that temperature T the anisotropic parameter Δ and the size of systemL have important effect on the entanglement. When Δ is fixed and the temperatureapproaching zero or very low, the entanglement of terminal spins is the larges and with thetemperature increasing, the entanglement decreases monotonically. There exists the criticaltemperatureTC above which the entanglement vanishes. With the decrease of Δ, the maximumvalue of entanglement decrease andTC increases. At certain temperature, there exist noentanglement when the anisotropic parameter Δ≥0. In the range of Δ <0, the entanglementincreases with Δ increasing and sharply decreases to zero in the vicinity of Δ=0. The maximalvalue of thermal entanglement and the critical temperatureTC decrease as the size of the systembecomes large. The entanglement shows stability and exists when L is very large.
By using the same method, we calculate the thermal entanglement of quantum Heisenbergspin system on the diamond-type hierarchical lattice with fractal dimension d f=2.32andd f=2.58. It is found that some similar properties of the entanglement on these two lattices andthe above lattice. The entanglement decreases with T increasing when Δ is certain and itincreases firstly and then decreases with Δ increasing at certain temperature. We also find thatthe entanglement of these two lattices compared with that of d f=2lattice exhibit differentfeatures that there occurs “entanglement crossing”. With the size of the system becoming large,the entanglement decrease much more slowly and the critical temperatureTC doesn’t decreasesbut increases and tends to the stable value. These results indicate that the entanglement is muchmore robust against the decoherence caused by temperature. The different fractal can affect different the energy level structure of system, which can lead to the different entanglementproperties.
引文
[1] E. Schr dinger. Die genewartige situation in der quantenmechanik [J]. Naturwissenschaften,1935,23,807-812.
[2] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physicalreality be considered complete?[J]. Phys. Rev.,1935,47,777-780.
[3] D. Bohm. Quantum Theory [M]. Prentice Hall, Englewood Cliffs, NJ,1951.
[4] J. S. Bell. On the Einstein-Podolsky-Rosen paradox [J]. Physics,1964,1,195-290.
[5] A. Aspect, J. Dalibard, and G. Roger. Experimental test of Bell's inequalities usingtime-varying analyzers [J]. Phys. Rev. Lett.,1982,49,1804-1807.
[6] A. Aspect. Bell's inequality test: more ideal than ever [J]. Nature,1999,398,189-190.
[7] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Quantum entanglement [J].Rev. Mod. Phys.,2009,81,865-942.
[8] A. K. Ekert. Quantum cryptography based on Bell’s theorem [J]. Phys. Rev. Lett.,1991,67,661-663.
[9] C. H. Bennett and S. J. Wiesner. Communication via one-and two-particle operators onEinstein-Podolsky-Rosen states [J]. Phys. Rev. Lett.,1992,69,2881-2884.
[10] C. H. Bennett, et al. Teleporting an unknown quantum state via dual classical andEinstein-Podolsky-Rosen channels [J]. Phys. Rev. Lett.,1993,70,1895-1899
[11] B. E. Kane. A silicon-based nuclear spin quantum computer [J]. Nature (Londond),1998,393,133-137.
[12] C. H. Bennett and D. P. Divincenzo. Quantum information and computation [J]. Nature(Londond),2000,404,247-255
[13] E. Knill, R. Laflamme, and G. J. Milburn. A scheme for efficient quantum computation withlinear optics [J]. Nature,2001,409,46-52.
[14] Y. Makhlin, G. Sch n, and A. Shnirman. Josephson junction quantum bits and logic gates [J].Physica B,2000,280,410-411.
[15] R. Hanson, et al. Spins in few-electron quantum dots [J]. Rev. Mod. Phys.,2007,79,1217-1265
[16] T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller. Decoherence, continuous observation,and quantum computing: a cavity QED model [J]. Phys. Rev. Lett.,1995,75,3788-3791.
[17] L. M. Duan, J. I. Cirac, and P. Zoller. Geometric manipulation of trapped ions for quantumcomputation [J]. Science,2001,292,1695-1697.
[18] J. H. Reina, L. Quiroga, and N. F. Johnson. NMR-based nanostructure switch for quantumlogic [J]. Phys. Rev. B,2000,62, R2267-2270.
[19] A. Peres. Separability Criterion for Density Matrices [J]. Phys. Rev. Lett.,1996,77,1413-1415.
[20] G. Vidal and R. F. Werner. Computable measure of entanglement [J]. Phys. Rev. A,2002,65,032314.
[21] W. K. Wootters. Entanglement of Formation of an Arbitrary State of Two Qubits [J]. Phys.Rev. Lett.,1998,80,2245-2248.
[22] S. Hill and W. K. Wootters. Entanglement of a Pair of Quantum Bits [J]. Phys. Rev. Lett.,1997,78,5022-5025.
[23] C. H. Bennett, et al. Exact and asymptotic measures of multipartite pure-state entanglement[J]. Phys. Rev. A,2000,63,012307.
[24] S. Wu and Y. Zhang. Multipartite pure-state entanglement and the generalizedGreenberger-Horne-Zeilinger states [J]. Phys. Rev. A,200063,012308.
[25] D. Loss and D. P. Divincenzo. Quantum computation with quantum dots [J]. Phys. Rev. A,1998,57,120-126.
[26] L. Amico, R. Fazio, A. Osterloh, and V. Vedral. Entanglement in many-body systems [J].Rev. Mod. Phys.,2008,80,517.
[27] M. A. Nielsen, Quantum Information Theory, in University of Mexico.1998, University ofMexico.
[28] M. C. Arnesen, S. Bose, V. Vedral, and S. Bose. Natural thermal and magnetic entanglementin the1D Heisenberg model [J]. Phys. Rev. Lett.,2001,87,017901.
[29] X. G. Wang. Entanglement in the quantum Heisenberg XY model [J]. Phys. Rev. A,2001,64,012313.
[30] G. L. Kamta and A. F. Starace. Anisotropy and magnetic field effect on the entanglement ofa two qubit Heisenberg XY chain [J]. Phys. Rev. Lett.,2002,88,107901.
[31] S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin. Entanglement of the Heisenberg chain with thenext-nearest-neighbor interaction [J]. Phys. Rev. A,2004,70,052302.
[32] F. Kheirandish, S. J. Akhtarshenas, and H. Mohammadi. Effect of spin-orbit interaction onentanglement of two-qubit Heisenberg XYZ systems in an inhomogeneous magnetic field [J].Phys. Rev. A,2008,77,042309.
[33] X. Hao and S. Zhu. Entanglement in a spin-s antiferromagnetic Heisenberg chain [J]. Phys.Rev. A,2005,72,042306.
[34] A. Saguia and M. S. Sarandy. Entanglement in the one-dimensional Kondo necklace model[J]. Phys. Rev. A,2003,67,012315.
[35] I. Bose and A. Tribedi. Thermal entanglement properties of small spin clusters [J]. Phys.Rev. A,200572,022314.
[36] Y.-C. Li and S.-S. Li. Quantum phase transitions in the S=1/2distorted diamond chain [J].Phys. Rev. B,2008,78,184412.
[37] S. Ghosh, T. F. Rosenbaum, G. Aeppli, and S. N. Coppersmith. Entangled quantum state ofmagnetic dipoles [J]. Nature,2003,425,48-51.
[38] T. Vértesi and E. Bene. Thermal entanglement in the nanotubular system Na2V3O7[J]. Phys.Rev.B,2006,73,134404.
[39] A. M. Souza, et al. Experimental determination of thermal entanglement in spin clustersusing magnetic susceptibility measurements [J]. Phys. Rev. B,2008,77,104402.
[40] A. Osterloh, L. Amico, G. Falci, and R. Fazio. Scaling of entanglement close to a quantumphase transition [J]. Nature (London),2002,416,608-610.
[41] T. J. Osborne and M. A. Nielsen. Entanglement in a simple quantum phase transition [J].Phys. Rev. A,2002,66,032110-032123.
[42] A. Langari. Quantum renormalization group of XXZ modle in a transverse magnetic field[J]. Phys. Rev. B,2004,69,100402.
[43] M. Kargarian, R. Jafari, and A. Langari. Renormalization of concurrence: The application ofthe quantum renormalization group to quantum-information systems [J]. Phys. Rev. A,2007,76,060304(R).
[44] M. Kargarian, R. Jafari, and A. Langari. Renormalization of entanglement in the anisotropicHeisenberg XXZ model [J]. Phys. Rev. A,2008,77,032346.
[45] M. Kargarian, R. Jafari, and A. Langari. Dzyaloshinskii-Moriya interaction and anisotropyeffects on the entanglement of the Heisenberg model [J]. Phys. Rev. A,2009,042319.
[46]杨展如.分形物理学[M].上海：上海科技教育出版社,1996,28-79.
[47] B. B. Mandelbort. Fractal: Form, Chance and Dimension [M]. Freeman, San Francisco,1977,15-23.
[48] Y. Gefen, A. Aharony, and B. B. Mandelbrot. Phase transitions on fractals. I. Quasi-linearlattices [J]. J. Phys. A,1983,16,1267-1278.
[49] Y. Gefen, et al. Phase transitions on fractals. II. Sierpinski gaskets [J]. J. Phys. A,1984,17,435-444.
[50] Y. Gefen, A. Aharony, and B. B. Mandelbrot. Phase transitions on fractals. III. Infinitelyramified lattices [J]. J. Phys. A,1984,17,1277-1289.
[51] Z. R. Yang. Family of diamond-type hierarchical lattices [J]. Phys. Rev. B,1988,38,728-731.
[52] M. Kaufman and R. B. Griffiths. Spin systems on hierarchical lattices. II. Some examples ofsoluble models [J]. Phys. Rev. B,1984,30,244-249.
[53] R. B. Griffiths and M. Kaufman. Spin systems on hierarchical lattices. I. Introductionandthermodynamic limit [J]. Phys. Rev. B,1982,26,5022-5032.
[54] A. S rensen and K. M lmer. Spin-spin interaction and spin squeezing in an optical lattice[J]. Phys. Rev. Lett.,1999,83,2274-2277
[55] R. Vrijen, et al. Electron spin resonance transistors for quantum computing inSilicon-Germanium Heterostructures [J]. Phys. Rev. A,2000,62,012306.
[56] E. Kane. A silicon-based nuclear spin quantum computer [J]. Nature,1998,393,133-137.
[57]杨展如.量子统计物理学[M].北京：高等教育出版社,2007,28-79.
[58] A. O. Caride, C. Tsallis, and S. I. Zanette. Criticality of the Anisotropic QuantumHeisenberg Model on a Self-Dual Hierarchical Lattice [J]. Phys. Rev. Lett.,1983,51,145-147.
[59] A. M. Mariz and C. Tsallis. Criticality of the D=2quantum Heisenberg ferromagnet withquenched random anisotropy [J]. Phys. Rev. B,1985,31,7491-7493.
[60] A. M. Mariz, C. Tsallis, and A. O. Caride. Criticality of the D=2bond-dilute anisotropicHeisenberg ferromagnet [J]. J. Phys. C,1985,18,4189-4209.
[61] J. R. D. Sousa, N. S. Branco, B. Boechat, and C. Cordeiro. Quantum spin-1/2two-dimensional XXZ model: an alternative quantum renormalization-group approach [J].Physica A,2003,328,167-173.
[62] A. M. Mariz, R. M. Zorzenon, C. Tsallis, and R. R. D. Santos. Critical of the anisotropicquantum Heisenberg model on a simple cubic lattice [J]. Phys. Lett. A,1985,108,95-98.
[63] T. Werlang, C. Trippe, G. a. P. Ribeiro, and G. Rigolin. Quantum Correlations in Spin Chainsat Finite Temperatures and Quantum Phase Transitions [J]. Phys. Rev. Lett.,2010,105,095702.
[2] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physicalreality be considered complete?[J]. Phys. Rev.,1935,47,777-780.
[3] D. Bohm. Quantum Theory [M]. Prentice Hall, Englewood Cliffs, NJ,1951.
[4] J. S. Bell. On the Einstein-Podolsky-Rosen paradox [J]. Physics,1964,1,195-290.
[5] A. Aspect, J. Dalibard, and G. Roger. Experimental test of Bell's inequalities usingtime-varying analyzers [J]. Phys. Rev. Lett.,1982,49,1804-1807.
[6] A. Aspect. Bell's inequality test: more ideal than ever [J]. Nature,1999,398,189-190.
[7] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Quantum entanglement [J].Rev. Mod. Phys.,2009,81,865-942.
[8] A. K. Ekert. Quantum cryptography based on Bell’s theorem [J]. Phys. Rev. Lett.,1991,67,661-663.
[9] C. H. Bennett and S. J. Wiesner. Communication via one-and two-particle operators onEinstein-Podolsky-Rosen states [J]. Phys. Rev. Lett.,1992,69,2881-2884.
[10] C. H. Bennett, et al. Teleporting an unknown quantum state via dual classical andEinstein-Podolsky-Rosen channels [J]. Phys. Rev. Lett.,1993,70,1895-1899
[11] B. E. Kane. A silicon-based nuclear spin quantum computer [J]. Nature (Londond),1998,393,133-137.
[12] C. H. Bennett and D. P. Divincenzo. Quantum information and computation [J]. Nature(Londond),2000,404,247-255
[13] E. Knill, R. Laflamme, and G. J. Milburn. A scheme for efficient quantum computation withlinear optics [J]. Nature,2001,409,46-52.
[14] Y. Makhlin, G. Sch n, and A. Shnirman. Josephson junction quantum bits and logic gates [J].Physica B,2000,280,410-411.
[15] R. Hanson, et al. Spins in few-electron quantum dots [J]. Rev. Mod. Phys.,2007,79,1217-1265
[16] T. Pellizzari, S. A. Gardiner, J. I. Cirac, and P. Zoller. Decoherence, continuous observation,and quantum computing: a cavity QED model [J]. Phys. Rev. Lett.,1995,75,3788-3791.
[17] L. M. Duan, J. I. Cirac, and P. Zoller. Geometric manipulation of trapped ions for quantumcomputation [J]. Science,2001,292,1695-1697.
[18] J. H. Reina, L. Quiroga, and N. F. Johnson. NMR-based nanostructure switch for quantumlogic [J]. Phys. Rev. B,2000,62, R2267-2270.
[19] A. Peres. Separability Criterion for Density Matrices [J]. Phys. Rev. Lett.,1996,77,1413-1415.
[20] G. Vidal and R. F. Werner. Computable measure of entanglement [J]. Phys. Rev. A,2002,65,032314.
[21] W. K. Wootters. Entanglement of Formation of an Arbitrary State of Two Qubits [J]. Phys.Rev. Lett.,1998,80,2245-2248.
[22] S. Hill and W. K. Wootters. Entanglement of a Pair of Quantum Bits [J]. Phys. Rev. Lett.,1997,78,5022-5025.
[23] C. H. Bennett, et al. Exact and asymptotic measures of multipartite pure-state entanglement[J]. Phys. Rev. A,2000,63,012307.
[24] S. Wu and Y. Zhang. Multipartite pure-state entanglement and the generalizedGreenberger-Horne-Zeilinger states [J]. Phys. Rev. A,200063,012308.
[25] D. Loss and D. P. Divincenzo. Quantum computation with quantum dots [J]. Phys. Rev. A,1998,57,120-126.
[26] L. Amico, R. Fazio, A. Osterloh, and V. Vedral. Entanglement in many-body systems [J].Rev. Mod. Phys.,2008,80,517.
[27] M. A. Nielsen, Quantum Information Theory, in University of Mexico.1998, University ofMexico.
[28] M. C. Arnesen, S. Bose, V. Vedral, and S. Bose. Natural thermal and magnetic entanglementin the1D Heisenberg model [J]. Phys. Rev. Lett.,2001,87,017901.
[29] X. G. Wang. Entanglement in the quantum Heisenberg XY model [J]. Phys. Rev. A,2001,64,012313.
[30] G. L. Kamta and A. F. Starace. Anisotropy and magnetic field effect on the entanglement ofa two qubit Heisenberg XY chain [J]. Phys. Rev. Lett.,2002,88,107901.
[31] S.-J. Gu, H. Li, Y.-Q. Li, and H.-Q. Lin. Entanglement of the Heisenberg chain with thenext-nearest-neighbor interaction [J]. Phys. Rev. A,2004,70,052302.
[32] F. Kheirandish, S. J. Akhtarshenas, and H. Mohammadi. Effect of spin-orbit interaction onentanglement of two-qubit Heisenberg XYZ systems in an inhomogeneous magnetic field [J].Phys. Rev. A,2008,77,042309.
[33] X. Hao and S. Zhu. Entanglement in a spin-s antiferromagnetic Heisenberg chain [J]. Phys.Rev. A,2005,72,042306.
[34] A. Saguia and M. S. Sarandy. Entanglement in the one-dimensional Kondo necklace model[J]. Phys. Rev. A,2003,67,012315.
[35] I. Bose and A. Tribedi. Thermal entanglement properties of small spin clusters [J]. Phys.Rev. A,200572,022314.
[36] Y.-C. Li and S.-S. Li. Quantum phase transitions in the S=1/2distorted diamond chain [J].Phys. Rev. B,2008,78,184412.
[37] S. Ghosh, T. F. Rosenbaum, G. Aeppli, and S. N. Coppersmith. Entangled quantum state ofmagnetic dipoles [J]. Nature,2003,425,48-51.
[38] T. Vértesi and E. Bene. Thermal entanglement in the nanotubular system Na2V3O7[J]. Phys.Rev.B,2006,73,134404.
[39] A. M. Souza, et al. Experimental determination of thermal entanglement in spin clustersusing magnetic susceptibility measurements [J]. Phys. Rev. B,2008,77,104402.
[40] A. Osterloh, L. Amico, G. Falci, and R. Fazio. Scaling of entanglement close to a quantumphase transition [J]. Nature (London),2002,416,608-610.
[41] T. J. Osborne and M. A. Nielsen. Entanglement in a simple quantum phase transition [J].Phys. Rev. A,2002,66,032110-032123.
[42] A. Langari. Quantum renormalization group of XXZ modle in a transverse magnetic field[J]. Phys. Rev. B,2004,69,100402.
[43] M. Kargarian, R. Jafari, and A. Langari. Renormalization of concurrence: The application ofthe quantum renormalization group to quantum-information systems [J]. Phys. Rev. A,2007,76,060304(R).
[44] M. Kargarian, R. Jafari, and A. Langari. Renormalization of entanglement in the anisotropicHeisenberg XXZ model [J]. Phys. Rev. A,2008,77,032346.
[45] M. Kargarian, R. Jafari, and A. Langari. Dzyaloshinskii-Moriya interaction and anisotropyeffects on the entanglement of the Heisenberg model [J]. Phys. Rev. A,2009,042319.
[46]杨展如.分形物理学[M].上海：上海科技教育出版社,1996,28-79.
[47] B. B. Mandelbort. Fractal: Form, Chance and Dimension [M]. Freeman, San Francisco,1977,15-23.
[48] Y. Gefen, A. Aharony, and B. B. Mandelbrot. Phase transitions on fractals. I. Quasi-linearlattices [J]. J. Phys. A,1983,16,1267-1278.
[49] Y. Gefen, et al. Phase transitions on fractals. II. Sierpinski gaskets [J]. J. Phys. A,1984,17,435-444.
[50] Y. Gefen, A. Aharony, and B. B. Mandelbrot. Phase transitions on fractals. III. Infinitelyramified lattices [J]. J. Phys. A,1984,17,1277-1289.
[51] Z. R. Yang. Family of diamond-type hierarchical lattices [J]. Phys. Rev. B,1988,38,728-731.
[52] M. Kaufman and R. B. Griffiths. Spin systems on hierarchical lattices. II. Some examples ofsoluble models [J]. Phys. Rev. B,1984,30,244-249.
[53] R. B. Griffiths and M. Kaufman. Spin systems on hierarchical lattices. I. Introductionandthermodynamic limit [J]. Phys. Rev. B,1982,26,5022-5032.
[54] A. S rensen and K. M lmer. Spin-spin interaction and spin squeezing in an optical lattice[J]. Phys. Rev. Lett.,1999,83,2274-2277
[55] R. Vrijen, et al. Electron spin resonance transistors for quantum computing inSilicon-Germanium Heterostructures [J]. Phys. Rev. A,2000,62,012306.
[56] E. Kane. A silicon-based nuclear spin quantum computer [J]. Nature,1998,393,133-137.
[57]杨展如.量子统计物理学[M].北京：高等教育出版社,2007,28-79.
[58] A. O. Caride, C. Tsallis, and S. I. Zanette. Criticality of the Anisotropic QuantumHeisenberg Model on a Self-Dual Hierarchical Lattice [J]. Phys. Rev. Lett.,1983,51,145-147.
[59] A. M. Mariz and C. Tsallis. Criticality of the D=2quantum Heisenberg ferromagnet withquenched random anisotropy [J]. Phys. Rev. B,1985,31,7491-7493.
[60] A. M. Mariz, C. Tsallis, and A. O. Caride. Criticality of the D=2bond-dilute anisotropicHeisenberg ferromagnet [J]. J. Phys. C,1985,18,4189-4209.
[61] J. R. D. Sousa, N. S. Branco, B. Boechat, and C. Cordeiro. Quantum spin-1/2two-dimensional XXZ model: an alternative quantum renormalization-group approach [J].Physica A,2003,328,167-173.
[62] A. M. Mariz, R. M. Zorzenon, C. Tsallis, and R. R. D. Santos. Critical of the anisotropicquantum Heisenberg model on a simple cubic lattice [J]. Phys. Lett. A,1985,108,95-98.
[63] T. Werlang, C. Trippe, G. a. P. Ribeiro, and G. Rigolin. Quantum Correlations in Spin Chainsat Finite Temperatures and Quantum Phase Transitions [J]. Phys. Rev. Lett.,2010,105,095702.