具有交错Dzyaloshinskii-Moriya相互作用XY模型的量子纠缠和相变
|
摘要
量子纠缠是量子理论中最显著的奇妙特性之一,也是实现量子信息的基本资源,已经被广泛地应用于量子计算、量子密钥分配和量子隐形传态中。近年来,人们发现在凝聚态物理中量子纠缠和量子相变存在着密切的联系,且纠缠可以描述自旋系统的量子临界性质。论文利用量子重整化群方法研究了具有交错Dzyaloshinskii-Moriya (DM)相互作用一维XY模型的量子纠缠和量子相变。
利用卡丹诺夫块-自旋方法,得到了系统的两个稳定不动点和不稳定不动点,它们分别对应着系统的自旋液相与Neel相和系统的临界点。把共生纠缠度作为对量子关联的一种度量,利用系统的约化密度矩阵求出了三格点模型的共生纠缠度的解析表达式,发现它为各向异性参数和DM相互作用的函数,并分析了它在临界点附近的变化趋势、非解析行为以及系统的标度行为。研究了共生纠缠度与各向异性参数和DM相互作用的关系,发现它们都能影响系统的相图。当DM相互作用为某一定值时,随着重整化迭代次数的增加,共生纠缠度逐渐趋于两个稳定的值,这两个值分别对应着系统的两个不同的相,即自旋液相和Neel相。研究还发现DM相互作用越大,其Neel相对应的各向异性参数的取值范围就越大。还计算了共生纠缠度的一阶偏导数,发现在热力学极限下,它在临界点处表现出了非解析行为,即发生了量子相变。在各向异性参数为某一定值时,也能够得到相似的结果。并且,共生纠缠度一阶偏导数的最值与系统尺度存在着线性关系,即系统的标度行为。最后得到了系统的临界指数与关联长度指数之间存在一种倒数关系。
Quantum entanglement is one of the most string features of quantum theory and is also regarded as a fundamental resource. It has been widely used in quantum computation, quantum cryptographic key distribution, quantum teleportation and so on. In recent years, the researchers have found that the quantum entanglement plays an important role in the quantum phase transition. The quantum critical property of the spin system can be described by the concept of entanglement. We study the quantum entanglement and quantum phase transition of the anisotropic S=1/2 model with staggered Dzyaloshinskii-Moriya (DM) interaction by using the quantum renormalization group (QRG) method.
With the help of the Kadanoff's block method, two stable fixed points and an unstable fixed point are obtained, which correspond with spin-fluid phase, Neel phase and the phase transition point, respectively. The concurrence is considered as a measure of the quantum correlation. The analytical expression of the concurrence can be gotten by the reduced density matrix of the system which is the function of anisotropy parameter and DM interaction. The variation tendency of the concurrence at the critical point, the nonanalytic behavior of the first derivative concurrence and the scaling behavior of the system can be also obtained. We study the concurrence how changes versus the anisotropy parameter and DM interaction, respectively. It is found that both of them can influence the phase diagrams. As the number of RG iterations increases, the concurrence develops two different values for a given value of DM interaction which correspond with the spin-fluid and Neel phases, respectively. For Nccl phase, the larger the value of DM interaction is, the wider the width of value of anisotropy parameter is. Further insight, the first partial derivative of the concurrence exhibits the nonanalytic behavior at the thermodynamic limit, it is shown that the system occurs quantum phase transition. For a given value of anisotropy parameter, we can also obtain the similar results. The scaling behavior can be obtained which characterizes how the critical point of the model is touched as the size of the system increases. At last, we get the relation between the critical exponent and correlation length exponent, and find that there exist the relationship of reciprocal.
利用卡丹诺夫块-自旋方法,得到了系统的两个稳定不动点和不稳定不动点,它们分别对应着系统的自旋液相与Neel相和系统的临界点。把共生纠缠度作为对量子关联的一种度量,利用系统的约化密度矩阵求出了三格点模型的共生纠缠度的解析表达式,发现它为各向异性参数和DM相互作用的函数,并分析了它在临界点附近的变化趋势、非解析行为以及系统的标度行为。研究了共生纠缠度与各向异性参数和DM相互作用的关系,发现它们都能影响系统的相图。当DM相互作用为某一定值时,随着重整化迭代次数的增加,共生纠缠度逐渐趋于两个稳定的值,这两个值分别对应着系统的两个不同的相,即自旋液相和Neel相。研究还发现DM相互作用越大,其Neel相对应的各向异性参数的取值范围就越大。还计算了共生纠缠度的一阶偏导数,发现在热力学极限下,它在临界点处表现出了非解析行为,即发生了量子相变。在各向异性参数为某一定值时,也能够得到相似的结果。并且,共生纠缠度一阶偏导数的最值与系统尺度存在着线性关系,即系统的标度行为。最后得到了系统的临界指数与关联长度指数之间存在一种倒数关系。
Quantum entanglement is one of the most string features of quantum theory and is also regarded as a fundamental resource. It has been widely used in quantum computation, quantum cryptographic key distribution, quantum teleportation and so on. In recent years, the researchers have found that the quantum entanglement plays an important role in the quantum phase transition. The quantum critical property of the spin system can be described by the concept of entanglement. We study the quantum entanglement and quantum phase transition of the anisotropic S=1/2 model with staggered Dzyaloshinskii-Moriya (DM) interaction by using the quantum renormalization group (QRG) method.
With the help of the Kadanoff's block method, two stable fixed points and an unstable fixed point are obtained, which correspond with spin-fluid phase, Neel phase and the phase transition point, respectively. The concurrence is considered as a measure of the quantum correlation. The analytical expression of the concurrence can be gotten by the reduced density matrix of the system which is the function of anisotropy parameter and DM interaction. The variation tendency of the concurrence at the critical point, the nonanalytic behavior of the first derivative concurrence and the scaling behavior of the system can be also obtained. We study the concurrence how changes versus the anisotropy parameter and DM interaction, respectively. It is found that both of them can influence the phase diagrams. As the number of RG iterations increases, the concurrence develops two different values for a given value of DM interaction which correspond with the spin-fluid and Neel phases, respectively. For Nccl phase, the larger the value of DM interaction is, the wider the width of value of anisotropy parameter is. Further insight, the first partial derivative of the concurrence exhibits the nonanalytic behavior at the thermodynamic limit, it is shown that the system occurs quantum phase transition. For a given value of anisotropy parameter, we can also obtain the similar results. The scaling behavior can be obtained which characterizes how the critical point of the model is touched as the size of the system increases. At last, we get the relation between the critical exponent and correlation length exponent, and find that there exist the relationship of reciprocal.
引文
[1]A. Einstein, B. Podolsky, and N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? [J]. Phys. Rev.,1935,47:777-780.
[2]M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Infomiation[M]. Cambridge:Cambridge University Press,2000.
[3]J. Preskill. Quantum Information and Quantum Computation[M]. Pasadena:California Institute of Technology,1998.
[4]周正威,郭光灿.量子纠缠态[J].物理,2000,29(11)：695-699.
[5]C. H. Bennett, G. Brassard, C. Crepeau, et al. Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels [J]. Phys. Rev. Lett.,1993,70: 1895-1899.
[6]D. Bouwmeester, J. W. Pan, K. Mattle, et al. Expermental Quantum Teleportation [J]. Nature,1997,390:575-579.
[7]M. Zukowski, A. Zeilinger, M. A. Home, et al. Event-ready-detectors" Bell experiment via entanglement swapping [J]. Phys. Rev. Lett.,1993,71:4287-4290.
[8]J. W. Pan, D. Bouwmeester, H. Weinfurter, et al. Experimental Entanglement Swapping: Entangling Photons That Never Interacted [J]. Phys. Rev. Lett.,1998,80:3891-3894.
[9]D. Deutsch. Quantum Computational Networks [J]. Proc. R. Soc. Lond. Ser. A,1989,425: 73-90.
[10]P. W. Shor. Scheme for Reducing Decoherence in Quantum Computer Memory [J]. Phys. Rev. A,1995,52:R2493-R2496.
[11]C. H. Bennett, G. Brassard. Quantum Cryptography:Public Key Distribution and Coin Tossing [J]. in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, (IEEE, New York,1984),1984, pp.:175-179.
[12]L. K. Grover. Quantum Mechanics Helps in Searching for a Needle in a Haystack [J]. Phys. Rev. Lett.,1997,79:325-328.
[13]A. Ekert and R. Josza. Quantum Computation and Shor's Factoring Algorithm [J]. Rev. Mod. Phys.,1996,68:733-753.
[14]G. Vidal and R. F. Werner. Computable Measure of Entanglement [J]. Phys. Rev. A,2002, 65:032314.
[15]A. Peres. Separability Criterion for Density Matrices [J]. Phys. Rev. Lett.,1996,77: 1413-1415.
[16]M. Horodecki, P. Horodecki, R. Horodecki. Separability of Mixed States:Necessary and Sufficient Conditions [J]. Phys. Lett. A,1996,223:1-8.
[17]S. Hill and W. K. Wootters. Entanglement of a Pair of Quantum Bits [J]. Phys. Rev. Lett., 1997,78:5022-5025.
[18]W. K. Wootters. Entanglement of Formation of an Arbitrary State of Two Qubits [J]. Phys. Rev. Lett.,1998,80:2245-2248.
[19]于渌,郝柏林,陈晓松.边缘奇迹-相变和临界现象[M].北京：科学出版社,2005.
[20]杨展如.量子统计物理[M].北京：高等教育出版社,2007.
[21]L. D. Landau. Zur Theorie der Phasenumwandlungen [J]. Phys. Z. Sowjetunion,1937,11: 26-35.
[22]L. Onsager. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition [J]. Phys. Rev.,1944,65:117-149.
[23]K. G. Wilson. Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture [J]. Phys. Rev. B,1971,4:3174-3183.
[24]D. V. Shopova and D. I. Uzunov. Some Basic Aspects of Quantum Phase Transitions [J]. Phys. Rep.,2003,379:1-67.
[25]S. Sachdev. Quantum Phase Transitions [M]. Cambridge:Cambirdge University Press, 1999.
[26]M. Vojta. Quantum Phase Transitions [J]. Rep. Prog. Phys.,2003,66:2069-2110.
[27]H. Barnum, E. Knill, G. Ortiz, et al. A Subsystem-Independent Generalization of Entanglement [J]. Phys. Rev. Lett.,2004,92:107902.
[28]N. Lambert, C. Emary, and T. Brandes. Entanglement and the Phase Transition in Single-Mode Superradiance [J]. Phys. Rev. Lett.,2004,92:073602.
[29]R. Somma, G. Ortiz, H. Bamum, et al. Nature and Measure of Entanglement in Quantum PhaseTransitions [J]. Phys. Rev. A,2004,70:042311.
[30]A. Osterloh, L. Amico, G. Falci, et al. Scaling of Entanglement Close to a Quantum Phase Transitions [J]. Nature,2002,416:608-610.
[31]T. J. Osborne and M. A. Nielsen. Entanglement in a Simple Quantum Phase Transition [J]. Phys. Rev. A,2002,66:032110.
[32]G. Vidal, J. I. Latorre, E. Rico, et al. Entanglement in Quantum Critical Phenomena [J]. Phys. Rev. Lett.,2003,90:227902.
[33]L. A. Wu, M. S. Sarandy, and D. A. Lidar. Quantum Phase Transitions and Bipartite Entanglement [J]. Phys. Rev. Lett.,2004,93:250404.
[34]Z. Friedman. Critical Exponents for the Three-Dimensional Ising Model from the Real-Space Renormalization Group in Two Dimensions [J]. Phys. Rev. Lett.,1976,36: 1326-1328.
[35]R. C. Brower. F. Kuttner, M. Nauenberg, et al. Renormalization Transformations for Quantum Spin Systems [J]. Phys. Rev. Lett.,1977,38:1231-1234.
[36]S. K. Ma. Renormalization Group by Monte Carlo Methods [J]. Phys. Rev. Lett.,1976,37: 461-464.
[37]S. R. White. Density Matrix Formulation for Quantum Renormalization Groups [J]. Phys. Rev. Lett.,1992,69:2863-2866.
[38]S. R. White. Density-matrix Algorithms for Quantum Renormalization Groups [J]. Phys. Rev. B,1993,48:10345-10356.
[39]K. G. Wilson. The Renormalization Group:Critical Phenomena and the Kondo Problem [J]. Rev. Mod. Phys.,1975,47:773-840.
[40]P. Pefeuty, R. Jullian and K. L. Penson. in Real-space Renormalization [M]. Berlin: Springer,1982.
[41]M. Kargarian, R. Jafari, and A. Langari. Renormalization of Concurrence:The application of the Quantum Renormalization Group to Quantum-information Systems [J]. Phys. Rev. A, 2007,76:060304.
[42]A. Langari. Quantum Renormalization Group of XYZ Model in a Transverse Magnetic Field [J]. Phys. Rev. B,2004,69:100402.
[43]R. Jafari, M. Kargarian, A. Langari, et al. Phase Diagram and Entanglement of the Ising Model with Dzyaloshinskii-Moriya Interaction [J]. Phys. Rev. B,2008,78:214414.
[44]M. Kargarian, R. Jafari, and A. Langari. Renormalization of Entanglement in the Anisotropic Heisenberg (XXZ) Model [J]. Phys. Rev. A,2008,77:032346.
[45]M. Kargarian, R. Jafari, and A. Langari. Dzyaloshinskii-Moriya Interaction and Anisotropy Effects on the Entanglement of the Heisenberg Model [J]. Phys. Rev. A,2009,79:042319.
[46]R. Jafari and A. Langari. Second Order Quantum Renormalization Group of XXZ Chain with Next-nearest Neighbour Interaction [J]. Phys. A,2006,364:213-222.
[47]R. Jafari and A. Langari. Phase Diagram of the One-dimensional S= 1/2 XXZ Model with Ferromagnetic Nearest-neighbor and Antiferromagnetic Next-nearest-neighbor Interactions [J]. Phys. Rev. B,2007,76:014412.
[48]C. R. Menyuk. Particle Motion in the Field of a Modulated Wave [J]. Phys. Rev. A,1985, 31:3282-3290.
[49]K. G. Wilson. Renormalization of a Scalar Field Theory in Strong Coupling [J]. Phys. Rev. D,1972,6:419-426.
[50]K. G. Wilson. Renormalization Group and Critical Phenomena. Ⅱ. Phase-Space Cell Analysis of Critical Behavior [J]. Phys. Rev. B,1971,4:3184-3205.
[51]R. Shankar, R. Gupta, and G. Murthy. Dealing with Truncation in Monte Carlo Renormalization-Group Calculations [J]. Phys. Rev. Lett.,1985,55:1812-1815.
[52]杨展如.分形物理[M].上海：上海科技出版社,1996.
[53]I. Dzyaloshinsky. A Thennodynamic Theory of "Weak" Ferromagnetism of Antiferromagnetics [J]. J. Phys. Chem. Solids,1958,4:241-255.
[54]T. Moriya. Anisotropic Superexchange Interaction and Weak Ferromagnetism [J]. Phys. Rev.,1960,120:91-98.
[55]T. Thio, T. R. Thurston, N. W. Preyer, et al. Antisymmetric Exchange and Its Influence on the Magnetic Structure and Conductivity of La2Cu04 [J]. Phys. Rev. B,1988,38:905-908.
[56]H. De Raedt, S. Miyashita, K. Michielsen, et al. Dzyaloshinskii-Moriya Interactions and Adiabatic Magnetization Dynamics in Molecular Magnets [J]. Phys. Rev. B,2004,70: 064401.
[57]T. Moriya. New Mechanism of Anisotropic Superexchange Interaction [J]. Phys. Rev. Lett., 1960,4:228-230.
[58]M. Oshikawa and I. Affleck. Field-Induced Gap in S=1/2 Antiferromagnetic Chains [J]. Phys. Rev. Lett.,1997,79:2883-2886.
[59]R. Jafari. Low-energy-state Dynamics of Entanglement for Spin Systems [J]. Phys. Rev. A, 2010,82:052317.
[2]M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Infomiation[M]. Cambridge:Cambridge University Press,2000.
[3]J. Preskill. Quantum Information and Quantum Computation[M]. Pasadena:California Institute of Technology,1998.
[4]周正威,郭光灿.量子纠缠态[J].物理,2000,29(11)：695-699.
[5]C. H. Bennett, G. Brassard, C. Crepeau, et al. Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels [J]. Phys. Rev. Lett.,1993,70: 1895-1899.
[6]D. Bouwmeester, J. W. Pan, K. Mattle, et al. Expermental Quantum Teleportation [J]. Nature,1997,390:575-579.
[7]M. Zukowski, A. Zeilinger, M. A. Home, et al. Event-ready-detectors" Bell experiment via entanglement swapping [J]. Phys. Rev. Lett.,1993,71:4287-4290.
[8]J. W. Pan, D. Bouwmeester, H. Weinfurter, et al. Experimental Entanglement Swapping: Entangling Photons That Never Interacted [J]. Phys. Rev. Lett.,1998,80:3891-3894.
[9]D. Deutsch. Quantum Computational Networks [J]. Proc. R. Soc. Lond. Ser. A,1989,425: 73-90.
[10]P. W. Shor. Scheme for Reducing Decoherence in Quantum Computer Memory [J]. Phys. Rev. A,1995,52:R2493-R2496.
[11]C. H. Bennett, G. Brassard. Quantum Cryptography:Public Key Distribution and Coin Tossing [J]. in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, (IEEE, New York,1984),1984, pp.:175-179.
[12]L. K. Grover. Quantum Mechanics Helps in Searching for a Needle in a Haystack [J]. Phys. Rev. Lett.,1997,79:325-328.
[13]A. Ekert and R. Josza. Quantum Computation and Shor's Factoring Algorithm [J]. Rev. Mod. Phys.,1996,68:733-753.
[14]G. Vidal and R. F. Werner. Computable Measure of Entanglement [J]. Phys. Rev. A,2002, 65:032314.
[15]A. Peres. Separability Criterion for Density Matrices [J]. Phys. Rev. Lett.,1996,77: 1413-1415.
[16]M. Horodecki, P. Horodecki, R. Horodecki. Separability of Mixed States:Necessary and Sufficient Conditions [J]. Phys. Lett. A,1996,223:1-8.
[17]S. Hill and W. K. Wootters. Entanglement of a Pair of Quantum Bits [J]. Phys. Rev. Lett., 1997,78:5022-5025.
[18]W. K. Wootters. Entanglement of Formation of an Arbitrary State of Two Qubits [J]. Phys. Rev. Lett.,1998,80:2245-2248.
[19]于渌,郝柏林,陈晓松.边缘奇迹-相变和临界现象[M].北京：科学出版社,2005.
[20]杨展如.量子统计物理[M].北京：高等教育出版社,2007.
[21]L. D. Landau. Zur Theorie der Phasenumwandlungen [J]. Phys. Z. Sowjetunion,1937,11: 26-35.
[22]L. Onsager. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition [J]. Phys. Rev.,1944,65:117-149.
[23]K. G. Wilson. Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture [J]. Phys. Rev. B,1971,4:3174-3183.
[24]D. V. Shopova and D. I. Uzunov. Some Basic Aspects of Quantum Phase Transitions [J]. Phys. Rep.,2003,379:1-67.
[25]S. Sachdev. Quantum Phase Transitions [M]. Cambridge:Cambirdge University Press, 1999.
[26]M. Vojta. Quantum Phase Transitions [J]. Rep. Prog. Phys.,2003,66:2069-2110.
[27]H. Barnum, E. Knill, G. Ortiz, et al. A Subsystem-Independent Generalization of Entanglement [J]. Phys. Rev. Lett.,2004,92:107902.
[28]N. Lambert, C. Emary, and T. Brandes. Entanglement and the Phase Transition in Single-Mode Superradiance [J]. Phys. Rev. Lett.,2004,92:073602.
[29]R. Somma, G. Ortiz, H. Bamum, et al. Nature and Measure of Entanglement in Quantum PhaseTransitions [J]. Phys. Rev. A,2004,70:042311.
[30]A. Osterloh, L. Amico, G. Falci, et al. Scaling of Entanglement Close to a Quantum Phase Transitions [J]. Nature,2002,416:608-610.
[31]T. J. Osborne and M. A. Nielsen. Entanglement in a Simple Quantum Phase Transition [J]. Phys. Rev. A,2002,66:032110.
[32]G. Vidal, J. I. Latorre, E. Rico, et al. Entanglement in Quantum Critical Phenomena [J]. Phys. Rev. Lett.,2003,90:227902.
[33]L. A. Wu, M. S. Sarandy, and D. A. Lidar. Quantum Phase Transitions and Bipartite Entanglement [J]. Phys. Rev. Lett.,2004,93:250404.
[34]Z. Friedman. Critical Exponents for the Three-Dimensional Ising Model from the Real-Space Renormalization Group in Two Dimensions [J]. Phys. Rev. Lett.,1976,36: 1326-1328.
[35]R. C. Brower. F. Kuttner, M. Nauenberg, et al. Renormalization Transformations for Quantum Spin Systems [J]. Phys. Rev. Lett.,1977,38:1231-1234.
[36]S. K. Ma. Renormalization Group by Monte Carlo Methods [J]. Phys. Rev. Lett.,1976,37: 461-464.
[37]S. R. White. Density Matrix Formulation for Quantum Renormalization Groups [J]. Phys. Rev. Lett.,1992,69:2863-2866.
[38]S. R. White. Density-matrix Algorithms for Quantum Renormalization Groups [J]. Phys. Rev. B,1993,48:10345-10356.
[39]K. G. Wilson. The Renormalization Group:Critical Phenomena and the Kondo Problem [J]. Rev. Mod. Phys.,1975,47:773-840.
[40]P. Pefeuty, R. Jullian and K. L. Penson. in Real-space Renormalization [M]. Berlin: Springer,1982.
[41]M. Kargarian, R. Jafari, and A. Langari. Renormalization of Concurrence:The application of the Quantum Renormalization Group to Quantum-information Systems [J]. Phys. Rev. A, 2007,76:060304.
[42]A. Langari. Quantum Renormalization Group of XYZ Model in a Transverse Magnetic Field [J]. Phys. Rev. B,2004,69:100402.
[43]R. Jafari, M. Kargarian, A. Langari, et al. Phase Diagram and Entanglement of the Ising Model with Dzyaloshinskii-Moriya Interaction [J]. Phys. Rev. B,2008,78:214414.
[44]M. Kargarian, R. Jafari, and A. Langari. Renormalization of Entanglement in the Anisotropic Heisenberg (XXZ) Model [J]. Phys. Rev. A,2008,77:032346.
[45]M. Kargarian, R. Jafari, and A. Langari. Dzyaloshinskii-Moriya Interaction and Anisotropy Effects on the Entanglement of the Heisenberg Model [J]. Phys. Rev. A,2009,79:042319.
[46]R. Jafari and A. Langari. Second Order Quantum Renormalization Group of XXZ Chain with Next-nearest Neighbour Interaction [J]. Phys. A,2006,364:213-222.
[47]R. Jafari and A. Langari. Phase Diagram of the One-dimensional S= 1/2 XXZ Model with Ferromagnetic Nearest-neighbor and Antiferromagnetic Next-nearest-neighbor Interactions [J]. Phys. Rev. B,2007,76:014412.
[48]C. R. Menyuk. Particle Motion in the Field of a Modulated Wave [J]. Phys. Rev. A,1985, 31:3282-3290.
[49]K. G. Wilson. Renormalization of a Scalar Field Theory in Strong Coupling [J]. Phys. Rev. D,1972,6:419-426.
[50]K. G. Wilson. Renormalization Group and Critical Phenomena. Ⅱ. Phase-Space Cell Analysis of Critical Behavior [J]. Phys. Rev. B,1971,4:3184-3205.
[51]R. Shankar, R. Gupta, and G. Murthy. Dealing with Truncation in Monte Carlo Renormalization-Group Calculations [J]. Phys. Rev. Lett.,1985,55:1812-1815.
[52]杨展如.分形物理[M].上海：上海科技出版社,1996.
[53]I. Dzyaloshinsky. A Thennodynamic Theory of "Weak" Ferromagnetism of Antiferromagnetics [J]. J. Phys. Chem. Solids,1958,4:241-255.
[54]T. Moriya. Anisotropic Superexchange Interaction and Weak Ferromagnetism [J]. Phys. Rev.,1960,120:91-98.
[55]T. Thio, T. R. Thurston, N. W. Preyer, et al. Antisymmetric Exchange and Its Influence on the Magnetic Structure and Conductivity of La2Cu04 [J]. Phys. Rev. B,1988,38:905-908.
[56]H. De Raedt, S. Miyashita, K. Michielsen, et al. Dzyaloshinskii-Moriya Interactions and Adiabatic Magnetization Dynamics in Molecular Magnets [J]. Phys. Rev. B,2004,70: 064401.
[57]T. Moriya. New Mechanism of Anisotropic Superexchange Interaction [J]. Phys. Rev. Lett., 1960,4:228-230.
[58]M. Oshikawa and I. Affleck. Field-Induced Gap in S=1/2 Antiferromagnetic Chains [J]. Phys. Rev. Lett.,1997,79:2883-2886.
[59]R. Jafari. Low-energy-state Dynamics of Entanglement for Spin Systems [J]. Phys. Rev. A, 2010,82:052317.