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横场中具有周期性各向异性的一维XY模型的量子相变

宋加丽 钟鸣 童培庆

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横场中具有周期性各向异性的一维XY模型的量子相变

宋加丽, 钟鸣, 童培庆

Quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field

Song Jia-Li, Zhong Ming, Tong Pei-Qing
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  • 通过解析和数值计算的方法研究了横场中具有周期性各向异性的一维XY自旋模型的量子相变和量子纠缠.主要讨论了周期为二的情况,即各向异性参数交替地取比值为的两个值.结果表明,与横场中均匀XY模型相比,=-1所对应的模型在参数空间的相图存在着明显的不同.原来的Ising相变仍然存在,没有了沿x和y方向的各向异性铁磁(FMx,FMy)相,即各向异性相变消失,出现了一个新的相,并且该相内沿x和y方向的长程关联函数相等且大于零,我们称新相为各向同性铁磁(FMxx)相.这是由于系统新的对称性所导致的.解析结果还说明系统在FMxx相中的单粒子能谱有两个零点,是一个无能隙的相.最后,利用冯诺依曼熵数值地研究了系统在新相内各点的量子纠缠,发现该相内每一点的冯诺依曼熵的标度行为与均匀XY模型在各向异性相变处的相似,即SL~1/3log2 L+Const.
    The quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field with the Hamiltonian where the anisotropy parameters i take and alternately, are studied. The Hamiltonian can be reduced to the diagonal form by Jordan-Wigner and Bogoliubov transformations. The long-range correlations Cx and Cy are calculated numerically. The phase with Cx Cy0 (or Cy Cx0) is referred to as the ferromagnetic (FM) phase along the x (or y) direction, while the phase with Cx=Cy=0 is the paramagnetic (PM) phase. It is found that the phase diagrams with the ratio -1 and =-1 are different obviously. For the case with -1, the line h=hc1=1-[(1-)/2]2 separates an FM phase from a PM phase, while the line =0 is the boundary between a ferromagnetic phase along the x direction (FMx) and a ferromagnetic phase along the y direction (FMy). These are similar to those of the uniform XY chains in a transverse field (i.e., =1). When =-1, the FMx and FMy phases disappear and there appears a new FM phase. The line h=hc2=1-2 separates this new FM phase from the PM phase. The new phase is gapless with two zeros in single particle energy spectrum. This is due to the new symmetry in the system with =-1, i.e., the Hamiltonian is invariant under the transformation 2ix 2i+1y,2iy 2i+1x. The correlation function between the 2i-1 and 2i lattice points along the x (y) direction is equal to that between the 2i and 2i+1 lattice points along the y (x) direction. As a result, the long-range correlation functions along two directions are equivalent. In order to facilitate the description, we call this gapless phase the isotropic ferromagnetic (FMxx) phase. Finally, the relationship between quantum entanglement and quantum phase transitions of the system is studied. The scaling behaviour of the von Neumann entropy at each point in the FMxx phase is SL~1/3log2L+ Const, which is similar to that at the anisotropic phase transition point of the uniform XY model in a transverse field, and different from those in the FMx and FMy phases.
      Corresponding author: Zhong Ming, mzhong@njnu.edu.cn;pqtong@njnu.edu.cn ; Tong Pei-Qing, mzhong@njnu.edu.cn;pqtong@njnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11575087).
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    Vtyurina N N, Dulin D, Docter M W, Meyer A S, Dekker N H, Abbondanzieri E A 2016 Proc. Nat. Acad. Sci. USA 113 4982

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    Babkevich P, Jeong M, Matsumoto Y, Kovacevic I, Finco A, Toft-Petersen R, Ritter C, Månsson M, Nakatsuji S, Rønnow H M 2016 Phys. Rev. Lett. 116 197202

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    Kenzelmann M, Coldea R, Tennant D A, Visser D, Hofmann M, Smeibidl P, Tylczynski Z 2002 Phys. Rev. B 65 144432

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    Lieb E, Schultz T, Mattis D 1961 Ann. Phys. NY 16 407

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    Pfeuty P 1970 Ann. Phys. NY 57 79

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    Osterloh A, Amieo L, Falci G, Fazio R 2002 Nature 416 608

    [15]

    Vidal G, Latorre J I, Rico E, Kitaev A 2003 Phys. Rev. Lett. 90 227902

    [16]

    Franchini F, Its A R, Korepin V E 2008 J. Phys. A:Math. Theor. 41 025302

    [17]

    Raoul D 2008 Phys. Rev. B 78 224413

    [18]

    Guo J L, Wei J L, Qin W, Mu Q X 2015 Quantum Int. Process 14 1429

    [19]

    Cheng W W, Li J X, Shan C J, Gong L Y, Zhao S M 2015 Quantum Int. Process 14 2535

    [20]

    Zanardi P, Paunkovic N 2006 Phys. Rev. E 74 031123

    [21]

    Quan H T, Song Z, Liu X F, Zanardi P, Sun C P 2006 Phys. Rev. Lett. 96 140604

    [22]

    Fisher D 1994 Phys.Rev.B 50 3799

    [23]

    Bunder J, McKenzie R 1999 Phys. Rev. B 60 344

    [24]

    Luck J M 1993 J.Stat.Phys. 72 417

    [25]

    Zhang Z J, Li W J, Zhu X, Xiong Y, Tong P Q 2015 Acta Phys. Sin. 64 190501(in Chinese)[张振俊, 李文娟, 朱璇, 熊烨, 童培庆2015物理学报 64 190501]

    [26]

    Tong P Q, Zhong M 2001 Physica B 304 91

    [27]

    Zhong M, Tong P Q 2010 J. Phys. A:Math. Theor. 43 505302

    [28]

    Tong P Q, Liu X X 2006 Phys. Rev. Lett. 97 017201

    [29]

    Zhong M, Liu X X, Tong P Q 2007 Int. J. Mod. Phys. B 21 4225

    [30]

    Latorre J I, Rico E, Vidal G 2004 Quantum Int. Comput. 4 48

    [31]

    Sachdev S 2011 Quantum Phase Transitions (Cambridge:Cambridge University Press) p133

  • [1]

    Suzuki S, Inoue J I, Chakrabarti B K 2013 Quantum Ising Phases and Transitions in Transverse Ising Models (Berlin:Springer-Verlag) p13

    [2]

    de Gennes P G 1963 Solid State Commun. 1 132

    [3]

    Bitko D, Rosenbaum T F, Aeppli G 1996 Phys. Rev. Lett. 77 940

    [4]

    Vtyurina N N, Dulin D, Docter M W, Meyer A S, Dekker N H, Abbondanzieri E A 2016 Proc. Nat. Acad. Sci. USA 113 4982

    [5]

    Fan B, Branch R W, Nicolau D V, Pilizota T, Steel B C, Maini P K, Berry R M 2010 Science 327 685

    [6]

    Shi Y, Duke T 1998 Phys. Rev. E 58 6399

    [7]

    Sornette D 2014 arXiv:1404.0243v1[q-fin.GN]

    [8]

    Jin B Q, Korepin V E 2004 I. Stat. Phys. 116 79

    [9]

    Islói F, Juhász R 2008 Europhys. Lett. 81 57003

    [10]

    Babkevich P, Jeong M, Matsumoto Y, Kovacevic I, Finco A, Toft-Petersen R, Ritter C, Månsson M, Nakatsuji S, Rønnow H M 2016 Phys. Rev. Lett. 116 197202

    [11]

    Kenzelmann M, Coldea R, Tennant D A, Visser D, Hofmann M, Smeibidl P, Tylczynski Z 2002 Phys. Rev. B 65 144432

    [12]

    Lieb E, Schultz T, Mattis D 1961 Ann. Phys. NY 16 407

    [13]

    Pfeuty P 1970 Ann. Phys. NY 57 79

    [14]

    Osterloh A, Amieo L, Falci G, Fazio R 2002 Nature 416 608

    [15]

    Vidal G, Latorre J I, Rico E, Kitaev A 2003 Phys. Rev. Lett. 90 227902

    [16]

    Franchini F, Its A R, Korepin V E 2008 J. Phys. A:Math. Theor. 41 025302

    [17]

    Raoul D 2008 Phys. Rev. B 78 224413

    [18]

    Guo J L, Wei J L, Qin W, Mu Q X 2015 Quantum Int. Process 14 1429

    [19]

    Cheng W W, Li J X, Shan C J, Gong L Y, Zhao S M 2015 Quantum Int. Process 14 2535

    [20]

    Zanardi P, Paunkovic N 2006 Phys. Rev. E 74 031123

    [21]

    Quan H T, Song Z, Liu X F, Zanardi P, Sun C P 2006 Phys. Rev. Lett. 96 140604

    [22]

    Fisher D 1994 Phys.Rev.B 50 3799

    [23]

    Bunder J, McKenzie R 1999 Phys. Rev. B 60 344

    [24]

    Luck J M 1993 J.Stat.Phys. 72 417

    [25]

    Zhang Z J, Li W J, Zhu X, Xiong Y, Tong P Q 2015 Acta Phys. Sin. 64 190501(in Chinese)[张振俊, 李文娟, 朱璇, 熊烨, 童培庆2015物理学报 64 190501]

    [26]

    Tong P Q, Zhong M 2001 Physica B 304 91

    [27]

    Zhong M, Tong P Q 2010 J. Phys. A:Math. Theor. 43 505302

    [28]

    Tong P Q, Liu X X 2006 Phys. Rev. Lett. 97 017201

    [29]

    Zhong M, Liu X X, Tong P Q 2007 Int. J. Mod. Phys. B 21 4225

    [30]

    Latorre J I, Rico E, Vidal G 2004 Quantum Int. Comput. 4 48

    [31]

    Sachdev S 2011 Quantum Phase Transitions (Cambridge:Cambridge University Press) p133

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出版历程
  • 收稿日期:  2017-02-18
  • 修回日期:  2017-05-18
  • 刊出日期:  2017-09-05

横场中具有周期性各向异性的一维XY模型的量子相变

    基金项目: 国家自然科学基金(批准号:11575087)资助的课题.

摘要: 通过解析和数值计算的方法研究了横场中具有周期性各向异性的一维XY自旋模型的量子相变和量子纠缠.主要讨论了周期为二的情况,即各向异性参数交替地取比值为的两个值.结果表明,与横场中均匀XY模型相比,=-1所对应的模型在参数空间的相图存在着明显的不同.原来的Ising相变仍然存在,没有了沿x和y方向的各向异性铁磁(FMx,FMy)相,即各向异性相变消失,出现了一个新的相,并且该相内沿x和y方向的长程关联函数相等且大于零,我们称新相为各向同性铁磁(FMxx)相.这是由于系统新的对称性所导致的.解析结果还说明系统在FMxx相中的单粒子能谱有两个零点,是一个无能隙的相.最后,利用冯诺依曼熵数值地研究了系统在新相内各点的量子纠缠,发现该相内每一点的冯诺依曼熵的标度行为与均匀XY模型在各向异性相变处的相似,即SL~1/3log2 L+Const.

English Abstract

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