两腿XXZ自旋梯子模型的量子相变与量子临界现象
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摘要
对于无限大尺寸两腿自旋1/2的XXZ自旋梯子模型,通过运用基于随机行走的张量网络(TN)算法数值模拟出基态波函数,首次尝试研究自旋梯子模型的约化保真度、普适序参量、纠缠熵等物理观测量,并系统研究基态保真度的三维挤点与二维分叉、约化保真度的分叉、局域序参量、普适序参量、纠缠熵和量子相变之间存在的关联关系.基于张量网络表示的算法在任意随机选择初始状态时,可以得到两腿XXZ量子自旋梯子系统简并的对称破缺基态波函数,该基态波函数是由于Z2对称破缺引起的.本文期望所提供的方法可为进一步研究凝聚态物质中热力学极限下的强关联电子量子晶格自旋梯子系统的量子相变和量子临界现象提供一种更有效的强大的工具.
In this study,a spin-1/2 two-leg XXZ spin ladder model with a double coupled infinite-size ladder system is systematically studied in terms of a tensor network( TN) algorithm. The algorithm is based on the tensor network representation of quantum many-body states,as an adaptation of the projected entangled pair states to the geometry of translationally invariant infinite-size quantum spin ladders. The TN algorithm provides an effective method by which to generate degenerate symmetry-breaking ground-state wave functions arising from the Z2 symmetry breaking,each of which results from a randomly chosen initial state. At the same time,we have also investigated an intriguing connection among fidelity per lattice site,reduced fidelity,local order parameter,universal order parameter,entropy and quantum phase transitions in the ground-state for the quantum two-leg XXZ model. The ground-state phase diagram of the ladder model exhibits a rich diversity of quantum phases. We expect that this approach may provide further insights into critical phenomena in quantum many-body lattice systems of condensed matter.
In this study,a spin-1/2 two-leg XXZ spin ladder model with a double coupled infinite-size ladder system is systematically studied in terms of a tensor network( TN) algorithm. The algorithm is based on the tensor network representation of quantum many-body states,as an adaptation of the projected entangled pair states to the geometry of translationally invariant infinite-size quantum spin ladders. The TN algorithm provides an effective method by which to generate degenerate symmetry-breaking ground-state wave functions arising from the Z2 symmetry breaking,each of which results from a randomly chosen initial state. At the same time,we have also investigated an intriguing connection among fidelity per lattice site,reduced fidelity,local order parameter,universal order parameter,entropy and quantum phase transitions in the ground-state for the quantum two-leg XXZ model. The ground-state phase diagram of the ladder model exhibits a rich diversity of quantum phases. We expect that this approach may provide further insights into critical phenomena in quantum many-body lattice systems of condensed matter.
引文
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[45]Yang J,Cong M Y,Huang Y X.Quantum entanglement and geometric discord in the Heisenberg spin chain with three-site interaction[J].J.At.Mol.Phys.,2017,34:919(in Chinese)[杨晶,丛美艳,黄燕霞.具有三体相互作用海森堡自旋链模型的纠缠与几何失协[J].原子与分子物理学报,2017,34:919]
[46]Chen Y,Zanardi P,Wang Z D,et al.Sublattice entanglement and quantum phase transitions in antiferromagnetic spin chains[J].New J.Phys.,2006,8:97.
[47]Hijii K,Kitazawa A,Nomura K.Phase diagram of S=1/2 two-leg XXZ spin-ladder systems[J].Phys.Rev.B,2005,72:014449.
[48]Crawford J D.Introduction to bifurcation theory[J].Rev.Mod.Phys.,1991,63:991.
[49]Gorin T,Prosen T,Seligman T H,et al.Dynamics of Loschmidt echoes and fidelity decay[J].Phys.Rep.,2006,435:33.
[50]Paunkovi8 N,Sacramento P D,Nogueira P,et al.Fidelity between partial states as a signature of quantum phase transitions[J].Phys.Rev.A,2008,77:052302.
[51]Nijs M D,Rommelse K.Preroughening transitions in crystal surfaces and valence bond phases in quantum spin chains[J].Phys.Rev.B,1989,40:4709.
[2]Batchelor M T,Guan X W,Oelkers N,et al.Integrable models and quantum spin ladders:comparison between theory and experiment for the strong coupling ladder compounds[J].Adv.Phys.,2007,56:465.
[3]White S R,Noack R M,Scalapino D J.Resonating valence bond theory of coupled Heisenberg chains[J].Phys.Rev.Lett.,1994,73:886.
[4]Shelton D G,Nersesyan A A,Tsvelik A M.Antiferromagnetic spin ladders:Crossover between spin S=1/2and S=1 chains[J].Phys.Rev.B,1996,53:8521.
[5]Nersesyan A A,Tsvelik A M.One-dimensional spin-liquid without magnon excitations[J].Phys.Rev.Lett.,1997,78:3939.
[6]Fouet J B,Mila F,Clarke D,et al.Condensation of magnons and spinons in a frustrated ladder[J].Phys.Rev.B,2006,73:214405.
[7]Meyer J S,Matveev K A.Wigner crystal physics in quantum wires[J].J.Phys.:Condens.Matter,2009,21:023203.
[8]Sheng D N,Motrunich O I,Trebst S,et al.Strongcoupling phases of frustrated bosons on a two-leg ladder with ring exchange[J].Phys.Rev.B,2008,78:054520.
[9]Strong S P,Millis A J.Competition between singlet formation and magnetic ordering in one-dimensional spin systems[J].Phys.Rev.Lett.,1992,69:2419.
[10]Watanabe H,Nomura K,Takada S.S=1/2 quantum heisenberg ladder and S=1 haldane phase[J].J.Phys.Soc.Jpn.,1993,62:2845.
[11]Narushima T,Nakamura T,Takada S.Numerical study on the ground-state phase diagram of the S=1/2 XXZ ladder model[J].J.Phys.Soc.Jpn.,1995,64:4322.
[12]White S R.Density matrix formulation for quantum renormalization groups[J].Phys.Rev.Lett.,1992,69:2863.
[13]Ueda M,Nagata T,Akimitsu J,et al.Superconductivity in the ladder material Sr0.4Ca13.6Cu24O41.84[J].J.Phys.Soc.Jpn.,1996,65:2764.
[14]Kolezhuk A K,Mikeska H J.Phase transitions in the Heisenberg spin ladder with ferromagnetic legs[J].Phys.Rev.B,1996,53:R8848.
[15]Kopinga K,Tinus A M C,de Jonge W J M.Magnetic behavior of the ferromagnetic quantum chain systems(C6H11NH3)Cu Cl3(CHAC)and(C6H11NH3)Cu Br3(CHAB)[J].Phys.Rev.B,1982,25:4685.
[16]Kopinga K,Tinus A M C,de Jonge W J M.Evidence for solitons in a quantum(S=1/2)XY system[J].Phys.Rev.B,1984,29:2868(R).
[17]Vekua T,Japaridze G I,Mikeska H J.Phase diagrams of spin ladders with ferromagnetic legs[J].Phys.Rev.B,2003,67:064419.
[18]Legeza O,Solyom J.Stability of the Haldane phase in anisotropic magnetic ladders[J].Phys.Rev.B,1997,56:14449.
[19]Lecheminant P,Orignac E.Magnetization and dimerization profiles of the cut two-leg spin ladder[J].Phys.Rev.B,2002,65:174406.
[20]Hung H H,Gong C D,Chen Y C,et al.Search for quantum dimer phases and transitions in a frustrated spin ladder[J].Phys.Rev.B,2006,73:224433.
[21]Liu G H,Deng X Y,Wen R.Phase diagram of the frustrated spin ladder with the next nearest intrachain couplings[J].Physica B,2012,407:2068.
[22]Zanardi P,Paunkovi8 N.Ground state overlap and quantum phase transitions[J].Phys.Rev.E,2006,74:031123.
[23]Li B,Li S H,Zhou H Q.Quantum phase transitions in a two-dimensional quantum XYX model:Ground-state fidelity and entanglement[J].Phys.Rev.E,2009,79:060101(R).
[24]Zhou H Q,Zhao J H,Li B.Fidelity approach to quantum phase transitions:finite-size scaling for the quantum Ising model in a transverse field[J].J.Phys.A:Math.Theor.,2008,41:492002.
[25]Zhou H Q,Orus R,Vidal G.Ground state fidelity from tensor network representations[J].Phys.Rev.Lett.,2008,100:080601.
[26]Zhao J H,Wang H L,Li B,et al.Spontaneous symmetry breaking and bifurcations in ground-state fidelity for quantum lattice systems[J].Phys.Rev.E,2010,82:061127.
[27]Wang H L,Zhao J H,Li B,et al.Kosterlitz–Thouless phase transition and ground state fidelity:a novel perspective from matrix product states[J].J.Stat.Mech.,2011,L10001.
[28]Rams M M,Damski B.Quantum fidelity in the thermodynamic limit[J].Phys.Rev.Lett.,2011,106:055701.
[29]Li S H,Shi Q Q,Su Y H,et al.Tensor network states and ground-state fidelity for quantum spin ladders[J].Phys.Rev.B,2012,86:064401.
[30]Li S H,Shi Q Q,Batchelor M T,et al.Groundstate fidelity phase diagram of the fully anisotropic two-leg spin-1/2 XXZ ladder[J].2017,ar Xiv:1704.05484.
[31]Chen X H,Cho S Y,Batchelor M T,et al.The antiferromagnetic cross-coupled spin ladder:Quantum fidelity and tensor networks approach[J].J.Kor.Phys.Soc.,2016,68:1114.
[32]Sachdev S.Quantum phase transitions[M].Cambridge:Cambridge University Press,1999.
[33]Wen X G.Quantum Field Theory of Many-Body Systems[M].Oxford:Oxford University Press,2004.
[34]Liu J H,Shi Q Q,Zhao J H,et al.Quantum phase transitions and bifurcations:reduced fidelity as a phase transition indicator for quantum lattice many body systems[J].J.Phys.A:Math.Theor.,2011,44:495302.
[35]Dai Y W,Cho S Y,Batchelor M T,et al.Degenerate ground states and multiple bifurcations in a two-dimensional q-state quantum Potts model[J].Phys.Rev.E,2014,89:062142.
[36]Li S H,Wu X B,Huang C F,et al.Ground-state fidelity and order parameter for quantum 3-state Potts model in 2-D[J].J.At.Mol.Phys.,2015,32:526(in Chinese)[李生好,伍小兵,黄崇富,等.二维3态Potts量子系统的保真度与序参量[J].原子与分子物理学报,2015,32:526]
[37]Li S H,Wu X B,Huang C F,et al.Optimization of the projected entangled pair state algorithm for quantum systems[J].Acta Phys.Sin.,2014,63:140501(in Chinese)[李生好,伍小兵,黄崇富,等.基于投影纠缠对态算法优化的研究[J].物理学报,2014,63:140501]
[38]Liu J H,Shi Q Q,Wang H L,et al.Universal construction of order parameters for translation invariant quantum lattice systems with symmetry breaking order[J].Phys.Rev.E,2012,86:020102.
[39]Tang Z R,Ruan X S,Chen G X,et al.Quantum entanglement and coherence in two coupled nonlinear oscillators[J].J.At.Mol.Phys.,2016,33:873(in Chinese)[唐柱荣,阮小爽,陈耿祥,等.用两耦合非线性振子模型研究量子纠缠和相干性[J].原子与分子物理学报,2016,33:873]
[40]Cong M Y,Yang J,Huang Y X.Effects of Dzyaloshinskii-Moriya interaction and intrinsic decoherence on quantum entanglement in a three-qubit XXZ Heisenberg system[J].J.At.Mol.Phys.,2017,34:318(in Chinese)[丛美艳,杨晶,黄燕霞.Dzyaloshinskii-Moriya相互作用和內禀退相干对三粒子XXZ海森堡系统的量子纠缠的影响[J].原子与分子物理学报,2017,34:318]
[41]Osborne T J,Nielsen M A.Entanglement in a simple quantum phase transition[J].Phys.Rev.A,2002,66:032110.
[42]Osterloh A,Amico L,Falci G,et al.Scaling of entanglement close to a quantum phase transition[J].Nature,2002,416:608.
[43]Huang C Y,Lin F L.Multipartite entanglement measures and quantum criticality from matrix and tensor product states[J].Phys.Rev.A,2010,81:032304.
[44]Shi Q Q,Wang H L,Li S H,et al.Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models[J].Phys.Rev.A,2016,93:062341.
[45]Yang J,Cong M Y,Huang Y X.Quantum entanglement and geometric discord in the Heisenberg spin chain with three-site interaction[J].J.At.Mol.Phys.,2017,34:919(in Chinese)[杨晶,丛美艳,黄燕霞.具有三体相互作用海森堡自旋链模型的纠缠与几何失协[J].原子与分子物理学报,2017,34:919]
[46]Chen Y,Zanardi P,Wang Z D,et al.Sublattice entanglement and quantum phase transitions in antiferromagnetic spin chains[J].New J.Phys.,2006,8:97.
[47]Hijii K,Kitazawa A,Nomura K.Phase diagram of S=1/2 two-leg XXZ spin-ladder systems[J].Phys.Rev.B,2005,72:014449.
[48]Crawford J D.Introduction to bifurcation theory[J].Rev.Mod.Phys.,1991,63:991.
[49]Gorin T,Prosen T,Seligman T H,et al.Dynamics of Loschmidt echoes and fidelity decay[J].Phys.Rep.,2006,435:33.
[50]Paunkovi8 N,Sacramento P D,Nogueira P,et al.Fidelity between partial states as a signature of quantum phase transitions[J].Phys.Rev.A,2008,77:052302.
[51]Nijs M D,Rommelse K.Preroughening transitions in crystal surfaces and valence bond phases in quantum spin chains[J].Phys.Rev.B,1989,40:4709.