一维与准一维量子多体系统的相变和临界现象
发布时间：2018-09-10 17:34
【摘要】：本文使用了以张量网络表述为背景的一维无限虚时间演化块算法(infinite time evolving block decimation,iTEBD)和准一维的沿梯度定向行走的梯子算法(projected entangled-pair states,PEPS对梯子系统的几何适应),对一维和准一维量子自旋多体系统进行了模拟研究。在热力学极限下,通过近似逼近的办法,获得了很好的一维无限链和准一维无限梯子的量子多体系统的基态波函数。然后由表示基态波函数的张量表示出了量子多体系统对应的约化密度矩阵。进而使用基态单位格点保真度理论和其它可观测物理量的算符作用在约化密度矩阵上,从而得到关于量子相变、量子相变点以及量子多体系统所处的态的序参量等这些让人感兴趣的物理信息。本文的第一章是绪论,介绍了和本课题密切相关的一些背景知识。首先以热力学相变为参照介绍了什么是量子相变的概念。其次介绍了基态能谱、保真度、自发对称性破缺、局域序参量、重整化群论和标度理论等基本概念。再次介绍了二分纠缠熵——冯诺伊曼熵,以及使用冯诺伊曼熵时必须满足的面积定理。本文第二章第一部分介绍了准一维沿梯度定向行走的梯子张量网络算法更新基态波函数、得到近似基态波函数的过程。第二部分运用梯度算法结合单位格点保真度理论,对“具有反铁磁交叉交换相互作用的海森堡受挫自旋梯子”的相图构成进行了模拟研究,研究的原因在于:该区域存在关于CD相是否存在的争议。单位格点保真度相图捕捉到了一个相变点和连续变化的过程,保真度之所以能够探测到相变点和相变变化的过程是因为:波函数内部结构本质上的不同,由于内部结构的不同导致了挤点(相变点)的出现;另一方面对于连续相变,当参考态的选择不是自身时,保真度的值总是沿着重整化群流的方向单调的减小(连续变化的原因)。同时,本文还首次将单位格点保真度探测简并基态的功能引入到了受挫两腿自旋梯子量子多体系统里面。结果表明:在反铁磁区域,没有属于Landau-Ginzberg-Wilson范式的由自发对称性破缺所引起的简并基态。最后从超越Landau理论非局域长程序——弦序参量,对系统的两个拓扑序进行了刻画,结果表明:只有Handane相和rung-singlet相的存在(CD相的局域序参量的值一直保持为零)。另外,本文还首次从哈密顿量对称性对相图的限制进行了讨论。几个方面的结果一致赞成所研究区域的相图的本质是:该区域仅由Handane相和rung-singlet相构成(没有CD相的存在),两相之间发生的是连续相变。本文第三章第一部分介绍了观测量——单位格点几何纠缠的实现过程。第二部分基于准一维沿梯度定向行走的梯子张量网络算法结合单位格点几何纠缠和基态单位格点保真度理论,对“具有反铁磁交叉交换相互作用的海森堡受挫自旋梯子模型”的相图相变级数进行了模拟研究。研究相变级数的原因在于:该模型在强弱耦合作用不同的条件下,相变级数存在着争议。为了让数据更具有说服力,我们从几何纠缠、弦序参量和单位格点基态保真度分别进行了研究。三个方面的结果一致支持:在弱耦合作用区,系统发生的是连续相变;而在强耦合作用区,系统发生的是非连续相变。目前整条相变线的数据还没有计算完,根据现有阶段性结果能判断出由于耦合作用强弱的不同,相图相变的级数会相应的变化。完整的相图会在下一步工作中去实现。本文第四章第一部分介绍了iTEBD算法。第二部分基于iTEBD算法结合基态波函数的约化密度矩阵导出的各种观测量模拟研究了扩展的量子罗盘模型(extended quantum compass model,EQCM)的相图。首次从序参量的角度深入研究了这个模型的相图。为了合适地描述相图当中相的序参量,我们套用了从保真度理论研究相变的三大步骤:第一,确定出相变点和各个相所在区域,用基态能量和导数确定了相变点和各个相所在区域。第二,刻画指定相中选出作为代表态的局域序参量(如果有任何可能的局域序参量),用两点关联和局域磁化强度导出了局域序参量,并用局域序参量刻画对应相。第三,刻画非局域长程序,用短程弦关联函数的单调行为和振荡行为定义了可能的弦序参量,然后做饱和验证,满足饱和验证就导出弦序参量,并用弦序参量刻画对应相。同时,我们对序参量在临界区域附近的临界行为、冯诺伊曼熵在临界点处的有限纠缠标度行为做了标度。临界指数?=1/8、中心荷c?0.5表明:连续相变线的相变类型属于Ising普适类。第五章介绍主要结论和后续研究工作展望。
[Abstract]:In this paper, the infinite time evolving block decimation (iTEBD) algorithm and the projected entangled-pair states (PEPS) algorithm for one-dimensional and quasi-one-dimensional quantum spin multibody systems are used. The ground state wave functions of quantum multibody systems with one-dimensional infinite chains and quasi-one-dimensional infinite ladders are obtained by approximate approximation under the thermodynamic limit. Then the reduced density matrices corresponding to the quantum multibody systems are shown by the tensors representing the ground state wave functions. Degree theory and operators of other observable physical quantities act on the reduced density matrix to obtain interesting physical information about quantum phase transitions, quantum phase transition points and the order parameters of the states in which quantum multibody systems are located. In this paper, the concept of quantum phase transition is introduced with reference to thermodynamic phase transition. Secondly, the basic concepts of ground state energy spectrum, fidelity, spontaneous symmetry breaking, local order parameter, renormalization group theory and scaling theory are introduced. In the second chapter, the process of updating the ground state wave function and obtaining the approximate ground state wave function by the quasi-one-dimensional gradient-oriented ladder tensor network algorithm is introduced. The simulation of the phase diagram is carried out because there is a controversy about whether the CD phase exists in the region. The unit lattice fidelity phase diagram captures a phase transition point and a continuous change process. The reason why the fidelity can detect the phase transition point and the phase transition process is that the internal structure of the wave function is essentially different. On the other hand, for the continuous phase transition, the fidelity decreases monotonously along the direction of the renormalization group flow (the cause of the continuous change). At the same time, the function of detecting degenerate ground state with the fidelity of the unit lattice is introduced for the first time. The results show that in the antiferromagnetic region, there is no degenerate ground state caused by spontaneous symmetry breaking in the Landau-Ginzberg-Wilson normal form. Finally, the two topological orders of the system are characterized by the nonlocal long program, the string order parameter, which transcends the Landau theory. It is shown that only Handane phase and rung-singlet phase exist (the value of local order parameter of CD phase remains zero all the time). In addition, the limitation of phase diagrams from Hamiltonian symmetry is discussed for the first time. The results from several aspects agree that the nature of phase diagrams in the studied region is that the region is composed only of Handane phase and rung-singlet phase. In the third chapter, we introduce the realization of the geometric entanglement of the unit lattice. In the second part, we combine the geometric entanglement of the unit lattice and the fidelity theory of the ground state unit lattice with the algorithm of the ladder tensor network based on the quasi-one-dimensional directional gradient walk. The phase transition series of Heisenberg frustrated spin ladder model with antiferromagnetic cross-exchange interaction is simulated. The reason for studying the phase transition series is that the phase transition series of the model is controversial under different coupling conditions. To make the data more convincing, we consider the geometric entanglement and the chord order parameter. In the weak coupling region, the continuous phase transition occurs; in the strong coupling region, the discontinuous phase transition occurs. In the fourth chapter, the iTEBD algorithm is introduced. In the second part, based on the iTEBD algorithm and the reduced density matrix of the ground state wave function, the extended quantum compass model is studied. The phase diagram of the extended quantum compass model (EQCM) is studied from the point of view of order parameters for the first time. In order to describe the order parameters of the phase diagram appropriately, three steps are applied to study the phase transition from the fidelity theory. Firstly, the phase transition point and the region of each phase are determined, and the ground state energy and derivative are used to determine the phase transition. Secondly, the local order parameter (if any possible local order parameter) selected from the specified phase is characterized, and the local order parameter is derived by two-point correlation and local magnetization, and the corresponding phase is characterized by the local order parameter. Thirdly, the non-local long program is described, and the short-range chord correlation function is used. The possible chord order parameters are defined for the monotonic and oscillatory behavior of numbers, and then saturation tests are performed to derive the chord order parameters and characterize the corresponding phase by chord order parameters. The exponent?=1/8 and the central charge c?0.5 indicate that the phase transition types of the continuous phase change lines belong to the Ising universal class.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O469
[Abstract]:In this paper, the infinite time evolving block decimation (iTEBD) algorithm and the projected entangled-pair states (PEPS) algorithm for one-dimensional and quasi-one-dimensional quantum spin multibody systems are used. The ground state wave functions of quantum multibody systems with one-dimensional infinite chains and quasi-one-dimensional infinite ladders are obtained by approximate approximation under the thermodynamic limit. Then the reduced density matrices corresponding to the quantum multibody systems are shown by the tensors representing the ground state wave functions. Degree theory and operators of other observable physical quantities act on the reduced density matrix to obtain interesting physical information about quantum phase transitions, quantum phase transition points and the order parameters of the states in which quantum multibody systems are located. In this paper, the concept of quantum phase transition is introduced with reference to thermodynamic phase transition. Secondly, the basic concepts of ground state energy spectrum, fidelity, spontaneous symmetry breaking, local order parameter, renormalization group theory and scaling theory are introduced. In the second chapter, the process of updating the ground state wave function and obtaining the approximate ground state wave function by the quasi-one-dimensional gradient-oriented ladder tensor network algorithm is introduced. The simulation of the phase diagram is carried out because there is a controversy about whether the CD phase exists in the region. The unit lattice fidelity phase diagram captures a phase transition point and a continuous change process. The reason why the fidelity can detect the phase transition point and the phase transition process is that the internal structure of the wave function is essentially different. On the other hand, for the continuous phase transition, the fidelity decreases monotonously along the direction of the renormalization group flow (the cause of the continuous change). At the same time, the function of detecting degenerate ground state with the fidelity of the unit lattice is introduced for the first time. The results show that in the antiferromagnetic region, there is no degenerate ground state caused by spontaneous symmetry breaking in the Landau-Ginzberg-Wilson normal form. Finally, the two topological orders of the system are characterized by the nonlocal long program, the string order parameter, which transcends the Landau theory. It is shown that only Handane phase and rung-singlet phase exist (the value of local order parameter of CD phase remains zero all the time). In addition, the limitation of phase diagrams from Hamiltonian symmetry is discussed for the first time. The results from several aspects agree that the nature of phase diagrams in the studied region is that the region is composed only of Handane phase and rung-singlet phase. In the third chapter, we introduce the realization of the geometric entanglement of the unit lattice. In the second part, we combine the geometric entanglement of the unit lattice and the fidelity theory of the ground state unit lattice with the algorithm of the ladder tensor network based on the quasi-one-dimensional directional gradient walk. The phase transition series of Heisenberg frustrated spin ladder model with antiferromagnetic cross-exchange interaction is simulated. The reason for studying the phase transition series is that the phase transition series of the model is controversial under different coupling conditions. To make the data more convincing, we consider the geometric entanglement and the chord order parameter. In the weak coupling region, the continuous phase transition occurs; in the strong coupling region, the discontinuous phase transition occurs. In the fourth chapter, the iTEBD algorithm is introduced. In the second part, based on the iTEBD algorithm and the reduced density matrix of the ground state wave function, the extended quantum compass model is studied. The phase diagram of the extended quantum compass model (EQCM) is studied from the point of view of order parameters for the first time. In order to describe the order parameters of the phase diagram appropriately, three steps are applied to study the phase transition from the fidelity theory. Firstly, the phase transition point and the region of each phase are determined, and the ground state energy and derivative are used to determine the phase transition. Secondly, the local order parameter (if any possible local order parameter) selected from the specified phase is characterized, and the local order parameter is derived by two-point correlation and local magnetization, and the corresponding phase is characterized by the local order parameter. Thirdly, the non-local long program is described, and the short-range chord correlation function is used. The possible chord order parameters are defined for the monotonic and oscillatory behavior of numbers, and then saturation tests are performed to derive the chord order parameters and characterize the corresponding phase by chord order parameters. The exponent?=1/8 and the central charge c?0.5 indicate that the phase transition types of the continuous phase change lines belong to the Ising universal class.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O469
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