低维量子系统中多方量子非定域性与量子相变研究

A quantum phase transition(QPT), which is the dramatic change in the ground state of a quantum system, has attracted much attention in condensed matter physics. Recently, measures of multipartite quantum correlation(MQC), such as global quantum entanglement, geometric quantum entanglement and global quantum discord, have been used to characterize QPTs in various low-dimensional quantum systems. The singularity of these quantum information quantities reveals the dramatic changes in the “strength” of MQC, thus has been widely used to detect QPTs. However, the changes in quantum states may also be reflected in the structure (or hierarchy) of MQC, especially in many-body systems. It would be interesting to characterize QPTs by the changes of the structure of MQC in quantum systems. In this project, we will use the generalized Bell-type inequalities and the so-called multipartite quantum nonlocality (MQN) to investigate the structure of MQC and QPTs in low-dimensional quantum models. .. In the calculation of MQN, one will face a multivariable global optimization problem, which is numerically difficult and has greatly restricted the research of MQN. In this project, we will propose a general numerical framework to calculate MQN in low-dimensional systems. First, with the help of infinite time-evolving block decimation (iTEBD) algorithm, we will express the object states (ground states for zero temperature and thermal equilibrium states for finite temperatures) of 1D quantum models in the form of matrix product states(zero temperature) and matrix product density operators(finite temperatures). Second, by using the recursive definition of the multi-site Bell operator, we will reduce the multivariable optimization problem in Bell-type inequalities into a serial of two-site optimization problems, thus can calculate MQN efficiently. The whole process can be integrated into standard DMRG frameworks, thus can be easily used by scientists in the field of condensed matter physics... Our studies will cover the topological QPTs(TQPTs) in a 1D p-wave superconductor and a bond-alternating spin-1/2 Heisenberg chain. In contrary with transitional QPTs, there is no symmetry-breaking in TQPTs, thus one cannot describe the TQPTs with any local order parameter. Instead, TQPTs are related to the topological order of the multi-particle systems. We will study these TQPTs with the help of MQN, and characterize the TQPTs with the change of the structure of MQC in the systems... We will also study MQN in a 1D quantum frustrated diamond chain. The system presents the so-called 1/3 magnetic plateaus under a magnetic field. The unit cell of the lattice consists of three spins, thus multipartite correlation plays a central role in the formation of the magnetic plateaus. In addition, the model undergoes QPTs between different magnetic plateau phases. Thus, we will use MQN to study these phases and figure out the possible change of the correlation structure in the QPTs... Furthermore, we will study the thermal behavior of MQN in quantum chains. Although QPTs occur at zero temperature, according to the third law of thermodynamics one can never achieve zero temperature in the laboratories. Thus, scientists always detect QPTs at finite temperatures in real experiments. We will investigate the effect of thermal fluctuations on MQN, and figure out the ability of MQN in estimating the QPT points at finite temperatures... Finally, we will extend our algorithm to 2D quantum system with tensor network states(TNSs), and study the behavior of MQN in the second-order QPT of the 2D quantum Ising model. We will also study several typical quantum states for 2D lattices that can be expressed exactly in the form of TNSs, including the 2D cluster states, the Toric Code model and the 2D resonating valence bond states. Special attention will be paid to the scaling behavior of MQN in these typical quantum phases and states.

近年来，全局量子纠缠等信息量被用来刻画低维系统中的量子相变(QPT)。这类量的奇异点可捕捉系统中量子关联“程度”的剧变，故可探测QPT。多粒子系统中，除关联“程度”可强弱变化外，关联“结构”亦可有丰富的变化；研究QPT中的关联结构，将是一个新颖有趣的课题。本项目将借助贝尔不等式，通过多方量子非定域性(MQN)的结构来刻画QPT。首先，应用矩阵积态、矩阵积密度算符、张量积态等工具，采取虚时演化算法，求解系统的基态和热力学平衡态；随后，运用Mermin-Svetlichny算符的迭代定义，分解贝尔不等式中的多变量优化难题，解出MQN。分析p波超导链、键交替的自旋链中的拓扑相变，量子阻搓棱形链中的磁化平台相变，二维横场伊辛模型中的相变；借助关联“结构”的变化描述QPT的直观图像；分析团簇态、Toric Code等典型二维量子相(态)；研究MQN的有限尺寸标度；研究热波动对探测QPT准确性的影响。

多体量子系统中的多方量子关联具有丰富的结构。对多方量子关联结构进行研究，有助于为量子相变提供更直观的物理图像。基于此，本项目开展了低维量子系统中多方量子非定域性与量子相变的研究，完成了以下几个方面的工作：一维量子系统中各种量子相变与多方非定域性的研究，二维及更高配位数的系统中量子相变与非定域性的研究，以及有限温度下多方非定域性的研究等。通过对横场伊辛链、自旋梯子、阻挫棱形链、2D横场伊辛格子以及Lipkin-Meshkov-Glick (LMG)模型、J1-J2模型等进行研究，我们发现：多方非定域性为我们理解这些模型及其量子相变提供了有益的视角。伊辛链中，在大N极限下，局域量的边界效应为零，但非定域性的边界效应并不会消失、且在量子相变点达到最大。在梯子链和棱形链中，在特定的量子相中，我们在子系统的“局部非定域性Sp”中观察到了一种新颖的磁致震荡，并发现其包络线与波函数的“全局非定域性Sg”具有相似的形貌；这揭示出Sg对Sp的调制作用。在J1-J2等模型的研究中，我们发现多方非定域性能够很好地指示拓扑量子相变。在2D横场伊辛格子中，我们发现子系统的“边界周长”这一几何特征对非定域性有重要影响；找到了两组证据证明非定域性与纠缠熵之间存在互补竞争关系。在LMG模型中，在某些参数区域，我们发现两方纠缠消失、但多方非定域性仍能指示该量子相变。我们还对非定域性进行了标度分析，观察到了链长、2D网格的边长与面积等几何因素的影响。考虑到热力学第三定律，我们也计算了低温下伊辛链的非定域性，也观察地找到了量子相变的信号；在LMG模型中，我们在非定域性的温度曲线中观察到了一个尖谷；这一反常的热响应行为揭示出了“基态-激发态贡献反转”这一反直觉的底层物理图像。这些研究极大地增加了我们对多方量子关联的认识。最后，我们还完成、完善了一整套计算多方非定域性的张量网络算法，为今后的工作打下了良好的基础。