张量积投影态在变分蒙特卡罗方法的理论及应用

Approximating ground states of strongly correlated quantum systems is one of the significant challenges in condensed matter physics. In recent years, there has been a class of tensor network states aimed at tackling this problem, e.g., tensor- and correlator-based wave functions. The former is based on a class of states which introduce the tensors with auxiliary degrees of freedom to generate correlations. For a quantum system with high dimensions and large lattice size, the tensors, living on the sites and being connected by a set of links, will give rise to considerable computational complexity. The latter explicitly correlates the physical degrees of freedom in trial wave functions, more suitable for practical calculations in complex many-body systems. However, it is unclear how well this ansatz can reproduce correlation functions for fermions. Thus, in order for tensor product approaches to be efficient and the correlator ansatz to be accurate, we further introduce correlations between fermions in real space by multiplying the tensor or correlator with the Slater determinant, which we call tensor- and correlator-product projected states (TPPS and CPPS). According to the variational ansatz of TPPS and CPPS, many important issues in strongly correlated electron systems can be re-examined. In the following we will focus on two interesting topics: (1) The entanglement measure in a specific Hamiltonian of interacting fermions in two dimensions; (2) The variational phase diagram of the multi-orbital Hubbard models with strong spin-orbit coupling. In part (1), using variational Monte Carlo method, we calculate the entanglement entropy of TPPS or CPPS in given fermionic Hamiltonians, e.g., Hubbard- and t-J-type models. We further examine how the entanglement entropy of these fermionic ground states is modified by correlations introduced by tensors or correlators. A fundamental question thus can be answered: Could TPPS and CPPS have ability to generate the long-range quantum correlations (topological entangled states), except the long-range classical correlations (symmetry broken states)? As for part (2), we will consider 5d transition metal oxide, 213 and 214 iridate compounds, which are recently found to be a candidate for a novel Mott insulator. Theoretically the insulating mechanism and the possibility of superconductivity after doping are still under debate. Therefore, the systematic study of the ground-state phase diagram using TPPS or CPPS is highly desired.

如何寻找强关联量子系统的基态一直是凝聚态物理学家的重大挑战。近年发展的张量网络波函数主要有两类：张量型与关联子型。前者透过带有辅助自由度的张量产生关联性，其缺点是：张量构成的网络结构会造成庞大的计算器负荷，因而限制应用范围。后者则引进真实的物理自由度，然而却引进过多的假设，造成描述实际系统关联的困难度。最近我们已成功且有效的结合传统平均场波函数与张量网络，增进试探波函数的准确性。 根据这些变分假设，本项目着重两大主题：（1）交互作用费米子哈密顿量的纠缠测量。利用变分蒙特卡罗法计算特定费米子哈密顿量的纠缠熵。一个基本问题可获得解答：除了远程经典关联外，张量与关联子是否也有能力产生远程量子关联？（2）多轨道Hubbard模型的基态相图。我们将关注一个新型Mott绝缘材料：铱系化合物，目前其绝缘机制及超导的可能性仍具争议，因此系统化的研究其基态相图将是刻不容缓的工作。

围绕强关联多体量子系统计算⽅法,发展了张量网络态求解基态以及低能激发态的有效且准确的算法,写出了完整的张量网络态算法程序,可以实现包括U(1) 和SU(2)对称的张量网络基态及其激发态的模拟,以及完成各种局部和全局观测量的测量的计算,如能量,总⾓角动量,平移算符镜像算符本征值等,对于判断基态相图极为重要。同时研究了⽅格子上的J1-J2模型的基态相图,利⽤激发态能级交叉现象来判断热力学极限下的基态相图,通过能隙随系统尺寸的指数变化关系推断在参数空间g介于0.46-0.52的范围内系统存在无能隙的自旋液体相。科研成果首次在二维量子系统中发现了由S=2和S=0的低能级交叉标记的二级量子相变,揭示了通过激发态研究基态及相变的新的思路和途径,开拓了张量网络态研究多体量子系统激发态和相变的新领域。