二维量子多体系统的张量网络态算法

发布时间：2017-12-31 18:16

本文关键词：二维量子多体系统的张量网络态算法　出处：《中国科学技术大学》2017年博士论文　论文类型：学位论文

【摘要】：发展求解强关联系统的高效的数值方法是现代物理最为核心的任务之一。在强关联的相互作用体系中,由于传统微扰论不再适用,研究强关联系统的物理性质主要依靠数值求解办法,包括严格对角化方法,量子蒙特卡洛方法和密度矩阵重整化群方法。这些数值方法已被广泛应用于研究强关联系统,并且取得了巨大的成功。然而,上述方法都有其局限性:严格对角化方法会遇到所谓的"指数墙"问题;量子蒙特卡洛方法在处理费米子问题和阻挫磁性问题时会遇到符号问题;而密度矩阵重整化方法主要用于处理一维或准一维系统,难以处理更高维系统。因此,发展新的高效的数值方法仍然是解决强关联问题的当务之急。近些年来,人们开始以量子信息理论的视角看待问题,通过对量子纠缠的深入理解,一种基于量子纠缠的张量网络态(TNS)理论,包括矩阵乘积态(MPS)理论和投影纠缠对态(PEPS)理论逐渐建立起来。MPS和PEPS分别描述一维和二维系统时都满足纠缠熵的面积定律和尺寸一致性,已经被证明是研究强关联系统的强有力的工具。基于MPS表示,人们建立起了描述一维量子多体系统的完善的理论。对于二维系统,基于PEPS的相关算法还处于非常初级的阶段。由于PEPS本身的复杂性和计算能力的限制,其在实际应用中受到了很大限制。我们希望能够发展一种高效地算法,使得PEPS可以能够真正解决一些长期以来难以求解的问题。基于二维量子多体系统基态的张量网络态表示,本论文讲述了我们发展的求解二维量子多体系统基态的方法,主要内容包括两部分:第一部分着重讲述了如何用副本交换的分子动力学方法来解决用TNS求解多体问题时遇到的局域极小值问题。用TNS做为变分波函数求解多体系统的基态的很关键一步是如何有效优化这个变分波函数,使得其尽量避免陷入局域极小值。我们发展了一种可以大规模并行的高效地副本交换分子动力学方法,用来解决这个问题。通过将TNS的元素看做广义坐标,我们把这个优化问题映射到一个经典力学问题。在优化时,我们首先设定一系列不同的温度,然后从随机态出发,根据这个经典系统的势能函数,采用分子动力学的方法对系统进行演化,最终会得到不同温度下的解,零温下的解就是这个优化问题的解。为了避免在分子动力学的演化过程中陷入局域极小值,可以采用副本交换的方法将不同温度下的构型进行充分交换来帮助其跳出局域极小值。第二部分着重讲述了如何用PEPS的变分波函数来有效地求解二维量子自旋系统。PEPS可以很好地描述二维系统的基态,但是由于其计算复杂度很高,在用它来模拟二维体系时受到很大限制。我们提出了用梯度优化结合蒙特卡洛采样的方法来优化PEPS变分波函数。首先我们采用一种虚实演化的SU(simple update)方法来得到一个粗糙的PEPS波函数做为出发点,然后通过梯度优化来进一步精确地优化这个波函数来得到基态。在计算梯度和能量时我们采用了蒙特卡洛采样的方法。与人们常用的方法相比,这种方法不仅大大地降低了计算复杂度,而且采用的梯度优化算法可以更加精确地优化变分波函数,使得用PEPS解决一些长期以来难以求解的多体系统成为可能。
[Abstract]:For the development of efficient numerical methods for solving strongly correlated systems is one of the core tasks of modern physics. The interaction system in the strong correlation, because the traditional perturbation theory is no longer applicable, the physical properties of strongly correlated systems rely mainly on numerical solution, including the exact diagonalization method, quantum Monte Carlo method and the density matrix renormalization group method. These numerical methods have been widely used in the study of strongly correlated systems, and achieved great success. However, these methods have their limitations: the strict diagonalization method will encounter the so-called "refers to the number of wall" problem; quantum Monte Carlo method will encounter problems in dealing with problems and symbolic fermion frustrated magnetic the problem; and the density matrix renormalization method is mainly used for processing one dimensional system, difficult to deal with higher dimensional systems. Therefore, the development of efficient numerical methods is still new To solve the problem of strong association a pressing matter of the moment. In recent years, people began to look at the problem in quantum information theory, through in-depth understanding of quantum entanglement, a tensor network state based on quantum entanglement (TNS) theory, including the matrix product state (MPS) theory and projection of entanglement state (PEPS) theory has been gradually established.MPS and PEPS respectively describe the one-dimensional and two-dimensional system satisfies the entanglement entropy area law and uniform size, has proven to be a powerful tool to study the strong correlation systems. Based on MPS, people established a description of a dimension quantum many body system theory. For the two-dimensional system, correlation algorithm based on PEPS at a very early stage. Due to the complexity of PEPS and calculation of capacity constraints, it is limited in practical application. We hope to develop an efficient algorithm to make PEPS Can really solve some long-standing problems. It is difficult to solve that tensor network states in two-dimensional quantum system based on ground state, this paper describes the method for solving the two-dimensional quantum we develop multibody system ground state. The main contents include two parts: the first part describes how to use a replica exchange molecular dynamics method to solve local minima encountered by TNS to solve the many body problem when the value problem. Using TNS wave function for the ground state of multi-body system as the variable is the key step is how to effectively optimize the variational wave function, making it as far as possible to avoid falling into local minimum. We developed an efficient parallel can copy the exchange of molecular dynamics method is used to solve this problem. The TNS elements as generalized coordinates, we mapped the optimization problem to a problem in classical mechanics. When we first set a series of different temperature, and then starting from the random state, according to the potential energy function of this classic system, by the method of molecular dynamics of the evolution of the system, can be obtained under different temperature solution is the optimization solution of zero temperature. In order to avoid falling into local minimum evolution in the process of molecular dynamics, the replica exchange method can be used under different temperature and the configuration of full exchange to help them escape from the local minima. The second part focuses on how to use PEPS variational wave function to effectively solve the two-dimensional quantum spin system.PEPS can well describe the ground state of two dimensional systems, but due to its high computational complexity, it is limited it is used to simulate the two-dimensional system. We put forward the optimization method combining Monte Carlo sampling to optimize PEPS gradient Wave function. First we use a virtual evolution (simple update) SU method to obtain a rough PEPS wave function as the starting point, and then through the gradient optimization to further optimize the exact wave function to get the ground. In the calculation of gradient and energy when we use Monte Carlo sampling method. Compared with the one commonly used method, this method not only greatly reduces the computational complexity, and the use of the gradient optimization algorithm can accurately optimize the variational wave function, makes the use of PEPS to solve some long-standing difficulty in solving multi body system becomes possible.

【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O413.3

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