用密度矩阵重正化群方法研究一维量子系统及量子杂质系统的动力学性质

While organizing old files, I realized I had lost my master’s thesis. After some effort, I finally found it online, but unfortunately, it is not very accessible. Upon reading it a bit, I realized my master’s thesis (written in Chinese) could be useful for others; therefore, I will post it here.

Abstract

Density matrix renormalization group method was firstly developed during the 90’s last century as a numerical method to study the properties of the ground state and low-lying states of the one dimensional quantum lattice systems. DMRG has been extended to precisely calculate the real-time dynamics (t-DMRG) and thermodynamics (FTDMRG) of 1D quantum lattice systems. We try to combine t-DMRG and FTDMRG to calculate the real-time dynamics of 1D quantum systems at finite temperature. On the other hand, recent experimental progresses demand theoretical and numerical methods to help experimentalists to study the real-time dynamics of quantum impurity systems. By borrowing some techniques in numerical renormalization group method(NRG), we extend the t-DMRG and FTDMRG method to studying the thermodynamics and real-time dynamics of quantum impurity models. Our first results show that DMRG is a very accurate and efficient method to study the real-time dynamics and thermodynamics of quantum impurity systems.

Keywords: DMRG, MPS, NRG, quantum impurity system, dissipative quantum system, time-dependent hamiltonian, dissipative Landau-Zener problem, resonant level model, spin-boson model

摘要

密度矩阵重正化群方法（DMRG）是上世纪九十年代发展起来的一种非常 精确的计算一维量子多体格点系统基态和低能激发态性质的数值方法。最近几 年DMRG已经被推广到可以分别精确计算一维量子格点系统的动力学演化(自 适应含时密度矩阵重正化群, t-DMRG)和热力学性质(有限温密度矩阵重正化 群, FTDMRG)。本文中我们将t-DMRG与FTDMRG结合起来尝试了计算一维 量子系统在有限温度下的动力学性质。 另一方面,最新实验技术的发展对在理论和数值上研究量子杂质系统的 动力学提出了迫切的要求。借助数值重正化群(NRG)中的一些技巧, 我们将tDMRG和FTDMRG推广到计算量子杂质系统的动力学和热力学性质。初步测 试结果表明，t-DMRG和FTDMRG能够很精确和有效率的研究量子杂质系统的 动力学和热力学性质。

关键词： 密度矩阵重正化群，矩阵乘积态，数值重正化群, 量子杂质系统，耗散 量子系统，含时哈密顿量，共振能级模型，耗散Landau-Zener模型，自旋-玻色子 模型