矩陣積態在一維量子自旋系統的應用

Abstract

在一維量子自旋系統中，矩陣積態可作為變量數值模擬的試驗波函數。在此研究中，我們展示了兩種建構矩陣積態的方法，這些方法源自於密度矩陣重整群與量子資訊理論。我們發展了兩種在一維量子系統中矩陣積態的演算法，分別為隨機最佳化的量子蒙地卡羅變量模擬 (Variational quantum Monte Carlo simulations with stochastic optimization) 與時間演化間隔消除法 (Time-evolving block decimation)。我們推廣了隨機最佳化的方法至開放邊界 (open boundary condition) 並且探討了伊辛模型加入橫向磁場與海森堡模型。另外，我們處理了無限長的伊辛模型加入橫向磁場，我們的結果顯示量子糾纏 (quantum entanglement)與量子相變息息相關。

In one-dimensional quantum spin systems, the matrix product states (MPS) can be used as a trail wave function for variational numerical simulations. In this thesis, we investigate the construction of MPS which is related to the density matrix renormalization group (DMRG) and the Quantum information theory (QIT). We develop two algorithms, variational quantum Monte Carlo (QMC) simulations with stochastic optimization [1] and time-evolving block decimation (TEBD) [2, 3], in one dimensional systems. We generalize QMC with stochastic optimization to the open boundary condition and study the transverse Ising model and Heisenerg model. We also applied the infinite TEBD algorithm [4] to the infinite transverse Ising model and demonstrate that entanglement is a key ingredient in the quantum phase transition.