自旋系統的量子糾纏量子相變與拓樸序

Abstract

這是一篇探討一維與二維量子自旋系統中基態波函數的性質.我們使用Matrix product state (MPS) 和Tensor product state (TPS) 描述基態波函數,並使用反覆投射(Iterative Projection)方法決定波函數的參數. 藉由計算物理量及基態的量子糾纏(entanglement)性質,發現量子糾纏可以用來描述量子相變的發生,並指出相變點. 接著我們討論臨界點附近entanglement的尺度性質(scaling behaviors). 一維系統中,許多物理量都遵守普遍(universal)性質.此外,我們也討論拓樸序(topological order)與奇異係數(singular values)簡併的關係.This thesis concerns the ground state property and the quantum entanglement in the one- and two-dimensional quantum spin systems. We use the Iterative Projection method to find the ground states numerically in the form of the tensor product states, and then evaluate their expectation values and their entanglement measures such as the geometric entanglement by the method of tensor renormalization group. We find that these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points. We also study the scaling behaviors of the entanglement measures near the quantum critical point by the ideas of quantum-state renormalization group transformations. We find some universal features for one-dimensional spin system. However, we fails to capture the area-law for two-dimensional spin system. We then study the connection between topological order and the degeneracy of the singular value spectrum by explicitly solving the two-dimensional dimerized quantum Heisenberg model in the form of tensor product state ansatz. By evaluating the topological entanglement entropy, we identify a new phase with topological order in this model, in which the singular value spectrum is doubly degenerate. Degeneracy of the singular value spectrum is robust against various types of perturbations, in accordance with our expectation for topological order.