比較不同張量網路演算法應用在二微多體量子物理系統之優劣

Comparative Studies of Tensor Network Algorithms for Two Dimensional Quantum Many-Body Systems

周昀萱(Yun-Hsuan Chou)

指導教授 ： 高英哲

國立臺灣大學/理學院/物理學研究所/碩士(2016年)

https://doi.org/10.6342/NTU201601956

摘要

如何判斷多體量子系統的相變化,且從微觀系統來得到巨觀上的物 理性質,在現代仍為凝態物理學中十分有趣的領域。
從 NRG 的為起點開始,多年來出現了許多突破性的演算法。其中 DMRG 在一維的系統的模擬中得到了相當好的結果。但在二微系統 中,因為 Area law 的關係使其表現不如在一微系統中精確,不僅如此, 在二維系統中,計算複雜度上升之速度也非一維系統可比擬。為了解 決這些問題,因而出現了許許多多不同的建立在張亮網路理論上的演 算法。
此篇論文,紀錄了幾個當今較為主流或新穎並用以模疑二維量子系 統的張亮網路演算法。一開始將簡單解釋張亮網路的基本理論; 再來會 介紹如何實做、優化演算法,以增加精確度和降低計算複雜度。章節 中也附上偽代碼,來說明實作中應注意之細節。最後會比較它們計算 二維易辛模型與海森堡模型的結果,來說明各演算法之優缺點。

關鍵字

矩陣基態 ； 投影糾纏對態 ； 投影糾纏單態 ； 無限時間演化區塊 ； 邊角移矩陣 ； 張量重整化群

並列摘要

Determining the phase transition of many body systems and the physi- cal properties of macroscopic systems from microscopic description are still challenging in condense matter physics.
Since the numerical renormalization group (NRG) came out, various al- gorithms sprang up like mushrooms for analyzing these problems . Among all, the density matrix renormalization group (DMRG) could be considered as the most remarkable outcome, which analyze accurately in one dimensional systems. However, it perform worse in two dimensional systems. Not only the physical reasons, such as the area law, but also the rapid increment of computational complexity which is much higher than in one dimensional sys- tems.
In order to study the phenomenals in twe-dimensional systems. First of all, we briefly introduce the tensor network theory. Secondly, we recorded some of popular tensor network algorithms which are developed for handling the problems in two-dimensional systems. Furthermore, the network diagrams and pseudo-code are presented, which gives the instruction of how to imple- ment these algorithms.

並列關鍵字

matrix product state(MPS) ； projected entangled pair state(PEPS) ； projected entangled simplex state ； infinite time-evolveing block-decimation ； corner transfer matrix ； tensor renormalization group

參考文獻

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