無序 S=1 海森堡鏈之強無序重整化群分析

Study of the S = 1 random Heisenberg chain by tree tensor network strong disorder renormalization group

蔡政霖(Zheng-Lin Tsai)

指導教授 ： 陳柏中

國立清華大學/理學院/物理學系/碩士(2019年)

摘要

本篇論文應用樹狀張量網路強無序重整化群來分析無序S=1 海森堡鏈之行為。耦合常數為均質的S=1 自旋鏈之基態為著名的Haldane 有能隙相態。這裡我們所探討的自旋鏈之鍵結為隨機無序的，其基態的能隙消失於強無序範疇。且當無序強度超過某一門檻，系統將進入隨機單態。強無序重整化群演算法搭配樹狀張量網路提供一可探索無序量子多體系統基態性質之高效能數值工具。利用此方法我們得以藉弦序參數找出介於無能隙Haldane 相態與隨機單態間之相變點，並進而求得在此相變點及隨機單態的一系列臨界指數；這些臨界指數分別關於端對端自旋關聯函數、整體自旋關聯函數、弦序參數。另外我們也藉能隙分佈探討能量及長度之關係。相較其他現有的數值計算結果，本論文的結果更吻合理論預測值。

關鍵字

強無序重整化群 ； 無序海森堡鏈 ； 隨機單態 ； 張量網路

並列摘要

In this thesis, we use a tree tensor network strong disorder renormalization group method to study spin-1 random Heisenberg chains.
The ground state of the clean spin-1 Heisenberg chain with uniform nearest-neighbor couplings is a gapped phase as known as the Haldane phase.
Here we consider disordered chains with random couplings, in which the Haldane gap closes in the strong disorder regime.
As the randomness strength is increased further and exceeds a certain threshold,
the random chain undergoes a phase transition to a random singlet phase.
The strong disorder renormalization group method formulated in terms of a tree tensor network provides an efficient tool for exploring ground state properties of disordered quantum many-body systems.
Using this method we detect the quantum critical point between the gapless Haldane phase and the random singlet phase via the disorder-averaged string order parameter.
We determine the critical exponents related to the average string order parameter, the average end-to-end correlation function and the average bulk spin-spin correlation function, both at the critical point and in the random singlet phase.
Furthermore, we study energy-length scaling properties through the distribution of energy gaps for a finite chain.
Our results are closer agreement with the theoretical predictions than what was found in previous numerical studies.

並列關鍵字

strong disorder renormalization group ； spin-1 ； random Heisenberg chain ； random singlet phase ； tensor network

參考文獻

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