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。

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.

摘要
Abstract
誌謝
Contents
1 Introduction -------------------------------------------------- 1
1.1 Disordered systems ------------------------------------------ 1
1.2 Numerical analysis ------------------------------------------ 1
2 Model and Methods --------------------------------------------- 3
2.1 S = 1 random Heisenberg chain ------------------------------- 3
2.2 Matrix product operator ------------------------------------- 5
2.3 Tree tensor network strong disorder renormalization group --- 7
3 String order parameter and Bulk correlation function --------- 13
3.1 String order parameter ------------------------------------- 13
3.2 Bulk correlation function ---------------------------------- 19
4 Distributions of energy gaps and End-to-end correlation function - 22
4.1 Distributions of energy gaps ------------------------------- 22
4.2 Distributions of end-to-end correlations ------------------- 29
4.3 End-to-end correlation function ---------------------------- 32
5 Conclusion and Outlook --------------------------------------- 34
Bibliographies ------------------------------------------------- 36

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