本篇論文中，我們利用樹狀張量網路強無序重整化群的數值方法對一維S=1與S=2 bond-alternating海森堡鏈分析。首先我們介紹了twist order parameter ZL以及VBS態分析，並計算在不同無序強度R及bond alternation強度δ下的ZL，利用有限長度擬合方法找出對應的相變點以及在(R,δ)平面的相圖。
對於S=1，我們的結果與其他現有數值模擬計算的結果大致相符。
對於S=2，我們利用S=1的方法來分析，並將以上結果進行總結，我們在現有理論預測的兩種相圖中，給出最有可能的其中一種相圖。

In this thesis, we use a tree tensor network strong disorder renormalization group method to study spin-1 and spin-2 bond-alternating antiferromagnetic Heisenberg chain (BAHC). First, we introduce the twist order parameter ZL and VBS picture, by calculating ZL under different strengths of disorder R and bond alternation δ, we could use finite size scaling to find out the corresponding phase transition points and the phase diagram in the (R,δ) plane.
For S=1, our results are in a good agreement with existing result
obtained by other numerical methods.
For S=2, there are currently two possible theoretical phase diagrams, by using the same analysis methods in spin-1 case, we summarize our results to give a conjecture that the phase diagram should belong to one of them

Contents
Abstract (Chinese) 1
Acknowledgements (Chinese) 2
Abstract 3
Contents 4
1 Introduction 6
1.1 Disorder system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 1-D bond-alternating Heisenberg model . . . . . . . . . . . . . . . . 7
1.3 VBS states and Twist order parameter . . . . . . . . . . . . . . . . 8
2 Numerical analysis 10
2.1 Matrix Product Operator . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Tree tensor network strong disorder renormalization group . . . . . 13
3 Spin 1 Result 18
3.1 Prediction for spin-1 phase diagram . . . . . . . . . . . . . . . . . . 18
3.2 Finite size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Compare the result . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Spin 2 Result 29
4.1 Prediction of spin-2 phase diagram . . . . . . . . . . . . . . . . . . 29
4.2 Finite size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Compare the result . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Conclusion 43
Bibliography 45

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