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研究生:顏敬哲
研究生(外文):Jing-Jer Yen
論文名稱:應用隨機採樣方法對量子蒙地卡羅資料做解析延拓
論文名稱(外文):Analytic Continuation of Quantum Monte CarloData by Stochastic Methods
指導教授:高英哲高英哲引用關係
指導教授(外文):Ying-Jer Kao
口試委員:林及仁林瑜琤
口試日期:2019-07-26
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:物理學研究所
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2019
畢業學年度:107
語文別:中文
論文頁數:58
中文關鍵詞:隨機解析延拓哈密頓蒙地卡羅
DOI:10.6342/NTU201903111
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量子蒙地卡羅是一個數值方法,可用來模擬量子多體系統,像是自旋模型以及強關聯電子系統。這個模擬可以得到在虛數時間軸上的兩點關聯函數,然而真實實驗卻只能量測在實數時間軸上的動態特徵,像是能量激發態的頻譜。為了更容易比較模擬與實驗的結果,利用解析延拓將虛數軸上的關聯函數延伸到實數軸上是一個常見且重要的過程。
在這篇論文中,我們將探討如何使用隨機採樣的方法來完成這個解析延拓。藉由設計不同的採樣過程,我們可以在不同型態的頻譜上都得到相當精準的解析結果,而我們更進一步使用這些方法來研究一維海森堡自旋模型的動態結構因子。
除此之外,我們提出了一個新的架構,將哈密頓蒙地卡羅用於採樣方法上,結果顯示這是一個值得未來繼續研究的方向。
Quantum Monte Carlo (QMC) is a useful numerical method for simulating quantum many body systems, such as spin models and strongly correlated electronic systems. Most QMC simulations provide two point correlation functions in imaginary time, however, most experiments only probe
real-time dynamical properties such as dynamical susceptibilities and elementary excitations in energy (or frequency) domain. To bridge the gap, analytic continuation is an essential tool.
In this thesis, we demonstrate how to perform analytic continuation by using stochastic methods. We get resulting spectrum in high precision through many strategies of proposing updates in the sampling process. Therefore, we further employ it to study the dynamical structure factor in Heisenberg spin chain under zero and non-zero magnetic field. Besides, we also demonstrated
a HMC-SAC scheme which exploits Hamiltonian Monte Carlo to generate global updates in the sampling process. Results show that this scheme is a promising direction for future study.
口試委員會審定書iii
誌謝v
摘要vii
Abstract ix
1 Introduction 1
2 Formalism 5
2.1 Analytic Continuation in QMC . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Formal Expression . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Bayesian Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Covariance of Measured Correlation Function . . . . . . . . . . . . . . . 10
2.4 Review on Maximum Entropy Method . . . . . . . . . . . . . . . . . . 12
2.5 Stochastic Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . 14
2.5.1 Choice of Temperature . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.2 Algorithmic Details . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Application of Hamiltonian Monte Carlo on SAC . . . . . . . . . . . . . 18
2.6.1 A Compact Tutorial of Hamiltonian Monte Carlo . . . . . . . . . 18
2.6.2 A Scheme of HMC-SAC . . . . . . . . . . . . . . . . . . . . . . 21
3 Numerical Experiments And Results 23
3.1 Preparation of Synthetic Data . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 General Stochastic Sampling . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Experimental Settings . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Case Study: Spectral Function With a Delta Function . . . . . . . . . . . 28
3.3.1 Free Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2 Restricted Sampling . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.3 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Case Study: Spectral Function With a Sharp Edge (Diverging Edge) . . . 33
3.4.1 Free Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.2 Free Sampling with a Lowest Boundary . . . . . . . . . . . . . . 34
3.4.3 Constraint and Single Update . . . . . . . . . . . . . . . . . . . 35
3.4.4 Chunk Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.5 Sharp Edge with Non-decaying Continuum . . . . . . . . . . . . 39
3.4.6 A Delta Peak or a Diverging Peak ? . . . . . . . . . . . . . . . 40
3.5 Numerical Results of Hamiltonian Monte Carlo Method . . . . . . . . . . 41
3.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5.2 Interacting Potentials . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Application to The Heisenberg Spin Chain 49
4.1 Sxx(q, ω) for h=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Sxx(q, ω) for h=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Conclusion and Outlook 53
Bibliography 55
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