應用隨機採樣方法對量子蒙地卡羅資料做解析延拓

Analytic Continuation of Quantum Monte Carlo Data by Stochastic Methods

顏敬哲(Jing-Jer Yen)

指導教授 ： 高英哲

國立臺灣大學/理學院/物理學研究所/碩士(2019年)

https://doi.org/10.6342/NTU201903111

摘要

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。

關鍵字

隨機解析延拓 ； 哈密頓蒙地卡羅

並列摘要

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.

並列關鍵字

analytic continuation ； Hamiltonian Monte Carlo

參考文獻

[1] Anders W. Sandvik. Stochastic series expansion method for quantum Ising models with arbitrary interactions. Phys. Rev. E, 68(5):056701, Nov 2003.

[2] N. Xu, C. E. Matt, E. Pomjakushina, X. Shi, R. S. Dhaka, N. C. Plumb, M. Radović, P. K. Biswas, D. Evtushinsky, V. Zabolotnyy, J. H. Dil, K. Conder, J. Mesot, H. Ding, and M. Shi. Exotic kondo crossover in a wide temperature region in the topological kondo insulator smb6 revealed by high-resolution arpes. Phys. Rev. B, 90:085148, Aug 2014.

[3] B. Dalla Piazza, M. Mourigal, N. B. Christensen, G. J. Nilsen, P. Tregenna-Piggott, T. G. Perring, M. Enderle, D. F. McMorrow, D. A. Ivanov, and H. M. Rønnow. Fractional excitations in the square-lattice quantum antiferromagnet. Nature Physics, 11(1):62–68, Jan 2015.

[4] K. S. D. Beach. Identifying the maximum entropy method as a special limit of stochastic analytic continuation. arXiv e-prints, pages cond–mat/0403055, Mar 2004.

[5] Agustin Nieto. Evaluating sums over the Matsubara frequencies. Computer Physics Communications, 92(1):54–64, Nov 1995.