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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 高英哲(Ying-Jer Kao) | |
| dc.contributor.author | Jing-Jer Yen | en |
| dc.contributor.author | 顏敬哲 | zh_TW |
| dc.date.accessioned | 2021-06-17T08:24:40Z | - |
| dc.date.available | 2020-08-20 | |
| dc.date.copyright | 2019-08-20 | |
| dc.date.issued | 2019 | |
| dc.date.submitted | 2019-08-13 | |
| dc.identifier.citation | [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. [6] Ryan Levy, J. P. F. LeBlanc, and Emanuel Gull. Implementation of the maximum entropy method for analytic continuation. Computer Physics Communications, 215:149 155, Jun 2017. 55 [7] Sebastian Fuchs, Thomas Pruschke, and Mark Jarrell. Analytic continuation of quantum Monte Carlo data by stochastic analytical inference. Phys. Rev. E, 81(5):056701, May 2010. [8] Anders W. Sandvik. Stochastic method for analytic continuation of quantum monte carlo data. Phys. Rev. B, 57:10287–10290, May 1998. [9] Hui Shao, Yan Qi Qin, Sylvain Capponi, Stefano Chesi, Zi Yang Meng, and Anders W. Sandvik. Nearly deconfined spinon excitations in the square-lattice spin-1/2 heisenberg antiferromagnet. Phys. Rev. X, 7:041072, Dec 2017. [10] Michael Betancourt. A Conceptual Introduction to Hamiltonian Monte Carlo. arXiv e-prints, page arXiv:1701.02434, Jan 2017. [11] Radford M. Neal. MCMC using Hamiltonian dynamics. arXiv e-prints, page arXiv:1206.1901, Jun 2012. [12] M. J. Betancourt, Simon Byrne, and Mark Girolami. Optimizing The Integrator Step Size for Hamiltonian Monte Carlo. arXiv e-prints, page arXiv:1411.6669, Nov 2014. [13] Michael Betancourt. Identifying the Optimal Integration Time in Hamiltonian Monte Carlo. arXiv e-prints, page arXiv:1601.00225, Jan 2016. [14] Anders W. Sandvik. Constrained sampling method for analytic continuation. Phys. Rev. E, 94(6):063308, Dec 2016. [15] Michael Karbach, Gerhard Müautller, A. Hamid Bougourzi, Andreas Fledderjohann, and Karl-Heinz Müauttter. Two-spinon dynamical structure factor of the onedimensional s= Heisenberg antiferromagnet. Phys. Rev. B, 55(18):12510–12517, May 1997. [16] J M P Carmelo, P D Sacramento, J D P Machado, and D K Campbell. Singularities of the dynamical structure factors of the spin-1/2xxxchain at finite magnetic field. Journal of Physics: Condensed Matter, 27(40):406001, sep 2015. 56 [17] Gerhard Müller, Harry Thomas, Hans Beck, and Jill C. Bonner. Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field. Phys. Rev. B, 24, 08 1981. [18] Michael Karbach and Gerhard Müller. Line-shape predictions via Bethe ansatz for the one-dimensional spin-12 Heisenberg antiferromagnet in a magnetic field. Phys. Rev. B, 62(22):14871–14879, Dec 2000. [19] Jean-Sébastien Caux, Rob Hagemans, and Jean Michel Maillet. Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime. Journal of Statistical Mechanics: Theory and Experiment, 2005(9):09003, Sep 2005. [20] Chun-Jiong Huang, Youjin Deng, Yuan Wan, and Zi Yang Meng. Dynamics of Topological Excitations in a Model Quantum Spin Ice. Phys. Rev. Lett., 120(16):167202, Apr 2018. [21] A. Reymbaut, D. Bergeron, and A. M. S. Tremblay. Maximum entropy analytic continuation for spectral functions with nonpositive spectral weight. Phys. Rev. B, 92(6):060509, Aug 2015. [22] Gernot J. Kraberger, Robert Triebl, Manuel Zingl, and Markus Aichhorn. Maximum entropy formalism for the analytic continuation of matrix-valued green’s functions. Phys. Rev. B, 96:155128, Oct 2017. [23] Mark Jarrell and J.E. Gubernatis. Bayesian inference and the analytic continuation of imaginary-time quantum monte carlo data. Physics Reports, 269(3):133 – 195, 1996. [24] M. B. Stone, D. H. Reich, C. Broholm, K. Lefmann, C. Rischel, C. P. Landee, and M. M. Turnbull. Extended quantum critical phase in a magnetized spin-1 2 antiferromagnetic chain. Phys. Rev. Lett., 91:037205, Jul 2003. [25] Lynton Ardizzone, Jakob Kruse, Sebastian Wirkert, Daniel Rahner, Eric W. Pellegrini, Ralf S. Klessen, Lena Maier-Hein, Carsten Rother, and Ullrich Köthe. An- 57 alyzing Inverse Problems with Invertible Neural Networks. arXiv e-prints, page arXiv:1808.04730, Aug 2018. 58 | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/74214 | - |
| dc.description.abstract | 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。 | zh_TW |
| dc.description.abstract | 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. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T08:24:40Z (GMT). No. of bitstreams: 1 ntu-108-R06222029-1.pdf: 3210270 bytes, checksum: 11dc5abb015ba66203938353ef25fba0 (MD5) Previous issue date: 2019 | en |
| dc.description.tableofcontents | 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 | |
| dc.language.iso | zh-TW | |
| dc.subject | 哈密頓蒙地卡羅 | zh_TW |
| dc.subject | 隨機解析延拓 | zh_TW |
| dc.subject | analytic continuation | en |
| dc.subject | Hamiltonian Monte Carlo | en |
| dc.title | 應用隨機採樣方法對量子蒙地卡羅資料做解析延拓 | zh_TW |
| dc.title | Analytic Continuation of Quantum Monte Carlo
Data by Stochastic Methods | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 107-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 林及仁,林瑜琤 | |
| dc.subject.keyword | 隨機解析延拓,哈密頓蒙地卡羅, | zh_TW |
| dc.subject.keyword | analytic continuation,Hamiltonian Monte Carlo, | en |
| dc.relation.page | 58 | |
| dc.identifier.doi | 10.6342/NTU201903111 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2019-08-13 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 物理學研究所 | zh_TW |
| 顯示於系所單位： | 物理學系 | |
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