簡易檢索 / 詳目顯示
| 研究生： |
何政緯 Ho, Zheng-Wei |
|---|---|
| 論文名稱： |
關於量子蒙地卡羅退火法 On quantum Monte Carlo annealing |
| 指導教授： |
林瑜琤
Lin, Yu-Cheng |
| 口試委員: |
林瑜琤
Lin, Yu-Cheng 楊志開 Yang, Chih-Kai 高英哲 Kao, Ying-Jer |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用物理研究所 Graduate Institute of Applied Physics |
| 論文出版年： | 2019 |
| 畢業學年度： | 107 |
| 語文別： | 中文 |
| 論文頁數： | 65 |
| 中文關鍵詞： | 模擬退火 、隨機級數展開量子蒙地卡羅演算法 、零溫投射蒙地卡羅演算法 、非均質量子易辛鏈 、三角反鐵磁 、Kosterlitz-Thouless 相變 |
| 外文關鍵詞： | simulated annealing, stochastic series expansion method, zero-temperature projector method, random quantum Ising chain, triangular Ising antiferromagnet, Kosterlitz-Thouless transition |
| DOI URL： | http://doi.org/10.6814/NCCU201901066 |
| 相關次數： | 點閱：198 下載：81 |
| 分享至: |
| 查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報 |
本論文檢驗以蒙地卡羅模擬退火來探討平衡態相變點定標分析之可能性。以量子易辛模型為例，我們分別探討動力學指數為 z = 1 的量子臨界點，具 z = ∞ 的無序量子臨界點，及 Kosterlitz-Thouless (KT) 相變。應用有限溫度隨機級數展開法及基態投射演算法，我們考慮的退火路徑涵蓋降溫、降橫場（量子擾動項）及同時降溫及降場三種情形。我們的計算結果顯示對於 z = 1 量子臨界點，上述後兩類量子退火過程在緩慢改變參數下均能正確反應臨界點位置及臨界指數。通過 KT 相變的退火過程亦可找出吻合理論的定標行為。唯 z = ∞ 的量子臨界點為退火過程的瓶頸，似乎任意緩慢的退火速率均很難突破這個瓶頸來達到無序系統近似靜態的極限。
This thesis examines the use of quantum Monte Carlo simulated annealing in the study of finite-size scaling for equilibrium phase transitions. For quantum Ising models, we study quantum critical points with the dynamic exponent z = 1, a disordered quantum critical point with z = ∞, and the Kosterlitz-Thouless (KT) transition approached through various annealing protocols in quantum Monte Carlo simulations using the stochastic series expansion method and a zero-temperature projector method. We demonstrate that annealing by decreasing a transverse field at zero temperature, or by decreasing the temperature and the transverse field simultaneously can correctly capture the critical scaling behaviors at z = 1 quantum critical points and the KT transition, if the rate of change is sufficiently slow. However, the z = ∞ quantum critical point is an annealing bottleneck and our approaches fail to reach the quasi-static limit of the random quantum Ising chain.
致謝 i
摘要 iii
Abstract v
Contents vii
1 模型概述 1
1.1 量子相變 1
1.2 量子易辛模型 2
1.3 有限尺度定標 4
1.4 無序效應 7
2 隨機級數展開量子蒙地卡羅方法 11
2.1 隨機級數展開法之推導 11
2.1.1 局域更新 15
2.1.2 叢集更新 19
2.2 無序系統的SSE方法 20
2.3 零溫投射量子蒙地卡羅方法 22
2.3.1 基態投射法 23
3 模擬結果 27
3.1 自旋鐵磁鏈及方晶格鐵磁 27
3.1.1 平衡態模擬 27
3.1.2 模擬退火與Kibble-Zurek機制 30
3.1.3 模擬退火 34
3.1.4 零溫模擬退火 46
3.2 三角反鐵磁 47
3.3 無序自旋鏈 52
3.3.1 有限溫度模擬退火 52
3.3.2 零溫模擬退火 57
4 結論 61
參考文獻 63
[1] S. Sachdev, Quantum Phase Transitions, Cambridge University Press., 2000.
[2] Y. Jiang and T. Emig, Phys. Rev. B 73,104452 (2006).
[3] G. H. Wannier, Phys. Rev. 79, 357 (1950).
[4] S. V. Isakov and R. Moessner, Phys. Rev. B 68, 104409 (2003).
[5] D. Blankschtein, M. Ma, A. N. Berker, G. S. Grest, and C. M. Soukoulis, Phys. Rev. B 29, 5250 (1984).
[6] J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
[7] D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977).
[8] M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28, 1516 (1972).
[9] V. Privman, Finite-size scaling theoy, volume 1, Singapore: World Scientific, 1990.
[10] J. Cardy, Scaling and Renormalization in Statistical Physics, volume 5, Cambridge University Press, 1996.
[11] K. Binder, Phys. Rev. Lett. 47, 693 (1981).
[12] K. Binder, Zeitschrift für Physik B Condensed Matter (1981).
[13] M. S. S. Challa and D. P. Landau, Phys. Rev. B 33, 437 (1986).
[14] J. M. Kosterlitz, Journal of Physics C: Solid State Physics (1974).
[15] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 63, 214503 (2001).
[16] R. B. Griffiths, Phys. Rev. Lett. 23, 17 (1969).
[17] S. Guo et al., Phys. Rev. Lett. 100, 017209 (2008).
[18] S. Ubaid-Kassis, T. Vojta, and A. Schroeder, Phys. Rev. Lett. 104, 066402 (2010).
[19] Y. Xing et al., Science 350, 542 (2015).
[20] T. Vojta, J. Phys. A 39, R143 (2006).
[21] D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992).
[22] D. S. Fisher, Phys. Rev. B 51, 6411 (1995).
[23] C. Pich, A. P. Young, H. Rieger, and N. Kawashima, Phys. Rev. Lett. 81, 5916 (1998).
[24] A. W. Sandvik and J. Kurkijärvi, Phys. Rev. B 43, 5950 (1991).
[25] D. C. Handscomb, Proc. Cambridge Philos. Soc. 58, 594 (1962).
[26] A. W. Sandvik, Phys. Rev. B 56, 11678 (1997).
[27] A. W. Sandvik, Phys. Rev. E 68, 056701 (2003).
[28] R. G. Melko, Stochastic Series Expansion Quantum Monte Carlo, pages 185–206, Springer, Berlin, Heidelberg, 2013.
[29] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).
[30] R. H. Swendsen and J. S. Wang, Phys. Rev. Lett. 58, 86 (1987).
[31] A. W. Sandvik and K. S. Beach, arXiv:0704.1469v1 (2007).
[32] A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005).
[33] L. P. Kadanoff et al., Rev. Mod. Phys. 39, 395 (1967).
[34] M. E. Fisher, Phys. Rev. 180, 594 (1969).
[35] H. W. J. Blöte and Y. Deng, Phys. Rev. E 66, 066110 (2002).
[36] R. Guida and J. Zinn-Justin, Nuclear Physics B ,Volume 489, Issue 3, Pages 626-652 (1997).
[37] S. Kirkpatrick, M. P. Vecchi, and C. D. Gelatt Jr., science 220, 671 (1983).
[38] T. W. Kibble, Physics Report 67, 183 (1980).
[39] W. Zurek, Nature 317, 505 (1985).
[40] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005).
[41] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).
[42] C.-W. Liu, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B 89, 054307 (2014).
[43] 張鎮宇, 三角晶格易辛反鐵磁之量子相變, Master’s thesis, 國立政治大學, 2017.
