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題名 三角晶格易辛反鐵磁之量子相變
Quantum phase transition in the triangular lattice Ising antiferromagnet作者 張鎮宇
Chang, Chen Yu貢獻者 林瑜琤
Lin, Yu Cheng
張鎮宇
Chang, Chen Yu關鍵詞 挫折性反鐵磁
零溫投射蒙地卡羅演算法
隨機序列展開演算法
絕熱量子模擬
模擬退火
動力學指數
Frustrated antiferromagnet
Zero-temperature projector algorithm
Stochastic series expansion
Adiabatic quantum simulation
Simulated annealing
Dynamical exponent日期 2017 上傳時間 9-四月-2018 15:51:34 (UTC+8) 摘要 量子擾動及挫折性兩者均可破壞絕對零溫的磁序，為近代凝態物 理關注的有趣現象。在外加橫場下的三角晶格易辛反鐵磁兼具量子臨 界現象(quantum criticality)及幾何挫折性，可謂量子磁性物質之一典 範理論模型。本論文利用平衡態及非平衡態量子蒙地卡羅(quantum Monte Carlo)方法探測三角晶格易辛反鐵磁之量子相變，其界定零溫 時無磁性的順磁態及具 Z6 對稱破缺的有序態(所謂時鐘態)。這裡的 量子蒙地卡羅方法為運用算符的零溫投射(zero-temperature projector) 及隨機序列展開(stochastic series expansion)演算法。在非平衡模擬 中，我們分別沿降溫過程及量子絕熱過程逼近量子相變點，藉此我們 得到動力學指數，及其它相關臨界指數。
The destruction of magnetic long-range order at absolute zero temperature arising from quantum fluctuations and frustration is an interesting theme in modern condensed-matter physics. The triangular lattice Ising antiferromag- net in a transverse field provides a playground for the study of the combined effects of quantum criticality and geometrical frustration. In this thesis we use quantum Monte Carlo methods both in equilibrium and non-equilibrium setups to study the properties of the quantum critical point in the triangular lattice antiferromagnet, which separates a disordered paramagnetic state and an ordered clock state exhibiting Z6 symmetry breaking; The methods are based on a zero-temperature projector algorithm and the stochastic series ex- pansion algorithm. For the non-equilibrium setups, we obtain the dynamical exponent and other critical exponents at the quantum critical point approached by slowly decreasing temperature and through quantum annealing.參考文獻 [1] G. H. Wannier, Phys. Rev. 79, 357 (1950).[2] J. Stephenson, Journal of Mathematical Physics 11, 413 (1970).[3] Y. Jiang and T. Emig, Phys. Rev. B 73, 104452 (2006).[4] R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. Lett. 84, 4457 (2000).[5] R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).[6] R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. B 64, 144416 (2001).[7] S. V. Isakov and R. Moessner, Phys. Rev. B 68, 104409 (2003).[8] D. Blankschtein, M. Ma, A. N. Berker, G. S. Grest, and C. M. Soukoulis, Phys. Rev. B 29, 5250 (1984).[9] H. F. Trotter, Proc. Am. Math. Soc. 10, 545 (1959).[10] M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976).[11] J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).[12] D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977).[13] J. Cardy, Scaling and renormalization in statistical physics, volume 5, Cambridgeuniversity press, 1996.[14] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev.B 63, 214503 (2001).[15] M. Žukovič, L. Mižišin, and A. Bobák, Acta Physica Polonica A 126, 40 (2014).[16] S. Liang, Phys. Rev. B 42, 6555 (1990).[17] A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005).[18] A. W. Sandvik and K. S. D. Beach, arXiv:0704.1469, (2007).[19] R. G. Melko, Stochastic Series Expansion Quantum Monte Carlo, pages 185–206, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.[20] A. W. Sandvik, Phys. Rev. E 68, 056701 (2003).[21] S. Inglis and R. G. Melko, New Journal of Physics 15, 073048 (2013).[22] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).[23] R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987).[24] K. Binder, Phys. Rev. Lett. 47, 693 (1981).[25] A. W. Sandvik, AIP Conf. Proc. 1297, 135 (2010).[26] E. Farhi et al., Science 292, 472 (2001).[27] C.-W. Liu, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B 87, 174302 (2013).[28] C.-W.Liu,A.Polkovnikov,andA.W.Sandvik,Phys.Rev.Lett.114,147203(2015).[29] M. W. Johnson et al., Nature 473, 194 (2011).[30] M. Born and V. Fock, Zeitschrift fur Physik 51, 165 (1928).[31] T. Kadowaki and H. Nishimori, Phys. Rev. E 58, 5355 (1998).[32] G. E. Santoro, R. Martoňák, E. Tosatti, and R. Car, Science 295, 2427 (2002).[33] G. E. Santoro and E. Tosatti, Journal of Physics A: Mathematical and General 39, R393 (2006).[34] S. Boixo et al., Nature Physics 10, 218 (2014).[35] C. De Grandi and A. Polkovnikov, Adiabatic Perturbation Theory: From Landau– Zener Problem to Quenching Through a Quantum Critical Point, pages 75–114, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.[36] 黃湘喻, 以模擬量子退火過程探索自旋系統的基態, Master’s thesis, 國立政治 大學, 2014.[37] B. Damski and W. H. Zurek, Phys. Rev. A 73, 063405 (2006).[38] T. W. Kibble, Journal of Physics A: Mathematical and General 9, 1387 (1976).[39] T. W. Kibble, Physics Reports 67, 183 (1980).[40] W. Zurek, Nature 317, 505 (1985).[41] W. H. Zurek, Physics Reports 276, 177 (1996).[42] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).[43] A. Polkovnikov, Phys. Rev. B 72, 161201 (2005).[44] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005).[45] A. W. Sandvik and J. Kurkijärvi, Phys. Rev. B 43, 5950 (1991).[46] A. W. Sandvik, Journal of Physics A: Mathematical and General 25, 3667 (1992).[47] D. C. Handscomb, Mathematical Proceedings of the Cambridge Philosophical So- ciety 58, 594–598 (1962).[48] D. C. Handscomb, Mathematical Proceedings of the Cambridge Philosophical So- ciety 60, 115–122 (1964).[49] A. W. Sandvik, Phys. Rev. B 56, 11678 (1997).[50] M. Hasenbusch and S. Meyer, Physics Letters B 241, 238 (1990). 描述 碩士
國立政治大學
應用物理研究所
102755004資料來源 http://thesis.lib.nccu.edu.tw/record/#G0102755004 資料類型 thesis dc.contributor.advisor 林瑜琤 zh_TW dc.contributor.advisor Lin, Yu Cheng en_US dc.contributor.author (作者) 張鎮宇 zh_TW dc.contributor.author (作者) Chang, Chen Yu en_US dc.creator (作者) 張鎮宇 zh_TW dc.creator (作者) Chang, Chen Yu en_US dc.date (日期) 2017 en_US dc.date.accessioned 9-四月-2018 15:51:34 (UTC+8) - dc.date.available 9-四月-2018 15:51:34 (UTC+8) - dc.date.issued (上傳時間) 9-四月-2018 15:51:34 (UTC+8) - dc.identifier (其他 識別碼) G0102755004 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/116779 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用物理研究所 zh_TW dc.description (描述) 102755004 zh_TW dc.description.abstract (摘要) 量子擾動及挫折性兩者均可破壞絕對零溫的磁序，為近代凝態物 理關注的有趣現象。在外加橫場下的三角晶格易辛反鐵磁兼具量子臨 界現象(quantum criticality)及幾何挫折性，可謂量子磁性物質之一典 範理論模型。本論文利用平衡態及非平衡態量子蒙地卡羅(quantum Monte Carlo)方法探測三角晶格易辛反鐵磁之量子相變，其界定零溫 時無磁性的順磁態及具 Z6 對稱破缺的有序態(所謂時鐘態)。這裡的 量子蒙地卡羅方法為運用算符的零溫投射(zero-temperature projector) 及隨機序列展開(stochastic series expansion)演算法。在非平衡模擬 中，我們分別沿降溫過程及量子絕熱過程逼近量子相變點，藉此我們 得到動力學指數，及其它相關臨界指數。 zh_TW dc.description.abstract (摘要) The destruction of magnetic long-range order at absolute zero temperature arising from quantum fluctuations and frustration is an interesting theme in modern condensed-matter physics. The triangular lattice Ising antiferromag- net in a transverse field provides a playground for the study of the combined effects of quantum criticality and geometrical frustration. In this thesis we use quantum Monte Carlo methods both in equilibrium and non-equilibrium setups to study the properties of the quantum critical point in the triangular lattice antiferromagnet, which separates a disordered paramagnetic state and an ordered clock state exhibiting Z6 symmetry breaking; The methods are based on a zero-temperature projector algorithm and the stochastic series ex- pansion algorithm. For the non-equilibrium setups, we obtain the dynamical exponent and other critical exponents at the quantum critical point approached by slowly decreasing temperature and through quantum annealing. en_US dc.description.tableofcontents 目錄摘要 i Abstract iii 目錄 v 1 三角量子易辛反鐵磁 12 零溫投射量子蒙地卡羅法 52.1 零溫投射法之基本概念.......................... 52.2 處理量子易辛模型的零溫投射法..................... 72.2.1 局域組態更新法則 ........................ 92.2.2 叢集更新法則........................... 112.3 零溫標度分析 ............................... 112.4 量子絕熱演化 ............................... 153 隨機級數展開量子蒙地卡羅方法 23 3.1 量子易辛模型的隨機級數展開法..................... 23 3.2 有限溫度下的平衡態模擬......................... 26 3.3 模擬退火.................................. 304 總結與展望 33參考文獻33 zh_TW dc.format.extent 869721 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0102755004 en_US dc.subject (關鍵詞) 挫折性反鐵磁 zh_TW dc.subject (關鍵詞) 零溫投射蒙地卡羅演算法 zh_TW dc.subject (關鍵詞) 隨機序列展開演算法 zh_TW dc.subject (關鍵詞) 絕熱量子模擬 zh_TW dc.subject (關鍵詞) 模擬退火 zh_TW dc.subject (關鍵詞) 動力學指數 zh_TW dc.subject (關鍵詞) Frustrated antiferromagnet en_US dc.subject (關鍵詞) Zero-temperature projector algorithm en_US dc.subject (關鍵詞) Stochastic series expansion en_US dc.subject (關鍵詞) Adiabatic quantum simulation en_US dc.subject (關鍵詞) Simulated annealing en_US dc.subject (關鍵詞) Dynamical exponent en_US dc.title (題名) 三角晶格易辛反鐵磁之量子相變 zh_TW dc.title (題名) Quantum phase transition in the triangular lattice Ising antiferromagnet en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [1] G. H. Wannier, Phys. Rev. 79, 357 (1950).[2] J. Stephenson, Journal of Mathematical Physics 11, 413 (1970).[3] Y. Jiang and T. Emig, Phys. Rev. B 73, 104452 (2006).[4] R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. Lett. 84, 4457 (2000).[5] R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).[6] R. Moessner, S. L. Sondhi, and P. Chandra, Phys. Rev. B 64, 144416 (2001).[7] S. V. Isakov and R. Moessner, Phys. Rev. B 68, 104409 (2003).[8] D. Blankschtein, M. Ma, A. N. Berker, G. S. Grest, and C. M. Soukoulis, Phys. Rev. B 29, 5250 (1984).[9] H. F. Trotter, Proc. Am. Math. Soc. 10, 545 (1959).[10] M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976).[11] J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B 16, 1217 (1977).[12] D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977).[13] J. Cardy, Scaling and renormalization in statistical physics, volume 5, Cambridgeuniversity press, 1996.[14] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev.B 63, 214503 (2001).[15] M. Žukovič, L. Mižišin, and A. Bobák, Acta Physica Polonica A 126, 40 (2014).[16] S. Liang, Phys. Rev. B 42, 6555 (1990).[17] A. W. Sandvik, Phys. Rev. Lett. 95, 207203 (2005).[18] A. W. Sandvik and K. S. D. Beach, arXiv:0704.1469, (2007).[19] R. G. Melko, Stochastic Series Expansion Quantum Monte Carlo, pages 185–206, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.[20] A. W. Sandvik, Phys. Rev. E 68, 056701 (2003).[21] S. Inglis and R. G. Melko, New Journal of Physics 15, 073048 (2013).[22] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).[23] R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987).[24] K. Binder, Phys. Rev. Lett. 47, 693 (1981).[25] A. W. Sandvik, AIP Conf. Proc. 1297, 135 (2010).[26] E. Farhi et al., Science 292, 472 (2001).[27] C.-W. Liu, A. Polkovnikov, and A. W. Sandvik, Phys. Rev. B 87, 174302 (2013).[28] C.-W.Liu,A.Polkovnikov,andA.W.Sandvik,Phys.Rev.Lett.114,147203(2015).[29] M. W. Johnson et al., Nature 473, 194 (2011).[30] M. Born and V. Fock, Zeitschrift fur Physik 51, 165 (1928).[31] T. Kadowaki and H. Nishimori, Phys. Rev. E 58, 5355 (1998).[32] G. E. Santoro, R. Martoňák, E. Tosatti, and R. Car, Science 295, 2427 (2002).[33] G. E. Santoro and E. Tosatti, Journal of Physics A: Mathematical and General 39, R393 (2006).[34] S. Boixo et al., Nature Physics 10, 218 (2014).[35] C. De Grandi and A. Polkovnikov, Adiabatic Perturbation Theory: From Landau– Zener Problem to Quenching Through a Quantum Critical Point, pages 75–114, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.[36] 黃湘喻, 以模擬量子退火過程探索自旋系統的基態, Master’s thesis, 國立政治 大學, 2014.[37] B. Damski and W. H. Zurek, Phys. Rev. A 73, 063405 (2006).[38] T. W. Kibble, Journal of Physics A: Mathematical and General 9, 1387 (1976).[39] T. W. Kibble, Physics Reports 67, 183 (1980).[40] W. Zurek, Nature 317, 505 (1985).[41] W. H. Zurek, Physics Reports 276, 177 (1996).[42] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).[43] A. Polkovnikov, Phys. Rev. B 72, 161201 (2005).[44] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005).[45] A. W. Sandvik and J. Kurkijärvi, Phys. Rev. B 43, 5950 (1991).[46] A. W. Sandvik, Journal of Physics A: Mathematical and General 25, 3667 (1992).[47] D. C. Handscomb, Mathematical Proceedings of the Cambridge Philosophical So- ciety 58, 594–598 (1962).[48] D. C. Handscomb, Mathematical Proceedings of the Cambridge Philosophical So- ciety 60, 115–122 (1964).[49] A. W. Sandvik, Phys. Rev. B 56, 11678 (1997).[50] M. Hasenbusch and S. Meyer, Physics Letters B 241, 238 (1990). zh_TW
