研究生: |
黃靜瑜 Ching-Yu Huang |
---|---|
論文名稱: |
自旋系統的量子糾纏量子相變與拓樸序 Numerical study on quantum entanglement, quantum phase transition, and topological order in spin system |
指導教授: |
林豐利
Lin, Feng-Li |
學位類別: |
博士 Doctor |
系所名稱: |
物理學系 Department of Physics |
論文出版年: | 2012 |
畢業學年度: | 100 |
語文別: | 英文 |
論文頁數: | 98 |
中文關鍵詞: | 量子糾纏 、量子相變 、拓樸序 、自旋系統 、MPS/ TPS |
英文關鍵詞: | quantum entanglement, quantum phase transition, topological order, spin system, MPS/ TPS |
論文種類: | 學術論文 |
相關次數: | 點閱:152 下載:20 |
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 |
這是一篇探討一維與二維量子自旋系統中基態波函數的性質.我們使用Matrix product state (MPS) 和Tensor product state (TPS) 描述基態波函數,並使用反覆投射(Iterative Projection)方法決定波函數的參數. 藉由計算物理量及基態的量子糾纏(entanglement)性質,發現量子糾纏可以用來描述量子相變的發生,並指出相變點. 接著我們討論臨界點附近entanglement的尺度性質(scaling behaviors). 一維系統中,許多物理量都遵守普遍(universal)性質.此外,我們也討論拓樸序(topological order)與奇異係數(singular values)簡併的關係.
This thesis concerns the ground state property and the quantum entanglement in the one- and two-dimensional quantum spin systems. We use the Iterative Projection method to find the ground states numerically in the form of the tensor product states, and then evaluate their expectation values and their entanglement measures such as the geometric entanglement by the method of tensor renormalization group. We find that these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points. We also study the scaling behaviors of the entanglement measures near the quantum critical point by the ideas of quantum-state renormalization group transformations. We find some universal features for one-dimensional spin system. However, we fails to capture the area-law for two-dimensional spin system. We then study the connection between topological order and the degeneracy of the singular value spectrum by explicitly solving the two-dimensional dimerized quantum Heisenberg model in the form of tensor product state ansatz. By evaluating the topological entanglement entropy, we identify a new phase with topological order in this model, in which the singular value spectrum is doubly degenerate. Degeneracy of the singular value spectrum is robust against various types of perturbations, in accordance with our expectation for topological order.
[1] S. Sachdev, Quantum Phase Transitions" (Cambridge University Press, Cambridge, England,
1999).
[2] Xiao-Gang Wen, Phys. Rev. B 40, 7387 (1989).
[3] Xiao-Gang Wen, Int. J. Mod. Phys. B 4, 239 (1990).
[4] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
[5] M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
[6] X. Chen, Z.-C. Gu, and X.-G. Wen Phys. Rev. B 83, 035107 (2011).
[7] X. G. Wen, Phys. Rev. B 44, 2664 (1991).
[8] T. H. Hansson, V. Oganesyan, and S. L. Sondhi, Annals Of Physics vol. 313, 497 (2004).
[9] A. Yu. Kitaev, Annals Phys. 303, 2 (2003).
[10] M. H. Freedman, M. Larsen and Z. Wang, Comm. Math. Phys. 227, 605 (2002).
[11] N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989), S. Sachdev, Phys. Rev. B 45,
12377 (1992).
[12] S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Phys. Rev. B 35, 8865 (1987), R. Moessner
and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001).
[13] G. J. Chen, R. Joynt, F. C. Zhang, and C. Gros, Phys. Rev. B 42, 2662 (1990).
[14] X.-G. Wen Phys. Rev. B 65, 165113 (2002)
[15] T.-P. Choy and Y. B. Kim Phys. Rev. B 80, 064404 (2009).
[16] F. Wang and A. Vishwanath Phys. Rev. B 74, 174423 (2006).
[17] F. Wang Phys. Rev. B 82, 024419 (2010).
[18] Y.-M. Lu and Y. Ran arXiv:1005.4229v1.
[19] M. A. Nielen and I. Chuang, \Quantum computation and quantum communication" (Cambridge
University Press, Cambridge, 2000).
[20] M. A. Nielen, PhD Dissertation, The University of New Mexico (1998), quant-ph/0011036.
[21] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
[22] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys.
Rev. Lett. 70, 1895 (1993).
[23] M. Hillery, V. B.uzek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999).
[24] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
[25] M. B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007).
[26] J. I. Latorre, A. Riera, J. Phys. A: Math. Theor. 42, 504002 (2009).
[27] R. Augusiak, F. M. Cucchietti, M. Lewenstein, arXiv:1003.3153 (2010).
[28] T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002).
[29] A. Osterloh, L. Amico, G. Falci, and R.Fazio, Nature (London) 416, 608 (2002).
[30] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
[31] J. I. Latorre, E. Rico, and G. Vidal, Quantum Inf. Comput. 4, 48 (2004).
[32] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54,
3824 (1996).
[33] S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997).
[34] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[35] W. K. Wootters, Contemporary Mathematics 305, 299 (2002).
[36] K. M. O'Connor, W. K. Wootters, Phys. Rev. A 63, 052302 (2001).
[37] M. C. Arnesen, S. Bose, V. Vedral, Phys. Rev. Lett. 87, 017901 (2001).
[38] D. Gunlycke, S. Bose, V. M. Kendon, V. Vedral, Phys. Rev. A 64, 042302 (2001).
[39] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865
(2009).
[40] S. Popescu and D. Rohrlich, Phys. Rev. A 56,3319 (1997).
[41] V. Vedral, Rev. Mod. Phys. 74, 197 (2002).
[42] D. Bru, J. Math. Phys. 43, 4237 (2002).
[43] W. K. Wooters, Quant. Inf. Comp. 1, 27 (2001).
[44] M. B. Plenio and V. Vedral, Contemp. Phys. 39, 431 (1998).
[45] V. Coman1, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306 (2000).
[46] T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys. Rev. Lett. 93, 167203
(2004).
[47] T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys. Rev. Lett. 94, 147208
(2005).
[48] B. Li, S.-H. Li, and H.-Q. Zhou, Phys. Rev. E 79, 060101(R) (2009).
[49] M. Haque, O. Zozulya, and K. Schoutens, Phys. Rev. Lett. 98, 060401 (2007).
[50] O. S. Zozulya, M. Haque, K. Schoutens, and E. H. Rezayi, Phys. Rev. B 76, 125310 (2007).
[51] M. Srednicki, Phys. Rev. Lett. 71, 666 (1993).
[52] C. H. Bennett, H. J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53, 2046 (1996).
[53] H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
[54] P. Calabrese and A. Lefevre, Phys. Rev. A. 78, 032329 (2008).
[55] F. Pollmann, and J. E. Moore, New Journal of Physics 12, 025006 (2010).
[56] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010).
[57] B. A. Friedman and G. C. Levine. Phys. Rev. B 78,035320 (2008).
[58] S. V. Isakov, M.B. Hastings, R. G. Melko, arXiv:1102.1721.
[59] S. Furukawa and G. Misguich, Phys. Rev. B 75, 214407 (2007).
[60] H.Yao and X.-L. Qi, Phys. Rev. Lett. 105, 080501 (2010).
[61] T.-C. Wei and P. M. Goldbart, Phys. Rev. A 68, 042307 (2003).
[62] D. A. Meyer and N. R. Wallach, J. Math. Phys. 43,4273 (2002).
[63] T.-C. Wei, D. Das, S. Mukhopadyay, S. Vishveshwara, and P. M. Goldbart, Phys. Rev. A
71, 060305 (2005).
[64] R. Orus, Phys. Rev. Lett. 100, 130502 (2008).
[65] A. Botero and B. Reznik, arXiv:0708.3391 (2007).
[66] H. Barnum, E. Knill, G. Ortiz, and L. Viola, Phys. Rev. A 68, 032308 (2003).
[67] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
[68] S. R. White, Phys. Rev. B 48, 10345 (1993).
[69] S. Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995).
[70] S. Ostlund and S. Rommer, Phys. Rev. B 55, 2164 (1997).
[71] F. Verstraete, J. I. Cirac, arXiv:cond-mat/0407066.
[72] Y.-Y. Shi, L.-M. Duan, and G. Vidal, Phys. Rev. A. 74, 022320 (2006).
[73] D. Perez-Garcia, F. Verstraete, M.M. Wolf, J.I. Cirac, Quantum Inf. Com. 7, 401 (2007).
[74] F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205(2004).
[75] Hans J. Briegel and Robert Raussendorf, Phys. Rev. Lett. 86,910 (2001).
[76] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,5188 (2001).
[77] R.Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003).
[78] I. Aeck, T. Kennedy, E. H. Lieb and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987).
[79] I. Aeck, T. Kennedy, E. H. Lieb and H. Tasaki, Commun. Math. Phys. 115, 477 (1988).
[80] D. Perez-Garcia, F. Verstraete, M.M.Wolf, J.I. Cirac, Quantum Inf. Comput. 7, 401 (2007).
[81] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
[82] G. Vidal, Phys. Rev. Lett. 93,040502 (2004).
[83] G. Vidal, Phys. Rev. Lett. 98,070201 (2007).
[84] R. Orus and G. Vidal, Phys. Rev. B. 78, 155117 (2008).
[85] F. Verstraete, and J. I. Cirac, Phys. Rev. B 73, 094423 (2006).
[86] H.-C. Jiang, Z.-Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008).
[87] J. Jordan, R. Orus, G. Vidal, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett 101, 220605
(2008).
[88] M. Suzuki, Phys. Lett. A 146, 319 (1990); J. Math Phys. 32, 400 (1991).
[89] F. Verstraete, J. I. Cirac, J. I. Latorre, E. Rico, and M. M. Wolf, Phys. Rev. Lett. 94,
140601 (2005).
[90] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007).
[91] G. Vidal, Phys. Rev. Lett. 101, 110501 (2008).
[92] M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007).
[93] V. Murg, F. Verstraete, and J. I. Cirac, Phys. Rev. A. 75, 033605 (2007).
[94] B. Bauer, G. Vidal, and M. Troyer, J. Stat. Mech. P09006 (2009).
[95] A. Isacsson and O. F. Syljuasen, Phys. Rev. E 74 02670 (2006).
[96] Z. C. Gu, M. Levin and X. G. Wen, Phys. Rev. B. 78, 205116 (2008).
[97] Roman Orus, A. C. Doherty, and G. Vidal, Phys. Rev. Lett. 102, 077203 (2009).
[98] V. Murg, F. Verstraete, and J. I. Cirac, Phys. Rev. B 79, 195119 (2009).
[99] C. V. Kraus, N.Schuch, F. Verstraete, and J. I. Cirac, Phys. Rev. A 81, 052338 (2010).
[100] P. Corboz, R. Orus, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010).
[101] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett 96, 220601
(2006).
[102] F. Verstraete, and J. I. Cirac, Phys. Rev. A 70, 060302 (2004).
[103] R. Orus and G. Vidal , Phys. Rev. B 80, 094403 (2009).
[104] H. H. Zhao, Z. Y. Xie, Q. N. Chen, Z. C. Wei, J. C. Cai, and T. Xiang, Phys. Rev. B. 81,
174411 (2010).
[105] L. Wang, Y.J. Kao, A. W. Sandvik, arXiv:0901.0214.
[106] Z. Y. Xie, H. C. Jiang, Q. N. Chen, Z. Y. Weng, T. Xiang, Phys. Rev. Lett. 103, 160601
(2009).
[107] F. Verstraete, J. I. Cirac, J. I. Latorre, E. Rico, and M. M. Wolf, Phys. Rev. Lett. 94,
140601 (2005).
[108] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007).
[109] X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B. 82, 155138 (2010).
[110] R. Feynman, Internat. J. Theoret. Phys., 21 467 (1982).
[111] D. Deutsch, Mathematical and Physical Sciences 400 97 (1985).
[112] E. Biham, G. Brassard, D. Kenigsberg, and T. Mor Theoretical Computer Science, 320,
1 (2004).
[113] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, arXiv:quant-ph/0001106v1.
[114] A. Yu. Kitaev, Annals Phys. 303, 2 (2003).
[115] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Rev. Mod. Phys. 80,
1083 (2008).
[116] D. Gottesman, I. L. Chuang, Nature 402, 390 (1999).
[117] R, Jozsa, arXiv:quant-ph/0508124.
[118] M.A. Nielsen, and T.I.Oborne, arXiv:quant-ph/0504097.
[119] R. Oliverira and B. M. Terhal, Quant. Inf, Comp. 8, 090 (2008).
[120] S. D. Bartlett and T. Rudolph, Phys. Rev. A 74, 040302 (2006).
[121] T. Grin and S. D. Bartlett, Phys. Rev. A 78, 062306 (2008).
[122] X. Chen, B. Zeng, Z. Gu, B. Yoshida, and I. L. Chuang, Phys. Rev. Lett. 102, 220501
(2009)
[123] D. Gross and J. Eisert, Phys. Rev. Lett. 98, 220503 (2007)
[124] D. Gross, J. Eisert, N. Schuch, and D. Perez-Garcia, Phys. Rev. A 76, 052315 (2007)
[125] G. K. Brennen and A. Miyake, Phys. Rev. Lett. 101, 010502 (2008)
[126] A. Miyake, arXiv: 1009.3491.
[127] J. Cai, A. Miyake, W. Dur, H. J. Briegel Phys. Rev. A 82, 052309 (2010)
[128] X. Chen, R. Duan, Z. Ji, and B. Zeng, Phys. Rev. Lett. 105, 020502 (2010)
[129] T. C. Wei, I. Aeck, and R. Raussendorf, Phys. Rev. Lett. 106, 070501 (2010)
[130] D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, Phys. Rev. A 68, 062303 (2003)
[131] Q.-Q. Shi, R. Orus, J. O. Fjaerestad, H.-Q. Zhou, New J.Phys. 12 025008 (2010) .
[132] H.-Q. Zhou, R. Orus and G. Vidal, Phys. Rev. Lett. 100, 080601 (2008).
[133] M. Henkel, \Conformal Invariance and Critical Phenomena" (Springer, Berlin, 1999).
[134] H. J. Schulz, Phys. Rev. B 34, 6372 (1986).
[135] Y. Hatsugai and M. Kohmoto, Phys. Rev. B 44, 11789 (1991).
[136] W. Chen, K. Hida, and B. C. Sanctuary, Phys. Rev. B 67, 104401 (2003).
[137] P.-C. Chen, C.-Y. Lai, and M.-F. Yang, J. Stat. Mech. P10001 (2009).
[138] H. W. J. Blote and Y. Deng, Phys. Rev. E 66, 066110 (2002).
[139] S. Yunoki, Phys. Rev. B 65, 092402 (2002).
[140] M. Kohno and M. Takahashi, Phys. Rev. B 56, 3212 (1997).
[141] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B 424, 443 (1994).
[142] L. Tagliacozzo, T. R. de Oliveira, S. Iblisdir, and J. I. Latorre, Phys. Rev. B 78, 024410
(2008).
[143] F. Pollmann, S. Mukerjee, A. M. Turner, and J. E. Moore, Phys. Rev. Lett. 102, 255701
(2009).
[144] T.-C. Wei, Phys. Rev. A 81, 062313 (2010).
[145] R. Orus, Phys. Rev. A 78, 062332 (2008).
[146] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P. A. Fisher, Science 303, 1490
(2004).
[147] Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009); X. Chen, Z.-C. Gu, and X.-G.
Wen, arXiv:1008.3745 and arXiv:1103.3323.
[148] X. Chen, B. Zeng, Z.-C. Gu, I. L. Chuang, and X.-G. Wen, Phys. Rev. B 82, 165119
(2010); Z.-C. Gu, Z. Wang, and X.-G. Wen, arXiv:1010.1517.
[149] N. Schuch, D. Perez-Garca, and I. Cirac, arXiv:1010.3732.
[150] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, arXiv:0909.4059.
[151] X. G. Wen and Q. Niu. Phys. Rev. B. 41, 9377 (1990).
[152] A. Yu. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003).
[153] S. Dusuel, M. Kamfor, R. Orus, K. P. Schmidt, J. Vidal, Phys. Rev. Lett. 106, 107203
(2011).
[154] J. I. Cirac, D. Poilblanc, N. Schuch, F. Verstraete, arXiv:1103.3427.
[155] R. R. P. Singh, M. P. Gelfand, and D. A. Huse, Phys. Rev. Lett. 61, 2484 (1988).
[156] S.E. Kruger, J. Richter, D. J. J. Farnell and R. F. Bishop, Phys. Rev. B 61, 14607 (2000).
[157] S. Wenzel, L. Bogacz, and W. Janke1, Phys. Rev. Lett. 101, 127202 (2008).
[158] F.-J. Jiang, arXiv:0911.0653.
[159] N.B. Ivanov, S.E. Kruger, and J. Richter, Phys. Rev. B 53, 2633 (1996).
[160] J. Jordan, R. Orus, G. Vidal, F. Verstraete, J. I. Cirac, Phys. Rev. Lett. 101, 250602
(2008).
[161] L. Wang, I. Pizorn, F. Verstraete, Phys. Rev. B 83, 134421 (2011).
[162] P. W. Leung and N.-w. Lam, Phys. Rev. B 53, 2213 (1996).
[163] R. R. P. Singh, Z. Weihong, C. J. Hamer,and J. Oitmaa, Phys. Rev. B 60, 7278 (1999).
[164] Yongmin Yu and G. Muller, V. S. Viswanath, Phys. Rev. B 54, 9242 (1996).
[165] C. Y. Huang and F. L. Lin, Phys. Rev. A 81, 032304 (2010).