Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Physical Review B - Condensed Matter and Materials Physics

Volumn 82, Issue 15, 2010, Pages

  Chen, Xie ^a   Gu, Zheng Cheng ^b   Wen, Xiao Gang ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 78149237743     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.82.155138     Document Type: Article

References (80)

1. L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937).
2. V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).
3. J. G. Bednorz and K. A. Mueller, Z. Phys. B: Condens. Matter 64, 189 (1986). 10.1007/BF01303701
4. P. W. Anderson, Science 235, 1196 (1987). 10.1126/science.235.4793.1196
5. G. Baskaran, Z. Zou, and P. W. Anderson, Solid State Commun. 63, 973 (1987). 10.1016/0038-1098(87)90642-9
6. I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988). 10.1103/PhysRevB.37.3774
7. D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988). 10.1103/PhysRevLett.61.2376
8. I. Affleck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745 (1988). 10.1103/PhysRevB.38.745
9. E. Dagotto, E. Fradkin, and A. Moreo, Phys. Rev. B 38, 2926 (1988). 10.1103/PhysRevB.38.2926
10. V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, 2095 (1987). 10.1103/PhysRevLett.59.2095
11. X.-G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, 11413 (1989). 10.1103/PhysRevB.39.11413
12. X.-G. Wen, Phys. Rev. B 40, 7387 (1989). 10.1103/PhysRevB.40.7387
13. X.-G. Wen, Int. J. Mod. Phys. B 4, 239 (1990). 10.1142/S0217979290000139
14. The name "topological order" is motivated by the low-energy effective theory of the chiral spin states, which is a topological quantum field theory (Ref.).
15. X.-G. Wen, Phys. Rev. B 43, 11025 (1991). 10.1103/PhysRevB.43.11025
16. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006). 10.1103/PhysRevLett.96.110404
17. M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006). 10.1103/PhysRevLett.96.110405
18. J. B. Marston and I. Affleck, Phys. Rev. B 39, 11538 (1989). 10.1103/PhysRevB.39.11538
19. X.-G. Wen, Phys. Rev. Lett. 66, 802 (1991). 10.1103/PhysRevLett.66.802
20. A. Y. Kitaev, Ann. Phys. (N.Y.) 321, 2 (2006). 10.1016/j.aop.2005.10.005
21. X.-G. Wen and Q. Niu, Phys. Rev. B 41, 9377 (1990). 10.1103/PhysRevB.41. 9377
22. F. D. M. Haldane and E. H. Rezayi, Phys. Rev. B 31, 2529 (1985). 10.1103/PhysRevB.31.2529
23. G. Kotliar and J. Liu, Phys. Rev. B 38, 5142 (1988). 10.1103/PhysRevB.38. 5142
24. N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991). 10.1103/PhysRevLett.66.1773
25. X.-G. Wen, Phys. Rev. B 44, 2664 (1991). 10.1103/PhysRevB.44.2664
26. P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621 (1992). 10.1103/PhysRevB.46.5621
27. A. Y. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003). 10.1016/S0003-4916(02) 00018-0
28. W. Rantner and X.-G. Wen, Phys. Rev. B 66, 144501 (2002). 10.1103/PhysRevB.66.144501
29. M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, and X.-G. Wen, Phys. Rev. B 70, 214437 (2004). 10.1103/PhysRevB.70.214437
30. X.-G. Wen, Phys. Rev. B 65, 165113 (2002). 10.1103/PhysRevB.65.165113
31. X.-G. Wen and A. Zee, Phys. Rev. B 66, 235110 (2002). 10.1103/PhysRevB.66.235110
32. X.-G. Wen, Phys. Lett. A 300, 175 (2002). 10.1016/S0375-9601(02)00808-3
33. The conjecture is for symmetry-breaking states with finite energy gaps. However, it is unclear whether it is possible to find a definition of "short-range quantum entanglement" to make the conjecture valid for gapless symmetry-breaking states.
34. S. Bravyi, M. Hastings, and S. Michalakis, J. Math. Phys. 51, 093512 (2010). 10.1063/1.3490195
35. M. B. Hastings and X.-G. Wen, Phys. Rev. B 72, 045141 (2005). 10.1103/PhysRevB.72.045141
36. For each Hamiltonian, we can introduce an algebra of local operators. If the energy gap for H (g) is finite for all g in the segment [0,1], Ref. shows that the algebra of local operators for different g 's are isomorphic to each other at low energies, using a quasiadiabatic continuation of quantum states.
37. E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972). 10.1007/BF01645779
38. S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006). 10.1103/PhysRevLett.97.050401
39. D. I. Tsomokos, A. Hamma, W. Zhang, S. Haas, and R. Fazio, Phys. Rev. A 80, 060302 (2009). 10.1103/PhysRevA.80.060302
40. A state is called a gapped state if there exist a local Hamiltonian such that the state is the gapped ground state of the Hamiltonian.
41. Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009). 10.1103/PhysRevB.80.155131
42. F. Verstraete, J. I. Cirac, J. I. Latorre, E. Rico, and M. M. Wolf, Phys. Rev. Lett. 94, 140601 (2005). 10.1103/PhysRevLett.94.140601
43. G. Vidal, Phys. Rev. Lett. 99, 220405 (2007) 10.1103/PhysRevLett.99. 220405
44. Understanding Quantum Phase Transitions, edited by, L. D. Carr, (Taylor & Francis, Boca Raton, 2010)).
45. S. Bravyi (personal communication)
46. B. Yoshida, arXiv:1007.4601 (unpublished).
47. F. D. M. Haldane, Phys. Lett. A 93, 464 (1983). 10.1016/0375-9601(83) 90631-X
48. F. Pollmann, E. Berg, A. Turner, and M. Oshikawa, arXiv:0909.4059 (unpublished).
49. C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 10.1103/PhysRevLett.95.226801
50. B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006). 10.1103/PhysRevLett.96.106802
51. C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). 10.1103/PhysRevLett.95.146802
52. J. Moore and L. Balents, Phys. Rev. B 75, 121306 (R) (2007). 10.1103/PhysRevB.75.121306
53. L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007). 10.1103/PhysRevLett.98.106803
54. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008). 10.1103/PhysRevB.78.195424
55. N. Read and D. Green, Phys. Rev. B 61, 10267 (2000). 10.1103/PhysRevB.61. 10267
56. W.-H. Ko, P. A. Lee, and X.-G. Wen, Phys. Rev. B 79, 214502 (2009). 10.1103/PhysRevB.79.214502
57. S.-P. Kou, M. Levin, and X.-G. Wen, Phys. Rev. B 78, 155134 (2008). 10.1103/PhysRevB.78.155134
58. S.-P. Kou and X.-G. Wen, Phys. Rev. B 80, 224406 (2009). 10.1103/PhysRevB.80.224406
59. S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Phys. Rev. Lett. 100, 156401 (2008). 10.1103/PhysRevLett.100.156401
60. Y. Zhang, Y. Ran, and A. Vishwanath, Phys. Rev. B 79, 245331 (2009). 10.1103/PhysRevB.79.245331
61. D. Pesin and L. Balents, Nat. Phys. 6, 376 (2010). 10.1038/nphys1606
62. B. Yang and Y. Kim, Phys. Rev. B 82, 085111 (2010). 10.1103/PhysRevB.82. 085111
63. J. Maciejko, X. Qi, A. Karch, and S. Zhang, arXiv:1004.3628 (unpublished).
64. B. Swingle, M. Barkeshli, J. McGreevy, and T. Senthil, arXiv:1005.1076 (unpublished).
65. M. A. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005). 10.1103/PhysRevB.71.045110
66. M. Aguado and G. Vidal, Phys. Rev. Lett. 100, 070404 (2008) 10.1103/PhysRevLett.100.070404
67. R. König, B. W. Reichardt, and G. Vidal, Phys. Rev. B 79, 195123 (2009). 10.1103/PhysRevB.79.195123
68. S. Hong, arXiv:0907.2204 (unpublished).
69. M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Ann. Phys. (N.Y.) 310, 428 (2004). 10.1016/j.aop.2004.01.006
70. A. Gendiar, N. Maeshima, and T. Nishino, Prog. Theor. Phys. 110, 691 (2003). 10.1143/PTP.110.691
71. Y. Nishio, N. Maeshima, A. Gendiar, and T. Nishino, arXiv:cond-mat/ 0401115 (unpublished).
72. N. Maeshima, J. Phys. Soc. Jpn. 73, 60 (2004). 10.1143/JPSJ.73.60
73. F. Verstraete and J. Cirac, arXiv:cond-mat/0407066 (unpublished).
74. F. Verstraete, J. I. Cirac, and V. Murg, Adv. Phys. 57, 143 (2008). 10.1080/14789940801912366
75. Z.-C. Gu, M. Levin, and X.-G. Wen, Phys. Rev. B 78, 205116 (2008). 10.1103/PhysRevB.78.205116
76. H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008). 10.1103/PhysRevLett.101.090603
77. M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007). 10.1103/PhysRevLett.99.120601
78. X. Chen, B. Zeng, Z. Gu, I. Chuang, and X. Wen, arXiv:1003.1774 (unpublished).
79. S. Lloyd, Science 273, 1073 (1996). 10.1126/science.273.5278.1073
80. E. Witten, Commun. Math. Phys. 121, 351 (1989). 10.1007/BF01217730