Symmetry protected topological orders and the group cohomology of their symmetry group

Physical Review B - Condensed Matter and Materials Physics

Volumn 87, Issue 15, 2013, Pages

  Chen, Xie ^a,b   Gu, Zheng Cheng ^c,d   Liu, Zheng Xin ^b,e   Wen, Xiao Gang ^b,e,f  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^c CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^d California Institute of Technology   (United States)

^e TSINGHUA UNIVERSITY   (China)

^f PERIMETER INSTITUTE FOR THEORETICAL PHYSICS   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84876169194     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.87.155114     Document Type: Article

References (111)

1. L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937).
2. V. L. Ginzburg and L. D. Landau, Zh. Ekaper. Teoret. Fiz. 20, 1064 (1950).
3. L. D. Landau and E. M. Lifschitz, Statistical Physics-Course of Theoretical Physics (Pergamon, London, 1958), Vol 5.
4. X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.40.7387 40, 7387 (1989).
5. X.-G. Wen and Q. Niu, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.41. 9377 41, 9377 (1990).
6. X.-G. Wen, Int. J. Mod. Phys. B IJPBEV 0217-9792 10.1142/ S0217979290000139 4, 239 (1990).
7. B. I. Halperin, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.25.2185 25, 2185 (1982).
8. X.-G. Wen, Int. J. Mod. Phys. B IJPBEV 0217-9792 10.1142/ S0217979292000840 6, 1711 (1992).
9. G. 't Hooft, arXiv: gr-qc/9310026.
10. L. Susskind, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.531249 36, 6377 (1995).
11. R. Jackiw and C. Rebbi, Phys. Rev. D PRVDAQ 1550-7998 10.1103/PhysRevD.13.3398 13, 3398 (1976).
12. J. M. Leinaas and J. Myrheim, Il Nuovo Cimento 37B, 1 (1977).
13. F. Wilczek, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.49.957 49, 957 (1982).
14. B. I. Halperin, Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.52.1583 52, 1583 (1984).
15. D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.53.722 53, 722 (1984).
16. M. Levin and X.-G. Wen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.110405 96, 110405 (2006).
17. A. Kitaev and J. Preskill, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.110404 96, 110404 (2006).
18. X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.82.155138 82, 155138 (2010).
19. F. Verstraete, J. I. Cirac, J. I. Latorre, E. Rico, and M. M. Wolf, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.140601 94, 140601 (2005). (Pubitemid 40620407)
20. M. A. Levin and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.71.045110 71, 045110 (2005). (Pubitemid 40418061)
21. G. Vidal, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.99.220405 99, 220405 (2007). (Pubitemid 350191565)
22. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.48.1559 48, 1559 (1982).
23. R. B. Laughlin, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.50. 1395 50, 1395 (1983).
24. V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.59.2095 59, 2095 (1987).
25. X.-G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.39.11413 39, 11413 (1989).
26. N. Read and S. Sachdev, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.66.1773 66, 1773 (1991).
27. X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.44.2664 44, 2664 (1991).
28. R. Moessner and S. L. Sondhi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.86.1881 86, 1881 (2001). (Pubitemid 32252800)
29. G. Moore and N. Read, Nucl. Phys. B NUPBBO 0550-3213 10.1016/0550- 3213(91)90407-O 360, 362 (1991).
30. X.-G. Wen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.66.802 66, 802 (1991).
31. R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. H. English, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.59.1776 59, 1776 (1987).
32. I. P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, and K. W. West, Science SCIEAS 0036-8075 10.1126/science.1157560 320, 899 (2008). (Pubitemid 351929581)
33. M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Ann. Phys. (NY) APNYA6 0003-4916 10.1016/j.aop.2004.01.006 310, 428 (2004). (Pubitemid 38457639)
34. Z.-C. Gu, Z. Wang, and X.-G. Wen, arXiv: 1010.1517.
35. Z.-C. Gu and X.-G. Wen, arXiv: 1201.2648.
36. Y.-M. Lu, X.-G. Wen, Z. Wang, and Z. Wang, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.81.115124 81, 115124 (2010).
37. F. D. M. Haldane, Phys. Lett. A PYLAAG 0375-9601 10.1016/0375-9601(83) 90631-X 93, 464 (1983).
38. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01218021 115, 477 (1988).
39. C. L. Kane and E. J. Mele, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.226801 95, 226801 (2005). (Pubitemid 41776999)
40. B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.106802 96, 106802 (2006).
41. C. L. Kane and E. J. Mele, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.146802 95, 146802 (2005). (Pubitemid 41777602)
42. J. E. Moore and L. Balents, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.75.121306 75, 121306 (2007).
43. L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.98.106803 98, 106803 (2007). (Pubitemid 46393379)
44. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.78.195424 78, 195424 (2008).
45. X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.65.165113 65, 165113 (2002).
46. X.-G. Wen, Phys. Rev. D PRVDAQ 1550-7998 10.1103/PhysRevD.68.065003 68, 065003 (2003). (Pubitemid 44652415)
47. S.-P. Kou, M. Levin, and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.78.155134 78, 155134 (2008).
48. S.-P. Kou and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.80.224406 80, 224406 (2009).
49. M. Levin and A. Stern, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.103.196803 103, 196803 (2009).
50. H. Yao, L. Fu, and X.-L. Qi, arXiv: 1012.4470.
51. X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.83.035107 83, 035107 (2011).
52. N. Schuch, D. Perez-Garcia, and I. Cirac, Phys. Rev. B 1098-0121 10.1103/PhysRevB.84.165139 84, 165139 (2011).
53. X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 1098-0121 10.1103/PhysRevB.84.235128 84, 235128 (2011).
54. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.81.064439 81, 064439 (2010).
55. A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.83.075102 83, 075102 (2011).
56. L. Fidkowski and A. Kitaev, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.83.075103 83, 075103 (2011).
57. T. Hirano, H. Katsura, and Y. Hatsugai, Phys. Rev. B 1098-0121 10.1103/PhysRevB.77.094431 77, 094431 (2008).
58. Y. Hatsugai, New J. Phys. 1367-2630 10.1088/1367-2630/12/6/065004 12, 065004 (2010).
59. X. Chen, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 1098-0121 10.1103/PhysRevB.84.235141 84, 235141 (2011).
60. A. Kitaev, arXiv: 0901.2686.
61. A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.78.195125 78, 195125 (2008).
62. M. Levin (unpublished).
63. H. Yao and S. A. Kivelson, Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.105.166402 105, 166402 (2010).
64. T (1) to indicate that the time-reversal symmetry in G has a nontrivial action on the U (1) group (see Appendix 1). When G contains no time-reversal symmetry, its action on the U (1) group is trivial and we use U (1) to denote the U (1) group.
65. J. Wess and B. Zumino, Phys. Lett. B 10.1016/0370-2693(71)90582-X 37, 95 (1971).
66. E. Witten, Nucl. Phys. B NUPBBO 0550-3213 10.1016/0550-3213(83)90063-9 223, 422 (1983);
67. E. Witten, Commun. Math. Phys. 0010-3616 10.1007/BF01215276 4, 455 (1984).
68. Z.-X. Liu and X.-G. Wen, Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett. 110.067205 110, 067205 (2013).
69. B. Blok and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.42. 8145 42, 8145 (1990).
70. N. Read, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.65.1502 65, 1502 (1990).
71. X.-G. Wen and A. Zee, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.46. 2290 46, 2290 (1992).
72. Michael Levin (private communication).
73. M. Levin and A. Stern, Phys. Rev. B 1098-0121 10.1103/PhysRevB.86.115131 86, 115131 (2012).
74. Y.-M. Lu and A. Vishwanath, Phys. Rev. B 1098-0121 10.1103/PhysRevB.86. 125119 86, 125119 (2012).
75. F. D. M. Haldane, Phys. Rev. Lett. 0031-9007 10.1103/PhysRevLett.74.2090 74, 2090 (1995).
76. T. Senthil, J. B. Marston, and M. P. A. Fisher, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.60.4245 60, 4245 (1999).
77. N. Read and D. Green, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.61. 10267 61, 10267 (2000).
78. R. Roy, arXiv: cond-mat/0608064.
79. X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.102.187001 102, 187001 (2009).
80. M. Sato and S. Fujimoto, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.79.094504 79, 094504 (2009).
81. Z.-X. Liu, X. Chen, and X.-G. Wen, Phys. Rev. B 1098-0121 10.1103/PhysRevB.84.195145 84, 195145 (2011).
82. M. B. Hastings and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.72.045141 72, 045141 (2005). (Pubitemid 41526568)
83. S. Bravyi, M. Hastings, and S. Michalakis, J. Math. Phys. 0022-2488 10.1063/1.3490195 51, 093512 (2010).
84. S. Bravyi (private communication).
85. A. Gendiar, N. Maeshima, and T. Nishino, Prog. Theor. Phys. PTPKAV 0033-068X 10.1143/PTP.110.691 110, 691 (2003).
86. Y. Nishio, N. Maeshima, A. Gendiar, and T. Nishino, arXiv: cond-mat/0401115.
87. N. Maeshima, J. Phys. Soc. Jpn. JUPSAU 0031-9015 10.1143/JPSJ.73.60 73, 60 (2004). (Pubitemid 41222227)
88. F. Verstraete and J. I. Cirac, arXiv: cond-mat/0407066.
89. J. Jordan, R. Orus, G. Vidal, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.250602 101, 250602 (2008).
90. Z.-C. Gu, M. Levin, and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.78.205116 78, 205116 (2008).
91. H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.090603 101, 090603 (2008).
92. http://en.wikipedia.org/wiki/Measurable-function.
93. E. Witten, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01217730 121, 351 (1989).
94. T.-K. Ng, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.50.555 50, 555 (1994).
95. H. Yao and D.-H. Lee, Phys. Rev. B 10.1103/PhysRevB.82.245117 82, 245117 (2010).
96. For an introduction to group cohomology, see http://en.wikipedia.org/ wiki/Group-cohomology and http://math.arizona.edu/~sharifi/groupcoh.pdf.
97. V. G. Turaev and O. Y. Viro, Topology 0040-9383 10.1016/0040-9383(92) 90015-A 31, 865 (1992).
98. F. Costantino, Math. Z. MAZEAX 0025-5874 10.1007/s00209-005-0810-0 251, 427 (2005). (Pubitemid 41261035)
99. David Joyner, arXiv: 0706.0549.
100. R. Dijkgraaf and E. Witten, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF02096988 129, 393 (1990).
101. For a review, see J. D. Stasheff, Bull. Am. Math. Soc. 0273-0979 10.1090/S0002-9904-1978-14488-7 84, 513 (1978).
102. Friedrich Wagemann and Christoph Wockel, arXiv: 1110.3304.
103. C. Schommer-Pries, Geom. Topol. 1465-3060 10.2140/gt.2011.15.609 15, 609 (2011).
104. Robert Greenblatt, Homol. Homotopy Appl. 8, 91 (2006)
105. http://www.intlpress.com/HHA/v8/n2/a5/pdf.
106. Chris Gerig (mathoverflow.net/users/12310), Kuenneth-formula for group cohomology with nontrivial action on the coefficient, http://mathoverflow.net/ questions/75485 (version: 2011-09-15).
107. Group Cohomology at http://planetmath.org/encyclopedia/GroupCohomology3. html.
108. T], http://mathoverflow.net/questions/75582 (version: 2011-09-16).
109. David Handel, Tohoku Math. J. (2) 0040-8735 10.2748/tmj/1178225952 45, 13 (1993).
110. Tye Lidman, Free Actions of GRoups Through Cohomology at http://math.berkeley.edu/~teichner/Courses/215B/Surveys/TyeLidman.pdf.
111. Neil Strickland (mathoverflow.net/users/10366), Group cohomology of compact Lie group with integer coeffient, http://mathoverflow.net/questions/ 75512 (version: 2011-09-15).