Symmetry-protected topological phases and orbifolds: Generalized Laughlin's argument

Physical Review B - Condensed Matter and Materials Physics

Volumn 88, Issue 7, 2013, Pages

  Sule, Olabode Mayodele ^a   Chen, Xiao ^a   Ryu, Shinsei ^a  

^a United States of America   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84884399416     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.88.075125     Document Type: Article

References (57)

1. X.-G. Wen, Quantum Field Theory of Many-Body Systems (Oxford University Press, Oxford, 2004).
2. The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer, New York, 1987).
3. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and Sankar Das Sarma, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.80.1083 80, 1083 (2008).
4. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.82.3045 82, 3045 (2010).
5. X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.83.1057 83, 1057 (2011).
6. C. L. Kane and E. J. Mele, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.146802 95, 146802 (2005); (Pubitemid 41777602)
7. C. L. Kane and E. J. Mele, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.226801 95, 226801 (2005). (Pubitemid 41776999)
8. B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.106802 96, 106802 (2006).
9. B. A. Bernevig, T. Hughes, and S.-C. Zhang, Science SCIEAS 0036-8075 10.1126/science.1133734 314, 1757 (2006). (Pubitemid 44969083)
10. J. E. Moore and L. Balents, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.75.121306 75, 121306 (R) (2007).
11. R. Roy, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.79.195322 79, 195322 (2009).
12. 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)
13. L. Fu and C. L. Kane, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.76. 045302 76, 045302 (2007).
14. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.78.195424 78, 195424 (2008).
15. 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).
16. S. Ryu, A. Schnyder, A. Furusaki, and A. W. W. Ludwig, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/12/6/065010 12, 065010 (2010).
17. A. Yu Kitaev, in Advances in Theoretical Physics: Landau Memorial Conference, Chernogolovka, Russia, 2008, edited by, V. Lebedev, and, M. Feigel'man, AIP Conf. Proc. No. 1134 (AIP, Melville, NY, 2009), p. 22.
18. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.81.064439 81, 064439 (2010).
19. X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.83.035107 83, 035107 (2011).
20. N. Schuch, D. Perez-Garcia, and I. Cirac, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.84.165139 84, 165139 (2011).
21. X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.84.235128 84, 235128 (2011).
22. L. Fidkowski and A. Kitaev, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.81.134509 81, 134509 (2010).
23. L. Fidkowski and A. Kitaev, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.83.075103 83, 075103 (2011).
24. A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.83.075102 83, 075102 (2011).
25. Zheng-Cheng Gu and Xiao-Gang Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.80.155131 80, 155131 (2009).
26. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.87.155114 87, 155114 (2013).
27. Zheng-Cheng Gu and Xiao-Gang Wen, arXiv: 1201.2648.
28. M. Levin and A. Stern, Phys. Rev. Lett PRLTAO 0031-9007 10.1103/PhysRevLett.103.196803 103, 196803 (2009).
29. T. Neupert, L. Santos, S. Ryu, C. Chamon, and C. Mudry, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.84.165107 84, 165107 (2011).
30. Y.-M. Lu and A. Vishwanath, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.86.125119 86, 125119 (2012).
31. Y.-M. Lu and A. Vishwanath, arXiv: 1302.2634.
32. J. Wang, and X.-G. Wen, arXiv: 1212.4863.
33. A. Kapustin and N. Saulina, Nucl. Phys. B NUPBBO 0550-3213 10.1016/j.nuclphysb.2010.12.017 845, 393 (2011).
34. M. Levin, Phys. Rev. X 10.1103/PhysRevX.3.021009 3, 021009 (2013).
35. Ling-Yan Hung and Yidun Wan, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.87.195103 87, 195103 (2013).
36. Meng Cheng and Zhen-Cheng Gu, arXiv: 1302.4803.
37. M. Levin and Z.-C. Gu, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.86. 115109 86, 115109 (2012).
38. Z.-C. Gu and M. Levin, arXiv: 1304.4569.
39. S. Ryu and S.-C. Zhang, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.85.245132 85, 245132 (2012).
40. R. B. Laughlin, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.23.5632 23, 5632 (1981).
41. L. Dixon, D. Friedan, E. Martinec, and S. Shenker, Nucl. Phys. B NUPBBO 0550-3213 10.1016/0550-3213(87)90676-6 282, 13 (1987).
42. L. Dixon, J. Harvey, C. Vafa, and E. Witten, Nucl. Phys. B NUPBBO 0550-3213 10.1016/0550-3213(85)90593-0 261, 678 (1985).
43. An orbifold projection usually refers to a symmetry projection which leads to a spacetime singularity in the string theory context. In this paper, we use the word orbifold for symmetry projections in general, which do not necessary generate a spacetime singularity. More specfically, the type of projections we discuss in this paper include "shifted orbifolds" that do not give rise to a singularity (but are still called orbifolds).
44. K. S. Narain, M. H. Sarmadi, and C. Vafa, Nucl. Phys. B NUPBBO 0550-3213 10.1016/0550-3213(87)90228-8 288, 551 (1987);
45. K. S. Narain, M. H. Sarmadi, and C. Vafa, Nucl. Phys. B NUPBBO 0550-3213 10.1016/0550-3213(91)90145-N 356, 163 (1991).
46. Xiao-Liang Qi, New J. Phys. 10.1088/1367-2630/15/6/065002 15, 065002 (2013).
47. Hong Yao and Shinsei Ryu, arXiv: 1202.5805.
48. A. Cappelli and G. R. Zemba, Nucl. Phys. B NUPBBO 0550-3213 10.1016/S0550-3213(97)00110-7 490, 595 (1997);
49. A. Cappelli, L. S. Georgiev, and G. R. Zemba, J. Phys. A 1751-8113 10.1088/1751-8113/42/22/222001 42, 222001 (2009);
50. A. Cappelli, G. Viola, and G. R. Zemba, Ann. Phys. (NY) APNYA6 0003-4916 10.1016/j.aop.2009.10.007 325, 465 (2010);
51. A. Cappelli and G. Viola, J. Phys. A 1751-8113 10.1088/1751-8113/44/7/ 075401 44, 075401 (2011).
52. Xiao-Liang Qi and Shou-Cheng Zhang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.086802 101, 086802 (2008).
53. Orbifolding by a symmetry group is also known as gauging since from the point of view of target space physics we are using labels in a covering space to label states in the orbifolded theory. Therefore there is a redundancy in the labeling and hence a gauge symmetry.
54. J. L. Cardy, Nucl. Phys. B NUPBBO 0550-3213 10.1016/0550-3213(86)90552-3 270, 186 (1986).
55. J. Polchinski, String Theory (Cambridge University Press, Cambridge, 1998).
56. X. Chen, O. M. Sule, and S. Ryu (unpublished).
57. C. Vafa, Nucl. Phys. B 10.1016/0550-3213(86)90379-2 273, 592 (1986).