Gapped quantum phases for the S=1 spin chain with D2h symmetry

Physical Review B - Condensed Matter and Materials Physics

Volumn 84, Issue 7, 2011, Pages

  Liu, Zheng Xin ^a   Liu, Min ^a   Wen, Xiao Gang ^a,b  

^a TSINGHUA UNIVERSITY   (China)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80052409812     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.84.075135     Document Type: Article

References (42)

1. X. G. Wen, Int. J. Mod. Phys. B IJPBEV 0217-9792 10.1142/ S0217979290000139 4, 239 (1990).
2. X. Chen, Z. C. Gu, and X. G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.82.155138 82, 155138 (2010).
3. F. D. M. Haldane, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett. 50.1153 50, 1153 (1983);
4. F. D. M. Haldane, Phys. Lett. PYLAAG 0375-9601 10.1016/0375-9601(83) 90631-X 93, 464 (1983);
5. I. Affleck and F. D. M. Haldane, Phys. Rev. B PYLAAG 1098-0121 10.1103/PhysRevB.36.5291 36, 5291 (1987);
6. I. Affleck, J. Phys. Condens. Matter JCOMEL 0953-8984 10.1088/0953-8984/1/19/001 1, 3047 (1989).
7. M. den Nijs and K. Rommelse, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.40.4709 40, 4709 (1989);
8. T. Kennedy and H. Tasaki, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.45.304 45, 304 (1992).
9. Z. C. Gu and X. G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.80.155131 80, 155131 (2009).
10. X. Chen, Z. C. Gu, and X. G. Wen, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.83.035107 83, 035107 (2011).
11. 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)
12. F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, e-print arXiv: 0909.4059 (2009);
13. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.81.064439 81, 064439 (2010).
14. H. Li and F. D. M. Haldane, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.010504 101, 010504 (2008).
15. C. L. Kane and E. J. Mele, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.226801 95, 226801 (2005); (Pubitemid 41776999)
16. e-print arXiv: cond-mat/0411737.
17. B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.106802 96, 106802 (2006)
18. e-print arXiv: cond-mat/0504147.
19. C. L. Kane and E. J. Mele, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.95.146802 95, 146802 (2005); (Pubitemid 41777602)
20. e-print arXiv: cond-mat/0506581.
21. J. E. Moore and L. Balents, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.75.121306 75, 121306 (2007);
22. e-print arXiv: cond-mat/0607314.
23. 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)
24. e-print arXiv: cond-mat/0607699.
25. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.78.195424 78, 195424 (2008);
26. e-print arXiv: 0802.3537.
27. X. Chen, Z. C. Gu, and X. G. Wen, e-print arXiv: 1103.3323.
28. 3→ correspond to three "trivial" phases; they are classified in Ref. by different phases α(g) given in Eq.
29. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.59.799 59, 799 (1987);
30. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01218021 115, 477 (1988).
31. Here a sMP state means that it is injective, ergodic, and invariant under the symmetry group. In literature, such a state is also called a valence bond solid state. Usually, a VBS state is a translational symmetry breaking state (for spin-1/2 system). To avoid this confusion, we call it the sMP state instead.
32. G. Vidal, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.91.147902 91, 147902 (2003);
33. G. Vidal, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.040502 93, 040502 (2004);
34. G. Vidal, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.98.070201 98, 070201 (2007).
35. H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.090603 101, 090603 (2008).
36. xy still satisfy angular momentum [or SU(2)] algebra, but they do not stand for an usual spin, because the y- and z-component operators do not change their signs under time reversal transformation.
37. Charles Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, New York, 2004).
38. L. L. Boyle and Kerie F. Green, Math. Phys. Schi. A 288, 237 (1978).
39. H. H. Tu, G. M. Zhang, T. Xiang, Z. X. Liu, and T. K. Ng, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.80.014401 80, 014401 (2009).
40. J. Q. Chen, M. J. Gao, and G. Q. Ma, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.57.211 57, 211 (1985);
41. J. Q. Chen, J. Ping, and F. Wang, Group Representation Theory For Physicists (World Scientific, Singapore, 2002).
42. M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.97.110403 97, 110403 (2006). (Pubitemid 44394855)