Competing charge-density wave, magnetic, and topological ground states at and near Dirac points in graphene in axial magnetic fields

Physical Review B - Condensed Matter and Materials Physics

Volumn 90, Issue 7, 2014, Pages

  Roy, Bitan ^a   Sau, Jay D ^a  

^a University of Maryland   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84926496834     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.90.075427     Document Type: Article

References (55)

1. C. X. Liu, X. L. Qi, H. J. Zhang, X. Dai, Z. Fang, and S. C. Zhang, Phys. Rev. B 82, 045122 (2010). PRBMDO 1098-0121 10.1103/PhysRevB.82.045122
2. P. Goswami and B. Roy, arXiv:1211.4023;
3. B. Roy and P. Goswami, Phys. Rev. B 89, 144507 (2014) 10.1103/PhysRevB.89.144507.
4. V. Braguta, M. N. Chernodub, K. Landsteiner, M. I. Polikarpov, and M. V. Ulybyshev, Phys. Rev. D 88, 071501 (R) (2013). PRVDAQ 1550-7998 10.1103/PhysRevD.88.071501
5. F. Guinea, M. I. Katsnelson, and A. K. Geim, Nat. Phys. 6, 30 (2010). 1745-2473 10.1038/nphys1420
6. N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A. H. Castro Neto, and M. F. Crommie, Science 329, 544 (2010). SCIEAS 0036-8075 10.1126/science.1191700
7. J. Lu, A. H. Castro Neto, and K. P. Loh, Nat. Commun. 3, 823 (2012). 2041-1723 10.1038/ncomms1818
8. K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. Manoharan, Nature (London) 483, 306 (2012). NATUAS 0028-0836 10.1038/nature10941
9. P. R. Wallace, Phys. Rev. 71, 622 (1947). PHRVAO 0031-899X 10.1103/PhysRev.71.622
10. G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984). PRLTAO 0031-9007 10.1103/PhysRevLett.53.2449
11. I. F. Herbut, Phys. Rev. B 78, 205433 (2008). PRBMDO 1098-0121 10.1103/PhysRevB.78.205433
12. Y. Aharonov and A. Casher, Phys. Rev. A 19, 2461 (1979). 0556-2791 10.1103/PhysRevA.19.2461
13. B. Roy, Ph.D. thesis, Simon Fraser University, 2011, http://theses.lib. sfu.ca/thesis/etd6841
14. B. Roy and I. F. Herbut, Phys. Rev. B 88, 045425 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.88.045425
15. I. F. Herbut and B. Roy, Phys. Rev. B 77, 245438 (2008). PRBMDO 1098-0121 10.1103/PhysRevB.77.245438
16. B. Roy and I. F. Herbut, Phys. Rev. B 83, 195422 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.83.195422
17. F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). PRLTAO 0031-9007 10.1103/PhysRevLett.61.2015
18. I. F. Herbut, V. Juričić, and B. Roy, Phys. Rev. B 79, 085116 (2009). PRBMDO 1098-0121 10.1103/PhysRevB.79.085116
19. P. Ghaemi, J. Cayssol, D. N. Sheng, and A. Vishwanath, Phys. Rev. Lett. 108, 266801 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.108.266801
20. B. Roy, F. F. Assaad, and I. F. Herbut, Phys. Rev. X 4, 021042 (2014). 2160-3308 10.1103/PhysRevX.4.021042
21. D. A. Abanin and D. A. Pesin, Phys. Rev. Lett. 109, 066802 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.109.066802
22. O. Motrunich, K. Damle, and D. A. Huse, Phys. Rev. B 65, 064206 (2002). PRBMDO 0163-1829 10.1103/PhysRevB.65.064206
23. Throughout this paper, (Equation presented), (Equation presented) are measured in units of (Equation presented).
24. In cylindrical graphene the top and the bottom edges are constituted by (Equation presented) and (Equation presented) sublattices, respectively (see Fig. 1). Therefore, two edges of the cylinder are inequivalent.
25. R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 98, 266402 (2007). PRLTAO 0031-9007 10.1103/PhysRevLett.98.266402 (Pubitemid 47139970)
26. B. Roy, Phys. Rev. B 84, 035458 (2011) PRBMDO 1098-0121 10.1103/PhysRevB.84.035458;
27. B. Roy, Phys. Rev. B 85, 165453 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.85.165453
28. F. de Juan, Phys. Rev. B 87, 125419 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.87.125419
29. S. Gopalakrishnan, P. Ghaemi, and S. Ryu, Phys. Rev. B 86, 081403 (R) (2012). PRBMDO 1098-0121 10.1103/PhysRevB.86.081403
30. Sublattice degeneracy of the zero-energy manifold in a finite-size graphene can be argued in the following way. As mentioned, in finite systems generically there is no state at exact zero energy. However, for any two states (Equation presented), placed at energies (Equation presented), the symmetric and the antisymmetric combinations of (Equation presented), say (Equation presented) and (Equation presented), reside on (Equation presented) and (Equation presented) sublattices, respectively, for example. In particular, when (Equation presented), and the graphene flake is strained, (Equation presented) is localized in the bulk, whereas (Equation presented) can only be found near the edge of the system.
31. C.-Y. Hou, C. Chamon, and C. Mudry, Phys. Rev. Lett. 98, 186809 (2007). PRLTAO 0031-9007 10.1103/PhysRevLett.98.186809 (Pubitemid 46701718)
32. C. Weeks and M. Franz, Phys. Rev. B 81, 085105 (2010). PRBMDO 1098-0121 10.1103/PhysRevB.81.085105
33. E. V. Castro, A. G. Grushin, B. Valenzuela, M. A. H. Vozmediano, A. Cortijo, and F. de Juan, Phys. Rev. Lett. 107, 106402 (2011) PRLTAO 0031-9007 10.1103/PhysRevLett.107.106402;
34. E. V. Castro, A. G. Grushin, B. Valenzuela, M. A. H. Vozmediano, A. Cortijo, and F. de Juan, Phys. Rev. B 87, 085136 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.87.085136
35. The time-reversal operator in real space is simply (Equation presented) (complex conjugation).
36. J. A. Gracey, Int. J. Mod. Phys. A 9, 727 (1994). IMPAEF 0217-751X 10.1142/S0217751X94000340
37. L. Rosa, P. Vitale, and C. Wetterich, Phys. Rev. Lett. 86, 958 (2001) PRLTAO 0031-9007 10.1103/PhysRevLett.86.958;
38. F. Hofling, C. Nowak, and C. Wetterich, Phys. Rev. B 66, 205111 (2002). PRBMDO 0163-1829 10.1103/PhysRevB.66.205111
39. Any translational symmetry-breaking order parameter of periodicity (Equation presented) can be accommodated in honeycomb lattice by enlarging the size of the unit cell, which now includes six sites and nine NN bonds. For such construction, see B. Roy and I. F. Herbut, Phys. Rev. B 82, 035429 (2010). PRBMDO 1098-0121 10.1103/PhysRevB.82.035429
40. M. Manzardo, J. W. F. Venderbos, J. van den Brink, and C. Ortix, arXiv:1402.3145.
41. M. Daghofer and M. Hohenadler, Phys. Rev. B 89, 035103 (2014). PRBMDO 1098-0121 10.1103/PhysRevB.89.035103
42. N. A. García-Martínez, A. G. Grushin, T. Neupert, B. Valenzuela, and E. V. Castro, Phys. Rev. B 88, 245123 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.88.245123
43. T. DJuric, N. Chancellor, and I.F. Herbut, Phys. Rev. B 89, 165123 (2014). PRBMDO 1098-0121 10.1103/PhysRevB.89.165123
44. T. O. Wehling, E. Sasioglu, C. Friedrich, A. I. Lichtenstein, M. I. Katsnelson, and S. Blügel, Phys. Rev. Lett. 106, 236805 (2011). PRLTAO 0031-9007 10.1103/PhysRevLett.106.236805
45. S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Phys. Rev. Lett. 100, 156401 (2008). PRLTAO 0031-9007 10.1103/PhysRevLett.100.156401 (Pubitemid 351549477)
46. M. Polini, F. Guinea, M. Lewenstein, H. C. Manoharan, and V. Pellegrini, Nat. Nanotechnol. 8, 625 (2013). 1748-3387 10.1038/nnano.2013.161
47. B. Roy, Z. X. Hu, and K. Yang, Phys. Rev. B 87, 121408 (R) (2013). PRBMDO 1098-0121 10.1103/PhysRevB.87.121408
48. A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, Nat. Phys. 3, 172 (2007). 1745-2473 10.1038/nphys547
49. B. Roy and V. Juričić, Phys. Rev. B 90, 041413 (R) (2014). 1098-0121 10.1103/PhysRevB.90.041413
50. D. W. Zhang, C. J. Shan, F. Mei, M. Yang, R. Q. Wang, and S. L. Zhu, Phys. Rev. A 89, 015601 (2014). PLRAAN 1050-2947 10.1103/PhysRevA.89.015601
51. B. Yang, arXiv:1402.0941.
52. Z. Y. Meng, H. H. Hung, and T. C. Lang, Mod. Phys. Lett. B 28, 1430001 (2014). MPLBET 0217-9849 10.1142/S0217984914300014
53. M. Laubach, J. Reuther, R. Thomale, and S. Rachel, arXiv:1312.2934.
54. S. Rachel, arXiv:1310.3159.
55. M. Yang, J. B. Qiao, Z. D. Chu, W. Y. He, and L. He, arXiv:1310.1671.