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1) K. S. Novoselov et al.: Nature438(2005)197; A. K. Geim and K. S. Novoselov: Nature Materials6(2007)183; A. H.

C astro Neto et al.: Rev. Mod. Phys.81(2009)109.

2) Y. Zhang, Y.W. Tan, H. L. Stormer, and P. Kim: Nature 438(2005)197.

3) D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. C hakraborty: Adv. Phys.59(2010)261.

4) ÄØKêN#¿{mu(Ò)F«åÑ¢ïm@\gpW ](CMCoÅ#2009)$

5) R. Mitsuhashi et al.: Nature464(2010)76.

6) T. Kosugi et al.: J. Phys. Soc. Jpn78(2009)113704.

7) T. E. Weller et al.: Nature Phys.1(2005)39; N. Emery et al.: Phys. Rev. Lett.95(2005)087003.

8) RmTO#4³óÔjg£X~K#iq®ô´

_fmM. Creutz: JHEP17(2008)0804#js

eC$

9) N. D. Mermin: Phys. Rev.176(1968)250.

10) J. C . Meyer, A. K. Geim, M. I. Katsnelson, K. S.

Novoselov, T. J. Booth, and S. Roth: Nature446(2007)60.

11) A. Fasolino, J. H. Los, and M. I. Katsnelson: Nature Materials6,(2007)858.

12) P. R. Wallace: Phys. Rev.71(1947)622.

13) W. H. Lomer: Roc. Roy. Soc.(London),330(1955)A227.

14) J. W. McClure: Phys. Rev.104(1956)666.

15) J. C . Slonczewsky and P. R. Weiss: Phys. Rev.109(1958) 272.

16) T. Nakajima and H. Aoki: Physica E40(2008)1354.

17) H. Hsu and L. E. Reichl: Phys. Rev. B72(2005)155413.

18) Y. Hatsugai: Proc. HMF_19, to be published arXiv:

1008.4653.

19) M. V. Berry: Proc. R. Soc. A392(1984)45.

20) Y. Hatsugai, T. Fukui, and H. Aoki: Phys. Rev. B74(2006) 205414.

21) Y. Hatsugai, T. Fukui, and H. Aoki: Euro. Phys. J. Spec.

Top.148(2007)133.

22) A. H. C astro Neto et al.: Rev. Mod. Phys.81(2009)109.

23) H. B. Nielsen and M. Ninomiya: Nucl. Phys. B185,(1981) 20.

24) Y. Hatsugai: New J. Physics12(2010)055014.

25) Y. Hatsugai: Solid State Comm.149(2009)1016.

26) A. Altland and M. R. Zirnbauer: Phys. Rev. B55(1997) 1142.

27) A. Ludwig, M. Fisher, R. Shankar, and G. Grinstein: Phys.

Rev. B50(1994)7526.

28) Y. Hatsugai, M. Kohomoto, and XG. Wen: Phys. Rev. B 56(1997)1061.

29) R. B. Laughlin: Phys. Rev. B25(1985)2185.

30) H. Watanabe, Y. Hatsugai, and H. Aoki: arXiv: 1008.0130.

31) V. P. Gusynin and S. G. Sharapov: Phys. Rev. Lett.95 (2005)146801.

32) Y. Zheng and T. Ando: Phys. Rev. B65(2002)245420.

33) T. Morimoto, Y. Hatsugai, and H. Aoki: Phys. Rev. Lett.

103(2009)116803.

34) D. J. Thouless, M. Kohmoto, P. Nightingale, and M. den

Nijs: Phys. Rev. Lett.49(1982)405.

35) Q. Niu, D. J. Thouless, and Y. S. Wu: Phys. Rev. B31 (1985)3372.

36) M. Kohmoto: Ann. Phys.(N.Y.)160(1985)355.

37) Y. Hatsugai: Phys. Rev. Lett.71(1993)3697.

38) Y. Hatsugai: J. Phys. C , C ondens. Matter9(1997)2507.

39) Y. Hatsugai: J. Phys. Soc. Jpn.73(2004)2604.

42) F. Wilczek and A. Zee: Phys. Rev. Lett.52(1984)2111.

43) T. Fukui, Y. Hatsugai, and H. Suzuki: J. Phys. Soc. Jpn.74 (2005)1674.

44) R. D. KingSmith and D. Vanderbilt: Phys. Rev. B47 (1993)1651.

45) M. Arai and Y. Hatsugai: Phys. Rev. B79(2009)075429.

46) Y. Aharonov and A. C asher: Phys. Rev. A19(1979)2461.

47) T. Kawarabayashi, Y. Hatsugai, and H. Aoki: Phys. Rev.

Lett.103(2009)156804.

48) B. I. Halperin: Phys. Rev. B25(1982)2185.

49) Y. Hatsugai: Phys. Rev. B48(1993)11851.

50) T. Kennedy: J. Phys. Condens. Matter2(1990)5737.

51) M. Arikawa, S. Tanaya, I. Maruyama, and Y. Hatsugai:

Phys. Rev.79(2009)205107.

52) V. W. Scarola and S. Das Sarma: Phys. Rev. Lett.98(2007) 210403.

53) Z. Wang, Y. D. C hong, J. D. Joannopoulos, and M. Soljacic:

Phys. Rev. Lett.100(2008)013905.

54) Z. Wang, Y. D. C hong, J. D. Joannopoulos, and M. Soljacic:

Nature461(2009)772.

55) C . L. Kane and E. J. Mele: Phys. Rev. Lett.95(2005) 146802.

56) B. A. Bernevig, T. L. Hughes, and S.C . Zhang: Science314 (2006)1757.

57) M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe:

J. Phys. Soc. Jpn.65(1996)1920.

58) S. Okada and A. Oshiyama: Phys. Rev. Lett.87(2001) 146803.

59) Y. Kobayashi et al.: Phys. Rev. B71(2005)193406.

60) S. Ryu and Y. Hatsugai: Phys. Rev. Lett.89(2002)077002.

61) C . Hu: Phys. Rev. Lett.72(1994)1526.

62) Y. Tanaka and S. Kashiwaya: Phys. Rev. Lett.74(1995) 3451.

63) M. Aprili, E. Badica and L. H. Greene: Phys. Rev. Lett.83 (1999)4630.

64) S. Ryu and Y. Hatsugai: Physica C388_389(2003)90.

65) S. Ryu and Y. Hatsugai: Physica E22(2003)779.

66) M. Matsumoto and H. Shiba: J. Phys. Soc. Jpn.64(1995) 3384.

67) M. Sigrist, B. Bailey, and R. B. Laughlin: Phys. Rev. Lett.

74(1995)3249.

68) M. Arikawa, Y. Hatsugai, and H. Aoki: Phys. Rev. B78 (2008)205401.

69) M. Arikawa, Y. Hatsugai, and H. Aoki: J. Phys. Conf. Series 150(2009)022003.

70) D. L. Miller and et al.: Science324(2009)924.

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