(extended state) L (2 L 1, O(1), d O(V), V = L d V V e 2 /h 1980 Klitzing

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1 [1] [2] 2.1 STM [3, 4, 5, 6] 2.1: 2 ( 3 [1] ) [7, 8] [9]( 2.2)

2 (extended state) L (2 L 1, O(1), d O(V), V = L d V V e 2 /h 1980 Klitzing

3 [7, 8] 2.2 [10] [8] 2.2: (a) (b) [9] [11] [12] [13] 1

4 [9] 2 [6] k 0 E C (k) =E C C C δk 2, E V (k) = E V + C V δk 2, (δk = k k 0 ) C C = 2mC, C V = 2m V mc,v E g = E V E C E g E g δk 2 E g δk 1 2 E(k) = ± c δk (2.1)

5 Maxwell (2.1) c Dirac Dirac Dirac Von Neumann-Wigner [14] 2 - k x, k y 2 2 Von Neumann-Wigner Berry [15] 2 2 H 2 H Tr H = R z R z ir y H =R x σ x + R y σ y + R z σ z = R x + ir y R = R σ (2.2) z H 2 = R 2 E 2, (E ), R = R 2 x + R 2 y + R 2 z H ±R R = 0 2 R x, R y, R z k x, k y 1 (k 0 ) k 0 E = ± δk E = ± δk 2 + m 2 m = 0

6 6 2 (K,K ) [18] [6] 2 [20] [21] (2.2) k R R(k) R(k k A = i R(k) k R(k) R = e iθ R A = A + k θ A C e iγ [15, 16] γ = C dk A k = S ds B k B = rot A γ C [6] , [1] 2 [4]

7 (a) (b) (c) k y 2.3: 2 [6] H = t (c i c j + c j c i ) i j i j 2.3 c i, c j [8, 6] k y k y k y H 2D = H1D Z (k y) = H1D A (k y) k y k y H1D Z, HA 1D (b) 2.3(c) 2 k y

8 H 1D 1 [6] H Γ {H, Γ} = HΓ + ΓH = 0, Γ 2 = 1 Γ : + E ψ ψ Γ = Γ ψ E ( ψ ψ Γ = 0) ψ Γ ψ, ψ 2 Γ 2.3 2

9 STM [4] k y 1 H 1D (k y ) [6] 1 H(k x, k y ) 2 2, H(k x, k y ) = R(k x, k y ) σ 1 ψ = R(k x, k y ) k x ({H(k x, k y ), Γ} = 0), (mod 2π ) 1 [6] π mod 2π π k y γ(k y ) = i dk x R(k x, k y ) R(k x, k y ) = π k x 0 k y γ(k y ) 1 γ(k y ) Zigzag = π k y > 2π 3 0, γ(k y) Armchair = 0 (k y 0) k y (1) (2) [22] [2, 17, 18, 19]

10 : n = 0 [22] n = 0 [7] n = 0 n = 0 n 0 [22] n = 0

11 STM n = [1], [23] [6] 2.5: ( [6] 2 k y 1

12 12 2 k y 1 [24] d- [6] [25]

13 13 [1] M. Fujita et al., J. Phys. Soc. Jpn. 65, 1920 (1999). [2] Novoselov et al., Nature. 438, 197 (2005). [3] Niimi et al, Phys. Rev. Lett. 97, (2006). [4] [5], vol. 63, 344 (2008). [6] S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, (2002), Physica E (2004), Y. Hatsugai, Solid State Comm (2009). [7] B. I. Halperin, Phys. Rev. B 25, 2185 (1982). [8] Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993). [9] W. P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979). [10] D.. Thouless et al., Phys. Rev. Lett. 49, 405 (1982). [11] Z. Wang et al, Phys. Rev. Lett. 100, (2008). [12] V. W. Scarola and S. Das Sarma, Phys. Rev. Lett. 98, (2007). [13] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, (2005). [14] J. Von Neumann and E. P. Wigner, Phys. Z, 30, 467 (1929) [15] M.V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984) [16] J. Zak, Phys. Rev. Lett. 62, 2747 (1989). [17] Y. Zheng and Y. Ando, Phys. Rev B 65, (2002) [18] Y. Hatsugai, T. Fukui and H. Aoki, Phys. Rev. B 74, (2006), E. Phys. J. Special Topics (2007). Dirac T. Morimoto, Y. Hatsugai, H. Aoki, Phys. Rev. B78, (2008). [19] M. Arai and Y. Hatsugai, Phys. Rev. B 79, (2009). [20] J. C. Meyer et al, Nature 46, 60 (2007). [21] T. Kawarabayashi, Y. Hatsugai and H. Aoki, submitted, arxiv: [22] M. Arikawa, Y. Hatsugai, and H. Aoki, Phys. Rev. B 78, (2008).

14 14 2 [23] S. Okada and A. Oshiyama, Phys. Rev. Lett. 87, (2001). [24] Y. Otsuka and Y. Hatsugai, Phys. Rev. B 65, (2002). [25] S. Katayama, A. Kobayashi, Y. Suzumura, J. Phys. Soc. Jpn. 75, (2006).