9, 10) 11, 12) 13, 14) 15) QED 16) , 19, 20) 21, 22) tight-binding Chern Hofstadter 23) Haldane 24) tight-binding Chern 25, 26) Chern 3 18, 1
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1 March 2015 A B A B 1 1) 1990 Hubbard Mielke 2, 3, 4) 1 8) 1 Hubbard 5) 6) 7) 1
2 9, 10) 11, 12) 13, 14) 15) QED 16) , 19, 20) 21, 22) tight-binding Chern Hofstadter 23) Haldane 24) tight-binding Chern 25, 26) Chern 3 18, 19, 20) Chern 2 Nielsen- Dagotto 17) 3 Chern Berry Chern Berry Berry 27) 2
3 Chern 28) Organometallic Framework 29) 30, 31) Chern 32, 33) tight-binding 4 34, 35) 36) 2 Tight-binding tight-binding 4 3
4 2.1 tight-binding Λ i = 1, 2,..., N i c i c i [c i, c j ] ± = [c i, c j ] ± = 0, [c i, c j ] ± = δ i,j (2.1) [A, B] ± = AB ± BA +/ / 5 tight-binding N H = t i,j c i c j (2.2) i,j=1 t i,j i = j i j ( ( 1) ) T = (t i,j ) i,j=1,...,n N 2.2 H T T k (= 1, 2,..., N) ε k φ (k) = (φ (k) 1, φ (k) 2,..., φ (k) N )T 6 N j=1 t i,j φ (k) j = ε k φ (k) i (2.3) 5 6 4
5 i φ (k) a k = N i=1 φ (k) i c i (2.4) [AB, C] = A[B, C] ± [A, C] ± B [H, a k ] = ε k a k (2.5) 0 Ha k 0 = [H, a k ] 0 = ε k a k 0 (2.6) a k 0 H ε k i c i 0 = 0 (2.6) a k ε k H (2.5) n 1, n 2,..., n N = (a 1) n 1 (a 2) (n 2) (a N )n N 0 (2.7) n k 1 (c i )2 = 0 Pauli H a k ( N ) H n 1, n 2,..., n N = n k ε k n 1, n 2,..., n N (2.8) k=1 H E = k n kε k k n k tight-binding 2.3 ε k T Λ 5
6 M N M/N (> 0) T t i,j 7 t i,j Λ G = (V, E) V E i, j V (i, j) e E e = (i, j) e i, j i j G p = (i 1, e 1, i 2, e 2,..., i k, e k, i k+1 ) (2.9) e l = (i l, i l+1 ) (l = 1, 2,..., k) p i 1 i k+1 k G V A, B A B G bipartite A, B A B bipartite 1(a) A B N M = (m i,j ) i,j=1,...,n N V i j m i,j e = (i, j) G = (V, E) M T tight-binding Λ T 0 t 0 t t t 0 t 0 0 T = 0 t 0 t t t 0 t 0 0 t 0 t 0 0 (2.10) 7 i j t i,j = 0 6
7 t, t T 1(a) T 2 1(b) 8 1(b) T bipartite T (t i,i = 0) T 2 G 1 = (A, E 1 ) G 2 = (B, E 2 ) T tight-binding T ε k T tight-binding 3 bipartite A B Sutherland 37) 3.1 Bipartite T G bipartite t i,i = 0 (i = 1, 2,..., N) T ε ε G i = 1, 2,..., A A i = A + 1,..., N B 9 T A B T AB T BA = (T AB ) 8 t, t (t /t = 1, 2) 9 S S 7
8 T = O T BA T AB O (3.1) O 0 T ε φ = φ A φ = φ A T ϵ φ B φ B T A B {}}{{}}{ C = diag( 1,..., 1, 1,..., 1) (3.2) CTC = T ε 0 ε ε = 0 φ φ = Cφ N ε = 0 N (N = 5) (2.10) 1(c) t /t ε = 0 T A B A B > A T 2 = T ABT BA O O (3.3) T BA T AB 1(b) A B S A = T AB T BA S B = T BA T AB T 2 S A, S B 1. S A, S B 0 2. S A S B 8
9 3. S B ( B A ) T T 2 3. T ( B A ) 1(a) 3 2 = 1 t, t S B = (T AB ) T AB B ψ ψ, S B ψ = T AB ψ, T AB ψ = T AB ψ S A 2. ψ S B λ > 0 T BA T AB ψ = λψ T AB S A ψ = λ ψ ψ = TAB ψ 11 S A λ S B S A 3. ( ) S B ψ 0 ψ 0, S B ψ 0 = T AB ψ 0 2 = 0 T AB ψ 0 = 0 T AB A B ( B A ) S B T B T T = T AA T BA T AB O (3.4) (T AA A ) ( B A ) B A bipartite 2(a) Lieb 38) N c B A = N c T 1/3 3(a) ( B A = N c ) 40) 4(a) Lieb ( B A = N c ) 10 d u, v u, v = d i=1 u i v i 11 ψ 0 ψ = 0 0 = ψ 2 = S B ψ, ψ = λ ψ, ψ = 0 9
10 t i,j ε = 0 t i,j T 39) t i,j 41) ε = bipartite bipartite Mielke 42) G = (V, E) V A E B G = (Ṽ, Ẽ) 12 G G Lieb ( 2(a)) G G ( 3(a)) G G bipartite G i, j t i,j 0 i, j t i,j = 0 T (3.1) S B = T BA T AB S B G L(G) 2 G L(G) 3 G L(G) S A = T AB T BA G S A, S B 2. G S A tight-binding L(G) S B ε = 0 ( ) tight-binding 3 (b) (c) 2 (b) (c) 12 Ising decoration-iteration transformation 43) 10
11 4(a) bipartite i, j t i,j = t i, j t i,j = 0 T S A S B 3) ε = t 2 S A, S B 2., 3. ε = 0, t 2 4(c) bipartite bipartite Lieb Mielke 44) 4 t i,j 4.1 5(a) 3 11
12 t ϕ 1, ϕ 2 0 te iϕ 1 t t te iϕ 2 te iϕ 1 0 t 0 0 T = t t (4.1) t t te iϕ t 0 ϕ 1, ϕ 2 ϕ 1, ϕ 2 ε = ±t 5 (c) T ϕ 1, ϕ 2 5(b) p q ϕ 1, ϕ 2 (p q 1) idle state 45) idle state 5(a) 2, 3 φ 1 = (0, 1, 1, 0, 0) T (4.1) T (2.3) ε k = t φ 1 i = 1 4, 5 φ 2 = (0, 0, 0, 1, 1) T ε k = t (2.3) i = 1 φ 1 φ 2 T i = 1 (4.1) φ = (1 + e iϕ 2 )φ 1 (1 + e iϕ 1 )φ 2 (4.2) i = 1 i (4.1) 13 ε k = t φ 3 = (0, 1, 1, 0, 0) T φ 4 = (0, 0, 0, 1, 1) T 5(b) (p 1)- (q 1)- 13 ϕ 1 = π ϕ 2 = π φ 1 φ 2 idle state 12
13 n ( ) mπ ε m (n) = 2t cos, n + 1 m = 1, 2,..., n (4.3) p q (2.3) idle state idle state ( 5(d)) 4.2 6(a) 5 i, j t i,j = t i, j t i,j = 0 T x, y T (4.1) ϕ 1 k x, ϕ 2 k y idle state ε = ±t 6(a) 7(c) ε = 0, ± 2t Lieb 7 14 α-graphyne 46, 47) 0 ansatz 14 40) 8) 13
14 T tight-binding 6(a) bipartite 6(b) tight-binding 15 tight-binding 48, 49) tight-binding 4.3 (4.2) ϕ 1 k x, ϕ 2 k y k a (k) = (1 + e ik y ){c 2(k) + c 3(k)} (1 + e ik x ){c 4(k) + c 5(k)} (4.4) Z(k) = cos k x + cos k y ã (k) = a (k)/z(k) [ã(k), ã (k )] ± = δ k,k ã (k) ε = t Wannier a (k) R, x, y e x, e y a (k) a (R) = c 2(R) + c 3(R) + c 2(R + e y ) + c 3(R + e y ) {c 4(R) + c 5(R) + c 4(R + e x ) + c 5(R + e x )} (4.5) c a(r) R a 6(a) 15 6(b) or 14
15 t i,j Hubbard 5 tight-binding t i,j bipartite 50) 51) Atiyah-Singer Witten 52) (3.1) T AB ( )=(Witten ) ADHM 53) 54, 55)
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20 1: (a) bipartite (2.10) T 1 5 t t (b) T 2 (c) T t t = 1 2: (a) Lieb (b) S A (c) S B (a) t = 1 3: (a) (b) S A (c) S B (a) t = 1 4: (a) Lieb (b) S A (c) S B (a) t = 1 5: (a) (b) idle state( ) (c) T 0 ϕ 1, ϕ 2 2π t = 1 (d) idle state 6: (a) 5 (a) (4.5) (b) (a) T 20
21 7: (a) (b) (c) (4.3) 21
22 図 1 (a) 4 3 (b) t (c) 4 2 e t
23 図 2 (a) (b) (c) ε ε ε k y k x k y k x k y k x
24 図 3 (a) (b) (c) ε ε ε k y k x k y k x k y k x
25 図 4 (a) (b) (c) ε ε ε k y k x k y k x k y k x
26 図 5 (a) (b) φ 1 φ 2 φ 1 φ (c) (d) ε φ 1 φ 2
27 図 6 (a) (b) y x
28 図 7 (a) (b) (c) ε ε ε k y k x ky k x ky k x
(extended state) L (2 L 1, O(1), d O(V), V = L d V V e 2 /h 1980 Klitzing
1 2 2.1 [1] [2] 2.1 STM [3, 4, 5, 6] 2.1: 2 ( 3 [1] ) [7, 8] [9]( 2.2) 2 2 2.1.1 (extended state) L (2 L 1, O(1), d O(V), V = L d V V 2.1.2 1985 2 e 2 /h 1980 Klitzing 2.1. 3 [7, 8] 2.2 [10] [8] 2.2: (a)
1: Pauli 2 Heisenberg [3] 3 r 1, r 2 V (r 1, r 2 )=V (r 2, r 1 ) V (r 1, r 2 ) 5 ϕ(r 1, r 2 ) Schrödinger } { h2 2m ( 1 + 2 )+V (r 1, r 2 ) ϕ(r 1, r 2
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Hubbard 2 1 1 Pauli 0 3 Pauli 4 1 Vol. 51, No. 10, 1996, pp. 741 747. 2 http://www.gakushuin.ac.jp/ 881791/ 3 8 4 1 1: Pauli 2 Heisenberg [3] 3 r 1, r 2 V (r 1, r 2 )=V (r 2, r 1 ) V (r 1, r 2 ) 5 ϕ(r
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1 1.1 hν A(k,ε)[ k ρ(ω)] [1] A(k,ε) ε k μ f(ε) 1/[1 + exp( ε μ k B T )] A(k,ε)f(ε) ρ(ε)f(ε) A(k,ε)(1 f(ε)) ρ(ε)(1 f(ε)) A(k,ε) σ(ω) χ(q,ω) k B T ev k
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62 2017 7 25-29 - ARPES ARPES ARPES - ARPES 1 1.1 1.2 1.3 1.4 1.4.1 1.4.2 1.4.3 1.5 2 2.1 2.2 2.3 BCS 2.4 1 1.1 hν A(k,ε)[ k ρ(ω)] [1] A(k,ε) ε k μ f(ε) 1/[1 + exp( ε μ k B T )] A(k,ε)f(ε) ρ(ε)f(ε) A(k,ε)(1
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(2016 ) Dept. of Phys., Kyushu Univ. 2017 8 10 1 / 59 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER 2 / 59 ( ) ( ) (Dirac, t Hooft-Polyakov)
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01UM1301 1 3 1.1 : : : : : : : : : : : : : : : : : : : : : : 3 1.2 : : : : : : : : : : : : : : : : : : : : 4 1.3 : : : : : : : : : : : : : : : : : 6 1.4 : : : : : : : : : : : : : : : 10 1.5 : : : : : :
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16 5 19 10 d (i) (ii) 1 Georges[2] Maier [3] 2 10 1 [1] ω = 0 1 [4, 5] Dynamical Mean-Field Theory (DMFT) [2] DMFT I CPA [10] CPA CPA Σ(z) z CPA Σ(z) Σ(z) Σ(z) z - CPA Σ(z) DMFT Σ(z) CPA [6] 3 1960 [7]
スケーリング理論とはなにか? - --尺度を変えて見えること--
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Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ
a L = Ψ éiγ c pa qaa mc ù êë ( - )- úû Ψ 1 Ψ 4 γ a a 0, 1,, 3 {γ a, γ b } η ab æi O ö æo ö β, σ = ço I α = è - ø çèσ O ø γ 0 x iβ γ i x iβα i
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2 2 T c µ T c 1 1.1 1911 Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ 58 213 email:takada@issp.u-tokyo.ac.jp 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B =
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1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
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1: (Emmy Noether; ) (Feynman) [3] [4] {C i } A {C i } (A A )C i = 0 [5] 2
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2003 1 1 (Emmy Noether 1) [1] [2] [ (Paul Gordan Clebsch-Gordan ] 1915 habilitation habilitation außerordentlicher Professor Außerordentlich(=extraordinary) 1 1: (Emmy Noether; 1882-1935) (Feynman) [3]
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
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