Hubbard t-j Heisenberg 19 8 9 Hubbard t-j Heisenberg Hubbard t-j Heisenberg 1 Hubbard Hubbard H = H t + H = i,j,σ t ij c iσ c jσ + i c i c i c i c i (1) t ij 1.1 0 H = H t = i,j,σ t ij c iσ c jσ () t ij = t R i σ 1 c iσ = c kσ e ik R i (3) k c 1 iσ = c e ik R i (4) kσ k k σ kσ t H = c c kσ k e ik R i e ik R j (5) σ c ( ) i,j.k,k,σ c ( ) iσ = c c t kσ k e ik R i e ik R j = c σ c kσ k σ k,k,σ i,j k,k,σ i,δ = c c t kσ k e i(k k ) R i e ik δ σ = c c kσ k σ k,k,σ i,δ k,k,σ = k,σ ɛ(k)c kσ c kσ t e ik R i e ik (R i +δ) t δ(k k )e ik δ δ (6) (7) (8) 1
ɛ(k) = t e ik δ (9) δ δ e Φ GS e/ Φ GS = c k c Φ i k vac (10) i i=1 k i 1 1. t 0 ( > 0) H = H = i c i c i c i c i (11) e ( e < ) e = + Φ GS ( ) ( ) Φ GS = Φ vac (1) i X c i i Y X Y X Y c i 1.3 Hubbard H t H Hubbard.1 Heisenberg Hubbard Heisenberg Heisenberg Heisenberg H H = i,j J ij S i S j (13) S 1 k
. S i = 1 σc is (14) σ σ = (σ x, σ y.σ z (15) ( ) ( ) ( ) 0 1 0 i 1 0 σ x =, σ y =, σ z = (16) 1 0 i 0 0 1 S z z M is M is = 1 (n i n i ) (17) n i n i M is S iz σ z σαβ z S iz = 1 (c i c i c i c i ) (18) S iz = 1 (c i σz c i + c i σz c i ) (19) = 1 σz ss c is (0) S i = (S ix, S iy, S iz ) (41).3 Heisenberg (13) S i S j = S ix S jx + S iy S jy + S iz S jz S ix S jx = 1 4 (c i c i + c i c i )(c j c j + c j c j ) (1) = c i c i c j c j + c i c i c j c j + c i c i c j c j + c i c i c j c j () S iy S jy = 1 4 ( ic i c i + ic i c i )( ic j c j + ic j c j ) (3) = c i c i c j c j + c i c i c j c j + c i c i c j c j c i c i c j c j (4) S iz S jz = 1 4 (c i c i c i c i )(c j c j c j c j ) (5) = c i c i c j c j c i c i c j c j c i c i c j c j + c i c i c j c j (6) S i S j = 1 4 (c i c i c j c j + c i c i c j c j + c i c i c j c j c i c i c j c j c i c i c j c j + c i c i c j c j ) (7) = 1 4 (c i c i c j c j + c i c i c j c j + n i n j n i n j n i n j + n i n j ) (8) 1/ 3
n i = n i + n i S i S j + 1 4 n in j = 1 4 (c i c i c j c j + c i c i c j c j + n i n j + n i n j ) (9) = 1 c is c js c js (30) 3 t-j Heisenberg t t/ 1 Hubbard t-j half-filled t-j Heisenberg Schrieffer-Wolff 3 t/ 1 Schrieffer-Wolff Fock S D S = [ n 1, n 1,, n : i, n i + n i 1] (31) D = [ n 1, n 1,, n : i, n i + n i = ] (3) i i i i S D S 3.1 Shrieffer-Wollf Hubbard H t H t = H t,h + H t,d + H t,mix (33) H t,h = i,j,s t ij (1 n i s ) c js(1 n j s ) (34) H t,d = i,j,s t ij n i s c jsn j s (35) H t,mix = i,j,s n i, s c js(1 n j s ) + (1 n i s ) c jsn j s (36) s s s s H t s j i j i s H t,h i j S H t,d i j D H t,mix i j 3 4
D S S D H = H + H t,h + H t,d + H t,mix (37) S D H t,d S H t,h D S-D H t,mix H = H 0 + H (1) (38) H 0 = H H (1) = H t,mix H t,mix S D D S H = e S He S = H + [H, S] + 1 [[H, S], S] + (39) H = H 0 + H (1) + [H 0, S] + [H (1), S] + 1 [[H 0, S], S] + (40) S H (1) + [H 0, S] = 0 (41) 4 H = H 0 + 1 [H(1), S] + O((t/) 3 ) (4) H eff = H t,h + H = H t,h + H + 1 [H t,mix, S] (43) H H 3. t-j H int = [H t,mix, S]/ f H int i = 1 f (H t,mixs SH t,mix ) i (44) ( ) = 1 f H t,mix α α S i f S α α H t,mix ) i (45) α f i S α H t,mix α D (41) 4 H t,mix = SH H S (46) β H t,mix γ = β SH γ β H S γ (47) β H t,mix γ = β S γ (E γ E β ) (48) β S γ = β H t,mix γ (E γ E β ) (49) 5
β γ E β(γ) H β(γ) = E β(γ) β(γ) β(γ) S E β(γ) H t,mix S β(γ) D E β(γ) = (45) f H int i = 1 ( α f H t,mix α α H t,mix i = f H t,mixh t,mix i ) f H t,mix α α H t,mix ) i H int = 1 H t,mixh t,mix (5) H int S S P S H int = 1 P S (H t,mixh t,mix ) P S (53) = 1 P S t ij t jk c jsn j n j c js c ks P S i,j,k, (54) H eff = H t,h 1 P S t ij t jk c jsn j n j c js c ks P S i,j,k,s,s (55) = P S t ij c js 1 t ij t jk jsn j n j c js c ks P S i,j,s i,j,k, (56) 6 t-j H (50) (51) 3.3 t-j t-j 56 c jsn j n j c js c ks = c i c j n j n j c j c k +c i c j n j n j c j c k +c i c j n j n j c j c k +c i c j n j n j c j c k (57) n j n j c js + c js c is = δ ijδ ss (58) n j n j S c i c k n j + c i c j c j c k n j + c i c j c j c k n j + c i c k n j (59) j c j c j n j n j j c j c j j 5 n j n j 6 H t,h S P S 6
c j c j n j c j c j n j ( ) c i c k n j + c i c j c j c k + c i c j c j c k + c i c k n j (60) (8) S i S j 1 4 n in j = 1 4 (c i c i c j c j + c i c i c j c j n i n j n i n j ) (61) = 1 (c i c i c j c j + c i c i c j c j c i c i c j c j c i c i c j c j ) (6) = 1 ( c i c j c j c i c i c j c j c i c i c i c j c j c i c i c j c j ) (63) H eff = P ( S t ij c js 1 t ij c jsn j n j c js c is + ) i k t ij t jk c jsn j n j c js c ks P S (64) i,j,s i,j, 63) H eff = P S t ij c js + 1 (J ij (S i S j 14 ) n in j 1 ) i k t ij t jk c jsn j n j c js c ks P S (65) i,j,s i,j J ij = 4t ij / S i = 1 σ ss c is (66) k k 1 σ ss c ks (67) H eff = P S (H t + H QHM + H J )P S (68) H QHM = 1 ( J ij S i S j 1 ) 4 n in j (69) H J = i,j i k i,j,k t ij t jk [ c i σc k c j σc j s ( c ksn j ) ] (70) 3.4 Heisenberg (68) t-j Heisenberg Half-filled S H t,h (68) H J i k Half-filled k k Half-filled Hubbard H eff = H QHM = 1 ( J ij S i S j 1 ) 4 n in j (71) i,j 7
1/ Heisenberg QHM Quantum Heisenberg model 4 Hubbard t-j 16 Hubbard Web http://www.gakushuin.ac.jp/ 881791/ Assa Auerbach, Interacting Electrons and Quantum Magnetism 8