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 ( )+V (r 1, r 2 ) ϕ(r 1, r 2
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1 Hubbard Pauli 0 3 Pauli 4 1 Vol. 51, No. 10, 1996, pp /
2 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 ( )+V (r 1, r 2 ) ϕ(r 1, r 2 )=Eϕ(r 1, r 2 ) (1) (1) ϕ(r 1, r 2 ) (1) r 1 r 2 ϕ(r 1, r 2 )=ϕ(r 2, r 1 ) ϕ(r 1, r 2 )= ϕ(r 2, r 1 ) ϕ(r 1, r 2 ) [4] 62 ϕ(r 1, r 2 ) = ϕ(r 1, r 2 ) = (2) [1] Hubbard [2] 5 ψ(r 1,σ 1 ; r 2,σ 2 )= (ϕ(r 1, r 2 ) δ σ1, δ σ2, ϕ(r 2, r 1 ) δ σ2, δ σ1, )/ 2 2
![r 2 ϕ(r 1, r 2 )=ϕ(r 2, r 1 ) ϕ(r 1, r 2 )= ϕ(r 2, r 1 ) ϕ(r 1, r 2 ) [4] 62 ϕ(r 1, r 2 ) = ϕ(r 1, r](/docs-images/41/22677730/images/page_2.jpg "2 ) = (2) [1] Hubbard [2] 5 ψ(r 1,σ 1 ; r 2,σ 2 )= (ϕ(r 1, r 2 ) δ σ1, δ σ2, ϕ(r 2, r 1 ) δ σ2, δ σ1,")
3 3 ϕ(r 1, r 2 ) ϕ(r, r) =0 (2) 4 2 ϕ(r 1, r 2 ) [5] Schrödinger (1) (r 1, r 2 ) 6 Schrödinger [4] 20 (1) 6 ϕ(r 1, r 2 ) > 0 (2) Heisenberg Heisenberg Hund 3
![Schrödinger [4] 20 (1) 6 ϕ(r 1, r 2 ) >](/docs-images/41/22677730/images/page_3.jpg "0 (2) 2 6 6 3 Heisenberg Heisenberg Hund")
4 a) b) c) d) 2: a) b) c) d) Hubbard Hubbard Hubbard minimum model [6, 7] 2 Λ N i =1, 2,...,N Pauli 3 N e 0 N e 2N 4
![minimum model [6, 7] 2 Λ N i](/docs-images/41/22677730/images/page_4.jpg "=1, 2,.")
5 3: Hubbard U 3U i j t i,j = t j,i (= ) 7 t i,j t i,j Λ t i,j ϕ ε Schrödinger N t i,j ϕ j = εϕ i (3) j=1 Λ 1 i j =1 t i,j = t t i,j =0 Schrödinger (3) ϕ j = N 1/2 e ikj ε(k) = 2t cos k k n =0, ±1,...,±(N 1)/2 k =2πn/N Hubbard U(> 0) 8 3 Hubbard 7 H hop = i,j,σ t i,jc i,σ c j,σ 8 5
6 t' t t' : Hubbard Hubbard Luttinger [2, 7, 8, 1] Hubbard Hartree-Fock UD F > 1 Stoner D F Stoner [7] 6 Hubbard 1 Hubbard N =3 N e =2 i j i, j 9 4 t 1,2 = t 2,1 = t 2,3 = t 3,2 = t > 0 t 1,3 = t 3,1 = t U U 9 0 i, j = c i, c 0 j, 6
7 Φ 13 t' t' Φ 12 Φ 23 t t Φ 32 Φ 21 t' t' Φ 31 5: U i, j i j ϕ = ϕ i,j i, j (4) i,j=1,2,3 (i j) ϕ i,j 2 ϕ(r 1, r 2 ) (2) 5 10 [9] t ϕ (4) ϕ 1,2 > ϕ 3,2 > 0, ϕ 3,1 < 0, ϕ 2,1 > 0, ϕ 2,3 > 0, ϕ 1,3 < 0 2 ϕ i,j i j ϕ i,j = ϕ j,i
![5 10 [9] t ϕ (4) ϕ 1,2 > 0 5 5 ϕ 3,2 > 0, ϕ 3,1 < 0, ϕ 2,1](/docs-images/41/22677730/images/page_7.jpg "> 0, ϕ 2,3 > 0, ϕ 1,3 < 0 2 ϕ i,j i j ϕ i,j = ϕ j,i 10 4 7")
8 E / t' U / t' 6: t = t /2 E sym E asym U U E sym >E asym (2) U, t<0 t ϕ ϕ 1,2 > 0, ϕ 3,2 < 0, ϕ 3,1 > 0, ϕ 2,1 < 0, ϕ 2,3 > 0, ϕ 1,3 < 0 ϕ i,j = ϕ j,i (2) U, t>0 U 6 t = t /2 > 0 E sym E asym U U =0 Pauli 1 E sym <E asym U E sym >E asym 4 4 t>0 2 t > s t<0 8
9 7: Hubbard 7 5 U = [10] Λ i, j t i,j = t>0 t i,j =0 12 U N e N N e = N 1 7 Hubbard 6 N e = N Hubbard 12 t i.j 9
10 1.4 E / t' U / t' 8: t = t E sym E asym U U > 0 E sym >E asym Hubbard U [2] 8 Mielke 6 U 8 t = t > 0 E sym E asym U U =0 E sym E asym 13 U>0 E sym >E asym 3 (4) ϕ i,j ϕ i,i =0 U E asym U 6 8 ϕ i,j U E sym U U =0 E sym = E asym U>0 E sym >E asym 13 Schrödinger (3) U =0 10
![[2] 8 Mielke 6 U 8 t = t > 0 E sym E asym U U =0 E sym E asym 13 U>0](/docs-images/41/22677730/images/page_10.jpg "E sym >E asym 3 (4) ϕ i,j ϕ i,i =0 U E asym U 6 8 ϕ i,j U E sym U U")
11 9: Mielke Hubbard Mielke [11] Λ N 9 14 i, j t i,j = t>0 t i,j =0 U Schrödinger (3) (N/3) + 1 N e =(N/3) + 1 S tot =1/2, 3/2,...,{(N/3) + 1}/2 15 U =0 Mielke U>0 Hubbard U Hubbard Mielke U =0 U>0 [12] 14 Mielke line graph 15 N 11
12 t t t t' t' t' t' t' t' s s 10: Hubbard U 4 Mielke [2] U =0 U 3 [13, 14, 15] 9 Hubbard 10 1 Λ={1, 2,...,N} N +1 1 i t i,i+1 = t i+1,i = t i t i,i+2 = t i+2,i = t i t i,i+2 = t i+2,i = s t i,j =0 t >0 s>0 t t = 2(t + s) U 10 6 Schrödinger (3) ε 1 (k) = 2t 2s(1+cos 2k), ε 2 (k) =2s+2t(1+cos 2k) n =0, ±1,...,±{(N/2) 1} /2 k =2πn/N N e = N/2 U =0 Pauli 1 U<4s U 12
13 t/s U/s 16 Hubbard [16] Hubbard U s =0 t/s 17 [15] Hubbard N/2 [15] U/s 27 t/s U/s t/s =1.6 1/
![Hubbard N/2 [15] 1 10 18 16](/docs-images/41/22677730/images/page_13.jpg "U/s 27 t/s 27 17 U/s t/s =1.")
14 [1] 31, 173 (1996). [2] 30, 769, 867 (1995), 31, 16, 100, 205 (1996). [3] W. J. Heisenberg, Z. Phys. 49, 619 (1928). [4] L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Nonrelavitistic Theory) (Pregamon, 1977). [5] E. H. Lieb and D. Mattis, Phy. Rev. 125, 164(1962) Introduction Wigner [6] J. Kanamori, Prog. Theor. Phys. 30, 275 (1963); M. C. Gutzwiller, Phy. Rev. Lett. 10, 159 (1963); J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963). Hubbard [7] [8] 46, 565 (1991); 48, 437 (1993); 49, 893 (1994). [9] Perron-Frobenius 58, 121 (1992) 5.1 [10] Y. Nagaoka, Phy. Rev. 147, 392 (1966); D. J. Thouless, Proc. Phys. Soc. London 86, 893 (1965); H. Tasaki, Phy. Rev. B40, 9192 (1989). [11] A. Mielke, J. Phys. A24, 3311 (1991); ibid. A25, 4335 (1992); Phys. Lett. A174, 443 (1993). 14
![30, 275 (1963); M. C. Gutzwiller, Phy. Rev. Lett. 10, 159 (1963); J. Hubbard, Proc. Roy. Soc. (London) A276, 238 (1963). Hubbard [7] 1992 1996 [8] 46, 565 (1991); 48, 437 (1993); 49, 893 (1994).](/docs-images/41/22677730/images/page_14.jpg "[9] Perron-Frobenius 58, 121 (1992) 5.1 [10] Y. Nagaoka, Phy. Rev. 147, 392 (1966); D. J. Thouless, Proc. Phys. Soc. London 86, 893 (1965); H. Tasaki, Phy. Rev. B40, 9192 (1989). [11] A. Mielke, J.")
15 [12] H. Tasaki, Phy. Rev. Lett. 69, 1608 (1992); A. Mielke and H. Tasaki, Commun. Math Phys. 158, 341 (1993). [13] K. Kusakabe and H. Aoki, Phys. Rev. Lett. 72, 144 (1994). [14] H. Tasaki, Phy. Rev. Lett. 73, 1158 (1994); J. Stat. Phys. 84, Numbs. 3/4(1996) in press. [15] K. Penc, H. Shiba, F. Mila and T. Tsukagoshi, preprint (condmat/ ); H. Sakamoto and K. Kubo, [16] H. Tasaki, Phy. Rev. Lett., 75, 4678 (1995). (Thorem I) H. Tasaki λ>λ c λ 0 15
![Stat. Phys. 84, Numbs. 3/4(1996) in press. [15] K. Penc, H. Shiba, F. Mila and T.](/docs-images/41/22677730/images/page_15.jpg "Tsukagoshi, preprint (condmat/9603042); H. Sakamoto and K. Kubo, [16] H. Tasaki, Phy. Rev. Lett., 75, 4678 (1995).")
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