,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising Model 1 Ising 1 Ising Model N Ising (σ i = ±1) (Free Boundary Condition) H(σ 1... σ N ) = J N 1 i σ i σ i+1 (1) J exp( βh N ) Z N = σ 1...σ N exp[ βh(σ 1... σ N )] = σ 1...σ N N 1 i=1 exp(βjσ i σ i+1 ) (2) N = 3 σ 1 Z 3 Z 3 = exp(βjσ 1 σ 2 ) exp(βjσ 2 σ 3 ) = 2 cosh βj exp(βjσ 2 σ 3 ) σ 1 σ 2 σ 3 σ 2 σ 3 = 2 cosh βj 2 cosh βj σ 3 1 = 2(2 cosh βj) 2 (3) 1
Z N = 2(2 cosh βj) N 1... 1 Ψ N (σ N ) = exp[ βh(σ 1... σ N )] = (2 cosh βj) N 1 (4) σ 1...σ N 1 Ψ N (σ N ) σ N h Ψ N (σ N ) σ N σ N Ψ N ( Ψ N (σ) ) 2N 1 Z 2N 1 = σ Ψ N (σ) Ψ N (σ) = σ (2 cosh βj) 2N 2 = 2(2 cosh βj) 2N 2 (5) 2N 1 Z 2N 1 N 1 Ψ N 2N 1 T (σ σ ) = exp(βjσσ ) (6) 2N Ψ N (σ) Z 2N = Ψ N (σ) T (σ σ ) Ψ N (σ ) σσ = σ Ψ N (σ) Ψ N+1 (σ) (7) Ψ N (σ) Ψ N+1 = T Ψ N (8) Z 2N 1 Z 2N λ = Z 2N Z 2N 1 = Ψ N T Ψ N Ψ N Ψ N (9) Ψ N Ψ N Ψ N T Ψ N 2
λ 1 λ N λ λ... 1 Z 2N /Z 2N 1 2 cosh βj N Ψ N (σ) 2 2 2 Ising Model Ising σ i 2 Ising i 2 H = J σ i σ j (10) {ij} {ij} Z = exp( βh) (11) {σ} {σ} 1 (2) Z 2? 2 Ising 4 45 Z W (σ a σ b σ c σ d ) = exp [βj(σ a σ b + σ b σ c + σ c σ d + σ d σ a )] (12) Z = σ a σ b σ c σ d W (σ a σ b σ c σ d ) = (cosh βj) 4 + (sinh βj) 4 (13) 3
4 1 Ising (6) 1 Ising T Z Z = Tr T 4 (14) 12 4 4 Z ( ) C(σ a σ b ) = σ c σd W (σ a σ b σ c σ d ) (15) C(σ a σ b ) W ( suchi suri chi-ri...) C Z = C(σ σ ) C(σ σ ) C(σ σ ) C(σ σ ) (16) σσ σ σ C 2 Z = Tr C 4 (17) (14) C (Corner Transfer Matrix, CTM) 40 4
CTM ( ) C P (σ a σ b σ c ) = σd W (σ a σ b σ c σ d ) (18) P 3 2 P σ b (... ) P C W C 2 C 2 (σ a σ a σ b σ b) = C(σ σ )P (σ a σ σ)p (σ σ σ b )W (σ σ σ aσ b) σσ σ σ (19) 40 Z = Tr (C 2 ) 4 84 24 C 3 (σ a σ aσ a σ b σ b σ b ) Z = Tr (C 3) 4 C 3 P P 2 (σ a σ a σ b σ c σ c) = σ P (σ a σ σ c )W (σσ aσ b σ c) (20) ( P W ) C 2 W C 3 = C 2 P P W (21) P C 5
1 2 (9) T 2N = P N P N, T 2N+1 = P N W P N (22) 2N 2N + 1 : T 7 = T 2 3+1 Ψ 2N = C N C N, Ψ 2N+1 = C N P N C N (23) λ 2N = Ψ 2N T 2N Ψ 2N Ψ 2N Ψ 2N, λ 2N+1 = Ψ 2N+1 T 2N+1 Ψ 2N+1 Ψ 2N+1 Ψ 2N+1 (24) λ 2N λ 2N+1 γ = λ 2N+1 λ 2N = (Ψ 2N+1 T 2N+1 Ψ 2N+1 )(Ψ 2N Ψ 2N ) (Ψ 2N T 2N Ψ 2N )(Ψ 2N+1 Ψ 2N+1 ) (25) γ 1? (Baxter ) 6
: N = 2 N γ N N C N 2 N?! 3 N σ σ σ σ... {σ σ σ σ...} η (26) η 1 m A(σ σ σ σ... η) A 7
C N P N C N (ξ η) = C N (σ a... σ b...)a(σ a... ξ)a(σ b... η) (27) P N (ξ σ b η) = P N (σ a... σ b σ c...)a(σ a... ξ)a(σ c... η) 2 ( ) A A C N = A C N A, PN = A P N A (28) C N C N P N P N A A 4 ρ = (C N ) 4, Tr ρ = Tr (C N ) 4 = Z (29) C N 2 Ising ρ O ( )[6, 7] ρ(σ a... σ b...) = α ζ O(σ a... ζ) O(σ b... ζ) (30) ζ α ζ m Z = Tr ρ = 2 N α ζ ζ=1 ζ=1 m α ζ = Z (31) (29) C N ρ ρ α ζ C N 4 α ζ m ζ m Z Z 8
O A... O(σ... ζ) ζ = 1 ζ = m A(σ... ζ) ρ ρ(ξ η) = A(σ a... ξ)a(σ b... η)ρ(σ a... σ b...) m = δ ξη α η (32) Z = Tr ρ = Tr (A ρa) = Tr (AA ρ) = Tr ( ˆP ρ) (33) ˆP = AA ˆP 2 = ˆP ( P N ˆP ) 4 ρ = (C N ) 4 A (22) T 2N = P N P N T 2N λ 2N = Ψ 2N T 2N Ψ 2N /Ψ 2N Ψ 2N Ψ 2N = C N C N N Ψ 2N T 2N Ψ 2N (σ a... ; σ c...) = C N (σ a... σ b...) C N (σ b... σ c...) (34) Ψ 2N 2 N Ψ 2N = (C N ) 2 = C N C N 9
1 η Ψ 2N = C N C N C N C N Φ 2N = C N AA C N = C N ˆP CN (35) Ψ 2N ˆP Φ 2N A Φ 2N Ψ 2N T 2N? λ = Φ 2N T 2N Φ 2N Φ 2N Φ 2N = [C N ˆP C N ] T 2N [C N ˆP CN ] [C N ˆP CN ] [C N ˆP CN ] (36) λ A Φ 2N Ψ 2N 2 A (36) λ A ( (36) ) Ψ 2N Φ 2N Ψ 2N Φ 2N Ψ 2N Ψ 2N Φ 2N Φ 2N (37) Φ 2N Ψ 2N 1 2 10
1 (Φ 2N Ψ 2N )(Ψ 2N Φ 2N ) (Ψ 2N Ψ 2N )(Φ 2N Φ 2N ) (38) 3 1 A ( ) 1 ( (31-33)) Φ 2NΨ 2N = [C N ˆP CN ] [C N C N ] = Tr (C N ) 4 ˆP = Tr ρ ˆP (39) 3 ( ) 2 A 3 5 2 Ising 2N L 2 (?) Ising Z = Tr (T 2N ) L = Tr (P N P N ) L (40) Z 2 L Tr (P N P N ) L = [ Tr (P N ) L ] [ Tr (PN ) L ] = Ψ NΨ N (41) 11
P N (σ... σ σ...) σ 2 N Ψ N L ( ) 2N + 1 ( ) τ τ [6, 7] λ = Tr (T 2N+1 )L Tr (T 2N ) L = Ψ N τ Ψ N Ψ N Ψ N = Ψ N Ψ N+1 Ψ N Ψ N (42) τ DMRG P N ( ) 2N Ψ N Tr (P N ) L P N P N = A P N A A P N ˆP Φ N = Tr (P N ) L ˆP (43) ˆP Tr (P N ˆP ) L Φ N ( ) Φ N Ψ N = Tr ρ ˆP ( ) Φ N τ Φ N /Φ N Φ N ( ) (Φ N Ψ N )2 /(Ψ N Ψ N )(Φ N Φ N )? Φ N Ψ N? ρ N 12
m ρ (?) ξ N L N τ τ 3 Ising [8, 2]... A Ψ N+1 = τψ N 2 3 ( ) N ( ) A Φ N τ λ = Φ N τ Φ N /Φ N Φ N [9] 13
6 A 4 A 2 4 A ( ) [?] 2 3 2 : http://quattro.phys.sci.kobe-u.ac.jp/nishi/publist.html DMRG http://quattro.phys.sci.kobe-u.ac.jp/dmrg.html [1] S. R. White: Phys. Rev. Lett. 69 (1992) 2863; Phys. Rev. B48 (1993) 10345. [2] Density-Matrix Renormalization, Springer Lecture Note in Physics 528, eds. I. Peschel, X. Wang, M. Kaulke and K. Hallberg, Springer 1999. [3] T. Nishino: J. Phys. Soc. Jpn. 64 (1995) 3598. [4] R.J. Baxter, J. Math. Phys. 9 (1968), 650; J. Stat. Phys. 19 (1978), 461; Exactly Solved Models in Statistical Mechanics (AcademicPress, London, 1982), p.363. 14
[5] T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 65 (1996) 891; J. Phys. Soc. Jpn. 66(1997) 3040. [6] X. Wang and T. Xiang: Phys. Rev. B56 (1997) 5061. [7] N. Shibata: J. Phys. Soc. Jpn 66 (1997) 2221; J. Phys. A: Math. Gen. vol.36 (2003) R381. [8] Ming-Chiang Chung, and Ingo Peschel; Phys. Rev. B 64 (2001) 064412 [9] in preparation (for ever?) 15