Part I Review on correlation functions of the XXZ spin chain (1) H. Bethe(1930): Exact solutions of the one-dimensional Heisenberg model (XXX spin cha
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1 Part I Review on correlation functions of the XXZ spin chain (1) H. Bethe(1930): Exact solutions of the one-dimensional Heisenberg model (XXX spin chain) (2) C.N. Yang and C.P. Yang (1966): the ground state of the XXZ spin chain (3) L.D. Faddeev and L. Takhtajan (1979): Algebraic Bethe-ansatz solution of the XXZ spin chain (1) and (2): Coordinate Bethe ansatz ( (3) Algebraic Bethe ansatz
2 ( ) The Hamiltonian of the spin-1/2 XXZ spin chain under P. B. C. (the Periodic Boundary Conditions) H XXZ = 1 L ( ) σ X 2 j σx j+1 + σy j σy j+1 + σz j σz j+1. Here σ a j j=1 (a = X, Y, Z) are Pauli matrices on the jth site. We define q by = (q + q 1 )/2 (q = exp η) Quantum phase transitions at = ±1: For 1 < 1, H XXZ is gapless. ( = cos ζ by q = e iζ, 0 ζ < π.) Low excited spectrum is consistent with CFT with c = 1 For > 1 or < 1, it is gapful. ( = ± cosh ζ by q = e ζ, 0 < ζ.)
3 M. Jimbo, K. Miki, T. Miwa and A. Nakayashiki, Phys. Lett. A 168 (1992) M. Jimbo and T. Miwa, J. Phys. A: Math. Gen. 29 (1996) N. Kitanine, J.M. Maillet and V. Terras, Nucl. Phys. B 567 [FS] (2000) cf. Exact form factors of the sine-gordon model (F. Smirnov, 1980 s)
4 Emptiness Formation Probability (EFP) Let us consider unit matrices e a, b for a, b = 0, 1. ( ) e 1, 1 j = 1 2 (1 σz j ) = For the XXZ spin chain (massless regime) we have τ(m) e 1,1 1 e 1,1 m ( = ( 1) m π ) m(m+1)/2 ζ m j=1 dλ 1 2π dλ m 2π sinh j 1 (λ j iζ/2) sinh m j (λ j + iζ/2) cosh m π ζ λ j cf. N. Kitanine et al., NPB 567, 554 (2000). a>b sinh π ζ (λ a λ b ) sinh(λ a λ b iζ)
5 II F. Göhmann, A. Klümper and A. Seel, J. Phys. A: Math. Gen. Vol. 37 (2004) (quantum transfer matrix) Cf. Trotter
6 (ζ(n): H. Boos and V.E. Korepin, J. Phys. A 34, 5311 (2001). J. Sato, M. Shiroishi, and M. Takahashi, Nucl. Phys. B 729, 441 (2005). Cf. M. Takahashi, J. Phys. C: Solid State Phys. 10, 1289 (1976) σ z 1 σz 3 J. Sato et al, PRL106, (2011) σ z 1 σz 2 = ln (T/J)2 σ z 1 σz 3 = ln 2 + 3ζ(3) + ( ) 1 9 π2 72 (T/J) 2
7 N. Kitanine, K.K. Kozlowski, J.M. Maillet, A.N. Slavnov and V. Terras, JSTAT (2009) P04003 σ1 z σz m+1 = σz 2 2Z2 π 2 m F σ 2 cos(2k F m) +. m 2Z2 F σ σ z form factor) Ψ g F σ = M σ z Ψ g M Z: the dressed charge form factor)
8 N. Kitanine, K.K. Kozlowski, J.M. Maillet, A.N. Slavnov and V. Terras, JSTAT (2011) P12010 σ z 1 σz m+1 = = n: all Bethe states n: all Bethe states σ z 1 n n σz m+1 σ z 1 n n σz 1 m 2 (n) σ1 z n, n σz m+1 : (form factors) n ρ(x, t)ρ(0, 0) N. Kitanine et al., arxiv:
9 Introduction to the lgebraic Bethe ansatz We define the R-matrix by b(u) c(u) 0 R 12 (λ 1, λ 2 ) = 0 c(u) b(u) [1,2] b(u) = sinh u/sinh(u + η), c(u) = sinh η/sinh(u + η) (1) a 2 a 1 λ 1 λ 2 b 1 b 2 Here u = λ 1 λ 2 and q = exp η. Figure 1: R(λ 1, λ 2 ) a 1,a 2 b 1,b 2
10 The Yang-Baxter equations: v u + v u 1 2 u = u + v 3 1 v 2 3 The R-matrices satisfy the Yang-Baxter equations. R 2, 3 (v)r 1, 3 (u + v)r 1, 2 (u) = R 1, 2 (u)r 1, 3 (u + v)r 2, 3 (v) Spectral parameter u is expressed by the angle between lines 1 and 2, where the interstion corresponds to R 1, 2 (u).
11 We introduce the monodromy matrix T 0,12 L (λ): T 0,12 L (λ) = R 0L (λ, w L )R 0L 1 (λ, w L 1 ) R 02 (λ, w 2 )R 01 (λ, w 1 ). Here w 1, w 2,..., w L are inhomogeneity parameters. a 1 a 2 a L c 1 = α c 2 c L c L + 1 = β b 1 b 2 b L Figure 2: Matrix element of the monodromy matrix (T α,β ) a 1,...,a L b 1,...,b L.
12 The operator-valued matrix element of the monodromy matrix give the creation and annihilation operators ( ) A(u) B(u) T 0,12 L (u) =. C(u) D(u) The transfer matrix, t(u), is given by the trace of the monodromy matrix with respect to the 0th space: t(u) = tr 0 ( T0,12 L (u) ) [0] = A(u) + D(u). (2) The logarithmic derivative of the transfer matrix gives the XXZ Hamiltonian: H XXZ = d du log t(u) u=0 Thus, the transfer matrix and the Hamiltonian share the eigenvectors.
13 Let 0 be the vacuum vector with all spins being up. 0 = The Bethe vector n B(λ k ) 0 = B(λ 1 ) B(λ n ) 0 k=1 becomes an eigenvector of the transfer matrix if rapidities λ 1,..., λ n satisfy the Bethe ansatz equations: ( ) sinh(λj + η/2) n sinh(λ j λ k + η) =, (j = 1, 2,..., n) sinh(λ j η/2) sinh(λ j λ k η) k=1; k j
14 Review: algebraic BA derivation of the multiple-integral representation of the spin-1/2 XXZ correlation functions Quantum Inverse Scattering Problem (QISP) Local spin-1/2 operators expressed by A, B, C, D (spin-1/2) QISP formula x n = n 1 j=1 For example, we have σ n = n 1 t(w j )tr 0 (x 0 T 0, 12 L (w n )) (A(w j ) + D(w j )) B(w n ) j=1 n j=1 t(w j ) 1 n (A(w j ) + D(w j )) 1 j=1
15 Scalar products of the BA: Suppose that {µ j } or {λ j } are Bethe roots, (i) the Gaudin-Korepin formula for the Bethe ansatz norm 0 C(λ 1 ) C(λ M ) B(λ 1 ) B(λ M ) 0 = det Φ Φ : the Gaudin matrix (ii) Slavnov s formula: 0 C(µ 1 ) C(µ M ) B(λ 1 ) B(λ M ) 0 = detψ Thus, we obtain the expectation values of local operators by calculating the ratio for Bethe roots {λ j } and arbitrary parameters {µ j }: 0 C(µ 1 ) C(µ M ) B(λ 1 ) B(λ M ) 0 0 C(λ 1 ) C(λ M ) B(λ 1 ) B(λ M ) 0 = det(ψ /Φ )
16 Integral equations for the matrix elements of Ψ /Φ : ( To evaluate the matrix elements of Ψ /Φ ) we solve integral equations for them (cf. Izergin) The thermodynamic property of the ground state is taken into accout by the density of the roots of the Bethe-ansatz equations.
17 Commutation relations (b jβ = b(λ j λ β ) c jβ = c(λ j λ β )) n n n 0 C(µ k ) D(µ 0 ) = d α c α0 b 1 αj 0 C(µ k ) 0 0 k=1 n k=1 n k=1 C(µ k ) A(µ 0 ) = C(µ k ) B(µ 0 ) = α=0 n α=0 n α=0 a α c 0α d α c α0 n β=0;β α j=0;j α j=0;j α j=0;j α a β c 0β b 1 jα 0 b 1 αj j=0;j α,β Here a α = a(λ α ) and d α = d(λ α ) are defined by A(λ) 0 = a(λ) 0, k=1;k α n k=1;k α b 1 jβ 0 D(λ) 0 = d(λ) 0. C(µ k ) n k=1;k α,β C(µ k )
18 Examples of multiple integrals (T.D. and C. Matsui, NPB(2010)). For s = 1 and m = 1 (w (2) 1 = ξ 1, w (2) 2 = ξ 1 η), we have 11 (2+) E1 = ψ g (2+) 11 (2+) E1 ψ g (2+) / ψ g (2+) ψ g (2+) = 2 ( +iϵ +iϵ + iζ+iϵ iζ+iϵ ) dλ 1 ( iϵ iϵ + iζ iϵ iζ iϵ ) dλ 2 Q(λ 1, λ 2 ) dets(λ 1, λ 2 ) (3) Q(λ 1, λ 2 ) = ( 1) φ(λ 2 w (2) 2 )φ(λ 1 w (2) 1 η) φ(λ 2 λ 1 + η + ϵ 2,1 )φ(η) and matrix S(λ 1, λ 2 ) is given by ( ρ(λ 1 w (2) 1 + η/2)δ(α(λ 1 ), 1) ρ(λ 1 w (2) 2 + η/2)δ(α(λ 1 ), 2) ρ(λ 2 w (2) 1 + η/2)δ(α(λ 2 ), 1) ρ(λ 2 w (2) 2 + η/2)δ(α(λ 2 ), 2) ) (4). (5)
19 Evaluating the integrals for the spin-1 one-point function (T. D. and J. Sato, SIGMA(2011)) Evaluating the multiple integrals explicitly, we have obtained all the onepoint function for the integrable spin-1 XXZ chain as E 2, 2 (2 p) = E 0, 0 (2 p) ζ sin ζ cos ζ = 2ζ sin 2, ζ E 1, 1 (2 p) cos ζ(sin ζ ζ cos ζ) = ζ sin 2. (6) ζ In particular, we have E 22 = E 00. (7) Through the direct evaluation of the multiple integrals we confirm the identity: E 22 + E 11 + E 00 = 1.
20 Figure 3: Comparison with the exact numerical diagonalization. The red and blue lines represent analytical results obtained by the multiple integrals for E 22 = E 00 and E 11, respectively. The black dotted lines represent those obtained by exact diagonalization with the system size N s = 8.
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