Typeset with jpsj2.cls <ver.1.2> Full Paper Exact calculation of correlation functions for spin-1/2 Heisenberg chain Masahiro Shiroishi * and Minoru Takahashi † Insitutite for Solid State Physics, University of Tokyo We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain, The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nonetheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical develop- ments have enabled us to derive the explicit analytical form of the correlation functions in some short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of XXZ chain up to four lattice sites. For the special case of XXX chain, the correlation functions are further calculated up to five lattice sites. KEYWORDS: spin-1/2 Heisenberg chain, Bethe ansatz, correlation functions, multiple integral formulas 1. Introduction The spin-1/2 Heisenberg XXZ chain is one of the most fundamental models in the study of quantum magnetism in low dimensions. In fact many quasi one-dimensional substances are well described by the model. The model also appears as as an effecticve Hamiltonian of more com- plicated systems. The Hamiltonian of the XXZ chain is given by H = N X j=1 £ S x j S x j+1 + S y j S y j+1 +ΔS z j S z j+1 / , (1) where S α = σ α /2(α = x, y, z) with σ α being Pauli ma- trices. Here N is the number of the lattice sites and Δ is the anisotropy parameter. We specially call XXX chain when Δ = 1. Under the periodic boundary condition S j+N = S j , the eigenvalues and eigenfunctions of the Hamiltonian (1) have been obtained by Bethe ansatz. 1–4 In fact by solving the Bethe ansatz equations e ikj N =(-1) M-1 Y l=j e i(kj +k l ) +1 - 2Δe ikj e i(kj +k l ) +1 - 2Δe ik l , (j =1, ..., M ), (2) where M is the number of the down spins, we can calcu- late the eigen energy E = N Δ 4 + M X j=1 (cos k j - Δ) . Especially the ground state is given by a solution in the sector M = N/2. In the critical region -1 < Δ < 1, its value per site in the thermodynamic limit N →∞ can be calculated as e 0 = Δ 4 - sin πν 2π Z ∞ -∞ dw sinh(1 - ν )w sinh w cosh νw = Δ 4 + sin πν 2π Z ∞+ i 2 -∞+ i 2 dx 1 sinh x cosh νx sinh νx . (3) * siroisi@issp.u-tokyo.ac.jp † mtaka@issp.u-tokyo.ac.jp Here and hereafter we parametrize the anisotropy param- eter as Δ = cos πν for the critical region -1 < Δ < 1. In the XXX limit (Δ → 1), the ground state energy per site (3) is given by 2 e 0 = 1 4 - ln 2. (4) Other physical quantities such as magnetic susceptibil- ity, elementary excitations, Drude weight, etc..., have been evaluated based on the Bethe ansatz equations (2). 3 Now even at finite temperature, we can evaluate the bulk physical quantities with an extreme high precision. On the other hand, the exact calculation of the correlation functions even at the ground state without magnetic field has been a very difficult problem. The exceptional case is Δ = 0, where the system reduces to the free-fermion model after the Jordan-Wigner transformation. In this case we can calculate arbitrary correlation functions by means of Wick’s theorem. 5, 6 For general Δ cases, there have been many attempts to evaluate the correlation functions exactly. Actually for the simplest nearest-neighbor correlation functions, we can obtain their values in the thermodynamic limit form the expression of of the ground state energy (3) as S z j S z j+1  = ∂e 0 ∂ Δ , S x j S x j+1  = 1 2 (e 0 - ΔS z j S z j+1 ). (5) Especially in the XXX case, we have S z j S z j+1  = 1 12 - 1 3 ln 2 = -0.14771572685..., (6) from Hulth´ en’s result (4). Further for the XXX chain, the explicit form of the second-neighbor correlation function S z j S z j+2  = 1 12 - 4 3 ln 2 + 3 4 ζ (3) =0.06067976995..., (7) was obtained by Takahashi in 1977 via the strong cou- pling expansion for the ground state energy of the half- filled Hubbard model. 7 The expression (??) was also ob- 1