Journal of Mathematical Sciences, Vol. 115, No. 1, 2003 TEMPERATURE CORRELATORS OF THE XXZ HEISENBERG MAGNET FOR Δ = -∞ N. I. Abarenkova and A. G. Pronko UDC 517.9 The one-dimensional XXZ Heizenberg magnet for Δ= -∞ is considered, and the time-dependent temperature correlation function of the z components of local spin operators is calculated. In the thermodynamic limit, the correlation function is expressed in terms of the Fredholm determinants of linear integral operators. Bibliography: 23 titles. Dedicated to the memory of A. G. Izergin 1. Introduction Calculation of correlation functions solvable by means of the Bethe ansatz is one of the most interesting and important problems in the theory of quantum integrable systems (see [1]). The most intensively investigated model on this line of inquiry is the one-dimensional XXZ Heisenberg magnet of spin 1/2 (see, e.g., [1, 2] and original papers [3, 4] with references therein). On a periodic chain of M sites in an external magnetic field, the quantum dynamics of the model h is governed by the Hamiltonian H XXZ = − M  m=1  σ − m σ + m+1 + σ − m+1 σ + m + Δ 2 ( σ z m σ z m+1 − 1 ) + h 2 σ z m  , (1.1) where σ ± m = 1 2 (σ x m ± iσ y m ) and σ x,y,z are the Pauli matrices σ x =  0 1 1 0  , σ y =  0 −i i 0  , σ z =  1 0 0 −1  . (1.2) The operators σ x,y,z m act nontrivially in the two-dimensional space H m , and the total space H = ⊗ M m=1 H m of states of model (1.1) is 2 M -dimensional. The periodic boundary conditions mean that σ x,y,z M+1 = σ x,y,z 1 . Two-point temperature correlators are defined in the standard way as temperature normalized mean values 〈 σ a m+1 (t) σ a 1 (0) 〉 (T,M) = Sp H  e −βHXXZ σ a m+1 (t) σ a 1 (0)  Sp H  e −βHXXZ  , a = x, y, z, (1.3) where β is the inverse temperature, β =1/T , and the time dependence is defined canonically as follows: σ a m (t)= e itHXXZ σ a m e −itHXXZ , a = x, y, z. (1.4) The temperature correlators are of most interest in the thermodynamic limit, 〈 σ a m+1 (t) σ a 1 (0) 〉 (T ) = lim M→∞ 〈 σ a m+1 (t) σ a 1 (0) 〉 (T,M) . (1.5) The most explicit expressions for the correlation functions can be obtained for the zero value of the anisotropy parameter, Δ = 0. This particular case of the XXZ magnet, known also as the XX0 (isotropic XY ) Heisenberg magnet, is the point of free fermions of the model. The exact representations for correlation functions (1.3) of the XX0 magnet are known both for the case of a finite lattice [5] and in the thermodynamic limit (1.5) (see [6–8]). In the latter case, the correlators are expressed as the Fredholm determinants of linear integral operators of a special type, the so-called “integrable” integral operators (see [9]). Such representations are important, because the Fredholm determinant of an “integrable” integral operator in the representation for correlation functions of Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 269, 2000, pp. 43–61. Original article submitted November 16, 2000. 1910 1072-3374/03/1151-1910 $25.00 c  2003 Plenum Publishing Corporation