Abstract. The XXZ Heisenberg chain is considered for two specific limits of the anisotropy parameter: Δ → 0 and Δ → −∞. The corresponding wave functions are expressed in terms of symmetric Schur functions. Certain expectation values and thermal correlation functions of the ferromagnetic string operators are calculated over the basis of N -particle Bethe states. The thermal correlator of the ferromagnetic string is expressed through the generating function of the lattice paths of random walks of vicious walkers. A relationship between the expectation values obtained and the generating functions of strict plane partitions in a box is discussed. An asymptotic estimate of the thermal correlator of the ferromagnetic string is obtained in the zero temperature limit. It is shown that its amplitude is related to the number of plane partitions. §1. Introduction
XXZ Heisenberg chain.A system of spin 1/2 particles occupying sites of a one-dimensional lattice, widely known as the quantum XYZ Heisenberg chain [1], has attracted considerable attention both in theoretical and mathematical physics, and it has been thoroughly investigated for a long time [2]- [8]. The quantum inverse scattering method, developed for solving integrable models of quantum field theory and statistical physics [9,10], has also been employed to investigate the XY Z Heisenberg chain [11, 12]. An important special case of the XY Z model, the so-called XXZ spin chain, also attracts considerable attention [13,14] The random walks is a classical problem both for combinatorics and statistical physics. The problem of enumeration of the paths made by the so-called vicious walkers on the one-dimensional lattice was formulated and investigated in detail by Fisher [21]. This problem still continues to attract considerable attention of physicists and mathematicians [22]-[34].
The XX0 Heisenberg model on a cyclic chain is considered. The representation of the Bethe wave functions via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The determinantal expressions of the form-factors and of the thermal correlation functions are obtained. The q-binomial determinants enable the connection of the form-factors with the generating functions both of boxed plane partitions and of self-avoiding lattice paths. The asymptotical behavior of the thermal correlation functions is studied in the limit of low temperature provided that the characteristic parameters of the system are large enough.
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