The correlation functions of the $XXZ$ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers

Abstract

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].

Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks

“…The N-particle state-vectors | Ψ(u N ) , the states with N spins "down", is convenient to express by means of the Schur functions [30,31]:…”

Phase structure of XX0 spin chain and nonintersecting Brownian motion

“…The correlation functions of XXZ and XX0 models in thermodynamic limit are studied in [51], [52], [53], [54], [50]. Using similarities between XX0 model and NIBM, correlation functions of XX0 model in low temperature limit are recently formulated and investigated by Bogoliubov, [16], [17], [18].…”

The Ground-State Vector of the XY Heisenberg Chain and the Gauss Decomposition

“…The correlation functions of certain quantum integrable models demonstrate connection with enumerative combinatorics and with the theory of symmetric functions [1][2][3][4][5]. For instance, random lattice walks and boxed plane partitions, as subjects of enumerative combinatorics [6], are related to the correlation functions of the XX model [7][8][9][10][11]. Various spin lattice models [12], including the XY Heisenberg chain model, as well as its isotropic limit, the XX model, provide a base for such actively developing subjects in the theory of quantum information [13] as random lattice walks [14] and entanglement entropy [15].…”