The Correlation Functions of the XXZ Heisenberg Chain for Zero or Infinite Anisotropy and Random Walks of Vicious Walkers

TL;DR

This paper analyzes the XXZ Heisenberg chain at specific anisotropy limits, expressing wave functions via Schur functions, and links correlation functions to vicious walkers and plane partitions, providing asymptotic estimates at zero temperature.

Contribution

It introduces explicit expressions for wave functions and correlation functions in the XXZ chain at special anisotropy limits, connecting quantum spin models with combinatorial structures.

Findings

Correlation functions expressed through Schur functions.

Thermal correlator linked to vicious walkers' lattice paths.

Asymptotic behavior related to plane partitions.

Abstract

The XXZ Heisenberg chain is considered for two specific limits of the anisotropy parameter: $\Dl\to 0$ and $\Dl\to -\infty$. The corresponding wave functions are expressed by means of the symmetric Schur functions. Certain expectation values and thermal correlation functions of the ferromagnetic string operators are calculated over the base 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. Asymptotic estimate of the thermal correlator of the ferromagnetic string is obtained in the limit of zero temperature. It is shown that its amplitude is related to the number of plane partitions.