Spin correlation functions, Ramus-like identities, and enumeration of constrained lattice walks and plane partitions

TL;DR

This paper explores the connection between spin correlation functions in the Heisenberg XX chain and combinatorial enumeration problems, providing new determinantal formulas and asymptotic analysis for plane partitions and lattice walks.

Contribution

It introduces a novel representation of correlation functions as non-intersecting lattice walks and plane partitions, linking quantum spin models with combinatorial enumeration.

Findings

Derived determinantal formulas for plane partition generating functions.

Established a connection between random turns walks and circulant matrices.

Calculated asymptotics of mean values in large-scale limits.

Abstract

Relations between the mean values of distributions of flipped spins on periodic Heisenberg XX chain and some aspects of enumerative combinatorics are discussed. The Bethe vectors, which are the state-vectors of the model, are considered both as on- and off-shell. It is this approach that makes it possible to represent and to study the correlation functions in the form of non-intersecting nests of lattice walks and related plane partitions. We distinguish between two types of walkers, namely lock step models and random turns. Of particular interest is the connection of random turns walks and a circulant matrix. The determinantal representation for the norm-trace generating function of plane partitions with fixed height of diagonal parts is obtained as the expectation of the generating exponential over off-shell N-particle Bethe states. The asymptotics of the dynamical mean value of the generating exponential is calculated in the double scaling limit provided that the evolution parameter is large. It is shown that the amplitudes of the leading asymptotics depend on the number of diagonally constrained plane partitions.