The XXZ Heisenberg model on random surfaces

TL;DR

This paper develops a framework for studying integrable models, like the XXZ Heisenberg model, on random surfaces using random Manhattan lattices and matrix models, enabling analysis of their averaged properties.

Contribution

It introduces a novel approach to analyze integrable models on random geometries via random matrix models and eigenvalue integration techniques.

Findings

Formulation of a matrix model reproducing the XXZ Heisenberg model average

Reduction of matrix integration to eigenvalue integration

Application to models on random Manhattan lattices

Abstract

We consider integrable models, or in general any model defined by an $R$-matrix, on random surfaces, which are discretized using random Manhattan lattices. The set of random Manhattan lattices is defined as the set dual to the lattice random surfaces embedded on a regular d-dimensional lattice. They can also be associated with the random graphs of multiparticle scattering nodes. As an example we formulate a random matrix model where the partition function reproduces the annealed average of the XXZ Heisenberg model over all random Manhattan lattices. A technique is presented which reduces the random matrix integration in partition function to an integration over their eigenvalues.