Quantum Crystals and Spin Chains

TL;DR

This paper explores the quantum analogs of melting crystal corners across different dimensions, revealing integrable models and proposing a conjecture for the mass gap in a complex coupled spin chain system.

Contribution

It introduces a quantum generalization of melting crystal corners, mapping them to spin chains, and analyzes their integrability and spectral properties.

Findings

Two-dimensional case is integrable and maps to the Heisenberg XXZ model.

Three-dimensional case involves coupled XXZ spin chains.

Numerical analysis supports a conjecture for the mass gap.

Abstract

In this note, we discuss the quantum version of the melting crystal corner in one, two, and three dimensions, generalizing the treatment for the quantum dimer model. Using a mapping to spin chains we find that the two--dimensional case (growth of random partitions) is integrable and leads directly to the Hamiltonian of the Heisenberg XXZ ferromagnet. The three--dimensional case of the melting crystal corner is described in terms of a system of coupled XXZ spin chains. We give a conjecture for its mass gap and analyze the system numerically.