New Integrable Systems from Unitary Matrix Models

TL;DR

This paper demonstrates the integrability of certain unitary matrix models and their reductions, leading to new generalized integrable particle systems with external potentials of sine and hyperbolic sine types.

Contribution

The authors establish the integrability of a class of unitary matrix models and their eigenvalue dynamics, generalizing the Sutherland model with external potentials.

Findings

Eigenvalue dynamics are integrable with sine and hyperbolic sine potentials.

Generalization of the Sutherland model with external potentials.

Positive-definite matrix models are also shown to be integrable.

Abstract

We show that the one dimensional unitary matrix model with potential of the form $a U + b U^2 + h.c.$ is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space dimension in an external potential of the form $a \cos (x+\alpha ) + b \cos ( 2x +\beta )$ and interacting through two-body potentials of the inverse sine square type. This system constitutes a generalization of the Sutherland model in the presence of external potentials. The positive-definite matrix model, obtained by analytic continuation, is also integrable, which leads to the integrability of a system of particles in hyperbolic potentials interacting through two-body potentials of the inverse hypebolic sine square type.