Rotationally invariant ensembles of integrable matrices

TL;DR

This paper introduces a new ensemble of random integrable matrices with rotational invariance, establishing a framework called integrable matrix theory (IMT) that parallels random matrix theory for quantum integrable systems.

Contribution

It develops a basis-independent parametrization of integrable matrices and derives their joint probability density, extending the random matrix theory framework to integrable models.

Findings

Constructed rotationally invariant ensembles of integrable matrices.

Formulated integrable matrix theory (IMT) as a counterpart to RMT.

Derived joint probability density for integrable matrices.

Abstract

We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M family of integrable matrices consists of exactly N-M independent commuting N-by-N matrices linear in a real parameter. We first develop a rotationally invariant parametrization of such matrices, previously only constructed in a preferred basis. For example, an arbitrary choice of a vector and two commuting Hermitian matrices defines a type-1 family and vice versa. Higher types similarly involve a random vector and two matrices. The basis-independent formulation allows us to derive the joint probability density for integrable matrices, in a manner similar to the construction of Gaussian ensembles in the RMT.