k-Fold Gaussian Random Matrix Ensembles I: Forcing Structure into Random Matrices

TL;DR

This paper extends Gaussian Random Matrix Ensembles to k-fold tensor product spaces, characterizing their structure, eigenvalue distributions, and applications to quantum systems with symmetries, using advanced representation theory techniques.

Contribution

It introduces a generalized framework for Gaussian ensembles on k-fold tensor products, with explicit eigenvalue distribution formulas and connections to quantum entanglement.

Findings

Eigenvalue distributions are computed exactly for these ensembles.

The 2-fold case links eigenvalue and entanglement spectra.

The framework applies to quantum systems with gauge symmetries.

Abstract

Random Matrix Theory is a powerful tool in applied mathematics. Three canonical models of random matrix distributions are the Gaussian Orthogonal, Unitary and Symplectic Ensembles. For matrix ensembles defined on k-fold tensor products of identical vector spaces we motivate natural generalizations of the Gaussian Ensemble family. We show how the k-fold invariant constraints are satisfied in both disordered spin models and systems with gauge symmetries, specifically quantum double models. We use Schur-Weyl duality to completely characterize the form of allowed Gaussian distributions satisfying k-fold invariant constraints. The eigenvalue distribution of our proposed ensembles is computed exactly using the Harish-Chandra integral method. For the 2-fold tensor product case, we show that the derived distribution couples eigenvalue spectrum to entanglement spectrum. Guided by representation theory, our work is a natural extension of the standard Gaussian random matrix ensembles.