Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition

TL;DR

This paper introduces a two-matrix random matrix model that interpolates between chiral and non-chiral ensembles, connecting to Wilson chiral perturbation theory and deriving spectral correlation functions relevant for lattice QCD.

Contribution

It develops a novel two-matrix model bridging chGUE and GUE, providing eigenvalue representations and spectral correlations in the weakly non-chiral limit.

Findings

Derived eigenvalue representation with Pfaffian structure.

Obtained spectral correlation functions for finite matrix size.

Connected random matrix results to Wilson chiral perturbation theory.

Abstract

We introduce a random two-matrix model interpolating between a chiral Hermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix without symmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE) and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-n limit in the vicinity of the chGUE (which we denote by weakly non-chiral limit) this theory is in one to one correspondence to the partition function of Wilson chiral perturbation theory in the epsilon regime, such as the related two matrix-model previously introduced in refs. [20,21]. For a generic number of flavours and rectangular block matrices in the chGUE part we derive an eigenvalue representation for the partition function displaying a Pfaffian structure. In the quenched case with nu=0,1 we derive all spectral correlations functions in our model for finite-n, given in terms of skew-orthogonal polynomials. The latter are expressed as Gaussian integrals over standard Laguerre polynomials. In the weakly non-chiral microscopic limit this yields all corresponding quenched eigenvalue correlation functions of the Hermitian Wilson operator.