The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices

TL;DR

This paper introduces and solves a complex non-Hermitian quaternion real matrix ensemble, deriving eigenvalue correlations and orthogonal polynomials, with applications to quantum field theory with non-zero chemical potential.

Contribution

It provides the first explicit solution of the complex Laguerre symplectic ensemble, including correlation functions and orthogonal polynomials, for finite and large matrix sizes.

Findings

Derived explicit eigenvalue correlation functions.

Proved orthogonality of complex Laguerre polynomials.

Mapped the model to the complex Dirac operator spectrum.

Abstract

We solve the complex extension of the chiral Gaussian Symplectic Ensemble, defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real matrices. This leads to the appearance of Laguerre polynomials in the complex plane and we prove their orthogonality. Alternatively, a complex eigenvalue representation of this ensemble is given for general weight functions. All k-point correlation functions of complex eigenvalues are given in terms of the corresponding skew orthogonal polynomials in the complex plane for finite-N, where N is the matrix size or number of eigenvalues, respectively. We also allow for an arbitrary number of complex conjugate pairs of characteristic polynomials in the weight function, corresponding to massive quark flavours in applications to field theory. Explicit expressions are given in the large-N limit at both weak and strong non-Hermiticity for the weight of the Gaussian two-matrix model. This model can be mapped to the complex Dirac operator spectrum with non-vanishing chemical potential. It belongs to the symmetry class of either the adjoint representation or two colours in the fundamental representation using staggered lattice fermions.