Distribution of complex eigenvalues for symplectic ensembles of non-Hermitian matrices

TL;DR

This paper analytically derives the eigenvalue density for symplectic non-Hermitian matrices, revealing a unique depletion near the real axis that differs from other ensembles and aligns with numerical QCD studies.

Contribution

It introduces a supermatrix sigma-model for symplectic non-Hermitian ensembles and provides explicit eigenvalue density formulas showing qualitative differences from other ensembles.

Findings

Eigenvalue density exhibits depletion near the real axis.

Derived explicit formulas for symplectic ensemble eigenvalues.

Results agree with previous numerical QCD studies.

Abstract

Symplectic ensemble of disordered non-Hermitian Hamiltonians is studied. Starting from a model with an imaginary magnetic field, we derive a proper supermatrix $\sigma $-model. The zero-dimensional version of this model corresponds to a symplectic ensemble of weakly non-Hermitian matrices. We derive analytically an explicit expression for the density of complex eigenvalues. This function proves to differ qualitatively from those known for the unitary and orthogonal ensembles. In contrast to these cases, a {\it depletion} of eigenvalues near the real axis occurs. The result about the depletion is in agreement with a previous numerical study performed for QCD models.