Eigenvalues as Quasiparticles

The characteristic polynomials of the pseudo-Hermitian Gaussian ensemble are studied by considering them as polynomials wave function. In this picture, the eigenvalues become quasiparticles moving in the complex plane. We start by deriving their equations of motion as a function of the parameter that breaks the Hermitian condition. It is shown that as the eigenvalues move in conjugate pairs into the complex plane their 2D trajectories form bubbles. By investigating the singularities in the phase of the determinantal wave function, it is shown that the eigenvalues are vortices, and the presence of saddles is identified.