Gradual eigenvector ergodization in coupled Ginibre matrices

TL;DR

This paper analyzes how eigenvectors of coupled non-Hermitian Ginibre matrices gradually become ergodic as the interaction strength increases, providing explicit formulas for the transition and eigenvalue density behavior.

Contribution

It offers a detailed, explicit characterization of eigenvector ergodization and spectral density transition in coupled Ginibre matrices as the coupling varies.

Findings

Eigenvectors spread over the full system with increasing coupling.

Explicit asymptotic formula for eigenvector spread as a function of coupling.

Eigenvalue density at the origin vanishes beyond a critical coupling, indicating spectral support split.

Abstract

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$ complex Ginibre matrices interacting via a deterministic matrix $c{\bf 1}_N$, where $c$ is the complex coupling parameter whose magnitude $|c|$ controls the interaction strength. We characterize quantitatively how the eigenvectors of the whole system, initially localized in one of the individual subsystems for $|c|=0$, eventually spread over the full system with growing interaction strength. The resulting asymptotic formula describing such spread in the limit $N\to \infty$ is very explicit and provides a full picture of the gradual ergodization of eigenvectors as a function of the coupling parameter $|c|$ in the whole transition regime. As a by-product of our method we also compute the mean eigenvalue density for our model at the origin of the spectral bulk $z=0$ in the fully ergodic regime, when the coupling is scaled with the matrix size as $c=\sqrt{N}\tilde{c}$. We find that as $N\to \infty$ the limiting density at the origin vanishes beyond the critical value $|\tilde{c}|=1$, signalling of a split of the density support in the complex plane into two disjoint domains.