Random-Matrix Approach to RPA equations. I

TL;DR

This paper models RPA equations using random matrices to analyze spectral properties, identifying conditions for system stability and spectral mixing through analytical and numerical methods.

Contribution

It introduces a novel random-matrix model for RPA equations, providing analytical insights into spectral behavior and stability criteria.

Findings

Spectral spectrum described by two semicircles deformed with increasing coupling.

Critical coupling strength for system instability derived analytically.

Numerical illustrations confirm theoretical predictions.

Abstract

We study the RPA equations in their most general form by taking the matrix elements appearing in the RPA equations as random. This yields either a unitarily or an orthogonally invariant random-matrix model which is not of the Cartan type. The average spectrum of the model is studied with the help of a generalized Pastur equation. Two independent parameters govern the behaviour of the system: The strength $\alpha^2$ of the coupling between positive- and negative-energy states and the distance between the origin and the centers of the two semicircles that describe the average spectrum for $\alpha^2 = 0$, the latter measured in units of the equal radii of the two semicircles. With increasing $\alpha^2$, positive- and negative-energy states become mixed and ever more of the spectral strength of the positive-energy states is transferred to those at negative energy, and vice versa. The two semicircles are deformed and pulled toward each other. As they begin to overlap, the RPA equations yield non--real eigenvalues: The system becomes unstable. We determine analytically the critical value of the strength for the instability to occur. Several features of the model are illustrated numerically.