Solutions of random-phase approximation equation for positive-semidefinite stability matrix

TL;DR

This paper proves that when the stability matrix in RPA is positive-semidefinite, all solutions are physical or NG modes, with NG modes forming at most two-dimensional Jordan blocks, ensuring their separation via canonical variables.

Contribution

It provides a rigorous mathematical proof characterizing RPA solutions for positive-semidefinite stability matrices, clarifying the nature of NG modes.

Findings

All RPA solutions are physical or NG modes.

NG modes can form Jordan blocks of dimension at most two.

NG modes can be separated using canonical conjugate variables.

Abstract

It is mathematically proven that, if the stability matrix $\mathsf{S}$ is positive-semidefinite, solutions of the random-phase approximation (RPA) equation are all physical or belong to Nambu-Goldstone (NG) modes, and the NG-mode solutions may form Jordan blocks of $\mathsf{N\,S}$ ($\mathsf{N}$ is the norm matrix) but their dimension is not more than two. This guarantees that the NG modes in the RPA can be separated out via canonically conjugate variables.