Solutions of random-phase approximation equation for positive-semidefinite stability matrix
Abstract
It is mathematically proven that, if the stability matrix 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 N\,S (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.
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