Physical and unphysical solutions of the random-phase approximation equation

Abstract

As an addendum to the previous paper [1], 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 as canonically conjugate variables. The random-phase approximation (RPA) is widely used as describing excitation properties on top of mean-field (MF) solutions. In Ref. [1], I mathematically argued properties of solutions of the RPA equation based on two types of dualities: UL-and LR-dualities. The solutions were classified into five categories, as disclosed by the dualities, in Prop. 2 of Ref. [1]. It was also reconfirmed that, if the stability matrix S is positive-definite, the solutions are all physical, belonging to Class (1) of Prop. 2, which has already been verified in Ref. [2]. Its opposite was also proven in Ref. [1]. However, spontaneous symmetry breakdown (SSB) necessarily occurs for the MF solution in localized self-bound systems like atomic nuclei [3]. Individual SSB leads to a Nambu-Goldstone (NG) mode, and therefore S is quite generally positive-semidefinite in physical cases, rather than positive-definite. A method to handle NG modes has been established [2,4,5], which seems valid as long as the NG mode corresponds to physical degrees of freedom (d.o.f.). This method presumes that each NG mode forms a two-dimensional Jordan block of N S, where N is the norm matrix. On the contrary, the dualities do not limit the dimension of the Jordan blocks, as exemplified in Appendix C.5 of Ref. [1]. To the best of my knowledge, there have been no rigorous arguments that elucidate the dimensions of Jordan blocks for NG modes, except for restricted cases [6]. In this addendum, I shall prove what dimensionality is possible for Jordan blocks associated with NG-mode solutions (i.e., Class (5) in Prop. 2 of Ref. [1]) when S is positive-semidefinite. The section, appendix, and proposition numbers of Ref. [1] will be referred to directly in the text. The RPA equation is expressed as S x ν = ω ν N x ν ; N := 1 0 0 −1. (1)