Sum-rules and Goldstone modes from extended RPA theories in Fermi systems with spontaneously broken symmetries

TL;DR

This paper extends the Self-Consistent RPA approach to systems with spontaneously broken symmetries, ensuring the preservation of key properties like sum rules and Goldstone modes, with theoretical and numerical validation.

Contribution

It develops a comprehensive SCRPA framework that maintains standard RPA properties in broken symmetry cases, including a simplified renormalised RPA version.

Findings

SCRPA preserves energy weighted sum rule

SCRPA correctly reproduces Goldstone modes

Numerical tests confirm theoretical predictions

Abstract

The Self-Consistent RPA (SCRPA) approach is elaborated for cases with a continuously broken symmetry, this being the main focus of the present article. Correlations beyond standard RPA are summed up correcting for the quasi-boson approximation in standard RPA. Desirable properties of standard RPA such as fullfillment of energy weighted sum rule and appearance of Goldstone (zero) modes are kept. We show theoretically and, for a model case, numerically that, indeed, SCRPA maintains all properties of standard RPA for practically all situations of spontaneously broken symmetries. A simpler approximate form of SCRPA, the so-called renormalised RPA, also has these properties. The SCRPA equations are first outlined as an eigenvalue problem, but it is also shown how an equivalent many body Green's function approach can be formulated.