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

Abstract

The Self-Consistent RPA (SCRPA) approach is elaborated for cases with a continuouslybroken symmetry, this being the main focus of the present article. Correlations beyondstandard RPA are summed up correcting for the quasi-boson approximation in standard RPA.Desirable properties of standard RPA such as fulfillment of energy weighted sum rule andappearance of Goldstone (zero) modes are kept. We show theoretically and, for a modelcase, numerically that, indeed, SCRPA maintains all properties of standard RPA forpractically all situations of spontaneously broken symmetries. A simpler approximate formof SCRPA, the so-called renormalised RPA, also has these properties. The SCRPA equationsare first outlined as an eigenvalue problem, but it is also shown how an equivalent manybody Green’s function approach can be formulated.