Collapse of the random phase approximation: examples and counter-examples from the shell model

TL;DR

This paper investigates the stability of the random phase approximation (RPA) in many-fermion systems, demonstrating through shell-model examples how RPA can collapse during phase transitions, with implications for its reliability.

Contribution

It provides the first non-trivial shell-model examples of both first- and second-order phase transitions affecting RPA stability, comparing these to exact solutions.

Findings

RPA remains stable during first-order transitions.

RPA collapses during second-order transitions.

Shell-model examples illustrate RPA limitations.

Abstract

The Hartree-Fock approximation to the many-fermion problem can break exact symmetries, and in some cases by changing a parameter in the interaction one can drive the Hartree-Fock minimum from a symmetry-breaking state to a symmetry-conserving state (also referred to as a ``phase transition'' in the literature). The order of the transition is important when one applies the random phase approximation (RPA) to the of the Hartree-Fock wavefunction: if first order, RPA is stable through the transition, but if second-order, then the RPA amplitudes become large and lead to unphysical results. The latter is known as ``collapse'' of the RPA. While the difference between first- and second-order transitions in the RPA was first pointed out by Thouless, we present for the first time non-trivial examples of both first- and second-order transitions in a uniform model, the interacting shell-model, where we can compare to exact numerical results.