Second random-phase approximation, Thouless' theorem and the stability condition reexamined and clarified

TL;DR

This paper reexamines the Second Random Phase Approximation (SRPA), clarifies its issues with spurious states and instabilities, and links these problems to violations of the stability condition, providing a more general theoretical framework.

Contribution

It demonstrates that SRPA's issues do not violate Thouless' theorem and attributes these problems to stability condition violations, offering a generalized theoretical understanding.

Findings

SRPA can produce finite-energy spurious states and instabilities.

Thouless' theorem remains valid despite SRPA issues.

Stability condition violations cause SRPA shortcomings.

Abstract

It has been revealed through numerical calculations that the Second Random Phase Approximation (SRPA) with the Hartree-Fock solution as its reference state results in 1) spurious states at genuinely finite energy, contrary to common expectation, and 2) unstable solutions, which within the first-order Random Phase Approximation correspond to real low-energy collective vibrations. In the present work, these shortcomings of SRPA are shown to not contradict Thouless' theorem about the energy-weighted sum rule, and their origin is traced to the violation of the stability condition. A more general theorem is proven. Formal arguments are elucidated through numerical examples. Implications for the validity of SRPA are discussed.