Nambu-Goldstone modes in the random phase approximation

TL;DR

This paper demonstrates that the kernel of the RPA matrix in stable mean field solutions decomposes into Nambu-Goldstone modes and other modes, providing a new proof independent of Nakada's recent results.

Contribution

It provides a novel proof of the decomposition of the RPA kernel into Nambu-Goldstone modes without relying on Nakada's general analysis.

Findings

Kernel decomposes into Nambu-Goldstone modes and other modes

Subspace with cyclic coordinates associated with Nambu-Goldstone modes

Complementary subspace where RPA matrix behaves as in zero-dimensional case

Abstract

I show that the kernel of the random phase approximation (RPA) matrix based on a stable Hartree, Hartree-Fock, Hartree-Bogolyubov or Hartree-Fock-Bogolyubov mean field solution is decomposed into a subspace with a basis whose vectors are associated, in the equivalent formalism of a classical Hamiltonian homogeneous of second degree in canonical coordinates, with conjugate momenta of cyclic coordinates (Nambu-Goldstone modes) and a subspace with a basis whose vectors are associated with pairs of a coordinate and its conjugate momentum neither of which enters the Hamiltonian at all. In a subspace complementary to the one spanned by all these coordinates including the conjugate coordinates of the Nambu-Goldstone momenta, the RPA matrix behaves as in the case of a zerodimensional kernel. This result was derived very recently by Nakada as a corollary to a general analysis of RPA matrices based on both stable and unstable mean field solutions. The present proof does not rest on Nakada's general results.