Goldstone Theorem, Hugenholtz-Pines Theorem and Ward-Takahashi Relation in Finite Volume Bose-Einstein Condensed Gases

TL;DR

This paper develops an approximate scheme for finite volume Bose-Einstein condensed gases that respects the Goldstone theorem, addressing zero and finite temperature cases without relying on the thermodynamic limit.

Contribution

It introduces a method that explicitly handles the Nambu-Goldstone mode in finite volume systems, ensuring the Goldstone theorem is satisfied without the usual Bogoliubov approximation.

Findings

Explicit treatment of Nambu-Goldstone mode in finite volume

Confirmation of vacuum unitary inequivalence in finite systems

Scheme applicable at both zero and finite temperatures

Abstract

We construct an approximate scheme based on the concept of the spontaneous symmetry breakdown, satisfying the Goldstone theorem, for finite volume Bose-Einstein condensed gases in both zero and finite temperature cases. In this paper, we discuss the Bose-Einstein condensation in a box with periodic boundary condition and do not assume the thermodynamic limit. When energy spectrum is discrete, we found that it is necessary to deal with the Nambu-Goldstone mode explicitly without the Bogoliubov's prescription, in which zero-mode creation- and annihilation-operators are replaced with a {\it c}-number by hand, for satisfying the Goldstone theorem. Furthermore, we confirm that the unitary inequivalence of vacua in the spontaneous symmetry breakdown is true for the finite volume system.