Self-consistent perturbation expansion for Bose-Einstein condensates satisfying Goldstone's theorem and conservation laws

TL;DR

This paper develops a systematic, self-consistent perturbation approach for Bose-Einstein condensates that respects Goldstone's theorem and conservation laws, ensuring accurate theoretical descriptions.

Contribution

It introduces a method to construct self-consistent approximations using the interaction energy or Hugenholtz-Pines relation, unifying gapless and conserving properties in condensate theories.

Findings

Third-order perturbation reproduces mean-field theory

Series becomes exact with infinite terms

Derived expressions for entropy and superfluid density

Abstract

Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.