Bose-Einstein condensation and symmetry breaking

TL;DR

This paper rigorously analyzes Bose-Einstein condensation and symmetry breaking in simplified interacting Bose gas models, establishing conditions under which condensation occurs and linking it to spontaneous gauge symmetry breaking.

Contribution

It provides a rigorous proof connecting Bose-Einstein condensation with spontaneous gauge symmetry breaking in simplified models, including mean field and imperfect Bose gases.

Findings

Condensation occurs if and only if gauge symmetry is spontaneously broken.

The method involves analyzing single-mode Hamiltonians and microcanonical ensembles.

The results imply the compressibility sum rule fails in the ground state of 1D mean-field Bose gas.

Abstract

Adding a gauge symmetry breaking field -\nu\sqrt{V}(a_0+a_0^*) to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis \lim_{V\to\infty}< a_0>/\sqrt{V}={\rm sgn}\nu \lim_{V\to\infty}\sqrt{< a_0^*a_0>/V} where the averages are taken in the ground state or in thermal equilibrium states. Letting \nu tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode (k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.