The Bogoliubov inequality and the nature of Bose-Einstein condensates for interacting atoms in spatial dimensions $D \le 2$

TL;DR

This paper explores how the Bogoliubov inequality constrains the existence and nature of Bose-Einstein condensates (BECs) for interacting atoms in low spatial dimensions, revealing conditions for BEC formation and similarities between disordered and noninteracting systems.

Contribution

It establishes the necessary conditions for BEC existence in low dimensions considering interactions and external potentials, highlighting the role of momentum state occupation.

Findings

BECs require macroscopic occupation of many momentum states with an accumulation point.

The nature of BECs in disordered potentials for noninteracting atoms matches that for interacting atoms without external potentials.

The Bogoliubov inequality imposes fundamental restrictions on BECs in low-dimensional systems.

Abstract

We consider the restriction placed by the Bogoliubov inequality on the nature of the Bose-Einstein condensates (BECs) for interacting atoms in a spatial dimension D </- 2 and in the presence of an external arbitrary potential, which may be a confining "box", a periodic, or a disordered potential. The atom-atom interaction gives rise to a (gauge invariance) symmetry-breaking term that places further restrictions on BECs in the form of a consistency proviso. The necessary condition for the existence of a BEC in D </- 2 in all cases is macroscopic occupation of many single-particle momenta states with the origin a limit point (or accumulation point) of condensates. It is shown that the nature of BECs for noninteracting atoms in a disordered potential is precisely the same as that of BECs for interacting atoms in the absence of an external potential.