Stability of flat-band Bose-Einstein condensation from the geometry of compact localized states

TL;DR

This paper introduces a real-space geometric approach to analyze the stability of Bose-Einstein condensation in flat-band models, revealing conditions under which condensation is feasible based on the geometry of localized states.

Contribution

It reformulates the mean-field energy minimization as a Euclidean geometry problem using compact localized states, providing new principles for designing flat-band models with stable Bose-Einstein condensation.

Findings

Triangular frameworks of localized states support condensation.

Square frameworks indicate condensation in a single mode is impossible.

Relation to quantum distance conditions in Bloch states.

Abstract

We consider Bose-Einstein condensation in flat-band models from a real-space perspective. Using a basis of compact localized states, we reformulate the minimization of the mean-field energy as a Euclidian geometry problem. Within Bogoliubov theory, we show that flat-band models where the solutions to this problem are frameworks consisting of triangles with nonzero area are promising for condensation, whereas for instance square frameworks indicate condensation in a single mode is impossible. When restricting the analysis to Bloch states, this approach can be related to a necessary condition for a non-vanishing quantum distance. This work provides a new perspective on how condensation in flat bands is destabilized, and offers principles for the construction of models where flat-band Bose-Einstein condensation is possible.