Infinite cycles of interacting bosons

TL;DR

This paper investigates the relationship between permutation cycles and Bose-Einstein condensation in interacting bosonic systems, revealing conditions under which infinite cycles occur and their connection to thermodynamic singularities.

Contribution

It demonstrates that infinite permutation cycles always accompany singularities in thermodynamic quantities and explores effects of long-range interactions on cycle formation.

Findings

Infinite cycles are linked to singularities in thermodynamic quantities.

Long-range interactions can suppress or induce infinite cycles depending on the system.

In three and four dimensions, infinite cycles are associated with strong singularities.

Abstract

In the first-quantized description of bosonic systems permutation cycles formed by the particles play a fundamental role. In the ideal Bose gas Bose-Enstein condensation (BEC) is signaled by the appearance of infinite cycles. When the particles interact, the two phenomena may not be simultaneous, the existence of infinite cycles is necessary but not sufficient for BEC. We demonstrate that their appearance is always accompanied by a singularity in the thermodynamic quantities which in three and four dimensions can be as strong as a one-sided divergence of the isothermal compressibility. Arguments are presented that long-range interactions can give rise to unexpected results, such as the absence of infinite cycles in three dimensions for long-range repulsion or their presence in one and two dimensions if the pair potential has a long attractive tail.