Bose-Einstein condensation of interacting bosons: A two-step proof

TL;DR

This paper provides a rigorous two-step proof demonstrating that in systems of interacting bosons with specific pair potentials, Bose-Einstein condensation occurs at sufficiently high densities, linking permutation cycles and long-range order.

Contribution

It offers a novel, rigorous proof connecting permutation cycles, long-range order, and Bose-Einstein condensation in interacting bosonic systems.

Findings

Self-organization into infinite permutation cycles occurs with off-diagonal long-range order.

Bose-Einstein condensation occurs when cycle lengths grow sufficiently fast.

Condensation appears above a density threshold depending on temperature.

Abstract

We prove two equilibrium properties of a system of interacting atoms in three or higher dimensional continuous space. (i) If the particles interact via pair potentials of a nonnegative Fourier transform, their self-organization into infinite permutation cycles is simultaneous with off-diagonal long-range order. If the cycle lengths tend to infinity not slower than the square of the linear extension of the system, there is also Bose-Einstein condensation. (ii) If the pair potential is also nonnegative, cycles composed of a nonzero fraction of the total number of particles do appear if the density exceeds a temperature-dependent threshold value. The two together constitute the proof that in such a system Bose-Einstein condensation takes place at high enough densities.