Why a Bose-Einstein condensate cannot exist in a system of interacting bosons at ultrahigh temperatures

TL;DR

This paper provides an approximate mathematical proof that a Bose-Einstein condensate cannot exist in an interacting bosonic system at ultrahigh temperatures, extending the known ideal gas result.

Contribution

It offers the first approximate proof for the non-existence of BEC in interacting bosons at ultrahigh temperatures.

Findings

BEC is impossible at ultrahigh temperatures for interacting bosons.

The main contribution to occupation numbers at high T comes from states with quasiparticles.

Ultrahigh temperature effectively destroys the zero-momentum condensate.

Abstract

It is well known that a Bose-Einstein (BE) condensate of atoms exists in a system of interacting Bose atoms at $T\lesssim T^{(i)}_{c}$, where $T^{(i)}_{c}$ is the BE condensation temperature of an ideal gas. It is also generally accepted that BE condensation is impossible at ``ultrahigh'' temperatures $T\gg T^{(i)}_{c}$. While the latter property has been theoretically proven for an ideal gas, no such proof exists for an interacting system, to our knowledge. In this paper, we propose an approximate mathematical proof for a finite, nonrelativistic, periodic system of $N$ spinless interacting bosons. The key point is that, at $T\gg T^{(i)}_{c}$, the main contribution to the occupation number $N_{0}=\frac{1}{Z}\sum_{\wp}e^{-E_{\wp}/k_{B}T}\langle \Psi_{\wp}|\hat{a}^{+}_{\mathbf{0}}\hat{a}_{\mathbf{0}}|\Psi_{\wp}\rangle$, corresponding to atoms with zero momentum, originates from the states containing $N$ elementary quasiparticles. These states do not contain the BE condensate of zero-momentum atoms, implying that an ultrahigh temperature should ``blur'' such a condensate.