Rigorous Upper Bound on the Critical Temperature of Dilute Bose Gases

TL;DR

This paper establishes rigorous upper bounds on the critical temperature for Bose-Einstein condensation in dilute Bose gases, matching expected theoretical values in 2D and providing bounds in 3D.

Contribution

It provides the first rigorous upper bounds on the critical temperature for Bose gases, aligning with theoretical predictions in 2D and offering new bounds in 3D.

Findings

Exponential decay of correlations in 2D when a^2 ρ is small and T exceeds a specific threshold.

In 3D, exponential decay when ΔT_c / T_c^0 > 5 √(a ρ^{1/3}), giving an upper bound on T_c.

Results match the expected critical temperature in 2D and set bounds in 3D.

Abstract

We prove exponential decay of the off-diagonal correlation function in the two-dimensional homogeneous Bose gas when a^2 \rho is small and the temperature T satisfies T > 4 \pi \rho / \ln |\ln(a^2\rho). Here, a is the scattering length of the repulsive interaction potential and \rho is the density. To leading order in a^2 \rho, this bound agrees with the expected critical temperature for superfluidity. In the three-dimensional Bose gas, exponential decay is proved when \Delta T_c / T_c^0 > 5 \sqrt{a \rho^{1/3}}, where T_c^0 is the critical temperature of the ideal gas. While this condition is not expected to be sharp, it gives a rigorous upper bound on the critical temperature for Bose-Einstein condensation.