$\Phi^4_2$ theory limit of a many-body bosonic free energy

TL;DR

This paper proves that the free energy of an interacting 2D Bose gas converges to the $\

Contribution

It provides a streamlined proof connecting quantum Bose gas free energy to the $\

Findings

Convergence of quantum free energy to $\

Use of variational and classical field methods

Revisiting and streamlining previous results

Abstract

We consider the quantum Gibbs state of an interacting Bose gas on the 2D torus. We set temperature, chemical potential and coupling constant in a regime where classical field theory gives leading order asymptotics. In the same limit, the repulsive interaction potential is set to be short-range: it converges to a Dirac delta function with a rate depending polynomially on the other scaling parameters. We prove that the free-energy of the interacting Bose gas (counted relatively to the non-interacting one) converges to the free energy of the $\Phi^4_2$ non-linear Schr{\"o}dinger-Gibbs measure, thereby revisiting recent results and streamlining proofs thereof. We combine the variational method of Lewin-Nam-Rougerie to connect, with controled error, the quantum free energy to a classical Hartree-Gibbs one with smeared non-linearity. The convergence of the latter to the $\Phi^4_2$ free energy then follows from arguments of Fr{\"o}hlich-Knowles-Schlein-Sohinger. This derivation parallels recent results of Nam-Zhu-Zhu.