Φ24 theory limit of a many-body bosonic free energy

We consider the quantum Gibbs state of an interacting Bose gas on the 2D torus T2. 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 Φ24 non-linear Schrödinger-Gibbs measure, thereby revisiting recent results and streamlining proofs thereof. We combine the variational method of Lewin–Nam–Rougerie to connect, with controlled error, the quantum free energy to a classical Hartree-Gibbs one with smeared non-linearity. The convergence of the latter to the Φ24 free energy then follows from arguments of Fröhlich–Knowles–Schlein–Sohinger. This derivation parallels recent results of Nam–Zhu–Zhu.