Derives the focusing Φ⁶₁ measure from quantum Gibbs states at the optimal classical mass threshold using variational methods and new trial states.
$\Phi^4_2$ theory limit of a many-body bosonic free energy
3 Pith papers cite this work. Polarity classification is still indexing.
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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\"odinger-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\"ohlich-Knowles-Schlein-Sohinger. This derivation parallels recent results of Nam-Zhu-Zhu.
fields
math-ph 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
In the large-mass limit with tuned chemical potential, quantum gases converge to classical interacting particles via Ginibre loop ensembles and cluster expansions.
The classical Gibbs measure with fractional Bessel interaction potential is derived from the renormalized grand-canonical quantum Bose gas for 3/2 < β ≤ 2 via convergence of relative free energy and reduced density matrices.
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