Classical field theory limit of 2D many-body quantum Gibbs states
Résumé
Nonlinear Gibbs measures play an important role in many areas of mathematics, including nonlinear dispersive equations with random initial data and stochastic partial differential equations. In statistical physics, they are believed to emerge for some quantum systems at criticality. In space dimensions two or larger, the measures are concentrated on singular distributions and they have to be be appropriately renormalized.
In this paper we provide the first rigorous derivation of these renormalized nonlinear Gibbs measures in two space dimensions, starting from a linear quantum system in a mean-field-type limit. More precisely, we consider the grand-canonical Gibbs state of a large 2D bosonic quantum system and prove that it converges to the renormalized Gibbs measure of a nonlinear Schrödinger-type classical field theory. Reduced density matrices of the quantum Gibbs state converge to their classical analogues. The quantum system is well defined without any renormalization. By only tuning its chemical potential, we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory.
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