Classical field theory limit of many-body quantum Gibbs states in 2D and 3D

• 成果类型: Article
• 署名作者: Lewin, Mathieu; Nam, Phan Thanh; Rougerie, Nicolas
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Universite Paris-Dauphine; University of Munich; Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); CNRS - Institute of Physics (INP)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-01010-4
• 发表日期: 2021
• 页码: 315-444
• 关键词: bose-einstein condensation; nonlinear schrodinger-equation; global well-posedness; statistical-mechanics; invariant-measures; scattering-theory; correlation inequalities; bogoliubov correction; phase-transitions; nobel lecture

摘要

We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schrodinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose-Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.