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

Mathieu Lewin (1) , Phan Thành Nam (2) , Nicolas Rougerie (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LMU - Ludwig Maximilian University [Munich] = Ludwig Maximilians Universität München
3 LPM2C - Laboratoire de physique et modélisation des milieux condensés

We prove that the grand-canonical Gibbs states of a large 2D bosonic system converge to the Gibbs measure of an interacting classical field theory, in a mean-field-type limit. Reduced density matrices of the quantum Gibbs state converge to their classical analogues, given by a nonlinear Schrödinger-Gibbs measure supported on distributions with low regularity. Tuning the chemical potential of the grand-canonical ensemble provides a counter-term for the diverging repulsive interactions, analogue to the Wick ordering of the limit classical theory.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Gaz Quantiques [cond-mat.quant-gas]

Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie. Classical field theory limit of 2D many-body quantum Gibbs states. Benjamin Schlein, Giuseppe Genovese and Vedran Sohinger. Gibbs measures for nonlinear dispersive equations, 2018., 2018, Oberwolfach, Germany. Oberwolfach Reports, 2018. ⟨hal-01898563v1⟩