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

Abstract

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 Schr{ö}dinger-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.

arXiv commentThis revised version (to appear in Inventiones Mathematicae) covers both the 2D and 3D cases. It replaces an older 2018 work that was limited to 2D. The older, non-refereed, version is accessible as v1-v2 of the preprint

References in corpus (39)

• Unknown Quantum States: The Quantum de Finetti Representation
• Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional
• Critical Point of a Weakly Interacting Two-Dimensional Bose Gas
• The transition temperature of the dilute interacting Bose gas
• Generalized Hartree-Fock Theory and the Hubbard Model
• One-and-a-half quantum de Finetti theorems
• Transition temperature of a dilute homogeneous imperfect Bose gas
• Two-Dimensional Weakly Interacting Bose Gas in the Fluctuation Region
• Second-order corrections to mean field evolution for weakly interacting Bosons. I
• The excitation spectrum for weakly interacting bosons in a trap
• Geometric methods for nonlinear many-body quantum systems
• Second-order corrections to mean-field evolution of weakly interacting Bosons, II
• Bose-Einstein transition in a dilute interacting gas
• A hint of renormalization
• Kosterlitz-Thouless transition of the quasi two-dimensional trapped Bose gas
• Quasicondensation and coherence in the quasi-two-dimensional trapped Bose gas
• Excitation spectrum of interacting bosons in the mean-field infinite-volume limit
• The Church of the Symmetric Subspace
• Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball I: the 2D case
• Bogoliubov corrections and trace norm convergence for the Hartree dynamics
• Asymptotic Completeness in Quantum Field Theory: Translation Invariant Nelson Type Models Restricted to the Vacuum and One-Particle Sectors
• Semiclassical theory of the quasi two-dimensional trapped Bose gas
• Remainder Terms for Some Quantum Entropy Inequalities
• A Perspective on Constructive Quantum Field Theory
• Superfluidity of an interacting trapped quasi-2D Bose gas
• Condensate density and superfluid mass density of a dilute Bose gas near the condensation transition
• Mean-field limit of Bose systems: rigorous results
• Rigorous Upper Bound on the Critical Temperature of Dilute Bose Gases
• Universal correlations and coherence in quasi-two-dimensional trapped Bose gases
• A semi-classical field method for the equilibrium Bose gas and application to thermal vortices in two dimensions
• Renormalization of the Regularized Relativistic Electron-Positron Field
• Free Energy of a Dilute Bose Gas: Lower Bound
• On quantum estimation, quantum cloning and finite quantum de Finetti theorems
• Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d\le 3$
• Free Energies of Dilute Bose gases: upper bound
• Examples of bosonic de Finetti states over finite dimensional Hilbert spaces
• Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
• A Correlation Estimate for Quantum Many-Body Systems at Positive Temperature

Cited by in corpus (6)

• Correlation Energy of a Weakly Interacting Fermi Gas
• Variational Approach to Quantum State Tomography based on Maximal Entropy Formalism
• Focusing ${\Phi }_3^4$-model with a Hartree-type nonlinearity
• On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint
• Multi-target quantum compilation algorithm
• Hasimoto frames and the Gibbs measure of periodic nonlinear Schrödinger Equation