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arXiv:1810.08370 · Inventiones mathematicae

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

Mathieu Lewin B, Phan Thành Nam, Nicolas Rougerie

Published Jan 1, 2021• 22 citations
arXivHTMLPDF
DOI
10.1007/s00222-020-01010-4

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In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus Td\mathbb T^dTd, d=2,3d=2,3d=2,3, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.
We consider the many-body quantum Gibbs state for the Bose–Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schrödinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.