The Euclidean $\phi^4_2$ theory as a limit of an inhomogeneous Bose gas

TL;DR

This paper proves that the grand canonical Gibbs state of an interacting 2D quantum Bose gas with a trapping potential converges to a complex Euclidean field theory with quartic interaction, involving new renormalisation techniques.

Contribution

It establishes convergence of the Bose gas to a Euclidean field theory with diverging counterterms, extending previous homogeneous results to trapped gases with novel renormalisation methods.

Findings

Convergence of the partition function and density matrices to the field theory.

Development of renormalisation involving diverging counterterm functions.

Quantitative bounds on Schrödinger operator Green functions.

Abstract

We prove that the grand canonical Gibbs state of an interacting two-dimensional quantum Bose gas confined by a trapping potential converges to the complex Euclidean field theory with local quartic self-interaction, when the density of the gas becomes large and the range of the interaction becomes small. We obtain convergence of the relative partition function and convergence in $L^1 \cap L^\infty$ of the renormalised reduced density matrices. The field theory is supported on distributions of negative regularity, which requires a renormalisation by divergent mass and energy counterterms. Unlike previous results in the homogeneous setting of the torus without a trapping potential, the counterterms are not given by a finite collection of scalars but by diverging counterterm functions. This leads to significant new mathematical challenges. For our proof, we also derive quantitative bounds on the Green function of Schr\"odinger operators and of its gradient, which might be of independent interest.