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

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 Schrödinger operators and of its gradient, which might be of independent interest.