A Condensation of Interacting Bosons in Two Dimensional Space

TL;DR

This paper develops a finite, renormalized mean field theory for two-dimensional interacting bosons, revealing a novel condensate state with exponentially growing energy as particle number increases.

Contribution

It introduces a new renormalization scheme and describes a unique bosonic condensate state that breaks translation symmetry in two dimensions.

Findings

Ground state energy grows exponentially with particle number.

Identifies a new type of bosonic condensate (soliton).

Provides a finite formulation for 2D boson interactions.

Abstract

We develop a theory of non-relativistic bosons in two spatial dimensions with a weak short range attractive interaction. In the limit as the range of the interaction becomes small, there is an ultra-violet divergence in the problem. We devise a scheme to remove this divergence and produce a completely finite formulation of the theory. This involves reformulating the dynamics in terms of a new operator whose eigenvalues give the {\it logarithm} of the energy levels. Then, a mean field theory is developed which allows us to describe the limit of a large number of bosons. The ground state is a new kind of condensate (soliton) of bosons that breaks translation invariance spontaneously. The ground state energy is negative and its magnitude grows {\it exponentially} with the number of particles, rather than like a power law as for conventional many body systems.