Proof of Bose condensation for weakly interacting lattice bosons

Abstract

A weakly interacting Bose gas on a simple cubic lattice is considered. We prove the existence of the standard or zero-mode Bose condensation at sufficiently low temperature. This result is valid for sufficiently small interaction potential and small values of chemical potential. Our method exploits infra-red bound for the suitable two-point Bogolyubov's inner product. We do not use the reflection positivity or some expansion methods.