Proof of phase transition in homogeneous systems of interacting bosons

TL;DR

This paper rigorously proves London's long-standing conjecture that Bose-Einstein condensation occurs during the superfluid transition in liquid helium, linking macroscopic permutation cycles to BEC and extending results to lattice spin models.

Contribution

It provides a rigorous proof of the phase transition and Bose-Einstein condensation in homogeneous interacting boson systems, and extends the methodology to lattice spin models.

Findings

Macroscopic permutation cycles appear at low temperatures.

Bose-Einstein condensation occurs during the superfluid transition.

Spin models exhibit magnetic ordering at low temperatures.

Abstract

Using the rigorous path integral formalism of Feynman and Kac we prove London's eighty years old conjecture that during the superfluid transition in liquid helium Bose-Einstein condensation (BEC) takes place. The result is obtained by proving first that at low enough temperatures macroscopic permutation cycles appear in the system, and then showing that this implies BEC. We find also that in the limit of zero temperature the infinite cycles cover the whole system, while BEC remains partial. For the Bose-condensed fluid at rest we define a macroscopic wave function. Via the equivalence of 1/2 spins and hard-core bosons the method extends to lattice models. We show that at low enough temperatures the spin-1/2 axially anisotropic Heisenberg models, including the isotropic ferro- and antiferromagnet and the XY model, undergo magnetic ordering.