Bose gas to Bose-Einstein Condensate by the Phase Transition of the Klein-Gordon equation

TL;DR

This paper derives a generalized Gross-Pitaevskii equation from the Klein-Gordon equation with a mexican-hat potential, linking symmetry breaking to Bose-Einstein condensation and superfluidity/superconductivity at finite temperature.

Contribution

It introduces a charged, finite-temperature generalization of the GP equation from the KG equation and connects symmetry breaking to phase transition into a BEC with superfluid and superconductor properties.

Findings

Derived a new GP-like equation from the KG equation with a mexican-hat potential.

Linked U(1) symmetry breaking to the phase transition into a BEC.

Identified conditions for superfluidity and superconductivity in the system.

Abstract

We rewrite the complex Klein-Gordon (KG) equation with a mexican-hat scalar field potential in a thermal bath with one loop contribution as a new Gross-Pitaevskii (GP)-like equation. We interpret it as a charged and finite temperature generalization of the GP equation. We find its hydrodynamic version as well and using it, we derive the corresponding thermodynamics. We obtain a generalized first law for a charged Bose-Einstein Condensate (BEC). We translate the breaking of the U(1) local symmetry of the KG field into the new version of the GP equation and demonstrate that this symmetry breaking corresponds to a phase transition of the gas into a BEC, and show the conditions for which this system naturally becomes superfluid and/or superconductor.