# Bose-Einstein Condensation and Symmetry Breaking of a Complex Charged Scalar Field

## TL;DR

This paper derives a charged scalar field equation analogous to the Gross-Pitaevskii equation, analyzes its thermodynamics, and explores symmetry breaking and phase transitions in a relativistic Bose-Einstein condensate with electromagnetic interactions.

## Contribution

It formulates a charged, finite-temperature Klein-Gordon equation as a Gross-Pitaevskii-like equation and investigates its thermodynamics and symmetry-breaking behavior.

## Key findings

- Condensation temperature calculated in the non-relativistic limit.
- Generalized energy conservation equation derived with electromagnetic fields.
- Insights into phase transition mechanisms related to symmetry breaking.

## Abstract

In this work the Klein-Gordon (KG) equation for a complex scalar field with U(1) symmetry endowed in a mexican-hat scalar field potential with thermal and electromagnetic contributions is written as a Gross-Pitaevskii (GP)-like equation. This equation is interpreted as a charged generalization of the GP equation at finite temperatures found in previous works. Its hydrodynamical representation is obtained and the corresponding thermodynamical properties are derived and related to measurable quantities. The condensation temperature in the non-relativistic regime associated with the aforementioned system within the semiclassical approximation is calculated. Also, a generalized equation for the conservation of energy for a charged bosonic gas is found when electromagnetic fields are introduced, and it is studied how under certain circumstances its breaking of symmetry can give some insight on the phase transition of the system not just into the condensed phase but also on other related systems.