# Spontaneous symmetry breaking and Higgs mode: comparing Gross-Pitaevskii and nonlinear Klein-Gordon equations

## TL;DR

This paper compares spontaneous symmetry breaking and Higgs modes in Bose gases described by Gross-Pitaevskii and nonlinear Klein-Gordon equations, highlighting differences in their excitation spectra and analyzing their relation to quantum phase transitions.

## Contribution

It provides a comparative analysis of phase and amplitude modes in non-relativistic and relativistic Bose gases, revealing how their excitation spectra differ and relate in various regimes.

## Key findings

- Non-relativistic case exhibits a single gapless Bogoliubov spectrum.
- Relativistic case shows two branches: one gapless, one gapped.
- Relativistic spectrum reduces to Bogoliubov spectrum in the non-relativistic limit.

## Abstract

We discuss the mechanism of spontaneous symmetry breaking and the elementary excitations for a weakly-interacting Bose gas at finite temperature. We consider both the non-relativistic case, described by the Gross-Pitaevskii equation, and the relativistic one, described by the cubic nonlinear Klein-Gordon equation. We analyze similarities and differences in the two equations and, in particular, in the phase and amplitude modes (i.e. Goldstone and Higgs modes) of the bosonic matter field. We show that the coupling between phase and amplitude modes gives rise to a single gapless Bogoliubov spectrum in the non-relativistic case. Instead, in the relativistic case the spectrum has two branches: one is gapless and the other is gapped. In the non-relativistic limit we find that the relativistic spectrum reduces to the Bogoliubov one. Finally, as an application of the above analysis, we consider the Bose-Hubbard model close to the superfluid-Mott quantum phase transition and we investigate the elementary excitations of its effective action, which contains both non-relativistic and relativistic terms.