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Symmetry 2018, 10(4), 80; https://doi.org/10.3390/sym10040080

Spontaneous Symmetry Breaking and Higgs Mode: Comparing Gross-Pitaevskii and Nonlinear Klein-Gordon Equations

1
Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
2
Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR), Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy
*
Author to whom correspondence should be addressed.
Received: 3 March 2018 / Revised: 18 March 2018 / Accepted: 19 March 2018 / Published: 23 March 2018
(This article belongs to the Special Issue Broken Symmetry)
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Abstract

We discuss the mechanism of spontaneous symmetry breaking and the elementary excitations for a weakly-interacting Bose gas at a 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. View Full-Text
Keywords: superfluidity; Gross-Pitaevskii equation; nonlinear Klein-Gordon equation; Higgs mode; Bose-Hubbard model superfluidity; Gross-Pitaevskii equation; nonlinear Klein-Gordon equation; Higgs mode; Bose-Hubbard model
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Faccioli, M.; Salasnich, L. Spontaneous Symmetry Breaking and Higgs Mode: Comparing Gross-Pitaevskii and Nonlinear Klein-Gordon Equations. Symmetry 2018, 10, 80.

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