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

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.PACS numbers:

Electromagnetic fluctuation and collective modes in relativistic bosonic superfluid in mixed dimensions

“…The asymptotic behaviors of the dispersion relations can be further elaborated in (2+1) dimensions by the method of duality. It is done by directly Legendre transforming the Gaussian Lagrangian equation (13). With a Hubbard-Stratonovich transformation, we first introduce a vector field…”

“…This velocity gives rise to the Bogoliubov mode in the non-relativistic weakly interacting bosons. In the case where the fluctuation at the scale of healing length is further omitted [13], the constant velocity can be identified with the ordinary speed of hydrodynamic sound v c s s…”

Gaussian quantum fluctuations in the superfluid–Mott-insulator phase transition

“…The 4×4 matrix M( q, iΩ m ) is the inverse propagator of Gaussian fluctuations with q the D-dimensional wavevector, Ω m = 2πm/β the Matsubara frequencies, and i = √ −1 the imaginary unit. See [32] for details on the derivation of M( q, iΩ m ) in the case of both non-relativistic and relativistic bosonic actions.…”

The Cauchy problem for the Klein-Gordon equation under the quartic potential in the de Sitter spacetime

“…The mechanism of the spontaneous symmetry breaking is used in the study of phase transitions (see [18]). Faccioli and Salasnich [12] considered it for the Gross-Pitaevskii equation and also for the cubic nonlinear Klein-Gordon equation, and studied the spectrum of the superfluid phase of bosonic gases. Honda and Choptuik [17] considered the monotonically increasingly boosted coordinates with n = 3 in (1.8) to study localized and unstable solutions like oscillon.…”