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.
Highlights
The mechanism of spontaneous symmetry breaking is widely used to study phase transitions [1].Usually the approach introduced by Landau [2,3] for second-order phase transitions is adopted, where an order parameter is identified and its acquiring a non-zero value corresponds to a transition from a disordered phase to an ordered one
In other words, when a nonlinearity is added to the symmetric problem, and its strength exceeds a critical value, there is a loss of symmetry in the system, that is called spontaneous symmetry breaking, alias self-trapping into an asymmetric state [4]
∇ σ + 2μσ which gives the spectrum: hω = 2μ +. This time we have a gapped spectrum, the gap being 2μ. This is consistent with what we would expect by the spontaneous symmetry mechanism: the breaking of the U (1) symmetry produces always a gapless mode, which is usually called Goldstone mode [17], and a gapped mode, which in Condensed Matter Physics is referred as Higgs mode [18,19]
Summary
The mechanism of spontaneous symmetry breaking is widely used to study phase transitions [1]. For weakly-interacting Bose gases, the spontaneous breaking of the U (1) group leads to the transition to a superfluid phase [1] In this normal-to-superfluid phase transition the order parameter is the mean value of the bosonic matter field both in the non-relativistic case [5,6] and the relativistic one [7]. In this review paper we compute and study the spectrum of elementary excitations for both non-relativistic and relativistic Bose gases in the ordered phase of the normal-to-superfluid phase transition. We calculate and analyze the spectrum of elementary excitations of this effective action
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