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arXiv:1701.04894 · The European Physical Journal C

Bose–Einstein condensation and symmetry breaking of a complex charged scalar field

Tonatiuh Matos, Elías Castellanos, Abril Suárez

Published Jul 28, 2017• 6 citations
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Abstract

In this work the Klein–Gordon 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

DOI
10.1140/epjc/s10052-017-5065-5

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In this work the Klein–Gordon 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
We rewrite the complex Klein-Gordon (KG) equation with a mexican-hat scalar field potential in a thermal bath with one loop contribution as a new Gross-Pitaevskii (GP)-like equation. We interpret it as a charged and finite temperature generalization of the GP equation. We find its hydrodynamic version as well and using it, we derive the corresponding thermodynamics. We obtain a generalized first law for a charged Bose-Einstein Condensate (BEC). We translate the breaking of the U(1) local symmetry of the KG field into the new version of the GP equation and demonstrate that this symmetry breaking corresponds to a phase transition of the gas into a BEC, and show the conditions for which this system naturally becomes superfluid and/or superconductor.
Starting with a scalar field in a thermal bath and using the one-loop quantum correction potential, we rewrite the Klein-Gordon equation in its thermodynamical representation and study the behavior of this scalar field due to temperature variations in the equations of motion. We find the generalization of a Gross-Pitaevskii–like equation for a relativistic Bose gas with finite temperature, the corresponding thermodynamic and viscosity expressions, and an expression for the postulate of the first law of the thermodynamics for this BECs. We also propose that the equations obtained might help to explain at some level the phase transition of a Bose-Einstein Condensate in terms of quantum field theory in a simple way.
We analyze the the remarkable analogy between the classical Klein–Gordon equation for a test scalar field in a flat and also in a curved background, and the Gross–Pitaevskii equation for a Bose–Einstein condensate trapped by an external potential. We stress here that the solution associated with the Klein–Gordon equation (KG) in a flat space time has the same mathematical structure, under certain circumstances, to those obtained for the Gross–Pitaevskii equation, that is, a static soliton solution. Additionally, Thomas–Fermi approximation is applied to the 3–dimensional version of this equation, in order to calculate some thermodynamical properties of the system in curved a space–time back ground. Finally, we stress the fact that a gravitational background provides, in some cases, a kind of confining potential for the scalar field, allowing us to remarks even more the possible connection between scalar fields and the phenomenon of Bose–Einstein condensation.
We analyze the consequences caused by a thermal bath upon the properties associated to the symmetry breaking of scalar fields with one-loop correction potential. Concerning the nonrelativistic regime associated with the aforementioned system, we calculate the shift in the condensation temperature caused by the thermal bath, assuming a harmonic oscillator type potential. We prove that the shift in the condensation temperature depends on the associated scale of the system. In addition, we obtain bounds associated to the scale that could lead to relevant corrections on the condensation temperature under typical conditions.
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.
We deduce the relation between the critical temperature associated to the symmetry breaking of scalar fields with one--loop correction potential immersed in a thermal bath and the condensation temperature of the aforementioned system, assuming a harmonic oscillator type potential. We show that these two temperatures are related through the \emph{scale} associated to the system. In this aim, we infer the order of magnitude for the \emph{scale} as a function of the corresponding healing length, in order to give a criterium to compare both temperatures. Additionally, we prove that the condensation temperature is independent of the thermal bath within the semiclassical approximation, for a positive coupling constant, assuming that the thermal bath contribution is the lowest energy associated to the system.