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arXiv:1110.5126 · Physical Review A

Composite-field Goldstone states and Higgs mechanism in dilute Bose gases

Fred Cooper, Chih-Chun Chien, Bogdan Mihaila, John F. Dawson, Eddy Timmermans

Published Feb 24, 2012• 9 citations
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Abstract

We show that a composite-field (diatom) Goldstone state is expected in a dilute Bose gas for temperatures between the Bose gas critical temperature where the atom Bose-Einstein condensate appears and the temperature where superfluidity sets in. The presence of superfluidity is tied to the existence of a U(1) charge-two diatom condensate in the system. By promoting the global U(1) symmetry of the theory to a gauge symmetry, we find that the mass of the gauge particle generated through the Anderson-Higgs mechanism is related to the superfluid density via the Meissner effect, and the superfluid density is related to the square of the anomalous density in the Bose system.

DOI
10.1103/physreva.85.023631

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We show that a composite-field (diatom) Goldstone state is expected in a dilute Bose gas for temperatures between the Bose gas critical temperature where the atom Bose-Einstein condensate appears and the temperature where superfluidity sets in. The presence of superfluidity is tied to the existence of a U(1) charge-two diatom condensate in the system. By promoting the global U(1) symmetry of the theory to a gauge symmetry, we find that the mass of the gauge particle generated through the Anderson-Higgs mechanism is related to the superfluid density via the Meissner effect, and the superfluid density is related to the square of the anomalous density in the Bose system.
We derive the Josephson relation for a dilute Bose gas in the framework of an auxiliary-field (AF) resummation of the theory in terms of the normal- and anomalous-density condensates. In the leading order of the AF loop expansion (LOAF) we find two critical temperatures in the phase diagram: the critical temperature Tc{T}_{c}Tc​, where the atom Bose-Einstein condensate (BEC) appears first, and the temperature ${T}^{\ensuremath{\star}}>{T}_{c}$, which indicates the onset of superfluidity and the appearance of a diatom condensate in the system. In this context, the Josephson relation shows that the superfluid density is related to the square of a second-order parameter, the anomalous density, $\ensuremath{\langle}A\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\varphi}\ensuremath{\rangle}$, which can be thought of as a diatom condensate analogous to the Cooper-pair condensate discussed in the BCS approach to dilute Fermi gases. This is in contrast with the corresponding result in the Bose gas theory without a diatom condensate, which predicts that the superfluid density is proportional to the square of the usual atom BEC condensate, $\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\rangle}$. In the region between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ the anomalous-density Green's function contains a zero energy-momentum state corresponding to a Goldstone state. This is a two-particle state with U(1) charge two. Our findings are consistent with the prediction that in the temperature range between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ a fraction of the system is in the superfluid state in the absence of the usual atom BEC condensate. This situation is similar to the case of dilute Fermi gases, where the superfluid density is proportional to the square of the gap parameter. The Josephson relation relies on the existence of zero energy and momentum excitations showing the intimate relationship between superfluidity and the Goldstone theorem.
We derive via diagrammatic perturbation theory the scaling behavior of the condensate and superfluid mass density of a dilute Bose gas just below the condensation temperature, T(c). Sufficiently below T(c) particle excitations are described by mean field (Bogoliubov). Near T(c), however, mean field fails, and the system undergoes a second order phase transition, rather than first order as predicted by Bogoliubov theory. Both condensation and superfluidity occur at the same T(c), and have similar scaling functions below T(c), but different finite size scaling at T(c) to leading order in the system size. A self-consistent two-loop calculation yields the condensate fraction critical exponent, 2beta approximately 0.66.
We derive a theoretical description for dilute Bose gases as a loop expansion in terms of composite-field propagators by rewriting the Lagrangian in terms of auxiliary fields related to the normal and anomalous densities. We demonstrate that already in leading order this nonperturbative approach describes a large interval of coupling-constant values, satisfies Goldstone's theorem, yields a Bose-Einstein transition that is second order, and is consistent with the critical temperature predicted in the weak-coupling limit by the next-to-leading-order large-N expansion.
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We discuss the mean-field theories obtained from the leading order in a large-NNN approximation for one- and two-component dilute Bose gases. For a one-component Bose gas, this approximation has the following properties: the Bose-Einstein condensation (BEC) phase transition is second order but the critical temperature Tc{T}_{c}Tc​ is not shifted from the noninteracting gas value T0{T}_{0}T0​. The spectrum of excitations in the BEC phase resembles the Bogoliubov dispersion with the usual coupling constant replaced by the running coupling constant which depends on both temperature and momentum. We then study two-component Bose gases with both inter- and intraspecies interactions and focus on the stability of the mixture state above Tc{T}_{c}Tc​. Our mean-field approximation predicts an instability from the mixture state to a phase-separated state when the ratio of the interspecies interaction strength to the intraspecies interaction strength (assuming equal strength for both species) exceeds a critical value. At high temperature this is a structural transition and the global translational symmetry is broken. Our work complements previous studies on the instability of the mixture phase in the presence of BEC.
We investigate an ultracold and dilute Bose gas by taking into account a finite-range two-body interaction. The coupling constants of the resulting Lagrangian density are related to measurable scattering parameters by following the effective-field-theory approach. A perturbative scheme is then developed up to the Gaussian level, where both quantum and thermal fluctuations are crucially affected by finite-range corrections. In particular, the relation between spontaneous symmetry breaking and the onset of superfluidity is emphasized by recovering the renowned Landau’s equation for the superfluid density in terms of the condensate one.