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Extended Ginzburg-Landau functional for two-band superfluid Fermi gases

Serghei N. Klimin, Jacques Tempere, J. T. Devreese

Published Sep 5, 2013• 0 citations
OpenAlex

Abstract

The standard Ginzburg-Landau theory is valid only near the critical temperature, where the order parameter is small. As a consequence it is not straightforward to develop a Ginzburg-Landau framework for two-bandgap superconductors such as magnesium diboride, where two order parameters exist with different intrinsic critical temperatures. We derive an extended Ginzburg-Landau functional, capable of describing both single-band and two-bandgap superfluids at all temperatures below the critical temperature. In the limit of small temperatures, we retrieve the ground state energy obtained from the microscopic path-integral theory for Fermi superfluids, and for temperatures near the critical temperature, we obtain the standard Ginzburg-Landau expressions. The present formalism is applied to reveal hidden criticality and the presence of two healing length scales in two-bandgap superfluids. The developed method is focused on strong-coupling superfluid systems, in particular, the ultracold atomic Fermi gases in the BCS-BEC crossover regime.

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The standard Ginzburg-Landau theory is valid only near the critical temperature, where the order parameter is small. As a consequence it is not straightforward to develop a Ginzburg-Landau framework for two-bandgap superconductors such as magnesium diboride, where two order parameters exist with different intrinsic critical temperatures. We derive an extended Ginzburg-Landau functional, capable of describing both single-band and two-bandgap superfluids at all temperatures below the critical temperature. In the limit of small temperatures, we retrieve the ground state energy obtained from the microscopic path-integral theory for Fermi superfluids, and for temperatures near the critical temperature, we obtain the standard Ginzburg-Landau expressions. The present formalism is applied to reveal hidden criticality and the presence of two healing length scales in two-bandgap superfluids. The developed method is focused on strong-coupling superfluid systems, in particular, the ultracold atomic Fermi gases in the BCS-BEC crossover regime.
The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is derived by the path-integral method. In addition to the standard Ginzburg-Landau equation, a second equation describing the total atom density is obtained. These two coupled equations are necessary to describe both homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near the transition temperature Tc{T}_{c}Tc​ on both sides of the crossover. In the weakly interacting BEC region, it is also accurate at zero temperature where the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP) equation. The applicability of GP equation at finite temperature is discussed. On the BEC side, the fluctuation of the order parameter is studied and the renormalization to the molecule coupling constant is obtained.
We consider the evolution of superfluid properties of a three-dimensional p-wave Fermi gas from a weak coupling Bardeen-Cooper-Schrieffer (BCS) to strong coupling Bose-Einstein condensation (BEC) limit as a function of scattering volume. At zero temperature, we show that a quantum phase transition occurs for p-wave systems, unlike the s-wave case where the BCS to BEC evolution is just a crossover. Near the critical temperature, we derive a time-dependent Ginzburg-Landau (GL) theory and show that the GL coherence length is generally anisotropic due to the p-wave nature of the order parameter, and becomes isotropic only in the BEC limit.
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the sss-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
Superconductivity in iron-based, magnesium diborides, and other novel superconducting materials has a strong multi-band and multi-gap character. Recent experiments support the possibillity for a BCS-BEC crossover induced by strong-coupling and proximity of the chemical potential to the band edge of one of the bands. Here we study the simplest theoretical model which accounts for the BCS-BEC crossover in a two-band superconductor, considering tunable interactions and tunable energy separations between the bands. Mean-field results for condensate fraction, correlation length, and superconducting gap are reported in typical crossover diagrams to locate the boundaries of the BCS, crossover, and BEC regimes. When the superconducting gap is of the order of the local chemical potential, superconductivity is in the crossover regime of the BCS-BEC crossover and the Fermi surface of the small band is smeared by the gap opening. In this situation, small and large Cooper pairs coexist in the total condensate, which is the optimal condition for high-Tc superconductivity. The ratio between the gap and the Fermi energy in a given band results to be the best detection parameter for experiments to locate the system in the BCS-BEC crossover. Using available experimental data, our analysis shows that iron-based superconductors have the partial condensate of the small Fermi surface in the crossover regime of the BCS-BEC crossover, supporting the recent ARPES findings.
The Ginzburg-Landau formalism is constructed for Fermi superfluidity based on the 2n-body condensation in parallel with the usual Ginzburg-Landau formalism for the Cooper pair condensation. By this formalism, the transition temperature of the 2n-body condensation is given compactly by the zero of coefficient of the quadratic term in the 2n-body condensed order parameter without counting complicated Feynman diagrams for the 2n-body condensation susceptibility. It is shown that the 2n-body condensed state is stabilized in the intermediate- or strong-coupling regime against the BCS state based on the Cooper pair condensation. This theory is applied to the case of the possible superfluid state of cold atomic gases such as 9^99Be and 173^{173}173Yb; the quartet state is possible in the former case, while the sextet state is expected in the latter case. It is predicted that the 173^{173}173Yb atomic gas so far attained satisfies the condition for the sextet condensed state to be realized.