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This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii–Kosterlitz–Thouless (BKT) temperature TBKT as a function of attractive pairing strength by associating it with the onset of 'quasi-condensation' in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS–BEC crossover theory for fermions, leads to non-monotonic behavior for TBKT as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKT has an absolute maximum of the order of 0.1EF. We discuss how our results compare with those derived from the Nelson–Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKT followed by an often dramatic decrease before the system enters the BEC regime. [ABSTRACT FROM AUTHOR]

We present fundamental constraints required for a consistent linear response theory of fermionic superfluids and address temperatures both above and below the transition temperature Tc. We emphasize two independent constraints, one associated with gauge invariance (and the related Ward identity) and another associated with the compressibility sum rule, both of which are satisfied in strict BCS theory. However, we point out that it is the rare many body theory which satisfies both of these. Indeed, well studied quantum Hall systems and random-phase approximations to the electron gas are found to have difficulties with meeting these constraints. We summarize two distinct theoretical approaches which are, however, demonstrably compatible with gauge invariance and the compressibility sum rule. The first of these involves an extension of BCS theory to a mean field description of the BCS-Bose Einstein condensation crossover. The second is the simplest Nozieres Schmitt-Rink (NSR) treatment of pairing correlations in the normal state. As a point of comparison we focus on the compressibility κ of each and contrast the predictions above Tc. We note here that despite the compliance with sum rules, this NSR based scheme leads to an unphysical divergence in κ at the transition. Because of the delicacy of the various consistency requirements, the results of this paper suggest that avoiding this divergence may repair one problem while at the same time introducing others. [ABSTRACT FROM AUTHOR]

We present a pairing fluctuation theory which self-consistently incorporates finite momentum pair excitations in the context of BCS-Bose-Einstein condensation (BEC) crossover, and we apply this theory to high T superconductors and ultracold Fermi gases. There are strong similarities between Fermi gases in the unitary regime and high T superconductors. Here, we address key issues of common interest, especially the pseudogap. In the Fermi gases, we summarize recent experiments including various phase diagrams (with and without population imbalance), as well as evidence for a pseudogap in thermodynamic and other experiments. [ABSTRACT FROM AUTHOR]

This review is written at the time of the twentieth anniversary of the discovery of high-temperature superconductors, which nearly coincides with the important discovery of the superfluid phases of ultracold trapped fermionic atoms. We show how these two subjects have much in common. Both have been addressed from the perspective of the BCS—Bose–Einstein condensation (BEC) crossover scenario, which is designed to treat short coherence length superfluids with transition temperatures which are “high” with respect to the Fermi energy. A generalized mean field treatment of BCS–BEC crossover at general temperatures T, based on the BCS–Leggett ground state, has met with remarkable success in the fermionic atomic systems. Here we summarize this success in the context of four different cold atom experiments, all of which provide indications, direct or indirect, for the existence of a pseudogap. This scenario also provides a physical picture of the pseudogap phase in the underdoped cuprates which is a central focus of high Tc research. We summarize successful applications of BCS–BEC crossover to key experiments in high Tc systems, including the phase diagram, specific heat, and vortex core STM data, along with the Nernst effect, and exciting recent data on the superfluid density in very underdoped samples. [ABSTRACT FROM AUTHOR]

In this paper we address the ratio of the shear viscosity to entropy density η/s in bosonic and fermionic superfluids. A small η/s is associated with nearly perfect fluidity, and more general measures of the fluidity perfection/imperfection are of wide interest to a number of communities. We use a Kubo approach to concretely address this ratio via low-temperature transport associated with the quasiparticles. Our analysis for bosonic superfluids utilizes the framework of the one-loop Bogoliubov approximation, whereas for fermionic superfluids we apply BCS theory and its BCS-BEC extension. Interestingly, we find that the transport properties of strict BCS and Bogoliubov superfluids have very similar structures, albeit with different quasiparticle dispersion relations. While there is a dramatic contrast between the power law and exponential temperature dependence for η alone, the ratio η/s for both systems is more similar. Specifically, we find the same linear dependence (on the ratio of temperature T to inverse lifetime γ(T)) with η/sαT/γ(T), corresponding to imperfect fluidity. By contrast, near the unitary limit of BCS-BEC superfluids a very different behavior results, which is more consistent with near-perfect fluidity. [ABSTRACT FROM AUTHOR]

We present fundamental constraints required for a consistent linear response theory of fermionic superfluids and address temperatures both above and below the transition temperature Tc. We emphasize two independent constraints, one associated with gauge invariance (and the related Ward identity) and another associated with the compressibility sum rule, both of which are satisfied in strict BCS theory. However, we point out that it is the rare many body theory which satisfies both of these. Indeed, well studied quantum Hall systems and random-phase approximations to the electron gas are found to have difficulties with meeting these constraints. We summarize two distinct theoretical approaches which are, however, demonstrably compatible with gauge invariance and the compressibility sum rule. The first of these involves an extension of BCS theory to a mean field description of the BCS-Bose Einstein condensation crossover. The second is the simplest Nozieres Schmitt-Rink (NSR) treatment of pairing correlations in the normal state. As a point of comparison we focus on the compressibility κ of each and contrast the predictions above Tc. We note here that despite the compliance with sum rules, this NSR based scheme leads to an unphysical divergence in κ at the transition. Because of the delicacy of the various consistency requirements, the results of this paper suggest that avoiding this divergence may repair one problem while at the same time introducing others. [ABSTRACT FROM AUTHOR]

We study the spontaneous formation of vortices during the superfluid condensation in a trapped fermionic gas subjected to a rapid thermal quench via evaporative cooling. Our work is based on the numerical solution of the time-dependent crossover Ginzburg-Landau equation coupled to the heat diffusion equation. We quantify the evolution of condensate density and vortex length as a function of a crossover phase parameter from BCS to BEC. The more interesting phenomena occur somewhat nearer to the BEC regime and should be experimentally observable; during the propagation of the cold front, the increase in condensate density leads to the formation of supercurrents toward the center of the condensate as well as possible condensate volume oscillations. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate the necessity of including the generally omitted collective-mode contributions in calculations of the Meissner effect for nonuniform superconductors. We consider superconducting pairing with nonzero center-of-mass momentum, as is possibly relevant to high transition temperature cuprates, cold atoms, and color superconductors in quantum chromodynamics. For the concrete example of the Fulde-Ferrell phase we present a quantitative calculation of the superfluid density, showing not only that the collective-mode contributions are appreciable but also that they derive from the amplitude mode of the order parameter. This latter mode is generally viewed as being invisible in conventional superconductors. However, our analysis shows that it is extremely important in pair-density-wave-type superconductors, where it destroys stable superfluidity well before the mean-field order parameter vanishes. [ABSTRACT FROM AUTHOR]

We address band engineering in the presence of periodic driving by numerically shaking a lattice containing a bosonic condensate. By not restricting to simplified band structure models we are able to address arbitrary values of the shaking frequency, amplitude, and interaction strengths g. For "near-resonant" shaking frequencies with moderate g, a quantum phase transition to a finite momentum superfluid is obtained with Kibble-Zurek scaling and quantitative agreement with experiment. We use this successful calibration as a platform to support a more general investigation of the interplay between (one particle) Floquet theory and the effects associated with arbitrary g. Band crossings lead to superfluid destabilization, but where this occurs depends on g in a complicated fashion. [ABSTRACT FROM AUTHOR]

In this Rapid Communication we derive the full gauge-invariant electromagnetic response beyond the BCS level using the fermionic superfluid path integral. In the process we identify and redress a failure to satisfy the compressibility sum rule; this shortcoming is associated with the conventional path-integral formulation of BCS-level electrodynamics. The approach in this paper builds on an alternative saddle point scheme. At the mean field level, this leads to the well known gauge-invariant electrodynamics of BCS theory and to the satisfaction of the compressibility sum rule. Moreover, this scheme can be readily extended to address arbitrary higher order fluctuation theories (for example, at the Gaussian level.) At any level this approach will lead to a gauge invariant and compressibility sum rule consistent treatment of electrodynamics and thermodynamics. [ABSTRACT FROM AUTHOR]

We investigate the effects of topological order on the transition temperature, Tc, and response functions in spin-orbit-coupled fermionic superfluids with a transverse Zeeman field in three dimensions. Our calculations include fluctuation effects beyond mean-field theory and are compatible with f-sum rules. In the topological phase we find that Tc can be as large as 10% of the Fermi temperature, which should be experimentally accessible in cold-gas experiments. At higher temperatures, above Tc, the spin and density response functions provide signatures of topological phases via the recombination or amplification of frequency-dependent peaks. [ABSTRACT FROM AUTHOR]