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.
We present a diagrammatic theory for determining consistent electromagnetic response functions in strongly correlated fermionic superfluids. While a gauge-invariant electromagnetic response is well understood at the BCS level, a treatment of correlations beyond BCS theory requires extending this theoretical formalism. The challenge in such systems is to maintain gauge invariance, while simultaneously incorporating additional self-energy terms arising from strong correlation effects. Central to our approach is the application of the Ward-Takahashi identity, which introduces collective mode contributions in the response functions and guarantees that the
f-sum rule is satisfied. We outline a powerful method, which determines these collective modes in the presence of correlation effects and in a manner compatible with gauge invariance. Since this method is based on fundamental aspects of quantum field theory, the underlying principles are broadly applicable to strongly correlated superfluids. As an illustration of the technique, we apply it to a simple class of theoretical models that contain a frequency-independent order parameter. These models include BCS-BEC crossover theories of the ultracold Fermi gases, along with models specifically associated with the high-
Tc cuprates. Finally, as an alternative approach, we contrast with the path integral formalism. Here, the calculation of gauge-invariant response appears more straightforward. However, the collective modes introduced are those of strict BCS theory, without any modification from additional correlations. As the path integral simultaneously addresses electrodynamics and thermodynamics, we emphasize that it should be subjected to a consistency test beyond gauge invariance, namely that of the compressibility sum rule. We show how this sum rule fails in the conventional path integral approach.
A diagrammatic method of obtaining exact gauge-invariant response functions in strongly correlated Fermi superfluids is implemented for several example condensed matter systems of current interest. These include: topological superfluids, high temperature superconductors, and superfluids with finite center-of-mass momentum pairing known as Fulde-Ferrell superfluids. Much of the literature on these systems has focused on single-particle properties or alternatively has invoked simple approximations to treat response functions. The goal is to show that, for this wide class of topical problems, one can compute exact response functions. This enables assessment of the validity of different physical scenarios and allows a very broad class of experiments to be addressed.,The method developed is based on deriving the full electromagnetic vertex, which satisfies the Ward-Takahashi identity, and determining the collective modes in a manner compatible with the self-consistent gap equation. In the condensed phase of a superfluid and a superconductor, where gauge invariance is spontaneously broken, it is crucial to determine the collective modes from the gap equation in a manner which restores gauge invariance. Our diagrammatic framework provides a very general and powerful method for obtaining these collective modes in a variety of strongly correlated Fermi superfluids. We show that a full electromagnetic vertex satisfying the Ward-Takahashi identity ensures the
f-sum rule is satisfied and thus charge is conserved. ,This diagrammatic method is implemented for both normal and superfluid phases. While there are no collective modes in the normal phase, the Ward-Takahashi identity plays a similarly important role. In particular, for the normal phase we study Rashba spin-orbit coupled Fermi gases with intrinsic pairing in the absence and presence of a magnetic field. Exact density and spin response functions are obtained, even in the absence of a spin conservation law, providing signatures of topological order. Another set of examples studied in the normal phase include a broad collection of models of the cuprate pseudogap. For these cases too, the exact response functions are obtained and signatures of each model are identified. A detailed discussion of the superfluid phase is provided by the study of Fulde-Ferrell superfluids. The generally omitted collective modes are computed and the exact gauge-invariant response is obtained. The first complete calculation of the superfluid density, incorporating the collective mode contribution and without any assumptions on the size of the gap, is provided. It is found that the amplitude mode causes the superfluid to become unstable at temperatures much lower than the mean-field transition temperature.,As an alternative to the diagrammatic approach, the functional path integral technique for Fermi superfluids is presented. While this approach is widely studied in the literature, it is of restricted utility, in comparison to the diagrammatic approach presented here, due to the inability to perform the functional integral except in very limited cases. A short coming in the current literature is identified, in that the standard method of deriving gauge-invariant electrodynamics leads to a violation of the compressibility sum rule. It is found that to satisfy this sum rule the amplitude mode of the order parameter must be incorporated. A reformulation of the path integral method for Fermi superfluids is derived, which has both gauge-invariant electrodynamics and thermodynamics compatible with the compressibility sum rule, to all orders of approximation.
Within a gauge-invariant microscopic kinetic theory, we study the electromagnetic response in the superconducting states. Both superfluid and normal-fluid dynamics are involved. We predict that the normal fluid is present only when the excited superconducting velocity
vs is larger than a threshold ${v}_{L}=|\mathrm{\ensuremath{\Delta}}|/{k}_{F}$. Interestingly, with the normal fluid, we find that there exists friction between the normal-fluid and superfluid currents. Due to this friction, part of the superfluid becomes viscous. Therefore a three-fluid model, normal fluid and nonviscous and viscous superfluids, is proposed. For the stationary magnetic response, at
vs<vL with only the nonviscous superfluid, the Meissner supercurrent is excited and the gap equation can be reduced to the Ginzburg-Landau equation. At ${v}_{s}\ensuremath{\ge}{v}_{L}$, with the normal fluid and nonviscous and viscous superfluids, in addition to the directly excited Meissner supercurrent in the superfluid, a normal-fluid current is also induced through the friction drag with the viscous superfluid current. Due to the normal-fluid and viscous-superfluid currents, the penetration depth is influenced by the scattering effect. In addition, a modified Ginzburg-Landau equation is proposed. We predict an exotic phase in which both the resistivity and superconducting gap are finite. As for the optical response, the excited
vs oscillates with time. When
vs<vL, only the nonviscous superfluid is present, whereas at ${v}_{s}\ensuremath{\ge}{v}_{L}$, normal fluid and nonviscous and viscous superfluids are present. We show that the excited normal-fluid current exhibits the Drude-model behavior, while the superfluid current consists of the Meissner supercurrent and Bogoliubov quasiparticle current. Due to the friction between the superfluid and normal-fluid currents, the optical conductivity is captured by the three-fluid model. Finally, we also study the optical excitation of the Higgs mode. By comparing the contributions from the drive and Anderson-pseudospin pump effects, we find that the drive effect is dominant at finite temperature whereas at zero temperature, both effects contribute.
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.
The Meissner effect is a hallmark of superconductivity arising from the interplay between charged superfluids and electromagnetic fields. However, superfluidity can also occur in systems of charge-neutral particles with magnetic or electric dipole moments, such as Bose-Einstein condensates of magnons or excitons. Despite this interest, the electromagnetic response of dipole superfluids, including potential analogues or contrasts to the Meissner effect, remains poorly understood. Here we develop a Ginzburg-Landau theory to describe dipole superfluids subjected to a pseudo-magnetic field, which is induced by a spatially modulating electromagnetic field. In magnetic dipole superfluids, the response differs strikingly from the conventional Meissner effect: the external field is enhanced by a positive feedback, forming a vortex lattice separated by singular domain walls with diverging electric charges. In contrast, electric dipole superfluids exhibit screening akin to superconductors, where the pseudo-magnetic field is suppressed except near vortices. The electromagnetic response of dipole superfluids, including potential analogues to the Meissner effect, remains poorly understood. This paper develops a phenomenological theory and shows that magnetic-dipole superfluids form vortex lattices separated by singular domain walls with concentrated electric charge, while electric-dipole superfluids exhibit vortex lattices resembling those in conventional superconductors.