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Abstract

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.

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