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Establishing a consistent theory of transport in strongly correlated Fermi superfluids

Rufus Boyack

Published Aug 1, 2017• 1 citation
Publisher

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.

DOI
10.6082/m17s7kw8

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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 fff-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.
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 fff-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{T}_{c}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.
Recent experimental progresses allow for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger-than-BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC) crossover, especially below the superfluid transition temperature TcT_cTc​. We adapt a particular ttt-matrix approach which is close to the G0GG_0GG0​G formalism to incorporate non-condensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities, QQQ-limit Ward identity are indeed satisfied.
Recent experimental progress allows for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity, and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger than BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC) crossover, especially below the superfluid transition temperature Tc . We adapt a particular t -matrix approach which is close to the G0G formalism to incorporate noncondensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities and Q -limit Ward identity are indeed satisfied.
In this work we consider the superfluid stiffness of a generically non-Galilean invariant interacting system and investigate under what conditions the stiffness may nonetheless approach the Galilean-invariant value n/mn/mn/m. Within Eliashberg theory we find that the renormalized stiffness is approximately given by n/mn/mn/m in the case when the l=0l=0l=0 and l=1l=1l=1 components of the effective Fermi-surface projected interaction are approximately equal over a range of frequencies. This holds, in particular, when the interaction is peaked at zero momentum transfer. We examine this result through three complementary lenses: the δ(ω)δ(ω)δ(ω) term in the conductivity, the phase dependence of the Luttinger-Ward free energy, and the coupling of the amplitude and phase sectors in the Hubbard-Stratonovich collective mode action. From these considerations we show that the value of the stiffness is determined by the strength of renormalization of the current vertex and that the latter can be interpreted as the shift of the self-consistent solution due to flow of the condensate, or alternatively as coupling of the phase mode to l=1l=1l=1 fluctuations of the order parameter. We highlight that even though the superfluid stiffness in some non-Galilean systems approaches the Galilean value, this is not enforced by symmetry, and in general the stiffness may be strongly suppressed from its BCS value. As a corollary we obtain the generic form of the phase action within Eliashberg theory and charge and spin Ward identities for a superconductor with frequency dependent gap function.
In this paper, we examine in a unified fashion dissipative transport in strongly correlated systems. We thereby demonstrate the connection between "bad metals" (such as the high temperature superconductors) and "perfect fluids" (such as the ultracold Fermi gases, near unitarity). One motivation of this work is to communicate to the high energy physics community some of the central unsolved problems in high TcT_cTc​ superconductors. Because of interest in the nearly perfect fluidity of the cold gases and because of new tools such as the AdS/CFT correspondence, this better communication may lead to important progress in a variety of different fields. A second motivation is to draw attention to the great power of transport measurements which more directly reflect the excitation spectrum than, say, thermodynamics and thus strongly constrain microscopic theories of correlated fermionic superfluids. Our calculations show that bad metal and perfect fluid behavior is associated with the presence of a normal state excitation gap which suppresses the effective number of carriers leading to anomalously low conductivity and viscosity above the transition temperature TcT_cTc​. Below TcT_cTc​ we demonstrate that the condensate collective modes ("phonons") do not couple to transverse probes such as the shear viscosity. As a result, our calculated shear viscosity at low TTT becomes arbitrarily small as observed in experiments. In both homogeneous and trap calculations we do not find the upturn in ηηη or η/sη/sη/s (where sss is the entropy density) found in most theories. In the process of these studies we demonstrate compatibility with the transverse sum rule and find reasonable agreement with both viscosity and cuprate conductivity experiments.
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.