Correcting inconsistencies in the conventional superfluid path integral scheme
In this paper we show how to redress a shortcoming of the path integral scheme for fermionic superfluids and superconductors. This approach is built around a simultaneous calculation of electrodynamics and thermodynamics. An important sum rule, the compressibility sum rule, fails to be satisfied in the usual calculation of the electromagnetic and thermodynamic response at the Gaussian fluctuation level. Here we present a path integral scheme to address this inconsistency. Specifically, at the leading order we argue that the superconducting gap should be calculated using a different saddle point condition modified by the presence of an external vector potential. This leads to the well known gauge-invariant BCS electrodynamic response and is associated with the usual (mean field) expression for thermodynamics. In this way the compressibility sum rule is satisfied at the BCS level. Moreover, this scheme can be readily extended to address arbitrary higher order fluctuation theories. At any level this approach will lead to a gauge invariant and compressibility sum rule consistent treatment of electrodynamics and thermodynamics.
There is a great interest from diverse physics communities in understanding superfluids Giorgini et al. (2008); Drummond et al. (2009); Hu (2012) and superconductors Hur and Maurice Rice (2009); Chen et al. (2005) with stronger than BCS correlations. These strong correlations are present in both high temperature superconductors and in ultra coldourcompanion Fermi superfluids. At the heart of probes of superfluidity are electrodynamic and thermodynamic responses. It is, therefore, important to have a consistent theory for addressing both of these. One consistency requirement is that of gauge invariance. This affects only the electrodynamics, and importantly introduces collective modes of the order parameter. Another consistency requirement involves the inter-connection between electrodynamics and thermodynamics. This is encapsulated in the compressibility sum rule Pines and Nozieres (1966).
The path integral scheme is particularly well suited to consistency checks related to this inter-connection because it simultaneously derives electrodynamics and thermodynamics. However, this scheme, as it is applied in the literature, is not consistent with the compressibility sum rule Boyack et al. (2016). Indeed, this inconsistency shows up at the lowest level of approximation needed to arrive at gauge invariant electrodynamics. Stated more concretely, the path integral approach raises a natural question: even at the strict BCS level, if fluctuations around the mean-field solution are necessary for gauge invariant electrodynamics, should these fluctuations yield additional contributions to thermodynamics beyond those of the fermionic quasiparticles? Such “gauge restoring” terms should have definite observable consequences. For example, in neutral superfluids (such as superfluid He-3 and atomic Fermi gases) these phonon modes would lead to power law contributions in measurable properties such as the specific heat. There seems to be no consensus about whether these non-BCS terms should or should be not be considered Yu et al. (2009).
Here we appeal to the compressibility sum rule to address this question. We define as the thermodynamic potential resulting from a calculation that uses Gaussian fluctuations () around mean field theory () to establish a BCS-level gauge invariant electrodynamic response. We consider particles having chemical potential . Within this formulation, which we call the gauge restoring Gaussian fluctuation (GRGF) theory, the number of particles has a leading order mean-field term and a fluctuation contribution . Similarly the electrodynamic kernel which derives from contains the counterpart mean-field and fluctuation terms, both of which combined lead to a proper gauge invariant BCS density-density correlation function . One can show that satisfies
This demonstrates an explicit violation Boyack et al. (2016) of the compressibility sum rule, which should read . It also demonstrates (at least at an empirically suggestive level) what assumptions need to be made to satisfy the compressibility sum rule within BCS theory.
In this paper we present a path integral framework modified from that outlined above. For both the lowest order mean-field, and Gaussian fluctuation levels, we will derive theories fully consistent with gauge invariance and the compressibility sum rule. Indeed, this consistency can in principle be achieved at all orders of approximation within our path integral re-formulation.
The GRGF approach leading to Eq. (1) was presented in a fairly extensive literature Altland and Simons (2006); Goryo and Ishikawa (1999); Lutchyn et al. (2008); Ojanen and Kitagawa (2013); Roy and Kallin (2008); Stone and Roy (2004); Goryo (1998), where fluctuations of the mean-field phase were used to restore gauge invariance. These fluctuations enter as a “dressed” vector potential , which is then expanded to quadratic order. Integration of the fluctuations resulted in the standard electromagnetic response kernel of strict BCS theory. We emphasize here Altland and Simons (2006); Goryo and Ishikawa (1999); Lutchyn et al. (2008); Ojanen and Kitagawa (2013); Roy and Kallin (2008); Stone and Roy (2004); Goryo (1998) that the focus was on electrodynamics while the thermodynamic implications were of no concern.
In contrast, understanding thermodynamics associated with Gaussian fluctuation theories (beyond the BCS level) was the focus of work by a different community, that studying ultracold Fermi superfluids Diener et al. (2008); He et al. (2015); Loktev et al. (2001); Ohashi and Griffin (2003); Perali et al. (2004); Pieri and Strinati (2000, 2005). In these neutral superfluids, soft bosonic collective modes arising from fluctuations were shown to provide new thermodynamic contributions in addition to those of the fermionic quasi-particles of BCS theory.
Yet another series of studies incorporated these Gaussian-level (beyond BCS) fluctuations to revisit electrodynamics in a higher level theory. By introducing a small phase twist in the thermodynamic potential, it was argued that one could determine the superfluid density Taylor et al. (2006); Fukushima et al. (2007); Taylor and Randeria (2010); moreover, this now contained bosonic contributions, not present in BCS theory. These were somewhat similar (but not equivalent) to contributions found Perali et al. (2004); Pieri and Strinati (2000) within a very different diagrammatic formalism.
All this previous literature relating to Gaussian fluctuations can be summarized by noting that there have been separate path integral studies of superfluid electrodynamics and of thermodynamics. What is missing is an analysis of the constraints which relate the two. In this paper we address this shortcoming.
Path integral and mean field.–
Here we consider a fermionic partition function for a neutral, attractive, Fermi gas with -wave pairing. The techniques presented here can be readily extended to higher order pairing, and Coulomb interactions can be included at the RPA level Lutchyn et al. (2008). The partition function is calculated using the Hubbard-Stratonovich (HS) path integral
where the HS action takes the usual form Altland and Simons (2006); Fradkin (1991), is an interaction constant, and includes a trace over both position and Nambu indices; throughout we set . The inverse Nambu Green’s function is constructed from a single particle Green’s function and a self-energy , with a vector of Nambu Pauli matrices. Throughout we use the notation to represent two real HS fields , with , consistent with previous literature 111This notation is equivalent to that in Refs. Kulik et al. (1981); Kosztin et al. (2000); Guo et al. (2013); Boyack et al. (2016). The reality condition is expressed in position space; a momentum-space parametrization has a different condition Guo et al. (2013). The conventional BCS self energy suggests an equivalent complex parametrization where the BCS gap is identified through . See the Supplemental Material sup () for details.. The single particle Green’s function is kept general, but we note that an electromagnetic vector potential has been explicitly included at this level.
We now calculate at the mean-field level using the saddle point approximation in the presence of . This is to be contrasted with previous work (belonging to the GRGF scheme) Altland and Simons (2006); Goryo and Ishikawa (1999); Lutchyn et al. (2008); Ojanen and Kitagawa (2013); Roy and Kallin (2008); Stone and Roy (2004); Goryo (1998) where the saddle point condition assumed . Here, explicit calculation produces the standard BCS gap equation, , in the presence of a non-zero vector potential . We define the solution to this gap equation as , which depends on . We note that a different community has exploited the advantages of considering alternative saddle point schemes Kamenev and Andreev (1999).
At the present mean-field (saddle point) level, we can write , where the mean-field action is the HS action evaluated at the solution to the saddle point equations. In general we cannot explicitly calculate the solution to the gap equation for . Instead, we will first use the self-consistent gap equation to find the variation of with respect to a variation in . We then take the limit, after which all quantities are calculated using . Thus, no additional computational difficulties arise when using this self-consistency condition compared to the GRGF formalism.
Response functions at saddle point level.–
Given an arbitrary “effective action” in the presence of a weak perturbation , the response kernel comes from the second functional derivative of the action in the limit Fradkin (1991). As such, we can expand to second order in the vector potential , where
is the response kernel for an arbitrary action .
We now calculate the mean-field response using the definition in Eq. (3) by including a nonzero vector potential in the saddle point condition, i.e., replace by . When taking a functional derivative with respect to , new terms arise from a “functional chain rule” Altland and Simons (2006) applied to the self-consistent gap . These terms, which do not not emerge for a gap calculated around as in GRGF, are crucial for maintaining gauge invariance. The full response kernel then takes the form:
where the limit is applied after taking all derivatives. In this equation we have introduced the notation and ; repeated subscript (superscript) indices () should be interpreted as an implied Einstein summation (integration.)
where parameterizes both gap and vector potential response. The kernel is the standard (non-gauge invariant) response as calculated with a gap ; the functions and come from “partial” derivatives in the functional chain rule. We note that the propagator is equivalent to a “” -matrix theory for a BCS self-energy, and therefore can be interpreted as an emergent bosonic propagator Chen et al. (2005); Diener et al. (2008). Using these definitions, the mean-field level gauge invariant response is compactly written
where we henceforth include an implicit integration over for every Einstein summation over . In Eq. (6) we have introduced the collective mode terms ; these explicitly restore gauge invariance beyond the “bubble” response kernel Kulik et al. (1981); Kosztin et al. (2000); Guo et al. (2013); Boyack et al. (2016). In the saddle point response, the third line in Eq. (4) vanishes.
Using the revised saddle point condition, along with the above definitions, the collective modes are where the inverse is taken over both position and Nambu indices (see Supplemental Material sup ()). We emphasize that these collective modes are associated with the mean-field level of approximation. Finally, after taking the limit, the momentum space response is
This is the usual gauge invariant response kernel in BCS theory Kulik et al. (1981) which includes both amplitude and phase collective modes.
Importantly, the response kernel , which is explicitly gauge invariant, was obtained without including Gaussian fluctuations, which are usually invoked in the GRGF literature. In this way the self-consistent treatment of the gap in the presence of a vector potential restores gauge invariance at the mean-field level. Because there are no accompanying bosonic degrees of freedom in the thermodynamics, the compressibility sum rule will be shown to be exactly satisfied using this method, in contrast to the more conventional path integral methodology.
Beyond saddle point.–
Often it is desirable to calculate the path integral beyond the saddle point approximation. In order to do this, one changes variables from the HS field to a fluctuation around the saddle point solution defined through . We note that since is a dynamical variable it does not have any dependence on . The full action is then expressed exactly as , where the action is or higher, since any term linear in vanishes by the saddle point condition. This definition allows for the exact factorization of the partition function , where
is the contribution due to fluctuations beyond mean field.
In calculations of response beyond saddle point, one uses Eq. (3) with an effective action , and the fluctuation action also depends on the self-consistent gap . The response kernel is linear in the action, so that , where the mean-field response is given in Eq. (7). The new contribution to the response, , has a form identical to Eq. (4), only with replaced by . Note, however, that the collective mode terms still arise from the mean field self-consistent gap condition; these collective modes are always constructed from the propagators, and not from an analogous .
This higher order fluctuation response again contains a “bubble” term that arises from bosonic fluctuations. On its own, is not gauge invariant. Analogous to the saddle-point response, the collective modes , along with the corresponding response functions, are necessary to restore gauge invariance. To show that this arbitrary fluctuation theory is fully gauge invariant, one can verify that is satisfied (see the Supplemental Material sup ().) In this way, gauge invariance holds term by term in the expansion of the action beyond mean-field. This calculation scheme for gauge invariant response beyond-BCS is a completely general sum rule consistent scheme and a central result of this manuscript.
Compressibility sum rule.–
Thermodynamic quantities can be calculated from derivatives of the thermodynamic potential, , which is the effective action up to the prefactor . Since electromagnetic response functions also come from derivatives of the effective action, it is clear that there should be an intimate connection between the two. An important requirement for consistency between electrodynamics and thermodynamics is contained in the compressibility sum rule: .
A formal derivation of this sum rule, for the exact action, arises from twice invoking the identity on the partition function in Eq. (2). A more intuitive derivation of this sum rule follows from the fermionic path integral, before applying the HS transformation. The atom number is , where is the local fermion density operator. A second derivative gives . On the other hand, the small momentum limit of the density-density correlation function is , where follows from Eq. (3). It is straightforward to see this response function is just as defined above. Therefore, the compressibility sum rule is an exact consequence of a path integral approach provided no approximations are made.
When considering only thermodynamics, it is not necessary to keep track of the vector potential in the self-consistent solution, and can be calculated for and . However, when simultaneously considering electrodynamics and thermodynamics it is important to calculate to the same level of approximation for both quantities. Due to the linear dependence of both electrodynamic and thermodynamic quantities on the effective action, any theory studying both quantities, which considers a consistent approximation scheme, will also satisfy the compressibility sum rule.
An exact calculation of is in general difficult and is frequently treated at the Gaussian level in the literature. We similarly consider response at this level: fluctuations about the saddle point solution are assumed small and the fluctuation action is expanded to quadratic order: . The path integral can then be solved exactly; integration of the fluctuation field gives a effective action at the Gaussian level. We emphasize that in the calculation of the fluctuation response kernel, , the propagator includes dependence on both explicitly, and through the mean-field solution. This is in contrast to previous literature which used the fluctuation propagator in Eq. (5).
It is clear that setting will reproduce beyond-BCS thermodynamics found in the literature Diener et al. (2008); He et al. (2015); Loktev et al. (2001); Ohashi and Griffin (2003); Perali et al. (2004); Pieri and Strinati (2000, 2005); Taylor and Randeria (2010). Similarly, a calculation of will reproduce the bosonic contribution to the superfluid density found in Refs. Taylor et al. (2006); Fukushima et al. (2007); Taylor and Randeria (2010). Therefore, our results reproduce and extend previous explorations of Gaussian fluctuations, now establishing consistency with the compressibility sum rule.
Amplitude and Phase fluctuations.–
While not explicitly discussed, amplitude fluctuations of the gap were implicitly included in the compressibility sum rule arguments presented in this paper. These are often ignored, although they have been introduced in the literature via an alternative parameterization of the gap, by writing , where and are respectively the amplitude and phase of the order parameter. Including amplitude fluctuations by setting and integrating out both and fluctuations results in a different gauge invariant formulation but one which is equivalent to the fluctuation used above. It should be noted that while amplitude fluctuations result in a contribution to electrodynamic (and thermodynamic) response, phase fluctuations alone are sufficient to restore gauge invariance at both the mean-field and fluctuation levels. We note, however, that by neglecting amplitude fluctuations, the compressibility sum rule will be violated and this violation is apparent even at the mean field level of strict BCS theory.
In this paper we have presented a path integral formulation for superfluids and superconductors which: (1) allows for a consistent calculation of (gauge invariant) electrodynamic and thermodynamic response at any desired level of approximation, and (2) gives the full gauge invariant response kernel for beyond mean-field physics. The consistency of our formulation is apparent in the compressibility sum rule which related electrodynamics and thermodynamics. This sum rule is not satisfied at the BCS level in the path integral formalism if Gaussian fluctuations are invoked as in GRGF; instead a consistent treatment involves finding the saddle point solution in the presence of a vector potential. Our way of introducing collective mode effects is closer in spirit to earlier work Rickayzen (1965) on BCS theory using the Kubo formalism.
We stress an important physical implication of the current scheme. Within the conventional path integral approach, Gaussian fluctuations are needed to arrive at gauge invariant electrodynamics. One might posit that there ought to be fluctuation contributions to thermodynamics. Specifically, in a neutral superfluid these collective modes would seem to require power law contributions, say in the specific heat. We argue here, despite some controversy in the literature Yu et al. (2009), including these correction terms in strict BCS theory is unphysical, as they are inconsistent with the compressibility sum rule.
Within the present formalism, the next level approximation, involving Gaussian fluctuations then emerges as a true beyond-BCS theory in which there are inter-related (by the compressibility sum rule) contributions to both thermodynamics and the electromagnetic response. This beyond-BCS level of approximation provides a starting point for studying strongly correlated superfluids. It should be viewed as an alternative to schemes which build on a correlation self energy and the Ward-Takahashi identity Boyack et al. (2016).
This approach provides a promising new route to bench marking beyond-BCS calculations derived from path integral approaches. There are indications from the superfluid density at the Gaussian level that possibly unphysical non-monotonicities appear Taylor and Randeria (2010). These may also be present when comparing with density correlation functions which are measured in Bragg scattering experiments. Nevertheless it will be interesting to look at these higher level (Gaussian) corrections in a variety of physical contexts, including, for example, their role in topological Goryo (1998); Goryo and Ishikawa (1999); Stone and Roy (2004); Roy and Kallin (2008); Lutchyn et al. (2008); Ojanen and Kitagawa (2013) or disordered superfluids Kamenev and Andreev (1999). Quite generally, this work should be viewed as providing a new paradigm for exploring beyond-BCS physics using path integral techniques.
We are grateful for illuminating discussions with A. Altland and A. Kamenev. This work was supported by NSF-DMR-MRSEC 1420709.
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