## Abstract

In this chapter, we present the details of the derivation of an effective field theory (EFT) for a Fermi gas of neutral dilute atoms and apply it to study the structure of both vortices and solitons in superfluid Fermi gases throughout the BEC-BCS crossover. One of the merits of the effective field theory is that, for both applications, it can provide some form of analytical results. For one-dimensional solitons, the entire structure can be determined analytically, allowing for an easy analysis of soliton properties and dynamics across the BEC-BCS interaction domain. For vortices on the other hand, a variational model has to be proposed. The variational parameter can be determined analytically using the EFT, allowing to also study the vortex structure (variationally) throughout the BEC-BCS crossover.

### Keywords

- fermionic superfluids
- superfluidity
- effective field theory
- solitons
- vortices

## 1. Introduction

When cooling down a dilute cloud of fermionic atoms to ultralow temperatures, particles of different spin type can form Cooper pairs and condense into a superfluid state. The properties and features of these superfluid Fermi gases have been the subject of a considerable amount of theoretical and experimental research [1, 2]. The opportunity to investigate a whole continuum of inter-particle interaction regimes and the possibility to create a population imbalance result in an even richer physics than that of superfluid Bose gases. In this chapter, we present an effective field theory (EFT) suitable for the description of ultracold Fermi gases across the BEC-BCS interaction regime in a wide range of temperatures. The merits of this formalism mainly lie in the fact that it is computationally much less requiring than the Bogoliubov-de Gennes method, and that, in some cases, it can provide exact analytical solutions for the problem at hand. In Section 2, we give a short overview of the path integral theory that forms the basis for the EFT. In Section 3, we study the associated mean field theory for the description of homogeneous superfluids. In Section 4, we go beyond the mean-field approximation and describe the framework of the EFT. Sections 5 and 6 are dedicated to the application of the EFT to two important topological excitations: dark solitons and vortices.

## 2. Path integral theory and bosonification

The effective field theory for fermionic superfluids presented in this chapter is based on the path integral formalism of quantum field theory. The advantage of this formalism lies in the fact that the operators are replaced by fields, which can yield a more intuitive interpretation for the physics of the system. Moreover, the fact that there are no operators make working with functions of the quantum fields a lot easier.

In this section, the path integral description for ultracold Fermi gases will be briefly introduced. Using the Hubbard-Stratonovich identity, the fermionic degrees of freedom can be integrated out, resulting in an effective bosonic action. This effective bosonic action is the object of interest of this chapter and will lie at the basis of the effective field theory. An extended discussion of this section and the mean-field theory of the next section are given in an earlier publication [3]. Comprehensive introductions to the path integral method include [4] (Quantum Field Theory with Path Integrals), [5, 6] (The “classical” Path Integral), and [7] (General review book on the Path Integrals and most of its applications).

### 2.1. A brief introduction to the path integral formalism

The partition function of a system described by the quantum field action functional

Here,

For systems with an Euclidean action which is at most quadratic in the fields, the path integral (1) can be calculated analytically. In particular, two distinct cases can be considered:

**Bosonic path integral:** The path integral sums over a bosonic (scalar, complex valued) field

For the case of a quadratic bosonic path integral, the integration over the complex field

**Fermionic path integral:** The path integral sums over a fermionic (Grassmann, complex valued) field

In the case of spin-dependent fermionic fields, the matrix

Using the trace-log formula, these results can also be rewritten as:

Partition functions with quadratic action functionals form the basis of the path integral formalism. The usual approach for solving path integrals with higher order action functionals is to reduce them to the quadratic forms given above by the means of transformations and/or approximations.

In this chapter, the system of interest is an ultracold Fermi gas in which fermionic particles of opposite pseudo-spin interact via an

where

For the remainder of the chapter, the units

will be used, meaning that we work in the natural units of

where the label

### 2.2. Bosonification: the Hubbard-Stratonovich transformation

Using the Hubbard-Stratonovich identity [11, 12, 13, 14],

it is possible to rewrite the action in a form that is quadratic in the fermionic fields

After applying the Hubbard-Stratonovich identity (11) to expression (10), the partition function becomes

### 2.3. The resulting bosonic path integral

Since the path integral over the fermionic fields

where the components of the inverse Green’s function matrix

Since *slow* fluctuations of the pair field

## 3. The mean field theory

At first sight, the introduction of the auxiliary bosonic fields

where

where

The parameter

After performing the Matsubara summation over

The saddle-point value

This is illustrated in Figure 2, which shows the thermodynamic potential

When working with a fixed number of particles, the chemical potential

Since in our units

## 4. The effective field theory

While the saddle-point approximation is a suitable model for the qualitative description of homogeneous Fermi superfluids, it does not account for the effects of fluctuations of the order parameter, nor does it include any excitations other than the single-particle Bogoliubov excitations. To study the properties and dynamics of non-homogeneous systems, one needs to go beyond the limitations of a mean field theory. In this section, we formulate an effective field theory (EFT) for the pair field

where

where

While, in general, this infinite sum over all powers of the pair field cannot be calculated analytically, there exist many possible approximations that lead to various theoretical treatments of the ultracold Fermi gas. For example, the mean field saddle-point approximation from the previous section can be retrieved by simply setting

in (26) and calculating the whole sum over

with *slow* variations in space and time around a constant bulk value. Since this is a weaker condition than the GL assumption of small variations, it is ultimately expected to lead to a larger applicability domain. The assumption of slow fluctuations is implemented through a gradient expansion of the pair field around its saddle-point value, similar to (28) but with

In every term of this sum, we replace (at most) two occurrences of

The EFT coefficients

where the functions

In general, each of these EFT coefficients depends on the modulus squared of the order parameter

The effective action functional (30) forms the basis of our EFT description of superfluid Fermi gases. The validity and limitations of the formalism are largely determined by the main assumption that the order parameter varies slowly in space and time, which corresponds to the condition that the pair field should vary over a spatial region much larger than the Cooper pair correlation length. A detailed study of the limitations imposed by this condition was carried out in [18]. In the following chapters, we will demonstrate some of the ways in which the EFT can be employed by applying it to the description of two important topological excitations of the superfluid: dark solitons and vortices.

## 5. Application 1: Soliton dynamics

In this section, we will use the EFT that was developed in Section 4 to study the properties of dark solitons in Fermi superfluids.

### 5.1. What is a dark soliton?

Solitons are nonlinear solitary waves that maintain their shape while propagating through a medium at a constant velocity. They are found as the solution of nonlinear wave equations and emerge in a wide variety of physical systems, including optical fibers, classical fluids, and plasmas. More recently, they have also become a subject of interest in superfluid quantum gases [19, 20, 21, 22, 23]. In these systems, solitons appear most often in the form of *dark* solitons, which are characterized by a localized density dip in the uniform background and a jump in the phase profile of the order parameter. The magnitude of this density dip and phase jump are intrinsically connected to the velocity

### 5.2. Solution for a one-dimensional dark soliton

For the case of a dark soliton in a one-dimensional (1D) Fermi superfluid with a uniform background, the EFT provides an exact analytical solution for the pair field [24]. To describe the dynamics of the system, it is necessary to move from the imaginary-time action functional (30) to the real-time one, using the formal replacements.

From the relation between the real-time action functional and the Lagrangian density

we subsequently find the following expression for

where the Hamiltonian density

As mentioned above, a dark soliton in a superfluid is mainly characterized by a jump in the phase profile and a dip in the amplitude profile of the order parameter. Therefore, it is convenient to write the pair field

Moreover, since a soliton is a localized perturbation, we write the modulus as a product of the constant background value

Substituting this form for the pair field in the field Lagrangian (42), we find

with

Here, we added

If we further drop the primes, the Lagrangian density (46) in the soliton frame of reference can be written as

with the modified superfluid density and quantum pressure

From the above expression for

Starting with the equation for the phase field, we easily find:

The integration constant

which yields

If we set

Next, we derive the equation of motion for

Inserting the solution for the derivative of the phase field (59) and defining

we find

While the above equation does not allow for a straightforward solution for

we find that (64) can be integrated, yielding:

Here,

For given values of the interaction parameter

### 5.3. Dark solitons in imbalanced Fermi gases

The dark soliton solution derived in the previous section has been employed in the description of various soliton phenomena in superfluid Fermi gases. For instance, adding a small two-dimensional perturbation to the exact 1D solution allows for a description of the snake instability mechanism [25], which makes the soliton decay into vortices if the radial width of the system is too large [23, 26]. We have also studied collisions between dark solitons by numerically evolving two counter-propagating 1D solitons in time [27]. As an example of an application, we will give a short description of the influence of *spin-imbalance* on dark solitons, a topic that was studied in detail in [18].

In ultracold Fermi gases, the amount of atoms in each spin population can be tuned experimentally, allowing for the possibility of having unequal amounts of spin-up and spin-down particles [28, 29]. In that case, when particles of different spin type pair up and form a superfluid state, an excess of unpaired particles will remain in the normal state, which in turn can have interesting effects on other phenomena in the system, including dark solitons. In the context of the EFT, we control the population imbalance by setting the value of the imbalance chemical potential

As the imbalance between the spin components in the Fermi gas increases, so does the amount of unpaired particles that cannot participate in the superfluid state of pairs. While some of these normal state particles can coexist with the pair condensate as a thermal gas, it is energetically favorable for the remaining excess to be spatially separated from the superfluid. In this context, the soliton dip is a very suitable location to accommodate the excess particles and consequently fills up with an increasing amount of unpaired particles as the imbalance gets higher. Also, the broadening of the soliton with increasing imbalance might be a way of providing the system with more space to store the excess component. The fact that a dark soliton in an imbalanced superfluid Fermi gas has to drag along additional particles changes its effective mass, which in turn influences its general dynamical properties [18]. Moreover, since a soliton plane provides more space to accommodate the excess component than a vortex core, the presence of spin imbalance has been found to stabilize dark solitons with respect to the snake instability [25].

## 6. Application 2: the vortex structure

As a second application, the time-independent version of the theory is considered in order to derive the stable vortex structure. For the description of the vortex, the quantum velocity field

where

The free energy was written in a more compact3 form using the hydrodynamical description (48), (49), (62) and (70). As an application of the effective field theory, the general structure of a superfluid vortex will be numerically determined and compared with the commonly used variational hyperbolic tangent. A more detailed description on vortices in superfluids and their behavior can be found in [30].

### 6.1. What is a vortex?

Both in the classical and the quantum sense, a vortex is defined as a line in the fluid around which there is a circulating flow. In order to quantify this rotation around an axis, the circulation

where

This quantization of the circulation can be derived using the definition of the velocity field (70). Upon substitution, the circulation (72) can be written as:

where the gradient theorem was used together with the fact that the phase field

As the bulk superfluid itself is irrotational, any loop with nonzero circulation must encircle a node in the superfluid order parameter. As a consequence, the superfluid pair density must go to zero along the entire vortex line, resulting in a vortex “core” region with a radius comparable to the healing length. Important to note is that vortices of a single circulation quantum are energetically more favorable than multiply quantized vortices in a homogeneous condensate (which is the type of condensate that will be considered in this chapter) [9]. For the remainder of this application, only singly quantized vortices will thus be studied.

### 6.2. About the structure of a quantum vortex

The most natural coordinate system to describe vortices are the polar coordinates

meaning that the superfluid density relaxes to the bulk value away from the vortex.

We factorize the amplitude function in a radial and an angular part6:

Since the structure is periodic, the general solution for

leading to a basis of angular modes for the vortex structure. In order to find the lowest energy state, one usually restricts the problem to one of the many possible modes:

which results in the velocity field and circulation (using (70) and (72)) for a single mode given by:

where the velocity field diverges in the point where the superfluid vanishes. It was noted before that for our case, the most energetic vortex states are those with the least circulation quanta. Since the object of interest is the vortex structure with a minimal free energy, the value of

where the “+” sign is for vortices and the “-” sign for anti-vortices.

Currently, there is no analytical solution available for the full vortex structure

### 6.3. A variational model for the vortex core

In order to speed up the vortex calculations, a variational model can be used to describe the vortex structure. First of all, the variational model should meet the required boundary conditions (74). Second, the variational model should contain the necessary information to describe the vortex physics. For example, in liquid helium, the vortex core sizes are of the order of nanometers [33], meaning that the vortex core structure will not play a prominent role in the vortex physics; in this case, a simple hollow cylinder is already a good variational model for the vortex core. For vortices in ultracold gases on the other hand, the vortex core size is of the order of micrometers [34], meaning that its structure becomes important8; a simple cylindric hole will no longer capture the entire vortex physics. In order to provide a more detailed description, different variational models are available [9, 30].

The variational model that will be discussed here is the hyperbolic tangent model:

where the quantity

where the value of the constant

The second term in the integrand of (81) causes a divergence, since

diverges logarithmically with increasing radius of the integration domain. The physical reason is clear: the velocity profile of a vortex decays as

The last term no longer contains a dependency on

The remaining derivative now acts on the boundary of the integration domain. Applying

to (85) we get

so that

With

we find a closed form result

The formula for the healing length (90) can also be plotted, this is done in Figure 6. In both limits, the healing length shows to be in a good agreement with the exact limits.

### 6.4. Comparison to the exact (numerical) solution

In order to check the validity of the variational model (80) (and thus the results it produces) is, the variational structure should be compared with the exact vortex structure. This exact vortex structure is easily obtained by a direct minimization of the free energy functional (71). As mentioned before, the direct minimization of the free energy is more suitable for the calculation of the vortex structures; the reason why this method is preferential lies in the fact that the velocity field diverges for

The numerical method that was used in order to determine the exact vortex structure for a given set of parameters

Once the exact structure is obtained, it can be analyzed and compared to the variational vortex structure. As an example, we can look at the relative difference in the free energy throughout the BEC-BCS crossover for different temperatures and polarizations. From the plots shown in Figure 7, it can be seen that the difference in free energy is around the order of 1%; this seems to suggest that the variational guess is a good guess.9 Moreover, the results in Figure 7 allow to provide an error bar on energy calculations using the variational structure. This error bar on the energy is useful for example when making phase diagrams including vortex structures. In order to be sure whether the variational model is indeed a good description of the vortex hole, other parameters were also tested and discussed in [38]. The conclusion from the numerical analysis was that the variational model is indeed a good fit for describing the vortex structure.

## 7. Concluding section

In this chapter, an effective field theory for the description of dilute fermionic superfluids was derived. The main advantages of an effective field theory are the gain in computational speed and the fact that it allows analytic solutions for dark solitons and the variational healing length of the vortex structure. Both the gain in computational speed and the availability of an analytic starting point contribute to the possibility to study several soliton/vortex phenomena throughout the entire BEC-BCS crossover at finite temperatures

On the subject of soliton dynamics, we specifically looked at 1D dark solitons, for which an exact analytical solution was derived. Using this solution, the effect of spin-imbalance on the soliton properties was studied, revealing that the unpaired particles of the excess component mainly reside inside the soliton core. Additionally, the EFT has also been employed in the study of the snake instability of dark solitons [25] and the dynamics of dark soliton collisions [27] in imbalanced superfluid Fermi gases.

For vortices, the structure of a vortex was studied, for which unfortunately no analytical solution is available at the moment. Using a variational model, an analytical solution for the vortex healing length was derived. The variational model was compared with the exact solution. From this analysis, the variational model was found to be a good fit for the exact vortex structure. Other EFT research on vortices includes the behavior of vortices in multiband systems [39] and the study of the “vortex charge” [40].

## Acknowledgments

The authors gratefully acknowledge the financial support provided by the Fund for Scientific Research Flanders (FWO), through the FWO project: G042915 N (Superfluidity and superconductivity in multi-component quantum condensates). One of us (N.V.) acknowledges a Ph.D. fellowship of the University of Antwerp (2014BAPDOCPROEX167). One of us (W.V.A.) acknowledges a Ph.D. fellowship from the FWO (1123317 N). We also acknowledge financial support from the Research Fund (BOF-GOA) of the University of Antwerp.

## Notes

- The matrix A can be thought of as an infinite matrix composed of either 2×2 or 4×4 matrices, depending on whether the spin-dependence of the fermionic field is considered in the theory.
- Of course, it is always possible, given sufficient computational resources and time, to calculate the partition sum numerically.
- Where again the free energy at infinity was subtracted to obtain a well behaved free energy.
- In the case of a superconductor, the quantized value is given by the magnetic flux.
- Note that the condition at r→∞ could be replaced by ∂rar→∞=0. This could however lead to numerical difficulties in the center of the vortex.
- This product decomposition is not generally valid in all coordinate systems [31].
- Note that this velocity field is the same as the elementary vortex flow known in classical hydrodynamics [32].
- The condensate size to vortex core size is typically in the range 10–50.
- In Figure 7b, the energy difference seems to blow up towards the BCS limit. This divergence is due to the fact that at that point superfluidity is lost due to polarization (Clogston limit); at this point superfluidity disappears.