Open access peer-reviewed chapter

An Effective Field Description for Fermionic Superfluids

By Wout Van Alphen, Nick Verhelst, Giovanni Lombardi, Serghei Klimin and Jacques Tempere

Submitted: October 3rd 2017Reviewed: December 12th 2017Published: May 30th 2018

DOI: 10.5772/intechopen.73058

Downloaded: 239

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 Sϕxtϕ¯xtcan be expressed as a path integral [7]:

Z=Dϕ¯x,τDϕx,τexpSEϕx,τϕ¯x,τ.E1

Here, Dϕx,τrepresents a sum over all possible space-time configurations of the field ϕxτ, and τ=itindicates imaginary times running from τ=0to τ=ħβwith β=1/kBT. The Euclidian action SEβof the system is found from the real-time action functional Stbtathrough the substitution

tStbtaiSEβ.E2

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 Ψxτ:

ZB=DΨ¯DΨexpdτdxdτdx'Ψ¯xτA(xτx'τ)Ψ(x'τ)=1detA,E3

For the case of a quadratic bosonic path integral, the integration over the complex field Ψreduces to a convolution of Gaussian integrals, which reduces to the inverse of the determinant of the matrix Acontaining the coefficients of the quadratic form.

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

ZF=Dψ¯Dψexpdτdxdτdx'ψ¯xτA(xτx'τ)ψ(x'τ)=detA,E4

In the case of spin-dependent fermionic fields, the matrix Abecomes slightly more complex since the spinor fields have multiple components1 to account for the spin degree of freedom. The spinors ψare described by anti-commuting Grassmann numbers [4, 8], thus satisfying ψ2=0. For the quadratic case, the fermionic path integral simply returns the determinant of the matrix A.

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

ZB=expTrlnA,E5
ZF=exp+TrlnA.E6

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 s-wave contact potential. The Euclidian action functional for this system is given by

SE=0βdτdxσψ¯σxτ22mxx2μσψσx+0βdτdxdyψ¯xψ¯yxyψyψx,E7

where σdenotes the spin components of the fermionic spinor fields, the chemical potentials μσfix the amount of particles of each spin population, and gis the renormalized interaction strength [9, 10], linking the interaction potential to the s-wave scattering length as:

1g=m4π2asdk2π3m2k2.E8

For the remainder of the chapter, the units

=kB=kF=2m=1E9

will be used, meaning that we work in the natural units of kF, EF, ωF=EF/, and TF=EF/kB. Consequentially, the partition function of the ultracold Fermi gas can be written down as

Z=Dψ¯σDψσexp0βdτdxσψ¯σxττx2μσψσ(xτ)+gψ¯xτψ¯(xτ)ψ(xτ)ψ(xτ),E10

where the label σwas explicitly added to the integration measure to show that the path integration is performed also over both spin components of the spinor ψ. As noted above, only quadratic path integrals can be solved analytically, meaning that an additional trick is needed2 to calculate the above partition sum (10). In the present treatment, this trick will be the Hubbard-Stratonovich transformation.

2.2. Bosonification: the Hubbard-Stratonovich transformation

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

expgd3xgψ¯ψ¯ψψ=DΨ¯DΨexpd3xΨ2g+ψ¯ψ¯Ψ+ψψΨ¯,E11

it is possible to rewrite the action in a form that is quadratic in the fermionic fields ψand ψ¯, allowing for the fermionic degrees of freedom to be integrated out. The price of this transformation is the introduction of a new (auxiliary) bosonic field Ψrτ, which can be interpreted as the field of the Cooper pairs that will form the superfluid state. Diagramatically, the Hubbard-Stratonovich identity removes the four-point vertex (quartic interaction term) and replaces it with two three-point vertices (quadratic terms), as illustrated in Figure 1. It is important to note that, although the Hubbard-Stratonovich transformation is an exact identity, further calculations will require approximations for which the choice of collective field (or “channel”) becomes important. Whereas the bosonic pair field is suitable for the superfluid state, it will fail when one tries to use it to take into account interactions in the normal state. It should therefore be pointed out that alternatives exist, notably Kleinert’s variational perturbation theory, in which a classical collective field rather than a quantum collective field is used. This allows for the simultaneous treatment of multiple collective fields [15], for example, the pair field and the density field. For our present purposes, however, it is sufficient to restrict ourselves to the superfluid state and describe it with a single collective field.

Figure 1.

A diagrammatic representation of the different terms in the Hubbard-Stratonovich identity (11).

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

Z=Dψ¯σDψσDΨ¯DΨexp0βdτdxσψ¯σxττx2μσψσ(xτ)Ψxτ2gψ¯xτψ¯(xτ)Ψ(xτ)ψ(xτ)ψ(xτ)Ψ¯(xτ)E12

2.3. The resulting bosonic path integral

Since the path integral over the fermionic fields ψand ψ¯is now quadratic, it can be performed analytically using formula (4), resulting in the effective bosonic path integral [3]

Z=DΨ¯DΨexp0βdxΨxτ2gTrlnG1,E13

where the components of the inverse Green’s function matrix G1are given by

G1xτ=τx2μΨxτΨ¯xττ+x2+μE14

Since G1depends on the bosonic field Ψxτ, the action in the exponent is not quadratic, and hence, the remaining bosonic path integral can still not be solved analytically. In order to obtain a workable solution, two different approximations will be considered. First, a mean field approximation (using a constant value for Ψ) will be discussed in Section 3. Subsequently, this mean field theory will form the basis for a finite temperature effective field theory, which also takes into account slow fluctuations of the pair field Ψxτ. This theory will be presented in Section 4.

3. The mean field theory

At first sight, the introduction of the auxiliary bosonic fields Ψxτand Ψ¯xτthrough the Hubbard-Stratonovich transformation seems to have been of little use; while the transformation enables us to perform the path integrals over the fermionic fields, we end up with path integrals for Ψxτand Ψ¯xτthat still cannot be calculated exactly. The advantage of switching to the bosonic pair fields, however, lies in the fact that they allow us to make a physically plausible approximation based on our knowledge of the system. If we want to investigate the superfluid state, we can assume that the most important contribution to the path integral will come from the configuration in which all the bosonic pairs are condensed into the lowest energy state of the system and form a homogeneous superfluid. This assumption is most easily expressed in momentum-frequency representation qm:

ΨqmβVδqδm,0×Δ,E15
Ψ¯qmβVδqδm,0×Δ,E16

where mcharacterizes the bosonic Matsubara frequencies ω˜m=2/β, and Vrepresents the volume of the system. This approximation, which is called the saddle-point approximation for the bosonic path integral, comes down to assuming that the pair field Ψxτtakes on a constant value Δ. By applying this approximation to the bosonic path integral in expression (12) (i.e., after performing the Hubbard-Stratonovich transformation but before performing the Grassmann integration over the fermionic fields), the resulting fermionic path integral can be solved analytically using formula (4) to find the saddle-point expression for the partition function:

Zsp=expΔ2gk,nlniωnEk+ζiωnEkζ.E17

where ωnare the fermionic Matsubara frequencies ωn=2n+1π/β. We have also introduced the single-particle excitation energy Ek=ξk2+Δ2with ξk=k2μ, and we have defined the average chemical potential μand the imbalance chemical potential ζas

μ=μ+μ2andζ=μμ2.E18

The parameter ζdetermines the population imbalance between the two spin populations. For ζ=0, the numbers of particles of each spin type are equal, while for non-zero values of ζ, there will be more spin-up than spin-down particles or vice versa. The saddle-point partition function can now be rewritten in terms of the saddle-point thermodynamic potential per unit volume Ωspas

Zsp=expβVΩsp.E19

After performing the Matsubara summation over n[3] and replacing the sum over kby a continuous integral in expression (17), we finally find for Ωsp:

Ωsp=Δ28πkFasdk2π31β2coshβEk+2coshβζξkΔ22k2E20

The saddle-point value Δspfor the pair field is found through the requirement that Δspminimizes Ωsp, which yields the gap equation:

Ωsp∂ΔT,μ,ζ=0E21

This is illustrated in Figure 2, which shows the thermodynamic potential Ωspas a function of Δfor several values of the imbalance chemical potential ζ. The superfluid state exists when Ωspreaches its minimum at a nonzero value of Δ. As ζis increased, the normal state at Δ=0develops and becomes the global minimum above a critical imbalance level. This transition from the superfluid to the normal state under influence of increasing population imbalance is known as the Clogston phase transition [16].

Figure 2.

The thermodynamic potential Ωsp in function of Δ for several values of the imbalance chemical potential ζ, at temperature T/TF=0.01 and chemical potential μ=1.3EF. The evolution of the normal state at Δ=0 as ζ increases illustrates the Clogston phase transition.

When working with a fixed number of particles, the chemical potential μand the imbalance chemical potential ζhave to be related to the fermion density nspand density difference δnsp(between the two spin populations) through the number equations

nsp=ΩspμT,ζ,ΔE22
δnsp=ΩspζT,μ,ΔE23

Since in our units kF=1, the particle density nspis fixed by nsp=1/3π2. Given the input parameters β, ζ, and as, the values Δand μcan then be found from the coupled set of Eqs. (21) and (22), while (23) fixes δnspas a function of ζ. Solutions for Δspand μacross the BEC-BCS crossover are shown in Figure 3a and b.

Figure 3.

Solutions for the pair field Δ and the average chemical potential μ in function of the interaction strength kFas−1 at temperature T/TF=0.01. The solution for Δ is shown for several values of the imbalance chemical potential ζ, illustrating the transition from the superfluid to the normal state under influence of population imbalance.

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 Ψrtthat can describe nonhomogeneous Fermi superfluids in the BEC-BCS crossover at finite temperatures. To this end, we return to the path integral expression (13) for the partition function, which was obtained after performing the Hubbard-Stratonovich transformation and integrating out the fermionic degrees of freedom. Since the exponent of this partition function only depends on the fields Ψrtand Ψ¯rt, we can define an effective bosonic action for the pair field given by

Seff=SBTrlnG1,E24

where SB=0βdxΨxτ28is the action for free bosonic fields. The inverse Green’s function matrix G1for interacting fermions, which was defined in expression (14), can be separated into its diagonal and off-diagonal components

G1(x,τ)=G01(x,τ)+F(x,τ)=(τx2μ00τ+x2+μ)+(0Ψ(x,τ)Ψ¯(x,τ)0),E25

where G01describes free fermionic fields, while Fdescribes the pairing of the fermions. Using this decomposition, we can write the effective bosonic action functional (24) as

Seff=SBTr[ln(G01+F)]=SBTr[ln(G01)]Tr[ln(1G0F)]=SB+S0+p=11pTr[(G0F)p].E26

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

FxτFsp=0ΔΔ¯0E27

in (26) and calculating the whole sum over p. In the Ginzburg-Landau (GL) treatment for ultracold Fermi gases, the action is approximated by assuming small fluctuations of the pair field Ψxτaround the normal state Ψ=0. This assumption comes down to keeping only terms up to p=2in the sum in (26) and approximating Fxτby the following gradient expansion

FxτF0+xx0xFx0+12i,j=x,y,zxix0,ixjx0,j2Fxixjx0+ττ0Fττ0+12ττ022Fτ2τ0,E28

with F00. The result is an effective field treatment which is valid close to the critical temperature Tcof the superfluid phase transition. Inspired by the GL formalism, we will now present a beyond saddle-point EFT that is capable of describing Fermi superfluids in the BEC-BCS crossover at finite temperatures. This theory is based on the assumption that the pair field Ψxτexhibits 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 F0Fsp. Subsequently, we consider the full infinite sum in (26):

p=11pTrG0Fp=p=11pTrG0FG0FG0Fpfactors.E29

In every term of this sum, we replace (at most) two occurrences of Fxτby its gradient expansion and substitute all remaining factors Fxτby Fsp. Afterward, the entire sum over pcan be calculated analytically. The result of this calculation, the details of which can be found in [17], is an explicit expression for the Euclidian action functional that governs the dynamics of the pair field Ψxτof a three-dimensional (3D) superfluid Fermi gas:

SEFT=0βdτdxD2Ψ¯∂ΨτΨ¯τΨ+Ωs+C2mxΨ¯xΨE2mxΨ22+QΨ¯τ∂ΨτRΨ2τ2.E30

The EFT coefficients Ωs, C, D, E, Qand Rare given by

Ωs=18πkFasΔ2dk2π31βln2coshβEk+2coshβζξkΔ22k2E31
C=dk2π3k23mf2βEkζE32
D=dk2π3ξkΨ2f1βξkζf1βEkζE33
E=2dk2π3k23mξk2f4βEkζE34
Q=12Ψ2dk2π3f1βEkζEk2+ξk2f2βEkζE35
R=12Ψ2dk2π3f1βEkζ+Ek23ξk2f2βEkζ3Ψ2
+4ξk22Ek23f3βEkζ+2Ek2Ψ2f4(βEkζ),E36

where the functions fpβεζare recursively defined as

f1βϵζ=12ϵsinhβϵcoshβϵ+coshβζE37
fp+1βϵζ=12pϵfpβϵζ∂ϵE38

In general, each of these EFT coefficients depends on the modulus squared of the order parameter Ψxτ2. In practice, however, we will assume that the coefficients associated with the second order derivatives of the pair field can be kept constant and equal to their bulk value, since retaining their full space-time dependence would strictly speaking lead us beyond the second-order approximation of the gradient expansion. This means that in expressions (32), (34), (35), and (36) for the coefficients C, E, Q, and R, we set Ψxτ2=Ψ2and Ek=ξk2+Ψ2, where Ψ2is the saddle-point value of the pair field for a uniform system. For the coefficients Ωsand Don the other hand, the full space-time dependence of Ψxτ2is preserved.

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 vswith which the soliton propagates through the superfluid, as illustrated in Figure 4. The higher the soliton velocity, the smaller the phase jump and soliton depth become. Above a certain critical velocity vc, the phase jump and density dip will disappear completely and a dark soliton solution no longer exists.

Figure 4.

Example of the density profile (upper row) and phase profile (lower row) of a dark soliton for different soliton velocities vs relative to the critical velocity vc.

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.

τitE39
SEFTβiSEFTtbta.E40

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

SEFTtbta=tatbdtdxL,E41

we subsequently find the following expression for L:

L=iD2Ψ¯ΨtΨ¯tΨΩsC2mxΨ¯xΨ+E2mxΨ22+QΨ¯tΨtRΨ2t2,E42

where the Hamiltonian density His defined as

H=Ωs+C2mxΨ¯xΨE2mxΨ22+QΨ¯tΨtRΨ2t2.E43

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 Ψxtas

Ψxt=Ψxtext.E44

Moreover, since a soliton is a localized perturbation, we write the modulus as a product of the constant background value Ψand a relative amplitude axtthat modifies the background value at the position of the soliton:

Ψxt=Ψaxt.E45

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

L=κaa2θtΩsaΩsa12ρqpaxa212ρsfaxθ2+Q4RΨ2a2Ψ2at2+QΨ2a2θt2,E46

with

κa=DaΨ2,E47
ρqpa=C4EΨ2a2mΨ2,E48
ρsfa=CmΨ2a2.E49

Here, we added Ωsato the original Lagrangian to obtain a regularized Lagrangian density in which energy values are considered with respect to the energy of the uniform system. The superfluid density ρsfdetermines how much the pair condensate resists gradients in its phase field, while the quantum pressure ρqpis a consequence of the fact that the condensate also resists gradients in the pair density. We will further limit ourselves to a 1D problem in which the soliton propagates with constant speed vsin the x-direction on a uniform background. This assumption can be implemented through the condition that the space-time dependence of the pair field satisfies the relation fxt=fxvst. We then perform a change of variables x=xvstand t=t, corresponding to a transformation to the frame of reference that moves along with the soliton and has its origin at the soliton center. It follows that

fxvst=fx,x=x,t=tvsx.E50

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

L=κaa2vsθxΩsaΩsa12ρ˜qpaax212ρ˜sfaθx2.E51

with the modified superfluid density and quantum pressure

ρ˜qpa=C4EΨ2a2mΨ22Q4RΨ2a2Ψ2vs2,E52
ρ˜sfa=CmΨ2a22QΨ2a2vs2.E53

From the above expression for Laθ, we can now find the equations of motion for the relative amplitude field axand the phase field θx:

tLta+xLxa=La,E54
tLtθ+xLxθ=Lθ.E55

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

xκaa2vsρ˜sfaθx=0E56
θx=κaa2vs+αρ˜sfa.E57

The integration constant αcan be determined through the boundary condition for a dark soliton:

θx0forx±.E58

which yields α=vsκwith κ=κaand thus

θx=vsρ˜sfaκaa2κ.E59

If we set θ=0, the phase profile of the superfluid is given by

θx=vsxκaxa2xκρ˜sfaxdx.E60

Next, we derive the equation of motion for ax:

xρ˜qpaax=aκaa2vsθxΩsa12ρ˜qpaax212ρ˜sfaθx2.E61

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

Xa=ΩsaΩsa,E62
Ya=κaa2κ22ρ˜sfa,E63

we find

12ρ˜qpaax2+ρ˜qpa2ax2=aXavs2Ya.E64

While the above equation does not allow for a straightforward solution for aas a function of the position x, it can be solved for xas a function of ainstead. Using the boundary conditions for a dark soliton

axx±=0andaxx±=1,E65

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

12ρ˜qpaax2=Xavs2Ya,E66
xa2=12ρ˜qpaXavs2Ya,E67
x=±12a0aρ˜qpaXavs2Yada.E68

Here, a0=ax=0is the relative amplitude at the center of the soliton, which is found as the solution of

Xa0vs2Ya0=0.E69

For given values of the interaction parameter kFas1, the temperature T/TF, the imbalance chemical potential ζ, and the soliton velocity vs, formulae (60) and (68) allow us to calculate the complete pair field profile of the dark soliton. For example, the soliton density and phase profiles in Figure 4 were calculated using the above expressions.

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 ζ, defined in (18). Figure 5a and b shows respectively the fermion particle density nxand spin-population density difference δnx(both with respect to the bulk density n) along a stationary dark soliton for kFas1=0(unitarity), T=0.1TF, and for different values of ζ. The density and density difference profiles are calculated using formulas (22) and (23) in a mean-field local density approximation. From the left figure, we observe that as we raise the imbalance chemical potential, the fermion density at the soliton center increases and the soliton broadens. However, we also know that, for a stationary dark soliton, the pair density at the center is always zero (as shown in the upper left panel of Figure 4), which means that the particles filling up the soliton are unpaired particles. This is confirmed by the right figure, which shows that the density difference between spin-up and spin-down particles in the soliton center increases with ζ. The same effects are observed across the whole BEC-BCS crossover.

Figure 5.

Fermion density (left figure) and density difference (right figure) profiles of a dark soliton for kFaS−1=0 at temperature T/TF=0.1, for different values of the imbalance chemical potential ζ. The densities are given with respect to the bulk density n∞.

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 vwill be used, defined as:

v=mxθ,E70

where θis the phase field from the hydrodynamical description (44). In the time-independent case, the action (30) reduces to the free energy (times the inverse temperature), which is given by:

F=drFaxaxwithFaxax=Xa+18ρsfav2x+12ρqpaxa2.E71

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 κis defined as:

κ=γvrds,E72

where γis a closed contour and vthe superfluid velocity field (70). A distinct feature of superfluids4 is that the circulation κis only allowed to take on values which are integer multiples of the circulation quantum h/m. In superfluids, circulation is always carried by quantized vortices.

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:

κ=mγxθds=nhmwithn,E73

where the gradient theorem was used together with the fact that the phase field θis a periodic function (period 2π).

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 x=rϕ. The origin of the polar coordinates will be chosen in the center of the vortex (at the point where the superfluid density reaches zero). In order to derive the vortex structure, a set of boundary conditions is required. In the radial direction, the boundary conditions are then given by5:

ar0=0andar=1,E74

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:

arϕ=frΦϕ.E75

Since the structure is periodic, the general solution for Φϕis thus given by:

Φϕ=n=anei,E76

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:

Φϕ=eiwithn,E77

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

vr=nmreϕκ=nhm,E78

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 nwill be restricted to n=±1. The state with n=1is known as the “vortex,” where the state with n=1is known as the “anti-vortex.” This means that the vortex velocity field is given by7:

vr=±mreϕ,E79

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

Currently, there is no analytical solution available for the full vortex structure fr. Calculations including vortices are therefore either done numerically (for the exact structure) or variationally. One way to numerically find the minimal structure is by writing down the equations of motion (the Euler-Lagrange equations for the free energy (71)), which is analogous to what was done for the soliton in the previous section. Directly solving the equations of motion, however, is a numerical challenge due to the divergence of the velocity field in the center of the vortex. A second numerical method is briefly discussed further on. The disadvantage of the full numerical approach is that it takes time. As an alternative, it is possible to work with a variational model. By working with a variational model, it is possible to retain a fair amount of accuracy while gaining several orders of magnitude in computational speed. The usage of variational models is discussed in the next subsection. A disadvantage of using variational models is however that a certain structure is proposed, meaning the variational guess can be wrong in certain situations. When using variational models, one should consequently always check the validity of the model and the range of application.

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:

fr=tanhr2ξ,E80

where the quantity ξis defined as the healing length. The hyperbolic tangent (80) is the exact solution of the Gross-Pitaevskii equation in 1D for a condensate with a hard wall boundary [9, 35]. Since the variational model describes the healing from a hole in the condensate, it is expected that this model will also sufficiently describe the vortex physics. The merit of the presented effective field theory in Section 4 is that an analytical solution can be derived for the vortex healing length ξ; this will be done in the remainder of this subsection. Using the definitions (71), the free energy of the variational vortex structure is given by:

F=0rdrXa+A2r2tanh2r2ξ+ρqptanhr2ξ4ξ2cosh4r2ξ,E81

where the value of the constant Ais defined as:

A=2CΔ2.E82

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

limRA20R1rtanh2r2ξdrlogRE83

diverges logarithmically with increasing radius of the integration domain. The physical reason is clear: the velocity profile of a vortex decays as 1/r, so that the kinetic energy of the superflow will grow as the logarithm of the container size. However, the derivative with respect to ξof this kinetic energy of the superflow does not diverge. This can be seen by first switching to a dimensionless variable x=r/ξ:

F=ξ20xXadx+A2limR0R/ξtanh2x2dxx+0ρqptanhx/24cosh4x/2dx.E84

The last term no longer contains a dependency on ξ, so its derivative with respect to ξvanishes. We obtain

dFdξ=2ξ0xXadx+A2limRddξ0R/ξtanh2x2dxxE85

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

ddξ0R/ξgxdx=limΔξ01Δξ0R/ξ+Δξgxdx0R/ξgxdx=limΔξ01ΔξRξ+ΔξRξgRξ=Rξ2gRξE86

to (85) we get

dFdξ=2ξ0xXadxA2ξlimRtanh2R2ξ,E87

so that

dFdξ=02ξ0xXadx=A2ξ.E88

With A=2CΔ2as above, and

B=0xXadxE89

we find a closed form result

ξ=12AB.E90

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.

Figure 6.

The vortex variational healing length (90) throughout the BEC-BCS crossover for the case β=100 and ζ=0. The dotted lines yield the exact solutions in the deep BEC [36] and BCS [37] limits. This plot made using the same data as in [38].

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 r0. The divergence of the velocity field in the origin will be strongly pronounced when solving the equations of motion. While in the case of a direct minimization, the same divergence will have less impact on the solution.

The numerical method that was used in order to determine the exact vortex structure for a given set of parameters βasζis discussed in full detail in [38]. In a nutshell, this method comes down to making a discretized version of the vortex structure: f1f2fN, where f1=0and fN=1due to the boundary conditions. During the minimization procedure, a program runs through the list of points fnn23N1, where it suggests a (random) new value; if the new value results in a lower energy, it is accepted as the new value of the vortex structure. The minimization program continues to run until a certain tolerance is reached and the structure is not changing any more.

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.

Figure 7.

The relative energy difference between the exact and variational solutions throughout the BEC-BCS crossover for different values of the temperature β and polarization ζ. These results were also shown and discussed in [38].

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 βfor a given polarization ζwithin a reasonable computational time span.

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.

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Wout Van Alphen, Nick Verhelst, Giovanni Lombardi, Serghei Klimin and Jacques Tempere (May 30th 2018). An Effective Field Description for Fermionic Superfluids, Superfluids and Superconductors, Roberto Zivieri, IntechOpen, DOI: 10.5772/intechopen.73058. Available from:

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