6. Application 2: the vortex structure

application, we will give a short description of the influence of spin-imbalance on dark solitons,

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 n xð Þ and spinpopulation density difference δn xð Þ (both with respect to the bulk density n∞) along a stationary dark soliton for ð Þ kFas �<sup>1</sup> <sup>¼</sup> 0 (unitarity), <sup>T</sup> <sup>¼</sup> <sup>0</sup>:1TF, and for different values of <sup>ζ</sup>. 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

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

Figure 5. Fermion density (left figure) and density difference (right figure) profiles of a dark soliton for ð Þ kFaS �<sup>1</sup> <sup>¼</sup> 0 at temperature T=TF ¼ 0:1, for different values of the imbalance chemical potential ζ. The densities are given with respect to

a topic that was studied in detail in [18].

22 Superfluids and Superconductors

same effects are observed across the whole BEC-BCS crossover.

the bulk density n∞.

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 v will be used, defined as:

$$\mathbf{v} = \frac{\hbar}{m} \nabla\_{\mathbf{x}} \theta\_{\mathbf{v}} \tag{70}$$

where θ is 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:

$$F = \int \text{d}\mathbf{r} \mathcal{F}(a, \nabla\_{\mathbf{x}} a, \mathbf{x}) \quad \text{with} \quad \mathcal{F}(a, \nabla\_{\mathbf{x}} a, \mathbf{x}) = X(a) + \frac{1}{8} \rho\_{\text{sf}}(a) \upsilon^2(\mathbf{x}) + \frac{1}{2} \rho\_{\text{app}}(a) |\nabla\_{\mathbf{x}} a|^2. \tag{71}$$

The free energy was written in a more compact<sup>3</sup> 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 κ is defined as:

$$
\kappa = \oint\_{\gamma} \mathbf{v}(\mathbf{r}) \cdot \mathbf{ds},\tag{72}
$$

where γ is a closed contour and v the superfluid velocity field (70). A distinct feature of superfluids4 is that the circulation κ is 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:

<sup>3</sup> Where again the free energy at infinity was subtracted to obtain a well behaved free energy. 4

In the case of a superconductor, the quantized value is given by the magnetic flux.

$$
\kappa = \frac{\hbar}{m} \oint\_{\gamma} \nabla\_{\mathbf{x}} \theta \cdot \mathbf{ds} = n \frac{h}{m} \quad \text{with} \quad n \in \mathbb{Z}, \tag{73}
$$

vð Þ¼ r n

is known as the "anti-vortex." This means that the vortex velocity field is given by7

where the "+" sign is for vortices and the "-" sign for anti-vortices.

check the validity of the model and the range of application.

order of micrometers [34], meaning that its structure becomes important<sup>8</sup>

description, different variational models are available [9, 30].

The condensate size to vortex core size is typically in the range 10–50.

6.3. A variational model for the vortex core

7

8

vð Þ¼� r

ℏ mr

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 n will be restricted to n ¼ �1. The state with n ¼ 1 is known as the "vortex," where the state with n ¼ �1

> ℏ mr

Currently, there is no analytical solution available for the full vortex structure f rð Þ. 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

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

hole will no longer capture the entire vortex physics. In order to provide a more detailed

Note that this velocity field is the same as the elementary vortex flow known in classical hydrodynamics [32].

e<sup>ϕ</sup> ) κ ¼ n

h

<sup>m</sup> , (78)

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058

eϕ, (79)

:

25

; a simple cylindric

where the gradient theorem was used together with the fact that the phase field θ is 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 <sup>x</sup> <sup>¼</sup> <sup>r</sup>; <sup>ϕ</sup> � �. 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 by<sup>5</sup> :

$$a(r \to 0) = 0 \quad \text{and} \quad a(r \to \infty) = 1,\tag{74}$$

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 part<sup>6</sup> :

$$a(r, \phi) = f(r)\Phi(\phi). \tag{75}$$

Since the structure is periodic, the general solution for Φ ϕ � � is thus given by:

Φ ϕ � � <sup>¼</sup> <sup>X</sup><sup>∞</sup> n¼�∞ ane <sup>i</sup>n<sup>ϕ</sup>, (76)

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:

$$\Phi(\phi) = e^{i n \phi} \quad \text{with} \quad n \in \mathbb{Z}, \tag{77}$$

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

<sup>5</sup> Note that the condition at r ! ∞ could be replaced by ∂rar!<sup>∞</sup> ¼ 0. This could however lead to numerical difficulties in the center of the vortex.

<sup>6</sup> This product decomposition is not generally valid in all coordinate systems [31].

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 25

$$\mathbf{v}(r) = n \frac{\hbar}{mr} \mathbf{e}\_{\phi} \Rightarrow \kappa = n \frac{\hbar}{m'} \tag{78}$$

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 n will be restricted to n ¼ �1. The state with n ¼ 1 is known as the "vortex," where the state with n ¼ �1 is known as the "anti-vortex." This means that the vortex velocity field is given by7 :

$$\mathbf{v}(r) = \pm \frac{\hbar}{mr} \mathbf{e}\_{\phi\nu} \tag{79}$$

where the "+" sign is for vortices and the "-" sign for anti-vortices.

Currently, there is no analytical solution available for the full vortex structure f rð Þ. 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

<sup>κ</sup> <sup>¼</sup> <sup>ℏ</sup> <sup>m</sup> <sup>∮</sup> <sup>γ</sup>

of this application, only singly quantized vortices will thus be studied.

6.2. About the structure of a quantum vortex

function (period 2π).

24 Superfluids and Superconductors

5

6

center of the vortex.

∇xθ � ds ¼ n

h m

where the gradient theorem was used together with the fact that the phase field θ is a periodic

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

The most natural coordinate system to describe vortices are the polar coordinates <sup>x</sup> <sup>¼</sup> <sup>r</sup>; <sup>ϕ</sup> � �. 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 by<sup>5</sup>

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

Φ ϕ

state, one usually restricts the problem to one of the many possible modes:

Φ ϕ � � <sup>¼</sup> <sup>e</sup>

This product decomposition is not generally valid in all coordinate systems [31].

a r; <sup>ϕ</sup> � � <sup>¼</sup> f rð Þ<sup>Φ</sup> <sup>ϕ</sup>

� � <sup>¼</sup> <sup>X</sup><sup>∞</sup>

n¼�∞

leading to a basis of angular modes for the vortex structure. In order to find the lowest energy

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

Note that the condition at r ! ∞ could be replaced by ∂rar!<sup>∞</sup> ¼ 0. This could however lead to numerical difficulties in the

ane

We factorize the amplitude function in a radial and an angular part<sup>6</sup>

Since the structure is periodic, the general solution for Φ ϕ

a rð Þ¼ ! 0 0 and a rð Þ¼ ! ∞ 1, (74)

:

� � is thus given by:

<sup>i</sup>n<sup>ϕ</sup> with n∈ ℤ, (77)

� �: (75)

<sup>i</sup>n<sup>ϕ</sup>, (76)

with n∈ ℤ, (73)

:

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 important<sup>8</sup> ; 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].

<sup>7</sup> Note that this velocity field is the same as the elementary vortex flow known in classical hydrodynamics [32]. 8 The condensate size to vortex core size is typically in the range 10–50.

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

$$f(r) = \tanh\left(\frac{r}{\sqrt{2}\xi}\right),\tag{80}$$

d dξ ð R=ξ

With <sup>A</sup> <sup>¼</sup> <sup>2</sup>Cj j <sup>Δ</sup><sup>∞</sup> <sup>2</sup> as above, and

we find a closed form result

to (85) we get

so that

0

dF <sup>d</sup><sup>ξ</sup> <sup>¼</sup> <sup>2</sup><sup>ξ</sup>

6.4. Comparison to the exact (numerical) solution

g xð Þdx ¼ lim

Δξ!0

¼ lim Δξ!0

> ð ∞

> > 0

dF

<sup>d</sup><sup>ξ</sup> <sup>¼</sup> <sup>0</sup> <sup>⇔</sup> <sup>2</sup><sup>ξ</sup>

B ¼ ð ∞

0

<sup>ξ</sup> <sup>¼</sup> <sup>1</sup> 2

limits, the healing length shows to be in a good agreement with the exact limits.

minimization, the same divergence will have less impact on the solution.

The formula for the healing length (90) can also be plotted, this is done in Figure 6. In both

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 r ! 0. 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

The numerical method that was used in order to determine the exact vortex structure for a given set of parameters β; as; ζ � � is discussed in full detail in [38]. In a nutshell, this method

ffiffiffiffi A B r

1 Δξ

1 Δξ

xX að Þd<sup>x</sup> � <sup>A</sup>

0 B@

ð R=ð Þ ξþΔξ

g xð Þdx �

ξ

<sup>R</sup>!<sup>∞</sup> tanh<sup>2</sup> <sup>R</sup>

xX að Þd<sup>x</sup> <sup>¼</sup> <sup>A</sup>

ð R=ξ

g xð Þdx

¼ � <sup>R</sup> ξ2 g

ffiffiffi 2 <sup>p</sup> <sup>ξ</sup> 1 CA

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058

> R ξ � �

� � � � , (87)

<sup>2</sup><sup>ξ</sup> : (88)

xX að Þdx (89)

: (90)

(86)

27

0

R ξ � �

0

R <sup>ξ</sup> <sup>þ</sup> <sup>Δ</sup><sup>ξ</sup> � <sup>R</sup>

� �<sup>g</sup>

<sup>2</sup><sup>ξ</sup> lim

ð ∞

0

where the quantity ξ is 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:

$$F = \int\_0^\kappa r \text{d}r \left[ X(a) + \frac{A}{2r^2} \tanh^2 \left( \frac{r}{\sqrt{2}\xi} \right) + \frac{\rho\_{\text{qp}} \left( \tanh \left( \frac{r}{\sqrt{2}\xi} \right) \right)}{4\xi^2 \cosh^4 \left( \frac{r}{\sqrt{2}\xi} \right)} \right],\tag{81}$$

where the value of the constant A is defined as:

$$A = 2\mathbb{C}|\Delta\_{\simeq}|^2. \tag{82}$$

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

$$\lim\_{R \to \infty} \frac{A}{2} \left[ \frac{1}{r} \tanh^2 \left( \frac{r}{\sqrt{2}\xi} \right) \mathrm{d}r \propto \log \left( R \right) \tag{83}$$

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 ξ of this kinetic energy of the superflow does not diverge. This can be seen by first switching to a dimensionless variable x ¼ r=ξ:

$$F = \xi^2 \int\_0^\infty \mathbf{x} X(a) \mathbf{dx} + \frac{A}{2} \lim\_{\mathbb{R} \to \infty} \left( \int\_0^{\mathbb{R}/\xi} \tanh^2 \left( \frac{\mathbf{x}}{\sqrt{2}} \right) \frac{\mathbf{dx}}{\mathbf{x}} \right) + \int\_0^\infty \frac{\rho\_{\eta\nu} \left( \tanh \left( \mathbf{x} / \sqrt{2} \right) \right)}{4 \cosh^4 \left( \mathbf{x} / \sqrt{2} \right)} \mathbf{dx}.\tag{84}$$

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

$$\frac{\mathrm{d}F}{\mathrm{d}\xi} = 2\xi \int\_0^\infty \mathrm{x}X(a)\mathrm{d}x + \frac{A}{2} \lim\_{R \to \infty} \frac{\mathrm{d}}{\mathrm{d}\xi} \left[ \int\_0^{R/\xi} \tanh^2\left(\frac{x}{\sqrt{2}}\right) \frac{\mathrm{d}x}{x} \right] \tag{85}$$

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

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 27

$$\begin{split} \frac{d}{d\xi} \int\_{0}^{R/\xi} g(\mathbf{x}) \mathbf{dx} &= \lim\_{\Delta\xi \to 0} \frac{1}{\Delta\xi} \left( \int\_{0}^{R/(\xi + \Delta\xi)} g(\mathbf{x}) \mathbf{dx} - \int\_{0}^{R/\xi} g(\mathbf{x}) \mathbf{dx} \right) \\ &= \lim\_{\Delta\xi \to 0} \frac{1}{\Delta\xi} \left( \frac{R}{\xi + \Delta\xi} - \frac{R}{\xi} \right) g\left(\frac{R}{\xi}\right) = -\frac{R}{\xi^2} g\left(\frac{R}{\xi}\right) \end{split} \tag{86}$$

to (85) we get

$$\frac{\mathrm{d}F}{\mathrm{d}\xi} = 2\xi \int\_0^\alpha \mathrm{x}X(a)\mathrm{d}x - \frac{A}{2\xi} \lim\_{R \to \infty} \left[ \tanh^2 \left( \frac{R}{\sqrt{2}\xi} \right) \right],\tag{87}$$

so that

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

definitions (71), the free energy of the variational vortex structure is given by:

<sup>2</sup>r<sup>2</sup> tanh<sup>2</sup> <sup>r</sup>

ffiffiffi 2 <sup>p</sup> <sup>ξ</sup> � �

<sup>A</sup> <sup>¼</sup> <sup>2</sup>Cj j <sup>Δ</sup><sup>∞</sup> <sup>2</sup>

tanh<sup>2</sup> <sup>r</sup>

ffiffiffi 2 <sup>p</sup> <sup>ξ</sup> � �

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 ξ of this kinetic energy of the superflow does not diverge. This can be seen by first switching to a

tanh<sup>2</sup> <sup>x</sup>

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

d dξ

ð R=ξ

2 6 4

0

A <sup>2</sup> lim R!∞

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

ffiffiffi 2 p � � dx x

1

CA <sup>þ</sup> ð ∞

0

tanh<sup>2</sup> <sup>x</sup>

ffiffiffi 2 p � � dx x

þ

4ξ<sup>2</sup>

ρqp tanh <sup>r</sup>

cosh <sup>4</sup> <sup>r</sup> ffiffi 2 <sup>p</sup> <sup>ξ</sup> � �

ffiffi 2 <sup>p</sup> <sup>ξ</sup> � � � �

3

: (82)

dr∝ log ð Þ R (83)

<sup>ρ</sup>qp tanh <sup>x</sup><sup>=</sup> ffiffiffi <sup>2</sup> � � � � <sup>p</sup> 4cosh <sup>4</sup> x= ffiffiffi

> 3 7

5, (81)

<sup>2</sup> � � <sup>p</sup> <sup>d</sup>x: (84)

<sup>5</sup> (85)

<sup>r</sup>dr Xað Þþ <sup>A</sup>

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

lim R!∞ A 2 ð R

A <sup>2</sup> lim R!∞

ð ∞

0

0

1 r

> ð R=ξ

0 B@

xX að Þdx þ

0

2 4

F ¼ ð ∞

26 Superfluids and Superconductors

dimensionless variable x ¼ r=ξ:

<sup>F</sup> <sup>¼</sup> <sup>ξ</sup><sup>2</sup> ð ∞

vanishes. We obtain

0

xX að Þdx þ

dF <sup>d</sup><sup>ξ</sup> <sup>¼</sup> <sup>2</sup><sup>ξ</sup>

0

where the value of the constant A is defined as:

f rð Þ¼ tanh <sup>r</sup>

where the quantity ξ is 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

ffiffiffi 2 <sup>p</sup> <sup>ξ</sup> � �

, (80)

$$\frac{\mathrm{d}F}{\mathrm{d}\xi} = 0 \Leftrightarrow 2\xi \int\_0^\infty \mathrm{x}X(a) \mathrm{d}x = \frac{A}{2\xi}.\tag{88}$$

With <sup>A</sup> <sup>¼</sup> <sup>2</sup>Cj j <sup>Δ</sup><sup>∞</sup> <sup>2</sup> as above, and

$$B = \bigcap\_{0}^{\infty} \mathbf{x} \mathbf{X}(a) \mathbf{d}x \tag{89}$$

we find a closed form result

$$
\xi = \frac{1}{2} \sqrt{\frac{A}{B}}.\tag{90}
$$

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 r ! 0. 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 β; as; ζ � � is discussed in full detail in [38]. In a nutshell, this method

to suggest that the variational guess is a good guess.<sup>9</sup> 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.

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 29

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 β for a given polarization ζ within a

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 imbal-

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

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

In Figure 7b, the energy difference seems to blow up towards the BCS limit. This divergence is due to the fact that at that

financial support from the Research Fund (BOF-GOA) of the University of Antwerp.

point superfluidity is lost due to polarization (Clogston limit); at this point superfluidity disappears.

7. Concluding section

reasonable computational time span.

anced superfluid Fermi gases.

Acknowledgements

9

systems [39] and the study of the "vortex charge" [40].

Figure 6. The vortex variational healing length (90) throughout the BEC-BCS crossover for the case β ¼ 100 and ζ ¼ 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].

comes down to making a discretized version of the vortex structure: f <sup>1</sup>; f <sup>2</sup>;…; f <sup>N</sup> , where f <sup>1</sup> ¼ 0 and f <sup>N</sup> ¼ 1 due to the boundary conditions. During the minimization procedure, a program runs through the list of points <sup>f</sup> <sup>n</sup>jn<sup>∈</sup> f g <sup>2</sup>; <sup>3</sup>;…; <sup>N</sup> � <sup>1</sup> , 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

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].

to suggest that the variational guess is a good guess.<sup>9</sup> 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.
