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 Ψð Þ r; t that 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 Ψð Þ r; t and Ψð Þ r; t , we can define an effective bosonic action for the pair field given by

Figure 3. Solutions for the pair field <sup>Δ</sup> and the average chemical potential <sup>μ</sup> in function of the interaction strength ð Þ kFas �<sup>1</sup> at temperature T=TF ¼ 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.

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 15

$$\mathcal{S}\_{\text{eff}} = \mathcal{S}\_{\text{B}} - \text{Tr}\left[\ln\left(-\mathbb{G}^{-1}\right)\right],\tag{24}$$

where SB ¼ � <sup>Ð</sup> β 0 Ð <sup>d</sup><sup>x</sup> j j <sup>Ψ</sup>ð Þ <sup>x</sup>;<sup>τ</sup> <sup>2</sup> <sup>8</sup> is the action for free bosonic fields. The inverse Green's function matrix �G�<sup>1</sup> for interacting fermions, which was defined in expression (14), can be separated into its diagonal and off-diagonal components

<sup>n</sup>sp ¼ �∂Ωsp ∂μ T, ζ,Δ

<sup>δ</sup>nsp ¼ �∂Ωsp

crossover are shown in Figure 3a and b.

4. The effective field theory

14 Superfluids and Superconductors

pair field given by

∂ζ T,μ,Δ

Since in our units kF <sup>¼</sup> 1, the particle density <sup>n</sup>sp is fixed by <sup>n</sup>sp <sup>¼</sup> <sup>1</sup><sup>=</sup> <sup>3</sup>π<sup>2</sup> . Given the input parameters β, ζ, and as, the values Δ and μ can then be found from the coupled set of Eqs. (21) and (22), while (23) fixes δnsp as a function of ζ. Solutions for Δsp and μ across the BEC-BCS

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 Ψð Þ r; t that 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 Ψð Þ r; t and Ψð Þ r; t , we can define an effective bosonic action for the

Figure 3. Solutions for the pair field <sup>Δ</sup> and the average chemical potential <sup>μ</sup> in function of the interaction strength ð Þ kFas �<sup>1</sup> at temperature T=TF ¼ 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.

(22)

(23)

$$\begin{aligned} -\mathbb{G}^{-1}(\mathbf{x},\tau) &= -\mathbb{G}\_0^{-1}(\mathbf{x},\tau) + \mathbb{F}(\mathbf{x},\tau) \\ &= \begin{pmatrix} \frac{\partial}{\partial\tau} - \nabla\_\mathbf{x}^2 - \mu\_\uparrow & 0 \\ 0 & \frac{\partial}{\partial\tau} + \nabla\_\mathbf{x}^2 + \mu\_\downarrow \end{pmatrix} + \begin{pmatrix} 0 & -\Psi(\mathbf{x},\tau) \\ -\overline{\Psi}(\mathbf{x},\tau) & 0 \end{pmatrix} \end{aligned} \tag{25}$$

where �G�<sup>1</sup> <sup>0</sup> describes free fermionic fields, while F describes the pairing of the fermions. Using this decomposition, we can write the effective bosonic action functional (24) as

$$\begin{split} \mathcal{S}\_{\rm eff} &= \mathcal{S}\_{\rm B} - Tr[\ln(-\mathbb{G}\_{0}^{-1} + \mathbb{F})] \\ &= \mathcal{S}\_{\rm B} - Tr[\ln(-\mathbb{G}\_{0}^{-1})] - Tr[\ln(1 - \mathbb{G}\_{0} \mathbb{F})] \\ &= \mathcal{S}\_{\rm B} + \mathcal{S}\_{0} + \sum\_{p=1}^{\prime\prime} \frac{1}{p} Tr[(\mathbb{G}\_{0} \mathbb{F})^{p}]. \end{split} \tag{26}$$

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

$$\mathbb{F}(\mathbf{x},\tau) \approx \mathbb{F}\_{\text{sp}} = \begin{pmatrix} 0 & -\Delta \\ -\overline{\Delta} & 0 \end{pmatrix} \tag{27}$$

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 Ψð Þ x; τ around the normal state Ψ ¼ 0. This assumption comes down to keeping only terms up to p ¼ 2 in the sum in (26) and approximating Fð Þ x; τ by the following gradient expansion

$$\begin{split} \left. \mathbb{F}(\mathbf{x}, \tau) \approx \mathbb{F}\_0 + (\mathbf{x} - \mathbf{x}\_0) \cdot \nabla\_{\mathbf{x}} \mathbb{F}|\_{\mathbf{x}\_0} + \frac{1}{2} \sum\_{i,j=\mathbf{x}\_i, y\_i \, z} (\mathbf{x}\_i - \mathbf{x}\_{0,i}) \left(\mathbf{x}\_j - \mathbf{x}\_{0,j}\right) \frac{\partial^2 \mathbb{F}}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \right|\_{\mathbf{x}\_0} \\ + (\tau - \tau\_0) \frac{\partial \mathbb{F}}{\partial \tau} \bigg|\_{\tau\_0} + \frac{1}{2} (\tau - \tau\_0)^2 \frac{\partial^2 \mathbb{F}}{\partial \tau^2} \bigg|\_{\tau\_0} \end{split} \tag{28}$$

with F<sup>0</sup> ! 0. The result is an effective field treatment which is valid close to the critical temperature Tc of 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 Ψð Þ x; τ exhibits 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 F<sup>0</sup> ! Fsp. Subsequently, we consider the full infinite sum in (26):

$$\sum\_{p=1}^{\bullet} \frac{1}{p} \text{Tr}[(\mathbb{G}\_0 \mathbb{F})^p] = \sum\_{p=1}^{\bullet} \frac{1}{p} \text{Tr}\left[\underbrace{\mathbb{G}\_0 \mathbb{F} \mathbb{G}\_0 \mathbb{F} \dots \mathbb{G}\_0 \mathbb{F}}\_{p \text{ factors}}\right].\tag{29}$$

where the functions <sup>f</sup> <sup>p</sup> <sup>β</sup>; <sup>ε</sup>; <sup>ζ</sup> � � are recursively defined as

(36) for the coefficients C, E, Q, and R, we set j j Ψð Þ x; τ

on the other hand, the full space-time dependence of j j Ψð Þ x; τ

excitations of the superfluid: dark solitons and vortices.

5. Application 1: Soliton dynamics

dark solitons in Fermi superfluids.

5.1. What is a dark soliton?

j j Ψð Þ x; τ 2 <sup>f</sup> <sup>1</sup> <sup>β</sup>; <sup>e</sup>; <sup>ζ</sup> � � <sup>¼</sup> <sup>1</sup>

<sup>f</sup> <sup>p</sup>þ<sup>1</sup> <sup>β</sup>; <sup>e</sup>; <sup>ζ</sup> � � ¼ � <sup>1</sup>

2e

2pe

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

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 secondorder approximation of the gradient expansion. This means that in expressions (32), (34), (35), and

j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> is the saddle-point value of the pair field for a uniform system. For the coefficients <sup>Ω</sup><sup>s</sup> and <sup>D</sup>

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

In this section, we will use the EFT that was developed in Section 4 to study the properties of

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 [19–23]. 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 vs with which the soliton propagates through the superfluid, as illustrated in Figure 4. The higher the soliton velocity, the smaller the phase jump and

sinh βe � �

<sup>∂</sup><sup>f</sup> <sup>p</sup> <sup>β</sup>; <sup>e</sup>; <sup>ζ</sup> � �

. In practice, however, we will assume that the coefficients associated with the second

cosh <sup>β</sup><sup>e</sup> � � <sup>þ</sup> cosh βζ � � (37)

<sup>2</sup> <sup>¼</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> and <sup>E</sup><sup>k</sup> <sup>¼</sup>

<sup>2</sup> is preserved.

<sup>∂</sup><sup>E</sup> (38)

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

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ2 <sup>k</sup> <sup>þ</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup>

, where

17

q

In every term of this sum, we replace (at most) two occurrences of Fð Þ x; τ by its gradient expansion and substitute all remaining factors Fð Þ x; τ by Fsp. Afterward, the entire sum over p can 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 Ψð Þ x; τ of a three-dimensional (3D) superfluid Fermi gas:

$$S\_{\rm EFT} = -\int\_0^\delta \mathbf{d}\tau \int \mathbf{dx} \left[ \frac{D}{2} \left( \overline{\Psi} \frac{\partial \Psi}{\partial \tau} - \frac{\partial \overline{\Psi}}{\partial \tau} \Psi \right) + \Omega\_s + \frac{\mathcal{C}}{2m} (\nabla\_\mathbf{x} \overline{\Psi} \cdot \nabla\_\mathbf{x} \Psi) - \frac{E}{2m} \left( \nabla\_\mathbf{x} |\Psi|^2 \right)^2 \right. \tag{30}$$

$$+ Q \frac{\partial \overline{\Psi}}{\partial \tau} \frac{\partial \Psi}{\partial \tau} - R \left( \frac{\partial |\Psi|^2}{\partial \tau} \right)^2 \Big]. \tag{31}$$

The EFT coefficients Ωs, C, D, E, Q and R are given by

$$\Omega\_{\rm s} = -\frac{1}{8\pi k\_{\rm F}a\_{\rm s}} |\Delta|^2 - \int \frac{d\mathbf{k}}{(2\pi)^3} \left\{ \frac{1}{\beta} \ln \left[ 2\cosh\left(\beta E\_{\rm k}\right) + 2\cosh\left(\beta \zeta\right) \right] - \xi\_{\rm k} - \frac{|\Delta|^2}{2k^2} \right\} \tag{31}$$

$$\mathbb{C} = \int \frac{d\mathbf{k}}{(2\pi)^3} \frac{k^2}{3m} f\_2\left(\boldsymbol{\beta}, E\_\mathbf{k}, \zeta\right) \tag{32}$$

$$D = \int \frac{d\mathbf{k}}{(2\pi)^3} \frac{\xi\_\mathbf{k}}{|\Psi|^2} \left[ f\_1(\boldsymbol{\beta}, \xi\_\mathbf{k}, \zeta) - f\_1(\boldsymbol{\beta}, E\_\mathbf{k}, \zeta) \right] \tag{33}$$

$$E = 2\int \frac{d\mathbf{k}}{\left(2\pi\right)^3} \frac{k^2}{3m} \,\xi\_\mathbf{k}^2 f\_4\left(\boldsymbol{\beta}, E\_\mathbf{k}, \zeta\right) \tag{34}$$

$$Q = \frac{1}{2|\Psi|^2} \left[ \frac{d\mathbf{k}}{(2\pi)^3} \left[ f\_1 \left( \beta, E\_\mathbf{k}, \zeta \right) - \left( E\_\mathbf{k}^2 + \xi\_\mathbf{k}^2 \right) f\_2 \left( \beta, E\_\mathbf{k}, \zeta \right) \right] \tag{35}$$

$$R = \frac{1}{2|\Psi|^2} \int \frac{d\mathbf{k}}{\left(2\pi\right)^3} \left[\frac{f\_1\left(\boldsymbol{\beta}, E\_\mathbf{k}, \zeta\right) + \left(E\_\mathbf{k}^2 - 3\xi\_\mathbf{k}^2\right) f\_2\left(\boldsymbol{\beta}, E\_\mathbf{k}, \zeta\right)}{3|\Psi|^2}\right]$$

$$+ \frac{4(\xi\_\mathbf{k}^2 - 2E\_\mathbf{k}^2)}{3} f\_3\left(\boldsymbol{\beta}, E\_\mathbf{k}, \zeta\right) + 2E\_\mathbf{k}^2 |\Psi|^2 f\_4\left(\boldsymbol{\beta}, E\_\mathbf{k}, \zeta\right)}\Big],\tag{36}$$

where the functions <sup>f</sup> <sup>p</sup> <sup>β</sup>; <sup>ε</sup>; <sup>ζ</sup> � � are recursively defined as

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 Ψð Þ x; τ exhibits 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

F<sup>0</sup> ! Fsp. Subsequently, we consider the full infinite sum in (26):

Tr ð Þ <sup>G</sup>0<sup>F</sup> <sup>p</sup> ½ �¼ <sup>X</sup><sup>∞</sup>

the pair field Ψð Þ x; τ of a three-dimensional (3D) superfluid Fermi gas:

� �

<sup>Ψ</sup> <sup>∂</sup><sup>Ψ</sup> <sup>∂</sup><sup>τ</sup> � <sup>∂</sup><sup>Ψ</sup> ∂τ Ψ

ð dk ð Þ 2π 3 1 β

ð dk ð Þ 2π 3 ξk

E ¼ 2

ð dk ð Þ 2π

ð dk ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup>

<sup>k</sup> � <sup>2</sup>E<sup>2</sup> k � � "

<sup>3</sup> <sup>f</sup> <sup>3</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � � <sup>þ</sup> <sup>2</sup>E<sup>2</sup>

C ¼

ð dk ð Þ 2π 3 k2

ð dk ð Þ 2π 3 k2 3m ξ2

<sup>3</sup> <sup>f</sup> <sup>1</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � � � <sup>E</sup><sup>2</sup>

<sup>f</sup> <sup>1</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � � <sup>þ</sup> <sup>E</sup><sup>2</sup>

The EFT coefficients Ωs, C, D, E, Q and R are given by

j j <sup>Δ</sup> <sup>2</sup> �

D ¼

<sup>Q</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup>j j <sup>Ψ</sup> <sup>2</sup>

<sup>R</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup>j j <sup>Ψ</sup> <sup>2</sup>

> þ 4 ξ<sup>2</sup>

p¼1

1 p

In every term of this sum, we replace (at most) two occurrences of Fð Þ x; τ by its gradient expansion and substitute all remaining factors Fð Þ x; τ by Fsp. Afterward, the entire sum over p can 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

þ Ω<sup>s</sup> þ

<sup>þ</sup><sup>Q</sup> <sup>∂</sup><sup>Ψ</sup> ∂τ ∂Ψ

ln 2cosh βE<sup>k</sup>

C 2m

<sup>∇</sup>xj j <sup>Ψ</sup> <sup>2</sup> � �<sup>2</sup> �

<sup>∂</sup><sup>τ</sup> � <sup>R</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> ∂τ !<sup>2</sup>

� � <sup>þ</sup> 2cosh βζ � � � � � <sup>ξ</sup><sup>k</sup> � j j <sup>Δ</sup> <sup>2</sup>

( )

<sup>k</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup> k

<sup>k</sup> � <sup>3</sup>ξ<sup>2</sup> k � �<sup>f</sup> <sup>2</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � �

<sup>3</sup>j j <sup>Ψ</sup> <sup>2</sup>

<sup>k</sup>j j <sup>Ψ</sup> <sup>2</sup>

2 6 4

Tr G0FG0F…G0F |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} p factors

3 7

<sup>∇</sup>x<sup>Ψ</sup> � <sup>∇</sup>x<sup>Ψ</sup> � � � <sup>E</sup>

2m

5: (30)

2k 2

, (36)

(31)

3

<sup>3</sup><sup>m</sup> <sup>f</sup> <sup>2</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � � (32)

<sup>k</sup> <sup>f</sup> <sup>4</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � � (34)

j j <sup>Ψ</sup> <sup>2</sup> <sup>f</sup> <sup>1</sup> <sup>β</sup>; <sup>ξ</sup>k; <sup>ζ</sup> � � � <sup>f</sup> <sup>1</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � � � � (33)

� �<sup>f</sup> <sup>2</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � � � � (35)

<sup>f</sup> <sup>4</sup> <sup>β</sup>; <sup>E</sup>k; <sup>ζ</sup> � �#

<sup>5</sup>: (29)

X∞ p¼1

SEFT ¼

16 Superfluids and Superconductors

ðβ 0 dτ ð dx D 2

<sup>Ω</sup><sup>s</sup> ¼ � <sup>1</sup>

8πkFas

1 p

$$f\_1\left(\beta,\epsilon,\zeta\right) = \frac{1}{2\epsilon} \frac{\sinh\left(\beta\epsilon\right)}{\cosh\left(\beta\epsilon\right) + \cosh\left(\beta\zeta\right)}\tag{37}$$

$$f\_{p+1}(\boldsymbol{\beta}, \boldsymbol{\epsilon}, \zeta) = -\frac{1}{2p\epsilon} \frac{\partial f\_p(\boldsymbol{\beta}, \boldsymbol{\epsilon}, \zeta)}{\partial \epsilon} \tag{38}$$

In general, each of these EFT coefficients depends on the modulus squared of the order parameter j j Ψð Þ x; τ 2 . 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 secondorder 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 j j Ψð Þ x; τ <sup>2</sup> <sup>¼</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> and <sup>E</sup><sup>k</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ2 <sup>k</sup> <sup>þ</sup> j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> q , where j j <sup>Ψ</sup><sup>∞</sup> <sup>2</sup> is the saddle-point value of the pair field for a uniform system. For the coefficients <sup>Ω</sup><sup>s</sup> and <sup>D</sup> on the other hand, the full space-time dependence of j j Ψð Þ x; τ <sup>2</sup> is 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.
