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 <sup>S</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>t</sup> ; <sup>ϕ</sup> � ð Þ� x; t can be expressed as a path integral [7]:

$$\mathcal{Z} = \int \mathcal{D}\overline{\phi}\_{\mathbf{x},\tau} \mathcal{D}\phi\_{\mathbf{x},\tau} \exp\left(-S\_E \left[\phi\_{\mathbf{x},\tau}, \overline{\phi}\_{\mathbf{x},\tau}\right]\right). \tag{1}$$

Here, Dϕx, <sup>τ</sup> represents a sum over all possible space-time configurations of the field ϕð Þ x; τ , and τ ¼ it indicates imaginary times running from τ ¼ 0 to τ ¼ ħβ with β ¼ 1=ð Þ kBT . The Euclidian action SE β � � of the system is found from the real-time action functional S tð Þ <sup>b</sup>; ta through the substitution

$$\mathbf{t} \to -\mathbf{i}\tau \Rightarrow \mathbf{S}(t\_b, t\_a) \to i\mathbf{S}\_E(\boldsymbol{\beta}). \tag{2}$$

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 Ψð Þ x; τ :

An Effective Field Description for Fermionic Superfluids http://dx.doi.org/10.5772/intechopen.73058 9

$$\mathcal{Z}\_{\mathcal{B}} = \int \mathcal{D}\overline{\Psi}\mathcal{D}\Psi \exp\left(-\int \mathrm{d}\tau \int \mathrm{d}\mathbf{x} \int \mathrm{d}\tau' \int \mathrm{d}\mathbf{x}' \left[\overline{\Psi}(\mathbf{x},\tau)\mathbb{A}\left(\mathbf{x},\tau;\mathbf{x}',\tau'\right)\Psi\left(\mathbf{x}',\tau'\right)\right]\right) = \frac{1}{\det(\mathbf{A})},\tag{3}$$

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

Fermionic path integral: The path integral sums over a fermionic (Grassmann, complex valued) field ψð Þ x; τ :

$$\mathcal{Z}\_{\mathbb{F}} = \int \mathcal{D}\overline{\psi} \left[ \mathcal{D}\psi \text{exp}\left(-\int \text{d}\tau \int \text{d}\mathbf{x} \int \text{d}\tau' \int \text{d}\mathbf{x}' \left[\overline{\psi}(\mathbf{x},\tau)\mathbb{A}\left(\mathbf{x},\tau;\mathbf{x}',\tau'\right)\psi\left(\mathbf{x}',\tau'\right)\right]\right) \right.\right] = \text{det}(\mathbb{A}), \quad \text{(4)}$$

In the case of spin-dependent fermionic fields, the matrix A becomes slightly more complex since the spinor fields have multiple components<sup>1</sup> to account for the spin degree of freedom. The spinors ψ are described by anti-commuting Grassmann numbers [4, 8], thus satisfying <sup>ψ</sup><sup>2</sup> <sup>¼</sup> 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:

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

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

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]

The partition function of a system described by the quantum field action functional <sup>S</sup> <sup>ϕ</sup>ð Þ <sup>x</sup>; <sup>t</sup> ; <sup>ϕ</sup> �

Dϕx, <sup>τ</sup>Dϕx, <sup>τ</sup>exp �SE ϕx, <sup>τ</sup>; ϕx, <sup>τ</sup>

Here, Dϕx, <sup>τ</sup> represents a sum over all possible space-time configurations of the field ϕð Þ x; τ , and τ ¼ it indicates imaginary times running from τ ¼ 0 to τ ¼ ħβ with β ¼ 1=ð Þ kBT . The

t ! �iτ ) S tð Þ! <sup>b</sup>; ta iSE β

For systems with an Euclidean action which is at most quadratic in the fields, the path integral

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

(1) can be calculated analytically. In particular, two distinct cases can be considered:

� � h i

� � of the system is found from the real-time action functional S tð Þ <sup>b</sup>; ta

: (1)

� �: (2)

(General review book on the Path Integrals and most of its applications).

2.1. A brief introduction to the path integral formalism

Z ¼ ð

ð Þ� x; t can be expressed as a path integral [7]:

Euclidian action SE β

Ψð Þ x; τ :

through the substitution

important topological excitations: dark solitons and vortices.

2. Path integral theory and bosonification

functions of the quantum fields a lot easier.

8 Superfluids and Superconductors

$$\mathcal{Z}\_{\mathbb{B}} = \exp\left(-\operatorname{Tr}[\ln\left(\mathbb{A}\right)]\right),\tag{5}$$

$$\mathcal{Z}\_{\mathbb{F}} = \exp\left( + \text{Tr}[\ln \left( \mathbb{A} \right)] \right). \tag{6}$$

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

$$\begin{split} S\_{\mathrm{E}} &= \int\_{0}^{\hbar\beta} \mathrm{d}\tau \int \mathrm{d}\mathbf{x} \sum\_{\boldsymbol{\sigma} \in \{\dagger,\downarrow\}} \left[ \overline{\psi}\_{\boldsymbol{\sigma}}(\mathbf{x}) \Big( \hbar \partial\_{\mathbf{r}} - \frac{\hbar^{2}}{2m\_{\mathbf{x}}} \nabla\_{\mathbf{x}}^{2} - \mu\_{\boldsymbol{\sigma}} \Big) \right] \psi\_{\boldsymbol{\sigma}}(\mathbf{x}) \\ &\quad + \int\_{0}^{\hbar\beta} \mathrm{d}\tau \int \mathrm{d}\mathbf{x} \int \mathrm{d}\mathbf{y} \overline{\psi}\_{\dagger}(\mathbf{x}) \overline{\psi}\_{\downarrow}(\mathbf{y}) \mathrm{g} \delta(\mathbf{x} - \mathbf{y}) \psi\_{\downarrow}(\mathbf{y}) \psi\_{\uparrow}(\mathbf{x}) .\end{split} \tag{7}$$

where σ∈f g ↑; ↓ denotes the spin components of the fermionic spinor fields, the chemical potentials μσ fix the amount of particles of each spin population, and g is the renormalized interaction strength [9, 10], linking the interaction potential to the s-wave scattering length as:

<sup>1</sup> 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.

$$\frac{1}{g} = \frac{m}{4\pi\hbar^2 a\_s} - \int \frac{d\mathbf{k}}{(2\pi)^3} \frac{m}{\hbar^2 k^2} \,. \tag{8}$$

For the remainder of the chapter, the units

$$
\hbar = k\_{\text{B}} = k\_{\text{F}} = 2m = 1 \tag{9}
$$

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

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

dx X σ ∈f g ↑;↓

<sup>g</sup> � <sup>ψ</sup>↑ð Þ <sup>x</sup>; <sup>τ</sup> <sup>ψ</sup>↓ðx; <sup>τ</sup>ÞΨðx; <sup>τ</sup>Þ � <sup>ψ</sup>↓ðx; <sup>τ</sup>Þψ↑ðx; <sup>τ</sup>ÞΨðx; <sup>τ</sup><sup>Þ</sup>

Since the path integral over the fermionic fields ψ and ψ is now quadratic, it can be performed

<sup>d</sup><sup>x</sup> j j <sup>Ψ</sup>ð Þ <sup>x</sup>; <sup>τ</sup>

2 <sup>g</sup> � Tr ln �G�<sup>1</sup> � � � � <sup>0</sup>

<sup>x</sup> � μ<sup>↑</sup> �Ψð Þ x; τ

∂ ∂τ <sup>þ</sup> <sup>∇</sup><sup>2</sup> <sup>x</sup> þ μ<sup>↓</sup>

0 @

ψσð Þ x; τ

∂ <sup>∂</sup><sup>τ</sup> � <sup>∇</sup><sup>2</sup>

<sup>x</sup> � μσ � �ψσðx; <sup>τ</sup><sup>Þ</sup>

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

> 1 CA

1

CCA

3 7

<sup>5</sup>, (13)

!#

(12)

11

(14)

ourselves to the superfluid state and describe it with a single collective field.

2 6 4

ð β

Figure 1. A diagrammatic representation of the different terms in the Hubbard-Stratonovich identity (11).

0 dτ ð

analytically using formula (4), resulting in the effective bosonic path integral [3]

B@

β

ð

0

where the components of the inverse Green's function matrix �G�<sup>1</sup> are given by

0

BB@

∂ <sup>∂</sup><sup>τ</sup> � <sup>∇</sup><sup>2</sup>

�Ψð Þ x; τ

Since �G�<sup>1</sup> depends on the bosonic field <sup>Ψ</sup>ð Þ <sup>x</sup>; <sup>τ</sup> , 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,

DΨDΨexp �

<sup>D</sup>ΨDΨexp � � <sup>ð</sup>

ð Þ¼ x; τ

�G�<sup>1</sup>

2 6 4

2

tion becomes

<sup>Z</sup> <sup>¼</sup> <sup>Ð</sup>

DψσDψσ

Ð

� j j <sup>Ψ</sup>ð Þ <sup>x</sup>; <sup>τ</sup>

2.3. The resulting bosonic path integral

Z ¼ ð

will be used, meaning that we work in the natural units of kF, EF, ω<sup>F</sup> ¼ EF=ℏ, and TF ¼ EF=kB. Consequentially, the partition function of the ultracold Fermi gas can be written down as

$$\begin{split} \mathcal{Z} = & \int \mathcal{D}\overline{\psi}\_{\sigma} \mathcal{D}\psi\_{\sigma} \exp\left[ -\oint\_{0} \text{d}\tau \int \text{d}\mathbf{x} \left( \sum\_{\sigma \in \{\uparrow,\downarrow\}} \overline{\psi}\_{\sigma}(\mathbf{x},\tau) \left( \frac{\partial}{\partial \tau} - \nabla\_{\mathbf{x}}^{2} - \mu\_{\sigma} \right) \psi\_{\sigma}(\mathbf{x},\tau) \right. \\ & \left. + g\overline{\psi}\_{\uparrow}(\mathbf{x},\tau) \overline{\psi}\_{\downarrow}(\mathbf{x},\tau) \psi\_{\downarrow}(\mathbf{x},\tau) \psi\_{\uparrow}(\mathbf{x},\tau) \right) \right], \end{split} \tag{10}$$

where the label σ was 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 needed<sup>2</sup> 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 [11–14],

$$\exp\left(-g\left[\mathbf{d}^3 \mathbf{x} \overline{\mathbf{g}} \overline{\boldsymbol{\psi}}\_{\uparrow} \overline{\boldsymbol{\psi}}\_{\downarrow} \boldsymbol{\psi}\_{\downarrow} \boldsymbol{\psi}\_{\uparrow}\right]\right) = \int \mathcal{D}\overline{\mathbf{V}} \mathcal{D}\mathbf{V} \mathbf{e} \exp\left(\int \mathbf{d}^3 \mathbf{x} \left[\frac{|\mathbf{V}|^2}{g} + \overline{\boldsymbol{\psi}}\_{\uparrow} \overline{\boldsymbol{\psi}}\_{\downarrow} \mathbf{V} + \boldsymbol{\psi}\_{\downarrow} \boldsymbol{\psi}\_{\uparrow} \overline{\mathbf{V}}\right]\right),\tag{11}$$

it is possible to rewrite the action in a form that is quadratic in the fermionic fields ψ and 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 Ψð Þ r; τ , 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

<sup>2</sup> Of course, it is always possible, given sufficient computational resources and time, to calculate the partition sum numerically.

Figure 1. A diagrammatic representation of the different terms in the Hubbard-Stratonovich identity (11).

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.

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

$$\begin{split} \mathcal{Z} = & \int \mathcal{D}\overline{\psi}\_{\sigma} \mathcal{D}\psi\_{\sigma} \int \mathcal{D}\overline{\mathbf{V}} \mathcal{D}\Psi \exp\left[ -\int\_{0}^{\delta} \text{d}\tau \int \text{d}\mathbf{x} \left( \sum\_{\sigma \in \{\uparrow,\downarrow\}} \overline{\psi}\_{\sigma}(\mathbf{x},\tau) \left( \frac{\partial}{\partial \tau} - \nabla\_{\mathbf{x}}^{2} - \mu\_{\sigma} \right) \psi\_{\sigma}(\mathbf{x},\tau) \right) \right] \\\\ & \quad - \frac{|\Psi(\mathbf{x},\tau)|^{2}}{g} - \overline{\psi}\_{\uparrow}(\mathbf{x},\tau) \overline{\psi}\_{\downarrow}(\mathbf{x},\tau) \Psi(\mathbf{x},\tau) - \psi\_{\downarrow}(\mathbf{x},\tau) \psi\_{\uparrow}(\mathbf{x},\tau) \overline{\Psi}(\mathbf{x},\tau) \Bigg) \Big] \end{split} \tag{12}$$

#### 2.3. The resulting bosonic path integral

1 <sup>g</sup> <sup>¼</sup> <sup>m</sup> 4πℏ<sup>2</sup> as �

ð β

2 6 4

2.2. Bosonification: the Hubbard-Stratonovich transformation

¼ ð

DΨDΨexp

it is possible to rewrite the action in a form that is quadratic in the fermionic fields ψ and 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 Ψð Þ r; τ , 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

Of course, it is always possible, given sufficient computational resources and time, to calculate the partition sum

ð d3 <sup>x</sup> j j <sup>Ψ</sup> <sup>2</sup> g

Using the Hubbard-Stratonovich identity [11–14],

xgψ↑ψ↓ψ↓ψ<sup>↑</sup> � � 0 dτ ð

For the remainder of the chapter, the units

DψσDψσexp �

<sup>Z</sup> <sup>¼</sup> <sup>Ð</sup>

10 Superfluids and Superconductors

Hubbard-Stratonovich transformation.

exp �g

2

numerically.

ð d3 ð dk ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup>

will be used, meaning that we work in the natural units of kF, EF, ω<sup>F</sup> ¼ EF=ℏ, and TF ¼ EF=kB. Consequentially, the partition function of the ultracold Fermi gas can be written down as

> dx X σ∈ f g ↑;↓

þgψ↑ð Þ x; τ ψ↓ðx; τÞψ↓ðx; τÞψ↑ðx; τÞ

where the label σ was 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 needed<sup>2</sup> to calculate the above partition sum (10). In the present treatment, this trick will be the

0 @

m ℏ2 k

ψσð Þ x; τ

ℏ ¼ kB ¼ kF ¼ 2m ¼ 1 (9)

∂ <sup>∂</sup><sup>τ</sup> � <sup>∇</sup><sup>2</sup>

> !# ,

<sup>x</sup> � μσ � �

þ ψ↑ψ↓Ψ þ ψ↓ψ↑Ψ

! " #

ψσðx; τÞ

(10)

, (11)

<sup>2</sup> : (8)

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

$$\mathcal{Z} = \int \mathcal{D}\overline{\Psi}\mathcal{D}\Psi \exp\left[ -\left( -\int\_0^\beta \left[ \text{dx} \frac{|\Psi(\mathbf{x},\tau)|^2}{\mathcal{S}} - \text{Tr}\left[ \ln\left(-\mathbb{G}^{-1}\right) \right] \right] \right) \right] \tag{13}$$

where the components of the inverse Green's function matrix �G�<sup>1</sup> are given by

$$-\mathbb{G}^{-1}(\mathbf{x},\tau) = \begin{pmatrix} \frac{\partial}{\partial \tau} - \nabla\_{\mathbf{x}}^2 - \mu\_{\uparrow} & -\Psi(\mathbf{x},\tau) \\\\ -\overline{\Psi}(\mathbf{x},\tau) & \frac{\partial}{\partial \tau} + \nabla\_{\mathbf{x}}^2 + \mu\_{\downarrow} \end{pmatrix} \tag{14}$$

Since �G�<sup>1</sup> depends on the bosonic field <sup>Ψ</sup>ð Þ <sup>x</sup>; <sup>τ</sup> , 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 Ψð Þ x; τ . This theory will be presented in Section 4.

## 3. The mean field theory

At first sight, the introduction of the auxiliary bosonic fields Ψð Þ x; τ and Ψð Þ x; τ through 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 Ψð Þ x; τ and Ψð Þ x; τ that 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 f g q; m :

$$
\Psi(\mathbf{q},m) \to \sqrt{\beta V} \delta(\mathbf{q}) \delta\_{m,0} \times \Delta,\tag{15}
$$

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

After performing the Matsubara summation over n [3] and replacing the sum over k by a

<sup>β</sup> 2cosh <sup>β</sup>E<sup>k</sup>

The saddle-point value Δsp for the pair field is found through the requirement that Δsp

This is illustrated in Figure 2, which shows the thermodynamic potential Ωsp as a function of Δ for several values of the imbalance chemical potential ζ. The superfluid state exists when Ωsp reaches its minimum at a nonzero value of Δ. As ζ is increased, the normal state at Δ ¼ 0 develops 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

When working with a fixed number of particles, the chemical potential μ and the imbalance chemical potential ζ have to be related to the fermion density nsp and density difference δnsp

Figure 2. The thermodynamic potential Ωsp in function of Δ for several values of the imbalance chemical potential ζ, at temperature T=TF ¼ 0:01 and chemical potential μ ¼ 1:3EF. The evolution of the normal state at Δ ¼ 0 as ζ increases

∂Ωsp ∂Δ � � � � T,μ,ζ

continuous integral in expression (17), we finally find for Ωsp:

ð dk ð Þ 2π 3 1

(between the two spin populations) through the number equations

<sup>Ω</sup>sp ¼ � j j <sup>Δ</sup> <sup>2</sup>

8πkFas �

minimizes Ωsp, which yields the gap equation:

known as the Clogston phase transition [16].

illustrates the Clogston phase transition.

<sup>Z</sup>sp <sup>¼</sup> exp �βVΩsp � �: (19)

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

> 2k 2

¼ 0 (21)

(20)

13

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

( )

volume Ωsp as

$$
\overline{\Psi}(\mathbf{q}, m) \to \sqrt{\beta V} \delta(\mathbf{q}) \delta\_{m, 0} \times \Delta^\*,\tag{16}
$$

where m characterizes the bosonic Matsubara frequencies ω~ <sup>m</sup> ¼ 2mπ=β, and V represents 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 Ψð Þ x; τ takes 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:

$$\mathcal{Z}\_{\rm sp} = \exp\left\{ \frac{|\Delta|^2}{g} - \sum\_{\mathbf{k},n} \ln\left[ (i\omega\_n - E\_\mathbf{k} + \zeta)(-i\omega\_n - E\_\mathbf{k} - \zeta) \right] \right\}.\tag{17}$$

where ω<sup>n</sup> are the fermionic Matsubara frequencies ω<sup>n</sup> ¼ ð Þ 2n þ 1 π=β. We have also introduced the single-particle excitation energy E<sup>k</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ2 <sup>k</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup> q with ξ<sup>k</sup> ¼ k <sup>2</sup> � <sup>μ</sup>, and we have defined the average chemical potential μ and the imbalance chemical potential ζ as

$$
\mu = \frac{\mu\_\uparrow + \mu\_\downarrow}{2} \quad \text{and} \quad \zeta = \frac{\mu\_\uparrow - \mu\_\downarrow}{2}. \tag{18}
$$

The parameter ζ determines the population imbalance between the two spin populations. For ζ ¼ 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 Ωsp as

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 Ψð Þ x; τ . This theory will be presented

At first sight, the introduction of the auxiliary bosonic fields Ψð Þ x; τ and Ψð Þ x; τ through 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 Ψð Þ x; τ and Ψð Þ x; τ that 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 assump-

tion is most easily expressed in momentum-frequency representation f g q; m :

<sup>Ψ</sup>ð Þ! <sup>q</sup>; <sup>m</sup> ffiffiffiffiffiffi

<sup>Ψ</sup>ð Þ! <sup>q</sup>; <sup>m</sup> ffiffiffiffiffiffi

<sup>Z</sup>sp <sup>¼</sup> exp j j <sup>Δ</sup> <sup>2</sup>

the single-particle excitation energy E<sup>k</sup> ¼

<sup>g</sup> �<sup>X</sup> <sup>k</sup>, <sup>n</sup>

average chemical potential μ and the imbalance chemical potential ζ as

<sup>μ</sup> <sup>¼</sup> <sup>μ</sup><sup>↑</sup> <sup>þ</sup> <sup>μ</sup><sup>↓</sup>

where m characterizes the bosonic Matsubara frequencies ω~ <sup>m</sup> ¼ 2mπ=β, and V represents 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 Ψð Þ x; τ takes 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

ln ½ � ðiω<sup>n</sup> � E<sup>k</sup> þ ζÞ �ð Þ iω<sup>n</sup> � E<sup>k</sup> � ζ

with ξ<sup>k</sup> ¼ k

( )

where ω<sup>n</sup> are the fermionic Matsubara frequencies ω<sup>n</sup> ¼ ð Þ 2n þ 1 π=β. We have also introduced

The parameter ζ determines the population imbalance between the two spin populations. For ζ ¼ 0, the numbers of particles of each spin type are equal, while for non-zero values of ζ, there

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ2 <sup>k</sup> <sup>þ</sup> <sup>Δ</sup><sup>2</sup>

<sup>2</sup> and <sup>ζ</sup> <sup>¼</sup> <sup>μ</sup><sup>↑</sup> � <sup>μ</sup><sup>↓</sup>

<sup>β</sup><sup>V</sup> <sup>p</sup> <sup>δ</sup>ð Þ <sup>q</sup> <sup>δ</sup>m, <sup>0</sup> � <sup>Δ</sup>, (15)

<sup>β</sup><sup>V</sup> <sup>p</sup> <sup>δ</sup>ð Þ <sup>q</sup> <sup>δ</sup>m, <sup>0</sup> � <sup>Δ</sup><sup>∗</sup>, (16)

: (17)

<sup>2</sup> � <sup>μ</sup>, and we have defined the

<sup>2</sup> : (18)

in Section 4.

12 Superfluids and Superconductors

function:

3. The mean field theory

$$\mathcal{Z}\_{\rm sp} = \exp\left\{-\beta V \Omega\_{\rm sp}\right\}.\tag{19}$$

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

$$\Omega\_{\rm sp} = -\frac{\left|\Delta\right|^2}{8\pi k\_{\rm P}a\_{\rm s}} - \int \frac{d\mathbf{k}}{\left(2\pi\right)^3} \left\{ \frac{1}{\beta} \left[2\cosh\left(\beta\mathcal{E}\_{\mathbf{k}}\right) + 2\cosh\left(\beta\zeta\right)\right] - \xi\_{\rm k} - \frac{\left|\Delta\right|^2}{2\zeta^2} \right\} \tag{20}$$

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

$$\left.\frac{\partial \Omega\_{\rm sp}}{\partial \Delta}\right|\_{T,\mu,\zeta} = 0\tag{21}$$

This is illustrated in Figure 2, which shows the thermodynamic potential Ωsp as a function of Δ for several values of the imbalance chemical potential ζ. The superfluid state exists when Ωsp reaches its minimum at a nonzero value of Δ. As ζ is increased, the normal state at Δ ¼ 0 develops 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].

When working with a fixed number of particles, the chemical potential μ and the imbalance chemical potential ζ have to be related to the fermion density nsp and density difference δnsp (between the two spin populations) through the number equations

Figure 2. The thermodynamic potential Ωsp in function of Δ for several values of the imbalance chemical potential ζ, at temperature T=TF ¼ 0:01 and chemical potential μ ¼ 1:3EF. The evolution of the normal state at Δ ¼ 0 as ζ increases illustrates the Clogston phase transition.

$$m\_{\rm sp} = -\frac{\partial \Omega\_{\rm sp}}{\partial \mu}\Big|\_{T,\zeta,\Lambda} \tag{22}$$

<sup>S</sup>eff <sup>¼</sup> SB � Tr ln �G�<sup>1</sup> � � � � , (24)

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

� 0 �Ψðx, τÞ �Ψðx, τÞ 0

�

� � (27)

� � <sup>∂</sup><sup>2</sup>

ð Þ τ � τ<sup>0</sup>

F ∂xi∂xj

2 ∂2 F ∂τ<sup>2</sup> � � � � τ0 ,

� � � � x0

, (25)

(26)

15

(28)

<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

1

CCA þ

<sup>0</sup> þ FÞ�

Tr½ðG0FÞ

�Δ 0

<sup>0</sup> describes free fermionic fields, while F describes the pairing of the fermions.

<sup>0</sup> Þ� � Tr½lnð1 � G0FÞ�

p �:

<sup>0</sup> ðx, τÞ þ Fðx, τÞ

<sup>x</sup> � μ<sup>↑</sup> 0

Sef f <sup>¼</sup> SB � Tr½lnð�G�<sup>1</sup>

¼ SB þ S<sup>0</sup> þ

from the previous section can be retrieved by simply setting

Fð Þ x; τ ≈ F<sup>0</sup> þ ð Þ� x � x<sup>0</sup> ∇xFj

<sup>¼</sup> SB � Tr½lnð�G�<sup>1</sup>

∂τ <sup>þ</sup> <sup>∇</sup><sup>2</sup> <sup>x</sup> þ μ<sup>↓</sup>

Using this decomposition, we can write the effective bosonic action functional (24) as

X<sup>∞</sup>

1 p

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

<sup>F</sup>ð Þ <sup>x</sup>; <sup>τ</sup> <sup>≈</sup> <sup>F</sup>sp <sup>¼</sup> <sup>0</sup> �<sup>Δ</sup>

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

> X i, <sup>j</sup>¼x, y, <sup>z</sup>

þð Þ τ � τ<sup>0</sup>

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

∂F ∂τ � � � � τ0 þ 1 2

ð Þ xi � x0,i xj � x0,j

x0 þ 1 2

p¼1

<sup>0</sup> <sup>∂</sup>

where SB ¼ � <sup>Ð</sup>

where �G�<sup>1</sup>

expansion

β 0 Ð

�G�<sup>1</sup>

¼

<sup>d</sup><sup>x</sup> j j <sup>Ψ</sup>ð Þ <sup>x</sup>;<sup>τ</sup> <sup>2</sup>

into its diagonal and off-diagonal components

∂ <sup>∂</sup><sup>τ</sup> � <sup>∇</sup><sup>2</sup>

0

BB@

<sup>ð</sup>x, <sup>τ</sup>޼�G�<sup>1</sup>

$$
\delta n\_{\rm sp} = -\frac{\partial \Omega\_{\rm sp}}{\partial \zeta}\Big|\_{T,\mu,\Delta} \tag{23}
$$

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 crossover are shown in Figure 3a and b.
