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

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

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.

Z<sup>B</sup> ¼ ð

Z<sup>F</sup> ¼ ð Dψ ð

the matrix A.

approximations.

1

for this system is given by

S<sup>E</sup> ¼

ð ℏβ

dx X σ∈ f g ↑;↓

0 dτ ð

þ ð ℏβ

0 dτ ð dx ð

the spin-dependence of the fermionic field is considered in the theory.

valued) field ψð Þ x; τ :

DΨDΨexp �

Dψexp �

ð dτ ð dx ð dτ<sup>0</sup> ð dx 0

ð dτ ð dx ð dτ<sup>0</sup> ð dx 0

of the matrix A containing the coefficients of the quadratic form.

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

Ψð Þ x; τ A x; τ; x

ψð Þ x; τ A x; τ; x

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

Fermionic path integral: The path integral sums over a fermionic (Grassmann, complex

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

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

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

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:

The matrix A can be thought of as an infinite matrix composed of either 2 � 2 or 4 � 4 matrices, depending on whether

ψσð Þ<sup>x</sup> <sup>ℏ</sup>∂<sup>τ</sup> � <sup>ℏ</sup><sup>2</sup>

2m<sup>x</sup> ∇2 <sup>x</sup> � μσ

dyψ↑ð Þx ψ↓ð Þ y gδð Þ x � y ψ↓ð Þ y ψ↑ð Þx ,

� � � � ψσð Þ<sup>x</sup>

� � h i � � <sup>¼</sup> <sup>1</sup>

0 ; τ<sup>0</sup> � �<sup>Ψ</sup> <sup>x</sup>

0 ; <sup>τ</sup><sup>0</sup> � �<sup>ψ</sup> <sup>x</sup>

Z<sup>B</sup> ¼ exp ð Þ �Tr ln½ � ð Þ A , (5)

Z<sup>F</sup> ¼ exp ð Þ þTr ln½ � ð Þ A : (6)

� � h i � � <sup>¼</sup> detð Þ <sup>A</sup> , (4)

0 ; τ<sup>0</sup>

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

> 0 ; τ<sup>0</sup>

detð Þ <sup>A</sup> , (3)

9

(7)
