**1. Introduction**

18 Will-be-set-by-IN-TECH

[41] Preseter, M. (1996). Dynamical exponents for the current-induced percolation transition

in high-Tc superconductors. *Phys. Rev. B*, Vol. 54 (No. 1), 606–618.

Before we start applying path integration to treat Cooper pairing and superfluidity, it is a good idea to quickly review the concepts behind path integration. There are many textbooks providing plentiful details, such as Feynman's seminal text [1] and Kleinert's comprehensive compendium [2], and other works listed in the bibliography [3–5]. We will assume that the reader is already familiar with the basics of path-integral theory, so if the following paragraphs are not merely reminders to you, it is probably better to first consult these textbooks.

Quantum mechanics, according to the path-integral formalism, rests on two axioms. The first axiom, the superposition axiom, states that the amplitude of any process is a weighed sum of the amplitudes of all possible possibilities for the process to occur. These "possible possibilities" should be interpreted as the alternatives that cannot be distinguished by the experimental setup under consideration. For example, the amplitude for a particle to go from a starting point "A" to a final point "B" is a weighed sum of the amplitudes of all the paths that this particle can take to get to "B" from "A". The second axiom assigns to the weight the complex value exp{*i*S/¯*h*} where S is the action functional. In our example, each path *x*(*t*) that the particle can take to go from A to B gets a weight exp {*i*S[*x*(*t*)]/¯*h*} since the action is the time integral of the Lagrangian. There is a natural link with quantum-statistical mechanics: in the path-integral formalism, quantum statistical averages are expressed as the same weighed averages but now the weight is a real value exp{−S[*x*(*τ*)]/¯*h*} and the path is taken in imaginary time *τ* = *it*.

In the example of the above paragraph, we considered a particle which could take many different paths from A to B. However, the same axioms can be applied to fields. As an example we take a complex scalar field *φ***x**,*t*, where **x** and *t* denote position and time respectively. Let us mentally discretize space-time, and to make things easy, we assume there are only five moments in time and five places to sit. In this simple universe, the field *φ***x**,*<sup>t</sup>* is represented by a set of 25 complex numbers, i.e. an element of **C**<sup>25</sup> if **C** is the set of complex numbers. Summing over all possible realizations of the fields corresponds to integrating over **C**25, a 25-fold integral over complex variables, or a 50-fold integral over real variables. Writing

©2012 Tempere and Devreese, licensee InTech. This is an open access chapter 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. ©2012 Tempere and Devreese, licensee InTech. This is a paper 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.

*φ***x**,*<sup>t</sup>* = *u***x**,*<sup>t</sup>* + *iv***x**,*<sup>t</sup>* with *u* and *v* real, the summation over all possible possibilities for *φ***x**,*<sup>t</sup>* is written as

$$\int \mathcal{D}\phi\_{\mathbf{x},t} := \prod\_{\mathbf{x}=1}^{5} \prod\_{t=1}^{5} \int du\_{\mathbf{x},t} \int dv\_{\mathbf{x},t} \,. \tag{1}$$

<sup>S</sup>[*ψ*¯**x**,*τ*,*σ*, *<sup>ψ</sup>***x**,*τ*,*σ*]/¯*<sup>h</sup>* <sup>=</sup> <sup>∑</sup>*<sup>σ</sup>*

Z = 

functionals.

to include this effect.

50 pairs of Grassmann elements). The result is

<sup>D</sup>*ψ*¯**x**,*τ*,*<sup>σ</sup>*

 *d***x** 

**2. The action functional for the atomic Fermi gas**

be investigated in a way that is thus far not possible in solids.

*dt* ∑ *σ*�

 *d***x**� 

where *ψ*¯**x**,*τ*,*<sup>σ</sup>* and *ψ***x**,*τ*,*<sup>σ</sup>* are different Grassmann variables (so in our example, we would need

Despite having only analytic results for quadratic action functionals, the path-integral technique is nevertheless a very versatile tool and has become in fact the main tool to study field theory [9]. The trick usually consists in finding suitable transformations and approximations to bring the path integral into the same form as that with quadratic action

In the study of superconductivity, ultracold quantum gases offer a singular advantage over condensed matter systems in that their system parameters can be tuned experimentally with a high degree of precision and over a wide range. For example, the interaction strength between fermionic atoms is tunable by an external magnetic field. This field can be used to vary the scattering length over a Feshbach resonance, from a large negative to a large positive value. In the limit of large negative scatting lengths, a cloud of ultracold fermionic atoms will undergo Cooper pairing, and exhibit superfluidity when cooled below the critical temperature. On the other side of the resonance, at large positive scattering lengths, a molecular bound state gets admixed to the scattering state, and a Bose-Einstein condensate (BEC) of fermionic dimers can form. With the magnetically tunable s-wave scattering length, the entire crossover region between the Bardeen-Cooper-Schrieffer (BCS) superfluid and the molecular BEC can

Also the amount of atoms in each hyperfine state can be tuned experimentally with high precision. Typically, fermionic atoms (such as 40K or 6Li) are trapped in two different hyperfine states. These two hyperfine spin states provide the "spin-up" and "spin-down" partners that form the Cooper pairs. Unlike in metals, in quantum gases the individual amounts of "spin-up" and "spin-down" components of the Fermi gas can be set independently (using evaporative cooling and Rabi oscillations). This allows to investigate how Cooper pairing (and the ensuing superfluidity) is frustrated when there is no equal amount of spin ups and spin downs, i.e. in the so-called "(spin-)imbalanced Fermi gas". In a superconducting metal, the magnetic field that would be required to provide a substantial imbalance between spin-up and spin-down electrons is simply expelled by the Meissner effect, so that the imbalanced situation cannot be studied. The particular question of the effect of spin-imbalance is of great current interest [10], and we will keep our treatment general enough

Finally, the geometry of the gas is adaptable. Counterpropagating laser beams can be used to make periodic potentials for the atoms, called "optical lattices". Imposing such a lattice in just one direction transforms the atomic cloud into a stack of pancake-shaped clouds,

*dt*� *ψ*¯**x**,*τ*,*σ***A**(**x**, *t*; **x**�

<sup>D</sup>*ψ***x**,*τ*,*<sup>σ</sup>* exp {−S [*ψ*¯**x**,*τ*,*σ*, *<sup>ψ</sup>***x**,*τ*,*σ*]} <sup>=</sup> det(**A**). (7)

, *t* � )*ψ***x**� ,*τ*�

Path-Integral Description of Cooper Pairing 385

,*σ*� , (6)

The notation with calligraphic D indicates the path-integral sum, and we keep this notation for actual continuous spacetime, that we may see as a limit of a finer and finer grid of spacetime points (our 5x5 grid is obviously very crude). Although this is, strictly speaking, no longer a sum over paths, it is still called a path integral because the description is based on the same axiomatic view as outlined in the previous paragraph.

Each particular realization of *φ***x**,*<sup>t</sup>* again gets assigned a weight, where now we need a functional of *φ***x**,*<sup>t</sup>* (or, in our example, a function of 25 complex variables). Again, we use the action functional

$$\mathcal{S}[\phi\_{\mathbf{x},t}] = \int \mathcal{L}(\phi\_{\mathbf{x},t\prime}\phi\_{\mathbf{x},t})dt\tag{2}$$

to construct the weight, where L is the Lagrangian of the field theory suitable for *φ***x**,*t*. A central quantity to calculate is the statistical partition sum

$$\mathcal{Z} = \int \mathcal{D}\phi\_{\mathbf{x},\tau} \, \exp\left\{-\mathcal{S}[\phi\_{\mathbf{x},\tau}]/\hbar\right\},\tag{3}$$

where *τ* = *it* indicates imaginary times required for the quantum statistical expression, running from *τ* = 0 to *τ* = *h*¯ *β* with *β* = 1/(*kBT*) the inverse temperature. Bose gases in condensed matter are described by complex scalar fields like *φ***x**,*t*, and the path integral can basically only be solved analytically when the action functional is quadratic in form, i.e. when

$$\mathcal{S}[\phi\_{\mathbf{x},\tau}]/\hbar = \int d\mathbf{x} \int dt \int d\mathbf{x}' \int dt' \,\phi\_{\mathbf{x},\tau} \mathcal{A}(\mathbf{x},t;\mathbf{x}',t') \phi\_{\mathbf{x}',\tau'}.\tag{4}$$

In our simple universe, **A** would be a 25×25 matrix, and the path integral would reduce to a product of 25 complex Gaussian integrals, leading to

$$\mathcal{Z} \propto \frac{1}{\det(\mathbb{A})}.\tag{5}$$

The proportionality is written here because every integration also gives a (physically unimportant) factor *π* that can be absorbed in the integration measure, if needed.

Fermionic systems, such as the electrons in a metal or ultracold fermionic atoms in a magnetic trap, cannot be described by complex scalar fields: fermionic fields should anticommute [6]. If we axiomatically impose anticommutation onto scalar variables, we obtain Grassmann variables [7]. Since there is also a spin degree of freedom, the fermionic fields require a spin index *σ* as well as spacetime indices **x**, *τ*: *ψ***x**,*τ*,*σ*. As Grassmann variables anticommute, we have *ψ*<sup>2</sup> **<sup>x</sup>**,*τ*,*<sup>σ</sup>* = 0. Integrals over Grassmann variables ("Berezin-Grassmann integrals" [8]) are defined by *dψ***x**,*τ*,*<sup>σ</sup>* = 0 and *ψ***x**,*τ*,*σdψ***x**,*τ*,*<sup>σ</sup>* = 1. As was the case for bosonic fields, also for fermionic fields there is only one generic path integration that can be done analytically, namely that with a quadratic action. A quadratic action functional in Grassmann fields is written in general form as

$$\mathcal{S}\left[\bar{\psi}\_{\mathbf{x},\tau,\sigma},\psi\_{\mathbf{x},\tau,\sigma}\right]/\hbar = \sum\_{\sigma} \int d\mathbf{x} \int dt \sum\_{\sigma'} \int d\mathbf{x}' \int dt' \,\bar{\psi}\_{\mathbf{x},\tau,\sigma} \mathbf{A}(\mathbf{x},t;\mathbf{x}',t')\psi\_{\mathbf{x}',\tau',\sigma'} \tag{6}$$

where *ψ*¯**x**,*τ*,*<sup>σ</sup>* and *ψ***x**,*τ*,*<sup>σ</sup>* are different Grassmann variables (so in our example, we would need 50 pairs of Grassmann elements). The result is

$$\mathcal{Z} = \int \mathcal{D}\bar{\psi}\_{\mathbf{x},\tau,\sigma} \int \mathcal{D}\psi\_{\mathbf{x},\tau,\sigma} \exp\left\{-\mathcal{S}\left[\bar{\psi}\_{\mathbf{x},\tau,\sigma}, \psi\_{\mathbf{x},\tau,\sigma}\right]\right\} = \det\left(\mathbb{A}\right). \tag{7}$$

Despite having only analytic results for quadratic action functionals, the path-integral technique is nevertheless a very versatile tool and has become in fact the main tool to study field theory [9]. The trick usually consists in finding suitable transformations and approximations to bring the path integral into the same form as that with quadratic action functionals.
