From Equilibrium to Transport Properties of Strongly Correlated Fermi Liquids
Abstract
We summarize recent results regarding the equilibrium and nonequilibrium behavior of cold dilute atomic gases in the limit in which the two body scattering length goes to infinity. In this limit the system is described by a Galilean invariant (nonrelativistic) conformal field theory. We discuss the low energy effective lagrangian appropriate to the limit , and compute low energy coefficients using an expansion. We also show how to combine the effective lagrangian with kinetic theory in order to compute the shear viscosity, and compare the kinetic theory predictions to experimental results extracted from the damping of collective modes in trapped Fermi gases.
1 Introduction
Over the last ten years there has been remarkable progress in the study of “designer fluids”, dilute, nonrelativistic Bose and Fermi gases in which the scattering length between the Bosons or Fermions can be continuously adjusted. In the following we are particularly interested in Fermi gases, since these systems are stable for both positive and negative values of the scattering length, including the strongly correlated limit in which the scattering length is taken to infinity.
The scattering length is controlled through a Feshbach resonance. Alkali atoms such as Li and K have a single valence electron. When a dilute gas of atoms is cooled to very low temperatures, we can view the atoms as pointlike particles interacting via interatomic potentials which depend on the hyperfine quantum numbers. A Feshbach resonance arises if a molecular bound state in a “closed” hyperfine channel crosses near the threshold of an energetically lower “open” channel. Because the magnetic moments of the open and closed states are in general different, Feshbach resonances can be tuned using an applied magnetic field. At resonance the twobody scattering length in the open channel diverges, and the cross section is limited only by unitarity, for low momenta . In the unitarity limit, details about the microscopic interaction are irrelevant, and the system displays universal properties.
Near a Feshbach resonance the scattering length behaves as
(1) 
where is the nonresonant value of the scattering length (typically on the order of the effective range of the interatomic potential), is the magnetic field, the position of the resonance, and the width. A small negative scattering length corresponds to a weak attractive interaction between the atoms. This case is known as the BCS (BardeenCooperSchrieffer) limit. On the other side of the resonance the scattering length is positive. In the BEC (BoseEinstein condensation) limit the interaction is strongly attractive and the fermions form deeply bound molecules. For this reason the unitarity limit is also known at the BCS/BEC crossover.
The unitarity limit is of interest to QCD practitioners for a for a number of reasons:

The unitarity limit provides an approximate description of dilute neutron matter. The neutronneutron scattering length is fm, and the effective range is fm. This means that there is a range of densities, relevant to the outer layers of neutron stars, for which the interparticle spacing is large compared to the effective range, but small compared to the scattering length.

The Fermi gas at unitarity is a high superconductor. There is an attractive interaction in the spin singlet channel which leads to swave superconductivity below some critical temperature . In the unitarity limit the only energy scale in the problem is the Fermi energy , and we must have with some numerical constant . Quantum Monte Carlo calculations (and experimental results) indicate that [1, 2], much larger than in ordinary (or even high ) electronic superconductors, but comparable to what might be achieved in color superconducting quark matter [3].

The limit corresponds to a nonrelativistic conformal field theory [4]. In the unitarity limit there is no scale in the problem (other than the thermodynamic variables temperature and density). Indeed, one can show that the theory is not only scale invariant, but invariant under the full conformal group. This raises the question whether there are any physical consequences of conformal symmetry that go beyond results that follow from scale invariance. It also raises the possibility that a holographic description, similar to the correspondence, can be obtained [5, 6].

Nonrelativistic fermions at unitarity behave as a very good fluid and show interesting transport properties, including a very small shear viscosity. Kinetic theory suggests that the shear viscosity is inversely proportional to the scattering cross section, and reaches a minimum at unitarity. This expectation is confirmed by experiments that demonstrate large elliptic flow and a very small damping rate for collective oscillations [7, 8].
2 Equilibrium Properties
We begin by analyzing equilibrium properties of the dilute Fermi gas at unitarity. If the temperature is large, , then the scattering cross section is regularized by the thermal wave length, and the effective interaction is weak. Here the Fermi energy is defined by , where is the density, and is the mass of the atoms. In the high temperature regime the equation of state is well described by the Virial expansion, and the system has single particle excitations with the quantum numbers of the fundamental fermions. In the regime the interactions are strong. As noted above, superfluidity occurs at . Below the critical temperature the excitations are Goldstone bosons. In following section we will discuss the effective theory of the Goldstone bosons, and relate the parameters in the effective lagrangian to static properties of the system.
2.1 Low Energy Effective Theory and Density Functional
The Goldstone boson field can be defined as the phase of the difermion condensate . The effective Lagrangian at nexttoleading order (NLO) in derivatives of and the external potential is [9]
(2) 
where we have defined
(3) 
Here, is the chemical potential and is an external potential. The functional form of the effective lagrangian is fixed by the symmetries of the problem, Galilean invariance, symmetry, and conformal symmetry. The NLO effective lagrangian is characterized by three dimensionless parameters, . These parameters can be related to physical properties of the system. The first parameter, , can be related to the equation of state. We have
(4) 
where determines the chemical potential in units of the Fermi energy, . The two NLO parameters are related to the momentum dependence of correlation functions. The phonon dispersion relation, for example, is given by
(5) 
where and . The static susceptibility
(6) 
involves a different linear combination of and ,
(7) 
Higher derivative terms in the effective lagrangian can also be used to compute the energy of inhomogeneous matter. At NLO in an expansion in derivatives of the density we find the following energy density functional [10]
The first two terms correspond to the local density approximation (LDA) and the terms proportional to and are the leading correction to the LDA involving derivatives of the density.
2.2 Epsilon Expansion
At unitarity the determination of and is a nonperturbative problem, and we will perform the calculation using an expansion around spatial dimensions [11, 12]. Our starting point is the lagrangian
(9) 
where is a twocomponent NambuGorkov field, are Pauli matrices acting in the NambuGorkov space, , is a complex boson field, and is a coupling constant. In dimensional regularization the fermionfermion scattering length becomes infinite for .
The epsilon expansion is based on the observation that the fermionfermion scattering amplitude near dimensions is saturated by the propagator of a boson with mass . The coupling of the boson to pairs of fermions is given by
(10) 
In the superfluid phase acquires an expectation value . We write the boson field as . The lagrangian is split into a free part
(11) 
and an interacting part , where
(12)  
(13) 
Note that the leading self energy corrections to the boson propagator generated by the interaction term cancel against the counterterms in . The chemical potential term for the fermions is included in rather than in . This is motivated by the fact that near the system reduces to a noninteracting Bose gas and . We will count as a quantity of . The Feynman rules are quite simple. The fermion and boson propagators are
(14)  
(15) 
where and . The fermionboson vertices are . Insertions of the chemical potential are . Both and are corrections of order .
In order to determine we have to compute three physical observables. We have studied , and the curvature terms in the phonon dispersion relation and the static susceptibility. The universal parameter was originally calculated by Nishida and Son. They computed the effective potential to NLO in the epsilon expansion, see Fig. 1. The derivative of the effective potential with respect to determines the density , and the relation between and fixes . The result is
(16) 
with . The phonon dispersion relation can be extracted from the scalar propagator. We introduce a twocomponent scalar field . The scalar propagator satisfies a DysonSchwinger equation [13]
(17) 
At LO in the epsilon expansion the self energy is determined by the diagrams shown in Fig. 2. NLO contributions were calculated in [10]. The phonon dispersion relation is
(18) 
We note that the dispersion relation curves up (unlike He, but similar to weakly interacting Bose gases). This implies that there is decay. Finally, we can determine the static susceptibility. Computing the diagrams in Fig. 3 we get [10, 14]
The coefficient follows from the result for ( at NLO in the expansion) using equ. (4). Matching equ. (18,2.2) against equ. (5,7) gives and . The corresponding energy density functional was studied in [10]. Compared to a free Fermi gas the local density term is reduced by a factor (the interaction is attractive), while the gradient correction proportional to is enhanced by a factor .
3 Transport Properties
In the following we will discuss transport properties of the Fermi gas at unitarity. The interest in nonequilibrium properties arises from the observation that transport coefficients are much more sensitive to the strength of the interaction than thermodynamic quantities. A renewed interest in transport properties was also sparked the AdS/CFT correspondence and the experimental limits on the shear viscosity of the quark gluon plasma obtained at RHIC. In the following we shall focus on the shear viscosity of the Fermi gas at unitarity. Close to equilibrium the (coarse grained) energy momentum tensor can be written as
(20)  
where and are the energy density and pressure, and is the local flow velocity. The first term is the ideal gas contribution, and is the leading order (in gradients of ) dissipative correction. The traceless part of is proportional to the shear viscosity .
3.1 Kinetic Theory
We first consider the case that the fluid is composed of weakly interacting quasiparticles. In the unitarity limited Fermi gas this is the case at (phonons) and (atoms). In these limits we can compute the shear viscosity using kinetic theory. In the following we will concentrate on the low temperature case discussed in [15]. In kinetic theory the stressenergy tensor is given by
(21) 
where is the distribution function of the phonons, is the speed of sound, is the momentum and the quasiparticle energy. Close to equilibrium , where is the BoseEinstein distribution and is a small departure from equilibrium. We write . In the case of shear viscosity we can further decompose
(22) 
Inserting equ. (22) into equ. (21) we get
(23) 
The nonequilibrium distribution is determined by the Boltzmann equation
(24) 
relating the rate of change of the distribution function to the collision operator . The collision integral is given by
where contains the distribution functions and is the scattering amplitude shown in Fig. 4. The three and fourphonon vertices are fixed by the effective lagrangian (2). Linearizing in one finds
(26) 
There are a variety of methods for solving the linearized Boltzmann equation. A standard technique is based on expanding in a complete set of functions. A nice feature of this method is that the truncated expansion gives a variational estimate
(27) 
where is a set of expansion coefficients, is a normalization integral, and are matrix elements of the linearized collision operator. For the best trial function we find [15]
(28) 
where is the universal parameter introduced in Sect. 2.1 and we have normalized the result to the entropy density of a weakly interacting phonon gas. A similar estimate can be obtained in the high temperature limit. In this case the relevant degrees of freedom are atoms, and the dominant scattering process is shown in Fig. 4b. The result is [16, 17]
(29) 
The high and low temperature limits of are shown in Fig. 5, together with the proposed lower bound [18] and experimental data which we will discuss in the next section.
3.2 Hydrodynamics
Hydrodynamics describes the evolution of longwavelength, slowfrequency modes. The hydrodynamic description remains valid even if there is no underlying kinetic theory. The hydrodynamic equations follow from conservation of mass (particle number), energy and momentum. In a nonrelativistic system the equations of continuity and of momentum conservation are given by
(30)  
(31) 
where is the number density, is the mass of the atoms, is the fluid velocity, is the pressure and is the external potential. In an ideal fluid the equation of energy conservation can be rewritten as conservation of entropy,
(32) 
A nonzero shear viscosity leads to dissipation, converting kinetic energy to heat and increasing the entropy. The shear viscosity of the dilute Fermi gas in the unitarity limit can be measured by studying the damping of collective modes in trapped systems [21]. The frequency of these modes agrees well the prediction of ideal hydrodynamics. The dissipated energy is given by
(33) 
The damping rate is given by the ratio of the energy dissipated to the total energy of the collective mode. The kinetic energy is
(34) 
If the damping rate is small both and can be computed using the solution of ideal hydrodynamics. We recently performed an analysis [22] which is based on measurements of the damping rate of the lowest radial breathing mode performed by the Duke group [8]. We showed that can relate the dimensionless ratio , where is the damping rate and is the trap frequency, to the shear viscosity to entropy density ratio
(35) 
Here is the total number of particles in the trap ( in [8]), is the universal parameter defined in Sec. 2.1, is the ratio of the total energy to the energy at (which can be extracted using a Virial theorem from the measured cloud size), and is the entropy per particle (which is measured using adiabatic sweeps to the BCS limit [20]). The results are compared to theoretical prediction in the high and low temperature limit in Fig. 5. The data show a minimum near . At the minimum . This should probably be considered as an upper bound, since dissipative mechanism other than shear viscosity may be present. In the high limit there is fairly good agreement with kinetic theory. The temperature dependence implied by the low prediction is not seen in the data. This is maybe not very surprising, since the mean free path in the low regime quickly exceeds the system size.
4 Outlook
There are many promising directions for further study. Clearly, it is desirable to obtain additional experimental constraints on the shear viscosity, and to improve the theoretical analysis of the existing data sets. It would also be interesting to confirm that the bulk viscosity vanishes in the normal phase, and to measure the thermal conductivity. We would also like to improve the theoretical tools for computing transport properties in the interesting regime near . There are some recent ideas for applying holography and the correspondence to Galilean invariant conformal field theories [5, 6], but there are also many purely field theoretic methods ( expansions, large methods) that have yet to be pursued.
Acknowledgments: Much of this work was carried out in collaboration with G. Rupak. The work is supported in part by the US Department of Energy grant DEFG0203ER41260.
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