A Viscosity of a Lennard-Jones fluid

# Viscosity and scale invariance in the unitary Fermi gas

## Abstract

We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of , thus invalidating a quasiparticle description. The ratio between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound by a factor of about seven.
PACS: 67.10.Jn; 67.85.Lm; 11.30.-j

## 1 Introduction

The remarkable derivation of a simple proportionality between the shear viscosity and the entropy per volume in a supersymmetric Yang-Mills theory in the limit of infinite ’t Hooft coupling by Policastro, Son and Starinets Policastro et al. (2001) and the conjecture by Kovtun, Son and Starinets (KSS) Kovtun et al. (2005) that the ratio is larger than the value found in this limit for all scale invariant, relativistic field theories have motivated the search for the ‘perfect fluid’ which realizes, or at least comes close to, this bound Schaefer and Teaney (2009). In spite of some theoretical counter-examples Cohen (2007); Son (2008); Brigante et al. (2008); Kats and Petrov (2009); Buchel et al. (2009), the KSS conjecture turns out to be valid for all real fluids that are known. In particular, since the velocity of light does not appear in the KSS bound, the conjecture is applicable to both relativistic and non-relativistic field theories and may even be extended to complex, classical fluids. In the case of water, for instance, the minimum value for the ratio is found close to its critical point at K and is only a factor of above the KSS bound Kovtun et al. (2005); Schaefer and Teaney (2009). Clearly, is irrelevant for the minimum value of of water at these temperatures. Yet, as shown in A, there is a simple argument which shows that the viscosity minimum of purely classical fluids is not far above that expected from the KSS bound. This begs the question what are the necessary conditions for a fluid to be ‘perfect’ in the sense of a minimum value of that is limited only by quantum mechanics and, moreover, what is the role of scale invariance in this context? As pointed out by Kovtun, Son and Starinets Kovtun et al. (2005) and discussed in more detail in a number of recent lecture notes Hartnoll (2009); McGreevy (2009); Sachdev (2010) on connections between holographic duality and many-body physics, a crucial requirement for a fluid to come close to the KSS bound is the fact that it is strongly interacting and thus has no well-defined quasiparticles. Indeed, in a situation with proper quasiparticles, transport coefficients like the shear viscosity can be computed using kinetic theory. Since the lifetime broadening is much less than the average energy for well-defined quasiparticles, the resulting is typically far above the KSS bound Schaefer and Teaney (2009). By contrast, for strongly coupled quantum field theories, the relaxation times are expected to be of order . Since is a characteristic time scale for shear relaxation, this immediately implies that is of order in the strongly coupled limit. A nontrivial example in this context is the standard Yang-Mills theory. The associated ratio for the pure gauge theory has been calculated numerically using lattice QCD Meyer (2007, 2008a, 2008b). Its minimum appears near the deconfinement transition temperature and turns out to be rather close to the KSS bound. The standard Yang-Mills theory near and its supersymmetric extension—for which is independent of temperature and no well-defined quasiparticles exist at arbitrary energies—thus have very similar values of . Adding fermions to the pure gauge theory, the ratio becomes an experimentally accessible quantity in high-energy, non-central collisions of heavy nuclei. The observed ratio of the quark-gluon plasma is around , i.e., a factor five above the KSS bound Schaefer and Teaney (2009).

In a condensed matter context, generic examples for strongly coupled, finite temperature field theories for which no quasiparticle description holds are provided by models that exhibit a zero temperature critical point Sachdev (1998). These systems are scale invariant at a critical value of the coupling . At finite temperature , there is a quantum critical regime above the critical point, which covers a finite window . In this regime, the thermal energy is the only energy scale and correlations of the order parameter exhibit incoherent relaxation with a characteristic time scale Sachdev (2010). Here, is a constant that only depends on the universality class of the quantum phase transition. In addition, universal behavior shows up in transport coefficients like the conductivity and also the shear viscosity in the hydrodynamic regime . A concrete example is the pseudo-relativistic theory of graphene where the ratio has recently been calculated within a Boltzmann equation approach. The marginally irrelevant Coulomb interaction in this case gives rise to a logarithmic temperature dependence of the prefactor Müller et al. (2009). This yields a monotonically increasing viscosity as the temperature approaches zero within the quantum critical regime. Logarithmic singularities as are also present in standard 2d Fermi liquids Novikov (2006).

Here, we consider the shear viscosity for the unitary Fermi gas, a system of attractively interacting Fermions at infinite scattering length. The unitary Fermi gas is realized experimentally with ultracold atoms in a balanced mixture of, e.g., the two lowest hyperfine levels of Li at a Feshbach resonance and has been studied quite extensively over the past few years Ketterle and Zwierlein (2008); Bloch et al. (2008); Giorgini et al. (2008). It provides an example of a non-relativistic field theory that is both scale- and conformally invariant Nishida and Son (2007). The underlying quantum critical point in this case is the zero density gas at unitarity, as was shown by Nikolić and Sachdev Nikolić and Sachdev (2007). As a consequence of scale invariance, pressure and energy density of the gas are related by Ho (2004). Moreover, the bulk viscosity vanishes at all temperatures Son (2007); Werner and Castin (2006). For the shear viscosity, quantitative results so far are only available in the high-temperature limit where a classical description in terms of a Boltzmann equation is possible Bruun and Smith (2007), and also deep in the superfluid regime . In the superfluid, a finite viscosity arises from phonon-phonon collisions in the normal fluid component, giving rise to a rapid increase of the viscosity as the temperature approaches zero Rupak and Schaefer (2007). Since also increases in the classical limit, both the viscosity and the ratio of the unitary gas necessarily exhibit a minimum, a behavior which is in fact typical for any fluid Schaefer and Teaney (2009). The major question which will be discussed in this paper, is where this minimum appears and what the associated value of the viscosity and entropy is. It turns out that the minimum in for the non-relativistic unitary gas is about a factor of seven above the KSS bound, rather close to the value that is found for the relativistic quark-gluon plasma. We also show that the minimum in implies a lower bound on the shear diffusion constant , which is about .

In detail, we determine the real part of the frequency-dependent shear viscosity from a diagrammatic method that evaluates the stress tensor correlation function in the exact Kubo formula. Within a conserving approximation that respects all symmetries and the associated conservation laws, we obtain in the normal phase and, in particular, the static viscosity to entropy density ratio . Our results for show a Drude-like transport peak around . Its width defines a viscous scattering rate which obeys at high temperatures, where kinetic theory is applicable. Near the transition temperature to the superfluid phase of the unitary gas, the width approaches , thus clearly invalidating a quasiparticle description of viscous transport in this regime. The weight of the Drude peak turns out to be equal to the pressure at all temperatures, consistent with a sum rule derived recently by Taylor and Randeria Taylor and Randeria (2010). For large frequencies , there is a crossover from the Drude peak to an inverse square-root tail , whose amplitude is proportional to the Tan contact density Tan (2008). In the high-temperature limit we complement our numerical solution of the transport integral equations by an analytical solution to leading order in the fugacity . We thus confirm that the Kubo formula yields exactly the same expression as the Boltzmann equation in this limit, and the vertex corrections, in particular the Aslamazov-Larkin contributions, are crucial even in the classical limit and increase the scattering time by a factor of almost three. This resolves an inconsistency between the Boltzmann equation result and a previous diagrammatic calculation in the high-temperature limit by Bruun and Smith Bruun and Smith (2007). Finally, we derive Ward identities associated with the scale invariance of the unitary gas and show that they are obeyed within our approximation for the bulk viscosity vertices at all frequencies and temperatures. As a consequence, the bulk viscosity vanishes identically Son (2007); Nishida and Son (2007); Taylor and Randeria (2010).

The paper is organized as follows: in section 2 we define the microscopic model for the unitary gas and discuss its scale invariance. The basic correlation functions that allow to calculate the shear viscosity from first principles via a Kubo formula are defined in section 3 together with a discussion of hydrodynamic relations and sum rules. In section 4 we give a qualitative discussion of the temperature dependence of the shear viscosity in the classical and in the deep superfluid limit, as well as near the superfluid transition temperature. The transport integral equations and their solution within a self-consistent Luttinger-Ward formalism are derived in section 5. An exact solution of these equations to leading order in the fugacity is presented in section 6, which turns out to reproduce the full solution of the Boltzmann equation in the high temperature limit. In section 7 we discuss our results for the frequency dependent viscosity in the low temperature—but still normal fluid—regime of the unitary gas, where the assumption of independent collisions of well defined quasiparticles that underlies the Boltzmann equation breaks down. In particular, we calculate the shear viscosity to entropy density ratio , which displays a minimum value slightly above the transition to the superfluid. Conclusions and open questions are presented in section 8. There are three appendices: in A, we give an argument based on purely dimensional analysis which explains why the inherently quantum mechanical KSS bound applies even to purely classical systems like a Lennard-Jones fluid. B is devoted to the derivation of the Ward identities that result from scale invariance and the associated vanishing of the bulk viscosity. Finally, in C we analytically compute the tail coefficient of the shear viscosity at large frequencies.

## 2 The unitary Fermi gas and scale invariance

The basic model Hamiltonian that describes a two-component Fermi gas with contact interactions of strength is given by

 ^H=∑pσεpc†pσcpσ+¯g(Λ)∑pp′qc†p+q↑c†p′−q↓cp′↓cp↑. (1)

Here is the kinetic energy, while denotes the two fermion species (hyperfine levels in the context of cold atoms). Since a contact interaction in three dimensions needs to be regularized, the coupling strength is cutoff-dependent. It is related to the renormalized physical coupling strength that is fixed by the scattering length via . The unitary gas corresponds to the special case of infinite scattering length. In order to see that the system is scale invariant at this point, it is convenient to model the contact interaction by an exchange of bosons. In a path integral language, the partition function at temperature can then be written as a double integral

 Z=∫D(¯ψσ,ψσ)D(¯ψB,ψB)e−S (2)

over Grassmann fields , with and a complex scalar field , . In the case of a Feshbach resonance, this field has a direct physical interpretation as a bound fermion pair in the closed channel. The action that describes the interacting Fermi gas in this two-channel description is

 S=∫β0dτ∫d3x[∑σ=↑,↓¯ψσ[ℏ∂τ−ℏ2∇22m]ψσ+¯ψB[ℏ∂τ−ℏ2∇24m+ν]ψB+~g(¯ψBψ↑ψ↓+h.c.)]. (3)

Here, is the detuning of the bosonic field, which allows to change the induced interaction between the fermions. In particular, if is held at the fixed point corresponding to zero effective range, is reached at resonance , where the bosonic field is massless. As was shown by Nikolić and Sachdev Nikolić and Sachdev (2007), the action at this point is invariant under scale transformations and . The fermionic field transforms with the canonical scaling dimension, while the bosonic field acquires an anomalous dimension. As a result, the dynamics of the bosons, which arises from the second term in equation (3), is irrelevant at unitarity. The description of a finite fermion density requires adding a contribution to the action. With a fixed value of the chemical potential , this term explicitly breaks scale invariance. For the unitary gas, however, the relation between the chemical potential and the Fermi wave vector is identical to that of an ideal Fermi gas, except for a universal constant that describes the lowering of the Fermi energy due to the attractive interactions Bloch et al. (2008); Giorgini et al. (2008). A scale transformation thus leaves the action invariant provided is changed accordingly to account for the change in density. Alternatively, one may introduce the chemical potential through a special choice for the (common) phase of the fields and Son and Wingate (2006). Scale invariance gives rise to a conserved dilatation current Nishida and Son (2007) which implies the relation and the vanishing of the bulk viscosity: during a uniform expansion the system remains self-similar and no entropy is produced Werner and Castin (2006); Son (2007). In fact, the bulk viscosity of the unitary gas vanishes for all frequencies Taylor and Randeria (2010), a result that will be derived from an exact Ward identity in B. A subtle point in this context is the issue of whether scale invariance of the action will be present in the full theory, i.e., whether the classical symmetry of survives quantum fluctuations Wess and Zumino (1971); Zee (2003). For the unitary gas with equal masses of the two spin components, which is the situation discussed in this work, this is believed to be the case. For different masses, however, scale invariance will be broken due to the appearance of an Efimov effect, i.e., three- and four-body bound states appear even in the absence of a two-body bound state Petrov (2003); Nishida et al. (2008). In particular, four-body bound states appear for mass ratios larger than a critical value Castin et al. (2010). The breaking of scale invariance through the appearance of an Efimov effect has been discussed so far mostly for bosons Braaten and Hammer (2006) and leads to limit cycle flows in a renormalization group treatment Moroz and Schmidt (2009).

## 3 Kubo formula for the viscosity

The unitary Fermi gas is in a fluid state at arbitrary temperatures and thus has well defined bulk and shear viscosities and . On a microscopic level, the viscosities are the zero frequency limits of linear response functions that may be calculated from first principles by the Kubo formula. Specifically, both the shear and bulk viscosities follow from the retarded correlation functions of the stress tensor (in the zero external momentum limit),

 χretij,kl(q=0,ω)=i∫dtd3xeiωtθ(t)⟨[Πij(x,t),Πkl(0,0)]⟩, (4)

and its imaginary parts , which are odd functions of . In particular, the real part of the frequency dependent shear viscosity is determined by the associated positive and even spectral function

 Reη(ω) =Imχretxy,xy(ω)ω (5)

and its static limit . A completely analogous expression exists for the bulk viscosity which involves the trace of the stress tensor,

 Reζ(ω)=Imχretii,jj(ω)9ω. (6)

The microscopic expression for the stress tensor operator that enters equation (4) for a non-relativistic quantum many-body system has been derived by Martin and Schwinger Martin and Schwinger (1959). In the case of a scale-invariant short-range potential, where , it is given by

 Πij(q=0,t)=ℏ2m∑kσkikjc†kσckσ+∫d3x∫d3rrirjr22V(r):n↑(x)n↓(x+r):. (7)

The second term explicitly involves the interaction and guarantees that the stress tensor satisfies local momentum current conservation as an operator equation (see appendix B in Ref. Nishida and Son (2007)). In the zero-frequency limit, the kinetic part of the stress tensor (7) is the one that appears in the viscous terms of the hydrodynamic equations Schaefer and Teaney (2009), and also as a source term in the Boltzmann equation Smith and Jensen (1989). In general, the two contributions to the stress tensor (7) describe physically different processes for momentum relaxation upon insertion into the Kubo formula (4). In particular, the correlation function of two interaction contributions describes collisional transport due to interparticle forces, which is in fact the dominant contribution in the liquid phase Hansen and McDonald (2006). By contrast, the kinetic part is associated with the transfer of transverse momentum due to free particle motion and dominates in the gaseous phase. As will be shown in section 5, for the unitary gas the interaction part of the stress tensor is important to guarantee that the bulk viscosity vanishes, however it gives no contribution to the shear viscosity due to the zero-range nature of the interaction.

Quite generally, momentum current conservation implies that the stress tensor correlation functions are directly related to the current correlation functions, which are defined by

 Missing or unrecognized delimiter for \bigl (8)

with

 ji(q)=∑kσℏkic†k−q/2,σck+q/2,σ (9)

the current operator. In the limit , these correlation functions acquire a simple form, dictated by hydrodynamics. In particular, the existence of a finite shear viscosity implies diffusive relaxation of the transverse currents Hohenberg and Martin (1965); Forster (1975). The transverse part of the current correlation spectral function thus has the generic form (for small )

 Imχret⊥(q,ω)ω=ηq2ω2+(Dηq2)2 (10)

with a diffusion constant that is directly proportional to the shear viscosity. Indeed, the sum rule

 limq→0∫dωπImχret⊥(q,ω)ω=ρn (11)

which quite generally defines the normal fluid density Hohenberg and Martin (1965); Forster (1975) immediately implies an Einstein relation

 η=ρnDη (12)

between the shear viscosity and the associated diffusion constant that was first derived by Hohenberg and Martin Hohenberg and Martin (1965). The shear viscosity can thus be defined both from the stress tensor correlation function and, alternatively, from the diffusion constant that appears in the transverse current response Forster (1975); Taylor and Randeria (2010). For non-superfluid systems, the diffusion constant is identical to the standard kinematic viscosity .

For strongly interacting fluids, an exact calculation of correlation functions like the ones in equation (4) is hardly possible. It is therefore of considerable interest to find constraints that allow to check the validity of approximate calculations. Such constraints are derived generically from a short-time expansion of the correlation functions that can be expressed in terms of equilibrium expectation values of certain commutators. For the viscosities of a Fermi gas with contact interactions, this has been achieved recently by Taylor and Randeria Taylor and Randeria (2010). As will be shown in section 5 below, a straightforward derivation of the relevant sum rule can be given using a Ward identity due to Polyakov Polyakov (1969), who has discussed the behavior of the shear viscosity near the critical point of a neutral superfluid within a microscopic approach. This Ward identity, which follows from momentum conservation, implies that the frequency-dependent shear viscosity at arbitrary values of the scattering length obeys the sum rule

 2π∫∞0dω[η(ω)−ℏ3/2C15π√mω]=2ε3−ℏ2C6πma=p−ℏ2C4πma. (13)

Here, in the second form of the equality, we have used the non-trivial relation between energy density and pressure away from unitarity, first derived by Tan Tan (2008). The sum rule (13) has precisely the form given by Taylor and Randeria Taylor and Randeria (2010), however their high-frequency tail coefficient is larger by a factor of and there is also a difference in the prefactor of the contribution proportional to away from unitarity. The sum rule implies that decays like at large frequencies with a prefactor that is determined by the Tan contact density , which is a measure for two fermions with opposite spin to be close together Tan (2008, 2008); Braaten and Platter (2008); Braaten et al. (2008). Moreover, the area under the frequency-dependent viscosity of the unitary gas is fixed by the equilibrium pressure, since the term is absent at unitarity. Both features are verified with high accuracy by our calculations of that are detailed in section 7.

## 4 Temperature dependence of the shear viscosity

For the unitary gas, the density fixes both the momentum and energy scale, which are the Fermi wavevector and the associated Fermi energy . Since the viscosity has dimensions , purely dimensional arguments require the static shear viscosity to be of the form

 η(T)=ℏnα(θ) (14)

where is the dimensionless temperature scale and a universal scaling function. Remarkably, this function is fixed up to a universal constant in the high temperature limit. This is based on the counter-intuitive fact that the viscosity of a classical gas is independent of its density Balian (1992). Since , this requires , i.e., a shear viscosity that increases like . The same qualitative result is obtained from the kinetic theory expression with an average momentum and a thermally averaged cross section. Since the differential cross section for a collision with a given relative momentum is at unitarity, this immediately gives with the thermal momentum. Note that scales like even in the classical limit, provided the assumption of zero range s-wave scattering remains valid in this regime.

At low temperatures , the unitary Fermi gas is a superfluid. Contrary to naive expectations, a superfluid is not a ‘perfect fluid’ despite the fact that there is a vanishing viscosity here. In particular, superfluids do not provide trivial counter-examples to the KSS conjecture. Indeed, according to the Landau two-fluid picture, the superfluid component has both zero viscosity and zero entropy, so is undefined at . At any finite temperature, however, a normal component appears, whose entropy and viscosity are non-zero. For the bosonic superfluid He this was discussed already by Landau and Khalatnikov in 1949 Landau and Khalatnikov (1949). In the low temperature, phonon-dominated regime, they found that the shear viscosity of the normal component grows like because the mean free path for phonon-phonon collisions, which are necessary for the relaxation of shear, diverges. Specifically, with the assumption that the phonon dispersion has negative curvature and thus no Beliaev decay is possible (this assumption is actually violated in superfluid He but is likely to hold for the unitary Fermi gas, see Diener et al. (2008)), the calculations of Landau and Khalatnikov predict the low temperature shear viscosity to be

 η(T→0)=ρn(T)ρ2c3sℏ2213(2π)79(13)!(u+1)4(ℏcskBT)9. (15)

Here is the normal fluid mass density and is the dimensionless strength of the non-linear corrections to the leading-order quantum hydrodynamic Hamiltonian which lead to phonon-phonon scattering. The viscosity of the normal fluid component thus asymptotically diverges like . As realized by Rupak and Schäfer Rupak and Schaefer (2007) the same behavior is expected in the unitary Fermi gas. Indeed, at temperatures far below , the microscopic nature of the superfluid is irrelevant and the linearly dispersing Bogoliubov-Anderson phonons are the only excitations that remain. As a result, equation (15) for the viscosity applies also to the unitary Fermi gas, provided the exact values of the sound velocity and coupling constant are inserted. At unitarity, the sound velocity is directly proportional to the Fermi velocity with a factor that is determined by the universal Bertsch parameter Bloch et al. (2008). Since as for an ideal Fermi gas, the dimensionless coupling constant that fixes the strength of the phonon-phonon scattering amplitude has the universal value . Together with the standard low-temperature expression for the entropy density of a scalar phonon field, the viscosity to entropy density ratio

 η(T→0)s=ℏkB2.15×10−5ξ5θ−8 (16)

of the unitary Fermi gas at temperatures far below the superfluid transition diverges rapidly as . The associated prefactor agrees within with that found from a diagrammatic calculation by Rupak and Schäfer Rupak and Schaefer (2007), which is based on the next-to-leading terms in the effective field theory of the unitary gas by Son and Wingate Son and Wingate (2006) (note the different prefactor of the entropy in Rupak and Schaefer (2007)). It is a pure number that only contains the Bertsch parameter. Note that due to at very low temperatures, the ratio is just the characteristic relaxation time for shear fluctuations. According to equation (15), this relaxation time diverges like as , much stronger than the pure thermal energy time scale . In fact, the latter would lead to a viscosity to entropy density ratio which approaches a constant at very low temperatures, as in the supersymmetric Yang-Mills theory, where no quasiparticles exist.

Near the superfluid transition temperature , the universal scaling function will be continuous, yet there will be singularities in higher derivatives. Within the conventional theory of critical dynamics, the non-analytical behavior of near only comes from the thermodynamic singularity in Hohenberg and Halperin (1977). Since with a positive constant and the universal critical exponent of the 3D XY-model, the conventional theory predicts that reaches its finite value at with an infinite slope as the critical temperature is approached from below Ferrell (1988). A non-analytical temperature dependence of the shear viscosity is indeed observed near the point of superfluid He. In this case, however, a singularity of the form sign appears on both sides of the transition Biskeborn and Guernsey Jr (1975). The associated exponent is consistent with the prediction of conventional theory below , however it has a different value above the critical temperature. This behavior can be explained within a semi-phenomenological approach Schloms et al. (1990) which is, however, genuine for a liquid state. Thus it cannot be carried over to the case of the unitary Fermi gas, for which the precise behavior of near the critical point remains an open problem.

## 5 Diagrammatic evaluation of the stress tensor correlation functions

In order to compute the stress correlation functions (4) we start from the single-channel model and derive the corresponding stress tensor in the two-channel model. This has the advantage that the potential (fermion interaction) term of the stress tensor becomes simply a detuning (mass term) of the bosonic field in the two-channel model. We then derive the exact expression (30) for the stress correlation function in the two-channel model, including both kinetic and potential terms, in the zero-range limit. This expression is evaluated in the self-consistent T-matrix approximation Baym and Kadanoff (1961); Baym (1962). In the two-channel model we compute the fermionic and bosonic self-energies self-consistently at the one-loop level but neglect the loop corrections to the Yukawa coupling between fermions and bosons which appear only at two-loop order (cf. Fig. 1). Variation of these coupled single-particle equations with respect to a time-dependent external field results in a set of transport equations for the viscosity response functions. A crucial feature of this procedure is that it respects the symmetries of the underlying model exactly, in particular it obeys scale invariance and thus leads to a vanishing bulk viscosity as required.

Since the continuum model is Galilean invariant, one may expand the stress tensor operator (7) of the single-channel model (1) into spherical harmonics with angular-momentum quantum number :

 Πℓ=∑pσ2εpYℓ(^p)c†pσcpσ+2δℓ0¯g(Λ)∫d3r:n↑(r)n↓(r): (17)

where the spherical harmonics depend on the angle of the vector as and . (For convenience we omit the standard normalization factor in the spherical harmonics.) For a quadratic dispersion the fermion kinetic term admits and . Note that for the scale-invariant, zero-range model at unitarity the interaction term contributes only for !

In order to compute linear response one has to add a perturbation to the Hamiltonian which couples the stress tensor to a time-dependent external field,

 ΔH(t)=∑ℓhℓ(t)Πℓ(t)=∑pσℓ2hℓ(t)εpYℓ(^p)c†pσcpσ+2hℓ=0(t)¯g(Λ)∫d3r:n↑(r)n↓(r):. (18)

This amounts to the replacement and in the full Hamiltonian, and likewise in the full action at unitarity

 S[h]=∫β0dτ[∑pσ[1+∑ℓ2hℓ(τ)Yℓ(^p)]εp¯cpσcpσ+~g¯ψBψ↑ψ↓+h.c.−~g2¯g(Λ)[1+2hℓ=0(τ)]¯ψBψB]. (19)

The change in the action due to the external field, , can be parametrized as

 ΔS=∑XX′[∑σ¯ψσ(X)Uσ,XX′ψσ(X′)+¯ψB(X)UB,XX′ψB(X′)] (20)

with the coefficient functions

 Uσ,XX′=∫dτ∑ℓhℓ(τ)Yℓ(^p)T(0)σℓ(τXX′), (21) UB,XX′=~g2∫dτ∑ℓhℓ(τ)1+2hℓ(τ)S(0)ℓ(τXX′). (22)

For convenience we use a short-hand notation for the real-space argument and for the Fourier argument, and as well as . Equations (21), (22) define the bare fermionic and bosonic viscosity response vertices at unitarity,

 T(0)σℓ(τX1X′1)=ℏ2∇1∇′1mδX1X′1δ(τ−τ1), (23) S(0)ℓ(τX1X′1)={2¯g(Λ)−1δX1X′1δ(τ−τ1)for ℓ=00for ℓ=2. (24)

The response of the grand potential to the external field in terms of the fermionic Green’s functions and bosonic Green’s functions is

 δΩ =tr[(δUσ)Gσ]+tr[(δUB)GB] (25) =tr[T(0)σℓGσδhℓ(τ)]−tr[S(0)ℓΓδhℓ(τ)(1+2hℓ(τ))2] (26)

where the trace includes the spin sum and in the second line additionally the sum. In the second line the bosonic Green’s function is replaced by the vertex function . Hence, we obtain

 −δΩδhℓ(τ)=−∑σXX′T(0)σℓ(τXX′)Gσ,X′X+1(1+2hℓ(τ))2∑XX′S(0)ℓ(τXX′)ΓX′X. (27)

In particular, for a static scaling perturbation we recover the Tan energy formula Tan (2008) with the correct UV regularization,

 Missing or unrecognized delimiter for \right (28)

where the local limit of the vertex function has been expressed in terms of the contact density (79). The Kubo formula (4) in imaginary time can be re-expressed in partial-wave components as

 χℓ(τ)=∫d3r⟨TΠℓ(r,τ)Πℓ(0,0)⟩, (29)

where denotes time ordering. This stress correlation function can be obtained from the second derivative of the grand potential (27),

 χℓ(τ) =−δ2Ωδhℓ(τ)δhℓ(0)∣∣∣hℓ=0 =−tr[T(0)σℓ(0)~Tσℓ(τ)]+tr[S(0)ℓ(0)(~Sℓ(τ)−4δ(τ)Γ(0))] (30)

where the last term in the square brackets comes from the explicit dependence in the second term of equation (27). Note also that this last term is crucial to obtain a vanishing bulk viscosity at unitarity , as we will show in B using the Ward identities that follow from scale invariance. and are the fermionic and bosonic viscosity response functions

 ~Tσℓ(τXX′)=δGσ,XX′δhℓ(τ)∣∣∣hℓ=0=⟨TΠℓ(τ)ψσ(X)ψ†σ(X′)⟩, (31) ~Sℓ(τXX′)=δΓXX′δhℓ(τ)∣∣∣hℓ=0=−~g2⟨TΠℓ(τ)ψB(X)ψ†B(X′)⟩. (32)

Note that for the case of the shear viscosity where , the viscosity response function can be expressed by the L function Baym and Kadanoff (1961) as , and the correlation function (30) reduces to the well-known form .

The formalism developed so far now allows to derive the sum rule given in equation (13), starting from the Ward identity for momentum conservation in the static limit of external , which reads (Polyakov, 1969, eq. (5.26))

 ~Txy=−px∂G/∂py. (33)

Here is the vertex function of the stress tensor (cf. equation (31)). Starting from equation (30) for the shear viscosity correlation function, we insert the bare shear viscosity vertex from equation (23) and the full shear viscosity vertex from the Ward identity (33) and obtain at zero external Matsubara frequency (note that due to the angular average of )

 χxy,xy(iωm=0) =−115∑pσ1β∑iϵnT(0)ℓ=2(p,iϵn)~Tℓ=2(p,iϵn) =115∑pσ1β∑iϵnℏ2p2mp∂G(p,iϵn)∂p =−215∑pσεpp∂np∂p

in terms of the fermionic momentum distribution . We now integrate by parts, with a boundary term at momentum cutoff , and employ the Tan energy formula Tan (2008) to express the sum over the momentum distribution by the internal energy density ,

 χxy,xy(iωm=0) =215[5∑pσεpnp−ℏ2p5np2π2m∣∣Λ0] =2ε3+4ℏ2CΛ15π2m−ℏ2C6πma. (34)

If the momentum cutoff is translated into a frequency cutoff (Taylor and Randeria, 2010, endnote 39) we arrive at the sum rule given in equation (13).

The derivation of the correlation function (30) so far has been completely general and exact. The remaining challenge then is to evaluate the viscosity response functions and within the microscopic model for the unitary Fermi gas. In the following we shall do this within the T-matrix approximation.

We start from the Dyson equation for the fermionic Green’s function in the Matsubara formalism,

 G−1σ,XX′=G−10,σXX′−Uσ,XX′−Σσ,XX′ (35)

with bare Green’s functions and the external field from equation (21). In the T-matrix approximation the fermionic self-energy describes how fermions scatter off pair fluctuations (cf. Fig. 1a),

 Σσ,XX′=G−σ,X′XΓXX′. (36)

The Bethe-Salpeter equation for the vertex function which mediates the resonant Fermi-Fermi interaction is

 Γ−1XX′=¯g(Λ)−1−UB,XX′+G↑,XX′G↓,XX′. (37)

It contains the inverse bare coupling , the external field from equation (22), and the bosonic self-energy (cf. Fig. 1b). As mentioned above, the dynamics of the pair field at unitarity only arises from the excitation of fermion pairs while the dynamics of the bosons is irrelevant. Since we consider a balanced gas with equal populations of fermion species we will henceforth drop the spin index .

The T-matrix approximation for the exact viscosity response vertices is then obtained by taking the derivative of the self-consistency equations (35)–(37) with respect to the external field (cf. Fig. 2); in this way it is guaranteed that the conservation laws are satisfied. The amputated viscosity response vertex

 Tℓ(τXX′)=−δG−1XX′δhℓ(τ)=∑YY′G−1XY~Tℓ(τYY′)G−1Y′X′ (38)

is given by the variation of the Dyson equation (35) with respect to ,

 Tℓ(τXX′)=T(0)ℓ(τXX′)+TMTℓ(τXX′)+TALℓ(τXX′) (39)

with the Maki-Thompson and Aslamazov-Larkin vertex corrections

 TMTℓ(τXX′) =δGX′Xδhℓ(τ)ΓXX′=~Tℓ(τX′X)Γ(XX′), (40) TALℓ(τXX′) =GX′XδΓXX′δhℓ(τ)=GX′X~Sℓ(τXX′). (41)

The amputated bosonic viscosity response vertex

 Sℓ(τXX′)=−δΓ−1XX′δhℓ(τ)=∑YY′Γ−1XY~Sℓ(τYY′)Γ−1Y′X′ (42)

is given by the derivative of the Bethe-Salpeter equation (37),

 Sℓ(τXX′)=S(0)ℓ(τXX′)−2GXX′~Tℓ(τXX′). (43)

The bosonic viscosity vertex describes the response of the pair field to a scaling or shear perturbation. The bare bosonic viscosity vertex serves to remove the UV divergence of the particle-particle loop (second term) which is present only for the bulk viscosity .

The self-consistent transport equations for the two-channel model are equivalent to those of the standard T-matrix approximation in a purely fermionic description. Specifically, the two fermionic vertex corrections and above are precisely those arising in the self-consistent solution of the integral kernel of the function, as introduced by Baym (Baym and Kadanoff, 1961, equation (60)). In physical terms, the first term describes the interaction between two fermions by exchange of a single pair while in the second term two pairs are exchanged. In the context of calculating the change in conductivity due to superconducting fluctuations, these terms are called the Maki-Thompson (MT) and Aslamazov-Larkin (AL) contributions respectively Aslamazov and Larkin (1968), a notation that will be used also in the present context. Note that the second term explicitly includes particle-hole fluctuations, and it is also referred to as the “box diagram” in a functional renormalization group approach to the thermodynamics of the unitary Fermi gas Floerchinger et al. (2008). These vertex corrections are in fact crucial to obtain an approximation which satisfies the conservation laws of the underlying model.

The self-consistent equations (35)–(43) have a structure similar to the equations of the Luttinger-Ward approach to the BCS-BEC crossover developed in our previous work Haussmann et al. (2007, 2009). In particular, using Fourier transforms and the convolution theorem they become algebraic equations which afford an efficient numerical solution. The numerical calculations are performed in three steps. In the first step the Green’s function and the vertex function are calculated by solving the self-consistent equations (35)–(37) iteratively. Without the external fields and the Dyson and Bethe-Salpeter equations are diagonal in Fourier space, while the fermionic and bosonic self-energies are local in real space. Hence, the coupled equations are solved efficiently by going back and forth between real and Fourier space.

In the second step and are used as input for the self-consistent equations (38)–(43) to calculate the viscosity response functions , . Again, the integral equations (38) and (42) become algebraic and are solved in Fourier space, while the other equations remain local in real space. Note that the spatial Fourier transform between radial distances and radial wavenumber for the partial-wave component is given by

 Tℓ(k) =4π(−i)ℓ∫∞0drr2jℓ(kr)Tℓ(r), (44) Tℓ(r) =iℓ2π2∫∞0dkk2jℓ(kr)Tℓ(k). (45)

In the third step the correlation function is computed from (30). It is continued analytically from the discrete imaginary Matsubara frequencies to the continuous real frequencies via both the Padé method and a model fit function (cf. section 7). We thus obtain the retarded correlation function . Finally, the real parts of the viscosities and are obtained from the correlation functions for and according to (cf. equations (5) and (6))

 Reη(ω) =Imχretℓ=2(ω)15ω, (46) Reζ(ω) =Imχretℓ=0(ω)9ω, (47)

where the prefactor of comes from the angular integration of the spherical harmonics . Alternatively, one may solve the integral equation directly for real frequencies where the limit can be taken analytically. In practice, this approach is useful at high temperatures, where self-consistency no longer plays a role.

## 6 Boltzmann-equation limit

In the high-temperature limit the integral equations (38)–(43) can be solved by expanding in powers of the fugacity

 z=eβμ=43√πθ−3/2+O(θ−3). (48)

To leading order in , the pair propagator and on-shell self-energy are given by

 Γret(k,Ω)=−i4πℏ3m−3/2√ℏΩ+2μ−εk/2+O(z) (49) Σret(p,ϵ=εp−μ)=i8εF3πerf(√πp/pT)p/pF+O(z). (50)

In the case of on-shell fermions with , the pair propagator reduces to the well-known scattering amplitude at infinite scattering length of two particles in vacuum, with relative momentum . Note that the exact leading-order result for the on-shell fermionic self-energy contains a non-trivial error-function dependence on the ratio of the momentum to its thermal value that was missing in previous studies Combescot et al. (2006). It is due to the square-root tail in the pair propagator and gives a noticeable correction at thermal momenta . Moreover, this form is indeed crucial to fulfill the condition of scale invariance, as will be discussed below.

The fermionic spectral function in the low fugacity, high temperature limit has most of the spectral weight concentrated in the coherent peak at . The peak width vanishes like for typical momenta , consistent with the assumption for the temperature dependence of the relaxation time introduced by Bruun and Smith Bruun and Smith (2007). This implies, in particular, that the fermionic quasiparticles become well-defined and thus a Boltzmann equation description is valid in the regime .

From a numerical, iterative solution of the integral equations (38)–(43) in the high-temperature limit we obtain . This fixes the constant in the asymptotic behavior at large values of of the universal function introduced in (14). Within the error bars, the numerical value agrees with that obtained from a variational solution of the full Boltzmann equation, using higher Sonine polynomials (Bruun and Smith, 2007, appendix). The prediction of a simple power-law dependence of the shear viscosity has recently been verified experimentally in a temperature range between and by measuring the expansion dynamics of a unitary gas released from an optical trap Cao et al. (2010). Very good agreement has been found also with the expected prefactor, thus considerably improving the situation compared to earlier measurements of the shear viscosity from the damping of the radial breathing mode Turlapov et al. (2008).

Remarkably, the solution of the transport integral equation at high temperatures and small frequencies can also be obtained by a completely analytical approach. In fact, in the low fugacity limit, one can terminate the iterative procedure after the first iteration step (correlation function to first order in the pair propagator) and resum via a memory function approach, a method that was developed in the context of electrical conductivities by Götze and Wölfle Götze and Wölfle (1972). The first-order correlation function contains the diagrams for self-energy, Maki-Thompson and Aslamazov-Larkin contributions shown in Fig. 3. These diagrams are obtained by evaluating the transport equations (38)–(43) with the bare viscosity vertices ,