Abstract
We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length a, including the unitarity limit a→∞. We perform the calculation in kinetic theory by studying the response to a non-trivial background metric. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is ω0≃(2ε)/(3η) where ε is the energy density and η is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of the order of ω0∼k(T/m)1/2.
Highlights
The dilute Fermi gas at unitarity is a strongly correlated scale invariant quantum fluid [1, 2]
Where in the last step we have evaluated the integrals in the high temperature limit. This result agrees with the hydrodynamic limit given in Eq (3.7) and the known formula for the shear viscosity of a dilute Fermi gas in kinetic theory [16]
We note that the sum rule agrees with the integral of the free spectral function given in Eq (4.20)
Summary
The dilute Fermi gas at unitarity is a strongly correlated scale invariant quantum fluid [1, 2]. The Kubo formula relates the shear viscosity to the zero momentum, zero frequency limit of the retarded stress tensor correlation function. The first observation is that the frequency and momentum dependence of the retarded correlator encodes the time and distance scales over which the viscous contribution to the stress tensor relaxes to the Navier-Stokes expression. These scales are important for understanding how hydrodynamics breaks down in the dilute corona of a trapped Fermi gas [7, 15]. The spectral function of the stress tensor in QCD is studied in [27,28,29,30,31]
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