In interaction-dominated two-dimensional electron gases at intermediate temperatures, electron transport is not diffusive as in the conventional Drude picture but instead hydrodynamic. The relevant transport coefficient in this regime is the shear viscosity. Here, we develop a numerically exact basis expansion to solve the Fermi liquid equation, and apply it to compute the shear viscosity of the electron gas with screened Coulomb interactions. Our calculations are valid at all temperatures and in particular describe the response beyond the asymptotic low-temperature limit, where perturbative approaches exist. We show that even in this low-temperature limit, there is a nonanalytic exchange contribution to the shear viscosity, highlighting the need for a full nonperturbative solution of the Fermi liquid equation. We hope that the techniques developed in this work will serve as a platform to determine the response of interacting Fermi liquids.
The shear viscosity of a two-dimensional Fermi gas interacting via a short-range potential with scattering length
a2d in kinetic theory is computed. It is found that classical kinetic theory predicts that the shear viscosity to entropy density ratio of a strongly interacting two-dimensional gas is comparable to that of the three-dimensional unitary gas. Results are applied to the damping of collective modes of a trapped Fermi gas, and compared to experimental data recently obtained by Vogt et al. [Phys. Rev. Lett. 108, 070404 (2012)].
In the first part of the thesis I derive a full quantitative formula which describes the amplitude and frequency of magnetic oscillations in two-dimensional Dirac systems. The investigations are on the basis of graphene, but they generally also hold for other two-dimensional Dirac systems. Starting from the Luttinger-Ward functional I derive an expression for the oscillatory part of the grand potential of graphene in a magnetic field. The amplitude of this ex- pression is usually called the Lifshitz-Kosevich (LK) formula. I perform the computation for the clean and the disordered system, and I study the effect of electron-electron interactions on the oscillations. I discuss my results by comparing them to the analogue expressions for the two-dimensional Fermi gas (2DEG) which have been derived by Adamov et al.. I find that, unlike in the 2DEG, a finite temperature and impurity scattering also affects the oscillation frequency. Further I find that in graphene, compared to the 2DEG, additional interaction induced damping effects occur: to two-loop order electron-electron interactions do lead to an additional damping factor in the amplitude of the LK-formula. Moreover the renormalization effects cannot fully be accounted for by renormalizations of the Fermi velocity but they also have to be described by field renormalizations. In the second part of the thesis I investigate the temperature dependence of the shear viscosity and spin diffusion in a two-dimensional, two-component Fermi gas, as realized in ultracold atomic gases. I implement a contact interaction that only acts between fermions in different hyperfine states. The transport coefficients are obtained within a kinetic approach. I solve the linearized Boltzmann equation by using a variational principle and present a full numerical solution for the degenerate gas. In contrast to previous works I take the medium effect due to finite density fully into account. This effect reduces the viscosity to particle density ratio, η/n, by a factor of four for strong interactions; and similarly for spin diffusion. The lowest value I obtain for the viscosity to entropy ratio is η/s = 0.15h ̄/kB, and it occurs close to the phase transition to the superfluid phase. This value is about twice the conjectured lower bound of η/s = 1/(4π) ̄h/kB, computed using the AdS/CFT correspondence. I compare my result for the shear viscosity to the measurements by Vogt et al., who measured the damping rate of the quadrupole mode of a trapped Fermi gas confined to two dimensions. This damping rate is related to the shear viscosity of the gas and our numerical results agree well with the experiment.
Using kinetic theory, we calculate the shear viscosity and the spin-diffusion coefficient as well as the associated relaxation times for a two-component Fermi gas in two dimensions, as a function of temperature, coupling strength, polarization, and mass ratio of the two components. It is demonstrated that the minimum value of the viscosity decreases with the mass ratio, since Fermi blocking becomes less efficient. We furthermore analyze recent experimental results for the quadrupole mode of a two-dimensional gas in terms of viscous damping, obtaining a qualitative agreement using no fitting parameters.
We calculate expressions for the state-dependent quasiparticle lifetime ${\ensuremath{\tau}}_{\ensuremath{\sigma}}$, the thermal conductivity $\ensuremath{\kappa}$, the shear viscosity $\ensuremath{\eta}$, and discuss the spin diffusion coefficient
D for Fermi-liquid films in two dimensions. The expressions are valid for low temperatures and arbitrary polarization. In two dimensions, as in three dimensions, the integrals over the transition rates factor into energy and angular parts. However, the angular integrations contain a weak divergence. This problem is addressed using the method of K. Miyake and W. J. Mullin [Phys. Rev. Lett. 50, 197 (1983); J. Low Temp. Phys. 56, 499 (1984)]. The low-temperature expressions for the transport coefficients are essentially exact. We find that ${\ensuremath{\kappa}}^{\ensuremath{-}1}\ensuremath{\sim}TlnT$, and ${\ensuremath{\eta}}^{\ensuremath{-}1}\ensuremath{\sim}{T}^{2}$ for arbitrary polarizations $0\ensuremath{\le}\mathcal{P}\ensuremath{\le}1$. These results are in agreement with earlier zero-polarization results of H. H. Fu and C. Ebner [Phys. Rev. A 10, 338 (1974).], but differ from the temperature dependence of the shear viscosity found by D. S. Novikov (arXiv:cond-mat/0603184). They also differ from the discontinuous change of temperature dependence in
D from zero to nonzero polarization that was discovered by Miyake and Mullin. We note that in two dimensions the shear viscosity requires a unique analysis. We obtain predictions for the density, temperature, and polarization dependence of $\ensuremath{\kappa},\phantom{\rule{0.16em}{0ex}}\ensuremath{\eta}$, and
D for second-layer
3He films on graphite, and thin $^{3}\mathrm{He}\ensuremath{-}^{4}\mathrm{He}$ superfluid mixtures. For
3He on graphite, we find roughly an order of magnitude increase in magnitude for $\ensuremath{\kappa}$ and $\ensuremath{\eta}$ as the polarization is increased from 0 to 1. For
D a similar large increase is predicted from zero polarization to the polarization where
D is a maximum $(\ensuremath{\sim}0.74)$. We discuss the applicability of
3He thin films to the question of the existence of a universal lower bound for the ratio of the shear viscosity to the entropy density.
We compute the shear viscosity, thermal conductivity, and spin diffusivity of a Fermi gas with short-range interactions in the Fermi liquid regime of the normal phase, that is, at temperatures T much lower than the Fermi temperature T_{F} and larger than the superfluid critical temperature T_{c}. In line with recent advances in the precision of cold atom experiments, we provide exact results up to the second order in the interaction strength. We extend the Landau-Salpeter equation to compute the collision amplitude beyond the forward-scattering limit, covering all collisions on the Fermi surface. We treat the collision kernel exactly, leading to significant corrections beyond relaxation-time or variational approximations. The transport coefficients, as functions of the s wave scattering length a and Fermi wave number k_{F}, follow (1+γk_{F}a)/a^{2} up to corrections of order O(a^{0}), with a positive coefficient γ for the viscosity and negative one for the thermal conductivity and spin diffusivity.