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.
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.
We investigate the temperature dependence of the shear viscosity and spin diffusion in a two-dimensional Fermi gas with contact interactions, as realized in ultracold atomic gases. We describe the transport coefficients in terms of a Boltzmann equation and present a full numerical solution for the degenerate gas. In contrast to previous works we take the medium effects due to finite density fully into account. This effect reduces the viscosity-to-entropy ratio $\ensuremath{\eta}/s$ by a factor of 3, and similarly for spin diffusion. The trap-averaged viscosity agrees well with recent measurements by Vogt et al. [Phys. Rev. Lett. 108, 070404 (2012)].
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.
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.
This thesis considers out-of-equilibrium dynamics of strongly interacting non-relativistic Fermi gases in several two and three dimensional geometries. The tools of second-order hydrodynamics and gauge-gravity duality will be utilized to address this system. Many of the themes of this work are motivated by the observed similarities in transport properties between strongly interacting Fermi gases and other very different strongly interacting quantum fluids such as the quark-gluon plasma, high temperature superconductors, and quantum field theories described by gauge-gravity duality. In particular, these systems all nearly saturate the conjectured lower bound on the ratio of shear viscosity to entropy density η/s≥ℏ/(4πkB) coming from the AdS/CFT correspondence. Among other things, this observation, in conjunction with current experiment and data analysis in atomic, condensed matter, and nuclear physics lends itself to the following questions: How perfect of a fluid is the strongly interacting Fermi gas, and can one find a more stringent constraint on η/s in Fermi gases? Do the similarities in transport properties among strongly interacting quantum systems extend beyond dynamics controlled by the hydrodynamical shear viscosity? In regards to the first question, by utilizing second-order hydrodynamics, it will be demonstrated that higher-order collective modes of a harmonically trapped Fermi gas may serve as a more sensitive probe of the shear viscosity. For the second question, both second-order hydrodynamics and a gravity dual theory are used to make predictions about dynamics occurring on short timescales where hydrodynamics is expected to break down. In particular the appearance of a class of "non-hydrodynamic" collective modes not contained within a Navier-Stokes description of the strongly interacting Fermi gas will be discussed.
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)].
After its first isolation by Geim and Novoselov in 2004, graphene has grown quickly into an independent subfield of contemporary condensed matter physics, counting thousands of publications. This material has attracted so much attention not only because it is truly two-dimensional -- a feature long thought to be impossible to be realized in nature -- but also because it is the prime representative of a new class of condensed matter systems, the Dirac materials, whose electronic spectrum disperses linearly around isolated points in the Brillouin zone. An outstanding consequence of this linear energy-momentum dispersion is the collapse of the Fermi surface to isolated points at charge neutrality, accompanied by a vanishing single-particle density of states. While this feature of the noninteracting model is what separates graphene from most other, more conventional condensed matter systems, giving rise to many of its astounding physical properties, it leads to severe computational complications when extending the model to contain two-body interactions and/or disorder. In this thesis we consider both extensions separately. After rigorously deriving the low-energy quantum field theory from a tight-binding model and establishing the one-particle irreducible (1PI) vertex functions as the fundamental building blocks of correlation functions, we construct a nonperturbative formalism for the calculation of these vertex functions using the well-established functional renormalization group (fRG). By combining the fRG with the Keldysh formalism, accounting for possible external electromagnetic fields and nontrivial initial correlations, this framework is capable of handling thermal equilibrium and true nonequilibrium alike. To circumvent some of the technical problems of the standard fRG formalism when treating fermionic systems at finite density, we also explore a variant of the fRG, where the chemical potential is interpreted as a flow parameter. We obtain hierarchical sets of flow equations, which describe the change of the 1PI vertex functions upon varying an artificial cutoff scale or the chemical potential, respectively. These nonperturbative formalisms have been used to calculate the renormalization of the Fermi velocity and the static dielectric function at finite temperature and density in the strong coupling regime as an explicit demonstration of their capabilities. Since the Keldysh fRG is an exact formulation of quantum field theory, in particular when external magnetic fields are present, it should be possible to describe the fractional quantum Hall effect in this framework. However, the typical truncation schemes that are currently available to approximately solve the fRG flow equation fail to account for the nontrivial correlation effects necessary to access the fractional quantum Hall regime. To circumvent this problem we consider a modified field theory, where these correlations are implemented ``by hand'' via a Chern-Simons gauge field, coupling to the fermions in addition to the Coulomb interaction term. This modified field theory is analyzed in a stationary phase approximation including Gaussian fluctuations. The electromagnetic response tensor is calculated in the random phase approximation, yielding the Hall conductivities. In the second part of this thesis we consider disordered Dirac fermions in the absence of two-particle interactions. After sketching how the most general disorder potential for Dirac fermions arises from a tight-binding model, we first consider an explicit physical scenario, a disordered graphene pn junction in the presence of a quantizing magnetic field perpendicular to the graphene sheet. We derive an effective one-dimensional theory for the chiral states that propagate along the junction interface and calculate the full conductance distribution in the crossover between the clean and the strong-disorder limit, via an exact solution of a Fokker-Planck equation. Lastly, we develop a nonperturbative approach to calculate the disorder induced self-energy that is based on exact Schwinger-Dyson equations and Ward identities, not only for graphene, but also for other paradigmatic semimetals with a nodal point dispersion.