Abstract
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
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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.
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