Two-dimensional weakly interacting Fermi gas in a magnetic field: Level splitting

We apply the diagram-technique formalism beyond the Hartree-Fock approximation to a two-dimensional nearly ideal electron gas in a weak perpendicular magnetic field. The case of an almost completely filled upper Landau level (filling factor ${\nu }_0\lesssim 1$) with a quantum number $N_0 \gg 1$ is considered. We uncover two regimes of renormalization by electron-electron interactions. In the first regime, where $N^{1/2}_0(1-\nu_0) \ll 1$, these interactions lead to a splitting of the Landau levels. In the second regime, where $N_0^{1\mathrm{/}2}(1-{\nu }_0)\gg 1$, apart of the splitting, a renormalization of the bare Zeeman splitting occurs. The intermediate case $N_0^{1\mathrm{/}2}(1-{\nu }_0)\approx 1$ cannot be studied within our approach. The applicability of the Fermi-liquid description is investigated for both regimes.