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 [for its filling factor ${\nu }_0$ holds $\left(1\mathrm{}-{\nu }_0\right)\ll 1]$ 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 $N_0^{1/2}\left(1\mathrm{}-{\nu }_0\right)\ll 1,$ these interactions lead to a splitting of the Landau levels. In the second regime, where $N_0^{1/2}\left(1\mathrm{}-{\nu }_0\right)\gg 1,$ apart of the splitting, a renormalization of the bare Zeeman splitting occurs. The intermediate case $N_0^{1/2}\left(1\mathrm{}-{\nu }_0\right)\approx 1$ cannot be studied within our approach. The applicability of the Fermi-liquid description is investigated for both regimes.