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

Abstract

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 ${\ensuremath{\nu}}_{0}$ holds $(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\ll}1]$ with a quantum number ${N}_{0}\ensuremath{\gg}1$ is considered. We uncover two regimes of renormalization by electron-electron interactions. In the first regime, where ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\ll}1,$ these interactions lead to a splitting of the Landau levels. In the second regime, where ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\gg}1,$ apart of the splitting, a renormalization of the bare Zeeman splitting occurs. The intermediate case ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\approx}1$ cannot be studied within our approach. The applicability of the Fermi-liquid description is investigated for both regimes.