Metals in a high magnetic field: A universality class of marginal Fermi liquids

Parquet equations, describing the competition between superconducting and density-wave instabilities, are solved for a three-dimensional isotropic metal in a high magnetic field when only the lowest Landau level is filled. In the case of repulsive interaction between electrons, a phase transition to the density-wave state is found at finite temperature. In the opposite case of attractive interaction, no phase transition is found. With decreasing temperature T, the effective vertex of interaction between electrons renormalizes toward a one-dimensional limit in a self-similar way with the characteristic length (transverse to the magnetic field) decreasing as ${\ln }^{\mathrm{-}1/6}$(${\mathrm{\omega }}_{\mathit{c}}$/T) (${\mathrm{\omega }}_{\mathit{c}}$ is a cutoff). The residue of the one-particle Green function vanishes at the Fermi surface indicating the marginal character of the Fermi liquid.