Two-dimensional electron liquid in a weak magnetic field

We present an effective theory describing the low-energy properties of an interacting two-dimensional electron gas at large noninteger filling factors ν≫1. Assuming that the interaction is sufficiently weak, ${\mathit{r}}_{\mathit{s}}$<1, we integrate out all the fast degrees of freedom, and derive the effective Hamiltonian acting in the Fock space of the partially filled Landau level only. This theory enables us to find two energy scales controling the electron dynamics at energies less than ħ${\mathrm{\omega }}_{\mathit{c}}$. The first energy scale, (ħ${\mathrm{\omega }}_{\mathit{c}}$/ν)ln(ν${\mathit{r}}_{\mathit{s}}$), appears in the one-electron spectral density as the width of a pseudogap. The second scale, ${\mathit{r}}_{\mathit{s}}$ħ${\mathrm{\omega }}_{\mathit{c}}$, is parametrically larger, it characterizes the exchange-enhanced spin splitting and the thermodynamic density of states.