Theory of fermion liquids

We develop a general theory of fermion liquids in spatial dimensions greater than 1. The principal method, bosonization, is applied to the cases of short- and long-range longitudinal interactions and to transverse gauge interactions. All the correlation funtions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau-Fermi fixed point. Non-Fermi liquid fixed points are found, however, in the cases of a super-long-range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the (2+1)-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction V(q)∝‖q ${\mathrm{\Vert }}^{\mathit{y}\mathrm{-}1}$, which controls the density, and hence gauge, fluctuations. For yy>0 the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case y=0 (Coulomb interaction), which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator, which is of a marginal Fermi-liquid form. Using Ward identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion nor vertex corrections alters our results: the fixed point is accessible by bosonization. As the two-point fermion Green’s function is not gauge invariant, we also invetigate the gauge-invariant density response function. Near momentum Q=2${\mathit{k}}_{\mathit{F}}$, in addition to the Kohn anomaly we find other nonanalytic behavior. In the appendies we present a numerical calculation of the spectral function for a Fermi liquid with Landau parameter ${\mathit{f}}_0$≠0. We also show how Kohn’s theorem is satisfied within the bosonization framework.