2-DIMENSIONAL FERMI LIQUIDS AT LOW-DENSITIES - THE SINGLE-PARTICLE SELF-ENERGY

The two-dimensional (2D) nature of the high-T-c materials has lead to renewed interest in the properties of fermions in two dimensions. Calc ulations of the corrections to the leading Fermi-liquid (FL) behaviour of the single-particle self-energy Sigma(p, E), and the specific heat have shown that the differences between 2D and three-dimensional ns a rise from the difference in phase space between two and three dimensio ns. Here we investigate the imaginary part Sigma ''(p, xi(p)) of Sigma (p, E) on shell at a low density which has a (p-p(F)) ln (p-p(F)) depe ndence in two dimensions where p(F) is the Fermi wavenumber. We fmd th at both particle-hole (ph) and particle-particle (pp) contributions ar e important in this limit so that in principle no class of diagrams ca n be neglected in the low-density limit in two dimensions, and one is effectively limited to second order in the interaction for reliable qu antitative results. This is in contrast with three dimensions where th e ph contribution becomes negligible compared with the pp contribution at low densities and only pp ladder diagrams survive. We argue, howev er, that the density dependence of Sigma(p, E) is unchanged by high or ders in perturbation theory and that in the limit p(F)-->0 the form of the dependence of Sigma(p, E) on p-p(F) is given by the second-order contribution. We find that the (p-p(F))(2) ln (p-p(F)) behaviour is li mited to a values of p-p(F)<p(F) so that the region over which FL beha viour is evident shrinks with decreasing density.