RPAE versus RPA for the Tomonaga model with quadratic energy dispersion

Recently, the damping of the collective charge (and spin) modes of interacting fermions in one spatial dimension was studied. It results from the nonlinear correction to the energy dispersion in the vicinity of the Fermi points. To investigate the damping, one has to replace the random-phase approximation (RPA) bare bubble by a sum of more complicated diagrams. It is shown here that a better starting point than the bare RPA is to use the (conserving) linearized time-dependent Hartree-Fock equations, i.e., to perform a random-phase approximation (with) exchange (RPAE) calculation. It is shown that the RPAE equation can be solved analytically for the special form of the two-body interaction often used in the Luttinger liquid framework. While (bare) RPA and RPAE agree for the case of a strictly linear dispersion, there are qualitative differences for the case of the usual nonrelativistic quadratic dispersion.