Fermi surface phenomena in the (2+1)D four-Fermi model

We study the Gross-Neveu model in 2+1 dimensions with a baryon chemical potential $\mu$ using both analytical and numerical methods. For $\mu$ greater than a critical value the model is chirally symmetric and has a Fermi surface with $k_F\simeq \mu .$ We have calculated the particle interaction in medium due to scalar meson exchange to leading order in $N_f^{-1},$ where $N_f$ is the number of flavors, in the hard dense loop approach. The result has been used to calculate the relation between $\mu$ and the Fermi momentum and velocity in the resulting Fermi liquid to ${O(N}_f^{-1}).\mathrm{}$ Simulation results from a ${32}^2\times 48$ lattice for fermion and meson dispersion relations and meson wave functions are then presented, showing qualitative and in some cases quantitative agreement with analytic predictions. In particular, the simulations show clear evidence for the in-medium modification of the scalar propagator, oscillatory behavior in the wave function consistent with a sharp Fermi surface, and tentative evidence for a massless pole in the vector meson channel resembling zero sound.