Low-energy properties of fermions with singular interactions

We calculate the fermion Green function and particle-hole susceptibilities for a degenerate two-dimensional fermion system with a singular gauge interaction. We show that this is a strong-coupling problem, with no small parameter other than the fermion spin degeneracy N. We consider two interactions, one arising in the context of the t-J model and the other in the theory of half-filled Landau level. For the fermion self-energy we show that the qualitative behavior found in the leading order of perturbation theory is preserved to all orders in the interaction. The susceptibility ${\mathrm{\chi }}_{\mathit{Q}}$ at a general wave vector Q≠2${\mathbf{p}}_{\mathbf{F}}$ retains the Fermi-liquid form. However, the 2${\mathit{p}}_{\mathit{F}}$ susceptibility ${\mathrm{\chi }}_{2\mathit{p}\mathit{F}}$ either diverges as T→0 or remains finite but with nonanalytic wave-vector, frequency, and temperature dependence. We express our results in the language of recently discussed scaling theories, give the fixed-point action, and show that at this fixed point the fermion-gauge-field interaction is marginal in d=2, but irrelevant at low energies in d≥2.