Low-density repulsive Fermi gas in two dimensions: Bound-pair excitations and Fermi-liquid behavior

We study the dilute-gas expansion for a two-dimensional Fermi system with arbitrary short-range repulsive interactions. In contrast with the three-dimensional case, we find an unusual pole in the vertex part in the particle-particle channel, for all center-of-mass momenta q<2${\mathit{k}}_{\mathit{F}}$. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2${\mathit{k}}_{\mathit{F}}$. We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.