We prove the existence of an Efimov sequence in an internal gap of the essential spectrum of a three-particle lattice Hamiltonian. The system consists of two identical spinless fermions and one distinguishable particle on
Z3, with repulsive pair interactions. For each fixed fermion-to-third-particle mass ratio
γ>γ∗≈13.6069657 and total quasi-momentum near
π1, the mixed zero-range coupling is tuned to a simultaneous upper-threshold resonance. Sufficiently strong nearest-neighbour repulsion between the fermions separates the upper fermion-pair two-cluster spectrum from the free three-particle band. Infinitely many discrete eigenvalues accumulate at the lower endpoint of the resulting gap, which coincides with the free upper edge. We establish logarithmic eigenvalue-counting asymptotics with the same positive coefficient as at zero fermion--fermion coupling. This coefficient also governs accumulation above the essential spectrum throughout the closed weak-coupling interval, including its critical endpoint. We classify the channel spectrum near the free upper edge for all nonnegative fermion couplings.
We analyze a system of two-component fermions, which interact via a Feshbach resonance in the presence of a three-dimensional lattice potential. By expressing a two-channel model of the resonance in the basis of Bloch states appropriate for the lattice, we derive an eigenvalue equation for the two-particle bound states, which is nonlinear in the energy eigenvalue. Compact expressions for the interchannel matrix elements, numerical methods for the solution of the nonlinear eigenvalue problem, and a renormalization procedure to remove ultraviolet divergences are presented. From the structure of the two-body solutions we identify the relevant degrees of freedom, which describe the resonance behavior in the lowest Bloch band. These degrees of freedom, which we call dressed molecules, form an effective closed channel in a many-body model of the resonance, the Fermi resonance Hamiltonian (FRH). It is shown how the properties of the FRH can be determined numerically by solving a projected lattice two-channel model at the two-particle level. As opposed to single-channel lattice models such as the Hubbard model, the FRH is valid for general
s-wave scattering length and resonance width. Hence, the FRH provides an accurate description of the BEC-BCS crossover for ultracold fermions on an optical lattice.