Bound state problem for two particles in separable and non-separable optical lattices

M. L. Glasser

Presents a non-separable one-dimensional two-body problem. It describes the reflection of a two-particle bound state off a discontinuity across which there is a steplike change in the interparticle potential. The model can be used as a primitive description of a pick-up or stripping reaction at an interface and also describes the diffusion of a particle in a plane containing an extended finite depth trap. The reflection coefficient for the discontinuity is obtained in closed form.

P. E. Kornilovitch

J. I. Abdullaev, A.I. Eshniyоzov, M. V. Dolgopolov

We study three-particle Schrödinger operators on the two-dimensional lattice Z 2 and show that a critical mass ratio γ c ≈ 2.75194 governs the existence of a bound trimer in the fermionic 2 + 1 configuration (two identical fermions and a third particle). For γ < γ c there is a topological prohibition (Pauli suppression) of a three-body bound state, whereas for γ > γ c a doubly degenerate eigenvalue emerges below the essential spectrum with the strong-coupling asymptotics z(γ, λ) = −λ+e 0 (γ)+O(λ −1 ). Within a unified framework based on the Birman–Schwinger principle and strong-coupling asymptotic analysis, we compare this behaviour with the bosonic case of three identical particles, where two bound states exist below the essential spectrum and the ground-state energy satisfies z 1 s (µ) = −3µ + C 2 + O(µ −1 ). The resulting second-order phase transition with respect to the mass ratio γ is relevant for the design of experiments on fermionic trimers in optical lattices and for modelling excitonic complexes and defect-bound states in two-dimensional nanomaterials, where the critical value γ c serves as a design guideline for the observability of three-body bound states. We also outline a modified three-particle lattice model with two competing interaction channels, for which the Birman–Schwinger analysis naturally leads to a Landau-type scenario of a first-order phase transition in the space of trimer bound states. In the bosonic case we prove a strong-coupling theorem describing the existence and asymptotics of trimer bound states, while in the fermionic 2+1 case we establish a spectral phase-transition theorem that identifies an explicit critical mass ratio γ c separating the trimer and non-trimer regimes.

Thomas M. Hanna, Eite Tiesinga, William James Mitchell +1 more

Johannes Hecker Denschlag

Michael L. Wall, Lincoln D. Carr

We numerically study the problem of two fermions in a three-dimensional optical lattice interacting via a zero-range Feshbach resonance and display the dispersions of the bound states as a two-particle band structure with unique features compared to typical single-particle band structures. We show that the exact two-particle solutions of a projected Hamiltonian may be used to define an effective two-channel, few-band model for the low-energy, low-density physics of many fermions at arbitrary s-wave scattering length. Our method applies to resonances of any width and can be adapted to multichannel situations or higher-ℓ pairing. In strong contrast to usual Hubbard physics, we find that pair hopping is significantly altered by strong interactions and the presence of the lattice, and the lattice induces multiple molecular bound states.