Lattice two-body problem with arbitrary finite-range interactions

Abstract

We study the exact solution of the two-body problem on a tight-binding one-dimensional lattice, with pairwise interaction potentials which have an arbitrary but finite range. We show how to obtain the full spectrum, the bound and scattering states, and the "low-energy" solutions by very efficient and easy-to-implement numerical means. All bound states are proven to be characterized by roots of a polynomial whose degree depends linearly on the range of the potential, and we discuss the connections between the number of bound states and the scattering lengths. "Low-energy" resonances can be located with great precision with the methods we introduce. Further generalizations to include more exotic interactions are also discussed.

Low-energy behavior of strongly interacting bosons on a flat-band lattice above the critical filling factor

“…This bound state traverses the crystal as if it were a single particle acting under a pure hopping Hamiltonian, and moves with a quadratic dispersion relation at low energies, in stark contrast to the situation at and below critical filling where kinetic energy is completely quenched. Repulsively bound pairs in the Hubbard model have been studied [21][22][23][24][25] and observed with ultracold atoms in optical lattices [26] and nonlinear optical systems [27], but the physical situations treated in those studies and experiments are completely different from the scenario analyzed here. For instance, in the above works the repulsively bound pairs exhibit large double occupancies for strong on-site interactions.…”

Quasi-one-dimensional scattering in a discrete model

“…for all n. The above relation expresses the fact that the energy of the virtual bound state [23] lowers the total energy to E with respect to E n . This point is not specific to our model, but appears in general multichannel scattering problems involving an energy gap to coupled excitations [24].…”

“…(i) The first term is the lowest-energy scattering solution (the analog to the zero-energy solution in the continuum) in one dimension [23] multiplied by the transversal ground state; it represents the quasi-1D solution we are looking for. (ii) Every term in the infinite sum corresponds to a virtual excitation to the nth trapped transversal state.…”

“…The symmetric bound states of the transversal Hamiltonian H y are readily obtained by calculating the roots of a certain polynomial of degree 2R + 1 if R 1 [23] or of degree 2 for the zero-range case [23,25]. The antisymmetric states do not contribute to the scattering length or amplitude.…”

“…The antisymmetric states do not contribute to the scattering length or amplitude. The scattering states, with their corresponding phase shifts are also easily calculated [23].…”