Two-particle bound states on a lattice

Abstract

Two-particle lattice states are important for physics of magnetism, superconducting oxides, and cold quantum gases. The quantum-mechanical lattice problem is exactly solvable for finite-range interaction potentials. A two-body Schrödinder equation can be reduced to a system of linear equations whose numbers scale with the number of interacting sites. For the simplest cases such as on-site or nearest-neighbor attractions, many pair properties can be derived analytically, although final expressions can be quite complicated. In this work, we systematically investigate bound pairs in one-, two-, and three-dimensional lattices. We derive pairing conditions, plot phase diagrams, and compute energies, effective masses and radii. Along the way, we analyze nontrivial physical effects such as light pairs and the dependence of binding thresholds on pair momenta. At the end, we discuss the preformed-pair mechanism of superconductivity and stability of many-pair systems against phase separation. The paper is a combination of original work and pedagogical tutorial.