Two particles on a chain with disordered interaction: Localization and dissociation of bound states and mapping to chaotic billiards

We consider two particles hopping on a chain with a contact interaction between them. At strong interaction, there is a molecular bound state separated by a direct gap from a continuous band of atomic states. Introducing weak disorder in the interaction, the molecular state becomes Anderson-localized. At stronger disorder, part of the molecular band delocalizes and dissociates due to its hybridization to the atomic band. We characterize these different regimes by computing the density of states, the inverse participation ratio, the level-spacing statistics, and the survival probability of an initially localized state. The atomic band is best described as that of a rough billiard for a single particle on a square lattice that shows signatures of quantum chaos. In addition to typical "chaotic states," we find states that are localized along only one direction. These "separatrix states" are more localized than chaotic states, and similar in this respect to scarred states, but their existence is due to the separatrix isoenergy line in the interaction-free dispersion relation, rather than to unstable periodic orbits.