Exact two-body bound states with Coulomb repulsion in a periodic potential

It is shown, through an elementary quantum mechanical calculation, that two particles interacting via a short range repulsive force in an external periodic potential can form a bound state. The two-particle wavefunction is labelled by a continuous centre-of-mass momentum. It is bounded and spatially localized in the centre-of-mass system; thus, the spatial wavefunction in the relative distance is square integrable and corresponds to a discrete energy. For instance, a combination of short-range (i.e. screened) binary Coulomb interactions and the periodic potential provided by the stationary ions, can create a two-electron bound state in a crystalline solid (Slater et al 1953 Phys. Rev. 91 1323 and Hubbard 1963 Proc. R. Soc. A 276 238). However, the phenomenon delineated here is universal in the sense that, under appropriate conditions, bound states are possible independent of the nature of the particles and/or the mechanism by which the external periodic potential is engineered. Our general wave mechanical result may explain experimental results presenting evidence of such bound pair states in solids (Gross et al 1971 JETP Lett. 13) and photonic lattices (Winkler et al 2006 Nature 441 853). It has many other potentially interesting consequences even for classical interacting wave systems (e.g. solitons) propagating in a periodic background. This result of wave mechanics and interference is remarkable in that two repulsively interacting particles cannot form a bound state when moving in vacuum. Two non-interacting particles moving in a periodic external potential can only ever form uncorrelated two-particle Bloch states and yet when both physical conditions are present they can move as a 'bound pair'.