Matter-wave gap vortices in optical lattices

We predict the existence of spatially localized nontrivial topological states of the Bose-Einstein condensate with repulsive atomic interactions confined by an optical lattice. These nonlinear localized states, matter-wave gap vortices, possess a phase dislocation and exist in the gaps of the matter-wave bandgap spectrum due to the Bragg scattering. We discuss the structure, stability, and formation dynamics of the gap vortices in the case of two-dimensional optical lattices.