Discrete model for partial defects

I will present a discrete two dimensional toy model for crystal defects based on nearest neighbours and next to nearest neighbours interaction, via a period potentials. In the asymptotic limit as the lattice spacing tends to zero, in terms of Gamma convergence, the model accounts for the formation and interaction of partial dislocations and stacking faults. The model falls into a class of models with topological (fractional) singularities. One of the key ingredients is the characterisation of the minimisers for the core energy of the singularities, the so called one-vortex solutions.