Mathematics

We describe and analyze a generalized two dimensional  model, whose study is motivated by the understanding of several topological singularities arising in Physics and Materials Science. Among them, disclinations and string defects in liquid crystals, fractional vortices and domain walls in micromagnetics, partial dislocations and stacking faults in crystal plasticity. We focus on nearest neighbor interactions on a square lattice and we assume that the interaction potential has  wells corresponding to the symmetries of the system. As the lattice spacing vanishes, we derive by Gamma-convergence the discrete-to-continuum limit of this model. In the energy regime we deal with, the asymptotic ground states exhibit fractional vortices, connected by string defects. The Gamma-limit takes into account both contributions, through a renormalized energy, depending on the configuration of fractional vortices, and a surface energy, proportional to the length of the strings. This is a joint work with R. Badal, M. Cicalese (both TU München), and M. Ponsiglione (University of Rome "La Sapienza").