This project addresses the asymptotic analysis via Gamma-convergence of specific energy-driven lattice systems on manifolds in which the energy concentrates on lower-dimensional sets. In particular, we are interested in Ising- and spin systems, where the concentration of energy on surface- and vortex scales leads (in dimension 2) to the emergence of line- and point singularities. In the current project we will focus on the first type of singularity. This will be a first step in the direction of treating discrete energies on manifolds which give rise to the emergence of both types of singularities simultanously.
To the best of our knowledge this is the first attempt to address the variational limit analysis via Gamma-convergence of discrete energies on manifolds, where the energy concentrates on a surface scale, thus giving rise to line singularities. As pointed out above, such a limit analysis is of independent interest and at the same time opens the way to treat discrete energies on manifolds where energy can concentrate on different scales. Besides the generalised XY model mentioned above, these can also be discrete energies which in the Gamma-limit typically lead to free-discontinuity functionals, such as weak-membrane models [17, 20] or phase-field models à la Ambrosio-Tortorelli [P2, P3].