Emergence of topological and geometric defects in the Gamma-limit of discrete energies

We introduce and analyse a variant of the two-dimensional $XY$-model energy which is suited to detect both topological defects and geometric defects in form of fractional vortices and domain walls, respectively. In contrast to previously introduced variants, the energies we consider here are defined without using an angular lifting of the ${\mathbb{S}}^1$-valued spin variables. Moreover, they combine in an explicit way the features of the $XY$-model energy on the one hand and weak-membrane energies on the other hand. This leads to simplified proofs of compactness and lower bound in the $\mathrm{\Gamma }$-convergence analysis.