Structure of curved crystals in the thermodynamic limit and the perfect screening condition

The dislocation and disclination density defining the structure of a crystal constrained on a general curved background is computed analytically in the thermodynamic limit, when the number of particles is arbitrarily large. It is shown that the minimum of the elastic energy, where strains are optimally close to zero, can be formulated in terms of a connection (a rule on how to parallel transport vectors), which allows to provide an explicit solution in the thermodynamic limit that consists of disclinations surrounded by scars (grain boundaries with variable spacing). The approach allows to compute the interaction potential. For a sphere, a full characterization of the scars is provided, and it is shown that the potential of interaction among disclinations "dressed" by scars is inversely proportional to the sinus of the geodesic distance and that the ground state consists of 12 dressed disclinations that display icosahedral symmetry. The case of a torus is also considered. More generally, the thermodynamic solution implements a "perfect screening" condition, where defects completely screen the Gaussian curvature. Implications for the problem of melting are discussed.