Mathematics

The N-clock model is a two-dimensional ferromagnetic spin model on the square lattice in which the spin field is constrained to take values in a set of N equi-spaced points of the unit circle. It is usually considered as an approximation of the XY model, for which instead the spin field is allowed to attain all the values of the unit circle. In the theory of superconductivity the latter models phase transitions mediated by the formation and the interaction of co-dimension 2 topological defects as in the well-known Ginzburg-Landau functional. A breakthrough result by Fröhlich and Spencer (CMP 1981) shows that the same kind of phase transitions appear in the N-clock model for N large enough. By a variational analysis we find the explicit rate of divergence of N (with respect to the number of interacting lattice points) for which low energy configurations of the N-clock model asymptotically behave like those of the XY model at zero temperature. We moreover exhaustively discuss all the other regimes of N and we show how Cartesian Currents can detect the energy concentration on sets of co-dimenion smaller or equal than 2.  The results presented are contained in a series of recent papers in collaboration with G. Orlando (Poliba) and M. Ruf (EPFL).