Study of melting in two dimensions

Techniques developed in studies of the two-dimensional planar model of magnetism are applied to the Kosterlitz-Thouless dislocation model of melting in two dimensions. Duality transformations relate the dislocation problem to Hamiltonians with short-ranged interactions amenable to a Migdal-Kadanoff recursion-relation analysis. In an approximation which neglects an angular dependence in the decay of correlations, the melting problem is equivalent to the statistical mechanics of two interpenetrating superfluids with coupled phases. The coupling between the phases is a marginal operator which (together with other marginal couplings) destroys the universality of critical exponents at the melting temperature $T_m$. Indeed melting seems to bear roughly the same relation to the planar model as the Baxter model does to the Ising model. A recursion scheme similar to that developed by Kosterlitz for the planar model is used to study the correlation length, specific heat, and Burger's-vector correlations near melting. For melting of a triangular lattice, the correlation length diverges, ${\xi }_{+}\sim \exp \left[C{(\frac{T}{T_m}-1)}^{-\frac{2}{5}}\right]$, as $T$ approaches $T_m$ from above. There is an analogous essential singularity in the specific heat.