Defect-Mediated Melting of Square-Lattice Solids

The Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory successfully explains the melting mechanism of two-dimensional isotropic lattices as a two-step process driven by the unbinding of topological defects. By considering the elastic theory of the square lattice, we extend the KTHNY theory to melting of square lattice solids. In addition to the familiar elastic constants that govern the theory -- the Young's modulus and the Poisson ratio -- a third constant controlling the anisotropy of the medium emerges. This modifies both the logarithmic and angular interactions between the topological defects. Despite this modification, the extended theory retains the qualitative features of the isotropic case, predicting a two-step melting with an intermediate tetratic phase. However, some subtle differences arise, including a modified bound on the translational correlation exponent and the absence of universal values for the Young's modulus at the solid-to-tetratic phase transition.