Dislocation-mediated melting in two dimensions

A theory of dislocation-mediated melting in two dimensions is described in detail, with an emphasis on results for triangular lattices on both smooth and periodic substrates. The transition from solid to liquid on a smooth substrate takes place in two steps with increasing temperatures. Dissociation of dislocation pairs first drives a transition out of a low-temperature solid phase, with algebraic decay of translational order and long-range orientational order. This transition is into a "liquid-crystal" phase characterized by exponential decay of translational order, but power-law decay of sixfold orientational order. Dissociation of disclination pairs at a higher temperature then produces an isotropic fluid. The behavior of the specific heat, structure factor, and various elastic constants near these transitions is worked out. We also discuss the applicability of our results to melting on a periodic substrate. Dislocation unbinding should describe melting of a "floating" (and, in general, incommensurate) adsorbate solid into a high-temperature fluid phase. The orientation bias imposed by the substrate can alter or eliminate the disclination-unbinding transition, however. Transitions from a floating solid into a low-temperature registered or partially registered phase can also be mapped onto the dislocation-unbinding transition, but only at certain special values of the coverage. Substrate reciprocallattice vectors play the role of Burger's vectors in this case.