Berezinskii – Kosterlitz – Thouless transition and two-dimensional melting

The main aspects of the theory of phase transitions in two-dimensional degenerate systems (Berezinskii–Kosterlitz–Thouless, or BKT, transitions) are reviewed in detail, including the transition mechanism, the renormalization group as a tool for describing the transition, and how the transition scenario can possibly depend on the core energy of topological defects (in particular, in thin superconducting films). Various melting scenarios in two-dimensional systems are analyzed, and the current status of actual experiments and computer simulations in the field is examined. Whereas in three dimensions melting always occurs as a single first-order transition, in two dimensions, as shown by Halperin, Nelson, and Young, melting via two continuous BKT transitions with an intermediate hexatic phase characterized by quasi-long-range orientational order is possible. But there is also a possibility for a first-order phase transition to occur. Recently, one further melting scenario, different from that occurring in the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young theory, has been proposed, according to which a solid can melt in two stages: a continuous BKT-type solid–hexatic transition and then a first-order hexatic-phase–isotropic-liquid phase transition. Particular attention is given to the melting scenario as a function of the potential shape and to the random pinning effect on two-dimensional melting. In particular, it is shown that random pinning can alter the melting scenario fundamentally in the case of a first-order transition. Also considered is the melting of systems with potentials having a negative curvature in the repulsion region—potentials that are successfully used in describing the anomalous properties of water in two dimensions.