Possible scenarios of a phase transition from isotropic liquid to a hexatic phase in the theory of melting in two-dimensional systems

A two-stage process consisting of two continuous Berezinskii–Kosterlitz–Thouless-type transitions with an intermediate anisotropic liquid, a hexatic phase, is a well-known scenario of melting in two-dimensional systems. A direct first-order transition, similar to melting in three-dimensional systems, is another scenario variant. We prove the possibility in principle of the existence of a third scenario according to which melting occurs via two transitions, but in contrast to predictions of the Berezinskii–Kosterlitz–Thouless theory, the transition from an isotropic liquid to a hexatic phase is a first-order transition. Such a scenario was recently observed in a computer simulation of two-dimensional systems and then in a real experiment. Our proof is based on an analysis of branching solutions of an exact closed nonlinear integral equation for a two-particle conditional distribution function.