The liquid-solid transition of hard discs: first-order transition or Kosterlitz-Thouless-Halperin-Nelson-Young scenario?

We consider the question of whether a two-dimensional hard-disc fluid has a first-order transition from the liquid state to the solid state as in the three-dimensional inciting-crystallization transition or whether one has two subsequent continuous transitions, from the liquid to the hexatic phase and then to the solid phase, as proposed by Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY). Monte Carlo (MC) simulations of the fluid that study the growth of the bond orientational correlation length, and of the crystal are discussed. The emphasis is on a recent consistency test of the KTHNY renormalization group (RG) scenario, where MC simulations are used to estimate the bare elastic constants and dislocation fugacities in the solid, as a function of density, which then are used as starting values for the RG flow. This approach was validated earlier for the XY model as well.