Shape and symmetry determine two-dimensional melting transitions of hard regular polygons

The melting transition of two-dimensional (2D) systems is a fundamental problem in condensed matter and statistical physics that has advanced significantly through the application of computational resources and algorithms. 2D systems present the opportunity for novel phases and phase transition scenarios not observed in 3D systems, but these phases depend sensitively on the system and thus predicting how any given 2D system will behave remains a challenge. Here we report a comprehensive simulation study of the phase behavior near the melting transition of all hard regular polygons with $3\leq n\leq 14$ vertices using massively parallel Monte Carlo simulations of up to one million particles. By investigating this family of shapes, we show that the melting transition depends upon both particle shape and symmetry considerations, which together can predict which of three different melting scenarios will occur for a given $n$. We show that systems of polygons with as few as seven edges behave like hard disks; they melt continuously from a solid to a hexatic fluid and then undergo a first-order transition from the hexatic phase to the fluid phase. We show that this behavior, which holds for all $7\leq n\leq 14$, arises from weak entropic forces among the particles. Strong directional entropic forces align polygons with fewer than seven edges and impose local order in the fluid. These forces can enhance or suppress the discontinuous character of the transition depending on whether the local order in the fluid is compatible with the local order in the solid. As a result, systems of triangles, squares, and hexagons exhibit a KTHNY-type continuous transition between fluid and hexatic, tetratic, and hexatic phases, respectively, and a continuous transition from the appropriate "x"-atic to the solid. [abstract truncated due to arxiv length limitations]

Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods

We report large-scale computer simulations of the hard-disk system at high densities in the region of the melting transition. Our simulations reproduce the equation of state, previously obtained using the event-chain Monte Carlo algorithm, with a massively parallel implementation of the local Monte Carlo method and with event-driven molecular dynamics. We analyze the relative performance of these simulation methods to sample configuration space and approach equilibrium. Our results confirm the first-order nature of the melting phase transition in hard disks. Phase coexistence is visualized for individual configurations via the orientational order parameter field. The analysis of positional order confirms the existence of the hexatic phase.Comment: 9 pages, 8 figures, 2 table