Unifying two-dimensional melting in diverse systems with repulsive potentials

Two-dimensional melting remains a central question in soft condensed matter physics, where hexagonal crystals can melt either continuously through an intermediate hexatic phase or discontinuously via a first-order transition. Despite extensive studies, the physical origin determining these distinct pathways has not been unified. Using large-scale molecular dynamics simulations of systems with diverse repulsive interaction potentials, we identify a dimensionless parameter that universally governs both the isothermal equation of state and the nature of the melting transition. This single quantity integrates the effects of density, interaction softness, and interaction range, allowing continuous and discontinuous melting behaviors to be clearly separated across all studied models. Our findings establish a unified framework for understanding two-dimensional melting in soft-matter systems and provide a predictive criterion linking microscopic interactions to macroscopic phase behavior.