Extended Data Fig. 5: Application of C(r) in two-dimensional melting of monodisperse crystal.
From: Ideal non-crystals as a distinct form of ordered states without symmetry breaking
Monodisperse crystals with the same interacting potential and packing fraction as our ideal-non-crystal system are heated and then cooled down using molecular dynamics simulations, with a constant heating (cooling) rate dT/dt = 10−9. a, Temperature dependence of potential energy per particle E during heating (red line) and cooling (black line) processes. The evolution of E during melting resembles that of ideal non-crystals, suggesting a similar physical mechanism of melting. Marginal hysteresis is observed between cooling and heating, differing from the ideal-non-crystal system. For state points indicated in a, the structural correlations are characterized in three different ways: The conventional correlation function of the hexatic order parameter \({G}_{6}(r)=\langle {\Psi }_{6}^{* }(r){\Psi }_{6}(0)\rangle\) (b), the correlation of Ψ6 along the coherent path \({G}_{6}^{p}(r)\) (c), and the path-integral-like correlation function C(r) defined in this work (d). The dashed line is the power-law scaling predicted by the KTHNY theory of two-dimensional melting from hexatic phase to liquids. A close comparison confirms prefect agreement between them, validating the efficacy of our methodology. In essence, our path-integral-like scheme, based on the definition of the coherent path and the corresponding correlation function, captures the structural coherence from the steric constraint of triangle units. For monodisperse systems, the steric order reduces to the hexatic orientational order, making C(r) and G6(r) give essentially the same information.
