Fig. 4: Physical nature of the non-equilibrium amorphous state and the phase transition.

a Mean-squared displacements (MSDs) (Δ²) in the glassy state exhibiting frustrated diffusive behavior. b Non-Gaussian parameters for MSD \(\alpha\) vs. temperature T. The two values at T = 2.83 correspond to the crystalline state (smaller one) and the glassy state (larger one). c Schematic illustration of the spin-ice-like state defined on a topologically triangular lattice with the ‘4-in-2-out’ ice-rule. The left panel shows the six neighboring atom-clusters (circled in blue) of the central cluster (circled in red) while the right one illustrates that the six atoms in the cluster exhibit the ‘4-in-2-out’ feature along the direction indicated by the orange arrows. d Bond-orientational order parameter \({\varPsi }_{6,{\mathrm{atom}}}\) vs. T, whose discontinuous change suggests a first-order nature of the melting of the spin-ice-like state. The inset is a small portion of the snapshot exhibiting the local packing scheme, where the purple lines and circles highlight the six neighboring atoms of the central atom shadowed in purple arranged in a nearly dense-packing way. The error-bars in this figure represent standard deviations, resulting from the thermal fluctuations in one simulation trajectory. Physical quantities are expressed in L-J units.