Temporal Renormalization and the Critical-like Behavior in Supercooled Liquids

Inspired by the Kadanoff transformation in the standard renormalization group theory, we propose a temporal renormalization scheme. A Boltzmann factor that explicitly depends on the renormalized timescale is constructed, permitting thermodynamic quantities to be evaluated self-consistently across different timescales. By applying the scheme to the long-time dynamics of supercooled liquids, we uncover critical-like behaviors in supercooled liquids with three characteristic renormalization timescales: At the first timescale s_\alpha, the system appears to be "thermodynamically frozen", i.e., the energy fluctuation becomes temperature-independent throughout the supercooled regime. At the second timescale s_\beta, the third-order moment of the energy distribution reaches a maximum, and s_\beta is nearly temperature-independent. At the third timescale s_\gamma, the third-order moment of the energy distribution passes through a minimum, and s_\gamma diverges as a power law, s_\gamma \sim |T-T_\rm c|^-\gamma. The scaling relations may reveal an intrinsic behavior in supercooled liquids, highlighting their unique features. The current findings also demonstrate that temporal renormalization provides a powerful lens for investigating timescale-specific dynamics.