High-dimensional theory of the glass transition revisited: hopping and local defects

Abstract

The replicated liquid theory provides a microscopic mean-field description of the glass transition by combining the density functional theory of liquids with the replica method originally developed for spin glasses. In the conventional replica liquid theory, a glassy state is described by assuming that particles in different replicas undergo vibrational motion around common centers of mass, thereby forming molecules that contain one particle from every replica. Here we revisit this assumption by allowing each molecule to contain only a subset of replicas. This generalized formulation describes particle-level replica mismatches, which may be associated with non-vibrational motions such as particle hopping. We apply the theory to high-dimensional hard and harmonic spheres, where the mean-field description is expected to become exact. For hard spheres, replica mismatches destabilize the glassy metastable state and shift the dynamical transition to a significantly higher packing fraction, while leaving the leading thermodynamic glass transition unchanged. The resulting transition density agrees, at leading order in high dimensions, with the recent rigorous lower bound for random sphere packings obtained by Campos, Jenssen, Michelen, and Sahasrabudhe by using a discretized version of greedy Random Sequential Absorption, suggesting an algorithmic interpretation of the transition: grandcanonical dynamics is more efficient in high dimensional spaces than canonical one. For harmonic spheres at finite temperature, the glassy state contains a finite replica-mismatch fraction even at the thermodynamic ideal-glass transition, thereby shifting the transition point from that predicted by the conventional replica ansatz.