Hierarchical and Ultrametric Barriers in the Energy Landscape of Jammed Granular Matter
Phys. Rev. Lett. 136, 198202 – Published 12 May, 2026
DOI: https://doi.org/10.1103/mtn5-s26m
Abstract
According to the mean-field glass theory, the (free) energy landscape of disordered systems is hierarchical and ultrametric if they belong to the full-replica-symmetry-breaking universality class. However, examining this theoretical picture in three-dimensional systems remains challenging, where the energy barriers become finite. Here, we numerically explore the energy landscape of granular models near the jamming transition using a saddle dynamics algorithm to locate both local energy minima and saddles. The multiscale distances and energy barriers between minima are characterized by two metrics, both of which exhibit signatures of an ultrametric space. The scale-free distributions of energy barriers confirm the landscape’s hierarchical organization and align with the distribution of elastic avalanches under athermal quasistatic shear.
