Rigidity of fragile matter: A gauge theoretic perspective

Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454

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Published online: 1 December 2025

Abstract

The process of jamming creates fragile matter, whose rheology is distinct from solids or liquids. The solidity of fragile matter emerges from the applied stress itself. Despite their disordered, fluid-like structure, they can exhibit elastic behavior under small perturbations. The rigidity of these solids emerge from internally organized contact networks created by external forces such as gravity or confinement. Unlike traditional elastic solids, these fragile solids do not possess a stress-free reference structure. This poses challenges for classical elasticity, whose formulation relies on strain, relative to a stress-free configuration. This has, for decades, motivated the search for a stress-only elasticity framework. Recent advances have provided a new perspective based on a gauge theoretic framework defined by a generalization of the familiar Gauss’s law of electromagnetism. For fragile, granular solids, the Gauss’s law represents the constraints of mechanical equilibrium. Elasticity emerges as a “dielectric” response to external stresses: internal, bound force dipoles, screen external forces. This paper presents a short review of this theoretical framework.