Schramm-Loewner evolution in 2d rigidity percolation

Abstract

Amorphous solids may resist external deformation such as shear or compression while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid, such materials acquire a high degree of symmetry hidden in the disorder fluctuations: their microstructure becomes statistically conformally invariant. In this Letter we exploit this finding to characterise the universality class of central-force rigidity percolation (RP), using Schramm-Loewner Evolution (SLE) theory. We provide numerical evidences that the interfaces of the mechanically stable structures (rigid clusters), at the rigidification transition, are consistently described by SLE$_\kappa$, showing that this powerful framework can be applied to a mechanical percolation transition. Using well-known relations between different SLE observables and the universal diffusion constant $\kappa$, we obtain the estimation $\kappa\sim2.9$ for central-force RP. This value is consistent, through relations coming from conformal field theory, with previously measured values for the clusters' fractal dimension $D_f$ and correlation length exponent $\nu$, providing new, non-trivial relations between critical exponents for RP. These findings open the way to a fine understanding of the microstructure in other important classes of rigidity and jamming transitions.