Universal scaling solution for a rigidity transition: renormalization group flows near the upper critical dimension

TL;DR

This paper identifies the upper critical dimension for rigidity transitions as 2, revealing universal scaling laws and corrections near this dimension through renormalization group analysis and mean-field theory.

Contribution

It introduces a universal scaling framework for rigidity transitions near the upper critical dimension, reducing it from 4 to 2, with explicit predictions for behavior in all dimensions.

Findings

Logarithmic corrections in 2D indicate an upper critical dimension of 2.

Universal scaling functions derived from CPA predict linear response in elastic systems.

Renormalization group analysis reveals a transcritical bifurcation in the flow near dimension 2.

Abstract

Rigidity transitions induced by the formation of system-spanning disordered rigid clusters, like the jamming transition, can be well-described in most physically relevant dimensions by mean-field theories. A dynamical mean-field theory commonly used to study these transitions, the coherent potential approximation (CPA), shows logarithmic corrections in $2$ dimensions. By solving the theory in arbitrary dimensions and extracting the universal scaling predictions, we show that these logarithmic corrections are a symptom of an upper critical dimension $d_{u}=2$, below which the critical exponents are modified. We recapitulate Ken Wilson's phenomenology of the $(4-\epsilon)$-dimensional Ising model, but with the upper critical dimension reduced to $2$. We interpret this using normal form theory as a transcritical bifurcation in the RG flows and extract the universal nonlinear coefficients to make explicit predictions for the behavior near $2$ dimensions. This bifurcation is driven by a variable that is dangerously irrelevant in all dimensions $d>2$ which incorporates the physics of long-wavelength phonons and low-frequency elastic dissipation. We derive universal scaling functions from the CPA sufficient to predict all linear response in randomly diluted isotropic elastic systems in all dimensions.