2024/07/19 by Thornton, Stephen J., Liarte, Danilo B., Cohen, Itai +1 · 1 citation
#FOS: Physical sciences #Soft Condensed Matter (cond-mat.soft) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2407.14586
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 du=2, below which the critical exponents are modified. We recapitulate Ken Wilson's phenomenology of the (4-ε)-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.