Kavli Affiliate: Itai Cohen
| First 5 Authors: Stephen J. Thornton, Danilo B. Liarte, Itai Cohen, James P. Sethna,
| Summary:
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.
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