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. However, we lack a
theoretical description of these kinds of rigidity transitions that can detect
an upper critical dimension below which mean-field theory fails and predict
corrections to the theory in and below this upper critical dimension. Towards
that end, we show that the critical exponents for a continuous isotropic
rigidity transition predicted by a simple dynamical effective medium theory
(the Coherent Potential Approximation, CPA) are not mean-field-like,
dimension-independent quantities. Instead, we find that the theory has an upper
critical dimension $d_{u}=2$, in which there are the usual logarithmic
corrections and 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
posit their explicit forms, incorporating a dangerously irrelevant variable
becoming marginal in $d=2$. We also derive universal scaling functions from the
CPA sufficient to predict all linear response in randomly diluted isotropic
elastic systems. These results indicate the types of corrections one expects to
disordered rigidity transitions in $2$ dimensions and provide a pathway to
understanding the critical phenomena in such systems.
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