Penrose tilings form lattices, exhibiting 5-fold symmetry and isotropic elasticity, with inhomogeneous coordination much like that of the force networks in jammed systems. Under periodic boundary conditions, their average coordination is exactly four
. We study the elastic and vibrational properties of rational approximants to these lattices as a function of unit-cell size $N_S$ and find that they have of order $sqrt{N_S}$ zero modes and states of self stress and yet all their elastic moduli vanish. In their generic form obtained by randomizing site positions, their elastic and vibrational properties are similar to those of particulate systems at jamming with a nonzero bulk modulus, vanishing shear modulus, and a flat density of states.
Driven surface diffusion occurs, for example, in molecular beam epitaxy when particles are deposited under an oblique angle. Elastic phase transitions happen when normal modes in crystals become soft due to the vanishing of certain elastic constants.
We show that these seemingly entirely disparate systems fall under appropriate conditions into the same universality class. We derive the field theoretic Hamiltonian for this universality class, and we use renormalized field theory to calculate critical exponents and logarithmic corrections for several experimentally relevant quantities.
We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the c
ollapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loop order and, where available, compare them to numerical results.
Long linear polymers in strongly disordered media are well described by self-avoiding walks (SAWs) on percolation clusters. The length-distribution of these SAWs encompasses to distinct averages, viz. the averages over cluster- and SAW-conformations.
For the latter average, there are two basic options, one being static and one being kinetic. It is well known for static averaging that if the disorder of the underlying medium is weak, differences to the ordered case appear merely in non-universal quantities. Using dynamical field theory, we show that the same holds true for kinetic averaging. For strong disorder, i.e., the medium being close to the percolation point, we employ a field theory for the nonlinear random resistor network in conjunction with a real-world interpretation of Feynman diagrams, and we calculate the scaling exponents for the shortest, the longest and the mean or average SAW to 2-loop order. In addition, we calculate to 2-loop order the entire family of multifractal exponents that governs the moments of the the statistical weights of the elementary constituents (bonds or sites of the underlying fractal cluster) contributing to the SAWs. Our RG analysis reveals that kinetic averaging leads to renormalizability whereas static averaging does not, and hence, we argue that the latter does not lead to a well-defined scaling limit. We discuss the possible implications of this finding for experiments and numerical simulations which have produced wide-spread results for the exponent of the average SAW. To corroborate our results, we also study the well-known Meir-Harris model for SAWs on percolation clusters. We demonstrate that this model leads back to 2-loop order to the renormalizable real world formulation with kinetic averaging if the replica limit is consistently performed at the first possible instant of the calculation.
We construct a three-dimensional lattice model for biological gels in which straight lines of bonds correspond to filamentous semi-flexible polymers and lattice sites, which are exactly four-fold coordinated, to crosslinks. With only stretching centr
al forces between nearest neighbors, this lattice is sub-isostatic with an extensive number of zero modes; but all of its elastic constants are nonzero, and its elastic response is affine. Removal of bonds with probability $1-p$ leads to a lattice with average coordination number less than four and a distribution of polymer lengths. When bending forces are added, the diluted lattice exhibits a rigidity threshold at $p=p_b<1$ and crossover from bending-dominated nonaffine to stretching-dominated affine response between $p_b$ and $p=1$.
Nematic elastomers with a locked-in anisotropy direction exhibit semi-soft elastic response characterized by a plateau in the stress-strain curve in which stress does not change with strain. We calculate the global phase diagram for a minimal model,
which is equivalent to one describing a nematic in crossed electric and magnetic fields, and show that semi-soft behavior is associated with a broken symmetry biaxial phase and that it persists well into the supercritical regime. We also consider generalizations beyond the minimal model and find similar results.
