Peano modes at the D = 2 delocalisation transition

Abstract

.We study the elasticity of the plane-filling Peano chain in terms of a spring network, modelling a flat and thin elastic strip, smoothly curved so as to follow the chain pattern.This network sustains normal modes, named Peano modes; in the same way as acoustic waves in crystals are sustained by the matching of translations with rotations, these modes arise when discrete dilatations match with rotations. The pattern shares this eightfold symmetry with the octagonal quasicrystal, but is much looser and prone to disruption; its relaxation marks the saddle point separating the solid from the liquid. In terms of the equivalent quantum mechanical problem the above corresponds to the D = 2 quantum delocalisation transition.Some experimental observations, regarding a priori uncorrelated systems such as colloids, bilayer water and chromosomal DNA come together, at least qualitatively, within the viewpoint proposed here. With reference to the D = 2 colloids the model accounts for the bosonic peak found in their phononic spectrum and for the relaxation of the excitations. The latter typically evolve towards patterns of increasingly large size, as in the inverse energy cascade of D = 2 turbulence. As for bilayer water, it undergoes a liquid to quasicrystal transition surprisingly similar to the phase transition found here. Finally, considering the chromosomal DNA, the model unveils the mixed nature of its compact conformations, lying at the boundary between a liquid and a solid, similar to colloids and quasicrystals.