Geometrical universality in vibrational dynamics

TL;DR

This paper rigorously proves that the spectral dimension, a key measure of vibrational dynamics in disordered systems, remains invariant under various local and coarse-graining transformations, enabling simplified modeling of complex geometries.

Contribution

The authors provide a rigorous analytical proof that the spectral dimension is geometrically universal and invariant under local rescaling, finite range, and coarse-graining transformations.

Findings

Spectral dimension is invariant under local rescaling of couplings.

Spectral dimension remains unchanged with addition of finite or rapidly decaying infinite range couplings.

Coarse graining transformations do not alter the spectral dimension.

Abstract

A good generalization of the Euclidean dimension to disordered systems and non crystalline structures is commonly required to be related to large scale geometry and it is expected to be independent of local geometrical modifications. The spectral dimension, defined according to the low frequency density of vibrational states, appears to be the best candidate as far as dynamical and thermodynamical properties are concerned. In this letter we give the rigorous analytical proof of its independence of finite scale geometry. We show that the spectral dimension is invariant under local rescaling of couplings and under addition of finite range couplings, or infinite range couplings decaying faster then a characteristic power law. We also prove that it is left unchanged by coarse graining transformations, which are the generalization to graphs and networks of the usual decimation on regular structures. A quite important consequence of all these properties is the possibility of dealing with simplified geometrical models with nearest-neighbors interactions to study the critical behavior of systems with geometrical disorder.