Topological regulation of activation barriers on fractal substrates

TL;DR

This paper investigates how the topological features of fractal substrates influence the dynamics of phase ordering in ferromagnetic systems, revealing a crossover in growth behavior governed by the substrate's properties.

Contribution

It introduces a scaling approach inspired by renormalization group ideas to analyze the crossover between different dynamical regimes on fractal graphs with distinct topological features.

Findings

Logarithmic growth of free energy barriers on finitely ramified graphs

Power-law growth of barriers on other structures

Excellent agreement between theory and numerical simulations

Abstract

We study phase-ordering dynamics of a ferromagnetic system with a scalar order-parameter on fractal graphs. We propose a scaling approach, inspired by renormalization group ideas, where a crossover between distinct dynamical behaviors is induced by the presence of a length $\lambda $ associated to the topological properties of the graph. The transition between the early and the asymptotic stage is observed when the typical size $L(t)$ of the growing ordered domains reaches the crossover length $\lambda $. We consider two classes of inhomogeneous substrates, with different activated processes, where the effects of the free energy barriers can be analytically controlled during the evolution. On finitely ramified graphs the free energy barriers encountered by domains walls grow logarithmically with $L(t)$ while they increase as a power-law on all the other structures. This produces different asymptotic growth laws (power-laws vs logarithmic) and different dependence of the crossover length $\lambda $ on the model parameters. Our theoretical picture agrees very well with extensive numerical simulations.