Interface pinning and slow ordering kinetics on infinitely ramified fractal structures

TL;DR

This paper studies the dynamics of phase ordering on fractal lattices, revealing domain pinning and slow growth due to lattice defects, contrasting with behavior on Euclidean lattices.

Contribution

It demonstrates that on infinitely ramified fractals, domain growth is hindered by pinning effects, leading to different asymptotic behaviors than in regular lattices.

Findings

Domain size grows as t^{1/d_w} initially

Pinning causes the system to enter a frozen state at zero temperature

Metastable minima and rough energy landscape prevent self-similar pattern formation

Abstract

We investigate the time dependent Ginzburg-Landau (TDGL) equation for a non conserved order parameter on an infinitely ramified (deterministic) fractal lattice employing two alternative methods: the auxiliary field approach and a numerical method of integration of the equations of evolution. In the first case the domain size evolves with time as $L(t)\sim t^{1/d_w}$, where $d_w$ is the anomalous random walk exponent associated with the fractal and differs from the normal value 2, which characterizes all Euclidean lattices. Such a power law growth is identical to the one observed in the study of the spherical model on the same lattice, but fails to describe the asymptotic behavior of the numerical solutions of the TDGL equation for a scalar order parameter. In fact, the simulations performed on a two dimensional Sierpinski Carpet indicate that, after an initial stage dominated by a curvature reduction mechanism \`a la Allen-Cahn, the system enters in a regime where the domain walls between competing phases are pinned by lattice defects. The lack of translational invariance determines a rough free energy landscape, the existence of many metastable minima and the suppression of the marginally stable modes, which in translationally invariant systems lead to power law growth and self similar patterns. On fractal structures as the temperature vanishes the evolution is frozen, since only thermally activated processes can sustain the growth of pinned domains.