Low temperature dynamics of kinks on Ising interfaces

TL;DR

This paper develops a continuum model for the anisotropic motion of Ising interfaces driven by curvature or external fields, deriving equations in 2D and 3D that connect microscopic kink dynamics to macroscopic interface velocity.

Contribution

It introduces a novel continuum evolution equation for kink density in 2D and extends the analysis to 3D interface velocities, incorporating anisotropic effects and impurity modeling.

Findings

Derived a nonlinear diffusion equation for kink density in 2D.

Validated the interface velocity law in 3D near <100> orientation.

Showed the smoothness of velocity despite singular stiffness and curvature tensors.

Abstract

The anisotropic motion of an interface driven by its intrinsic curvature or by an external field is investigated in the context of the kinetic Ising model in both two and three dimensions. We derive in two dimensions (2d) a continuum evolution equation for the density of kinks by a time-dependent and nonlocal mapping to the asymmetric exclusion process. Whereas kinks execute random walks biased by the external field and pile up vertically on the physical 2d lattice, then execute hard-core biased random walks on a transformed 1d lattice. Their density obeys a nonlinear diffusion equation which can be transformed into the standard expression for the interface velocity v = M[(gamma + gamma'')kappa + H]$, where M, gamma + gamma'', and kappa are the interface mobility, stiffness, and curvature, respectively. In 3d, we obtain the velocity of a curved interface near the <100> orientation from an analysis of the self-similar evolution of 2d shrinking terraces. We show that this velocity is consistent with the one predicted from the 3d tensorial generalization of the law for anisotropic curvature-driven motion. In this generalization, both the interface stiffness tensor and the curvature tensor are singular at the <100> orientation. However, their product, which determines the interface velocity, is smooth. In addition, we illustrate how this kink-based kinetic description provides a useful framework for studying more complex situations by modeling the effect of immobile dilute impurities.