Anomalous self-diffusion in the ferromagnetic Ising chain with Kawasaki dynamics

TL;DR

This paper studies the anomalous diffusion of a tagged spin in a ferromagnetic Ising chain with Kawasaki dynamics, revealing exact temperature-dependent prefactors and different growth regimes at equilibrium and during coarsening.

Contribution

It provides exact results for the temperature dependence of displacement variance and analyzes both symmetric and asymmetric Kawasaki dynamics in the Ising chain.

Findings

Displacement variance grows as $A t^{1/2}$ at equilibrium.

At low temperature, growth as $(t/\xi^2)^{2/3}$ during coarsening.

In asymmetric dynamics, variance grows as $B t^{2/3}$, with exact prefactors derived.

Abstract

We investigate the motion of a tagged spin in a ferromagnetic Ising chain evolving under Kawasaki dynamics. At equilibrium, the displacement is Gaussian, with a variance growing as $A t^{1/2}$. The temperature dependence of the prefactor $A$ is derived exactly. At low temperature, where the static correlation length $\xi$ is large, the mean square displacement grows as $(t/\xi^2)^{2/3}$ in the coarsening regime, i.e., as a finite fraction of the mean square domain length. The case of totally asymmetric dynamics, where $(+)$ (resp. $(-)$) spins move only to the right (resp. to the left), is also considered. In the steady state, the displacement variance grows as $B t^{2/3}$. The temperature dependence of the prefactor $B$ is derived exactly, using the Kardar-Parisi-Zhang theory. At low temperature, the displacement variance grows as $t/\xi^2$ in the coarsening regime, again proportionally to the mean square domain length.