Limiting Shapes of Ising Droplets, Ising Fingers, and Ising Solitons

TL;DR

This paper analytically investigates the deterministic limiting shapes of Ising droplets, fingers, and solitons under zero-temperature dynamics, revealing universal geometries and velocity relations in two and three dimensions.

Contribution

It provides the first analytical determination of the limiting shapes and velocities of Ising droplets, fingers, and solitons on square and cubic lattices.

Findings

Derived the universal shape of Ising droplets in 2D.

Calculated the velocity and shape of Ising fingers on the square lattice.

Connected the shape of Ising solitons to 2D droplet geometries.

Abstract

We examine the evolution of an Ising ferromagnet endowed with zero-temperature single spin-flip dynamics. A large droplet of one phase in the sea of the opposite phase eventually disappears. An interesting behavior occurs in the intermediate regime when the droplet is still very large compared to the lattice spacing, but already very small compared to the initial size. In this regime the shape of the droplet is essentially deterministic (fluctuations are negligible in comparison with characteristic size). In two dimensions the shape is also universal, that is, independent on the initial shape. We analytically determine the limiting shape of the Ising droplet on the square lattice. When the initial state is a semi-infinite stripe of one phase in the sea of the opposite phase, it evolves into a finger which translates along its axis. We determine the limiting shape and the velocity of the Ising finger on the square lattice. An analog of the Ising finger on the cubic lattice is the translating Ising soliton. We show that far away from the tip, the cross-section of the Ising soliton coincides with the limiting shape of the two-dimensional Ising droplet and we determine a relation between the cross-section area, the distance from the tip, and the velocity of the soliton.