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

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.

7 pages, 3 figures; version 2: two figures and references added

References in corpus (7)

• Exact results for curvature-driven coarsening in two dimensions
• Domain growth morphology in curvature driven two dimensional coarsening
• Zero-Temperature Relaxation of Three-Dimensional Ising Ferromagnets
• Zero-Temperature Freezing in Three-Dimensional Kinetic Ising Model
• Dirichlet sigma models and mean curvature flow
• Growth Inside a Corner: The Limiting Interface Shape
• Critical behavior of collapsing surfaces

Cited by in corpus (11)

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• Zero-temperature 2D Ising model and anisotropic curve-shortening flow
• Dynamics of the two-dimensional directed Ising model: zero-temperature coarsening
• Limiting Shapes in Two-Dimensional Ising Ferromagnets
• Aggregation in non-uniform systems with advection and localized source
• Crystal Growth Inside an Octant
• Random walk through a fertile site
• Generic phase coexistence in the totally asymmetric kinetic Ising model
• Exact field-driven interface dynamics in the two-dimensional stochastic Ising model with helicoidal boundary conditions
• Localization and melting of interfaces in the two-dimensional quantum Ising model