Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An efficient geometric integrator for thermostatted anti-/ferromagnetic models

Teijo Arponen, Ben Leimkuhler

TL;DR

The paper develops a geometric, split-operator integrator for semi-discrete Landau-Lifshitz(-Gilbert) spin dynamics that include nonlinear dissipation and a Bulgac–Kusnetsov–type thermostat. By decomposing the vector field into explicitly solvable subflows and composing them symmetrically, the method preserves spin length and, in the thermostatted case, exhibits time-reversibility while enabling energy exchange with a thermal bath. Numerical experiments on a $50\times50$ lattice demonstrate stable energy dissipation in the dissipative case and rich, canonical-ensemble-like dynamics (e.g., creeping boundaries and wandering vortices) in the thermostatted case, with superior stability and larger allowable time steps compared to projected RK4. The approach offers a practical, structure-preserving tool for efficient long-time micromagnetic simulations at fixed temperature, with potential for further extensions to more general thermostat models.

Abstract

(Anti)-/ferromagnetic Heisenberg spin models arise from discretization of Landau-Lifshitz models in micromagnetic modelling. In many applications it is essential to study the behavior of the system at a fixed temperature. A formulation for thermostatted spin dynamics was given by Bulgac and Kusnetsov which incorporates a complicated nonlinear dissipation/driving term while preserving spin length. It is essential to properly model this term in simulation, and simplified schemes give poor numerical performance, e.g. requiring an excessively small timestep for stable integration. In this paper we present an efficient, structure-preserving method for thermostatted spin dynamics.

An efficient geometric integrator for thermostatted anti-/ferromagnetic models

TL;DR

The paper develops a geometric, split-operator integrator for semi-discrete Landau-Lifshitz(-Gilbert) spin dynamics that include nonlinear dissipation and a Bulgac–Kusnetsov–type thermostat. By decomposing the vector field into explicitly solvable subflows and composing them symmetrically, the method preserves spin length and, in the thermostatted case, exhibits time-reversibility while enabling energy exchange with a thermal bath. Numerical experiments on a lattice demonstrate stable energy dissipation in the dissipative case and rich, canonical-ensemble-like dynamics (e.g., creeping boundaries and wandering vortices) in the thermostatted case, with superior stability and larger allowable time steps compared to projected RK4. The approach offers a practical, structure-preserving tool for efficient long-time micromagnetic simulations at fixed temperature, with potential for further extensions to more general thermostat models.

Abstract

(Anti)-/ferromagnetic Heisenberg spin models arise from discretization of Landau-Lifshitz models in micromagnetic modelling. In many applications it is essential to study the behavior of the system at a fixed temperature. A formulation for thermostatted spin dynamics was given by Bulgac and Kusnetsov which incorporates a complicated nonlinear dissipation/driving term while preserving spin length. It is essential to properly model this term in simulation, and simplified schemes give poor numerical performance, e.g. requiring an excessively small timestep for stable integration. In this paper we present an efficient, structure-preserving method for thermostatted spin dynamics.

Paper Structure

This paper contains 9 sections, 1 theorem, 67 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background: Review of splitting methods
  3. The original Landau-Lifshitz model as a Poisson system
  4. Dissipated version
  5. Thermostatted version
  6. Numerical results
  7. Dissipated system
  8. Thermostatted system
  9. Discussion

Key Result

Lemma 2.1

Any system of the form $\dot{z}=J(z)v(z)$ with J and $v$ an arbitrary vector function, conserves the spin lengths in time:

Figures (9)

  • Figure 1: Energy of dissipated system.
  • Figure 2: Snapshots of $z$-components (black is down spin, white is up spin) of the dissipated system, example 1.
  • Figure 3: Energy, alpha, and norm of Laplacian of thermostatted system.
  • Figure 4: Snapshots of $z$-components (black is down spin, white is up spin) of thermostatted system, Example 2.
  • Figure 5: Energy and alpha of "wandering vortices", Example 3.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma 2.1
  • proof