Cartesian coordinates solver for Bloch-like equations. Application to symplectic and metriplectic dynamics of 1d magnetic chain

Abstract

Avoiding the use of either spherical coordinates or stereographic projection representation for the spin we develop a new algorithm for solving the Bloch equation for a spin precessing in a magnetic field. The length of the spin is rigorously preserved and the algorithm is unconditionally stable. The usefulness of the algorithm is illustrated on solving the one-dimensional model of the Heisenberg ferromagnetic chain, and analyzing its soliton-like excitations with and without the coupling to the chain elastic degress of freedom. The algorithm is also generalized to include Gilbert-Landau damping of spins. Implementation of our algorithm in the C + + language is discussed pointing out the usefulness of abstract data type definitions and function overloading inherent to that language.