Spinor Formulation of the Landau–Lifshitz–Gilbert Equation With Geometric Algebra

The Landau–Lifshitz–Gilbert (LLG) equation for magnetization dynamics is recast into spinor form using the real-valued Clifford algebra (geometric algebra) of three-space. We show how the undamped case can be explicitly solved to obtain componentwise solutions, with clear geometrical meaning. Generalizations of the approach to include damping are formulated. The implications of the axial property of the magnetization vector are briefly discussed.

I. Introduction

Recent advancements in fabrication and theoretical understanding of spin-dependent heterostructures near nanoscale have opened up a plethora of new technological abilities, applicable in the data and energy sectors [1], [2]. Short time scales of spin flipping and low energy cost of spin transport allow for the fabrication of faster next-generation technological components, with significantly lowered power consumption compared to conventional ones [3]. Strong spin-orbit coupling in various condensed matter systems plays a key role and has allowed for the realization of multiple types of quasi-particles, which are stable fermion-like excitations, such as skyrmions [4] and Majorana zero modes [5]. Both have topological properties and hold promise in next-generation information processors: skyrmions as low-power memory and logic devices [6] and Majorana zero modes as qubits for quantum computation [7].