On quantifying the topological charge in micromagnetics using a lattice-based approach

Abstract

The topological charge or skyrmion number Q is a useful measure to characterize the topology of spin textures such as vortices and skyrmions in two-dimensional systems. When the magnetization field m(r) is projected onto the unit sphere, Q measures the number of times the magnetic moments wrap around this sphere. For vortices and merons, Q = ±1/2, while for skyrmions, Q = ±1. Higher-order half- and full-integer charges are also possible.In numerical micromagnetism, a common approach involves solving the equations of motions for m(r) that is discretized on a regular grid. The underlying assumption is that cell-to-cell variations in m are sufficiently small such that the exchange energy, proportional to the square of the magnetization gradient, remains meaningful. However, issues can arise under certain conditions, such as in the nucleation and annihilation of vortices and skyrmions, or in the stochastic dynamics under random fields, where large cell-to-cell variations in the magnetization can reduce the accuracy of the finite-difference approximations of the spatial derivatives used to estimate Q, which can result in spurious deviations from half- and full-integer values of Q.Here, we will describe a lattice-based implementation for the open source micromagnetics code MuMax3 [1] that allows an accurate determination of Q, particularly under Langevin dynamics used to model finite temperatures and in nucleation and annihilation processes [2]. The implementation is based on the approach of Berg and Lüscher [3], which involves summing over the solid angle subtended by three spins, forming elementary signed triangles, that are constructed from the finite difference cells. The implementation accounts for both periodic boundary conditions and finite-sized systems.A comparison between the finite-difference and lattice-based approaches is given in Fig. 1, where the time variation of the total topological charge is shown for a system hosting a single skyrmion under a temperature of 300 K. The blue circles represent the Q computed using finite-difference derivatives, which exhibit large fluctuations about the expected value of Q = -1 that are about as large as Q itself. Moreover, the time-averaged mean Q does not coincide with -1. The solid red line represents the result of the lattice-based approach, which remains constant near the expected value of -1. More precisely, the result is limited primarily by the single-precision of the floating-point arithmetic used, e.g., we find Q = −1.0000001, −1.0000004, −1.0000008, −1.0000002, and −0.9999996 over 20-ns intervals at 400 K. We also show how the lattice-based approach is vastly superior to the finite-difference approximation in other test cases, which involve skyrmion-antiskyrmion pair nucleation due to spin-transfer torques and skyrmion fluctuations in finite dots.While the results do not necessarily call into question the validity of published work (since the topological charge is often only used as a proxy for magnetization gradients), they do highlight the care with which noninteger values of Q should be interpreted, particularly when processes such as nucleation, annihilation, and thermal fluctuations are at play.This work was partially supported by the Agence Nationale de la Recherche under contract no. ANR-17-CE24-0025 (TOPSKY) and Fonds WetenschappelijkOnderzoek (FWO-Vlaanderen) through Project No. G098917N. **