Fig. 7: Pairfinder illustrations and energy derivative consistency test.

a, b Schematic illustration of the Pairfinder algorithm utilized to calculate the neighbor relations for a specific HOI term for a 3 by 3 segment of the hexagonal lattice. Exemplary the derivation of the 4-site HOI term is shown. The moments i and j participate in the first scalar product and k, l in the latter. The sites marked by blue triangles satisfy the adjacency to i and j during the generation of the neighbor \(k,{k}^{{\prime} }\). For \(k,{k}^{{\prime} }\) on the right only the site marked by the green diamond satisfies the adjacency of the fourth moment l to the others. b The final set of four moments with pink lines as the scalar products is illustrated in addition to the indices of the atomic sites. c Histogram of via Euler’s theorem numerically calculated 4-spin energy differences within the atomistic spin simulation code SPINAKER. The difference between the energy and the energy via the gradient in Eq. (27) (red) and the energy via the matrix of second-order derivatives in Eq. (28) (blue) of 1000 states with random magnetization is compared.