Spin dynamics algorithms for systems with exchange interactions beyond nearest neighbors

Fast spin dynamics algorithms for classical spin systems with nearest neighbor exchange interactions ($J_1 \neq 0$, and $J_i = 0$ for $i^{th}$ nearest neighbors with $i \geq 2$ ) were studied extensively years ago\footnotemark[2].\footnotetext[2]{M. Krech, A. Bunker, and D. P. Landau, Comput. Phys. Commun. {\bf 111}, 1 (1998).} For some realistic magnetic systems, such as Fe, $J_i$ can not be neglected for several shells of neighbors. To study dynamic properties of such systems, fast algorithms are still applicable; however, with $n$ shells of interacting neighbors, a lattice needs to be decomposed into $2^n$ sublattices and there can be as many as $(5^{2^{n}}-3)/2$ factors for the fourth order Suzuki-Trotter decompositions of exponential operators. In comparison, only $2^{n+1}-1$ factors are needed for second order decompositions. Consequently, only second order decompositions are practical for $n\geq 2$. Examples are given showing the implementation of the algorithms for systems in which as many as four shells of near neighbors play a significant role.