Modulated multipolar structures in magnetic arrays

Abstract\n Electric and magnetic multipole moments are important \nquantities in studies of intermolecular forces, electrostatic and magnetostatic potentials. The experimental determination of multipole moments in multipole-multipole coupling is difficult and therefore \nthe theoretical prediction of these quantities is important. \nThe aim of the present review is to \ngive a general theoretical description of multipolar ordering on \ntwo-dimensional periodic and aperiodic lattices. \nAfter an introduction to the role of multipolar interactions in magnetic nanoarrays in the first part of this manuscript the static multipole expansion in cartesian and spherical coordinates are outlined. Next, the established numerical approach for the calculation of \nmultipolar ground states; i.e., Monte Carlo simulations are summarized. Special emphasis is put on the review of ground states in multipolar systems consisting of moments of odd parity relevant for the magnetized or polarized particle ensembles. We demonstrate that higher-order interactions considerably change the dipolar ground states of in-plane magnetized arrays. While in periodic triangular, square and kagome arrays the higher order interactions induce three or four-fold in-plane anisotropy, on a Penrose tiling with ten-fold symmetry the multipolar terms do not seriously affect the dipolar order.