The perturbative construction of the effective soft-spin Hamiltonian of the system of magnetized nano-loops

The effective large-scale Hamiltonian of a planar system of nano-loops in a weakly excited flux-closed magnetized state has been constructed by means of a perturbative technique based on micromagnetic theory. The Hamiltonian is written by means of two classes of collective variables: the continuous soft spins and discrete vorticity charges. Analytical and numerical calculations of the inter-loop magnetostatic energy are compared for a pair of magnetic nano-loops. The transformation from small-scale to collective variables is performed for intra-loop exchange-coupling, magnetostatic and Zeeman energy terms. Evidence of correlations of uniform vortex charges in low-energy configurations is uncovered numerically. The generalization of the perturbative method that deals with more realistic out-of-plane excitations is also considered.