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The state-of-the-art theoretical description of magnetic materials relies onsolving effective Heisenberg spin problems or their generalizations torelativistic or multi-spin-interaction cases that explicitly assume thepresence of local magnetic moments in the system. We start with a generalinteracting fermionic model that is often obtained in ab initio electronicstructure calculations and show that the corresponding spin problem can beintroduced even in the paramagnetic regime, which is characterized by a zeroaverage value of the magnetization. Further, we derive a physical criterion forthe formation of the local magnetic moment and confirm that the latter existsalready at high temperatures well above the transition to the ordered magneticstate. The use of path-integral techniques allows us to disentangle spin andelectronic degrees of freedom and to carefully separate rotational dynamics ofthe local magnetic moment from Higgs fluctuations of its absolute value. Italso allows us to accurately derive the topological Berry phase and relate itto a physical bosonic variable that describes dynamics of the spin degrees offreedom. As the result, we demonstrate that the equation of motion in the caseof a large magnetic moment takes a conventional Landau-Lifshitz form thatexplicitly accounts for the Gilbert damping due to itinerant nature of theoriginal electronic model.

Spin dynamics of itinerant electrons: local magnetic moment formation and Berry phase