  The state-of-the-art theoretical description of magnetic materials relies on
solving effective Heisenberg spin problems or their generalizations to
relativistic or multi-spin-interaction cases that explicitly assume the
presence of local magnetic moments in the system. We start with a general
interacting fermionic model that is often obtained in ab initio electronic
structure calculations and show that the corresponding spin problem can be
introduced even in the paramagnetic regime, which is characterized by a zero
average value of the magnetization. Further, we derive a physical criterion for
the formation of the local magnetic moment and confirm that the latter exists
already at high temperatures well above the transition to the ordered magnetic
state. The use of path-integral techniques allows us to disentangle spin and
electronic degrees of freedom and to carefully separate rotational dynamics of
the local magnetic moment from Higgs fluctuations of its absolute value. It
also allows us to accurately derive the topological Berry phase and relate it
to a physical bosonic variable that describes dynamics of the spin degrees of
freedom. As the result, we demonstrate that the equation of motion in the case
of a large magnetic moment takes a conventional Landau-Lifshitz form that
explicitly accounts for the Gilbert damping due to itinerant nature of the
original electronic model.
