Extended Landau--Lifshitz equation for nanomagnets: a path-integral derivation of surface-induced magnetization nutation
An effective dynamical equation for the magnetization of a nanomagnet with surface anisotropy is derived from an atomistic spin Hamiltonian using the spin coherent-state path integral formalism. The derivation proceeds in two steps. First, the continuum Euclidean action for the many-spin nanomagnet is obtained, including the Wess--Zumino--Witten (Berry phase) term, as well as exchange, Zeeman, and core/surface anisotropy contributions. Second, the local magnetization density is decomposed into a slowly varying macrospin component and transverse spin-misalignment fluctuations driven by surface effects. A systematic expansion of the action is then performed up to quadratic order in the transverse-fluctuation variables. Under the adiabatic approximation, in which transverse modes relax much faster than the macrospin, these modes are eliminated by using their static Green's function solution. This results in a closed, extended Landau--Lifshitz equation for the macrospin, featuring an effective field with nontrivial corrections from spin misalignment. These corrections renormalize both the Zeeman and anisotropy fields and introduce additional terms that act as nutation- and damping-like contributions. ... Together, these results establish a microscopic foundation for surface-induced magnetization nutation in nanomagnets and provide a framework to estimate corrections to the precession frequency and effective damping. The corresponding shift in the ferromagnetic-resonance frequency and linewidth is measurable with standard GHz spectrometers, and the underlying adiabatic-elimination mechanism is expected to generalize to any slow magnetic variable coupled to a bath of fast-fluctuating modes.