Local spin magnetization in itinerant non-collinear magnets: The local spin Berry curvature

Conventionally, the local spin magnetization in itinerant magnets is determined from the equilibrium local spin density. Here, we propose a thermodynamic approach in which the local spin magnetization is defined from the response of the system to an infinitesimal external magnetic field. The predictions of the two theories are identical for collinear magnets, but differ qualitatively and quantitatively for non-collinear magnets. In the present thermodynamic approach, the spin coherences determine an alternative distribution of local spin magnetization due to the field-induced deformation of the energy eigenstates. This effect is captured by a Berry-curvature-like contribution reminiscent of orbital magnetization and has several distinct observable consequences. We explore the differences between the conventional and thermodynamic approaches in several test cases.