  The geometrical spin torque mediates an indirect interaction of magnetic
moments, which are weakly exchange coupled to a system of itinerant electrons.
It originates from a finite spin-Berry curvature and leads to a non-Hamiltonian
magnetic-moment dynamics. We demonstrate that there is an unprecedentedly
strong geometrical spin torque in case of an electron system, where
correlations cause antiferromagnetic long-range order. The key observation is
that the anomalous torque is strongly boosted by low-energy magnon modes
emerging in the two-electron spin-excitation spectrum due to spontaneous
breaking of SU(2) spin-rotation symmetry. As long as single-electron
excitations are gapped out, the effect is largely universal, i.e., essentially
independent of the details of the electronic structure, but decisively
dependent on the lattice dimension and spatial and spin anisotropies. Analogous
to the reasoning that leads to the Mermin-Wagner theorem, there is a lower
critical dimension at and below which the spin-Berry curvature diverges.
