Hedgehog lattices induced by chiral spin interactions

We analyze a classical Heisenberg spin model on the simple cubic lattice which is invariant under time reversal and contains multiple chiral spin interactions. The modelled dynamics is appropriate either for local moments coupled to itinerant Weyl electrons, or localized electrons with a strong spin-orbit coupling that would produce a Weyl spectrum away from half filling. Using a Monte Carlo method, we find a robust $4Q$ bipartite lattice of hedgehogs and antihedgehogs which melts through a first order phase transition at a critical temperature in certain segments of the phase diagram. The density of hedgehogs is a non-linear function of the Dzyaloshinskii-Moriya interaction, and a linear function of the multiple-spin chiral interaction which plays the fundamental role of a ``magnetic flux'' or a hedgehog chemical potential. These findings are related to the observations of hedgehog lattices in MnGe, MnSi$_{1-x}$Ge$_x$ and SrFeO$_3$, and indirectly support the possible existence of incompressible quantum-disordered hedgehog liquids.