This paper studies how magnetic impurities interact inside a class of materials called Weyl semimetals. The authors show that the pattern, size, and directional character of the indirect magnetic exchange — known as the RKKY interaction — remain largely unchanged when pairs of Weyl nodes in the electronic spectrum annihilate each other and the material moves into a different semimetal phase.

Weyl semimetals host pairs of special points in momentum space called Weyl nodes. Near those nodes the electrons behave like relativistic particles and carry a small, quantized topological charge (sometimes pictured as a “Berry curvature monopole”). The RKKY interaction is the effective coupling between two localized magnetic moments that arises because those moments polarize the itinerant electrons. The authors derive an exact, closed-form expression for the full RKKY exchange tensor for arbitrary two-band, spinful lattice models. Their formula reduces the problem to a few energy integrals of real-space Green’s functions taken over the whole Brillouin zone, with the band edge providing a natural high-energy cutoff.

They use that framework on a two-band tight-binding model that smoothly interpolates from a Weyl phase with separated chiral nodes to a quadratic band-touching semimetal where the nodes have annihilated. Numerically integrating their expressions across the full Brillouin zone, they find that short- and intermediate-range RKKY couplings keep the same spatial form, similar magnitudes, and the same anisotropic tensor structure across the node-annihilation transition. In other words, these real-space magnetic interactions do not rely only on the local low-energy physics near Weyl nodes.

The paper gives a physical reason for this robustness. Short- and intermediate-range exchange are dominated by the global geometry of the filled valence band across the entire Brillouin zone. The authors identify the quantum metric — a measure of how electronic states change with momentum — as the relevant band property, rather than just the local Berry curvature near Weyl nodes. They also report that the antisymmetric Dzyaloshinsky–Moriya (DM) term, which can twist magnetic order, is strongly sensitive to the Fermi level: it is suppressed at charge neutrality by a parity cancellation in the valence-band integral, but grows quickly when the Fermi level is moved past a Van Hove energy scale.