Spin Stiffness and Domain Walls in Dirac-Electron Mediated Magnets

We consider the problem of ordering of classical magnetic impurities which are indirectly coupled by two-dimensional Dirac electrons, as might be present in graphene or at the surface of a topological insulator. For chemical potential at a Dirac point, the magnetic order parameter develops an emergent long-range form of the spin stiffness, becoming truly long-range as the magnetization density becomes very small. It is demonstrated that this leads to screened Coulomb-like interactions among domain walls. A transfer matrix analysis demonstrates that the interaction takes this form from a subtle cancellation between energy contributions from in-gap bound states and phase shifts of scattered electrons. Detailed studies of graphene and a topological crystalline insulator surface are used to illustrate the behavior. The non-analytic behavior of the stiffness on magnetization density is shown to have interesting consequences for the phase diagram of these systems.