  Spin interactions of magnetic impurities mediated by conduction electrons is
one of the most interesting and potentially useful routes to ferromagnetism in
condensed matter. In recent years such systems have received renewed attention
due to the advent of materials in which Dirac electrons are the mediating
particles, with prominent examples being graphene and topological insulator
surfaces. In this paper, we demonstrate that such systems can host a remarkable
variety of behaviors, in many cases controlled only by the density of electrons
in the system. Uniquely characteristic of these systems is an emergent
long-range form of the spin stiffnes when the Fermi energy resides at a Dirac
point, 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, via a subtle mechanism in which the topology of the Dirac
electrons plays a key role: the combination of attraction due to bound in-gap
states that the topology necessitates, and repulsion due to scattering phase
shifts, yields logarithmic interactions over a range of length scales. We
present detailed results for domain walls in a particularly rich system, the
(111) surface of a model topological crystalline insulator. This hosts two-fold
and six- fold degenerate groundstates, with either short-range or emergent
long-range interactions among the spins. In the latter case we demonstrate in
detail the presence of in-gap states associated with domain walls, and argue
that this stabilizes a pseudogap regime at finite temperature. Thus the
topological nature of these systems, through its impact on domain wall
excitations, leads to unique behaviors distinguishing them markedly from their
non-topological analogs.
