Abstract
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.
Highlights
The study of magnetism hosted by dilute impurities in a non-magnetic metal has a long history in physics, both for its fundamental interest and for possible applications such systems might host
Estimates of J for the timereversal symmetry (TIs) Sb2Te3 with vanadium impurities[50,51] yields a length scale of ∼ 0.3μm, and for the Topological crystalline insulators (TCIs) (Sn/Pb)Te with manganese impurities[52,53] yield ∼ 1.0μm. ) Beyond this distance scale, we find that the gradient energy becomes non-analytic in the amplitude of the magnetization
The numerical calculations we present below demonstrate that one may understand the number of conducting modes hosted by a given domain walls (DWs), as well as their chirality, from the change in Chern numbers summed over all the Dirac points on the surface
Summary
The study of magnetism hosted by dilute impurities in a non-magnetic metal has a long history in physics, both for its fundamental interest and for possible applications such systems might host. In a course-grained theory, the spins appearing in the Si · Sj coupling will each be proportional to spin density, leading to energies that are quadratic in the magnetic impurity density for DW’s in systems governed by shortrange effective exchange interactions This should be reflected most directly in a critical temperature for thermal disordering that scales linearly rather than quadratically with impurity density, as we explain below. For short-range spin interactions these modes disperse linearly with wavevector[55], but if the stiffness changes to the long-range form above some wavevector scale, one expects a crossover to Q1/2 behavior This crossover should occur only in these systems when the Fermi energy is adjusted to be near the Dirac point energy, allowing for in-principle tunable behavior.
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