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Role of localized magnetic moments in metal-insulator transitions lies at theheart of modern condensed matter physics, for example, the mechanism of highT$_{c}$ superconductivity, the nature of non-Fermi liquid physics near heavyfermion quantum criticality, the problem of metal-insulator transitions indoped semiconductors, and etc. Dilute magnetic semiconductors have been studiedfor more than twenty years, achieving spin polarized electric currents in spiteof low Curie temperatures. Replacing semiconductors with topologicalinsulators, we propose the problem of dilute magnetic topologicalsemiconductors. Increasing disorder strength which corresponds to the sizedistribution of ferromagnetic clusters, we suggest a novel disordered metallicstate, where Weyl metallic islands appear to form inhomogeneous mixtures withtopological insulating phases. Performing the renormalization group analysiscombined with experimental results, we propose a phase diagram in$(\lambda_{so},\Gamma,T)$, where the spin-orbit coupling $\lambda_{so}$controls a topological phase transition from a topological semiconductor to asemiconductor with temperature $T$ and the distribution for ferromagneticclusters $\Gamma$ gives rise to a novel insulator-metal transition from eithera topological insulating or band insulating phase to an inhomogeneouslydistributed Weyl metallic state with such insulating islands. Sinceelectromagnetic properties in Weyl metal are described by axionelectrodynamics, the role of random axion electrodynamics in transportphenomena casts an interesting problem beyond the physics of percolation inconventional disorder-driven metal-insulator transitions. We also discuss howto verify such inhomogeneous mixtures based on atomic force microscopy.

Dilute magnetic topological semiconductors: What's new beyond the physics of dilute magnetic semiconductors?