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Constructing an effective field theory in terms of doped magnetic impurities(described by an O(3) vector model with a random mass term), itinerantelectrons of spin-orbit coupled semiconductors (given by a Dirac theory with arelatively large mass term), and effective interactions between doped magneticions and itinerant electrons (assumed by an effective Zeeman coupling term), weperform the perturbative renormalization group analysis in the one-loop levelbased on the dimensional regularization technique. As a result, we find thatthe mass renormalization in dynamics of itinerant electrons acquires negativefeedback effects due to quantum fluctuations involved with the Zeeman couplingterm, in contrast with that of the conventional problem of quantumelectrodynamics, where such interaction effects enhance the fermion mass morerapidly. Recalling that the applied magnetic field decreases the band gap inthe presence of spin-orbit coupling, this renormalization group analysis showsthat the external magnetic field overcomes the renormalized band gap, allowedby doped magnetic impurities even without ferromagnetic ordering. In otherwords, the Weyl metal physics can be controlled by doping magnetic impuritiesinto spin-orbit coupled semiconductors, even if the external magnetic fieldalone cannot realize the Weyl metal phase due to relatively large band gaps ofsemiconductors. Furthermore, we emphasize that quasiparticles do not exist inthis emergent disordered Weyl metal phase due to correlations with strongmagnetic fluctuations. This non-Fermi liquid type Weyl metal state may beregarded to be a novel metallic phase in the respect that a topologicallynontrivial band structure appears in the vicinity of quantum criticality.

Emergence of Weyl metals driven by doped magnetic impurities in spin-orbit coupled semiconductors