Introducing both Berry curvature and chiral anomaly into Landaus Fermi-liquid theory, we construct a topological Fermi-liquid theory, applicable to interacting Weyl metals in the absence of time reversal symmetry. Following the Landaus Fermi-liquid t
heory, we obtain an effective free-energy functional in terms of the density field of chiral fermions. The density field of chiral fermions is determined by a self-consistent equation, minimizing the effective free-energy functional with respect to the order-parameter field. Beyond these thermodynamic properties, we construct Boltzmann transport theory to encode both the Berry curvature and the chiral anomaly in the presence of forward scattering of a Fermi-liquid state, essential for understanding dynamic correlations in interacting Weyl metals. This generalizes the Boltzmann transport theory for the Landaus Fermi-liquid state in the respect of incorporating the topological structure and extends that for noninteracting Weyl metals in the sense of introducing the forward scattering. Finally, we justify this topological Fermi-liquid theory, generalizing the first-quantization description for noninteracting Weyl metals into the second-quantization representation for interacting Weyl metals. First, we derive a topological Fermi-gas theory, integrating over high-energy electronic degrees of freedom deep inside a pair of chiral Fermi surfaces. As a result, we reproduce a topological Drude model with both the Berry curvature and the chiral anomaly. Second, we take into account interactions between such low-energy chiral fermions on the pair of chiral Fermi surfaces. We perform the renormalization group analysis, and find that only forward scattering turns out to be marginal above possible superconducting transition temperatures, justifying the topological Fermi-liquid theory of interacting Weyl metals with time reversal symmetry breaking.
Role of localized magnetic moments in metal-insulator transitions lies at the heart of modern condensed matter physics, for example, the mechanism of high T$_{c}$ superconductivity, the nature of non-Fermi liquid physics near heavy fermion quantum cr
iticality, the problem of metal-insulator transitions in doped semiconductors, and etc. Dilute magnetic semiconductors have been studied for more than twenty years, achieving spin polarized electric currents in spite of low Curie temperatures. Replacing semiconductors with topological insulators, we propose the problem of dilute magnetic topological semiconductors. Increasing disorder strength which corresponds to the size distribution of ferromagnetic clusters, we suggest a novel disordered metallic state, where Weyl metallic islands appear to form inhomogeneous mixtures with topological insulating phases. Performing the renormalization group analysis combined 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 a semiconductor with temperature $T$ and the distribution for ferromagnetic clusters $Gamma$ gives rise to a novel insulator-metal transition from either a topological insulating or band insulating phase to an inhomogeneously distributed Weyl metallic state with such insulating islands. Since electromagnetic properties in Weyl metal are described by axion electrodynamics, the role of random axion electrodynamics in transport phenomena casts an interesting problem beyond the physics of percolation in conventional disorder-driven metal-insulator transitions. We also discuss how to verify such inhomogeneous mixtures based on atomic force microscopy.
