Resorting to the Schwinger-boson representation for the description of a localized magnetic-impurity state, we develop an RG-improved (renormalization group) perturbation theory for the Kondo effect. This Schwinger-boson based RG-improved perturbatio
n theory covers the whole temperature range from a decoupled local moment state to a local Fermi-liquid state through the crossover temperature regime, shown from the specific heat and spin susceptibility of the magnetic impurity. The Schwinger-boson based RG-improved perturbation theory makes the strong coupling fixed point at IR (infrared) accessible from the gaussian one at UV (ultraviolet) within the perturbation framework, regarded to be complementary to the Schwinger-boson based NCA (non-crossing approximation) self-consistent theory [Phys. Rev. Lett. {bf 96}, 016601 (2006)]. The existence of the perturbatively accessible strong coupling fixed point implies the nature on the statistics of spinons, not determined by hands but chosen by the nature of strongly coupled systems.
The history of modern condensed matter physics may be regarded as the competition and reconciliation between Stoners and Andersons physical pictures, where the former is based on momentum-space descriptions focusing on long wave-length fluctuations w
hile the latter is based on real-space physics emphasizing emergent localized excitations. In particular, these two view points compete with each other in various nonperturbative phenomena, which range from the problem of high T$_{c}$ superconductivity, quantum spin liquids in organic materials and frustrated spin systems, heavy-fermion quantum criticality, metal-insulator transitions in correlated electron systems such as doped silicons and two-dimensional electron systems, the fractional quantum Hall effect, to the recently discussed Fe-based superconductors. An approach to reconcile these competing frameworks is to introduce topologically nontrivial excitations into the Stoners description, which appear to be localized in either space or time and sometimes both, where scattering between itinerant electrons and topological excitations such as skyrmions, vortices, various forms of instantons, emergent magnetic monopoles, and etc. may catch nonperturbative local physics beyond the Stoners paradigm. In this review article we discuss nonperturbative effects of topological excitations on dynamics of correlated electrons. ......
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.
Landaus Fermi-liquid theory is the standard model for metals, characterized by the existence of electron quasiparticles near a Fermi surface as long as Landaus interaction parameters lie below critical values for instabilities. Recently, this fundame
ntal paradigm has been challenged by physics of strong spin-orbit coupling although the concept of electron quasiparticles remains valid near the Fermi surface, where the Landaus Fermi-liquid theory fails to describe electromagnetic properties of this novel metallic state, referred to as Weyl metal. A novel ingredient is that such a Fermi surface encloses a Weyl point with definite chirality, referred to as a chiral Fermi surface, which can arise from breaking of either time reversal or inversion symmetry in systems with strong spin-orbit coupling, responsible for both Berry curvature and chiral anomaly. As a result, electromagnetic properties of the Weyl metallic state are described not by conventional Maxwell equations but by axion electrodynamics, where Maxwell equations are modified with a topological-in-origin spatially modulated $theta(bm{r}) bm{E} cdot bm{B}$ term. This novel metallic state has been realized recently in Bi$_{1-x}$Sb$_{x}$ around $x sim 3%$ under magnetic fields, where the Dirac spectrum appears around the critical point between the normal semiconducting ($x < 3%$) and topological semiconducting phases ($x > 3%$) and the time reversal symmetry breaking perturbation causes the Dirac point to split into a pair of Weyl points along the direction of the applied magnetic field for such a strong spin-orbit coupled system. In this review article, we discuss how the topological structure of both the Berry curvature and chiral anomaly (axion electrodynamics) gives rise to anomalous transport phenomena in Bi$_{1-x}$Sb$_{x}$ around $x sim 3%$ under magnetic fields, modifying the Drude model of Landaus Fermi liquids.
Weyl metal is regarded as a platform toward interacting topological states of matter, where its topological structure gives rise to anomalous transport phenomena, referred to as chiral magnetic effect and negative magneto-resistivity, the origin of w
hich is chiral anomaly. Recently, the negative magneto-resistivity has been observed with the signature of weak anti-localization at $x = 3 sim 4 ~ %$ in Bi$_{1-x}$Sb$_{x}$, where magnetic field is applied in parallel with electric field. Based on the Boltzmann-equation approach, we find the negative magneto-resistivity in the presence of weak anti-localization. An essential ingredient is to introduce the topological structure of chiral anomaly into the Boltzmann-equation approach, resorting to semi-classical equations of motion with Berry curvature.
We uncover that the competition between electron-electron correlations and electron-phonon interactions gives rise to unexpectedly huge enhancement of the superconducting transition temperature, several hundreds percent larger ($geq$ 200 K) than that
of the case when only one of the two is taken into account ($sim$ 30 K). Our renormalization group analysis claims that this mechanism for the enhancement of the critical temperature is not limited on superconductivity but applied to various Fermi surface instabilities, proposing an underlying universal structure, which turns out to be essentially identical to that of a recent study [Phys. Rev. Lett. {bf 108}, 046601 (2012)] on the enhancement of the Kondo temperature in the presence of Rashba spin-orbit interactions. We also discuss the stability of superconductivity against nonmagnetic randomness.
