Disorder such as impurities and dislocations in Weyl semimetals (SMs) drives a quantum critical point (QCP) where the density of states at the Weyl point gains a non-zero value. Near the QCP, the asymptotic low energy singularities of physical quanti
ties are controlled by the critical exponents $ u$ and $z$. The nuclear spin-lattice relaxation rate, which originates from the hyperfine coupling between a nuclear spin and long-range orbital currents in Weyl fermion systems, shows intriguing critical behavior. Based on the self-consistent Born approximation for impurities, we study the nuclear spin-lattice relaxation rate $1/T_1$ due to the orbital currents in disordered Weyl SMs. We find that $(T_1T)^{-1}sim E^{2/z}$ at the QCP where $E$ is the maximum of temperature $T$ and chemical potential $mu(T)$ relative to the Weyl point. This scaling behavior of $(T_1T)^{-1}$ is also confirmed by the self-consistent $T$-matrix approximation, where a remarkable temperature dependence of $mu(T)$ could play an important role. We hope these results of $(T_1T)^{-1}$ will serve as an impetus for exploration of the disorder-driven quantum criticality in Weyl materials.
We study the nuclear magnetic relaxation rate and Knight shift in the presence of the orbital and quadrupole interactions for three-dimensional Dirac electron systems (e.g., bismuth-antimony alloys). By using recent results of the dynamic magnetic su
sceptibility and permittivity, we obtain rigorous results of the relaxation rates $(1/T_1)_{rm orb}$ and $(1/T_1)_{rm Q}$, which are due to the orbital and quadrupole interactions, respectively, and show that $(1/T_1)_{rm Q}$ gives a negligible contribution compared with $(1/T_1)_{rm orb}$. It is found that $(1/T_1)_{rm orb}$ exhibits anomalous dependences on temperature $T$ and chemical potential $mu$. When $mu$ is inside the band gap, $(1/T_1)_{rm orb} sim T ^3 log (2 T/omega_0)$ for temperatures above the band gap, where $omega_0$ is the nuclear Larmor frequency. When $mu$ lies in the conduction or valence bands, $(1/T_1)_{rm orb} propto T k_{rm F}^2 log (2 |v_{rm F}| k_{rm F}/omega_0)$ for low temperatures, where $k_{rm F}$ and $v_{rm F}$ are the Fermi momentum and Fermi velocity, respectively. The Knight shift $K_{rm orb}$ due to the orbital interaction also shows anomalous dependences on $T$ and $mu$. It is shown that $K_{rm orb}$ is negative and its magnitude significantly increases with decreasing temperature when $mu$ is located in the band gap. Because the anomalous dependences in $K_{rm orb}$ is caused by the interband particle-hole excitations across the small band gap while $left( 1/T_1 right)_{rm orb}$ is governed by the intraband excitations, the Korringa relation does not hold in the Dirac electron systems.
Based on an exact functional form derived for the three-point vertex function $Gamma$, we propose a self-consistent calculation scheme for the electron self-energy with $Gamma$ always satisfying the Ward identity. This scheme is basically equivalent
to the one proposed in 2001, but it is improved in the aspects of computational costs and its applicability range; it can treat a low-density electron system with a dielectric catastrophe. If it is applied to semiconductors and insulators, we find that the obtained quasiparticle dispersion is virtually the same as that in the one-shot $GW$ approximation (or $G_0W_0$A), indicating that the $G_0W_0$A actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner.
