Weyl node with random vector potential

We study Weyl semimetals in the presence of generic disorder, consisting of a random vector potential as well as a random scalar potential. We derive renormalization group flow equations to second order in the disorder strength. These flow equations predict a disorder-induced phase transition between a pseudoballistic weak-disorder phase and a diffusive strong-disorder phase for a sufficiently strong random scalar potential or for a pure three-component random vector potential. We verify these predictions using a numerical study of the density of states near the Weyl point and of quantum transport properties at the Weyl point. In contrast, for a pure single-component random vector potential, the diffusive strong-disorder phase is absent.

Figure 1

Top panel: Density of states $\nu \left(0\right)$ at the nodal point and per Weyl node for various disorder types ($K_0,K_x,K_y,K_z$) as a function of disorder strength $K$, as indicated in the figure. Bottom panel: $\nu \left(\varepsilon \right)$ vs energy $\varepsilon$ for scalar disorder (left) and for two-component vector disorder with $K_x=K_y=K_{xy}$ (right). The disorder strengths $K_0$ or $K_{xy}$ are 0, 4, 8, 12, 16 (bottom to top). The density of states ${\nu }_0\left(\varepsilon \right)={\varepsilon }^2/2{\pi }^2{\left(\hslash v\right)}^3$ of the clean Hamiltonian (1) is shown dashed. The numerical calculation was performed for a cube geometry with linear size $L=200a$. The number of random vectors used for calculating the trace in the kernel polynomial method is 20 [28]. The energy resolution at the nodal point is $\mathrm{\Delta }\varepsilon \simeq 0.024\hslash v/\xi$. The data represent an average over approximately ten disorder realizations. The finite offset value for $\nu \left(0\right)$ that is observed even for $K=0$ is a finite-size effect.

Figure 2

Cube conductance $G_{\mathrm{cube}}$ and Fano factor $F$ as a function of sample size $L$ for a single Weyl node with various types of weak-disorder potentials in the pseudoballistic transport regime, with disorder strengths ($K_0,K_x,K_y,K_z$) as indicated in the figure. Numerical data are from an average over at least ten disorder realizations and over periodic and antiperiodic boundary conditions. The asymptotic values $G_{0,\mathrm{cube}}$ and $F_0$ for transport in a clean, isotropic Weyl node are denoted by dashed lines.

Figure 3

Conductivity $\sigma$ as a function of disorder strength $K$ for a single Weyl node with various types of disorder potentials ($K_0,K_x,K_y,K_z$) as indicated in the figure. The inset depicts cube conductance $G_{\mathrm{cube}}$ as a function of sample size $L$. The data represent an average over at least ten disorder realizations and over periodic and antiperiodic boundary conditions. The finite offset value for $\sigma$ that is observed even for $K=0$ is a finite-size effect.