Quantum critical exponents for a disordered three-dimensional Weyl node

Three-dimensional Dirac and Weyl semimetals exhibit a disorder-induced quantum phase transition between a semimetallic phase at weak disorder and a diffusive-metallic phase at strong disorder. Despite considerable effort, both numerically and analytically, the critical exponents $\nu$ and $z$ of this phase transition are not known precisely. Here we report a numerical calculation of the critical exponent $\nu =1.47\pm 0.03$ using a minimal single-Weyl node model and a finite-size scaling analysis of conductance. Our high-precision numerical value for $\nu$ is incompatible with previous numerical studies on tight-binding models and with one- and two-loop calculations in an $\epsilon$-expansion scheme. We further obtain $z=1.49\pm 0.02$ from the scaling of the conductivity with chemical potential.

Figure 1

Probability distribution of the difference $\delta G=G-G_{K=0}$ of the conductance with and without disorder potential. The aspect ratio $r=W/L=5$, periodic boundary conditions were applied in the transverse direction, and the sample length $L=(2\pi /\mathrm{\Lambda })\times 11/r$. The dimensionless disorder strength is indicated in the figure. The number of disorder realizations is 4000. Dots indicate the value of the median $m$, which is used as the scaling variable. Dashed lines show fits to a log-normal probability distribution. The data for $K>4.25$ is offset in vertical direction for clarity.

Figure 2

Scaling plot for the logarithm of the median $m$ of the distribution of $\delta G=G-G_{K=0}$ for a disordered Weyl node with $W/L=5$ and periodic transverse boundary conditions. For better visibility we plot $log\left(m\right)-K$ vs. $K$ on the vertical axis. The solid curves show the results of a least-squares fit to a scaling form $m=m_r\left(L^{1/\nu }k\right)$, expanded in a fourth-order polynomial (for details, see Ref. [35]). The estimates for the most important fit parameters and their standard error are $K_{\mathrm{c}}=5.024\pm 0.004, log\left(m_{\mathrm{c}}\right)=0.12\pm 0.004, \nu =1.47\pm 0.02$. The quality of the fit is ${\chi }^2/N=0.93$. The sample lengths are indicated in the figure. The gray vertical line indicates the position of the estimated critical disorder strength. The inset shows a scaling collapse of the data in the main panel using the estimated values of $K_{\mathrm{c}}$ and $\nu$.

Figure 3

Scaling of the conductivity $\sigma \left(K,\mu \right)$ for (near) critical disorder strengths $K=5.0$ (dots) with chemical potential $\mu$. The solid line is a power-law fit $\sigma \propto {\mu }^{1/z}$ to the data points, with $z=1.49\pm 0.02$. The dashed line indicates a $\sigma \propto {\mu }^2$ power law as expected from Drude transport theory when scaling breaks down. The sample width $W=(2\pi /\mathrm{\Lambda })\times 29$ and periodic boundary conditions were applied. For chemical potentials below the values shown in the figure a bulk conductivity could not be reliably obtained from the calculated conductance data.