Tilted disordered Weyl semimetals

Although Lorentz invariance forbids the presence of a term that tilts the energy-momentum relation in the Weyl Hamiltonian, a tilted dispersion is not forbidden and, in fact, generic for condensed matter realizations of Weyl semimetals. We here investigate the combined effect of such a tilted Weyl dispersion and the presence of potential disorder. In particular, we address the influence of a tilt on the disorder-induced phase transition between a quasiballistic phase at weak disorder, in which the disorder is an irrelevant perturbation, and a diffusive phase at strong disorder. Our main result is that the presence of a tilt leads to a reduction of the critical disorder strength for this transition or, equivalently, that increasing the tilt at fixed disorder strength drives the system through the phase transition to the diffusive strong-disorder phase. Notably this obscures the tilt-induced Lifshitz transition to an overtilted type II Weyl phase at any finite disorder strength. Our results are supported by analytical calculations using the self-consistent Born approximation and numerical calculations of the density of states and of transport properties.

Figure 1

Illustration of a tilted Weyl dispersion, projected to two momentum dimensions. The left panel shows the isotropic case $\left(a=0\right)$; the right panel shows a dispersion tilted in the negative $x$ direction with tilt parameter $a=0.4$.

Figure 2

Phase diagram for the tilted Weyl cone obtained from the SCBA. The numerical uncertainty for $K_{\mathrm{c}}$ is shown as error bars (see the appendix for a discussion).

Figure 3

Effective SCBA tilt $\tilde{a}$ vs dimensionless disorder strength $K$ in the weak-disorder phase $K<K_{\mathrm{c}}$.

Figure 4

Top: Density of states $\nu \left(\varepsilon \right)$ as a function of energy $\varepsilon$ without tilt (left, $a=0)$ and with tilt (right, $a=0.5)$, as calculated by the kernel polynomial method for disorder strengths $K=0,2,4,6,8$ (bottom to top). The dashed lines denote the SCBA results, Eq. (6). [For $K=0$ the SCBA coincides with the zero-disorder density of states of Eq. (3).] Bottom: Density of states $\nu \left(0\right)$ at the nodal point versus disorder strength for different values of the tilt, $a=0,0.25,0.4,0.5$ (right to left). The calculations were carried out for an average over 20 realizations of the random potential, for a cubic lattice of size $L/b=200$, with 30 random vectors for calculating the trace in the KPM and an expansion order of roughly 1000. The density of states is normalized to a single Weyl node.

Figure 5

Dimensionless conductance $g$, rescaled for a system of size $L^3$ (top) and Fano factor $F$ for subcritical disorder strength $K=1$ and for tilt parameter $a=0$, 0.25, and 0.5 (bottom to top). The data points show results from numerical calculations, as described in the text. The solid and dashed line indicate the values expected without disorder and for a clean system with disorder-renormalized tilt $\tilde{a}$, respectively. We have not shown data for $L<4$, which are dominated by finite-size effects. The numerical curves are based on an average over at least 10 disorder realizations. The error bars show the residual statistical error.

Figure 6

Dimensionless conductance $g$ as a function of system size $L$ for disorder strength $K=4$ and tilt parameters $a=0$, 0.25, 0.5, and 0.75 (bottom to top). The data points show results from numerical calculations after averaging over at least 10 disorder realizations. The error bars show the residual statistical error. We have not shown data for $L<4$, which are dominated by finite-size effects. The conductance is bounded for $a=0$ and $a=0.25$ (bottom two data sets), which is characteristic of the quasiballistic weak-disorder phase. For $a=0.5$ and $a=0.75$ the conductance is proportional to the system size $L$ for large $L$ (top two data sets), characteristic of the diffusive strong-disorder phase. The dashed lines indicate a linear increase with $L$ and are shown as a guide to the eye.

Figure 7

Dimensionless conductance $g$ and Fano factor $F$ as a function of system size $L$ for disorder strength $K=30$ and tilt parameters $a=0$, 0.25, and 0.5 (red, blue, green; i.e., top to bottom in the left panel). The data points show results from numerical calculations after averaging over at least 10 disorder realizations. The error bars show the residual statistical error. Data for $L<4$ are dominated by finite-size effects and are not shown. The dashed black line in the right panel indicates the Fano factor $F=1/3$ expected in the diffusive regime.

Figure 8

Linear fit to $D$ in the region where $D>D_{\mathrm{threshold}}$ to determine $K_c$.