Phase diagrams of disordered Weyl semimetals

Weyl semimetals are gapless quasitopological materials with a set of isolated nodal points forming their Fermi surface. They manifest their quasitopological character in a series of topological electromagnetic responses including the anomalous Hall effect. Here, we study the effect of disorder on Weyl semimetals while monitoring both their nodal/semimetallic and topological properties through computations of the localization length and the Hall conductivity. We examine three different lattice tight-binding models which realize the Weyl semimetal in part of their phase diagram and look for universal features that are common to all of the models, and interesting distinguishing features of each model. We present detailed phase diagrams of these models for large system sizes and we find that weak disorder preserves the nodal points up to the diffusive limit, but does affect the Hall conductivity. We show that the trend of the Hall conductivity is consistent with an effective picture in which disorder causes the Weyl nodes move within the Brillouin zone along a specific direction that depends deterministically on the properties of the model and the neighboring phases to the Weyl semimetal phase. We also uncover an unusual (nonquantized) anomalous Hall insulator phase which can only exist in the presence of disorder.

Figure 1

Phase diagram of the layered Chern insulator model given in Eq. (2.4). The color map represents ${\sigma }_{xy}$ and the solid lines representing phase boundaries are determined by the localization length. We show the phase diagram for two different values of $t_3$: $t_3=0.5$ (top) and $t_3=1$ (bottom). The system size is $30\times 30\times 12$. Black triangles on the $x$ axis mark the clean phase boundaries. The dashed line in the lower panel connects maxima of ${\sigma }_{xy}$ as a guide for the eye. We note that there is no difference in meaning between the solid black and solid yellow lines, the color difference is just to add contrast.

Figure 2

The Hall conductivity and localization length versus disorder strength for layered Chern insulator model with $t_3=0.5$. In the lower panel of each subfigure, the colors show different cross-section sizes $L=8$ (blue), 10 (red), 12 (orange), and 14 (cyan). The dashed vertical lines represent phase boundaries, the dashed horizontal lines indicate the value of ${\sigma }_{xy}$ in the clean limit using the analytical expressions explained in the text, and the system sizes for ${\sigma }_{xy}$ are $30\times 30\times 12$ (red), $70\times 70\times 16$ (blue), and $90\times 90\times 20$ (green).

Figure 3

The effective shifting directions of the Weyl nodes inside BZ for the layered CI model as the disorder strength is increased (but below the diffusive metal regime). The Chern number of the momentum slices in each section of the BZ is shown above them. Weyl cones of different colors represent opposite chiralities. This motion is determined by the SCBA calculation and is consistent with the numerical results. For (c), the Weyl nodes do not move.

Figure 4

Phase diagram of the dWSM model given by Eq. (2.8). The color map represents ${\sigma }_{xy},$ and the solid lines are phase boundaries determined by the localization length. The black triangle on the $m$ axis marks the clean phase boundary. The system size is $30\times 30\times 14$. We note that there is no difference in meaning between the solid black and solid yellow lines, the color difference is just to add contrast.

Figure 5

The phase diagram of a WSM generated by broken TRS in a 3D Dirac model given by Eq. (2.12). The color map represents ${\sigma }_{xy}$ and solid lines are phase boundaries determined by localization length. The AHI phase is contained in a closed region of the phase diagram and the NI-AHI and AHI-DM boundaries eventually intersect at $(m,W)=(2.6,9.5)$ (this lies outside the graph on the top right). The black triangles on the $m$ axis mark the clean phase boundaries. $b_3=0.6$ and the system size is $30\times 30\times 10$. We note that the STI phase would have a quantized surface Hall conductivity if we had chosen open boundaries in the $z$ direction instead of periodic. We checked that this was the case, but did not include both diagrams.

Figure 6

The Hall conductivity and localization length versus disorder strength for the WSM generated by broken TRS in a 3D Dirac model with $b_3=0.6$. The dashed vertical lines represent phase boundaries and the dashed horizontal lines indicate the value of ${\sigma }_{xy}$ in the clean limit using Eq. (2.14). For ${\sigma }_{xy}$, the system size is $30\times 30\times 10$ (red) and $50\times 50\times 16$ (blue). For $\xi /L$, the colors show different cross-section sizes $L=6$ (blue), 8 (red), 10 (orange), and 12 (cyan). We note that in (a) the STI phase would have a quantized surface Hall conductivity if we had chosen open boundaries in the $z$ direction instead of periodic. We checked that this was the case, but did not include both diagrams. Also, CR in panels (b) and (c) refers to the critical STI phase near the critical line between STI and AHI phases, where the nonzero bulk ${\sigma }_{xy}$ could be due to finite-size effects (i.e., ${\sigma }_{xy}$ varies smoothly from AHI to STI).

Figure 7

The localization length as a function of disorder strength in the WSM generated by TRS-broken 3D Dirac model with $b_3=0.6$ for $m=0.1$ (a) and $m=1.8$ (b). The dashed vertical lines represent phase boundaries. The colors show different cross-section sizes $L=6$ (blue), 8 (red), 10 (orange), and 12 (cyan). In panel (a), we should note that there is a hint of the scale-invariant point between AHI-STI.

Figure 8

Phase diagram of a disordered 2D Chern insulator that serves as a building block of the 3D WSM phase with (a) isotropic ($r_1=r_2=t$) and (b) anisotropic ($r_1=1.3t, r_2=0.7t$) Wilson-Dirac mass terms. The chemical potential is set to zero.