Quantitative analytical theory for disordered nodal points

Disorder effects are especially pronounced around nodal points in linearly dispersing band structures as present in graphene or Weyl semimetals. Despite the enormous experimental and numerical progress, even a simple quantity like the average density of states cannot be assessed quantitatively by analytical means. We demonstrate how this important problem can be solved employing the functional renormalization group method, and, for the two-dimensional case, we demonstrate excellent agreement with reference data from numerical simulations based on tight-binding models. In three dimensions our analytic results also improve drastically on existing approaches.

Figure 1

Top: Density of states ${\nu }_2$ for a two-dimensional disordered Dirac node as a function of energy $E$ as calculated by the KPM for various disorder strengths $K$ (for values of $K$ cf. bottom panel). The dashed line denotes the analytic result for the clean case. Bottom: The zero energy density of states ${\nu }_2(E=0)$ from KPM (dots) compared to the self-consistent Born approximation (SCBA) (blue line) and the fRG (red line). The parameters for the simulation are $\xi =3a$ (except for the two largest $K$, where $\xi =4a$), linear system size $L=2000\xi$, 20 random vectors for calculating the trace, and an expansion order of up to 15 000 moments. The data represent an average over 20 disorder realizations and are normalized to a single node. The dashed lines denote fits to the white noise forms of the density of states from the SCBA and the RG as discussed in the main text.

Figure 2

Top: Density of states ${\nu }_3$ for a three-dimensional disordered Weyl node as a function of energy $E$ as calculated by the KPM for disorder strengths $K=0,1,2,...,11$ (bottom to top). The dashed line denotes the analytic result for the clean case. Bottom: The zero energy density of states ${\nu }_3(E=0)$ from the KPM (dots) compared to the SCBA (blue line) and the fRG (red line). The parameters for the simulation are $\xi =4a$, linear system size $L=180\xi$, 20 random vectors for calculating the trace, and an expansion order of up to 2000 moments. The data represent an average over 40 disorder realizations and are normalized to a single node. The semitransparent data points for $K\le 4$ suffer from finite-$L$ effects and overestimate the true bulk DOS.

Figure 3

Diagrammatic representation of the self-consistency equation (8) for the disorder-induced self-energy $\mathrm{\Sigma }$ as obtained from the $\mathcal{O}\left(K^2\right)$ truncation of the fRG. Diagram (i) is the first-order term equivalent to SCBA while diagrams (ii.a) and (ii.b) are of second order in $K$. Dashed lines denote disorder correlators and double lines self-energy-dressed Green functions.