Delocalization of topological surface states by diagonal disorder in nodal loop semimetals

The effect of Anderson diagonal disorder on the topological surface (drumhead) states of a Weyl nodal loop semimetal (SM) is addressed. Since diagonal disorder breaks chiral symmetry, a winding number cannot be defined. Seen as a perturbation, the weak random potential mixes the clean exponentially localized drumhead states of the SM, thereby producing two effects: (i) the algebraic decay of the surface states into the bulk and (ii) a broadening of the low energy density of surface states of the open system due to degeneracy lifting. This behavior persists with increasing disorder, up to the bulk SM-to-metal transition at the critical disorder $W_c$. Above $W_c$, the surface states hybridize with bulk states and become extended into the bulk.

Figure 1

Exact ($p$, red) and estimated ($\tilde{p}$, black) number of surface states of the clean system for different system sizes containing $L^3$ unit cells. $N_m=2^{10}$.

Figure 2

Averaged density of surface states (DOSS). $\langle \mathrm{\Delta }\rho \left(\epsilon \right)\rangle >0$ defines the energy window $[-E_w,E_w]$, which is found to grow with increasing $W$. Results averaged over 20 disorder realizations and $L=100$. Inset: ${\rho }_{\text{bulk}}\left(E\right)$ and ${\rho }_{\text{open}}\left(E\right)$ for two values of $W$.

Figure 3

Logarithmic plot of $L\int \mathrm{\Delta }\rho =\tilde{p}/L^2$ for different $L$. A reasonable collapse is observed for $W<6$, indicating $L^{-1}$ scaling of $L\int \mathrm{\Delta }\rho$. For $W>6$, the collapse of the curves is not clear.

Figure 4

Local density of states (LDOS) on the state defined in Eq. (10) for $L=100$, averaged over 40 disorder realizations.

Figure 5

Log-log plot of the probability of the eigenstate projected onto sublattice $\alpha =1$, for the state with energy closest to $E=0$, for $L=26$ and $W=0.5$. In this example, the probability decays approximately as $z^{-1.8}$ into the bulk.

Figure 6

$L\times {\mathrm{IPR}}_z^{\alpha }$ for $L=24–28$ and the two eigenstates closer to $E=0$, averaged over 400 disorder realizations.