Quantum Hall Plateau Transition in Graphene with Spatially Correlated Random Hopping

We investigate how the criticality of the quantum Hall plateau transition in disordered graphene differs from those in the ordinary quantum Hall systems, based on the honeycomb lattice with ripples modeled as random hoppings. The criticality of the graphene-specific $n=0$ Landau level is found to change dramatically to an anomalous, almost exact fixed point as soon as we make the random hopping spatially correlated over a few bond lengths. We attribute this to the preserved chiral symmetry and suppressed scattering between $K$ and $K^{\prime }$ points in the Brillouin zone. The results suggest that a fixed point for random Dirac fermions with chiral symmetry can be realized in freestanding, clean graphene with ripples.

Figure 1

An example of the spatial landscape of the random components, $\delta t/\sigma$, in the hopping for a spatial correlation length $\eta /a=5$.

Figure 2

The density of states for the spatially correlated random bonds for various values of the correlation length $0\le \eta /a\le 2.0$ with the strength of disorder fixed at $\sigma /t=0.12$ in a magnetic field $\varphi /{\varphi }_0=1/50$. The system size is $L_x/(\sqrt{3}a/2)=5000$, $L_y/(3a/2)=100$, and $ϵ/t=6.25\times {10}^{-4}$. Inset: The density of states for various values of the disorder strength $\sigma$ with a fixed $\eta /a=3$.

Figure 3

The Hall conductivity (Chern number) against the Fermi energy $E/t$ for spatially correlated random bonds for the correlation length $\eta /a=1.5$ (a) and $\eta /a=0$ (b). We have a disorder strength $\sigma /t=0.12$, a magnetic field $\varphi /{\varphi }_0=1/50$, a system size $L_x/(\sqrt{3}a/2)=L_y/(3a/2)=20$, $N=10$ , and an average over 300 samples.

Figure 4

The Hall conductivity (Chern number) as a function of the Fermi energy $E/t$ for spatially correlated random bonds with $\eta /a=1.5$ for two system sizes, $L_x/(\sqrt{3}a/2)=L_y/(3a/2)=20$ (a) and $=10$ (b). We have a disorder strength $\sigma /t=0.12$, a magnetic field $\varphi /{\varphi }_0=1/50$ with an average over 300 samples. Insets: Plateau transitions for $n=0$ (c) and $n=-1$ (d) when we add a potential disorder introduced as random site-energies uniformly distributed over $[-w/2,w/2]$ and spatially uncorrelated, and the curves are for $w/t=0$, 0.05, 0.1, 0.2, 0.3, 0.4 with $L_x/(\sqrt{3}a/2)=L_y/(3a/2)=10$.