Theory of anomalous quantum Hall effects in graphene

We develop a theory of anomalous quantum Hall effects in graphene. We demonstrate that the Landau level structure by itself is not sufficient to determine the form of the quantum Hall effect. It is only a special symmetry of disorder that gives rise to anomalous quantization of Hall conductivity in graphene. We analyze the symmetries of disordered single- and double-layer graphene samples in magnetic field and identify the conditions for anomalous Hall quantization.

Figure 1

(Color online) Renormalization group flow of ${\sigma }_{xx}$ and ${\sigma }_{xy}$ in graphene with decoupled and mixed valleys. Dotted (dashed) lines are separatrices of the flow for graphene with decoupled (mixed) valleys. Open circles are unstable fixed points corresponding to quantum Hall transitions. Stable fixed points (plateaus) are shown as disks. Two solid curves demonstrate a possible flow toward even- and odd-plateau fixed point for a model with weakly mixed valleys. Each curve has a cusp when the running scale reaches $l_{\mathrm{mix}}$.

Figure 2

(Color online) Quantum Hall effect in graphene with smooth disorder at zero temperature. Hall conductivity as a function of the filling factor: odd (decoupled valleys, dashed line) vs normal (weak valley mixing, solid line) quantization. The inset shows the energy dependence of the density of states. The state in the center of Landau level is delocalized (dashed lines) when the valleys are decoupled. The valley mixing splits this delocalized state (solid lines).

Figure 3

(Color online) Quantum Hall transition at finite temperature. A double step in ${\sigma }_{xy}$ and a double peak in ${\sigma }_{xx}$ (solid lines) require low temperature, $T\lesssim \hslash ∕{\tau }_{\mathrm{mix}}$. Otherwise, a single broadened quantum Hall transition is seen (dashed lines).

Figure 4

(Color online) “Classical” QHE in graphene with chiral disorder (random vector potential). Chiral symmetry protects degeneracy of the lowest Landau level (inset: delta-function in the density of states). Hall conductivity is a linear function of carrier concentration while the lowest Landau level is being filled. In Abelian case (ripples), only odd plateaus appear away from zero energy (dashed line). Non-Abelian gauge disorder (dislocations) split quantum Hall transitions as shown by solid line.

Figure 5

(Color online) QHE in a double-layer graphene with smooth disorder (decoupled valleys). Degeneracy of the lowest Landau level is twice larger than for other levels. Double step at zero filling factor (dashed line) is split when the disorder has finite correlation length $d$. The inset shows the density of states and positions of delocalized states (solid lines). Two such states within the lowest Landau level remain degenerate (dashed line) in the limit $d\to \infty$.