Half-integer contributions to the quantum Hall conductivity from single Dirac cones

While the quantum Hall effect in graphene has been regarded as a realization of the anomaly associated with the massless Dirac particle carrying half the usual topological integer, this is hidden due to the doubling of the Dirac cones. In order to confirm the half-integer contribution from each Dirac cone, here we theoretically consider a lattice model in which the relative energy between the two Dirac points is systematically shifted. With an explicit calculation of the topological (Chern) number, we have demonstrated that each Dirac cone does indeed contribute to the Hall conductivity as the half-odd-integer series $\left(\dots ,-3/2,-1/2,1/2,3/2,\dots \right)$ when the Fermi energy traverses the (shifted sets of) Landau levels. The picture is also endorsed, via the bulk-edge correspondence, from the edge mode spectrum for the present model.

Figure 1

(Color online) (a) The honeycomb lattice model considered here with nearest-neighbor hopping $t_0$ (solid lines) and second-neighbor hopping which is $i{t}_1$ along each dashed arrow, $-i{t}_1$ in the opposite direction. A unit cell and $A$, $B$ sublattices are indicated. (b) The dispersion relation in the present model, seen horizontally with Dirac cones shifted in energy by $\Delta =\sqrt{3}{t}_1$.

Figure 2

(Color online) (a) For the ordinary graphene $\left(t_1=0\right)$ and (b) for the present model with a shifted Dirac cones $\left(t_1\ne 0\right)$ we display the numerically calculated Chern numbers against $E_F$ (upper panels), along with the Landau levels for each of the two Dirac cones, where the Chern numbers are indicated for $E_F$ in each of the gaps (lower panels). The result is for the flux $\varphi =1/100$.

Figure 3

(Color online) Numerical result (black curves) for the Landau-level spectrum when the shift ($\Delta \propto t_1$; the horizontal axis) is varied with the Chern number indicated for each of the Landau gap. The shading represents the Chern numbers expected from the effective theory (see text), where the boundaries between different shades (Chern numbers) almost exactly coincide with the numerical result. Accumulation of lines around $E=\pm 1$ corresponds to the Van Hove singularity, which shifts quadratically with $t_1$. The result is for a magnetic field of $\varphi =1/100$, and two vertical lines indicate, respectively, the situation for Figs. 2a, 2b.

Figure 4

(Color online) Energy spectrum against $k_2$ in the present model for a finite sample with zigzag-bearded edges (Refs. 17, 18). Horizontal solid (dashed) lines correspond to bulk Landau levels corresponding to the shifted Dirac fermions by $\Delta$ ($-\Delta$). Curves are the edge modes whose amplitudes are confirmed to be localized along the zigzag right (red) or bearded left (blue) edge. The result is for a magnetic field $\varphi =1/48$ with $t_1/t_0=0.02$.