Topological analysis of the quantum Hall effect in graphene: Dirac-Fermi transition across van Hove singularities and edge versus bulk quantum numbers

Inspired by a recent discovery of a peculiar integer quantum Hall effect (QHE) in graphene, we study QHE on a honeycomb lattice in terms of the topological quantum number, with two interests. First, how the zero-mass Dirac QHE around the center of the tight-binding band crosses over to the ordinary finite-mass fermion QHE around the band edges. Second, how the bulk QHE is related with the edge QHE for the entire spectrum including Dirac and ordinary behaviors. We find the following. (i) The zero-mass Dirac QHE [with ${\sigma }_{xy}=\mp (2N+1)e^2∕h,N$: integer] persists, surprisingly, up to the van Hove singularities, at which the ordinary fermion behavior abruptly takes over. Here a technique developed in the lattice gauge theory enabled us to calculate the behavior of the topological number over the entire spectrum. This result indicates a robustness of the topological quantum number, and should be observable if the chemical potential can be varied over a wide range in graphene. (ii) To see if the honeycomb lattice is singular in producing the anomalous QHE, we have systematically surveyed over square $\leftrightarrow$ honeycomb $\leftrightarrow \pi$-flux lattices, which is scanned by introducing a diagonal transfer $t^{\prime }$. We find that the massless Dirac QHE [$\propto (2N+1)$] forms a critical line, that is, the presence of Dirac cones in the Brillouin zone is preserved by the inclusion of $t^{\prime }$ and the Dirac region sits side by side with ordinary one persists all through the transformation. (iii) We have compared the bulk QHE number obtained by an adiabatic continuity of the Chern number across the square $\leftrightarrow$ honeycomb $\leftrightarrow \pi$-flux transformation and numerically obtained edge QHE number calculated from the whole energy spectra for sample with edges, which shows that the bulk QHE number coincides, as in ordinary lattices, with the edge QHE number throughout the lattice transformation.

Figure 1

A honeycomb lattice with a unit cell and a extra transfer $t^{\prime }$ indicated.

Figure 2

(Color online) Energy dispersions for (a) $t^{\prime }∕t=-1$ ($\pi$-flux lattice), (b) $t^{\prime }∕t=0.5$, (c) $t^{\prime }∕t=0$ (honeycomb), (d) $t^{\prime }∕t=0.5$, and (e) $t^{\prime }∕t=1$ (square).

Figure 3

Density of states for (a) $t^{\prime }∕t=-1$ ($\pi$-flux lattice), (b) $t^{\prime }∕t=0.5$, (c) $t^{\prime }∕t=0$ (honeycomb), (d) $t^{\prime }∕t=0.5$, and (e) $t^{\prime }∕t=1$ (square).

Figure 4

Hofstadter’s diagram (energy spectrum against the magnetic flux $\varphi$) for (a) $t^{\prime }∕t=-1$ ($\pi$-flux lattice), (b) $t^{\prime }∕t=0.5$, (c) $t^{\prime }∕t=0$ (honeycomb), (d) $t^{\prime }∕t=0.5$, and (e) $t^{\prime }∕t=1$ (square).

Figure 5

(Color online) Blowup of Hofstadter’s diagram in a weak magnetic field region for (a) $t^{\prime }∕t=-1$ ($\pi$ flux), (b) $t^{\prime }∕t=0.5$, (c) $t^{\prime }∕t=0$ (honeycomb), (d) $t^{\prime }∕t=0.5$ and $t^{\prime }∕t=1$ (square). The arrows indicate the positions of van Hove singularities. The blue lines indicate positions of a flux $\varphi =1∕31$, which corresponds to the one in Fig. 6.

Figure 6

(Color online) The Chern number (Hall conductivity in units of $-e^2∕h$) for magnetic field $\varphi =1∕31$ is plotted against the Fermi energy $E_F$: for (a) $t^{\prime }∕t=-1$ ($\pi$ flux), (b) $t^{\prime }∕t=0.5$, (c) $t^{\prime }∕t=0$ (honeycomb), (d) $t^{\prime }∕t=0.5$, and (e) $t^{\prime }∕t=1$ (square). The red lines indicate the steps of two in the Chern number sequence [${\sigma }_{xy}=\pm (2N+1)e^2∕h,N$: integer], while the blue lines step of one $({\sigma }_{xy}=\pm N{e}^2∕h)$.

Figure 7

A cylindrical system with zigzag and bearded edges.

Figure 8

A Riemann surface ${\Sigma }_{2q-1}\left(k_2\right)$ for the Bloch function which is a complex energy surface for defining $\rho \left(E\right)$ consistently. Its genus is $2q-1$, which coincides with the number of energy gaps.

Figure 9

(Color online) The energy spectra for the cylindrical system in Fig. 7 with the flux $\varphi =1∕5$ for (a) $t^{\prime }∕t=-1$ ($\pi$-flux), (b) $t^{\prime }∕t=0.625$, (c) $t^{\prime }∕t=0$ (honeycomb), (d) $t^{\prime }∕t=0.5$, and (e) $t^{\prime }∕t=1$ (square). Blue (red) lines are edge states localized at the bearded (zigzag) edges, while the bulk energy bands are shown as grey regions. The topological number $I\left(E_F\right)$, when $E_F$ is in a major gap in the negative energies are (a) $2$, $-1$; (b) $1$, $2$, $3$, $-1$; (c) $1$, $2$, $3$, $-1$; (d) $1$, $2$, $3$, (ill-defined); and (e) $1$, $2$, $3$, $4$. A topological transition accompanied with a discrete Chern number change by $\mathrm{\Delta }c=-5$ (from $-4$ to +1) at the fourth major gap occurs between $t^{\prime }∕t=0.5$ $\to 1$. The green lines in the right are the topological numbers $I\left(E_F\right)$ obtained by counting the edge states.

Figure 10

(Color online) The energy spectra of the honeycomb lattice with zigzag and bearded edges for (a) $\varphi =1∕5$ and (b) $\varphi =1∕21$. (c) is a blowup of (b) and (d) is for $\varphi =1∕51$. The topological number $I\left(E_F\right)$ when $E_F$ is in a major energy gaps is also indicated next to the vertical axis.