High Chern number quantum anomalous Hall phases in single-layer graphene with Haldane orbital coupling

We investigate possible phase transitions among the different quantum anomalous Hall (QAH) phases in single-layer graphene under the influence of the exchange field. The effective tight-binding Hamiltonian for graphene is made up of the hopping term, the Kane-Mele and Rashba spin-orbit couplings as well as the Haldane orbital term. We find that the variation of the exchange field results in bulk gap-closing phenomena and phase transitions occur in the graphene system. If the Haldane orbital coupling is absent, the phase transition between the chiral (antichiral) edge state $\nu =+2$ ($\nu =-2$) and the pseudoquantum spin Hall state ($\nu =0$) takes place. Surprisingly, when the Haldane orbital coupling is taken into account, an intermediate QSH phase with two additional edge modes appears in between phases $\nu =+2$ and $\nu =-2$. This intermediate phase is therefore either the hyperchiral edge state of high Chern number $\nu =+4$ or antihyperchiral edge state of $\nu =-4$ when the direction of exchange field is reversed. We present the band structures, edge state wave functions, and current distributions of the different QAH phases in the system. We also report the critical exchange field values for the QAH phase transitions.

Figure 1

(a) A segment of a zigzag graphene ribbon with its unit cell marked by the red dashed lines. (b) Illustrations of Haldane phase factors ${\nu }_{ij}^z$, vectors ${\mathbf{d}}_{ij}$, bulk basis vectors $({\mathit{a}}_1,{\mathit{a}}_2)$, and bond length $a$. (c) The first Brillouin zones of bulk graphene (gray region) and the zigzag graphene ribbon (red double arrow) in the 2D $k$ space.

Figure 2

Calculated energy bands (left panels), edge state probability (middle panels), and charge current (right panels) distributions in the zigzag ribbon in the presence of the exchange field. The Fermi level (the dashed line in the left panels) $E_F=0.05t$. (a) Kane-Mele system ($\lambda =0.06t$ and $\gamma =0.2t$). (b) Rashba system ($\alpha =0.2t$ and $\gamma =0.2t$). (c) Haldane system ($\beta =-0.07t$ and $\gamma =0.2t$).

Figure 3

Expectation value $\langle y\rangle$ and Chern number as a function of $\gamma /t$. (a) The Kane-Mele-Rashba system with $\lambda =0.06t$, $\alpha =0.05t$, and $\gamma$ ranging from $-0.5t$ to $0.5t$. (b) The Haldane-Rashba system with $\beta =-0.1t$, $\alpha =0.5t$, and $\gamma$ ranging from $-t$ to $t$.

Figure 4

The band structure (a), the edge state probability and current distributions (b), bulk energy bands (c), and Berry curvature (d) of the intermediate state in ${\gamma }_1^c<\gamma <{\gamma }_2^c$ ($\alpha =0.5t$, $\beta =-0.1t$, and $\gamma =0.6t$) (see text) in the graphene ribbon. The Fermi level $E_F$ is at $0.01t$ [the red dashed line in (a)]. In (a), the eight edge states are marked as A, B, C, D, A${}_2,{\text{B}}_2,{\text{C}}_2$, and ${\text{D}}_2$, respectively. In (c) and (d), the bulk energy bands and Berry curvature are plotted along the $k_y=0$ profile.

Figure 5

Calculated spin polarizations of edge states B and D in a graphene ribbon with (a) the Kane-Mele Hamiltonian ($\lambda =0.06t$) and (b) the Kane-Mele-Rashba Hamiltonian ($\lambda =0.06t,\alpha =0.05t$).

Figure 6

The Kane-Mele-Rashba system $\lambda =0.006t$ and $\alpha =0.005t$. The critical field is ${\gamma }^c=0.0293t$. (a) The expectation value $\langle y\rangle$ and (b) the variation of Chern number.