High Chern Number Quantum Anomalous Hall States in Haldane-Graphene Multilayers

Abstract

We consider a rhombohedral-stacked $N$-layer graphene coupled to a monolayer of Haldane model. We show that high order Dirac points in multilayer graphene can be gapped out by topological proximity effect of the Haldane model layer, leading to total Chern number $|C|=N+1$ quantum anomalous Hall states. This provides a new way to construct high Chern number quantum anomalous Hall states in realistic crystalline graphene systems.

Figures (4)

• Figure 1: (a) Schematic of heterostructure composed of a layer of Haldane model \ref{['eq_Haldane']} in blue color and rhombohedral-stacked $N=3$ layer graphene in red color. The black vertical lines indicate interlayer hoppings. (b1) and (b2) show phase diagrams in parameter space $\left(\phi,M/t^\prime\right)$ for the total Chern number $C$ in \ref{['eq_Fukui']} of the heterostructure \ref{['eq_Htotal']} for $N=1$ and $N=3$ respectively. In the calculation, we choose $t=1$, $t^\prime=0.1$, $t_\perp=0.5$, and $t^\prime_\perp=0.1$. Phase diagrams are very similar for other values of $N$. (c1) and (c2) are the calculated dispersions with zigzag boundary in cylinder geometry. The red line shows edge modes. The stripe width is $L=10$ with $M=0$ and $\phi=\frac{\pi}{2}$. Other parameters are same as (b1) and (b2).
• Figure 2: (a) shows the gapless dispersions $E(q)\sim\pm{q}^3$ near the Dirac points $K_\pm$ at $N=3$ when $t^\prime_\perp=0$. Other parameters are chosen as $t^\prime=0.1$, $t_\perp=1$, $M=0$ and $\phi=\pi/2$. (b) shows the degeneracy at Dirac points is lifted by the Haldane layer when $t^\prime_\perp$ is turned on.
• Figure 3: (a) illustrates the band structure in the large $t^\prime_\perp$ limit. The blue lines indicates the low(high)-energy bands formed by (anti-)bonding states. The effective theory are based on the red bands near the Fermi surface whose degrees of freedom come from $c_{0,k,A}$ and $c_{N,k,B}$. (b) shows the phase diagram of \ref{['eq_Htv']}. The topological phase boundaries are given by \ref{['eq_boundary']}, where the $C=\pm(N+1)$ phases are meet at $\phi=\pm\pi$, $M=-3$ and $\phi=0$, $M=3$.
• Figure 4: (a) shows the exchange of Chern number between two occupied bands with $t^\text{H}_\perp$ increasing at $N=3$. Other parameters are set as $t_\perp=1$, $t^\prime=0.1$, $M=0$ and $\phi=\frac{\pi}{2}$. The dashed line shows the gap $W$ between occupied and vacant bands, which verifies the system being an insulator at finite $t^\prime_\perp$. The full line shows that the gap $W^\prime$ between two occupied bands closes, at which the exchange of Chern number happens, as shown by the blue stars and dots corresponding to the Chern numbers of the upper and lower occupied bands. (b) shows the phase diagram of \ref{['eq_Heff0']} at $N=3$. The total Chern number is always being $C=4$ and boundaries of two limit regimes regarding to different Haldane model parameters are labeled by colors, which are denoted in the phase diagram of Haldane monolayer as shown in the inner figure.