Magic angle hierarchy in twisted graphene multilayers

When two monolayers of graphene are stacked with a small relative twist angle, the resulting band structure exhibits a remarkably flat pair of bands at a sequence of “magic angles” where correlation effects can induce a host of exotic phases. Here we study a class of related models of $n$-layered graphene with an alternating relative twist angle $\pm \theta$ which exhibits magic angle flat bands coexisting with several Dirac dispersing bands at the moiré $K$ point. Remarkably, we find that the Hamiltonian for the multilayer system can be mapped exactly to a set of decoupled bilayers at different angles, revealing a remarkable hierarchy mathematically relating all these magic angles to the TBG case. For the trilayer case ($n=3$), we show that the sequence of magic angle is obtained by multiplying the bilayer magic angles by $\sqrt{2}$, whereas the quadrilayer case ($n=4$) has two sequences of magic angles obtained by multiplying the bilayer magic angles by the golden ratio $\phi =(\sqrt{5}+1)/2\approx 1.62$ and its inverse. We also show that for larger $n$, we can tune the angle to obtain several narrow (almost flat) bands simultaneously and that for $n\to \infty$, there is a continuum of magic angles for $\theta \lesssim 2^{\circ }$. Furthermore, we show that tuning several perfectly flat bands for a small number of layers is possible if the coupling between different layers is different. The setup proposed here can be readily achieved by repeatedly applying the “tear and stack” method without the need of any extra tuning of the twist angle and has the advantage that the first magic angle is always larger than the bilayer case.

Figure 1

A schematic illustration of the alternating-twist multilayer graphene: Trilayer (left) and quadrilayer (right).

Figure 2

Magic angles families for the alternating-twist multilayer graphene in the chiral limit ($\kappa =0$). Upper panel: Numerical values of magic angles for alternating-twist $n$-layered systems ($n=2,3,\cdots$). The magic angle parameters ${\alpha }_i=w_{\text{AB}}/\left(v_F{k}_D{\theta }_i\right)$ designate the twists under which the lowest bands become perfectly flat. For each $n$, there are $\lfloor n/2\rfloor$ sequences of magic angles (denoted by different colors) obtained by dividing the bilayer magic angles by $2cos\frac{\pi k}{n+1}$, $k=1,\cdots ,\lfloor n/2\rfloor$ as illustrated schematically in the lower panel.

Figure 3

Band structure at the first two magic angles for the trilayer $n=3$ and quadrilayer $n=4$ cases. We show the spectrum for the chiral limit $\kappa =0$ (red, solid) as well as the realistic lattice relaxation value for $\kappa$ at the corresponding angle [46] (blue, dashed). In the chiral limit, we can observe a perfectly flat band coexisting with a single Dirac cone at the $K$ point for $n=3$. For $n=4$, the chiral flat band coexists with another tBG spectrum at nonmagic angle. We can see that the flat bands for the first magic angle for $n=3$ (upper left) and the first two magic angles for $n=4$ (lower left and right) are stable to the addition of intrasublattice interlayer coupling $\kappa \ne 0$, whereas the flat band at the second magic for $n=3$ (upper right) gets destroyed. All energies are measured in units of $\hslash v_F{k}_D\theta =w_{\mathrm{AB}}/\alpha$.

Figure 4

Band structure for the models with $n=5$ layers with $\alpha =2.2$ and $n=6$ layers with $\alpha =1.275$. In both cases, the vicinity to two very close magic angles leads to the appearance of two pairs of almost perfectly flat bands. All energies are measured in units of $\hslash v_F{k}_D\theta =w_{\mathrm{AB}}/\alpha$.

Figure 5

Band structure for the quadrilayer problem with unequal layer couplings ${\alpha }_{12}={\alpha }_{34}=1.14$, ${\alpha }_{23}=1.64$. We notice the appearance of a pair of perfectly flat bands which are relatively stable to the inclusion of intrasublattice intralayer coupling $\kappa =w_{\text{AA}}/w_{\text{AB}}$. All energies here are measured in units of $\hslash v_F{k}_D\theta$.

Figure 6

Bandwidth in units of $\hslash v_F{k}_D\theta$ as a function of the displacement vector $\mathit{D}$ within the moiré unit cell formed by the vectors ${\mathit{a}}_{1,2}=\frac{4\pi }{3}(\frac{\sqrt{3}}{2},\pm \frac{1}{2})$ for $n=3$ and $n=4$ at the first magic angle for $\kappa =0.8$ (left panel) together with the band structures for one selected point ${\mathit{D}}^{\prime }=(1,0)$ at the border of the blue (narrow band) region (middle panel). The density of states exhibits a very large split peak close to charge neutrality due to the flat band (left panel).