Quasiperiodicity, band topology, and moiré graphene

A number of moiré graphene systems have nearly flat topological bands where electron motion is strongly correlated. Though microscopically these systems are only quasiperiodic, they can typically be treated as translation invariant to an excellent approximation. Here we reconsider this question for magic angle twisted bilayer graphene that is nearly aligned with a hexagonal boron nitride (hBN) substrate. We carefully study the effect of the periodic potential induced by hBN on the low energy physics. The combination of this potential and the moiré lattice produced by the twisted graphene generates a quasiperiodic term that depends on the alignment angle between hBN and the moiré graphene. We find that the alignment angle has a significant impact on both the band gap near charge neutrality and the behavior of electrical transport. We also introduce and study toy models to illustrate how a quasiperiodic potential can give rise to localization and change in transport properties of topological bands.

Figure 1

(a) Case 1, ${\theta }_{\text{BN}}>0$ and ${\theta }_G>0$. (b) Case 2, ${\theta }_{\text{BN}}<0$ and ${\theta }_G>0$. The angles are exaggerated for illustration purpose.

Figure 2

(a) Dependence of $|{\theta }_{\text{BN}}|$ and $|{\theta }_G|$ on the change of the ratio $a_G/a_{\text{BN}}$ for satisfying perfect alignment conditions. $\mathrm{\Delta }(a_G/a_{\text{BN}})$ denotes the change of $a_G/a_{\text{BN}}$ from $a_G/a_{\text{BN}}=2.46/2.504$. (b) The points on blue and orange lines are $|{\theta }_{\text{BN}}|$ and $|{\theta }_G|$ taken from (a). The blue and orange dots are data from two experimental samples from the Stanford group [7] and the UCSB group [8].

Figure 3

Dispersion for (a) Case 1 and (b) Case 2. ${\theta }_G=1.2^{\circ }$.

Figure 4

Berry curvature distribution of valence band for (a) $m_z$ only, (b) Case 1, and (c) Case 2. ${\theta }_G=1.2^{\circ }$. The black line is the boundary of the first BZ and the black dot is the $\mathrm{\Gamma }$ point.

Figure 5

Labels of the hopping terms from site $A$ (orange circle) to other nearby sites. For example, hopping from $A$ to an orange site labeled ‘2’ corresponds to $t_{2AA}$ and from $A$ to a green site labeled $3_1$ corresponds to $t_{3AB1}$.

Figure 6

Dispersion of the effective tight-binding model.

Figure 7

Density of states at (a) ${\theta }_{\text{BN}}=0^{\circ }$ and (b) ${\theta }_{\text{BN}}=0.8^{\circ }$ for a $71\times 71$ grid in $\vec{k}$ space. The dotted line indicates the energy of the middle state in the spectrum.

Figure 8

PR at (a) ${\theta }_{\text{BN}}=0^{\circ }$ and (b) ${\theta }_{\text{BN}}=0.8^{\circ }$ for different states throughout the entire energy spectrum. The dotted line indicates the energy of the middle state in the spectrum. Different color indicates different system size. From bottom to top: $N=30,50,70$ and the system size is $N\times N$ in $\vec{k}$ space.

Figure 9

${\sigma }_{xy}$ vs chemical potential at (a) ${\theta }_{\text{BN}}=0^{\circ }$ and (b) ${\theta }_{\text{BN}}=0.8^{\circ }$ for a $50\times 50$ grid in $\vec{k}$ space. The dotted line indicates the energy of the middle state in the spectrum.

Figure 10

Schematic of the line integrals in Eq. (14).

Figure 11

Dispersion for (a) only $m_z$, (b) Case 1, and (c) Case 2. ${\theta }_G=1.{15}^{\circ }$.

Figure 12

Berry curvature distribution of the valence band for (a) only $m_z$, (b) Case 1, and (c) Case 2. ${\theta }_G=1.{15}^{\circ }$.

Figure 13

Density of states, PR, and ${\sigma }_{xy}$ for ${\theta }_G=1.{15}^{\circ }$ and ${\theta }_{\text{BN}}=-0.6^{\circ }$.

Figure 14

IPR for single band model with dispersion. (a), (b) trivial case. (c), (d) nontrivial case. In (a), (b), Log(IPR) is plotted; the $x$ axis labels different eigenstates in ascending order of their energies and the $y$ axis labels different QP potential $V$ from 0 to 4. The system size is $60\times 60$. In (b), (d), green dots with different brightness connected by the same line label different QP potential $V$, from 0 to 4 (top to bottom), with 0.2 interval. The black line is the linear fitting for $V=4$. The linear system sizes are from 20 to 60.

Figure 15

IPR for single band model with nonuniform Berry curvature. (a) trivial ($B_0=0$) and (b) nontrivial ($B_0\approx 7.26$). Green dots with different brightness connected by the same line label different $B_1$, from 0 to 4 (top to bottom), with 0.2 interval. The black line in (a) is the linear fitting of the average IPR. The black line in (b) is the linear fitting of the average IPR for $B_1\ge 2$ and the blue line in (b) is the linear fitting for $B_1=0.2$.

Figure 16

IPR for the two band model. (a) Color plot of Log(IPR). The $x$ axis labels different eigenvalues. (b) Dependence of Log(IPR) with linear system size. The green lines from bright to dark label different $V_0$ from 0 to 1. The black line is the linear fitting to the Log(IPR) at $V_0=1$.

Figure 17

IPR of the states that is in the middle of the spectrum. The $x$ axis is Log(L) and the $y$ axis is Log(IPR) for (a) small $V_0$, (b) intermediate $V_0$, and (c) large $V_0$.