Moiré butterflies in twisted bilayer graphene

The Hofstadter butterfly spectral patterns of lattice electrons in an external magnetic field yield some of the most beguiling images in physics. Here we explore the magnetoelectronic spectra of systems with moiré spatial patterns, concentrating on the case of twisted bilayer graphene. Because long-period spatial patterns are accurately formed at small twist angles, fractal butterfly spectra and associated magnetotransport and magnetomechanical anomalies emerge at accessible magnetic field strengths.

Figure 1

Moiré unit cell. The magnitude and phase of the $AA$ hopping amplitude (represented by arrows) is plotted as a function of position (in units of $k_{\theta }^{-1}$). The lattice that corresponds to the moiré pattern is marked by the red dots and the lattice on which the interlayer hopping Hamiltonian is periodic is denoted by the green circles. It is this second lattice that determines the moiré unit cell area ${\Omega }_{\mathrm{M}}$.

Figure 2

Spectrum support. The support of the spectrum as a function of $\alpha$ for $\theta =2^{\circ }$ ($w=110$ meV). The periodic interlayer hopping amplitude results in a Hofstadter-like sub-band structure. The integers denote the Hall conductivity associated with the larger energy gaps between $\nu =1$ and $\nu =-1$ for $\alpha \approx 0.3$.

Figure 3

Hall conductivity. The energy gaps are depicted as a function of filling factor $\nu$ and $p/q$ for $\theta =2^{\circ }$ ($w=110$ meV). The conspicuous straight lines satisfy the Diophantine gap equation (9) and determine both ${\sigma }_{\mathrm{H}}$ and $s$. The Hall conductivity corresponds to the intercept of the line with the $y$-axis whereas $s$ is given by its slope.