Optical properties of the Hofstadter butterfly in the moiré superlattice

We investigate the optical absorption spectrum and the selection rule for the Hofstadter butterfly in twisted bilayer graphene under magnetic fields. We demonstrate that the absorption spectrum exhibits a self-similar recursive pattern reflecting the fractal nature of the energy spectrum. We find that the optical selection rule has a nested self-similar structure as well, and it is governed by the conservation of the total angular momentum summed over different hierarchies.

Figure 1

(a) Band structure at $B=0\mathrm{T}$ and (b) energy spectrum as a function of $B$, calculated for TBG with $\theta =2.{65}^{\circ }$. (c) Intensity map of the optical conductivity for right circularly polarized light (Re ${\sigma }_{+}$) of the same TBG.

Figure 2

(a) Energy spectrum and (b) optical conductivity for right circularly polarized light (Re ${\sigma }_{+}$) of TBG with $\theta =2.{65}^{\circ }$, near the $n=0$ Landau level around $\Phi /{\Phi }_0=1/3$ [the regions marked by dashed boxes in Figs. 1 and 1].

Figure 3

(a) Band structure at $B=0\mathrm{T}$ and (b) energy spectrum as a function of $B$, calculated for TBG with $\theta =1.{47}^{\circ }$. (c) Intensity map of the optical conductivity (Re ${\sigma }_{+}$) of the same TBG.