Fig. 2: Hofstadter butterflies of the atomistic tight-binding model of TBG.

a, b Hofstadter spectrum of TBG a without and b with interlayer coupling calculated by using the kernel polynomial method (see Methods). Red dashed lines indicate the system’s half and full flux periodicity. Blue dashed lines indicate quasi periodicities at \(\phi =\frac{p}{14}\Phi (p=1,2,\ldots ,6)\). In b, the red and green solid lines denote the highest occupied and lowest unoccupied states, respectively. c–j Magnified view of b at specific flux values c ϕ = 0, d \(\phi =\frac{1}{2}\Phi\), e \(\phi =\frac{1}{14}\Phi\), f \(\phi =\frac{1}{7}\Phi\), g \(\phi =\frac{3}{14}\Phi\), h \(\phi =\frac{2}{7}\Phi\), i \(\phi =\frac{5}{14}\Phi\), and j \(\phi =\frac{3}{7}\Phi\), where \(\varepsilon =\frac{1}{1680}\Phi\).