Hofstadter quasicrystals, hidden symmetries and irrational quantum oscillations

Landau levels perturbed by a periodic potential is a prime setting to design quantum systems with exotic fractal spectra. Motivated by recent advances in twistronics, we introduce Hofstadter quasicrystal' problem describing Landau levels perturbed by a set of incommensurate cosine waves. We illustrate the underlying physics for moiré quasicrystals with octagonal and dodecagonal symmetries, finding spectra that are vastly more complex than the Hofstadter spectrum. Surprisingly, due to the high spatial symmetry, the quasicrystal problem exhibits hidden inner’ symmetry arising at special `magic’ values of the magnetic field. The $1\mathrm{/}B$-periodic pattern of magic field values explains striking wide-range oscillations in the observed spectra that have irrational periodicity incommensurate with the Aharonov-Bohm and Brown-Zak periodicities. The prominent character of these oscillations makes them readily accessible in state-of-the-art moiré graphene systems.