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We investigate the electronic structure and the optical absorption onset ofhexagonal boron nitride bilayers with twist angles in the vicinity of30$^\circ$. Our study is carried out with a tight-binding model that wedeveloped on purpose and validated against DFT simulations. We demonstrate thatapproaching 30$^\circ$ (quasicrystal limit), all bilayers sharing the samemoir\'e supercell develop identical band structures, irrespective of theirstacking sequence. This band structure features a bundle of flat bands layingslightly above the bottom conduction state which is responsible for an intensepeak at the onset of independent-particle absorption spectra. These resultsreveal the presence of strong, stable and stacking-independent opticalproperties in boron nitride 30$^\circ$-twisted bilayers. By carefully analyzingthe electronic spatial distribution, we elucidate the origin of these states asdue to interlayer B-B coupling. We take advantage of the the physicaltransparency of the tight-binding parameters to derive a simple triangularmodel based on the B sublattice that accurately describes the emergence of thebundle. Being our conclusions very general, we predict that a similar bundleshould emerge in other close-to-30$^\circ$ bilayers, like transition metaldichalcogenides, shedding new light on the unique potential of 2D materials.

Emergence of flat bands in the quasicrystal limit of boron nitride twisted bilayers