Topological gaps by twisting

It is shown that twisted $n$-layers have an intrinsic degree of freedom living on $2n$-tori, which the phason supplied by relative slidings layers and twist generates pseudo magnetic fields. As a result, host higher dimensional topological phases those characterized second Chern numbers can be found in bi-layer. Indeed, our investigation phononic lattices with interactions modulated lattice reveals Hofstadter-like spectral butterflies terms angle, whose gaps carry predicted invariants. Our work demonstrates how multi-layered systems are virtual laboratories for studying physics quantum Hall effect to generate edge chiral modes simply sliding each other. In context classical metamaterials, both photonic phononic, these findings open path engineering pumping via simple twisting sliding.