Gap labeling theorem for multilayer thin film heterostructures

Quasiperiodic systems show a universal gap structure due to quasiperiodicity which is analogous openings at the Brillouin zone boundary in periodic systems. The integrated density of states (IDoS) below those energy gaps are characterized by few integers, known as ``gap labeling theorem'' (GLT) for quasiperiodic In this study, focusing on multilayer thin film such twisted bilayer graphene and stacked transition metal dichalcogenides, we extend GLT arbitrary dimensions number layers, using an approach based algebra called ``a noncommutative torus''. We find that associated IDoS generally $_{DN}C_D$ integer labels $N$ layer $D$ dimensions, when effect interlayer coupling can be approximated intralayer each layer. demonstrate generalized holds 1D tight binding models numerical simulations.