Gap labeling theorem for multilayer thin film heterostructures

Quasiperiodic systems show a universal gap structure due to quasiperiodicity which is analogous to gap openings at the Brillouin zone boundary in periodic systems. The integrated density of states (IDoS) below those energy gaps are characterized by a few integers, which is known as the “gap labelling theorem” (GLT) for quasiperiodic systems. 

Here, we extend the GLT for multilayer systems of arbitrary dimensions and number of layers, using an approach based on the noncommutative torus. Consequently, we identified the number of integer labels to label the D-dimensional N-layer system.