Gap labeling theorem for multilayer thin film heterostructures

TL;DR

This paper extends the gap labeling theorem to multilayer quasiperiodic systems like twisted bilayer graphene, providing a mathematical framework to characterize energy gaps and their associated states.

Contribution

It generalizes the gap labeling theorem for multilayer systems of arbitrary dimensions using noncommutative torus algebra, applicable to various 2D materials.

Findings

Energy gaps characterized by $_{DN}C_D$ integer labels in N-layer systems

Generalized GLT holds for quasiperiodic 1D tight binding models

Applicable to multilayer 2D materials like twisted bilayer graphene

Abstract

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 labeling theorem'' (GLT) for quasiperiodic systems. In this study, focusing on multilayer thin film systems such as twisted bilayer graphene and stacked transition metal dichalcogenides, we extend the GLT for multilayer systems of arbitrary dimensions and number of layers, using an approach based on the algebra called ``a noncommutative torus''. We find that the energy gaps and the associated IDoS are generally characterized by $_{DN}C_D$ integer labels in $N$ layer systems in the $D$ dimensions, when the effect of the interlayer coupling can be approximated by a quasiperiodic intralayer coupling for each layer. We demonstrate that the generalized GLT holds for quasiperiodic 1D tight binding models by numerical simulations.