Classically forbidden regions in the chiral model of twisted bilayer graphene. With an appendix by Zhongkai Tao and Maciej Zworski

Abstract

We establish exponential decay, as the angle of twisting goes to $ 0$, of eigenstates in a model of twisted bilayer graphene (TBG), near the hexagon connecting stacking points. That is done by adapting microlocal methods Kawai-Kashiwara and Sj\"ostrand used to establish analytic hypoellipticity by Tr\'epreau and Himonas. That replaces ellipticity, absent here, which is the usual mechanism behind classically forbidden regions. We also discuss numerical evidence of exponential decay in other regions (the center of the hexagon) and analytic complications involved in establishing that decay.