Localization in the Incommensurate Systems: A Plane Wave Study via Effective Potentials

In this paper, we apply the effective potentials in the localization landscape theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the incommensurate systems. We uniquely develop a plane wave framework for the effective potentials of the incommensurate systems. And utilizing the effective potentials represented by the plane wave, the location of the electron density can be inferred. Moreover, the spectral distribution can be obtained from the effective potential version of Weyl’s law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for investigating the localization and spectral properties of the incommensurate systems, without solving the eigenvalue problem explicitly.