Identifying and classifying localized states in gapless systems using pseudospectral methods

Abstract

For decades, band theory has been used to understand the physical properties of crystals by leveraging these systems’ translational symmetries to determine the spectral range and distribution of their Bloch eigenstates. However, many important phenomena only manifest at a crystal’s defects or boundaries where this translational symmetry is broken, such as defect-localized states that appear at color centers, or boundary-localized states whose existence is guaranteed by topology. Although it is possible to extend band theoretic approaches to study these classes of states, these extensions are only applicable in gapped systems, where any localized states are spectrally isolated from the crystal’s bulk states.