2025/09/07 by Xinyu Guo, Chen, Jin-Rong, Guo, Xin-Yu +10
Computer Science · Materials Science · Mathematics · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Graph theory and applications #Quasicrystal Structures and Properties #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2509.05950
openalex publication_date 2025/09/07 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
The physics of localized states in quasiperiodic lattices has been extensively studied for decades, but still lacks an comprehensive theoretical framework. Recently, we developed a incommensurate energy band (IEB) theory, which extends the concept of energy bands to quasiperiodic systems lacking translational symmetry, thereby achieving a breakthrough in elucidating extended states. Here, we demonstrate that, due to the inherent duality between momentum and real space, the IEB theory also offers a comprehensive framework for elucidating localized states. Specifically, via a so-called spiral (module) mapping, the energy spectrum of localized states can be represented as a function defined on a compact circular manifold-akin to the Brillouin zone-whose form resembles conventional energy bands. These localized state energy bands (LSEBs) fully characterize all the properties of the localized states. Moreover, we show that quasiperiodic systems with mobility edges exhibit a unique hybrid band structure: the IEB for extended states (momentum space) and LSEB for localized states (real space), separated by mobility edges. Our theory thus establishes a comprehensive framework for analyzing the localized states in quasiperiodic lattices.