Dual-Space Invariance as a Definitive Signature of Critical States in Anderson Localization

Abstract: Critical states represent a fundamental and fascinating research frontier in Anderson localization physics, known for their non-ergodic properties, including multifractal structure and self-similarity. However, exactly characterizing critical states continues to pose a significant challenge up to now. In this work, we establish a universal mechanism demonstrating that critical states must maintain dual-space invariance in both position and momentum representations, leading to delocalized dynamics in both spaces. Therefore, our discovery soundly answers this long-standing unsolved puzzle regarding the definition of the critical state and its rigorous characterization. Furthermore, keeping pace with the idea of Liu-Xia criterion, we prove rigorously that physical quantities being directly observed in experiments, such as the inverse participation ratio and information entropy, exhibit invariance in both position and momentum spaces as expected. Subsequent numerical simulations provide the smoking gun for the correctness of the dual-space invariance, thereby not only highlighting the universality of the rigorous mechanism, but also establishing a robust foundation for future experimental validation of critical states.