Dual-spaces invariance as a concise criterion for multifractal critical states

Within the framework of Anderson localization, eigenstates of quantum disordered systems are commonly classified as extended, localized or critical, distinguished by their distinct spatial structures. While critical states are known to exhibit multifractal characteristics, a precise and operational criterion for their identification remains challenging. In this work, we revisit criticality from a dual-space perspective and investigate the interplay between position and momentum representations. Building on the Liu–Xia criterion, which characterizes critical states by the simultaneous vanishing of Lyapunov exponents in both spaces, we show that critical states display closely related localization properties in position and momentum space. We further demonstrate, through analytical arguments and numerical simulations, that experimentally accessible quantities such as the inverse participation ratio exhibit comparable scaling behavior in the two dual spaces for critical states. This dual-space correspondence distinguishes critical states from extended or localized ones, which exhibit pronounced asymmetry between the two representations. Our results establish position–momentum correspondence as a robust and experimentally relevant framework for characterizing multifractal critical states in one-dimensional quasiperiodic systems, and provide guidance for their identification in current quantum simulation platforms.