Dual-Space Invariance as a Universal Criterion for Multifractal Critical States

In Anderson localization, eigenstates of disordered quantum systems are broadly classified as extended, localized, or critical. Although critical states exhibit multifractal character, a precise and operational criterion for their identification remains an open challenge, as Lyapunov exponents in real space cannot uniquely distinguish them from extended states. Here we address this challenge by asserting that critical states are uniquely characterized by an emergent dual-space invariance between position and momentum space. Building on the Liu--Xia criterion of the simultaneous vanishing of Lyapunov exponents ($\gamma=\gamma_m=0$), we show that this dual-space invariance extends beyond Lyapunov exponents and governs wavefunction scaling, revealing a fundamental property inaccessible from either space alone. Through numerical simulations, we demonstrate that the inverse participation ratio exhibits matching scaling behavior in position and momentum space for critical states, in sharp contrast to extended and localized states, which display a pronounced asymmetry between the two spaces. This dual-space invariance provides a direct, robust, and universal criterion for identifying multifractal critical states. Our results establish a fundamental principle of Anderson criticality and open new avenues for its detection in modern quantum simulation platforms.