Dual-Spaces Invariance as a Universal Criterion for Identifying Multifractal Critical States

Abstract

In Anderson localization physics, eigenstates of disordered quantum systems are commonly classified as extended, localized, or critical, distinguished by their distinct spatial structures in real space. While critical states are known to exhibit multifractal characteristics, a precise and operational criterion for characterizing critical states remains an open challenge. In this work, we address this challenge by revisiting criticality from a dual-spaces perspective that treats position and momentum representations on equal footing. Building on the Liu--Xia criterion, which characterizes critical states by the simultaneous vanishing of Lyapunov exponents ($γ=γ_m=0$) in both spaces, we show that this dual-spaces characterization captures an essential feature of critical states that is not limited to Lyapunov exponents. In particular, through numerical simulations, we demonstrate that the inverse participation ratio exhibits closely related scaling behavior in position and momentum space for critical states. This position-momentum correspondence clearly distinguishes critical states from extended or localized ones, which instead display a pronounced asymmetry between the two representations. Our results establish a robust and universal framework for precisely characterizing multifractal critical states in disordered quantum systems, and provide practical guidance for their identification in current quantum simulation platforms.

Figures (3)

• Figure 1: (Color online) Schematic illustration of typical eigenstates in position and momentum space. Panels (a), (c), and (e) show extended, critical, and localized states in position space, respectively, while panels (b), (d), and (f) show their corresponding momentum-space distributions. In contrast to conventional approaches that emphasize real-space multifractality, a dual-space perspective highlights that critical states are distinguished by the absence of exponential localization in both representations.
• Figure 2: (Color online) Panels (a1) and (a2) depict the wave function of the AAH model (with $V=2$, $E=0.0009$) in real and momentum spaces, respectively. Panels (b1) and (b2) display the analogous results for the QNE model at $V=1$, $E=0.4$. The system size is fixed at $L=987$ lattice sites. Notably, the AAH model exhibits self-dual symmetry, under which the critical states remain invariant between position and momentum spaces. By contrast, the QNE model does not possess self-dual symmetry; as a result, a critical state in position space is mapped onto a distinct critical state in momentum space.
• Figure 3: Mean inverse participation ratios in position space (MIPR) and momentum space (MIPR$_m$) for quasiperiodic models. (a) AAH model: the two quantities exhibit markedly different finite-size scaling behaviors away from the critical point $V=2$, while they become comparable at criticality. The data are obtained from all eigenstates for system sizes $L = 500,\;1000,\;2000$. (b) QNE model: in the regime $0<V<2$, both MIPR and MIPR$_m$ show strong fluctuations and similar scaling tendencies, consistent with critical behavior over an extended parameter range. The data are obtained from six eigenstates near $E = 0.00001$ for system sizes $L = 500,\;1000,\;2000$.