Lyapunov Exponents as Duality-Invariant Signatures of Critical States

Abstract

Critical eigenstates are usually identified through wave-function geometry in a chosen basis, such as participation ratios, multifractal spectra, or finite-size scaling. Here we formulate criticality instead as a dual-space Lyapunov property. We prove a Fourier exclusion principle: exponential localization in one representation is incompatible with exponential localization in its Fourier-dual representation. This turns the Liu--Xia condition, \(\gamma_x(E)=\gamma_m(E)=0\), from a phenomenological criterion into a rigorous length-scale statement: a critical state is characterized by the simultaneous absence of exponential confinement in real and momentum space. The criterion is invariant under bounded local gauge transformations of the transfer matrix and remains compatible with conventional single-space multifractal diagnostics. More importantly, it is exactly predictive. In analytically tractable quasiperiodic models, the same condition yields closed-form critical lines, an exact finite critical region with an additional critical branch, and a complex critical surface in a non-Hermitian non-self-dual spectrum. Thus the Liu--Xia condition provides not only a diagnostic of critical states, but an exact solvability principle for locating critical sets across distinct microscopic structures.