Lyapunov Exponents as Duality-Invariant Signatures of Critical States

TL;DR

This paper introduces a dual-space Lyapunov criterion for identifying critical states, providing a rigorous, invariant, and predictive method that extends beyond traditional wave-function geometry diagnostics.

Contribution

It formulates criticality as a dual-space Lyapunov property, establishing a rigorous length-scale criterion that is invariant and exactly solvable in quasiperiodic models.

Findings

Proves a Fourier exclusion principle linking localization in dual representations.

Derives exact critical lines and regions in non-Hermitian quasiperiodic models.

Provides a length-scale criterion that predicts critical states across microscopic structures.

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.