How to predict critical state: Invariance of Lyapunov exponent in dual spaces

TL;DR

This paper introduces a universal criterion based on the invariance of the Lyapunov exponent in dual spaces to predict and analyze critical states in disordered systems, supported by numerical verification.

Contribution

It proposes a novel criterion for critical states based on Lyapunov exponent invariance and predicts systems hosting critical states for the first time.

Findings

Lyapunov exponent remains invariant under Fourier transform for critical states

Numerical verification confirms the predicted critical states and their self-similarity

Conjecture of a connection between Lyapunov invariance and conformal invariance

Abstract

The critical state in disordered systems, a fascinating and subtle eigenstate, has attracted a lot of research interest. However, the nature of the critical state is difficult to describe quantitatively. Most of the studies focus on numerical verification, and cannot predict the system in which the critical state exists. In this work, we propose an explicit and universal criterion that for the critical state Lyapunov exponent should be 0 simultaneously in dual spaces, namely Lyapunov exponent remains invariant under Fourier transform. With this criterion, we exactly predict a specific system hosting a large number of critical states for the first time. Then, we perform numerical verification of the theoretical prediction, and display the self-similarity and scale invariance of the critical state. Finally, we conjecture that there exist some kind of connection between the invariance of the Lyapunov exponent and conformal invariance.