Predicted Critical State Based on Invariance of the Lyapunov Exponent in Dual Spaces

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, and in general, it cannot predict a system that hosts the critical state. In this work, we propose an explicit criterion that the Lyapunov exponent of the critical state should be 0 simultaneously in dual spaces, namely Lyapunov exponent remains invariant under the Fourier transform. With this criterion, we exactly predict a one-dimensional quasiperiodic model which is not self-duality, but hosts a large number of critical states. Then, we perform numerical verification of the theoretical prediction and display the self-similarity of the critical state. Due to computational complexity, higher dimensional models are not performed calculations, however, since the description of extended and localized states by the Lyapunov exponent is universal and dimensionless, utilizing the Lyapunov exponent of dual spaces to describe critical states should also be universal. Finally, we conjecture that there exists some kind of connection between the invariance of the Lyapunov exponent and conformal invariance, which can promote the research of critical phenomena.

Real eigenvalues determined by recursion of eigenstates

“…while E is a pure imaginary number, γ m (E) = 0, all detailed calculations can be found in Ref. [25]. Since the energy E exactly satisfies the duality relation (γ = 0, γ m > 0 and γ > 0, γ m = 0) indicated by Thouless formula between Lyapunov exponent in position space and momentum space, we conjecture that E is the eigenvalue of Eq.…”

“…In fact, real eigenvalues in many complicated models [23][24][25] also originated from the recursion of eigenstates. However, due to the complexity of models, obtaining rigorous mathematical solutions is very difficult.…”