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

Critical states in disordered systems, fascinating and subtle eigenstates, have attracted a lot of research interests. However, the nature of critical states is difficult to describe quantitatively, and in general, it cannot predict a system that hosts the critical state. We propose an explicit criterion whereby the Lyapunov exponent of the critical state should be 0 simultaneously in dual spaces, namely the Lyapunov exponent remains invariant under the Fourier transform. With this criterion, we can exactly predict a one-dimensional quasiperiodic model which is not of 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, calculations are not performed for higher dimensional models. 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 some kind of connection exists between the invariance of the Lyapunov exponent and conformal invariance, which can promote the research of critical phenomena.