Structural constraints on mobility edges in one-dimensional quasiperiodic systems

TL;DR

This paper reveals that mobility edge positions in one-dimensional quasiperiodic systems are structurally constrained across related Hamiltonians, leading to a reduced set of possible energies and specific critical scaling behaviors.

Contribution

It demonstrates that mobility edge positions are not independent but are constrained by an exact identity for Lyapunov exponents derived from isospectral duality.

Findings

Mobility edge positions are restricted to a reduced set of energies.

In the self-dual limit, mobility edges coincide at a single transition.

Lyapunov spectrum exhibits linear critical scaling with a universal exponent.

Abstract

Mobility edges commonly arise in one-dimensional quasiperiodic systems once exact self-duality is broken, yet their origin is typically understood only at the level of individual Hamiltonians. Here we show that mobility edge positions are not independent spectral features of individual Hamiltonians, but are structurally constrained across quasiperiodic Hamiltonians related by an isospectral duality. Using a bichromatic Aubry--Andr\'e model as a minimal setting, we demonstrate that this constraint is encoded in an exact identity for Lyapunov exponents derived from the Thouless formula. As a consequence, the mobility edge positions are restricted to a reduced set of energies. In the self-dual limit, these mobility edge positions coincide at a single localization--delocalization transition. This structural constraint enforces a linear critical scaling of the physical Lyapunov spectrum near the self-dual point. Numerical results confirm a critical exponent consistent with the standard Aubry--Andr\'e value of $\nu = 1$, while simultaneously revealing a novel, non-universal energy-dependent prefactor.