Exact mobility edges in Aubry-Andr\'{e}-Harper models with relative phases

TL;DR

This paper analytically derives exact mobility edges in a generalized Aubry-Andrb9-Harper model with a relative phase, revealing their dependence on phase and extending understanding of localization physics.

Contribution

It introduces a generalized model with a relative phase and provides exact analytical expressions for mobility edges using Avila's global theory, verified by numerical simulations.

Findings

Exact mobility edges depend on the relative phase.

The analytical results are verified numerically.

The approach applies to a broad class of Aubry-Andrb9-Harper models.

Abstract

Mobility edge (ME), a critical energy separating localized and extended states in spectrum, is a central concept in understanding the localization physics. However, there are few models with exact MEs. In the paper, we generalize the Aubry-Andr\'{e}-Harper model proposed in [Phys. Rev. Lett. 114, 146601 (2015)] and recently realized in [Phys. Rev. Lett. 126, 040603 (2021)], by introducing a relative phase in the quasiperiodic potential. Applying Avila's global theory we analytically compute localization lengths of all single-particle states and determine the exact expression of ME, which both significantly depend on the relative phase. They are verified by numerical simulations, and a physical perception of the exact expression is also provided. We further demonstrate that the exact expression of ME works for an even broad class of generalized Aubry-Andr\'{e}-Harper models. Moreover, we show that the exact ME is related to the one in the dual model which has long-range hoppings.