Scale-free localization versus Anderson localization in unidirectional quasiperiodic lattices

Scale-free localization emerging in non-Hermitian physics has recently garnered significant attention. In this work, we explore the interplay between scale-free localization and Anderson localization by investigating a unidirectional quasiperiodic model with generalized boundary conditions. We derive analytical expressions of Lyapunov exponent from the bulk equations. Together with the boundary equation, we can determine properties of eigenstates and spectrum and establish their exact relationships with the quasiperiodic potential strength and boundary parameter. While eigenstates exhibit scale-free localization in the weak disorder regime, they become localized in the strong disorder regime. The scale-free and Anderson localized states satisfy the boundary equation in distinct ways, leading to different localization properties and scaling behaviors. Generalizing our framework, we design a model with exact energy edges separating the scale-free and Anderson localized states via the mosaic modulation of quasiperiodic potentials. Our models can be realized experimentally in electric circuits.