Exact Mobility Edges in One-Dimensional Mosaic Lattices Inlaid with Slowly Varying Potentials

TL;DR

This paper analytically determines exact mobility edges in one-dimensional mosaic lattices with slowly varying potentials, revealing the transition points between different localization regimes with high accuracy.

Contribution

It introduces a semi-analytical method combining heuristic and trace map theory to precisely identify mobility and pseudo-mobility edges in these lattice models.

Findings

Exact semi-analytical expressions for mobility edges

Excellent agreement between numerical and theoretical results

Clear distinction of eigenstate types via multiple diagnostics

Abstract

We propose a family of one-dimensional mosaic models inlaid with a slowly varying potential $V_n=\lambda\cos(\pi\alpha n^\nu)$, where $n$ is the lattice site index and $0<\nu<1$. Combinating the asymptotic heuristic argument with the theory of trace map of transfer matrix, mobility edges (MEs) and pseudo-mobility edges (PMEs) in their energy spectra are solved semi-analytically, where ME separates extended states from weakly localized ones and PME separates weakly localized states from strongly localized ones. The nature of eigenstates in extended, critical, weakly localized and strongly localized is diagnosed by the local density of states, the Lyapunov exponent, and the localization tensor. Numerical calculation results are in excellent quantitative agreement with theoretical predictions.