[Paper Review] Exact anomalous mobility edges in one-dimensional non-Hermitian quasicrystals

Xiang-Ping Jiang, Weilei Zeng+2 more|arXiv (Cornell University)|Sep 5, 2024

Quasicrystal Structures and PropertiesMaterials Science3 citations

TL;DR

This paper analytically derives exact anomalous mobility edges (AMEs) in two one-dimensional non-Hermitian quasicrystalline models using Avila’s global theory, demonstrating that robust critical states emerge at these AMEs. The study reveals a topological origin for AMEs linked to a real-to-complex spectrum transition, with winding numbers distinguishing localized, critical, and extended states in the complex energy plane.

ABSTRACT

Recent research has made significant progress in understanding localization transitions and mobility edges (MEs) that separate extended and localized states in non-Hermitian (NH) quasicrystals. Here we focus on studying critical states and anomalous MEs, which identify the boundaries between critical and localized states within two distinct NH quasiperiodic models. Specifically, the first model is a quasiperiodic mosaic lattice with both nonreciprocal hopping term and on-site potential. In contrast, the second model features an unbounded quasiperiodic on-site potential and nonreciprocal hopping. Using Avila's global theory, we analytically derive the Lyapunov exponent and exact anomalous MEs. To confirm the emergence of the robust critical states in both models, we conduct a numerical multifractal analysis of the wave functions and spectrum analysis of level spacing. Furthermore, we investigate the transition between real and complex spectra and the topological origins of the anomalous MEs. Our results may shed light on exploring the critical states and anomalous MEs in NH quasiperiodic systems.

Motivation & Objective

• To investigate the existence and characterization of critical states and anomalous mobility edges (AMEs) in non-Hermitian (NH) quasiperiodic systems.
• To determine whether AMEs separating critical and localized states can emerge robustly in NH quasicrystals with nonreciprocal hopping.
• To explore the connection between the real-complex spectrum transition and the topological origin of AMEs in NH quasiperiodic models.
• To confirm the stability of critical states through multifractal and level spacing analysis in finite-size systems.

Proposed method

• Application of Avila’s global theory to analytically compute the Lyapunov exponent and exact AMEs in two NH quasiperiodic models with nonreciprocal hopping and quasiperiodic on-site potentials.
• Use of finite-size multifractal analysis (MFD) to characterize the spatial structure of wave functions and identify critical states.
• Level spacing analysis of eigenvalues to distinguish between localized, critical, and extended states based on spectral statistics.
• Numerical computation of the winding number via the spectral trace formula to identify topological transitions associated with AMEs.
• Diagonalization of Hamiltonians under periodic boundary conditions to study the real-complex spectrum transition as a function of quasiperiodic potential strength.
• Definition of the winding number as a topological invariant: $ w(E_B) = rac{1}{2 au i} ext{Tr} ig[ ext{d} ext{ln det}(H( heta) - E_B) ig] $, tracking spectral evolution under phase variation.

Experimental results

Research questions

• RQ1 Do robust critical states and anomalous mobility edges (AMEs) exist in non-Hermitian quasiperiodic systems with nonreciprocal hopping?
• RQ2 Can the AMEs separating critical and localized states be analytically derived in NH quasicrystals using Avila’s global theory?
• RQ3 Is the transition between real and complex energy spectra correlated with the localization-critical transition and the emergence of AMEs?
• RQ4 What is the topological origin of AMEs in NH quasiperiodic models, and how is it encoded in the winding number of the spectrum?

Key findings

• Exact anomalous mobility edges (AMEs) are analytically derived for two non-Hermitian quasiperiodic models using Avila’s global theory, confirming the existence of critical states at these boundaries.
• The AMEs in model I are stable across all quasiperiodic potential strengths $ heta $, indicating a persistent topological phase with coexisting localized, critical, and extended states.
• For model II, the winding number changes from 0 to 1/2 and back to 0 as $ heta $ increases from -10 to 10, signaling a topological phase transition from trivial localized to topological AME phase.
• The emergence of critical states at AMEs is confirmed by multifractal analysis showing non-ergodic, scale-invariant wave functions with intermediate multifractal dimensions.
• Level spacing statistics show that eigenstates at AMEs exhibit intermediate statistics (neither Poisson nor Wigner-Dyson), confirming their critical nature.
• The real-to-complex spectrum transition co-occurs with the localization-critical transition, and the winding number $ w = 1/2 $ at the AME energy $ E_{ ext{mid}} $, establishing a topological invariant for the AME phase.