[Paper Review] Localization and mobility edges in non-Hermitian disorder-free lattices

Rui Qi, Junpeng Cao+1 more|arXiv (Cornell University)|Jun 6, 2023

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This paper investigates non-Hermitian localization and mobility edges in disorder-free 1D lattices under electric fields, revealing that a critical electric field strength governs the transition between non-Hermitian skin effect (NHSE) existence and suppression. For κ=1, NHSE vanishes above a field strength inversely proportional to lattice size; for κ=2, a non-Hermitian mobility edge emerges, and skin states persist even in strong fields, demonstrating tunable control via electric fields.

ABSTRACT

The non-Hermitian skin effect (NHSE) is a significant phenomenon observed in non-Hermitian systems under open boundary conditions, where the extensive bulk eigenstates tend to accumulate at the lattice edges. In this article, we investigate how an electric field affects the localization properties in a non-Hermitian mosaic Stark lattice, exploring the interplay between the Stark localization, mobility edge (ME), and the NHSE induced by nonreciprocity. We analytically obtain the Lyapunov exponent and the phase transition points as well as numerically calculate the density distributions and the spectral winding number. We reveal that in the nonreciprocal Stark lattice with the mosaic periodic parameter $κ=1$, there exists a critical electric field strength that describes the transition of the existence-nonexistence of NHSE and is inversely proportional to the lattice size. This transition is consistent with the real-complex transition and topological transition characterized by spectral winding number under periodic boundary conditions. In the strong fields, the Wannier-Stark ladder is recovered, and the Stark localization is sufficient to suppress the NHSE. When the mosaic period $κ=2$, we show that the system manifests an exact non-Hermitian ME and the skin states are still existing in the strong fields, in contrast to the gigantic field can restrain the NHSE in the $κ=1$ case. Moreover, we further study the expansion dynamics of an initially localized state and dynamically probe the existence of the NHSE and the non-Hermitian ME. These results could help us to control the NHSE and the non-Hermitian ME by using electric fields in the disorder-free systems.

Motivation & Objective

• To understand the interplay between non-Hermitian skin effect (NHSE), Stark localization, and mobility edges in non-Hermitian, disorder-free lattices.
• To investigate how electric fields control the existence and suppression of NHSE in non-reciprocal, mosaic Stark lattices.
• To identify the emergence of exact non-Hermitian mobility edges in systems with mosaic period κ=2.
• To demonstrate dynamical signatures of NHSE and mobility edges through wave packet expansion.

Proposed method

• Analytical calculation of the Lyapunov exponent using Avila’s global theory to study localization transitions.
• Numerical computation of density distributions and spectral winding number to characterize topological and real-complex transitions.
• Use of periodic boundary conditions (PBC) to identify phase transitions via spectral winding number and real-complex transition points.
• Study of wave packet dynamics under open boundary conditions (OBC) to probe NHSE and mobility edge behavior.
• Definition and analysis of mean-square displacement σ²ₜ to detect dynamical spreading and localization.
• Comparison of Hermitian and non-Hermitian cases for κ=1 and κ=2 to isolate non-Hermitian effects.

Experimental results

Research questions

• RQ1 How does an electric field influence the existence and suppression of the non-Hermitian skin effect (NHSE) in a non-reciprocal, disorder-free lattice?
• RQ2 What is the critical electric field strength that separates the presence and absence of NHSE, and how does it scale with system size?
• RQ3 Does a non-Hermitian mobility edge emerge in the non-Hermitian mosaic Stark model with κ=2, and how does it differ from the κ=1 case?
• RQ4 Can dynamical wave packet expansion reveal the presence of NHSE and mobility edges in non-Hermitian systems?
• RQ5 How do the spectral winding number and real-complex transition under PBC correlate with the NHSE transition under OBC?

Key findings

• For the κ=1 case, a critical electric field F_c1 ∝ 1/L governs the transition between NHSE existence and suppression, with weak fields sufficient to suppress NHSE in the thermodynamic limit.
• The NHSE transition in the κ=1 case is consistent with the real-complex transition and topological transition characterized by the spectral winding number under PBC.
• In the κ=2 case, a non-Hermitian mobility edge emerges in the weak-field regime, and localized states appear gradually above a critical weak field.
• Even in strong fields, skin states persist in the κ=2 case, in contrast to the κ=1 case where strong fields suppress NHSE.
• Dynamical wave packet expansion reveals distinct behaviors: asymmetric localization and biased oscillation under non-Hermitian conditions, with σ²ₜ showing saturation in ME phase and later increase due to mobility edge.
• The system exhibits Bloch oscillation in the Hermitian case, while non-Hermitian systems show asymmetric oscillation and persistent boundary localization, confirming NHSE and ME signatures.