[Paper Review] Exceptional points of any order in a generalized Hatano-Nelson model

Julius T. Gohsrich, Jacob Fauman+1 more|arXiv (Cornell University)|Mar 18, 2024

Nonlinear Waves and Solitons4 citations

TL;DR

This paper introduces a generalized Hatano-Nelson model with long-range hoppings that enables the realization of exceptional points (EPs) of arbitrary order, independent of system size. By exploiting generalized chiral symmetry and a $(l+r)$-partite lattice structure, the authors demonstrate that EPs are robust against generic hopping perturbations and specific on-site disorder, with eigenstates localized on specific sublattices and exhibiting the non-Hermitian skin effect.

ABSTRACT

Exceptional points (EPs) are truly non-Hermitian (NH) degeneracies where matrices become defective. The order of such an EP is given by the number of coalescing eigenvectors. On the one hand, most work focuses on studying $N$th-order EPs in $(N\leq4)$-dimensional NH Bloch Hamiltonians. On the other hand, some works have remarked on the existence of EPs of orders scaling with systems size in models exhibiting the NH skin effect. In this work, we introduce a new type of EP and provide a recipe on how to realize EPs of arbitrary order not scaling with system size. We introduce a generalized version of the paradigmatic Hatano-Nelson model with longer-range hoppings. The EPs existing in this system show remarkable physical features: Their associated eigenstates have support on a subset of sites and exhibit the NH skin effect, which can be tuned to localize on the opposite end of the chain compared to all remaining states. Furthermore, the EPs are robust against generic perturbations in the hopping strengths as well as against a specific form of on-site disorder.

Motivation & Objective

• To address the lack of systematic methods for generating non-Hermitian exceptional points (EPs) of arbitrary order in finite-dimensional systems.
• To explore the emergence of EPs whose order does not scale with system size, contrasting with typical EPs in unidirectional models.
• To establish the role of generalized chiral symmetry in stabilizing EPs of fixed order regardless of system size.
• To investigate the robustness of EPs against hopping perturbations and on-site disorder on specific sublattices.
• To demonstrate that eigenstates associated with EPs are localized on a subset of sites and exhibit the non-Hermitian skin effect.

Proposed method

• Generalizing the standard Hatano-Nelson model to include $l$-site left and $r$-site right hoppings, with $\gcd(l,r) = 1$, to form a $(l+r)$-partite lattice structure.
• Using periodic boundary conditions (PBCs) to analyze spectral winding and rotational symmetry, which pin EPs to the center of rotation in the complex energy plane.
• Applying open boundary conditions (OBCs) to observe the non-Hermitian skin effect and localization of eigenstates on specific sublattices.
• Constructing Jordan chains for EPs via generalized eigenvectors and analyzing their stability under perturbations.
• Introducing generic hopping perturbations and on-site disorder on specific sublattices to test robustness of EPs.
• Mapping the spectrum and eigenvectors of the full Hamiltonian $H_{lr}$ to those of its parent Hamiltonian $\mathcal{H}^{l+r=n}$ to reveal spectral topology.

Experimental results

Research questions

• RQ1 Can exceptional points of arbitrary order be realized in a non-Hermitian system without their order scaling with system size?
• RQ2 How does generalized chiral symmetry protect EPs of fixed order in a finite-sized system?
• RQ3 What is the spatial localization pattern of eigenstates associated with high-order EPs in the generalized Hatano-Nelson model?
• RQ4 To what extent are EPs robust against generic hopping perturbations and on-site disorder on specific sublattices?
• RQ5 How does the non-Hermitian skin effect coexist with high-order EPs in a system with long-range hoppings?

Key findings

• The generalized Hatano-Nelson model supports exceptional points of arbitrary order $m$, where $m$ is independent of the system size $N$, achieved by tuning $l$ and $r$ with $\gcd(l,r) = 1$.
• EPs appear at the center of rotational symmetry in the complex energy plane due to generalized chiral symmetry, which stabilizes their existence for specific system sizes.
• Eigenstates associated with EPs are localized on a specific subset of sites corresponding to particular sublattices, and this localization is robust against changes in hopping strengths.
• The non-Hermitian skin effect persists under generic hopping perturbations, with eigenstates of the EP and the rest of the spectrum localized on opposite ends of the chain when hopping asymmetry is tuned.
• EPs remain stable or are only reduced in order under on-site disorder applied to sublattices hosting generalized eigenvectors, indicating topological protection via spatial structure.
• The model can be experimentally realized in photonic ring systems, topoelectric circuits, and synthetic frequency dimensions, enabling direct observation of robust high-order EPs.