[Paper Review] Symmetry-protected topological exceptional chains in non-Hermitian crystals

Ruo-Yang Zhang, Xiaohan Cui+3 more|arXiv (Cornell University)|Apr 17, 2022

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This paper introduces symmetry-protected topological exceptional chains (ECs) in non-Hermitian crystals by proving a source-free principle for exceptional lines (ELs) via a generalized Fermion doubling theorem. It identifies three distinct symmetry mechanisms—mirror, mirror-adjoint, and $C_2\mathcal{T}$—that stabilize robust ECs, and demonstrates their realization in photonic crystals through numerical simulations.

ABSTRACT

In non-Hermitian systems, the defective band degeneracies, so-called exceptional points (EPs), can form robust exceptional lines (ELs) in 3D momentum space in the absence of any symmetries. Here, we show that a natural orientation can be assigned to every EL according to the eigenenergy braiding around it, and prove the source-free principle of ELs as a corollary of the generalized Fermion doubling theorem for EPs on an arbitrary closed oriented surface, which indicates that if several ELs flow into a junction, the same number of outflow ELs from the junction must exist. Based on this principle, we discover three different mechanisms that can stabilize the junction of ELs and therefore guarantee the formation of various types of exceptional chains (ECs) under the protection of mirror, mirror-adjoint, or ${C}_2\mathcal{T}$ symmetries. Furthermore, we analyze the thresholdless perturbations to a Hermitian nodal line and map out all possible EC configurations that can be evolved. By strategically designing the structure and materials, we further exhibit that these exotic ECs can be readily observed in non-Hermitian photonic crystals. Our results directly manifest the combined effect of spatial symmetry and topology on the non-Hermitian singularities and pave the way for manipulating the morphology of ELs in non-Hermitian crystalline systems.

Motivation & Objective

• To understand the stability and formation mechanisms of exceptional chains (ECs) in 3D non-Hermitian systems, which are not generically stable without symmetry protection.
• To establish a topological principle governing the orientation and connectivity of exceptional lines (ELs) in momentum space, ensuring source-free configurations.
• To classify and realize distinct types of ECs protected by specific non-Hermitian spatial and spatiotemporal symmetries, such as mirror-adjoint and $C_2\mathcal{T}$.
• To map the evolution of ECs from Hermitian nodal rings under thresholdless non-Hermitian perturbations, revealing topological phase transitions.
• To demonstrate the experimental feasibility of ECs in non-Hermitian photonic crystals through designed structures and full-wave simulations.

Proposed method

• Proves a generalized Fermion doubling theorem for exceptional points (EPs) on arbitrary closed oriented surfaces, leading to the source-free principle for directed exceptional lines (ELs).
• Assigns a topological orientation to each EL based on eigenvalue braiding around it, using the discriminant number and monodromy of energy bands.
• Identifies three distinct symmetry mechanisms—mirror, mirror-adjoint, and $C_2\mathcal{T}$—that enforce balance between inflowing and outflowing ELs at chain junctions.
• Introduces four types of Berry phases ($\theta^{LL}, \theta^{RR}, \theta^{LR}, \theta^{RL}$) along loops in momentum space, with quantization protected by double-antisymmetry (DAS) point groups.
• Constructs a perturbation roadmap from Hermitian nodal rings to ECs by applying thresholdless non-Hermitian terms, preserving topological features.
• Designs and simulates non-Hermitian photonic crystals to realize three distinct EC configurations: linked orthogonal networks, planar ECs with four chain points, and double-earring ECs.

Experimental results

Research questions

• RQ1 How can exceptional lines (ELs) in 3D non-Hermitian systems be topologically oriented, and what topological principle governs their connectivity?
• RQ2 What symmetry mechanisms stabilize the junction of multiple ELs into robust exceptional chains (ECs) in the absence of generic EL stability?
• RQ3 How do non-Hermitian spatiotemporal symmetries—such as mirror-adjoint and $C_2\mathcal{T}$—protect quantized Berry phases and distinct EC morphologies?
• RQ4 What is the topological evolution pathway from a Hermitian nodal ring to various EC configurations under non-Hermitian perturbations?
• RQ5 Can symmetry-protected ECs be experimentally realized in photonic crystals, and what structural designs enable their observation?

Key findings

• The source-free principle for ELs is rigorously proven via a generalized Fermion doubling theorem for EPs on closed oriented surfaces, ensuring that the number of incoming and outgoing ELs at any junction must balance.
• A natural orientation is assigned to each EL based on eigenvalue braiding, with the discriminant number determining the directionality of the EL in momentum space.
• Three distinct symmetry mechanisms—mirror, mirror-adjoint, and $C_2\mathcal{T}$—are identified as sufficient to stabilize different types of ECs, including topologically nontrivial configurations like eigenenergy Hopf links.
• Quantized Berry phases are protected by DAS point groups, with $\theta^{LR} = \theta^{RL}^*$ and $\theta^{LL} + \theta^{RR} = 2\theta^{LR}$, ensuring topological robustness.
• Three distinct EC configurations are numerically realized in non-Hermitian photonic crystals: a pair of linked orthogonal EC networks under three mirror-adjoint symmetries, a planar EC with four non-defected chain points, and a double-earring EC protected by mirror-adjoint and $C_2\mathcal{T}$ symmetries.
• The transition from a Hermitian nodal ring to ECs is shown to be continuous under thresholdless non-Hermitian perturbations, with all possible EC configurations mapped out systematically.