Theory of Remaining Exceptional Points from Nongeneric Splitting in Non-Hermitian Systems

In non-Hermitian physics, high-order exceptional points(HOEPs) with eigenvalues and eigenvectors coalesce are known for their enhanced sensitivity to perturbations. Typically, they exhibit eigenvalue splitting that scales as ε^(1/n), which is referred to as the generic response. However, under certain conditions, a nongeneric response of HOEPs occurs where the splitting follows a lower order ε^(1/m) (m<n). A nongeneric response of HOEPs with a lower order splitting lead to the remaining EPs. While the presence of these remaining EPs is acknowledged, a thorough elucidation of their fundamental properties has yet to be achieved. In this work, we demonstrate those unsplit eigenvalue points must constitute remaining EPs in a perturbed n-orders HOEPs system. Combining graph theory and topological analysis, the number and splitting order of the remaining EPs is studied. This framework not only resolves a fundamental challenge in HOEPs but also paves the way for exploiting remaining EPs in applications such as anisotropic sensing and the design of Dirac exceptional points.

💡 Research Summary

This paper presents a comprehensive theoretical framework for understanding “Remaining Exceptional Points (REPs),” which emerge in non-Hermitian systems when a high-order exceptional point (HOEP) undergoes a nongeneric perturbation response. Typically, an nth-order EP exhibits eigenvalue splitting that scales as ε^(1/n) under a generic perturbation. However, under specific conditions, a nongeneric response occurs, characterized by a lower-order splitting scaling as ε^(1/m) where m < n. This incomplete splitting leaves behind a set of unsplit eigenvalues that themselves form exceptional points, termed Remaining EPs. While their existence has been noted, a systematic theory for their properties was lacking.

The core of the work establishes two pivotal methodologies: one for determining the number of REPs and another for predicting their splitting order under further perturbation.

First, the authors mathematically prove that a rank deficiency in the perturbed Hamiltonian (H = H0 + εH1, where H0 hosts an nth-order EP) is a necessary and sufficient condition for the emergence of REPs. The unsplit eigenvalues correspond to zero eigenvalues of H, and their geometric multiplicity defines the order of the remaining degeneracy.

To determine the number of REPs, the study employs graph theory. The Hamiltonian matrix is represented as a weighted directed graph. The key tool is the concept of a Linear Subdigraph (LSD), a subgraph where each vertex has exactly one incoming and one outgoing edge. For a system under multiple perturbations, the number of eigenvalues that participate in splitting (k) is identified by finding an LSD that maximizes the number of vertices (k) while minimizing the number of perturbation edges (l) included. A specific formula involving the product of edge weights and the number of connected components in this optimal LSD is used. If the result is non-zero, then (n - k) gives the number of REPs. This generalizes a simpler rule applicable to single perturbations on a specific off-diagonal of a Jordan block.

Second, and more innovatively, the paper introduces a topological eigenvalue trajectory analysis to determine the splitting order of the REPs themselves when subjected to an additional perturbation. The perturbation strength is treated as a complex parameter, ε = η e^(iθ), with a fixed small η and a phase θ varying from 0 to 2π. As θ evolves, the eigenvalues trace out continuous paths in a 3D space (Re(λ), Im(λ), θ). Projecting these trajectories onto the complex plane (the “top view”) reveals their topological structure. The analysis focuses on identifying closed loops formed either by a single eigenvalue (“self-exchange”) or by a cyclic exchange of multiple eigenvalues (“mutual-exchange”). The critical quantity is the winding number (w) of these closed trajectories around the base point (0,0) in the complex plane. The splitting order is then given by ε^(1/(s*w)), where ’s’ is the number of eigenvalues involved in the exchange loop. A practical method for determining ‘w’ by counting intersections with a ray from the origin is described.

The theory is rigorously demonstrated through several concrete examples of 8x8 Jordan block matrices (labeled A, B, C, D) subjected to carefully chosen perturbation patterns. Using 3D and 2D visualizations of the eigenvalue trajectories, the authors show how REPs can split with diverse orders, such as ε^(1/2), ε^(1/1) (linear), and even ε^(1/4), depending on the perturbation configuration and the resulting winding number. A summary table clearly links the initial state, applied perturbations, observed winding number, exchange type, and final splitting order.

Finally, the authors note that the same analytical principles apply when the secondary perturbation is purely real-valued, making the framework relevant for practical sensing scenarios where perturbations are often real. This work resolves a fundamental puzzle in the behavior of HOEPs and provides a powerful design tool. It opens new avenues for applications, such as engineering systems with specific REP configurations for anisotropic sensing or creating novel spectral features like “Dirac exceptional points” by combining winding numbers and splitting orders.