Universal Topology of Exceptional Points in Nonlinear Non-Hermitian Systems

TL;DR

The paper addresses how nonlinearities affect exceptional points (EPs) in non-Hermitian systems, showing that a second-order linear EP reorganizes into a universal elliptic umbilic topology in the nonlinear parameter space. Using catastrophe theory and a Lyapunov potential, the authors map eigenvector degeneracies to degenerate critical points, revealing a cone-like EP structure with deltoid cross sections that persists across a broad class of nonlinearities. They demonstrate this universality in a representative $2\times2$ model with Kerr-like nonlinearity and extend it to general nonlinearities, asymmetric matrices, complex couplings, and larger matrix dimensions, arguing that the elliptic umbilic topology is robust and potentially extends to higher-order EPs. The findings provide a canonical blueprint for predicting EP locations under nonlinear perturbations, enabling rigorous bounds on sensitivity enhancements and guiding experimental designs across photonics, atomic, and condensed-matter platforms.

Abstract

Exceptional points (EPs) are non-Hermitian degeneracies where eigenvalues and eigenvectors coalesce, giving rise to unusual physical effects across scientific disciplines. The concept of EPs has recently been extended to nonlinear physical systems. We theoretically demonstrate a universal topology in the nonlinear parameter space for a large class of physical systems that support 2nd order EPs in the linear regime. Knowledge of this topology (called elliptic umbilic singularity in bifurcation theory) deepens our understanding of 2nd order linear EPs, which here emerge as coalescence of 4 nonlinear eigenvectors. This helps guide future experimental discovery of nonlinear EPs and their classification, establish rigorous bounds of sensitivity enhancement of EPs in nonlinear systems, and helps envision and optimize technological applications of nonlinear EPs. Our theoretical approach is general and can be extended to nonlinear perturbations of 3rd and higher-order EPs.

Figures (4)

• Figure 1: Non-Hermitian two-state system.(a) Sketch of a non-Hermitian and nonlinear two-state (or two-mode) system, with linear coupling $\beta$, energy splitting $2\delta=\delta_x-\delta_y$, and loss/gain difference $2\gamma$. (b) Sketch of hypothetical location of exceptional points as a function of physical parameters. Without a nonlinearity, $\alpha=0$, isolated linear EPs are at $( \beta, \delta) = ( \pm \gamma, 0)$; in the nonlinear regime, $\alpha \neq 0$, EPs are implied on the colored surfaces. To illustrate the problem our study is solving, the surfaces we show here are one of infinitely many hypothetically possible (but generally wrong) EP neighborhoods. The correct geometric shape is shown in Fig. \ref{['fig:3D-elliptic-umbilic']}. Panels (c) and (d) show the eigenvalues in the linear case ($\alpha = 0$) with two isolated EPs at $(\beta,\delta) = (\pm \gamma, 0)$.
• Figure 2: Elliptic umbilic singularity set.Surface of EP locations as a function of physical parameters (singularity set) explained in Fig. \ref{['fig:sketch-incorrect-neighborhood']}. In the weakly nonlinear regime, $\alpha \ll \gamma$, the shape approaches that of an elliptic umbilic bifurcation ('catastrophe'). The shape of each singularity set consists of two infinite three--cusped conical surfaces, the apices of which meet at the two linear EPs $(\alpha, \beta,\delta) = (0, \pm \gamma, 0)$. At large $\alpha$, the three--cusped shape remains but is deformed guaranteeing that the two cone-like surfaces originating at $\beta= \pm 1$ do not overlap; a see Fig. \ref{['fig:elliptic-umbilic-cross-section']}.
• Figure 3: Cross section of singularity set.Cross section of EP surface for fixed nonlinearity $\alpha$; axis scaled with scale factor $s=2( \sqrt{1+ \alpha^2 / \gamma^2} - 1 )$. Here, $\beta' = \beta - \gamma$. Small $\alpha \ll \gamma$ (vicinity of linear EP, Eq. \ref{['EU-1.equ']}): deltoid or three-cusped hypocycloid broecker-lander.1975poston-stewart.1978saunders.80 of the elliptic umbilic surface with exact scaling $s= \alpha^2 / \gamma^2$. Large $\alpha$ (farther away from linear EP, Eq. \ref{['Lyapunov.equ']}): exact scaling is lost, but three--cusped shape remains. The half-infinite line where PT symmetry is unbroken is indicated by the blue line.
• Figure 4: Selected complex eigenvalue traces.Complex eigenvalues (real and imaginary parts) of $H$ in Eq \ref{['eigen-1.equ']} at fixed, nonzero $\alpha/\gamma =1$ across lines in the $(\beta',\delta)$ plane shown as insets. Regardless of the lines' directions, we have two and four eigenvalues outside and inside the cone, respectively, in agreement with the number of values of critical points of the Lyapunov potential (see Supplementary Figure S3 for examples of the potential landscape). At the smooth parts of the deltoid boundary (fold lines), two states are created/annihilated, and two states are unaffected by the boundary (a,b,e,f). Crossing the cusp, (c,d), three states coalesce at a 3rd-order nonlinear EP, denoted as EP3, and one state is unaffected.