Exceptional nodal rings emerging in spinful Rice-Mele chains

TL;DR

The paper tackles the topology of non-Hermitian nodal-ring semimetals by showing that dissipation splits a 3D nodal ring into two exceptional rings, interpretable as vortex filaments in momentum space via a spectrum-derived field. It introduces a 1D spinful Rice-Mele chain with complex spin-orbit coupling that reproduces the same EP topology in parameter space, offering a practical route to measure the winding invariant using only single-energy levels. The key contributions are the vortex-field characterization of exceptional rings, a concrete low-dimensional model linking to the 3D case through a parameter mapping, and demonstrated robustness to perturbations. This work provides a concrete framework for probing non-Hermitian topological features in both high- and low-dimensional systems with potential experimental realizations. The findings advance understanding of how dissipation reshapes topological structures and how to access their invariants in simpler settings.

Abstract

The Weyl exceptional nodal lines usually occur in 3D topological semimetals, but also emerge in the parameter space of 1D systems. In this work, we study the impact of dissipation on the nodal ring in a 3D topological semimetal. We find that the energy spectrum becomes fully complex in the presence of dissipation, and the original nodal ring is split into two exceptional rings. We introduce a vortex field in the momentum space, which is generated from the spectrum, to characterize the topology of the exceptional rings. This provides a clear physical picture of the topological structure. The two exceptional rings act as two vortex filaments of a free vortex flow with opposite circulations. In this context, the 3D topological semimetal is the boundary separating two quantum phases identified by two configurations of exceptional rings. We also propose a 1D model that has the same topological feature in the parameter space. It provides a simple way to measure the topological invariant in a low-dimensional system. Numerical simulations indicate that the topological invariant is robust under the random perturbations of the system parameters.

Figures (5)

• Figure 1: Schematic of the exceptional rings described by Eq. (\ref{['semimetalER']}) in three-dimensional momentum space, together with the gradient field of the eignenenrgy argument defined in the Eq. (\ref{['field']}) and the analogous field of the RM modal under the parameter transformation of Eq. (\ref{['Ptransformation']}). Panels (a1), (b1) and (c1) correspond to $\alpha=0.6$ and $\beta =0.6$, $0$, and $-0.6$, respectively. Panels (a2), (b2) and (c2) show the corresponding field lines of $\mathbf{P}$ (black) and the approximate field lines of the RM model (blue) near the exceptional points in the $k_{y}-k_{z}$ plane; arrows indicate the directions. For the semimetal model, the field $\mathbf{P}$ forms a free vortex flow whose circulation follows the right-hand screw rule, marked by green or red circles.
• Figure 2: Exceptional rings viewed in the $k_{x}-k_{y}$ plane, given by Eq. (\ref{['RMepring']}) with $\alpha= \beta=0.6$. Each loop represent a pair of exceptional rings corresponding $\sin k_{\text{c}z}=\pm \beta$.
• Figure 3: Contour plots of $\phi = \phi_{+}+\phi_{-}$ from Eq. (\ref{['argument']}) ($\Phi = \Phi_{+}+\Phi_{-}$ from Eq. (\ref{['Phi']})) on the $k_{y}-k_{z}$ ($V-k$) plane with $k_{x}=0$ ($\delta=0$) and $\alpha=\beta=0.1$. Green dots mark the exceptional points, symmetric about both axes. O (O$^{\prime }$) denotes the origin. Black rectangles in (a1) and (a2) encircle single exceptional points; (c1) and (c2) show the corresponding values along the arrows, yielding a winding number of 1. Panels (b1) and (b2) magnify the regions within the green-dashed frames of (a1) and (a2).
• Figure 4: Plot of $\Phi$ extracted from the real space energy spectrum of a ring model along the black rectangle shown in Fig. \ref{['fig3']}(a2) with the perturbation described in Eq. (\ref{['perturbation']}); here $k$ can be interpreted as quasi-momentum. Other parameters are $a=0.1$, $\alpha=\beta=0.1$, and $N=200$.
• Figure A 1: Schematic of the cylindrical coordinate system in momentum space. Here $\rho =\sqrt{k_{x}^{2}+k_{y}^{2}}$ is the radial distance, $\varphi$ is the azimuthal angle, and $\left( \mathbf{e}_{\rho },\mathbf{e} _{\varphi },\mathbf{e}_{z}\right)$ are the orthonormal basis vectors.