Exceptional nodal rings emerging in spinful Rice-Mele chains

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.

💡 Research Summary

In this paper the authors investigate how non‑Hermitian dissipation modifies the nodal‑ring structure of a three‑dimensional topological semimetal and they establish a concrete connection to a one‑dimensional spinful Rice‑Mele (RM) ladder. Starting from a Hermitian nodal‑ring Hamiltonian H(k)=k_x s_x + k_y τ_y s_y + k_z s_z + m τ_x s_x, they promote the mass term to a complex quantity m=α+iβ. The system retains chiral symmetry ΛHΛ⁻¹=−H, but the energy eigenvalues become fully complex: ε_{μ,ν}=μ√(k_z²+q k_x²+k_y²+ν m²). When β≠0 the zero‑energy condition no longer yields a single ring; instead it produces two concentric circles in the k_x–k_y plane displaced along k_z by ±β. These are identified as two exceptional rings (EP rings) where eigenvectors coalesce.

To characterize the topology of the EP rings the authors introduce the phases ϕ_{±}=arg(ε²_{μ,±}) of the squared eigenvalues and define a vector field P=∇k(ϕ+ + ϕ_−). This field describes a free vortex flow in momentum space; the EP rings act as vortex filaments with opposite circulations. The winding number w=∮_L P·dk/(2π) computed over any closed loop L is quantized to 0, ±1, depending on whether the loop encloses none, one, or both EP rings. As β changes sign the two rings exchange positions, causing a topological phase transition signaled by a change in w.

Realizing the 3D model experimentally would require a complicated lattice. Therefore the authors propose a 1D spinful Rice‑Mele ladder with Hamiltonian
H = Σ_j