Exceptional concentric rings in a non-Hermitian bilayer photonic system

We establish the bilayer non-Hermitian theory with interlayer coupling between two graphenelike lattices described by non-Hermitian Hamiltonians. Two exceptional rings (ERs) and an exceptional concentric ring (ECR) in this system are discovered. A topological classification has been identified for such non-Hermitian Hamiltonians via their global topological invariants. We find that two ERs and the ECR correspond to the global ${\mathbb{Z}}_3$ topological invariant $V=+1$ and $V=-1$, respectively, whose topological properties are determined by the four-level bands and go beyond the topological properties of a single ER. It is shown that in contrast to the Hermitian case, the topological surface arcs with gain and loss are both present and redistributed in the two ERs and the ECR systems. We further propose an experimental design with bilayer photonic crystals (PCs) to realize the ECR. Our work can potentially pave the way for unveiling the unusual topological properties in such stacked two-dimensional materials and trigger further theoretical and experimental investigations.

Figure 1

(a) Schematic of a bilayer coupling system with particle gain or loss at four sites (four sites constitute a unit cell). (b) and (c) Type I bands with two ERs in red lines (${\gamma }_1={\gamma }_3=0.025, {\gamma }_2={\gamma }_4=-0.025$). (d) and (e) Type II bands with the ECR in green lines (${\gamma }_1=-{\gamma }_3=0.025, {\gamma }_2=-{\gamma }_4=-0.025$). (f) and (g) Type III bands with both two ERs in red lines and the ECR in green lines (${\gamma }_1=0.025, {\gamma }_2=-0.025, {\gamma }_3=0.01, {\gamma }_4=-0.01$).

Figure 2

(a) Schematic of a closed trajectory enclosing the single EP (red point). (b) Real parts of the spectra (type I) with EPs (red points) and two ERs (red lines) in $\delta k_y=0$ plane. (c) Real parts of the spectra (type II) with EPs (red points) and ECR (green lines) in $\delta k_y=0$ plane. (d) Real parts of the spectra (type III) with two ERs (red lines) and the ECR (green lines) in $\delta k_y=0$ plane. The value at the red point represents the quantized Berry phase $C$.

Figure 3

Phase diagram of the non-Hermitian system with periodic boundaries. The red line (yellow line) represents type I (type II) phases characterized by the $V=+1$ ($V=-1$), while green regions represent type III phases characterized by the $V=-1$. For simplicity, we assume that the benchmark for gain and loss is zero, ${\gamma }_1=-{\gamma }_2$ and ${\gamma }_3=-{\gamma }_4$.

Figure 4

(a) Schematic of a bilayer coupling system with particle gain or loss at four sites (${\kappa }_1,{\kappa }_2,{\kappa }_3,{\kappa }_4$). (b) and (c) Real and imaginary parts of the type II bands with the ECR (green lines). (d) and (e) Real and imaginary parts of the eigenvalues, when the open boundary condition is imposed along the $x$ direction and periodic boundaries in the $y$ direction. The red lines are the surface arcs with gain, while the blue lines are the surface arcs with loss. (f) Surface states for A and B shown in (d).