Bulk and edge-state arcs in non-Hermitian coupled-resonator arrays

We describe the formation of bulk and edge arcs in the dispersion relation of two-dimensional coupled-resonator arrays that are topologically trivial in the Hermitian limit. Each resonator provides two asymmetrically coupled internal modes, as realized in noncircular open geometries, which enables the system to exhibit non-Hermitian physics. Neighboring resonators are coupled chirally to induce non-Hermitian symmetries. The bulk dispersion displays Fermi arcs connecting spectral singularities known as exceptional points and can be tuned to display purely real and imaginary branches. At an interface between resonators of different shape, one-dimensional edge states form that spectrally align along complex arcs connecting different parts of the bulk bands. We also describe conditions under which the edge-state arcs are freestanding. These features can be controlled via anisotropy in the resonator couplings.

Figure 1

Bulk Fermi arcs in a two-dimensional array of evanescently coupled non-Hermitian resonators. (a) Each resonator supports a clockwise (CW) and a counterclockwise (CCW) internal mode, which are coupled by asymmetric backscattering amplitudes $A$ and $B$, as obtained, e.g., from a small nonspherical deformation of open dielectric resonators. The resonators are placed on a square lattice and are coupled evanescently with coupling coefficients $W_x$ and $W_y$, which convert CW waves into CCW waves. This coupling configuration introduces a chiral symmetry into this non-Hermitian system. (b)–(d) Real part $\mathrm{Re}\mathrm{\Omega }$ of the bulk dispersion for $B=-2.5+0.2i, W_x=1.0+0.1i, W_y=1.0+0.5i$, and the three values $A=1.5+0.1i$ (b), $A=1.5+0.2i$ (c), and $A=1.5+0.3i$ (d). In each case, the dispersion consists of two sheets ${\mathrm{\Omega }}_{+}$ (yellow, top surface) and ${\mathrm{\Omega }}_{-}=-{\mathrm{\Omega }}_{+}$ (blue, bottom surface) that are related by the chiral symmetry. White lines indicate Fermi arcs and lines with $\mathrm{Re}\mathrm{\Omega }=0$, corresponding to intersections of the two sheets. The arcs terminate at exceptional points (EPs), which are the non-Hermitian counterparts of Weyl points in topological insulators. In (b), four EPs are connected by two arcs. In (d), the EPs are reconnected by two arcs with a different topology, while a closed Fermi line is also present. (c) The reconnection point between these two scenarios, which is mediated by two smaller closed Fermi lines.

Figure 2

(a) Real and (b) imaginary parts of the bulk dispersion for $A=1.0, B=-1.0$, and $W_x=W_y=1$, representing the $\mathcal{PT}$-symmetric case (symmetry class BDI) where the band structure displays purely real and imaginary branches, and the exceptional points degenerate into lines.

Figure 3

Edge-state arcs in an array with an interface joining resonator arrays with opposite backscattering. (a) Horizontal slice through the array, where the dotted line indicates the interface between resonators with backscattering amplitudes $A$ and $B$ as in Fig. 1 (blue resonators at the left) and resonators where the values of these backscattering amplitudes are interchanged (green resonators at the right). (b) Density plot of the intensity of a representative edge state in a finite square array of $40\times 40$ resonators, with $A=-B=W_x=W_y$. (c) Quasi-one-dimensional band structure in the infinite version of this array, where $k_y$ is a good quantum number. In this representation, the bulk bands form sheets, which here lie in the real and imaginary plane as all parameters are real ($\mathcal{PT}$-symmetric symmetry class BDI; see Fig. 2). Thick black curves are the edge-state arcs, which connect the different sheets.

Figure 4

(a) Traces of the edge-state arcs in the real section of their effective parameter space $(\mathcal{A},\mathcal{B})$ [defined in Eq. (11)]. Traces are horizontal lines of length $4W_y/W_x$, which are centered at $\mathcal{A}=(A+B)/2W_x, \mathcal{B}=(A-B)/2W_x$. The solid and dashed curves denote the termination conditions at the real and imaginary branches of the bulk bands, where the edge states (green region) turn into extended scattering states (red region) or into nonnormalizable, unphysical states (blue region). The three representative traces correspond to the quasi-one-dimensional band structures shown in (b)–(d), which display edge-state arcs (thick green curves), their scattering predecessors (red curves in the real bands), and unphysical states [blue curves connecting the imaginary bands in (b)]. (b) $A/W_x=0.55, B/W_x=-0.55, W_y/W_x=1.175$, for which the trace crosses both termination lines and the arcs connect the real and imaginary branches of the bulk bands. (c) $A/W_x=-0.9, B/W_x=0.9, W_y/W_x=1$, for which the trace only reaches the real termination line so that the arcs loop back to the real branches. (d), $A/W_x=-2.0, B/W_x=2.0, W_y/W_x=1$, for which the trace remains confined in the edge-state region so that the arcs are freestanding.

Figure 5

(a) Edge-state arcs for complex backscattering amplitudes $A/W_x=0.55+0.02i, B/W_x=-0.55+0.02i$, and $W_y=1.175$ [close to the real values in Fig. 4]. All arcs still terminate on the bulk bands, which now are no longer real or imaginary. (b, d) Propagation factors $|{\lambda }_l|, 1/|{\lambda }_l|$ of potential edge states as determined by Eq. (16). In (b), $A=-B=0.55/W_x, W_y/W_x=1.175$, corresponding to the real values of Fig. 4. In (c) and (d), the parameters take the complex values given above. For complex parameters the region of scattering states is replaced by regions of physical and unphysical states. Furthermore, the termination points of different arcs now appear at separate values of $k_y$, as shown in detail in (d) and (e), which zoom in on the termination region at the formerly purely imaginary and real branches of the bulk dispersion, respectively.