Higher-order topological bound states in the continuum in a topoelectrical lattice with long-range coupling

TL;DR

This work shows that corner-bound states in the continuum, a hallmark of higher-order topology, can persist in a 2D SSH topoelectrical lattice even when long-range, symmetry-preserving couplings are introduced. By mapping LC circuits to a tight-binding framework at a characteristic frequency $\omega_0$, the authors demonstrate that symmetry protections from chiral symmetry and $C_{4v}$ symmetry pin zero-admittance corner modes to zero energy, existing as bound states within the bulk continuum. The introduction of next-nearest-neighbor couplings yields new off-site corner-like states (Type-II and Type-III) and edge precursors while leaving conventional corner BICs intact under off-diagonal disorder; diagonal disorder can lift zero-energy pinning, highlighting the role of symmetry. Overall, the study confirms the robustness of higher-order topological corner states in circuitRealizations and provides impedance-spectroscopy fingerprints for experimental verification, offering a controllable platform for exploring BICs and HOTIs in classical systems.

Abstract

Linear electric circuits composed of inductors and capacitors can serve as analogues of tight-binding models that describe the electronic band structure of materials. This mapping provides a versatile approach for exploring topological phenomena within engineered electrical lattices. In this work, the two-dimensional Su-Schrieffer-Heeger model is examined through electric circuit analogues to study the interplay between higher-order topology, bound states in the continuum, and disorder. Building upon this model, the effect of introducing next-nearest-neighbour interactions that preserve chiral and spatial symmetries of the system is analyzed. The results reveal that even without Hamiltonian separability, corner-localized bound states in the continuum remain protected by symmetry in the long-range coupled lattice. This robustness highlights the potential of circuit-based platforms for probing advanced topological phenomena in a highly controllable setting.

Figures (10)

• Figure 1: Periodic 2D SSH circuit.(a) Schematic of the unit cell of the 2D SSH circuit. Each unit cell hosts four sites, labeled 1-4. $L_1$ and $L_2$ represent the intracell and intercell couplings, respectively, corresponding to the NN hopping. The ratio $\lambda = L_1/L_2$ defines the condition for a nontrivial topological phase, occurring when $\lambda <1$. Each node is grounded by an on-site capacitor $C$. (b) Bulk band structure with open boundary conditions for $\lambda =0.1$ and $C=100\;\hbox{pF}$. (c, d) Bulk admittance bands along a high-symmetry path in the Brillouin zone for the circuit with periodic boundaries for $\lambda=0.1$ (c) and $\lambda=0.8$ (d).
• Figure 2: Layout of the finite-sized 2D SSH circuit. Inductors $L_1$ (green) and $L_2$ (brown) represent intracell and intercell couplings, respectively. All nodes (bulk, edge, and corner) are grounded via capacitors $C$ (yellow), which are omitted from the schematic for clarity except within the highlighted unit cell. The circuit contains $N^2$ nodes, with labels shown in orange.
• Figure 3: Finite-sized 2D SSH circuit. Left panel: Numerically computed resonant admittance eigenvalues for the 2D SSH circuit comprising $196$ nodes ($14 \times 14$ node configuration), in the topological phase with $\lambda = 0.1$. In-gap edge-localized states are marked in green, while the four corner-localized modes in the continuum are highlighted in red (see also Fig. S1-S2, and Table S1, Supplementary Material). Insets show enlarged views of these bound states. Right panel: Normalized spatial distribution of the amplitude of a corner-localized eigenmode, $|\psi(\omega)|^2$. The colourmap quantifies the degree of localization across the circuit layout. The ground capacitance is set to $C=1\;\hbox{F}$ for numerical convenience, scaling the admittance spectrum to the range $[-1,\:1]$ without affecting the physical interpretation.
• Figure 4: Amplitude of the corner modes as a function of the coupling ratio $\lambda$ for $\lambda\ll1$. Panel (a) shows modes supported on sublattice $A$, and panel (b) shows modes on sublattice $B$ for a $4\times4$ lattice. The amplitudes scale as $\lambda, \: \lambda^2$, consistent with exponential localization. The area of each circle representing a node is proportional to its amplitude for $\lambda=0.1$ , illustrating the strong localization at the corners. (Supplementary Material, section 'Analytical derivation of corner modes')
• Figure 5: Normalized two-point impedance spectra. Calculated impedances for three representative configurations: Bulk, between nodes $(N/2, N/2)$ and $(N/2, N/2+1)$; Corner, between nodes $(1,1)$ and $(N,N)$; and, Edge, between nodes $(1,N/2)$ and $(N/2, N/2)$. Computations used $L_1=10\: \mu \hbox{H}$, $L_2=100\: \mu \hbox{H}$, $C=10\:\hbox{nF}$, and $N=14$.