Large-chiral-number corner modes in $\mathbb{Z}$-class higher-order topolectrical circuits

Topological corner states are exotic topological boundary states bounded to zero-dimensional geometry even when the dimension of bulk systems is larger than one. So far, all previous realizations of higher-order topological insulators (HOTI) phases are hallmarked by ${\mathbb{Z}}_2$ topological invariants and therefore have only one corner state at each corner. Here, we report an experimental demonstration of $\mathbb{Z}$-class HOTI phases in electrical circuits, characterized by multipole chiral numbers $N$, hosting large-number corner modes at each corner. By measuring the impedance spectra and distributions, we clearly observe that the multipole corner modes in $\mathbb{Z}$-class HOTI phases feature scalable mode areas. Moreover, we find that the local density of states (LDOS) at each corner is maximally distributed at $N$ corner unit cells, differing conspicuously from the ${\mathbb{Z}}_2$-class case, where the LDOS only dominates over one corner unit cell, allowing us to probe the topological number $N$ and reveal the corresponding fractional corner charges. Our results extend the observation of HOTIs from the ${\mathbb{Z}}_2$ class to the $\mathbb{Z}$ class and the coexistence of spatially overlapping large numbers of corner modes that may enable exotic topological devices that require high-degeneracy boundary states.

Figure 1

(a) The ${\mathbb{Z}}_2$-class HOTI phases, characterized by quantized quadrupole moments, featuring one corner mode at each corner. (b) The $\mathbb{Z}$-class HOTI phases, protected by multipole chiral numbers, hosting multipole corner modes at each corner. (c) A diagram of the tight-binding lattice model with nontrival multipole chiral numbers $N$, where $J_1$, $J_2$, $J_{1}x$, $J_{2}x$, $J_{1}y$, and $J_{2}y$ represent different hopping strengths and the dashed lines represent hopping terms with negative sign. (d) The circuit diagram for implementing the long-range-coupling lattice model hosting $\mathbb{Z}$-class HOTI phases. The gray square lists the specific circuit elements and the corresponding lattice parameters. (e) An experimentally realized circuit for a unit cell on a printed circuit board. (f) The topological phase diagram in terms of the MCNs in the parameter space of $J_{x1}$ and $J_{x2}$ for $J_{y1}=J_{x1}$, $J_{y2}=J_{x2}$, and $J_2=-0.5J_1$.

Figure 2

The large-number corner modes that emerge in circuit Laplacian spectra. (a)–(d) Simulated admittance spectra as a function of the driving frequency for different MCNs: (a),(e) $N=4$; (b),(f) $N=3$; (c),(g) $N=2$; (d),(h) $N=1$. At the resonant frequencies (gray solid lines), the spectra respect chiral symmetry (red dashed lines) and host large-number ($4N$) zero modes that correspond to corner modes, as manifested by sorting them out in (e)–(h). (i)–(l) The simulated distributions of the zero-energy corner modes, revealing that the corner modes in $\mathbb{Z}$-class HOTI phases have extended mode areas.

Figure 3

The observation of large-number corner modes featuring scalable mode areas. The (a)–(d) simulated and (e)–(h) measured impedances between the corner, edge, and bulk circuit nodes and the ground via scanning the driving frequency in different $\mathbb{Z}$-class higher-order topological circuits: (a),(e) $N=1$; (b),(f) $N=2$; (c),(g) $N=3$; (d),(h) $N=4$. Both results clearly verify that the corner-mode resonances occur at the resonant frequencies (black dashed lines). (i)–(l) The measured impedance distributions for the corner modes, demonstrating that the mode areas in the $\mathbb{Z}$-class HOTI phases scale up with the MCNs.

Figure 4

Spatial maps of simulated and measured LDOSs for $N=1$ and $N=4$ integrating over (a),(b),(e),(f) the lower band and (c),(d),(g),(h) an in-gap spectral range covering all corner modes: (a),(e) simulated; (b),(f) measured; (c) four corner modes, simulated; (d) four corner modes, measured; (g) 16 corner modes, simulated; (h) 16 corner modes, measured. Each circuit node is represented as a circle with a radius proportional to the LDOS. Each number gives the LDOS in the corresponding unit cell. The topological invariants $N$ and fractional corner charges are, respectively, detected and manifested by the LDOSs in the shaded corner unit cells.