In this work, we experimentally implemented square-root higher-order topological insulators (HOTIs) in topological electric circuits with decorated Su–Schrieffer–Heeger (SSH) lattice. Due to the square-root operation, the corner states of the square-root HOTIs typically appear on the finite energy. Through theory and experiments, the corner states in both the parent and square-root HOTIs models, as well as their spectrum connection, were demonstrated. Our work not only experimentally substantiates the existence of square-root higher-order topology in decorated SSH lattice model, from the perspective of electric circuits, but also extends the possibilities for observing various topological phases in acoustic and mechanical systems.

Topological circuits become a powerful platform for realizing and observing topological states with significant flexibility and tunability. In this work, we experimentally realized three-dimensional pyrochlore topological circuits with a third-order topological corner state. The generalized chiral symmetry enables the corner state to exist in the band gap or the continuum, and the corner state in our circuits is localized on a single corner due to passing through the Wannier center. Our work provides a paradigm for exploring various topological phenomena in higher-dimensional systems by using topological circuits, which may have potential applications in the fields of future communication and signal processing.

The valley degree of freedom, like the spin degree of freedom in spintronics, is regarded as a new information carrier, promoting the emerging valley photonics. Although there exist topologically protected valley edge states which are immune to optical backscattering caused by defects and sharp edges at the inverse valley Hall phase interfaces composed of ordinary optical dielectric materials, the dispersion and the frequency range of the edge states cannot be tuned once the geometrical parameters of the materials are determined. In this paper, we propose a chirped valley graphene plasmonic metamaterial waveguide composed of the valley graphene plasmonic metamaterials (VGPMs) with regularly varying chemical potentials while keeping the geometrical parameters constant. Due to the excellent tunability of graphene, the proposed waveguide supports group velocity modulation and zero group velocity of the edge states, where the light field of different frequencies focuses at different specific locations. The proposed structures may find significant applications in the fields of slow light, micro–nano-optics, topological plasmonics, and on-chip light manipulation.

We propose graphene plasmonic crystals (GPCs) with a Kagome lattice, and investigate the properties of the flat band (FB) in the plasmonic system. By modulating the arrangement of the chemical potentials, a FB is obtained. Furthermore, the authenticity of the FB is confirmed by comparing the band structures and the eigen field distributions obtained from using the tight-binding modeled Hamiltonian with numerical calculations. The proposed Kagome-type GPCs could be of great significance for the study of novel effects in strong interaction systems in the field of plasmonics.