Evolution of planar topological phases in topocircuits

Liberation of topological phases from the realm of electronic materials has been recently initiated by their realizations in highly tunable matematerials, such as phononic and photonic systems, as well as in electric circuits. These classical systems facilitate engineering of various seemingly complex phases of matter, and unveil their topological or geometric properties. We here present a systematic evolution of two-dimensional topological insulators, starting from the analog of quantum anomalous and spin Hall insulators, supporting one-dimensional edge states in electric circuits, also known as topocircuits. Subsequently, we construct rotational symmetry breaking higher-order topological insulators (HOTIs), supporting pointlike corner modes protected by an antiunitary symmetry. The systemic reduction in localization of the topological boundary modes from the edge to the corners is detected from site and orbital selective measurement of the impedance. We also present a simple circuit model for three-dimensional HOTI.