Topological Phases and Anomalous Phase Transitions in Topolectrical Circuit

Distinct topological phases and anomalous phase transitions are investigated in a 2D topolectrical circuit system. The topological phases of the system are characterized by the Chern number and fractional wave polarization, and the phase-transition lines (PTLs) obtained from the phase diagrams are consistent with those derived from the zero-energy condition. It is shown that different PTLs emerge gapless points at different high-symmetry points of the first Brillouin zone. Moreover, gapless points appear simultaneously at two different high-symmetry points when two PTLs intersect. In topological phases with a nonzero Chern number, the chiral edge states are present when periodic boundary conditions (PBCs) are applied along any one direction. In topological phases with a nonzero fractional wave polarization, there are in-gap edge states when PBCs are applied along one direction and near-bulk edge states along another direction. The edge states are present at the anomalous PTLs between two nontrivial phases and are absent at the normal PTLs between the trivial and nontrivial phases. Furthermore, an efficient method is proposed to identify desirable topological edge states based on the impedance of the topolectrical circuit system. The work reveals that the presence of anomalous phase transitions is crucial for understanding the essentials of topological phase transitions.