Edge state mimicking topological behavior in a one-dimensional electrical circuit

For one-dimensional (1D) topological insulators, the edge states always reside in the bulk bandgaps as isolated modes. The emergence and vanishing of these topological edge states are always associated with the closing/reopening of the bulk bandgap and changes in topological invariants. In this work, we discover a special kind of edge state in a 1D electrical circuit, which can appear not only inside the bandgap but also outside the bulk bands with the changing of bulk circuit parameters, resembling Tamm states or Shockley states. We prove analytically that the emergence/vanishing of this edge state and its position relative to the bulk bands depends on the intersections of certain critical frequencies. Specifically, the edge mode in the proposed circuit can be mathematically described by polynomials with roots equal to some critical frequencies in the bulk circuit. From this point of view, the transition of the edge state is uniquely determined by the order of the critical frequencies in the bulk circuit. Such topological behaviors shown by the edge state in the proposed electrical circuit may indicate, in a broader sense, the presence of certain type of topology.