Type-II corner modes in topolectrical circuits

We study the rich properties of a topolectrical (TE) circuit array consisting of lossless basic electrical components, such as capacitors and inductors, which can be designed to exhibit higher-order topological phases (HOTP). The HOTP of the circuit exhibits the characteristics of higher-order topology, i.e., unconventional bulk-boundary correspondence with strongly localized corner modes, and higher winding numbers. More interestingly, a type-II corner mode emerges in the presence of long-range interaction, which is realized in the TE circuit by the introduction of next-nearest neighbor (NNN) coupling capacitances. Unlike the usual (i.e., ``type-I'') corner modes that are localized at a particular sublattice node due to the chiral symmetry, the type-II corner modes possess a spatial extent with an exponential decay length. We analytically derive this decay length as a function of the circuit parameters. The NNN coupling is also associated with the tilt parameter in the admittance spectrum of the circuit. The admittance spectrum is reminiscent of that of Dirac fermions. Changing the tilt parameter can lead to a transition from the type-I to the overtilted type-II Dirac dispersion. This overtilting results in a hybridization of the bulk and corner modes in which the distinct corner modes disappear. Furthermore, the type-I and type-II corner modes can be distinguished by their impedance readout. By virtue of their flexibility, the TE circuits provide an ideal platform to demonstrate unusual features of HOTPs arising from long-range interactions, and to engineer different types of robust topological corner modes.