Non-Hermitian topological phases and exceptional lines in topolectrical circuits.

We propose a scheme to realize various non-Hermitian topological phases in a topolectrical (TE) circuit network consisting of resistors, inductors, and capacitors. These phases are characterized by topologically protected exceptional points and lines. The positive and negative resistive couplings Rg in the circuit provide loss and gain factors which break the Hermiticity of the circuit Laplacian. By controlling Rg, the exceptional lines of the circuit can be modulated, e.g., from open curves to closed ellipses in the Brillouin zone. In practice, the topology of the exceptional lines can be detected by the impedance spectra of the circuit. We also considered finite TE systems with open boundary conditions, the admittance spectrum of which exhibits highly tunable zero-admittance states demarcated by boundary points (BPs). The phase diagram of the system shows topological phases which are characterized by the number of their BPs. The transition between different phases can be controlled by varying the circuit parameters and tracked via impedance readout between the terminal nodes. Our TE model offers an accessible and tunable means of realizing different topological phases in a non-Hermitian framework, and characterizing them based on their boundary point and exceptional line configurations.