  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.
