  Higher-order topological insulators are a new class of topological phases of
matter, originally conceived for electrons in solids. It has been suggested
that $\mathbb{Z}_N$ Berry phase (Berry phase quantized into $2\pi/N$) is a
useful tool to characterize the symmetry protected topological states, while
the experimental evidence is still elusive. Recently, topolectrical circuits
have emerged as a simple yet very powerful platform for studying topological
physics that are challenging to realize in condensed matter systems. Here, we
present the first experimental observation of second-order corner states
characterized by $\mathbb{Z}_3$ Berry phase in topolectrical circuits. We
demonstrate theoretically and experimentally that the localized second-order
topological states are protected by a generalized chiral symmetry of tripartite
lattices, and they are pinned to "zero energy". By introducing extra capacitors
within sublattices in the circuit, we are able to examine the robustness of the
zero modes against both chiral-symmetry conserving and breaking disturbances.
Our work paves the way for testing exotic topological band theory by
electrical-circuit experiments.
