  Long-range interactions (LRIs) are ubiquitous in nature. Higher-order
topological (HOT) insulator represents a new phase of matter. A critical issue
is how LRI dictates the HOT phases. In this work, we discover four
topologically distinct phases, i.e., HOT phase, the bound state in the
continuum, Dirac semimetal (DSM) phase, and trivial insulator phase in a
breathing kagome circuit with tunable LRIs. We find an emerging type-III corner
state in the HOT phase, which splits from the edge state continuum and
originates from the strong couplings between nodes at different edges. We
experimentally detect this novel state by impedance and voltage measurements.
In the DSM phase, the Dirac cone exhibits an itinerant feature with a tunable
position that depends on the LRI strength. Our findings provide a deeper
understanding of the LRI effect on exotic topological states and pave the way
for regulating interactions in topolectrical circuits.

Mesoscale and Nanoscale Physics