Measuring the topological Euler charge of triply degenerate points in topolectrical circuits.

Triply degenerate points represent the simplest solid-state bands crossing without any quantum field theory counterpart. Triple points can emerge as a simple crossing of three bands resulting in a Chern number χ=±2, as well as from the crossing of two degenerate bands with a spectator band in PT-symmetric systems. In the latter case, the Chern number is ill-defined, and the topological classification relies on the Euler charge.

Here, we present the experimental realization of a topolectrical circuit network that emulates a PT-symmetric Bloch Hamiltonian hosting triple points with non-trivial Euler charge. The key component of our setup is a complex-phase circuit element, which simulates complex hopping terms and allows us to map the Hamiltonian across the Brillouin zone (BZ) onto the circuit Laplacian. By systematically scanning the first BZ, we identify the triple points and classify them according to their Euler charge using the Wilson loop and Euler curvature methods. Our results highlight the versatility of topolectrical circuits in simulating advanced topological phenomena and offer a compact, accessible platform for directly probing exotic topological invariants.

This approach opens the door to advanced explorations of topological matter in engineered systems.