Fig. 3: Experimental results on hyperbolic band topology with second Chern numbers in artificial circuit networks with PBCs.

a The photograph image of the fabricated hyperbolic circuit sample under PBCs. Right charts present the front and back sides of enlarged views. White dash circles labeled from ‘a’ to ‘l’ label twelve units of the finite hyperbolic lattice model. Group generators \({\gamma }_{1}\) to \({\gamma }_{4}^{-1}\) are marked by eight colored arrows. b The schematic diagram for the realization of different inter-cell couplings. c, d The recovered circuit frequency spectra with the mass term being (m = 0.7, a = 0.2) and (m = 0.7, a = 3.2) under PBCs. e, f The measured impedance responses of four bulk nodes marked by four white stars in the ‘a’ unit cell with mass terms being (m = 0.7, a = 0.2) and (m = 0.7, a = 3.2), respectively. g and h Simulation results of frequency-dependent impedance responses with mass terms being (m = 0.7, a = 0.2) and (m = 0.7, a = 3.2). i, j Measured spatial impedance distributions at 0.607 MHz (m = 0.7, a = 0.2) and 0.464 MHz (m = 0.7, a = 3.2), respectively. Orange and blue shaded regions correspond to non-trivial and trivial bandgaps, respectively.