Fig. 2: Experimental observation of the unconventional BFA with topological circuit.
a Implementation of the Hamiltonian. Left panel: The system includes three nodes A, B, and C. The hoppings between A and B, and between A and C are nonreciprocal hoppings, implemented using an impedance converter with a current inversion (INIC) circuit in series with capacitors (C1, Cx). The hopping between B and C is reciprocal, realized with a pure capacitor (Cy). The hoppings between A and C and between B and C implement synthetic dimensions fx and fy, and the hopping between A and B is a linear function of fx and fy, g(fx,fy) = 0.343-fx + b. Here, b in the linear function denotes the perturbation term. Rag, Rbg, and Rcg are all grounded resistors; Cag, Cbg, and Ccg are all grounded capacitors. Right panel: INIC structure, with two equal resistors Ra in both the positive and negative feedback paths, and capacitor Ci in series. b–d Experimental measurements of admittance eigenvalues for the cut planes with perturbation terms b = 0 (b), b = 0.0299 (c), and b = 0.0569 (d), respectively. Left panels: Degeneracies in the synthetic 2D parameter space on these cut planes. The ELs, DTP, and BFAs are denoted by the orange lines, red star, and blue line, respectively. The experimentally identified degeneracies are marked with solid dots. Dashed lines in different colors correspond to the measured lines (fy = t) on the right panels. Right panels: Real parts of the dispersion of eigenvalues as a function of fx for different fy lines (fy = t) on the corresponding cut planes. The eigenvalues are ordered in exact phases from small to large. The experimentally measured admittance eigenvalues are marked in circles, and all the degeneracies (EL, DTP, EP3, and BFA) are indicated by arrows. The experimental error bars are provided in Supplementary Note 1. The unconventional BFA can be found to stably connect the paired EP3s. Increasing the perturbation simply enlarges the interspace between the two EP3s, but cannot eliminate the BFA, demonstrating the stability of the BFA. Experimental raw data for plotting (b–d) is provided in the Supplementary Data.
