Fig. 4: Experimental results of the three-generation non-Hermitian fractal braiding with a circular Lissajous trajectory.

a The k-space lattice model for the three-generation fractal braiding with a circular Lissajous trajectory, containing a total of 27 lattice sites within a unit cell. The k-dependent unidirectional couplings \({t}_{2n}(k)\) are established between the first node and the 2n-th node. b Circuit design for simulating a three-generation fractal braiding structure based on a circular Lissajous trajectory, containing a total of 27 nodes. c Schematic and photograph of the constant nonreciprocal coupling between adjacent nodes, achieved by connecting a voltage follower in series with a capacitor. The scale bar represents 5 mm. d Configuration and photograph for the circuit realization of the tunable k-dependent nonreciprocal coupling. This tunable coupling is implemented using two multipliers connected in parallel, with a capacitor and a resistor as loads. External input voltages control the real and imaginary parts of the coupling, respectively. The scale bar represents 5 mm. e–g Experimental reconstruction of the admittance eigen-spectrum sections for the circuit under various values of \({V}_{a,3n}\) and \({V}_{b,3n}\), implementing the k-dependent couplings. The solid lines represent the simulation results by the spice software LTspice (https://www.analog.com/cn/resources/design-tools-and-calculators/ltspice-simulator.html).