Observation of flat-band localization and topological edge states induced by effective strong interactions in electrical circuit networks

Flat-band topologies and localizations in noninteracting systems are extensively studied in different quantum and classical-wave systems. Recently, the exploration on the novel physics of flat-band localizations and topologies in interacting systems has aroused great interest. In particular, it is theoretically shown that the strong interaction could drive the formation of nontrivial topological flat bands; even dispersive trivial bands dominate the single-particle counterparts. However, the experimental observation of those interesting phenomena is still lacking. Here, we experimentally simulate the interaction-induced flat-band localizations and topological edge states in electrical circuit networks. We directly map the eigenstates of two correlated bosons in one-dimensional Aharonov-Bohm cages to modes of two-dimensional circuit lattices. In this case, by tuning the effective interaction strength through circuits’ groundings, the two-boson flat bands and topological edge states are detected by measuring frequency-dependent impedance responses and voltage dynamics in the time domain. Our finding suggests a flexible platform to simulate the interaction-induced flat-band topology, and may possess potential applications in designing novel electronic devices.

Figure 1

The electric circuit for simulating interaction-induced localizations and topology of two bosons. (a) Schematic diagram of a quasi-1D lattice with open boundary conditions. Each sublattice is marked with $\left(l,\alpha \right)$, where $\alpha =A,B,C$. A synthetic flux with $\theta =\pi$ is applied in each closed loop. (b) Schematic diagram of quantum model of two correlated bosons. (c) Schematic diagram of the designed 2D circuit lattice for simulating 1D two-boson model. (d),(e) Illustrations of complex-valued and real-valued hopping rates in circuits between two sites enclosed by yellow and green blocks in (c). (f),(g) The detailed ground setting of circuit nodes enclosed by frames with consistent colors.

Figure 2

Numerical results of interaction-induced two-boson localizations and topological edge states in electric circuits. (a) Eigenfrequencies of the designed electric circuit with $m\times n=55\times 55$, and other parameters are set as $C=1nF,C_U=40nF,L_g=3.3uH$. (b) Enlarged view of eigenfrequencies with interactions. (c) Aharonov-Bohm cage. In the system with $\theta =\pi /2$, the time evolution of the state $|1,A,1,A\rangle$ never allows the particle to leave the Aharonov-Bohm cage centered on that site (shaded in dark purple). (d) The voltage distribution at $f=0.897$ MHz. (e)–(i) The voltage distribution diagram; the corresponding frequency is consistent with (b).

Figure 3

The fabricated circuit sample to simulate the interaction-induced localizations and topological edge states. (a) The photograph image of the front of the fabricated circuit simulator. (b) The photograph image of the back of the fabricated circuit simulator. (c),(d) Enlarged views of the green and red boxes in (a).

Figure 4

Experimental results of frequency-dependent impedance spectra on the simulation of interaction-induced two-boson flat bands and topological edge states. (a),(b),(d),(f) The experimental impedance responses of $\left(2,A\right)\left(2,A\right), \left(2,B\right)\left(2,B\right), \left(1,C\right)\left(2,B\right)$, and $\left(1,A\right)\left(1,A\right)$ circuit nodes. (c),(e) The simulated impedance responses of different bulk and edge nodes.

Figure 5

Experimental results of voltage dynamics. (a),(c),(e) The experimental results of the time dynamics of $\left(2,A\right)\left(2,A\right), \left(1,C\right)\left(2,B\right)$, and $\left(1,A\right)\left(1,A\right)$ nodes in the fabricated circuit, respectively. (b),(d),(f) The distribution of the voltage amplitude at $t=3$ ms related to results in (a)–(c), respectively.