Fig. 1: Theoretical and experimental results of (1 + 1)-dimensional topolectrical space-time circuits.

a Numerical results of the quasi-energy spectrum for the (1 + 1)-dimensional topolectrical space-time crystal with \({J}_{0}=0.5\Omega\), \(\Delta=0.5\Omega\), \({k}_{\delta }=0.81\pi\), and L = 31. The colormap quantifies the boundary localization of all eigenmodes. b The spatial profile of a midgap space-time topological boundary state at \(\varepsilon=0.5\Omega\). c The quasi-energy spectra of the finite chains as a function of \({k}_{\delta }\). Other parameters are the same as that used in (a). d The schematic diagram for the implementation of a time-varying INIC connecting two adjacent circuit nodes, where the external voltage \(V\left(t\right)\) is used to manipulate the detailed time behavior. e The scheme of the (1 + 1)-dimensional topolectrical space-time circuit. The position-dependent initial phase \(\varphi \left(x\right)\) of the external voltage \({v}_{x}\left(x,\,t\right)\) is realized by an array of signal generators. f The photograph image of the fabricated circuit sample with L = 11. The inset presents an enlarged view around a single node. g, h Measured and simulated voltage waveforms of all nodes in the circuit with red and black lines showing the results of edge and bulk nodes. i Black and red lines show FT frequency spectra of measured voltages at boundary and bulk circuit nodes. j, k Measured spatial distributions of \({FT}[{{{\boldsymbol{V}}}}_{x}\left(t\right)]\) at f = 267 Hz for a bulk mode and 133.5 Hz for the midgap topological boundary mode.