Fig. 2: Theoretical and experimental results of (2 + 1)-dimensional topolectrical space-time circuits.

a Numerical results of the quasi-energy spectrum for the finite (2 + 1)-dimensional topological space-time crystal with \({J}_{0}=0.3\varOmega\), \(\Delta=0.5\varOmega\), \({k}_{\delta }^{x}=0.81\pi\), \({k}_{\delta }^{y}=0.81\pi\), and L = 15. The color map quantifies the boundary localization of all eigenmodes. b The edge-state dispersion under mixed boundary conditions, where the periodic boundary condition is applied in the x-direction, while the open boundary condition with fifteen lattice sites is used in the y-direction. c The spatial profile of a chiral topological boundary state. d The phase diagram of the (2 + 1)-dimensional topological space-time crystal in the (\({k}_{\delta }^{x}\), \({k}_{\delta }^{y}\)) space. Other parameters are the same as that used in (a). e The schematic diagram of the (2 + 1)-dimensional topological space-time circuit with L = 5. The right-top inset presents the structure of the time-varying coupling element, where the color map quantifies the position-dependent initial phase of external voltages. The right-bottom inset shows the values of various grounding INICs and resistors. f The photograph image of the fabricated circuit sample. The inset displays an enlarged view of a time-varying INIC along the y-axis. g–j The measured spatial distributions of \({{{\boldsymbol{V}}}}_{x,y}\left(t\right)\) at t = 0, 3.7 ms, 6.3 ms, and 9.4 ms.