Topolectrical space-time circuits.

Topolectrical circuits have emerged as a pivotal platform for realizing static topological states that are challenging to construct in other systems, facilitating the design of robust circuit devices. In addition to spatial dimensionality, synergistic engineering of both temporal and spatial degrees in circuit networks holds tremendous potential across diverse technologies, such as wireless communications, non-reciprocal electronics and dynamic signal controls with exotic space-time topology. However, the realization of space-time modulated circuit networks is still lacking due to the necessity for flexible modulation of node connections in both spatial and temporal domains. Here, we propose a class of topolectrical circuits, referred to as topolectrical space-time circuits, to bridge this gap. By designing and applying a time-varying circuit element controlled by external voltages, we can construct circuit networks exhibiting discrete space-time translational symmetries in any dimensionality, where the circuit dynamical equation is in the same form with time-dependent Schrödinger equation. Through the implementation of topolectrical space-time circuits, three distinct types of topological space-time crystals are experimentally demonstrated, including the (1 + 1)-dimensional topological space-time crystal with midgap edge modes, (2 + 1)-dimensional topological space-time crystal with chiral edge states, and (3 + 1)-dimensional Weyl space-time semimetals. Our work establishes a solid foundation for the exploration of intricate space-time topological phenomena and holds potential applications in the field of dynamically manipulating electronic signals with unique space-time topology.