Abstract
A space-time crystal is defined as a quantum-mechanical system with both spatial and temporal periodicity. Such a system can be described by the Floquet-Bloch (FB) theory. We first formulate a semiclassical theory by constructing a wave packet through the superposition of the FB wave functions, and we derive the equations of motion of FB electrons subjected to slowly varying external fields (not to be confused with the fast-changing Floquet drive), revealing behaviors similar to ordinary Bloch electrons but with quantities modified in the Floquet context. Specifically, we study the local magnetic moment due to the self-rotation of the wave packet, a contribution to total magnetization from the Berry curvature in $\mathbf{k}$-space, and the polarization of a fully occupied FB band. Based on the semiclassical theory, we can also show the fingerprint of the energy flow in such an energy-nonconserved system. We then discuss the density matrix of a FB system attached to a thermal bath, which allows us to investigate quantities involving many electrons in the noninteracting limit. As an application, we calculate the intrinsic current response in an oblique space-time metal showing the nonequilibrium nature of the FB system. The current response can also be related to the acoustoelectric effect. Overall, we develop a systematic approach for studying space-time crystals, and we provide a powerful tool to explore the electronic properties of this exotic system with coupled space and time.
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