  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 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 local magnetic moment
due to the self-rotation of the wave-packet, a contribution to total
magnetization from the Berry curvature in 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-non-conserved 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
non-interacting limit. As an application, we calculate the intrinsic current
response in an oblique spacetime metal showing the non-equilibrium 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 provide a powerful tool to explore the electronic properties of
this exotic system with coupled space and time.
