  Spontaneous symmetry breaking can lead to the formation of time crystals, as
well as spatial crystals. Here we propose a space-time crystal of trapped ions
and a method to realize it experimentally by confining ions in a ring-shaped
trapping potential with a static magnetic field. The ions spontaneously form a
spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate
persistently at the lowest quantum energy state in magnetic fields with
fractional fluxes. The persistent rotation of trapped ions produces the
temporal order, leading to the formation of a space-time crystal. We show that
these space-time crystals are robust for direct experimental observation. We
also study the effects of finite temperatures on the persistent rotation. The
proposed space-time crystals of trapped ions provide a new dimension for
exploring many-body physics and emerging properties of matter.
