A time crystal is a system that exhibits time-periodic behavior in its ground state (GS) or in a nonequilibrium steady state, in the absence of an external periodic driving force. In particular for the GS, it exhibits time-periodic behavior, meaning that if you measure the state of the time crystal at different times, it will repeat itself periodically. This is in contrast to a normal system, whose ground state is time-independent and does not change over time.
The condition for a system to be a time crystal is that it must break time-translation symmetry. In other words, the system must exhibit a preferred time scale that is not present in the Hamiltonian.
Let us focus on the GS. Is there a simple discrete spin chain example of such a time crystal?
I have been suggested the following Hamiltonian: $$H = -J_1 (\sigma_{x} \otimes \sigma_{x} + \sigma_{y} \otimes \sigma_{y}+\sigma_{z} \otimes \sigma_{z}) - J_2 (\sigma_{x} \otimes \sigma_{x} + \sigma_{y} \otimes \sigma_{y}-\sigma_{z} \otimes \sigma_{z}),$$ but struggle to see how it is relevant.
