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What's the difference between a time crystal and a system undergoing periodic motion? My understanding of a crystal is that it is a rigid body with a spatially periodic structure. Is any system undergoing periodic motion a time crystal, or is there some other requirement that makes it special? In other words, what (if anything) excludes the ideal simple harmonic oscillator from the category of time crystal?

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A crystal is a system which spontaneously breaks the translation symmetry of the underlying physical laws.

Similarly, a time crystal is a system which is subject to a periodic driving with period $T$, but which however does not oscillate with the same period but shows oscillations which a different period $T'=kT$ with $k>1$ integer, i.e., which spontaneously breaks the (discrete) time translation symmetry.

This answer provides a very nice explanation of time crystals.

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