Abstract
In the present chapter we concentrate on periodically driven systems. Although such systems possess already discrete time translation symmetry, we will see that this symmetry can be spontaneously broken into another discrete time translation symmetry and a system is able to spontaneously reorganize its motion and move with a period which is different from the period of the driving. These phenomena are dubbed discrete (or Floquet) time crystals because discrete time translation symmetry becomes spontaneously violated. Discrete time crystals attract considerable attention: they are examples of quantum many-body systems that break ergodicity and they are believed to be resistant to decoherence. Moreover, discrete time crystals have been already demonstrated in the laboratory and further experiments are being planned.
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