Time crystals: a review.

@article{Sacha2018TimeCA,
  title={Time crystals: a review.},
  author={Krzysztof Sacha and Jakub J. Zakrzewski},
  journal={Reports on progress in physics. Physical Society},
  year={2018},
  volume={81 1},
  pages={
          016401
        }
}
Time crystals are time-periodic self-organized structures postulated by Frank Wilczek in 2012. While the original concept was strongly criticized, it stimulated at the same time an intensive research leading to propositions and experimental verifications of discrete (or Floquet) time crystals-the structures that appear in the time domain due to spontaneous breaking of discrete time translation symmetry. The struggle to observe discrete time crystals is reviewed here together with propositions… 
Fractional time crystals
Time crystals are quantum systems that are able to reveal condensed matter behavior in the time domain. It is known that crystallization in time can be observed in a periodically driven many-body
Classical Many-Body Time Crystals.
TLDR
This work provides a simple and pedagogical framework by which to obtain many-body time crystals using parametrically coupled resonators and presents a clear distinction between single-mode time-translation symmetry breaking and a situation where an extensive number of degrees of freedom undergo the transition.
Topological time crystals
By analogy with the formation of space crystals, crystalline structures can also appear in the time domain. While in the case of space crystals we often ask about periodic arrangements of atoms in
Controlled preparation of phases in two-dimensional time crystals
The study of phases is useful for understanding novel states of matter. One such state of matter are time crystals which constitute periodically driven interacting many-body systems that
Time Crystal Platform: From Quasicrystal Structures in Time to Systems with Exotic Interactions.
TLDR
Here it is shown that condensed matter problems ranging from single particles in potentials of quasicrystal structure to many-body systems with exotic long-range interactions can be realized in the time domain with an appropriate periodic driving.
Time crystals: Analysis of experimental conditions
Time crystals are quantum many-body systems which are able to self-organize their motion in a periodic way in time. Discrete time crystals have been experimentally demonstrated in spin systems.
A colloidal time crystal and its tempomechanical properties
The spontaneous breaking of symmetries is a widespread phenomenon in physics. When time translational symmetry is spontaneously broken, an exotic nonequilibrium state of matter in which the same
Condensed matter physics in big discrete time crystals
We review the application of discrete time crystals created in a Bose-Einstein condensate (BEC) of ultracold atoms bouncing resonantly on an oscillating atom mirror to the investigation of condensed
Time crystal minimizes its energy by performing Sisyphus motion
TLDR
Motion and minimal energy seem to contradict each other but Shapere and Wilczek showed that if the kinetic energy of a particle on a ring is a quartic function of its velocity, it may happen that the minimal energy corresponds to a particle moving along a ring with non-zero velocity.
Boundary Time Crystals.
TLDR
This work introduces boundary time crystals and analyzes in detail a solvable model where an accurate scaling analysis can be performed.
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References

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TLDR
A simple model for a one-dimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied is considered and a blueprint based upon a one dimensional chain of trapped ions is proposed.
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TLDR
It is pointed out that time crystals can generally exist in systems without disorder, and a series of clean quasi-one-dimensional models under Floquet driving are proposed to demonstrate this unexpected result in principle.
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TLDR
A no-go theorem is proved that rules out the possibility of time crystals defined as such, in the ground state or in the canonical ensemble of a general Hamiltonian, which consists of not-too-long-range interactions.
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Time crystals are proposed states of matter which spontaneously break time translation symmetry. There is no settled definition of such states. We offer a new definition which follows the traditional
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TLDR
The experimental observation of a discrete time crystal, in an interacting spin chain of trapped atomic ions, is presented, which opens the door to the study of systems with long-range spatio-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions.
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TLDR
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We show that both discrete and continuous time-translation symmetry can be broken in the pre-thermal regime of quantum systems that eventually thermalize. We prove a theorem that states that such
Boundary Time Crystals.
TLDR
This work introduces boundary time crystals and analyzes in detail a solvable model where an accurate scaling analysis can be performed.
Fate of a discrete time crystal in an open system
Following the recent realisation that periodically driven quantum matter can support new types of spatiotemporal order, now known as discrete time crystals (DTCs), we consider the stability of this
Discrete Time-Crystalline Order in Cavity and Circuit QED Systems.
TLDR
A phenomenology of dissipative discrete time crystals is established by generalizing the Landau theory of phase transitions to Floquet open systems and finding clear signatures of a transient discrete time-crystalline behavior, which is absent in the isolated counterpart.
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