Spontaneous symmetry breaking happens when some parts of the system interact with each other in a certain way. For example, electrons' spins can interact in a large atom can interact in such a way that they split the energy levels in a certain way. These interactions are complex and are an ongoing subject of Condensed Matter Physics research. As for similar violation of temporal symmetry, it is an even more unexplored territory.
Time crystals are hypothetical structures that repeat themselves in time without expending energy, like a mechanical clock without clockwork. The sequence repeats in time as the atoms repeat in the crystal lattice. Frank Wilczek (2004 Nobel Prize in Physics for describing the interaction between quarks and gluons) came up with a concept of temporal crystals as a way to break time symmetry. In 2012, he began to wonder
why the temporal symmetry is never broken spontaneously (that is, due to random interactions between the elements of the system), and whether it is possible to make it happen.
Thus has appeared the concept of time crystal. One way to visualize it is to imagine a ring made of atoms, isolated from the environment and rotating, regularly returning to its original state. Its properties would be eternally synchronized in time, just as the position of the atoms in a salt crystal. Now pause for a second and imagine such a system.
A careful consideration will expose a serious problem. Indeed, the temporal crystal definition requires that such a system would be in a state with the lowest energy, so it would not require energy from outside. In a sense, a time crystal must be a perpetuum mobile, except that it would not produce any useful work.
Most of the scientific community considered this idea at best provocative. Nevertheless, Frank Wilczek stood his ground, confident that the problem is more subtle than it seems, and time crystals represent a new type of order. Not to forget that perpetual motion has precedents in the quantum world: superconductors are supposed to conduct the electric current forever. It is a degenerate case of a time crystal though: the electrical current is homogeneous, so it does not show variations in time.