Laws of nature are beautiful, and this beauty goes hand in hand with the symmetry. Physicists understand symmetry in a wider sense: for them, essentially, it means that no matter how you twist the object or the Universe, the laws of physics will remain the same. More specifically, it means that a property remains unchanged under a transformation: rotation or a shift in space, or coordinate system change.

Symmetry can be continuous or discrete, regardless the type of symmetry. For example, if you place a metal sphere in a well-heated oven long enough for the heat exchange to be complete the temperature inside the sphere will be the same everywhere anytime. This means that no matter how you turn the sphere, or which coordinate system you choose, the temperature will remain unchanged until you turn off the oven. This is an example of the continuous space-time symmetry. And if you place a cube in the oven, the spatial symmetry is broken, because the cube can be rotated in different ways, and it "repeats itself" only when you rotate it by a certain angle. This is an example of discrete rotational symmetry.

The interesting physics begins where the symmetry ends or, rather, breaks down. A great example of broken spatial symmetry is a "usual" crystal (say, salt). Atoms sit in the nodes of the so-called crystal lattice with clearly defined distances (= broken spatial symmetry) and angles (= broken rotational symmetry) between each other. Crystal properties are defined by the number of electrons on the outer shell of atoms (= which chemical elements they are made of), their magnetic moments, medium temperature, which defines the crystal structure and its properties.

Temporal symmetry can also be broken. An example can be found if you ask "will this system behave differently if I reverse the time?" This is not such a pointless question as it may seem. For example, two identical particles move along the same trajectory, but in opposite directions (so their momentum is reversed) so they are connected through a time reversal operation. In the presence of the magnetic field, given that the trajectory is curved, and the particles are charged (so they create a magnetic field of their own, perpendicular to the trajectory and the momentum), what happens with a particle cannot be reproduced by simply hitting the reverse button for its counterpart.

The paradox of time crystals began to haunt Haruki Watanabe after an oral exam during his postgraduate studies at Berkeley. He defended his work on breaking of spatial symmetry, and his supervisor asked him what would be the wider implication of Wilczek's idea of a temporal symmetry breaking. "I could not answer that question in that exam, but it interested me" Watanabe recalls, skeptical about the existence of such a phenomenon, "I wondered, 'how can I convince people that it's not possible?"

Together with the physicist Masaki Oshikawa from the University of Tokyo, Watanabe began to derive a mathematically rigorous proof of his intuitive hypothesis. In 2015, they have proved a theorem according to which creation of a time crystal in the state with the lowest energy was impossible. Moreover, such crystals consisting of any system in the state of equilibrium at any value of energy cannot exist. Research community considered the case of time crystals solved and closed.

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.

Concept of crystalline time haunted Haruki Watanabe since his postgraduate studies at Berkeley. During an oral exam, after they discussed spontaneous symmetry breaking in crystals, his supervisor asked him about wider implications of Wilczek time crystals and temporal symmetry breaking. Watanabe could not answer this question during the exam but he had a strong feeling that such system was not physically possible. Together with his colleague from Tokyo University, he has embarked on a mission to prove that his intuition was right. Their theorem has proven that time crystals for any system in its ground or equilibrium state were not possible. For the majority of the scientific community the case of time crystals was solved and closed.

However, the proof left a loophole: it did not exclude a possibility of time crystal in a non-equilibrium system.

A breakthrough unexpectedly came from the field of physics where researchers did not think about time crystals at all. Theoretician Shivaji Sondi and his colleagues from the University of Princeton studied the behavior of an isolated quantum system consisting of a "soup" of interacting particles, regularly "kicked" energetically. According to the textbooks, such a system should heat up and eventually become completely chaotic, but the Sondi group has shown that, under certain conditions, the particles should be grouped together and form a state of matter that does not exist in the equilibrium state: a system of weakly correlated particles following a certain pattern that repeats in time.

This assumption attracted the attention of Chetan Nayak, one of the former Wilczek's students. Nayak and his colleagues thought that the strange nonequilibrium state of matter could be a kind of time crystal, although not quite like the one Wilczek originally had in mind. The difference is that such a system is not in the lowest energy state, so it requires an external energy kick to pulsate (in this case, laser radiation). Amazingly, such a "soup" would pulsate with its own rhythm, different from the pumping frequency, which in fact does mean a violation of temporal symmetry.

When Christopher Monroe of the University of Maryland at College Park heard about time cristals, he was skeptical: "The more I read about it, the more intrigued I became," he recalls. The physicist tried to create a time crystal with help of "improvised" cold atoms. An intricate recipe contained three main ingredients: a force periodically affecting the system, the interaction between the atoms, and an element introducing random disorder. This combination limited the particles in the amount of energy they could absorb, allowing them to remain in an ordered state.

In practice, a chain of ten ytterbium ions was alternately illuminated by two lasers. The first laser flipped magnetic moments of atoms, and the second forced them to interact with each other in a random way. This combination led to fluctuations in the projection of the magnetic moment, and the period of these oscillations turned out to be twice as long as the period of laser-induced flipping of the spins. More importantly, even if the first laser "strayed" from the set frequency, the oscillation period of the system would not change. Just as the conventional crystals resist attempts to move atoms from their positions in the crystal lattice, so did Monroe's time crystal retain its periodicity in time.

Mikhail Lukin, a physicist from Harvard University, took the other path and set to realize a time crystal of his own, using a diamond. For this purpose, a special sample was synthesized, containing about a million defects, each with its own magnetic moment. In addition, random inclusions in the crystal introduced disorder to the system. When the sample was subjected to the microwave radiation, capable of flipping the spins, the response frequency of the system was the fraction of the frequency of the exciting radiation.

"At first glance, it seems that this idea is completely wrong," says Norman Yao, a theorician who participated in both experiments, "by definition, a system in the lowest energy state should not change in time." Otherwise, it would mean that they have extra energy that they can expend, and ultimately the movement must stop. The result of the experiments can be compared to a rope: the hand makes two turns, and the rope only one," Yao explained. "Although this is a weaker symmetry violation than originally conceived by Wilczek, as in his vision the rope would oscillate by itself."

The results of both experiments were published in Nature and are certainly interesting, but the time crystal definition in both cases is a little forced. Physicists agree that both systems spontaneously violate temporal symmetry in some way, and therefore formally can be called time crystals. Nevertheless, whether they can really be considered as such is still a subject to scientific debate. To begin with, it would be useful to narrow down the definition of a time crystal in order to exclude phenomena that have already been studied and therefore show no new physics.

If what Monroe and Lukin made was time crystals or not, only the time will tell. In any case, these experiments are interesting because they have discovered a new state of matter in a relatively unexplored domain of nonequilibrium systems. This new phase of matter is composed of quantum particles, and it continuously changes, never reaching equilibrium. Stability is achieved through random interactions that would destroy equilibrium in other systems. Norman Yao remarks that "it's less strange than the original idea, but still very strange." Vedika Khamani of the Harvard Group was more enthusiastic: "this is a new kind of order that was considered impossible."

Moreover, these results can have practical significance. Time crystals can be useful as super-current sensors. Defects in diamond are already used to record the slightest changes in temperature and magnetic fields. But this approach has its limitations, because when too many defects "crowd" a small volume, the interactions between them destroy the quantum states. In the time crystal, on the contrary, the system is stabilized, so millions of defects can be used together to amplify the signal and to study living cells and materials of atomic thickness.

Another possible application of such systems is a high temperature quantum computer. Quantum computers are a promising and long-awaited technology, which is still confined to the lab space. Fragile quantum bits need to be isolated from the thermal motion that destroys quantum states and other "side effects" of the real world. At the same time, they need to be able to encode and readout information. Physicists use intricate quantum particles at nanodegrees above the absolute zero to attempt quantum computing. One of the goals of such calculations would be to model the behavior of materials that are too difficult to predict by means of conventional computations. Time crystals are an inherently quantum system that exist at substantially higher temperatures, in the case of Lukin's diamond even at room temperature, bringing a promise of quantum computers at non-cryogenic temperatures.