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Surprising New States of Matter Called Time Crystals Show the Same Symmetry Properties in Time That Ordinary Crystals Do in Space by Frank Wilczek

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Surprising New States of Matter Called Time Crystals Show the Same Symmetry Properties in Time That Ordinary Crystals Do in Space by Frank Wilczek PHYSICS CRYSTAL S IN TIME Surprising new states of matter called time crystals show the same symmetry properties in time that ordinary crystals do in space By Frank Wilczek Illustration by Mark Ross Studio 28 Scientific American, November 2019 © 2019 Scientific American November 2019, ScientificAmerican.com 29 © 2019 Scientific American Frank Wilczek is a theoretical physicist at the Massachusetts Institute of Technology. He won the 2004 Nobel Prize in Physics for his work on the theory of the strong force, and in 2012 he proposed the concept of time crystals. CRYSTAL S are nature’s most orderly suBstances. InsIde them, atoms and molecules are arranged in regular, repeating structures, giving rise to solids that are stable and rigid—and often beautiful to behold. People have found crystals fascinating and attractive since before the dawn of modern science, often prizing them as jewels. In the 19th century scientists’ quest to classify forms of crystals and understand their effect on light catalyzed important progress in mathematics and physics. Then, in the 20th century, study of the fundamental quantum mechanics of elec- trons in crystals led directly to modern semiconductor electronics and eventually to smart- phones and the Internet. The next step in our understanding of crystals is oc- These examples show that the mathematical concept IN BRIEF curring now, thanks to a principle that arose from Al- of symmetry captures an essential aspect of its com- bert Einstein’s relativity theory: space and time are in- mon meaning while adding the virtue of precision. Crystals are orderly timately connected and ultimately on the same footing. states of matter in which the arrange- Thus, it is natural to wonder whether any objects dis- Rotational Symmetry ments of atoms take play properties in time that are analogous to the prop- on repeating pat- erties of ordinary crystals in space. In exploring that terns. In the language question, we discovered “time crystals.” This new con- of physics, they are cept, along with the growing class of novel materials said to have “sponta- that fit within it, has led to exciting insights about neously broken physics, as well as the potential for novel applications, spatial symmetry.” including clocks more accurate than any that exist now. Time crystals, a newer concept, Perfect symmetry Partial symmetry are states of matter SYMMETRY whose patterns Before I fully explaIn this new idea, I must clarify repeat at set intervals what, exactly, a crystal is. The most fruitful answer for A second virtue of this concept of symmetry is that of time rather than scientific purposes brings in two profound concepts: it can be generalized. We can adapt the idea so that it space. They are sys- symmetry and spontaneous symmetry breaking. applies not just to shapes but more widely to physical tems in which time In common usage, “symmetry” very broadly indi- laws. We say a law has symmetry if we can change the symmetry is sponta- neously broken. cates balance, harmony or even justice. In physics and context in which the law is applied without changing The notion of time mathematics, the meaning is more precise. We say the law itself. For example, the basic axiom of special crystals was first pro- that an object is symmetric or has symmetry if there relativity is that the same physical laws apply when posed in 2012, and are transformations that could change it but do not. we view the world from different platforms that move in 2017 scientists dis- That definition might seem strange and abstract at at constant velocities relative to one another. Thus, covered the first new first, so let us focus on a simple example: Consider a relativity demands that physical laws display a kind of materials that fully fit circle. When we rotate a circle around its center, through symmetry—namely, symmetry under the platform- this category. These any angle, it remains visually the same, even though changing transformations that physicists call “boosts.” and others that fol- lowed offer promise every point on it may have moved—it has perfect rota- A different class of transformations is important for the creation of tional symmetry. A square has some symmetry but less for crystals, including time crystals. They are the very clocks more accurate than a circle because you must rotate a square through simple yet profoundly important transformations than ever before. a full 90 degrees before it regains its initial appearance. known as “translations.” Whereas relativity says the 30 Scientific American, November 2019 Illustrations by Jen Christiansen © 2019 Scientific American same laws apply for observers on moving platforms, Physicists say that in a crystal the translation sym- spatial translation symmetry says the same laws apply metry of the fundamental laws is “broken,” leading to for observers on platforms in different places. If you a lesser translation symmetry. That remaining sym- move—or “translate”—your laboratory from one place metry conveys the essence of our crystal. Indeed, if we to another, you will find that the same laws hold in the know that a crystal’s symmetry involves translations new place. Spatial translation symmetry, in other through multiples of the distance d, then we know words, asserts that the laws we discover anywhere ap - where to place its atoms relative to one another. ply everywhere. Crystalline patterns in two and three dimensions Time translation symmetry expresses a similar can be more complicated, and they come in many va- idea but for time instead of space. It says the same rieties. They can display partial rotational and partial laws we operate under now also apply for observers translational symmetry. The 14th-century artists who in the past or in the future. In other words, the laws decorated the Alhambra palace in Granada, Spain, we discover at any time apply at every time. In view of discovered many possible forms of two-dimensional its basic importance, time translation symmetry de- crystals by intuition and experimentation, and mathe- serves to have a less forbidding name, with fewer than maticians in the 19th century classified the possible seven syllables. Here I will call it tau, denoted by the forms of three-dimensional crystals. Greek symbol τ. Without space and time translation symmetry, ex- Comple Crystalline Pattern xamples periments carried out in different places and at differ- ent times would not be reproducible. In their everyday work, scientists take those symmetries for granted. In- deed, science as we know it would be impossible with- out them. But it is important to emphasize that we can test space and time translation symmetry empirically. Specifically, we can observe behavior in distant astro- nomical objects. Such objects are situated, obviously, in different places, and thanks to the finite speed of light we can observe in the present how they behaved in the past. Astronomers have determined, in great detail and with high accuracy, that the same laws do in fact apply. SYMMETRY BREAKING for all theIr aesthetIc symmetry, it is actually the way Two dimensions (from the Alhambra palace) Three dimensions (diamond crystal structure) crystals lack symmetry that is, for physicists, their de- fining characteristic. In the summer of 2011 I was preparing to teach this Consider a drastically idealized crystal. It will be elegant chapter of mathematics as part of a course on one-dimensional, and its atomic nuclei will be located the uses of symmetry in physics. I always try to take a at regular intervals along a line, separated by the dis- fresh look at material I will be teaching and, if possi- tance d. (Their coordinates therefore will be nd, where ble, add something new. It occurred to me then that n is a whole number.) If we translate this crystal to the one could extend the classification of possible crystal- right by a tiny distance, it will not look like the same line patterns in three-dimensional space to crystalline object. Only after we translate through the specific patterns in four-dimensional spacetime. distance d will we see the same crystal. Thus, our ide- When I mentioned this mathematical line of inves- alized crystal has a reduced degree of spatial transla- tigation to Alfred Shapere, my former student turned tion symmetry, similarly to how a square has a re- valued colleague, who is now at the University of Ken- duced degree of rotation symmetry. tucky, he urged me to consider two very basic physical questions. They launched me on a surprising scientif- Translational Symmetry ic adventure: What real-world systems could crystals in space- time describe? Might these patterns lead us to identify distinctive Atomic nucleus states of matter? d The answer to the first question is fairly straight- forward. Whereas ordinary crystals are orderly ar- rangements of objects in space, spacetime crystals are orderly arrangements of events in spacetime. As we did for ordinary crystals, we can get our bearings by considering the one-dimensional case, in November 2019, ScientificAmerican.com 31 © 2019 Scientific American which spacetime crystals simplify to purely time crys- breaks spatial translation symmetry “spontaneously.” tals. We are looking, then, for systems whose overall An important feature of crystallization is a sharp state repeats itself at regular intervals. Such systems change in the system’s behavior or, in technical lan- are almost embarrassingly familiar. For example, guage, a sharp phase transition. Above a certain criti- Earth repeats its orientation in space at daily inter- cal temperature (which depends on the system’s vals, and the Earth-sun system repeats its configura- chemical composition and the ambient pressure), we tion at yearly intervals. Inventors and scientists have, have a liquid; below it we have a crystal—objects with over many decades, developed systems that repeat quite different properties.
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