When water freezes and forms a crystal, it creates a periodic pattern in space. Could there be something that crystallizes to form a periodic pattern in *time*? Frank Wilczek, who won the Nobel Prize for helping explain why quarks and gluons trapped inside a proton or neutron act like freely moving particles when you examine them very close up, dreamt up this idea and called it a **time crystal**:

• Frank Wilczek, Classical time crystals.

• Frank Wilczek, Quantum time crystals.

‘Time crystals’ sound like something from Greg Egan’s *Orthogonal* trilogy, set in a universe where there’s no fundamental distinction between time and space. But Wilczek wanted to realize these in *our* universe.

Of course, it’s easy to make a system that behaves in an approximately periodic way while it slowly runs down: that’s how a clock works: tick tock, tick tock, tick tock… But a system that keeps ‘ticking away’ without using up any resource or running down would be a strange new thing. There’s no telling what weird stuff we might do with it.

It’s also interesting because physicists love symmetry. In ordinary physics there are two very important symmetries: spatial translation symmetry, and time translation symmetry. Spatial translation symmetry says that if you move an experiment any amount to the left or right, it works the same way. Time translation symmetry says that if you do an experiment any amount of time earlier or later, it works the same way.

Crystals are fascinating because they ‘spontaneously break’ spatial translation symmetry. Take a liquid, cool it until it freezes, and it forms a crystal which does not look the same if you move it any amount to the right or left. It only looks the same if you move it certain discrete steps to the right or left!

The idea of a ‘time crystal’ is that it’s a system that spontaneously breaks time translation symmetry.

Given how much physicists have studied time translation symmetry and spontaneous symmetry breaking, it’s sort of shocking that nobody before 2012 wrote about this possibility. Or maybe someone did—but I haven’t heard about it.

It takes real creativity to think of an idea so radical yet so simple. But Wilczek is famously creative. For example, he came up with anyons: a new kind of particle, neither boson nor fermion, that lives in a universe where space is 2-dimensional. And now we can make those in the lab.

Unfortunately, Wilczek didn’t know how to make a time crystal. But now a team including Xiang Zhang (seated) and Tongcang Li (standing) at U.C. Berkeley have a plan for how to do it.

Actually they propose a ring-shaped system that’s periodic in time and also in space, as shown in the picture. They call it a **space-time crystal**:

Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.

The new paper is here:

• Tongcang Li, Zhe-Xuan Gong, Zhang-Qi Yin, H. T. Quan, Xiaobo Yin, Peng Zhang, L.-M. Duan and Xiang Zhang, Space-time crystals of trapped ions.

Alas, the press release put out by Lawrence Berkeley National Laboratory is very misleading. It describes the space-time crystal as a clock that

will theoretically persist even after the rest of our universe reaches entropy, thermodynamic equilibrium or “heat-death”.

*NO!*

First of all, ‘reaching entropy’ doesn’t mean anything. More importantly, as time goes by and things fall apart, this space-time crystal, assuming anyone can actually make it, will also fall apart.

I know what the person talking to the reporter was trying to say: the cool thing about this setup is that it gives a system that’s truly time-periodic, not gradually using up some resource and running down like an ordinary clock. But nonphysicist readers, seeing an article entitled ‘A Clock that Will Last Forever’, may be fooled into thinking this setup is, umm, a clock that will last forever. It’s not.

If this setup were the whole universe, it might keep ticking away forever. But in fact it’s just a small, carefully crafted portion of *our* universe, and it interacts with the rest of our universe, so it will gradually fall apart when everything else does… or in fact much sooner: as soon as the scientists running it turn off the experiment.

### Classifying space-time crystals

What could we do with space-time crystals? It’s way too early to tell, at least for me. But since I’m a mathematician, I’d be happy to *classify* them. Over on Google+, William Rutiser asked if there are 4d analogs of the 3d crystallographic groups. And the answer is *yes!* Mathematicians with too much time on their hands have classified the analogues of crystallographic groups in 4 dimensions:

• Space group: classification in small dimensions, Wikipedia.

In general these groups are called **space groups** (see the article for the definition). In 1 dimension there are just two, namely the symmetry groups of this:

— o — o — o — o — o — o —

and this:

— > — > — > — > — > — > —

In 2 dimensions there are 17 and they’re called **wallpaper groups**. In 3 dimensions there are 230 and they are called **crystallographic groups**. In 4 dimensions there are 4894, in 5 dimensions there are… hey, Wikipedia leaves this space blank in their table!… and in 6 dimensions there are 28,934,974.

So, there is in principle quite a large subject to study here, if people can figure out how to build a variety of space-time crystals.

There’s already book on this, if you’re interested:

• Harold Brown, Rolf Bulow, Joachim Neubuser, Hans Wondratschek and Hans Zassenhaus, *Crystallographic Groups of Four-Dimensional Space*, Wiley Monographs in Crystallography, 1978.