## Time Crystals

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.﻿

### 12 Responses to Time Crystals

1. Arrow says:

It’s an interesting idea but is there anything new here beyond using the usual terminology in a novel way?

It seems to me there are plenty of such “time crystals” around, planetary systems for example, they don’t consume energy.

I also suspect that the system they describe does consume energy to maintain strong enough magnetic field, cooling and vacuum.

• John Baez says:

Arrow wrote:

It seems to me there are plenty of such “time crystals” around, planetary systems for example, they don’t consume energy.

Good point! A single planet orbiting a star will exhibit periodic motion, and it’s a system we can describe well using classical mechanics, but Shapere Wilczek has a stronger definition in mind in their paper Classical time crystals. From the abstract:

We consider the possibility that classical dynamical systems display motion in their lowest energy state, forming a time analogue of crystalline spatial order. Challenges facing that idea are identified and overcome. We display arbitrary orbits of an angular variable as lowest-energy trajectories for nonsingular Lagrangian systems. Dynamics within orbits of broken symmetry provide a natural arena for formation of time crystals. We exhibit models of that kind, including a model with traveling density waves.

So the challenge they pose is to find classical systems that display periodic motion in their lowest-energy state. This seems to be impossible in the Hamiltonian formalism, where we write down equations of motion starting from the Hamiltonian (the energy). After all, a smooth Hamiltonian $H(q,p)$ that takes a minimum along some periodic trajectory will have vanishing derivatives along that trajectory, so by Hamilton’s equations

$\displaystyle{\frac{dq^i}{d t} = \frac{\partial H}{\partial p_i} }$

$\displaystyle{\frac{dp_i}{d t} = -\frac{\partial H}{\partial q^i} }$

we get

$\displaystyle{\frac{dq^i}{d t} = 0 }$

$\displaystyle{\frac{dp_i}{d t} = 0 }$

The system stands still.

What’s the way out? Start with the Lagrangian formalism, and use a smooth Lagrangian such that the corresponding Hamiltonian is not smooth at its minima.

The game seems to be quite different in the quantum version, and I don’t understand that version very well. For one thing, in quantum theory, a system in a definite energy state

$H \psi = E \psi$

will only change phase as time passes:

$\psi(t) = e^{-it H/\hbar} \psi$

The phase has no observable effect for an isolated system, so we say the system is in a stationary state. In particular, if a system is in its state of least energy (its ground state), it’s in a stationary state.

This is familiar in the quantum analogue of the classical planetary system: the hydrogen atom. Classically, two particles attracting each other by an inverse-square force have no state of least energy. Quantum-mechanically, they do: it’s called the 1s state of the hydrogen atom; it’s spherically symmetric and there’s no motion in this state.

So, getting quantum time crystals sounds impossible, unless we change our attitude a bit about what they should be, or find quantum systems that can be described by the Lagrangian formalism but not the Hamiltonian formalism.

2. grlcowan says:

You can tell they’re onto something by the spatial deformation the keyboard near the image is suffering.

3. John Baez says:

Of course, time is different than space, but we can also imagine doing chemistry in a world where space has more or fewer than 3 dimensions, and since chemistry is a highly mathematical subject one can study it using equations rather than test tubes. I sometimes hope that when the world has worked out a lot of its less urgent problems, chemists can have fun studying higher-dimensional chemistry.

But there are basic questions about higher-dimensional chemistry that need to be settled before we start having have fun. For example: in 4 dimensions, instead of $1/r$ potential, electrostatic forces have a $1/r^2$ potential

Classically a force with a $1/r^2$ potential has strange properties, first discovered by Newton.

But for chemistry we need quantum mechanics! It turns out that if the attractive force between charged particles blows up sufficiently intensely as they get close, atoms are quantum-mechanically unstable, so chemistry doesn’t exist.

Mathematically, the question is this: is the operator

$\displaystyle{ -\nabla^2 - \frac{k}{r^2} }$

self-adjoint in 4 dimensions? I used to know this stuff, but I forget now, and the relevant book is still packed away—I just got back from Singapore. I remember that in 3 dimensions,
the operator

$\displaystyle{ -\nabla^2 - \frac{k}{r^p} }$

is self-adjoint for $p < 3/2$, while for $p = 3/2$ it’s only self-adjoint when the number $k$ is smaller than (or equal to?) some constant. Chemistry in 3 dimensions works fine because it uses $p = 1$.

I taught a course about this once and you can see proofs of some of these facts here:

• John Baez, Quantum Theory and Analysis, around page 55.

The beautifully delicate borderline case $p = 3/2$ can be found in the book I have packed away somewhere:

• Reed and Simon, Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness, Academic Press, 1978

But the exact power $p$ at which

$\displaystyle{ -\nabla^2 - \frac{k}{r^p} }$

ceases to be self-adjoint depends strongly on the dimension of space, and I’m forgetting the exact formula. The answer will settle whether chemistry based on electrostatic attraction of oppositely charged particles can exist in 4d space.

4. Aaron Denney says:

How does any of this change with Minkowski space, rather Gallilean or Euclidean?

• John Baez says:

You spotted a weak spot in my post. I was talking about ‘space groups’. These are certain discrete subgroups of the symmetries of Euclidean space. So, it’s somewhat artificial to classify spacetime crystals in terms of their space groups. You can do it, but it probably makes more sense to look at subgroups of the symmetries of Galilean spacetime or Minkowski spacetime. I don’t know of work that’s classified ‘crystals’ in this way.

I might as well take this excuse to quote Wikipedia about ‘space groups’:

In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain. Bieberbach (1911, 1912) proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert’s 18th problem. Zassenhaus (1948) showed that conversely any group that is the extension of Zn by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Zn by a finite group acting faithfully.

It is essential in Bieberbach’s theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z3

Since space groups are subgroups of the isometries of Euclidean space it may seem odd to classify them up to conjugation by affine transformations (which aren’t necessarily isometries). The reason is that if we have something like a triclinic crystal:

we don’t want to say its space group changes whenever we change the angles $\alpha, \beta, \gamma$ slightly… though of course it does change, by getting suddenly bigger, when some of these angles take the value 90°. Then we say the crystal isn’t triclinic anymore: it belongs to some other, more symmetrical crystal system.

5. Uncle Al says:

Chemists can have fun studying higher dimensional chemistry” Great – now the ACS has a replacement for its BS-degree idiot German requirement. “8^>)

Organic chemistry is topologically 2-D, less an epsilon. Molecules’ skeletal Schlegel diagrams are planar non-crossing in about 36 million cases. So-called “K_5” molecules, about a dozen catalogued, are more interesting. Excluded from that catalog, for reasons of noisome cleverness, are [6.6]chiralane and its analogues,

Terahedral core, octahedral periphery, point group T (not T_h or T_d).

The five central carbon atoms are inescapably chiral, but are utterly unnamable as to hand, even in principle.
http://www.berezin.com/3d/3dprism.htm
http://www.amazon.com/3-D-Stereo-Prism-Glasses-3Dphotoscope/dp/B00465OY3Y
For stereogram easy viewing.

The same strategy allows creation of planar carbon atoms (e.g., sp^2 hybridized in acyclic olefins) that are chiral centers (no S_n improper symmetry axes) in 3-D. Neither the ACS nor IUPAC were amused. Party poopers.

6. mfrasca says:

Sorry, but Frank Wilczek did not explained why and how gluons and quarks are confined. He and other proposed a theory, that works, but we are looking for an answer yet at this specific question.

Marco

7. Daniel Walsh says:

In order to answer the question of the “spacetime-groups” maybe should study the 3D regular space group. It’s thus tempting to believe that the spacetime-groups will be some kind of modification of the 3D crystallographic group when we add in the temporal dimension. As an analogy, consider the symmetry group of the infinite cylinder. It is useful to decompose the cylinder as a cartesian product of a circle and a line, so the symmetry group of the infinite cylinder is generated by rotations and translations about the axis.

I wonder if a similar thing happens with spacetime-groups.

• John Baez says:

It’s true that working on a spacetime cylinder, as shown in the photo in the blog article, eliminates Lorentz transformations: the only symmetries are translations of the time line and rotations of the space circle, and reflections in time, and reflections in space (and combinations of all of these).

So, the symmetries of a spacetime cylinder are the same as the symmetries of an ordinary cylinder in space.

I once read an incredibly detailed book on crystallography, which classified the things like ‘space groups’ for cylinders. These are important for structures like nanotubes:

So, this math is already out there, waiting for people who want to build spacetime crystals using rings of atoms.

8. With all due respect to Frank Wilczek’s outstanding talent and achievements, his “proof” of existence of a “quantum time crystal” is incorrect.

In short, Wilczek proposed a model consisting of attractively coupled particles on an Aharonov-Bohm ring, and found a solution to the corresponding Schrödinger which (for non-zero Aharonov-Bohm flux) consists of a rotating lump, thus breaking symmetry with respect to time translation.

However, Wilczek did not bother to prove that the solution he has found is really the ground state, i.e., that there exist no other states of lower energy.

Furthermore, one can show that, if correct, Wilczek result would lead to several paradoxical (unphysical) consequences. In particular, the classical behavior would not be recovered in the limit of infinitely large interactions, as it should, and the system would be able to radiate energy while being in its ground state, thereby violating the principle of energy conservation.

The true ground state can be found, is perfectly stationary, and does not lead to any unphysical paradox; this is discussed in detail in the following Comment: http://arxiv.org/abs/1210.4128

Patrick BRUNO
ESRF