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As is usually the case with any advanced physics / astrophysics article I read, I can't get into the article because I'm stuck mulling over a premise. Article states:

> The laws of physics are symmetric through space ... But in a crystal, this gorgeous symmetry gets broken. The molecules of a crystal arrange themselves in a preferred direction, creating a repeating spatial structure. In the jargon of physicists, a crystal is a perfect example of "spontaneous symmetry breaking" — the fundamental laws of physics remain symmetric, but the arrangement of the molecules is not.

I don't understand how crystals break spatial symmetry. Are we talking about some absolute spatial directional bias? If it's just relative the crystal lattice itself I can't see how that breaks symmetry.

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There's two things that can have symmetry here: The laws of physics themselves, and the system under investigation. The symmetries of the laws of physics don't get broken, but those of the system do. Compare a crystal to a gas: In a gas, the atoms are all bouncing around pretty randomly, so at any given point in space, there's roughly the same chance of finding an atom. Shift the gas to the left by distance x, and the local probability distribution of atom positions looks pretty much the same. In a crystal on the other hand, the atoms are still moving around and vibrating (so there's still some uncertainty in the positions of the atoms), but they tend to stay pretty close to their proper position in the crystal lattice. So the atoms are more likely to be in positions that line up with the rest of the crystal lattice than anywhere else. This breaks the symmetry. Shift by distance x and the peaks of that probability distribution no longer line up. The exception to this is if x is a multiple of the spacing of atoms in the crystal. Then you're shifting the peaks by exactly the right amount that they line up again when you're done. So a crystal doesn't completely break the symmetry of space, but it reduces it from a continuous symmetry (you can translate by any amount in any direction) to a much weaker discrete symmetry (only certain translations of space will preserve the symmetry).

A time crystal is similar to an ordinary crystal except that instead of reducing symmetry of translations in space from a continuous symmetry to a discrete symmetry, it reduces symmetry of translations in time from a continuous symmetry to a discrete symmetry.

EDIT: It's a little ironic that if you ask most people, they would say that a crystal is more symmetric that a gas, since a gas will look completely random and asymmetric if you take a snapshot of the positions of all the atoms at a single time. But since physicists care about the probability distribution of atom positions, they say that the gas is more symmetric than the crystal.

What you've described is a repeating loop. What makes a time crystal more than that?

It's about being periodic in a lowest-energy configuration.

A spatial crystal freezes into a spatially-periodic configuration at its lowest energy: you don't need to add energy to keep a crystalline solid's microscopic components arranged in lattice-like form.

A time crystal freezes into a periodic configuration at its lowest energy: you don't need to add energy to keep a time crystal arranged in its temporally periodic arrangement. If at t_0 we have one spatial configuration, at t_1 another spatial configuration, ... at t_n-1 we have yet another spatial configuration, and at t_n we have the same spatial configuration as at t_0, and we have no net flow of energy into the spatial configuration at any t_x, we have a time crystal. The spatial configurations at any t_x need not be crystalline, they just have to differ at different points in their cycle.

An analogue clock is not a time crystal because even though the configuration of the hands at 12:00->12:01->...->11:59->12:00 is temporally periodic, you have to wind a clock (or power it in some other way) or it gets stuck at some arbitrary configuration -- it stops cycling unless "disturbed" with added energy. The clock's lowest-energy configuration has its hands always pointing to one hh:mm time, and no different time is shown over the course of a day.

A time crystal, being in its lowest-energy state, cycles through all its configurations endlessly until energy is added.

Does this mean you could use a time crystal to efficiently measure time accurately, by counting the number of cycles? I know we do that with quartz oscillators but those need energy input to keep oscillating.

Good question. I was about to edit this into my comment, but now it works better as a reply.

The act of "reading" the configuration of a time crystal disturbs the time crystal. So you either a set of maximally-similar time crystals that you read at various times during a day, or you need to re-freeze your single disturbed time-crystal each time you read it.

There are ordinary crystals which literally melt out of their crystalline state when handled / measured-by-bright-light. The organized pattern is broken with the additional energy. Time crystals are patterened over time, and that pattern breaks when they are handled / measured-by-bright-light.

You could think of it as having to shine a flashlight (or laser) through the time crystal to figure out which way it twists the light at a given time t_x. If you know the temporally-periodic structure, you can predict the different twisting when you turn on the light at t_x versus t_x+1 or t_x-1. But lighting up the crystal breaks the lowest-energy condition of the time crystal -- it's melted by the light it twists -- so you have to re-freeze it back into its predictable periodic structure, otherwise you might get the same twisting (or none) at t_{measured}+1, t_{measured}+2, ..., t_{measured}+n.

(It is fairly literally re-freezing: you have to do laser cooling or the like. And it takes energy to run the cooler, which removes energy from the not-lowest-energy-state broken time crystal, so thermodynamics isn't violated.)

Will it be possible to read the crystal without disturbing it, or is it a physical quantum limit that prevents it?

That's a very good question. I started but abandoned a fairly deep answer, mostly because this is an area far from my expertise and in which it is easy to be howlingly wrong. (To be fair to me, subject matter experts have been arguing about this in the literature for some twenty years.)

Instead I'll direct you Sean Carroll's "Quantum Interrogation" blog posting as a starting point: <https://www.preposterousuniverse.com/blog/2006/02/27/quantum...>. ("how you can detect something without actually looking at it")

There is also the 2021 work by the Google & Stanford team (decent press release: <https://news.stanford.edu/2021/11/30/time-crystal-quantum-co...> open-access paper: <https://www.nature.com/articles/s41586-021-04257-w>, the PDF version of which (click in top right) is legible) which is at least suggestive that certain types of time crystals can be interrogated without breaking them.

Thank you! Reading it now!

I don't get why it's surprising. Any regular structure has resonance modes: struck a bridge or a violin and it will keep vibrating in a time-periodic fashion. Most structures are leaky and lose that initial energy quickly. Crystals are super-regular, nearly perfect macroscopic structures, so they lose energy slowly, and if the periodic motion only involves electrons in the crystal, the motion will conserve energy almost perfectly and may keep going for billions of years.

> struck [strike]

Energy in.

> will keep vibrating ... lose energy slowly

Dissipative process: energy out. Eventually stop vibrating.

Time crystals cycle through their configurations until you put energy in. Otherwise, they are in lowest energy state, so no energy out. Ever.

Practically speaking, this means we couldn't use the cyclic process to do work, because that would require it to have excess energy it could transfer. If a time crystal's periodic behavior were to spin, you couldn't use the rotation to push something. Right?

Right. Lowest-energy means that you can't pull energy out of a time crystal; there's no excess. Anything you try to attach to a time crystal will transfer energy into the crystal.

Practically all we can do with a time crystal is to measure it with the lightest possible touch and hope it doesn't break the periodicity. (So far, afaik, the periodicity has always been broken by the measurement process's energy input).

I don't know what we could do with large numbers of time crystals, though. One can't pick up a snowflake in one's bare hands and use it like a buzz saw to cut a sheet of paper (the snowflake melts on contact), but an avalanche of snowflakes can snap trees. Maybe for time crystals that rotate light a predictable amount at a given time t_x, we could create some sort of interesting lens from a large cloud of such time crystals arranged at different distances from a bright light source -- a sort of "anti-fog".

If the periodicity is broken by any measurement we have been able to make so far then how do we know the periodicity is there in the first place? Just theory? If it’s not too much of a burden could you outline the proof of the existence of time crystals?

> how do we know the periodicity is there ... Just theory?

Theory guides us, but experimentally you can for example make a whole bunch of time crystals (especially straightfoward for driven time crystals, which have a period that's an integer multiple of the driver, the driving force being laser light or microwaves) with an expected set of states it cycles through, and you can test those states once per time crystal. If you reliably get the states theory predicts, that's good evidence.

> could you outline the proof of the existence

The excellent <https://physics.aps.org/articles/v10/5> and decent <https://en.wikipedia.org/wiki/Time_crystal#Experiments> (the former is [5] in that wikipedia article) are good starting points, with ample references.

Great explanations, thanks!

This reminds me of the anti-zeno effect, in which a particle evolution is increasingly delayed with more frequent observations.

I feel like there must be more to it that this. Wouldn't any physical structure that's not homogeneous and isotropic also break symmetry in the same way. Does the room I'm in break symmetry as there's a different likelihood of hitting a wall depending on the direction I travel?

> Wouldn't any physical structure that's not homogeneous and isotropic also break symmetry in the same way. Does the room I'm in break symmetry as there's a different likelihood of hitting a wall depending on the direction I travel?

Uhuh, but enough to be physically detectable?

The x in "distance x" (grandparent comment) is very small.

"Very small" in this context is less than the lattice spacings, which for a typical crystal can be on the order of the wavelength of an X-ray (i.e., there's ~ 0.1-100 ångströms between the crystal's diffracting planes, so "distance x" must be a fraction of that length).

A typical room is effectively a <https://en.wikipedia.org/wiki/Gas_in_a_box>. If the walls of your box are good X-ray detectors then an isotropically-radiating X-ray source somewhere near the middle of the room will evidence an essentially uniform energy loss at the detectors, and and weak reflection from air molecules back towards the source. However, if you substituted the air in the room with a crystal lattice, the energy loss would be much stronger at detectors in some directions, and there would be strong reflections back towards the source along some directions. See <https://physicsopenlab.org/2018/01/18/bragg-diffraction/> for some details.

> Does the room I'm in break symmetry

Symmetry breaking is very important to the study of condensed matter physics, including solid matter, and arguably it’s the reason your room exists at all.

https://simple.wikipedia.org/wiki/Higgs_field https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking

> continuous symmetry (you can translate by any amount in any direction)

Thanks for the definition. That is close to the math idea of automophic, that is, can map onto itself. So, what physics means is that translations are essentially automorphisms. Simple now that we have a clear definition!

> I don't understand how crystals break spatial symmetry.

Imagine doing some experiment in a vacuum. It will work the same no matter which direction you orient the experiment or where in space you put it.

Now imagine doing the same experiment inside a crystal. Now it won't work the same no matter which direction you orient the experiment (because some directions will cause something in the experiment to hit one of the atoms of the crystal, and other directions won't) or where in space you put it (because there are crystal atoms in some places but not in others).

That's how the crystal breaks spatial symmetry.

A minor addition to clarify: if one did the same experiment inside an ideal gas, rather than a crystal, there are still atoms in the way (compared to vacuum) but there is no preferred direction through the atoms. Crystalline structure gives a preferred direction along the principal axes. This might mean there is less optical extinction along the principal axes from deep inside the crystal, for example, whereas from deep inside the ideal gas optical extinction is the same in every direction.

That is, the view varies depending on direction one looks within a crystal. "Symmetrical" means that the view should not vary that way. If we freeze our gas into a crystal lattice, we break this symmetry.

Symmetry becomes much more easy to grasp if you think of it only in terms of transformations - in this case coordinate transformations. If you for example rotate your system by a few degrees, does it look the same if you were to overlay it with the initial state (imagine the crystal as an infinite lattice). If yes, you have found a symmetry. For a crystal structure, you usually only have some discrete symmetries, i.e. you can maybe rotate by multiples of 90 degrees or shift axes by multiples of a certain length, but apart from these things the "inherent" rotational and translational symmetry of empty space is gone. What they're calling "spontaneous symmetry breaking" here is technically correct, but in this context it's a pretty trivial observation (I mean, yeah, it is a lattice after all) without any deep insight, as opposed to the Higgs mechanism for example.

It’s counterintuitive because most people are more familiar with the mathematical concept of symmetry whereby a lattice has more symmetry than a random set of points (which almost certainly has no symmetry at all). However from the physics point of view, the random set of points is more symmetrical than the lattice because there’s no way of telling which way a random set of points is oriented.

Symmetry in physics is not just about orientations in space. This notion rather comes from the fact that large systems of discrete points are often described using continuum mechanics or field theory, which naturally has a lot more symmetries. But even a set of randomly distributed point-like particles can have symmetries. If the set was made up of identical bosons for example, it would have a wave function symmetry under particle permutations. And real particles would also exhibit a whole lot of other discrete symmetries, like CPT symmetry at the very least.

Your question reminded me of this video about "Homochirality: Why Nature Never Makes Mirror Molecules"[1] - even though it's not directly related I think it may be interesting to you.

[1] https://www.youtube.com/watch?v=SKhcan8pk2w

If you imagine every molecule in the crystal can be oriented randomly, then there is a very large number of possible global configurations that are equally likely and we say the crystal is "symmetric" with respect to these outcomes. If the orientations become ordered in some fashion as the article is saying we say the symmetry is broken.

Consider a light source (flashlight) pointed to a wall, it would be symmetrical (uniform)

But in case a crystal gets into the way, it won't be a uniform as a whole.

Similarly an event happening all the time or in some frequency, would be disrupted by a time crystal. (changing the frequency)

> If it's just relative the crystal lattice itself I can't see how that breaks symmetry.

Yeah, just relative to the crystal lattice.

Yes, I went to grad school in math. The first test was a pop quiz in real analysis and was about the little axiomatic set theory the prof had started the course with.

When the prof handed back the papers, he'd given me a zero. At the end of class, he asked that I stay. We went over my test paper: I had used little omega as the first infinite ordinal, essentially the same as the set of natural numbers. In an NSF course in axiomatic set theory the previous summer, that notation was standard. I'd thought of my proof, that is, my solution on the test, only at the last moment so just used little omega without defining it. When I gave the prof the definition, he saw that my solution was correct and, indeed, one step shorter than his. So, I got credit on the test.

Then I asked the prof why he had wanted to see me, and he smiled and said that he no longer did.

It was a class of 20+ students, and I doubt that all the other students got a good proof. So, why was the prof picking on me?????

He was regarded as a bright prof. In later years, he did have some fame. So much for such a bright math prof. I smelled I was being dumped on. I didn't know that prof at all.

His lectures were not very clear. The book he had selected was just in typing which is awful for the math of real analysis -- he could have used, say, Royden. And he'd just tried to dump on me for no good reason. Also, I was correct and he hadn't known that -- my meaning for little omega is and was standard.

At times I'd been dumped on in grades 1-12. I did well on standardized tests in math and physics, and that saved me. I was well out of patience being dumped on so walked out of the course and never saw that prof again.

In high school plane geometry the teacher believed that I refused to do any homework. Well, I didn't bother with her homework assignments -- they were too easy. Instead in the back of the book there were lots of more difficult exercises, and I made sure I worked 100% of those, never missed even one. One of those took me the weekend, and when I mentioned it in class on Monday, my first and last class participation, 20 minutes later the teacher was close to screaming exhorting the class to "think". Not wanting to be accused of ruining the class, as I started to give the steps to a solution, the teacher cut me off and screamed "You knew how to do it all along." Guilty as charged! Gee, she was not interested in getting the solution from me!

I've come to suspect: Students with some flair for originality and creativity can look different, not the same as the image of a good student -- nose to the grindstone, ear to the ground, shoulder to the wheel, and from that position dot all the i's and cross all the t's -- and less good than desired instead of better.

The other two courses they had me in were close to what I'd already studied carefully in ugrad school. One of these two was from Kelley, General Topology that I had lectured from to a prof.

I did well at my ugrad teaching, started violin, met my wife, and got a Ph.D. in applied math later at another university.

There, too, at times I did work comparable with that of the brilliant profs.

Net, I'm not so impressed by the super brains of so called brilliant profs.

Or I believe in the standard, "Everyone puts pants on one leg at a time."

More generally, my experience is, given the basic context of data and background, a lot of people can work through and find the immediate consequences.

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