Crystals theres been a lot of newspaper headlines, making this out to be a major breakthrough. Huge sensational headlines, like google may have created an unruly new state of matter. This could be a great thing for quantum computing googles time crystal discovery is so big. We cant fully comprehend it googles time crystal could be the greatest scientific achievement of our lifetimes, which is really some huge claims, so uh. What were going to do today is to talk a little bit about, so what is a time crystal to talk through a little bit of googles? Recent results, the story of time crystal starts around about 10 years ago. Back in 2012, it was introduced by frank wilcek hes, a nobel prize winner at mit and, as you can see already from this first paper, its actually closely related to spontaneous symmetry breaking and the basic idea, its spontaneous symmetry breaking of time. So to first understand this, we need to understand a little bit more about spontaneous symmetry breaking. So lets just recap that the archetypal example of spontaneous symmetry breaking is the ferromagnet. So, as you know, in a ferromagnet, basically, you have little spins, which are the magnets of the atoms and they all like to point in the same direction. And so we learn in undergraduate physics that we can describe this with an ising model and we have the minus sign here in the hamiltonian because they want to point in the same direction.

So the thing about a ferromagnet is actually you can have two energy configurations which are both lowest energy states right. So you can have all the spins pointing up, because each spin wants to be facing the same direction or you can have all the spins pointing down, and you can have some other configurations where you can have like domains of ferromagnetic regions. But these are higher energy states and so were mainly concerned with the like the low energy states of the system, so heres. Our first problem that actually the general ground state of the hamiltonian that you might write is actually, in general, a superposition of those two states. So all spins up or all spins down, but we never really see such a state. So why dont? We see this kind of state the way that people understand. This is a little bit different depending upon what community you are talking to. So in the condensed metaphysics or particle physics. Community people say that this is an example of spontaneous symmetry breaking, and so basically, the idea here is that the hamiltonian has some particular symmetry in this case, saying that all the spin flip so up spins, all up or all down, are basically the same. But then the ground state doesnt obey that it just goes. Okay, i dont care about your symmetry im, just going to have all the spins pointing up, and so that symmetry is broken. So this is spontaneous symmetry breaking now you might ask like okay, but like how did it get into a particular configuration to start with well the way that people understand? That is by saying that.

Well, it just depends upon the past. History of whats happened to the fero magnet in the past, so this very magnetic thing that was in a rock from the beginning of time and then dinosaurs were there and then eventually you came along and found it, but you found it in this up configuration and Just depends upon what happened to it in the past, but if you talk to a quantum information person, then they would have a pretty different answer, so they would say that, well, you can have a superposition of those two states, but youll have the coherence and then It will just decohere into one of these states so now lets talk about crystals, which is getting closer to what we are trying to talk about. So in a spatial crystal, we also have this kind of spontaneous symmetry effect, so the all the atoms in the crystal they have bonds holding them together. So they all want to be separated by some distance. But then, like the crystal configuration at the top or the bottom, these are equally valid configurations say you find the crystal in the top position. Why would it be in that one rather than the bottom one? Well again, you would explain this by spontaneous symmetry break theres. No real preference for this crystal to be in any position, but due to again some historical reasons of this crystal being in some some position, it just happens to be in the top position rather than the bottom position.

Okay, so thats spontaneous symmetry breaking in space, okay, so whats a time crystal. Well, if you look at the crystal in terms of position, so this top graph, then youll see that the atoms are in arranged in some periodic way. If you plot the atomic density within the crystal itll, go up and down right, so thats oscillations and position so a time crystal would be some oscillations, not in position but in time right: okay, uh that sounds fine, but uh thats, just an oscillator. So what whats? So amazing about that, instead of just being an oscillator, we need this extra ingredient to have spontaneous symmetry breaking. So basically it should be something thats oscillating, but it shouldnt be oscillating because you are shaking around. It should just be oscillating because thats the natural state that the system prefers to be in so the oscillation should arise, despite the fact that the hamiltonian is completely independent of time. You write down your hamiltonian theres, no time varying parameters. Yet somehow the system likes to oscillate time, symmetry is like spontaneously broken another thing: it should also be a kind of a stable state, so it shouldnt be some kind of transient state where it just sort of decays away or something that happens at the beginning. It should be a stable state of the system because if we go back to the original case of the spontaneous symmetry breaking of a crystal, that was like a ground state of the system right.

So this means that in the time crystal case, it should also be a stable state that the system would be in and it should just basically keep on oscillating forever, so basically thats. What the time crystal is, as defined by will check, but what people are talking about more these days are actually discrete time crystals, and this time we say that its okay, that we are gon na drive the system were gon na actively oscillate the system in some Way: okay, but then in that case, how do we have this idea of spontaneous symmetry breaking? So in this case, we have a slightly different idea of if we are driving the system at some frequency say with period t. Then, if the system responds with a different frequency to the driving frequency, then the system is responding with a frequency that you didnt put in there right. So if you didnt put that frequency in there, then it spontaneously come up with this different frequency of oscillation. So people talk of a discrete time crystal if this driving frequency is different to how the system actually responds. Specifically, the period is usually some integer, multiple of the driving frequency, so if youre driving it originally at 1 hertz, then the expectation value would be at a frequency which is like lower than 1 hertz, so 0.5 hertz or maybe a third of a hertz or something Like that, okay, so i think thats the crash course in time crystals.

So we can go back to what this google paper is talking about so ill hand it over to junghang, now, okay yeah, so im going to give more uh detailed explanation of this paper. Okay, so so the idea that the system will have uh expected value that oscillates like twice the spirit of the system is because that they have a against state order that they will flee. You know from one organ state to the other and then flick back so so then you have the twice the period of the oscillating system. So a typical example of this kind of system is like icing model, for example. This is on the left that you can find in the in this google paper that typical operations that there are three operations in place, so one is the driving force driving field which oscillates and the other one. The second one is the interaction, and also you have the rotation so which will flip the every qubit. So, in order to have this kind of time crystal, you need all these three steps. The way to observe such a tiny crystal is to measure the expecting value of sigma that which is the poly operator on the z axis. If you drive the system at the frequency of the period of t, and you will see that the expectation value of sigma x and x y does not oscillates it just simply decays right, but but the sigma that oscillates in the frequency of two at two t.

So this one is uh, not googles results. No, this is a theoretical result. Yes, yes, so for them on the right side, they do a free transform of sigma x, symbol y and second, that and you can see this distinct uh peak of sigma debt as 0.5. Right, which is means that the frequency of the response is half of the frequency of the driving for driving fields. So this is the very distinct uh a characteristic of the discrete time chris. And if we look at the google paper – and so they didnt really show that you know its like twice the period because maybe they think this is commonly accepted, so they dont show it more, but this is their results. So this kind of time, discrete time crystal, is realized in the case that we all have out of equip equilibrium uh phase, because if you have a thermal thermophase, for example, if this g is very small, this icing model will just go to a thermal phase where You simply have decay you, you will not have this kind of if you have, for example, this g equal to zero points 97, which is close to one. You have this phase as auto equivalent and with oscillates among two eigen states. So, and in this way you you can see that the expectation value here it also oscillates. You know because theres a decay right there whats the reason of the decay, because there are two reasons: decay one is the external decoherence and one is the internal summarization and the the the black line is to show that what would be like for uh, external intercoherence And so on, on the right on the left side, where you have a thermal phase, you can see that this decays much faster than this black line.

So in indication of this because of the internal thermalization, but on the right side, you can find that this the degree of the envelope is exactly because of the external decoders. So so internally, they dont have uh similarization. So when you normalize it on the – and you will see that this, when you normalize with the envelope, you will see that the oscillates you know very, very well in the prior theoretical walks have shown. The model is expected to have this kind of discrete time. Crystal phase in a range where g is bigger than g gc, where gc equals approximately to 0.84. So this is the phase transition between these two phases and the other one important thing is because you see that there are three steps right, one is rotation, one in action and then you have a discord. So these two, these two graphics, is from the 2017 paper, and this will show that the importance of the interactions example graph, a only the flip, is on. I think the rotation and second one you have disco disorder and then on the right side. You will turn on the interactions and you can see maybe not easily from this uh time graph, but from the free transform uh the graph. You can see thats the this 0.5. You know frequency uh, its distinct. Only for when the interaction is on and the interactions off theres i mean theres, actually, two peaks and theyre both around 0.

5 right, no, not already. No, they are so at the 0.5 exactly 0.5. This, oh theres, no peak right, oh okay, so i mean on the right side on the other side on the left side. Well, yes, but uh. I thought that, even if its like close to 0.5, its still a time crystal its not a time crystal, if its not exactly no twice, no, yes, you have to be 0.5 or maybe its transient or something is this transient based on the ic model. You need, you must have there just have one half of the original frequency. Otherwise, you know you cannot explain, because the theoretic model says well, okay, so and this graph to show you the importance of the disorder and also another important of time crystal uh is because that you have to show this kind of uh sub harmonic. You know response. No matter what your initial state is like whatev, whatever the choice of the initial state, you must have this and you can see that um. So, on the on the left side that this disorder value for each qubit is randomly chosen right on the right side, where you have a constant. So, here that the test on three uh initial state, one is the new state and the one is the polarized and one in london and for if you choose the choose this, this order value to be randomly distributed among certain value. And you will see that all these initial states doesnt affect how you know how you behave, how this expected value evolved during time, but on the right side, where you have a constant disorder value, you will see that it behaves totally different.

So, and these two are the most uh this – you know important uh characteristic of the time crystal its kind of robust against yes situation, yes, and it should be robot against the disorder variation well, my main question is like so it looks like it actually does decay Off right, yes, because uh, i just dont, know a decoration uh, sorry, because yes, so so thats apparently okay yeah, but but this one is like because they do did have like some pre uh thermal. You know you had have some uh oscillations, but in the end, then they would go to general state, but the google, why it it seems that does not have that property, so its somehow better uh, i said is this: why something? But i think its attracting a lot of attention yeah, but i didnt think they explained very well. You know why the their work is like somehow much better than the other ones, but um yes, yeah, i mean you know its kind of surprising everybody is going so crazy over it, considering theres already two papers in 2017. Yes, that actually said they observed right whats. The difference between the normal slater youre, not in the camera, Music, just Music, normal yeah, so uh – i mean in wilchecks original paper. The kind of nearly time crystal example that he gave was just like a super fluid that was revolving in a ring, because that has some kind of oscillation in a sense and its also a stable kind of ground state right.

So i think i think it needs to you know, at least in the original definition it should. It should be kind of stable forever. So i guess most oscillators would just decay and in that sense, at least in wilczeks kind of idea of a time crystal it doesnt really quite satisfy. But on the other hand, these do decay right well, yeah. But another thing is that, because, for this kind of uh many body localization, they are robust against the publications right right and and the with external paths. I think the robot, whether its for so, if theres, some kind of disorder. Yes, it would uh start to oscillate in different frequencies.