 if the person who's doing the video is, okay, because otherwise I suspect people, there will be lots of complaints about how small my handwriting is. Yeah, okay, well, you know, there it is. Perfect, thank you so much, thank you, thank you. Okay, good, okay, so let's continue. So we ended the last discussion, just a super quick recap. We sort of, you know, have been zooming in continuously on sort of, you know, the context of the problem where things are very non-trivial. We emphasized, you know, at some level that there were three ingredients that one might want to have for the definition of a time crystalline phase of matter. And we said that in general, those three ingredients can be satisfied quite relatively naturally in dynamical map systems where the dynamics themselves are given by a contractive dynamics. Actually, I should emphasize just here, at the end of the lecture, there were two questions that were kind of related, which means that I probably didn't do a good enough job. There was a question, essentially, you know, in the beginning, I was talking about time translation symmetry and I was saying we had a flow case system. So there was a time dependent Hamiltonian H of T and it came back to itself every period. And then at some point, I dropped that notation and I just went to classical maps where there's an update rule given by phi. And the question that was asked was what happened to the time translation symmetry? It is the time translation symmetry. You know, time translation symmetry in this language of the dynamical update rule is literally apply phi at every time step and just don't worry about there being intermediate times at all. So the discrete time translation symmetry is just applying phi at each time step. And if there's an observable that only comes back to itself every three periods, four periods, two periods, that is the discrete time translation symmetry breaking. I'm sorry if that was unclear to people. Good, so, and then we zoomed in to basically emphasize that even in Hamiltonian systems, actually subharmonic oscillations are not so few and far between. I gave the example of Faraday waves and what we're going to do now is try to understand at some level a feeling for what the obstruction might be for getting to infinitely long-lived discrete time translation symmetry breaking in such systems. And to do so, we're going to introduce a different way of thinking about discrete time translation symmetry breaking in terms of emergent symmetries and what I'll call an effective flow-K Hamiltonian. And to do that, we will go from our pretty picture of Faraday waves to the case, to start off, of a single parametrically driven, a single parametrically driven non-linear oscillator. And along the way, I really will emphasize this already as just a road map. I want us to, it doesn't have to be done this way, but I think there's a very, very powerful way of understanding time translation symmetry breaking from a different perspective. Perspective. And really, it'll be understanding time translation symmetry from the perspective of the breaking of usual spontaneous symmetries breaking. So single parametrically driven non-linear oscillator, we wrote down basically a 4A version of this already. So imagine that we have an oscillator whose position is given by q momentum p and clearly we have time dependence because we have parametric driving. Set the mass equal to one. The Hamiltonian that describes such a system is p squared over two plus omega naught squared over two multiplied by one plus delta cosine omega d, the frequency of the parametric drive q squared and it's non-linear. So we'll have a small non-linearity epsilon over four q to the four. You can imagine that at some level, one of the k equations that we wrote down in the linearized math two equation basically corresponds to this piece of the Hamiltonian. So from the Faraday wave discussion, you already know, you already know that there exists period doubled solutions whenever the driving frequency is somewhat close to two times the natural frequency, you already know that there exists period double solutions or trajectories of my equations of motion that take the form, where root of two capital P of t, e to the i omega d over two t, here's the period doubled sub harmonic. In general sub harmonic, I'm often I'm focusing on period doubling because it's the easiest thing to draw and the easiest thing to write down. You already know that when omega d is close to two times omega naught, there is a stable parametric resonance. We discussed in the context of Faraday waves and this period double solution of trajectory takes the form in general oscillations. Of course, the amplitude of the oscillation is the momentum, the phase of the oscillation is the position. So square root of two capital P of t, e to the i omega over two t, plus let's say capital Q of t. This is just on physical grounds. I'm just saying that by definition, I'm looking for a period double solution that's this and on totally general grounds, coefficient of this should be the momentum, the phase is the position. And of course, if I write down the solution in this way, I have a period, a nearly period doubled solution. It turns out that as a general rule of thumb, you will find that this capital P of t and capital Q of t are now slowly varying in comparison to the original driving frequency. And should be super familiar to any of you who've thought about parametric resonance. Classical parametric resonator, parametrically parametric resonance of a nonlinear oscillator. The period double solution takes this form on physical grounds and now there may still be time dependence but the time dependence is now slowly varying in comparison to your original driving frequency. Not unlike some of the things that Anatoly was thinking about. So of course, at some level, going from the original degrees of freedom, little Q and little P to this transformed degrees of freedom, capital P and capital Q, which are now kind of in this period double language, there is very naturally, we can imagine, a transformation that takes us, takes us from the original degrees of freedom, Q and P to these new objects, capital Q and capital P unsurprisingly from what you guys have already heard. This is, this transformation precisely corresponds to or is achieved via some time dependent, time dependent canonical transformation, canonical transformation that I will call script K naught. So it's achieved via some time dependent canonical transformation, classical system, K naught of T. And of course, or a, if we were thinking about the physics of a purported time crystal, we're thinking about the physics of a purported time crystal since the oscillations are subharmonic, the oscillations are subharmonic. Well in general have that canonical transformation is periodic with the subharmonic periodicity. And it turns out that at some level, the intuition for what this canonical transformation is doing should be super obvious. So again, you have a periodic, you have a periodically driven system, a parametrically driven system, there are period double solutions. And now we want to transform to a frame via canonical transformation where there are kind of new momenta and positions which are slowly varying, what's the way to do that? Literally to go into the rotating frame that's rotating at the same frequency as the subharmonic, it's totally obvious thing. So the intuition, good, I see nodding. The intuition of course is very simple. It's in fact that really what we're doing is we're going into a rotating frame that's rotating together with what we think should be the emergent subharmonic oscillation. Nothing very complicated, but this is the machinery that's very natural to analyze the problem. And of course, this example, in this example we would have M equals two because we have period doubling. It turns out as I've kind of already implicitly been writing down in terms of the form of the trajectories, in the rotating frame, in the rotating frame, U Hamiltonian, which is written in terms of the variables capital Q and capital P is in general still time dependent, is in general still time dependent, but as I emphasized, for those familiar with parametric resonances, the time dependences of capital P and capital K, I mean you're rotating at the frame that is the subharmonic response of the system, that's the dominant basically frequency, and P and Q are now going to be varying slowly in comparison to omega D. So what this means that now for this system in the rotating frame, the natural frequencies of the system are now kind of far off resonant from omega D, which is precisely not the case that we had in the beginning, because that we had in the beginning was when omega D was approximately two omega naught, that's where you got the parametric resonance, but once you go into the rotating frame that knows about the subharmonic response that knows about the subharmonic oscillation, it's just a canonical transformation, sort of basis change, now you've rotated away most of the time dependence of the situation, and you're now left with slowly varying Q of T and slowly varying P of T, because they're slow, we can say that the natural frequencies or the time scale that governs those dynamics are now off resonant relative to the original driving frequency, and this off resonance precisely implies that it's natural to do some type of a perturbation theory and the type of perturbation theory that people typically do in this context is called a Magnus expansion. Won't get into too much detail there, but now in this context, it's natural to imagine, to ask, and one sort of perturbatively, perturbatively can one perturbatively rotate away, way, all of the remaining, all of the remaining time dependence, and the idea is somehow very simple, right? Physically on physical grounds, the idea is very simple. Now you're in a new situation, Q of T and P of T are still varying, but they're varying slowly, so you have some oscillation frequency that's very high, and you might imagine that if you kind of want a coarse grain or average over some description, that that averaging basically means that you sort of average out whatever the slowly varying oscillations are, cancels that out, and so we're looking for an effective static description of the physics in some new canonical transformation. So question is, can one perturbatively rotate away all of the remaining time dependence in order to get, in order to get a completely, completely time independent, time independent, what I'll call effective flow K Hamiltonian. I may forget to say the word effective later, but usually I say the flow K Hamiltonian is the actual underlying Hamiltonian, H of T equals H of T plus T, but here I'll say we're imagining whether or not the dynamics in an appropriate rotating frame is actually equivalent, canonically equivalent, canonically via canonical transformation equivalent to an effective time independent Hamiltonian and effective flow K Hamiltonian H F. They forget to say the word effective, but whenever I write down H F, I mean the time independent one after the pair of canonical transformations that we're imagining. And the second transformation, this second transformation that we're thinking about here, this second canonical transformation that we're thinking about here makes the form of something called a magnet expansion. Although I won't go, I don't have time to go into very much detail about explicitly working through a magnet expansion, one of the last reference that I'd written down from Mori certainly does it in a lot of detail. So the idea now is very simple. The intuition is super, super simple. How do we want to analyze parametric resonance, right? We just said that we have, we know that there are sub harmonic oscillations even in Hamiltonian systems, they look like parametric resonances as a classic example of that. Now I want to analyze parametric resonance and try to understand stability, time scales and things like that. How do I do that? Well, I look at the period-doubled solution. I go into a rotating frame at the sub harmonic oscillation frequency, which means that then I have slowly varying degrees of freedom and in a magnet expansion or kind of a perturbative expansion at some level in powers of one over omega D, I ask myself, is it actually possible to have another canonical transformation that have a composite with the first rotating frame one would essentially take me from my time-dependent initial H of t to some effective flow-K Hamiltonian that's entirely time-independent. That's the question. Again, to very, very quickly summarize, our current strategy, the sort of strategy for our current analysis, analysis is the following. We'd like to find some composite, canonical transformation, K of t, which is K-magnus of t composed with the rotating frame transformation, where, of course, almost by definition, since we're analyzing the sub harmonic solutions of the differential equation, where Km of t equals K of zero. So the canonical transformation doesn't come back to itself every single period, but comes back to itself at the sub harmonic of the drive. So the strategy for our current analysis, and it's just a strategy at this point, is trying to find a composite canonical transformation K of t, such that K of t takes in my original degrees of freedom, q and p, and takes me to a basis-transformed degrees of freedom, capital Q and capital P, and most importantly, such that the Hamiltonian in terms of these new variables, capital Q and p, hopefully is completely time-independent. And when this is possible, sometimes this is referred to in the literature as crypto-equilibrium. Crypto in the sense of hidden, and equilibrium in the sense that your original degrees of freedom in your original system is via canonical transformation equivalent to a time-independent, and hence equilibrium effective flow K Hamiltonian. At the moment I haven't actually analyzed anything about symmetry breaking, anything about time crystal, I'm just telling you how one would generally analyze the physics of stuff near parametric resonance. Yeah, sorry? Yeah, the solutions are not stable, that's right. I don't understand the question, sorry, in what rotated frame? What do you mean, I still don't understand, sorry, maybe let me say something and then you tell me if you understand or if it's actually, in terms of the denominator, we solve just like Faraday waves. There is a solution in the linearized equation with epsilon set to zero, where there's basically, you have a period double solution, but it has this exponential blowup. When you have this Q to the fourth term, it turns out it regulates the blowup, in some sense the amplitude can spread to other K-modes so it doesn't blow up anymore. But now what's the, in this case now it's still, that solution is still at the sub-harmonic frequency, it's just that the exponential blowup is now regulated. So here, going into the rotating frame over here, I'm still going into the same rotating frame at omega du over two. What do you mean by the resonance? Well, what's the resonance case? Yeah, but it's okay, it's no longer actually so, let me say how that's happening. So you can, it's like a, I think I understand the question, but it feels like a chicken and egg question now. So the point is that, there are period double solutions when omega d is near two times omega naught, for example. And then you say, okay, but wait Norm, if I now transform myself basically, and I'm now rotating at omega d over two, now I have to add that and now it's no longer resonant. But the whole point is that, that's not quite the right way to think about it because you are actually saying that, the way you're thinking about it is actually, you're looking, you know that the dynamics of the system take this form and now your frame is sort of rotating at that rate. So at the end of the day, the description of this at the end of the day is that omega d is close to two omega naught. But then because you know the solutions take this form, now in the rotating frame, when you're rotating at this, now you see that the effective degrees of freedom P and Q are slowly varying. Is that clear? Okay, good, good, good, sorry, good, good, good. Yeah, okay, very good, other questions. Again, at this stage, nothing really about time translation symmetry breaking, just as a general rule of thumb, if you had harmonically changing solutions, how would you try to analyze it? You go into rotating frame to try to remove that time dependence because it's trivial. And then after it, if there was some additional time dependence, but you happen to have some small parameter for perturbation theory, you try to do perturbation theory. It's basically what every physicist tries to do. Good, okay. So now the caveat of course, so I've given you a very, I think, clear strategy for how one would naturally, I think sort of intuitively, even at first year physics approach the problem. And I would say that in fact, we have, we have indeed introduced, we've indeed introduced in broad strokes some broad strokes strategy for how to analyze the system, but we certainly have not proved that it works. What do I mean by that? Well, the first thing that you can kind of already guess that I'm going to say, given that I'm talking about expansions, is that there's no statement that I've made that says the madness expansion, this frequency expansion, the madness expansion need not converge. So it's a very natural strategy for analyzing the problem. And we will precisely try to understand in detail when it works, when it doesn't work, what are the subtleties for when it works. But although we've introduced this broad stroke strategy, we've never proved that it works. And one way it might fail is that the madness expansion would not converge. And if the madness expansion doesn't converge, we would say that in which case the pair of objects, basically the canonical transformation K of T and the effective flow K Hamilton that gets out, HF would also not exist. The roadmap at this point, just to be super clear, there's a strategy. I haven't said that the strategy is always going to work. We'll analyze in detail when it does work and the subtleties of when it doesn't work. But for a little while, I'm going to assume that it does work or at least it works for sufficiently long times that you can think about the fact that it's useful up to those time scales. And under the assumption that it does work out, although it will not always work out, under the assumption that it does work out, I will try to give you a different perspective on how to think about discrete time translation symmetry breaking. And the idea conceptually is that I'll want to equate discrete time translation symmetry breaking to the effective breaking of an internal symmetry of this time independent Hamiltonian. Super powerful actually. This idea is quite important in general in flow K physics actually. So what I'm trying to do ultimately here, again with caveats about existence or not, is if I do assume that HF exists, I will now try to give you a perspective that one way or a different point of view of understanding discrete time translation symmetry breaking is in fact understanding it as spontaneous symmetry breaking of an internal symmetry of this effective flow K Hamiltonian. So you can either think about it as spontaneous time translation symmetry of the entire flow K problem, or in the appropriate rotated frame as spontaneous symmetry breaking of an internal emergent symmetry of an effective time independent Hamiltonian. And that's very beautiful, okay. But before we get there, just a couple of remarks, given that we've introduced this broad strategy, a couple of remarks. The first remark, which is something that we'll again come back to towards the end of this lecture where we start talking about different strategies to break ergodicity and different types of time crystals. The first remark is whether and how this Magnus expansion converges, whether and how this Magnus expansion converges is precisely the difference, a little bit of foreshadowing, the difference between what some people call true time crystals, which correspond to the situation in which the Magnus expansion does actually converge has a finite radius of convergence. And what are oftentimes called exponentially good exponentially good time crystals, crystals where in fact the Magnus expansion may not fully converge, but maybe converges out to very, very long time scales. And again, this was something that I was emphasizing at the end of the lecture. There is a sharp physics question from a statistical mechanics perspective here about whether or not the lifetime of a time crystal diverges exponentially in the system size or naturally as a thermodynamic limit is taken. On the other hand, from an experimental perspective, it's not really clear that one can distinguish between these two things at all if the lifetime is exponential in some parameter, whether or not it diverges a system size, maybe more of a pedantic point than a very, very practical point. But the distinction between these two is precisely whether and how the Magnus expansion converges. And I would say that in particular, we will see in this lecture, we will see the multiple cases, multiple very nice cases where it turns out there is some residual time dependence, let's call it V of t, that doesn't get fully rotated away even after we go into the rotating frame and then we also try the Magnus expansion. We're never able to fully converge the Magnus expansion. And in this case, we would say that in fact, we would have, for example, we might say that following the Magnus expansion, we now have an effective Floke Hamiltonian but we have some residual time dependence left over so it's not fully rotated away. And the idea is that depending on how small V of t is, if V of t is extremely small, that it's in fact possible that Hf, this effective time independent effective Floke Hamilton, describes the dynamics for extremely long time scales, and that depends on the size of V of t. And it turns out for many of the cases, and in particular, the cases of exponentially good time crystals, it is precisely that in some sense, the one norm of this V of t can be made exponentially small in some parameter. It's not exponentially small in system size, which would be kind of what we would want from a thermodynamic perspective, but maybe it can be made exponentially small in temperature and frequency. So there's some way that you can control the amplitude, the size of the piece of the time dependence that you can't rotate away, so you can make it very, very small. And if that is indeed the case, then of course this Hf describes or captures the dynamics, can capture the dynamics to extremely long times, extremely late times. Can capture the dynamics to extremely late times. And this will precisely be the example, I think there are a couple of questions already after the lecture is on this. This will precisely be the difference between what are oftentimes called pre-thermal time crystals and depending a little bit on questions of its existence on many body localized time crystals. Okay, maybe just because we've been using the language of classical mechanics for so much of this, I should emphasize that in the quantum setting, in the quantum case, at some level, this effective flow K Hamiltonian always exists because I can always take a logarithm of the flow K unitary. So we should really imagine that in this case we want to replace the word exist from the caveat over there with local. In general, in the quantum case, the question of whether or not the flow K Hamiltonian exists is really more question of whether or not it's local or quasi local as opposed to it being strictly defined. Now that we're talking about the quantum case, I should just for completeness. In the quantum case, just to get, again, maybe the statement is to get the nomenclature right. In the quantum setting, the quantum setting, what I want to do is essentially, intuitively, the exact same strategy. But now I'd like to change from my canonical transformation to a unitary transformation just to write down a set of equations for completeness. In the quantum case, the dynamics of the system are generated, for example, by time evolution under this flow K unitary. In this case, it's the time ordered exponential e to the minus i integral from initial time t naught to final time t one of whatever my Hamiltonian is h of t prime dt prime. And now it's truly the same language. Now what we'd like to find is to find some, I'll keep the same language, I'll keep exactly the same notation, but you should remember now that K is gone from a canonical transformation to a unitary transformation. We need to find some K of t, a unitary transformation, a unitary transformation such that u of t one t zero as that ideally we can describe the evolution as K inverse t one e to the minus i from t one to t naught an effective time independent flow K Hamiltonian K. All I'm saying now is that the way you think about canonical transformations over there, in this case, a unitary transformation corresponds to conjugation basically. And so since we have two times on the right, we have K of t naught on the left, we have K inverse of t one, that's what we expect. And again, of course, if we had a sub harmonic response or an M fold version of discrete time translation symmetry breaking, then the unitary would have the property that K of MT is equal to K of zero. And I suspect that there are some low K quantum efficient autos, can somebody remind me in the literature on quantum flow K systems, what is this K of t usually called? What is this transformation usually called? What? The one more minute, what? Kick top, I'm not sure, don't exactly think so. I haven't heard that language, it might be that language, but I think, really? I say, oh, you've heard this, okay. Well, maybe then maybe the kicked operator, I think I normally think about it as what's called the micromotion. It is, the symmetry basically, well, there is at some level of symmetry, it's time translation symmetry. If you broke the time translation symmetry, then you would not have the pre-thermal time crystal. But it turns out, I think it'll become more clear in just a second, that in some sense, the emergent symmetry that exists, that will emerge in HF is really protected by time translation symmetry or generated. It is, I think the right word to say is the manifestation of time translation symmetry. Super good question, actually, very, very good. So, building upon that particular perspective, we'll now exactly try to unpack the question that was just asked. How, I've emphasized that somehow, there will be a way to think about discrete time translation symmetry breaking, or S-Tow-B, as a version of spontaneous symmetry breaking of the effective Flo-K Hamiltonian. So, how do we do that? Let's unpack that. So, thinking about S-Tow-B as spontaneous symmetry breaking of the time independent effective Flo-K Hamiltonian, and I should say, again, if it exists, because we haven't said that it always has to, or if it's useful for sufficiently long time. I'm just emphasizing, again, these kind of two contexts. So, over here, okay? Well, let's assume it exists. And when it exists, it turns out there is a beautiful way to understand discrete time translation symmetry breaking. So, let's first start with the obvious statement. Clearly, if, and we'll just use the same notation over here, clearly, if your micromotion, if this particular unitary transformation that corresponds to your micromotion, which is not kind of your stroboscopic, not your kind of slow stroboscopic dynamics that are governed by HF, but the harmonic, but the piece of the dynamic that have harmonics of omega D over M, which are now the micromotion. Clearly, if this micromotion is periodic, if your unitary transformation is periodic, such that K of a single period T equals K of zero, then there is no S tau B period, right? Because you know that we said that K was the thing that got us to this effective equilibrium thing. And if this itself is now periodic, and the other thing doesn't have any time dependence, there's no way to get anything subharmonic. There's no S tau B. But of course, that is the case that precisely we're not studying and we have not been talking about. But of course, but I would say crucially here, but crucially, in a time crystal, in a time crystal, the frame, the rotating frame, or the rotating frame plus magnet expansion, the frame indeed rotates subharmonically, indeed it rotates subharmonically, but this is true. And what that means, I'll write it here, it's very, very important, so I maybe should have started over there, but I'll write it here anyways, because it's so natural. But crucially, in a time crystal, the frame rotates subharmonically. And what this means is that if you look at the transformation that corresponds to X equals K of zero, K inverse of T, this is not equal to the identity. If in fact, K of T is equal to K of zero, then they're the inverse of each other. So in this case, it turns out that when the frame rotates subharmonically, you can define an object X, we'll understand the meaning of that object in a lot of depth in just a second as K of zero, K inverse of T, and now it acts non-trivial. It acts non-trivial. And what's the intuition? You already can almost guess what the intuition is from looking at the form of what the unitary transformation does. The intuition is that this object X permutes, it permutes through cycles of the subharmonic response. And of course, already by definition, X to the M then will be equal to the identity. Kind of obvious, you know? The magnet's expansion phase is a little confusing, but if you think about just the rotating frame piece of it, if M is equal to three, what the rotating frame is, it's rotating along with whatever the subharmonic piece of it is. And precisely because you have a subharmonic piece of it, the thing that you do at zero and at the period afterwards does not cancel, and the piece of it that does not cancel is precisely the piece of it that moved with the subharmonic oscillation. So this object X is really like a symmetry operator at some level in the sense that it permutes through the M cycles of the subharmonic response and has X to the M equals one. Plugging in the definition of K over there, plugging in for the definition of K into the definition of a time translation symmetry, which is that U at from zero to T is the same as U from NT to N plus one T. If you literally plug in the definition that we wrote over there into the manifestation of time translation symmetry, you will find that with respect to the flow K, the effective flow K Hamiltonian, you will find X inverse E to the minus I, a period HF, X is equal to the minus I, T, H. This is something that actually has been known for a little while, but was first pointed out in this context by Dominic Else, Bela Bauer, and Chayton Hayek in 2017. And actually this kind of idea of these frame-dependent transformations goes back to Holt Haus and Flatt 30, 40 years ago, but in the specific context of thinking about, now we're discussing specifically the quantum flow K setting, at least we're using the language of unit Harries. This was pointed out by Else Bauer and Hayek in 2017. So what in words does this equation mean? This equation means is that X is an internal ZM symmetry, symmetry of HF, right? So go back to remember, at some point what we talked about with symmetries, symmetries are transformations that leave the equations of motion invariant. This is the description, if HF is time dependent, then this is the generator of stroboscopic unitary flow K dynamics, conjugating by X leaves it invariant. So that precisely means that X is an internal ZM symmetry of HF. And you might have thought, wow this is kind of weird, like what's going on? Well we did all this crazy stuff so far, you felt kind of natural, I mean you just follow around your nose every single time, we were trying to rotate away time dependencies, but now we said that if we get to the point where we've successfully rotated away all the time dependencies, and we have this effective time independent flow K in Hamiltonian, now somehow it is a symmetry? Where did the symmetry emerge from? At some level it's super obvious, it's just a manifestation of the original time translation symmetry. It's just literally the rotating frame manifest in terms of this internal symmetry. So I would say that it's simply a re-expression, a re-expression of the original original time translation symmetry, time translation symmetry, but rotated by X at each time step. And this now is very related to the pre-thermal question that was just asked a little while ago. In some sense it is protected by, I'm not sure it's quite the right word I would use, but I think it's okay to say that, in some sense the ZM symmetry is protected by or only exists because of the time translation invariance of the original flow K system. If you broke the time translation invariance of the original flow K system, if you had some jitter so that your system wasn't truly flow K, it didn't come back to itself every period, you would also not have this exact ZM symmetry of the flow K Hamiltonian. So this is just at some level machinery, it's just algebra that we've gone through. But now let's really take a conceptual step back and in particular, harken back to the first lecture that I gave and in particular suppose that we have spontaneous symmetry breaking of X in the equilibrium effective flow K Hamiltonian. In the rotating frame. By definition, what we have in this case is that the state space breaks up. If it broke up, if you were breaking a discrete ZM symmetry you'd expect there to be M different ground states. That's what I mean by the state space breaks up. The state state breaks up into M sectors just like any good symmetry operator thinking back to the classic case of a transverse field ising model. There are two states of the ferromagnet. They are literally the thing that moves one from the other or permutes me across the different sectors. The two sectors being all up and all down is the symmetry operator. In that case, I've represented it as X here but in that case it is actually X. It is just Sigma X along the entire chain, right? Flipping all spins by 180 degrees which is also the generator of the ising symmetry in that case. In this case, the state space breaks up into M sectors and the sectors are indeed permuted by the action of X in this case. So X plays the role, as we've said here, of the symmetry operator. X plays the role of the symmetry operator. So the sectors are commuted by action by the action of X but crucially, as we've already thought about many times in the context of spontaneous symmetry breaking where the parent Hamiltonian has a larger symmetry than the state space, the sectors are permuted by the action of X but are not connected otherwise by the dynamics of HF. The sectors are permuted by the action of X but are not connected otherwise by dynamics under H. So again, recasting this in slightly older language that we've seen already in my previous lectures, we would say that in fact, in this case, there exists M different, there exists M different ensembles of initial conditions, ensembles of initial conditions in this case, if we have the spontaneous symmetry breaking of a ZM symmetry of HF, there exists M different ensembles of initial conditions. Let's call those different initial conditions YI. So I runs from one to M describing the M sectors. Such that Y, let's say, I plus one is equal to XYI. This is just writing down what I mean when I say that X permutes me through the different sectors. And of course, by virtue of symmetry breaking, there exists some local observable O such that, again, using the averaging notation but I don't think that's required. The average of O within one of these initial condition ensembles, which let's say by definition is averaging the observable to tau, the operator evaluated in this ensemble of states time evolved stroboscopically. This would say not equal to, for example, the average of the observable in any of the other sectors of the initial ensemble conditions of the ensemble of initial conditions, you can almost guess what's going to happen. So, again, what has happened here? What's happened here is we've said, ultimately, and just follow our nose, if we take a transformation and we try to get to this effective time independent flow K Hamiltonian, we'll find that time translation symmetry imbues this flow K Hamiltonian with a ZM symmetry. And in fact, where the ZM symmetry where the M corresponds to the subharmonic associated with the drive, the subharmonic oscillation frequency associated with the drive, and then if we now, for a second, forget about any of the transformations that got here, forget about the rotating frame. Now, we just analyzed the static problem of spontaneous symmetry breaking of a ZM symmetry in HF, that everything I've said here is true. Spontaneous symmetry breaking corresponds to the either finite temperature or ground state breaking up into M different sectors. They're commuted by the symmetry operator AX, but not connected by the actual Hamilton in HF. And because they're symmetry breaking, there's a local order parameter, and that local order parameter will look different in each of the different sectors. It's a sigma Z for this and sigma Z for this. One is pointing up, one is pointing down, but in the general case here, there could be more. And the whole point here is now very, very simple. Principle, now as soon as you transform, transferring back to the lab frame, so undoing either the canonical transformation or the unitary transformation, and then plugging into the literal definition of S tau B that we had in the middle of the last lecture, you will find precisely that that's satisfied. But the way to think about it is really even much simpler. I mean, this is a mathematical description, but all we're saying is that in principle, if you had an effective Flo K Hamiltonian and you had symmetry breaking, so let's say it found out that it was in the all downstate. If the rotating frame is flipping your head by 180 degrees, then in the lab frame when you go back, if in the rotating frame it looks like this, when you undo the rotating frame and you go back to the lab frame, then it looks like this. So it's all it's saying is that it's very natural from this rotating frame perspective to think about discrete time translation symmetry breaking in terms of spontaneous symmetry breaking of an emergent ZM symmetry in the Flo K Hamiltonian. And it turns out this will be very important for a couple of reasons, but it's not required to understand time crystals. One doesn't have to use this perspective, but to analyze the stability of time crystals and also to see distinctions between pre-thermal time crystals of different versions, it's very, very helpful to have this picture in mind. Sorry, please. Can't. So it turns out that essentially basically the, so I can say exactly, so at some level it's a very natural question. So you're saying, what he's saying is like, okay, there's something that happens in this time translation, and now I've said that basically, there's this ZM discrete symmetry that emerges for HF. But what happens if instead the symmetry that emerges for HF is a continuous symmetry? Turns out it's impossible. And the reason is because the time translation symmetry, remember the table that we had was already discreet. Because we're in the Flo K setting, we have a discrete time translation symmetry and that discrete time translation symmetry can only be broken discreetly. And that discreteness is precisely the discreteness of this subgroup over here. Questions? Excellent, excellent, excellent, excellent. So the general principle, the general principle that I'd like us to keep in mind at this point is that in fact, in essentially all of the time crystalline phases that people in modern research have discovered or explored, this M fold S tau B is essentially equivalent or manifests in the rotating frame as spontaneous symmetry breaking of some emergent ZM symmetry is in the rotating frame. So the general principle is that the way you can understand discrete time translation symmetry breaking is as spontaneous symmetry breaking of an emergent internal discrete symmetry of a time independent Flo K Hamiltonian, which may or may not exist in the rotating frame. So I've put off this question of may or may not exist many times. I've said all the times I've kept saying, well, it may or may not exist. Let's assume it exists. It still may or may not exist. So now let's finally, at least in one case, talk about a situation where it exists. In particular, let's go back to the case of the, well, okay, we won't just do single. So back to the case of a parametrically driven nonlinear oscillator. So now I will make one sharp existence statement. There were many caveats and I will make one sharp existence statement, which is the following for a single oscillator, a single nonlinear parametrically driven oscillator, or a single parametrically driven nonlinear oscillator, clearly not what we would call a phase of matter in the sense that there's no thermodynamic limit, but I can make sharp statements here. For a single nonlinear parametrically driven oscillator, it was indeed proven that H, that HF exists for a finite volume of phase space and HF is stable to arbitrary small perturbations, arbitrary small perturbations. Not much, we're getting there. So, yes, sorry. Yeah, sorry, say it one more time. I just couldn't hear the first part of the question. It's my fault. Yeah, exactly, exactly, exactly. Generator of that symmetry. Exactly, yeah, exactly. I don't know what we mean by exactly remaining, I would say. Like, at some level, there was no internal, like right, at the starting point, there is H of t equals H of t plus t. It's a flow case system, but there's no internal symmetries with that Hamiltonian whatsoever. And then I did a series of transformations knowing that I'm close to a subharmonic response. And if there is a subharmonic response by doing this transformation, which may or may not converge, I got to a Hamiltonian HF, which was time independent, and that Hamiltonian now has an internal ZM symmetry. So it's not, I wouldn't use the word sort of, it's the one thing that remains, it's the thing that emerges basically from not having any internal symmetry, but it is just a manifestation of the time translation symmetry in this rotating frame. And X is the generator of that, exactly, yeah, please. Whoa, let me think about it. Okay, can't imagine it. I mean, I'd have to think that you, maybe one way is to have like two time translation symmetries, so you can imagine somehow, like I'm immediately thinking about like, kind of like quasi periodic drives, although I'm not immediately sure. I mean, you kind of have to have some way to not have stuff commute. So one is not good enough, so I need at least two. So then I don't know exactly, I suspect at least in the simple case of quasi periodic drives, I still don't think it's not a billion, but I'm not 100% sure that would be the starting point where I would look for stuff like that, but you'd want at least something where like, yeah, there's like kind of two time directions that are maybe incommensurate in some way so that you can imagine symmetries, some generators of the two that don't commute with each other, but it's hard for me to imagine how that would work immediately. Good question though. So it turns out that indeed, yeah. Ah, sorry, sorry, so you're saying like, what happens with the fractional sub-harmonic? So like if you had a sub-harmonic response that was like, yeah, N over M or something like that times omega D, then what would the internal symmetry of that look like? I think it would still be a ZM symmetry, but then the way you would break the symmetry is not to fully break the symmetry down to nothing, but to a sub-group of that. I think it's gotta be something like that, although I'm not 100% sure. So it turns out that I keep talking about HF and whether it exists or not, and it turns out that for one non-linear parametrically driven oscillator, it was proven that HF exists for a finite volume of phase space and that it is stable to arbitrary small perturbations. This proof was kind of started, but I think not fully done by Chirikov in 1979. And then I think it was put on pretty rigorous footing by Zounds and Rand in 2002. And in fact, in the Zounds and Rand paper, they call this object HF of Q comma P, they call it the Hamiltonian. Yes, it is a play on words. So funny story is I was trying to write a paper on this and the editor kept thinking that I spelled Hamiltonian wrong. I was like, no, it really is Hamiltonian. It was a play on words. But yes, so Zounds and Rand prove and construct HF of Q and P. They call it the Hamiltonian and the reason they call it the Hamiltonian is indeed because they use KM, the Carnold-Kolmogorov-Arnold-Moser theorem to prove stability. They rely on the KM theorem, which again is a property of stability in classical few-body dynamical systems. They use that to prove the stability of this flow K effective flow K Hamiltonian. Yeah, okay, I see. Oh God, okay, sorry. When did this happen? I hope it wasn't like. Okay, okay, I see, I see, I see, I see. Okay, all right, no problem. Okay, good, I see. I've been mostly saying the words that I'm saying, writing, but nonetheless. So for a single parametrically driven nonlinear oscillator, it was proven that this effective flow K Hamilton exists for a finite volume of phase space and a stable to arbitrary perturbations started by Cherikov in 79. I would say formally airtight by Zounds and Rand in 2002 and Zounds and Ryan called HF of Q and P, the Hamiltonian, because they used KM to be able to prove stability. So the relevant question now is at this point, we know that it works at least HF exists. We have, it's not thermodynamically, there's no well-defined thermodynamic limit, but at least we have the other properties, you know, discrete S tau B that's stable and rigid and that we understand in this language for a single nonlinear oscillator. Perfect, fantastic, excellent. And the question now we should ask is going back to the Faraday wave case, what about the kind of many body generalization? You have to make an executive decision at some point about what to skip. What about the many body generalization? So in this case, just to kind of explain a little bit, a little bit of this, we would say that again, there's now many nonlinear harmonic oscillators that are all parametrically driven. So the oscillators are indexed by I, we imagine we have PI squared over two plus omega naught squared over two, one plus delta cosine omega DT, so parametrically driven, QI squared plus epsilon over four, QI to the fourth. And now let's say that we can imagine that there's a coupling between all of the oscillators, could be nearest neighbor coupling. Of some form, just position wants to be aligned with each other, QI minus QJ squared. So again, we've gone through the construction of trying to go into a rotating frame, get to an effective Floke Hamiltonian. We understand the physics of discrete time translation symmetry breaking in terms of spontaneous symmetry breaking of the emergent ZM of HF. And it turns out that at least for a single parametrically driven oscillator, HF exists. And the existence means that the magnus expansion converged, which means that there is no time dependence that you cannot rotate away. It turns out that for the many body generalization, the system that I've written down here, that will not be the case. So of course, if one considers the exactly uniform, perfectly period-doubled initial condition, exactly uniform initial condition, if one considers the exactly uniform initial condition such that QI and PI are let's say, the period-doubled solutions are Q star and P star, then clearly this just decouples into a bunch of single oscillators because everything will move together because there's no interaction term. So this reduces to the one body problem to well, N copies of the one body case, one body problem. And we already know that this one is has KM stabilized, stabilized S tau B. But now, usual the fly in the ointment is, but as one now considers finite volume of initial conditions, you consider QI equal to Q star plus delta QI. What do these different initial conditions, delta QI correspond to? At some of those uniform here, I just have the K equals zero mode. Again, thinking back to sort of, you know, this K analogy. As soon as you consider a finite volume of initial conditions, your initial condition immediately has weight in the higher K modes, Q sub K. It's very similar to the obstruction that we were talking about when we were thinking intuitively about Faraday waves. You have weight in higher K modes, Q sub K that are all coupled, that are essentially all coupled by this Q to the fourth non-linearity. So you can try, you can attempt, you can attempt to construct, you can attempt to construct either the unitary or the canonical transformation K of T. And oftentimes you can try to do it the most natural way since you know that you have stability for the single particle case, the single oscillator case, is to do this perturbatively in the strength of a coupling G. But we'll find that as a general rule of thumb, the Magnik's expansion does not converge. So let me, let me just say a couple words. You all have access to my lecture notes. It turns out that I'm relatively far behind where I want it to be, but such as life. So what I was going to do at this stage was to kind of, I'm telling you from a perturbative line where we've now discussed everything in terms of this Magnik's expansion. And I keep saying that Magnik's expansion doesn't converge and that's this representation of the fact that you cannot rotate away all the time dependence and you can't rotate away all the time dependence that at some sufficiently late times there will be something that happens to the system that is not going to be just discrete time translation symmetry breaking. It'll be some physics that's governed by the small v of t that we were talking about. And here I was going to introduce kind of the ergodicity perspective for this obstruction, but I'll just say it in words in 30 seconds. There is kind of a very nice, I think it's essentially the same physics but a very different way to think about this. There's the ergodicity. So one way to think about the obstruction is that literally you try to construct K and the Magnik's expansion doesn't converge. Super mathematical. And nice. Another way from this perspective of ergodicity is actually the fact that in general, you expect, I'll say it in words, in general the simple expectation for a many body system that's periodically driven or periodically shaken is that it does what? I'm sure many people know this, is that you shake a system, eventually the system has to absorb some energy from you as you're doing work on the system. So what ultimately would you expect to happen? Exactly, that's exactly right. So it turns out that there's another way to see the obstruction of this in general in the many body. In a single particle problem, you can't really heat up. You know, you kind of give back as much heat as you take in. In the many body problem, you heat up and you can spread that heat out diffusively maybe Anatoly into the full system. But there's sort of an ergodicity plus flow K version of the obstruction, which is really that a periodically driven, that a periodically driven many body system, any body system, whether classical or quantum mechanical will ultimately absorb energy energy from the driving field. Driving field and at late times will heat up to infinite temperature such that the reduced density matrix of any subsystem, I guess the quantum language here is effectively infinite temperature. Ergodicity or in the language of what Anatoly was talking about thermalization, thermalization says that if you have a generic interacting system at sufficiently late times, subsystems will look like the Gibbs's ensemble. The statement of thermalization or ergodicity in a flow K system where energy is not conserved is that because energy is not conserved, there is no E to the minus of beta H to look at. The only thing that you can get to is that the subsystem will at the end of the day be infinite temperature and this infinite temperature you can understand as lack of energy conservation or as a manifestation of the fact that you're absorbing energy from the periodic drive. So there was quite a number of pages of lectures on this but nonetheless that's at some very, very high level and ergodicity based picture for sort of how to think about that obstruction. Also valid to think about it from a magnum expansion picture, not converging. Okay, but in the last couple of minutes of lecture, let's actually try to make the table of time crystals. Get this stuff. So again, what I like to do now is just very, very quickly just remind ourselves about the different dynamical classes, the different dynamical classes that we kind of have talked about already as being relevant for physical systems. So we said there was classical closed. That's Hamiltonian, Hamilton's equations. There's classical open finite temperature. We said that was Langevin. That there was quantum closed which we thought of as unitary. There was quantum open finite temperature which we thought of as Linblad. Going to each of these four different cases. Okay, so now say a couple of words that are very important now. So the entire time I've tried to show you that the description of time translation symmetry breaking just the symmetry breaking aspect of it is not quantum mechanical at all. It's just some property of the breaking of some emergent flow-K Hamiltonian that could happen just as well in a classical, so there's nothing intrinsically quantum mechanical about it. But at the end of the day, it turns out that there is an obstruction to getting infinitely long lived or stable discrete time translation symmetry breaking and that obstruction can be understood as ergodicity. But it turns out that I would say in these four different dynamical classes, there may or may not be distinct ways of breaking ergodicity that are either intrinsically classical or intrinsically quantum. I would say one way of breaking ergodicity that's intrinsically classical of few body, KAM. Certainly there's no, well, at the moment there's no known analog of quantum KAM. So the way I would think about this is as follows. One, if we're thinking about sort of doing research on time crystals. What is the idea? One, you sort of start by identifying the dynamical class that your discrete time update rule lives in. And again, I've emphasized there are some classes where it turns out that just getting the symmetry breaking is relatively easy. So that all to all interacting systems is one example, purely dissipative system, contractive maps is another system. But for most of the stuff that people are looking at, they're looking at these four systems, but you pick which one you're looking at. You identify this dynamical class and now you try to find some strategy to evade fully or to delay ergodicity. But again, by ergodicity, I don't just mean kind of thermalization or that subsystems look like Gibbs ensembles. I mean specifically in the flow K context, you avoid this slow heating to infinite temperature. So you find some strategy to evade or declare ergodicity, but specifically in the flow K system, you could equate this ergodicity and flow K. What we really mean by this is you avoid or delay the kind of heating that you naturally imagine should happen in such a many body, periodically driven system. And as soon as you find some strategy to evade or delay, if you find if successful, if successful, I promise you, you can call me and we will find a time crystal. So what I'm really emphasizing here is that in a way, the newness, the interesting piece of this particular time crystal as a flow K phase of matter, and the reason why there's a tremendous amount of modern research on it, isn't necessarily just the fact that the breaking of time translation symmetry is something that people hadn't thought about. It's really that there's a lot of interest as you heard from the last four and a half hours of lectures from Anatoly. There's a lot of interest and somewhat advances in identifying new strategies to evade or delay ergodicity first in equilibrium systems, and then maybe also even a little bit harder in flow K systems. So now let's go through once. I can promise you already, I'm going to go a little bit over. Sorry about that. Okay, so what I'd like to do is now to just offer, what I'd like to do is to slowly fill in what I call, and it will not be an exhaustive list, but I will slowly fill this in, what I call the great big table of time crystals. And so for the table of time crystals, we'll slowly fill this in. The first column is going to be what exactly is the strategy that one's considering to break or delay eating? Eating. Once we identify that strategy, that will lead to some version of a time crystal. We can ask ourselves, what dimensions does the strategy work in? We can ask ourselves, for the resulting time crystals, what are the initial states that exhibit the persistent oscillations? And for those states that exhibit those oscillations, what is the lifetime of the time crystalline order? How is it parametrically controlled? And maybe comments over here. So I would say the first kind of discussion of these discrete time crystals were specifically, and that's why I think there's a lot of people that feel like time crystals are intrinsically quantum mechanical objects, but I hope I've disabused you of this notion. People started thinking about it in this class three type of system, closed quantum unitary evolution. So the first strategy, which applies specifically to class three, and again, what this means is not that the time translation symmetry breaking is quantum, but there was a strategy for evading or breaking ergodicity that was quantum, and you will all no doubt guess that this started to be discussed in the context of many body localization. Dangerous, I mean, I'm sure you will hear more about many body localization soon in some of the lectures from people. The context of many body localization without going to any real detail is essentially that if you have extremely strong disorder, so if you have, for example, some potential energy landscape that's extremely disordered, or for those of you that know about glasses, that looks like it might want to drive a system into a glass, that many body localization started off and mostly I think about in kind of non-flow case systems, and these non-flow case systems, the claim if many body localization is true, and I think there is truly an if there, in these non-flow case systems is that the system is unable to thermalize, and the reason why it's unable to thermalize is it's essentially very, very difficult to move energy across this extremely disordered potential energy landscape. Things get stuck in local minima. It's unable to thermalize and thus breaks ergodicity. I think the best understanding at the moment is that probably if at all it exists, only exists in one dimensional systems, in higher dimensional systems there are, I mean it's not proven, but there are obstructions that seem like it's very hard to find arguments around the obstruction, and if many body localization in this non-flow K context survives into the flow K context, if MBL persists, persists in non-flow K context, periodically driven in driven 1D systems, then this lack of thermalization also implies an ability to evade, strictly to evade, this kind of problem of heating. So at least want to write down one time crystal and draw a phase diagram, so I'll do that, maybe as just one example very, very quickly. So in a 1D spin chain, I would say maybe this is the kind of eye-using model of time crystals or at least many body localized time crystals, we can imagine that there is a flow K system where the time dependent Hamiltonian corresponds to applying H1 for a time period T1 and H2 for a time period T, and H1 we can take to be a very general Hamiltonian, it's eye-using interactions, nearest neighbor eye-using interactions, and then some field in a random direction, excuse me, I should make sure to put in disorder here, HZI, sigma ZI, HYI, sigma YI, HXI, sigma XI. So again, we're in the MBL strategy and the disorder is coming from the fact that there is a site index on all of the parameters, so those coefficients are drawn from some random distribution. Let's imagine H2 is equal to G, a uniform transverse field sigma XI. There is a limited case of this model, which is so simple. If you imagine, it's really, truly simple, if you imagine that there's no interaction, so J or J is zero, and there's no field in the Y or the Z direction, then your initial field is just in the Z direction, so pick some product state in the Z direction, some random state up, down, up, up, whatever it is. Let's pick the timing T2 to be equal to pi over two times the coupling strength G. What this does is it implies that when you evolve H2 for a time T2, you do a 180 degree rotation around the X axis, and what that does is it implements what we would have called kind of the utterly trivial but certainly not stable time crystal from the beginning of the lecture in the morning. It would correspond to flipping each spin up to down to up to down. But of course, we know that that is not stable without interactions because if we had, for example, change from G goes to G times one minus epsilon, then those oscillations would immediately defaze on a time scale one over epsilon, just like the original problem we were talking about in terms of maps. The magic is that when you turn on interactions and all the other fields, those don't matter so much, we turn on interactions, it turns out that as far as people can tell numerically, although there is no proof, as far as people can tell numerically, it is possible to get both many body localization in this flow case system, as well as a period doubled or sub harmonic response, which is now stable, has a finite radius of convergence with respect to arbitrary perturbations so long as those perturbations respect time translation symmetry. The statement there was from some work from myself, Ashvin, Drew Potter, and Dragosh, was that effectively, if you think about the phase diagram where this epsilon plays the role of the perturbation, J is the strength of the interaction, that given the amount of disorder that one has, for strong enough interactions, it turns out that you basically have ergodicity, let's say ergodic, but for relatively weak interactions, you have this whole phase over here being MBL, but then in fact, there's a second phase boundary over here such that for large enough epsilon where you're very strongly perturbative away from the period doubled solution, you have a MBL non-time crystal for no symmetry breaking, no symmetry breaking, and in this regime over here, you have an MBL discrete time crystal. That was the intuitive phase diagram. Okay, so I already realized I'm over, so we're just gonna fill out the table and then I'll let us go to coffee. So, strategy that I've just said over here is many body localization. I said that the only, it seems like it works only dimension D equals one, at least if it does work, which there's a big if, let's put a question mark over here, we believe that it should work for all initial states and the lifetime in this case would diverge if it exists as E to the L. As E to the system size, this would be assuming that flow K MBL stable, an example of discrete time translation symmetry breaking with a lifetime that truly scales as E to the L, so in the thermodynamic limit already exponential. Something that I now apologize to your next speaker about, not covering enough, is that there is a pre-thermal, a pre-thermal version of the time crystal where essentially the intuition is super simple. At some level, if your driving frequency is very large, if it's larger than the local energy scales of your system, I said that the problem that we're trying to avoid is heating, but in order to absorb one photon's worth of energy from the drive, if that driving frequency is very large, you have to make many, many, many local rearrangements because there's no way to be on shell if your local interactions are much smaller than that driving frequency. So in that sense, pre-thermalization corresponds to a situation where essentially the time scale for heating the system for energy absorption for this challenge of ergodicity in a flow case system that time scale diverges as E to the frequency of the drive or any finite omega D, it will be cut off as you take the thermodynamic limit, but it's still scaled exponentially with something. Pre-thermal time crystals we believe should work in D equals one with long range interacting systems and for D equals two, D greater than or equal to two for short range interacting systems. The initial states here, because it's pre-thermal, it ultimately has to be, again, related to the ability to break this internal ZM symmetry. So it turns out that it's only a finite subset of states below the transition temperature associated with the ZM symmetry breaking. There's strategies that people have recently started to explore based on error correction. In this case, if there is an ability to essentially implement error correction within a Hamiltonian formalism, this works for any dimension, well, okay, I'll have to, it works for any initial state, any dimension, and here the lifetime would also diverge as E to the L. There's questions here that are related to whether or not it's possible to truly embed this type of error correction into a Hamiltonian slash Lange of formalism. So that's sort of unknown, but if you are able to use that strategy, it's very, very robust in general. And then my favorite case that we've spent so much time talking about, which is the case of the many body parametric residence, in this case, it turns out that in a closed system, it doesn't work very well, but if you couple this many body parametric residence to a finite temperature bath, the lifetime of the time crystal will decrease as E to the one over temperature. So as you go towards zero temperature of the bath, that lifetime gets extremely long. And this, again, works in principle for any dimension, but you need initial configurations, initial configurations kind of near the parametric residences. And truly, there are many, many more. Essentially, as I promised, if there is a way to slow down ergodicity in a parametric way, or even better to break ergodicity in this flow case setting, then there is in general a way to stabilize time translation symmetry breaking because it's not necessarily the symmetry breaking that's really tough, but rather it's the ergodicity breaking or the ability to evade that in a many body system that's extremely tough. Okay, I apologize sincerely for going over, but I wanted to at least write down a little piece of the table. Thanks for your attention. Thank you, Norm, for the nice lectures. So I will allow only for one question because we have to go to coffee break and you can ask normal questions you want. Yes, absolutely. During the coffee break and afterwards. So who wants to ask one question? Nobody wants to ask? I'm sure. Yes, that's right. That's why you can actually, it's more of the third class. The second and fourth, second and fourth. The second and fourth, exactly. Yeah, well, it depends on what we say. So flow K integrability, I mean, we have to be very careful about that. I'm not really sure. I mean, I think there are, integrable, non-flow K systems that if you split them apart, remain integrable in a flow K setting. I don't know that there are actually any truly unique flow K integrable systems that don't stem from integrability in a non-flow K setting. But even that case, I would say, if it's kind of related to integrability, it's probably not as robust as the things that I want to have here. But sure, at least in terms of heating, one can imagine that the fact that you have extensive conservation laws prevents the heating from happening. All right, so we reconvene at five past four. So five more minutes, five minutes delay, but let's try to be on time.