 of matter in periodically driven flow case systems. Please. Hello, hello, yeah, that's better. Can people hear me well? Okay, fantastic. Great, so, yeah, it's great to see so many of you here, some familiar faces, some not so familiar faces. Anatoli and I tried to collaborate on our lectures a little bit, so some of the discussions and themes that I'll try to reinforce were hopefully already introduced a little bit in Anatoli's lectures from this morning. And yes, as Alessandro mentioned, I'll generally be trying to tell you a story, although very slowly and carefully, about a modern research topic, which is really about thinking about and exploring new phases of matter in periodically driven or flow case systems. Just a couple of references, probably not today, but maybe sort of, you know, in the next lecture, I'll mention some stuff that I first learned in a summer school, from a Boulder summer school, exactly 10 years ago from some lecture notes from David Hughes, there's an annual review of condensed matter physics from 2020 from myself and some of our friends, and then a more recent review of modern physics on this also, this broad topic of phases of matter and periodically driven systems just from this year. And there's a very, very nice annual review of condensed matter physics from Takahajimori, also about periodically driven systems, as well as open systems. So the broad focus of the lecture today and probably tomorrow mostly as well is going to be on a topic that has received quite a lot of recent research interest that goes under the name of time crystals. Whether or not that's an apt name we'll soon see for ourselves. And what I really hope to be able to do kind of very slowly and precisely is to explain carefully what the questions surrounding time crystals actually is. When that particular question is interesting to help answer the question that I always get in lectures and talks, which is, is blank a time crystal? And I'll also, I have a love much like your previous lecture, Anatoly, for classical, dynamical, nonlinear systems. So I'll try to very, very precisely connect the modern question on time crystals to really old school questions in classical, nonlinear, dynamical systems. So very broadly speaking, the school is about out of equilibrium physics, many body quantum physics. There certainly will be some quantum, there will certainly be many body and there will be some out of equilibrium. But I suspect that more truthfully, the topic of time crystals and the one that I'll try to explain in detail over the next couple of lectures is really an intersection between three different subsets, each of which could be sort of their own, line of research and are their own line of research. Broadly that being flow K systems, so periodically driven systems. The question of phases of matter is inevitably correlated with the notion of spontaneous symmetry breaking. I'll try to very precisely explain what that is. And then, thanks to Anatoly, we've already seen questions about when a many body system or even a few body system is able to forget its initial conditions or in a broad sense be able to reach ergodicity. And it turns out the stability of time crystals or phases of matter in flow K systems in general will require some robust mechanism for ergodicity breaking, which by itself is not an easy question. So I should emphasize just in the title of the school that my picture for sort of how non-equilibrium, so let me sort of, if I had an axis for non-equilibrium-ness, I would say at one fixed point there are truly equilibrium systems, so static for example systems. And I would argue that really where we're going to sit for most of the lecturers not that far away from equilibrium, it's kind of the simplest example of a non-equilibrium system. And those are of course, as I've been mentioning flow K systems where essentially the equations of motion return back to themselves after some purity. We'll get into a lot more detail about this, but just to put some context in mind. So I think it's always really, really important, especially I wish this was the case for me when I was a younger student. I want to emphasize and we're gonna be talking for almost five hours about this broad topic, but I think when always it's extremely important as a young student to ask two questions whenever somebody like myself is introducing a modern research topic to you. So two questions that one should ask really of any research topic. And I promise to try to address each of these as we go, but they're very, very important. And I think really this is kind of, yeah at some level when matured in research one really develops a taste for the types of questions to ask. And it's very helpful to contextualize always in these two questions. The first is why now? Why is it that this particular set of questions is being asked at this present moment in time? And in the context of periodically driven systems I would say this question is particularly apt because flow case systems or in general periodically driven systems. And I apologize, I think the writing's up there anyways, but if something's unclear I'm not known for either the quality or size of my handwriting, so please just call out and I'll try to clarify. So periodically driven systems have been around, been around for a very long time. So why is it that people are asking about days of matter in these systems? Why has it become such a hot topic in terms of modern research? And I would say this is true, even kind of experimentally this is true, kind of classic NMR experiments could be thought of as many body, somewhat quantum versions of periodically driven questions. And the second question that I think is also really important to ask and I will also try to ask is that if the title of a particular series of lectures is of the form, is of the form something interesting, is of the form something interesting in the law, law could be anything. So clearly my titles of this form, phase of matter, interesting things that one learns about in physics, but I'm talking about them in the context of periodically driven flow case systems. It's always, you should always ask the question, is the in necessary? Very important. And what I mean by that is couldn't one get the same physics, couldn't one get the same physics in a non-flow case system? It feels very much like extremely broad philosophical-ish questions, but it's really important, it's really important. When people are telling you about modern research, you should ask the question, why is that particular question asked now? It could be because of an insight, it could be because the tools that one has allows one to ask that question now? One should also ask the question, if people are studying a particular phenomenon in an open system, in a monitored system, in a periodically driven system, could the same phenomenon, in a quantum system, could the same phenomenon occur without that particular qualifier? Or what is intrinsically important about the qualifier that enables the particular phenomenology that one's looking at? So I claim that generically, the answer to question two, and we'll see this in the series of lectures, the answer to the second question can be one of two cases. One case, which I think is reasonably true for time crystals, is context. What I mean by that is that the question is only reasonable, is only valid, the question's only well-defined to ask in, for example, a flow case system. So that's a case where the question is necessary in the sense that the context of the question is required for the definition, and a second answer, which is not any more interesting, but it's distinct, is that there is something intrinsically flow-k about the phenomenon that one talks about, and what I mean there is that, for example, it's somehow, for some reason, forbidden, or there's some generic obstruction to realizing the phenomenon in equilibrium. So again, and I promise I will try to always contextualize as I go where we are and how we think about this question to be able to answer these two questions to give you a flavor of why people are asking this particular question now. So with that, I think we're essentially ready to start our foray into the first topic that will make up certainly all of this lecture, and I suspect a lot of the next one, which is the topic of classical and quantum time crystals. And of course, just like in Anaheim's lecture, please, now that you're full of energy after lunch, ask, feel free to call out questions, raise your hand, if I don't see you, just call out. Okay, so I'll try to start by making sure we're all on the same page. I suspect that this will be a review for many of you, but nonetheless, I think it's important to get the language straight. I'll start by trying to ignore the word time and just introduce the idea of crystals or crystalline order in space, which is something that you're very intuitively familiar with. And I suspect many of you are also familiar with thinking about crystals as a phase of matter in the context of spontaneous symmetry breaking and in the particular context of a spontaneous spatial translation symmetry breaking. And I absolutely promise that I will unpack all of these words as we continue. So it's a very general rule of thumb and this is also true in Flo K systems. In equilibrium, we often have to think of phases of matter as defined with respect to the notion of spontaneously breaking a symmetry. Specifically in the context of crystals, we often think about it in terms of the breaking of a spatial translation symmetry and we'll sort of carry this analogy through when we start to ask the question about what a time crystal is. So I guess this board is probably still okay, right? Maybe the four boards of the center, okay, fantastic. Good. So in the context of symmetry breaking, I would say as a very, very general statement that in equilibrium, so we won't get to non-equilibrium for a little bit, but in equilibrium, phases of matter are often described and even classified, classified according to a particular framework, according to a particular framework, which is oftentimes called the Landau-Binsberg framework, of spontaneous symmetry breaking, which I'll oftentimes abbreviate as SSB. And conceptually, this framework has a very, very simple idea which will illustrate in a very, very easy way in just a little bit, but just to write out the English words. Conceptually, the idea is that in fact, the low energy states, for example, a many body system exhibit, exhibit a lower symmetry, and again, I'll be precise about what that means in just a second, a lower symmetry or a less symmetric behavior than parent Hamiltonian, than the equations of motion that describe them in the first place. So normally, we think about a collection of many particle system, a collection of interacting particles to describe by some equation of motion. Some of you might like the Lagrangian formalism, some of you might like the Hamiltonian formalism, but there's some description of the equations of motion in the system, and I'll define exactly what I mean by symmetry, but it's possible that the low energy states of that many body system, for example, that are accessed as one decreases the temperature, exhibit properties that are less symmetric than their parent Hamiltonian, and we'll illustrate this concept very simply via a very simple example, and in particular, we're going to discuss what I would say is really one of the classic kind of textbook examples, textbook examples to illustrate this concept. So we'll work this through together just to understand the notion of spontaneous symmetry breaking, and in particular to define a spatial crystal. In particular, let's imagine together, imagine together that we are writing down some theory, writing down some theoretical description of a complex scalar field, phi of x and time. So x could be in three space, could be in two space, but some theory described a complex scalar field phi of x and t. Intuitively, you can imagine that this scalar field is, for example, the field associated with the creation slash annihilation, is the field associated with the creation slash annihilation of some types of bosonic particles. If you're a cold atom aficionado, rubidium 87, if you're a more solid state old school helium four. But you would just have some scalar field as you would write down thinking about a very simple field theory, the fields associated with the creation annihilation of some bosonic particles, or just for the more magnetism inclined folks. This could also be very simply describing the distribution of some planar spin density. Oh, for example, we imagine some spin density in the xy plane, then that complex scalar field can tell me where in the xy plane that object points. So our starting point for illustrating this very simple example, and why we think about phases of matter in the language corresponding to symmetry breaking is thinking about writing down a theory of some complex scalar field. Physically it could be anything, very general, could be the creation annihilation of bosonic particles, or the distribution of a planar spin density, or the creation annihilation of cooper pairs. And when we talk about symmetry, symmetry is also a supernatural idea that you already have some intuition or some gut feeling for, I would say. A symmetry of the theory, or a symmetry of the equations of motion is some transformation, some transformation that does not change the theory. So again, just thinking about the definition, the literal definition of a symmetry, that makes sense. It's some transformation that does not change, does not change the nature of the equations of motion or the nature of the theory. And in the example, in this kind of textbook example, there are maybe two extremely common examples of transformations that one could imagine, just literally from noting the fact that we have a complex scalar field in the first place, perhaps my favorite, I guess, is a phase transformation where we can imagine that the scalar field, for example, is invariant under some phase, phi goes to e to the i lambda phi. Again, super general statement. We're not making any microscopic statement, just saying, okay, what could be an allowed symmetry given that we have a complex scalar field that we're trying to describe. And one that will be very, very relevant, very relevant for the language of time crystals is spatial translation symmetry. You might expect that, for example, the Hamiltonian or the equation of motion is the same at all points in space. So these are two examples of symmetries, transformations that could, for example, leave the nominal equation of motion to the nominal theory describing my complex scalar field the same. And we'll see why symmetries are so important in thinking about the physics of low energy configurations of our scalar field in just a second. And it turns out that in this case, you might argue that the kind of simplest equations of motion or if we're really used to thinking about the language of Hamiltonians, the simplest energy functional, the simplest energy functional that is, in fact, consistent with the transformations above, with the symmetries above, write down a potential energy of the form some coefficient, which is a real value coefficient, but I haven't told you more than that. Let's call the coefficient a2 by squared, very natural naming convention a4, phi to the four, say b2 grad phi squared, gradients are taken in space, derivatives in space, and b4 grad phi to the four. I suspect many of you have seen this as sort of a really classic textbook example, but important to get the language straight. One might write down a very simple energy functional before my complex scalar field phi, a2 phi squared, a4 phi to the four, and some gradients in space, b2 grad phi squared plus b4 grad phi to the four. I guess if you're Lagrangian aficionado, this is minus the Lagrangian. And before going further and analyzing a couple of limiting cases here, let me also just very quickly note for a second that one can also go the other way. So here I'm kind of starting from thinking about what I want the symmetries of the theory to be, and then writing down an energy functional or a Hamiltonian that's consistent with said symmetries, but oftentimes it also goes the other way, where one has, for example, some phenomenological description that's given to oneself of the Hamiltonian. So, for example, oftentimes one is given a Hamiltonian, and then in order to think about symmetry breaking or phase of matter, one then identifies the symmetries, identifies the symmetries of the Hamiltonian. And in some sense, the low temperature phases of matter that are allowed are those that correspond to symmetry broken states of that Hamiltonian that one is given. So just either way. As usual, the question that one asks when one has an energy functional like this or has written down a free energy or a Hamiltonian of some form is what are the field configurations, what are the field configurations of my complex scalar field, phi? What are the field configurations that are, in fact, low energy and thus as one, for example, decreases temperature are favored as one decreases the temperature. So I think hopefully I'm keeping everyone with me, but just in case, are there any questions at this point again? I think most of you hopefully have seen something like this. If not, it should be in some, in a few seconds a very, very intuitive way to think about why people, especially in stat mech, describe phase of matter in the context of this notion of symmetry breaking. It's important actually, the higher order terms are important actually to make the theory sort of not free, to make the theory interacting. Sure, that's actually correct, but it turns out I'll make it, when I want to make an analogy of this kind of like Mexican hat potential, I'll actually want to have that term to be able to get my Mexican hat. But you'll see in a second. Good, so the question at this stage that we're asking whenever we have an energy functional this form is what are the field configurations that are low energy? And the answer is, of course, that it depends the particular parameters that one's chosen for describes the Hamiltonian, A2B2, A4B4. And if one focuses for a second, first, on the non gradient terms on just the phi squared and the phi four term, and one considers the following case where A2 is less than zero, but A4 is greater than zero, exactly an analogy to the question that I was just asked. It turns out, you can kind of immediately see if you plotted the potential, the potential energy as a function of phi, that in fact states with phi non zero are energetically favorable, energetically favorable, and a very, very simple idea. So if we were to plot the energy as a function of phi, we'd see that there's, or some drawing just one two dimensional cut, that the energy is a function of phi, it turns out has this shape, and the minimum of the energy happens at finite values of phi, finite values in the x direction. And we say in this case, we precisely say in this case that there is, in fact, that because of this, there's the spontaneous breaking of this phase rotation symmetry, this phase rotation symmetry. Once I pick a particular value for phi in my Mexican hat, for example, now I'm no longer going to be able to rotate in this kind of xy plane in either the i lambda, because I picked a particular direction. And so in the thermodynamic limit, we say that there's a spontaneous breaking of this phase rotation symmetry, either in the language of bosons or in the language of magnets, can somebody yell out for me, what exactly the type of days of matter this corresponds to? But no one said that. Confidence. What? Yes, exactly, exactly. Perfect, so beautiful. So in the case of a, in the case of magnetism, we would say this is like an xy ferromagnet. So the particular phase of matter, so the phase of matter, beautiful, well answered, is an xy ferromagnet or, for example, in the context of bosons, a super fluid, let's say helium-4, for example, if phi was corresponding to the creation and annihilation of helium-4 particles. Good, so the concept is very, very simple. At some level, we think about writing down some equation of motion, some Hamiltonian, we look at the symmetries of that and we ask what are the low energy configurations of a field? And it turns out, as you can already see here, that in certain cases, depending on the parameter regime, the system will win a non-zero energy per unit volume in order to be in a state that is less symmetric. And thus, when one decreases temperature in the thermodynamic limit, the only states that are accessible by the many-body system are, in fact, those that spontaneously break that particular symmetry of the parent Hamiltonian. For each type of symmetry breaking, there is an associated nomenclature for the particular phase of matter that this describes in the context of this phase rotation symmetry, xy ferromagnets or plainer, easy plain ferromagnets or super fluidity is the right word. Okay, let's quickly analyze a second case and this will help answer the question why I wanted to sort of have that analogy. So in the second case, I wanted you to think about exactly the same thing, but now, essentially, don't worry about the non-gradient terms, just focus on the gradient terms and keep b2 less than zero, b4 greater than zero. And in this case, you will have that states with grad phi being non-zero are energetically favorable. Energetically favorable. States with grad phi non-zero energetically favorable and again, you can think by analogy in this kind of Mexican hat picture. And what this corresponds to is, we would say, in words that in fact, there is now a spontaneous breaking of the spatial translation symmetry. We'd say in words that there is now a spontaneous breaking of the spatial translation symmetry. Previously, we had that h, the equation motion, the description of the system, the theory, was the same at all points in space, but if the gradient, the derivative of the complex scalar field is non-zero, this means it has to be changing in space. So by definition, we now say that we have the spontaneous breaking of a spatial translation symmetry. And oftentimes, this is also called the emergence of spatial. And now let me really use the word crystals here. Spatial, for example, crystalline patterns. So the spontaneous breaking of the spatial translation symmetry indeed leads to a phase of matter that is a crystal. Because now one has some spatial pattern or some spatial crystalline pattern. It's very subtle, I understand the question, very subtle. So the question is basically, okay, if you wanted to have a super solid phase, this is a very good question, a super solid phase, it seems like basically you want to break both translation symmetry and you'd also want to break, for example, phase rotation symmetry. So I think one answer, I'm not sure this will be, this is the answer that I know the best, but I'm not sure it fully answers the most non-trivial case of this, is that oftentimes it turns out at least super solids that I know of, especially in cold atomic systems, are in different directions. So oftentimes one has, for example, stripy super solids where one has, for example, spatial translation symmetry in sort of a striped pattern in one direction and super fluid order in the sense of a phase, basically stiffness in the other direction. I don't actually know, maybe other people do, what exactly the answer is in the totally generic case. I think it's still possible to actually have a super solid where basically, in fact, there is no distinction between this, but the clearest one in my mind is where you literally have a distinction between the two. Good, other questions. So I hope what this example has sort of walked us through in a very, very slow and simple way is exactly, as I said, the conclusion is, in fact, by reducing the symmetry of the particular field configuration, of the particular field configuration of phi, this system has, in fact, gained or lowered its energy by some non-zero energy per unit volume. Sorry, I can even tell my handwriting is getting smaller, but hopefully on the projected screen, it's still very big. Good, and again, this means that in the thermodynamic limit, the symmetry of the system at low temperatures, at low energy, wants to be less than the symmetry of the original equation of motion, the Hamiltonian that we read down. And in fact, what we've talked about thus far, this is precisely the mechanism of spontaneous symmetry breaking in equilibrium. Just illustrated via this example. Yeah, please. Yeah, because there's no time dependence on the Hamiltonian, good question, good question, good question. Here, I'm assuming that there's no time dependence on the Hamiltonian, it's sort of everything static is just described by equations of motion that don't change in time. Good, so now let's build on this analogy and try to define, at least conceptually, what we would want to think about as a time crystal. And in particular, we'll again use the language of spontaneous symmetry breaking, but now instead of spatial translation symmetry, we'd like to ask about the spontaneous breaking of a time translation symmetry. So the broad thing I'd like you to keep in mind as that we'd like to think about time crystals with respect to the spontaneous time translation symmetry breaking, a mouthful to say and a handful to write. So oftentimes I'll abbreviate the spontaneous breaking of a time translation as S tau B. Yeah, please, sorry, why am I saying, say it, I just can't hear the question, say it one more time, why am I saying, yeah, I heard the first part, just the second part. Okay, perfect, even better. So why are you telling when this expectation value of phi not equals to zero to be phase rotational symmetry and when this expectation value of this derivative of phi not equals to zero to be spatial translation? Sorry, I must have not explained the picture well enough. Good, so this phase, we have a complex field phi, right? And you can imagine that complex field is literally just some polar angle, e to the i theta, r e to the i theta in the x, y plane, right? And if I say that it's invariant under this, it means it's invariant under any phase rotation basically in this plane, right? But if I've picked a particular value for phi, then I've picked a particular angle in this complex, in this complex plane and that means that that particular angle that I've picked, if I multiply by e to the i lambda, it'll certainly rotate, but it will not be invariant anymore unless lambda happened to be the trivial case of two pi. So that's what I mean by if you have a quantum expectation value, but classically just having states that have that property, then you can say that basically that phase rotation is broken. For the translation symmetry, I think it's even nicer, it's even more intuitive. So spatial translation symmetry says that the equation of motion, the description of what happens is at every single point in space the same. And in order for a state to have that same property, there must be no gradient. It must be that for example the scalar field, this complex scalar field phi, grad phi has to equal zero because if there was a gradient that means that at any epsilon x, there's some change and that change would mean that at some later location, just following it, you would see that basically it's no longer the same phi. So knowing that the gradient of phi is non-zero means that you have to have broken the statement that h is the same at all points in space. Clear? Excellent. Okay, so let's now talk about sort of the exact analogy for spontaneous time translation symmetry breaking. And this I think is precisely the language that Frank Wilczek, Frank Wilczek and also Al Shapir and Frank Wilczek wrote I think in a pair of papers back 11 years ago in 2012. And in fact, it is perhaps super natural to imagine generalizing, generalize the formalism that we wrote down there either Hamiltonian or Lagrangian to try to put space and time on the same footing, space and time on the same footing. And in particular, one can imagine by literal analogy to consider adding the leading order kinetic terms to the Lagrangian. So things that have now derivatives in time. And you might say, okay, by direct analogy, let's now add a couple of extra terms. We know what spatial translation symmetry leading to crystals look like. Let's think about time translation symmetry and what types of terms you would want to have in such a Lagrangian. You might say it's something like a coefficient C2, dt phi squared plus C4 dt phi to the four. Almost building like literally a direct analogy. And the analogy would state for example, that in fact, if you had C2 less than zero, C4 greater than zero, then it would in fact seem that we have energetically favorable field configurations. It's energetically favorable for states to have dt, excuse me, dt phi not equal to zero. And literally almost just replacing the word space with time, we would say here that again, if we have equations of motion that do not change in time. Here, given the fact that dt phi is non-zero, we have the spontaneous breaking of a time translation symmetry, of a time translation symmetry. And physically, of course, physically what this corresponds to, much like we had spatial crystalline patterns emerging over there, here we would have physically repetitions or repetitive patterns of the field configuration of the configuration of psi. And I'm using a lot of words, repetitive pattern of a configuration in time, but really the most intuitive way to think about this is that just like you have periodicity in space for crystalline pattern, here what we really mean just intuitively here is that we would have periodic oscillations in time, oscillations in time, periodic oscillations in time. That at some level will be what most of the discussion that we'll work on together for the next couple of days will be about when we can get interesting periodic oscillations in time that are intrinsically many body, maybe intrinsically quantum mechanical, but that are very non-trivial, is the analogy I'm trying to sort of build up sort of this intuitive analogy very, very slow. Yeah, please. I'm sorry, one more time, one more time, I apologize. So far we have focused on SSP, Landau paradigm. What about the topological phase of matter? Yeah, okay, this is a very, very good question. So at the moment we forget about topology, so for the moment we're only talking about, at the moment with time crystals we're only really discussed in the language of symmetry breaking, but I promise that in the third lecture we will do an example of a symmetry protected topological phase that's unique in the case of flow case systems. So in that case we'll sort of get both sides of the coin, SSB and topology as well. And then if there's really time, if I manage to fly through, we'll also do a topological band structure. Good, other questions? Yes, please. Yeah, good. So in principle at the moment basically over here when I think about this type of symmetry breaking I haven't appealed to any type of spatial translation symmetry breaking. So in principle I'd like to design time crystals, at least the symmetry breaking associated with time translation symmetry independent of everything else. So you can imagine that you have equation of motion where there's no spatial translation symmetry, no phase rotation, no other symmetry except time translation symmetry and a time crystal is defined most cleanly in that context as breaking of that time translation symmetry. I guess the point is that here the word crystal don't think about crystal as like a crystal in space think about a crystal as a crystal in time. Like the point is that basically these periodic oscillations I think that's what you mean I think you're sort of saying like oh well if you have a crystal is that what you mean? Just saying that the word crystal sort of makes us think basically about objects that are sort of crystalline in space. Just what I mean by that is don't worry about that. I can imagine basically something that's totally uniform in space but just oscillates in time and that periodicity in time is what one means by a time crystal. It's not like time by itself and the crystal by itself is one word. Why is it sort of like that? I think what he means that a crystal has a discrete symmetry, right? So you can translate forward by a certain. Yeah, in general that's right. Yeah, he said sorry. You break basically, yes exactly. So maybe this is something that I'll get to in just one second, but here we will actually get there in just one second. When I talk about spatial translations I should have really said that it's continuous spatial translations because I'm imagining the fact that I could for example move an epsilon amount in space an arbitrary small amount and the Hamilton is the same and normally crystals are the breaking of a continuous spatial translation symmetry down to a discrete one. And here in fact I'm thinking about the same thing. I want you to start imagining basically a continuous time translation symmetry and this time crystal in this case would break that down to a discrete time translation symmetry. But it turns out that actually won't be the relevant question for almost all of this but it's a very good question. Yeah, please. It is coming from did you start by saying that there is a crystalline structure and then with where is the periodicity and repetition coming is what I missed? The periodicity it's a good, good, good. The periodicity is coming from energetic is coming from energetic penalties. It's coming from the fact that if I had an energy functional like that the configuration that's low energy has for example a finite value in this example of a phase rotation of phi over here and that finite value is picked spontaneously in the thermodynamic limit to be something but it's picked spontaneously because of the fact that that energy functional has this type of a shape. You could imagine a situation where the energy functional doesn't have this type of shape as totally symmetric and has its lowest point at phi equals zero. In that case we would have no symmetry breaking, no phase rotational symmetry breaking because then we would know that states at low energy want to have phi equal to zero. So it's sort of an emergent property of the many body system, this notion of spontaneous symmetry breaking. Clear? Beautiful, maybe you can end this and then this. Purely classical. Yeah, nothing quantum. Haven't gotten there yet. So I will claim already right now that I'll say these are words for now but they'll make Anatolia very happy or depends on Anatolia's mood. But I would say really the nature of the symmetry breaking of a time crystal is not quantum mechanical at all. The nature of the order associated with time translation symmetry is not quantum mechanical at all. But as I had drawn somewhere over there in the past, I emphasized that to really think about time crystals as phases of matter, we needed to have some mechanism for ergodicity breaking. And it turns out that my perspective on this is that there may be uniquely quantum mechanical strategies to break or delay ergodicity or may not be. But in that sense basically the order where the symmetry breaking is not quantum mechanical but the ergodicity breaking could be quantum mechanical. So we haven't started talking about this but I just want to clarify. If we're talking about like time crystalline order in a flow case system, so we're already imposing, like we're already breaking the continuous. Yes, yes, yes, you're one step ahead of me. Exactly, it's prescient but just one step ahead we'll get there in a second, yes, absolutely. Yeah, please, call out, call out, I can't. It's a good question. I think yeah, so when you break a continuous symmetry in general you have goldstone modes. I think I guess the point that I'll make in just one second is that I don't really think it's possible to break continuous time translation symmetry. At least I'll say this in just one second. So in that sense, sort of irrelevant. I don't think it's, in the end it turns out we will only ever as was sort of alluded to by the question, we will only ever be breaking discrete time translation symmetries and in that case there's no goldstone whatever. Yeah, are there any time scales associated with this symmetry because if we wait long enough then the system will eventually thermalize or am I missing something? A super, super good question. So what I would like in general when I define phases of matter and we'll see that in one second it's a point that I'll emphasize a lot actually is that I'd like to take the thermodynamic limit, as the system size goes to infinity and in that limit I would like that the lifetime of the order diverges for example, I mean in the strictest case exponentially as the system size diverges. So for any finite size system and maybe Anatoly will talk about this in the next lecture, for any finite size system as you, if you fix a finite size and you go to infinitely late times, exponentially late times, you get point-car-ray recurrences, there's lots of stuff that kills the order. But if I take the limit as L goes to infinity I would like in the strictest case to define a situation with order or symmetry breaking as the lifetime of that order or the autocorrelation time of that order diverges exponentially in the system size. But it turns out that may or may not actually be possible in the context of time translation symmetry breaking. So for a lot of the lectures I'll actually be quite satisfied as long as you have parametric control over how long the lifetime is even if the parameter isn't e to the L. Super good question. Other questions? I have a question. Amazing. Yeah, so thinking from the perspective of Anderson, let's say, and symmetry breaking as some sort of rigidity, let's say, what would be the equivalent of the rigidity in this case? Yeah, so the equivalent, we'll see again in just a second is that we'd like whatever the nature of the order parameter, the nature of the ordering to kind of be stable, to small generically locality preserving perturbations to either the initial state or the Hamiltonian or the equations of motion. And we'll see, I mean, I'll sort of define basically what time translation symmetry breaking should be, sort of formally, mathematically, and then we'll add kind of these two notions of rigidity exactly as you're saying to really elevate that concept of symmetry breaking to well-defined phase of matter in the sort of stat-mecky sense. Very good question. Exhausted the questions. Okay, good. So let's continue for a second. So it turns out that as I was sort of just saying over there, in fact, at this moment, at least by this analogy, it seems this analogy seems simple enough. It seems simple enough to be able to get, for example, periodic oscillations in time for some ensemble of initial states, just thinking about minimizing some energy functional. But again, if you think about this intuitively, what I've said is really quite confusing. It's that I have a many particle system that's interacting, whose constituents are interacting, but the equations of motion are not time dependent. Equations of motion themselves have a continuous time translation symmetry, but then from that many particle system at low temperatures, for example, all of a sudden the system starts to spontaneously oscillate and this should feel to many of you very wrong if you've sort of thought about physics, because in general you might say, well, if there is an oscillation that spontaneously arises when I'm not causing any oscillation in the first place, shouldn't I be able to tether some little wheel to that oscillation and be able to extract work from that? And that kind of idea of this feeling very much similar to perpetual motion was initially back in 2012 and 2013, initially why people felt like this notion of getting spontaneously emergent periodic oscillations was sort of a little bit tough to imagine. It seems simple enough from this naive mathematical formulation, but I would say there are, there are and were, there are obvious and then not so obvious. There were kind of obvious and not so obvious obstructions, obstructions to this. I think a little bit more precisely than appealing to things feeling like, feeling like perpetual motion machines is for example, the fact that if we're thinking about a classical system, Hamilton's equations immediately imply, Hamilton's equations immediately imply that the many body system must be stationary, must be stationary at any energy minima. Here's kind of a very simple to state of destruction. Hamilton's equations immediately imply that the system must be stationary at any energy minima. And for example, this already kind of tells us that we shouldn't be getting periodic oscillations in the ground state. And I think it was already kind of accepted early on that in the ground state, these periodic oscillations could not occur in either classical many particle or quantum many particle systems. But it turns out, I would say more generally, there was very pretty work from Haruki Watanabe and Masaki Oshikawa, a couple years later back in 2015, where they essentially proved or showed that persistent oscillations, cannot arise any system that is in equilibrium. So here I'm thinking in thermal equilibrium is some thermal state. Cannot arise in any system that is in equilibrium with respect to a local Hamiltonian. So already you can kind of feel in your gut there are some obstructions, there's some obstructions that you can immediately write down that are obvious at energetic minima. And more generally, a couple years later, Watanabe and Oshikawa proved that persistent period oscillations cannot arise in any system that is in equilibrium with respect to a local Hamiltonian. So here is at some level our starting point for getting into the main topic of the school, both classical and quantum, exactly. Yeah, very good, yep. Now the question for the next section of the lecture is indeed kind of what loopholes do out of equilibrium physics give us? And that will be the focus of the next section. But before we get to the out of equilibrium context and specifically flow K, I would like to argue that there is in fact another challenge having this type of order or this type of time crystalline order and this challenge is directly connected to Anatoly's lectures, Anatoly's lectures where he emphasized to you that ergidicity breaking was somewhat inevitable. And in particular, okay, maybe I'll go over here again. This is going to be a short subsection, but important enough that I want to elevate it to its own subsection. It turns out that although it's not often taught this way, it's very natural to think of spontaneous symmetry breaking or in general the emergence of a phase of matter as a particular form, as a specific form of ergidicity breaking, ergidicity breaking. But here I won't be so worried about subtleties between what's the difference between chaos and ergidicity. At the moment here, all I want to emphasize in terms of what I mean by ergidicity breaking is that in particular it's important for any phase of matter and in particular in the context of spontaneous symmetry breaking that I don't want my system, I don't want my system to forget its quote unquote initial condition. This seems patently obvious. If you had a ferromagnet, for example, an eising ferromagnet, you'd like that if you were in the all upstate for a macroscopic chunk of a magnet, that it wouldn't just at some point forget that it was in the all upstate and go to the other low energy configuration which is all down. But that by definition is some form of ergidicity breaking because the system doesn't want to forget its initial condition. So it's very, very natural to think of spontaneous symmetry breaking as a form of ergidicity breaking and to think about the symmetry as playing the role of what separates the different pieces of the phase space. Yeah, finite temperature is also fine. Finite temperature is also fine. So actually, you know, yeah, I'm, yeah. Finite temperature is even better. It doesn't have to be just grafting. If you start from an integrable system, okay, fine. Yeah, so they're sort of different at some level, I would say. Integrability is way stronger at some level because there's an extensive number of conserved quantities. So in that sense, you know, there are many, many integrals of motion that constrain the dynamics. And here in the context of symmetry breaking, it's really the symmetry that's playing that role and it's way, way less stringent than the integrable case. So we can have spontaneous symmetry breaking that's a rather stable or rigid concept. But integrability, typically we think about as relatively fine tuned that most generic perturbations all generic perturbations will essentially destroy integrability. Good. So just maybe in words one more time because I like this. For example, in the case, I think someone called out earlier of the x, y magnet. In the case of the x, y magnet, one has to have, one must have, in fact, ergodicity breaking. One must have ergodicity breaking so that, so that the system remembers, precisely remembers specific, for example, the specific complex phase, phi that it's ordered to out to infinite times. And again, here I'm thinking in general in the thermodynamic. So I'll define time translation symmetry breaking in slightly more equations in just a second. But let me, at this point just in words, define what I think is the proper way to think about this topic, which is kind of an ergodicity centric definition of a time crystal. This ergodicity centric definition of time crystal, I would say is that we have the spontaneous breaking of a time translation symmetry where some ensemble of initial states already working in a little bit the language of radinity here, where some ensemble of initial states exhibits persistent oscillations, persistent periodic oscillations, persistent periodic oscillations with a temporal phase shift. And I'll recast those three words in just a second with a temporal phase shift that, and here is the play on the word ergodicity that really remembers its initial condition. What I mean here is very, very simple. What we've been saying thus far, although it's not possible, what we've been saying thus far is that if we think about a very, very natural definition of time crystal, we have a many particle system whose interacting description has a continuous time translation symmetry, the description at the level of equation of motion is unchanged in time. If there are some emergent periodic oscillations, for example, in the configuration of the field phi, then in order for you to remember your initial condition, you don't want to just have random phases, hop at random phase in the oscillation, you want to know where in the phase you started so that you're always oscillating at the same phase. If you forgot about that initial condition, sort of where in the cosine t plus phi you were, then that would precisely be the case where you're no longer spontaneously breaking the time translation symmetry, where you'd average out that oscillating behavior. So this temporal phase shift, what I mean here is just where in the oscillation you are, literally. Where in the oscillation you are? So now we're ready to really get into the setting kind of is related to the slightly, the more non-equilibrium side of things. And in particular, we're going to now enter the discussion of flow k time. I'll put in parentheses here. It's not often called this way, although you'll see in just a little bit why I put this in parentheses, what are called flow k time crystals. Although, again, I put in parenthetical density wave. And what I mean here is that the last two topics have been about the spontaneous breaking of first a spatial translation symmetry, and then the spontaneous breaking of a time translation symmetry. And here, we're going to be discussing the spontaneous breaking of a discrete time translation symmetry, something that I will call discrete S tau B. And again, because it was, I think, I don't remember, but a very nice question. Let's start, since we've been doing the whole time with our spatial analogy. Spatial analogy. You've already seen, you've already seen how states with grad phi non-zero can be favored energetically. And this was a very, very good question that was asked over here. This is indeed, it is truly indeed spontaneous symmetry breaking. But if we were a little bit more precise, but if we were a little bit more precise, it's the spontaneous symmetry breaking of a continuous, of a continuous space translation symmetry. In the sense that when I was thinking about the description of my equations of motion, I had stated that, for example, the Hamiltonian, I was imagining being invariant under arbitrarily small translations in space. So we have a continuous group that's formed by the generator of such translations. So again, before getting to time, let's do space first. And this is very much related to a couple of questions that were asked. You can ask, what about the same physics but in a lattice? So again, for those cold atom aficionados, you can imagine these are atoms in an optical lattice or those solid state aficionados. You could imagine that these are electrons in some solid state material. You really want to get fancy and have a larger scale electrons in a more ray potential. But just pictorially, what I mean by this question is now the following. Pictorially, what I mean by this is that one might imagine that the potential energy landscape underlying atoms could be something that itself is already periodic in space. And if we have a lattice spacing given by a knot over here, then the continuous translation symmetry of space has already been broken down. So this means that the spatial translation symmetry has already been broken down to only discrete translation symmetry. You can clearly see that it's discrete because the only thing that this is unchanged by is translation by, for example, the underlying direct lattice vector, a knot. And again, keeping with the atomic physics analogy, we can imagine two physical scenarios if we had atoms, atomic density, atoms living on top of this optical lattice potential. We could have a scenario where the atomic, slightly larger piece of chalk, you can imagine a scenario where the atomic density, atomic density could be uniform. It'd be uniform on top of this optical lattice potential. And in this case, this would correspond to there being no discrete symmetry breaking. And we would say that the atomic density is the same at every lattice site within the optical lattice potential. But you could also imagine a scenario where the atomic density, atomic density, or the electron density, or the atomic density could further, further break the already discrete, the already discrete spatial translation symmetry. So you could imagine that we have repelling atoms. And now, instead of wanting to have a uniform density on every lattice site, they repel and they form some sort of a solid, some sort of a crystal on top of the underlying lattice that it lives in. And it turns out that the phase of matter, it's very, very conceptually, pictorially, extremely similar to thinking about the emergence of a crystal where we said, okay, we could have grad phi being non-zero. So now there's a crystalline pattern that now has a discrete symmetry, which is a subgroup of the continuous symmetry of the Hamiltonian. In this case, you have that the state of the system is now governed by a discrete symmetry group, a discrete subgroup, which is a subgroup of still the underlying, but already discretized spatial translation symmetry of the underlying lattice itself. And instead of calling these phases of matter, in this case, when we discreetly break the spatial translation symmetry, a crystal, these are oftentimes called density waves, density wave patterns. So for example, you'll hear in the literature that these are called in the context of electrons or charges, charged density waves, in the context of spins, spin density waves, that's the concept. So you can now already see a little bit of my naming convention over there, as was kind of presciently alluded to. In the flow K context, we're really going to be talking about the breaking of a discrete time translation symmetry to an even smaller subgroup, which probably more properly would really be called a time density wave pattern, but for whatever reason, for whatever reason the word time crystals are stuck. Okay, so just very, very quickly, just summarizing in a little table, we can have the breaking of a continuous spatial translation symmetry. And this one usually calls crystalline order. We can have, imagine, the breaking of a discrete spatial translation symmetry. This oftentimes is called a density wave phase of matter, or a density wave pattern. We can imagine trying to put space and time on the same footing, and imagine the spontaneous symmetry breaking of a continuous time translation symmetry. But as I've already emphasized, there are, except in kind of specific cases of singular Hamiltonians, in general, not allowed. And the remaining focus of our discussion is going to be on the case of discrete time translation symmetry breaking. And here precisely is where flow K physics will enter. So enter flow K. And this, I think, probably would be more properly called a time density wave, but really these days is often called a discrete time crystal or a flow K time crystal. And I want to emphasize that already with this kind of a pseudo table, you already see a little bit the answer to one of the two questions that I emphasized in the very beginning that you should always ask when someone's trying to introduce a modern research topic to you, which is why flow K? And the answer to this in some level is context. Really context is, you know, the answer, let's say, to kind of question two. It's that really, because we're thinking about discrete time translation symmetry breaking, we need to break down the continuous time translation symmetry into a discrete subgroup in the first place because we know that on quite general ground it's very hard to get continuous time translation symmetry breaking, so flow K is kind of where the rules of the game allow one to ask this question. So intrinsically flow K in a way, another way to say it, it's only a reasonable or a relevant question in the specific context. Good, so let's continue for a second. Again, now I think you guys all see the analogy very, very clearly. What about the same kind of a question? What about the time? What about a lattice in time? And indeed, much of kind of the modern research on time crystals, or specifically S tau B, the spontaneous breaking of time translation symmetry is focused on flow K systems, periodically driven systems. Again, let me write this down, where really the equation of motion describing the interacting multi-particle system come back to themselves, to themselves, themselves after some discrete period of time. And again, I had already written this in one of the very first boards that I used, but what I mean by that is that I have a Hamiltonian, for example, if I'm thinking about that as the description of the equation of motion, such that h of t plus 2 pi over omega d, where omega d here is the frequency of the periodic drive, where this is the driving frequency, comes back to itself, is equal to h of t. Again, omega d is the driving frequency, and t naught equals 2 pi over omega d is the driving period, the driving period. Again, as Anatoli said, you will remember from your first year physics course that the generic expectation, expectation, the generic expectation, and this really is first year physics, then generic expectation that you should already have about such periodically driven systems, perhaps the simplest setting where I'm almost certain you've seen this, either in the context of linear oscillators or, as we'll soon see, nonlinear oscillators, the generic expectation is that, in fact, a simple harmonic oscillator driven frequency omega d responds at frequency omega d, and this is kind of independent of its natural frequency, very, very intuitive. You drive a system, you shake a system at a particular frequency, you expect it to respond at that frequency, and I'm just emphasizing the example of a simple harmonic oscillator because I suspect that all of you have already thought about this, but it turns out, again, I think it's still pretty unsurprising. It turns out that, in fact, this fact that we have here, in fact, this is also true, this is also true of many, I would say, almost all of many complex nonlinear systems, complex nonlinear systems, that independent of the details of their underlying microscopic structure that if such a system is driven at frequency omega d, in general, its observable properties will respond at frequency omega d or integer harmonics of omega d. The idea is independent, independent of kind of microscopic details, microscopic details, the observable properties, these properties will respond at frequency omega d or integer harmonics of, and this generic expectation that observable properties, that if you drive a system at frequency omega d, that the observable properties of the system will respond at frequency omega d. This corresponds, I hope people can see this. This corresponds, this corresponds to the no symmetry breaking case. I would have drawn that pictorially, but it sort of makes sense. Symmetry breaking every single time corresponds to the system exhibiting properties that have a smaller symmetry group than the underlying equations of motion. In this case, the underlying equations of motion are drive at frequency omega d. So if your observable properties are also responding at frequency omega d, that certainly looks like you are consistent with the symmetries of the underlying, although now periodically driven equations of motion. An infinite amount of time, but let me try to get through to a good stopping point. I apologize if I go over just slightly. By contrast, let me at least try to define this. We say that discrete s tau b, or the spontaneous breaking of a discrete time translation symmetry, is said to occur, is said to occur if the system exhibit properties, exhibits properties that oscillate at a subharmonic of the driving frequency. At a subharmonic of the driving frequency, and what I mean by that is that some omega d over m for integer m greater than one. Again, drawing the analogy to space, we say that spontaneous breaking of a discrete time translation symmetry is said to occur if the system exhibits properties that oscillate at a subharmonic of the driving frequency. And just to, again, kind of make the picture analogy, instead of the optical lattice potential that we imagine on the board just to my right, imagine that we're looking at time. And we say, for example, clearly if we have a flow K Hamiltonian, the equations of motion are coming back to themselves every period. So 3T0, 4T0, 5T0, et cetera. And if we have a property that's oscillating at a subharmonic of the drive, it's tupling the period. It is a subharmonic, so it's a higher harmonic from the perspective of a period, a higher tupling from this perspective of a period. In general, again, using this kind of a picture, the subharmonic response of associated with discrete S tau B breaks the symmetry down, down from the initial already discretized symmetry group, which is T to T plus, N T0 where N is in Z. So this is the symmetry group corresponding to the original discrete time translation symmetry. T gets translated to any integer, so T0, 2T0, 3T0, et cetera. That is the group associated with the equation of motion, the symmetry group associated with the equation of motion. It breaks it from this, for example, to T goes to T plus N prime T0 where now we have N prime is in M times Z. So if you have discrete S tau B, M fold discrete time translation symmetry breaking, we say that a system has properties that oscillate at a subharmonic of the drive, omega D over M for some specific M, and this means that the symmetry group is broken down from T plus N T0 where N is an integer to T plus N prime T0 where M prime is in M times the integers. Super obvious. Pictorially, for example, we could give the example of period doubling, of period doubling oscillations, which are a subharmonic omega D over 2, M equals 2 in this case, and this would, for example, correspond to a situation where observables kind of in this time lattice come back to themselves every two driving periods, although the underlying equations of motion are symmetric, come back to themselves every driving period. So very, very similar to the spatial density wave pattern, but now looking at some observable measured in time. One last line. So at some level, the goal that we'll start with tomorrow morning is we'd like to get, we would like to get, and this again goes to a question over here, we'd like to get some form of stable or rigid discrete S tau B, discrete time translation symmetry breaking in a many body system. Again, in this case I will, at this point, discuss quantum and classical and pretty equal footings. In the classical case, much like Anatoly, we'll go through canonical transformations that try to rotate away the period doubling. In the quantum case, it'll be unitary transformations that attempt to do exactly the same thing. But if we're able to get stable, rigid time translation symmetry breaking, discrete time translation symmetry breaking, this is kind of the beginning of having a well-defined discrete time crystal as a particular phase of matter. But it turns out even with these caveats about requiring discrete S tau B and requiring some notion of stability and rigidity, still, we'll have to sharpen the question a little bit further before we get into the really, really non-trivial aspects of when you could have this type of a time crystal and that will be deeply connected to what Anatoly's been talking about, which is a way to break ergodicity in a many particle system. So I think that's probably a pretty good stopping point and we'll continue tomorrow. Thanks. Thank you. Are there some urgent questions? You defined the time lattice there with this driving frequency. Yeah, exactly. But then there it's a different frequency or is it... No, no, it's the same frequency. Sorry, so I imagine, sorry, I should have been more clear about this. So my flow case system is defined by being driven at a particular frequency and that particular frequency defines a period with which the equations of motion come back to itself. So with symmetries, I would say the equations of motion have a particular symmetry, a transformation that renders them unchanged and that transformation is moved by period t0 because it always comes back to itself. And then if I think about trying to imagine time and writing down what exactly, when exactly the equations of motion come back to themselves, they come back to themselves every period t0, 2t0, 3t0, 4t0. And all I was emphasizing over here was, again, much like we have an optical lattice potential breaking down a continuous spatial translation symmetry to a discrete spatial translation symmetry, here the periodic drive breaks down the continuous time translation symmetry to an already discrete time translation symmetry. But on top of that, if you want to imagine symmetry breaking on top of that already discretized time translation symmetry breaking, already discretized time translation symmetry, then you can have a scenario where you have a sub-harmonic response or a period tupled response where the systems observables come back to themselves at some larger multiple of the period as opposed to just responding at the underlying period itself. Is that clear? It's very complicated. So this is something that we'll unpack in a little bit, but let me help to explain the subtlety of your question because you asked a really beautiful question. So in some sense, what we emphasized over here that was stated was that, well, it might be interactions. How do interactions manifest? They manifest as a term that corresponds to how the particles talk to each other in the Hamiltonian. And so interactions can change the configuration of my field because it lowers the energy, for example, for them to be further apart. So at the end of the day, symmetry breaking, when we think about this in equilibrium, it's always related at some level to energy and energy penalties that prevent fluctuations from messing up order. But in this case, energy isn't conserved, so I can't actually appeal to the same thing. So it's sort of more of a locking phenomenon, but we'll understand exactly why it happens. We'll have to wait a little bit. But it's a very, very good point that you're making. I do. Sorry, I can't. One more time, one more time. For this case, you got m is greater than 1, right? So why it can't be m less than 1? But if m is less than 1, then it's again a higher harmonic. Let's look at this for a second. If m is less than 1, for example, then basically it's sort of a higher harmonic response of the system, and that higher harmonic response of the system would not break down from a period perspective like this, would not break me down from a particular group to a smaller subgroup. So it's actually very natural. If you literally take a piece of jello and you tap on the jello with a, maybe you did this, I did this in high school, but you take a piece of jello and you tap on it with a little mallet, and then you basically put it through a synthesizer and then you Fourier transform basically the kind of oscillation response of the jello. You'll find that it responds to the frequency that you're tapping at two times the frequency, four times the frequency, all of the higher harmonics, but it doesn't have any peaks at subharmonics. So it turns out the higher harmonics are still not that hard to get, but you don't usually get it from basically tapping or shaking a system. Yeah, so that's another case. So you can imagine there are situations indeed where you can have subharmonics, for example, but that are not integers, for example, so you could have a situation where M is five thirds or something like that. You can get those types of situations. Oftentimes what that corresponds to is essentially having kind of like five regions of stability and a period three cycle through the five regions of stability. So you can get those things, there's not, I wouldn't say it's not super non-trivial why those emerge, like sometimes in like quasi crystals it's very, very non-trivial why you have that type of a structure, but most often you can indeed get subharmonics or time crystals with fractional frequency responses, but oftentimes there's some underlying structure that sort of gives that response and it doesn't usually happen in a totally spontaneous way. Great. So if there are no more questions, let's thank Norm.