 OK, then thanks for coming back. So let me now continue, in particular, to essentially starting to address what I already see a question before starting. Yeah, please. Yes? Yeah. No, they all have. But yes, but not on your physical axis. So when you replace now time by a temperature here, you study now your Ising model as a function of temperature, the partition function of an Ising model as a function of temperature. Now you extend temperature to the complex plane. Actually, we see it in two, three slides. That in that case, indirectly you will see it on the next two, three slides, that away from your physical axis, you will actually have zeros. But they will not cross your real-time axis necessarily in one dimension, for example, because you know that you don't have a phase transition at the finite temperature for a classical Ising model, say. So they lie somewhere in the complex plane and don't touch your physical axis where your temperature is real. But they're always there. But they also don't matter so much if they're somewhere in the complex plane and don't touch your real axis, you also don't care so much about them. No, that's when these zeros now cut, touch your real temperature axis in some way. Yes, now I would like to start now discussing a bit what these non-analyticities now actually mean. So although I've used this terminology of a dynamical free energy density, it's important to emphasize that this is only a formal kind of identification and does not mean that these rate functions behave as free energies in the sense that you can derive measurable quantities from that. For example, if you take the second derivative of a free energy with respect to temperature, you get a specific heat, for example, so a measurable quantity. So it means if your free energy is non-analytic, you also know that the measurable quantity like a specific heat is non-analytic. These rate functions or dynamical free energies as in quotation marks, as I was using them, they don't have this property that by taking derivatives, you get other measurable quantities out. So which immediately leads to the question, so what? Does it then mean if to have non-analytic behavior in this quantity, does it mean anything about other physical properties that the system might have or quantities which we are typically measuring like magnetizations or some local observables? Do these non-analyticities have some impact at all? And these are things I would like now to address. And I would like to start with one example, a very simple to be honest, but a very instructive example where you can show that these dynamical transitions can exhibit scaling and universality in the very equilibrium sense. In that, I will show you a model where you can construct an exact renormalization group transformation which allows you to show that this dynamical quantum phase transition is associated to an unstable fixed point, which necessarily has to lead to a scaling and universality. And I will do that for the working horse of many fields, which is the transverse field ising model. Initially, some derivations I will show are very general. They are valid on any graph, on any dimension you would like to have. So that's the model I would like to study now. That's the overall Hamiltonian. It has two terms. The one is ferromagnetic coupling of spins, which would like to induce some magnetic order between spins along the sigma z direction. These brackets, so here's a small restriction. Actually, it's not really necessary for what comes in the following denote. I use for denoting nearest neighbor interactions. And the second part here is a transverse field which tries to counteract or compete with this magnetic ordering tendency of the first term and would like to align the spins along transverse field direction. And in equilibrium, now these two terms compete with each other. So depending on whether the transverse field is large or small, you can have magnetic order or not. And also thermal fluctuations that you might have would like to destroy magnetic order. It can lead to rich phase diagrams depending on the graph you're studying or the type of couplings you use, ferromagnetic, anti-ferromagnetic, and so on. Now let us study a non-equilibrium scenario for this model and a quantum quench and a very particular one. Namely, a quantum quench between two very extreme limits initially in this model from infinite field to zero external field. So it means infinite fields in the sense that the initial Hamiltonian is dominated only by the transverse field term here. And our final Hamiltonian does not involve any transverse field and only has this spin-spin coupling. Later on, one can also go away from this extreme limit, but this one will be particularly instructive. So what does it mean if this is the initial Hamiltonian? We choose as the initial Hamiltonian, the ground as the initial state, the ground state of our initial Hamiltonian, which for this one here is very simple. All spins pointing along the magnetic field along the sigma-axe direction because I assume the magnetic field to be positive here. So all spins point along the sigma-axe direction of the Bloch sphere. Now, let me write this initial condition in a slightly different form. So first of all, this initial condition is a product state. So each of the n's, the l spins I'm considering initially they're all uncorrelated to each other. It's a product state. Overall, n and l, I'm sorry, now I'm using different symbols, n and l. So just take them to be the same. So both of them give you the number of spins I'm using here. So it's a product state over all the n lattice sites. And now let me represent the spin polarized along sigma-axe in the sigma-z basis. If you do that in the sigma-z basis, it's locally the equal superposition of spin pointing to the north pole and to the south pole. This product, if you multiply this out, like explicitly evaluating this product here, you can immediately see that, recognize that this initial condition is somehow special in that when s denotes a spin configuration in the sigma-z basis, this initial condition is nothing but the equal superposition with equal weight of all spin configurations. That will be very important in the form. And now time evolution, because this is our final Hamiltonian. The time evolution of our initial condition, we can, of course, formally write in this form. And now we can use a second property that we have in this rather extreme kind of quantum quench I'm discussing here, in that all the terms like this Hamiltonian is, to some extent, classical. In the sense that all the operators that appear here in this Hamiltonian, all of them mutually commute with each other. So that means that the exponential of that Hamiltonian can be factorized as a product of this only involving these two body terms. And now these two properties of the initial condition here and of the time evolution operator for this specific Hamiltonian can be used to show the following, which, in light also of what I discussed in the first part of today's lecture, is very interesting. Namely that now this Lo-Schmidt amplitude for this initial state and this final Hamiltonian is exactly, you can rewrite exactly as the equilibrium partition function, so not a boundary partition function, but really an equilibrium partition function of a classical Ising model, which lives on the same graph as the Hamiltonian I was studying before. The only difference to equilibrium here is that the effective coupling that appears, this k, is now complex and not real. That's the only difference. But apart from that, it's a conventional partition function of a classical Ising model studied not along the real temperature, but on the imaginary temperature axis. And the proof is very simple. I see it's rather, it's a bit small to see. I hope you can decrypt it. So proof is, as I said, it's just essentially a one-liner. So this is the definition of our amplitude, as you see also up there. Now we use that this initial condition is the equal superposition of all spin configurations, which gives us initially two sums, like some normalization pre-factor, one over two to the power L, and then two summations over all spin configurations for the left and right boundary state here. But now you immediately can see that the Hamiltonian, which is doing the time evolution in between, is diagonal in the sigma z basis. It does not induce any kind of transitions. And that means that this here only gives a non-zero contribution when s is equal to s prime. That's the only way, only non-zero contribution. And that's what the formula here to the right-hand side is. So only a single sum over one spin configuration ss. But that's nothing than a trace in the end. And so like sum over our spin configurations and the same spin configuration on the left and right-hand side is nothing else than a trace. And therefore, we have arrived already at this identification that this Loschmann amplitude is a partition function of a classical Ising model, because it's a simple trace. And that is, of course, now useful because we know a lot about classical Ising models. And even in the complex plane, and we can now use this knowledge to study dynamical phase transitions. Now, as I said, actually the duration on the previous slide is completely general. Put your Ising model on any graph you would like to do. But now let me take the most simple example one can imagine here. And that is now a nearest neighbor Ising model in one dimension, which has this form. So only spins sitting on a line and interacting and then this nearest neighbor way long sigma z direction. So if you now this, you can imagine that one can solve this exactly. And this is the result. You can focus here on this brownish solid line. And you see that there are times here where the associated dynamic of energy shows these kink-like structures, so these non-analytic points. So you have these dynamical transitions. And now we would like to understand them in a bit more detail. And for that, we will now use this mapping onto this classical partition function. So the partition function of the one-dimensional classical Ising model in general can be solved using the transfer matrix technique. I quickly outlined it here. So here, that's our expression that we derived on the previous slide. Now let's write down this trace explicitly as a sum of all spin configurations. This means nothing but like summing all the L spins over their two possible orientations, plus or minus one. And depending on whether the spins get straightforward we replace now this operator as we had here by the respective numbers corresponding to the spin configuration we are summing over here. And from this expression here, you see that you can define now for each pair of spins a matrix, which is called T in general, the transfer matrix, between two nearest neighboring spins. And you can do this for every lattice site. And these summations in the end are nothing but giving you a different representation or equivalent to essentially doing a matrix product of all these transfer matrices. So in the end, you can rewrite this classical partition function in this form of the transfer matrix T in their L of them because we have L links. I'm using periodic boundary conditions here. And all of them are the same. And we have L of them because we have L bonds, L lattice sites. So we take them to the power L. And each of these on this transfer matrix is only a 2 by 2 matrix given by these expressions. And since this matrix T is only a 2 by 2 matrix, you can solve this partition function in the thermodynamic limit easily just by searching for the eigenvalue of the transfer matrix of largest magnitude. This one will dominate in the limit of L to infinity. So that's how you can solve this exactly. And the exact solution you have seen actually on the previous slide. So that in this case, the Lois-Schmidt amplitude or the dynamical free energy gets these non-analyticities. But now we want to use this representation of the partition function in terms of the transfer matrix to construct an exact renormalization group transformation, a very easy one. In particular, a simple decimation, a real space decimation transformation where we integrate out every second spin in one RG step. So initially, you have a set of spins indicated by the blue dots here. And they are coupled initially by some coupling k. And now we want to get rid of every second of these spins and to find a new effective theory for the remaining spins, which are now described in general by a new coupling k prime. And if you are interested in more details, you can, for example, have a look at this extensive review on real space RG transformations for classical systems. Now, how can we do this here? And that's very easy. As soon as we have our transfer matrix representation, how can we implement this renormalization group transformation? So we have that the partition function is trace transfer matrix to the power L. Now let's just define a new matrix T prime, which we get by multiplying two of these transfer matrices T together. If we do that, we can write our partition function in terms of these T prime matrices in this form, but now having only half of the lattice sites available, because two of them we just glued together. In this way, we have eliminated every second that the site by just doing a simple matrix multiplication of this T matrix. OK? Now, we have seen on the previous slide that this T matrix can be directly or is uniquely determined by this coupling k. So by multiplying these two matrices, we can associate to this new transfer matrix T prime, a new coupling T prime, a k prime, which obeys the following equation. So given some coupling k initially, we get the new coupling k prime here of our reduced spin chain by solving this equation. Importantly, that's something very special about this one-dimensional nearest neighbor IC model is that this RG transformation is closed and that you can solve it exactly. OK, so now we have this exact recursion relation. And actually, that's not given a priori. It turns out to be exactly the same recursion relation, no matter you are in equilibrium, so meaning real numbers, real couplings k, or whether you have imaginary or complex couplings k as they appear in this non-equilibrium context. So this recursion relation is the same. So at this point, we don't see any difference between equilibrium and non-equilibrium. OK. Now before continuing, let's now study the fixed points of this RG transformations. So in equilibrium, you are of course only interested along real temperature line. But now we are actually interested on the imaginary temperature line and more in the complex plane. So it's not clear a priori like whether there can be, for example, new fixed points appearing somewhere in the complex plane. Now if you want to determine fixed points of this RG equation, fixed points are those points where the coupling does not change upon acting with the RG transformation on. So it means that you take your recursion relation and put the same claim on the left and right hand side. OK. This gives you the fixed point. That's the defining equation. When you solve this equation, you see that there are still only two possible solutions for fixed points. And they turn out to be exactly the same you find in equilibrium. So there are no new fixed points in the complex plane. So this does not need to be the case in general. For this model and this RG transformation, it seems to be the case. So one solution here of this equation is, of course, when the tanh of k star is equal to 0. That is a solution here. And this means that your coupling k star is equal to 0. The other possible solution to this equation is when the tanh is equal to 1. And this corresponds, again, to a real coupling, but a real coupling which is infinity. So what do these two different points now mean? So first of all, I said they are purely real, although we started initially from complex numbers. So no matter what we do, if we start to do the RG, we will always end up in those two equilibrium fixed points. And what do these numbers now mean? In equilibrium, this k you would have the following expression for the coupling k. It would be the coupling divided by temperature. So the k star equals 0. Fixed point corresponds to the infinite temperature 1, which is the boring and uninteresting stable fixed point of the IC model. And the k star infinity means it's equivalent to the zero temperature fixed point, and therefore the interesting unstable one. The phase, you mean? So this is the 1D classical icing chain. So it is always a paramagnet at any non-zero temperature, only at zero temperature. The ground state has ferromagnetic order, has a fully polarized state as its ground state. So this unstable fixed point corresponds precisely to the point where in equilibrium, all spins will be aligned, would be magnetically ordered. So this is not a true phase here, because it's only a singular point. So you have magnetic order only at a singular point. So it does not change whether you have complex numbers or not. Is that what you mean? That's true. Let me just point this out. But there can be some other interesting thing that can happen, which is not relevant here. But something in general interesting when you have complex RGs, you can get limit cycles. So you can find actually solutions to this equation. So solutions to the fixed points of the RG equation, which only appear when you do two or three RG steps at once. So between, for example, when you do RG steps like this, which are jumping between two different points in complex space, this can actually happen. But it's not relevant here. That's why I don't go into detail. But something interesting which cannot appear in the case of real complex. But these are the only fixed points here for that equation. And these are the ones which are relevant for what I will discuss in the photo. OK. So now coming back to the data I've shown you. So here I told you that you should look at this brown solid line. And there were points in time here and here where you observe kinks from the exact solution. And these points are located, or the first one here, is located at a value of the dimensionless coupling k i pi over 4, turns out. So now let's study what happens to this coupling when we do the RG transformation. So when you take i pi over 4 and compute the tanche of it, you find it's i. So that's our starting point. Now let's do the RG transformation. The RG transformation is taking just the square of this tanche to get the new tanche. So the square of i is minus 1. So after one RG step, we are at the new coupling which is given by tanche equal to minus 1. And now we do a second RG step. And the square of minus 1 is 1. So we get that tanche of k flows to plus 1. And when we do more of RG steps, we will stay there, of course. And remarkably, it is the value of the unstable fixed point. So in other words, the dynamical quantum phase transition, which happens at this kc equal to i pi over 4, is a critical point which flows to the unstable fixed point of the one-dimensional Ising model. And in this very equilibrium sense, it's an unstable fixed point that has to show scaling. And in the universality, you can take all the knowledge you have from equilibrium here. And for example, you can check whether the universal scaling form of the dynamic free energy is consistent. So let's define a distance to the critical point, which is here, not distance in terms of temperature distance to the critical point, but rather a temporal distance. And when you take the prediction from universal scaling, you would say that this dynamical free energy is given by this expression. Dimension d is equal to 1. The exponent lambda, you can compute from your IG equation. You find this also one. So you will find that this G, or the singular contribution to the free energy, has this absolute value tau behavior, where tau is the distance to the critical point. And that's precisely the kink we have observed from the exact solution. So everything is consistent. Please, there was a question. Can you go back to the previous slide? So that's the result from the exact calculation that you find that there is a non-analytic behavior at this point. Is there a critical point? Yes, and now we study how this critical point behaves under the RG. Yes, yes. What changes remarkably is the structure of these limit cycles that I mentioned. But the other things do not change. So now what I told you somehow establishes on a rigorous way that these dynamical transitions, at least for our particular models, can obey scaling and universality. Here for a one-dimensional IC model. Now actually, the mapping of the displacement amplitude to a classical partition function was much more general. For, say, two dimensions or three dimensions, we cannot construct an exact RG anymore for that. Cannot be done. However, what we can do still is we can use the Onsaga solution for the 2-Di-Sync model. But now adapt it to complex couplets. You have to be careful at a few steps particular when you select the dominant eigenvalue. But you can compute then the partition function for the energy density exactly for the two-dimensional case. And you find that there is, again, a critical point in time, a dynamical quantum phase transition in the 2-D IC model. And from the exact solution, you find that the non-analytic behavior has this form, tau squared log tau, where tau is the distance to the critical point. And remarkably, that's exactly the same scaling as the two-dimensional IC model at its finite temperature critical point, which also has scaling and universality. We cannot prove anything further. We only can show that at least we get the same scaling of the non-analytic behavior as for the equilibrium finite temperature critical point of the 2-D IC model. We also tried to do three dimensions, but we failed. We cannot say anything about that. OK. Good. So yes, please. You have to be careful in the following way that this Lusch-Mitt amplitude looks like a partition function of a particular system. But this is only formal. It does not mean that this is the action like, I should say that, for example, here we are mapping the Lusch-Mitt amplitude to a classical partition function of the same dimension, not the dimension d plus 1, as you would typically do, mapping quantum classical. Here it's really the same dimension. We are, the actual quantum dynamics is not one-dimensional classical IC model. There's more to that. So for example, it does not make sense now to, if I were now to measure spin-spin correlations, the dynamics of my spin-spin correlation function in this quantum quench protocol, I would not find a divergent correlation length. It's only that this Lusch-Mitt amplitude has a behavior as a one-dimensional classical IC model with a divergent correlation length. It's only the Lusch-Mitt amplitude. If I study the dynamics in this quantum quench of correlations or local observers, that's a different issue. OK, so I'm all clear. And that's also something which I will discuss later on in more detail. One has to be careful to distinguish those two things a bit. And I will also discuss a bit in tandem, the connection to entanglement later on. I will tell you later. I will tell you later precisely why I think one should associate or think of it as a quantum phase transition precisely in the sense that you can think of it as a dynamical analog to a quantum phase transition. Or let me make a short statement. Why? In that, what the Lusch-Mitt amplitude is doing for you, it's taking the full-time evolved state and projects it onto the initial state. The initial state is a ground state by construction in this quantum quench scenario. So since it's the ground state, the Lusch-Mitt amplitude brobes the ground state like the dynamics of your time evolved state in some ground state of some Hamiltonian. And I will show you how you can use this to define analogs of critical regions, which draw a direct analogy to what you would typically think of quantum phase transitions. But tomorrow. Yes? So which kind of fluctuation theorems are you thinking about? Jajinsky or? Jajinsky, there is some relation in the sense that the Fourier transform of the Lusch-Mitt amplitude is actually the buck distribution function. So in some way or like the Jajinsky relation is related or I think there's not a deep relation in the end. But maybe as a small, nice analogy because I was discussing complex partition functions here, the Jajinsky equality is actually nothing but the Lusch-Mitt amplitude somewhere in the complex time plane. And in this way, also the Jajinsky relation is hidden in all this complex partition function structure. But we have tried to connect or tried to see whether signatures of these dynamical transitions appear in buck distribution functions, but it's tricky. And therefore, there's nothing which relates them to Jajinsky or anything like this. Yes, you can, again, not in general, but we know for classes of models how to define order parameters which change as a function of time at this particular point. For example, there are also actually, I think precisely, that there are three or four experiments which use that. And what they show is that you can, that when you have study quantum quenches in systems which involve Hamiltonians that have topological non-trivial ground states, that you can develop some, for example, some vortices in some phase profiles. I will come to that. And these vortices appear suddenly precisely at dynamical phase transitions. And the number of vortices you can take then as an order parameter. You will see them. You can think of them then as an order parameter. So you monitor this particular phase profile, and then you see at some particular point in time suddenly pairs of vortices appear in this phase profile. And you can measure them and use them as an order parameter. But for that particular dynamical transition I was discussing here, I cannot tell you an order parameter. I don't know it. Ah, so for this one-dimensional transfer speed icing model, that's overall exactly solvable. But even without, I've discussed only a very particular point quenching from infinite to zero field. You can solve it exactly by Jordan Wigner transformation for any value of fields and couplings. Oh, sorry. And that's actually data you see up here. Not only seed for one value of the transverse fields, but for different ones. And for this particular model, you see that these non-analyticities occur periodically on and on and on until infinity. But that's, I guess, my guess is that this is a consequence of the exact solvability of the model as soon as you add sufficiently generic interactions. At some point, this has to change. So at short times, they can still survive that's somehow related to a certain robustness of these transitions. But at long times, they might not be robust. It might change the nature. Yeah, please. So maybe let me take the chance here to make some actual further analogy here. I think what these dualities are also doing, one way of thinking maybe about them is to match high and low temperature expansions. I think for this one dimension, I don't I cannot do a general statement here. But I think it like high and low temperature would correspond like a short time and some time here or some later time here expansion about one or the other points. And I think I would guess there should be also a duality. But I've not thought deeply about that. One would have to check it. So actually, so since there are only a few minutes left, I'm done with this one part. I think it does not make sense to start with the next one, which is now on the experiment that I was showing you before. I think I would stop now and let you have a longer afternoon for yourself. Thank you very much.