 And that brings us to the last talk of today. So it's my pleasure to announce John Chalker, but we'll talk about many body localizationists. Okay, thanks. So obviously this topic is a bit of an outlier at the conference. It's very nice to be here. And I'm grateful to the organizers for sanctioning the subject. Actually, yesterday, Piers Goldman in the introduction to his talk mentioned that there's a school going on in the main building, which is more quantum information focused. And so perhaps you could think of this as news from the other side. But if Piers is listening, then I'd like to reassure him that two of the lecture courses at that summer school are firstly on Fermi liquids and secondly on frustrated magnetism. So these two branches haven't completely separated yet. So I want to talk about many body localization and the transition to an egotic phase. And I guess I'm assuming that people will know what I mean by those words. And my main point is going to be that whereas many body localization and this phase transition is generally held up as being quite distinct from say the thermal phase transition so we're familiar with, there is actually a way of looking at things in the language of symmetry breaking. And it's that approach that I want to talk about. I felt I should say a little bit to set things in a broader context. So I'll be talking about random quantum circuits and I'll spend a couple of minutes first introducing quantum circuits and also mentioning other things that people have been using them to study. But then the particular content of the talk is thinking about the spectral form factor which is a way of discussing energy level correlations or eigenvalue correlations and how it behaves as you go through the many body localization transition. And then it sounds rather technical but I hope I can get it across. I want to talk about a transfer matrix which generates this spectral form factor and how the behavior of the transfer matrix as you go across the transition involves exactly the features that one usually associates with symmetry breaking. And then finally I'll try and explain what the symmetry breaking involves. And the limitation is that it's very much something internal to the calculation. Although I'll argue in the introduction that it's a kind of phenomenon that we're quite familiar with in a lot of different contexts. It involves the Feynman-Pals that contribute to the spectral form factor that I'll introduce. So I think there have been real advances in thinking about dynamics in many body quantum systems in the high temperature, highly excited regime. And the point is that, well, traditionally we've thought about what happens at long times in terms of hydrodynamics for conserved quantities. We now have ways of thinking about equilibration of an isolated system evolving under the usual unitary dynamics of quantum mechanics in terms of other slow degrees of freedom aside from the ones that are connected with local conservation laws. And this involves the dynamics of entanglement. And alternatively you can think of it in terms of operator spreading and the dynamics of quantum information if you disturb the system at one point and ask how the effects of the disturbance spread out. So I think the new understanding has come partly from constructing some very simple tractable models and these models are the quantum circuits and really the motivating ideas are quite few. I mean, the first one is that you could say that one of the great lessons from random matrix theory is that although no individual generic quantum system is solvable, we can calculate properties quite easily if we're willing to discuss averages over some kind of random ensemble. So the strategy that's used here is to combine random matrices with some kind of spatial structure so that you have models which have the notion of locality built into them from the spatial structure and which use the randomness to give you something solvable. And then the third point that's maybe not obvious but turns out to be quite important is that since for the problems that we're trying to understand, we're really interested in time evolution. It's actually useful to model the time evolution operated directly so write down some model for a unitary rather than a model for the Hamiltonian and that takes us to quantum circuits. So to explain what they are, we can think in terms of this sort of space time diagram. So suppose we have a one-dimensional system although it doesn't have to be one-dimensional and we think of a lattice and then at each lattice site, we have some kind of generalized spin with in the general case, a q-dimensional local Hilbert space and then time evolution involves a series of steps so like trotterized time evolution but without any intention of making the time steps infinitesimal and so we apply in this example two site unitary gates alternately across say the odd bonds and the even bonds in the lattice and so build up some dynamics and then you need randomness to have something that has a chance of being tractable and there are two ways of doing things. You can either represent a problem where the Hamiltonian is random as a function of time and what this picture is supposed to represent is each gate being chosen independently of all of the others and so the colors are all intended to be independent of each other but alternatively you could think about a blockade system so that you have some fixed time evolution operator and then each gate acting on a given bond will be the same realization of the randomness and in both cases under certain conditions you can do exact calculations in the agotic phase so that's going to be my starting point although I'll also be interested in the transition to the MBL phase where we don't have ways of doing things exactly and the particular language that I'm going to want to use when I talk about symmetry breaking is in terms of Feynman paths in folk space and I've put paths in inverted commas because they're not really paths in phase space the way Feynman introduced them. I simply mean that if we have the evolution operator say over T time steps of the blockade system between some initial state and the final state and we expand all of this out so by W of T I simply mean the evolution operator for one blockade step raised to the power T if we expand all this out then obviously I can express things in terms of sum over intermediate states and I want to think of one sequence of labels in that sum as being a path in folk space. So there are lots of situations where you think about contributions from paths and if I think not about the amplitude for a transition but rather the probability then of course I have a double sum over pairs of paths and the diagonal term in that sum is necessary positive and so if we can drop the fluctuations for some reason or other then we've got something simpler that we can focus on and there are lots of situations where making that diagonal approximation has been very useful. So I think one of the first was in semi-classical calculations in quantum mechanics of course developed by Goodfiller and applied to chaotic systems by Barry where you have a theory of energy levels based on periodic orbits treated within this diagonal approximation and if we think about this order of conductors then of course all of the understanding based on diffusons and coupons again is in terms of this diagonal approximation. So my point is partly that in these many body systems based on quantum circuits for example everything is set up actually in order that the diagonal approximation is a controlled treatment and these fluctuations are either zero after averaging over the random matrix ensemble or otherwise at least under control. Now I want to put that idea of Feynman paths on one side for a bit and talk first of all about this way that I want to use to characterize the MVL transition and then about symmetry breaking in the language of the transfer matrix and unless I run out of time badly I'll come back to Feynman paths and symmetry breaking at the end. So I'm going to think about flow case systems from now on. So I have a unitary matrix which has some eigenvalues and I want to think about the spectral statistics of these eigenvalues and I can do it in terms of what's called the spectral form factor which is just the Fourier transform with T the Fourier transform variable of the two point correlation function of these eigenvalues and I can write it as the trace of the evolution operator over T time steps where I'm using this notation here to make it clear that T is not a matrix transport. And to remember what to expect in the behavior of the spectral form factor as a function of time if the eigen phases are completely uncorrelated then the only term in this double sum that survives the average is the diagonal one and if we have N uncorrelated phases then the spectral form factor is just a constant N but to have something to contrast with that we can think about random matrices and if we take a uniformly distributed ensemble of N by N unitary random matrices then large values of time we have the same behavior as for a Poisson distributed set of energy levels but at shorter times at times smaller than the Heisenberg time the inverse of the level spacing you in fact have a spectral form factor that's simply linear in T and in fact equal to T in the units that I'm using. So those two examples are in fact good proxies for what happens in the two phases of a system with an NBL transition. So in the NBL phase the levels are almost Poisson distributed and the spectral form factor behaves as for Poisson levels after a time set by some microscopic localization time and in an egotics system beyond some Dallas time we get the random matrix behavior of the spectral form factor. So the idea is that the distinction between these two types of behavior is a pretty straightforward clear cut way of distinguishing between the two phases. Sure, this probably doesn't I mean I agree that that would be a good definition of an NBL phase and I'd be happy to accept it. I probably ought to say that that's it's not obvious that this connects with that and then to explain why it should connect with being non-egotic I think you probably want to go via the L bit picture of an NBL phase and then according to the L bit picture you get to this behavior of the spectral form factor. So I'd say there's something higher level which includes what you're saying about non-egotics egoticity and that leads to that behavior. Yeah. In the previous slide for the unitary random matrix why does it have a distinction between these two? Well, that's because of level repulsion really which says that the short wavelengths components in the spectral density fluctuate much less than if there weren't any correlations between the levels. Okay, so now I want to take it as I accepted that the spectral form factor is a good diagnostic for the transition and I want to talk a little bit about calculating the spectral form factor. And so here's some pictorial notation. So this is the picture in space and time that I had of quantum circuits. And I said the spectral form factor is the modular squared of the trace of the evolution operator over T time step. So taking the trace, we wrap this circuit on a cylinder in the time direction and that's what I'm representing here without of course the details of the gates. And so the spectral form factor, that's the modular squared of the trace. So I need two of these cylinders and I'm drawing one in blue and one in green. And then when I think about the details that I've left out here in contracting these gates, I can think about evaluating the spectral form factor either the way that I did originally contracting the gates in the time direction. So going around the circumference of the cylinder or alternatively I could operate in the space direction taking slices. And if I choose to do it second way, then once I've averaged over disorder, then I'll have a translation invariant transfer matrix which gives me the spectral form factor. So this is the central object in what I once talked about and it's partly the spectrum of this transfer matrix that gives an analogy with problems in statistical mechanics where you have phase transitions. So of course as in statistical mechanics, it'll be the leading eigenvalues of this transfer matrix which tell us the spectral form factor. And what generically happens in a transfer matrix description of a problem when you have symmetry breaking is that you get degeneracy of the leading eigenvalues and it's that feature which you also find in the MBL transition in these Flocke models. So actually we can see what we should find in the spectrum of the transfer matrix in each of the two phases just on the basis of what I said happens to the spectral form factor in the two phases. So I mean, I'm ordering the eigenvalues by their magnitude and in the usual way, the spectral form factor in a system of length L is given by a sum over these eigenvalues raised to the power L. So in the agotic phase on time smaller than the Heisenberg time, I said that the spectral form factor is just equal to the integer valued time and then of course, that's independent of the system length. So in order for that to work out, we need the leading eigenvalues just to be unity and then we have some sub leading eigenvalues but they're less than one. So if L is large, they don't contribute and that having eigenvalues equal to one that's enough to have a spectral form factor that's independent of the length of the system but to have the value of the spectral form factor given by the number of block A periods T, we need to have T of these eigenvalues degenerate and equal to one. Whereas in the NBL phase, the spectral form factor is simply the dimension of the Hilbert space and if I have a local Hilbert space dimension Q, then in an outside system, the overall Hilbert space dimension is Q to the L and what we require from the eigenvalues to reproduce that behavior is that the leading eigenvalue is simply Q so that raising it to the L power gives the behavior we're looking for. And so in the agotic phase, we have these degenerate eigenvalues, a signature of symmetry breaking and it's the NBL phase that's the high symmetry phase. Okay, so that's what we should expect in principle. What's the evidence that it actually happens? Well, we've got two bits of evidence I'm only going to tell you about one of them. In the agotic phase, there's a way of constructing these quantum circuits that's exactly solvable in the limit of large local Hilbert space dimension and you can verify that that gives exactly the behavior I've talked about from the transfer matrix but you can also go to a problem that's more generic in the sense that it doesn't use this large Q limit and that importantly it gives you access to both phases and then you can do numerics and it's that evidence that I will talk about. So the specific problem that we studied to do numerics on it's a spin a half chain and it's basically a kicked in the sense of a floquet system random field Heisenberg spin chain. So the floquet operator is defined pictorially by this arrangement of gates as in the systems I talked about in the introduction and an individual, one of these gates has two factors to it. Partly we evolve the pair of sites that the gate couples under a standard Heisenberg interaction with some strength J and then for the other part of the time we evolve the two sites independently in random fields and these random fields chosen to generate you to rotations of the spinner halves and as you'd expect if the Heisenberg coupling is small then you're in an MBL phase and if the coupling is large, you're in an agotic phase. So you can study this model in all the ways that people do for MBL transitions but I want to talk specifically about this transfer matrix approach and in particular you can look at the behavior of the leading eigenvalue of this transfer matrix as a function of the number of block A steps for different values of the coupling that take you between the MBL phase up here in blue and the agotic phase down here in red. So what you see is the intermediate times there's behavior that's not so clear cut but if you go to long times then in the MBL phase the leading eigenvalue is simply the dimension of the local Hilbert space so that's two and there's been a half model and on the other hand in the agotic phase, thermal phase the leading eigenvalue is one and then you can look at the degeneracy of these eigenvalues basically by looking at the spectral form factor which has constant contributions from all of the eigenvalues divided by the leading eigenvalue and so if there's only one leading eigenvalue then this ratio will be unity which is what happens in the MBL phase but if there are T degenerate eigenvalues which is what I argued should happen in the agotic phase then this ratio will be proportional to T and again that's what we find at long times. So the argument is that exactly the behavior that we anticipated comes out in the numerics for the transfer matrix in this model. So then the remaining thing is to talk a little bit about what the symmetry breaking actually is that's implied by the degeneracy of the eigenvalue of the transfer matrix in the agotic phase. So for that I want to go back to the diagonal approximation the idea that if you work out the probability state to get between an initial and a final state you can expand it in terms of amplitudes associated with paths in folk space and pick out a diagonal term where the phases from the two contributions cancel. So in the simplest case of a transition between initial and the final state the paths that pair in the diagonal approximation exactly the ones where this label Q is the same as P but this kind of phase cancellation can sometimes involve not just a unique path but rather a set of paths and actually that's exactly what happens if we think about a calculation of the spectral form factor. So these pictures are supposed to represent the calculation of the spectral form factor and of course that's the modular squared of the trace of the evolution operator raised to a certain power. So the green loop is supposed to represent a particular contribution to the trace and it's of course a closed loop like the cylinders that I had in the other pictures because I'm taking the trace and then the blue path is supposed to represent a contribution to the complex conjugate and I get phase cancellation if the two paths are the same but I also get phase cancellation if I offset the origin for one of the traces by either one time step or two time steps or more time steps and it's exactly that kind of constructive interference that leads to a linear in T behavior for the spectral form factor. Now those pictures are I think a faithful representation of what happens in for instance random matrix theory but when we go to a spatially extended system then you have a possibility of one particular kind of pairing between the paths in one domain and a different kind of pairing in another domain and the question is whether the system breaks up or the contributions to the spectral form factor break up into many separate domains or not and the kind of long range order in the agotic phase is exactly the formation of a single domain for this pairing and the absence of long range order in the NBL phase I think is exactly a proliferation of these domain walls. So the last point I want to make is to do with some more numerics and so we can try and introduce a local order parameter which tells us about the pairing of these paths. So there's a bit of notation here but I hope it is digestible. So let's focus on what the spin is doing at a particular site X in the system as a function of time in some path in Fox space that contributes to the trace of the Flock A operator over T time steps. So for a given path in Fox space then I'll have a sequence of orientations for the spin in whatever basis I'm using and if I think about a contribution to the complex conjugate that I need to combine with this first trace in order to get the spectral form factor then I have some other path in Fox space at that site and if I have the kind of pairing that I was representing either here or over here then I can recognize it by saying that the paths are the same and I can ask what the offset is between the two paths in other words whether these starting points are the same or different by looking at these two sequences of spins and gauging what this offset is. So what we want to do is construct an order parameter that tells us whether the two sequences are the same and if they are what this offset is and basically you can write something down with delta functions and so on which has a phase that tells you the offset and a magnitude that tells you the fraction of the states in one path that match states with a given offset in the other path. So this is a local order parameter but as usual unless you apply a symmetry breaking field which we're not going to to probe long range order you need to look at a two point function so we make a two point function out of this local order parameter in the usual way and then these averages are averages over the paths in Fox space that contribute to the spectral form factor and the basic result is that this correlation function shows long range order in the agotic phase and only short range order in the MBL phase. So this is data admittedly from very short time systems but we see exactly this difference between order at least over the length of the system in the thermal phase and only short range correlations in the MBL phase. So that's what I wanted to tell you so the messages are that the MBL transition is signaled in this transfer matrix spectrum and the idea of the diagonal approximation that's familiar say in treatments of single particle problems in this ordered conductors in terms of diffusons and coupons that gives us a description of the agotic phase and when you think about a quantity such as the spectral form factor then you have some freedom in the pairing of the Feynman paths that contribute to the spectral form factor within the diagonal approximation and that pairing is a form of long range order and it's exactly this long range order that disappears at the MBL transition, thanks. Questions, do you have a feeling for whether this transition occurs continuously or discontinuously? Well, I think there's a general strong presumption that it should be continuous. We so far anyway have deliberately stayed away from the transition because of course it seems to be very hard to get reliable results on it from numerics in the available system sizes. So, I mean, our approach has been to try and nail down what's going on in the two phases and postpone talking about what's happening actually at the transition. I mean, I think there's a kind of qualitative picture that you can imagine in terms of these domain walls proliferating as you go through the transition and that's sort of what happens at continuous thermal transitions. But I don't think, I haven't got anything more solid to say about why the transition should be continuous except that it's quite hard to have, it's harder to have best order transitions in random systems. What happens if one considers unitaries that are specially extended but the number of gates is larger than two or introduces, let's say, unitaries with random number of gates or some flow case states, time periodic flow case states for unitaries that are like multi leg type. But I guess what I'm asking is how universal are MBL transitions with respect to a special extent of unitaries? Well, I guess what I can say is mostly prejudice rather than based on specific calculations. But I mean, I suppose what you'd expect is that if you have gates that couple more than two sites at a time, it tends to make the Goddix phase more stable and the MBL phase less stable. But if the coupling that the gates induces sufficiently weak then you should be able to have an MBL phase with three-site or four-site gates. But I mean, I think that's reasonable prejudice but it's more prejudiced in calculation. In some toy models, which are actually not interacting, there is a situation where the eigenstates are ergodic but in a squeezed part of the space, of all available space. For instance, this Rosnoprak for the model is such. And what kind of behavior would you expect if this would take place for some quantum circuits? So you have two type of behavior. Either it goes to one or it is going at large time to Q. Suppose that you have a situation where not every corner of human space is visited but on the manifold, sub-manifold of the space you have totally ergodic distributes. Well, I think, I mean, the understanding is that the states in the MBL phase are partially extended in Fox space in the way that you say. I mean, they involve a divergent number of basis states in Fox space but a vanishing fraction of the total set of basis states in Fox space. But in the MBL problem, that's consistent with the behavior that I showed for the spectral form factor which reaches possible behavior after a finite time. So it may be that that's different from what happens in random regular graphs. Okay. I don't see more questions here. For some reason, I think there's something in the chat that doesn't show up on my computer. So we'll have to check it here. Yeah. Do you want to read it or should I? Sorry, I'm scrolling in the wrong direction. Yeah, Pes said he explains why he can't come. Is a glass, well, this is Pes to everyone. So is a glass an MBL phase? If not, what's the distinction? And then he says, if need be, think about a quantum glass. Right. So does anyone else want to say something about, I mean, it's addressed to everyone. Or maybe it's addressed to me, I'm not sure. Yeah, well, I guess one point in a spatially extended system would be that you could imagine having something that you'd be happy to call glassy because particles didn't move but it would still conduct heat. Whereas in a MBL phase, you want all excitations to be localized. And yeah, well, Pes was terrifying that meant that to me. And yeah, then there was a question about time crystals and could we use the transformatrix spectrum to find a time crystal phase? So really the question is, what happens if you think about time crystals in the language that I was using? In other words, the language of the spectral form factor. So time crystals do have a characteristic feature in the spectrum of the evolution operator but it's to do with correlations between eigenvalues. On opposite sides of the unit circle. So I don't think the kind of transfer matrix approach the spectral form factor that I was talking about would particularly shed any light on that. Perhaps you could devise a different kind of transfer matrix that would be useful. All right, I don't see any more questions neither here or in the chat. So let's thank John again. I guess we pass the question.