 ddiddordeb nhw'n gwelwch i'ch gweithio i'r ddweud ac yn rhan o'r ffordd a'r ffordd. Rydyn ni'n byw, ac yn ddiddordeb nhw'n gweld rhywbeth i'w dweud o'r ffordd a'r ffordd, a'i'r ffordd gyda Andrew, y ddweud ac'r ffordd. Y twbl fyddion ymlaen gyda'i gael y ddiddordeb nhw'n gweld yr hyn oeson fwyaf o mynedd a'r mynedd mewn yma, ac yn y sylfaen o'r system hwn yn ymweld, yn oed eich storio i wych. Fy fyddwn i'r ffordd eich hwn yn ei gael ystod i'r ffordd am unrhywgol yng Nghymru, ac mae'n gwybod wedi'i meddwl mewn cyfnod i'r cyfnod wedi'i'r rhan o bwysig o'r bod yn iawn o'r ffawr. rydych chi'n gael ei gael i'r ffwrdd, ac mae'n gweithio'r cyfrifiadau, ac mae'n gweithio'r cyfrifiadau, ac mae'n gweithio'r cyfrifiadau. I feel it's good to get everyone on the same page, and then I'll kind of go through some details. What is Eigenstate Firmalisation Hypothesis, and how this is related to Firmalisation? I will explicitly show that the Eigenstate Firmalisation Hypothesis is violated in the non-integrable easing field theory, and then I'll discuss some consequences for non-equilibrium and give some conclusions. So this is the final talk of the conference, the very first talk of the conference by Gabor. Also discussed maybe some of these topics or in passing. Yes, OK. So the kind of problem that we're thinking about is how does a generic quantum system time evolve? So I'm thinking of how to compute quantities such as this, where psi is not an Eigenstate of the Hamiltonian. So I'm going to be considering the case where we have isolated quantum systems, so it's unitary time evolution, reasonable Hamiltonians, so the lattice model that I will explicitly consider is a local lattice model. We're going to prepare the system in some state, the ground state, or an Eigenstate of some Hamiltonian, and then change the Hamiltonian and time evolve, and then we're going to compute expectation values and see what happens. So there's kind of all sorts of questions that one can ask about such a problem. So we have our time evolved state, and you can ask questions such as, does the system relax? Do expectation values have a well-defined long-time limit? And if they do have a well-defined long-time limit, can it be described by a statistical ensemble and if so, which one? So some of these questions are really easy to answer, just by very basic arguments. So, for example, does the system relax? Well, we have a unitary time evolution. If we construct the density matrix, there are always oscillatory terms so the system as a whole cannot relax. However, it's widely believed that subsystems can relax, so that's if I focus on some small region of my system and integrate out its complement, the system as a whole can relax in the sense that the reduced density matrix relaxes to something that's constant in time in the long-time limit. Do expectation values have a long-time limit? Well, it's really, really easy to construct operators that don't have a well-defined long-time limit. So the most obvious example is if we construct a permission operator that's just the projector between two different eigenstates, this will always oscillate. So there's some provisos to this. Generally, it's believed that local operators, their expectation values have a long-time limit and this basically coincides with the fact that a local subsystem has a long-time limit. And then I'll talk a little bit more about this statistical ensemble. So why do we kind of ask these kind of questions? I'll go back to some motivation from 12 years ago now. So in 2006, there was a really elegant and beautiful experiment by Kinoshita Manger and Weiss called the Quantum Newton's Cradle. So this is a pretty well-known experiment now and it kind of kicked off the whole field of study of non-equilibrium quantum systems. And one of the really nice things about this experiment is it's really extremely simple and it really revealed a kind of profound lack of understanding of formalisation and what happens in non-equilibrium. So I just want to kind of very briefly sketch what they do in the experiment. I'm not an experimentalist so this will be a kind of a fierce take on what the experimentalists do but I think it captures essentially the correct picture. Okay, so David Weiss's group studies cold atomic gases. So they take some bosons that interact essentially via a delta function potential, a purely local interaction and cool them down to low temperatures and place them in a trap. So they have some kind of confining parabolic trap that their gas sits in. So in equilibrium, they have some gas distributed in the bottom of their potential. Then they apply a brag pulse. So what the brag pulse does is it splits this gas into two. It makes wave packets of momentum K and minus K and those wave packets move away from the origin and go up the sides of the potential. So they start their experiments with some collection of bosons here and collections of bosons here. So after this brag pulse, they separate, they go up the sides of the potential and when their potential energy, when their kinetic energy is completely converted to potential energy, they fall back down. So when they do this, they fall down and collide at the bottom of the trap. So they collide and then one wants to see what happens. So the grand insight of this experiment is that they tried this in two particular scenarios. They tried it when this gas was confined to a 3D parabolic trap and they also tried it when it was confined to a one-dimensional parabolic trap. So in 3D, the result is really very simple. So they start with some momentum distribution function of their particles that strongly peak to separate and equal opposite values of the momenta and after about three collisions, so three oscillations in this experiment where they move back and forth in the trap, they get something that looks like a thermal blob, so they get something that looks nice, which looks pretty similar to what they would get if they just prepared their gas in the trap to begin with and didn't do this brag pulse. So that's after about three collisions in 3D. In 1D, they did precisely the same. So they started with nice, sharply-peaked momentum distribution functions and then they did this experiment and they waited and they kept examining what was going on and I think, for example, in their paper, they show some plot from after 400 scatterings roughly and they end up with some momentum distribution function that looks really non-fermal, it has extra peaks. Basically, these peaks remain for really long times, basically as long as they could run their experiments. These experiments being in cold atomic gases really last milliseconds or seconds, so it's really an extremely long timescale for a quantum system. So 400 scatterings and this was a lower bound on possibly the thermalisation time. So one of the major questions is, okay, this seems like a simple experiment and all we've done is restrict the dimensions. Why does 3D thermalise really rapidly within three collisions? But 1D seems to not thermalise at least for any timescales we can reach in experiments. One of the central insights was to look at 1D and consider what they're actually doing. They have delta function interacting bosons in 1D or an integrable system. So it's an integrable system, this means that there are many, many conservation laws, there are as many local conservation laws as there are degrees of freedom and this strongly restricts the dynamics and this avoids thermalisation. And there's been really kind of 10, 12 years of work to understand all the intricacies of this statement. So integrability avoids thermalisation. Is it like fine-tuned interoperability? So that's actually a really good question. I mean, this is a real experiment, right? They break integrability in lots of manners. So for example, the 1D bos gas is integrable, but it's integrable when it's translationally invariant. Here they've put it in a trap, integrability's broken. Somehow it seems that certain types of integrability breaking matter and certain types don't. And this is also still something that's under a large amount of study. In fact, myself, Robert, maybe Andreas has also studied these issues. Also Gabor, sorry. Don't you mean that they will never thermalise or just the time scales are very long? So that's one of the questions. So in an actually integrable system, it won't thermalise. We understand what it relaxes to and I'll mention this in just a second. So this set of experiments really kind of led to an understanding that somehow there's some kind of dichotomy in quantum systems where we split them into non-integrrable and integrable, and then we can say various things about them. So let me just kind of summarise what our kind of basic knowledge of these systems is. So it's widely believed that non-integrrable systems thermalise in the sense that if I dump energy into my system and wait long enough, the energy redistributes in such a manner that subsystems and expectation values look thermal. So the long-time limit of some local expectation value is reproduced by some trace over a thermal density matrix. Where, for example, if you only have energy conservation, this is your typical thermal density matrix. And this temperature, so beta 1 upon T, is fixed by the energy of the initial state that you start in. So you do something, you dump energy into your system, you start in some non-agon state, but energy has to be conserved by unitary time evolution. So we can fix the Lagrange multiplier here by measuring the expectation value of the Hamiltonian that's doing the time evolution in the initial state and computing. So we completely fix everything. This is in a system where I'm just assuming energy is conserved. OK, so in integrable systems, things are different. We've already seen in this quantum Newton's Cradle experiment that the system doesn't thermalize, but we do believe, and we have explicit calculations of that by many people in the room, so they equilibrate. Expectation values have a well-defined long-time limit and we also understand, after some debate and effort, what this statistical ensemble that describes the expectation values or the reduced density matrix of a subsystem is the reduced density matrix of this density matrix. And here, I'm just going to show you what the statistical ensemble describes is the reduced density matrix of this density matrix. And here, this is what's called the generalized Gibbs ensemble. The thermal ensemble is known as the Gibbs ensemble. So it's generalized because now you have to include all of your conservation laws. So basically you introduce Lagrange multipliers for every local or quasi-local conservation law in your system and so integrable systems possess a set of conserved quantities Q. They commute with one another and with the Hamiltonian and you can construct the density matrix. And these Lagrange multipliers are fixed in exactly the same manner as for the thermal ensemble. Can I ask you something? Why do you care whether they're local or not? Yeah, that's a good question. Is it because you actually have a physical environment and you're assuming that somehow the local ones are more likely to be... Well, any quantum system has an exponentially large number of conserved quantities, right? The projectors onto the eigen states. So one should ask, like, can't we just do that for the thermal case? It has to be correct as well, right? However, yeah, okay. This is not 100% something I want to get into, but yeah, you can say if I'm interested in local observables I'm going to have some intuition that local conservation laws are going to be what's important. It actually turned out, and Gabor was very important. And understanding this is that in certain systems new classes of conserved quantities and integrable systems, quasi-local, they actually have some decay in finite extent, which were not really appreciated or realised previously, and you have to include these in certain scenarios as well. So there's some subtleties, but it's not really the point of my talk, so maybe we can chat afterwards. And yeah, okay, so we can fix these Lagrange multipliers in an identical manner. These are conserved quantities, and we compute the expectation value of these conserved quantities on the initial state, and we can, and this fixes these Lagrange multipliers. We can construct the generalised Gibbs Ensemble, and we can predict what long-time expectation values of operators are. So this has been checked quite, quite thoroughly. This was how it was realised you have to include quasi-local conservation laws, et cetera. So okay, so this seems like a pretty nice story, and I should mention that there were, there were really kind of seminal contributions to this by people like Marcus Regal, and some of his collaborators, where this was really appreciated and realised over kind of 2007, 2008. So okay, so non-integrable systems, we can thermalise, and this is what we see in 3D. Integrable systems don't, but they equilibrate, and we can understand what this equilibration is. So you can just ask quite simply, is this the full story? Because to be honest, it would seem a little bit surprising that there's a single partition into integrable and non-integrable and all non-integrable systems, no matter how different they are, behave similarly. So this has been something that I've worked on for quite a while. We understand what happens if you weakly break integrability in some sense of weak, and now I'm going to present you an example where integrability is broken, but in some cases the system doesn't thermalise. So part of this story was actually understood quite a long time ago with the Eigenstate thermalisation hypothesis, so I just want to mention this here. So the Eigenstate thermalisation hypothesis is a hypothesis, as it says in the name, and it's a set of conditions on Eigenstates and also observables that we believe leads to observables looking like they're thermal. So this came out of work by Deutchen91 and also Sredniki in 94, and I'm going to write this on a board and keep it up here because it's important for the rest of the talk. So the statement is the following. So for some arbitrary state, expectation values of an operator will approach their thermal values, if, and those two conditions. So the first is diagonal matrix elements of the operator in the Eigenbasis. If these diagonal matrix elements are smooth functions of the energy of the Eigenstates. The second point is the off-diagonal matrix elements should be exponentially small in the system size in the Eigenbasis. In the basis you mean energising? Yes, the Eigenstates of the Hamiltonian under which you're doing time evolution. So this can be summarised in a simple formula. So does that mean for a conditional one it makes sense you need a continuous spectrum? You can't have a discrete spectrum of energy. You can think of this in a discrete spectrum as well. So when this is usually studied on the lattice, because this is a statement about generic Eigenstates, you do exact diagonalisation, and what one looks for is that the spread of operator expectation values that are given energy is sharpening up with system size or Hilbert space dimension. So this is believed to be sufficient for formalisation. And there's at least examples in lattice models where this holds and we can compute non-equilibrium dynamics and see that long-time expectation values in systems that satisfy ETH are formal. So you're not saying anything about O bar alpha beta. You're not making additional requirements because some people, you know, this is a random matrix parameterisation of this. Yeah, so I'm not going to say anything strongly about it and in fact I'm not going to talk about off-diagnome matrix elements at all. I'm going to show the first condition is violated in the non-integral using theory. But yeah, there are all sorts of other things you can look at with off-diagnome matrix elements. So okay, so this... You can go away and compute. So what I'm going to do is I'm going to use truncated spectrum methods to compute this. Why are these conditions easier to check than just what they imply? Well, how do you compute the full non-equilibrium time evolution and go to the long-time limit? Although I'll show you some way in which we can check these as well. So there's no statement about the locality of these operators but my general belief is that these are non-local operators, these conditions won't be satisfied. But I have no rigorous statement for that. This is for local operators from the off-diagnome matrix elements to go to zero in the infinite volume that the sets of the assistants. Yes. But then so can I construct a local operator that doesn't change the energy? So what I will show you is this question I haven't thought carefully about. So I have computed off-diagnome matrix elements in eigenbasis and you can show that for reasonable operators they can be small. I haven't done super detailed finite size scaling analysis. So the focus of my talk and actually most people who do exact diagonalisation on lattices is checking point one. So I'm going to show you what I'm going to show you because this is checking point one. OK. So now I want to show you that this is violated in the non-integral easing field theory. And then I'll say a little bit about why I'm interested in the non-integral easing field theory as well. So in a particular case I'll be considering I think was partially shown by Andrew in the previous talk. So here I'm just going to explicitly include the speed of light. So I'm going to consider massive easing so perturbed away from the critical point also perturbed by the spin operator. So this is non-integral away from the massless limit and away from the absence of the spin operator. So this was partially mentioned by Andrew but one of the reasons why I'm interested in this as a condensed matter physicist is that maybe I'm interested in going the opposite way to many of the talks that have been given at this conference where I'm interested in using the field theory to understand a bit more about the lattice. So there's a related lattice model to this field theory which is a simple spin chain which is very simple. It's just a transverse field easing model with an additional longitudinal field and one of the reasons I'm interested in this is that this model is really realised to a good procedure where there's a rather beautiful realisation of this model in cobalt niobiot where various aspects of the field theory were kind of examined in neutron scattering so they saw meson masses they can see that the masses of the first two mesons the ratio of them is the golden ratio as you approach the critical point and there's also some material that I won't really consider too much where there's kind of an antiferomagnetic realisation of this model as well. So the field theory that we're considering emerges in some scaling limit close to the quantum critical point of this simple lattice model. And as a condensed matter physicist I have much more intuition about the lattice so I can just sketch out quickly what adding a longitudinal field does Andrew covered this quite nicely so in the absence of a longitudinal field you have de-confined spin-ons so in the ordered phase on which I will focus the rest of the talk you can form domains we've just a single cost of essentially violating two bonds energy approximately related to 2J and then they can move around because you can move it and you only break two bonds when you move these around. However for H non zero including even the case of kind of infinitesimally small H one has a kind of profound restructuring of the theory so now we have these domain walls which in the H equals zero case are our low energy quality particles but now there is an energy cost to moving domain walls apart proportional to the distance between the two domain walls this leads to a confinement transition and in terms of the spectrum in the field theory for example we move from a case where we have a nice continuum of excitations from twice the excitation mass to where we only have a continuum from four times the excitation mass and then we have discrete meson like modes that come from the linearly confining potential. This still holds in the abstinence of integral ability. Okay, so returning to the field theory the kind of path forward is clear as I already mentioned to Slava we will solve this with truncated spectrum methods so here we'll really use a massive non-interacting theory as our computational basis matrix elements of the spin operator in this basis are well known for 20 or 25 years I think sorry 15 years and we do supplement some of these truncated spectrum methods with numerical renormalisation group methods which haven't really been discussed so I thought I would just briefly outline NRG which is complementary to many of the NLO techniques that have been discussed already. So numerical renormalisation group is really easy to add on to truncated spectrum methods it requires a little bit of motivation so you imagine ordering your massive fermion basis via energy and usually what we do in truncated spectrum is we introduce some energy cut off lambda disregard all the states above lambda and diagonalise our Hamiltonian numerical renormalisation group techniques which Robert and some of his collaborators have used extensively with truncated spectrum methods allow one to take into account states above the cut off in a kind of well defined numerical way so the idea is you introduce some number of states you want to keep in your Hamiltonian or some energy cut off and then you divide the rest of your Hilbert space up into into groups of basis states and you use the fact that the perturbation is strongly relevant means that states down here are coupled progressively weaker and weaker to the higher and higher states then one does a simple in step one you take an M by N matrix which is generally dense and diagonalise it in step two you say let's throw away some delta N of the highest energy eigen states so so we've thrown away the highest delta N eigen states constructed in our truncated spectrum methods and now we add in the next delta N basis states then you end up with a Hamiltonian that's kind of dense you diagonalise and you can rinse and repeat this procedure so you can just iteratively do this step after step after step and really include very very many basis states this allows you to get convergence of your low lying spectrum it allows you to convert many many low lying states so you just progressively construct the matrix you throw away the higher states introduce the next number of basis states re-diagonalise and rinse and repeat and you're motivated by Wilson's anarchy approach to the condo problem in the condo problem you have a natural separation of energy scales because you map the condo problem to a spin that's coupled to electrons with hopping that decreases the further you go from your spin here we kind of have a natural emergent separation of energy scales from the relevancy of the operator but it's a similar general question about this method you're interested in highly excited states by definition is so I mean where's this method is good for as you say for the low states I mean so you can get convergence of really many states hundreds thousands are completely possible I mean of course it's a good question to ask because we're dealing with a field theory we're dealing with truncated spectrum methods we aren't so fixed system size we're at finite energy density but if we want to extrapolate to the infinite volume limit we're not really however I'm going to not address this issue but yeah it is something one has to think about another thing to think about is that this is a relevant perturbation right so when we go to the UV it vanishes so we already know in the UV the theory is free so ETH can't be true because it won't formalise because free system is integrable so it only really makes sense to look at the low energy sector for this theory for formalisation but yeah I agree there's some some some subtleties one has to think about so ok yes so we do anarchy we can construct many low lying states and then we can compute expectation values of operators and see whether expectation values of operators in the eigen states obey ETH or not so address point one elements in the eigen basis smooth functions of energy so if I could get the screen oh it's the big switch ok so ok so I had a brief reminder of ETH but it's on the board so yes as I say the path forward is clear we construct eigen states with truncated spectrum methods use these to compute expectation values of observables and we can also use the eigen states to construct the micro canonical ensemble and see whether expectation values agree with the thermal result so the micro canonical ensemble if ETH is true just says as my diagonal matrix elements are a smooth function of energy I should be able to average over eigen states within a small window and find the thermal result so ok so what kind of things do we actually get so one should ignore the right hand side of these plots essentially because you get truncated truncation effects and these are for a number of system sizes which are pretty large I'm really going to focus on this one where everything is clear you have to do detailed finite size scaling in the higher ones so what do we really see so we construct really a large number of eigen states I think there are three and a half thousand in this plot and you should probably ignore beyond energies here where there are strong truncation effects and we see well we see two things what is this is for g is 0.2 and the mass is equal to 1 ok so yeah we construct states and we can measure eigen state expectation values so the diagonal part of of this expression and we are measuring matrix elements of the diagonal matrix elements in the eigen basis of the spin operator so I call these EEV the eigen state expectation values so what we can really see is that there is a broad continuum as we expect where essentially we can compute the micro canonical ensemble this is shown quite nicely in the higher system sizes so it's a dense continuum where we can construct the micro canonical ensemble so the thermal result and we see that essentially this continuum here has eigen state expectation values that are consistent with the micro canonical ensemble but we see kind of a line of states that persist above the continuum so yeah let me see so these parts basically eigen state expectation values and the thermal result agree and the error bars on this are showing the standard deviation of data averaged over whereas there are these kind of lines of states above which are maybe most convincingly in the smallest system size but we have some better data for this where they're really well separated from the continuum as you increase your cut-off energy they remain well separated from the continuum and you might ask well what's going on here because this is indirect violation to ETH because this should be a nice smooth function of energy and clearly there's some other stuff going on above so most states seem to be thermal but there are some states that are well outside what one would expect for states that are going to be thermal so okay so it's actually we know about those sub-leading terms can you say something about them is it important? it has to converge exponentially so this is something that I'm currently looking at I haven't done the finite size scaling yet so for now I think it's sufficient to show that one of these is violated okay so when one looks at this plot there's kind of a pretty natural conjecture that one can make about the nature of these states low-lying spectrum as I mentioned here are mesons so I should have said we're in the zero momentum sector of the Hamiltonian so we're really just examining along this vertical axes one should always work in a given symmetry sector when checking ETH otherwise the results are a mess so we see so we understand what the lowest eigen states of the system are there are these meson excitations and I mean it's pretty natural considering they lie in a pretty straight line to just conjecture that these are probably the higher meson excitations to but notice that these mesons are above the multi-meson continuum and they remain well separated so I don't expect anything I'm just this is a completely unfounded conjecture where you say well they're going in a straight line maybe we should think about whether these states here as they appear to go in a straight line are related to these states here I'm going to show you some evidence that this is yes so we can construct them I mean we construct them exactly although just looking at for me looking at truncated spectrum wave functions is not so useful but I'll give you some evidence for what these are I'm going to support this conjecture so yes so the lowest lying states are mesons so they're pairs of domain rules with momentum p and minus p and yes it's natural to guess that these states are just a continuation of the low lying mesons so as we construct the eigen states we can directly check this history so in this plot I have two pieces of evidence for them being the meson states the first is these arrows are drawn at the energies at which I can semi analytically compute from some logical from Fonseca the energies of the mesons and you can see basically they lie very nicely on top the second thing I can check is I have the the states so I can just see are these two particle states essentially because the mesons I expect to be majority two particle states and this is what these green blobs here show so what one should see is that these green blobs match up with the position of the arrows and also the positions of the eigen state expectation values that are well separated from the thermal continuum that's a good question that's something that we're looking at because it's not obvious why they should be well separated from the continuum so I think so we're doing the lifetime calculation ourselves at the moment I think we believe they have extremely long lifetimes but maybe not infinite but so that's a question so when you look at the decay mechanism you have delta functions right and in the finite volume can you satisfy these delta functions so it may be in the finite volume they have infinite lifetime or they have very large lifetime in the finite in the finite volume that's true in the infinite volume I'm not sure I have anything sensible to say I think what you can say is that the lifetime is very long and reasonably long because of some face-face suppression because there's a high momentum what are they I would say it's a pressure of the matrix elements that would hybridise them with other so one thing that I would say is that so if one was to say take your lattice model so take your lattice model and do neutron scattering so it's precisely what they do in cobalt myobate and you compute dynamical spin-spin correlation functions so that's what I would say so if one was to say take your lattice model dynamical spin-spin correlation functions and then you evaluate the lemon spectral representation these are like one term and then you're summing over a ton of thermal states so my belief would be that you wouldn't see these in equilibrium correlation functions it's a single state and they're washed out by the thermal continuum what I will show you is that we have evidence and Gabor also has evidence that in non-equilibrium you can project strongly onto these states and see them in non-equilibrium okay okay so yes so so far we've just looked at equilibrium or eigenstate expectation values ETH is somehow a statement about for some arbitrary state if we time evolve expectation values will approach thermal values if these things are satisfied they aren't satisfied so we should look at what happens for non-equilibrium so okay so can I get the yes so I just want to briefly mention what we do in non-equilibrium so now we're really going to consider non-equilibrium dynamics within the truncated space approach and this was this is not something new Gabor presented a method a year or two ago 2016 I think for doing this very nicely for computing real-time dynamics I may run out of time to talk about real-time dynamics so I'll say what we do to check formalisation so I will present you evidence that in the long-time limit expectation values do not relax to their formal result and how do I compute the long-time limit of expectation values I'm going to compute the diagonal ensemble so the diagonal ensemble is essentially motivated in the following way I time-evolve my wave function I want to compute expectation values on this time-evolved state so yeah I'm just going to insert a double a double sum over the Hilbert space so I end up with some oscillatory pieces that take care of the time evolution operators and then I have this expression and then one can maybe time-average I'm just going to wave my hands a little and say in the long-time limit all the oscillatory pieces average or fluctuate away to zero I don't really like time-averaging so so let's say it's postulated that in the long-time limit these expectation values can be described by the single sum of just the diagonal terms so this is something that we can explicitly construct within TCSA because we have at least to some cut-off our eigen states and we can check convergence with cut-off et cetera and we find that diagonal ensemble converges quite nicely so we can plot this and then we also have our eigen states so we can of course construct the micro-canonical ensemble by averaging over some energy window our eigen states so we are going to compare the diagonal ensemble to the thermal prediction that's given by the micro-canonical ensemble so if I can get the screen back please and yes I should have used it Is there any way you can check whether this diagonal ensemble Yes if I get to it I will show you that we compute real-time dynamics and you can see at least for low-lying states the diagonal ensemble is consistent with the values that the real-time evolution is approaching Is there something you are going to say what O is? Yes for me it's always going to be a spin operator ok so this is the kind of plot we can get so first I should say neglect the right-hand side of the plot again because there are strong truncation effects here what I do is I I consider starting in the ground states so in eigen states of my Hamiltonian with mass one and some small spin perturbation and then I time evolve each of those eigen states so there are 32,000 states I time evolve each of them according to the Hamiltonian with the same mass different magnetic field value and I compute the diagonal ensemble and the micro-canonical ensemble now this plot looks pretty much identical to the previous plot so what we see is for the vast majority of states that one starts from they look thermal in the sense that the diagonal ensemble result so the blue and the black results coincide with the micro-canonical average however there are these bars of states above the above this thermal continuum where you've started in some eigen states and that projects strongly on to rare states and they're nicely well separated so quenches that start from certain sets of eigen states don't thermalize okay so and you see that yes we have various truncation effects, we get convergence at low energies one has to work harder at higher energies but this separation between the thermal and non-fermal remains and I believe Gabor also has calculations that are similar basically these meson states seem to converge very quickly so okay so this is the diagonal ensemble and the micro-canonical ensemble you can also ask what about real-time dynamics so for real-time dynamics we basically follow the procedure that Gabor outlined in 2016 we do some Chevy Chevy expansion of the time evolution operator and compute dynamics in this plot I show basically just a few states from this plot I choose some particular energy density and I plot a rare the time evolution of a rare state and I plot time evolution of states that are within this continuum so the purple result here is the time evolution of one of the non-fermal states now here we see there are really large slow oscillations which actually I think implies the off diagonal matrix elements for this state are large and we see basically thermal states have much smaller amplitude oscillations and they really relax much more quickly to their kind of diagonal ensemble results sorry this plot hasn't come out crazily well but there's a dotted yellow line here and that's the diagonal ensemble result for the yellow for the yellow time evolve the yellow time evolution so we see that basically the diagonal ensemble and the time average of the yellow are in pretty good agreement the time evolution of the blue and this blue is not so well but there's likely some convergence issues and then there are some states which are on the kind of edge of the thermal spectrum that seem to time decay much seem to decay in time much more slowly and with large amplitude oscillations which probably implies that these are states that somehow have intermediate off diagonal matrix elements when compared to these states and these states that settle very nicely and this shaded region shows what we expect for the thermal result plus the standard deviation they all have the approximately the same energy within some small epsilon so if these were all thermalizing one should expect them all to lie within this gray region and I should say this purple line here is the diagonal ensemble result for the rare state maybe actually the diagonal ensemble should be a bit higher but it's hard to tell okay so I have a couple of minutes so maybe I'll just mention something kind of nice about doing these real time dynamics that was for me motivated by Gabor's work on the lattice which is that one can take say the time evolution so here I start from the ground states and I do a small quench so I mostly project on to low lying mesons you see there's a lot of structure to this time evolution very many frequencies of oscillation essentially I have to cut off energies here and the results are very well converged for low lying states you can see the diagonal ensemble prediction agrees pretty well with the time average and what one can do is just take this time evolve it for long times and do a freer transform and this allows you really to do kind of spectroscopy on your mesons so you get these very strong peaks in your power spectrum of your time evolution and this allows you to pick out what the meson masses are this was also done on the lattice by Gabor very nicely so with that let me just see yes okay so I have one minute so okay so we get certain quenches that don't formalize even though this is a non-indigable model one question that one can ask is does this carry you through or to into the lattice or is this some funniness of the field theory so this is a really hard question to address because in the field theory you're in some particular scaling limit, Gabor spoke about this he has some very nice results but it's really tough to get the correct limit of the lattice so we kind of took a slightly different approach which is let's we kind of want to know whether this is generic so can one see this away from the scaling limit so we just did some lattice simulations actually motivated by some work by Maria Karma from 2011 so there they had a very nice result about absence of formalization on relatively short time scales where they examined the reduced density matrix of the free body reduced density matrix of some of the time evolved state and showed there was an approaching the free body reduced density matrix of the thermal ensemble personally I don't have too much intuition about free body reduced density matrices so we just did some ITBD calculations from an initially polarized state along the X axes which is precisely the quench that was considered in this work time evolved with a particular lattice Hamiltonian and we looked to see okay is there any signs that in this lattice model which is completely non-integrable that there might be an absence of formalization and what we find is okay we compute essentially the same observable so we start in a completely polarized along the X axes state so we start from sigma z is equal to zero we see we get time evolution that takes us beyond the formal result that we compute quite simply with exact diagonalization and we believe this is at least some supporting evidence that formalization on the lattice may be avoided in certain scenarios okay so with that I will finish so to conclude I think this dichotomy of non-integrable being formalizing, integral being non-fermalizing needs to be re-thought or is at least not completely the full story Hamiltonian truncation is a nice tool and technique to tackle some of these problems and interesting non-integrable theories there's a question of how to go to finite energy in the dynamic limits that I think is still outstanding I've presented you some evidence that the non-integrable ising field theory violates ETH and this has real consequences for non-equilibrium dynamics and yeah I think there's still plenty to understand about formalization for example ETH in a field theory I still am not sure I entirely understand why because you might say that this is a statement for generic states states that are mid-spectrum on the lattice but of course there's no concept of mid-spectrum in a field theory I get two particle states infinitely far into my theory I don't know how to interpret ETH in this case this absence of formalization is it a feature of the field theory or is it a feature of confinement and if it is a feature of confinement in another confined theories so here it was I think it's important that the interaction is relevant and that you have three guys in the event for this feature QCT already does not show formalization in your definition two protons come in and jets come out it's of course non-integral okay so I think there's still plenty to understand about formalization and thanks for listening questions? well I guess the first question is it's clear that this you understand that there are these meson states if these meson states were exactly stable then clearly on the one hand it's clear that they would violate the exact termization on the other hand it's also clear that such things should not occur so if they do occur in your theory then perhaps you should just concentrate on why this happens in this theory yeah sure I mean this is something that we're working on understanding although I would say to me actually personally it's non obvious that two meson states which are what we probe in the low energy part of the spectrum are formal like why should two meson states be formal? because I thought that it's only meaningful to ask this sort of questions for states which have finite energy density if you have two meson states I think what we see is that this is a reasonable question to ask because we do see ETH being valid for the vast majority of states now I don't know why it's valid for the vast majority of states I think that's something that needs to be understood naively I would agree with you like at the low line part of the spectrum I don't know generically that one should expect ETH to hold and we can of course only target the low line part of the spectrum but the fact is that ETH does appear to hold for the low line part of the spectrum in this theory and I think that's something that we need to understand like I would like to see whether this happens in other cases but it's a non obvious statement I completely agree I was not expecting to see ETH to be valid there are people playing a strong version of ETH applying to all of the states so I didn't mention this but yes so there's a weak version of ETH and a strong version of ETH so a weak version of ETH says you have rare states but the number of rare states to the number of thermal states vanishes in the thermodynamic limits it's widely believed and I think there's some evidence from Vincenzo Alba and others that this applies in integrable models this statement that there are many states that are thermal in integrable models and you have a spread of non-fermal states but there are more thermal than non-fermal states then there's a strong version of ETH which is supposed to be sufficient to prove formalization which is that there are no rare states in the thermodynamic limits now there are some lattice models where it seems suggestive that strong ETH can apply in certain scenarios but so for example if you take this easing model and you go to some particularly weird couplings where these couplings are irrational they seem to see strong ETH but if you go to a different set of parameters you see weak ETH or signs of weak ETH so it's kind of a mess I don't know what much else to say other than like this is a tough problem because you need to study eigenstates so what do you do you have to do ED you're restricted to low energy or you're restricted to tiny systems maybe you have some comments I think some of these effects we actually have seen I mean not in the same language but in some study of policy hybrid models and we had a work where we looked at also at overlaps for the particular question we looked at that we had strong peaks of our signal on states which we also called rare typical ones I think that's exactly what Gabor sees as well strong overlap on to rare states in certain scenarios but the point is a bit that if you in the particular spectrum we looked at these are kind of these states are finite size there are states on a finite volume but if you start to think even if you cannot solve it what happens with these bands and these structures if you actually conceptually start to think if you're at finite density then you actually start to see these states start to overlap in energy with states close by which actually are of a different nature and you have to start worrying what the matrix element is so I think at some point if you start to think in some even if it's not very specific in terms of the quantum Boltzmann equation what would these meson states at finite density start to do you have to work out what its interaction is and even if there's a small matrix element I think it's surprising that there are small matrix elements perhaps a long game but ultimately I think these systems would thermalize that's my opinion I cannot prove it but I think at some point if there's not a reason why there's no matrix element to really protect them I think ultimately they will have a very long time but ultimately they will thermalize but it's very hard to see and in the disposal of the Harvard model I think this question is still open I think we also understand why it seems non-termal especially if the matrix element is small then you can show that it's going to start thermalizing in perturbation theory it's theoretically it's advantageous when the matrix element is not 0 but small I think the key is that the matrix element is small about the meson states and the full theory do not hybridize strongly with the continuum and will sort of maintain this two-meson this two-court character that they have even if that's what you want to target for a calculation I want to see I understand kind of what you're saying but I'm saying suppose that you can show that this meson is EG with a very small with a very long black line by meson then the meson is computed by the two-court approximation in the stem alive I mean you can't construct the full excitation that's one of the issues you want to compute the lifetime of the mesons what's the meson wave function that you want to compute for you only know that approximately so is the lifetime that you compute a result of that approximate realisation of the wave function or is it a if one included contributions to all-order do you get a finite or an infinite lifetime I don't know what the answer to that question is I think it's generically a tough one I mean we construct well-converged low-lying eigen states by definition eigen states have an infinite lifetime now how that happens is you increase truncation and finite size is a tough problem of course you have to be really careful I mean the states we construct have infinite lifetime within our truncation scheme but I agree is this a finite volume effect I think is a subtle question also you don't know if the absolute is stable sorry first mesons would be absolutely stable of course and I also have a comment about the relevance of the few-particle states namely there's another parameter in the quencies in this game is the density how large is the density as compared to correlation lengths because if the density is too small I mean the average separation of two particles is too large compared to the correlation length then even a finite density quench is just a few particle dynamics on average a sort of like exponentiates your system into one large boxes in which you have only a few particles occurring such that these boxes are already very very large compared to correlation lengths and in a field theory that means that finite size effects must have an exponentially small it's not clear that these finite size effects could turn over your things I mean could initiate penalisation being exponentially small already very very very small extremely small and also that this is what you see in our original DCS simulation of time evolution of which there are more to come actually very soon so that finite few particle dynamics dominates lots of quencies actually so these low-lying states they are not dominated directly by few particles but these few particle configurations separated by very large distances so in a sense the few particle contribution is what appears in the result itself but isn't that just saying that the time scale is again longer because it takes longer to actually I had a question whether it is just saying that I am in two minds about that whether it is just saying that the time scale is where it is longer than the lifetime of the universe you would never see it in an experiment by the way right so at some point it becomes about practically infinite long time scales maybe theoretically your time scales are infinite but if it's too long then you might as well forget about it I think that the question of the elements of these results does not depend on whether it is strictly thermalising or not I mean it is a dynamical fact we see this in simulation we see this in the experiment that seasons do not immediately thermalise on this short time scale that's a reality they ultimately thermalise or not that's a subtle and tough question I completely agree with integrability breaking we know from the quantum Newton's Cradle experiment integrability is broken absolutely there is a trap there are free body losses free body losses are disastrous they break even energy conservation yet they see something that is integrable for all the Q4 time scales I have always taken a practical approach to these things I am not sure that it is meaningful to worry so much as Gabor says if things are exponentially long does it really matter I mean we actually have worked on the spring chain which we say is not equal but actually I think that there is a nice playground where you can actually continuously change your energy between the strongly thermalising regime where even with current numerical methods we see very nice thermalisation but then you can slightly change into some of your other states like this x state I think that you have looked at which actually is like effectively integrable I think that some there is a paper by Alexey on his Q4 where they look at some of these channels and they see this like low quasi particle density that might be that it arrives on Boltzmann equation where they see ultimately but there are very long time scales on which it looks I think that might be a general picture depending on how much energy you are pumping and which energy range of your problem you are I mean the thermalisation length personally with the problems I've worked on in the past with pre-fermalisation and thermalisation this is exactly it I mean we have some systems that we know are not integrable but for any achievable time scales we don't see thermalisation if you really pump up the energy you can sometimes see faster thermalisation you can tune it's I would say like there aren't so many generic behaviours as such like it's really a case by case basis of understanding for me at least at the moment Do I understand correctly that there is no idea on how this idealisation for this equation is reaching in terms of time so one over two correction Yeah okay that's I don't have a good feel off the top of my head I mean I mean I have examples of system by system basis where I think I understand thermalisation in those systems but yeah I don't know that this gives you a generic question by some sort of conjecture is this an output of some sort of a complication because I don't know Yeah okay so one would need to check the Deutsch and Schradniki papers but I mean this is it's not rigorous and I mean they have some arguments but yeah I don't think it's on a rigorous footing but maybe you have To the conjecture why a closed one system would thermalise at all So the conditions that you need One of the locked in Yeah Okay if you have no further questions let's thank Neil again and let's thank all the speakers of the workshop