 Thank you for the invitation to this meeting at this very nice place, and I want to present sort of like a status report of ongoing research including also some extremely recent stuff. Some of it is just few days old actually, so But let's start with some introduction So what I, what my talk will be about quantum quenches in QFT, and more specifically about using Hamiltonian truncation approach While not just that but but it's a very big part of the thing, time we're going to talk about To understand these quantum quenches, okay, so what is a quantum quench? Probably many people are familiar with it, but anyway some quick introduction Doesn't do much harm, so we take a Hamiltonian of a system which is Even the system has this Hamiltonian until some time say time zero and the time zero sort of like we turn a switch We change something abruptly. I mean there's something could be a coupling in the Hamiltonian It could be some local flip of some degree of freedom It could be some change in boundary condition. It could be a lot of things. I'm going to talk about here Mostly, well not mostly, I think exclusively this time about global quantum quenches Which means that I will suppose that H0 is translationary invariant H is translationary invariant It's a in principle. It's an infinite system Which is not the case for the numerical calculations obviously, but in principle it's an infinite system and So this is called the global quantum quench translationary invariant global quantum quench and both of these are Translationary invariant and what we are doing actually is we are adding some interaction integral local operator to the Hamiltonian Okay, what are we interested in So in general, this is a very big field with lots of different systems in west being investigated But this touches some very fundamental cornerstones of quantum statistical mechanics It's basically what what what you are saying here is that I have a closed quantum system And I bring it out of equilibrium. What I want to know is whether it equilibrates Whether it becomes thermal by itself And I won't go into specific details about why do you think it is but this is possible But there is something called the eigenstate thermalization hypothesis Which is basically it states a mechanism whereby non-integrable systems can become thermal even without an Environment so it's a system is large enough then for its subsystems it can act as its own environment sort sort of and if the system is integrable then some other interesting things could happen like Non-termalization or thermalization in a more general sense Equilibration I should say in a more general sense. So the first question is Whether there is a equilibration whether the equilibrium if it exists whether it's thermal Or not and if there is a equilibration and The next question is how does it reach equilibration? So it's the question of relaxation So how does the time evolution actually happens towards equilibrium? You know, what are the relevant time scales? so actually the I will say time evolution time scales and related stuff and Well, you might actually be if there is no relaxation that could even be more interesting you could You could have some very interesting phenomena some unexpected things. I mean normally I should say whenever I turned on a new system on The computer just to look at its behavior numerically right away if I turn on a brand new system I always found something which I would which I didn't expect. Okay, so this was happening for quite some time So I should say there are lots of interesting things if if one just looks into these behaviors. There's a variety of behaviors so But it's also interesting to ask whether there are some universal features in these non-equilibrium behaviors for example, such as the so-called Lycone behavior which is I think Systems with nearest neighbor or local rather I would say local interactions Statistical mechanics. There's there are nice theorems saying that the Libro means on theorems. Basically, there are several versions of them Saying that there's an upper limit to the Propagation speed of excitation and that means that correlations entanglement whatever can only propagate inside the so-called light cone Well, the light cone is some sort of characteristic speed speed here Not necessarily light, but everybody calls it the light home because it just looks like a lot of stick on Okay, so so these are the things now Next question. So this is a statistical mechanical question. So why why do we think you FD? Well, maybe the simplest way to come to this idea is that QFT as we know in equilibrium It's sort of like a universal description of statistical mechanical system. So you might want to look at universal Description however, I'm already giving here a giving here way part of the game That is this is dangerous Because if you have a quantum quench Then an abrupt quantum quench, which you can actually do in a laboratory So this is really an experimental relevant question time talking about okay So the abrupt quantum quench has a timescale tau, which is very short. Basically, you never do anything instantly there's some timescale tau and your QFT as Applied to a statistical mechanical system has a cut of lambda Okay, so this is sort of like the timescale under which the switch happens from 8 0 to H and It's obviously a problem if or at least it seems so when one over tau is larger than lambda I should it doesn't necessarily need to be much larger Even if it's just comparable, that's a problem because you you are expecting to excite degrees of freedom Which are over the cutoff where your field theory description is not valid Okay, I'm giving about part of the game I'm telling you already that this turned out to be not such a big problem as it seems first so the argument is is is very I mean Clever and this this argument is fine, but for some reason that I'm going to tell you about this is this is this is not such a big problem However, there's another problem Which I think is still there is that the QFT is not something like a statistical mechanical system in the sense The statistical mechanical systems always as we treat them always have atoms by atoms I mean not just atoms in the sense but of having some particles, but If you have a spin chain your atom is basically a site and on your own a site You already only have only have some discrete degrees of freedom, but the field theory a local field theory is a full continuum. Okay a Full continuum degrees of freedom for which you can you can you can go any fine resolution down down to any scale It's a question whether relaxation can happen in such a in such a case and actually the real problem Is that what is a field theory? The field theory? It describes a critical point here plus its environment So from the statistical mechanical point of view you are doing a very crazy thing It's an infinitesimal environment and actually what you are doing you are sitting on argy trajectories and scaling down to the critical point. Okay, and The problem is You are changing parameters in such a way that you are keeping gaps finite and interaction quantities finite and all that and the problem With that is the quantum quench from statistical mechanical point of view happens that you are looking at the QFT happens in the vicinity of a critical point Actually infinitely close vicinity and it's an infinitely small quench from statistical mechanical point of view So it's sort of like a very very non-trivial limit And there are There are some indications at least that this this changes things I think new we'll talk about something like that as well Partially new Robinson later. So that this or ideas what we what we have from standard statistical mechanics may not be Directly or seem simply applicable to this situation. Okay, so this is interesting and Last point why is it interesting because this is experimentally realizable at least so it's claimed Yeah, I understand the first point about the cutoff The second point is basically I can I can phrase it in the same in relaxation times Okay, if you think about relaxation times in the vicinity of a critical point, they tend to blow up So basically what what what what you do is relaxation time correct Is characterizes your your sort of like transitional period while you are transiting between the behavior that you start with to the behavior At the end point and your relaxation time is you are just scaling your theory and continuum limit They can blow up That's that's the way I like to think about it So it could happen that the field theory is actually describing the transitional region. It's not describing the relaxation itself to some To the equilibrium because because you just blow up this this this transitional region basically to occupy infinite long time At least this could be the explanation for some behaviors that we see Okay, that we directly see Obviously if it's a non-integral theory, I can't I can't watch for this behavior to be exact in the sense It's it's it's numerically observed. It's calculated with certain expansions. Okay, but still It looks it looks very my very very different from what you do from what you have in standard Statistical systems and the basic problem is really that you can go up to arbitrarily small length scales and you scale your theory to that so I mean, I don't say I understand it So I just I just want to say there is another danger when you think about this as universal description Okay, there's another danger that you are scaling in a very very very very Intricate limit in which you can just lose part of the behavior that characterizes your statistical mechanical system simply because you are scaling it out of Oranges more or less Okay, and they are also claimed to be experimental realizable. So for example, York Schmidt Meyer in Vienna, which is very close to Budapest where I am Has this very nice system where whereby there are two seagull like Bosonic condensates which are described. I mean, these are these are basically some Bosonic atoms and They are described by a Libri-Nigar like theory But in certain limits the relative phase between the condensates which depends on the coordinate Obeys a sign-gordon dynamics At least so it's so it's claimed naively. So this five variable obeys sign-gordon dynamics and Indeed they have been able for example to see solitonic configurations in this condensate and Some of the behavior which is related to sign-gordon seems to be there but I spoke to Jörgen and And they also have some recent paper out which says that there are actual doubts whether this is actually described by sign-gordon Some some things that are cannot be understood from sign-gordon It's not clear whether this is because it's not described by a field theory or because it's not exactly sign-gordon, but it's Okay, but any case There are there there are at least attempts to realize really these sort of quantum field theory quenches in a lab Okay, so if this is really quantum field theory then maybe our methods could be applied to that. All right So what do we want to do? More more detail Or just punch. Oh, yeah. Yeah, it's just I have to find everything next speaker would have it much easier That's fine So now with a zero I just said there's a starting Hamiltonian is zero Well, I can choose lots of states to start with I can start from the ground state of this Hamiltonian I can start from thermal excited state I can start from Other excited states of this Hamiltonian, but I'm just doing it very simple. So I just say that I'm taking the simplest Possible starting point when I'm just taking the ground state of the Hamiltonian, which I take to be zero energy, okay? So it's Bound it from below by zero and that's it and what we can do is that we can expand this state on the Eigen states of the post quench Hamiltonian and And then we can obviously write down the time evolution of any observable. It's very simple. There's an observable Time evolution is simply Just Om okay, it's simple quantum mechanics. You just do the standard Hamiltonian time evolution and you send with your state with the evolving state okay, so First thing that it seems obvious That if there is any relaxation if it goes to a time independent case supposing that there are no degeneracies then the T goes to infinity Equilibrium should be something called a diagonal ensemble. Okay, so there is a there is this density matrix and What you think is that the equilibrium is just this because Basically because the diagonal terms are time independent. Okay The question is what is this diagonal ensemble whether this can be replaced by something more statistical mechanical thing because this is very microscopically it contains all the Overlaps microscopic overlaps whether this is actually equivalent to a thermal ensemble in some sense in a proper sense Which I'm not going into because that would take a lot of time But there is there is you have to specify this very carefully whether it or whether it is Equivalent in us in a sense to some more general general so-called generalized example or something Okay But that actually Lends itself to an approach by form factors when you consider field theory. So this approach is Very interesting I Hope to convince you that There is a third one thanks, but I wouldn't be able to reach more than two of them anyway, so I Just decided doesn't make sense because I'm not I'm just simply not tall enough to reach free blackboards Okay, so what you can do in a field theory these states are just Some multi particle states. I'm talking about one dimensional one theory here. So one plus one dimensions Okay, obviously you can do all this in higher dimensions, but my computations that I going to present are in one plus one dimensions And so these are so called rapidity parameters which characterize the momentum. So the particles have momentum For simplicity, I'm writing formula with just one type of particles I mean in the modus that I'm considering sometimes there are even eight times eight types of particles But okay, but I'm just I'm just neglecting all this complication because it's unnecessary detail and Basically if you have H the pulse bench Hamiltonian Then it's eigenvalue is thus is just the sum of the energies Okay Then what you can do you can always you can also expand your state on this basis So then it becomes Something like There are some amplitudes here and Okay, so this case would replace the C's These states would replace the ends That thing there Is something called the phone factor of a local operator in many theories many integrable theories we have explicit formula for that okay No Not yet. I don't I don't this is this this is very generic Well, the application needs Normally if you want to apply this analytically then you need integrability because you need an explicit idea About these matrix elements It's asymptotic state asymptotic states, but I need actually is a gap If there's a mass gap in the theory then I have this the basis of asymptotic states, okay? Yeah, I can I can choose any I mean in states and now states are just all particular ordering of rapidities in this formalism So it's and then I can extend the ordering analytically to any ordering of the rapidities basically Yeah, actually there should be a one over n factorial here if I do this extension and I just integrate freely over all the rapidities Then there should be a one over n factorial properly normalize it Okay Yes, so I did not understand your answer. You're saying if the theories gap then you are granted to have a good basis Yeah, asymptotical asymptotical scattering states in infinite volume. You have the scattering states So that's when it becomes a free theory Well every gap theory has this asymptotic states the theory is not free I mean you have an entrepreneurial matrix, but but asymptotic but asymptotically asymptotically I mean you have in states and out states you consider the dynamics you consider infinitely Past or infinite past or in the future then then you are basically guaranteed to have there are even theorems long time ago I think there's axiomatic field theory there are theorems proven about existence of these states, okay? So it's if it's gapped if it's not gapped. That's a tricky question But I suppose that my field theory is gapped, which would be the case anyway for my for what I'm considering explicitly Very good question. Yes, that's I'm trying to convince you about that And this this this turns out to be a good basis for at least for for very interesting questions that you would really like to consider It's not it's not a trivial question at all. Yeah, it's not a trivial question whether such a simplistic Because if you think about quench and now come now comes the next issue a quench is generating a finite energy density Simply because of translational invariance your state psi zero in terms of the post-quench Hamiltonian will have finite energy density So in principle at an infinite particle state Contains infinitely many particles whether it is useful to expand it in this way You will be amazed I I promise, okay It's it this this works extremely well, but and I will try to explain why it works But it's a very good question. It's it's it's it's not an obvious question whether this whether this should work at all Okay No for this question integrability is not important for integrability is important for being able to actually compute that some there and the tricky part of that some By very bad by integrability comes in is this You want to know these matrix elements and in integrable field theories you in many of them, you know explicitly these matrix elements They are computed. They were computed long time 20 30 years ago by by this all these people form factor bootstrap all that for many many theories Okay, also, I should give away another part of the game for this because of these questions is that is that see Knowing see I will come to buy. I will come back to that Or knowing these key functions is a very non-trivial proposition Okay, so I will come back and I will come back to that. I must come back to that at some point There is an infinite volume and then we have the symptotic states and if you go to a finite volume Then we know that everything changes Dramatically the eigenstates are not You know, you cannot just write it like this. Yes, so and also, you know that in finite volume There's going to be finitely many particles on average and if it's only going to be infinitely many particles So why are you writing all these terms? Why is it not the term with any cool infinity which is important? I Will explain Just try I can't explain it before you see before you see something more. Okay. Yeah, very good I mean, these are very good questions. Exactly. I have to say I have to say something about that But I can't say it at this point. Yes, but just just bear with me. This is the expansion You put it in. Oh, we know all the obvious difficulties. I already listed a lot of them you add this difficulty to those Okay, I says this is this is this is this is this is this is all the problem. Okay, I agree So what so what we actually do? Okay going to finite volume is not such a big problem because in because there's a theory There's a theory of finite size effects Okay, so we know we know we know how the finite size effects go for energies That's Lüscher's theory, but if it's an integrable model then we have even better description for for for this infinite volume, okay For the finite volume dependence of these quantities We have formally which were worked out by Balazs posh guy and me something like ten years ago that work Exponentially well in the volume So we have all captured all power like corrections in the volume in field theory So we know exactly so it's basically up to exponential corrections we know we know the finite volume dependence of these of the amplitudes and That that that's a formula is that has that that have been ordered that already had lots of different applications to calculation of different things so it's like thermal correlators and One point functions with boundaries and all sorts of things. So so we understand that The real problem is for finite energy density how we have how we get around that problem Yes, so actually first just suspend believe and do it and see what we get, okay Okay, so that's that that that that's where I'm going to need to project something because plotting it all With the chalk would be I mean very awkward. So the idea is that you do simulations With the truncated conformal space approach, but truncated space approach, which is very simple Okay, I I realize I have to describe this. This is a what is up there is a very simple model at At least the upper side. We just take the ising Quantum field theory which is basically just a free Majorana fermion one plus one dimension and we do Temperature quench or transfer transfers field quench if you are if you think about it on the spring chain or in the Majorana term in the Majorana field theory This is just a mass quench. Basically free free quench in a mass in a free mass quench. Okay, and We just we just take this Majorana theory which has this mass term here. So something like the Hamiltonian is It's one over. I think it's 2 pi DX. That's the standard. I think there's an eye here Yeah, and we basically what we do is that we take some m zero before the quench we quench to some m You can calculate everything explicitly the quench is described by a boggle above transformation Okay, it's a little tricky to calculate magnetization in this model because magnetization operator is non Is non-local in terms of the fermion, but you can do that. Okay, there are tricks I mean, this is the analytic results here are actually by Esler and sure it using some form factor using the same sort of form factor expansion Okay, you need it actually in the form factor expansion to get the analytic results You need to do a resummation of certain terms. I will I will come back to that later Okay, so you need some sort of resummation of infinite sets of contributions, but that's done So this is this is how it works So what we do is that we just take the Hilbert space of this theory Truncated at some energy level keep those states and do the numerics. That's the truncated space approach in its simplicity And these are these are lines with different truncations m is the mass lambda is basically truncation level in Energy so m is unit of energy here. This is time in units of mass or 1 over m and This is this is the so-called log Schmidt echo first log Schmidt echo is simply that you take the initial state Overlap with the time evolving state and take its absolute value square. That's log Schmidt echo for you It's how well the initial state is preserved in time. What is the overlap with it after as time goes and You see here that it depends on the cutoff But we understand a lot about RG which was also Worked by differ by many different people so people in this and this room like Slavarichko and his group and We can basically apply this machinery of RG to the log Schmidt echo and we can simply predict the exponent of lambda with Which it scares we can predict the exponent with the cutoff Okay, that's simple. We could actually compute the coefficient as well, but it doesn't help too much So what see so we just simply predicted the exponent which turns out to be minus one So it took scale as I like slays like lambda to the minus one and then we just extrapolate it and The curve here the red curve is the result of the extrapolation But because this theory can be solved exactly. It's a free theory after all we And apart from this little part here, which we know what it is. It's a trunk It's a it's a truncated conformal space Artifact there are some because of the cutoff There are some there are some little wiggles in the line and the extrapolation just doesn't work It cannot catch these wiggles for short times But is the extrapolation is going a little wrong and in here, okay? We see the extrapolation doesn't doesn't fit quite well. So we understand that what what is that but the rest is fine Is that because ensure an answer truncate the expansion that's written there So do you think that that difference at short times is due to that truncation and that expansion? It's actually what okay. How I know it's a truncation effect It's basically I can see these little wiggles which you don't see because the it's not magnified there But I see these little wiggles and then I look at their frequency and their frequency is actually the cutoff frequency And if I increase my cutoff the frequency goes down But the problem is that the wiggles don't fit together if I increase the cut of them the wiggles will become We will become with the other frequencies So they just interfere with the extrapolation basically But this only happens at short times where the wiggles are still there at long times You already have the coherence of modes and the wiggles get because I get washed out and at long times the extrapolation works much better That's that that's the thing that's happening here, okay How big is the system this system this system is is 40 in in size of one one over M So and this is the infinite volume result that I'm plotting here Okay So also I can do magnetization again these curves are the truncation curves. So The dashed curves How go back you'd eventually see the periodicity of the system No, because no no no no no because there are some other effects obscuring the periodicity So actually are just going to have to have the volume to be safe because basically because of the light cone light cone effect If you have any if you have a finite volume the light cone starts out At L over two because my unit of speed of light is one at L over two it is the boundary Then my that then my time evolution really deviates from infinite volume Okay, so then there that I would have a problem. Oh but We are going to just to the half half the volume Okay, and here you see here you see the relaxation the relaxation of the of the magnetization This is actually an exponential curve. It's just the first part of it But it doesn't seem to curve too much But we could actually fit the relaxation exponent from this curve Directly and it works. So this this is relaxation of magnetization. This is in a ferromagnetic regime So where you start with the spontaneous magnetization after a quench it relaxes and The way it relaxes is the sky better relaxation exponent and you can fit it from this curve And this works also quite well. We get quite quite well the relaxation exponent that that is predicted analytically Okay, but here Okay, here is a cheat line, which I don't say anything about this blue line if you want you can ask later They just don't want to go into where what what we cheat here. We are we are cheating the TCS a little but the really vanilla TCS a It's just a truncated Hamiltonian approach is just these lines extrapolate it here with the appropriate exponent that we know from our G gives you the red curve and then and then these dots are the analytic results Okay, but this may not Be so convincing because after all this is just a Free model with a bogoly ball transformation You would say maybe the bogoly ball makes like so simple that I mean after all there is not much error in doing this You know doing all this truncation. So the next is do non integrable quenches And the non integrable quenches are basically the following We are quenching from a free major on a fermion But the end point to which we are quenching also includes in this Hamiltonian another term Which is an external magnetic field or if you like it in the ising spin chain Quantumizing spin chain people would call this longitudinal Magnetic field. This is the transverse magnetic field on the quantum chain And this is the longitudinal magnetic field in the in the quantum spin chain Okay, so you can do this either in photomagnetic and paramagnetic regime and Then we saw these plots so the actual actual the lines here are the truncation results So these are really non trivial. It's already non integrable when we saw these plots. We we really Said there should be something wrong. Maybe we are not going to sufficiently long times here because we saw no relaxation here and Actually, it turns out that here the relaxation is is real problematic I don't want to talk about the details of that again because that that would lead us too far away But in the ferromagnetic case relaxation, there is confinement and confinement really interferes with relaxation That's that's a topic. That will be a topic of a completely different talk and it's in a completely different paper anyway and this one But the point is that you have this and at first you say this this just can't be right. This doesn't relax anywhere I mean if I continue this you would see that this this wiggling continues and then it comes back and then it comes down But that's already quite long times So what we did is that we asked a guy who already had the so-called ITBD Which is infinite war infinite volume time of all block decimation I think some of the actual experts are sitting and sitting in this audience probably at least they are on the list and so so Mario Collura from CISA to produce Spin chain results directly on the chain using ITBD, which are the dots okay here and You that that's very tricky because in order to compare with the filter You have to scale yourself as close to the critical point as you can be scaling or the parameters Whatever where ITBD do those or the goes all sorts of heavy Okay, so actual the filter the calculation here takes something like an like a like like a few minutes and The ITBD calculation here takes days Okay, so filter here is much more effective obviously. It's not it's in what's on turf, right? But you see that the blue line Filtory line is very well on this line and again these little vigils now you see the vigils there Those little because are actually truncation artifacts coming magnified by the extrapolation Okay, they would be much smaller if I showed you the non extrapolated line But this is just the extrapolated result and this is the paramagnetic phase where you see some very nice Oscillations with a frequency of almost. I mean the period is almost 2 pi the frequency is almost 1 and That's actually the particle mass the frequency So this is just one particle excitation that does this and there is a very very slow decay here Which is not obvious between but if I if I plotted more of this I mean the ITBD results are not are not available for longer times the TCSA results are available for longer time If I plotted it for longer time you would have seen the the decay Okay, so now you now one grows more confident as is this is really going to work come on Okay, so that's Yes, so in your results when you truncate in the filthy recite what it is that we are truncating the number of states Yes So basically we use the same approach in the sense that we have the post quench Hillbert space Postman she'll bird space, which is which is the Hillbert space of this Hamiltonian without age and We just truncated in energy and we represent the pre quench state on this Actually, we can do it numerically or in some cases We know the exact representation so that we can put in the exact vector But we but but but we did both and it doesn't really matter So that the evolution is always by a free theory. No, that's the whole point We are doing it on this Hillbert space, but we are to the Hamiltonian that we are actually doing I mean the basis is of the free space. We are adding this We are using the full hammer that that's the truncated conformal space idea Normally that you are doing it in a sort of non-interacting conformal space and then you are adding the interaction Which you can't solve but you know the matrix elements of this operator because if you can solve for the exact matrix elements so you can represent the may Hamilton as a matrix on this space and And and and then and then you just solve numerically for this time evolution There are some there are some arcane tricks if you really want to get if you really want to be Effective on in this but otherwise it's a very simplistic idea and then you have to take care about the cutoff That's another thing. You have to add the RG without the RG. It wouldn't work All right, okay, you want okay, okay that little blackboard is more than sufficient Yeah, so how many how many people do how many people need a lightning introduction to truncated Hamiltonian approach That's quite a lot. Okay, so let's do it. It's extremely simple. You just take a Hamiltonian Okay, this Hamiltonian, you know exactly you put it in a finite volume the spectrum of this Hamiltonian is then ground state Excited states, whatever is discrete because it's in a finite volume is discretized Okay, so basically there is some zero one two is this states of this Hamilton Okay, this is not the pre-quench Hamiltonian this case Okay, this is just some Hamiltonian of which you know the exact Solution so you know it in finite volume, you know the spectrum in finite volume What you do is that you just take a cutoff lambda in some units, whichever units some upper energy cutoff If you put this in because you have a discrete spectrum you retain only finitely non finite number of states Next assumption that you really must have is that you must know the exact matrix elements of this operator V in the space In many cases, you know this from conformal field theory from form factor approach from different things people computed this You don't even have to understand this you just take the book from the shelf and take the formula or whatever Okay, so you know so so so here are these states and and you have V and N And what you do is just then this becomes a huge matrix 5,000 times 5,000 10,000 times 10,000 whatever it just depends on where you put your cutoff and Then the ninth thing is just to take this as a matrix this approximation For for the Hamiltonian in the finite volume, and then you just say okay Well, we just do numerical quantum mechanics with that matrix one matrix quantum mechanics Okay, it's very simple now where all the tricky things starts for which this this lecture wouldn't suffice it That would that would be a completely different lecture, but is that is that then you want to get real results for really complicated models for which for example this Hilbert space grows too fast So that you cannot put a very high cutoff or for which the cutoff dependence is very slow So you would have to put a very high cutoff in order to get it well And then you have to improve all this stuff by the normalization group methods Basically what you what you compute is the dependence of your results on the cutoff If you can have analytical predictions Predictions come in different flavors. Sometimes you just know that your results like an like an expectation value Would be like the expectation that infinite cutoff plus something like C to lambda minus kappa kappa is an exponent And you know, what is the exponent? Sometimes you can do such a detailed calculation that you even know see Then you can even then you can even just really really get rid of this You just subtract this dependence from your from your results and see that they scale on top of each other In in the in the cutoff and all that so this is really yes as simple as it gets simple mind you think the I mean all the trick comes in Really first of all implementing Efficient approach to do this may take computation and The second is implementing doing the RG which predicts you this exponents and possibly if this is possible also the coefficients of the cutoff dependence okay, but But this is but but but this is all you have to think about you don't need to do really all the other intricate details So so in your example, it's key that the initial state was well represented in that subspace. Yes Very good. Yeah Actually that that that that that already comes down to an interesting question is that these K This K amplitudes they depend on the rapidities basically they depend on energy and So they should be decaying with energy in order for it to be very very represented and This is actually what happens. That's the whole point why it works Because it's counter intuitive at first because the energy scale could be at infinity in the southern quench But what happens in normally in models is that is that this thing here? Normally first of all in many cases, which we call integrable quenches. This thing here is just factorized into two particle overlaps So this is just product of two particle overlaps That's an that that's a long trivial thing whether it happens or not It's not enough that the initial and the end Hamiltonian is integrable The quench there it is happens for specific quenches for example for the free-to-free quenches It's always happens. That's a multiple transformation why this happens Okay, and then the question is that these two particle overlap Let's call it K2 whether it decays with energy fast enough now Energy is basically e to the theta because because because it's cost theta. I'm neglecting the e to the minus theta, okay? So what what what happens basically is that this K2? Normally the power of this an inverse power of this say in many cases. This is just more or less one over e squared Energy squared one over one over e to the two theta Okay, so it's a power like decay That's tricky. That's why you need extrapolation also because because because it's not it's not it's not fast enough So you are really leaving out sizable chunks of your state from your Hubert space But fortunately the chunks that you are leaving out have some very simple dependence in the energy So you can extrapolate using using the simple dependence So yes, you are you do leave some some parts of your state out of your Hubert space But you can but but but you can get around that That's that's right. Yes How is the map work from the multi particle states that you're showing onto this onto this? Yeah, so the so the Normally if this happens if this this happens at all this happens only then when you only have amplitudes Like this. So this is the so-called pair state structure When your quench state is just is just consisting of opposite momentum particle pairs This is what is drawn in if if someone if anyone is familiar with quantum with quantum filter quench for a reason by Cardian collaborator this is what drawn like this that the initial state in a quantum quench Which is here time is going this way is basically emitting particle pairs of opposite momentum If this is true Then you can prove this is a theorem which we prove which we which we proved with Spiros so to the Addis and Giuseppe Mussallo. I think yeah Or maybe with David, I don't remember which paper what the theorem actually was in We have three papers about this this problem and I don't remember which paper we actually have the proof of the theorem That's a theorem the theorem is that if this is if if if this is particle pairs Then basically your state exponentiates then your initial state can be written as Integral of k theta creation amplitude of particle pairs like a squeeze state This is what you would call an integrable quench an Integrable quench is not just that the initial and the end Hamilton is integrable in integrable quench is there's a recent paper by one of my former students and the collaborators on this Balash posh guy who That this is what you would like this is what you would really call an integrable quench This is like factorization it means that the multi particle overlaps are factorized in terms of two particle overlaps Basically, there are products of independent two particle overlaps is that it's like s matrix factorization integrable models It's like factorized scattering more or less Okay, if this is true, then there's a lot of things you can do with this With with this with this for example some tba like approach that Joseph Moussard and David if you're at the introduce long some time ago And and things that you can do if this is if this happens Not in the numerics directly you need it because because because you don't trust your numerics to start with because of all the issues That I told you in the beginning so you want to get an alternative approach I'm getting to that point So you really what you really want you want to have something to compare with and if you want to have something to compare with it It should rather be analytically calculable Yes Okay, yes, so this is basically this is basically this is basically the the result of the following Your interacting field theory let me suppose that it is a relevant perturbation of a free boson Or a free fermion it works the same way so it is basically at the ultraviolet it's free and Then there's some relevant operator perturbing that leads you out of the critical point the relevant operator at high energies is Small so your quench at very high energy is basically just quenching free to free So you expect to reproduce at high energies. You expect to reproduce the asymptotic behavior of the free amplitude That's it It's the question is how high Obviously, so this can be far away and then your and then your truncation method is not very effective because you can't extrapolate if it didn't get set in Right. It's the question of quench Of the quench itself. I can also tell you that and and and it turns out that The truncated method the relevant small parameter That should be small is is actually the particle density post quench So you should have a small post quench density in the units of the natural units natural units is the mass again The mass-related units of the smallest particle and that is basically telling you that this amplitude pair amplitude squared When you integrate it over all the rapidity range this integral should be small enough So we work this out in many cases and for example in the icing case It turns out that in principle our truncated conformal space could work out to 300 units in volume In field theory, that's huge because all finite size effects decay exponentially Okay, so this is just a limitation of the time evolution But the finite volume effects you can already forget So the time a time evolution is limited because of the light cone effect But otherwise but otherwise as as regarding the as regarding the volume This is fine. Okay, so this turns out to be the small pyramid the real small parameter in the game and This means that your quenches in a sense small This seems to be a real limitation However, I would like to draw your attention that most theoretical approaches are limited to that For example, there is the so-called Semiclassical approach to quantum quenches in field theory, which is also limited to to small densities And the form factor expansion approaches are also limited to small densities Do you only do you only have Excuse me. Do you only have an even number of particles? No, that's also possible to have a odd number Actually, I'm going to show you plots from a theory which has odd numbers. What could happen is that in this state? besides this integral There is some Some term like this So because the translation of invariance if it's an even if it's a single particle. It can only be zero momentum So you can you can have overlaps with the single particle states and then they have the round amplitude G we call it G normally with some good Okay, and and that's and that's that's also interesting Actually, there are some all sorts of complications related to that We are working on some of these complications right now because they are plugging all sorts of analytical approaches to quenches So we want to understand more about this about how this work How how the quenches analytical approaches work in the presence of this coupling? It's not it's tricky. Yes So at the beginning you said that that you expected the method to work for any get massive No, I didn't expect it to work at all But I expected to work for massive. That's right massive with sort of it with now certain qualification Which I order which I basically already spelled out the basic qualification is that this sort of integral is small But you also now said that This would work if the UV that there was a CFT asymptotically a CFT I said free theory I if it's if it's a CFT that it's a tricky it's an interacting CFT then it's a tricky question It seems it seems so Because in an interacting CFT you don't know what the overlaps would be So you don't have the argument here depends very crucially that for the for that free theory We have an explicit solution for the overlap so we know that it decays for an interacting CFT I can't watch for that Okay, that's basically the problem Yeah, yeah, yeah, I will show okay. Yeah, so basically you can I mean I told you about the damping here and We can also extract the damping and the damping calculate damping in this in this in this Paramagnetic oscillations the data points numerical data points are the are the color dots and there is an explicit prediction for the case when this is integrable But the problem is that we are looking at the non integrable So we are taking small magnets small small values values of this magnetic field and we see as we scale this magnetic field to zero then They really try to fall on that line Okay, so this is none. This isn't bad So so so even that even this damping this very small damping that seems so small there can be extracted with very high Precision from the curve the curve is so precise that even a small damping can be extracted from it and and then the grease with theory right, yeah, so next I I Next I next I have to tell you what to do if you if you don't have this Simple-minded free boggling of quench background in order, you know in order to have a Theory behind this so then basically what you do It's on it's it's on the blackboard You just evaluate this sum if you want to do analytics you just evaluate this sum up to some terms You have to be extremely careful there. You have to resum certain orders in order to actually get Get get to Real results, but this can but but this at can be done in some approaches So there are actually two approaches on the market one was By Chesualdo del Fino in Sissa and his approach is basically is that you quench from an integrable model from integrable To non integrable and on this side because it's non integrable You don't actually know these phone factors. You don't know the energies. You don't know nothing But if it's a small quench from an integrable you can think about this quench being perturbative So you are doing perturbation theory around the pre quench You are doing a perturbation theory coupled with this expansion Around the pre quench so you can do expansion around pre quench and actual that expansion predicts very well the amplitude on that plot the frequency are is wrong and it cannot get the Decay rate well, but the amplitude is okay. There's a reason for that actually maybe I won't have time to go into that so The other poor the other possibilities is is by a slur and sure it and In combination with some people like Bruno Bertini and maybe Axel Kuberl this time and but basically the now now It seems that dirk is the stable person whose name you should watch if you want to get all these papers and That is when you quench to an integrable theory and In that case you can use the post quench exact form factors and everything The only thing is that the integrable theory doesn't tell you the C's So there are differences between the two approaches namely first of all is what is the end state that they can treat The other is that hail the K function, which is the C of a lapse K is input But this approach is better in the sense that K is actually calculated perturbatively I mean, this is not what Aldo really does but it is effectively the same as calculating K as well perturbatively Okay, so that because because because he's expressing everything in the pre-quench and do it does a perturbation theory He actually captures all the all the things in this expansion in perturbation to end the pre-quench The problem with his formalism that is already apparent is that it's going to be is going to be using the pre-quench energies Which means the pre-quench frequencies and actually the system oscillates in the post quench frequencies that that that already gives a sizable shift to his results And there are there there are also some others Such a form factor approach because you are basically doing what you are summing one particle two particle three particle states Is the low frequency part of that? So it should work first of all for T long enough So T must be long enough T must be larger than the relevant gap So all these truncated expansions work for a long time for short time. They wouldn't work This is this this is true for the second expansion. Let's call it the sure it actually actually sure it whatever But I just say I just choose one of the names. Let's go the sure it expansion for the Delphino expansion There's another limitation is that because he's doing it. He's doing it perturbatively in quench In quench size, let me call this quench size parameter lambda There is some parameter lambda which specifies how big the quenches because it's perturbative then he also has another limitation That he must stay below one over lambda because otherwise these these things would would come okay, so We did what we called we did what we call the calculation of so-called E8 quenches and Here is what we have so E8 quenches are quenches in this theory Which are already much different from this M is zero We are taking H. That's a free famous E8 integrable model of Zamolochikov and we are quenching in H So pre-quenchant post quench are both integrable both both both methods are applicable to this and I think the results speak for themselves So the TCSA curve in the upper figures is the red curve This is the S-Lashivity expansion and this is Aldo's expansion their phenos expansion Okay, actually, this is a very small quench if I put in a much larger quench The result would be the result would be even even more pronounced But but but but it's just nothing that that figure is not in the production stage yet Okay, so I don't have production quality figure about that yet okay, and and this is and This these are basically this is magnetization operator and this is and this is the density of psi bar psi This is so-called epsilon and energy density operator in the ising model Okay, so you see that this is this is another quench So you see that first of all What you see is that the TCSA is very good in the sense in the following sense It follows very soon first of all should it expand is also very good in the sense that Even at very short times the shooting expansion is very precise. We are only keeping one particle pieces Okay, we are not even going to two particles in these expansions. We are just keeping the first one particle pieces here Okay, so to get if you take together the truncated conformal space and the shooting expansion We basically describe this quench all to infinite times because the shooting expansion it gets even better if you go to If you go to higher times, that's not a problem for it Okay, so we basically and and the only this the only difference here is Is this little tiny part which you which you barely can barely see that the TCSA results starts from here but actually shooting expansion starts as well and and and and Surely, sorry, Delfino's expansion starts as well And surely the expansion come goes goes here at t0 time if you please it together then you then then you have basically an infinite time description of this quench Okay, and for and and and and for other quenches as well So yeah If we quench away Yeah Okay, just let me just let me tell you about the upper line because that's easy What you can also do is that you keep age the same and you switch on M That breaks integrability that means quen we call it quenching away from the E8 line Then we don't have the shooting expansion because it uses post-wrench integrability But then we have all those expansion which again doesn't perform quite quite well, okay And The the upshot is what I would say is not that it's not that we should throw it out Maybe you shouldn't throw out to the baby with the bath water, right? The point is that this is for the time being this is the only expansion that would work for a post quench Non-integrable case, so I think it it has to be improved and I think it can be improved We have some ideas about how to How to go around that so actually so actually I'm hoping that this that that instead instead of saying that this It is just disagrees. I'm hoping that we can get this curve To to line up with that with some improvement of the of the actual expansion It just needs to be needs to be done much in a much more Comprehensive way including including lots of other stuff in it Okay, yeah So it's so that's the end. So let me just conclude So I think that for field theory quenches first of all they are interesting because we know much less about them then statistical mechanical quenches, I think this is true because and and and That the relaxation doesn't seem to happen so easily in field theories and there are and there are less approaches To really calculate field theory quenches What I what I first want to say is that truncated Hamiltonian metal seem to be effective at least for low density quenches. This is the real small parameter if it's a low density quench Then then the truncation method is applicable Even even after all its limitations that it has to work in finite volume that it has a that it has a cut off whatever even after all these limitations it is applicable and I Think that on the theory side. I mean having analytical expansions. There are some very nice works But but a lot more needs to be done because of their limitations So basically at this point we don't have a good good analytical calculation when the post quench theory is non integrable And this is one of the most outstanding questions for that post insurance integrable I think the surety expansion is Apart from little details may be that but but the idea is fine But but but but for the post quench you are non integrable, which is actually very interesting case Obviously the non integrable case. We really need to do much more in the theory. Okay, so thank you for your attention questions I didn't understand very well this part about the structure of the initial state So so for example in this case where you quench to a final amount of which is not integrable Do you still claim that it has this perfect? No, no, I don't claim however However, there is an interesting observation Is that if the density is small? You can imagine that these particles are created far away from each other So you can imagine that the leading part is still a more or less factorized particle per creation It's just that it's not exactly factorized So for small densities you can imagine that this is still a good approximation even if the quench is not not integrable And you can measure these k-factors experimentally by the way experimentally I would say T. C. S. A Experimentally truncated truncated Hamiltonian you can not measure numerically compute it Okay, so the k-factors you can numerically compute in truncated Hamiltonian approach. You can compute Just at the expense of here. Here is such a computation with a theory on it. Okay Like this is an integrable quench But but what you can do it in an integrable case so you can measure these overlaps So you can check whether your actual assumptions about this integral being small is valid or not You can check it in right right inside the approach itself It's an internal check. You don't need to know it from advance from from something else What do you show in these plots? It's just it's just an overlap calculation a k-function absolute value of a k-function as a as as as as well as momentum You see you see the power line decay. It's an overlap between what and what it's a it's actually it's an overlap in sign word on theory after a quench between the initial state and The two particles two particle state the lowest two particle state as the function of the of the momentum of the two particle state And the blue line is a theoretical curve that we got together with spiro sotriades and Giuseppe Moussardo The red line would be free theory and the dots are the measured values of the overlaps Okay Just I don't want to I just want to say that this can be measured inside the approximation itself So you can you can you can validate this approximation from inside You don't need to know anything from outside about the overlaps Actually you have to remember you have to remember that smallness can also be checked from the following the total the total Overlap should be one because it's a normalized state So if you check that your vacuum overlap is still sizable that means that all the rest should be small Okay, so that that validates your numerics that it's it's it may be an even easier check I told I showed you something which is which is more So to say theoretically based but but but actually if you just check that in the given volume your vacuum overlap is still sizable Then then then then then you are on the safe ground Actually could the question so okay quenches are of course fascinating, but When you were working on quenches Did you from the from the methodological point of view? Did you? Learn something about the method Which you know could be used to improve It's functioning even in other situations not just as far as quenches are concerned But in the classic applications of the method we just put trust could be and then extending it to other theories not just Ising safety Yeah, I learned more about the details of its validity It turns out that validity for these for these questions is the same as this one So the validity criterion of TCSA is actually is the smallness of this sort of integrals Because even then you there you are not doing quenches, but you have your you have your a zero theory From which you start from and that has a ground state and then you are looking at the overlaps of this ground of the post of this ground state with the post I mean the perturbed excited Excited states and they should decay fast enough basically and you can and you can you can even specify this in more details You can even give the highest volume to which you can go with the given cut off They you can you can give I mean for example in the icing theory in the standard icing Truncation I think one what we can do is 300 in the world. That's what I thought it 300 in the volume That that is not a quench related thing that is just TCA truncated Hamilton as it's functioning It's its precision is and basically we also learned but But you already I remember we discussed this so you already guessed this yourself or observe this yourself We learned about how the multi-particle overlaps are scaling with with parameters and they are while they are scaling. It's not too Okay, I'm there's not much time, but we can what I can tell you in private But it's it's it's not too surprising But we confirmed all that they are scaling in the proper way that you would expect in order for all these Overlaps should some up to one. They should scale with the volume in some non-trivial way in order for this to be possible Yeah