 Katja will tell us about rule 54, reversible pseudo-automaton, as is written there. Thank you, Katja. Go ahead. Okay, so I would like to thank the organizers for organizing this nice event and for inviting me. And today I would like to share some of my enthusiasm about this really nice model where we can really get many exact results. So the idea is that we really focus on, I mean, it's much less general than what people have been talking about, so this really applies only to this model so far, but the nice thing about it is the fact that basically everything can be done exactly. Obviously, this everything should be understood, should be understood, I mean, it's not literally everything, but the point is that really many, many things can be done in much greater detail than elsewhere. Okay, so it's about, the model was introduced at the beginning of the 90s, and then as far as I can tell, not much was done until it became popular in recent years, and I just put there some recent references, I think I put everything I could find if I forgot someone, I really apologize. So the dynamics is defined on a one-dimensional lattice where every site can be either free or occupied. And so the dynamics is defined in two time steps, in each time step, only half of the sites changes deterministically, so in the first time step we take this saw like chain, and we find new values of the sites at the bottom, so of all the even sites, and we get this other saw, and then in the next time step we do the same thing, only now we're evolving odd sites, so in two time steps then everything repeats again. Okay, these discrete maps are given as deterministic, so each configuration of three sites each configuration of three sites uniquely determines the new value of the middle one, so this is the formal rule which can be graphically represented as these diagrams at the bottom. So for example 0 0 0 maps into 0 0 0 1 into 1 and so on. Okay now this definition so far seems a bit abstract and not so concrete, but we can now try to take a configuration on the bottom and then just evolve it in time and see what happens. And as we can see this model describes dynamics of solitons that move with fixed velocities either plus one or minus one, so see this is a right mover, this is a left mover. Whenever two solitons meet they interact pairwise as is here and when they scatter they get delayed for one side, so for example this soliton was supposed to be here, but because of the scattering it's delayed, so it's one side back. Okay now this solitonic interpretation of the model has many really concrete and not so trivial consequences, so the first one is that due to this soliton dynamics we are able to explicitly express time evolution of all local observers in an efficient way. So the first thing we do is we map the problem of the time evolution of local observers, we can imagine these local observers to be just density as some position, so this is our this is the simplest observable but it's actually not because it's a classical model so everything is diagonal and so we map the problem of evolving this density in time onto a problem of counting solitons in a section of the lattice of length 2t plus one at time t. So the idea is that we take a configuration of 2t plus one sites and in this configuration we just have to and then we have to determine whether at time zero, so at this point at the bottom the middle the central site was occupied or empty so either either white or black and the nice the nice concept the nice thing about this is that computational complexity of doing this kind of of doing this kind of procedure grows as t squared so it grows polynomially instead of exponentially so why so the idea is that we have to identify all the solitons that are in here and then we can just look at how many times they scatter so let's say that we start with this orange one here and if you want to see if the soliton came from zero or not we just have to identify three solitons that scatter because we need three scattering to move it here like but at the same time we have to make sure that that all the solitons that we're looking at are actually causally connected so for for example this orange soliton scattered with this left mover here but it couldn't have scattered with the green one okay so the basic idea is really simple now there are a few technical details that one has to work out but the bottom line is that due to this due to these two possible speeds of solitons the only thing that the soliton can never overtake another one so everything we need is already included here and this can be done quite efficiently and so what we do is we describe we encode this procedure in an MPS so we write that matrix product states that's using this information here as the input and now that we have this MPS we can try to see if it gives us any if it's in any way useful and so what we can do is we can look at correlation functions or two-point correction functions at different different points and at different times and so this is the density density correlation function this is the exact result so this is valid for any time t and any x so it's just like just this binomial sum and if we plot it we see that we get two politically moving peaks one going to the left with with velocity one-half one going to the right with velocity one-half and these two peaks spread diffusively so it's a kind of a situation that Mark was describing before okay so this is quite a remarkable property of the model this in general can't really be done but now let me show you another consequence of these soliton dynamics so another thing that we can do is we can efficiently express all multi-time correlation functions at the same position so let's imagine that we start with a little equilibrium state here at the bottom and then we see here in the middle of the chain and just try to keep track of what configuration we see here or only at one point so we average over everything else and so the idea is if we think of this in a generic model what we would do we would probably start with some equilibrium state here and then we would add we would try to evolve it in time while keeping the information only here and if we think in terms of some some kind of t b d or d material so if we try to think about this computationally we would imagine at one point so the question is if we can really reduce this exponential complexity and so it turns out that generically no generically this is this still grows exponentially because whether or not we're going to see a particle at the next time step depends on the whole history of on the whole observed history however in this specific case of rule 54 we can do it and we can write these probability distributions in terms of again a matrix product state with a constant one dimension so one dimension three instead of growing with time growing exponentially with time okay so the main reason for why this is possible is the fact that the equilibrium in these equilibrium states all the solitons are statistically statistically independent this means that that the probability of observing a soliton at the next time does not really depend on all this history from before it only depends on a few time steps so hence this constant one dimension now one of the remarkable things about this is that if we if we take for example an equilibrium state to be a maximum entropy state so the probability of any configuration is equally likely the the configuration in time so the probability distribution in time will not be infinite temperature so will not be maximum entropy so there are still some short range correlations going on all the time so so maximum entropy state does not map into much maximum entropy state here okay now this then motivates us to try to think about this time space duality so what happens when the roles of space and time are reversed what happens if we rotate the model and try to see how this evolves in space not the motivation is are these recent works on dual unitary circuits which are the quantum many body models where this can be done and rotating the circuits gives you another unitary model so in our case what we would want to get is another deterministic model for example now if we just look at the soliton picture it seems that this could be done since if we if we look at the time dynamics here what happens is the soliton scatter and slow downwards while scattering so this means that the space dynamics if we can formulate it would be solitons moving with fixed velocities again but when they scatter they speed up instead of slowing down so for example we can look at the scattering of these two so as it is this guy jumps over and goes here and this time okay jumps over goes here okay now to try to formulate this let us look at the circuit representation of these dynamics so we can encode our three side rule into this deterministic matrix so this time evolution matrix that acts on three sides uh on on the left the left side is left untouched so data function here the right side on the right side nothing happens while the central side changes according to the the whole configuration um at the same time so at the same time step what this gets commutable with themselves so we can just imagine to smash them together and we can then write this really nicely symmetric um circuit here so this is one gate this is another gate third gate and so on and so uh so these gates are applied uh first on even side then on outside whether center to even center to not this now looks really symmetric and now let's just see what happens if we rotate it so we rotate it uh just by definition so we define this rotation in a way that we we get the same the same diagram the unequivalent diagram um so again the top side and the bottom side are not changing while the middle side changes according to to this three the time that the misty group that goes from these three sides up here and so this is the rotated picture so far we haven't done much and this is really uh general and so we haven't put in any knowledge about our system yet so this is just done by definition now at the beginning I was saying that we would like to get this you to be deterministic however this three side gate is clearly not deterministic so we can see that we have these two ones in the same row and the same column there are also these two rows and columns with only zeros however it turns out that we can we can introduce the notion of allowed and forbidden subspaces so we can project out all the forbidden configurations and then rewrite this the whole space evolution as a deterministic evolution on a reduced subspace so we introduce this project of p here so this p is like a diagonal eight by eight matrix that has ones everywhere except on this uh on this entry and this entry here where we have on the zeros and for this it clearly holds uh you uh that your p and equals pu and so also it's invariant under it okay now we can and now to see how this becomes deterministic I will I will not show everything just a bit of how to reformulate it so if we start with these three spatial space evolution layers the first thing we do is we just space this space these gates slightly apart so that we see what is going on then we introduce these projectors um we can do this by by this identity I showed you on the previous slide so this thing you can just introduce these projectors we move these projectors around because they're all diagonal so we can move them around and now we define this u e tilde or u odd tilde on the for the other party of sites and um my statement is that now this u e tilde is deterministic on this on this subspace that is invariant under all these projectors now this can be it can be shown and I will not show it here but it can be shown that in this subspace in this reduced subspace u e tilde and u odd tilde can be both expressed in terms of deterministic gates but they have uh support seven so instead of being three by three they are sorry seven acting on three sides they non-trivial act on seven sides um okay uh so now we know that uh what we showed I mean if we what I mentioned so far tells us that now we have a well-defined dynamic of system also in the space direction uh but now let us take a look at um at this time multi-tank correlation functions again just to just to see if these dual gates or these space evolution gates can be useful to really give us some some non-trivial results some more more explicit results okay so let us look at this multi-tank correlation function that I was mentioning before with this when we have this equilibrium state and we just look at solutons in the middle so um by definition this is defined as starting with an infinite temperature state at the bottom so these gray sides are all one side if uh maximum entropy states where uh one and zero are both equally likely then we add a layer of time of one layer of time evolution so one layer of the normal gate uh and then we add another and then we put one observable here one here and so on so um so we add all these layers of time evolution and we put our observable thing so this is by definition the expectation so the expectation value of observable one at time one observable two as time two and so on and these are all in these two sides in the in the middle at the end we have to take an inner product again infinite temperature states uh which is just due to the definition of expectation value for for these classical stochastic models well deterministic but general stochastic okay so now we simplify this a bit and the first thing that we do is we note that u is deterministic so this means that u has to map infinite temperature states into itself and also from the other side because it's um it's um symmetric uh which gives us this light construction so this means that I can just remove all this layer all these gates here because um they because of the first relation and the same thing goes to the top and so then I do this layer by layer until I left it with this causal light construction here so this rectangle is tilted rectangle and now the only thing that I do is uh so this is the thing that was so I rewrite this thing by just um evaluating all these inner products here and what they get is this and then I just by definition as before I just rotate this diagram and use dual gates here on on the right so to go from the left to the right I just use the same definition as before okay now at this point I haven't used any property of the dual evolution this is again general so here we know we haven't used anything but we want to simplify the right hand side of this expression now we can't just remove these gates because they're not deterministic however they have non-trivial structure and then there are these two non-trivial relations that hold for them and this means that I can just using the first relation I can just remove this gate here and this gate here because also the transpose relations hold so this and then I can go on until I am left with only two layers so in this way what I do is I can rewrite my multi-time correlation function into these two layers of gates and align and all these observers squeezed between them and now as I was saying before for infinite temperature state we don't get an infinite temperature state in time so we get really short-range correlated state here and we can now check that this is indeed the same thing because this really agrees with the other result that I mentioned and it's really so it's equivalent okay so this was all down for the infinite temperature state because it's conceptually the simplest but the same thing can be repeated for a class of equilibrium states of of Gibbs states let's say or you could think of it in terms of like a sort of a grand canonical states because we have chemical potentials corresponding to left and right movers okay so this was just like I was just trying to convince you that that it's possible to produce a more algebraic formulation of all these quantities that we might obtain in this model and the nice thing about this is that it does not explicitly depend on the quasi-partic interpretation of dynamics so in principle it makes it more generalizable so some output questions there are obviously many open questions but so the first one is how far can we push can we push this can we so maybe the most urgent question to answer here is if we can look at two-point correlation functions so the spatial temporal correlation functions can we get a nicer or equivalent expression to the one that I showed at the beginning and obviously we want to generalize these two other models so a really good candidate for a model for which this could be done is the model that we just very recently introduced in this preprint so other cellular automata could be used I mean this same thing could be repeated for other cellular automata we could think about stochastic generalizations of the stuff and most probably the nicest thing would be to find some quantum generalizations and by these quantum generalizations I there are two things obviously we can treat the same model as a quantum model but what we would like to do is get some models that really treat that don't treat the diagonal of the diagonal entries of the density matrix separately so we really want to get something that doesn't have like a that can be treated as a classical system at all and at this point I would like to finish and thank you for your attention thank you Katia um are there any questions I have a question go ahead what's so special about this particular rule um what is so special I we could look at the other rules but the main the main point of this one is that we have these quasi particles well-defined quasi particles with fixed velocities and this scattering rule so I mean for obviously we could also look at the one that I mentioned just just at the end which has similar structures but other than that there is another one would be like a free dynamics which is not what which is in some ways a bit more trivial well not I mean a bit simpler from the point of view of this multi-time correlation functions for example but then other rules if we just scan them look doesn't don't really have this nice quasi particle structure and so it's not clear if they are interesting or not or if something nice can be done or not thank you thank you very much I had a similar question but now let me ask a different one I mean people study the lot this box ball system you know which is sort of solid tones of different sizes which is also discrete sort of dynamic did you ever try to look at that um yeah I mean a bit not seriously but it seems to me that there um well I don't really have anything really concrete or to say about it it's I think this other one is a bit the point is there that this quasi this solid tones have different speeds so from so it can't be directly generalized there so we would have to think about it some more I mean there is no direct link can I also ask yeah so I was wondering did you look at the transfer matrix in the not the usual direction the transverse direction and maybe try to look for small values of time the structure of the agon states maybe the leading agon state because that would give you also essentially all the information you need for the quantum case kind of like what Bruno did for dwelling interior circuits I mean the short answer is no I mean it would be nice to to look at it but it's a fairly new result and so we haven't really thought about everything that could be done I mean that is especially in the case of say infinite temperature correlation function is fairly natural to look at some folded picture and then look at the leading agon state but yeah so thanks no but okay so if if the model is truly truly do a unitary what you would see you would see an infinite temperature state also in the time correction so you yeah well okay but then so maybe here some something different happens for example the leading agon state might be as one dimension agon state sorry so mps with a small band dimension since everything is so simple I mean it's not absurd to think that maybe you have something of course going to be a quantum state because the npo is not deterministic but I think it would be worth it to to to try I mean it's an interesting thing to think of I should think about it a bit more okay thank you can I ask a question yeah so okay I got a bit lost how did you get like a propagator that was not deterministic I missed a bit the step there so you're talking about this from here to here so basically what happens is okay so implicitly if you look at this diagram for example you can imagine this to be some sort of vertex model where you have these small balls that force all four sides to be the same and these big balls that give them one one that give you one weight depending on what is on the other four lines so if you look at this time evolution what you do is you get you you get this s1 s2 so you just from the time evolution you can imagine there to be nothing and so there is just one line here one right here one right here and so the point is that what you do here that this weight of this one should be psi or it should be this delta s s2 prime psi of s1 s2 s3 so this just follows from the definition of time evolution but then this means that if you go to the to the other direction what you do is you have the same weight but only now you put a data function this data functions here and here so it's it's the same as before so u s1 sorry so s1 s2 s3 s1 s2 s3 s2 prime is the same as u s2 s1 s2 prime s3 okay I should probably include a figure I don't think I can be much more clear about that I'm sorry no thanks and there I have a quick question you have a formula for correlation function would you try to compare with some kind of hydrodynamic projection formulas not explicitly I mean I I looked at some values of diffusion constants yeah but I never did it really seriously to really compare it from the beginning so I think I don't really even remember how well it agreed or not because then I always this this it depends how we scale time and so on so yeah because that could be nice because you have an explicit calculation here sure let's see okay so if there's a quick question otherwise I mean we thank you again