 lectures given by Tomas Prasen on the dual unitaries and chaos. As you probably noticed on the program, there's no lectures in the afternoon. Somehow we shortened the program for one lecture because Romain Vassar canceled due to illness, so you can go and swim. But before that, we look forward to the last part given by Tomas. OK, so sorry to hear. I'm the last one standing between you and the locations. OK, good morning, everyone. So I'll continue where I stopped yesterday. So I still owe you some explanation of the magic that happens when one requires the conditions of dual unitarity. So I remind you just to remind yourself, I mean, we are working now with these operator gates, which I didn't notice W, which are folded gates for states. So you could think of one on top of the other, but this one is upside down. The first one is upside down, so this is, again, you transpose tensor u dagger. So this one is your transpose and the one behind is your dagger. OK, so now once you translate our conditions of unitarity and dual unitarity to these W gates, we have four very elegant constraints. I will just write them on this part of the board because we will now be needing them. I should not use too much space. So which means two bullets. So this I will sometimes call a bullet. When you press two bullets from the bottom, they just go through. When you press two bullets from the top, they again just go through to the condition one, condition two, then the condition three. You do this from the sides as well. Condition four, you do it from the other side. OK, so now let's see what happens. Now if you try to contract some correlation function, let's do as an exercise, I mean you see how it goes immediately, but let's do like three by two circuit. This, as I explained to you yesterday, each two-point correlation can be reduced to a partition sum of a rectangle or a lattice. Right, in this case, it's lattice like this rectangle. So the initial operator, so now let's assume that the initial operator is here and final operator may be here. So they are not exactly on the same side. They are displaced a little bit. And everywhere else, because we have initial condition, which is completely mixed state, right? We put a press bullets. Completely mixed state means trace. It means just contracting all wires from the bottom. So we get bullets everywhere else. So now what happens? I don't know if you allow me to do this. It will be faster if I just use my hand, right? But of course for you it will be a mess, but sorry for that. So now what I will do now, I will try to contract this. How I contract, I use this rule, the bullets here. I can do this from the other side. You see, I already now got detachment of the diagram. So diagram is kind of trivial. There is no correlations. As soon as I have two disjoint tensor networks, there could be no correlations, but we go even further. Now we can detach this guy, so then we get this and this. Whenever you have this guy, this is one, right? Because I normalize this with one over square root of D, otherwise it would be D. This is one, I forget about it. Then this one I can contract this side or this side doesn't matter like this. Again, I do like this, and I do like this, and I get this. So at the end of the day, I get trace of A times trace of B divided by square root of D, square root of D, which is equal to zero. So now you can ask what remains, the only thing that remains, and of course that's easy to show formally, they're not going to be very picky. The only thing that remains are two types of correlators. In particular, only non-zero correlators. These guys, when the two field bullets that is the observables are along the light rays, which means that the time, so if this is X, so this is observable A, but this is position X, this is observable B, but this is position Y. So that is time. So when Y minus X is equal to T, and then these guys, of course, get contracted like this. So that is C of, of Y, which is Y, which is X plus T, comma X, T. And moreover, if you remember how I defined my lattices, I mean, this would be odd lattice number, right? So because my lattice was always defined that I started with the gate one, two, right? So this would be odd. There is a staggering. There is naturally a staggering in this problem, so we have to discriminate between all two. If you're thinking this problem, so we have to discriminate between odd and even sides. If you place an observable on the odd side, then it will go right. Excitation will go right. If you would place it on the even side, like here, as you will see, excitation could only go left. So in this case, it goes right. So this is what I will refer to as, well, sometimes we put it like a C plus, like a, I mean, because correlations really split into two contributions, the left mover and left and the right correlation. This is the right correlation. I will write it as m plus to power t. Okay, here it's still ba. As you will see, no, maybe I'll wait a second with this definition, but let me at the moment just define it as c plus ba of t. Okay? Because it only depends on t, right? It doesn't depend on x. It only depends on the fact that x is odd, and then it only depends on how many steps I do. Now, t really counts a number of half steps, so the full floc period is two steps, but here it's one, two, three, four. It's four steps, two floc periods. Now, this is one of the two. The other is when I place the observable on the odd, on the even side, like here, then the only way I can get not killed is if I contract in this way and then place the last observable here on this edge. Whatever else I do, for example, if I would place this last observer here, I could use my rule whatever three. I could do this way and it would immediately detach the circuit and this would be trace b. Trace b is zero. So the only way to get something on zero is in this diagonal. Okay, so now then I will call this, now this x is even, and then there is y. This is observable a, this is observable b. I will call this cda. Now here, y minus x is minus t. So now y is x minus t, comma x t. This I will call c minus ba. So now you already see what's going on. I mean, the dynamics, all the other correlations are zero. So now I can write a compact expression for the correlation function of arbitrary pair of local observables. If it's odd, this will be one. There is delta y minus x t. C plus ba of t plus 1 half 1 plus minus 1 of x delta y minus x minus t, c minus. So I mean, even before anything else, I mean we already got something quite remarkable for some dynamics which is not integrable. It is fine to use but it's some, rather still as you will see later, quite generic class of dynamics. We've been able to compute explicitly, I mean express explicitly the two-point function, which is a hard, looks like a hard object to compute. We evaluated it in terms of something that is essentially 1D. So I mean, now already the structure of this tensor network suggests that this could be computed like a 1D transfer matrix, right? It's like a partition sum of a 1D vertex model. So I will now define what I will call a transfer matrix or as you will see it's just a quantum channel, which will be this map which I will just iterate while I will define the quantum channel from consistent with my notation. Okay, but before doing that I wanted to do one more explanation. So how do I intuitively explain this result? So now, as you will see, we kind of shown that correlations can only spread along light rays. So even though there is Leigh-Robinson theorem which says that correlations should spread within the light cone, we have now found that they cannot be actually non-trivial within light cone, but they can be only on the light edge, right? So how do I intuitively explain that? Well, the point is now I have a strange system which has two types of causalities. It has spatial causality and temporal causality. Because I told you locality plus unitarity give me causality. And now I have unitarity plus locality in both directions, in space and time. So I can claim I have this t, this is x. On one hand, I have a statement that due to spatial temporal unitarity and locality, correlations can only move within this light cone. But because I have, so this is temporal unitarity, but since I have a spatial unitarity, correlations can only move. So correlation between this two point and this point, or the origin and any other point of space time can only be non-zero within this light ray. So the intersection of these two guys is x is plus minus t. The correlations can only move with the speed of light. There is maximal speed in our model. Okay. So now let me now be more precise. Let me now define the object in which I will evaluate this two pieces of correlation function C plus and C minus. I already sketched their tensor network diagrams. I have to now take one step and see how it goes. So I define what I would call a quantum channel. I mean, this is the terminology from quantum information theory. Basically, you could think of this as an evolution on a density matrix. Even though our observer is not a density matrix, I mean, or if you want, this is like a dual evolution to a quantum channel. It doesn't really matter. The structure is essentially the same. So you could think of let me just sketch now one thing. I mean, this is like m plus. I will define m plus as this black box. There will be input and there will be output. So let me now try to be precise. m plus operates in an operator. And now it speeds out this. So this is the object m plus of an operator. This is the transpose and this is the dagger. And if I now write it in terms of formula, what is this? This is just taking operator tens of product identity. Then you multiply from the make sure I'm correct. Then you multiply from the right by u. So that's why I write u transpose because it's multiplication from the right. But from the left, I'm multiplied by u dagger. And then I take at the end, I take a trace with respect to the first space. Partial trace, right? You all know, I guess, what partial trace means, right? So I take a partial trace and I get an object which is supported on a second hybrid space. I mean, it's an operator which acts on the second on one copy, let's say on H1. On a one queued state space. So and this is exactly what people in quantum info would call the representation of a quantum channel, right? It's a particular type of quantum channel. It's a unital quantum channel because it maps identity to identity because I put unit density matrix here. I forgot one thing. I think I forgot 1 over d. Because this has to be a density matrix, right? So it has to be 1 over d. But then this is an honest, proper parameterization of a quantum channel. It's a particular type of quantum channel because u has to be dual unitary. So it's not just any quantum channel. It has all the features that quantum channels can have. In particular I mean, it sometimes in some other community it would be called a quantum Markov chain. So it's like what some people would call it a quantum stochastic operator or quantum stochastic matrix. I mean, if you think of it as a matrix which acts on a vectorized operator, it's a stochastic matrix. But it has a little bit more structure, so it could think of it as a quantum stochastic matrix, but it has again satisfies the conditions of it. So its spectrum is in the unit disk and there is always an eigenvalue 1 which is associated in this case to a unit operator, right? So it is that its unitality is that also lambda 0 is equal to 1. The corresponding eigenvector v0 is equal to 1. I mean, it's the unit vector. I mean, please, I mean this is a bit, I don't know going to a slightly different field. So if there is anything needed to be explained and I didn't explain, please ask. I mean, I assume that most of you have seen something around this. If not, I can probably say something more. Now, for the completeness, then I define also the other channel m minus. m minus is the other diagram which goes left over d. I just flip 1 and 2. So here I make a trace over the second space to dagger and 1. Okay, now I have to pause a little bit and explain what we've achieved. I mean, I think for some people this kind of this finding was quite remarkable because you know, some people, I mean there was some quest to find many body dynamics which is its own perfect Markovian reservoir. And this is it. I mean, you usually you know, again, now I return to several of the lectures of this school. I mean, people have discussing thermalization which is a very important phenomenon in many body physics. And there we want, I mean, we want to explain thermalization. Basically what we are saying is that local state of a system is like a keep state, right? It becomes like a keep state. So why is this because the rest of the system acts on it as a perfect reservoir, right? And this is, I mean, there can be nothing better than this. Where the rest of the system acts as a perfect Markovian reservoir. So the reduced dynamics is a Markovian dynamics. This is this is the Markovian dynamics paraxonals. I mean, it cannot be better than that, right? It's a single qubit. You could think, I mean, if it doesn't work for a single qubit, I mean, maybe you could say, okay, it works for sufficiently large subsystem, yeah, but so that the correlation length is smaller than the end of the subsystem, then you can maybe argue in the same way for genetic dynamics. But this is for which it works even for a single qubit. Okay, so again, I mean, I try to advocate. I mean, this is useful if you want to have, you know, an example of a system which is perfect reservoir for, I mean, for, for itself. So I mean, you can test some of your ideas on this model, maybe much easier than non-generic models. Yeah, I mean, well, that's, I don't know if I would call this a reason, but yeah, that's of course consistent with that. Yeah, I mean, this transformatics, the fact that this transformatics is stochastic matrix, I mean, that's kind of necessity, I think, I mean, for the consistency of everything, right? Because, I mean, I guess you could formulate everything in terms of some sort of, yeah, I mean, I'm not sure if I can make a more clear statement, but, well, it's the lack of, it's a kind of lack of correlations, I would say, I mean, that correlations this somehow as you see, I mean, again, there is there are correlations which survive, of course, I mean, but they only could survive in one direction. Again, yeah, I don't know, I mean, I don't know if I can make a very, I mean, I mean, it should not be seen as something unexpected, but it should be somehow taken as surprisingly good, I mean, that it can work so good, I mean, one would not be, should not be surprised that something like that can happen. So you have a unitary many-body dynamics and the dynamics on a subsystem is Markovian. Well, it's usually not the case, but it's stochastic, I mean, it's it is equal, the fact that the dynamics on the local subsystem is a quantum, it's a quantum map, like CP, really positive phase preserving map, that should be obvious, right? But that map is Markovian so that it can be written in terms of iteration of a simple Krause if you want, I mean, this can be written in terms of Krause representation, for example, I mean, this is like, simple, people have discussed, some people discuss Limbladion in this school, I mean, this is like a disk or time version of Limbladion dynamics, right, so it's really as simple as it can get. I don't know if I can give you a more intuitive picture, I mean, one should just get to use to it, I mean, it's like, I'm more like, any other question? Relation between the two? Yes and no, they both contain the same U, but they have slightly different aspects, so, for example, I will now discuss the spectra, I mean, spectral decomposition of the two guys and see what we can say about correlation functions and in principle, these are two independent spectra. So you can have, for example, well, let me continue and then I will see, right, okay, so, right. So now we are basically, aha, now I basically touch the point of, no, before going there, I would now, okay, so before going there, I would now maybe characterize these duality gates for qubits, so, now the question is, so there are two things I want to do. Second, I will try to discuss then the general spectral classification of spectral features of these quantum channels, but before going there, maybe I will discuss first the representation for qubits, full parameterization for qubits, optimization of duality gates for qubits, which means d equal to 2. I already mentioned yesterday, I mean, the general question of how to characterize these duality is a hard question, but for qubits we can say a lot, I mean, we can say everything, we can basically characterize completely this manifold, this set of unitary. So before going there, I would like to make a little detour, and I would like to pose a little exercise, show the following. I mean, I decided on purpose not to show it myself, even though it's like a two-line calculation, but it would be useful to use it as an exercise, and then you think during the break. So if you is dual unitary, then, sorry, yeah, if you is dual unitary, if and only if, if and only if composition of you and the swap is unitary after partial transpose. So, for example, do a partial transpose irrespective of each subspace. I don't know, for those who don't know these terms, maybe I will not explain them even because it's not so crucial for our, but most of you, I guess, are familiar with these notions, like partial transpose, right? So if I take, if I compose you with the swap and do partial transpose, then this can be unitary or not. And if this remains to be unitary, it's, these gates are sometimes called T-dual, right? So gates, U times P are called T-dual if they have unitary partial transpose. And it turns out that T-duality is equivalent to dual unitarity after multiplication with a swap. Okay, that's a thing to show. Then another claim. Dual unitarity is preserved under multiplication with a single qubit gate, single qubit gates. So there is a simple proof. Now let's suppose that this guy is dual unitary. Then I claim this guy, right, put where I decorate this gate with a single qubit arbitrary quadruple of single qubit gates is still dual unitary. That's the most general thing I can do, right? So if this is U, let's call this U1, U2, V1, V2. So then this is U1, 10s or U2 on U, V1, 10s or V2. What I claim is that if this guy is dual unitary, this guy is dual unitary as well. How to see that should be obvious. Well, how do we know it's obvious? Well, unitary is obvious, right? We should only have to show dual unitary, but remember when we show dual unitary it was something like that. Now I'll do it in terms of gates, in terms of state gates. Now this is U, UBagger, U1, V1, sorry, sorry, this is Dagger, so this is U1, U2, V1, V2. This is V1, Dagger, V2, Dagger, because we have to transpose this is U1, Dagger, U2, so little U. Dagger, little U1, Dagger. Now what do we have to do for dual unitary? We have to check this guy, the wire, this wire, right? Then you see here it's V and V Dagger, against each other, so they annihilate. Here it's U1 and U1 Dagger, it's annihilating because it's unitary, right? So it's like nothing were there, right? So then dual unitary means I can erase this guy, but then this guy collides against it's Hermitian dual, Hermitian adjoint cancelling and this guy is colliding with itself. So dual unitary is kind of symmetric. I mean when you decorate your gate with arbitrary single qubit unitary, dual unitary survives just as unitary. And then again another reason why we like to paint these wires under 45 degree angles because it just says that it goes this way, so it goes and then its orientation is important, right? You have to decide on your direction of time or direction of evolution of the movie, right? And then the arrow should go along the movie direction then it's the gate, if it goes against it's the gate Hermitian adjoint, right? It has to be time reversed, right? That's why it's good to have this 45 degree because then you can always decide whether you have to use U or U Dagger. But again, I mean it's like direct notation, right? I mean you have consistency in your diagrammatics then everything can be again done automatically, right? Okay. Okay, now once we have this then it's very easy to classify dual unitaries. So we start with a general representation of U4. There is this statement, again I will not quote the original source because I don't know it, but there is at least two or three papers in the last 20 years which reported this and will use this. So it's well known in the quantum information community how to parametrize arbitrary U4 so four-dimensional unitary complex matrix can be parametrized as U now is a phase then there is two gates which are on the left, two single qubit gates V1, V2, then there is what people call Heisenberg gate and then there is U1 across U2 where these guys are arbitrary SU2 matrices. So arbitrary two by two matrices with diagonal one V1, V2, U1, U2 are arbitrary SU2s. So this you can parametrize each of them with three parameters as you like, maybe Euler angles or whatever the unit vector in the angle. So each one of them has three parameters. In this business it's usually nice to kind of do the dimension counting so how many independent parameters you have to specify a given manifold. Now it's one, two, three, four SU2s which means three plus three plus three is three times four, 12 parameters and it's one, two, three it's 15 plus one is 16 and 16 is equal to four square and that's what mathematicians tell you that it should be the dimension of the unitary group four it's four square. So it's fine now what we have to figure out is how constrained is now the condition of the unitary on this structure, on this parametrization. Now we can easily forget about this guy because we have just shown that we can always undo single cubic rotations so we only have to worry about these Heisenberg parameters and that's a trivial calculation again I give you as an exercise to figure out that if you want to respect the unitary two out of these three guys have to be equal to pi over four and since I can choose arbitrary since I can do cyclic permutation of the Cartesian axis by appropriate choice of single cubic rotations that is a global rotation if you want, can be written as a product of single cubic rotations I can always choose that these two guys will be x and y I mean the components which I will fix to pi over four will be x and y but the gate is dual unitary is dual unitary if and only if jx jy is equal to pi over four so I have what again mathematicians will call a manifold of co-dimension 2 and u4 so I have constrain just by fixing two parameters, so now you can say now you can say up to your taste whether you find this very constrained or not so very constrained, I have 16-dimension manifold I have to fix two out of 16 parameters and I arrive to something which is much more fancy in a sense at least it allows me to do much more so not so terrible, right? okay, then there is another parameterization which is sometimes used now we can ask ourselves that's for qubits but I have spin one system or spin seven over two system so can I do something can you propose me an interesting set of dual unitary with that the answer is yes I mean this inspires actually general or general parameterizations which are not complete by the way the only one which we know which is complete so complete means that any dual unitary in this Hilbert space can be written in this way what I will say next will be just a subset will be just an interesting subset okay, so so now the first statement is that again a claim now I will make a couple of claims which I will not prove because we have a limited amount of time it's Friday but long maybe you can try to prove all these claims or try to clarify them for you no it will be all SU2 representations as you will see I mean at least the examples I will make of course you can try to to go to spin n representation for n level systems sorry, yeah, I mean to go to SUM, let's say, rather than SU2 but no I will just remain within SU2 so higher spin representations of SU2 but now the next claim is that this Heisenberg gate with two parameters well, one thing which I remind you is that if I take all the three parameters equal to pi over 4 then this is just a permutation gate now I'm not sure whether there is some face factor in front of it or not I never remember that so I will just put it like this but that's just a permutation I think it's just a swap maybe there is an eye or something sticking out, maybe I should subtract some items and entity here and then it's like an eye sticking out but it doesn't really matter so I will just put it like this or proportional to okay so that means that I can take this gate so that means that whatever is left here, whatever face is left here I can absorb it to the global face here so it doesn't matter only parameter which survives now is 13 plus every face plus 12 parameters here, so 14 free parameters so what I will do now I will say okay now this is just a swap times a ZZ gate that's a diagonal gate that's very nice right so it's a swap times a diagonal matrix in the canonical representation in computational basis if you want now this J is related to JZ I think is just pi over 4, one is pi over 4 so you can take pi over 4 out so JZ is J plus pi over 4 I mean these three operators commute so I can just take this guy out okay so now it means that I have canonical representation canonical representation of dual unitaries CU, global face V1, V2 station into the IJ and now it turns out that this parameter J is probably the most important one right if J was 0 this gate does not entangle because it's just a swap essentially it's a swap times local cubits, local cubit operations but they cannot entangle right and swap is just exchanges so if you have product states it maps product states to product states so basically it turns out that people who study again quantum info I mean they already knew about this kind of gates and they knew that this J is just what they call entangling power so J or something, some function of J I mean sine J square or something is just entangling power of this gate so the larger the J is equal to pi over 4 is the most entangling gate so that's good to know so now when we will see some transitions from regularity to chaos these transitions will be mainly most easily observed when we will so if you need some good examples taking this gate, change J, J is the best parameter and this v1, v2 and u1, u2 should be there because they have to break integrability because without those guys this gate is integrable, remember this belongs to a class of gates I discussed in example yesterday which are integrable betans that's integrable right, this is xxz this is basically just a special degenerate case of xxz model so I have to scramble this with some local fields, these are like local fields single qubit unitaries are local magnetic fields and if they are all different then this will surely break integrability so we have some generic non-integrable dynamics which has some entangling power it could be a nice example of chaotic many-body quantum dynamics as you will see ok so now that's that was it, so now we are ready basically to discuss some ergodic theory on example of dual unitaries so this is really very satisfactory because now we can go to a textbook, we can open the textbook of Arnold from 40 years ago, 50 years ago and he explains to us very clearly what mathematicians mean by ergodicity, he says even more I mean he says that ok there is ergodicity but there is also mixing but there is weak mixing, I mean there are different aspects of thermalization if you want in mathematics stronger or weaker, ergodicity and then there is mixing which means that correlation functions decay and so on, so I will illustrate this on this example so for those of you who have never seen that it's also useful because it might ring some bells and stimulate you to do some further reeling, ergodic hierarchy dual unitaries so now let's take back our channels m plus and minus remember our correlation functions were like oh I should have left the formula which gives me the correlation function or maybe I'll just quickly rewrite it because I will have to rewrite it in terms of a and b, ok so now I have these two channels m plus and minus and they completely specified the k of correlation so now what I will do now is I will diagonalize I will leave this page still empty, I will put it in here so our first step is diagonalize m plus and m minus m plus and m minus are linear operators so they can be written in terms of matrices m plus and m minus are d squared by d squared matrices they act on vectorized single site operators they are d squared of them one of them is unit operator and you can take all the others to be traceless so it's good to use some sort of canonical operators basis like Pauli basis or in general generalizing them on basis so taking unit operator and the rest traceless so Hilbert-Schmidt orthogonal to unit operator well no matter what basis you use I will not go into that detail here but you can write left and right eigenvectors of these guys so uj lambda plus minus comma j so now I will use two indices for the eigenvectors the first index will be which channel plus or minus and the second will be the number which will go from zero to d squared minus one so zero will always refer to a unit operator and to eigenvalue one and now you aside from the right eigenvectors you have also left eigenvectors which you can write like this I will call them v plus minus here j plus minus j v is the same I sorry here it's u plus minus j right so now we can write a spectral decomposition of m plus m minus which is sum over u plus minus j lambda plus minus j cross v plus minus j okay and I will I mean I have to choose this left and right eigenvectors such that they are that they are I mean these are operators right so they have to be 1 meter or so right or to normal okay so now I can take arbitrary power of this now here I mean of course you could now scream right that this is not the most general spectral decomposition and it is not I assume now that the eigenvalues are not degenerate and if they were degenerate there could be some non-trivial Jordan structure which I have now neglected but never mind I mean I don't want to get now too messy but yeah if you want to conclude that point is now once you have diagonalize the problem then it's easy to find iterations right and then the correlation function which you want to compute right would be like trace of b m plus minus to power t of a and what would that be now I have to be a little bit careful because now this is vectorized right but now it's again so basically it's something like trace sum over j lambda plus minus j to power t and here it is trace b times u plus minus j times trace of a times v plus minus j okay so there are some scalars right there are some scalars which depend on the eigenvectors of the channel and on the observables a and b but the point is this is the crucial part which is the exponential decay right so basically what you get is that the only thing I mean the key thing to worry about is now what is how does the spectrum of this m plus and minus look there's always an eigenvalue there's always an eigenvalue one which corresponds to a unit operator right which we took out somewhere there's always this guy the question is where's the rest right this guy is irrelevant because eigenvalue one does not contribute because these guys are traceless because one of these guys is unit operator and this will give us this term to vanish so there is no worries about this isolated eigenvalue one but there could be other eigenvalues one or there could be other eigenvalues on unit circle or there could be all eigenvalues within unit circle so that's the key of classification of ergodic error right now once you have an access to to dynamics like this then you can immediately say what can at worst happen with correlation functions but then we have I will have four different levels of ergodicity or non-ergodicity depending on the structure of the spectrum of these quantum channels so the worst or the most non-regarded situation is non-interacting dynamics so dynamics can be even non-interacting and this means that all two d squared minus one so there is d squared minus one non-trivial eigenvalues for each of the channels one is always trivial right so d squared minus one for each so two times d squared minus one are non-trivial but it could happen for some miraculous reason that all of them are equal to one so all the two d squared minus one non-trivial eigenvalues this means that correlation functions stay constant nothing can happen this means that correlation function c, b, a try to plot a picture for each case so picture means all eigenvalues are here so that you see that there are many so this is the the spectrum and the correlation function so of course this is totally boring but we have such an example already on the previous board this would happen if j is equal to zero if j is equal to zero this is just swaps well we need more we need j equal to zero but we need also that this u and v sorry this u and v are trivial because if this u and v would not be trivial they could produce some phases this would be the next case I will discuss so really this means just a swap circuit as a swap circuit we could think of I mean paradigm of free dynamics right example yeah so I used my board here second case non-ergodic and interacting so there could be there could be some eigenvalues which remain one there are larger or equal to one but less than do this less than all non-trivial eigenvalues equal to one so that means that there are terms here in this spectral sum spectral the composition where was it here keep this formula under your sight so when there are few terms here which are one which have non-zero coefficients here this means that there is a term in correlation function which doesn't want to decay which means that correlation function will do something like this and then it will freeze it will freeze to what some people would call a druda weight so non-zero time average of the correlation function so for that we have the following situation we have here a bunch of eigenvalues at one but then the rest of the spectrum is inside unit isk so we have a generic okay now let's just read next to it so now next is ergodic but not mixing that means that there are eigenvalues which are not at one the non-trivial eigenvalues which are not at one so again always when I say this I mean j different from zero of course so there are eigenvalues so for all non-trivial eigenvalues all non-trivial eigenvalues are different from one but there exists an eigenvalue which modulus is equal to one right which means that there will be just one eigenvalue here but then it could be eigenvalues like here now again the nature of the problem is such that the characteristic polynomial of stagastic matrix is a real polynomial and the problem has to have conjugate pairs so there is eigenvalue here there has to be one across the real line so there could be at least one pair of eigenvalues which are all in a circle and then the problem cannot be cannot be mixing but it can be ergodic it will be ergodic if this eigenvalue will not be rationally related to pi which means that it will not return to one after finitely many after finitely many steps so this will if it's like pi over three then it returns to one after six steps right it returns to two pi three steps so say again exactly I said that already there but I stress it again I always when I say non-trivial eigenvalue I mean j different from zero then you would have what some people would call a tan crystal like in lectures of Norma and Yao this would be called tan crystal because that would be exactly sub-supharmonic response now we would have now what I have here is what some people would call a tan quasi crystal right because the evolution is a periodic but it's you know it's yeah it's quasi periodic I mean so that's exactly what the definition is called quasi periodic this means that correlation function will never will never die this is just giving you the frequency of oscillation of the correlation function because this is the phase I mean this is e to the i phi if you want and this e to the i phi happens here to be e to the i phi t right so this phi is just a frequency with which this I mean t for us is discrete so you can think of snapshots of the harmonic function stroboscopically sliced no no phi over 11 is rational right suppose I have square root of 5 minus 1 over 2 which is golden mean then I have quasi periodic any rational of course will return after some denominator right so if i would be 2 pi over k over l then after l steps steps I will have you know lambda to l equal to 1 no I mean I will discuss in the last lecture I will discuss chaos what I believe is the the best definition of chaos which has to do with random matrix theory so far I'm not using the word chaos at all I mean I'm just trying I mean if I say chaotic I mean it's just like a teaser right I mean I can only discuss notion of rigidity and mixing right the way that Arnold's taught us or mathematician's taught us right so it's there's not there's nothing really related to chaos yet well they cannot because it's a it's a it's a Markov chain it's a stochastic matrix and there you can easily show that a spectrum is bounded to unidisc this would be a completely I mean this would be instability right actually this could happen for bosonic systems I mean if you with the right let's say Linnblatt equation for bosonic systems with some strange drive this could happen right but for systems with finite dimensional hybrid space this can never have this non-mixing can you define it yeah I will define it's very clear when I will define mixing at the end right then you could do the reverse right you will go back and see why I call this mixing which is the best possible or the most generic but also the best possible situation which is mixing ergodic and mixing if I was a mathematician I would call it just mixing because for mathematicians it's obvious that mixing implies ergodicity but since physicists preferred the word ergodicity I say okay ergodic and mixing okay so that situation where all non-trivial eigenvalues are within unidisc are strictly less than one that means that now here of course there could be eigenvalues inside always here there is just one unique eigenvalue here and all the other spectrum is inside so basically you can draw a circle which is strictly smaller than unit circle which contains the all non-trivial eigenvalues so then you call the maximal circle sorry the minimal the gap between the minimal circle which enclose all the rest of the eigenvalues and the unit circle you call a gap spectral gap and that gives you the rate of decay of correlation so I mean it gives you a bound on the rate of decay of correlations right so that's a gap but then you can easily show I leave it as an exercise but it's almost trivial that all the correlation functions in this case can be bounded by a constant k times e to the times whatever let me call this delta no I still prefer to call this delta but I have to be a little careful so then it's e to the minus delta t okay so there is such delta which I might call a gap such that correlation function has exponentially k with exponent delta I mean there are some logarithms here which one has to fish out okay but then this is what mathematicians would call mixing right when they can prove that all correlation functions decay now this is something that mathematicians call exponential mixing and it's the best form of mixing right there are not many systems for which they can prove exponential mixing there are some maps so called Arnold cat map which is a simple classical dynamical system like a logistic map for logistic map by the way they can also prove exponential mixing at the extreme point in the chaotic point right and now we have a quantum many body dynamical system for which we can prove exponential mixing so that's great right okay so what is the time pretermization pretermization is a transient effect so it's not really a face of matter it's not really a regime of mathematics right yeah right so and preterm is I mean it could just well be that pretermal time crystal or pretermal face of matter is just the mixing chaotic and mixing is just that the time scale can be very long I mean here we cannot distinguish right I mean this is just this is now really I mean that's the problem right it's a general philosophical problem right this many body physics many body dynamics is a very hard subject right it's very hard to prove anything and then physicists you know like to talk about but in a very vague way right so then they invented terms like pretermalization but what does it mean in terms of rigorous mathematical physics have no idea I mean probably it means just that this is just a long time scale which is related to some interesting physical mechanisms which are again really hard to explain sometimes but you know here I cannot comment on that yeah I mean first of all algebraic decay of correlations are very hard to explain simple terms I mean it will not first of all I mean the fact that we have reduced the problem to a finite dimensional stochastic matrix means that we cannot explain algebraic decay of correlation full stop right whenever you have algebraic decay of correlations you need infinite dimensional Markov chains because you have to have a gap gap closing but not exactly zero but closing in some limits and so on so yeah so we know right from the start that we would only be able to to care now I'm I have a question can you speak up a little but remember the individual expectation values are zero I mean this is already connected correlation function the product of the average of A and average of B are zero oh right because you're taking traceless observables yeah yeah so the plateau in correlation function is not a trivial effect but like for case number one for example they're not interacting but the correlation is still like non-zero and constant so like how is that possible if it's a non-interacting system well correlations cannot decay if it's non-interacting right time equal to zero correlation is non-zero I mean it's okay so it's like a property of like this sort of state that you started with yeah I see it's an infinite temperature state but you are looking at two observables yeah I mean it's just right and there's some no it's okay it's okay I mean the point is yeah right I mean this would be of course non-zero only when the two observables will be on the same light ray right but since there is no dynamics it is just the product of two observables trace B dagger A I mean it's just overlap between two observables so if this is like auto correlation function it's just trace A square there's no dynamics right so it's like I mean this is a free circuit right so you place one observer here and the other there it's just forget about everything else this is just this tensor network right which is just this observable contracted with this endpoint but there's no dynamics in between so then there's no decay right it's a trivial thing but you know it's alright thank you for trace A and B it's only when it's auto correlation if A and B are trace less then it's only non-zero if A equals B if they are trace less it's not no it could be non-zero also for some non-trivial and well depends yeah I mean if you took if you take A and B as Pauli matrices right which are assumed to be Hilbert-Schmidt orthogonal yes that's true different there is no correlation yeah sorry maybe a question about the second case so here now ergodic and interacting does it immediately imply like integrable systems like interacting integrable or do you also include other mechanism for ergodicity breaking no it is does not mean integrable so non-ergodic non-ergodic interacting yeah this one this is what you mean no this in general means non-integrable as well I mean we have we can cook up examples of that for dueling three circuits right I mean in our paper we have examples of this parameterization just specifying parameters which give either of this and this is not in general integrable so it's another mechanism for ergodicity breaking if you want yes right do you know if it's related to any other known mechanism for ergodicity breaking that people talk about these things I don't know this at least is completely clear right I mean will Hilbert space fragmentation but it could be possibly but yeah okay thank you sorry regarding the same regime so if I understood well this basically non-zero and no one again values give you information on the early time dynamics is this true early time yes yeah well I mean okay and so what about if all the again values are either one or zero is that a situation that can happen yes it can if all are either one or zero yes that situation can happen sometimes it does happen and in which of these four groups will you put them because it's the second one because if there are more than one which is equal to one it's non-ergodic right no matter the rest is zero and in this case how does it would like sorry how does it look like the dynamics at early time at early time it can have high yeah it's a good point so of course in the simplest possible of cases right it could be just like this right so if there was no non-trivial Jordan structure it would be like this right but in the examples that I know there is always almost always non-trivial Jordan structure when there is macroscopically the general eigenvalue zero it's usually comes with non-trivial Jordan blocks and then you could have some dynamics which extends to times of the measure of the the largest non-trivial Jordan blocks okay thank you Hello question regarding the second class so if you start from an integrable system whatever the symmetries you have and let's say you have flow came any body localization for that you have some sort of not quasi local kind of integrals of motions so the eigenvalues which are non-zero so that symmetry structure will not reflect in those eigenvalues means somehow what is the ergodicity breaking mechanism that is not reflected in the non-zero non-zero means eigenvalues which are not equals to one that structure will not reflect the symmetries what is the mechanism of ergodicity breaking I am not following you so what I am saying that for the second class you have a chunk of eigenvalues which are one and some are non okay these eigenvalues right I mean just since you reminded me I forgot to say these eigenvalues of course are related to conserved operators right these operators which the eigenmodes which give eigenvalue one are conserved right so if I take now these are local operators but if I take translation this is translation variant system so if I take translation variant sums on every other side because this is staggered system so if I take this repeat it on every other side and take a sum this is exactly conserved after the full flow case step it will be exactly translated it is conserved so the question is yes I mean this would immediately imply that there could be no well there could be exact I mean there could be no I mean there is ergodicity breaking also because there are non-trivial conserved quantities again might not necessarily mean integrability I mean sometimes it's a trivial symmetry okay yeah thanks for questions I think I have now like 15 minutes more right until the break so let me now start with the rest because we'll need some time to explain the second main thing I wanted to do in these lectures so before I go there I mean let me stress one more thing I mean this is not everything of course we can tell about dual unitary circuits about the correlation decay in dual unitary circuits I have to stress again that what we assumed here is that observers were ultra local now I can relax this assumption and I can take let's say observers with large and larger support like so I mean I just want to quickly comment on what would happen if I would consider more complicated observers right could I still use this quantum channels or should I expand I mean question so now what I have now is an observer which is near local now imagine I take an observer which has some support R right so now what can I do well I can do one thing which is totally straightforward I can take my circuit and I replace now this is operatorial gate and I can replace one gate with a super gate which will have support two sides I mean this is a philosophy of RG if you want but in space time so I take instead of now I take I consider now this as a box and now this box has a thick wire which has dimension D square or D to the fourth right before each wire was D square now it's D to the fourth but my claim is that this big box is also a dual unitary it's easy to show right if these little boxes are dual unitary this tensor product box which has four pieces is also dual unitary which means now I can propagate observers of support two with this right so I can now define M plus and M minus for observers of support two using this big box now I have quantum channel which has dimension not D square but D to the fourth so stochastic with a matrix of dimension D to the fourth but everything remains the same all my classification remains the same now what can happen what can happen now the spectrum becomes more rich so that there could be eigenvalues which go closer to to unit circle so there are observers there could be observers which are less relaxing and so on so all this can happen and we know it's very hard to investigate this systematically but you know we can try at least first few steps but again I mean qualitatively the same remains for any local observer no yes I mean in general for unitary yes but of course yes of course of course this is true for any unitary but what I'm saying is also dual unitary survives yeah and that's crucial for us to make progress okay so now that's it that's what I wanted to say about the care of correlations and now yeah maybe before I go to the next chapter let me just comment on what else people could compute using these tricks which I will not go into any detail but there are papers by us or by other groups like Peter Clay's Oster Lamacraft Lorenzo Piroli so on I mean where people computed entanglement dynamics operator entanglement tripartite information it turns out that essentially almost anything right any dynamical quantity which characterize correlations or quantum correlations or entanglement or complexity or thermalization can be computed within dual unitary so that's really autox for example you know some people are really like autox I don't know why so I will now conclude my lecture so I mean there's a whole lecture but as you will see it's because this will be kind of hard because to me this is the most exciting thing you can do about dual unitary personally I don't know why but again probably has to do with my background but you know it's really I would say the best proof of many body chaos you know given interacting dynamics so it has to do with spectral correlations right so now we have spectral correlations in dual unitary systems in unitary circuits and I will discuss just one object which is turns out to be already difficult and interesting enough which is the spectral form factor so I will now spend like yeah almost the full lecture plus a few minutes of this lecture discussing spectral form factor first I will discuss definition and how it behaves in certain limiting cases like free dynamics and random matrix theory and then I will discuss spectral form factor for some special examples of dual unitary circuits I will not even go to most general maybe if the time permits I will maybe go to the general case using my slides because otherwise the time will be too short but I prefer to give you the main ideas rather than cover everything so I mean there is this question of what is quantum quantum anybody chaos what is quantum chaos right I mean different people understand it differently there is and this question has become really one of the central questions also because people from high energy entered here and their ideas are usually quite influential so then quantum chaos kind of became possible but it's usually I mean there in those ideas people try to understand Lyapunov exponents and this type of definitions of quantum dynamics which relates to Lyapunov exponents and this is usually require some sort of large parameter right which gives you a sort of essentially classical limit but the question is really I mean if you can define something meaningful which could be a definition of quantum chaos anybody chaos which doesn't require any smaller large parameter so it could be defined also for spin one half or a qubit circuits and to that I think the best the best definition would be comparing to random matrix theory I mean you already yesterday he mentioned you know reference to random matrices in the context of ETH ETH is basically just a version of random matrix theory right or comparison of random matrix inspired by random matrix random matrices right it's like a maximum ignorance I mean usually this is related to a principle which is sometimes phrased as maximum ignorance principle in physics right I mean when you don't know anything about your problem then you assume it's structure less right you assume that everything is possible everything that could be possible is possible and then you assume that you do calculation and you compare to your energy and sometimes it agrees and that's the best definition of chaos I mean that's to me the best definition of chaos right if the system does what the random system does even though it's very structured then it's chaotic so now this is the puzzle right we have these models which are extremely structured right they are composed of Lego bricks and they are only neighboring bricks are touching each other so the correlation spread in very particular ways as you can see and as people saw really long time ago I mean these problems have spectral correlations which are indistinguishable from random matrices so now how can this be I mean to me personally this was a question which inspired me for the last 25 years I mean how can this be this is a really I think very important question people ask this question you know and this is what used to be field called quantum chaos so this is a question for simple models like billiards individual atoms in strong fields hydrogen atom in magnetic strong magnetic fields were such a paradigm examples and they found that these simple models have random matrix spectral statistics and they explained this they explained this in terms of semi-classical methods because there was a small parameter there effective Planck constant so there was I mean I will briefly touch into that but then I will go to trying to explain a little bit about what I mean how can random matrix spectral correlations be explained for some examples of many body dynamics and it turns out that dual unitary dynamics is the only in only class of models for which we can actually accomplish something and that's for me personally the reason why I find it so exciting okay but now let me just go slowly I mean I would reduce spectral form factor so I will introduce a spectral form factor for floccy dynamics even though people have studied it for for arbitrary Hamiltonian dynamics but for Hamiltonian dynamics to have additional complications because spectral density of states could be the function of energy so things could be modulated with energy so it could be hard so I but since anyway what we have in mind are floccy systems and for floccy systems things are easier there is no preferred energy there is no ground state all the energies are kind of democratic equally important so there is no reason to expect some modulation of density of states so let's just say we have our many body evolution and now let's for some given system size we can diagonalize this called this spectrum quasi energy spectrum that we call curly n the dimension of the Hilbert space for us is L well Qt d to the L sorry we always have a many body system in mind but what I will save in the next 10-15 minutes will be complete general so we just use this curly n to denote the number of quasi energy levels and so then what I will now assume is that you have this spectrum and you can arrange the spectrum as a set of points on a circle right because this is unitary matrix but spectrum is a circle so you can think of this as a gas of particles in a box with periodic boundaries and this is a very fruitful analogy I mean this is something that was first time advocated by Dyson sometimes referred to as Dyson gas to think of eigenvalues of random matrix as a gas right and to write statistical mechanics of this gas I mean to study statistical mechanics of a gas and that's what spectrum form factor is about so it's basically a pair correlation function it's fully a transform of a pair correlation function of this gas of quasi energy particle so let me now slowly define it precisely okay so first we define density one point function so now we have a given so I mean and then also the I mean this is statistical physics right so now we'll take a given model but this is a statistical model so we then use an ensemble of models I mean we have to argue because we have to do certain averages right but now for the moment this is a given model so it has a given spectrum for a given spectrum you write one point function which is the density or of phi which I will define such that it will be correctly normalized so 2 pi over n sum over little n sum over delta centered at the eigenvalues so basically this is a sum of delta spikes define the average and the average now I will mean the average over the quasi energy so 1 over 2 pi rho phi d phi so now this term has n terms the sum has n terms so this is n and over n is cancelling and 2 pi over 2 pi is cancelling so this is 1 so the average this is normalized such that the average density is 1 right and now we define the pair correlation function the standard object density density correlation function the standard object in statistical physics we call it r of theta also in random matrix theory people use this r of theta I mean this is just as you will see it's just a Fourier transform of what I will be interested in but you know it depends it's the same object I mean it has the same formation it's just a Fourier transform in between so r of theta is the two point function of density density so we start a density which you will displace you center it around phi and you displace it by theta so we have the sum and the difference coordinates and then you subtract the product of averages and these are equal to 1 by definition and then you define spectral form factor K of t as a Fourier transform now there are different ways in which you want to normalize it I prefer to normalize it such as you will see later it's convenient but in the literature we find different normalizations so that's not so crucial the point is it's just the Fourier transform of the pair correlation function now Fourier this is a Fourier analysis on a circle with periodic boundaries so the Fourier transform of a function on a circle is a sequence it's an integer sequence it's not a function because the time is discrete so now the term in the sequence is an integer time it's a K of t t is integer okay so now this is I think I should probably not do all the calculations to the very end I invite you to do this exercise it's really something that is totally straightforward from what we have on the board when you do it carefully what you find is this is just product of two terms one term is e to the i phi and t in the other term e to the minus i phi n prime t minus n square delta t comma 0 this is what you get the exercise okay so now that's really cool right I mean what is it this is just sum of phases right which are weighted by t and this is just trace of u to this u which I called curly u because it's the full many body map to power t and this is trace of u to minus t or this guy complex conjugate is the same and this is some well some something which I will not even care about because I will assume my time is non-zero it's positive so when time is zero of course um this guy is n this guy is n so everything blows up right but the way I defined it it should be zero times zero so it should cancel right but now I will just define spectrum form factor and I will assume that t is not equal to zero just trace of u to the t model square I leave some space here I have to comment on this okay so now that's kind of nice right I mean you have a Fourier transform of a spectral correlation function so this knows about correlations between levels right I mean either on small or large distances I mean it has all the correlations all the pair two point correlations among levels and this is a Fourier transform so it also has all the correlations but short quasi energy correlations are hidden now in long times and vice versa I mean short times in code correlations across let's say short time two pi energy distance quasi energy distance so shortest possible time mean correlations between quasi energies which are in the opposite sides of unit circle but anyway I mean this encodes all the correlations the problem is that this object is not well defined unless one does not do an additional averaging and this averaging means you have to average over some ensemble of systems or over some parameter with which you can sample the spectrum and so on and so forth I mean the reason why you have to do it is that this object itself is not what mathematicians would call people who work in random matrices would call self averaging self averaging means that if you don't this object basically has fluctuation which is of the order of its of its value of its mean so the expected fluctuation of this object because it's a statistical object the expected fluctuation is of the same order as its value so if you just calculate this for a given dynamics then you will find plot like this right and it's rubbish right you can't make use of it now you can do two things you can do moving time average right and then you get something out or you can do really for analytical work the most honest averaging is to introduce some disorder even very tiny one and then you average over this disorder and then you get a smooth curve and this curve then can be compared to whatever model you want to compare to and usually will compare to random matrices so I think well now it's a good time to stop I already running over time but maybe we can now take a break and continue after so any question so far no no no this is completely general this is completely general I'm just throwing the definition which is has been around for decades and of course there are lots of measures of spectral correlations in textbooks on random matrices like meta big collection of all possible k-point functions that people can compute and I will not go into that if you want to show that some model has a random matrix spectral correlations you have to show that all k-point functions agree with random matrices that's usually very hard but this is the simplest measure which is the easiest to work with and already quite contrived all right then let's meet at 11.05 and thank Tomas for this first lecture