 Thank you for remaining after lecture yesterday. Today as I edition, will be the day that we discuss construction of the Entanglement entropy by means of fish theory. But before making this in order to try to let the larger number of you understanding what I am telling. Mamo, da pokazimo zelo, da pošle lem, da je sahne karatistice in taj finača teori, ki bomo zelo všeč mneza, ki najske zelo, in izvah da ne bomo zelo v Madrih Protoštjevnih, in zelo se zelo vzelo še spremno teori. In v Madrih Protoštjevnih, zelo se da vzelo vzelo spremno teori. Vzelo spremno teori je, da je vzelo vse spremne a tensor, a, da bom je, da bom... a, da bom je nabdaj bord v posjetinu, parno Hyundai S1000. Cевой for the component plus of the spin, one-half of the component minus of the spin, one-half of the spin, okay? So this is a Chi times Chi tensor, of matrix if you want. Let's say, a1 S1000 on the spin, 1 a2 S1000 on the other, et cetera. In to je tenzor, kaj, kaj, tenzor. Ok? Tenzor je pravda denoted s tem symbolem. Ok? To je počkaj, kaj je lež, ko je počkaj počkaj, je vzelo vzelo vzelo vzelo vzelo vzelo vzelo vzelo vzelo. So, this leg means that there is an index that is not contracted, that can be either plus or minus one, while the leg that appears here is i, and this one is j, where i and j are the two indices of the kaj times kaj in matrix. Yes. Didn't consider anything yet. I see that to each side, I give you a, see, a tenzor, A tensor, kaj je začal, to je začal, pa je začal drugo, a2, a3, a3. In naredil sem, da je vsega vsega, da ima vsega, da ima vsega vsega vsega vsega. Vsega? Vsega? Vsega? Vsega? Vsega? Vsega? Vsega? jaz bibene na poputarnosti izbrat. Srečume ki so dve zašli w epsilon du. Jaz te ne vidimo, da pa ne zelo nemunim, ne bo pa poplizniti lačiki. Čo se bo zelo? Lonsej, ne bi bo popliznel, kako se bo poplizniti lačiki zašli zašlok. Ni, ja ne zelo, ja nezelo igrat, ja ne zelo igrat... Rečimo, konflikta na dobro, kaj je zelo, tako da počekamo 1, s1, a2, s2, okej? Kaj je tukaj, če je matrič, kaj je naprezentacija? Tukaj je tukaj reprezentacija. Tukaj, če je tukaj, če je kontraktat, in nekaj je pozor, posledaj, da arejevačne in degetre, ni, da smo se vziste, da smo odročili kot Ratik. Spokoj, naredilem njič. Prevaj, pozor je kratko, ko bom se vzac in vzrona. Da sem začala med Kajор 1, 2, 3, 4, 5. Zato je zaraznja, v srečnih to zesaraj. Ne kaj je to dicočaj, da imamo krajega v začiji. OK, let's assume that the system is infinite so we don't have to wonder whether we should close it or we should put some state on the boundary. It's not important for where I want to go, OK? Imagine that just this guy goes over all the way here, it continues all the way here. Like for the expert could be an ITBD, OK? It's clear what this plot means. You will find these things everywhere, so better to say it once if you never saw them. So these are the physical indices, OK? The basis in which we are writing the thing. And these objects represent the tensor that is one over every side. Which states I can approximate or I can write in this notation? And this is the first question. OK, will depend on the dimension of the tensor chi as your colleague said, OK? I can imagine that clearly. Let's go to make it easy to the case where all these tensor are equal, like that we have a translational invariant system. Infinite, all equal. The number of terms that is inside each center is chi-square, OK? So I have chi-square parameter and with chi-square parameter I cannot describe all possible state of nature, OK? What you can easily show is that if chi is equal to 2 to the n where n is the number of spin eventually is infinity, you can write exactly, write any state as an MPS, matrix product state. But this is not the game that you are usually playing. Usually you want a chi much smaller than 2 to the n and still have a faithful representation of the state of interest. And the key to understand which state you can better represent with some finite chi, OK? Is again the entanglement. Because, but it's not the maximum entanglement of a chi and chi tensor, OK? MPS is low rate of chi. This happened when all the maximum entanglement is when the values again values are all the same because if you start decreasing one, obviously you get less entanglement and when they are all the same, since it's a chi and chi matrix, each again value should be 1 over chi because there should be some 2 1 and the corresponding entanglement entropy is low chi, OK? So this is really follows from what we have done already yesterday with no calculation. So if with a given chi you can capture the entanglement entropy of, let's say, of the worst partition of your state, like half the partition, then it means that the state is representable with a matrix product state. For example, we say that in one dimension there is the area low, OK? The area low means the entanglement saturates to some value, it's finite. So you have to fix chi to be the exponential of whatever, OK? For a gap 1D system, SA equal constant. So it's enough that log chi is equal to this constant, which means chi equal the exponential of this constant, which could be 3, 4, 5, 10, but still it's a finite number, OK? And so with the finite chi you can represent this state, OK? For a one dimension, a gapless system, we say that the entanglement entropy was C over 3 log L. If you match this with log chi, then you follow that chi should go like L to the C over 3. So it means that the dimension of your bone dimension, which is called chi, must grow polynomial in L in subsystem size, or system size, which is still something it's doable, OK? And that's why matrix product states works in 1D. In D greater than 1, entanglement entropy grows like the area, which means L to the D minus 1. If this must be equal to log chi, then that chi must be equal to the exponential of L to the D minus 1. And in this case, the bone dimension should grow like the exponential of your system or subsystem size, and this is not possible, OK? At some point, very soon when you increase your system size, this thing will break down, and in fact, MPS works only in 1D. This is not the reason why I introduced this state, but I introduced this state to write the reduced density matrix now. OK, this was a good detour to let you know one of the most important aspects that the theory of entanglement explained in the last ten years. I'm talking about the ground state. I don't know anything about a state like that. I mean, it's not part of what we want to discuss here. I'm talking about the ground state of locality. Then we can say what happened in time evolution. Let's fix to what we are talking about. If you want something else, discuss out of the lecture. If you want to discuss, I don't know, let's discuss another moment. It doesn't depend on subsystem size. That's what we did yesterday. Doesn't depend on subsystem size. That's what we discussed yesterday quite a lot. If every day I should repeat the things, we don't move from here. Are you every day, day by day, checking what we are doing here, because it makes little sense. This is the state. Our state psi, let me rewrite, is in this way. We want to write the reduced density matrix in this notation. First of all, we should write the density matrix, which is nothing but psi, the projector on the state. In this notation, you write the density matrix. In this notation, it's like this guy here is the ket, and this other guy here is the bra of the reduced density matrix. And you see that there are non-contractive indices corresponding to the row and the column of the density matrix. These indices here correspond to s i, as it was written here, and are the non-contractive element of the density matrix of the state, and if you write two copies of these, it means that are the non-contractive indices of the row and the column. We should write the reduced density matrix. What you should do, you should decide which part is a of the system. Let's imagine that a are these two spin here in the middle. And what you have to do is that in the rest of the system you must contract, because you have to make trace over the b degrees of freedom. Rho a is equal to the trace over the b degrees of freedom rho, which means that the sum over the s i belonging to b of the rho sum over s i means contract an index. If I rewrite this guy here, it means that I'm contracting these indices and all the other that are maybe on this other side and this other side, and I'm leaving un-contractive just the indices corresponding to a. And this is a representation of reduced density matrix. Now, if I want to calculate trace of rho a to the square, I should just take q of these matrices. This is one, this is the other, where there are these open indices. And then I should make the product of these two matrices. What does it mean to make the product of these two matrices? If this is like... Let's abuse with notation. Let's say that this is i1 and i2, meaning it's not the i of before, but it's the column index of this matrix. This is j1 and j2, the column index. This matrix, again, here I have i1 and i2, j1 and j2. And if I make the product, what I should do, I should identify the rho of one matrix with the column of the other. If this is the first, it means that I should connect with this j1, should identify with this i1, so I should contract in this way, and the same on the other side. And then I want to take the trace, it means that I should even contract this guy with this one down here and this one with this one down here. I hope you understand this plot. What happened here is that the spin belongings to b are just contracted between themselves, inside the same system, while the spins belonging to a are contracted between the two copies of your system. This plot is the essence of the replica trick that we use in the field theory. That's why I wanted to introduce in this other context because most of you are not familiar with field theory. Then maybe there is someone of you that knows anything about spin and knows only field theory and this is just a bad detour. And this concept of the replica trick is very important because it has been first introduced in field theory as we will use in a second. And okay, then it's the same story with the model process state, but most importantly, recent experiment in cold atoms were able to measure entanglement related quantity exactly by making this story. They prepared two copies of the system, the couple in some particular way and they were able to extract the entanglement entity. So this can sound a very theoretical and artificial construction, but actually it's what they do in real lab. Do you have any question? All the indices are separated. Trace of rho to anything is a number, which means that there is no left over index. Open indices means that there is still a tensor structure. So every time you find a number, you have to eliminate all the open legs. But trace of rho to the n, I can write you. Let's make only one spin that is easy. If there are many of them, you just... Here it means that the rest is contracted. It should do you understand this object? No, so this is one row, this is another row and this is a third row. What you should do, you should connect the first line of the first with the second, this one with this and then you should close secretly the first and the last. And okay, you can go on up to any whatever you want. It's always the same story. You multiply one times the second and the last one for the trace, you close. But we will see this abundantly even in field theory, it's the same story. Clear to everyone? Who knew mathematics for the state before, not that you know after. Okay, so it was a good idea to start with this. Who knows the replicatric in field theory in state? Okay, so much less, so the order of explanation was good for the majority of people. Now I hope you as student understand that teaching to such a variety of people is extremely complicated, okay? You have very different background with... So I'm just trying to optimize the understanding, but okay, obviously, you're not a uniform class like in normal school. Okay, so now what we will try to do is to repeat this kind of construction for field theory, reverting the historical order where first came field theory, but okay, since you are most familiar with this, probably this is the best thing to do. Okay, and for sake of simplicity, I will just write down things in one dimension, but what everything that I'm writing is valid in arbitrary dimension. Okay, so just to write and to draw things nicely, we use one d, but okay, everything we do today valid in any d. Just you have to draw and to have more indices in your formulas. Reduce density matrix in quantum field theory. That's the first step. So to have everything well-defined, let's start with a lattice quantum field theory, which means that your field theory has been discretized on a lattice so that we don't have... All right, so exactly like the spin chain of before, you have your one-dimensional lattices and your field theory defined on it. To define a field theory on this lattice means that we denote by phi to dx a set of local commuting observable, actually a complete set, which means a set of observable, which completely characterizes the theory. Like in the case of a spin chain, here you have just Sz. Okay, so that when you specify you know which states you have on that side. Depending on which field theory you have, you can have different fields. You know the story. If you don't know just every mind that there are set of observable, whose eigenvalues span all the... Hilbert space of the system. And in fact, I will denote pi x the eigenvalues and with the root of x pi x the eigenvector. If you don't understand this notation, just think for one second to the spin one-half. The observable is sigma z, the eigenvalues are plus or minus one, one for each side, plus, minus, plus, minus, minus, minus. This is a possible eigenstate. So this is what these three guys mean. Pi. Pi as operator. Pi as eigenstate and pi as state. There is a set of observable, which is complete in the sense that all these eigenvalues span the Hilbert space, local Hilbert space in one lattice side, and then the generic state of the theory can be written as a superposition of the state that have the values possibility in each place. So this is completely trivial for the case of the spin one-half, but in the most general case can be something extremely complicated. So are you with me with this notation that's the most important thing when you understand the notation, it's easier. So it's x, because this will become continuous. This is x, space direction. Let's say that lattice spacing is a, that at the end of the day will go to zero. So you have your x that is 1a, 2a, 3a, 4a, et cetera, et cetera, et cetera. Exactly the i of before, but now since we want to make the continuous limit, we just measure everything in units of a, and at some point we will make this limit. Actually we will make the limit and we will not even realize that we make the limit because we're writing so nicely that the limit will be automatic. It's clear, tensor product over x of phi phi is the eigenvalue of the position x which means this guy here. OK, when I write the state is plus, minus, I, it means that it's the tensor product of the plus at site a1, plus at site a2. OK, what does mean this notation? This notation means a tensor product. We are so used to this notation that we don't think anymore about it. This notation here stands for this guy. Every time that I write a formula when you don't understand it, think in your mind what will be back for the spin chain and then, OK, you just understand that this is a generalization to an arbitrary thing. If you still don't understand, ask me. OK, we want to work in field theory. So, we'll say that our system has Hamiltonian H to which it corresponds to a cladian Lagrangian that I will denote by L e. I will work in imaginary time. That's why in a cladian space. OK, and you know how to get from the Hamiltonian and the Lagrangian. You just write the Lagrangian transform. A cladian action S e, which is equal to the integral between zero and tau in d tau of the Lagrangian. Zero and tau, zero in beta. Because we will be at temperature beta and then send beta to infinity. OK, so beta is the inverse temperature equal one over t. And in the case of infinity the integral is just up to infinity as usual. OK, so in this field theory we should write what is written in the reduced density matrix. So, before taking the reduced density matrix let's write the density matrix in field theory. The density matrix at finite temperature is rho equal to minus beta h divided by z partition function to have it normalized to one. As you know, it's nothing to do with field theory. So let's write this object in this pat integral formula in a pat integral formulation in field theory. Questions? No, it's OK. Just ask even stupid questions. I'm very happy. You know what is the Lagrangian. OK, this is usually in real time. T goes in IT and this is equilibrium. Yes. So how everywhere is imaginary time equal IT in real time. As you well know if to beta here you put IT you get time evolution operator and everything at finite temperature correspond to imaginary time. OK, that's I didn't stress too much this because I told most of you know, but just to be clear on the notation let's repeat it. OK, you see that if you put IT instead of beta you get evolution in real time. So if you want to work at finite temperature usually what you do is to work in imaginary time. And that's the approach that is usually followed in field theory, but not all in field theory. OK, thank you for the question. At least clarify something. Other questions? OK, so to understand what is this density matrix in our formulation density matrix tautology we write that density matrix in our basis, in our notation. OK, so let's write the element of the density matrix between two states of our basis. The states of our basis are identified by this phi. OK, so let's let me write in this way product over x1 to say one point phi of x1 this is the first this is one element and this is product over x2 phi of x2 OK, and this is clearly the expectation value on the state what I'm writing now is tautology but it's good to understand what will go on divide the partition function. OK, what I'm doing is that I'm just telling that the element of my operator is just the operator evaluated on the states that I'm interested in. OK? It's really a tautology but better to rewrite it. And now what is this object? It's already written there but what is in a pati integral formulation? This is nothing but let me first draw and then explain and you can. I take a strip of width beta divided by z every time. I take a strip of width beta and on top I put the state let's say phi2 of x2 and now I put phi1 of in each point. OK, so look at this expression. You start from phi1 at x1 for example at tau equal to 0 you let it evolve for imagine in this pati integral formulation here is space and here it's time horizontal line is space vertical line is time you start from a given configuration of phi1 of x1 for the spin chain plus minus plus minus plus minus here phi1 of x1 whatever they are you let it evolve for imagine time in beta that's what is written here and then you make the ket with phi2 of x2 after that time so you start from some configuration you evolve for some time and then you impose that should arrive after this time which would be another configuration plus minus plus minus in your spin chain so this is the formula this is what you will write in pati integral for the same thing if you have something that you don't understand in this thing just ask now there is not much to understand just to be familiar with the concept but if you are not familiar can be complicated so tell me what makes you uncomfortable I will try to make to explain it to you you all have seen pati integral in one way or another yes who never saw pati integral in his life so more or less you understand this notation to understand it a bit better let's see something very stupid what happen if I take in this notation the trace of rho what is the trace of rho it must be 1 that's how we can see it so this is the element of the reduced derivative taking the trace means that we have to take the diagonal element and sum over them correct? what are the diagonal elements in this notation one of you that answers so 5 1 is equal to 5 2 that means diagonal so this is the diagonal element and then I have to sum over them and in in a plot what does it mean that I should take this strip identify the point on the opposite direction and then sum over them so it means I make a pati integral over on that on that as well and what I get if I do this operation I take a strip I close it it's a cylinder of circumference beta so this is the standard construction that I think at least 80% of you heard that the partition function at fat temperature is the pati integral on a cylinder already heard this story rise ends don't be shy I don't know how you prove it probably in this way or probably you went through Matsubara that is something a bit crazy but this is the easiest way to see the partition function on the cylinder the partition function is the pati integral on the cylinder in fact the trace of rho is 1 so it means that z is equal to the partition function on the cylinder so this is the partition function now we are interested in the reduced density matrix rho a if you understood what does it mean taking the trace for making this plot it should be now automatic to understand what happened to rho a what I should do I should make a let's say for simplicity a is an interval let's denote by u v between the point u v at distance l u minus v equal l the interval between two points u and v and their distance itself I have my rho where there are all the points on this rho I can identify where is a a is here to get rho a what I should do I should make the trace only over b which is outside a so it means that I should take this strip close it to form a cylinder but then sum over only the part b and leave a unsummed so if we do this I can get this graphical part integral which is just a cylinder always of circumference beta where I sum everywhere but here is a between u and v and here I am not summing up there are still steps to show that there is a there is a cut here let's use another color which is nicer there is a cut here where on top I still have phi 1 of x1 which is an index of the matrix and here I have still phi q of x2 which were the plus and minus of the spin g to make an analogy with what we have done before that's why I am doing I have done this story I just erased the part row A for the tensor network we wrote in this way where we left open indices on the part A and we contracted the part B the analogy here is just obvious we took the trace over the part B so we closed the part integral we don't contract indexes but we closed the part integral so that we leave open indices there on the part A so I hope the analogy that's analogy the equivalence of the two things should be clear and this is the main reason why I made the two re-matic proto-state because I guess that many of you are familiar with that but not familiar with filtering obviously here it's missing still the normalization if you want a formula for this guy this is the part integral over all the field configuration in imaginary time over the Euclidean action as usual where you impose the boundary condition right like that product over x prime delta of phi x prime at t equal to zero should be equal to what we call phi 1 of x1 while so we take the part integral over all the field configuration where the field has the constraint that imaginary time equals to zero here should be equal to phi 1 then I let imaginary time beta pass rather than the other end of the object and I impose the condition that phi takes phi time in imaginary time beta is equal to phi 2 and this is the formula in part integral that gives this plot the only thing that matters is this plot but this plot the formula means this I should probably write it bigger this x prime is a dummy indices so it doesn't matter if it's x prime, y, z call as you want but what I want is just at any point imaginary time equal to zero must be equal to phi 1 for any point in A is where the cut is, in B you are summing over all the configuration you are taking the trace which means you are doing the part integral B becomes like all the other points here it's A that still has an open cut so x prime belonging to A and reminding me this times product over x second belonging to A of delta phi x second beta minus phi q thank you for the question because I forgot to write that x belongs to A and that's very important this is the element of the reduced density matrix then when you write for each element you get the entire reduced density matrix you want expectation value product over x prime phi 1 of x prime rho A product over x second to be as clear as possible here there is the part the normalization and here there is the normalization thank you do you have questions? I think this is the right moment to have a break because you can think the break usually is 10 minutes let's make people happy we make a 15 minute break but for 5 minutes you don't go out of the room and think to this formula if you don't understand something you have the question ready for aft because if you don't get this point and what is the reduced density matrix it's really useless that I continue so I give you 5 minutes more break who understood just have fun who didn't understand thinks to this formula and make proper question where we reconvene 10-15 ok? who goes? student no, he doesn't then we spend the first conference with him did we play? so everyone is in place did you think to questions? yes what you mean to prove what it means that this notation think what it's here ok? you are just saying that obviously you are in imaginary time but imagine this the pat integral ok? so the integral over e to the minus ac provide you your state that in nps is not written but it's just this guy you can imagine this is true for any state but we are at final temperature we are at zero temperature ok? it means that you should take the imaginary time evolution for infinite time ok? and so instead of having the cylinder you just have asked plane top to give you the state so it's an infinite cylinder ok? for beta that goes to infinity this plot for the density matrix become a plane where you have phi1 here and phi2 here ok? the cylinder becomes a plane ok? because what you do you do imaginary time evolution for infinite time and you project over the ground state ok? if you take e to the minus beta h on an arbitrary state in the limit beta that goes to infinity this project on the ground state ok? with some normalization ok? it's clear this ok? so this is true for any state not for spin1R so the ground state is given by applying the imaginary time operation for infinite time now this is the ground state then I have to take the density matrix but I make exactly what is written here I close on one side and I leave this state on the other side ok? so if I this object is written here is the expression plus minus plus minus minus minus whatever you want ok? ground state begins this is what it is the density matrix at zero temperature and another state here minus minus minus plus and these two states here so it's just not that to prove anything you just have to rewrite this formula in terms of spin1R what I'm writing here you see what I did these two are identical object yes sure sure this is an infinite system infinite in which sorry which infinite the state are infinite yes and then in the general case also the fields can have whatever configuration you want just plus and minus what I denote here by phi1 of x prime and phi second of x second are just configuration of your system in that position and the configuration depends on the system for a spin1R phi1 and phi2 can just be plus or minus nothing else if you have a spin1 can be plus 1, 0 and minus if you have a bosonic field would be the local mode occupation within 0 and infinity and then there are many other circumstances the notation here is true for an arbitrary theory you can specify to spin1R in the ground state and finding exactly the construction we have written before but thank you it was a clarificatory point long story in general you cannot write a fundamental state with an MPS you can use MPO matrix product operator for the expert it's just trivial for the other I'm just making random words that detach from the main object of the lecture so if you want to discuss we can discuss it but it's definitely not important I won't especially a simple question like the one before was quite clarificatory so don't be ashamed that it's stupid the most stupid it is the best it is if you don't tell anything it means that you understood everything and you will not be allowed to say to ask things in the future about this like in a wedding talk now or be silent forever could be that what you don't understand will be clearer while making something else ask even after if something will be not clear ok so we arrived to reduce density matrix this is the expectation how to write the expectation value on a state on a basis of the reduced density matrix in patinte and now we have to use this expression to get the entangomen entropy I remind you first of all that the entangomen entropy is given by first trace of rho a log rho a evaluating this logarit ok for an operator it's complicated thing to do as we already said yesterday so what we will do we will consider the moments trace of rho a to the n yesterday we say that on that side we consider consider trace of rho a to the n for n integer this will be easier ok and then we will analytically continue arbitrary real value if we manage what this means will be clearer during the course and then the entangomen entropy minus trace of rho a log rho a can be written as minus the derivative with respect to n of trace of rho a to the n or equivalently as the limit the limit ok this is for n that goes to 1 or equivalently as the limit for n that goes to 1 of the reny entropy that we wrote yesterday as 1 minus n of trace of rho a to the n so this is we are not able to compute this object directly ok because to get the logarithm of an operator what you should do you should diagonalize first the operator, find the again values and then put back in the original basis but this will be too complicated ok no one knows how to do this ok and this maybe if some of you is familiar with theory of disorder system this is exactly the same story when you make the replica trick to get the average partition function that is logarithm of rho and not the rho or rho and that's why in that case you send n to 0 because you don't have this rho here here instead we have trace of rho or rho and that's why we send n to 1 ok this is just a parenthesis for the people that know disorder system if you don't know not important ok but it's exactly the same logic calculating logarithm of operator is complicated and so you just calculate the moments ok and then try to make an analytic continuation if you manage because it's not always easy now how we calculate trace of rho a to the n from what we have written should be almost evident to all of you ok if you understood this what is rho a and if you understood how to make the multiplication of matrices like it's written here for example for an NPS you just put in your mind together the tuning and you should see it automatically I'm just waiting for you to see it and ok if you didn't see yet let's for example make trace of rho a cube ok which means that we have 3 of this cylinder to have more space ok so I have 3 of this infinite cylinder each of them is one of the rho a this cylinder have this open cuts on a on this side I have phi 1 on this side I have phi 2 which are my row and column index ok here let's say I have phi 3 here I have phi 4 here I have phi 5 and here I have phi 6 ok imagine just how you will write in normal trace of rho a to the N you will write in terms of the indices sum over i j k rho a i j rho a j k rho a k i ok so you contract rho and column sequentially and then you close the trace the indices in this notation are just this field phi 1, phi 2, phi 3 etcetera so what you should do you should identify phi 2 with phi 3 and sum over them which is equivalent to sum over j here you should identify phi 4 with phi 3 and finally identify phi 1 with phi 6 ok and notice the analogy with that row that is still there and that's why I draw it ok now you should imagine again you make trace of rho a cube and there are no left over indices and things so just everything that is on top of this cut is identified with the bottom of this cut everything that is on the other side instead is identified with the other and this close cyclically in this way in field theory of this guy is exactly the same t, ok that's why I didn't erase all the time this guy to have it available and actually to make it let me redraw it in the case of beta that goes in field so now we take we go to the ground state and we take the in field plane I have three planes with these open cuts and now this guy is identified with the top here the down of this is identified with the top here and the bottom of this guy goes back cyclically so what does it mean that we have a pat integral that is defined on this surface what does it mean on this surface usually if I move continuously I just move like that in this plane but what does this convention let's say that I move at fixed x in imaginary time that is easier if I move at fixed x in imaginary time on the b part of the system I always remain on this plane nothing happen let's see instead what happen if I move at fixed x on the part a of the system I start from x negative I arrive to the cut here but when I arrive to the cut the bottom of this cut is connected with the top of here I don't go out on the same plane but I go out on this other plane so this blue line make this gay then I continue I arrive to infinity I reemerge from here if you wish let's just do another color if I continue if I move on this side and arrive to the cut I will reemerge here so continuously this point here and finally if I move here I reemerge here if you want continuity equation on this field is such that phi of x belonging to a let's phi of x and tau equal 0 minus it's phi of x belonging to the sheet 1 x belonging a in sheet 1 must be equal to phi of x tau equal 0 plus but with x belonging to a in sheet 2 so when you move here you don't end up here like in B part that seems to use something completely crazy did you ever see this thing in your life who has seen this thing and nothing felt here who has seen it at second year of university or third year maybe someone could have seen it in high school in high school in Russia we have seen it in high school for sure in Italy we have seen it in third year of university for sure in English even before where do you have seen this strange construction I hear Riemann sheet that's the right answer so I think any of you studied the square root you all know the square root especially you know the square root in complex number square root you know from 12 years old square root on the complex plane a bit later but when you study the square root you say let's make square root of z you say that on the complex plane the square root of z there's a cut here a branch cut that's exactly what we have there and actually the square root is not defined on the plane but is defined on a two-sheeted Riemann surface which means that you have two planes and when you go around the branch point when you arrive here exactly as I said before you don't go out on this plane but you go out here on the second sheet of the Riemann surface and you have to turn two times around the branch point in order to come to the original pose this is n equal to for the square root we are just in suite that you have to turn twice here you have to turn three times around the branch point in order to come back to the original place which means that is not the square root to the plot but is z to the one third and anyhow for any n trace of ray to the n to the Riemann surface of the power one over a ok, so it's not it's not rocket science that you can see in the beginning it's something that you already know in a more complicated language is it clear? yes I took the beta equal infinity limit for a cylinder obviously there is beta in this plot I put beta equal to infinity to have things easier this was beta and here I took the limit which I mean I'm on the ground state thank you for the question and ok these kind of questions are very much appreciated no, no, it's not similarity these are branch cuts these objects these things that I wrote here is a Riemann surface it's not a similarity it's a three-sheeted Riemann surface in fact let me continue here what we obtain is that trace of ray to the n except the z that get lost everywhere there is one over z to the n here and there is one over z to the n here trace of ray to the n is equal to let's write like that then n depending on a divided z 1 to the a z1 without a to the n ok, where z n of a is the partition function of the field theory in an n-sheeted Riemann surface with branch cut in the interval uv we are I read it if you don't read trace of ray to the n is equal to the ratio of zn depending on a which means the interval uv divide z1 to the n which means the partition function on the plane to the power n because each of this cylinder brings one z the partition function was one over z cylinder so we have n of them so there is one over z to the n so zn of a that appear in the numerator is the partition function of the quantum field theory in an n-sheeted Riemann surface with branch cut on the interval a that is equal to uv questions? yes, or do you talk loudly or I come up? ok, these are trivial modifications you should just put some defect line somewhere and things can be done but ok, it's really a detour that will not be fundamental for understanding you can take into account whatever open boundary condition many things of this kind it's just a small modification of this there are some insertion of operator somewhere ok, but it's easy that's what we will do friday to get something that's exactly the way to make calculation then yes, or? ok, everything I wrote here as I wrote at the beginning is by the arbitrary dimension the only thing that you should take care that the surface, what you get in higher dimension is not called Riemann surface just as not a name because there are no complex numbers etc but there is the same thing object connected on surfaces in a strange way perfectly done this is used in many circumstances it's not called Riemann surface but actually we can decide to call higher dimension on Riemann surface and we are ok apart from the name, everything is the same you don't have the nice interpretation of the complex plane because in higher dimension ok, you don't have a complex plane apart from that it's valid basically with this construction what we show is that trace of rho a to the n that's what we want it's a partition function on some strange surface ok, you agree on this if you didn't make question it means that you agree on this calculating partition function is something that is in many circumstances especially in field theory but also for lattice model you can think of making the same thing in fact we had this lattice spacing at the beginning that in the course of the writing it just disappeared because it doesn't matter too much but this is valid in general our trace of rho a to the n is a partition function on an n sheeted Riemann surface and calculating partition function is much easier than calculating an object like trace of rho a log rho a that you need the operatoral knowledge of the density matrix instead in this case you don't need anything just need to calculate the partition function ok, so this is a real major simplification ok so if you don't have question I will move to introduce another axillary concept that in some cases is very useful and maybe it helps before making any calculation it helps digesting what this formula means ok, and the concept I want to introduce is the one of the twist field it's a strange name that goes back to the 80s when it was introduced in string theory and it reappears here because what string theories do is to calculate partition function of strange Riemann surfaces ok, that should be maybe you have heard about this story they have Kalabiya, many fold that are very complicated, we have just a Riemann surface but the same kind of structure emerge so if you want we are making applied string theory so the partition function on the n sheeted Riemann surface is a of a where a again is just in case u v the interval this object we say this is a partition function that means I can write over integral over all the field configuration phi of x and tau where now this field lives on this n sheeted Riemann surface that let me denote by Rn ok so with this notation I mean that I should consider all the field configuration that live on this Riemann surface Rn which is that surface over there so the field does not live on the plane like before but live on the Riemann surface and then e to the minus integral over this Riemann surface of dx d tau lagrangian density of the field phi at x and tau ok and local because of the construction that we made the lagrangian density is the same as our original model ok on the plane we had a model on the plane we didn't change model in making this thing we just calculate the partition function of the very same model on this strange surface so lagrangian density is exactly the same as before ok so this is the same that we wrote the only thing that is complicated is that all x and t all x and tau x and tau as couple belongs to this n sheeted Riemann surface ok and that's what is your colleague here said you can map this to the to put the branch cut in the in the real negative axis you can do this but then you need to know what happened to this out lagrangian transform in general this is very complicated but there is one case where under conformal transformation things transform very well that are the conformal field theory ok this is what we will do during the next last lecture to see in one case you can really play with this object to get an answer in general it's much more complicated because it's not natural object don't transform nicely under conformal transformation ok so this is what we want to calculate and as written in this formula I repeat we have a field only one field phi that leaves on this strange surface ok now for for understanding what we are doing and for many application also it's useful to think to divide that partition function ok to think this partition function here as an expectation over n different fields one of which leaves on a single sheet of the Riemann surface ok we can do this ok we can take single field where x goes from minus infinity to plus infinity and tau goes on 3 different sheets ok let's think of having n field we have phi 1 ok let's try to rewrite this in terms of n different fields phi 1 of x tau phi 2 of x tau phi n phi n of x tau each of them ok where x and tau now we live on the complex play it's just exactly the same thing instead of having one field that goes from one place to another I want different fields each one leaves on one of the sheet of the Riemann surface this field cannot do whatever they want they should satisfy if I want to map that object to this plot they should satisfy this condition so they should satisfy some condition to map that and our partition function that n a as we noted it will be equal to the path integral over d phi 1 is living on the complex plane d phi 2 living on the complex plane up to d phi n on the complex plane time e to the minus integral over dx d tau always on the complex plane lagrangian of phi 1 plus the lagrangian of phi 2 et cetera et cetera lagrangian of phi 1 lagrangian of phi 2 are always the same as before the only thing is that this this path integral here now is a constraint that I will call c u v because depends on the interval a that is defined between u and v and the constraint is the following c u v is the constraint that phi i of x and 0 plus is equal to phi i plus 1 x of 0 minus if x belong to a where i is defined more than so that the last one is equal to the first if you think I'm just rewriting I didn't make any I just change basis I wrote except to the same object but instead of a single field in terms of n fields but these n fields cannot do whatever they want they have to satisfy this constraint in order to be in order to be equivalent to the Riemann surface question because we are going to finish soon so appear here i, I just say is defined mod n which means that n plus 1 is even easier n plus 1 is equal to 1 that's what it means because obviously when I arrive to n here I get phi of n plus 1 it's c-click it means 1 thank you for the question because if you don't understand these obviously you don't understand anything we didn't get yet there it's the last 5 minutes will be that but first make question we are almost there notice that the total Lagrangian that appear here which we can call ln which depends on all the fields phi 1 to phi n is nothing but the sum of the individual Lagrangian that are all the same so it's Lagrangian not additive it means that even energy is additive because it's just a general transform of this guy also the energy is just additive between the values surfaces so it seems we have been playing and doing nothing but we have done something because now this Zn as it's written here is not defined anymore on the Riemann surface but is defined on the plane then in this way is defined on the plane but is it a partition function of the theory on the plane the answer is no because this will be trivial the partition function of the theory on the plane the partition function of the system this constraint makes the things non-trivial and this constraint up and all at two point u and v so it's something on the plane where something then is defined on the plane in which there is some non-trivial action the non-trivial action is at u and v something non-trivial happens only there usually you call this thing this is a correlation function on the plane we usually write this thing as tau and u tau tilde and v on the complex plane u and v we are on the complex plane tau and u and tau and v tau and are called twist field and they are defined by this relation don't ask me what are the twist field we can make thing a bit more precise and constructive and rigorous from the mathematical point of view but it's not that we are aiming in this lecture because it will be too complicated but to understand that here the action happens only at two points what is special about this function is that the constraint is imposed at some point and if you think what is a correlation function is when you put something at one point usually you put an operator now instead of the operator I put a constraint this constraint defines me an operator a field that I call twist why it's called twist it's a long story it will take us away but the important thing that I want you to understand is that by rewriting this object in terms of the copies of the field defined on the complex plane you end up in an object that is identical to a correlation function on the plane of some object that are defined through this equation and actually as every time in filteratical language there never exist only the correlation function of the operator so if one wants to be regular this is completely fine maybe you have this story in filteratical operator don't really exist exist only the correlation function of the operator this is even this twist field can have many nice properties but we are not interested in the mirror we are just interested in the rewriting of the partition function ok, so let's write the final story I just rewrite so I finish and then we make all the question the partition function on the end sheet and the reman surface is equivalent the correlation function of some field that we call twist fields of twist fields of twist fields in a theory in a replicated theory in the sense that now we have n fields ok, so I re-read the sentence because maybe I wrote too fast the partition function on the end sheet and reman surface which is trace of rho e to dn is equivalent to the correlation function of twist fields in a replicated theory that's the message of this twist field it's equivalent by construction and by construction we define the correlation function of these fields ok, now the lecture of today is finished so go on with all the questions twist field is something else it's not a field of the theory it's something defined to satisfy that relation ok you can write if you wish and this is quite complicated you know field theory ok so you can write an operator product expansion for two of these guys to get one of the regular fields of the theory this you can do but the field itself is not directly related to it so it's like in any OP you have the OP that becomes other fields doesn't mean that the field itself is the other field so the OP of this tau can be written in terms of the field of theory but the field itself is not a field of the theory that's the for the people that understand what is an OP that's a very general statement ok the people that don't know OP just ignore the sentence sorry I didn't hear tilda is just what's happening in U is different from what's happening in V ok, because it's two different fields it's like field and anti field if you want and this is the usual notation this is not the same field in U and V because what's happening in U is different from what's happening in V yes there is but it's too complicated to explain here there is a relation it's really the inverse field in some sense but ok, this will take me what what I've written here shows you that this object should be written as a correlation function of two fields that in general are different obviously then in order for this correlation function to be non-zero it means that this object should have the same scaling dimension and some other things but ok just for the sake of this lecture I can write this this define the two fields these two fields should be in some relation that for the moment stay unspecified good question imagine you are on this line and then you have U here and V here now let's explain we answer two questions at the same time so since you ask you didn't realize that is the same question as your project but it is the same question so when you are in B the field is just continuous so there is no phi you arrive to U so you move along X now you arrive to the point U at the point U you meet this tau and U tau and U tells you that when you now move along this line field phi i here should be identified with field phi i plus 1 always verse I don't remember which notation I use but it doesn't matter so this is tau and then after it doesn't happen anything it continues to be true until you arrive at V when you get tau tilde and V which makes the opposite operation and tells you so since it's opposite basically annihilate the action of this guy and after everything is continuous again so in this sense tau and V tau tilde and V is the inverse of this guy and that's why it's applied only in U and V you move along this line and you see that the field here is continuous the first time that you made this twist field you go up in you change the field from i to i plus 1 so the action of tau tilde is to change i to i plus 1 so what tau tilde and U does is to change phi i is a den of x 0 plus is equal to phi i plus 1 x 0 minus or vice versa this doesn't matter too much for for all x larger than U is a tau n tilde of whatever of V makes phi i x 0 plus to be equal to phi i minus 1 x 0 minus for all x smaller larger than V so if you are to a point here you add in the back both tau n and tau n tilde this brought you up this brought you down on 1 and in total you stay in the same place clear why it's only in U and V clear why tau n and tau n tilde are 1 for you it's clear now why they are the opposite of the other so I answer two questions in one and ok we are happy I can do it I said the branch point in phi imagine the square root I said the branch point in phi it means ok probably this trace of rho to the n will be just goes to 0 obviously clear next lecture will happen but ok but there is nothing pathological in doing that ok just no no no I think the best thing to do you can come to me apart from that the best thing to do is that you I introduced several difficult concepts so think to that if you want we can start Friday lecture with the question that you can have write down and I'm very happy to answer that ok