 z Profesor Calabresem. Zato, da se je, da se prišli, posledaj se na dnevno, v tom, da je vsega tudi, vsega tudi, prišli. Čakaj, da se prišli. Čakaj, da se prišli, vsega tudi, da se prišli, da se prišli, da se prišli, Došliš imaš všeč nekaj prav, kako smo vsega vsega lečenja? Zelo nekaj vsega, nekaj vsega? Tak. To je vsega lečenja. V nekaj prav, nekaj prav je nekaj prav. Vsega. Vsega. Vsega. Vsega. Vsega. Vsega. Vsega. Vsega. Vsega. Vsega. Vsega. So it's a point. Ok, it's always points. And so the action of this twist field, of this twist object is local and you can call them field. But if you are in higher dimension, like in 2D, ok, you have your system, you have a subsystem in 2D, you see the entanglement surface, which means the boundary between one, ok, the boundary between A and B is now a one-dimensional object. And the equivalent of a twist field, ok, would be not a field, but a string. Ok, so in 1D you have a twist string. In higher dimension you have twist whatever you want. The difference is that while fields or local operators are well-behaved objects and we know how to deal with them, this object that has some higher dimensional manifold are just weird objects that we don't know what to do. Ok, so this concept is very fruitful in 1 plus 1 dimension, but it's instead not really useful in higher dimension even if there are many people that are trying, that are working in some, in the direction of taking some meeting out of this idea, but for the moment there is really nothing. Ok, maybe some of you in the future will do something in this direction. It's still an open problem how to extract useful information by the concept of twist higher dimensional object. Ok, other questions? Ok, if there are no questions, I will go on with the lecture. The goal of this last lecture, for me this hour and the hour after are the same lecture, is to provide you one example of the calculation to make happy your friend there, where we can work out the calculation of the entanglement entropy up to the end, and the example that I will discuss is the calculation of the entanglement entropy in a conformal field theory. I'm very, but I'm pretty sure that within this room not all of you are familiar with CFT. So who knows conformal field theory here? Wow, I would have expected even less. Ok, so anyhow for all the other people that are not familiar, I prepared a very short introduction to conformal field theory, after an hour just to fix the basic concept, everything will be very basic and not neither rigorous, neither precise, but it's mainly done to fix the names and concept that we can use. So for the people that know it, don't take it too deeply. For the people that don't know conformal field theory this will be just an introduction, not even an introduction, just a collection of concepts that should be done much more precisely and rigorous if you won't really to understand and use this thing. But still, I guess that can be useful. So a conformal field theory is a quantum field theory, QFT, that is conformal invariant. Conformal invariant means that it's invariant under the conformal transformation and the conformal transformation are those space-time transformations, which means like x go in x prime, which is a function of x. It's a space-time transformation, where the vector x is tau x1 xd, where d is the space dimension, and we work in equity and time just to have easy metric. So the conformal transformation of the space-time, which conserve the angles, that's the way. So you map your space-time in other space-time, but the angle should be locally the same, like I could deform a square in something like that, where all these angles are locally 90 degree, but the overall geometry is different. This is a standard conformal transformation and I don't want to write down even what it is. So, what are usually the conformal transformations? Obvious conformal transformations include translation for sure. If you translate your system, you don't change the angle. Rotation, if you rotate the space-time angle, don't change. Kail transformation or dilatation, if you just change your scale of the system, angles don't change. And it is almost everything, actually. What is missing is just something called special conformal transformation, which is something I don't want even to write, it's a special transformation, but basically it's a combination of rotation and dilatation, so nothing really special. These are the set of transformations that generically conserve the angle. So, if your theory is invariant under this transformation, it's a conformal field theory in arbitrary dimension. This is in any d. By the way, every time that I will use small d, for me it means space dimension, capital D is space-time dimension, so it's d plus one. Just to have a standard symbol that we don't mess up. But something special happens in two-dimension. It's a space-time dimension. Qd is special. Why is it special? Because all the morphic functions are conformal. What I mean is that if you write, if instead of using the coordinate tau and x, you move to z and z bar, which is equal to x plus i t and x minus i t. So, just a change of space-time variable, name. Any function, any change of coordinate z, because in w equal f of z, where with f allomorphic function, any of this transformation is conformal. And this is trivial. You studied in complex analysis. You studied in complex analysis, that because of Riemann-Koši theorem, which means that the derivative with dx of the function is equal to derivative with respect to y, et cetera, the angles of transformation, the angles are conserved. If you don't remember, maybe you don't remember how to prove it, but you may remember the thing, that the cross derivative are equal. And this is exactly the only condition that is required to preserve the angles. But what are the consequences of these invariants? The algebra formed by the allomorphic function, the algebra, allomorphic, is infinite dimension. You can expand your identity function in power series. You can see each z to the n as a generator of your algebra and you have infinite of them. These are independent objects, because you cannot write z cubed in terms of z squared. If you want to define analytic function, you have infinite parameter, all the coefficients of the series. So you have infinite many of this transformation, not just a rotation. Usually it's a three parameter family of three dimension. In two dimension will be only one parameter family. Instead, all analytic functions are infinite dimensional family. But if the algebra is infinite dimensional, it means that there should be something really big behind it. In fact, this is more or less equivalent to the fact that these theories are exactly solvable. Yes, they are integrable and even more, actually. Definitely they are integrable. If you consider a non-neuritary conformable theory, even what does it mean integrable, it's not clear. If it's not unitary, so that's not going. This could be, but let's not go in that direction. Unitary conformable theory are obviously integrable, because you have an infinite number of constraints, actually even more than what you need. So sometimes people refer to integrable. Conformable theory is super integrable, because they are really nice. That you can solve them. That you can solve everything about them. Solve. I have a theory, whatever theory you have. These are interacting theories, so there are not three theories. A theory is completely specified, for example, by all correlation functions. You know very well that if I have a generic five to the fourth theory, not that I can write all correlation functions. You are lucky if you write perturbative expansion but you point function and this is not even XR. What I'm telling you in this case, you can in principle reconstruct all the correlation function of the theory. In some cases that's been done. It's just more complicated things you want, more work you have to do, but you know the way you. People know the way how to deal with this problem. So you can reconstruct all correlation function. These things that I just told you were known from many years, but is it of any use for real words this set of conformal transformations relevant to something? That's the first question you should ask. And there is a very old result due to Poliakov 73 which showed that under very mild assumption, there are some assumptions but they are met by almost everything. If a system, if a theory, if a QFT is invariant under rotation, translation, dilatation, it is automatically conformal invariant. It's quite very strong argument. This is in arbitrary dimension in any D but this is particularly important in 2D. So you only need to have something that is invariant under rotation, dilatation and translation. Automatically you generate, this is not really to understand it, but just trust me, all these much bigger invariants. If you think in 2D translation are just two objects, rotation is just one and make three, dilatation is another operation, so it's four in total. So you need invariant under only a four-dimensional algebra and you generate by dynamical symmetry an infidimensional algebra. The expert knows it's not difficult, it would take 20 minutes to show how it works, but we have so much time, but probably all the people that raised the end before know how to do it and it's quite a remarkable result. In those days this became suddenly something extremely important. In 73 you can realize it's also the same time of renormalization group and this happened exactly at the right time when scaling variance, dilatation, was understood to be the main characteristic of critical system, system that are in a gapless phase or undergoes a quantum phase transition. Dilatation, scaling variance, is equivalent to critical system or if you want, which is the same as gapless points and phases. So if you have a critical point in one special dimension or in two or in three special dimension or phases like Latvian liquid, this automatically is conformal invariance. With the caveat that you should keep in mind, the system should be also invariant under rotation and translation. The fact that it is invariant under rotation in spacetime means that x and tau should go in the same way. So if you remember you must have this notation invariance in a kledijan spacetime, notation invariance implies z equal to 1. Remember I defined this critical exponent last time which means basically that it should be a relativistic field theory. It should not be a gailean field theory with different. Clearly for quantum many body system is important representative of this conformal field theory of Latvian liquid, Latvian and Fermi liquid in one day. So this includes CFT, includes Latvian. By the way, for your information are just the simplest CFT you can have. The other one are more complicated. This is just the simplest example of CFT is the free boson, which means Latvian and Fermi liquid. Do you have questions? I mean not the people that raised the end before that know really about CFT, but the other one. Fermi liquid is a special case of a Latvian liquid first of all. And anyhow it's a free bosonic theory. You can just use bosonization if you ever heard about this concept. You can use bosonization to map the Fermi liquid into a Latvian liquid. Into a free boson theory with the Latvian liquid parameter k equal to 1. It's a special case of a Latvian liquid. Yeah, this is 1 plus 1 d. Was exactly that the point. Yes, CFTs are except the free boson and the free Majorana. It's equal to one half. All the other cases are interacting in conformal field theory. So in four dimension. Conformal field theory is not exactly solvable in four dimension. But you may have heard, for example in recent time, there was a huge activity on the bootstrap program. People got huge money, huge prizes, et cetera, because they rediscovered the bootstrap, if you ever heard this name, it's just the practical implementation of conformal field theory in higher dimension. Since the algebra is not in free dimension, you should use this finite dimensional algebra to constrain your theory and take out information. As much information as you can. It's not in free so you cannot really solve exactly, but you can get very close to the solution. This is the bootstrap program. If you ever heard about this bootstrap, that's exactly what it's about. The bootstrap program starts using even in particle physics. It's a very fashionable thing. Actually, Polyakov made all these things for particle physics. Then there was the bootstrap program started in the 70s, thanks to Polyakov and many other people, including people in Italy. Then it got blocked for a long while. In 2D was solved by better in Polyakov and some logic ones and everything. Then was in higher dimension where it stayed silent and sleeping up to like five years ago when Slavaricikov and others really discovered the bootstrap and understood that a few more things can be done with the tools that we have in this program nowadays and they are using it now. No, no, no. This is a CFT and one thing that changes surely from the 70s is the computational power that we have nowadays. They can use the same equation on the computer much easier than to get something, but the only fact that you can put on your computer and get some initial solution allows to discover a lot of relations that just by ends would have been very difficult. I think the fact that there are computers allows to have a big theoretical understanding also. It's not a computational thing, it's really a theoretical story that went on. This is definitely much more complicated than the question. E is the conjugate variable of time. K is the conjugate variable of space. If x and t should scale in the same way, their conjugate variable should scale in the same way. If x goes like tau squared, this is not a convolutional invariance. This will give a different convolutional invariance. I need that x should scale like tau, which implies automatically that e should scale like k. Thank you for the question, because this is the kind of question that there is a difference in making the calculation. Yes, in the end of the day, no. You can make both approaches. You get to slightly different things. If you then make a quick rotation, you get to the other. It's much more intuitive to work in a gradient space. But yes, for many things, it's important to work in a state of Lorentzian signature. In fact, you can find in the literature both that they are both very important. Absolutely not. Any massive field theory usually eats invariant under rotation and translation, but it's not you need scale invariance. If not, let's see something that scale invariance and conformal field theory provide us. First of all, let's start with scale invariant. If you have a scale invariant theory, you all know basic concept of normalization group, but you know that each field, phi j under a scale transformation of factor b, will just go to b to the minus xj phi of r, where this xj is called the scaling dimension. This is what scaling invariant means. The various fields under a scale transformation are invariant, which means they get a factor out. They are not really invariant. They are covariant. They get a factor. And this factor defines the scaling dimension of the field. This is probably you all know by conformal field theory and by normalization group. This object is connected to critical exponent, et cetera, et cetera, et cetera. I assume that all of you are familiar with this. In particular, this equation implied that the two-point function of the field must be equal to b to the minus x2xj phi j r, phi j r prime. And you see that the only solution of this equation is that the two-point function must be an homogeneous function of r minus rj, which means that this implies that the two-point function of ij r, ij r prime must be proportional to 1 over r minus r prime to the qxj. You can obviously plug this form into here and see that this satisfies, but you can even easily convince yourself that this is the only one that satisfies this object. I already used, you know that using translation and rotation invariant, the two-point function should be a function of the absolute value of the distance. And this absolute value of the distance must be a power if it should satisfy this function. This is basic of normalization group. Probably you all know there is nothing new. There is something new happen when we promote a scale transformation to a conformal transformation. Let's focus, let's stay in 1 plus 1 d without making it. And we wonder what happened for under a general conformal transformation that goes in w equal f of z, because we say f is an anamorphic function. First of all, you can see f of z locally, which means just close to one point, is a combination, translation, rotation, dilatation. As you see, for example, from this picture, this point here went to this point here and to go from here to here, you made a small translation, a rotation and a bit of dilatation. If you remember what were the conformal transformation, you know that you can prove. And by the way, this fact is at the basis of the theorem of Polyakov. You use the fact that locally a conformal transformation is always that stuff, and then you reconstruct the global transformation and you end up that is invariant under that. And this is a parenthesis that is not important. But locally, this operation is that. So you know very well that if I apply a rotation and a translation, not much change to the correlation function, because it just changed r minus r prime, so it's nothing, what changed something is the dilatation. It's a local dilatation. So one would expect that the local scale factor is b of z, it's the derivative of f. The local scale factor is just the derivative of the function. The first thing that one should wonder is whether I can, that's where everything starts, everything conformal field theory, if I can promote this global transformation to the local one, if I can say that since this transformation is a combination of the three, may I tell that phi of w of z is actually the local scale factor f prime of z to the minus xj, I'm taking the same convention, yes, times phi of z. Is it true? So this would be a first guess. You say locally is just that, so I imagine that locally only the dilatation, the scaling variant transformation matter and there is not a global effect. This is the first guess. What people have proved is that there are some operators, some fields, for which this is true, for some field this is true and they are called primaries. This is a very important category of fields and one thing that we will show in today, I don't know if now or later, is that primary field are primaries and then we will use all the nice properties, primaries and the most important property of primaries is this one, the equivalent of this one, of this one, so if this formula is true it's trivial to say that phi of w of z1, phi of w of z2 must be equal f prime of z1, f prime of z2 xj times phi of z1 of z2. If this equation is valid, then automatically follow that the two point function should transform under conformal transformation in that way. This seems something very trivial and easy, but will be extremely useful. Just before, just to give you some idea and example of what you can do with this very simple object, imagine you have a two point function on the plane of some operator on the plane z. You can map the plane to the cylinder with the conformal transformation that is just the logarithm. Think a bit about it. And so from the result of the two point function in the ground state you get automatically the two point function at final temperature. If a theory is not conformal invariant getting final temperature, it's a mess. It's a lot of work. In a conformal invariant theory it's just trivial. If you have the two point function by the way it's written there the two point function on the plane the two point function of a primary operator at final temperature is just I can write you what happened under this any two point function at final T will be just 1 over beta is the inverse temperature pi hyperbolic sin beta pi L over beta minus 2xj L is the distance. Let's write x1 minus x2 This is equal time two point function. You can get even different time correlation function without problem. In a conformal field theory final temperature correlation are trivial because you can use this very simple relation. This is one example of the power of CFT. If you ever try to calculate a final temperature correlation function you know how difficult it is in a generic theory. You can see the difference in action here. Having all these invariants and all this freedom in playing with transformation allows you to get so many results that you cannot even imagine now. Yes? This is an example I can write. I don't have it written here so I could have done even some My memory could be wrong but I don't think so. Beta divided by pi hyperbolic sin pi x1 minus x2 divided by beta to the power 2x minus no, it's already down so 2xj to power. You can find this in any book and in notes in everything about CFT. There is an infinite quantity in CFT. They are complicated. The algebra of the CFT is called Virasoro-argebra. There are some generators of this algebra that are called Virasoro-generator and some proper combination of the Virasoro-operator are the charges. This will take The amyhtonian will be L0 plus L0 bar that are two-virasoro. This is the easiest Virasoro-operator associated with the allomorphic part and then you can construct many of them you have all these generators that are infinite from L0 to L1, L2, L3, L4 each of them correspond to the power series of this guy. That's the idea and in fact minus L0 bar is the momentum because you see that the constant in the among the allomorphic functions that are invariant the constant correspond to a translation if you add to a function something you are just translating the function in space time and the generator of translation in space time are called amyhtonian and momentum. Z correspond to the rotation and this is that the strange transformation the strange invariance and to each invariance correspond to a quantity as usually a constant density this is the logic writing formula wouldn't really take too long yes they don't form a basis in the sense of linear algebra but the other field are the sentence in the sense that it can be obtained by application of this what you get by applying a operator to one primary field is an independent field linearly independent field but it's okay you can just get through this operation so in this sense they are not a basis but from them to the application of the operator you can generate all the algebra and actually since at some point if you want to impose that your theory is unitary but in algebra you impose some relation that some are linearly independent like if you get to order 3 when you start applying some of this you have to you have some equation that constrain your theory and makes that actually the number of operator that you have in your theory is numerable and not too many instead of non-unitary theory there are not these constraints and things get much more complicated but this is a very good point because from them you can generate the other but not linearly in another way yes there are a set of exactly like for scaling variance only also for scaling variance there are local scaling operators that have this property in the same way first of all the same operator that satisfies this for a scaling variance organization group so I'm giving for granted the wrong organization group and this kind of ideas the same scaling operator of your scaling variant theory are the one that can be eventually promote to primary and not all of them will be in fact now this morning lecture will finish with an example of a non-scaling of a scaling operator that is not primary so this is not at all automatic you expect wrong and I don't see what is much strong first of all this relation in a constructive filter it doesn't mean anything but if you accept this relation the correlation function is just a trigger consequence of this relation I take this object for two different point, apply guinea and they get this so if you accept this the relation of the operator for the correlation function is just a trigger consequence as you see this and actually since in a constructive way of thinking filter this relation doesn't mean anything because you cannot redefine the operator in any tracking theory this relation in any book everywhere means that it should be seen as when introduced in any correlation function this relation holds this is the definition but not this relation is valid only when inserted in a correlation function because outside of a correlation function make very little sense of saying in a field theory because of all divergence, et cetera what this means I don't see why you don't expect this because it's really if you say that you don't accept this okay we can start discussing but if you accept this the two point function is just a trigger consequence of the operator I wrote the two because I'm interested in the two point function but the 20 point function is just I put 20 of them and I have 20 derivative give me your example and I will tell you why it doesn't have anything to do with this it's probably the best thing to do it's a scaling variant theory distance doesn't matter anything you don't have to think to your microscopic model of spin to get close in a scaling variant theory distance doesn't exist distance one of distance 300 this is what it means scale invariance you scale infinite time and distance not all then if you are thinking to apply this to a spin chain where there is a lattice spacing and by scaling, scaling, scaling you arrive to lattice spacing obviously this relation doesn't hold at some point you will break down this theory is not good it's because the field theory doesn't describe your microscopic model in that regime you have to apply this formula to the microscopic model where it's valid don't if this is what you had in mind then this is the answer let's see if in the remaining seven minutes I want to write down a specific example which will be important which we will use in the afternoon on primary field ok because one of the most important operator ok, which is scale invariant a scale invariant operator is the energy momentum tensor ok often in conspatter lecture is called stress tensor I hope you just you know what we are talking about in QD after in one plus one dimension resistance of T mu nu T mu nu is a q by q matrix with space and time coordinate ok, in a CFT only components non-zero which are t, t, t and txx which sometimes are called t00 ok, I'm just writing to make you you are saying this thing just to join with what you can already know without defining them ok and ok, txt and ttx are 0 by scale invariant, so you can easily convince yourself ok, don't let me go through this and this object are basically, ok, are function of x and t x and t are just the density of energy and momentum, ok that's what they mean that's why it's called energy momentum tensor they didn't give a name by some craziness ok like the Hamiltonian of your system is just the integral in dx of t00 of x that's what I mean t00 and momentum is the same with t11 now I'm putting indices in random places t00 and momentum that's what they mean the theory is scale invariant and so you can write a concept quantity as integral of density which is the essence of another theorem ok, and these are the only two that is not mixed term because we are scale invariant so it's easier in this case in two-dimensional conformal theory it's convenient to rewrite t00 and t11 in terms of this coordinate as usual z that is x plus psi tau and x minus psi tau and so the two independent component of t mu nu are t of z, denoted as t of z and t bar of z bar ok which are we just come from the one is t00 plus t11 and the other one is t00 minus t11 and you can show by making this combination that this linear combination the sum of the two is just a function of anomophic function of z ok, while the other one is anomophic function of z bar that are called t and t bar usually this object is a function of x and t and the function of x and t is a function of z bar, not of the two separately ok, if you make the algebra don't let me do this, you find in any book ok, you find that stress energy and that's why all conformal theory is nice this t of z is anomophic function of z and t bar of z bar is anomophic function of z bar and they are called usually in fact anomophic component of the stress energy dense how t of z transforms does, let's use proper English t of z transforms under conformal transformation question the answer is not very easy and was discovered by Belavin-Polyakov enzomologico in D84 in a seminal paper in this equation is the equation that started conformal t of z t of z under conformal transformation as usual z that goes in w equal f of z transform in the following way there is a term w prime of z square times t of w and this just tell you that the stress energy tensor is a scaling field with dimension 2 it's scales, it's a scaling field but there is an anomaly and that's the important piece c over 12 Varsian derivative of w with respect to z and Varsian derivative is some object introduced by the famous mathematician many years before which reads like that I don't want to show where it comes from but okay it's a combination of the first, the second and the third derivative of the function as you see notice that if the function is linear if the function is linear Varsian derivative is zero because w third is zero and w second is zero so for linear function there is not this term here and I already told you that translation rotation and dilatation are linear functions okay so for for a scale transformation this term is not there so that t is a scaling field as it should be but as soon as I promote to a global conformal transformation there is an anomaly term that is important which has this form that has been calculated again a better problem as I'm allergic of it's difficult but it's not you don't have to worry how to derive this the interested people can just take any book or lecture notes on CFT and this is where things start okay and okay it tells you that it is anomalous and this quantity c is extremely important this is called central charge and the central charge is the most important quantity of a conformal filter it's what distinguish okay almost one conformal filter is from another there is also something else but this is the most important thing like criticalizing model conformal filter is central charge equal to one half glating a liquid is central charge equal to one three state pots model is central charge equal to seven over ten and so on, so forth it's a very important quantity that when you know it you know what you are talking about and it enters there this is the if you want this one of the possible definition the central charge even operatively not very interesting that okay it's a possible one and basically what we will do in next hour we will use this relation to make your friend happy and calculate the entanglement entropy not generic conformal filter this relation is enough and then if we have time we will make some generalization by using this relation here so the the only thing of conformal filter that you need to know this one and that one but I made this one hour lecture just to give you an idea of what we are talking about I could have written this two-formal and go on but that would have been too rough so go to have lunch me too and see you in two hours