 Dobro, da se počuče. As I promised you before of the break, I will show you how to calculate the entangomen entropy in conformal field theory by using this formula here. We are considering the entangomen entropy of one interval, a between two points, u and v, and the distance between u and v is equal to l. As we already discussed at long, calculating the rays of rho a to the n in this geometry is same as the partition function on an n-sheeted Riemann surface with branch cut on the interval. And you remember it was this geometry here, where all these objects were joined in this cyclical way. We should remember what it means. It was already suggested by some of you, I don't remember the room, it was sitting here, but I don't remember the face now, that you can use conformal transformation to make this geometry nicer. But making a conformal transformation we just have seen that gives rise to an easy operation within conformal field theory like for the stress energy test, sorry, for the two point function. In general for an arbitrary field theory making a conformal transformation will not be a very useful operation. But so what we should find, we should find a conformal transformation that takes this surface, let's call it Rn, and map it to some geometry in which, for example, we know the result, for example the complex plane, and then using the conformal mapping, like this formula here, to get results in our geometry. So what is the transformation that map this surface into the complex plane? Let's just focus on one of the sheets. So this is our z plane, it's one of the sheets of the Riemann surface, then imagine there are many. The first transformation that you can make was suggested yesterday is to take this branch cut here and map where usually you have in making the roots, which means real negative axis. So you take this z plane and you map to some zeta plane where now this branch cut here, we have colors, this helps, this brown branch cut here, now it's here. You know the transformation that makes this game? You want to map q to zero and v infinity, so the transformation that makes this game is zeta equal z... In my note, I call this guy w, so let's use the same notation as I will mess up, set equal w minus u divided w minus v. And you see that with this transformation when w is equal to zero, when w is equal to u, zeta is equal to zero, while when w is equal to v, zeta is equal to minus infinity. So infinity is a single point, so whatever it is, is the same thing. So this transformation maps the interval uv in zero minus infinity in each sheet of the Riemann surface. It means now that this is one of the sheets, but there are n of them. And in this geometry it means that when I'm turning around this branch point, after I make one turn around the branch point, I'm not on the same sheet on the Riemann surface, but I'm on the shift above, for example. Then I turn around the time, and I'm on the shift above. How many times I should turn around the branch point to get back to the original plane? End times. You can see that very trivially the operation that mapped the same sheet surface in just in a stupid plane with nothing is the power of order 1 over n. By making this operation, each of the sheet of Riemann surface is a corner of opening angle, for example, q pi over n. Like this is for n equal to 6. You can imagine is something like that. So each sheet of Riemann surface is a corner of opening angle, q pi over n. You see it, no? What is happening? This is basic conformal transformation, but better, too. So overall the transformation that map our n-sheeted Riemann surface to the complex plane, I plug this formula here, is w minus u divided w minus v, everything to the power 1 over n. What I do with this formula? I follow Bellarino-Polerovitz homologico and I calculate this guy. I want to calculate, according to this formula, so I don't erase, because it's already written. I want to calculate the stress energy tensor on the Riemann surface knowing the stress energy tensor on the plane. In that way I will calculate the expectation value of the stress energy tensor. T of w on the stress energy tensor knowing the expectation value of the stress energy tensor on the plane. What is the expectation value of the stress energy tensor on the plane? OK, that's the main. It's flat means it is, it must be independent on z. It's translational invariance and rotational invariance. Is integral should be the energy density that must be finite. The only number that integrated gives finite number is zero. It's zero. So expectation value T of z in the complex plane is zero. So the only thing that matters is the Zvacian derivative. I leave you as an exercise to make the first, the second and the third derivative as written here. After I don't make here the exercise, just ask Matematica to make the three derivatives. And you get here that the anomalous term gives c over 12 not c over 24, sorry. c over 24 1 minus 1 minus n square w minus u square. OK, this is what you get. Sorry, especially soon after the light I understand that this is very difficult to read. And if you remember when we define the twist field, this formula here, the expectation value of any operator on the n-sheet driven surface can be thought as the expectation value of twist field, so of applied by the operator with the multiple Lagrange on the complex plane, Lagrange of n fields, divided by tau n and this is just the normalization. This was the definition of twist field that the insertion of the twist field in a correlation function give the expectation function, the expectation value of whatever is remaining on the n-sheet driven surface. There is one thing that we should mention. Here we inserted the stress energy tensor only in one point, but there are n-sheets. So if we want to relate to the total energy density of the theory one with the ln here, better to compute, remember that the Lagrangian was the sum of Lagrangian and the Hamiltonian was the sum of the Hamiltonian. This is, we said, I explained this on Tuesday. You may remember that this ln that appeared here was the sum from i to n of the Lagrangian of the single theory just for each field of the sheets, no? Remember the story and we said also then the total Hamiltonian should be the sum of the single Hamiltonian, no? Since the stress energy tensor is just the density of the energy, the total energy tensor, the total, which means where I insert one stress energy tensor on each sheet is just the sum of n of this guy. So the total introduced in matrix Nw in Rn is just n time this guy, so c over 24 n minus 1 over n the same formula as before. Ok, now if you know conformal field theory by looking at this formula it would come immediately to your mind even without thinking. Ok? The so called conformal word identity. How many of you know what is conformal word identity? Very few probably. Ok, 0. The conformal word identity is just the word identity for conformal field theory that is the way you construct conformal field theory and I can write you the formula and you see why it should come to your mind that if you have a primary field the expectation value of the stress energy tensor on the two point function ok, so the ratio between two point function with the insertion of the primary field and two point function itself is just given by the scaling dimension of the field times z1 minus z2 divide z minus z1 square z minus z2 This is the conformal word identity that tells you ok, what happen to a primary field when you make a conformal transformation? Ok? Because this stress energy tensor makes locality conformal transformation If you compare these two formulas Ok? If you actually, not if you compare if you look at these two formulas imagine that you knew this and now you calculate and you get this the formula is exactly the same there is no difference just written in different letters but it's exactly the same formula so this correspondence tell us that ok, this equivalence that under conformal transformation with fields like, I don't want to say equivalent they are equivalent if you want like primary field with scaling dimension delta n equal c over 24 and minus 1 over n ok, you understand the logic no, the insertion of a stress energy tensor on a core tube and function tells you, what happen to the tube and function when you make an infinitesimal a local conformal transformation ok? If the expectation this expectation value is identical to that of the primary field then under infinitesimal conformal transformation the transform is the same and then you can integrate this transformation as I didn't show you as it's known how to do to get that for any transformation you get exactly the same thing infinitesimal is the same story cannot be different globally so this twist field that we introduce to characterize the branch points are under conformal transformation they are in conformal field theory primary operators if they are primary field or primary operator they are two point function tau n u tau tilde n v that's what we want to calculate will just be equal proportional to u minus v times minus four delta n ok the four instead of the two is just because in this conformal identity delta is half of the x that I was mentioning in the previous hour ok the delta that appear in this formula is half of x that I wrote before if you want in this the delta that appear here it's x that I wrote before divided by two that's why there is Fourier this two point function was exactly trace of rho a to the n we were searching for so we have that rho a to the n is equal to some constant c n that is the constant here that we don't know but doesn't matter too much l to the minus c divided by six minus one over a this is the final result for the moments of the reduced density matrix within a conformal field theory ok the constant c n here means that is independent on l in l is all this one and then there is some constant that is not specified by the calculation ok now we calculated this for integer n what is the analytic continuation to non integer n yes exactly you just promote n to be non integer again the same formula just n doesn't matter if it's integer or not sometimes the formula for integer n inter explicitly the fact that n is integer like if you get a sum over something up to n then you should continue in some way but in this case it's just trivial the analytic continuation is the function itself just that you promote n to be a non integer number ok so the entanglement entropy will be minus the derivative with respect to n of trace of rho a to the n ok, let's make the calculation it's very simple but let's do over the other side trace of rho a to the n is equal to c n we have to take the derivative with respect to n so better to write this object as c n e to the minus c over 6 n minus 1 over n is all gal ok so that taking the derivative is easy ok by the way c n is unknown but c1 is equal to 1 ok 1, because c1 is 1 e to the minus c over 6 ok, but notice that this object for n equal to 1 is 1 ok, let's write everything c n e to the minus c over 6 n minus 1 over n rho gal times rho gal times the derivative of this guy that is c over 6 1 plus 1 over n square corrects, times minus sign somewhere no minus sign just erase and this is the the final thing ok, plus derivative of c n for the function e to the minus blah blah blah but e to the, now we should calculate this object for n equal to 1 for n equal to 1 c n is equal to 1 the exponential is equal to 1 so we have c over 3 log gal plus the derivative of this c n calculated in 1 that I denoted c1 prime so the final result is that for large gel the entangon entropy behave like c over 3 log gal and this is a result that I anticipated in the very first lecture ok, a constant but for large gel the constant is smaller than the logarithm yes before some trivial question please, 2 plus 1 dimensional is just another c1 is 1 that's what I said c1 is 1 because trace of rho is 1 please, this kind of question first then we do generalization uvika tof is it then actually when I said good question where I wrote here the tupe and function is not just proportional and actually if you want this cn to be dimensionless you put like a uvika tof here all length are measured in terms of uvika tof and you go over blah blah blah blah but you can even alternatively say that uvika tof is absorbing this non universal constant of care ok, so if you want in the final formula you can write like this where if you write in this way this c1 prime is dimensionless number if you state you don't put a this one is dimensionless but ok, doesn't matter too much choose one of the two that you prefer it's completely equivalent obviously the uvika tof give a scale to the length because length distances are not measured in neither in meter, nor in fermi there is a scale and the scale is given by uvika tof very good question more of this kind of question 3 comes from the fact that there is 12, etc. this is a long story for 12 comes from the fact 12 comes from the fact yes, now I remember it's complicated but you ask the dimension of the twist field of the stress energy tensor is 2 so the two point function of the of the stress energy tensor goes like distance of the two points to the power 4 to get to that formula to get this two point function so you get a 4 and a 3 and the product of 4 and 3 gives 12 but ok so it's a consequence of the fact that the scale the physics is a consequence of the scaling dimension of the stress energy tensor is 2 that's where it comes from then it's algebra but if the dimension of the stress energy tensor would not have been 2 but another one you would have got another number but the physics is that that 2 to get to that 12 is algebra there is no physics if you follow the derivation of this formula the only ingredient that matter is that 4 if the 2 that then becomes 4 if that would have been different you would get a completely different number if you start taking more complicated geometry like instead of one interval 2 this is a different topology then the formula that you get is a complete different mess and ok actually we have a formula for any n you can check the literature and no one is able to make the continuation so the formula is so much complicated that this operation it was you see here in the other case impossible I hope one of you will solve the problem it's an open problem you are young, fresh energy you can do it the central charge there is too many physical interpretation now depends on from which side you are looking at I am telling you some then now you will see if you like usually say that the central charge counts the number of degrees of freedom of your theory ok like central charge is equal to one corresponds to one boson and in terms of boson one free boson and in terms of how many of that you have, you have a theory but this can be even, it's not integer ok so not it must be only 2, 3, 4, 5 if you have 5 boson, central charge is 5 but like if you have a nice model central charge is one half but in fact if you think properly you can bosonize your rising model realize that it's a Majorana fermion that is half of a Dirac fermion and the Dirac fermion is equivalent to a free boson so everything fits ok this is the easy way provide the most physical way of thinking the central charge from the point of view of field theory this formula shows you that it is an anomaly if you had a classical field theory you will have just this relation but when you quantize the theory the fact that there are commutators generate this strange guy and this is what in field theory is called an anomaly generated by the quantization ok for Rafil theories this is completely physical I imagine that for a condensed matter this is not really some physical thing ok and if we have time and I think we have you will see how this counting will enter in a very nice formula no it's c1, I'm taking the derivative c1 is equal to 1 c1 is equal to 1 rho is equal to 1 let me write down the derivative of c in 1 is not 0 and it's a p-area there is something called Carlson theorem that dates back to 1913 that tells you that if you know a function in all integer plus you give some condition at infinity you have a unique continuation and condition at infinity doesn't come from the math that comes from the physics basically we we are requiring without telling that the function is not essential to write it in infinity what could make let's give an example if I have a function defined only on the integer if to this function I add sin pi n multiply by whatever function of n on the integer this function will get exactly the same number ok because pi n is but outside of the integer is different and we give in a different derivative in 1 but what this guy introduce introduce essential singularity at infinity ok so basically we are asking by the physics that there is no essential singularity at infinity all the non uniqueness that you have in a continuation can be mapped not only in this but in many variations of this essential singularity but that was a very good question and actually sometimes can happen that you write a formula that is not so easy you have a bunch of gamma function you don't understand anything you take the continuation without thinking and you get the wrong answer because that bunch of gamma function that you wrote if you make the expansion close to infinity you get an essential singularity in that case you should be very careful to get complicated objects more question here there is no reference to the unitary the point is what is the entanglement entropy from non-unitary CFT ok this tells you that the entanglement entropy of the vacuum of non-unitary CFT is that guy but as you may know if you know non-unitary CFT the vacuum of non-unitary CFT ok that's the trick more question before going to the 2 plus 1 dimension ok so what you want to know about 2 plus 1 what you want to know about the 2 plus 1 dimension absolutely not even the first part I don't know how to make the transformation I made that transform I used a conformal transformation to uniformize the entanglement surface to a plane in higher dimension there is not such thing no transformation there is nothing as I anticipated but completely different reasoning in higher dimension the in scaling variant in higher dimension sometimes you have a realo sometimes you have a realo plus logarithmic violation it's still not clear if there is a complete rationale for this ok but that's what you expect either a realo or a realo with logarithmic which is the same as in 1D you have always times log in higher dimension sometimes you don't have this log appearing if there is a reason why sometimes it's like that or like the other still unclear in higher dimension there is not one way to get all the answers like in 1D you have to make a case by case calculation and there are several calculations and no one knows yet what is universal and what is specific more questions? if not let me make the last small calculation I want to use the fact that the twist field behaves like primary fields to use this formula for its two-point function remember this formula it's still here before the break the two-point function of a primary field under a general conformal transformation ok transform in this way with the derivatives of the transformation acting as local scale factors ok I want to use now this formula to relate the results that we got on the plane to the results that we got in other geometries ok and the first geometry to consider is the finite temperature in the case of finite temperature we have a cylinder of circumference beta we have the two-point u and v where there are this twist field inserted or if you want this arranged sheet of dream as a question but we can just think as trace of rho a to the n as the two-point function of this twist field ok so we can forget that there is these n-sheeted stories just we have to calculate the two-point function of primary object on the cylinder but to do this we should just map the cylinder to the plane ok in my here I use the letter which letter z for the coordinate on the cylinder and w for the conference on the plane what is the transformation that map the cylinder to the plane is just w equal e to the pi q pi z divided by beta you see that when z is an integer multiple of beta you just get the same value ok this function is periodic x-axis like the cylinder it is the minimum of periodicity so this is the function that map one into the other you see that this function map the cylinder to the plane no or as opposite the logarit map the plane to the cylinder because it introduce a logaritmic branch cut that makes periodic now the only thing we need to calculate this option is to calculate the derivative of the exponential which is ok not so difficult just the exponential itself times q pi over beta ok so let's use the formula and get raise of rho e to the n on the cylinder will be just the derivative in the two points e to the q pi u divided by beta e to the 2 pi v divided by beta q pi over beta square ok these are the two derivatives power c over 12 n minus 1 over n ok time the two point function on the plane which we wrote as ok so I just rewrote that formula the two derivative I wrote explicitly ok and I rewrote the two point function in terms of the variable on the cylinder ok now there is something funny that I want to let you notice when you make this conformal transformation and you write as the derivative in the intermediate passage you lost the symmetry of this geometry ok like this object is translation on invariant so the final result should be just a function of u minus v not of u and v separately but the derivative in u and the derivative in v are function of u and v separately ok so in this intermediate step you lose this symmetry in this case the translation on symmetry in other case other symmetries but the final result must be in fact this trivial put this guy in and let's see what happen by putting this guy inside the power you get again the cn beta over 2 pi e to the 2 pi u minus v divided by beta minus e to the 2 pi v minus u divided by beta ok yes let's put parenthesis to the power minus c over 6 n minus 1 over n and this is explicitly a function of u minus v by the way that object there is just the hyperbolic sign so this is hyperbolic sign sorry the 2 here goes away you see when you put this guy inside this is c over 12 this was c over 6 so you subtract one of the 2 pi and you get only one sorry I messed up you don't get the 2 ok check the algebra it's not really result is correct step in the middle are correct just don't bother to understand them immediately now but then the entanglement entropy this guy is exactly of this form just that instead of having L you have the other function so the entanglement entropy also is very simple so this is just c over 3 logite of sinh pi let's write it properly so you don't read, it's a nice formula ok this is the final formula for entanglement entropy at the temperature in a CFT very simple formula this formula can take two limits first of all it's just a function of L divided by beta as you see that's the main thing that matters that doesn't matter too much if the temperature is low which means beta is i or if you want for L much smaller than beta you can expand the hyperbolic sign ok and you just get the first order pi L over beta pi and beta cancel and so for L much smaller than beta you get back the c over 3 log L plus c1 prime that we just calculated but for i of temperature instead what we get or if you want for very large subsystem size independent of the temperature ok you get an hyperbolic sign for very large argument is like an exponential correct here there is an exponential the logarithm of the exponential is the argument itself so entanglement entropy is just c over 3 pi L over beta if you want you can write like c over 3 pi L t where t is the temperature ok what it is this formula it's surely a volume low so it's extents it's very high temperature so the entropy is the entropy entanglement effect just went away there is a volume low what else so L we understand this linearity in t what tell us yeah the volume low means extents if it's synonymous if you want yes it is the thermodynamic entropy this we already said what is the thermodynamic entropy of a conformal field theory ok let's put like that which thermodynamic entropy of a mass less particle have you calculated in your life I heard the right one the first one who told it and so what is the thermodynamic the black body radiation how it goes for large temperature like t to the to the depends on the dimension ok it goes like t to the d space dimension this is the one dimensional Stefan Boltzmann this is one d Stefan Boltzmann the integral that you make for the black body radiation you make it in one d instead of making d3d you get this formula for the entropy t to the 4 is the energy that you have to make one derivative to get the entropy but someone was asking what about this c you remember that in the Stefan Boltzmann load is how many polarization as the photon how many of this particle you have what is the generation you see in this formula so in high temperature the central charge is counting you how many degrees of freedom you have and by the way if you make the calculation of Stefan Boltzmann load in one d even the pi and the 3 are all there just that guy that is the generation tube because of the polarization of the photon but that's not the small factor at the right one so let's see what it is at this point I will just leave the left 5 minutes for question without stressing you with other small calculation just ask whatever you want you are confusing me this is phenomenon entropy of the reduced density matrix of a thermal ensemble since the entropy in this regime very high temperature in this regime since the entropy is extensive taking L is large in this formula taking a large system you will get always the same this is extensivity so it doesn't matter if it's the entire system or it's a subsystem but only in this regime here when in fact entanglement that's put like that the entropy of the reduced density matrix is not a good measure of entanglement anymore here you are just measuring the entropy in this case in fact we say that very first lecture that the entanglement entropy is a good measure of entanglement for pure state so we are now calculating the entropy of the reduced density matrix but it's not a measure of entanglement anymore at t equal to 0 in the limit of L much smaller than beta you get a measure of entanglement at different temperature you get a number which has nice thermodynamic interpretation but it's not a measure of entanglement the entropy what you have been calculating for a life before don't forget entanglement entropy is anyhow an entropy it's always an entropy and with all the nice property of the entropy etc etc etc it measures the entanglement always of a pure system of bipartite pure system but it remains of a normal entropy of a density matrix anyhow yes in the idea it's ex-satellite this you need to know safety and I don't think anyone many people know but yes you need a black body a black hole in the center of your ideas to get the final temperature that was the next calculation that I would have done but I didn't do in that case the cylinder you have to make the map to a cylinder that is in this direction where now here is space x and here is imaginary time and it's infinite and you take an interval here u and v this corresponds to the same guy as before just by sending beta in I L where L is the the circumference make the substitution you get S A C over 3 log of L divided by pi sin not sin pi small L divided by bigger ok and you see very nicely this formula is periodic in L it is symmetric in L that goes in capital L minus small L which means A that goes to B should be symmetric and there is all nice property and this is this is the very important formula because in your computer you always have a finite system and that is what you will use this would be another lecture it's not a question if I would have had 4 hours probably I would have thought death but ok it's not something just to you have to understand what is a quench in CFT what is an initial state this is another finite size in fact temperature corresponds to a torus and in this case the torus has topologies as the difference from a plane so it's more complicated to make a calculation and it does not depend only on C depends also on the detail of the theory and for some theory it has been done for others in principle you know how to do but it's extremely complicated ok the calculation on the torus of the entanglement entropy exist and some are much more difficult you don't with a conformal mapping you can map only geometry with the same topology you know there is this koshityrem that says that topology of one the genus is invariant and conformal mapping the torus has an old and it's not mappable to the plane and so you have to make a calculation that are not obtainable in this easy way it's not a measure of entanglement so it cannot be linked to any quantum thing it's linked to the entropy of a classical system that's what do you mean ok don't mix up I'm using now CFTs to study one dimensional system at quantum critical point or quantum critical phases ok not a classical two dimensional system at temperature which is ok things should be adapted but it's the same stuff but don't mix up that you think they are related but not the same thing yes sure in that you should use massive field theory the formula is c over 6 log item psi ok if you won't divide by a and ok it's another calculation for a massive field theory it's we did 14 years ago there would have been another calculation that I would have done with more time but you can understand this formula is valid in the regime where the correlation length is much smaller than subsystem size no? sorry much larger than subsystem size when you don't see anymore so when you have what happen for the entanglement entropy of a system that has a final correlation length if it's infinity just grows logaritmic as we plot and then when you have a final correlation length it just saturates and this saturation value is proportional to log psi divided by a ok and how to do it ok the calculation it will require at least half an hour in conformal field theory for any conformal field theory just what was written here the transformation of this field tells you that it is a primary for a one dimension of one plus one dimension conformal field theory we have shown that it's a primary in unitary we didn't use anything of this field we just checked how it transformed it transformed like a primary full stop what you mean because a photon is a 50 a massless particle it's a massless boson you must recover not the letter I but just the photons are a CFT it's a in any dimension not in 1D you can do it in 4D it's a conformal electrodynamics without f mu nu f mu nu that one is a CFT full stop if you add the electron with the mass then it's not anymore a CFT but that's that formula it's before you follow the same but then at some point you cross over to this other regime breaking the symmetry means giving a small correlation length to the system and that's what happened ok it seems there is no more question thank you for your attention