 So, first of all I thank the organizer for giving me the opportunity to talk. Not in Triestry, because that's quite normal. But to this audience that's quite different from the audience which I am talking about this thing. And indeed it is been asked me to give an overview of what is known, general, about entangohen entropy, entangohen in many body systems, from the subtitle that I put, is from Field Theory from Condes Matter. z bistvenim prijevama, kaj počutim, ki v tudi obriga na zelo, ne znovu, da sam sem tukaj. Zelo nedaj ta k 마ča, ne znovu je. To je forno k mentalna začeliš, prav Queen prej ne bo, ki nije vse prijevama, da sporo laje napravo, ki počutim pa, da je, da jefasto. Na vse jest do vrne, tudi, da bomo izgledati, da imamo spetanje, včasno, v Vincenco Alba, Maritio Fagotti in Derek Tone. Čak je to? Čak je to? Včasno, da bomo izgledati, da imamo načine, ki so nekaj, da in tudi, in tango in entrobi, zelo vsega vsega vsega. Vsega vsega vsega vsega vsega vsega vsega, zelo vsega vsega vsega, ki lahko se predetilo s kosmologijami, pa nekako se vidijo kosmologije, zato si je zelo v izgledu, da je mi je prišlišil. Svetkoz, inkubija je to koncept, to je zelo, zemlji, informacije sa začeli, nekaj na konceptu, zelo, butj, kar je tako preprine na zelo energijovacij, teoretico, začeli, vrati, v tezboj, in tajto, je to zelo umožil vsega, kaj, da je tudi verejdovno dobro, či bomo, in naj različ z Zeiro v vednih, z nekaj dobro, ki so, selo, o 11, 13, 0, 4 bil mi malo. Zelo je zelo obrobilo V 220 in 2015, in postupa v 2016 so dobro,hire that is about 500. I will try to motivate here, why has all this is interesting in comets matter in all this fields which motivated also the a very big workshop at KTP two years ago, and even as you know are ten million grans of Simon Foundation like the one for woodstrap we had yesterday, and I will start the story really from the zero point, ki najbolj zelo, je, da, se ki počke kot zelo, naredimo počke, da nosimo všetko, izgleda počke, da ne všetko ne več, tudi počke počke, ‎kej Fermeon, bozon in bi pristop, da je tako zelo, zelo, da bozon na metalj, pri Fermeon, v superfluidu, počke bozon. Ne všetko všeč, je všeč superfluid, in ki je kako vzelo. Svega vzela je, da je bilo vzelo. Kako je učinila, kako pošla vzelo, da je bilo vzelo. Je to vzelo. Nesi je bilo vzelo, kako je vzelo, da je bilo vzelo. Svega vzela je, da je bilo vzelo. Spodnije, da je bilo našli. Da je bilo vzelo. so many different phases and so many different things in the world, among which we can... I made a list of things here, which goes from typical statistical physics object, like multinsulate or topological state, or high energy things, like confined phases in quantum chromodynamics, of which we had a lot in this talk. And the reason why I think this talk, even not talking about the normalization group at all, in this conference, is because all these very strange phases of matter of the interacting system have been tackled with the normalization group very successfully. The normalization group nowadays is one of the best tools to understand at least part of this thing. And what I will try to tell during this talk is that entanglement entropy can tell a lot again about all the different phases of matter in a way that I will enter in the next few slides. So, what is the problem in describing this interacting system? One cool thing just... OK, you have the Hamiltonian. We are not talking about using strong interaction or weak interaction. We just... Basically, with QED we could describe everything that we were interested in that slide. Actually, much less than QED. We just, for example, would like to know what happened to a system of spin that interacts. And still, we are not able to do this. One very easy approach would be just put on the computer, make your simulation and try to get results. But this is not possible, because the problem is too complex. Imagine that you have a very stupid problem in one of a spin chain. That means a chain where on each side there is a spin. If I want to describe a state of the system, I should write in the basis of the spin, which is, let's say, it's one-half spin, so it's just plus or minus one, the value of the spin. Plus or minus one-half, OK, I'll just plus or minus. But the basis, if I have n spin, obviously I need to specify two n coefficients in order to specify a state in the Hilbert space. And two to the n coefficient is a big number, when n start being already of the order of magnitude of those n's. Now, there is the most powerful computer that can make a sub-diagonization, which means calculate the value of this huge matrix for something like 30, 35 spins, not more than that, on a computer, not on this small computer here, on a big cluster, on the largest computer on Earth, we can at most manage something like 35 spin and 35 spin are very far from thermodynamic limits. One dimension 35 could be also already something, but you understand that in three dimension 35 it means nothing. It's three times three times three makes 27. It's too little to be stored on a computer. OK, so we cannot tackle this problem numerically. And it's clear that if we want to gain something in this game, what we could try to search is to find a criterion that said physical states apart from the other. This is a new thing that emerged in the last ten years, probably was not clear before, that not all this two to the n coefficient states are really important to describe a physical state. They describe states of the Hilbert space that will never be interested in. Physical state, in a sense that I will discuss later, could be different from the other. OK, and we want to understand what makes physical state different from the other. And I will try to push forward the idea that entanglement is, at least, it's a criterion, maybe it's not the only one, but it's a criterion that puts physical state different from all the other states of Hilbert space in which we will never be interested in. Let's see first of all what is this entanglement entropy. OK, now that I gave you an introduction, I'm still introducing you, just defining quantity, but for the people that never saw the quantity before, it's better to give proper introduction. So let's consider a quantum state, where it is, which means I mean a pure state psi. OK, from this pure state, I can construct, obviously, density matrix as the projector on this pure state. And I consider the Hilbert space where the system leaves, and I bipartite it, which means I divide in two parts. OK, as the tensor product of HA times HB, usually in quantum information language people say that there are two guys, one called Alice, one called Bob, which one has access to some degrees of freedom, the other one has access to other degrees of freedom, and they can make measurement only on one part of the system, but this is not at all important. OK, and in a general state, as we already know, just from basic quantum mechanics, the state shared between Alice and Bob is entangled, which means that the measurement can depend on what the other one does. This is encoded in this midde composition, which tells us that for each state there exist two bases, one in A, one in B, that are not just by psi here. OK, such that the state can be written as a superposition of these bases, with only one set of coefficient, a set of coefficient here C, and that depends only on one index, and not on two, like would be for an arbitrary basis. OK, the statement is not trivial, but it's quite easy theorem of linear algebra, probably most of you have seen in the first theorem of linear algebra, but OK, if not just, it's a theorem of 19th century, so just believe it. And, obviously, it's much easier to see that this coefficient can be chosen by rotating a phase to be positive, and there some must be one, because the initial state we were talking about is pure, just for normalization of the state. Now, what is the entanglement? If the state is a product state, it's not entangled. If a state is a product state, it means that there is only one coefficient that is equal to one and all the other are zero. This is a non entangled state. A state is entangled and being different from one. OK, so there is, for example, there could be a bell pairs with two states having one over square root of two. This is something that we have studied, but more coefficients in the submitted composition appear, more the state is entangled. OK, but we would like, being physicists, we would like to have a measurement that say not just yes or no, we would like this is more entangled than that. Intuition would say, for example, clearly this state is completely unentangled, but if all the coefficients are non zero and they are only equal, this would be a maximum entangled state with some intuition. So more the state is spread among all the parties and more the state is entangled. It's an intuitive picture, but we would like to have one number that measures this. Now, in quantum information theory, many years have been and are still working out measurement of entanglement. OK, so object that can measure the entanglement. There is nowadays a quite precise definition of a measurement of entanglement which is called entanglement monotone because obviously you cannot measure the entanglement in meter or in seconds. You cannot have an absolute measurement of entanglement. An entanglement monotone is an object that says if given an illber space and a big partition one state is more entangled than another. This in this sense is monotone. This can be the statement can be made rigorous, but what they say is that entanglement is a quantity cannot reuse under local operation in classical communication which translates in rigorous terms to quantity that is able to say if one state is more entangled than another. So it's a comparison measure. In this sense is a monotone. And there are many entanglement monotone in the literature. One that is very good and that I will consider probably is the only one that I will consider in this talk probably depending on where I will arrive is the entanglement entropy. Entanglement entropy is a quite natural object. It's defined in the following way. Let's take the reduced density matrix of the part A of the system. So by taking the trace over the B degrees of freedom this means is the reduced density matrix that Alice can access. So this is whatever Alice can do. And excluding whatever Bob can do. This is probably something that you have seen doing in the first course of statistical mechanics when you say in a subsystem and a bat take the trace over the bat. We are doing the same thing, but now I am not saying anywhere that the bat is much larger than the system and I am not saying anywhere. I have no other assumption. I trace out something that is of freedom. I define the entanglement entropy as the fornojman entropy corresponding to this reduced density matrix. Since you can write explicit expression of this entanglement entropy in the smith basis that we wrote up here and you can see just by linear algebra, probably you see immediately that in this basis the entanglement entropy is the expression minus cn squared, long cn squared. So cn squared are like probability of being in a state and obviously the fornojman entropy is given by this expression. But ok there are very few very important properties. This is the expression in smith basis that is important, but from the definition it is clear that entanglement entropy is basis independent. And this is something that as a physicist we like a lot from the quantum information perspective from the information perspective as a basis dependent something that they don't care, but many entanglement measurements in fact are on the basis, but as a physicist we like something that you don't need to go in one basis to calculate, but it is the same whatever basis I go and this we like. Ok, and there is another very important property that you can see going to the smith basis that depending on your cn squared you have that either you take the trace of A and make the over B and make the fornojman entropy or you take the trace over A defining some row B and making the fornojman entropy of B you will get exactly the same result because this object is completely symmetrical in A and B in the smith composition basis. So you have that the entanglement entropy of A is equal to the entanglement entropy of B independently of their volume. These two objects have very different volume, but they have the same value. So this entropy is not like the classical entropy we are talking about, that is an extensive object this cannot be extensive. And this was already at the basis of some totes in the 90s. In fact, based on the fact that it should be the same one can reason in a very bad way, it's not a good reasoning but the only thing that where is the point that A and B are sharing is just the area between them. So the idea that came in mind to people that start now to be correct but that should be taken really with a grain of salt is that the entanglement entropy would be proportional to the area that separates A and B. This is true grand state of a local Hamiltonian. Nowadays this was pointed out by Shredkin in 1993, probably even before by Bambelli et al. in the context of black hole entropy. And it's proved nowadays to some it's rigorously proved in one dimension it's proved in physical but not rigorously that it's it's true even in higher dimension that entanglement entropy satisfies an area low if the Hamiltonian is local and it's gapped. These are the conditions. In fact first this idea was proposed in 1993 and a year after to find a counter example that's what we all like to do in this conjecture. Also in our friend Wilczek calculated exactly what happened to the entanglement entropy in a one plus one dimension conformal field theory that will be a good part of my talk here showing that the entanglement entropy is equal to C over three logarithm of value. Who are this quantity entanglement here? In a one dimensional system you can be part of the system choosing as A one interval of a given length and B the rest. It's just a number, it's two. It's the number of boundary points between the system. If the area low will be satisfied entanglement entropy should not depend on L for large enough L. And this is in fact true for gapped system as it has been proved by many people and then proved rigorously by asking back in 2007. Instead for conformal field theory which is not gapped or in a more field theoretical language which is massless, so for a massive theory you have an area low. For a massless theory you don't have an area low and in conformal field theory you have these log L divergences with subsystem size and the prefectoria that you see C is the central charge of the theory that people in the odds I want just to comment without entering in any details that nowadays in this formula is the best, most effective way to our disposal of all physicists to measure the central charge. If you have a theory and you don't know the central charge just calculate this quantity which is very easy there will be no numerical problem in the sense that I want to discuss here but if you are someone that are interested in this theory don't try other methods that were popular in the past like skilling of the gap I just measure the entanglement entropy you will get the result. Why this is important now? I want just to make this to close the story of the introduction and then entering in something more technical like showing from how that formula C over 3 log L can be get. The importance is that if I imagine now this full Hilbert space with 2 to the end state ok and I pick up randomly a state whatever this means, this sentence I would expect on the basis of physical intuition that I will get a state that satisfies a volume low not an area low Now this statement that I am telling and that each of you if think properly should say yes it is correct it has been proved rigorously by a mathematical physicist obviously what you have to define so define a measure in the Hilbert space what does that mean taken a state rigorously obviously with some measurement this works very well but this you can make you can make the statement that I am telling rigorously ok and what it is that you have this huge 2 to the end state that is full of states that are volume low and only have small corner in this space satisfy the area low so if I am interested in knowing what is the ground state of a physical system which is what we are interested in knowing for example to solve the problem of ITC superconductivity we want to know if the ground state of the 2 dimensional above model can have this superconductor or not we are not interested in all the spectrum of the superconductor ok that will be the at least the first starting point to make this thing if I am interested in this I should not scan the full Hilbert space with my numerical code 2 to the end state I can just reduce to a very tiny fraction ok now if we can find a way to numerically reduce for example the search in this tiny fraction of state we are in we are in good shape to solve the problem and in fact one meaning of the entanglement entropy from information perspective is exactly that the entanglement entropy gives the amount of classical information I need to store to specify the state on a classical machine like this laptop ok if a state has very high entanglement entropy I need a lot of memory in my computer if a state has very low entanglement entropy like if it is a product state I just give to to give two elements of the basis ok so this is quite intuitive but also this information perspective can be done can be made more serious than the word I am telling ok so you understand why all the circle closes that being the entanglement entropy related to the amount of information I need to store classically a quantum system obviously if the entanglement entropy is low for a physical state that I am interested in I can try to make an algorithm that is low state this is not just a dream nowadays there are these algorithms based on tetson network which is a very new and powerful set new it is not so new nowadays but probably to many people it is not known yet numerical based on the entanglement contents of system ok that carry very strange name like MPS which means math explore state entanglement error my position project entanglement state so it is just a lot of acronyms here in fact so will such as say that we are in front of an alphabet super proposals ok but independently of what this name says these are states which naturally encode these encode the area low in one dimension it is the only one that I will try to explain so these object encode the divergence of the entanglement entropy from the structure you see 3 is like the normalization 3 these object encode the peps the area low in 2D so it is good to make two dimensional system because of their shape and ok they are quite good people are using actually the one dimensional we are using back from 92 without knowing it in fact these mathical states have been understood only later in 2000 something that this famous density matrix organization group that is a numerical argument that many of you could have heard about works because the entanglement is low and because can be written in terms of magic process state what is a magic process state? I want just to be quickly this maybe most of you are not interested but it is important to know what is going on in the world ok you write the famous coefficient that were 2 to the n I am writing randomly I write this trace of some matrices here ok one for each one for each side this one to n and one for plus and minus depending on the basis ok I didn't make anything by writing this if the matrix is big enough I am ok indeed let's assuming that the system of variance all these matrix are the same and I should just put a to the n and let's do this so I have to specify one matrix to specify the state but for a genetic state if I want to describe these matrix should be 2 to the n times 2 to the n that means I didn't gain anything ok but let's say instead of taking a 2 to the n matrix I take a matrix of dimension chi ok that's what people do chi is a number and in my matrix of chi times chi you understand that in greasing chi more and more entango can be stored in this state actually by a simple counting of states you see that the maximum entango in entropy you can store in nps like this is logarithm of chi ok so obviously if this chi is 2 to the n the maximum entango in entropy that you can store is n and this is the maximum entropy possible so you are ok you can surely define anything but the point is who makes numerical method fix chi to be some number 3, 10, 10, 20, 100 then takes this state as an ansatz like a variational ansatz minimize the energy in this multiparameter space using the same, et cetera this is more complicated which best describe the state this is what is a matrix for the state and the famous and known dmr density matrix normalization group that you could have heard is just a practical way to find the variational meaning of this problem this has been just written down in all mathematical details the two things are completely equivalent so there is a famous review by willis worldwork that is called density matrix the normalization group in the times of matrix of matrix protostate and you can find all the proper derivation of what I am writing here but now let's go to the physical state in one dimension the area is a number as we have seen before so the entango in entropy must be constant so a given matrix protostate with a finite chi can describe it so if I arrive to chi I try with chi equals 3, 4, 5, 10 at some point I will see that the energy I am calculating does not change anymore so I converge and I am at right chi to stop my simulation if the state in state is critical we have seen that it grows like log L the entango in entropy the chi should grow algebraically and having putting on a computer an algebraic growth as people in information theory say that polynomial growth of the complexity is very easy if you ever run something you know that you can do what you cannot do is an exponential but power law you can deal with with the better and better computer you can do with my stupid laptop you can make simulation up to extreme precise simulation up to system sizes of thousands without problem which will be impossible as I was telling in the most powerful computer in the world in higher dimension what happened the error law says that entango in entropy grows n to the d minus 1 so the chi that you should put is exponentially big still this is not good in fact the matrix protostate does not work but still it is a reduction of the problem before we had exponential of n to the d now we have exponential of n to d minus 1 we went down we reduced the complexity of the problem exponentially it's already something but again as I was telling if you go to this state here you can reduce even more the complexity of the problem and that's what are doing people in 2d in 3d there are still not concrete and working algorithm ok, the expert on these are working and probably in a few years I will say there will be there will be some also in this direction and by the way there is another simon foundation program exactly to finance this thing for fermions so it seems that all the money of some foundation are giving to one of the topics covered by this conference so that's good up to now it has been an introduction just to give a motivation why to do these things I hope I motivate well enough why to do things but at some point we should start doing things not only saying why and ok, I will give my own perspective on how to do things which is how we can calculate this entanglement entropy and find these nice results that are at the basis of what I have been telling plus many other developments that some of them I will try to explain some I will not have time to explain as field theorist here we are all field theorist so I am really easy to go on with what we should do is trying to define this entanglement entropy integral and that's what we did back many years ago in 2004 and ok being sure that you are all field theorist I think I can go relatively fast on this so what is the density matrix in pat integral formalism it's just the pat integral on a slab let's say at fine temperature beta it's the pat integral on a slab with a given equation action ok, the slab is with beta inverse temperature and the element of the density matrix is given just by the boundary that on the boundary of this club that define the element of the matrix that here I am calling phi 1 and phi 2 ok, so the pat integral it just means the pat integral over the clinical range with boundary condition that field is equal to phi 2 tau equal to imaginary time equal to 0 and is equal to phi 1 at imaginary time equal to beta ok, what is the effect of the trace the effect of the trace is trace means take the argument and sum over there in a continuum theory this amount to take the cylinder fold take the slab, fold to form a cylinder of circumference beta and sum over this diagonal element you may have heard you all have heard that the partition function of a system is just pat integral on a cylinder at fine temperature most of the time this is shown in filthy lecture by Matsubara frequencies are crazy difficult things this is a nice way to make things wringles but it is much easier just to go in a pat integral and make explaining what is a trace for some reason this thing is not found too much on the textbook but in some textbook is there but not in all of them it is much more intuitive than making Matsubara and it helps in understanding many other things what is the reduced density matrix in in this pat integral formula what I should do, I should take this slab I should take the trace only in part of the system which means while I am closing to this slab to form a cylinder I should sum on some part of the system the B part but I should not sum on the part A of the system because I am not tracing over there so I take this slab here I take this slab I close to form a cylinder but I leave some open cuts on the part A of the system you have the part A in this plot for some reasons I put three different intervals so it means this is B where I close to form a cylinder and this I left open this is a reduced density matrix this is the element of the reduced density matrix ok now from this density matrix I want to make the entanglement so take minus trace of rho a log rho a taking a log of an operator is some difficult object difficult operation that all the people here working with this order system should know very well that it is difficult to make the log of an operator but always in the from this order system we know that there is a way out through this to make this average of the log8 which is using a replica trick minus trace of rho a to the a minus trace of rho a log rho a is formate the same as taking the minus the limit for end that goes to one of the real thing with respect to end of trace of rho a to the end this is just a trivial algebraic expression just write on the again values of rho a and check that this object is correct ok but now what is the simplicity that when the end is integer and only when end is integer trace of rho a to the end is a simple object is a simple object because is just taking end of the cylinder that I plot before sew them together in a cyclical fashion because I have to make the trace by column matrix operation I mean making a cyclical operation ok and if I do this you will realize just by drawing that this is nothing but the partition function on an n-sheeted reman surface because I take end of this cylinder I couple them together let's take the limit that beta goes to infinity cylinder become a plane so I have end of this cylinder each sheet here is the reduced density matrix then I should identify the diagonal element the element of one matrix with the element of the next one because ok, just let's be clear the trace for example of rho a cubed is just rho i j, rho j k rho k i ok, so I should identify j with k which means that the first here should be equal to the the like rho element of this sheet should be equivalent to the column element of this one rho 1 of this should be equal to the column of this one and basically of the trace the first one should be equal to the last one ok, that's what I'm drawing here so this is trace of rho a to the n which it's a partition function and as you know calculating partition function in free theories is easy calculating operator is difficult but it's easier than calculating horrible object like this and that's why things will become easier and become even easier in a conformal field theory when a Riemann surface is a natural object a Riemann surface can be obtained by conformal mapping from the plane that's the fact I will use and this is what help us to make things even easier in conformal field theory where you can use conformal mapping to make everything you like ok, on passing and this is important that not only this replica trick gives the entanglement entropy as the finalist for Riemann entropy but just even for any integer this object trace of rho a to the n define in this way 1 divided 1 minus n log item of trace of rho a to the n are called rainy entropy of order n and these are very interesting objects for two different reasons the first reason is that they are entanglement monoton also so they are very good measurement of entanglement from the information perspective so if you are more from an information perspective you are just satisfied with n equal to 3, 4, you don't want to make this limit n equal to 1 after you are satisfied with one of them from a physical point of view from more field theory and statistical physical point of view you could prefer still the entanglement entropy ok, I just mention on passing why and the reason is that for example, the von Neumann entropy satisfies sub-subdivitivity while this object here does not satisfy sub-divitivity that's not very nice and we will see later why also we prefer the entanglement entropy but the rainy entropy is a very important thing that it became measurable just last year so it's something that can be measured in experiment which is very surprising for how old field went ok this being said, let's see this trick I have been doing the entanglement entropy defined as the replica limit of the partition function on an unsheated surface is independent of dimension theory lattice or no lattice this is just a general statement it's based just on sum stuff together as now I draw things in one dimension because I can draw it imagine you have either dimensional object connected in higher dimensional way and this you can do up to this point is valid for any system and it's used by many people for any system in most of the cases it's the only way to access the entanglement entropy then what I will do after I will start to specify to something that I know better but keep in mind up to this point this is valid for anything and there is no assumption on nothing what I want to show you is in one slide how you can get this famous silver trilogel and it's very easy actually if you know a bit of CFT what you have to do is to take this entanglement surface where is the guy this entanglement surface you try to map to the plane but this is very easy in complex analysis you can imagine that you just take this reman surface with a branch cut between u and v first of all you map the branch cut between minus infinity and zero and this you made with a simple mobius of bilinear transformation I don't know how you call this transformation here which map the w plane in this zeta plane like w minus u divide w minus v if you don't remember this from your complex analysis just notice that when w is equal to u you go to infinity and what map is take the branch cut and you put in the negative real axis then you have this surface here what does mean this surface that I start going around this branch point and every time that I cut the branch cut I go on the other sheet of the reman surface up to n times and after n times I'm back to the original point I cannot believe that there is one person in this room that didn't see this object and this is just the surface in which is defined in complex analysis the root of order n so you must have seen it if you know the square root and this object is uniformized on the complex plane just by the root of order 1 over n so I take just zeta through 1 over n and I get into a complex plane a complex plane in which each corner of opening angle 2 pi over n was one of the sheet on the original surface each of this sheet it's mapped in a corner of opening angle 2 pi over n and sheet make 2 pi ok, when you have this transformation what you have to do to get the trace of rhoe to the n is just perform this transformation for the stress energy tensor where one can use the famous formula of Bella Vipolakova examinological to relate the stress energy tensor to the plane to the stress energy tensor in this new surface one make this algebra that is extremely easy and I don't show just make first, second and third derivative of this transformation this stress energy tensor gives you the reaction to scale transformation as Slava was explaining yesterday of the system so which means this is nothing but derivative with respect to l of trace of rhoe to the n this is the partition function ok, then you can integrate this equation and extremely just elementary algebra you get the trace of rhoe to the n is equal to some constant that you cannot determine times u minus v to this power minus c over 6 and minus 1 over n c is the central charge now you put u minus v equal to l and you get that the entanglement by the replica limit is just c over 3 log n ok and this is a very easy way to get calculation if there is anyone in the audience that no conformal field theory just understood the logic of my calculation without showing details because it's extremely easy for the people that don't know conformal field theory probably even showing the details that's why I didn't write that one important thing to mention is that trace of rhoe to the n is equivalent by this construction to the two-point function of some operators which in the literature called branch point is filled and this is extremely useful for making many interesting things that I will not have the time to do nowadays this is nice but for many years I think in a very not fair way people have been saying ok but what you are doing, you not me your community is not measurable it's just uninteresting because it's not measurable I disagree on this statement but it's very interesting one of them I always say is the way function we cannot measure the way function but it's not a reason why not to calculate it and I think on the entanglement entropy there is a similar stories but it will be too long to go into all the details but what turned out last year in 2015 there was this interesting paper by the group of Marcus Grein at Harvard which measured the entanglement entropy in a cold-draffum experiment entanglement entropy as the fundamental entropy but he measured the rainy entropy for n equal to 2 and how he did that he construct the replica he made two copies of his cold-draffum system he coupled in a way to mimic what I was explaining obviously in real time it cannot make imaginary time but ok if we made this replica construction back in 2004 then some people worked out what is the equivalent in real time it's a bit of thinking with the operator but it's doable so you just make two copies identical of the same system coupled them in a proper way it can do for a relatively small number of stuff that's what he's done here and then from that measure the free energy in this system that gets the rainy entropy that he can do and he gets these results for example for this system which is a Bose Abarth model matter really much what it is Bose Abarth model obviously this was the first experiment he made the system of six pins no, here it's more than six actually this one is one, two, three, four five, six, seven, eight spins eight atoms, sorry eight sides and it be partied in several different ways ok, that's what he can do but they can do you can worry but eight is a small number and ok the important thing is a proof of concept this experiment is doable and things are scalable, oh only things are scalable so are interesting so what he did he made this experiment and what he found nice is that this system was known since ages to have a multinsulating phase and a superfluid the entanglement entropy you see that is constant here because it's increasing subsystem size because it's up, then when you pass the transition you go into a fluid phase which is a conformal theory and you see that it grows here with system size and if you put this continuous line at the theoretical numerical prediction for this small system and the points with the error bars are the experimental data for several division for several things that he did everything agrees between theory and experiment and so the entanglement entropy is measurable that's extremely interesting so it's not anymore something for us the rotation to play but it's something that can be measured and in a not too difficult way ok, since there is not much time I prefer to skip to this more difficult problem that was, what happened when you have more this joint interval it's nice to see that in this case we have Riemann surfaces with non-genous surface and this makes the calculation much more complicated you end up in objects like these that are very nice especially if some of you has to do comes or knows string theories of the early days conformal field theory on higher genus string on surface towards the everyday meals of these people and ok, what I can tell is that with this new all these, all the results of the early days of conformal field theory found an application that actually is measured, so I'm going in the wrong direction found an application that is even checkable on on your computer and nowadays probably also on the experiment ok I won't just to advertise quickly that there is a booklet of reviews that we wrote down in 2009, it's not up to date so many things up in these 7 years that it's impossible that they were all here but still there are the main theoretical tools to tackle the problem were really present at the time so if you are interested not in the growth of the result that this is probably outdated but if you are interested in how to make a replica trick and any continuation and all these kind of things you can find a series of reviews here where you can find many things and I want to mention that there have been many many further developments of these like it became a standard theory a standard way to detect criticality in general it's a way to detect topological order ok, even through the entanglement spectrum which became a very popular tool but I don't want to enter in this and there are C-theorem analogos of entanglement entropy that's a very popular thing in the energy community there is this F-theorem you can have heard in many places there is a geographic prescription for the entanglement entropy another field ok, what I wanted to discuss is two developments this one I will not have time so I will just explain to you what entanglement entropy help us to understand the non-equilibrium processes and this is very important because we are out of time what I am doing is to consider one of this many-body system out of equilibrium now, not anymore in the ground state of the system where there is an aerial law I increase the I take a system in a given state that I call psi-0 so prepare a system in a many-body state it's not an against state let this evolve according to an amytoin a different one psi-t e to the minus e ht psi-0 the question that we wonder is does it exist a stationary state in which sense exist a stationary state first of all psi-t it's obviously a pure state so we cannot have termitization but still we won't termitization we expect termitization, there should be a way to get termitization and the way out to this paradox is again through the reduced density matrix as so natural entanglement emerges because while psi-t is a time-dependent pure state the reduced density matrix rhoe of t equal to trace of b of rho t is not anymore a pure object is a mixed state and I will say that a stationary state exist if taking an infinite before the infinite volume limit for the full system and only after the infinite time limit there is a limit of this reduced density matrix that I call rhoe infinity if this limit exist we define a stationary state to exist and this can be proved in some cases to be the cases now this rhoe infinity is equal to reduced density matrix that you get from a thermal state then I will say that the system thermalize clear the logic? so it's not the full system that thermalize any subsystem thermalize so for all practical purposes the system thermalize but I don't enter in the paradox that the system is pure and cannot be thermal so any final subsystem of an infinite system is thermal what thermalization means but explaining why all this will be another story talk I want to continue to talk about entanglement entropy I make a very short long story very short telling that entanglement entropy just if you make the calculation on Riemann surface you play you make many of these things you get the entanglement entropy gross linearly up to time equal to l divided by 2 and then it became asymptotic ok so what you get is that first gross linearly and then subsystem there is this horizon effect which is ok I don't want to discuss this other thing which can be understood in terms of quasi particle picture which is very nice in the sense that the initial state emit pairs of quasi particle which move in opposite direction with the velocity vk quasi particle emitted from the same point entanglement entangled while particle emitted from different points don't know about each other there is a light cone only point separation l become entangled and even correlated when left and right movers originate from the same point reach them if there are not particles reaching this two point this two point cannot be entangled either correlated and if particle move all at the same speed entanglement and correlation are frozen afterward so that's why in conformal field theory you go back in this way you have this plateau straight because the particle move all at the same speed in a real system you have slow positive particle still arrive and don't make abrupt ok I'm done almost so so clear this argument because it's very important hopefully yes now if you understand this argument you can say first of all is it true that correlation are frozen for times smaller than that this has been measured in experiment numerical simulation that show this that was the case numerics we don't care too much nowadays we won't experiment ok there was this famous experiment by the group of manual block in Munich in 2012 where it showed that the light cone spreading of correlation was indeed correct as it's here but what does it mean for the entanglement I want to go to the entanglement not on the correlation if I assume this quasi particle picture etcetera I would say that the entanglement is proportional of the interval of length L proportional to number of pairs of particles that emitted from arbitrary points one reach A and one reach B ok in formula the entanglement is equal to the integral of all the particles emitted with some probability Fp that then from many point X in all the system minus infinity to infinity that from X travel at speed Vp one with positive and one with negative speed and one is in X prime and one in X second where X prime is in A and one is in B ok that's what I'm telling I'm putting in formula in a very easy formula what I've said even by word this integral is an integral of delta function you don't need to be very you don't need even mathematical to integrate it and get this very simple result ok which tells you that the entanglement for short time which means for T smaller the velocity if it exist the entanglement just grows linearly with time because this quantity is exactly zero then for larger time both of the integral contribute if the velocity is not all equal ok but for very large time this piece becomes zero and the start becomes extensive ok so this formula in a very easy way encode all everything I have been saying this has been checked explicitly in calculation I don't want to show I want to go to the interpretation of this interpretation is that the extensive value that we get at infinite time of for the entanglement entropy ok I have been starting by saying entanglement entropy is not thermodynamic entropy something else completely different but in a non-equilibrium process after we make the time evolution ok a time equal to infinity we get an entanglement entropy it is extensive it is extensive in the subsystem of an infinite system so we have no paradox s a equal s b I am talking about a finite system in an infinite subsystem so the other part will not be true because obviously it is infinite ok so it is the thermodynamic entropy of the mixed state just because in a thermodynamic system the entropy must be extensive by definition of thermodynamic the fundamental entropy of this trace of rho infinity is just extensive so it is the same wherever I put it this is what extensively means and should be the same of the full mixed state from which I started so when I make this non-equilibrium evolution the entanglement entropy become the the thermodynamic entropy and this is understood in more complicated situations that I want to discuss but not only in more complicated situations have been measuring experiments again this is another experiment published just not even one month ago in science by the group of Markus Greiner in Harvard where he repeated the experiment that we were talking before of coupling the replicas but in a lot of equilibrium what he is telling he makes this very intuitive picture and if this is the full system while time evolves there is entanglement created by many pairs of particles and this thing is what gives the mixedness to the subsystem that this is the picture but don't think to picture we repeat this experiment you make the replicas as you see in a very different way you make measurement and you got this you got the first the entropy increase and more or less saturates why does not perfectly saturates because obviously he has a finite system as a science system of like of 16 so you see revival effects you cannot have perfect saturation but you see that already with 16 you see this linear increase of the entropy and followed by saturation and this is very it's quite amazing that this is measured in experiments showing how thermalization entanglement so how we get thermodynamics out of entanglement this can be used even as a tool but I don't want to discuss this I will just to conclude the conclusion are just some the core messages entanglement is a very useful concept one important things and I gave only the example of the central charge because I skipped on many more complicated things but there are more complicated things and codes for example all universal properties of one dimensional many body system something that to people making RG should be very interesting ok and from the universal point of view this is also very interesting conceptually because all these is in the vacuum I only need to know the ground state of the theory I don't need to know the spectrum I don't need to calculate other things in the ground state of the theory in the universal information about the theory if you are about many body organizations this is a completely different story because there is nothing universal we are talking about universality and you just need to know the vacuum which is not vacuum at all from the conceptual point of view is plenty of information and all information is there as I was telling it's a tool to design better performing numeric algorithms this for many practical purpose will be more than in after going on it provides a very natural mechanism for thermalization which was not taught before all these ideas people are thinking about thermalization since the time of Von Neumach in 1929 he started by saying what is the thermalization of a quantum system and I should say that this mechanism despite of almost 90 years now of working on the field was not taught before and there is many other things as I was telling you that I would like to say but already now I am out of time so I just stop here