 So what I would like to do in this talk is to discuss to what extent we can understand quantum phase transitions within the non-perturbative or exact or functional logic. So the outline of the talk will be something like this. I will start with a general introduction to quantum phase transitions. This should be understandable by anybody here in the audience, even if you're not in the field. And then I will turn to the thermodynamics near quantum critical point. And in the last part of the talk I will discuss more recent work about response functions, so correlation functions, namely the scalar susceptibility and the conductivity. So let me start with the quantum phase transitions. So quantum phase transitions are transitions which occur at zero temperature. So they are not driven by thermal fluctuations, but by quantum fluctuations and they are induced by a change in a parameter, typically in a parameter of the Hamiltonian. So experimentally this could be an external field or this could be the chemical composition in a solid and so on. So here I give you two simple examples on quantum phase transitions. The first one is the transverse field ising model. So you're probably familiar with the Hamiltonian. So you have the ising model plus a field H which is in the x direction. And when you change the ratio J over H you go from one grand state to another one. Namely you go from a prime magnetic grand state with all spins aligned in the x direction, so along the field, to a grand state which is doubly degenerate, which is ferromagnetic. And it has all spins along the z direction pointing either up or down. So this is the first example. The second example is based on the Borser-Bart model. So the Borser-Bart model is a model of bosons hopping in a lattice. There are three parameters in this model, T which is the hopping amplitude for a tunneling between sides. Mu is a chemical potential so that the mean density of bosons is fixed on average and nu is a local onsite interaction. This is the phase diagram of this model in the plane T over U and nu over U. So of course this T over U is large, the kinetic energy dominates, and then you expect Bose-Einstein condensation, so a super free grand state. This is what you see here. If T over U is small, then you see that each time the densities commensurate, for instance if you have one boson per site, you will have an insulator with one boson per site on average. This is known as a mod insulator. So what you get here is a series of mod insulating phase with different densities. So these are lattice models, but of course if you're interested in the low energy physics, and in particular in the universal physics, you can start from a quantum field theory. For instance for the transverse field ising model, this will be just the quantum 01 model, which is nothing else but a quantum 54 theory with ising symmetry. And for the Bose-Einstein model, there are actually two universality classes depending on whether the transition occurs at fixed density of bosons or not. So basically you can induce the transition either by varying the density or by varying the strength of the local interactions. And this gives rise to those two different universality class, which are the universality class of the Bose-Gas orbit of the quantum O2 model. So in the rest of the talk, I will discuss the quantum 01 model. It's a very common model in condensed matter physics. So it describes for instance cold atoms in optical lattice when n equal to, this is for both of our model I just discussed. It also describes quantum ontiform magnets when n is equal to 3. And there are many other examples. The action is quite simple. It's just the action of the phi for a theory. So here phi is an n component field. What's special here is that the field depends not only on space but also on imaginary time. This is where quantum mechanic comes in. And the only place where you have a temperature dependence is here in the integration over the imaginary time. In particular, air not here and U not are supposed to be non-thermal parameters. They do not depend on temperature because they just come from the Hamiltonian of your system. So of course you see that at zero temperature, you just recover the classical D plus 1 dimensional 01 model. We know that this model has a phase transition which is governed by the D plus 1 dimensional Wilson Fisher fixed point. So you have a diverging correlation length with exponent U. You have a diverging scale in time as well with some exponent Z times U. And of course here for obvious reason you have a Lorentz invariant action Z is equal to 1. So now what happens at finite t? So you see that the only effect of a finite t is here. So it's as if you would put your system in a box of finite size in one direction, the direction of imaginary time. And therefore you can just very simply understand the phase diagram by comparing the size of a box which is beta 1 over temperature with the correlation length in that direction at zero temperature. And this is why you get two crossover lines here. And this gives rise to three regimes that I will discuss very briefly. The first one is the quantum disordered regime. So here at zero t you have already a disordered system because of quantum fluctuations. Therefore if you add temperature on top of this it doesn't really matter and the correlation length is essentially temperature independence. The second regime is the regime here in the middle. It's called the quantum critical regime. So to get here you tune the system to the critical point and then you add temperature. So you start from a scale invariant system at finite t. Therefore there is only one energy scale which is the temperature and everything has to scale as a power law of the temperature. And in particular the correlation length will scale as one over t because that is equal to one. And finally this is this regime here the renormalized classical regime. So at finite at zero t there is long range order and so if you go to finite t you will have a disordered system because of thermal fluctuations. So here I forgot to say that I was considering a two-dimensional system and I assume that n is equal to three at least. Although it doesn't really matter. So in this regime you have just the usual physics of thermal fluctuations destroying long range order in agreement with Mermin-Legner theorem. Okay so what do we want to calculate? What do we want to understand? Of course the first thing is to understand what's it's happening at zero temperature but in the end what we really want to understand are the properties at finite temperature because in any real experiment you have a finite t. So what we do want to understand is how does the presence of a quantum critical point modify the physical properties at finite temperature. Properties that might be experimentally accessible. So one thing for instance we would like to calculate is the thermodynamics. Another thing is time dependent correlation function and of course you want this in real time so this is a real issue as you will see. In each case if you are close enough to the quantum critical point there will be scaling forms and if you have scaling form it means that essentially what you want to calculate is a scaling function fn for the pressure here. So universal scaling function which depends on n the number of components of the other parameter and another scaling function here. Well so now how do we address all these questions within the non-pertributive functional Rg? So we do what we are used to do as theorists. So we had to reaction an infrared regulator term which is essentially a mass term for the fluctuations. It is this mass is q and omega a dependent. The typical shape of this cutoff function is shown here. So if you look at momenta or frequencies larger than k essentially you do nothing. This r function is very small but at low energy if q or omega is smaller than k then you have a large which is pretty large and therefore it freezes out the fluctuations at low energy. So now the partition function is k dependent. You define as usual the scale dependent effective action which is the Legendre transform of log of z the free energy to which you add for you subtract for convenience this term but this is not very important. So we know that there is an exact equation for this function and if the initial value of k which I call lambda is large enough then you can assume that all fluctuations are frozen in the system and therefore you start from mean field theory which is defined in this context by gamma lambda being equal to the to the action. So I remind you that from the scale dependent effective action which is essentially the Gibbs free energy of the system you get not only the thermodynamics but also the correlation functions because this object is a generating functional of one particle irreducible vertices. Okay so how do we proceed to compute the thermodynamics because this will be my first point. It's not possible of course to solve exactly the exact rg equation so one possibility is to use a derivative expansion. Essentially this is an and that's which is based on the derivative expansion of the effective action. So here I show you the most general and that's the second order in a ingredient. So there are various terms there is one term which is poorly locally without derivative this is the effective potential and from the effective potential you get the pressure very easily. So you do this of course numerically because you have to solve coupled equations for the z function the y functions and the effective potential u. Once you know this effective potential numerically you can check the scaling form of a pressure and if it's satisfied you also get this universal scaling function fn. So here I should say maybe in more detail what I call delta so delta is a characteristic zero temperature energy scale so typically if you're on the disordered side you will take delta to be the excitation gap of a system at zero temperature. If you're on the ordered side then you look at the symmetric point symmetrically located with respect to the quantum critical point and you take the excitation gap at this point okay. So this is the calculation we did a few years ago with Adam Van Son and others and this is what we got for the scaling function as a function of n. So for large n it is exact we get the exact result so it's something which is non-monotonous and for small n we get something which is non-monotonous for instance the red curve here when n is equal to 2 we can do the same jump for the entropy of course and any thermodynamic function. So when we did this calculation we thought it was the only calculation for this and in some sense it was but then Adam Van Son with somewhere in the room realized that some people have done the same calculation but in a completely different context so what Adam realized is that if you use the usual quantum classical mapping which is from here then you go from this 2d quantum system at finite t to a three-dimensional system in with a finite thickness here and in this kind of situation you have a casimir effect and so I'm not going to tell you more about this because Adam's talk will be on this so he will give you much more detail but this casimir effect so this casimir force has a scaling form and it turns out that the scaling function of this casimir force which is poorly classical here is essentially the scaling function which appears in the equation of state of the quantum system so therefore we have a way to compare what we got in the quantum setting to what people have computed using essentially three-dimensional Monte Carlo simulations of these models and here you see what the comparison gives so this is for n equal one and equal to n equal three the continuous line is what we calculated with a non perturbative or g and the symbols are data from various Monte Carlo simulations so you see that the comparison works very well and so again it works very well so I should mention also that the non perturbative or g is the to the best of my knowledge is the only or g approach which allows us to get this universal scaling function I'm not going to tell you more about this because Adam will tell you more okay so now I turn to the last part of the talk correlation functions so examples would be for instance the order parameter susceptibility which is just the two point functions another example is the scalar or x susceptibility which is the correlation function of the square of the other parameter I will tell you why we we consider this object and the last example would be the conductivity which is essentially a current current correlation functions so now it turns out that computing correlation functions is much more difficult than computing thermodynamics for three reasons actually mainly two the first reason is that we are looking at the strongly interacting theory because we are interested in the vicinity of the quantum critical point actually this is not too strong a problem because we have Rg and Rg does the job quite well the first real problem is that we are usually interested in the full frequency or full momentum dependence of the correlation functions and therefore the derivative expansion is not sufficient we have to do something better and the last problem which is a problem common to many fields not only Rg actually is that we have to perform an analytical continuation from imaginary frequencies to real frequencies so I will discuss all these problems so first a yeah I will discuss only the scalar susceptibility and the conductivity I will not discuss the other parameter susceptibility so why are we interested in the scalar susceptibility so first let me briefly remind you about the difference between the longitudinal and the scalar susceptibility if you look at the dynamics of this model in a mean field picture so since you have this Mexican hat shape potential which is shown here you expect that there will be a gap less fluctuations corresponding to the Gaussian modes of the system and you also expect that there will be a gap excitation corresponding to the amplitude mode of the system and indeed if you do a mean field calculation this is what you find you find n minus 1 Gaussian bosons and you find a longitudinal propagator which has a gap now it turns out that this result the gap in the longitudinal correlation is not correct if you go beyond mean field you will see that there is coupling between longitudinal and transverse fluctuations and in particular a longitudinal fluctuations can emit two Gaussian bosons and the outcome of this is that the longitudinal propagator is completely dominated by Gaussian bosons and for these reasons it is not gap that is diverges like that in the infrared limit with this exponent 3 minus t divided by 2 so for a while people were quite pessimistic they thought there is no way to observe this to take the amplitude situations because the longitudinal propagator is dominated by goldstone mode so this is what we see but then there was a very interesting paper by but then there was a very interesting paper by Podolski and coworkers telling us that maybe we were too pessimistic too pessimistic here if we look at the different susceptibility namely the scalar susceptibility which is the correlation function of phi square we might see the amplitude the amplitude mode as a well-defined excitation so this amplitude mode is also called the x mode because there is some analogy between condensed matter here that is called physics and high energy physics even though the analogy is not perfect however nowadays it's the terminology which is used so I will also use this terminology so what we want to do is to compute this spectral function it has a scaling form here and we want to compute this universal scaling function so how to compute the scalar susceptibility so we need to compute the frequency dependence of the correlation function the derivative expansion is not sufficient but there is a nice method which was proposed by a Blaiseau Mendez and Schreborgs two of them are in the room I'm not going to tell you in detail what it is about but essentially the alley is the following if you want to compute the propagator for instance you need an equation for gamma 2 which is the inverse propagator but these couples to gamma 3 and gamma 4 and in turn gamma 3 will couples in your j equations to gamma 4 and gamma 5 and so on so you get an infinite hierarchy of equations that you have to close in some way if you want to solve this hierarchy of equations and what BMW noticed is that if you consider gamma 3 and if you put some external momenta to 0 then it's simply related to gamma 2 just by a field derivative and this allowed them to close the equations so of course you have to justify why you are allowed to put some momenta to 0 to set some momenta to 0 in the RG equations but they did this very nicely so I refer to to their work so we apply this method to compute the scalar susceptibility so the scalar susceptibility it's a four point correlation function so we cannot exactly directly apply the BMW scheme apparently so what we have to do is that we have to introduce a bilinear source h which couples to phi square and therefore the effective action will be a function of h now and then you can relate the scalar susceptibility to the vertices of the theories and now the vertices of the theories they are kind of generalized vertices because you can derive the effective action with respect to the field or with respect to the external bilinear source so therefore you have vertices gamma and n and you end up with this expression and it turns out that you can use the BMW approximation to compute these vertices it works very well and you get closed equations with the momentum dependence so now last problem analytical continuation the problem is the following what we compute is the correlation function on the imaginary axis for that is for Matsubara frequencies that what we want is the retarded part of the correlation functions that is the correlation function for real frequency and this is done formally at least by some analytical continuation from this axis to the real axis so there is a relation which is written here between chi of i omega n and the spectral function that we would like to calculate the problem is that it's very hard to invert this relation so what we do in practice we use a Pade approximate method it's a method which works very well at zero temperature and if you have data without numerical noise which is our case here so the idea is quite simple you compute numerically chi of z for a set of Matsubara frequencies and then you assume a continuous fraction expression for your correlation function so you have what you have to determine is a one up to a m so you do this easily because you know chi of z at the end point so you can rewrite this as a Pade approximate and once you have this continuous fraction expression you can easily do explicitly the analytical continuation and so I said it worked quite well there is a recent application of this method by colleagues in Paris which shows that it really works very well okay so what did we get so here I showed what we got for the spectral function of a scalar susceptibility from n power g and we compare with Monte Carlo simulation so this is n equal to to be compared here so the scale is not the same but if you rescale everything properly you will see that these two scaling functions are very close together so we have done the calculation for any values of n and the outcome is that there is a well defined in this model x resonance for n equal to you see it here also for n equal to 3 but for higher values of m the resonance is not there the coupling to Boston boson is too strong for the resonance to survive okay so one thing which is interesting is to compute what would be the Higgs mass is in high energy physics which is the position of the resonance if you scale this position by delta the zero temperature energy scale you get something which is universal and here I showed you what people have obtained in the last years so the mean field result is square root of two so it's very far away of what we finally get here I showed you our recent result using n pair g bmw I think this is the best we can do with this non perturbative Rg because bmw is quite an elaborate approximation scheme and then I show various results obtained by various groups either with Monte Carlo simulations or quantum Monte Carlo simulations or exact diagonalizations and these results were also obtained from various models one model one model on a lattice both about model for n equal to x y model for n equal to or isambert model for n equal 3 and as you see I think that there is a nice convergence of the most recent works and you see once again that the n pair g is quite a reliable here nice agreement between all these results okay so now I turn to the last part of the talk which is the calculation of the other conductivity so it's a very challenging problem in this field in condense matter understanding the conductivity in the vicinity of the quantum critical point so essentially there are two methods which have been used so far to compute the conductivity the first one is quantum Monte Carlo simulation so quantum Monte Carlo is a very powerful method to compute things in imaginary time in imaginary frequencies but then the problem is to do this analytical continuation and in particular I don't think there is any reliable method to do this when the frequency is smaller than the temperature and in particular as you know in quantum Monte Carlo simulations you have statistical noise on the data so it makes things even more difficult the second method is based on holographic models which come from this ADS CFT correspondence so it's hard to say whether this will teach us a lot but I think the main problem is that the relationship between these models and the models we are interested in condense matter is not always very clear so some people are now trying to mix both approaches so recently with Felix Hose we'll talk this afternoon we have tried to use the non perturbative RG to understand the conductivity and this is what I would like to discuss now in a few words Felix has a talk this afternoon and he will give you much more technical detail so essentially we start from the action of the ON model and we make the ON rotation this global invariance a local one by adding a gauge field so it's a non-abelian gauge field so you can write it like this with TA the generators of the S1 group we also use it's very important in this approach a regulator which is a gauge invariant by using a covariant derivative here and then we can define the current in the usual way and compute the conductivity which is essentially the current current correlation function so unfortunately it was not possible to use the BMW approximation because it turns out that the BMW approximation does not satisfy the gauge invariance of a theory and we were not able to go around this problem so what we did is that we use the derivative expansion which means that so far we are able only to compute the low energy limit of the conductivity so this is the most general expression of the effective action to second order in derivatives so you see that compared to the previous case where I discussed thermodynamics there are two new terms so you have two new functions x1 and x2 f here is the field strength and these x1 and x2 functions essentially determines the conductivity so this is what we have to calculate so here I would like to stress that this gauge field is not a dynamical gauge field it's an external field we use it just to compute the response function however we include it in the action from scratch because it helps us to satisfy gauge invariance so it's a crucial point in in this approach so what would we like to calculate so the real challenge in this field is to compute the finite t connectivity in this regime why because as I said before so here you are at the critical point and at finite temperature so there are no no well defined quasi particles in the system so it's not possible to use any type of Boltzmann like approaches equation of a transport equation so you really have to do many body calculations but this is a very long time goal I don't think we really understand what's going on here with any method so here I tell you what we do understand so this is mostly the zero temperature limit we do understand that on the disorder side the system is insulating so it behaves like a capacitance see here on this side the system is ordered and the response is essentially a super free like response so it behaves like a perfect inductance and in between exactly at the quantum critical point the conductivity is universal which means that if you measure the conductivity in units of quantum and conductivity q square over h you get a universal number so that's a no prediction by fischer and collaborators so what we have computed with felix or mostly two things so far we have computed this universal ratio sorry I forgot to say that this ratio c divided by l so the c which is here divided by the inductance air is also universal if you look at two points which are symmetrically located with respect to the quantum critical point so we have computed this so it's shown here so here and here so the ratio is shown here so in the large end limit we got we recover the exact result which is known in the larger limit and it turns out that for n equal to there is a Monte Carlo simulations by this group gazette and coworkers and we get something which is in a reasonable agreement with this so another point I would like to discuss is the conductivity in the ordered phase so if you look at the ordered phase with n equal to 3 or higher there are two independent elements in the conductivity tensor which we call sigma a and sigma b so basically depends on whether the generator ta rotate the other parameter or not so we distinguish between sigma a and sigma b sigma a is the object that has been looked at most of the time so far and it's it's only the only component which remains when n is equal to 2 so typically it's as I said before it's essentially a super free like response and we have looked at sigma b which has been mostly overlooked and we found something which is quite interesting so first of all it's universal it goes to a finite number but what is very interesting is that this universal number is kind of super universal in the sense that it does not depend on n and this comes from the fixed point structure of this function x1 and x2 in the ordered phase so x2 actually goes to zero so it's not very interesting so here you see x1 at the fixed point of the ordered phase in dimensionless unit of course it varies with n but if you look at the minimum of the effective potential which corresponds to the physical case the case you want to study eventually then all the curves cross take the same value which means that this sigma b is a independent of n so there is a kind of a super universality here okay so I will conclude that how much time you have now I'm right so I have time to conclude it's time for question okay so the conclusion so I do think that the non-perturbative algae is a promising tool to study quantum critical point I think that the finite temperature thermodynamics near quantum critical point at least with this kind of relativistic symmetry is fully understood because we know the scaling the universal scaling function so we know the pressure we know the the entropy we know the specific heat so in principle we are able to explain or to compare with any experiments computing correlation function is much more difficult so for the heat susceptibility I think we were quite successful at zero temperature but still there is this problem to do finite temperature calculations and to do this we have to get away to find a way to do this analytical continuation so the paddy approximate method or the maximum entropy method cannot be used I think but there might be ways using the algae to circumvent this difficulty so here I cite a work by uh triple strodof and co-workers where they propose to do simplified calculations with simplified propagators so that we can explicitly do the matzubarassons and therefore perform explicitly the analytical continuation uh yeah so regarding the the dynamic me call conductivity it's a very challenging problem but there are very few methods that can address this I said before that quantum Monte Carlo simulations or the ADS CFT approach have their own problems so I think it's worth looking at this problem in more detail I will stop here and I thank you for your attention