 Or does it work like this? Well, first, thank you all for being here to attend to my presentation here. And I also would like to thank the organizers for giving me the opportunity to speak here at Exactergy. My name is Felix Rose, and I work in Paris with Nicolas Dupé, who speak here this morning about quantum phase transitions. So the topic of this talk is the following. I'm going to speak about the conductivity in the vicinity of a quantum critical point. So as Nicolas told you this morning, several quantum systems undergo a quantum phase transition at zero temperature when an external parameter is tuned. And he will be interested in continuous quantum phase transitions, where systems exhibit universal properties. And in particular, we'll be dealing with the universal property of the conductivity in the quantum oven model in two plus one dimensions. So the outline of this talk is the following. The first part I'll recall to everybody what is the quantum oven model and how is the conductivity defined and general results about the conductivity. Then, as this is the ERG conference, I'll speak about an Exactergy of Functional Non-Perturbative As-You-Wish scheme to compute the conductivity. And in the last part, I'll present some results about this. So as an introduction, the oven model has this action. The action is defined with phi, which is an n-component scalar field. And since this is a quantum model, the temperature dependence is contained in a space, in a time dimension, which has a finite length beta, the inverse temperature. And the system is defined in d-dimensions, d-space dimensions. So he will be looking into the two-dimensional problem, with two space dimensions. Because at zero temperature, the quantum phase position is controlled by the three-dimensional Wilson Fisher fixed point, which is interesting. The model has also experimental relevance for small m, for n equals 2, it corresponds to cold atoms, actually cold bosons in lattices. And for n equals 3, it corresponds to spin systems. So how is the conductivity defined for this model? Well, I recall to everybody that the one symmetry is a continuous symmetry. So we can make this global continuous symmetry local by adding a gauge field. That is, we change the derivative in the action into a covariant derivative, which is the derivative minus a mu, which is the gauge field. A mu belongs to the line of one, which is s1, which is the algebra of skew symmetric matrices. So we can decompose the gauge field this way over the generators of the skew symmetric matrices. Since we have n times n minus 1 over 2 generators of the skew symmetric of the group, we have that many current densities, which has these definitions. So maybe this definition is a little, how can I say, arid, for those not used to dealing with it. But for the case of small n, when you have bosons, and actually the auto symmetry is the u1 symmetry, you recall the standard expression in quantum mechanics for the current in quantum mechanics. And the concept of charge is obviously the electrical charge. So linear response theory predicts this form for the conductivity. It is rated to this canal k, which is mostly the correction function of two currents. But it is also the second derivative of free energy with respect to gauge fields. So it would have been possible for me not to use gauge fields in this formulation, since we simply need to compute this. And we could do so with external sources coupling to the current. But here we'll be interested in preserving the gauge symmetry. Simply because it provides us with useful identities, because it constrains the problem furthermore. So what can be said about the conductivity? Well, the conductivity tensor is diagonal in position space. This stems from the isotropy of the theory. But also in the space of the generators of the rotations. And the conductivity only depends on the rotation you consider. It has at most two independent components, which can be heuristically found this way. If you consider a rotation of the one space, it can either rotate the other parameter or not rotate it. So rotations that do actually rotate the other parameter will be said to be linked to i-class, whereas rotations that do not will be said to be linked to b-class. I note that in the disordered phase of the quantum critical point, since the value of the other parameter is zero, all rotations are equivalent with respect to the other parameter. And we have only one independent conductivity. So for n equals 2, which is an interesting case, as it describes bosons, you have only one generator, which always rotates the other parameter. So the conductivity sigma b is not defined. So generics things can be said about the low frequency behavior of the system. In the symmetric phase, the system is insulating. And the conductivity behaves like a capacitance. Whereas in the symmetry phase, the broken symmetry phase, excuse me, the conductivity sigma h assumes these forms. Actually, it behaves like an ectecton, which would be the case for a perfect metron. Also, at the critical point, this conductivity takes a universal value, sigma star, which is a constant with respect to omega. Sigma b, the behavior of sigma b has been a lot less studied, mostly because it does not exist for bosons, but we'll also discuss it later. So you see that already in the low frequency behavior, you have some interesting universal physics to study. The first one is this universal conductivity sigma star, but also the ratio of c over l, the capacitance over the andectrons at two symmetric points with respect to quantum phase transition, is a universal quantity. What we want to study is the universal scaling form of the conductivity in the whole momentum range, but we have also interesting things to study at low frequencies. So several approaches have been devised in the past to study this, both quantum Monte Carlo and the ADS CFT, as Nicolas spoke about in this morning. We'll also present right now exact energy. So all these approaches have advantages and drawbacks, and are complementary. So before devising an exact energy scheme, we need an effective action formulation of theory, which is the following. So you are all familiar with the effective action formulation of statistical mechanics. The only subtlety here is the presence of the gauge field, which is the source field, which is needed to derive the conductivity, which is present in the partition function and will be present in the effective action after we perform the Legendre transform. So you can define these generalized vertices, which are the derivatives of the gauge of the, excuse me, the effective action with respect to either the five field or the gauge field. And the kernel k, which is related to the conductivity, is related to these vertices, gamma naught two, gamma one one, and gamma two naught, which is the inverse propagator, actually. So what we need to do with non-partitive energy is to devise a scheme to compute these lower vertices. So I'm not going too much on this because everybody this afternoon has been talking about this scheme. We'll add an infrared regulator, which will interpolate between mean field and the exact theory. But this regulator, as usual for the one model with NPRG, is akin to a master, which depends on both q and k. And the problem here is that I said that I wished to compute, excuse me, to preserve gauged values of the theory. So how do we do so? Well, it's as simple as this. We make the regulator gauge dependent. So this idea has already been present in the past in the QCD community, but it's less useful for QCD than for us because here we are in the simple case where the gauge field is an external source. It's not dynamical, and we simply take the radius with respect to the gauge field before setting it to zero. You see that this regulator term in position space has this form, and with this function r, taken as a function of the gradient. Well, we see we promote this gradient to a covariant derivative to make this term, delta sk, explicitly dependent on the gauge field and explicitly gauge invariant. Because of this, when you take functional derivatives of the flow equation with respect to the gauge field, you have additional terms which actually preserve gauge invariance and help you recover what it entities and basically use symmetries. So that's the urge scheme, and we now need an approximation to actually compute the flow of the vertices gamma of k. So for this approximation, our first thought was to use the Blaiseau-Mendez-Gallon-Scheber approximation because what we wanted was, what we want, excuse me, is the full momentum dependence of vertices. So we've already used this scheme to obtain the, to study the Higgs amplitude model in the vicinity of the quantum phase formation, and we thought we'd be able to extend it easily to the problem of conductivity. But the issue is that this fails. Basically, it is, since in the current you have a gradient, when you take three transforms, you have momenta, and the vertices with derivative with respect to the gauge field has a non-trivial momenta dependence. So either you set the momenta to zero and you obtain like is done in the MWX approximation as evidenced by Leoni right before me. If you set the momenta to zero, you obtain trivial equations. So we tried to be a little smarter than this and we tried to work on it, but our solutions actually broke down the gauge invariance, even with the derivative. So it was not a solution. So we had to fall back on some less precise scheme, the derivative expansion scheme, which will allow us to obtain the low momenta, the low momenta range of the model. So what is the derivative expansion? Well, we project the flow equation onto a gauge invariant and that's, and what is important here is that we are able to explicitly devise a gauge invariant scheme. So we are using the standard derivative expansion at all the two in the fields with these three functions, the potential U and these two randomizations of the self energy. And we can construct two additional terms with the gauge field that respect the gauge invariance. So basically we use this F mu, which is the sort of generalization here of the electromagnetic tensor. So within this ansatz, the conductivity has an exact, well, has an expression, which is certainly not exact, which is the expression of the conductivity within the derivative expansion. Well, we can integrate the flows to obtain these values, values, and that's what we did. So the results go as follows. We recover most of the low momenta, low momenta physics. So earlier I said that the ratio of the capacitance over the inductance at two symmetric points with respect to the phase transition is a universal ratio. Well, we're about to compute it for all values of n and that large n will recover the exact value that you can compute, well, the exact value. And that's more for small values of n, we have a good agreement with Monte Carlo. So this is quite convincing. However, for the critical conductivity, sigma star, we have something more complicated. And this stems from the fact that the derivative expansion is only valid at small frequencies. So sometimes you are about to extrapolate, but this is not the case here. Actually, one of the vertices involved in the flow diverges like one over p. So the derivative expansion is a small p expansion of the effective action, and it fails here. Actually, you see that the conductivity takes this form for small k, with a fixed point value times a function of omega over k, the ratio of the frequency over the reigning scale k. Where is this fixed point value for x1 tilde critical? So something that Calvin does here is to say that we cut the flow by the frequency omega to have an estimate of the conductivity. It is of the order of x1 quite critical. So this is in no way an exact computation. x1 critical is not the universal value of sigma star, but it gives us an idea, and it is also consistent. It is a proof that we can see sort of in the derivative expansion scheme that the conductivity reaches a plateau value. So that's it for the universal conductivity, but something more interesting even happens in the ordered phase for the conductivity sigma b. Actually, the same thing happens. And the quantity x1, which is the conductivity sigma b, reaches a fixed point value in the ordered phase. So from this we can infer that sigma b, the conductivity sigma b, is a universal quantity in the whole ordered phase. This is also, this also can be verified by computation for n equals infinity, but here it is a conjecture because it is based with this idea of using the fixed point value of x1 and cutting the flow by the frequency omega. But something even more interesting happens is that the fixed point value in the ordered phase does not seem to depend on the value of n. So here I represent with the, when integrating the flow, this is the value of x1 tilde, the dimensionless quantity. In the full line represents what happens at the quantum critical point. So you see the fixed point values and the dash lines at different points in the ordered phase. And you see that for all values of n, all curves go to the same fixed point. So numerically we see that x1 tilde in the ordered phase, the value of the fixed point does not seem to depend on that. So we make this flowing conjecture. The conductivity sigma b is universal for all n's and this is what we sort of call the super universality with quotes. So to sum things up, we here devised a gauge invariant exacter g-scheme to compute the conductivity and with as n and z as simple, an approximation as simple as the divisional expansion, we recover most of the low momenta physics. These results also are us to make a conjecture on the universal bf and the conjecture sigma b in the ordered phase. But this is only a conjecture and to confirm it, we actually need to develop a momentum-dependent scheme. But actually, we are on way into it. The long-term goal for this study would be to see what happens at finite temperature because actually none of the methods we've seen before have been able yet to tell us what happens at finite temperature. So it's an unsolved question. If you're more interested into this, I think that in the next few weeks we'll publish a pre-print on the active. So that's it for my talk. Are there any questions?