 All right, so I think we are on. Perfect, thank you very much. Well, I would like to start by thanking the organizer to give me the opportunity to present our work on dynamical in field theory applied on molecules. It's a slightly different topic. So I am aware that we are moving from entanglement, quantum solids to molecules. But I'd like to introduce that topic as an opportunity to present a recent method we have been developing. And you can see that as a beginning of something that can be applied to a wide class of systems. So my name is Cedric Weber. I am at King's College. Oh, so hopefully I can use your clicker. All right, Apologies for the technical issues. So this is the outline of the talk. So I'll start by discussing and explain a bit where I've come from, which is my background. And as you will see, actually, I'm a condensed matter physicist. I'll come from the field of superconductivity. And we got interested in the field of transition metal system. And I'll explain a little bit the motivations why we started to look at molecular system. And then I'll link with the introduction of the chairman, chairwoman, who actually mentioned quantum entanglement. And I'll discuss briefly the role of quantum fluctuations and quantum physics in molecular system. And I'll highlight two examples. If I have time, one is myoglobin, that we all know, and the other one is hemocyanin. So of course, we are all aware in this room that DFT has had a tremendous success. It has been applied to all kinds of system with a very simple idea that we are dealing with a wave function, which is a complicated object with free and coordinate variables. And it can be tremendously simplified if we just map that problem to a simpler problem, which depends only on the density. And my slide stops here at 2003, but the number of citations with DFT in the abstracts have steadily increased through the years since the implementation of DFT has been done in most of the available codes. And up to the points where we can ask ourselves, is there any limiting cases where DFT is failing? Is there anything left to do? And the short answer is yes, of course. So there are still some problems with DFT and need for improvement in the case of transition metal system and F-systems, so essentially any system which is involving D or F atoms. So essentially, we are looking at the center of the periodic table with many possible examples which are known in solids such as, for example, copper oxide, the superconductor. So we're very interested in applications with vanadium oxides, which are interesting because of their metal insulator transition, and so on and so forth. The cobaltates are nice thermal electrics. So each time you look at a 3D or a D element and you combine it with ligands, with oxygens, usually you get interesting properties. But at the same time, you get into trouble because DFT predicts qualitatively wrong properties, such as, for example, you get a metal instead of an insulator, and so on and so forth. And in our community, canonical or standard theory is a Hubble model, which is a very simplified picture where you have electrons hopping on a lattice with an interaction term U, which is the effect that is missing in DFT that when states are very localized with narrow bandwidth, electrons feel each other by strongly interacting, and that leads to very interesting physics, which is also difficult to treat at the same time. Although this is a very simple theory because of the size of the Hilbert space that grows exponentially, we are still struggling to get an exact answer to this simple problem. We do, however, know a little bit the physics, what we expect from that type of theory. So if you start from U equals 0, so when you have weak interaction and you have typically density of states in a metallic state, as you increase you as the strength of correlations, you will end up in a situation that we expect that we open a mod gap, which essentially is associated with a concept that we all know which is called localization, that essentially when electrons or objects are correlated, they tend to behave with emergent and collective properties. One of them is a picture from, I think, suggested by Antoine Georges that I find quite striking, which is the effect we all know in the morning at 8 in the morning, we get traffic jams on the highway. We all behave in the same way. So we all collide. We have a direct appreciation of one effect of localization, mostly true in Paris and London, probably. But what is less known, maybe, is what happens in between. So when you're in the intermediate range of correlation, you actually have an additional structure called quasi-particles. These are fermions that excitation that behave like fermions with a renormalized mass. But essentially, those are quite important because you can see there is a sharp resonance, a sharp increase of the density of states as a fermi level. So this will dramatically affect most of the properties you're interested in. And this is the exact effect that is missing in DFT. So DFT will capture very nicely the two limits. It might, with DFT, you might open a gap. You might get the metal. But what is in between is essentially the quasi-particle excitations that are missing in your theory. So progress has been made. And we had a few talks this morning about possible approach that can capture this. One of them is dynamical mean field theory. And we had very nice introductions already so I can skip the details. But maybe just to link with what I'm going to discuss next, essentially, one simple idea is that you have a complicated problem to solve where you have a lattice. And the idea of mean field is that you simplify it. So you replace that complicated problem by a simpler one where you have taken out one side of the lattice and you connect it with a bar for reservoir. And that bar for reservoir mimics the effect of the rest of the lattice. So you have now a quantum correlated problem, but which is much simpler because it is a zero dimension problem. And the only question left is how can you describe the bar or define the bar such that you can recover the effect of the rest of your solids. And I won't say much more on DMFT. Besides that, one thing that wasn't mentioned maybe yet is that if you look at your solids and think of your solid as a potato with a few correlated atoms, if you want to go to DMFT, which is defining an auxiliary problem, which is called the Anderson impurity problem, you will have to do a mapping from a crystal to this impurity problem. And that involves an operation called projection or projectors. So you can think of that as your quantum space and you will have to go from your quantum space to the Anderson impurity model. We have this projection operation and this projection operation is essentially dependent on your DMFT code. So if you use a plane wave code, for example, you will have a plane wave site K mu and you will want to go to a local correlated problem, but typically it would be your d orbitals. Sorry about that, so I'm going to solve the problem. And this is essentially the manifold subspace that is correlated, so typically f or d, and that's an arbitrary choice. So you have already one choice to make here, but once this choice is made, essentially, you can define these projectors as the overlap of your plane waves and your hydrogenic orbitals and that's defined once and for all and that will allow you to go from the quantum space to your Anderson impurity model. And I don't want to upset anyone here. This is just a small selection of application of DMFT and the only purpose here is to show that DMFT has been applied to a wide range of systems. We have the mud transition in the hot-burn model, superconductors, chalcogenides, F-materials, cold atoms, vanadates, also molecular systems such as nanocontax and again the list is goes on and on. Maybe my only point here was to highlight that in all the applications of the DMFT, usually people look at solids which are a periodic system. So the one category of typical system that has been a bit overlooked in the past by our community is molecular system. At the same time, these systems have been covered quite in details by both the DMFT community and the quantum chemistry community which traditionally has looked at these problems. And here again, I cannot mention everyone but Nicolama Zary is a pioneer of that for example with DFT plus you. Garnet Shan has recently started an activity on DMFT coming from the quantum chemistry side. What was left is from our community to step in and to see if we can bring something new to this type of approach. So that's what we have to do to go from a periodic system to a gamma system. And well, there are many applications. Most of them have in common that you need now to do DMFT and DMFT on a system that has between a thousand to a few thousand atoms. So that's now the difficulty. And to do that, we first need to do a DFT calculation which is quite difficult in itself. How do you do DFT when you have 2000, 3000 atoms? I mean, you probably know that DFT scales like the cube of the number of orbitals. So that's a challenge in itself. And at the same time, there's been recent development in the DFT community regarding DFT approaches that are able to do that via approximation. Those are called linear scaling DFT approach. One is called OneTap, developed in the UK. And we have some of the developers in the audience. The project started with Mike Payne, Peter Haynes, Arash Mostofi, Nicolas Hine, and with very active, I would say, and sparse developers community and user community as well. So I won't say much about OneTap. Besides that, the approximation is relatively well-defined. You essentially will assume that you look at an insulator or molecule, and electrons sitting on one end of the molecule might not feel or not know about the electrons on the other side of the molecule. This is called the near-sighted approximation. More formally, what you are doing is a truncation of the density kernel. You will choose and impose a cutoff in the density kernel. This one approximation. The other, I would say, is more of an optimization. You will use a compact basis set, which is optimized during the DFT procedure. To do that, of course, if you want to optimize your basis set, you need support functions to write your basis set in the first place, which is in OneTap cardinal sine function basis, essentially localized basis. So you both optimize the basis at the same time as you truncate correlation on large distances. And if you do that, essentially you can recover a linear scaling in terms of computer time versus the number of atoms. As you can see here, this has gone up to 7,000 atoms, and here the blue curve is a traditional DFT scaling. So it's a very powerful method, but it unlocks a lot of problems that we might be interested in at the cost of these two approximations, which are quite, but break down from metallic system. So of course, this is good for molecules or mod insulators, not so good for good metals, which is not the aim here. If you want to do DMFT, once you have done the OneTap calculation, things are already well defined. So I will just spend a couple of minutes here before moving to the results. So in DMFT, the main object is a Green's function, which is essentially just the inverse of an Hamiltonian. So you have the inverse here. Times the scaling factor, this is a Matsubara frequency or real frequency. S is a novel of matrix because you are now dealing with an optimized basis set that is non-orthogonal. And sigma is your self energy that contains information about many body effects. So essentially what you have to do is inverting a big matrix. And once you have the big matrix in the quantum space, define this Anderson impurity model, which is obtained by projecting the Green's function via these projectiles that I just introduced a few minutes ago. And once you project the Green's function, you obtain, and I will skip here some of the maths, one function called viabilization that defines once and for all your Anderson impurity model, which is again coming from this projected Green's function. The maths are a bit rapid. I'm happy to discuss with anyone more in details some of these equations, but the idea is really simple. Define a Green's function in a quantum space, project it. You can then define this Anderson impurity model, you have this habillization function. And the last step is to solve that problem and get a self energy. And this was already discussed in some of the earlier talk. And essentially you have now all the ingredients to apply this to a system of relatively large size that contains transition metal ions. So here we were interested in oxygen transport. And there are mostly four molecules responsible, well, linked to the role of oxygen transport. So we know all of one of them, hemoglobin, which is based on iron, which is negating to oxygen. And there are three others. Hemocyanin is based on copper. And all of these molecules of course have a very important functions and have been extensively studied. And if I start now just with hemoglobin or heme, which is the kernel of hemoglobin containing the iron center, which is also negating or binding to oxygen two. Essentially that system is very important and people have been interested to understand how it can bind to oxygen two or carbon monoxide, which are the two main candidates for binding to heme. And the very first step you could do is apply DFT to that system. And this has been done a long time ago. And unfortunately you get a slightly disappointing answer that DFT predicts that the binding to carbon monoxide is one electron volt larger than the binding to oxygen two. Meaning that the affinity of binding to carbon monoxide is thousands of times larger than oxygen two. That's a bit disappointing because we all know it's not possible. In reality actually there is a very slight imbalance of our carbon monoxide on the order of one kilo-cal per mole which is about 0.02 electron volt. That's why carbon monoxide is toxic because it sticks to heme, but definitely not with one electron volt larger affinity. So that's what we get with DFT. And many things have been tried and what we have been doing is not the only answer to that problem. But we were interested in looking if electronic correlations might be responsible for that failure of DFT. So can we solve that problem with DMFT and essentially recover more realistic and possible binding energies? So this is your heme molecule. I run in the center about 240 orbitals. It's quite large to solve with quantum Monte Carlo. So even for such a small system you are entering the regime of challenging problems to solve exactly. However, it is a system that is very similar to our Anderson impurity model. So an impurity I run connected to above. So we can use these projectors that we introduced a bit earlier to do this mapping between that problem to that one and we know how to solve this one with any choice of interaction or vertex that we want. And let's have a look at the results. And before I go to the results, this is designed for undergrad students. So this is just the effect of the wound. What is the wounds we're coupling? And in England we do this analogy with a bus feeling rule that if you enter in any bus in London, people tend to occupy one seat at a time. You have singly occupied seats in the bus. So that's what our electrons are doing. If you introduce a wound's rule, essentially you want to occupy one orbital at a time to create a magnetic moment. So that's something that wasn't done so far or studied in him. So we start here to look at him by choosing our interaction vertex. So the value of U was mentioned before. The value of the wound's coupling was not so much discussed. So the wound's rule is another of the coefficient of this later coaster interaction. So here on the horizontal axis, you have the wound's coupling. When the wound's coupling is large, you want to align the electron in the same direction and create a large magnetic moment. On the other axis you have here the occupation of your iron atom. So the iron atom is inside him. And of course, if you want to create a large magnetic moment, ideally you would like to have five electrons. That's what you get in the limit of a very large wound's coupling. You create the largest possible moment that you can make. So you have here a high spin limit. When you don't include the wound's coupling, when J is equal to zero, you get a low spin regime where you have a larger number of electrons. And of course, you have somewhere in between a transition between low spin to high spin. Now in reality, in him, you have iron and there's only one value, sensible value of the wound's coupling. It's not something you can change. Iron has a screen value for its interactions and if you do some calculation, you would find that it is somewhere around 0.7 which places him in this regime here which is interestingly just as a transition between the high spin to low spin system. So most studies I would have mentioned that a lot of studies have been done at J equals zero and U equals four. Actually, not so much has been done with the wound's coupling. So we found this interesting and carrying on, you also will find without surprise that if you increase the wound's coupling, you increase the magnetic moment going for different steps and here this place is actually our him, somewhere in between the triplet and the quartet. Quite interestingly here, I should say one thing is that in the MFT, you are not actually breaking by hand the spin symmetry. So you are not polarizing your electrons. You are doing the full calculation with the same number of up and down electrons which is quite realistic. A molecule is not a magnet. Somehow you have a fluctuating moment but you are not breaking the spin symmetry. So this you can do in the MFT because you are looking at a ground state that is in a quantum superposition of all the possible contribution to the multi-plat. So you are building a multi-plat by not breaking the spin symmetry. This is why you have to think a little bit how to define the magnetic moment but in this case it's defined by the fluctuations of the S operator. You have to skip a few things. So since we have now this transition between low spin to high spin, another quantity you might want to look at is the quantum entanglement of this quantum entropy. So essentially the question is do you have only one state for him? Is it a pure classical state or is it a true quantum state that is in a superposition of many different contribution? This can be defined formally and calculated and measured by what we call the von Neumann entropy which is essentially simply the entropy coming from the density matrix. But because now you want to look at your iron atom that contains the information about the magnetic moment, you have first to trace out the bath degrees of freedom. So here what we are doing is a trace over all the bath quantum states and we are left with a density matrix that only contains the information about your d orbitals. From this density matrix you can obtain the eigenvalues and define the entropy in the usual way which is called the von Neumann entropy. And looking at what we get here, again the same thing. So we have now this transition as you increase the Hohn's coupling and what we realize is that the entropy, the entanglement, the quantum entropy which is the maximum just that in the physical regime where you would actually have the hemolecule at 0.7 and the entropy is smaller on either side which makes sense. So in the very high spin limit the entropy goes down again. In the low spin limit the entropy is lower as well. So in some way we are placing him somewhere in this transition and thinking of it a little bit as a Schrodinger cat rather than a pure state. However then we have to rethink a bit to be careful about what we call entanglement entropy. The reason why we have there are two contributions to the entanglement. One is the accumulation of possible quantum states. We are building a multiplet. Multiplet has many different contributions and you might have different multiplets which are all contributing to the partition function. However, these numbers we are looking at are the magnetic moment and charge of the iron atom which is in itself, those are not quantum number. The iron atom is hybridized to the rest of the molecule so a contribution to this valence fluctuations comes from the hybridization between the iron and the rest of the molecule. And that's not coming from a pure quantum effect it's just coming from the definition itself of what we call valence fluctuations or entropy. However still it holds that the system is somewhere in this transition to a maximum entropy. And now the last question is did we solve the problem? So did I with DMFT solve this imbalance of binding energies? And here you have the difference of binding energies between carbon monoxide and oxygen too. You can see that when the wounds coupling is small essentially the molecule has a very large tendency to bind to CO, actually five electrons more so it's actually even worse than DMFT. But if you increase the wounds coupling it goes down and somewhere near the value of 0.7 you are around one EV and it stays there. It saturates there for large value so essentially we have not solved the problem. So DMFT does not bring you towards the point where we can explain why carbon monoxide and O2 binds in the same way to him. So what is missing here? What did we miss in this calculation? Well, obviously things are in life more complicated. I did show you a very small model of this molecule and a question in biology or quantum chemistry is actually what model shall we consider? So what I just showed you is just this little portion of him and you can see now here the rest of the molecule a few more residues. Actually the full molecule is much larger than that. This contains now 53 residues about a thousand atoms but we did essentially look at a larger model. This larger model as you can see is important because it links him here to the rest of the molecule and there is a histidine group called the fifth ligand that is pushed up just under the iron atom and many people claim that actually that fifth ligand and the stress tensor in the molecule itself are actually important to capture correctly the electronic properties of in this case myoglobin. So long story short we extended the calculation with the same method to a larger system and we looked if we could find any differences. Interestingly one of the main results that came out is that this time when you look at, this is now the charge of the iron atom for oxygen two and CO. We saw that actually the wounds coupling had very little effect for the binding to carbon monoxide system, for the bound system to carbon monoxide. Actually you can see this, the iron atom keeps the same charge for relatively large values of the wounds coupling and actually that makes sense. This is coming from the fact that carbon monoxide is having a strong covalent bond with iron. So we expect to have quite a large gap in this case and for the case of oxygen two, actually the charge of iron, the charge transfer is changing gradually as you increase the wounds coupling and that comes from the fact that you affect the charge transfer between the iron and oxygen to ligand. So we obtain in some way now a binding discrimination that actually was missing in the previous, was not observed in the previous calculation. And looking now at the energy, you have now the binding energies for carbon monoxide in O2. The blue curve is carbon monoxide. So you can see that when the wounds coupling is zero you still bind to carbon monoxide. But now you can see that the red curve crosses the blue curve at around 0.7. And then actually we enter in a different regime where the binding is much more in favor of O2. So essentially it's quite interesting picture. It tells you that in this hemolecule, just looking at the interaction vertex, there are not so many choices of the interaction that allows actually reasonable binding. So that goes a little bit to a philosophical question why did nature choose iron in hemoglobin? One simple example, if you would go down iron you would find roughenate, which has the same dilence electrons, but with wounds coupling twice as small, twice smaller. And indeed so this, the molecule exists with roughenate instead of iron but does not bind to oxygen too. So there are some clue that this might not be completely wrong, although the situation is more complex for sure. And I am now reaching my time. So it's a good time to conclude. I did not have time to mention a mocianin. Let me just say one last thing, is that I did mention about the magnetic properties. If you want to understand now the magnetic properties of this molecule because you are looking at a quantum system you have to actually do a histogram. And here in this case you have the histogram of the multiplet component which contributes to the reduced density matrix. So if you are asking me what is the spin of him, the molecule, I would have to tell you why it depends on your measurement. And here he is a statistic. So you see that for the carbon monoxide which are these light marble colors the dominant contribution is a singlet but you see that there is a large contribution from the doublet and triplet. For oxygen two, the largest component is triplet but you have satellites as well. So although we are not breaking the spin symmetry in the MFT the new challenge is that now to understand your problem you have to do an analysis in terms of distributions which is actually probably realistic. The system is, the state of the iron atom is not necessarily entirely defined as a pure state. So I will conclude here. I will have to skip the Emocyanin topic but let me just finish by listing my collaborators. So Mohamed Al-Badri at KCL did most of the calculation for the Emocyanin and Edward Linscott in Cambridge for myoglobin. Of course in the theory part many people were involved but the project started in collaboration with Mike Payne originally in Antoine George and Daniel Cole and Nick Hine very much contributed to the design of the project and especially Danny is really our biological expert. So I will leave you with my conclusion. Thank you.