 So good morning, everyone. I first would like to thank the organizers for this opportunity to talk to you this morning in this very nice conference. It's actually very good to be back here at ICTP. I was last time I was here, I think it was 10 years ago, so I'm quite happy to, and also enjoying the conference, the talks in general. You will quickly see that I'm not a machine learning expert, but I'm learning a lot and well I have a couple of ideas that perhaps we could collaborate in somehow, right? So today I want to show you more or less how we are using entanglement measures and that was a very nice combination of talks because Benoit did most of the job of defining entanglement quantities and so on, so this will help, helps a lot, but I will give a general idea of how we are using this to point phase transitions in many body systems and then I will focus on this specific title that is going, specifying pretty much the system in a superfluid insulator transition for disorder systems, okay? Let's just see, no, this should be here, okay. So here's the outline of the talk, I don't have any, so it's not working the laser pointer, right? Okay, so that's true. I think it's finished now, but that's fine, I can use the mouse. Alright, no problem. So first I will just describe the system, the general idea about the, our system, the Hubbard model, fermionic one, then I will give a general idea of the methodology since I didn't see any talk on this respect up to now that is in the dense functional theory calculations and then show general results of our group in the combining entanglement with current phase transitions and then finally come to this specific system in this specific transition, superfluid to insulator transition, okay? So there were a couple of talks describing the Hubbard model. I just would like to give a general talk about it. So in particular what we can say about the Hubbard model first is that it's the basic model that you can use to describe itinerant and interacting electrons in a chain, right? So it's the most simplest one that is considering that you can have a certain probability, oh I have no laser, so right, it would be tough, let me try to use here. No, nothing there, okay. Is there any laser pointer that you could borrow me? Would help a lot. No, I thought it was, oh thank you. Thank you. Just know. Alright, thank you. And now where's the laser? Here, okay. Alright, so the idea is that the basic idea is if you consider a 1D Hubbard model then essentially you can have, there is a certain probability that this fermion can move to next neighbor and if they are lying together in the same site they pay a price in energy, that means you have some column interaction in the system that's the basic then Hamiltonian for the Hubbard. And although it's quite basic you can describe a lot of physics already with this simple model, right? So for example if you go for this limit where you have repulsive interaction you can, both, you can describe metals if you have a not have field in the system so you have any smaller than one and you can also describe insulators, mod insulators at this limit. If, instead of repulsive interaction you consider attractive you, you could describe for example conventional, BCS, superconductivity and if you add some imbalance defined like that, if you add some magnetization in your system you could also describe somehow at least qualitatively exotic superconductivity properties. I like a very state but that is the basic one. Now if you also add to this model a certain external potential V then you have now many other nice interesting things to explore as for example disorder chains. So if you turn off the interaction and turn on the disorder so randomly impurities along your chain then you can find and can, you can see Anderson localization, describable Anderson localization and other, any other spatial inhomogeneity could be for example explored with this V as superlatives, interface, harmonic confinement so, so and the other thing that I think it's pretty nice to study the Harvard model is because it's essentially a very paradigmatic model in the sense that you can add terms or extend somehow for example for multiband, multiband levels in your, in our model you can also specify regimes in go to very other nice models that we all know very popular in condensed matter physics, in chemistry and material science in general right? So but okay it's pretty nice but it's still very hard to to calculate it. So now I'm trying to, I start to specify the kind of properties we are measuring so and that was very nice that we just saw this entanglement definitions right now. In our case we are always looking at this average ground state single site entanglement. What is this essentially? So I fixed all, I choose only a single site A that I call A here and the remaining system is my system D. So it's not a block as less, less talk but now I have then entanglement between one site and the remaining but of course this choice which is, which A I choose will pretty much depend on this in homogeneity that is in there in the system. So what we do, we choose, we do the same calculation for all the sites and then take an average then we have this average single site entanglement okay for a specific system. So that's the the idea so it's definitely a bipartite per state. We are in the ground state and then we have a very good definition proper measure of entanglement which is the von Neumann entropy here right and we also have the linear entropy which is quite related to your Rene entropy in the sense that it's also related to the purity of the reduced density row A here. So that just this piece trace row A square is would be the purity of this sub system of this ITA okay. Well in this respect I would like to mention that this true we recently investigate the compare these two quantities for the particular Hubbard model and we found situations limits in the regime of parameters where actually it fails to predict even qualitatively the von Neumann entropy and so on but in this talk I will essentially use the linear for there results I will show you most of them are related to this quantity here so the linear entropy one okay. Well in the ninth thing about it and then in this I can connect with my methodology is the following these W's here that appear here I simply the occupation probabilities of a single site so that means I can just have two particles it's a single band Hubbard model so I can have at most two particles in each side so the probabilities are either I find found two particles one particle with one of these things or no particles at all so these are the probabilities and if you look closer to these formulas they are dependent on this parameter so that's the total day the average density this is the magnetization so a certain balance that you can put between these pieces and the energy so everything is dependent on the density so that means they are dense functionals so I have expressions for the Hubbard model for this quantities for the homogeneous case that are dependent on the on the density and that will make that will make much easier to calculate DFT calculations I will get there in this slide so how we solve this one D Hubbard problem well there are several ways to do that for example if you have a homogeneous system so it's really in the thermodynamic limit no boundaries then you have the exact solution from better answers now if you are interested in a more realistic description of the system then I necessarily need this external potential where is it now here we really necessarily this this special in homogeneity to describe for example these structures here so like superlases structures with a periodic modulation in our case of the most most of the results will be here with a randomly distributed local impurities and then harmonic confinement so on so you need anything else to calculate to get to solve the one D Hubbard model so with exact diagonalization the problem is you are limited to small systems for sure then you could do some DMRG calculations that it's quite exact so to say but you will have a high computational cost that are older methods of course but I'm now I would like to emphasize the one I'm using for most of these results that is approximate one DFT calculations but with a very local computational cost compared comparison to DMRG of course and so now in this respect I would like to call the attention to this recent paper of us we just put there in the archive exact presented as a poster on Tuesday I believe so anyway where we simply compare these two methodologies for several in homogeneities in the one D Hubbard model right so now I'd like to give you a brief introduction on the just the basic basic ideas behind all these DFT calculations so essentially with this Hall-Hemmel-Contail-Hem we know that there is a unique relation between wave function and ground state density right so that allows us to as a consequence any observable that in principle of course is obtained like that by the wave function will be then always dependent or functional of the ground state density okay so these are of course this if I can then based on this theorem replace any calculation that I used to do with the wave function many body wave function if I can do that with this other object the one single particle density then of course I have a great simplification from the computational point of view now and if this is true then essentially all quantities in principle could be obtained via the density without any knowing about the wave function the wave function all right so there are of course this is exact theoretically but there are several steps to to get in there so the first one is how can we actually obtain this density and it was solved quite fast just after the original theorem and which is one of the possibilities is to use the conscious skin so basically the idea is okay I have a very complex system which is an interacting system many body interacting system but I could try to use a map to a fictitious one there's no interaction and if I managed to do that I solve the simpler one single single particle Schrodinger equation and then I try to extract information from the from the the realistic one and how can I do that well I construct the effective potential in this fictitious part precisely as these terms here for the realistic system and then I solve a self-consistent cycle so essentially I solve the simpler system I obtained this single body wave function so this is called the conscious orbitals then I get the density for this system but they are were forced to represent the same density so I get the density of the interactive system by solving a much simpler problem fictitious one with all the interaction there so that's the basic idea but there's something here that is the the the problem part and only here the ft becomes actually approximate is the fact that we in general don't know this exchange correlation potential to do this conscious scheme precisely right so essentially here at this point we put some approximate these functionals for this object for this exchange correlation potential and then the all the method becomes approximate and of course depending on the quality of your functional the quality your results will be better or worse right so for this case I would like to discuss to present and then I will focus back to entanglement but the most simplest approach that you can do for that exchange correlation potential is called the local dense approximation right so just the general idea in general if you have then then our density in a homogeneous system that we know it will not be a constant density if you think again in 1g system that means you have a certain density profile there depending on the impurities and boundaries and whatever you have there so you have some inhomogeneous distribution for this object and then you could say if you know if you have information about the homogenous system so then you could estimate something about what you don't know so the inhomogeneous especially in homogeneous guy by doing a LDA which is given by this expression but essentially here in the graph means you are locally let me put this here locally considering it's a good approximation to consider uniform right so you get something that you know you have to know something about the system but for the homogenous and then you extract approximate solution for that quantity any observable by doing this local approximation here okay so that's the general idea from what we do in our calculations here you see that only this LDA for entanglement is sufficient of course for other properties people generally doing the FT goes beyond with more sophisticated approximations and I'll just give you some ideas of how we get this homogenous object here which is necessary as an input inside this LDA approach and we do that then for the exchange correlation energy of the homogeneous carbon model it's possible to do via exact numerical solution just the better than such solutions here is a nice example where they did this there that are also some analytical expressions which are approximated for the same set of data instead of using the numerical you can simply get a formula and use this and here is one that we have proposed some years ago and recently we also proposed something with a neural network which we call simply in and functional for for this this guy right so there are several possibilities to implement this thing here and now the question is okay if you do this LDA within the FT approach the first question is are they reliable somehow for yours for our system and without to specify much of the physics at the moment I will come to this later but the idea is here is that we have very good results compared to exact the MRG data so the red dots are the MRG and the blue curve is the FT so for entanglement proposed this type of entanglement I've mentioned before it is fairly well obtained and via our methodology okay all right so then let me see if I so the first thing that I would like to mention is how we are connecting entanglement with different one phase transitions so I will give some slides in several of them and then I will focus on on the one from the title right so I mentioned to you that Robert could describe for example conventional and exotic FFLO stuff so one question that when we addressed years ago already is can we distinguish between a normal phase a magnetic phase when we are plugging this magnetization and a superconductor with coexisting with this magnetic phase and then we look at this type of entanglement with this type of approach and we saw that actually whenever you have the FFLO phase in your parameter in your set of parameters you really found a non-monotonic behavior so it's a signature in this type of entanglement of a different phase in comparison to this black curve which is essentially non-monotonic for the normal phase then we also look at the repulsive interaction so for example if you go from density smaller than one from metal to a insulator by varying than the density one of the question we address was entanglement can how is the signature of this on the entanglement measure and this that was the result so entanglement has a function of the average density when you plug in with depends on the interaction it becomes there is a quite strong minima at the critical point and equal to one and so as an answer to that so it's substantially the reduce it at the critical point what's quite consistent with the fact that we are localizing the system charging localize the system but we also see that even for infinite interaction it never is with zero entanglement what means that we have a kind of partial localization in this state which is remain which relate to the fact that we still have spin degrees of freedom in the in this case of the mod now if you turn off the interaction if you turn off the interaction in plugging this disorder the random impurities in your system then we are in the regime of Anderson localization we also have address some questions there and so what we see from entanglement point of view is that in general entanglement decrease when you turn on the disorder but for any density but that is that I there are specific regimes that I there is a specific day cities where you reach a full localized the state meaning zero entanglement auto at all so you define your state here in the thing and and it was the number called this fully localized the state okay so these are general things that I'd like to show there was one more sorry and then we also ask how could we consider the interplay between interaction and disorder so Anderson is with zero interaction now if I plug in everything together so mot and then there's some physics one of the possible question is how is this interplay from the point of view of entanglement signatures and it's a mess but I will just spoke briefly the main results here so essentially we see that the minima look related to the mot is displaced so the mot density where we find the signature with a minima entanglement is displaced here we also see that the minima related to zero entanglement relate to Anderson localization also requires a stronger V to reach that full localization so you see that for example for a certain intensity here you need stronger disorder to really get zero entanglement now and that's it I think I just like to go that I'm more features if you want I can discuss later I can answer any question in this respect but that was the general idea so essentially up to here just the general signatures we are finding in the last years by using these entanglement measures in regimes where we face of one phase transition okay so now focus on the results I've prepared to show to you now these are the three main works that I'm showing this beyond this point and they are essentially performed by my former PAG student Guilherme Canella and with our collaborators Krissia Zavadsky and Irene D'Amico from New York and so let me now a little bit not so fast describe what kind of system we are describing here right so we have here attractive interactions so in principle in a superfluid regime described by the fermionic hub and we also added the impurities okay and of course there are many ways to implement this disorder stuff you can choose the specific format for this impurities and what we have chosen here is just this point-like disorder so what it's what is the idea behind this so I really have some specific sites with impurity in there so with a specific intensity of external potential V VI right so and we have chosen to be zero in the other sites so with that I define a certain type a certain concentration of impurities that can be written like that so okay so depending on how many impurities I plug in there all of them have the same intense TV for this model we had described here and I have a certain type of certain intensity for the concentration when we do that imagine that is randomly placed in the along the chain along the 1d chain and that means I have to do many realizations to actually get read of specific positions where these impurities lie and therefore I we are considering here 100 samples for each point so for a given set of parameters I need the one I'm using here in the following results 100 samples and an average over that so here is just an illustration of how we see entanglement changing with the number of samples we consider in the calculations we see that it's changed a lot here in the very beginning and it kind of saturates if you see the scale is quite small and then we consider 100 would be enough for this particular system which also has 100 sites it's mentioned here true okay so that would be definitely impossible to do with the MRG you cannot imagine doing the MRG exact data with so many data right okay so the first question we address in this specific system was is that any minimum disordering things you required it to for the localization because there was at the time there was a discussion in the literature whether it was necessary a minimum V to get the localization or any amount of view would alright enough and from the entanglement point of view via analyzing where is my figure analyzing the the signature from the entanglement point of view what we saw is that any intense TV for the superfluid system any intense of V actually the breaks the or decrease drops a lot the degree of entanglement suggesting that we localize with any small amount of disorder so that was already 50% for any of the concentration case I read for this very small V okay so now we also look at how the the double occupation in the impurity and no impurity sites at the impurity ones is just one and so it's the red point how it goes to one so how you concentrate in the impurity sites since it was attractive impurities there how we concentrate in the impurity sites and it was quite smooth there was not like a sudden move of the particles for those things and so with that and also analyzing the energy the total ground state the energy of the system we didn't see any signature of a first order phase one phase transition so that should be then a second order phase transition we cannot really verify that numerically by numerically the derivating this quantity or maybe a crossover right so it's a smooth transition by turning on the V by varying V we see a smooth transition but quite fast for a small V now we also try to look at how the concentration affects this superfluous insulator transition and in this case we saw a much different picture we really see something strong there at a certain concentration specifically here so by varying the concentration for even any given V we find that if V is strong enough we see a clearly something that's more related to a first order one phase transition which was also confirmed by the energy curves now now we see a red from the total energy that we really have some one phase transition happening here in this specific concentration I won't give you the details about the why is 40% but we can discuss it later and so on but the point is if you also look at the same same type of graph that I showed before with the occupation probability double occupation probability we see that actually there are that if V is not enough you kind of frustrate this strong first order one phase transition here so that means for example for this is V equal to minus one so it's related to this blue curve here we see that there is a is move move what part move of the particles for the attractive sites due to the impurities but if it is strong enough that is quite abrupt so you have some sudden moving feeling of the impurities sites there so what is also constant with a first order one phase transition then we also now I'm get to the end yes so we also try to connect with work so we analyze other moments too but here I'm in focus on the average work and the question was is there for this system any relation with the work that I can extract from the system in comparison to this entanglement signatures and to analyze then work we could we do this quaint two quaintian protocols one of them was by fixing the same concentration so let's say if I have one impurities we will be always one impurity the initial and the final state but I I vary the intensity of V in the second protocol was the opposite so I keep fixed the intensity so they they are the same but I vary the concentration and I vary in this very specific way so I compare a concentration like we have one a certain number of impurity sites and then I compare with one extra only one extra impurity site in the system right and so by doing this protocols then for the sudden quaint now what we've the idea was then any of these two protocols can give some connection with these entanglement measures that I showed already before so for the first protocol we do find some signature for some depending on the the initial value of our disorder intensity we do find some signature at the same place where the entanglement show the first order quaint phase transition but not for all of these V's and not so directly connected to entanglement but for this second protocol we find some very much closer connection between entanglement and work extraction we see it's positive for all of the the V considers here and that means we could extract work let me just plug in here so that means we we have for the critical point a great work extraction for this system right so with that I come to the general conclusion so here the conclusions only for this last part that okay that is this is connection at least for this type of quaint in C there is this connect direct connection between work extraction entanglement and at the critical point so you could imagine something like a quantum critical battery in the sense that you can easily get energy from there and also that the minimal entanglement of the initial allows for a maximum work extraction and we also conclude that this quaint in C is more effective than the quaint in V I just would like to do a little propaganda that is just because we are just about to open a postdoctoral position in our group and also to present the group itself we are in the Sao Paulo state but not in the Sao Paulo city so it's in Araraquara the name of this we are located and we are interested in this kind of things well many body properties with different numerical methods with that I think you will for your things interesting talk do we have we have time for one quick question okay thanks for the great talk and the great results so I have a question regarding your initial slides where you talked about low density so how low is your low density and why for considering the Hubbard model like if you have low density you can also go to sign Jordan TV yeah oh right well this low density is 0.6 it's not that low but 0.6 up to 0.6 we have some discrepancies between a linear entropy and von Neumann one and and if you go further for smaller density then it becomes even qualitatively wrong so that's not the work I presented but the idea is that for example these signatures that you typically see in a confined transition like minima or no monotonic behavior maximum entanglement and so on this kind of things is incorrectly predicted by the linear entropy if you are in this very low regime of density and if you are not so low like close to the boundary of this 0.6 then you see a small deviations like not precisely for in the first derivative you see difference but the most complicated regime becomes smaller than 0.4 in density okay so in my system we still keep this linear entropy because for the disordered case as you vary the impurity and you do this 100 samples essentially you average out this difficult of the linear entropy to predict the entanglement from low density okay okay so when you are having the transition from superfluid to moat so like experimentally seeing the so what is this how much is the trap depth of this lattice you are considering for the sorry I think I didn't get your question so for the lattices you are seeing so what is the lattice depth which you are considering for the coupling strength well well I'm considering just one G chain so you are asking how can I combine with experimentally it's right yeah means experimentally saying that like it matters for us for to see what is the depth of the lattice in order to see this well I think I should discuss later thanks so much Liz it's in the sticker again