 Okay, so the last talk for this session is by Jonas Rigo from Yulik Who will be talking about The ground state of the Anderson impurity model in terms of our current neural network, please Yeah, thank you very much. Thanks for the organizers to give me this opportunity and Yeah, today I would like to talk about whether they were whether the Anderson impurity model grounds They can be represented as a current neural network This is a project. I'm working on with Marcus Smith with Vadim Slavko-Krinitin and Disclaimer, this is very much work in progress. So because we have not heard it enough today I will tell you one more time what constitutes variational Monte Carlo So essentially defining the ground state of a many body Hamiltonian is a formidably hard task but luckily we can formulate it as a variational problem Which simply constitutes minimizing the energy expectation value of the Hamiltonian But usually doing this exactly is intractable So what we do is we parameterize the wave function in some way such we such that we can carry out this Minimization problem over some parameters that are much lower dimensional than actually the entire Hilbert space So in 2017 Kalle and Joy have of course proposed their in their seminal paper the idea to use there to use neural networks to Parameterize the wave function and this essentially works as follows So we take a basis state from our Hilbert space give it to the neural network and in return it gives us The amplitude of this different the wave function amplitude corresponding to this state So this is what constitutes our neural quantum states is simply the sum over all of the amplitudes that give The neural network gives us right So this idea as you've heard in the last talks has gained a lot of traction in the field of quantum Quantum anybody spin problems in fact they constitute the best known ground state Ground states for the J1J2 Hamiltonian as the previous speaker actually has taken great part in showing So you can find a lot of people that belong to these names here in the audience in fact But the question I want to ask is what about fermions? So we've heard already today in the first lecture that the current trend for tackling Fermions is to make a first-order approach So that means we manually Antisymmetrized the wave function which would exactly mean that we make a sum of an infinite amount There's a very large amount of state of determinants, which in practice is untrackable So there are very clever tricks like backflow correlations just refactors and very recently hidden fermions To use only a single slater determinant and then introduce the extra interactions by means of these tricks right here But what about working in second quantization when a second quantization? There's a work that stands out in particular, and that's by Pasetti a work that tried to find the ground state of a SYK model, which is of course a volume law entanglement state But they've but they found that this cannot be done efficiently However, this work was then swiftly rebuted by that is a company who showed well this can in fact be done Now the questions that we are asking is still the same Can we do it? Can we do fermionic models in second quantization with NQS? And instead of using something complicated like the SYK We are looking at a more reasonable model like the Anderson impurity model And Anderson's impurity model is very special in the way that it has this RG derived Structure what you said very similar to how our current neural network Works and so we post a question whether record neural networks do indeed to present a ideal representation of the ground state wave function of this model Right, so let me say some words about Anderson's impurity model to those are not familiar with with it It consists of two major contributions essentially first. We have the impurity which is in the name essentially It's a very simple single spin full fermionic orbital, but it is interacting and therefore it can model localization of magnetic moments in matter and then second we have a See of free fermions in the thermodynamic limit, but it has a special boundary condition because it Because it hybridizes with the impurity basically changing the boundary condition and introducing these these density oscillations now If you take these two Contributions together what we get is the condor effect the condor effect is a highly non-perturbative effect And it manifests itself in the ground state as a Microscopically large many-body singlet essentially we get this so-called condor screening loud condor screening cloud We're all electrons in the screening cloud Participate in magnetic screening of the impurity so because this is non-perturbative we need a very powerful method to actually solve this model and Canonically, this has been done using the numerical randomization group and that works roughly as follows So we take the spectral function, which you can see here from our non interacting System which we can obtain very straightforwardly and then we perform a logarithmic discretization of the energy domain Now these intervals that we get from the discretization are then mapped to a semi infinite chain the so-called Wilson chain The crucial property of the Wilson chain is that the hoppings from one side to the next do decrease with the distance from the impurity meaning that the further down I May I truncate this chain the smaller the error and the error is always exponentially small compared to the next correction to the spectrum Now the way we actually then go about to solve this model is as follows We start out with a very small Hamiltonian diagonalize it then we add one more site Diagonalize it again add a site again diagonalize it and so on and so forth to get the idea So the point is that we can actually discard high energy states Because they don't contribute really to the ground state physics and that means that the dimension of the Hamiltonian remains constant and Regardless of how long we make this chain the amount of memory to find the ground state remains constant It's independent of the length of the chain now That's a property that they would like to maintain when we decide this construct our variational ground state approach Right now speak of variational ground state approach as we've heard a lot about different architectures today And in fact there are a lot of this list that you can see here is by no means extensive but it's very important to choose this in With some physical intuition because after all there's no free lunch so the better choose our network the better this will work out in the end for us and The enders-in-purity model is a very specific model with very specific structure So we can actually formulate some design criteria that a good And did a good network has to satisfy in order well in order to be probably a good ansatz so the first So the first design criteria is that if we have found Optimized parameters for a enders-in-purity model with a Wilson chain of a certain length Then these parameters should be very close to the optimized parameters of any longer Wilson chain, right? so next the an optimized wave function for a certain length of the Wilson chain should generalize to shorter and longer Wilson chains and Finally similar to the numerical realization group. We ask that the number of parameters of The variational wave function does not increase beyond a certain length of the Wilson chain so that it saturates for a given target Precision. All right. So those are design criteria So let's go and look at the network that promises to satisfy all of them and that is the recurrent network Now why is that well very simple answer because of the way it processes inputs It does not ingest it all at once and then gives us an output, but rather it scans It scans the input and that goes as follows So we start with the last side of our of our of our State and we look at the state of the site there and then we produce a so-called hidden vector Which goes on to come to give us the contribution to the to the wave function amplitude But it also gets carried forward so such that in the next step We combine the information from the current side with the previous hidden vector which gives us the new hidden vector But also the contribution to the wave function amplitude We do this until the very end where we get the last contribution to the wave function amplitude and in the very last step We get the phase which is somewhat important, right? So now that we have our variational wave function at hand we can actually look at some results and variate and And validate our design criteria So first design criteria what I have done here is essentially plot The length of a Wilson chain for which I found the ground state energy and here you can see the ground state energy now because this model is exactly so This is very precisely solvable. We can compare the energy to the exact energy And you can see our target position was something like 5 to the 10 to the minus 5 Now the way we have carried out this optimization is in an iterative fashion. Essentially what we've done is we found The first parameters and then we have used these parameters once they were optimized In the net for the next longer Wilson chain in that way We create a iterative optimization scheme where we always reutilize The optimized parameters from the previous step and you can see that this actually works very well And the reason is very clearly you can see here for length of 31 We can see the ground state energy that is predicted By the way by the wave function that has been optimized on length 31 for 41 And this is something like 10 to the minus 2 which shows that in fact these parameters that we've optimized for length 31 are Already really close to the ground state of length 31. So that's I would say Well ticks off our first design criteria. Let's look at the second design criteria the generalization of the wave function so what you can see here on the y-axis is Essentially the length of our Wilson chain on which we have trained the variational wave function and on the y-axis You can see the length of the Wilson chain for which we predict the ground state energy, right? So clearly on the on the Wilson chain length on which the system has been trained on this is the variation wave function does clearly perform the best Right, so that was obvious But the very Interesting thing is that it also performs really well Around in the neighborhood of these Wilson chain lengths This again goes to show that the parameters that have that are optimal for a certain Wilson chain lengths are very close to Optimal for the next longer, but also previous the shorter Wilson chain length, but that's the neighborhood What about the much much longer Wilson chains? Well in fact it does a fairly good job. It's quite impressive that I was that the model has been trained in the Wilson length Wilson chain length of 11 Can still give a ground state a precision of the ground set of 10 to the minus 2 at the length of 51 So with 51 spinful sites down road still performs alright Good, so let's check our last design criteria Which is whether or not the number of parameters does saturate with the Wilson chain lengths so what you can see here on the left is the number of parameters in our variation wave function and on The x-axis here you can see the length of the Wilson chain now Here we have the target positions. Oh dear god. Oh here it is Well, you can see the target positions of our ground state energy that we want to reach and Let's look at the green curve right here So the green curve is a target position of 10 to the minus 4 so at the beginning Like say seven sites 305 parameters do suffice to find the ground state to this granted energy to this precision But when we make it longer So a larger system clearly requires more parameters and then we get a last jump to 1377 parameters, but then that's it It does not grow any further regardless of how long we make the Wilson chain This that's the amount of parameters that we need to find the ground set which is quite neat because essentially now Well, you could just go on making the Wilson chain longer if you do need that for your simulation right now I've talked about These design criteria, so let's look at some actual physical observables So as I've said before the ground state is a many-body singlet, right? So we can actually check what it really is in our variational approach by looking at the Spin spin correlators, so this is the impurity spin and this is the bulk spin of side N Essentially because it's a singlet There's a sum rule that the that the wave function has to satisfy in order to be in order to be a singlet, right? So if we sum this correlator up because of as you to Spin symmetry we can just multiply by three here all the spin correlators are equivalent We should get the minus three-quarter Now we do this for different Hamiltonian parameters that you can see this right here on from your perspective on the right Yes, on the right and this is in fact fairly close to minus three-quarters. All right But let's carry out the most stringent test and that would be extracting tk So the condo temperature is Essentially the condo temperature below which we form this the condo singlet You can see here on the right this very nice plot where we count the ground state the generacy of the impurity alone so this is only the impurity we ignore the bath and it goes here from The local moment fixed point where we have where the up and down configurations of the impurity are degenerate it crosses over to log one When now this the impurity is completely screened by the surrounding electrons and this happens on an energy scale of The condo temperature which is the universal energy scale in the system now because it's the universal energy scale basically everything is controlled in the system by the condo temperature and so also yes This there's there's a real data in the next slide Yeah, but yeah, this is very I mean this is where it comes to to real data artists impression, okay So because this is the universal energy scale Basically everything is controlled by it's and so is also the extent the the the size of the screening clouds So it's inversely proportional to this condo temperature That means we can actually extract the condo temperature from the ground state by simply Looking how big this free screening cloud is so the way we do this is again looking at the spin spin correlator and The screening cloud essentially ends where this spin spin correlator goes to zero So you can see here on the right different curves as Marin Marin asked nicely Yes, so you can see here This is actual extra numerical curves as obtained from the numerical realization group where we can compute this and where it drops From and where it drops roughly to 0.5. That's it. That's where we define the condo temperature but if we look at But if we look at this at the scale because in in in beauty physics We can actually draw one to one comparison between the length of the Wilson chain and the energy of the system So we can one to one compare this and in fact Where this drops to zero point five roughly as well we find again the condo temperature So the data that you can see in blue this comes from NPS. So this is a very close to exact. I would say These are best data and for these values here our variational Our variational way function does a really good job of capturing the condo temperature. So I Unfortunately, I cannot really say that about the top plots for different parameters It seems to unfortunately break that break down. So our variational approach. Sorry, it's blue. The The NPS results are orange and the variational way function is blue So he does unfortunately not as good as of a job as predicting the condo temperature But as I've said before this is very much working progress and I'm confident that we can iron out these minus issues in due course Right and with that I'm actually at the end of the presentation just to reiterate one more time Our record no network is capable of of doing the iterative Doing the iterative Optimization as we as we were looking for it can also generalize to different lengths of the Wilson chain and finally the number of required Parameter does in fact saturate for at a certain point for certain Wilson chain length just as we wanted and then so look some outlook what we really want to do with this is is to study some Studies of transport phenomena some non-equilibrium phenomenon phenomena and of course go to system configurations that are hitted to on well that are hidden to Untractable with method methods like NPS numerical organization group or even sophisticated continuous time quantum Monte Carlo Continuous time quantum Monte Carlo approaches Right. Thank you very much Thanks a lot Jonas. So we have plenty of time for questions So maybe two questions The precision target that's something you can you're varying and then it changes the number of parameters accordingly. Is that yes indeed? Well into if intuitively speaking the more precise I want my way function to be the more parameters I need I think that's a fairly intuitive concept And so what I do is just set a certain target precision and then when it's reached with this number of parameters I call it a day and it goes on the plot So then in the last grass where you're showing this sort of call it a discrepancy between the NPS and NRG Yes, what was the precision target and as you change it? Do you see that they actually get closer? I can already tell you that you got the one to the wrong direction I cannot I tell you yes, they do get closer However as of now we have not gone beyond the precision of 10 to the minus 5 for the ground set energy and that seems to not suffice So unfortunately, I couldn't run any higher precision Calculations yet very confident that if we just go to higher positions, this will work out just fine Yes, I see it's 5 times 10 to the minus 5 as of now But if we go precisely, I'm confident that this will just coincide. I wouldn't see why not There any other questions? If not, let's thank Jonas and all the speakers of this morning again I think it's at 2 p.m.