 this work? Yes? Okay, so good morning everybody. So first of all I want to thank the organizer for the nice invitation and for for me it's the first time I come I'm coming to Trieste so it's very nice. And so today I will talk about Amofus quantum magnet in two-dimensional with Bergeret and I hope that by the end of the talk you will understand the two parts which is really this Amofus quantum magnets and how we can or we hope to simulate them with Rydberg atoms. And so to present myself I'm leading the quantum simulation team at Pascal where we are working on basically proposing like so we are the reticent first and we want to propose like experimental schemes that can be realized on a platform at Pascal which are like Rydberg atoms and one of the example if the Samofus material but we are also interested in other materials and so to do so we also develop like a numerical benchmark with like different methods such as tensor network, quantum Monte Carlo, so many analytical methods as we will see today in our spin wave. So to outline a bit the talk I will first like discuss what Amofus materials are and I will come back from crystals and discuss a bit the difference between these materials and like a crystalline structure or quasi-crystalline structure that maybe some of you are familiar with. Then I will discuss a bit more on how we can actually engineer like Amofus quantum magnets that are compatible with our quantum processing units. We have one of where we have van der Waals interaction that are decaying one over our six. And then let's after that I will discuss about the physics. So first in the ferromagnetic case where we have like interaction of the ferromagnetic case and there what I will show is some of the results that we have done with like perturbative approach like the linear spin wave theory. And then I will discuss like the anti-ferromagnetic case that is also quite interesting as in these materials we have like geometric frustration and so we have like this interplay between the anti-ferromagnetism and the geometry that we have here. And finally I will finish with the conclusion and the outline. So let's go from crystal to Amofus materials. So like crystalline structures are like materials that have like translational invariance they have long range order in the system. But this long range order can also appear in material that do not have like translational invariance. And this is for example the case of quasi crystals where you have no translational invariance but you still have like long range order in the system which means that when you look at the static structure factor of the system which is really the Fourier transform of your lattice you will see that there are peaks that are well defined in your materials that are actually there. Then there are Amofus materials and in these materials you don't have long range order so they look disorder like at the long range. Nevertheless when you look like in the short range you still have an order and so this will see that this will reflect for example on a well-defined like nearest neighbors like number of nearest neighbors that you're in the system for example. And so this is really the focus that we will take here is like how can we actually like engineer this type of materials. So to be a bit more like detailed here so what I've plot here is like different lattices so we start really like with a square lattice that we have here. And so what I plot here is the radial distribution function which is really like the density of atoms that you have in a radius around the given like atom in your material. And so in the square lattice because everything is super order what you see is that you have perfect peaks that are corresponding to the nearest neighbors that you have in your material the next nearest neighbor etc. And so you see that even at long range you still have like disorder that is well defined. So the the thing that you can look at is you take your square lattice you put a bit of disorder in your system and you start to look at how these peaks are evolving. Here due to the fact of the disorder you have a broadening of the peak that you're observing you still observe them but you have like a broadening of these peaks. Then there is the other extreme case is I draw like a random disorder point from a uniform distribution and we obtain these materials like this. When I look at the like the radial distribution function actually I really see that they're like everything is flat so it's really uniform like in the radius. In the case of amorphous and this is one example of amorphous solids what you will have is that in the long range you will observe like this flat like Q which is really like typical for like random distribution. Nevertheless in the short range you observe that we have like well defined peaks that are actually corresponding to like nearest neighbor or next nearest neighbor. So in this sense you have like short range order but no long range order in the material. And so one of the nice thing is that you can also relate like the so the radial distribution function to the so-called coordination number. The coordination number is basically the number of nearest neighbor that you have in the system. So this defines like the short range. So there is a lot of interest in amorphous solids. First of all and this is quite interesting is that all materials when you prepare them like in experiments I mean like really by cooling them down actually like they will become amorphous if you call them fast enough. So this is interesting. So like in they were also like quite a lot of interest in like semiconductors because these materials can have gaps so that these amorphous semiconductors have quite a lot of interest and also like for superconductors because they might have effect also on the critical temperature that you can observe like the superconductor. And then more recently they have been interested in like a regain of interest like in the last five to ten years I would say because of these properties for like topological materials where you can ask yourself whether like this is this short range order sufficient to actually like generate like topology in the system and the answer is that yes you can observe for example like topological states in these materials. Also very recently like interest in like spin liquids and like all these spin liquids could also arise in like amorphous materials. Now I will go to amorphous quantum magnets on our neutral atom QPU and so like here the idea is really that these materials are in a sense disorder they are not easy to simulate and you could really think okay from a piece of material that I have like let's say theoretically and from my microscopy Hamiltonian I have two paths no one is to really like go from my microscopic Hamiltonian and use some methods that I have that can be perturbative in some regimes or some other methods like tensor network or quantum Monte Carlo neural quantum state that we have heard a lot yesterday to try to study for example like some observable such as the grand state or the dynamics of the system or the other way around which is really like the quantum simulation way that would be let's use like a device and here we would use like neutral atoms trapped in optical tweezers to actually study like the grand state or the dynamics of the system. So in the case of the platform that we have at Pascal's these are Riedberg atoms in optical tweezers I mean this has this advantage of the flexibility that you can really like put the atoms wherever you want and so for example this is one famous example that they did in the group of Antoine Brouwer where they are actually able to draw like famous paintings and so this is very nice because in the case of all materials for amorphous we could really like place the atoms wherever we want and like generate like very complicated or complete geometry that we could do. So the other thing is that it's scalable so in the sense that we can go to hundreds of atoms or thousands of atoms and so it's very good for us because here we are actually looking at a phenomenon that is in between so we are looking at a phenomenon that is in between which means okay so it's back which which means that we will have like finite size effects but we want to observe something that has short range but no long range so I mean we need to have like systems that are sufficiently large to not have only effects from the boundaries that would actually maybe give like spurious correlation in the system. So this is a very nice thing and so for example like this is one of the experiment that they did like also in the group of Antoine Brouwer where they were actually able to to measure like the correlation of the antiferromagnetic phase and show that this is like reproducing very nicely what you would expect like in terms of the structure factor. Then what type of interaction can you have in the system so here we really have like atoms that are trapped in optical tweezers and then you can by shining a laser like tandem to like the Riedberg state and there you can have two type of regimes that you could have really that would be like one that would be the Ising model with Van der Waals interaction decaying 1 over R6 or like also if you address like the interaction between two Riedberg state you can have like the type of XY model that we deliver with like depolar interaction that are decaying 1 over R3. So in the case of this proposal and for the platform that we have at Pascal it's really this type of Ising model that we will look in at but now like what was interesting when we started to look at your literature is that typically when people study amorphous materials they consider the so-called like Voronoi desolation where they start from a random lattice and then there is a procedure to go to like a lattice that is actually amorphous with this kind of Voronoi desolation. Nevertheless as you can see here like the distances some of the let's say the lattice that you have is actually amorphous but the distances are not like respecting and in our case for example the fact that all these distances are not actually like the same means that we will have like foreign or interaction that is decaying 1 over R6 this is actually a bit harder. So there are ways to actually like get better results in this direction with for example this Lloyd desolation where you can know like actually squeeze this distribution to have like a better ordering in terms of the distances that you would have in the system for the nearest neighbors but still you see like it's interesting to have other algorithms to generate like shapes one of the reason being also that here you are really limited to actually tree nearest neighbors that you would have in the system so like this Voronoi desolation always gives you like tree nearest neighbors like if you look at these graphs. So what we did is to come up to with an algorithm that would actually allow us to do with a variational approach to actually choose like an arbitrary number of nearest neighbors in our system and that can be tailored to the type of interaction that we have here that are actually like quickly decaying in 1 over R6 and so the idea is that we start with a system that is disorder and then we apply some more kind of a gradient descent procedure to like the minimization of a loss function that is defined here which is composed of three terms so basically the first term is the term that you would like to optimize that is that we fix someone like the number of nearest neighbors that we would like to have in our normal force material and we try to minimize that but we have two other terms because we don't want the atom to get too close to each other so this is this term that is actually avoiding that and then the last time is that we don't want like the atom to actually like generate clusters of points that would be detached now so we have to force them to go like together so this is why you have these three terms and so these parameters for example are hypermaradameters that you have to fine tune but by doing so you can like minimize this function and you are able to actually generate like different types of amorphous material and here are some examples of materials that you can generate for example with coordination number equals to 3 where you see that when you zoom in in the material for example here in the center can really see this kind of hexagonal structure that you would expect for coordination and equal to 3 but you see that you also observe this kind of defects that appear here with like placate that are like larger loops actually no then you can also generate like coordination number equals to 4 which would be like really the type of square lattice equivalent where you also observe this plated to 4 or like even like numbers that are actually not exact existing in crystals such as 3.5 that you can appear or 5 that also like once yes so actually you can do both so I think for the Voronite oscillation typically you can impose like periodic boundary conditions but it's like I would say it's the way you construct it but then you look at the finite piece of it no but it's very I would say it's very similar to what you do in quasi crystals quasi crystals also typically when you construct it you construct it through the periodicized version that you go like larger and larger so in our case actually in the so it's not yet it will be for the second version on the archive we actually also studied like periodic so this is like all open boundary conditions and so basically what we do is that we run our algorithm then we look like basically something in the bulk of the algorithm to avoid like the boundary effects but we also have a version that can be done with periodic boundary condition that is actually useful for like some of the results that I will show later on okay and so so once you have that what you can do is that you can look at the the the status which are factor which is really like the Fourier transform of your of your lattice in in case space and so when you look for example at this C equals 3 so like three nearest neighbors what you observe is that like the the status factor actually like resemble to what you would have in the exact one and lattice but you have like this kind of nice ring that appears which is actually also like reflecting the fact that you don't have like I mean you don't have like a preferred direction in your material anymore no and for C equal 4 like you have the same type of property you see that it's like a circle around like the typical square lattice that you expect for the zone 2 what is interesting is for example in the case of 3.5 what you observe is that also ring which is compatible actually to this hexagonal one and here distance interesting thing is that if you are looking at the lattice actually you will have like a mixing if you want between like some hexagonal rings and Kagome type of rings so in this sense like this structure factor is like compatible with that so as you can imagine these materials are relatively disorder which means that you don't have really topology in the systems it's difficult to do like it's quite challenging also for like numerical methods to actually like find so typically what people do like to study for example dynamics in in the system like it's to use for example matrix product states and but then you have to like take your two-dimensional system and to map it to a one-dimensional system and for regular lattices you can know what is the optimal way to do this mapping whereas in this system it's much more difficult how to define like the indexing of the system the other thing is that you cannot impose like the typical symmetries that you would use such as translation and variance or like other symmetries of this deal the other thing is that for this system as I mentioned before there are like important like boundaries effect so you would need to have large systems which means like this kind of calculation even more difficult and then what is also very interesting at the grand state level is indeed anti-ferromagnetic case you can have like local frustration in the system and this can make actually like also simulation difficult like for grand state finding with like classical methods so now I want to to go a bit to the ferromagnetic case I want to discuss a bit what we did which is like preliminary study with like linear spin-wave analysis to understand a bit like the phases that we expect in in the system and so here here what we did is to consider some more like the ferromagnetic interactions in the model see if it comes back so you have an Hamiltonian here like with like this interaction that is decaying like it's easy interaction decaying in one over our seats and basically like this type of model is so if you look at the original Hamiltonian the Riedberg Hamiltonian as anti-ferromagnetic interaction nevertheless like during the dynamics due to the time reversal symmetry you can consider like this anti-ferromagnetic interaction or like the ferromagnetic interaction because it's like a time reversal property so this is why here we can study like some more like the ferromagnetic regime and so basically what we do here to compare like these different lattices is that we set the minimum distance between the two sides some more to be like a constant no okay to actually be like able to compare these different ones and so we consider some of the transverse field here like and we will change a transfer field and see a bit how the phase diagram is changing so here what you expect like by increasing the interaction here is to so in the in the regime where interaction are somehow low you would expect to to be in the ferromagnetic phase and when you increase this interaction sorry when you increase like the Rabi frequency here that is a sigma x term of your model you expect to go to a paramagnetic phase now where the spins will be aligned in the like the alien vector of sigma x so what you can first do is to do like a mean field study of this this this model by doing some more like perturbative approach like assuming that your like your spin are still close to sigma z you already see here that this is perturbative in the sense that when hx will be large you don't expect the spins to point in this direction and so if you do that then you will arrive to this effective mean fin energy which is really like a product state one that you would like to minimize and this you can do like with like a classical minimizer no and so from that you can obtain like the the mean field energy and the mean field ground state which you can characterize through like the order parameter of the system which is the magnetization and so here you expect that in the ferromagnetic phase as all the spins are pointing in the each in the z direction you expect to see like a magnetization of one or 0.5 depending on your convention here we define like the spin as one half of the sigma z matrix whereas in the paramagnetic phase because you are like an alien state of sigma x you expect to see like a magnetization of zero and so this is what we did here we like studied the model for different values of the coordination number and so what you see is that the transition appear a certain value which is different also from the a bit different from the one that you would expect for the regular lattice which is the vertical line that you're observing here okay so then if you want to go like beyond some of this mean field approximation you can add a correction which is the linear spin wave theory which is basically what you do is that you start from your mean field ground state and you add the perturbation so it's represented by this Holstein-Primacheff mapping when you basically rewrite your spins in terms of the mean value plus a correction to it and so if you do that and you write now like the Hamiltonian that you would obtain you will obtain the Hamiltonian now that will be quadratic in terms of this new operator a and a dagger and so the nice thing of this Hamiltonian it's again quadratic so you can still get analyze it through the help of the Borrel-Uber transformation and so this allows you to obtain like the spectrum and you can also like obtain some more like an idea of the excitation that you would have like on top of the ground state so one of the nice thing that you can do is that for example to cross check that when you do this type of analysis you can look at the energy gap between the ground state that we had the reason I mean field and the first excited state that you obtain through this linear spin-wave analysis and what you see is that at the phase transition point you always observe that there is like a gap closing in the system so another thing that one could wonder is what happens with like some amount of the fact that the system has disorder how can it compare actually to like the disorder that you would add for example to a square lattice and so in this direction there was a nice work from Thomas Orochilde that is here where they were like considering actually like a system of a Riedberger array no in a square lattice but with disorder and I wanted to see whether the like the system has some localization or not appearing due to the fact that you have some intrinsic disorder in the Riedberger array like on the position on the tweezers for example and so what we wanted to do here is also to see in our system somehow like whether we can have like excitations that are like beyond what you would observe in localized so like typically like malnones or like what kind of physics we will have here no and so the the first part of the study that we did is to compute this inverse participation ratio which is like an indicator on how localized are your wave functions in your system and actually you can see like in this plot here is that that basically when the the H it is sufficiently large which I insist again it's like beyond this perturbation perturbative analysis that in this case you can actually have also like delocalized particles no and so this is also reflected in the so-called dynamical structure factor which is basically like a Fourier transform of your two-point correlators in position and in time which basically gives you information about the energy structure of your like the energy bands of your system if you want in the many body case no and so what you can observe is that through these linear spin-wave analysis we still see that there is a band that is appearing somehow like beyond like the the ground state in the system and so we expect to observe like this kind of malnony dynamics in the system when H is sufficiently large okay then I want to go to the the other case which is like the antiferromagnetic case which is really the one that we would have in the system if we would study the ground state of the system um so the first thing that is interesting here is so I told you before we can generate like a system with coordination number for example four but this does not tell you anything about the angles in your system no so you would have like for example for coordination four you have both like for example like a square type of let's say amorphous material or you can also have like kagome type of amorphous materials where you see like these plaquettes and these three angles around like um but you see already here that these two materials will have very different ground states so let's say that if I come back to regular lattices what you expect to have for square lattice is as the system is well-ordered you expect to have like a nice antiferromagnetic order in the system whereas in the kagome lattice let's say already at the classical level you expect to have a huge degeneracy of the ground state which is really like the type of spin liquid that you would expect there and so here one of the questions that we wanted to address here is a bit what we will observe here between these two lattices and especially in the case of the kagome lattice whether we could say something about the glassy nature or not or this type of amorphous materials no and so here it's again like interplay between the geometry the fact that we have frustration to the geometry and the antiferromagnetic interaction that we have in the system so to do so and we use an approach to have like a first benchmark in this direction which is like using like classical simulated annealing which i want to remind a bit how it works so the idea is that we do like some simulated annealing and the procedure of simulated annealing is that you do some random signal split the spin flip in your model then you do a metropolis update you know model at a given temperature and then little by little by step you actually reduce the temperature in the model and so so what you can do with that is that then you can study some more like different replicas so the different initial conditions that you're utilizing here like for your your metropolis algorithm and you can study like different quantities so just the energy that you would obtain through this simulated annealing all other parameters for example one that is commonly used for spin glass which is called the edward-anderson parameter which is basically written in terms of this qsa of alpha beta which is basically like for two like simulated annealing ground steps you are comparing some of the spins of these two replicas no like for the ground state locally and then you can also compute some more like this q square of sa which is like the mean over like this different replica no and so the other thing that you can also do is you can look at this distribution of probability and typically in the spin glass this is what people use to actually like characterize what type of spin glass you have like in your system how glass is your system to start with that and to give a bit more of intuition because these quantities are not that intuitive we started to look at paradigmatic models and so actually in the square lattice as I said before you expect to have like antiferromagnetic ordering um and actually there are two ground states here so like uh so let's say like for the one in case it would be like up down up down up down but you can also have down up down up down up no so it's really like a full like rotation of all the spins of your model and so what you expect in this case is that you expect to have like in your probability of this qsa alpha beta to watch a left two peaks one at one and one at minus one because it's exactly like the opposite pattern of the other one so this is what you observe actually like through this immunity in the case of the Kagome lattice because the ground state is like totally degenerate you expect to see somehow a peak at zero like in this probability distribution um then what you can do also is to study like the same type of uh to do the same type of analysis in the paradigmatic edward anderson like uh model which is a model with like basically like random coupling like ising model with random coupling um and there you can also study two types of models so one that would be like um Gaussian where you have like the Gaussian distribution for your coupling that can be positive or negative so it's center on zero so one over by model and so what is known for literature is that these two models have actually like different behaviors um so here I have to insist that simulated annealing is not a good method to study spin glass um because basically you are not converging to the ground state of the system so in this case we're actually converging to the ground state of the system because the system is small and so if you let the system evolve sufficiently long you will actually reach the ground state but in general you like a single kind of Monte Carlo update will not allow you to actually like explore properly like the low energy landscape of these spin glasses nevertheless and like in general this kind of spin glass analysis is very difficult and so we believe that this simulated annealing is a first step towards this direction to understand whether this phase is could be glassy or not but it would require like a proper analysis afterwards so we did the same type of analysis for amorphous material so we considered like a system we see equal for with 400 sites for both like the square and Kagome geometries that I showed you before um and so the first nice thing that you can see here what I plot is the energy spectrum for different number of steps in your simulated annealing and what you see is that in both these models and for the different replicas you see that there is not a nice convergence in terms of the energy which really reflects that some more like this algorithm didn't converge down like so this is the first thing like that you can observe you so the so like on the left you have actually the square type of lattice and on the right you have like the Kagome type of lattice and the different colors are different number of steps in your algorithm then the other thing that you can see is that there is actually like a difference when you look at this this probability of QSA in the case of square-ish kind of amorphous solids and Kagome type of amorphous solid where you actually see that in the case of square you still see that you are coming from these two peaks that were well localized in the like lattice version but here you see that they are broadened whereas in the case of the Kagome everything like is around zero no so this is like a first indication that these materials could be glassy at zero temperature but this would require like a like a further analysis and so i mean with that i want to to go to the conclusion and the outlook of the talk so like i presented a new algorithm to generate like amorphous layout and which is actually well suited for like our physical platform which is like this read by rathoms the resulting Hamiltonian that we have are actually hard to tackle numerically and even with state-of-the-art like numerical methods and we showed like some first benchmark with perturbative methods so here like the linear spin wave theory but it would be nice to actually do benchmark with like more like evolve techniques such as tensor network orders so in this case like it's really an example where neutral atom quantum simulator could really represent an avenue to study these materials both for the dynamics or for the ground state properties of the model and i mean there is a lot of potential interest in physics that could appear because like for example in the last part i really showed like what happens in the classical regime but you could imagine that the interplay also is like a quantum term of the type of Rabi frequency of sigma x would be quite interesting because you might have competition between spin glass and spin liquids or other phases that would appear here and so like the work that i presented today is on the archive the last part is not yet on the archive because we will add it in the salon version of the paper and these are my collaborators on this work and with that i would like to thank you for your attention. Thank you Alexander for the beautiful talk please. So you showed that simulated annealing is not working very well for these classical simulations have you tried using simulated quantum annealing like in a path integral Monte Carlo? So here all the results that i showed were classical. Yes but also simulated quantum annealing is classical is a way of simulating quantum annealing through a Marco Czee Monte Carlo basically it's by using path integral Monte Carlo approach and i don't know there are some papers showing that for some cases you do have an advantage over simulated annealing like on binary neural networks for instance you do have an exponential advantage so you don't get stuck in order and metastable states but you can kind of almost converge to the low minimum. Okay so now we didn't try that so i mean one of the things that is difficult for the spin glass study and i have to say we are not experts on spin glass so we it was not easy for us but one of the difficult things is that typically in 2D spin glass is expected to be only at zero temperature and so we tried simulated annealing then we tried a bit also of like parallel tempering but all this is actually quite difficult because you really want to go to this zero temperature. Yeah but in SQA you anneal on the field on the transverse field that are simulated so maybe just an idea to try. Yes thanks. Okay are there questions? I would have a question maybe more on the experimental side like what are you thinking about doing with these amorphous materials would you really try to target say the ground state via some maybe a diabetic protocol or would you more like target some actual dynamics some dynamics you start from some product state which is probably easy to prepare and then looking at dynamics yeah so what do you have in mind a bit in this I mean for the moment both aspects are open no so it could be like to study a bit like this phase diagram and try to prove this kind of glassy physics but it could also be for dynamics so where you prepare like your initial state which is typically in these Riedberg setups where all atom are in the ground state so it's a product state then you could study for example point dynamics in this system which is non-trivial or to probe also like a dynamical structure factor because this should be possible also like on quantum simulators is it like now for you also possible maybe to analyze random in an initialized random configuration so would they be always be identical for each spin like all spins in the ground state so I will answer in two steps so the first step is what do we have now no so what we have now is we have global addressability so you can prepare states but through global policies no so it's not random in this sense in the future it's in principle we will have also local addressability still you cannot prepare any arbitrary state because you're the underlying Hamiltonian is the one that you have in the system which is this Riedberg Hamiltonian but it allows you to do like local preparation for like for example okay thanks a lot okay other questions moments I have one you have mentioned that the inverse participation ratio of the excitations yes how does it scale with the sides what the kind of localization properties do you find yes so this is what was plotted here so we did the scaling analysis here and we observed that in some regime you actually see that you go to a localized regime so when it shifts is too small in a sense but then it changes when like the system is large also this is not kind of transition I mean you will have a transition from someone localized to delocalized yes yes thank you very much so let's thank Alexander again and let's go to lunch